18
Research Article Robust Inventory System Optimization Based on Simulation and Multiple Criteria Decision Making Ahmad Mortazavi and Alireza Arshadi khamseh Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran 15719 14911, Iran Correspondence should be addressed to Alireza Arshadi khamseh; [email protected] Received 21 February 2014; Revised 13 July 2014; Accepted 21 July 2014; Published 2 September 2014 Academic Editor: Hongyi Li Copyright © 2014 A. Mortazavi and A. Arshadi khamseh. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Inventory management in retailers is difficult and complex decision making process which is related to the conflict criteria, also existence of cyclic changes and trend in demand is inevitable in many industries. In this paper, simulation modeling is considered as efficient tool for modeling of retailer multiproduct inventory system. For simulation model optimization, a novel multicriteria and robust surrogate model is designed based on multiple attribute decision making (MADM) method, design of experiments (DOE), and principal component analysis (PCA). is approach as a main contribution of this paper, provides a framework for robust multiple criteria decision making under uncertainty. 1. Introduction Supply chain is complicated system that includes many components such as suppliers, factories, distribution centers, and retailers. ese components are connected to each other by three streams of financial, information, and material flow. In many references, there is a fundamental hypothesis that financial flow is upstream from customers to suppliers and material flow is downstream from suppliers to customers while information flow is mutual. By the way, sometimes, this hypothesis is neglected about material flow (e.g., reveres logistic). Material flow plays dominant role in supply chain and is defined as inventory problem. Inventory actually is the bridge that connects material handling and production planning to the supply chain [1]. On the other hand, retailers in highly competitive market face dynamic change of demand including seasonal cyclic change and long-term trend. ese situations caused multiproduct inventory systems to change into complex multicriteria systems. Complex systems are challenging in the case of modelling and computation. In most of the problems, modelling of complex systems is very time consuming and also is not error free. Furthermore, modelling of complex systems needs too much computational effort to solve and sometimes they are not solvable in reasonable time. In this situation, importance of data-driven methods emerges [24]. Although model-based approach (i.e., simulation model) has many advantages, such as what if analysis and ability of scenario generation, data-driven approach is less time consuming and also easy to implement. So, these approaches can be combined to achieve tailored approach which inherits advantages of model-based and data-driven approaches. In this research we used discrete event simulation model and data-driven methods like principal component analysis and multiple attribute decision making to design an efficient framework for robust and multiobjective optimization of retailer inventory system with multiproduct as complex system. e rest of the paper is structured as follows. Litera- ture review is presented in Section 2. Problem statement is presented in Section 3, proposed framework which contains simulation modeling and surrogate design is described in Section 4, Section 5 provides numerical result, and finally Section 6 is dedicated to conclusion. 2. Literature Review As the investigated problem encompasses two issues of simu- lation optimization and inventory problem, related literature Hindawi Publishing Corporation Modelling and Simulation in Engineering Volume 2014, Article ID 305120, 17 pages http://dx.doi.org/10.1155/2014/305120

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Research ArticleRobust Inventory System Optimization Based on Simulation andMultiple Criteria Decision Making

Ahmad Mortazavi and Alireza Arshadi khamseh

Department of Industrial Engineering Faculty of Engineering Kharazmi University Tehran 15719 14911 Iran

Correspondence should be addressed to Alireza Arshadi khamseh alirezaarshadikhamsehgmailcom

Received 21 February 2014 Revised 13 July 2014 Accepted 21 July 2014 Published 2 September 2014

Academic Editor Hongyi Li

Copyright copy 2014 A Mortazavi and A Arshadi khamseh This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Inventory management in retailers is difficult and complex decision making process which is related to the conflict criteria alsoexistence of cyclic changes and trend in demand is inevitable in many industries In this paper simulation modeling is consideredas efficient tool for modeling of retailer multiproduct inventory system For simulation model optimization a novel multicriteriaand robust surrogate model is designed based on multiple attribute decision making (MADM) method design of experiments(DOE) and principal component analysis (PCA) This approach as a main contribution of this paper provides a framework forrobust multiple criteria decision making under uncertainty

1 Introduction

Supply chain is complicated system that includes manycomponents such as suppliers factories distribution centersand retailers These components are connected to each otherby three streams of financial information and material flowIn many references there is a fundamental hypothesis thatfinancial flow is upstream from customers to suppliers andmaterial flow is downstream from suppliers to customerswhile information flow is mutual By the way sometimesthis hypothesis is neglected about material flow (eg revereslogistic)

Material flow plays dominant role in supply chain and isdefined as inventory problem Inventory actually is the bridgethat connects material handling and production planningto the supply chain [1] On the other hand retailers inhighly competitive market face dynamic change of demandincluding seasonal cyclic change and long-term trend Thesesituations caused multiproduct inventory systems to changeinto complex multicriteria systems

Complex systems are challenging in the case ofmodellingand computation In most of the problems modelling ofcomplex systems is very time consuming and also is not errorfree Furthermore modelling of complex systems needs toomuch computational effort to solve and sometimes they are

not solvable in reasonable time In this situation importanceof data-driven methods emerges [2ndash4]

Although model-based approach (ie simulation model)has many advantages such as what if analysis and abilityof scenario generation data-driven approach is less timeconsuming and also easy to implement So these approachescan be combined to achieve tailored approach which inheritsadvantages of model-based and data-driven approaches Inthis research we used discrete event simulation model anddata-driven methods like principal component analysis andmultiple attribute decision making to design an efficientframework for robust and multiobjective optimization ofretailer inventory system with multiproduct as complexsystem

The rest of the paper is structured as follows Litera-ture review is presented in Section 2 Problem statement ispresented in Section 3 proposed framework which containssimulation modeling and surrogate design is described inSection 4 Section 5 provides numerical result and finallySection 6 is dedicated to conclusion

2 Literature Review

As the investigated problem encompasses two issues of simu-lation optimization and inventory problem related literature

Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2014 Article ID 305120 17 pageshttpdxdoiorg1011552014305120

2 Modelling and Simulation in Engineering

is categorized in two separated parts for providing moresupportive literature review

21 Inventory Problem Basic problems of inventory systemare studied thoroughly by [5ndash7] Economical order quantity(EOQ) is the simplest model of inventory problem inthis model demand of each period is constant and timeindependent EOQ does not consider lost sale back orderand other cost for reason of simplification and emphasizesonly on holding and ordering cost Cheng [8] developeda model of inventory system with cost-dependent demandand included production cost in proposed model Chen etal [9] studied back order by fuzzy technique and Zhao etal [10] suggested analytic model with demand according totime seriesThey concluded that in time-dependent demandefficiency of EOQmodel increases with shorter lead time andweaker autocorrelation

Basicmodels thatwere offered for inventory problemonlyhave one objective function including different inventorycosts while advancement in technology and intensification ofbusiness competition caused necessity of other criteria To thebest of our knowledge in recent years service level has beenobserved in variety of supply chain and inventory problem asperformance criterion Adding service level criterion classicdefinition of inventory problem is changed into multiobjec-tive optimization In such problem inventory costs shouldbe minimized while service level should be maximizedAvailable models for this problem are divided in two groupsof deterministic and stochastic Both of these models can besolved by three approaches of analytic methods for examplemathematical programming metaheuristic methods andsimulation Lau et al [11] benefited from simulation to com-pare four inventory management policies with two criteriaof cost and service levels They also surveyed preorder andinformation sharing impacts on their models Xu and Zhao[12] used fuzzy rough simulation to optimize multiobjectiveproblem to minimize wasted cost and maximize expectedvalue of revenue Hnaien et al [13] Surveyed two-level as-sembly system with two objectives of service level andmaintenance cost They considered stochastic lead time andapplied genetic algorithm to solve this problem

Although cost and service levels are important criteria forinventory system performance there are other criteria thatshould be considered such as amount of systems inventorywhich is important factor with significant impact on inven-tory system behavior Because of lead time uncertainty thatorigins from natural disasters and transportation problemsorganizations face delay in delivery so most of them holdsafety stocks This phenomenon is the main cause that leadsto the increase of inventory in hand As inventory in handincreases the inventory system is faced with holding cost andother problems like decrease in quality lack of flexibility andso on So amount of systems inventory can be consideredas performance factor and inventory in hand should beminimized as mentioned in the just in time (JIT) philosophyPurnomo et al [14] researched about influence of periodicreplenishment and continue replenishment inventory poli-cies on supply chain and considered both inventory in handand work in progress as performance factors

Now in recent yearsmodeling of inventory system asmul-tiobjective and stochastic problem is an interesting area forresearch

22 Simulation Optimization One of the well-known simu-lation methods is discrete event simulation that is based onstochastic processes and could be efficient tool to capturestochastic behaviour of different systems While discreteevent simulation has several advantages it is not optimizationtool individually [15] By the way because of its flexibilitysimulation can be coupled with other techniques such asmetaheuristic algorithms or stochastic methodsThis synthe-sizes makes the powerful and advantageous approach of sim-ulation optimization with vast area of research Simulationoptimization is powerful arsenal for optimization of complexsystems such as military aerospace and supply chain [16]To the best of our knowledge there are three main opti-mization techniques that were reported as suitable techniquesfor simulation optimization These considered methods aremetaheuristic optimization stochastic approximation (SA)methods and surrogate models

Fu [17] extensively described role of applied methodsin simulation optimization and also surveyed techniquesemployed in optimization package of simulation softwareWang [18] used hybrid approach including genetic algorithmand artificial neural network for simulation optimization andKeskin et al [19] applied discrete event simulation and scattersearch algorithm for optimization of inventory system andvendor selection Mazhari et al [20] developed a simulationoptimization framework based on hybrid simulation model(system dynamic and agent based model) and metaheuristicalgorithm Also Duan and Liao [21] applied metaheuristicapproach for developing simulation optimization frameworkin order to optimize replenishment policy of inventorysystem in capacitated supply chain

Although using metaheuristic optimization algorithms isstraight forward approach for simulation optimization it istime consuming and needs high level of computational effortSo it is inefficient in case of simulation optimization withmore than one objective function

While metaheuristic algorithms use stochastic searchingmethods SA is based on gradient search Because of noisysituation of observations SA algorithms consider expectedvalue of objective function SA family includes attractivemethods because their convergence is guaranteed theoret-ically Simultaneous perturbation stochastic approximation(SPSA) is noteworthy algorithm of SA familyThe theoreticalaspects of SPSA are deeply described by Spallrsquos [22ndash25]proposed simulation optimization framework for inventorycontrol in supply chain based on SPSA

In contrary to two former methods surrogate modellingis postprocessing method so it is less time consuming Insurrogate modelling the main idea is to fit single surface tothe decision space and use this surface instead of simula-tion model for optimization In this area response surfacemethodology (RSM) [26] and supervised learning methods(eg artificial neural network or support vector machine) areconsiderable For instance Can and Heavey [27] applied arti-ficial neural network to develop surrogate model for discrete

Modelling and Simulation in Engineering 3

event simulation Azadeh et al [28] used artificial neural net-work for the designing of simulation optimization frameworkand they applied proposed framework for optimization ofwaiting time in tandem queue systems Bornatico et al [29]proposed a surrogate model based on redial basis functionfor simulation optimization of energy systems Wan et al[30] designed simulation optimization framework using leastsquare support vector machine (LSSVM) for optimizationof inventory level in three-stage supply chain They alsoshowed that proposed framework leads to better solutionwith less number of simulation runs in comparison withSPSA algorithm Surrogate modelling is less time consumingin comparison to metaheuristic or SPSA approaches butthis approach loses accuracy in multiple objectives problemsolving

Although a bunch of papers published in simulationoptimization area to the best of our knowledge a tiny numberof them are dedicated to the multiobjective optimization [31]and robustness [32 33] In this case using metaheuristicand SPSA approaches is very time consuming and it is noteconomic for optimization of simulationmodel with accuratedetails On the other hand all of the reviewed approacheslose their accuracy when there are multiple objectives Withthese considerations this paper purposes a framework foroptimization of detailed simulation model of inventory sys-tem with multiple objectives Proposed framework is lesstime consuming in comparison with metaheuristic or SPSAapproaches while it provides robust and accurate solutionsSo the proposed framework is relatively new and contribu-tion of this research entails threefold as follows

(i) We modelled cyclic and long-term demand basedon nonparametric time series modelling for morerealistic consideration

(ii) We proposed surrogate model for robust and multi-objective optimization ofmultiproduct inventory sys-tem based on discrete event simulation full factorialdesign of experiments (DOE) and multiple attributedecision making (MADM) technique

(iii) Due to the stochastic nature of objective functionwe employed principal component analysis (PCA) asstatistical method to improve MADM performance

3 Problem Statement

The problem is concerning retailer who sales office furnitureand facility The retailer sales four products respectively119860 119861 119862 and 119863 the aim is the optimization of inventorysystem according to information which is adapted from localbusiness Key features of retailer products from inventoryview point are as shown in Table 1

In this table second column gives average demand ofeach product type in a year third column provides holdingcost of each product in a planning period fourth columnis dedicated to ordering cost of each type of products andfinally numbers of fifth column are cost of lost sales which areincurred to retailers when they cannot satisfy the demand ofcustomers for each type of products

The fundamental assumption that should be consideredin this problem is as follows

(1) Order cost for each type of products includes trans-portation and order registration cost

(2) There is no backlog inventory so inventory level isnonnegative all the time

(3) Profit of each product is considered as lost sale costbecause unavailable products incur lost profits thatare interpreted as cost of lost sale

(4) According to the adapted information these productshave five years life cycle and then will be substitutedwith new products

(5) Planning periods for system under study are as longas 20 days

The notations that will be used to describe the problems areas follows

Indices Consider the following

119905 index of planning periods 119905 = 1 2 3 119879119894 index of demands in planning period 119894 = 1 2 3 119873119895 index of orders in planning period 119895 = 1 2 3 119872

Parameters Consider the following

119862119903 reorder cost

119862ℎ holding cost of each product in planning period

119862119897 cost of lost sale for each product

Variables Consider the following

119868119897

119905 inventory level in 119905th planning period119868119901

119905 inventory position in 119905th planning period

119889119894119905 quantity of 119894th demand in 119905th planning period

119899119905 number of reorder in 119905th planning period

119909119894119905 1 if 119889

119894119905is less than 119868119897

119905 0 otherwise

119876119895119905 quantity of 119895th order in 119905th planning period

119871 lead time for organized orders119877119901 reorder point

Demand of each product in the planning period (119889119894119905) is

stochastic variable and is generated by nonhomogeneousPoisson distribution So total number of arrived demandin planning period (119873) is probabilistic 119862

119903 119862ℎ and 119862

119897are

different costs of inventory system according to Table 1 119868119897119905is

inventory level and refers to physical quantity of inventorywhich is available in retailer while 119868119901

119905is the position of inven-

tory and includes quantity of on-order inventory in additionto inventory level in planning period 119909

119894119905is binary variable

which is one if inventory level is greater than the arriveddemand So if 119909

119894119905is one 119889

119894119905can be satisfied and otherwise

4 Modelling and Simulation in Engineering

Table 1

Product type Average demand Holding cost in planning period Reorder cost Lost sale cost119860 987 30000 50000 20000119861 1520 30000 50000 15000119862 1598 30000 50000 25000119863 1569 30000 50000 18000

it is lost sale 119877119901is the reorder point for organizing of new

order In other words if inventory level reaches 119877119901or less

a new order would be organized with quantity of 119876119895119905 Each

organized order reaches the retailer and increases inventorylevel after passing of lead time (119871) Considering assumptionsand described notations the following equations are themainobjective functions of defined problem

Min119879

sum

119905=1

119862119903119899119905+

119879

sum

119905=1

119862ℎ119868119897

119905+

119879

sum

119905=1

119873

sum

119894=1

119862119861(1 minus 119909

119894119905) 119889119894119905

(1)

Maxsum119879

119905=1sum119873

119894=1119909119894119905119889119894119905

sum119905

119905=1sum119873

119894=1119889119894119905

(2)

Minsum119879

119905=1119868119901

119905

119879 (3)

In (1) the objective is minimizing the total cost includingcosts that depend on reordering handling and lost salesLost sales not only incur excess cost but also decreaseretailer credit So (2) is considered to maximize service levelindependently In (2) the objective function increases where119909119894119905is 1 for 119894th demand in 119905th period and such situation is

possible if 119868119897119905is greater than 119889

119894119905 In fact (2) causes increase

in inventory level while (1)ndash(3) causes decrease in inventorylevel State of inventory level depends on number of ordersin each planning period (119899

119905) quantity of orders (119876

119895119905) and

quantity of demands (119889119894119905) while state of inventory position

depends on inventory level and lead time (119871) so (3) isresponsible for minimizing average of inventory positionincluding inventory in hand (119868119897

119905) and on-order inventory

Average of inventory position should be minimized in orderto improve flexibility of retailer and approach to the just intime (JIT)

Demand of each product follows different pattern withboth long-term and cyclic trend So mentioned objectivefunctions are considered individually for each product type

4 Proposed Framework

In the defined problem demand of products exposes highlydynamic pattern and as time passes demand and its variationincrease hence multiresolution method is employed fordemand modelling Also three different policies for inven-tory control are considered which are reordered based onfixed quantity (FQ) fixed interval (FI) and demand forecast-ing (DF) Simulation of developed model is implemented inArena 135 Optimization of simulation model is performed

by surrogate model that is based on full factorial design ofexperiment (DOE) For construction of decision space DOEfactors include inventory policy reorder point and lead timewith three levels for each of them So there are 33 = 27

different combinations of decision variables to form feasiblescenarios Ranking of produced scenarios is accomplished byMADM technique For ranking of scenarios three objectivefunction values are considered (ie cost service level andaverage of inventory position) and in addition robustness ofservice level against demand fluctuation is considered AlsoPCA is applied for more realistic weighting of objective func-tion values based on their statistical influence on improve-ment of other objectives Finally interacting plot is employedfor sensitivity analysis and investigation of solutions in detail

41 Simulation Modelling Simulation modelling of problemconsists of two parts which are modelling of demand andmodelling of inventory policies In this paper demand is non-homogeneous Poisson process and three different inventorypolicies based on continues reviewing periodic reviewingand periodic reviewing with forecasting of future demand areconsidered

411 Modelling of Demand For customers demand model-ing multiresolutionmethod is applied Kuhl andWilson [34]developed this method for simulation of nonhomogeneousPoisson process with trend and cyclic changes This methodestimates mean intensity function and the nonparametricnature of thismethod is one of themost important advantagesin comparison with other methods So it is independent ofstatistical parameters and applicable in variety of problemsFurthermore multiresolution method can support combi-nation of multiple cyclic changes simultaneously Anotheradvantage of multiresolution is its capability in the modellingof nonsymmetric cyclic pattern As in our case demandhas nonsymmetric pattern with cyclic changes and long-term trend and multiresolution approach is a reasonablechoice More theoretical and application of used method areprovided in [34]

412 Modelling of Inventory Policies As the main effort ofthis paper is inventory system optimization modelling ofinventory policies plays an important role in this problemIn this problem optimization is manipulated by selection ofappropriate inventory policy and configuration of its param-eters to the way that leads to the optimal state of inventorysystem In the inventory management three approaches arecommon strategies which are fixed order quantity fixed time

Modelling and Simulation in Engineering 5

interval and forecasting methods [35] In the first strategyinventory level should be reviewed continuously until itreaches below predetermined quantity (reorder point) thenorder would be organized with fixed quantity of inventory Inthe second strategy reviewing period is a fixed time intervalbut quantity order is variable for each order that is based onconsumption rate While first strategy needs more effort forcontinuously reviewing of inventory level the second strategyis easier to handle but the risk of shortage in fixed intervalstrategy is more in comparison to fixed order quantity Sodue to themitigation of shortage risk in fixed interval strategyorder quantity is slightly more than fixed order quantity [35]

In the third strategy reviewing period is fixed as sec-ond policy but reorder quantity is based on forecasting offuture demands As thementioned strategies are fundamentalin inventory management literature and are also commonamong retailers of office furniture in this study three policiesbased on fixed order quantity fixed time interval anddemand forecasting are developed as follows

(1) Continuous reviewing with economic quantity order(2) Periodic reviewing with order quantity based on

demand confidence interval during the consumptionperiod

(3) Periodic reviewing with order quantity based onforecasting of future demands

Policy 1 (Fixed Quantity) Based on this policy each demandwill be satisfied if there is sufficient inventory in handAfter satisfaction of each demand inventory in hand willbe checked to see if the inventory level reaches to reorderpoint (119877

119901) If inventory level has reached to reorder point

economic order quantity (119876119895119905)would be organized otherwise

system waits for next demand Economic order quantity isderived by Wilson formula [36] If there is not sufficientinventory to satisfy arrived demand quantity of demandis considered as lost sale and lost profit treated as costIf no order has been organized system reorder inventoryotherwise waits for arriving of organized order accordingto adjusted lead time (119871) Logic of this policy is visible inFigure 1(a) and is labelled as FQ policy

Policy 2 (Fixed Interval) In this policy criterion for reorder-ing is fixed time interval that is known as planning periodAfter this period inventory level would be examined andreorder will be organized on condition that inventory levelhas reached to reorder point (119877

119901) For more realistic consid-

eration order quantity (119876119895119905) is calculated based on demand

cumulative distribution function in planning period Forexample in the inventory system that is planned for 10 lostsale with demand which is distributed based on exponentialdistribution function it should be ordered as much ascumulative probability of exponential distribution equals to09 In this policy inventory level would be updated after leadtime (119871) Logic of this policy is shown in Figure 1(b) and islabelled as FI policy

Policy 3 (Demand Forecasting) This policy is similar to policy2 with some differences In policy 2 probability distribution

function of demand in planning period is estimated byhistorical data and then reorder is organized based on servicelevel (cumulative probability of demand satisfaction) But inthis policy after fixed interval forecasting of future demandwill be performed If inventory level has reached to reorderpoint (119877

119901) order would be organized and inventory level

would be updated after lead time (119871) otherwise only infor-mation would be updated If forecasted demand is less thaninventory capacity order quantity (119876

119895119905) would be as much as

forecasted quantity otherwise inventory orderwill be asmuchas inventory capacity for each product As demands followtime series with autocorrelation forecasting is implementedby exponential smoothing method [37]

In this procedure 119860119905is defined as demand estimation

of 119905th planning period This term is calculated according toreal demand of period 119905 indicated by 119863

119905and estimation of

demand in previous period which is indicated by 119860119905minus1

plustrend adjustment value of previous period which is indicatedby 119879119905minus1

Impact of current real demand is considered byparameter 120572 that is defined between zero and one Trendadjustment value is derived based on 119860

119905and 119860

119905minus1plus trend

adjustment value of previous period with consideration of120573 as weight parameter which is defined between zero andone Equations related to 119860

119905and 119879

119905are expressed in (4)

and (5) respectively Finally demand estimation plus trendadjustment form of future demand forecasting is indicated by119865119905+1

which is described in (6) Logic of this policy is shown inFigure 1(c) and is labelled as DF policy Consider

119860119905= 120572 times 119863

119905+ (1 minus 120572) times (119860

119905minus1+ 119879119905minus1) (4)

119879119905= 120573 times (119860

119905minus 119860119905minus1) + (1 minus 120573) times 119879

119905minus1 (5)

119865119905+1= 119860119905+ 119879119905 (6)

42 Surrogate Modelling Surrogate model is designed foroptimization of simulated inventory system In this frame-work design of experiments is responsible for producingdifferent scenarios Each scenario has four criteria includ-ing cost service level average of inventory position androbustness against demand fluctuation Importance of eachcriterion is determined by PCA and finally these scenariosare ranked by MADM technique as multiple criteria decisionmaking tool

421 Design of Experimentation Although DOE roots backto the statistical quality control nowadays it is a powerfultool for analysing complex systems DOE is statisticalmethodand organizes structured experiments with several factors[38] This method not only determines effect of each factoron response variable but also considers multiple effectssimultaneously DOE is extended technique that includesseveral designs such as full factorial fractional factorial andnested design Specific design varies for each problem andshould be selected based on problem condition Our problemconsists of tree effective factors namely inventory policyreorder point and lead time these factors have nonlineareffect on objective function values As the aim of DOE inthis research is producing decision space with few factors (in

6 Modelling and Simulation in Engineering

Demand arrival

Sufficientinventory isavailable

True

False

Reorderproducts

True

False

Demand satisfaction

Make an order

Passing of lead time

Demand dispose

Lost customer

Is there anyorder in way

True

False

Updating of inventory level

Demand satisfaction

(a) FQ policy

Demand arrival

Sufficientinventory isavailable

Demand satisfaction Lost customer

FalseTrue

Demand dispose

System start

Make order

Make an order

Passing of lead time

TrueFalse

Passing of reviewing period

Updating of inventory level

(b) FI policy

Demand arrival

Sufficientinventory isavailable

Lost customer

System start

Make orderFalse True

Estimatedquantity ispossible

True

False

True False

Update of estimation

Update of information

Order maximum quantity

Order estimated quantity

Passing of lead time

Updating of inventory level

Demand satisfaction

Passing of reviewing period

Demand dispose

(c) DF policy

Figure 1

Modelling and Simulation in Engineering 7

Table 2 DOE configuration

ProductsFactors

IP 119877119905

119871

minus 0 + minus 0 + minus 0 +119860 FI FQ DF 10 20 30 3 5 7119861 FI FQ DF 15 30 45 3 5 7119862 FI FQ DF 15 30 45 3 5 7119863 FI FQ DF 15 30 45 3 5 7

Table 3 Experimental design

Number of trial 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27119877119901

minus minus minus minus minus minus minus minus minus 0 0 0 0 0 0 0 0 0 + + + + + + + + +119871 minus minus minus 0 0 0 + + + minus minus minus 0 0 0 + + + minus minus minus 0 0 0 + + +IP minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 +

this research there are 3 factors) using full factorial design ispreferable On the other hand because nonlinear behaviourof response variables (ie objective functions) centre pointis considered Each factor has high level ldquo+rdquo low level ldquominusrdquoand centre point ldquo0rdquo Factors and levels configurations areshown in Table 2 while Table 3 shows experimental designIn both of these tables minus sign is representative of lowlevel of related factors while plus sign is indicator of high levelof factors and zero determines centre point For examplein the case of lead time (119871) 3 is low level 7 is high leveland 5 is centre point Lead time is considered with daysas measuring unit and begins from time order is organizeduntil the arrival of inventory Reorder point is labelled as 119877

119901

and is intended to satisfy product demand during lead time(119871) Eventually inventory policy is labelled as IP and refersto inventory policies that were described in Section 412The mentioned design includes 27 runs and there are fourresponse variables cost service level average of inventoryposition and robustness In fact first three response variablesare expected values of defined objective functions (ie resp(1) (2) and (3)) but robustness is a defined criterion whichguarantees that optimal solution remains valid in situationwhere demand variation and intensification increase Forevaluation of robustness demands are multiplied by randomvariable with normal distribution (120583 = 1 120590 = 15)This random variable is also restricted to positive values toprevent negative demands Expected value of 4000 generatednumbers with mentioned constraint is equal to 148 thisresult implies that multiplied random number increases bothdemand variation and intensification simultaneously Thenthe percent of decrease in service level is considered asrobustness for all 27 scenarios produced byDOE It is obviousthat a lower decrease in service level is desirable

422 Weight Assignment Based on PCA Method PCA isthe abbreviation of ldquoprincipal component analysisrdquo and isknown as powerful data-driven method successfully appliedin several areas [2] PCA analysis can determine key variablesthat govern most of the variability This technique helps toremove less important variables to reduce dimension of data

sets PCA considers two criteria for each variable in datasets the first is variability and the second is correlation withother variables So input for this method consists of variancecovariance matrix [39 40] Although other methods likeentropy is applicable in multiple attribute decision makingproblem PCA advantage is considerable because variablescorrelations are not ignored As our problem is constructedof highly correlated variables PCA is the preferable choiceFor correlation justification remember that improvementof inventory position criterion causes negative influence onshortage cost or improvement of service level criterion causesnegative influence on inventory in hand and holding cost Inthese circumstances there is no reason for applying variationbased methods like entropy so in this paper PCA is appliedfor weight assignment These weights are used by MADMtechnique (described in Section 423) to improve qualityof ranking Because of relative utility of scenarios linearscaleless method employed to scaleless data obtained fromsimulation

(1) Calculate scaleless matrix of 119863 = (1198891 1198892 119889

119901)119899times119901

which 119889119894119895is scaleless value of 119894th scenario in the view

point of 119895th criterion

(2) Calculate sample mean vector 119889 = (1198891 1198892 119889

119901)119901

and covariance matrix based on (7) and (8) respec-tively

119889119896=1

119899

119899

sum

119894=1

119889119894119895 (7)

119878 = (119904119896119902)119901times119901

=1

119899 minus 1(119863 minus 119889)

119879

(119863 minus 119889) (8)

(3) 1198621radic119904119896119896 is diagonal 119901 times 119901 matrix 119896th diagonal

element is 1radic119904119896119896 and sample correlation matrix (119877)is calculated as mentioned in the following equation

119877 =1198621

radic119904119896119896

sdot 119878 sdot1198621

radic119904119896119896

(9)

8 Modelling and Simulation in Engineering

(4) Solve the following equation where 119868119901is a 119901 times 119901

identity matrix10038161003816100381610038161003816119877 minus 120582119868

119901

10038161003816100381610038161003816= 0 (10)

Solving (10) results in 119875 individual eigenvaluesregarding 120582

1ge 1205822ge sdot sdot sdot ge 120582

119901and sum119901

119896=1120582119896=

119875 These eigenvalues have 119875-specific eigenvectors(119897119896

1 119897119896

2 119897

119896

119901) and 119896 = 1 119901 These vectors create

principal components as expressed in the followingequation

PC119896=

119901

sum

119902=1

119897119896

119902119889119895

119902 (11)

(5) Select sufficient number of principal components Inthis problem first119872 components whose cumulativepercentage of their eigenvalues (119862

119872) is greater than

95 are selected Cumulative percentage is calculatedbased on the following equation

119862119898=sum119898

119896=1120582119896

sum119901

119896=1120582119896

=sum119898

119896=1120582119896

119875 (12)

(6) Finally weights vector that is indicated by119885 is derivedfrom the mathematical relation in (13) while 119882

119896is

equal to 120582119896119875 if the entire elements of PC

119896are

positive otherwise minus120582119896119875 if entire component are

negative

119885 =

119872

sum

119896=1

119882119896PC119896 (13)

In other cases positive or negative 119882119896should be defined

in the order that final weight vector is positive For moredetailed discussion about 119882

119896 refer to [41] Finally for each

product there are four 119885 parameters that determine relativeimportance for the following criteria cost service levelaverage of inventory position and robustness each elementof 119885 vector is divided by the sum of vector elements fornormalization

423 VIKOR Method This method was developed by Opri-covic and is the abbreviation of expression that in Serbianmeans ldquomultiple criteria optimization and compromisingsolutionsrdquo [42] VIKOR method is based on compromisingsolutions which is the result of Yu [43] and Zeleny [44]studies This approach considers closeness to ideal solutionIn comparison with other MADM methods VIKOR is ableto present compromising solutions and substitutes thesesolutions with best one in the case of necessity [45] VIKORmethod is applicable for problems with multiple discretecriteria Generally this method can be considered as rankingtool for scenarios which are generated by DOE Ranking isbased on L-P metric function according to (14) with defini-tion of 119891

119894119895as the value of 119894th criterion for 119895th scenario In

construction of L-P metric function 119891lowast119894indicates best value

of all scenarios in the view point of 119894th criterion and 119891minus119894

represents the worst value of all scenarios with considerationof 119894th criterion 119908

119894is the weight parameter for 119894th criterion

and is derived by PCA method Consider

119871119901=

119899

sum

119894=1

[

119908119894(119891lowast

119894minus 119891119894119895)

(119891lowast

119894minus 119891minus

119894)]

119901

1119901

1 ≪ 119875 ≪ infin 119895 = 1 2 3 119869

119871infin= max

119894[

119908119894(119891lowast

119894minus 119891119894119895)

(119891lowast

119894minus 119891minus

119894)]

(14)

VIKOR method only involves 1198711and 119871

infinwhich are known

as 119878119895and 119877

119895 respectively 119878 and 119877 index refer to scenarios

so both of these functions are regarded for each scenarioVIKOR steps are as follows

(1) Calculate 119891minus119894and 119891lowast

119894for entire of criteria

(2) Calculate 119878119895and 119877

119895for entire of scenarios

(3) With the assumption that 119878lowast and 119877lowast are the smallestvalues among all of 119878

119895and 119877

119895 respectively and also

119878minus and 119877minus are the greatest values in comparison withall of 119878

119895and 119877

119895 respectively 119876

119895is calculated for 119895th

scenario based on (15) In this equation V is strategicweight that determines the importance of individualcriterion (119878

119895) against group importance of criteria

(119877119895) for each scenario Consider

119876119895=

V (119878119895minus 119878lowast

)

119878minus minus 119878lowast+ (1 minus V)

(119877119895minus 119877lowast

)

119877minus minus 119877lowast (15)

(4) Scenarios should be sorted in descending order basedon 119878119877 and119876 specifically that result in three differentranking lists

(5) According to 119876 the scenario which is indicated by 1198861015840is the best scenario if two conditions namely (C1)and (C2) are satisfied

(C1) Acceptable Advantage This condition implies that thesecond best solution (11988610158401015840) should be far enough from thebest solution (1198861015840) to be accepted as unique solution Themathematical equation is defined in (16) considering 119869 as thetotal number of scenarios Consider

119876(11988610158401015840

) minus 119876 (1198861015840

) ≫1

(119869 minus 1) (16)

(C2) Acceptable Stability This condition validates 1198861015840 as thebest stable solution if 1198861015840 is also the best solution based on 119878 or119877

In case that one or both of the mentioned conditions arenot satisfied compromised solutions are involved VIKORmethod suggests that in such case compromising solutionshave equal value to the best solution and could be consideredinterchangeably by decision makers

Modelling and Simulation in Engineering 9

If (C1) is not satisfied scenarios 1198861015840 11988610158401015840 119886119872 are consid-ered 1198861015840 is the best solution and 11988610158401015840 119886119872 are compromisingsolutions with priority of 119876 on the other hand 1198861015840 11988610158401015840are considered if (C2) is not satisfied 119872 superscript isdetermined by (17)

119876(119886119872

) minus 119876 (1198861015840

) ≫1

(119869 minus 1) (17)

5 Numerical Result

Simulation process was executed on laptop with 18 GH CPUand 4GB of RAM Running of each replication with anima-tion and maximum speed takes 7 minutes and 35 secondsAs simulation of each scenario contains 10 replications andthere are 27 scenarios the total simulation time will be 2046minutes and 36 seconds

51 Multiresolution Result for Demand Modelling Result ofdemand modelling is presented for product 119861 because thisanalysis for other products is similar For this problempolynomial functions are applied for yearly and monthlydemand rates Equations (18) and (19) are estimated bynonlinear regression model In these functions 119905

119910and 119905119898are

time variables

119891 (119905119910) = 0008942 times 119905

3

119910minus 003884 times 119905

2

119910

+ 01867 times 119905119910minus 008018

(18)

and 95 confidence interval is presented in Table 4 for eachcoefficient of (18)

In this table coefficients of (18) are visible in the firstcolumn while lower bund and upper bund of each coefficientare presented in the second and third columns respectivelyThere is no interval including zero so it is concluded thatcoefficient estimation is valid

Estimation error for 119891(119905119910) is reported based on the sum

of square errors (SSE) which is 1504 times 10minus6 Consider

119891 (119905119898) = 4768 times 10

minus5

times 1199055

119898minus 0001692 times 119905

4

119898+ 00214

times 1199053

119898minus 01159 times 119905

2

119898+ 03474 times 119905

119898minus 02286

(19)

Monthly demand rate function that is indicated by 119891(119905119898)

and its coefficients confidence interval are presented inTable 5 SEE is equal to 00005825 Based on local businessinformation total demand of product 119861 is estimated to beabout 6822 units in five years According to the informationabout product 119861 Figure 2 shows final multiresolution resultfor this product

52 Simulation Parameters and Configuration For simula-tion of problem it is necessary to configure inventory policyparameters and variables We have three different policiesnamely FQ FI and DF Variables of FQ policy are reorderpoint (119877

119901) lead time (119871) and economic quantity orders (119876

119895119905)

First two variables are configured based on Table 2 and thirdvariable is derived fromWilson formula according to average

Table 4 95 confidence interval for estimated coefficient of 119891(119905119910)

Coefficients Lower bund Upper bund0008942 0004835 001305minus003884 minus007604 minus000165201867 008654 0287minus008018 minus01568 minus0003523

Table 5 95 confidence interval for estimated coefficient of 119891(119905119898)

Coefficients Lower bund Upper bund4768 times 10minus5 4451 times 10minus5 5086 times 10minus5

minus0001692 minus0001795 minus00015900214 00202 00226minus01159 minus01221 minus0109703474 0334 03608minus02286 minus02371 minus022

Table 6 Estimation of distribution functions of demand and reor-ders quantity

Product type Distribution function 119875 value Order quantity119860 278 lowast BETA(113 328) gt015 100119861 GAMMA(678 118) gt015 92119862 EXPO(818) gt015 100119863 446 lowast BETA(0957 321) gt015 115

of demand holding cost (119862ℎ) and ordering cost (119862

119903) The

calculated values are 57 71 72 and 72 for product 119860 119861 119862and119863 respectively

Orders quantity is unknown for FI policy and is cal-culated based on probability distribution function whichis obtained by demand simulation of each product for 20daysrsquo time interval Distribution fitting is performed by inputanalyser of Arena software and graphical result for product 119861is shown in Figure 3 Results for other products are reportedin Table 6

In this table fitted probability distribution function foreach product is presented in second column while 119875 valueof fitting which is greater than 015 for fitted distributiongrantees goodness of fitting in 95 confidence level Finallyaccording to fitted distribution functions for FI policyproper ordering amount of each product is presented in thefourth column

DF policy has two unknown parameters namely 120572 and120573 The first parameter is weight of current real demand in (4)and second one is the weight of current trend in (5) Theseparameters are adjusted by simulation of five years demandDemand for the entire four products simultaneously is con-sidered and an average of lost sale percentage is consideredas response variable Different values of response variables arereported in Table 7 According to the different values of 120572 and120573 the best values for 120572 and 120573 are 09 and 01 respectivelywhich results in minimum lost percentage in average

10 Modelling and Simulation in Engineering

Table 7 Average of lost sale percentage for different values of 120572 and 120573

120572 and 120573 values 01 02 03 04 05 06 07 08 09Response to 120572 0380 0356 0334 0320 0306 0292 0284 0276 0268Response to 120573 0296 0302 0306 0306 0306 0304 0305 0304 0302

50

100

150

200

250

300

350Demand fit for each month

Dem

and

Fitted valueReal demand

00 10 20 30 40 50 60

Month

Figure 2 Comparison between real and estimated demand

100 200 300 400 500 600 700

50

100

150

200

Distribution summary

Distribution gammaExpression minus0001 + GAMM(00)Square error 0006825

Chi-square testNumber of intervals = 5Degrees of freedom = 2Test statistic = 148

Kolmogorov-Smirnov testTest statistic = 00733

Corresponding P value = 0483

Corresponding P value gt 015

Figure 3 Distribution function of customer demand for reorderinterval of product 119861

53Weight Assignment Asmentioned before PCA is appliedforweight assignment For this purposeDOE result is neededin the form of decision matrix that is shown in Table 8 Inthis table four criteria cost average of inventory positionservice level and robustness are evaluated for each scenarioof product 119861 Decision matrix should be scaleless that isperformed by linear method Then the result of eigenvalues

Table 8 Decision matrix for product 119861

Scenarionumber Cost Average of

inventory positionServicelevel

Robustness

1 32984000 4352 76 62 29214710 397514 79 33 37114636 339623 69 24 34840720 410948 71 55 36820570 344738 72 66 39191837 331091 67 27 37037180 391012 68 48 40994680 298712 64 69 41407045 319678 63 110 28282930 524462 83 711 26422145 489243 85 512 31023829 416049 79 313 30395190 487146 79 614 35393680 395812 74 515 32838055 395646 77 316 33285615 463341 76 617 39652020 335668 66 718 35269609 383944 73 219 26227680 625612 87 720 25565820 586688 88 621 28711069 49449 85 422 27301580 585472 85 723 34016415 470961 78 724 30601546 463975 82 325 29752815 560421 82 726 41024410 382994 68 627 31604741 443053 78 3

is calculated and shown in Table 9 Also result of principalcomponents is presented in Table 10

As it is visible in Table 9 cumulative proportion ofthe first and second components is 0966 which is greaterthan 095 so the first two components are considered asprincipal components Their relative weights are 078 and0186 respectively Final weights are normalized and resultsare reported in Table 11

54 Ranking and Selection of Scenarios Weights of criteriaobtained by PCA method in addition to decision matrix areinput of VIKOR method 119878 119876 and 119877 values are calculatedbased on (14) and (15) while strategic weight indicated byV is equal to 05 so individual and group utility have same

Modelling and Simulation in Engineering 11

Table 9 Result of eigenvalues

Eigenvalues 31204 07428 01111 00257Proportion 0780 0186 0028 0006Cumulative 0780 0966 0994 1000

Table 10 Result of principle components

Variables PC1 PC2 PC3 PC4Inventory 0543 0089 minus0818 0167Cost minus0538 minus0277 minus0511 minus0610Service level minus0551 minus0210 minus0230 0774Robustness 0335 minus0933 0126 0023

Table 11 Weights of criteria for different products

Criteria Products119860 119861 119862 119863

Average of inventory position 01012 00590 01942 00462Cost 02852 01836 01742 03081Service level 01195 01392 00966 01206Robustness 04941 06182 05350 05251

importance Calculated values are reported in Table 12 forentire scenario as VIKOR output

Product 119861 According to Table 12 values of 119877 119878 and 119876 arezero for scenario 18 Regarding (C1) (Acceptable Advantage)the difference between the best scenario and second rankedscenario should be more than 00038 The second rankedscenario regarding 119876 value is scenario 3 and its value is0051044 that satisfies both (C1) and (C2) Final result forproduct 119861 is unique scenario that is number 18 and consistsof the following configuration Inventory policy is DFwith119877

119901

being equal to 30 products and 7 days lead time According todecision matrix which is shown in Table 8 the performanceof this scenario for product 119861 results in 383944 products asaverage of inventory position and expected cost is equal to35269609 for product life cycle 73 of customer demandwillbe satisfied and in the case of increasing demand intensityand variation 2 decrease is expected in service level so itwill change into 71 Analyses for other products are similarso only final results are presented in this paper

Product 119860 In this case VIKORmethod ranks scenario num-ber 15 as the best one This scenario consists of DF inventorypolicy 20 products for 119877

119901and 5 days for lead time Perfor-

mance of this scenario results in 374778 products as averageof inventory position Expected cost is 31603294 and servicelevel will be 66 This scenario is robust against demandintensification and variation because changes in demandaffect service level as small as 1 But scenario number 15 doesnot satisfy (C1) so compromising solutions are consideredAlthough the best scenario is number 15 VIKOR methodimplies that compromising solutions have the same values

Table 12 VIKOR output for product 119861

Run number 119878 119877 119876

1 0634311 0796269 0715292 0003184 0185077 0094133 0059804 0042283 00510444 0536946 0592539 05647425 0718862 0796269 07575656 0116278 0089887 01030827 0428814 0388808 04088118 086078 0796269 08285259 002043 0140653 008054110 0677386 1 083869311 0266254 0592539 042939612 0043962 0185077 01145213 0571695 0796269 068398214 0515097 0592539 055381815 009173 0185077 013840416 0648693 0796269 072248117 1 1 118 0 0 019 0630698 1 081534920 04239 0796269 061008421 0139527 0388808 026416722 06581 1 08290523 0819338 1 090966924 0022245 0185077 010366125 0726183 1 086309126 0849516 0796269 082289327 0072768 0185077 0128922

So in real situation compromising solution can be substi-tuted This solution regarding their priority is as follows

Scenario number 24 inventory policy is DF 119877119901is

equal to 30 products and lead time is 5 daysScenario number 18 inventory policy is DF 119877

119901is

equal to 20 products and lead time is 7 daysScenario number 12 inventory policy isDF119877

119901is equal

to 20 products and lead time is 3 days

Product 119862 For this product scenario number 1 is ranked asthe best one This scenario consists of FI inventory policy 119877

119901

is 15 products and lead time is 3 days Result of this scenario is4387 products as the average of inventory position Expectedcost is 66143000 and service level will be 61 with robustnessof 1 Compromising solutions are as follows

Scenario number 6 inventory policy is DF 119877119901is 15

products and lead time is 5 daysScenario number 3 inventory policy is DF 119877

119901is 15

products and lead time is 3 daysScenario number 7 inventory policy is FI 119877

119901is 15

products and lead time is 7 days

12 Modelling and Simulation in Engineering

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 4 Interaction plot for product 119862 average of inventory position

Scenario number 4 inventory policy is FI 119877119901is 15

products and lead time is 5 days

Product119863 Scenario number 3 has the best rank for this prod-uct This scenario consists of DF inventory policy 119877

119901is 15

products and lead time is 3 days This scenario leads to

363896 products as average of inventory position with62648337 expected cost and 61 service level with 1robustness Compromising solutions are as follows

Scenario number 6 this scenario is mentioned beforeas a compromising solution for product C

Modelling and Simulation in Engineering 13

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 5 Interaction plot for product 119862 cost

14 Modelling and Simulation in Engineering

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

3

5

7

L

3

5

7

L

3

5

7

L

L

(b)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 6 Interaction plot for product 119862 service level

Scenario number 17 inventory policy is FI119877119901is equal

to 15 products and lead time is 7 days

55 Sensitivity Analysis As mentioned in the previous sec-tion all scenarios related to product 119860 prefer DF inventory

policy so it is concluded that DF policy is unique optimalpolicy while lead time includes values of 3 5 and 7 in selectedscenarios This situation implies that product 119860 is insensitiveto the delivery time It can be wise to decide lead time about 5days but there is no need for strict control on lead time while

Modelling and Simulation in Engineering 15

119877119901should be controlled strictly to prevent inventory in hand

from reaching lower than 20 productsIn the case of product 119861 (C1) is satisfied so there is a

unique scenario while there are four compromising solutionfor product 119862 In this situation based on VIKOR methodcompromising solutions have same value while interactionplot can be applied for investigation of existing differencebetween compromising solutions As mentioned before thebest scenario suggests FI inventory policy for this product butin compromising solutions there are both FI and DF policiesLead time varies from 3 to 7 days So this confusing situationis resolved by deeper analysis and applying interaction plotFigure 4 shows interaction plot for DOE factors and averageof inventory position as response variables

Figure 4 shows that with either DF or FI policy there isno significant difference among different lead times in theview point of inventory position but DF leads to less averageof inventory position With 119877

119901equaling 15 there is less

sensitivity to different lead times So configuring 119877119901which

is equal to 15 products the average of inventory positioncriterion will be robust against lead time variations

Figure 5 regards cost criterion and shows that FI andDF policies are equal and make least sensitivity related tolead time On the other hand three plots of 119877

119901against lead

time are parallel The parallel plots are interpreted as lackof relation between 119877

119901and lead time With 119877

119901equaling 15

increase in inventory cost could be mitigated when lead timesets to 3 days

In Figure 6 with 3 days for lead time there is no signifi-cant difference between inventory policies in the view pointof service level119877

119901and lead time are independent and the best

service level is related to FI policy which is also very sensitiveto 119877119901 As 119877

119901increases service level will grow According to

the interaction plot analyses optimal decision for product119862 is FI inventory policy 3 days lead time and 15 or moreproducts for 119877

119901

Decision about product 119863 is simple Based on the in-formation obtained from solutions it is recommended toemploy DF inventory policy with 119877

119901being equal to 15

products and 3 days for lead time

6 Conclusion

In this paper a simulation optimization framework is pro-posed for robust optimization of retailer inventory systemThis framework is less time consuming in comparison withmetaheuristic or SPSA approach it also provides robustsolution for multiple objective functions In this research alsotrend and cyclic change of customers demand are consideredfor more realistic view of problem Proposed framework con-sists of discrete event simulation and surrogate model In thisframework full factorial design of experiment is employedfor producing decision space Based on the three decisionfactors 27 different scenarios are produced and simulated Insimulation modelling multiresolution method is applied forsimulation of demand with highly dynamic pattern Becausethere are multiple criteria for inventory system performanceVIKORmethod is employed asmulticriteria decisionmaking

technique In addition robustness is considered as perfor-mance criterion and PCA is used for improving performanceof VIKOR method Finally developed framework gave thebest ranked and compromising solutions so interaction plotapplied for more investigation and sensitivity analysis ofobtained solutions

Due to the novelty of the proposed framework it isrecommended that future studies encompass optimization ofinventory system of integrated supply chain Also improve-ment of developed framework can be considered as futurestudy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers for theirsignificant roles in improvement of this research quality

References

[1] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[2] S Yin S XDingAHaghaniHHao andP Zhang ldquoA compar-ison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[3] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[4] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[5] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall International Series in Management New YorkNY USA 1963

[6] E Naddor Inventory Systems John Wiley amp Sons New YorkNY USA 1966

[7] R Peterson and E A Silver Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 2nd edition 1985

[8] T C E Cheng ldquoEconomic order quantity model with demand-dependent unit production cost and imperfect productionprocessesrdquo IIE Transactions vol 23 no 1 pp 23ndash28 1991

[9] S H Chen C CWang and A Ramer ldquoBackorder fuzzy inven-tory model under function principlerdquo Information Sciences vol95 no 1-2 pp 71ndash79 1996

[10] X Zhao J Xie and J C Wei ldquoThe value of early order com-mitment in a two-level supply chainrdquo European Journal of Op-erational Research vol 180 no 1 pp 194ndash214 2007

[11] R S M Lau J Xie and X Zhao ldquoEffects of inventory policyon supply chain performance a simulation study of critical

16 Modelling and Simulation in Engineering

decision parametersrdquo Computers and Industrial Engineeringvol 55 no 3 pp 620ndash633 2008

[12] J Xu and L Zhao ldquoA multi-objective decision-making modelwith fuzzy rough coefficients and its application to the inventoryproblemrdquo Information Sciences vol 180 no 5 pp 679ndash6962010

[13] F Hnaien X Delorme and A Dolgui ldquoMulti-objective opti-mization for inventory control in two-level assembly systemsunder uncertainty of lead timesrdquo Computers amp OperationsResearch vol 37 no 11 pp 1835ndash1843 2010

[14] H D Purnomo H M Wee and Y Praharsi ldquoTwo inventoryreview policies on supply chain configuration problemrdquo Com-puters and Industrial Engineering vol 63 no 2 pp 448ndash4552012

[15] B L Nelson ldquo50th anniversary article stochastic simulationresearch in management sciencerdquoManagement Science vol 50no 7 pp 855ndash868 2004

[16] J Boesel R O Bowden Jr F Glover J P Kelly and E WestwigldquoFuture of simulation optimizationrdquo inProceedings of theWinterSimulation Conference pp 1466ndash1469 Arlington Va USADecember 2001

[17] M C Fu ldquoOptimization for simulation theory vs practicerdquoINFORMS Journal on Computing vol 14 no 3 pp 192ndash2152002

[18] L Wang ldquoA hybrid genetic algorithm-neural network strategyfor simulation optimizationrdquoApplied Mathematics and Compu-tation vol 170 no 2 pp 1329ndash1343 2005

[19] B B Keskin S H Melouk and I L Meyer ldquoA simulation-optimization approach for integrated sourcing and inventorydecisionsrdquo Computers and Operations Research vol 37 no 9pp 1648ndash1661 2010

[20] E Mazhari J Zhao N Celik S Lee Y Son and L HeadldquoHybrid simulation and optimization-based design and oper-ation of integrated photovoltaic generation storage units andgridrdquo Simulation Modelling Practice and Theory vol 19 no 1pp 463ndash481 2011

[21] Q Duan and T W Liao ldquoOptimization of replenishment poli-cies for decentralized and centralized capacitated supply chainsunder various demandsrdquo International Journal of ProductionEconomics vol 142 no 1 pp 194ndash204 2013

[22] J C Spall ldquoMultivariate stochastic approximation usinga simultaneous perturbation gradient approximationrdquo IEEETransactions on Automatic Control vol 37 no 3 pp 332ndash3411992

[23] J C Spall ldquoImplementation of the simultaneous perturbationalgorithm for stochastic optimizationrdquo IEEE Transactions onAerospace and Electronic Systems vol 34 no 3 pp 817ndash8231998

[24] J C Spall ldquoStochastic optimization and the simultaneousperturbation methodrdquo in Proceedings of the Winter SimulationConference (WSC rsquo99) pp 101ndash109 December 1999

[25] J D Schwartz W Wang and D E Rivera ldquoSimulation-based optimization of process control policies for inventorymanagement in supply chainsrdquo Automatica vol 42 no 8 pp1311ndash1320 2006

[26] H G Neddermeijer G J V Oortmarssen and N P R DekkerldquoA framework for response surface methodology for simulationoptimizationrdquo in Proceedings of the Winter Simulation Confer-ence vol 1 Orlando Fla USA December 1999

[27] B Can and C Heavey ldquoA comparison of genetic programmingand artificial neural networks in metamodeling of discrete-event simulation modelsrdquo Computers amp Operations Researchvol 39 no 2 pp 424ndash436 2012

[28] A Azadeh Z S Faiz S M Asadzadeh and R Tavakkoli-Moghaddam ldquoAn integrated artificial neural network-comput-er simulation for optimization of complex tandem queuesystemsrdquoMathematics and Computers in Simulation vol 82 no4 pp 666ndash678 2011

[29] R Bornatico J Hussy A Witzig and L Guzzella ldquoSurrogatemodeling for the fast optimization of energy systemsrdquo Energyvol 57 pp 653ndash662 2013

[30] X Wan J F Pekny and G V Reklaitis ldquoSimulation-basedoptimizationwith surrogatemodels application to supply chainmanagementrdquoComputers and Chemical Engineering vol 29 no6 pp 1317ndash1328 2005

[31] W Shi Z Liu J Shang and Y Cui ldquoMulti-criteria robust designof a JIT-based cross-docking distribution center for an autoparts supply chainrdquo European Journal of Operational Researchvol 229 no 3 pp 695ndash706 2013

[32] G Dellino J P C Kleijnen and C Meloni ldquoRobust optimiza-tion in simulation taguchi and Response Surface Methodol-ogyrdquo International Journal of Production Economics vol 125 no1 pp 52ndash59 2010

[33] T Yang Y Wen and F Wang ldquoEvaluation of robustness of sup-ply chain information-sharing strategies using a hybrid Taguchiand multiple criteria decision-making methodrdquo InternationalJournal of Production Economics vol 134 no 2 pp 458ndash4662011

[34] M E Kuhl and J R Wilson ldquoModeling and simulating Poissonprocesses having trends or nontrigonometric cyclic effectsrdquoEuropean Journal of Operational Research vol 133 no 3 pp566ndash582 2001

[35] P Narayan and J Subramanian Inventory ManagementmdashPrinciples and Practices Excel Books New Delhi India 2008

[36] E A Silver and R Peterson Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 1985

[37] R G Brown Smoothing Forecasting and Prediction of DiscreteTime Series Courier 2004

[38] D C Montgomery Design and analysis of experiments JohnWiley amp Sons New York NY USA 6th edition 2005

[39] J Zhu ldquoData envelopment analysis vs principal componentanalysis an illustrative study of economic performance ofChinese citiesrdquo European Journal of Operational Research vol111 no 1 pp 50ndash61 1998

[40] I M Premachandra ldquoA note on DEA vs principal componentanalysis an improvement to Joe Zhursquos approachrdquo EuropeanJournal of Operational Research vol 132 no 3 pp 553ndash5602001

[41] D Slottje G W Scully G J Hirschberg and K Hayes Mea-suring the Quality of Life across Countries A MultidimensionalAnalysis Westview Press Boulder Colo USA 1991

[42] S Opricovic Multicriteria optimization of civil engineeringsystems [PhD thesis] Faculty of Civil Engineering BelgradeSerbia 1998

[43] P L Yu ldquoA class of solutions for group decision problemsrdquoManagement Science vol 19 no 8 pp 936ndash946 1973

Modelling and Simulation in Engineering 17

[44] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[45] S Opricovic and G H Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

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Page 2: Research Article Robust Inventory System Optimization ...downloads.hindawi.com/journals/mse/2014/305120.pdf · Robust Inventory System Optimization Based on Simulation and Multiple

2 Modelling and Simulation in Engineering

is categorized in two separated parts for providing moresupportive literature review

21 Inventory Problem Basic problems of inventory systemare studied thoroughly by [5ndash7] Economical order quantity(EOQ) is the simplest model of inventory problem inthis model demand of each period is constant and timeindependent EOQ does not consider lost sale back orderand other cost for reason of simplification and emphasizesonly on holding and ordering cost Cheng [8] developeda model of inventory system with cost-dependent demandand included production cost in proposed model Chen etal [9] studied back order by fuzzy technique and Zhao etal [10] suggested analytic model with demand according totime seriesThey concluded that in time-dependent demandefficiency of EOQmodel increases with shorter lead time andweaker autocorrelation

Basicmodels thatwere offered for inventory problemonlyhave one objective function including different inventorycosts while advancement in technology and intensification ofbusiness competition caused necessity of other criteria To thebest of our knowledge in recent years service level has beenobserved in variety of supply chain and inventory problem asperformance criterion Adding service level criterion classicdefinition of inventory problem is changed into multiobjec-tive optimization In such problem inventory costs shouldbe minimized while service level should be maximizedAvailable models for this problem are divided in two groupsof deterministic and stochastic Both of these models can besolved by three approaches of analytic methods for examplemathematical programming metaheuristic methods andsimulation Lau et al [11] benefited from simulation to com-pare four inventory management policies with two criteriaof cost and service levels They also surveyed preorder andinformation sharing impacts on their models Xu and Zhao[12] used fuzzy rough simulation to optimize multiobjectiveproblem to minimize wasted cost and maximize expectedvalue of revenue Hnaien et al [13] Surveyed two-level as-sembly system with two objectives of service level andmaintenance cost They considered stochastic lead time andapplied genetic algorithm to solve this problem

Although cost and service levels are important criteria forinventory system performance there are other criteria thatshould be considered such as amount of systems inventorywhich is important factor with significant impact on inven-tory system behavior Because of lead time uncertainty thatorigins from natural disasters and transportation problemsorganizations face delay in delivery so most of them holdsafety stocks This phenomenon is the main cause that leadsto the increase of inventory in hand As inventory in handincreases the inventory system is faced with holding cost andother problems like decrease in quality lack of flexibility andso on So amount of systems inventory can be consideredas performance factor and inventory in hand should beminimized as mentioned in the just in time (JIT) philosophyPurnomo et al [14] researched about influence of periodicreplenishment and continue replenishment inventory poli-cies on supply chain and considered both inventory in handand work in progress as performance factors

Now in recent yearsmodeling of inventory system asmul-tiobjective and stochastic problem is an interesting area forresearch

22 Simulation Optimization One of the well-known simu-lation methods is discrete event simulation that is based onstochastic processes and could be efficient tool to capturestochastic behaviour of different systems While discreteevent simulation has several advantages it is not optimizationtool individually [15] By the way because of its flexibilitysimulation can be coupled with other techniques such asmetaheuristic algorithms or stochastic methodsThis synthe-sizes makes the powerful and advantageous approach of sim-ulation optimization with vast area of research Simulationoptimization is powerful arsenal for optimization of complexsystems such as military aerospace and supply chain [16]To the best of our knowledge there are three main opti-mization techniques that were reported as suitable techniquesfor simulation optimization These considered methods aremetaheuristic optimization stochastic approximation (SA)methods and surrogate models

Fu [17] extensively described role of applied methodsin simulation optimization and also surveyed techniquesemployed in optimization package of simulation softwareWang [18] used hybrid approach including genetic algorithmand artificial neural network for simulation optimization andKeskin et al [19] applied discrete event simulation and scattersearch algorithm for optimization of inventory system andvendor selection Mazhari et al [20] developed a simulationoptimization framework based on hybrid simulation model(system dynamic and agent based model) and metaheuristicalgorithm Also Duan and Liao [21] applied metaheuristicapproach for developing simulation optimization frameworkin order to optimize replenishment policy of inventorysystem in capacitated supply chain

Although using metaheuristic optimization algorithms isstraight forward approach for simulation optimization it istime consuming and needs high level of computational effortSo it is inefficient in case of simulation optimization withmore than one objective function

While metaheuristic algorithms use stochastic searchingmethods SA is based on gradient search Because of noisysituation of observations SA algorithms consider expectedvalue of objective function SA family includes attractivemethods because their convergence is guaranteed theoret-ically Simultaneous perturbation stochastic approximation(SPSA) is noteworthy algorithm of SA familyThe theoreticalaspects of SPSA are deeply described by Spallrsquos [22ndash25]proposed simulation optimization framework for inventorycontrol in supply chain based on SPSA

In contrary to two former methods surrogate modellingis postprocessing method so it is less time consuming Insurrogate modelling the main idea is to fit single surface tothe decision space and use this surface instead of simula-tion model for optimization In this area response surfacemethodology (RSM) [26] and supervised learning methods(eg artificial neural network or support vector machine) areconsiderable For instance Can and Heavey [27] applied arti-ficial neural network to develop surrogate model for discrete

Modelling and Simulation in Engineering 3

event simulation Azadeh et al [28] used artificial neural net-work for the designing of simulation optimization frameworkand they applied proposed framework for optimization ofwaiting time in tandem queue systems Bornatico et al [29]proposed a surrogate model based on redial basis functionfor simulation optimization of energy systems Wan et al[30] designed simulation optimization framework using leastsquare support vector machine (LSSVM) for optimizationof inventory level in three-stage supply chain They alsoshowed that proposed framework leads to better solutionwith less number of simulation runs in comparison withSPSA algorithm Surrogate modelling is less time consumingin comparison to metaheuristic or SPSA approaches butthis approach loses accuracy in multiple objectives problemsolving

Although a bunch of papers published in simulationoptimization area to the best of our knowledge a tiny numberof them are dedicated to the multiobjective optimization [31]and robustness [32 33] In this case using metaheuristicand SPSA approaches is very time consuming and it is noteconomic for optimization of simulationmodel with accuratedetails On the other hand all of the reviewed approacheslose their accuracy when there are multiple objectives Withthese considerations this paper purposes a framework foroptimization of detailed simulation model of inventory sys-tem with multiple objectives Proposed framework is lesstime consuming in comparison with metaheuristic or SPSAapproaches while it provides robust and accurate solutionsSo the proposed framework is relatively new and contribu-tion of this research entails threefold as follows

(i) We modelled cyclic and long-term demand basedon nonparametric time series modelling for morerealistic consideration

(ii) We proposed surrogate model for robust and multi-objective optimization ofmultiproduct inventory sys-tem based on discrete event simulation full factorialdesign of experiments (DOE) and multiple attributedecision making (MADM) technique

(iii) Due to the stochastic nature of objective functionwe employed principal component analysis (PCA) asstatistical method to improve MADM performance

3 Problem Statement

The problem is concerning retailer who sales office furnitureand facility The retailer sales four products respectively119860 119861 119862 and 119863 the aim is the optimization of inventorysystem according to information which is adapted from localbusiness Key features of retailer products from inventoryview point are as shown in Table 1

In this table second column gives average demand ofeach product type in a year third column provides holdingcost of each product in a planning period fourth columnis dedicated to ordering cost of each type of products andfinally numbers of fifth column are cost of lost sales which areincurred to retailers when they cannot satisfy the demand ofcustomers for each type of products

The fundamental assumption that should be consideredin this problem is as follows

(1) Order cost for each type of products includes trans-portation and order registration cost

(2) There is no backlog inventory so inventory level isnonnegative all the time

(3) Profit of each product is considered as lost sale costbecause unavailable products incur lost profits thatare interpreted as cost of lost sale

(4) According to the adapted information these productshave five years life cycle and then will be substitutedwith new products

(5) Planning periods for system under study are as longas 20 days

The notations that will be used to describe the problems areas follows

Indices Consider the following

119905 index of planning periods 119905 = 1 2 3 119879119894 index of demands in planning period 119894 = 1 2 3 119873119895 index of orders in planning period 119895 = 1 2 3 119872

Parameters Consider the following

119862119903 reorder cost

119862ℎ holding cost of each product in planning period

119862119897 cost of lost sale for each product

Variables Consider the following

119868119897

119905 inventory level in 119905th planning period119868119901

119905 inventory position in 119905th planning period

119889119894119905 quantity of 119894th demand in 119905th planning period

119899119905 number of reorder in 119905th planning period

119909119894119905 1 if 119889

119894119905is less than 119868119897

119905 0 otherwise

119876119895119905 quantity of 119895th order in 119905th planning period

119871 lead time for organized orders119877119901 reorder point

Demand of each product in the planning period (119889119894119905) is

stochastic variable and is generated by nonhomogeneousPoisson distribution So total number of arrived demandin planning period (119873) is probabilistic 119862

119903 119862ℎ and 119862

119897are

different costs of inventory system according to Table 1 119868119897119905is

inventory level and refers to physical quantity of inventorywhich is available in retailer while 119868119901

119905is the position of inven-

tory and includes quantity of on-order inventory in additionto inventory level in planning period 119909

119894119905is binary variable

which is one if inventory level is greater than the arriveddemand So if 119909

119894119905is one 119889

119894119905can be satisfied and otherwise

4 Modelling and Simulation in Engineering

Table 1

Product type Average demand Holding cost in planning period Reorder cost Lost sale cost119860 987 30000 50000 20000119861 1520 30000 50000 15000119862 1598 30000 50000 25000119863 1569 30000 50000 18000

it is lost sale 119877119901is the reorder point for organizing of new

order In other words if inventory level reaches 119877119901or less

a new order would be organized with quantity of 119876119895119905 Each

organized order reaches the retailer and increases inventorylevel after passing of lead time (119871) Considering assumptionsand described notations the following equations are themainobjective functions of defined problem

Min119879

sum

119905=1

119862119903119899119905+

119879

sum

119905=1

119862ℎ119868119897

119905+

119879

sum

119905=1

119873

sum

119894=1

119862119861(1 minus 119909

119894119905) 119889119894119905

(1)

Maxsum119879

119905=1sum119873

119894=1119909119894119905119889119894119905

sum119905

119905=1sum119873

119894=1119889119894119905

(2)

Minsum119879

119905=1119868119901

119905

119879 (3)

In (1) the objective is minimizing the total cost includingcosts that depend on reordering handling and lost salesLost sales not only incur excess cost but also decreaseretailer credit So (2) is considered to maximize service levelindependently In (2) the objective function increases where119909119894119905is 1 for 119894th demand in 119905th period and such situation is

possible if 119868119897119905is greater than 119889

119894119905 In fact (2) causes increase

in inventory level while (1)ndash(3) causes decrease in inventorylevel State of inventory level depends on number of ordersin each planning period (119899

119905) quantity of orders (119876

119895119905) and

quantity of demands (119889119894119905) while state of inventory position

depends on inventory level and lead time (119871) so (3) isresponsible for minimizing average of inventory positionincluding inventory in hand (119868119897

119905) and on-order inventory

Average of inventory position should be minimized in orderto improve flexibility of retailer and approach to the just intime (JIT)

Demand of each product follows different pattern withboth long-term and cyclic trend So mentioned objectivefunctions are considered individually for each product type

4 Proposed Framework

In the defined problem demand of products exposes highlydynamic pattern and as time passes demand and its variationincrease hence multiresolution method is employed fordemand modelling Also three different policies for inven-tory control are considered which are reordered based onfixed quantity (FQ) fixed interval (FI) and demand forecast-ing (DF) Simulation of developed model is implemented inArena 135 Optimization of simulation model is performed

by surrogate model that is based on full factorial design ofexperiment (DOE) For construction of decision space DOEfactors include inventory policy reorder point and lead timewith three levels for each of them So there are 33 = 27

different combinations of decision variables to form feasiblescenarios Ranking of produced scenarios is accomplished byMADM technique For ranking of scenarios three objectivefunction values are considered (ie cost service level andaverage of inventory position) and in addition robustness ofservice level against demand fluctuation is considered AlsoPCA is applied for more realistic weighting of objective func-tion values based on their statistical influence on improve-ment of other objectives Finally interacting plot is employedfor sensitivity analysis and investigation of solutions in detail

41 Simulation Modelling Simulation modelling of problemconsists of two parts which are modelling of demand andmodelling of inventory policies In this paper demand is non-homogeneous Poisson process and three different inventorypolicies based on continues reviewing periodic reviewingand periodic reviewing with forecasting of future demand areconsidered

411 Modelling of Demand For customers demand model-ing multiresolutionmethod is applied Kuhl andWilson [34]developed this method for simulation of nonhomogeneousPoisson process with trend and cyclic changes This methodestimates mean intensity function and the nonparametricnature of thismethod is one of themost important advantagesin comparison with other methods So it is independent ofstatistical parameters and applicable in variety of problemsFurthermore multiresolution method can support combi-nation of multiple cyclic changes simultaneously Anotheradvantage of multiresolution is its capability in the modellingof nonsymmetric cyclic pattern As in our case demandhas nonsymmetric pattern with cyclic changes and long-term trend and multiresolution approach is a reasonablechoice More theoretical and application of used method areprovided in [34]

412 Modelling of Inventory Policies As the main effort ofthis paper is inventory system optimization modelling ofinventory policies plays an important role in this problemIn this problem optimization is manipulated by selection ofappropriate inventory policy and configuration of its param-eters to the way that leads to the optimal state of inventorysystem In the inventory management three approaches arecommon strategies which are fixed order quantity fixed time

Modelling and Simulation in Engineering 5

interval and forecasting methods [35] In the first strategyinventory level should be reviewed continuously until itreaches below predetermined quantity (reorder point) thenorder would be organized with fixed quantity of inventory Inthe second strategy reviewing period is a fixed time intervalbut quantity order is variable for each order that is based onconsumption rate While first strategy needs more effort forcontinuously reviewing of inventory level the second strategyis easier to handle but the risk of shortage in fixed intervalstrategy is more in comparison to fixed order quantity Sodue to themitigation of shortage risk in fixed interval strategyorder quantity is slightly more than fixed order quantity [35]

In the third strategy reviewing period is fixed as sec-ond policy but reorder quantity is based on forecasting offuture demands As thementioned strategies are fundamentalin inventory management literature and are also commonamong retailers of office furniture in this study three policiesbased on fixed order quantity fixed time interval anddemand forecasting are developed as follows

(1) Continuous reviewing with economic quantity order(2) Periodic reviewing with order quantity based on

demand confidence interval during the consumptionperiod

(3) Periodic reviewing with order quantity based onforecasting of future demands

Policy 1 (Fixed Quantity) Based on this policy each demandwill be satisfied if there is sufficient inventory in handAfter satisfaction of each demand inventory in hand willbe checked to see if the inventory level reaches to reorderpoint (119877

119901) If inventory level has reached to reorder point

economic order quantity (119876119895119905)would be organized otherwise

system waits for next demand Economic order quantity isderived by Wilson formula [36] If there is not sufficientinventory to satisfy arrived demand quantity of demandis considered as lost sale and lost profit treated as costIf no order has been organized system reorder inventoryotherwise waits for arriving of organized order accordingto adjusted lead time (119871) Logic of this policy is visible inFigure 1(a) and is labelled as FQ policy

Policy 2 (Fixed Interval) In this policy criterion for reorder-ing is fixed time interval that is known as planning periodAfter this period inventory level would be examined andreorder will be organized on condition that inventory levelhas reached to reorder point (119877

119901) For more realistic consid-

eration order quantity (119876119895119905) is calculated based on demand

cumulative distribution function in planning period Forexample in the inventory system that is planned for 10 lostsale with demand which is distributed based on exponentialdistribution function it should be ordered as much ascumulative probability of exponential distribution equals to09 In this policy inventory level would be updated after leadtime (119871) Logic of this policy is shown in Figure 1(b) and islabelled as FI policy

Policy 3 (Demand Forecasting) This policy is similar to policy2 with some differences In policy 2 probability distribution

function of demand in planning period is estimated byhistorical data and then reorder is organized based on servicelevel (cumulative probability of demand satisfaction) But inthis policy after fixed interval forecasting of future demandwill be performed If inventory level has reached to reorderpoint (119877

119901) order would be organized and inventory level

would be updated after lead time (119871) otherwise only infor-mation would be updated If forecasted demand is less thaninventory capacity order quantity (119876

119895119905) would be as much as

forecasted quantity otherwise inventory orderwill be asmuchas inventory capacity for each product As demands followtime series with autocorrelation forecasting is implementedby exponential smoothing method [37]

In this procedure 119860119905is defined as demand estimation

of 119905th planning period This term is calculated according toreal demand of period 119905 indicated by 119863

119905and estimation of

demand in previous period which is indicated by 119860119905minus1

plustrend adjustment value of previous period which is indicatedby 119879119905minus1

Impact of current real demand is considered byparameter 120572 that is defined between zero and one Trendadjustment value is derived based on 119860

119905and 119860

119905minus1plus trend

adjustment value of previous period with consideration of120573 as weight parameter which is defined between zero andone Equations related to 119860

119905and 119879

119905are expressed in (4)

and (5) respectively Finally demand estimation plus trendadjustment form of future demand forecasting is indicated by119865119905+1

which is described in (6) Logic of this policy is shown inFigure 1(c) and is labelled as DF policy Consider

119860119905= 120572 times 119863

119905+ (1 minus 120572) times (119860

119905minus1+ 119879119905minus1) (4)

119879119905= 120573 times (119860

119905minus 119860119905minus1) + (1 minus 120573) times 119879

119905minus1 (5)

119865119905+1= 119860119905+ 119879119905 (6)

42 Surrogate Modelling Surrogate model is designed foroptimization of simulated inventory system In this frame-work design of experiments is responsible for producingdifferent scenarios Each scenario has four criteria includ-ing cost service level average of inventory position androbustness against demand fluctuation Importance of eachcriterion is determined by PCA and finally these scenariosare ranked by MADM technique as multiple criteria decisionmaking tool

421 Design of Experimentation Although DOE roots backto the statistical quality control nowadays it is a powerfultool for analysing complex systems DOE is statisticalmethodand organizes structured experiments with several factors[38] This method not only determines effect of each factoron response variable but also considers multiple effectssimultaneously DOE is extended technique that includesseveral designs such as full factorial fractional factorial andnested design Specific design varies for each problem andshould be selected based on problem condition Our problemconsists of tree effective factors namely inventory policyreorder point and lead time these factors have nonlineareffect on objective function values As the aim of DOE inthis research is producing decision space with few factors (in

6 Modelling and Simulation in Engineering

Demand arrival

Sufficientinventory isavailable

True

False

Reorderproducts

True

False

Demand satisfaction

Make an order

Passing of lead time

Demand dispose

Lost customer

Is there anyorder in way

True

False

Updating of inventory level

Demand satisfaction

(a) FQ policy

Demand arrival

Sufficientinventory isavailable

Demand satisfaction Lost customer

FalseTrue

Demand dispose

System start

Make order

Make an order

Passing of lead time

TrueFalse

Passing of reviewing period

Updating of inventory level

(b) FI policy

Demand arrival

Sufficientinventory isavailable

Lost customer

System start

Make orderFalse True

Estimatedquantity ispossible

True

False

True False

Update of estimation

Update of information

Order maximum quantity

Order estimated quantity

Passing of lead time

Updating of inventory level

Demand satisfaction

Passing of reviewing period

Demand dispose

(c) DF policy

Figure 1

Modelling and Simulation in Engineering 7

Table 2 DOE configuration

ProductsFactors

IP 119877119905

119871

minus 0 + minus 0 + minus 0 +119860 FI FQ DF 10 20 30 3 5 7119861 FI FQ DF 15 30 45 3 5 7119862 FI FQ DF 15 30 45 3 5 7119863 FI FQ DF 15 30 45 3 5 7

Table 3 Experimental design

Number of trial 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27119877119901

minus minus minus minus minus minus minus minus minus 0 0 0 0 0 0 0 0 0 + + + + + + + + +119871 minus minus minus 0 0 0 + + + minus minus minus 0 0 0 + + + minus minus minus 0 0 0 + + +IP minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 +

this research there are 3 factors) using full factorial design ispreferable On the other hand because nonlinear behaviourof response variables (ie objective functions) centre pointis considered Each factor has high level ldquo+rdquo low level ldquominusrdquoand centre point ldquo0rdquo Factors and levels configurations areshown in Table 2 while Table 3 shows experimental designIn both of these tables minus sign is representative of lowlevel of related factors while plus sign is indicator of high levelof factors and zero determines centre point For examplein the case of lead time (119871) 3 is low level 7 is high leveland 5 is centre point Lead time is considered with daysas measuring unit and begins from time order is organizeduntil the arrival of inventory Reorder point is labelled as 119877

119901

and is intended to satisfy product demand during lead time(119871) Eventually inventory policy is labelled as IP and refersto inventory policies that were described in Section 412The mentioned design includes 27 runs and there are fourresponse variables cost service level average of inventoryposition and robustness In fact first three response variablesare expected values of defined objective functions (ie resp(1) (2) and (3)) but robustness is a defined criterion whichguarantees that optimal solution remains valid in situationwhere demand variation and intensification increase Forevaluation of robustness demands are multiplied by randomvariable with normal distribution (120583 = 1 120590 = 15)This random variable is also restricted to positive values toprevent negative demands Expected value of 4000 generatednumbers with mentioned constraint is equal to 148 thisresult implies that multiplied random number increases bothdemand variation and intensification simultaneously Thenthe percent of decrease in service level is considered asrobustness for all 27 scenarios produced byDOE It is obviousthat a lower decrease in service level is desirable

422 Weight Assignment Based on PCA Method PCA isthe abbreviation of ldquoprincipal component analysisrdquo and isknown as powerful data-driven method successfully appliedin several areas [2] PCA analysis can determine key variablesthat govern most of the variability This technique helps toremove less important variables to reduce dimension of data

sets PCA considers two criteria for each variable in datasets the first is variability and the second is correlation withother variables So input for this method consists of variancecovariance matrix [39 40] Although other methods likeentropy is applicable in multiple attribute decision makingproblem PCA advantage is considerable because variablescorrelations are not ignored As our problem is constructedof highly correlated variables PCA is the preferable choiceFor correlation justification remember that improvementof inventory position criterion causes negative influence onshortage cost or improvement of service level criterion causesnegative influence on inventory in hand and holding cost Inthese circumstances there is no reason for applying variationbased methods like entropy so in this paper PCA is appliedfor weight assignment These weights are used by MADMtechnique (described in Section 423) to improve qualityof ranking Because of relative utility of scenarios linearscaleless method employed to scaleless data obtained fromsimulation

(1) Calculate scaleless matrix of 119863 = (1198891 1198892 119889

119901)119899times119901

which 119889119894119895is scaleless value of 119894th scenario in the view

point of 119895th criterion

(2) Calculate sample mean vector 119889 = (1198891 1198892 119889

119901)119901

and covariance matrix based on (7) and (8) respec-tively

119889119896=1

119899

119899

sum

119894=1

119889119894119895 (7)

119878 = (119904119896119902)119901times119901

=1

119899 minus 1(119863 minus 119889)

119879

(119863 minus 119889) (8)

(3) 1198621radic119904119896119896 is diagonal 119901 times 119901 matrix 119896th diagonal

element is 1radic119904119896119896 and sample correlation matrix (119877)is calculated as mentioned in the following equation

119877 =1198621

radic119904119896119896

sdot 119878 sdot1198621

radic119904119896119896

(9)

8 Modelling and Simulation in Engineering

(4) Solve the following equation where 119868119901is a 119901 times 119901

identity matrix10038161003816100381610038161003816119877 minus 120582119868

119901

10038161003816100381610038161003816= 0 (10)

Solving (10) results in 119875 individual eigenvaluesregarding 120582

1ge 1205822ge sdot sdot sdot ge 120582

119901and sum119901

119896=1120582119896=

119875 These eigenvalues have 119875-specific eigenvectors(119897119896

1 119897119896

2 119897

119896

119901) and 119896 = 1 119901 These vectors create

principal components as expressed in the followingequation

PC119896=

119901

sum

119902=1

119897119896

119902119889119895

119902 (11)

(5) Select sufficient number of principal components Inthis problem first119872 components whose cumulativepercentage of their eigenvalues (119862

119872) is greater than

95 are selected Cumulative percentage is calculatedbased on the following equation

119862119898=sum119898

119896=1120582119896

sum119901

119896=1120582119896

=sum119898

119896=1120582119896

119875 (12)

(6) Finally weights vector that is indicated by119885 is derivedfrom the mathematical relation in (13) while 119882

119896is

equal to 120582119896119875 if the entire elements of PC

119896are

positive otherwise minus120582119896119875 if entire component are

negative

119885 =

119872

sum

119896=1

119882119896PC119896 (13)

In other cases positive or negative 119882119896should be defined

in the order that final weight vector is positive For moredetailed discussion about 119882

119896 refer to [41] Finally for each

product there are four 119885 parameters that determine relativeimportance for the following criteria cost service levelaverage of inventory position and robustness each elementof 119885 vector is divided by the sum of vector elements fornormalization

423 VIKOR Method This method was developed by Opri-covic and is the abbreviation of expression that in Serbianmeans ldquomultiple criteria optimization and compromisingsolutionsrdquo [42] VIKOR method is based on compromisingsolutions which is the result of Yu [43] and Zeleny [44]studies This approach considers closeness to ideal solutionIn comparison with other MADM methods VIKOR is ableto present compromising solutions and substitutes thesesolutions with best one in the case of necessity [45] VIKORmethod is applicable for problems with multiple discretecriteria Generally this method can be considered as rankingtool for scenarios which are generated by DOE Ranking isbased on L-P metric function according to (14) with defini-tion of 119891

119894119895as the value of 119894th criterion for 119895th scenario In

construction of L-P metric function 119891lowast119894indicates best value

of all scenarios in the view point of 119894th criterion and 119891minus119894

represents the worst value of all scenarios with considerationof 119894th criterion 119908

119894is the weight parameter for 119894th criterion

and is derived by PCA method Consider

119871119901=

119899

sum

119894=1

[

119908119894(119891lowast

119894minus 119891119894119895)

(119891lowast

119894minus 119891minus

119894)]

119901

1119901

1 ≪ 119875 ≪ infin 119895 = 1 2 3 119869

119871infin= max

119894[

119908119894(119891lowast

119894minus 119891119894119895)

(119891lowast

119894minus 119891minus

119894)]

(14)

VIKOR method only involves 1198711and 119871

infinwhich are known

as 119878119895and 119877

119895 respectively 119878 and 119877 index refer to scenarios

so both of these functions are regarded for each scenarioVIKOR steps are as follows

(1) Calculate 119891minus119894and 119891lowast

119894for entire of criteria

(2) Calculate 119878119895and 119877

119895for entire of scenarios

(3) With the assumption that 119878lowast and 119877lowast are the smallestvalues among all of 119878

119895and 119877

119895 respectively and also

119878minus and 119877minus are the greatest values in comparison withall of 119878

119895and 119877

119895 respectively 119876

119895is calculated for 119895th

scenario based on (15) In this equation V is strategicweight that determines the importance of individualcriterion (119878

119895) against group importance of criteria

(119877119895) for each scenario Consider

119876119895=

V (119878119895minus 119878lowast

)

119878minus minus 119878lowast+ (1 minus V)

(119877119895minus 119877lowast

)

119877minus minus 119877lowast (15)

(4) Scenarios should be sorted in descending order basedon 119878119877 and119876 specifically that result in three differentranking lists

(5) According to 119876 the scenario which is indicated by 1198861015840is the best scenario if two conditions namely (C1)and (C2) are satisfied

(C1) Acceptable Advantage This condition implies that thesecond best solution (11988610158401015840) should be far enough from thebest solution (1198861015840) to be accepted as unique solution Themathematical equation is defined in (16) considering 119869 as thetotal number of scenarios Consider

119876(11988610158401015840

) minus 119876 (1198861015840

) ≫1

(119869 minus 1) (16)

(C2) Acceptable Stability This condition validates 1198861015840 as thebest stable solution if 1198861015840 is also the best solution based on 119878 or119877

In case that one or both of the mentioned conditions arenot satisfied compromised solutions are involved VIKORmethod suggests that in such case compromising solutionshave equal value to the best solution and could be consideredinterchangeably by decision makers

Modelling and Simulation in Engineering 9

If (C1) is not satisfied scenarios 1198861015840 11988610158401015840 119886119872 are consid-ered 1198861015840 is the best solution and 11988610158401015840 119886119872 are compromisingsolutions with priority of 119876 on the other hand 1198861015840 11988610158401015840are considered if (C2) is not satisfied 119872 superscript isdetermined by (17)

119876(119886119872

) minus 119876 (1198861015840

) ≫1

(119869 minus 1) (17)

5 Numerical Result

Simulation process was executed on laptop with 18 GH CPUand 4GB of RAM Running of each replication with anima-tion and maximum speed takes 7 minutes and 35 secondsAs simulation of each scenario contains 10 replications andthere are 27 scenarios the total simulation time will be 2046minutes and 36 seconds

51 Multiresolution Result for Demand Modelling Result ofdemand modelling is presented for product 119861 because thisanalysis for other products is similar For this problempolynomial functions are applied for yearly and monthlydemand rates Equations (18) and (19) are estimated bynonlinear regression model In these functions 119905

119910and 119905119898are

time variables

119891 (119905119910) = 0008942 times 119905

3

119910minus 003884 times 119905

2

119910

+ 01867 times 119905119910minus 008018

(18)

and 95 confidence interval is presented in Table 4 for eachcoefficient of (18)

In this table coefficients of (18) are visible in the firstcolumn while lower bund and upper bund of each coefficientare presented in the second and third columns respectivelyThere is no interval including zero so it is concluded thatcoefficient estimation is valid

Estimation error for 119891(119905119910) is reported based on the sum

of square errors (SSE) which is 1504 times 10minus6 Consider

119891 (119905119898) = 4768 times 10

minus5

times 1199055

119898minus 0001692 times 119905

4

119898+ 00214

times 1199053

119898minus 01159 times 119905

2

119898+ 03474 times 119905

119898minus 02286

(19)

Monthly demand rate function that is indicated by 119891(119905119898)

and its coefficients confidence interval are presented inTable 5 SEE is equal to 00005825 Based on local businessinformation total demand of product 119861 is estimated to beabout 6822 units in five years According to the informationabout product 119861 Figure 2 shows final multiresolution resultfor this product

52 Simulation Parameters and Configuration For simula-tion of problem it is necessary to configure inventory policyparameters and variables We have three different policiesnamely FQ FI and DF Variables of FQ policy are reorderpoint (119877

119901) lead time (119871) and economic quantity orders (119876

119895119905)

First two variables are configured based on Table 2 and thirdvariable is derived fromWilson formula according to average

Table 4 95 confidence interval for estimated coefficient of 119891(119905119910)

Coefficients Lower bund Upper bund0008942 0004835 001305minus003884 minus007604 minus000165201867 008654 0287minus008018 minus01568 minus0003523

Table 5 95 confidence interval for estimated coefficient of 119891(119905119898)

Coefficients Lower bund Upper bund4768 times 10minus5 4451 times 10minus5 5086 times 10minus5

minus0001692 minus0001795 minus00015900214 00202 00226minus01159 minus01221 minus0109703474 0334 03608minus02286 minus02371 minus022

Table 6 Estimation of distribution functions of demand and reor-ders quantity

Product type Distribution function 119875 value Order quantity119860 278 lowast BETA(113 328) gt015 100119861 GAMMA(678 118) gt015 92119862 EXPO(818) gt015 100119863 446 lowast BETA(0957 321) gt015 115

of demand holding cost (119862ℎ) and ordering cost (119862

119903) The

calculated values are 57 71 72 and 72 for product 119860 119861 119862and119863 respectively

Orders quantity is unknown for FI policy and is cal-culated based on probability distribution function whichis obtained by demand simulation of each product for 20daysrsquo time interval Distribution fitting is performed by inputanalyser of Arena software and graphical result for product 119861is shown in Figure 3 Results for other products are reportedin Table 6

In this table fitted probability distribution function foreach product is presented in second column while 119875 valueof fitting which is greater than 015 for fitted distributiongrantees goodness of fitting in 95 confidence level Finallyaccording to fitted distribution functions for FI policyproper ordering amount of each product is presented in thefourth column

DF policy has two unknown parameters namely 120572 and120573 The first parameter is weight of current real demand in (4)and second one is the weight of current trend in (5) Theseparameters are adjusted by simulation of five years demandDemand for the entire four products simultaneously is con-sidered and an average of lost sale percentage is consideredas response variable Different values of response variables arereported in Table 7 According to the different values of 120572 and120573 the best values for 120572 and 120573 are 09 and 01 respectivelywhich results in minimum lost percentage in average

10 Modelling and Simulation in Engineering

Table 7 Average of lost sale percentage for different values of 120572 and 120573

120572 and 120573 values 01 02 03 04 05 06 07 08 09Response to 120572 0380 0356 0334 0320 0306 0292 0284 0276 0268Response to 120573 0296 0302 0306 0306 0306 0304 0305 0304 0302

50

100

150

200

250

300

350Demand fit for each month

Dem

and

Fitted valueReal demand

00 10 20 30 40 50 60

Month

Figure 2 Comparison between real and estimated demand

100 200 300 400 500 600 700

50

100

150

200

Distribution summary

Distribution gammaExpression minus0001 + GAMM(00)Square error 0006825

Chi-square testNumber of intervals = 5Degrees of freedom = 2Test statistic = 148

Kolmogorov-Smirnov testTest statistic = 00733

Corresponding P value = 0483

Corresponding P value gt 015

Figure 3 Distribution function of customer demand for reorderinterval of product 119861

53Weight Assignment Asmentioned before PCA is appliedforweight assignment For this purposeDOE result is neededin the form of decision matrix that is shown in Table 8 Inthis table four criteria cost average of inventory positionservice level and robustness are evaluated for each scenarioof product 119861 Decision matrix should be scaleless that isperformed by linear method Then the result of eigenvalues

Table 8 Decision matrix for product 119861

Scenarionumber Cost Average of

inventory positionServicelevel

Robustness

1 32984000 4352 76 62 29214710 397514 79 33 37114636 339623 69 24 34840720 410948 71 55 36820570 344738 72 66 39191837 331091 67 27 37037180 391012 68 48 40994680 298712 64 69 41407045 319678 63 110 28282930 524462 83 711 26422145 489243 85 512 31023829 416049 79 313 30395190 487146 79 614 35393680 395812 74 515 32838055 395646 77 316 33285615 463341 76 617 39652020 335668 66 718 35269609 383944 73 219 26227680 625612 87 720 25565820 586688 88 621 28711069 49449 85 422 27301580 585472 85 723 34016415 470961 78 724 30601546 463975 82 325 29752815 560421 82 726 41024410 382994 68 627 31604741 443053 78 3

is calculated and shown in Table 9 Also result of principalcomponents is presented in Table 10

As it is visible in Table 9 cumulative proportion ofthe first and second components is 0966 which is greaterthan 095 so the first two components are considered asprincipal components Their relative weights are 078 and0186 respectively Final weights are normalized and resultsare reported in Table 11

54 Ranking and Selection of Scenarios Weights of criteriaobtained by PCA method in addition to decision matrix areinput of VIKOR method 119878 119876 and 119877 values are calculatedbased on (14) and (15) while strategic weight indicated byV is equal to 05 so individual and group utility have same

Modelling and Simulation in Engineering 11

Table 9 Result of eigenvalues

Eigenvalues 31204 07428 01111 00257Proportion 0780 0186 0028 0006Cumulative 0780 0966 0994 1000

Table 10 Result of principle components

Variables PC1 PC2 PC3 PC4Inventory 0543 0089 minus0818 0167Cost minus0538 minus0277 minus0511 minus0610Service level minus0551 minus0210 minus0230 0774Robustness 0335 minus0933 0126 0023

Table 11 Weights of criteria for different products

Criteria Products119860 119861 119862 119863

Average of inventory position 01012 00590 01942 00462Cost 02852 01836 01742 03081Service level 01195 01392 00966 01206Robustness 04941 06182 05350 05251

importance Calculated values are reported in Table 12 forentire scenario as VIKOR output

Product 119861 According to Table 12 values of 119877 119878 and 119876 arezero for scenario 18 Regarding (C1) (Acceptable Advantage)the difference between the best scenario and second rankedscenario should be more than 00038 The second rankedscenario regarding 119876 value is scenario 3 and its value is0051044 that satisfies both (C1) and (C2) Final result forproduct 119861 is unique scenario that is number 18 and consistsof the following configuration Inventory policy is DFwith119877

119901

being equal to 30 products and 7 days lead time According todecision matrix which is shown in Table 8 the performanceof this scenario for product 119861 results in 383944 products asaverage of inventory position and expected cost is equal to35269609 for product life cycle 73 of customer demandwillbe satisfied and in the case of increasing demand intensityand variation 2 decrease is expected in service level so itwill change into 71 Analyses for other products are similarso only final results are presented in this paper

Product 119860 In this case VIKORmethod ranks scenario num-ber 15 as the best one This scenario consists of DF inventorypolicy 20 products for 119877

119901and 5 days for lead time Perfor-

mance of this scenario results in 374778 products as averageof inventory position Expected cost is 31603294 and servicelevel will be 66 This scenario is robust against demandintensification and variation because changes in demandaffect service level as small as 1 But scenario number 15 doesnot satisfy (C1) so compromising solutions are consideredAlthough the best scenario is number 15 VIKOR methodimplies that compromising solutions have the same values

Table 12 VIKOR output for product 119861

Run number 119878 119877 119876

1 0634311 0796269 0715292 0003184 0185077 0094133 0059804 0042283 00510444 0536946 0592539 05647425 0718862 0796269 07575656 0116278 0089887 01030827 0428814 0388808 04088118 086078 0796269 08285259 002043 0140653 008054110 0677386 1 083869311 0266254 0592539 042939612 0043962 0185077 01145213 0571695 0796269 068398214 0515097 0592539 055381815 009173 0185077 013840416 0648693 0796269 072248117 1 1 118 0 0 019 0630698 1 081534920 04239 0796269 061008421 0139527 0388808 026416722 06581 1 08290523 0819338 1 090966924 0022245 0185077 010366125 0726183 1 086309126 0849516 0796269 082289327 0072768 0185077 0128922

So in real situation compromising solution can be substi-tuted This solution regarding their priority is as follows

Scenario number 24 inventory policy is DF 119877119901is

equal to 30 products and lead time is 5 daysScenario number 18 inventory policy is DF 119877

119901is

equal to 20 products and lead time is 7 daysScenario number 12 inventory policy isDF119877

119901is equal

to 20 products and lead time is 3 days

Product 119862 For this product scenario number 1 is ranked asthe best one This scenario consists of FI inventory policy 119877

119901

is 15 products and lead time is 3 days Result of this scenario is4387 products as the average of inventory position Expectedcost is 66143000 and service level will be 61 with robustnessof 1 Compromising solutions are as follows

Scenario number 6 inventory policy is DF 119877119901is 15

products and lead time is 5 daysScenario number 3 inventory policy is DF 119877

119901is 15

products and lead time is 3 daysScenario number 7 inventory policy is FI 119877

119901is 15

products and lead time is 7 days

12 Modelling and Simulation in Engineering

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 4 Interaction plot for product 119862 average of inventory position

Scenario number 4 inventory policy is FI 119877119901is 15

products and lead time is 5 days

Product119863 Scenario number 3 has the best rank for this prod-uct This scenario consists of DF inventory policy 119877

119901is 15

products and lead time is 3 days This scenario leads to

363896 products as average of inventory position with62648337 expected cost and 61 service level with 1robustness Compromising solutions are as follows

Scenario number 6 this scenario is mentioned beforeas a compromising solution for product C

Modelling and Simulation in Engineering 13

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 5 Interaction plot for product 119862 cost

14 Modelling and Simulation in Engineering

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

3

5

7

L

3

5

7

L

3

5

7

L

L

(b)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 6 Interaction plot for product 119862 service level

Scenario number 17 inventory policy is FI119877119901is equal

to 15 products and lead time is 7 days

55 Sensitivity Analysis As mentioned in the previous sec-tion all scenarios related to product 119860 prefer DF inventory

policy so it is concluded that DF policy is unique optimalpolicy while lead time includes values of 3 5 and 7 in selectedscenarios This situation implies that product 119860 is insensitiveto the delivery time It can be wise to decide lead time about 5days but there is no need for strict control on lead time while

Modelling and Simulation in Engineering 15

119877119901should be controlled strictly to prevent inventory in hand

from reaching lower than 20 productsIn the case of product 119861 (C1) is satisfied so there is a

unique scenario while there are four compromising solutionfor product 119862 In this situation based on VIKOR methodcompromising solutions have same value while interactionplot can be applied for investigation of existing differencebetween compromising solutions As mentioned before thebest scenario suggests FI inventory policy for this product butin compromising solutions there are both FI and DF policiesLead time varies from 3 to 7 days So this confusing situationis resolved by deeper analysis and applying interaction plotFigure 4 shows interaction plot for DOE factors and averageof inventory position as response variables

Figure 4 shows that with either DF or FI policy there isno significant difference among different lead times in theview point of inventory position but DF leads to less averageof inventory position With 119877

119901equaling 15 there is less

sensitivity to different lead times So configuring 119877119901which

is equal to 15 products the average of inventory positioncriterion will be robust against lead time variations

Figure 5 regards cost criterion and shows that FI andDF policies are equal and make least sensitivity related tolead time On the other hand three plots of 119877

119901against lead

time are parallel The parallel plots are interpreted as lackof relation between 119877

119901and lead time With 119877

119901equaling 15

increase in inventory cost could be mitigated when lead timesets to 3 days

In Figure 6 with 3 days for lead time there is no signifi-cant difference between inventory policies in the view pointof service level119877

119901and lead time are independent and the best

service level is related to FI policy which is also very sensitiveto 119877119901 As 119877

119901increases service level will grow According to

the interaction plot analyses optimal decision for product119862 is FI inventory policy 3 days lead time and 15 or moreproducts for 119877

119901

Decision about product 119863 is simple Based on the in-formation obtained from solutions it is recommended toemploy DF inventory policy with 119877

119901being equal to 15

products and 3 days for lead time

6 Conclusion

In this paper a simulation optimization framework is pro-posed for robust optimization of retailer inventory systemThis framework is less time consuming in comparison withmetaheuristic or SPSA approach it also provides robustsolution for multiple objective functions In this research alsotrend and cyclic change of customers demand are consideredfor more realistic view of problem Proposed framework con-sists of discrete event simulation and surrogate model In thisframework full factorial design of experiment is employedfor producing decision space Based on the three decisionfactors 27 different scenarios are produced and simulated Insimulation modelling multiresolution method is applied forsimulation of demand with highly dynamic pattern Becausethere are multiple criteria for inventory system performanceVIKORmethod is employed asmulticriteria decisionmaking

technique In addition robustness is considered as perfor-mance criterion and PCA is used for improving performanceof VIKOR method Finally developed framework gave thebest ranked and compromising solutions so interaction plotapplied for more investigation and sensitivity analysis ofobtained solutions

Due to the novelty of the proposed framework it isrecommended that future studies encompass optimization ofinventory system of integrated supply chain Also improve-ment of developed framework can be considered as futurestudy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers for theirsignificant roles in improvement of this research quality

References

[1] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[2] S Yin S XDingAHaghaniHHao andP Zhang ldquoA compar-ison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[3] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[4] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[5] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall International Series in Management New YorkNY USA 1963

[6] E Naddor Inventory Systems John Wiley amp Sons New YorkNY USA 1966

[7] R Peterson and E A Silver Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 2nd edition 1985

[8] T C E Cheng ldquoEconomic order quantity model with demand-dependent unit production cost and imperfect productionprocessesrdquo IIE Transactions vol 23 no 1 pp 23ndash28 1991

[9] S H Chen C CWang and A Ramer ldquoBackorder fuzzy inven-tory model under function principlerdquo Information Sciences vol95 no 1-2 pp 71ndash79 1996

[10] X Zhao J Xie and J C Wei ldquoThe value of early order com-mitment in a two-level supply chainrdquo European Journal of Op-erational Research vol 180 no 1 pp 194ndash214 2007

[11] R S M Lau J Xie and X Zhao ldquoEffects of inventory policyon supply chain performance a simulation study of critical

16 Modelling and Simulation in Engineering

decision parametersrdquo Computers and Industrial Engineeringvol 55 no 3 pp 620ndash633 2008

[12] J Xu and L Zhao ldquoA multi-objective decision-making modelwith fuzzy rough coefficients and its application to the inventoryproblemrdquo Information Sciences vol 180 no 5 pp 679ndash6962010

[13] F Hnaien X Delorme and A Dolgui ldquoMulti-objective opti-mization for inventory control in two-level assembly systemsunder uncertainty of lead timesrdquo Computers amp OperationsResearch vol 37 no 11 pp 1835ndash1843 2010

[14] H D Purnomo H M Wee and Y Praharsi ldquoTwo inventoryreview policies on supply chain configuration problemrdquo Com-puters and Industrial Engineering vol 63 no 2 pp 448ndash4552012

[15] B L Nelson ldquo50th anniversary article stochastic simulationresearch in management sciencerdquoManagement Science vol 50no 7 pp 855ndash868 2004

[16] J Boesel R O Bowden Jr F Glover J P Kelly and E WestwigldquoFuture of simulation optimizationrdquo inProceedings of theWinterSimulation Conference pp 1466ndash1469 Arlington Va USADecember 2001

[17] M C Fu ldquoOptimization for simulation theory vs practicerdquoINFORMS Journal on Computing vol 14 no 3 pp 192ndash2152002

[18] L Wang ldquoA hybrid genetic algorithm-neural network strategyfor simulation optimizationrdquoApplied Mathematics and Compu-tation vol 170 no 2 pp 1329ndash1343 2005

[19] B B Keskin S H Melouk and I L Meyer ldquoA simulation-optimization approach for integrated sourcing and inventorydecisionsrdquo Computers and Operations Research vol 37 no 9pp 1648ndash1661 2010

[20] E Mazhari J Zhao N Celik S Lee Y Son and L HeadldquoHybrid simulation and optimization-based design and oper-ation of integrated photovoltaic generation storage units andgridrdquo Simulation Modelling Practice and Theory vol 19 no 1pp 463ndash481 2011

[21] Q Duan and T W Liao ldquoOptimization of replenishment poli-cies for decentralized and centralized capacitated supply chainsunder various demandsrdquo International Journal of ProductionEconomics vol 142 no 1 pp 194ndash204 2013

[22] J C Spall ldquoMultivariate stochastic approximation usinga simultaneous perturbation gradient approximationrdquo IEEETransactions on Automatic Control vol 37 no 3 pp 332ndash3411992

[23] J C Spall ldquoImplementation of the simultaneous perturbationalgorithm for stochastic optimizationrdquo IEEE Transactions onAerospace and Electronic Systems vol 34 no 3 pp 817ndash8231998

[24] J C Spall ldquoStochastic optimization and the simultaneousperturbation methodrdquo in Proceedings of the Winter SimulationConference (WSC rsquo99) pp 101ndash109 December 1999

[25] J D Schwartz W Wang and D E Rivera ldquoSimulation-based optimization of process control policies for inventorymanagement in supply chainsrdquo Automatica vol 42 no 8 pp1311ndash1320 2006

[26] H G Neddermeijer G J V Oortmarssen and N P R DekkerldquoA framework for response surface methodology for simulationoptimizationrdquo in Proceedings of the Winter Simulation Confer-ence vol 1 Orlando Fla USA December 1999

[27] B Can and C Heavey ldquoA comparison of genetic programmingand artificial neural networks in metamodeling of discrete-event simulation modelsrdquo Computers amp Operations Researchvol 39 no 2 pp 424ndash436 2012

[28] A Azadeh Z S Faiz S M Asadzadeh and R Tavakkoli-Moghaddam ldquoAn integrated artificial neural network-comput-er simulation for optimization of complex tandem queuesystemsrdquoMathematics and Computers in Simulation vol 82 no4 pp 666ndash678 2011

[29] R Bornatico J Hussy A Witzig and L Guzzella ldquoSurrogatemodeling for the fast optimization of energy systemsrdquo Energyvol 57 pp 653ndash662 2013

[30] X Wan J F Pekny and G V Reklaitis ldquoSimulation-basedoptimizationwith surrogatemodels application to supply chainmanagementrdquoComputers and Chemical Engineering vol 29 no6 pp 1317ndash1328 2005

[31] W Shi Z Liu J Shang and Y Cui ldquoMulti-criteria robust designof a JIT-based cross-docking distribution center for an autoparts supply chainrdquo European Journal of Operational Researchvol 229 no 3 pp 695ndash706 2013

[32] G Dellino J P C Kleijnen and C Meloni ldquoRobust optimiza-tion in simulation taguchi and Response Surface Methodol-ogyrdquo International Journal of Production Economics vol 125 no1 pp 52ndash59 2010

[33] T Yang Y Wen and F Wang ldquoEvaluation of robustness of sup-ply chain information-sharing strategies using a hybrid Taguchiand multiple criteria decision-making methodrdquo InternationalJournal of Production Economics vol 134 no 2 pp 458ndash4662011

[34] M E Kuhl and J R Wilson ldquoModeling and simulating Poissonprocesses having trends or nontrigonometric cyclic effectsrdquoEuropean Journal of Operational Research vol 133 no 3 pp566ndash582 2001

[35] P Narayan and J Subramanian Inventory ManagementmdashPrinciples and Practices Excel Books New Delhi India 2008

[36] E A Silver and R Peterson Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 1985

[37] R G Brown Smoothing Forecasting and Prediction of DiscreteTime Series Courier 2004

[38] D C Montgomery Design and analysis of experiments JohnWiley amp Sons New York NY USA 6th edition 2005

[39] J Zhu ldquoData envelopment analysis vs principal componentanalysis an illustrative study of economic performance ofChinese citiesrdquo European Journal of Operational Research vol111 no 1 pp 50ndash61 1998

[40] I M Premachandra ldquoA note on DEA vs principal componentanalysis an improvement to Joe Zhursquos approachrdquo EuropeanJournal of Operational Research vol 132 no 3 pp 553ndash5602001

[41] D Slottje G W Scully G J Hirschberg and K Hayes Mea-suring the Quality of Life across Countries A MultidimensionalAnalysis Westview Press Boulder Colo USA 1991

[42] S Opricovic Multicriteria optimization of civil engineeringsystems [PhD thesis] Faculty of Civil Engineering BelgradeSerbia 1998

[43] P L Yu ldquoA class of solutions for group decision problemsrdquoManagement Science vol 19 no 8 pp 936ndash946 1973

Modelling and Simulation in Engineering 17

[44] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[45] S Opricovic and G H Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

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Page 3: Research Article Robust Inventory System Optimization ...downloads.hindawi.com/journals/mse/2014/305120.pdf · Robust Inventory System Optimization Based on Simulation and Multiple

Modelling and Simulation in Engineering 3

event simulation Azadeh et al [28] used artificial neural net-work for the designing of simulation optimization frameworkand they applied proposed framework for optimization ofwaiting time in tandem queue systems Bornatico et al [29]proposed a surrogate model based on redial basis functionfor simulation optimization of energy systems Wan et al[30] designed simulation optimization framework using leastsquare support vector machine (LSSVM) for optimizationof inventory level in three-stage supply chain They alsoshowed that proposed framework leads to better solutionwith less number of simulation runs in comparison withSPSA algorithm Surrogate modelling is less time consumingin comparison to metaheuristic or SPSA approaches butthis approach loses accuracy in multiple objectives problemsolving

Although a bunch of papers published in simulationoptimization area to the best of our knowledge a tiny numberof them are dedicated to the multiobjective optimization [31]and robustness [32 33] In this case using metaheuristicand SPSA approaches is very time consuming and it is noteconomic for optimization of simulationmodel with accuratedetails On the other hand all of the reviewed approacheslose their accuracy when there are multiple objectives Withthese considerations this paper purposes a framework foroptimization of detailed simulation model of inventory sys-tem with multiple objectives Proposed framework is lesstime consuming in comparison with metaheuristic or SPSAapproaches while it provides robust and accurate solutionsSo the proposed framework is relatively new and contribu-tion of this research entails threefold as follows

(i) We modelled cyclic and long-term demand basedon nonparametric time series modelling for morerealistic consideration

(ii) We proposed surrogate model for robust and multi-objective optimization ofmultiproduct inventory sys-tem based on discrete event simulation full factorialdesign of experiments (DOE) and multiple attributedecision making (MADM) technique

(iii) Due to the stochastic nature of objective functionwe employed principal component analysis (PCA) asstatistical method to improve MADM performance

3 Problem Statement

The problem is concerning retailer who sales office furnitureand facility The retailer sales four products respectively119860 119861 119862 and 119863 the aim is the optimization of inventorysystem according to information which is adapted from localbusiness Key features of retailer products from inventoryview point are as shown in Table 1

In this table second column gives average demand ofeach product type in a year third column provides holdingcost of each product in a planning period fourth columnis dedicated to ordering cost of each type of products andfinally numbers of fifth column are cost of lost sales which areincurred to retailers when they cannot satisfy the demand ofcustomers for each type of products

The fundamental assumption that should be consideredin this problem is as follows

(1) Order cost for each type of products includes trans-portation and order registration cost

(2) There is no backlog inventory so inventory level isnonnegative all the time

(3) Profit of each product is considered as lost sale costbecause unavailable products incur lost profits thatare interpreted as cost of lost sale

(4) According to the adapted information these productshave five years life cycle and then will be substitutedwith new products

(5) Planning periods for system under study are as longas 20 days

The notations that will be used to describe the problems areas follows

Indices Consider the following

119905 index of planning periods 119905 = 1 2 3 119879119894 index of demands in planning period 119894 = 1 2 3 119873119895 index of orders in planning period 119895 = 1 2 3 119872

Parameters Consider the following

119862119903 reorder cost

119862ℎ holding cost of each product in planning period

119862119897 cost of lost sale for each product

Variables Consider the following

119868119897

119905 inventory level in 119905th planning period119868119901

119905 inventory position in 119905th planning period

119889119894119905 quantity of 119894th demand in 119905th planning period

119899119905 number of reorder in 119905th planning period

119909119894119905 1 if 119889

119894119905is less than 119868119897

119905 0 otherwise

119876119895119905 quantity of 119895th order in 119905th planning period

119871 lead time for organized orders119877119901 reorder point

Demand of each product in the planning period (119889119894119905) is

stochastic variable and is generated by nonhomogeneousPoisson distribution So total number of arrived demandin planning period (119873) is probabilistic 119862

119903 119862ℎ and 119862

119897are

different costs of inventory system according to Table 1 119868119897119905is

inventory level and refers to physical quantity of inventorywhich is available in retailer while 119868119901

119905is the position of inven-

tory and includes quantity of on-order inventory in additionto inventory level in planning period 119909

119894119905is binary variable

which is one if inventory level is greater than the arriveddemand So if 119909

119894119905is one 119889

119894119905can be satisfied and otherwise

4 Modelling and Simulation in Engineering

Table 1

Product type Average demand Holding cost in planning period Reorder cost Lost sale cost119860 987 30000 50000 20000119861 1520 30000 50000 15000119862 1598 30000 50000 25000119863 1569 30000 50000 18000

it is lost sale 119877119901is the reorder point for organizing of new

order In other words if inventory level reaches 119877119901or less

a new order would be organized with quantity of 119876119895119905 Each

organized order reaches the retailer and increases inventorylevel after passing of lead time (119871) Considering assumptionsand described notations the following equations are themainobjective functions of defined problem

Min119879

sum

119905=1

119862119903119899119905+

119879

sum

119905=1

119862ℎ119868119897

119905+

119879

sum

119905=1

119873

sum

119894=1

119862119861(1 minus 119909

119894119905) 119889119894119905

(1)

Maxsum119879

119905=1sum119873

119894=1119909119894119905119889119894119905

sum119905

119905=1sum119873

119894=1119889119894119905

(2)

Minsum119879

119905=1119868119901

119905

119879 (3)

In (1) the objective is minimizing the total cost includingcosts that depend on reordering handling and lost salesLost sales not only incur excess cost but also decreaseretailer credit So (2) is considered to maximize service levelindependently In (2) the objective function increases where119909119894119905is 1 for 119894th demand in 119905th period and such situation is

possible if 119868119897119905is greater than 119889

119894119905 In fact (2) causes increase

in inventory level while (1)ndash(3) causes decrease in inventorylevel State of inventory level depends on number of ordersin each planning period (119899

119905) quantity of orders (119876

119895119905) and

quantity of demands (119889119894119905) while state of inventory position

depends on inventory level and lead time (119871) so (3) isresponsible for minimizing average of inventory positionincluding inventory in hand (119868119897

119905) and on-order inventory

Average of inventory position should be minimized in orderto improve flexibility of retailer and approach to the just intime (JIT)

Demand of each product follows different pattern withboth long-term and cyclic trend So mentioned objectivefunctions are considered individually for each product type

4 Proposed Framework

In the defined problem demand of products exposes highlydynamic pattern and as time passes demand and its variationincrease hence multiresolution method is employed fordemand modelling Also three different policies for inven-tory control are considered which are reordered based onfixed quantity (FQ) fixed interval (FI) and demand forecast-ing (DF) Simulation of developed model is implemented inArena 135 Optimization of simulation model is performed

by surrogate model that is based on full factorial design ofexperiment (DOE) For construction of decision space DOEfactors include inventory policy reorder point and lead timewith three levels for each of them So there are 33 = 27

different combinations of decision variables to form feasiblescenarios Ranking of produced scenarios is accomplished byMADM technique For ranking of scenarios three objectivefunction values are considered (ie cost service level andaverage of inventory position) and in addition robustness ofservice level against demand fluctuation is considered AlsoPCA is applied for more realistic weighting of objective func-tion values based on their statistical influence on improve-ment of other objectives Finally interacting plot is employedfor sensitivity analysis and investigation of solutions in detail

41 Simulation Modelling Simulation modelling of problemconsists of two parts which are modelling of demand andmodelling of inventory policies In this paper demand is non-homogeneous Poisson process and three different inventorypolicies based on continues reviewing periodic reviewingand periodic reviewing with forecasting of future demand areconsidered

411 Modelling of Demand For customers demand model-ing multiresolutionmethod is applied Kuhl andWilson [34]developed this method for simulation of nonhomogeneousPoisson process with trend and cyclic changes This methodestimates mean intensity function and the nonparametricnature of thismethod is one of themost important advantagesin comparison with other methods So it is independent ofstatistical parameters and applicable in variety of problemsFurthermore multiresolution method can support combi-nation of multiple cyclic changes simultaneously Anotheradvantage of multiresolution is its capability in the modellingof nonsymmetric cyclic pattern As in our case demandhas nonsymmetric pattern with cyclic changes and long-term trend and multiresolution approach is a reasonablechoice More theoretical and application of used method areprovided in [34]

412 Modelling of Inventory Policies As the main effort ofthis paper is inventory system optimization modelling ofinventory policies plays an important role in this problemIn this problem optimization is manipulated by selection ofappropriate inventory policy and configuration of its param-eters to the way that leads to the optimal state of inventorysystem In the inventory management three approaches arecommon strategies which are fixed order quantity fixed time

Modelling and Simulation in Engineering 5

interval and forecasting methods [35] In the first strategyinventory level should be reviewed continuously until itreaches below predetermined quantity (reorder point) thenorder would be organized with fixed quantity of inventory Inthe second strategy reviewing period is a fixed time intervalbut quantity order is variable for each order that is based onconsumption rate While first strategy needs more effort forcontinuously reviewing of inventory level the second strategyis easier to handle but the risk of shortage in fixed intervalstrategy is more in comparison to fixed order quantity Sodue to themitigation of shortage risk in fixed interval strategyorder quantity is slightly more than fixed order quantity [35]

In the third strategy reviewing period is fixed as sec-ond policy but reorder quantity is based on forecasting offuture demands As thementioned strategies are fundamentalin inventory management literature and are also commonamong retailers of office furniture in this study three policiesbased on fixed order quantity fixed time interval anddemand forecasting are developed as follows

(1) Continuous reviewing with economic quantity order(2) Periodic reviewing with order quantity based on

demand confidence interval during the consumptionperiod

(3) Periodic reviewing with order quantity based onforecasting of future demands

Policy 1 (Fixed Quantity) Based on this policy each demandwill be satisfied if there is sufficient inventory in handAfter satisfaction of each demand inventory in hand willbe checked to see if the inventory level reaches to reorderpoint (119877

119901) If inventory level has reached to reorder point

economic order quantity (119876119895119905)would be organized otherwise

system waits for next demand Economic order quantity isderived by Wilson formula [36] If there is not sufficientinventory to satisfy arrived demand quantity of demandis considered as lost sale and lost profit treated as costIf no order has been organized system reorder inventoryotherwise waits for arriving of organized order accordingto adjusted lead time (119871) Logic of this policy is visible inFigure 1(a) and is labelled as FQ policy

Policy 2 (Fixed Interval) In this policy criterion for reorder-ing is fixed time interval that is known as planning periodAfter this period inventory level would be examined andreorder will be organized on condition that inventory levelhas reached to reorder point (119877

119901) For more realistic consid-

eration order quantity (119876119895119905) is calculated based on demand

cumulative distribution function in planning period Forexample in the inventory system that is planned for 10 lostsale with demand which is distributed based on exponentialdistribution function it should be ordered as much ascumulative probability of exponential distribution equals to09 In this policy inventory level would be updated after leadtime (119871) Logic of this policy is shown in Figure 1(b) and islabelled as FI policy

Policy 3 (Demand Forecasting) This policy is similar to policy2 with some differences In policy 2 probability distribution

function of demand in planning period is estimated byhistorical data and then reorder is organized based on servicelevel (cumulative probability of demand satisfaction) But inthis policy after fixed interval forecasting of future demandwill be performed If inventory level has reached to reorderpoint (119877

119901) order would be organized and inventory level

would be updated after lead time (119871) otherwise only infor-mation would be updated If forecasted demand is less thaninventory capacity order quantity (119876

119895119905) would be as much as

forecasted quantity otherwise inventory orderwill be asmuchas inventory capacity for each product As demands followtime series with autocorrelation forecasting is implementedby exponential smoothing method [37]

In this procedure 119860119905is defined as demand estimation

of 119905th planning period This term is calculated according toreal demand of period 119905 indicated by 119863

119905and estimation of

demand in previous period which is indicated by 119860119905minus1

plustrend adjustment value of previous period which is indicatedby 119879119905minus1

Impact of current real demand is considered byparameter 120572 that is defined between zero and one Trendadjustment value is derived based on 119860

119905and 119860

119905minus1plus trend

adjustment value of previous period with consideration of120573 as weight parameter which is defined between zero andone Equations related to 119860

119905and 119879

119905are expressed in (4)

and (5) respectively Finally demand estimation plus trendadjustment form of future demand forecasting is indicated by119865119905+1

which is described in (6) Logic of this policy is shown inFigure 1(c) and is labelled as DF policy Consider

119860119905= 120572 times 119863

119905+ (1 minus 120572) times (119860

119905minus1+ 119879119905minus1) (4)

119879119905= 120573 times (119860

119905minus 119860119905minus1) + (1 minus 120573) times 119879

119905minus1 (5)

119865119905+1= 119860119905+ 119879119905 (6)

42 Surrogate Modelling Surrogate model is designed foroptimization of simulated inventory system In this frame-work design of experiments is responsible for producingdifferent scenarios Each scenario has four criteria includ-ing cost service level average of inventory position androbustness against demand fluctuation Importance of eachcriterion is determined by PCA and finally these scenariosare ranked by MADM technique as multiple criteria decisionmaking tool

421 Design of Experimentation Although DOE roots backto the statistical quality control nowadays it is a powerfultool for analysing complex systems DOE is statisticalmethodand organizes structured experiments with several factors[38] This method not only determines effect of each factoron response variable but also considers multiple effectssimultaneously DOE is extended technique that includesseveral designs such as full factorial fractional factorial andnested design Specific design varies for each problem andshould be selected based on problem condition Our problemconsists of tree effective factors namely inventory policyreorder point and lead time these factors have nonlineareffect on objective function values As the aim of DOE inthis research is producing decision space with few factors (in

6 Modelling and Simulation in Engineering

Demand arrival

Sufficientinventory isavailable

True

False

Reorderproducts

True

False

Demand satisfaction

Make an order

Passing of lead time

Demand dispose

Lost customer

Is there anyorder in way

True

False

Updating of inventory level

Demand satisfaction

(a) FQ policy

Demand arrival

Sufficientinventory isavailable

Demand satisfaction Lost customer

FalseTrue

Demand dispose

System start

Make order

Make an order

Passing of lead time

TrueFalse

Passing of reviewing period

Updating of inventory level

(b) FI policy

Demand arrival

Sufficientinventory isavailable

Lost customer

System start

Make orderFalse True

Estimatedquantity ispossible

True

False

True False

Update of estimation

Update of information

Order maximum quantity

Order estimated quantity

Passing of lead time

Updating of inventory level

Demand satisfaction

Passing of reviewing period

Demand dispose

(c) DF policy

Figure 1

Modelling and Simulation in Engineering 7

Table 2 DOE configuration

ProductsFactors

IP 119877119905

119871

minus 0 + minus 0 + minus 0 +119860 FI FQ DF 10 20 30 3 5 7119861 FI FQ DF 15 30 45 3 5 7119862 FI FQ DF 15 30 45 3 5 7119863 FI FQ DF 15 30 45 3 5 7

Table 3 Experimental design

Number of trial 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27119877119901

minus minus minus minus minus minus minus minus minus 0 0 0 0 0 0 0 0 0 + + + + + + + + +119871 minus minus minus 0 0 0 + + + minus minus minus 0 0 0 + + + minus minus minus 0 0 0 + + +IP minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 +

this research there are 3 factors) using full factorial design ispreferable On the other hand because nonlinear behaviourof response variables (ie objective functions) centre pointis considered Each factor has high level ldquo+rdquo low level ldquominusrdquoand centre point ldquo0rdquo Factors and levels configurations areshown in Table 2 while Table 3 shows experimental designIn both of these tables minus sign is representative of lowlevel of related factors while plus sign is indicator of high levelof factors and zero determines centre point For examplein the case of lead time (119871) 3 is low level 7 is high leveland 5 is centre point Lead time is considered with daysas measuring unit and begins from time order is organizeduntil the arrival of inventory Reorder point is labelled as 119877

119901

and is intended to satisfy product demand during lead time(119871) Eventually inventory policy is labelled as IP and refersto inventory policies that were described in Section 412The mentioned design includes 27 runs and there are fourresponse variables cost service level average of inventoryposition and robustness In fact first three response variablesare expected values of defined objective functions (ie resp(1) (2) and (3)) but robustness is a defined criterion whichguarantees that optimal solution remains valid in situationwhere demand variation and intensification increase Forevaluation of robustness demands are multiplied by randomvariable with normal distribution (120583 = 1 120590 = 15)This random variable is also restricted to positive values toprevent negative demands Expected value of 4000 generatednumbers with mentioned constraint is equal to 148 thisresult implies that multiplied random number increases bothdemand variation and intensification simultaneously Thenthe percent of decrease in service level is considered asrobustness for all 27 scenarios produced byDOE It is obviousthat a lower decrease in service level is desirable

422 Weight Assignment Based on PCA Method PCA isthe abbreviation of ldquoprincipal component analysisrdquo and isknown as powerful data-driven method successfully appliedin several areas [2] PCA analysis can determine key variablesthat govern most of the variability This technique helps toremove less important variables to reduce dimension of data

sets PCA considers two criteria for each variable in datasets the first is variability and the second is correlation withother variables So input for this method consists of variancecovariance matrix [39 40] Although other methods likeentropy is applicable in multiple attribute decision makingproblem PCA advantage is considerable because variablescorrelations are not ignored As our problem is constructedof highly correlated variables PCA is the preferable choiceFor correlation justification remember that improvementof inventory position criterion causes negative influence onshortage cost or improvement of service level criterion causesnegative influence on inventory in hand and holding cost Inthese circumstances there is no reason for applying variationbased methods like entropy so in this paper PCA is appliedfor weight assignment These weights are used by MADMtechnique (described in Section 423) to improve qualityof ranking Because of relative utility of scenarios linearscaleless method employed to scaleless data obtained fromsimulation

(1) Calculate scaleless matrix of 119863 = (1198891 1198892 119889

119901)119899times119901

which 119889119894119895is scaleless value of 119894th scenario in the view

point of 119895th criterion

(2) Calculate sample mean vector 119889 = (1198891 1198892 119889

119901)119901

and covariance matrix based on (7) and (8) respec-tively

119889119896=1

119899

119899

sum

119894=1

119889119894119895 (7)

119878 = (119904119896119902)119901times119901

=1

119899 minus 1(119863 minus 119889)

119879

(119863 minus 119889) (8)

(3) 1198621radic119904119896119896 is diagonal 119901 times 119901 matrix 119896th diagonal

element is 1radic119904119896119896 and sample correlation matrix (119877)is calculated as mentioned in the following equation

119877 =1198621

radic119904119896119896

sdot 119878 sdot1198621

radic119904119896119896

(9)

8 Modelling and Simulation in Engineering

(4) Solve the following equation where 119868119901is a 119901 times 119901

identity matrix10038161003816100381610038161003816119877 minus 120582119868

119901

10038161003816100381610038161003816= 0 (10)

Solving (10) results in 119875 individual eigenvaluesregarding 120582

1ge 1205822ge sdot sdot sdot ge 120582

119901and sum119901

119896=1120582119896=

119875 These eigenvalues have 119875-specific eigenvectors(119897119896

1 119897119896

2 119897

119896

119901) and 119896 = 1 119901 These vectors create

principal components as expressed in the followingequation

PC119896=

119901

sum

119902=1

119897119896

119902119889119895

119902 (11)

(5) Select sufficient number of principal components Inthis problem first119872 components whose cumulativepercentage of their eigenvalues (119862

119872) is greater than

95 are selected Cumulative percentage is calculatedbased on the following equation

119862119898=sum119898

119896=1120582119896

sum119901

119896=1120582119896

=sum119898

119896=1120582119896

119875 (12)

(6) Finally weights vector that is indicated by119885 is derivedfrom the mathematical relation in (13) while 119882

119896is

equal to 120582119896119875 if the entire elements of PC

119896are

positive otherwise minus120582119896119875 if entire component are

negative

119885 =

119872

sum

119896=1

119882119896PC119896 (13)

In other cases positive or negative 119882119896should be defined

in the order that final weight vector is positive For moredetailed discussion about 119882

119896 refer to [41] Finally for each

product there are four 119885 parameters that determine relativeimportance for the following criteria cost service levelaverage of inventory position and robustness each elementof 119885 vector is divided by the sum of vector elements fornormalization

423 VIKOR Method This method was developed by Opri-covic and is the abbreviation of expression that in Serbianmeans ldquomultiple criteria optimization and compromisingsolutionsrdquo [42] VIKOR method is based on compromisingsolutions which is the result of Yu [43] and Zeleny [44]studies This approach considers closeness to ideal solutionIn comparison with other MADM methods VIKOR is ableto present compromising solutions and substitutes thesesolutions with best one in the case of necessity [45] VIKORmethod is applicable for problems with multiple discretecriteria Generally this method can be considered as rankingtool for scenarios which are generated by DOE Ranking isbased on L-P metric function according to (14) with defini-tion of 119891

119894119895as the value of 119894th criterion for 119895th scenario In

construction of L-P metric function 119891lowast119894indicates best value

of all scenarios in the view point of 119894th criterion and 119891minus119894

represents the worst value of all scenarios with considerationof 119894th criterion 119908

119894is the weight parameter for 119894th criterion

and is derived by PCA method Consider

119871119901=

119899

sum

119894=1

[

119908119894(119891lowast

119894minus 119891119894119895)

(119891lowast

119894minus 119891minus

119894)]

119901

1119901

1 ≪ 119875 ≪ infin 119895 = 1 2 3 119869

119871infin= max

119894[

119908119894(119891lowast

119894minus 119891119894119895)

(119891lowast

119894minus 119891minus

119894)]

(14)

VIKOR method only involves 1198711and 119871

infinwhich are known

as 119878119895and 119877

119895 respectively 119878 and 119877 index refer to scenarios

so both of these functions are regarded for each scenarioVIKOR steps are as follows

(1) Calculate 119891minus119894and 119891lowast

119894for entire of criteria

(2) Calculate 119878119895and 119877

119895for entire of scenarios

(3) With the assumption that 119878lowast and 119877lowast are the smallestvalues among all of 119878

119895and 119877

119895 respectively and also

119878minus and 119877minus are the greatest values in comparison withall of 119878

119895and 119877

119895 respectively 119876

119895is calculated for 119895th

scenario based on (15) In this equation V is strategicweight that determines the importance of individualcriterion (119878

119895) against group importance of criteria

(119877119895) for each scenario Consider

119876119895=

V (119878119895minus 119878lowast

)

119878minus minus 119878lowast+ (1 minus V)

(119877119895minus 119877lowast

)

119877minus minus 119877lowast (15)

(4) Scenarios should be sorted in descending order basedon 119878119877 and119876 specifically that result in three differentranking lists

(5) According to 119876 the scenario which is indicated by 1198861015840is the best scenario if two conditions namely (C1)and (C2) are satisfied

(C1) Acceptable Advantage This condition implies that thesecond best solution (11988610158401015840) should be far enough from thebest solution (1198861015840) to be accepted as unique solution Themathematical equation is defined in (16) considering 119869 as thetotal number of scenarios Consider

119876(11988610158401015840

) minus 119876 (1198861015840

) ≫1

(119869 minus 1) (16)

(C2) Acceptable Stability This condition validates 1198861015840 as thebest stable solution if 1198861015840 is also the best solution based on 119878 or119877

In case that one or both of the mentioned conditions arenot satisfied compromised solutions are involved VIKORmethod suggests that in such case compromising solutionshave equal value to the best solution and could be consideredinterchangeably by decision makers

Modelling and Simulation in Engineering 9

If (C1) is not satisfied scenarios 1198861015840 11988610158401015840 119886119872 are consid-ered 1198861015840 is the best solution and 11988610158401015840 119886119872 are compromisingsolutions with priority of 119876 on the other hand 1198861015840 11988610158401015840are considered if (C2) is not satisfied 119872 superscript isdetermined by (17)

119876(119886119872

) minus 119876 (1198861015840

) ≫1

(119869 minus 1) (17)

5 Numerical Result

Simulation process was executed on laptop with 18 GH CPUand 4GB of RAM Running of each replication with anima-tion and maximum speed takes 7 minutes and 35 secondsAs simulation of each scenario contains 10 replications andthere are 27 scenarios the total simulation time will be 2046minutes and 36 seconds

51 Multiresolution Result for Demand Modelling Result ofdemand modelling is presented for product 119861 because thisanalysis for other products is similar For this problempolynomial functions are applied for yearly and monthlydemand rates Equations (18) and (19) are estimated bynonlinear regression model In these functions 119905

119910and 119905119898are

time variables

119891 (119905119910) = 0008942 times 119905

3

119910minus 003884 times 119905

2

119910

+ 01867 times 119905119910minus 008018

(18)

and 95 confidence interval is presented in Table 4 for eachcoefficient of (18)

In this table coefficients of (18) are visible in the firstcolumn while lower bund and upper bund of each coefficientare presented in the second and third columns respectivelyThere is no interval including zero so it is concluded thatcoefficient estimation is valid

Estimation error for 119891(119905119910) is reported based on the sum

of square errors (SSE) which is 1504 times 10minus6 Consider

119891 (119905119898) = 4768 times 10

minus5

times 1199055

119898minus 0001692 times 119905

4

119898+ 00214

times 1199053

119898minus 01159 times 119905

2

119898+ 03474 times 119905

119898minus 02286

(19)

Monthly demand rate function that is indicated by 119891(119905119898)

and its coefficients confidence interval are presented inTable 5 SEE is equal to 00005825 Based on local businessinformation total demand of product 119861 is estimated to beabout 6822 units in five years According to the informationabout product 119861 Figure 2 shows final multiresolution resultfor this product

52 Simulation Parameters and Configuration For simula-tion of problem it is necessary to configure inventory policyparameters and variables We have three different policiesnamely FQ FI and DF Variables of FQ policy are reorderpoint (119877

119901) lead time (119871) and economic quantity orders (119876

119895119905)

First two variables are configured based on Table 2 and thirdvariable is derived fromWilson formula according to average

Table 4 95 confidence interval for estimated coefficient of 119891(119905119910)

Coefficients Lower bund Upper bund0008942 0004835 001305minus003884 minus007604 minus000165201867 008654 0287minus008018 minus01568 minus0003523

Table 5 95 confidence interval for estimated coefficient of 119891(119905119898)

Coefficients Lower bund Upper bund4768 times 10minus5 4451 times 10minus5 5086 times 10minus5

minus0001692 minus0001795 minus00015900214 00202 00226minus01159 minus01221 minus0109703474 0334 03608minus02286 minus02371 minus022

Table 6 Estimation of distribution functions of demand and reor-ders quantity

Product type Distribution function 119875 value Order quantity119860 278 lowast BETA(113 328) gt015 100119861 GAMMA(678 118) gt015 92119862 EXPO(818) gt015 100119863 446 lowast BETA(0957 321) gt015 115

of demand holding cost (119862ℎ) and ordering cost (119862

119903) The

calculated values are 57 71 72 and 72 for product 119860 119861 119862and119863 respectively

Orders quantity is unknown for FI policy and is cal-culated based on probability distribution function whichis obtained by demand simulation of each product for 20daysrsquo time interval Distribution fitting is performed by inputanalyser of Arena software and graphical result for product 119861is shown in Figure 3 Results for other products are reportedin Table 6

In this table fitted probability distribution function foreach product is presented in second column while 119875 valueof fitting which is greater than 015 for fitted distributiongrantees goodness of fitting in 95 confidence level Finallyaccording to fitted distribution functions for FI policyproper ordering amount of each product is presented in thefourth column

DF policy has two unknown parameters namely 120572 and120573 The first parameter is weight of current real demand in (4)and second one is the weight of current trend in (5) Theseparameters are adjusted by simulation of five years demandDemand for the entire four products simultaneously is con-sidered and an average of lost sale percentage is consideredas response variable Different values of response variables arereported in Table 7 According to the different values of 120572 and120573 the best values for 120572 and 120573 are 09 and 01 respectivelywhich results in minimum lost percentage in average

10 Modelling and Simulation in Engineering

Table 7 Average of lost sale percentage for different values of 120572 and 120573

120572 and 120573 values 01 02 03 04 05 06 07 08 09Response to 120572 0380 0356 0334 0320 0306 0292 0284 0276 0268Response to 120573 0296 0302 0306 0306 0306 0304 0305 0304 0302

50

100

150

200

250

300

350Demand fit for each month

Dem

and

Fitted valueReal demand

00 10 20 30 40 50 60

Month

Figure 2 Comparison between real and estimated demand

100 200 300 400 500 600 700

50

100

150

200

Distribution summary

Distribution gammaExpression minus0001 + GAMM(00)Square error 0006825

Chi-square testNumber of intervals = 5Degrees of freedom = 2Test statistic = 148

Kolmogorov-Smirnov testTest statistic = 00733

Corresponding P value = 0483

Corresponding P value gt 015

Figure 3 Distribution function of customer demand for reorderinterval of product 119861

53Weight Assignment Asmentioned before PCA is appliedforweight assignment For this purposeDOE result is neededin the form of decision matrix that is shown in Table 8 Inthis table four criteria cost average of inventory positionservice level and robustness are evaluated for each scenarioof product 119861 Decision matrix should be scaleless that isperformed by linear method Then the result of eigenvalues

Table 8 Decision matrix for product 119861

Scenarionumber Cost Average of

inventory positionServicelevel

Robustness

1 32984000 4352 76 62 29214710 397514 79 33 37114636 339623 69 24 34840720 410948 71 55 36820570 344738 72 66 39191837 331091 67 27 37037180 391012 68 48 40994680 298712 64 69 41407045 319678 63 110 28282930 524462 83 711 26422145 489243 85 512 31023829 416049 79 313 30395190 487146 79 614 35393680 395812 74 515 32838055 395646 77 316 33285615 463341 76 617 39652020 335668 66 718 35269609 383944 73 219 26227680 625612 87 720 25565820 586688 88 621 28711069 49449 85 422 27301580 585472 85 723 34016415 470961 78 724 30601546 463975 82 325 29752815 560421 82 726 41024410 382994 68 627 31604741 443053 78 3

is calculated and shown in Table 9 Also result of principalcomponents is presented in Table 10

As it is visible in Table 9 cumulative proportion ofthe first and second components is 0966 which is greaterthan 095 so the first two components are considered asprincipal components Their relative weights are 078 and0186 respectively Final weights are normalized and resultsare reported in Table 11

54 Ranking and Selection of Scenarios Weights of criteriaobtained by PCA method in addition to decision matrix areinput of VIKOR method 119878 119876 and 119877 values are calculatedbased on (14) and (15) while strategic weight indicated byV is equal to 05 so individual and group utility have same

Modelling and Simulation in Engineering 11

Table 9 Result of eigenvalues

Eigenvalues 31204 07428 01111 00257Proportion 0780 0186 0028 0006Cumulative 0780 0966 0994 1000

Table 10 Result of principle components

Variables PC1 PC2 PC3 PC4Inventory 0543 0089 minus0818 0167Cost minus0538 minus0277 minus0511 minus0610Service level minus0551 minus0210 minus0230 0774Robustness 0335 minus0933 0126 0023

Table 11 Weights of criteria for different products

Criteria Products119860 119861 119862 119863

Average of inventory position 01012 00590 01942 00462Cost 02852 01836 01742 03081Service level 01195 01392 00966 01206Robustness 04941 06182 05350 05251

importance Calculated values are reported in Table 12 forentire scenario as VIKOR output

Product 119861 According to Table 12 values of 119877 119878 and 119876 arezero for scenario 18 Regarding (C1) (Acceptable Advantage)the difference between the best scenario and second rankedscenario should be more than 00038 The second rankedscenario regarding 119876 value is scenario 3 and its value is0051044 that satisfies both (C1) and (C2) Final result forproduct 119861 is unique scenario that is number 18 and consistsof the following configuration Inventory policy is DFwith119877

119901

being equal to 30 products and 7 days lead time According todecision matrix which is shown in Table 8 the performanceof this scenario for product 119861 results in 383944 products asaverage of inventory position and expected cost is equal to35269609 for product life cycle 73 of customer demandwillbe satisfied and in the case of increasing demand intensityand variation 2 decrease is expected in service level so itwill change into 71 Analyses for other products are similarso only final results are presented in this paper

Product 119860 In this case VIKORmethod ranks scenario num-ber 15 as the best one This scenario consists of DF inventorypolicy 20 products for 119877

119901and 5 days for lead time Perfor-

mance of this scenario results in 374778 products as averageof inventory position Expected cost is 31603294 and servicelevel will be 66 This scenario is robust against demandintensification and variation because changes in demandaffect service level as small as 1 But scenario number 15 doesnot satisfy (C1) so compromising solutions are consideredAlthough the best scenario is number 15 VIKOR methodimplies that compromising solutions have the same values

Table 12 VIKOR output for product 119861

Run number 119878 119877 119876

1 0634311 0796269 0715292 0003184 0185077 0094133 0059804 0042283 00510444 0536946 0592539 05647425 0718862 0796269 07575656 0116278 0089887 01030827 0428814 0388808 04088118 086078 0796269 08285259 002043 0140653 008054110 0677386 1 083869311 0266254 0592539 042939612 0043962 0185077 01145213 0571695 0796269 068398214 0515097 0592539 055381815 009173 0185077 013840416 0648693 0796269 072248117 1 1 118 0 0 019 0630698 1 081534920 04239 0796269 061008421 0139527 0388808 026416722 06581 1 08290523 0819338 1 090966924 0022245 0185077 010366125 0726183 1 086309126 0849516 0796269 082289327 0072768 0185077 0128922

So in real situation compromising solution can be substi-tuted This solution regarding their priority is as follows

Scenario number 24 inventory policy is DF 119877119901is

equal to 30 products and lead time is 5 daysScenario number 18 inventory policy is DF 119877

119901is

equal to 20 products and lead time is 7 daysScenario number 12 inventory policy isDF119877

119901is equal

to 20 products and lead time is 3 days

Product 119862 For this product scenario number 1 is ranked asthe best one This scenario consists of FI inventory policy 119877

119901

is 15 products and lead time is 3 days Result of this scenario is4387 products as the average of inventory position Expectedcost is 66143000 and service level will be 61 with robustnessof 1 Compromising solutions are as follows

Scenario number 6 inventory policy is DF 119877119901is 15

products and lead time is 5 daysScenario number 3 inventory policy is DF 119877

119901is 15

products and lead time is 3 daysScenario number 7 inventory policy is FI 119877

119901is 15

products and lead time is 7 days

12 Modelling and Simulation in Engineering

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 4 Interaction plot for product 119862 average of inventory position

Scenario number 4 inventory policy is FI 119877119901is 15

products and lead time is 5 days

Product119863 Scenario number 3 has the best rank for this prod-uct This scenario consists of DF inventory policy 119877

119901is 15

products and lead time is 3 days This scenario leads to

363896 products as average of inventory position with62648337 expected cost and 61 service level with 1robustness Compromising solutions are as follows

Scenario number 6 this scenario is mentioned beforeas a compromising solution for product C

Modelling and Simulation in Engineering 13

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 5 Interaction plot for product 119862 cost

14 Modelling and Simulation in Engineering

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

3

5

7

L

3

5

7

L

3

5

7

L

L

(b)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 6 Interaction plot for product 119862 service level

Scenario number 17 inventory policy is FI119877119901is equal

to 15 products and lead time is 7 days

55 Sensitivity Analysis As mentioned in the previous sec-tion all scenarios related to product 119860 prefer DF inventory

policy so it is concluded that DF policy is unique optimalpolicy while lead time includes values of 3 5 and 7 in selectedscenarios This situation implies that product 119860 is insensitiveto the delivery time It can be wise to decide lead time about 5days but there is no need for strict control on lead time while

Modelling and Simulation in Engineering 15

119877119901should be controlled strictly to prevent inventory in hand

from reaching lower than 20 productsIn the case of product 119861 (C1) is satisfied so there is a

unique scenario while there are four compromising solutionfor product 119862 In this situation based on VIKOR methodcompromising solutions have same value while interactionplot can be applied for investigation of existing differencebetween compromising solutions As mentioned before thebest scenario suggests FI inventory policy for this product butin compromising solutions there are both FI and DF policiesLead time varies from 3 to 7 days So this confusing situationis resolved by deeper analysis and applying interaction plotFigure 4 shows interaction plot for DOE factors and averageof inventory position as response variables

Figure 4 shows that with either DF or FI policy there isno significant difference among different lead times in theview point of inventory position but DF leads to less averageof inventory position With 119877

119901equaling 15 there is less

sensitivity to different lead times So configuring 119877119901which

is equal to 15 products the average of inventory positioncriterion will be robust against lead time variations

Figure 5 regards cost criterion and shows that FI andDF policies are equal and make least sensitivity related tolead time On the other hand three plots of 119877

119901against lead

time are parallel The parallel plots are interpreted as lackof relation between 119877

119901and lead time With 119877

119901equaling 15

increase in inventory cost could be mitigated when lead timesets to 3 days

In Figure 6 with 3 days for lead time there is no signifi-cant difference between inventory policies in the view pointof service level119877

119901and lead time are independent and the best

service level is related to FI policy which is also very sensitiveto 119877119901 As 119877

119901increases service level will grow According to

the interaction plot analyses optimal decision for product119862 is FI inventory policy 3 days lead time and 15 or moreproducts for 119877

119901

Decision about product 119863 is simple Based on the in-formation obtained from solutions it is recommended toemploy DF inventory policy with 119877

119901being equal to 15

products and 3 days for lead time

6 Conclusion

In this paper a simulation optimization framework is pro-posed for robust optimization of retailer inventory systemThis framework is less time consuming in comparison withmetaheuristic or SPSA approach it also provides robustsolution for multiple objective functions In this research alsotrend and cyclic change of customers demand are consideredfor more realistic view of problem Proposed framework con-sists of discrete event simulation and surrogate model In thisframework full factorial design of experiment is employedfor producing decision space Based on the three decisionfactors 27 different scenarios are produced and simulated Insimulation modelling multiresolution method is applied forsimulation of demand with highly dynamic pattern Becausethere are multiple criteria for inventory system performanceVIKORmethod is employed asmulticriteria decisionmaking

technique In addition robustness is considered as perfor-mance criterion and PCA is used for improving performanceof VIKOR method Finally developed framework gave thebest ranked and compromising solutions so interaction plotapplied for more investigation and sensitivity analysis ofobtained solutions

Due to the novelty of the proposed framework it isrecommended that future studies encompass optimization ofinventory system of integrated supply chain Also improve-ment of developed framework can be considered as futurestudy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers for theirsignificant roles in improvement of this research quality

References

[1] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[2] S Yin S XDingAHaghaniHHao andP Zhang ldquoA compar-ison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[3] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[4] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[5] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall International Series in Management New YorkNY USA 1963

[6] E Naddor Inventory Systems John Wiley amp Sons New YorkNY USA 1966

[7] R Peterson and E A Silver Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 2nd edition 1985

[8] T C E Cheng ldquoEconomic order quantity model with demand-dependent unit production cost and imperfect productionprocessesrdquo IIE Transactions vol 23 no 1 pp 23ndash28 1991

[9] S H Chen C CWang and A Ramer ldquoBackorder fuzzy inven-tory model under function principlerdquo Information Sciences vol95 no 1-2 pp 71ndash79 1996

[10] X Zhao J Xie and J C Wei ldquoThe value of early order com-mitment in a two-level supply chainrdquo European Journal of Op-erational Research vol 180 no 1 pp 194ndash214 2007

[11] R S M Lau J Xie and X Zhao ldquoEffects of inventory policyon supply chain performance a simulation study of critical

16 Modelling and Simulation in Engineering

decision parametersrdquo Computers and Industrial Engineeringvol 55 no 3 pp 620ndash633 2008

[12] J Xu and L Zhao ldquoA multi-objective decision-making modelwith fuzzy rough coefficients and its application to the inventoryproblemrdquo Information Sciences vol 180 no 5 pp 679ndash6962010

[13] F Hnaien X Delorme and A Dolgui ldquoMulti-objective opti-mization for inventory control in two-level assembly systemsunder uncertainty of lead timesrdquo Computers amp OperationsResearch vol 37 no 11 pp 1835ndash1843 2010

[14] H D Purnomo H M Wee and Y Praharsi ldquoTwo inventoryreview policies on supply chain configuration problemrdquo Com-puters and Industrial Engineering vol 63 no 2 pp 448ndash4552012

[15] B L Nelson ldquo50th anniversary article stochastic simulationresearch in management sciencerdquoManagement Science vol 50no 7 pp 855ndash868 2004

[16] J Boesel R O Bowden Jr F Glover J P Kelly and E WestwigldquoFuture of simulation optimizationrdquo inProceedings of theWinterSimulation Conference pp 1466ndash1469 Arlington Va USADecember 2001

[17] M C Fu ldquoOptimization for simulation theory vs practicerdquoINFORMS Journal on Computing vol 14 no 3 pp 192ndash2152002

[18] L Wang ldquoA hybrid genetic algorithm-neural network strategyfor simulation optimizationrdquoApplied Mathematics and Compu-tation vol 170 no 2 pp 1329ndash1343 2005

[19] B B Keskin S H Melouk and I L Meyer ldquoA simulation-optimization approach for integrated sourcing and inventorydecisionsrdquo Computers and Operations Research vol 37 no 9pp 1648ndash1661 2010

[20] E Mazhari J Zhao N Celik S Lee Y Son and L HeadldquoHybrid simulation and optimization-based design and oper-ation of integrated photovoltaic generation storage units andgridrdquo Simulation Modelling Practice and Theory vol 19 no 1pp 463ndash481 2011

[21] Q Duan and T W Liao ldquoOptimization of replenishment poli-cies for decentralized and centralized capacitated supply chainsunder various demandsrdquo International Journal of ProductionEconomics vol 142 no 1 pp 194ndash204 2013

[22] J C Spall ldquoMultivariate stochastic approximation usinga simultaneous perturbation gradient approximationrdquo IEEETransactions on Automatic Control vol 37 no 3 pp 332ndash3411992

[23] J C Spall ldquoImplementation of the simultaneous perturbationalgorithm for stochastic optimizationrdquo IEEE Transactions onAerospace and Electronic Systems vol 34 no 3 pp 817ndash8231998

[24] J C Spall ldquoStochastic optimization and the simultaneousperturbation methodrdquo in Proceedings of the Winter SimulationConference (WSC rsquo99) pp 101ndash109 December 1999

[25] J D Schwartz W Wang and D E Rivera ldquoSimulation-based optimization of process control policies for inventorymanagement in supply chainsrdquo Automatica vol 42 no 8 pp1311ndash1320 2006

[26] H G Neddermeijer G J V Oortmarssen and N P R DekkerldquoA framework for response surface methodology for simulationoptimizationrdquo in Proceedings of the Winter Simulation Confer-ence vol 1 Orlando Fla USA December 1999

[27] B Can and C Heavey ldquoA comparison of genetic programmingand artificial neural networks in metamodeling of discrete-event simulation modelsrdquo Computers amp Operations Researchvol 39 no 2 pp 424ndash436 2012

[28] A Azadeh Z S Faiz S M Asadzadeh and R Tavakkoli-Moghaddam ldquoAn integrated artificial neural network-comput-er simulation for optimization of complex tandem queuesystemsrdquoMathematics and Computers in Simulation vol 82 no4 pp 666ndash678 2011

[29] R Bornatico J Hussy A Witzig and L Guzzella ldquoSurrogatemodeling for the fast optimization of energy systemsrdquo Energyvol 57 pp 653ndash662 2013

[30] X Wan J F Pekny and G V Reklaitis ldquoSimulation-basedoptimizationwith surrogatemodels application to supply chainmanagementrdquoComputers and Chemical Engineering vol 29 no6 pp 1317ndash1328 2005

[31] W Shi Z Liu J Shang and Y Cui ldquoMulti-criteria robust designof a JIT-based cross-docking distribution center for an autoparts supply chainrdquo European Journal of Operational Researchvol 229 no 3 pp 695ndash706 2013

[32] G Dellino J P C Kleijnen and C Meloni ldquoRobust optimiza-tion in simulation taguchi and Response Surface Methodol-ogyrdquo International Journal of Production Economics vol 125 no1 pp 52ndash59 2010

[33] T Yang Y Wen and F Wang ldquoEvaluation of robustness of sup-ply chain information-sharing strategies using a hybrid Taguchiand multiple criteria decision-making methodrdquo InternationalJournal of Production Economics vol 134 no 2 pp 458ndash4662011

[34] M E Kuhl and J R Wilson ldquoModeling and simulating Poissonprocesses having trends or nontrigonometric cyclic effectsrdquoEuropean Journal of Operational Research vol 133 no 3 pp566ndash582 2001

[35] P Narayan and J Subramanian Inventory ManagementmdashPrinciples and Practices Excel Books New Delhi India 2008

[36] E A Silver and R Peterson Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 1985

[37] R G Brown Smoothing Forecasting and Prediction of DiscreteTime Series Courier 2004

[38] D C Montgomery Design and analysis of experiments JohnWiley amp Sons New York NY USA 6th edition 2005

[39] J Zhu ldquoData envelopment analysis vs principal componentanalysis an illustrative study of economic performance ofChinese citiesrdquo European Journal of Operational Research vol111 no 1 pp 50ndash61 1998

[40] I M Premachandra ldquoA note on DEA vs principal componentanalysis an improvement to Joe Zhursquos approachrdquo EuropeanJournal of Operational Research vol 132 no 3 pp 553ndash5602001

[41] D Slottje G W Scully G J Hirschberg and K Hayes Mea-suring the Quality of Life across Countries A MultidimensionalAnalysis Westview Press Boulder Colo USA 1991

[42] S Opricovic Multicriteria optimization of civil engineeringsystems [PhD thesis] Faculty of Civil Engineering BelgradeSerbia 1998

[43] P L Yu ldquoA class of solutions for group decision problemsrdquoManagement Science vol 19 no 8 pp 936ndash946 1973

Modelling and Simulation in Engineering 17

[44] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[45] S Opricovic and G H Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

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Page 4: Research Article Robust Inventory System Optimization ...downloads.hindawi.com/journals/mse/2014/305120.pdf · Robust Inventory System Optimization Based on Simulation and Multiple

4 Modelling and Simulation in Engineering

Table 1

Product type Average demand Holding cost in planning period Reorder cost Lost sale cost119860 987 30000 50000 20000119861 1520 30000 50000 15000119862 1598 30000 50000 25000119863 1569 30000 50000 18000

it is lost sale 119877119901is the reorder point for organizing of new

order In other words if inventory level reaches 119877119901or less

a new order would be organized with quantity of 119876119895119905 Each

organized order reaches the retailer and increases inventorylevel after passing of lead time (119871) Considering assumptionsand described notations the following equations are themainobjective functions of defined problem

Min119879

sum

119905=1

119862119903119899119905+

119879

sum

119905=1

119862ℎ119868119897

119905+

119879

sum

119905=1

119873

sum

119894=1

119862119861(1 minus 119909

119894119905) 119889119894119905

(1)

Maxsum119879

119905=1sum119873

119894=1119909119894119905119889119894119905

sum119905

119905=1sum119873

119894=1119889119894119905

(2)

Minsum119879

119905=1119868119901

119905

119879 (3)

In (1) the objective is minimizing the total cost includingcosts that depend on reordering handling and lost salesLost sales not only incur excess cost but also decreaseretailer credit So (2) is considered to maximize service levelindependently In (2) the objective function increases where119909119894119905is 1 for 119894th demand in 119905th period and such situation is

possible if 119868119897119905is greater than 119889

119894119905 In fact (2) causes increase

in inventory level while (1)ndash(3) causes decrease in inventorylevel State of inventory level depends on number of ordersin each planning period (119899

119905) quantity of orders (119876

119895119905) and

quantity of demands (119889119894119905) while state of inventory position

depends on inventory level and lead time (119871) so (3) isresponsible for minimizing average of inventory positionincluding inventory in hand (119868119897

119905) and on-order inventory

Average of inventory position should be minimized in orderto improve flexibility of retailer and approach to the just intime (JIT)

Demand of each product follows different pattern withboth long-term and cyclic trend So mentioned objectivefunctions are considered individually for each product type

4 Proposed Framework

In the defined problem demand of products exposes highlydynamic pattern and as time passes demand and its variationincrease hence multiresolution method is employed fordemand modelling Also three different policies for inven-tory control are considered which are reordered based onfixed quantity (FQ) fixed interval (FI) and demand forecast-ing (DF) Simulation of developed model is implemented inArena 135 Optimization of simulation model is performed

by surrogate model that is based on full factorial design ofexperiment (DOE) For construction of decision space DOEfactors include inventory policy reorder point and lead timewith three levels for each of them So there are 33 = 27

different combinations of decision variables to form feasiblescenarios Ranking of produced scenarios is accomplished byMADM technique For ranking of scenarios three objectivefunction values are considered (ie cost service level andaverage of inventory position) and in addition robustness ofservice level against demand fluctuation is considered AlsoPCA is applied for more realistic weighting of objective func-tion values based on their statistical influence on improve-ment of other objectives Finally interacting plot is employedfor sensitivity analysis and investigation of solutions in detail

41 Simulation Modelling Simulation modelling of problemconsists of two parts which are modelling of demand andmodelling of inventory policies In this paper demand is non-homogeneous Poisson process and three different inventorypolicies based on continues reviewing periodic reviewingand periodic reviewing with forecasting of future demand areconsidered

411 Modelling of Demand For customers demand model-ing multiresolutionmethod is applied Kuhl andWilson [34]developed this method for simulation of nonhomogeneousPoisson process with trend and cyclic changes This methodestimates mean intensity function and the nonparametricnature of thismethod is one of themost important advantagesin comparison with other methods So it is independent ofstatistical parameters and applicable in variety of problemsFurthermore multiresolution method can support combi-nation of multiple cyclic changes simultaneously Anotheradvantage of multiresolution is its capability in the modellingof nonsymmetric cyclic pattern As in our case demandhas nonsymmetric pattern with cyclic changes and long-term trend and multiresolution approach is a reasonablechoice More theoretical and application of used method areprovided in [34]

412 Modelling of Inventory Policies As the main effort ofthis paper is inventory system optimization modelling ofinventory policies plays an important role in this problemIn this problem optimization is manipulated by selection ofappropriate inventory policy and configuration of its param-eters to the way that leads to the optimal state of inventorysystem In the inventory management three approaches arecommon strategies which are fixed order quantity fixed time

Modelling and Simulation in Engineering 5

interval and forecasting methods [35] In the first strategyinventory level should be reviewed continuously until itreaches below predetermined quantity (reorder point) thenorder would be organized with fixed quantity of inventory Inthe second strategy reviewing period is a fixed time intervalbut quantity order is variable for each order that is based onconsumption rate While first strategy needs more effort forcontinuously reviewing of inventory level the second strategyis easier to handle but the risk of shortage in fixed intervalstrategy is more in comparison to fixed order quantity Sodue to themitigation of shortage risk in fixed interval strategyorder quantity is slightly more than fixed order quantity [35]

In the third strategy reviewing period is fixed as sec-ond policy but reorder quantity is based on forecasting offuture demands As thementioned strategies are fundamentalin inventory management literature and are also commonamong retailers of office furniture in this study three policiesbased on fixed order quantity fixed time interval anddemand forecasting are developed as follows

(1) Continuous reviewing with economic quantity order(2) Periodic reviewing with order quantity based on

demand confidence interval during the consumptionperiod

(3) Periodic reviewing with order quantity based onforecasting of future demands

Policy 1 (Fixed Quantity) Based on this policy each demandwill be satisfied if there is sufficient inventory in handAfter satisfaction of each demand inventory in hand willbe checked to see if the inventory level reaches to reorderpoint (119877

119901) If inventory level has reached to reorder point

economic order quantity (119876119895119905)would be organized otherwise

system waits for next demand Economic order quantity isderived by Wilson formula [36] If there is not sufficientinventory to satisfy arrived demand quantity of demandis considered as lost sale and lost profit treated as costIf no order has been organized system reorder inventoryotherwise waits for arriving of organized order accordingto adjusted lead time (119871) Logic of this policy is visible inFigure 1(a) and is labelled as FQ policy

Policy 2 (Fixed Interval) In this policy criterion for reorder-ing is fixed time interval that is known as planning periodAfter this period inventory level would be examined andreorder will be organized on condition that inventory levelhas reached to reorder point (119877

119901) For more realistic consid-

eration order quantity (119876119895119905) is calculated based on demand

cumulative distribution function in planning period Forexample in the inventory system that is planned for 10 lostsale with demand which is distributed based on exponentialdistribution function it should be ordered as much ascumulative probability of exponential distribution equals to09 In this policy inventory level would be updated after leadtime (119871) Logic of this policy is shown in Figure 1(b) and islabelled as FI policy

Policy 3 (Demand Forecasting) This policy is similar to policy2 with some differences In policy 2 probability distribution

function of demand in planning period is estimated byhistorical data and then reorder is organized based on servicelevel (cumulative probability of demand satisfaction) But inthis policy after fixed interval forecasting of future demandwill be performed If inventory level has reached to reorderpoint (119877

119901) order would be organized and inventory level

would be updated after lead time (119871) otherwise only infor-mation would be updated If forecasted demand is less thaninventory capacity order quantity (119876

119895119905) would be as much as

forecasted quantity otherwise inventory orderwill be asmuchas inventory capacity for each product As demands followtime series with autocorrelation forecasting is implementedby exponential smoothing method [37]

In this procedure 119860119905is defined as demand estimation

of 119905th planning period This term is calculated according toreal demand of period 119905 indicated by 119863

119905and estimation of

demand in previous period which is indicated by 119860119905minus1

plustrend adjustment value of previous period which is indicatedby 119879119905minus1

Impact of current real demand is considered byparameter 120572 that is defined between zero and one Trendadjustment value is derived based on 119860

119905and 119860

119905minus1plus trend

adjustment value of previous period with consideration of120573 as weight parameter which is defined between zero andone Equations related to 119860

119905and 119879

119905are expressed in (4)

and (5) respectively Finally demand estimation plus trendadjustment form of future demand forecasting is indicated by119865119905+1

which is described in (6) Logic of this policy is shown inFigure 1(c) and is labelled as DF policy Consider

119860119905= 120572 times 119863

119905+ (1 minus 120572) times (119860

119905minus1+ 119879119905minus1) (4)

119879119905= 120573 times (119860

119905minus 119860119905minus1) + (1 minus 120573) times 119879

119905minus1 (5)

119865119905+1= 119860119905+ 119879119905 (6)

42 Surrogate Modelling Surrogate model is designed foroptimization of simulated inventory system In this frame-work design of experiments is responsible for producingdifferent scenarios Each scenario has four criteria includ-ing cost service level average of inventory position androbustness against demand fluctuation Importance of eachcriterion is determined by PCA and finally these scenariosare ranked by MADM technique as multiple criteria decisionmaking tool

421 Design of Experimentation Although DOE roots backto the statistical quality control nowadays it is a powerfultool for analysing complex systems DOE is statisticalmethodand organizes structured experiments with several factors[38] This method not only determines effect of each factoron response variable but also considers multiple effectssimultaneously DOE is extended technique that includesseveral designs such as full factorial fractional factorial andnested design Specific design varies for each problem andshould be selected based on problem condition Our problemconsists of tree effective factors namely inventory policyreorder point and lead time these factors have nonlineareffect on objective function values As the aim of DOE inthis research is producing decision space with few factors (in

6 Modelling and Simulation in Engineering

Demand arrival

Sufficientinventory isavailable

True

False

Reorderproducts

True

False

Demand satisfaction

Make an order

Passing of lead time

Demand dispose

Lost customer

Is there anyorder in way

True

False

Updating of inventory level

Demand satisfaction

(a) FQ policy

Demand arrival

Sufficientinventory isavailable

Demand satisfaction Lost customer

FalseTrue

Demand dispose

System start

Make order

Make an order

Passing of lead time

TrueFalse

Passing of reviewing period

Updating of inventory level

(b) FI policy

Demand arrival

Sufficientinventory isavailable

Lost customer

System start

Make orderFalse True

Estimatedquantity ispossible

True

False

True False

Update of estimation

Update of information

Order maximum quantity

Order estimated quantity

Passing of lead time

Updating of inventory level

Demand satisfaction

Passing of reviewing period

Demand dispose

(c) DF policy

Figure 1

Modelling and Simulation in Engineering 7

Table 2 DOE configuration

ProductsFactors

IP 119877119905

119871

minus 0 + minus 0 + minus 0 +119860 FI FQ DF 10 20 30 3 5 7119861 FI FQ DF 15 30 45 3 5 7119862 FI FQ DF 15 30 45 3 5 7119863 FI FQ DF 15 30 45 3 5 7

Table 3 Experimental design

Number of trial 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27119877119901

minus minus minus minus minus minus minus minus minus 0 0 0 0 0 0 0 0 0 + + + + + + + + +119871 minus minus minus 0 0 0 + + + minus minus minus 0 0 0 + + + minus minus minus 0 0 0 + + +IP minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 +

this research there are 3 factors) using full factorial design ispreferable On the other hand because nonlinear behaviourof response variables (ie objective functions) centre pointis considered Each factor has high level ldquo+rdquo low level ldquominusrdquoand centre point ldquo0rdquo Factors and levels configurations areshown in Table 2 while Table 3 shows experimental designIn both of these tables minus sign is representative of lowlevel of related factors while plus sign is indicator of high levelof factors and zero determines centre point For examplein the case of lead time (119871) 3 is low level 7 is high leveland 5 is centre point Lead time is considered with daysas measuring unit and begins from time order is organizeduntil the arrival of inventory Reorder point is labelled as 119877

119901

and is intended to satisfy product demand during lead time(119871) Eventually inventory policy is labelled as IP and refersto inventory policies that were described in Section 412The mentioned design includes 27 runs and there are fourresponse variables cost service level average of inventoryposition and robustness In fact first three response variablesare expected values of defined objective functions (ie resp(1) (2) and (3)) but robustness is a defined criterion whichguarantees that optimal solution remains valid in situationwhere demand variation and intensification increase Forevaluation of robustness demands are multiplied by randomvariable with normal distribution (120583 = 1 120590 = 15)This random variable is also restricted to positive values toprevent negative demands Expected value of 4000 generatednumbers with mentioned constraint is equal to 148 thisresult implies that multiplied random number increases bothdemand variation and intensification simultaneously Thenthe percent of decrease in service level is considered asrobustness for all 27 scenarios produced byDOE It is obviousthat a lower decrease in service level is desirable

422 Weight Assignment Based on PCA Method PCA isthe abbreviation of ldquoprincipal component analysisrdquo and isknown as powerful data-driven method successfully appliedin several areas [2] PCA analysis can determine key variablesthat govern most of the variability This technique helps toremove less important variables to reduce dimension of data

sets PCA considers two criteria for each variable in datasets the first is variability and the second is correlation withother variables So input for this method consists of variancecovariance matrix [39 40] Although other methods likeentropy is applicable in multiple attribute decision makingproblem PCA advantage is considerable because variablescorrelations are not ignored As our problem is constructedof highly correlated variables PCA is the preferable choiceFor correlation justification remember that improvementof inventory position criterion causes negative influence onshortage cost or improvement of service level criterion causesnegative influence on inventory in hand and holding cost Inthese circumstances there is no reason for applying variationbased methods like entropy so in this paper PCA is appliedfor weight assignment These weights are used by MADMtechnique (described in Section 423) to improve qualityof ranking Because of relative utility of scenarios linearscaleless method employed to scaleless data obtained fromsimulation

(1) Calculate scaleless matrix of 119863 = (1198891 1198892 119889

119901)119899times119901

which 119889119894119895is scaleless value of 119894th scenario in the view

point of 119895th criterion

(2) Calculate sample mean vector 119889 = (1198891 1198892 119889

119901)119901

and covariance matrix based on (7) and (8) respec-tively

119889119896=1

119899

119899

sum

119894=1

119889119894119895 (7)

119878 = (119904119896119902)119901times119901

=1

119899 minus 1(119863 minus 119889)

119879

(119863 minus 119889) (8)

(3) 1198621radic119904119896119896 is diagonal 119901 times 119901 matrix 119896th diagonal

element is 1radic119904119896119896 and sample correlation matrix (119877)is calculated as mentioned in the following equation

119877 =1198621

radic119904119896119896

sdot 119878 sdot1198621

radic119904119896119896

(9)

8 Modelling and Simulation in Engineering

(4) Solve the following equation where 119868119901is a 119901 times 119901

identity matrix10038161003816100381610038161003816119877 minus 120582119868

119901

10038161003816100381610038161003816= 0 (10)

Solving (10) results in 119875 individual eigenvaluesregarding 120582

1ge 1205822ge sdot sdot sdot ge 120582

119901and sum119901

119896=1120582119896=

119875 These eigenvalues have 119875-specific eigenvectors(119897119896

1 119897119896

2 119897

119896

119901) and 119896 = 1 119901 These vectors create

principal components as expressed in the followingequation

PC119896=

119901

sum

119902=1

119897119896

119902119889119895

119902 (11)

(5) Select sufficient number of principal components Inthis problem first119872 components whose cumulativepercentage of their eigenvalues (119862

119872) is greater than

95 are selected Cumulative percentage is calculatedbased on the following equation

119862119898=sum119898

119896=1120582119896

sum119901

119896=1120582119896

=sum119898

119896=1120582119896

119875 (12)

(6) Finally weights vector that is indicated by119885 is derivedfrom the mathematical relation in (13) while 119882

119896is

equal to 120582119896119875 if the entire elements of PC

119896are

positive otherwise minus120582119896119875 if entire component are

negative

119885 =

119872

sum

119896=1

119882119896PC119896 (13)

In other cases positive or negative 119882119896should be defined

in the order that final weight vector is positive For moredetailed discussion about 119882

119896 refer to [41] Finally for each

product there are four 119885 parameters that determine relativeimportance for the following criteria cost service levelaverage of inventory position and robustness each elementof 119885 vector is divided by the sum of vector elements fornormalization

423 VIKOR Method This method was developed by Opri-covic and is the abbreviation of expression that in Serbianmeans ldquomultiple criteria optimization and compromisingsolutionsrdquo [42] VIKOR method is based on compromisingsolutions which is the result of Yu [43] and Zeleny [44]studies This approach considers closeness to ideal solutionIn comparison with other MADM methods VIKOR is ableto present compromising solutions and substitutes thesesolutions with best one in the case of necessity [45] VIKORmethod is applicable for problems with multiple discretecriteria Generally this method can be considered as rankingtool for scenarios which are generated by DOE Ranking isbased on L-P metric function according to (14) with defini-tion of 119891

119894119895as the value of 119894th criterion for 119895th scenario In

construction of L-P metric function 119891lowast119894indicates best value

of all scenarios in the view point of 119894th criterion and 119891minus119894

represents the worst value of all scenarios with considerationof 119894th criterion 119908

119894is the weight parameter for 119894th criterion

and is derived by PCA method Consider

119871119901=

119899

sum

119894=1

[

119908119894(119891lowast

119894minus 119891119894119895)

(119891lowast

119894minus 119891minus

119894)]

119901

1119901

1 ≪ 119875 ≪ infin 119895 = 1 2 3 119869

119871infin= max

119894[

119908119894(119891lowast

119894minus 119891119894119895)

(119891lowast

119894minus 119891minus

119894)]

(14)

VIKOR method only involves 1198711and 119871

infinwhich are known

as 119878119895and 119877

119895 respectively 119878 and 119877 index refer to scenarios

so both of these functions are regarded for each scenarioVIKOR steps are as follows

(1) Calculate 119891minus119894and 119891lowast

119894for entire of criteria

(2) Calculate 119878119895and 119877

119895for entire of scenarios

(3) With the assumption that 119878lowast and 119877lowast are the smallestvalues among all of 119878

119895and 119877

119895 respectively and also

119878minus and 119877minus are the greatest values in comparison withall of 119878

119895and 119877

119895 respectively 119876

119895is calculated for 119895th

scenario based on (15) In this equation V is strategicweight that determines the importance of individualcriterion (119878

119895) against group importance of criteria

(119877119895) for each scenario Consider

119876119895=

V (119878119895minus 119878lowast

)

119878minus minus 119878lowast+ (1 minus V)

(119877119895minus 119877lowast

)

119877minus minus 119877lowast (15)

(4) Scenarios should be sorted in descending order basedon 119878119877 and119876 specifically that result in three differentranking lists

(5) According to 119876 the scenario which is indicated by 1198861015840is the best scenario if two conditions namely (C1)and (C2) are satisfied

(C1) Acceptable Advantage This condition implies that thesecond best solution (11988610158401015840) should be far enough from thebest solution (1198861015840) to be accepted as unique solution Themathematical equation is defined in (16) considering 119869 as thetotal number of scenarios Consider

119876(11988610158401015840

) minus 119876 (1198861015840

) ≫1

(119869 minus 1) (16)

(C2) Acceptable Stability This condition validates 1198861015840 as thebest stable solution if 1198861015840 is also the best solution based on 119878 or119877

In case that one or both of the mentioned conditions arenot satisfied compromised solutions are involved VIKORmethod suggests that in such case compromising solutionshave equal value to the best solution and could be consideredinterchangeably by decision makers

Modelling and Simulation in Engineering 9

If (C1) is not satisfied scenarios 1198861015840 11988610158401015840 119886119872 are consid-ered 1198861015840 is the best solution and 11988610158401015840 119886119872 are compromisingsolutions with priority of 119876 on the other hand 1198861015840 11988610158401015840are considered if (C2) is not satisfied 119872 superscript isdetermined by (17)

119876(119886119872

) minus 119876 (1198861015840

) ≫1

(119869 minus 1) (17)

5 Numerical Result

Simulation process was executed on laptop with 18 GH CPUand 4GB of RAM Running of each replication with anima-tion and maximum speed takes 7 minutes and 35 secondsAs simulation of each scenario contains 10 replications andthere are 27 scenarios the total simulation time will be 2046minutes and 36 seconds

51 Multiresolution Result for Demand Modelling Result ofdemand modelling is presented for product 119861 because thisanalysis for other products is similar For this problempolynomial functions are applied for yearly and monthlydemand rates Equations (18) and (19) are estimated bynonlinear regression model In these functions 119905

119910and 119905119898are

time variables

119891 (119905119910) = 0008942 times 119905

3

119910minus 003884 times 119905

2

119910

+ 01867 times 119905119910minus 008018

(18)

and 95 confidence interval is presented in Table 4 for eachcoefficient of (18)

In this table coefficients of (18) are visible in the firstcolumn while lower bund and upper bund of each coefficientare presented in the second and third columns respectivelyThere is no interval including zero so it is concluded thatcoefficient estimation is valid

Estimation error for 119891(119905119910) is reported based on the sum

of square errors (SSE) which is 1504 times 10minus6 Consider

119891 (119905119898) = 4768 times 10

minus5

times 1199055

119898minus 0001692 times 119905

4

119898+ 00214

times 1199053

119898minus 01159 times 119905

2

119898+ 03474 times 119905

119898minus 02286

(19)

Monthly demand rate function that is indicated by 119891(119905119898)

and its coefficients confidence interval are presented inTable 5 SEE is equal to 00005825 Based on local businessinformation total demand of product 119861 is estimated to beabout 6822 units in five years According to the informationabout product 119861 Figure 2 shows final multiresolution resultfor this product

52 Simulation Parameters and Configuration For simula-tion of problem it is necessary to configure inventory policyparameters and variables We have three different policiesnamely FQ FI and DF Variables of FQ policy are reorderpoint (119877

119901) lead time (119871) and economic quantity orders (119876

119895119905)

First two variables are configured based on Table 2 and thirdvariable is derived fromWilson formula according to average

Table 4 95 confidence interval for estimated coefficient of 119891(119905119910)

Coefficients Lower bund Upper bund0008942 0004835 001305minus003884 minus007604 minus000165201867 008654 0287minus008018 minus01568 minus0003523

Table 5 95 confidence interval for estimated coefficient of 119891(119905119898)

Coefficients Lower bund Upper bund4768 times 10minus5 4451 times 10minus5 5086 times 10minus5

minus0001692 minus0001795 minus00015900214 00202 00226minus01159 minus01221 minus0109703474 0334 03608minus02286 minus02371 minus022

Table 6 Estimation of distribution functions of demand and reor-ders quantity

Product type Distribution function 119875 value Order quantity119860 278 lowast BETA(113 328) gt015 100119861 GAMMA(678 118) gt015 92119862 EXPO(818) gt015 100119863 446 lowast BETA(0957 321) gt015 115

of demand holding cost (119862ℎ) and ordering cost (119862

119903) The

calculated values are 57 71 72 and 72 for product 119860 119861 119862and119863 respectively

Orders quantity is unknown for FI policy and is cal-culated based on probability distribution function whichis obtained by demand simulation of each product for 20daysrsquo time interval Distribution fitting is performed by inputanalyser of Arena software and graphical result for product 119861is shown in Figure 3 Results for other products are reportedin Table 6

In this table fitted probability distribution function foreach product is presented in second column while 119875 valueof fitting which is greater than 015 for fitted distributiongrantees goodness of fitting in 95 confidence level Finallyaccording to fitted distribution functions for FI policyproper ordering amount of each product is presented in thefourth column

DF policy has two unknown parameters namely 120572 and120573 The first parameter is weight of current real demand in (4)and second one is the weight of current trend in (5) Theseparameters are adjusted by simulation of five years demandDemand for the entire four products simultaneously is con-sidered and an average of lost sale percentage is consideredas response variable Different values of response variables arereported in Table 7 According to the different values of 120572 and120573 the best values for 120572 and 120573 are 09 and 01 respectivelywhich results in minimum lost percentage in average

10 Modelling and Simulation in Engineering

Table 7 Average of lost sale percentage for different values of 120572 and 120573

120572 and 120573 values 01 02 03 04 05 06 07 08 09Response to 120572 0380 0356 0334 0320 0306 0292 0284 0276 0268Response to 120573 0296 0302 0306 0306 0306 0304 0305 0304 0302

50

100

150

200

250

300

350Demand fit for each month

Dem

and

Fitted valueReal demand

00 10 20 30 40 50 60

Month

Figure 2 Comparison between real and estimated demand

100 200 300 400 500 600 700

50

100

150

200

Distribution summary

Distribution gammaExpression minus0001 + GAMM(00)Square error 0006825

Chi-square testNumber of intervals = 5Degrees of freedom = 2Test statistic = 148

Kolmogorov-Smirnov testTest statistic = 00733

Corresponding P value = 0483

Corresponding P value gt 015

Figure 3 Distribution function of customer demand for reorderinterval of product 119861

53Weight Assignment Asmentioned before PCA is appliedforweight assignment For this purposeDOE result is neededin the form of decision matrix that is shown in Table 8 Inthis table four criteria cost average of inventory positionservice level and robustness are evaluated for each scenarioof product 119861 Decision matrix should be scaleless that isperformed by linear method Then the result of eigenvalues

Table 8 Decision matrix for product 119861

Scenarionumber Cost Average of

inventory positionServicelevel

Robustness

1 32984000 4352 76 62 29214710 397514 79 33 37114636 339623 69 24 34840720 410948 71 55 36820570 344738 72 66 39191837 331091 67 27 37037180 391012 68 48 40994680 298712 64 69 41407045 319678 63 110 28282930 524462 83 711 26422145 489243 85 512 31023829 416049 79 313 30395190 487146 79 614 35393680 395812 74 515 32838055 395646 77 316 33285615 463341 76 617 39652020 335668 66 718 35269609 383944 73 219 26227680 625612 87 720 25565820 586688 88 621 28711069 49449 85 422 27301580 585472 85 723 34016415 470961 78 724 30601546 463975 82 325 29752815 560421 82 726 41024410 382994 68 627 31604741 443053 78 3

is calculated and shown in Table 9 Also result of principalcomponents is presented in Table 10

As it is visible in Table 9 cumulative proportion ofthe first and second components is 0966 which is greaterthan 095 so the first two components are considered asprincipal components Their relative weights are 078 and0186 respectively Final weights are normalized and resultsare reported in Table 11

54 Ranking and Selection of Scenarios Weights of criteriaobtained by PCA method in addition to decision matrix areinput of VIKOR method 119878 119876 and 119877 values are calculatedbased on (14) and (15) while strategic weight indicated byV is equal to 05 so individual and group utility have same

Modelling and Simulation in Engineering 11

Table 9 Result of eigenvalues

Eigenvalues 31204 07428 01111 00257Proportion 0780 0186 0028 0006Cumulative 0780 0966 0994 1000

Table 10 Result of principle components

Variables PC1 PC2 PC3 PC4Inventory 0543 0089 minus0818 0167Cost minus0538 minus0277 minus0511 minus0610Service level minus0551 minus0210 minus0230 0774Robustness 0335 minus0933 0126 0023

Table 11 Weights of criteria for different products

Criteria Products119860 119861 119862 119863

Average of inventory position 01012 00590 01942 00462Cost 02852 01836 01742 03081Service level 01195 01392 00966 01206Robustness 04941 06182 05350 05251

importance Calculated values are reported in Table 12 forentire scenario as VIKOR output

Product 119861 According to Table 12 values of 119877 119878 and 119876 arezero for scenario 18 Regarding (C1) (Acceptable Advantage)the difference between the best scenario and second rankedscenario should be more than 00038 The second rankedscenario regarding 119876 value is scenario 3 and its value is0051044 that satisfies both (C1) and (C2) Final result forproduct 119861 is unique scenario that is number 18 and consistsof the following configuration Inventory policy is DFwith119877

119901

being equal to 30 products and 7 days lead time According todecision matrix which is shown in Table 8 the performanceof this scenario for product 119861 results in 383944 products asaverage of inventory position and expected cost is equal to35269609 for product life cycle 73 of customer demandwillbe satisfied and in the case of increasing demand intensityand variation 2 decrease is expected in service level so itwill change into 71 Analyses for other products are similarso only final results are presented in this paper

Product 119860 In this case VIKORmethod ranks scenario num-ber 15 as the best one This scenario consists of DF inventorypolicy 20 products for 119877

119901and 5 days for lead time Perfor-

mance of this scenario results in 374778 products as averageof inventory position Expected cost is 31603294 and servicelevel will be 66 This scenario is robust against demandintensification and variation because changes in demandaffect service level as small as 1 But scenario number 15 doesnot satisfy (C1) so compromising solutions are consideredAlthough the best scenario is number 15 VIKOR methodimplies that compromising solutions have the same values

Table 12 VIKOR output for product 119861

Run number 119878 119877 119876

1 0634311 0796269 0715292 0003184 0185077 0094133 0059804 0042283 00510444 0536946 0592539 05647425 0718862 0796269 07575656 0116278 0089887 01030827 0428814 0388808 04088118 086078 0796269 08285259 002043 0140653 008054110 0677386 1 083869311 0266254 0592539 042939612 0043962 0185077 01145213 0571695 0796269 068398214 0515097 0592539 055381815 009173 0185077 013840416 0648693 0796269 072248117 1 1 118 0 0 019 0630698 1 081534920 04239 0796269 061008421 0139527 0388808 026416722 06581 1 08290523 0819338 1 090966924 0022245 0185077 010366125 0726183 1 086309126 0849516 0796269 082289327 0072768 0185077 0128922

So in real situation compromising solution can be substi-tuted This solution regarding their priority is as follows

Scenario number 24 inventory policy is DF 119877119901is

equal to 30 products and lead time is 5 daysScenario number 18 inventory policy is DF 119877

119901is

equal to 20 products and lead time is 7 daysScenario number 12 inventory policy isDF119877

119901is equal

to 20 products and lead time is 3 days

Product 119862 For this product scenario number 1 is ranked asthe best one This scenario consists of FI inventory policy 119877

119901

is 15 products and lead time is 3 days Result of this scenario is4387 products as the average of inventory position Expectedcost is 66143000 and service level will be 61 with robustnessof 1 Compromising solutions are as follows

Scenario number 6 inventory policy is DF 119877119901is 15

products and lead time is 5 daysScenario number 3 inventory policy is DF 119877

119901is 15

products and lead time is 3 daysScenario number 7 inventory policy is FI 119877

119901is 15

products and lead time is 7 days

12 Modelling and Simulation in Engineering

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 4 Interaction plot for product 119862 average of inventory position

Scenario number 4 inventory policy is FI 119877119901is 15

products and lead time is 5 days

Product119863 Scenario number 3 has the best rank for this prod-uct This scenario consists of DF inventory policy 119877

119901is 15

products and lead time is 3 days This scenario leads to

363896 products as average of inventory position with62648337 expected cost and 61 service level with 1robustness Compromising solutions are as follows

Scenario number 6 this scenario is mentioned beforeas a compromising solution for product C

Modelling and Simulation in Engineering 13

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 5 Interaction plot for product 119862 cost

14 Modelling and Simulation in Engineering

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

3

5

7

L

3

5

7

L

3

5

7

L

L

(b)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 6 Interaction plot for product 119862 service level

Scenario number 17 inventory policy is FI119877119901is equal

to 15 products and lead time is 7 days

55 Sensitivity Analysis As mentioned in the previous sec-tion all scenarios related to product 119860 prefer DF inventory

policy so it is concluded that DF policy is unique optimalpolicy while lead time includes values of 3 5 and 7 in selectedscenarios This situation implies that product 119860 is insensitiveto the delivery time It can be wise to decide lead time about 5days but there is no need for strict control on lead time while

Modelling and Simulation in Engineering 15

119877119901should be controlled strictly to prevent inventory in hand

from reaching lower than 20 productsIn the case of product 119861 (C1) is satisfied so there is a

unique scenario while there are four compromising solutionfor product 119862 In this situation based on VIKOR methodcompromising solutions have same value while interactionplot can be applied for investigation of existing differencebetween compromising solutions As mentioned before thebest scenario suggests FI inventory policy for this product butin compromising solutions there are both FI and DF policiesLead time varies from 3 to 7 days So this confusing situationis resolved by deeper analysis and applying interaction plotFigure 4 shows interaction plot for DOE factors and averageof inventory position as response variables

Figure 4 shows that with either DF or FI policy there isno significant difference among different lead times in theview point of inventory position but DF leads to less averageof inventory position With 119877

119901equaling 15 there is less

sensitivity to different lead times So configuring 119877119901which

is equal to 15 products the average of inventory positioncriterion will be robust against lead time variations

Figure 5 regards cost criterion and shows that FI andDF policies are equal and make least sensitivity related tolead time On the other hand three plots of 119877

119901against lead

time are parallel The parallel plots are interpreted as lackof relation between 119877

119901and lead time With 119877

119901equaling 15

increase in inventory cost could be mitigated when lead timesets to 3 days

In Figure 6 with 3 days for lead time there is no signifi-cant difference between inventory policies in the view pointof service level119877

119901and lead time are independent and the best

service level is related to FI policy which is also very sensitiveto 119877119901 As 119877

119901increases service level will grow According to

the interaction plot analyses optimal decision for product119862 is FI inventory policy 3 days lead time and 15 or moreproducts for 119877

119901

Decision about product 119863 is simple Based on the in-formation obtained from solutions it is recommended toemploy DF inventory policy with 119877

119901being equal to 15

products and 3 days for lead time

6 Conclusion

In this paper a simulation optimization framework is pro-posed for robust optimization of retailer inventory systemThis framework is less time consuming in comparison withmetaheuristic or SPSA approach it also provides robustsolution for multiple objective functions In this research alsotrend and cyclic change of customers demand are consideredfor more realistic view of problem Proposed framework con-sists of discrete event simulation and surrogate model In thisframework full factorial design of experiment is employedfor producing decision space Based on the three decisionfactors 27 different scenarios are produced and simulated Insimulation modelling multiresolution method is applied forsimulation of demand with highly dynamic pattern Becausethere are multiple criteria for inventory system performanceVIKORmethod is employed asmulticriteria decisionmaking

technique In addition robustness is considered as perfor-mance criterion and PCA is used for improving performanceof VIKOR method Finally developed framework gave thebest ranked and compromising solutions so interaction plotapplied for more investigation and sensitivity analysis ofobtained solutions

Due to the novelty of the proposed framework it isrecommended that future studies encompass optimization ofinventory system of integrated supply chain Also improve-ment of developed framework can be considered as futurestudy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers for theirsignificant roles in improvement of this research quality

References

[1] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[2] S Yin S XDingAHaghaniHHao andP Zhang ldquoA compar-ison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[3] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[4] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[5] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall International Series in Management New YorkNY USA 1963

[6] E Naddor Inventory Systems John Wiley amp Sons New YorkNY USA 1966

[7] R Peterson and E A Silver Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 2nd edition 1985

[8] T C E Cheng ldquoEconomic order quantity model with demand-dependent unit production cost and imperfect productionprocessesrdquo IIE Transactions vol 23 no 1 pp 23ndash28 1991

[9] S H Chen C CWang and A Ramer ldquoBackorder fuzzy inven-tory model under function principlerdquo Information Sciences vol95 no 1-2 pp 71ndash79 1996

[10] X Zhao J Xie and J C Wei ldquoThe value of early order com-mitment in a two-level supply chainrdquo European Journal of Op-erational Research vol 180 no 1 pp 194ndash214 2007

[11] R S M Lau J Xie and X Zhao ldquoEffects of inventory policyon supply chain performance a simulation study of critical

16 Modelling and Simulation in Engineering

decision parametersrdquo Computers and Industrial Engineeringvol 55 no 3 pp 620ndash633 2008

[12] J Xu and L Zhao ldquoA multi-objective decision-making modelwith fuzzy rough coefficients and its application to the inventoryproblemrdquo Information Sciences vol 180 no 5 pp 679ndash6962010

[13] F Hnaien X Delorme and A Dolgui ldquoMulti-objective opti-mization for inventory control in two-level assembly systemsunder uncertainty of lead timesrdquo Computers amp OperationsResearch vol 37 no 11 pp 1835ndash1843 2010

[14] H D Purnomo H M Wee and Y Praharsi ldquoTwo inventoryreview policies on supply chain configuration problemrdquo Com-puters and Industrial Engineering vol 63 no 2 pp 448ndash4552012

[15] B L Nelson ldquo50th anniversary article stochastic simulationresearch in management sciencerdquoManagement Science vol 50no 7 pp 855ndash868 2004

[16] J Boesel R O Bowden Jr F Glover J P Kelly and E WestwigldquoFuture of simulation optimizationrdquo inProceedings of theWinterSimulation Conference pp 1466ndash1469 Arlington Va USADecember 2001

[17] M C Fu ldquoOptimization for simulation theory vs practicerdquoINFORMS Journal on Computing vol 14 no 3 pp 192ndash2152002

[18] L Wang ldquoA hybrid genetic algorithm-neural network strategyfor simulation optimizationrdquoApplied Mathematics and Compu-tation vol 170 no 2 pp 1329ndash1343 2005

[19] B B Keskin S H Melouk and I L Meyer ldquoA simulation-optimization approach for integrated sourcing and inventorydecisionsrdquo Computers and Operations Research vol 37 no 9pp 1648ndash1661 2010

[20] E Mazhari J Zhao N Celik S Lee Y Son and L HeadldquoHybrid simulation and optimization-based design and oper-ation of integrated photovoltaic generation storage units andgridrdquo Simulation Modelling Practice and Theory vol 19 no 1pp 463ndash481 2011

[21] Q Duan and T W Liao ldquoOptimization of replenishment poli-cies for decentralized and centralized capacitated supply chainsunder various demandsrdquo International Journal of ProductionEconomics vol 142 no 1 pp 194ndash204 2013

[22] J C Spall ldquoMultivariate stochastic approximation usinga simultaneous perturbation gradient approximationrdquo IEEETransactions on Automatic Control vol 37 no 3 pp 332ndash3411992

[23] J C Spall ldquoImplementation of the simultaneous perturbationalgorithm for stochastic optimizationrdquo IEEE Transactions onAerospace and Electronic Systems vol 34 no 3 pp 817ndash8231998

[24] J C Spall ldquoStochastic optimization and the simultaneousperturbation methodrdquo in Proceedings of the Winter SimulationConference (WSC rsquo99) pp 101ndash109 December 1999

[25] J D Schwartz W Wang and D E Rivera ldquoSimulation-based optimization of process control policies for inventorymanagement in supply chainsrdquo Automatica vol 42 no 8 pp1311ndash1320 2006

[26] H G Neddermeijer G J V Oortmarssen and N P R DekkerldquoA framework for response surface methodology for simulationoptimizationrdquo in Proceedings of the Winter Simulation Confer-ence vol 1 Orlando Fla USA December 1999

[27] B Can and C Heavey ldquoA comparison of genetic programmingand artificial neural networks in metamodeling of discrete-event simulation modelsrdquo Computers amp Operations Researchvol 39 no 2 pp 424ndash436 2012

[28] A Azadeh Z S Faiz S M Asadzadeh and R Tavakkoli-Moghaddam ldquoAn integrated artificial neural network-comput-er simulation for optimization of complex tandem queuesystemsrdquoMathematics and Computers in Simulation vol 82 no4 pp 666ndash678 2011

[29] R Bornatico J Hussy A Witzig and L Guzzella ldquoSurrogatemodeling for the fast optimization of energy systemsrdquo Energyvol 57 pp 653ndash662 2013

[30] X Wan J F Pekny and G V Reklaitis ldquoSimulation-basedoptimizationwith surrogatemodels application to supply chainmanagementrdquoComputers and Chemical Engineering vol 29 no6 pp 1317ndash1328 2005

[31] W Shi Z Liu J Shang and Y Cui ldquoMulti-criteria robust designof a JIT-based cross-docking distribution center for an autoparts supply chainrdquo European Journal of Operational Researchvol 229 no 3 pp 695ndash706 2013

[32] G Dellino J P C Kleijnen and C Meloni ldquoRobust optimiza-tion in simulation taguchi and Response Surface Methodol-ogyrdquo International Journal of Production Economics vol 125 no1 pp 52ndash59 2010

[33] T Yang Y Wen and F Wang ldquoEvaluation of robustness of sup-ply chain information-sharing strategies using a hybrid Taguchiand multiple criteria decision-making methodrdquo InternationalJournal of Production Economics vol 134 no 2 pp 458ndash4662011

[34] M E Kuhl and J R Wilson ldquoModeling and simulating Poissonprocesses having trends or nontrigonometric cyclic effectsrdquoEuropean Journal of Operational Research vol 133 no 3 pp566ndash582 2001

[35] P Narayan and J Subramanian Inventory ManagementmdashPrinciples and Practices Excel Books New Delhi India 2008

[36] E A Silver and R Peterson Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 1985

[37] R G Brown Smoothing Forecasting and Prediction of DiscreteTime Series Courier 2004

[38] D C Montgomery Design and analysis of experiments JohnWiley amp Sons New York NY USA 6th edition 2005

[39] J Zhu ldquoData envelopment analysis vs principal componentanalysis an illustrative study of economic performance ofChinese citiesrdquo European Journal of Operational Research vol111 no 1 pp 50ndash61 1998

[40] I M Premachandra ldquoA note on DEA vs principal componentanalysis an improvement to Joe Zhursquos approachrdquo EuropeanJournal of Operational Research vol 132 no 3 pp 553ndash5602001

[41] D Slottje G W Scully G J Hirschberg and K Hayes Mea-suring the Quality of Life across Countries A MultidimensionalAnalysis Westview Press Boulder Colo USA 1991

[42] S Opricovic Multicriteria optimization of civil engineeringsystems [PhD thesis] Faculty of Civil Engineering BelgradeSerbia 1998

[43] P L Yu ldquoA class of solutions for group decision problemsrdquoManagement Science vol 19 no 8 pp 936ndash946 1973

Modelling and Simulation in Engineering 17

[44] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[45] S Opricovic and G H Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

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Page 5: Research Article Robust Inventory System Optimization ...downloads.hindawi.com/journals/mse/2014/305120.pdf · Robust Inventory System Optimization Based on Simulation and Multiple

Modelling and Simulation in Engineering 5

interval and forecasting methods [35] In the first strategyinventory level should be reviewed continuously until itreaches below predetermined quantity (reorder point) thenorder would be organized with fixed quantity of inventory Inthe second strategy reviewing period is a fixed time intervalbut quantity order is variable for each order that is based onconsumption rate While first strategy needs more effort forcontinuously reviewing of inventory level the second strategyis easier to handle but the risk of shortage in fixed intervalstrategy is more in comparison to fixed order quantity Sodue to themitigation of shortage risk in fixed interval strategyorder quantity is slightly more than fixed order quantity [35]

In the third strategy reviewing period is fixed as sec-ond policy but reorder quantity is based on forecasting offuture demands As thementioned strategies are fundamentalin inventory management literature and are also commonamong retailers of office furniture in this study three policiesbased on fixed order quantity fixed time interval anddemand forecasting are developed as follows

(1) Continuous reviewing with economic quantity order(2) Periodic reviewing with order quantity based on

demand confidence interval during the consumptionperiod

(3) Periodic reviewing with order quantity based onforecasting of future demands

Policy 1 (Fixed Quantity) Based on this policy each demandwill be satisfied if there is sufficient inventory in handAfter satisfaction of each demand inventory in hand willbe checked to see if the inventory level reaches to reorderpoint (119877

119901) If inventory level has reached to reorder point

economic order quantity (119876119895119905)would be organized otherwise

system waits for next demand Economic order quantity isderived by Wilson formula [36] If there is not sufficientinventory to satisfy arrived demand quantity of demandis considered as lost sale and lost profit treated as costIf no order has been organized system reorder inventoryotherwise waits for arriving of organized order accordingto adjusted lead time (119871) Logic of this policy is visible inFigure 1(a) and is labelled as FQ policy

Policy 2 (Fixed Interval) In this policy criterion for reorder-ing is fixed time interval that is known as planning periodAfter this period inventory level would be examined andreorder will be organized on condition that inventory levelhas reached to reorder point (119877

119901) For more realistic consid-

eration order quantity (119876119895119905) is calculated based on demand

cumulative distribution function in planning period Forexample in the inventory system that is planned for 10 lostsale with demand which is distributed based on exponentialdistribution function it should be ordered as much ascumulative probability of exponential distribution equals to09 In this policy inventory level would be updated after leadtime (119871) Logic of this policy is shown in Figure 1(b) and islabelled as FI policy

Policy 3 (Demand Forecasting) This policy is similar to policy2 with some differences In policy 2 probability distribution

function of demand in planning period is estimated byhistorical data and then reorder is organized based on servicelevel (cumulative probability of demand satisfaction) But inthis policy after fixed interval forecasting of future demandwill be performed If inventory level has reached to reorderpoint (119877

119901) order would be organized and inventory level

would be updated after lead time (119871) otherwise only infor-mation would be updated If forecasted demand is less thaninventory capacity order quantity (119876

119895119905) would be as much as

forecasted quantity otherwise inventory orderwill be asmuchas inventory capacity for each product As demands followtime series with autocorrelation forecasting is implementedby exponential smoothing method [37]

In this procedure 119860119905is defined as demand estimation

of 119905th planning period This term is calculated according toreal demand of period 119905 indicated by 119863

119905and estimation of

demand in previous period which is indicated by 119860119905minus1

plustrend adjustment value of previous period which is indicatedby 119879119905minus1

Impact of current real demand is considered byparameter 120572 that is defined between zero and one Trendadjustment value is derived based on 119860

119905and 119860

119905minus1plus trend

adjustment value of previous period with consideration of120573 as weight parameter which is defined between zero andone Equations related to 119860

119905and 119879

119905are expressed in (4)

and (5) respectively Finally demand estimation plus trendadjustment form of future demand forecasting is indicated by119865119905+1

which is described in (6) Logic of this policy is shown inFigure 1(c) and is labelled as DF policy Consider

119860119905= 120572 times 119863

119905+ (1 minus 120572) times (119860

119905minus1+ 119879119905minus1) (4)

119879119905= 120573 times (119860

119905minus 119860119905minus1) + (1 minus 120573) times 119879

119905minus1 (5)

119865119905+1= 119860119905+ 119879119905 (6)

42 Surrogate Modelling Surrogate model is designed foroptimization of simulated inventory system In this frame-work design of experiments is responsible for producingdifferent scenarios Each scenario has four criteria includ-ing cost service level average of inventory position androbustness against demand fluctuation Importance of eachcriterion is determined by PCA and finally these scenariosare ranked by MADM technique as multiple criteria decisionmaking tool

421 Design of Experimentation Although DOE roots backto the statistical quality control nowadays it is a powerfultool for analysing complex systems DOE is statisticalmethodand organizes structured experiments with several factors[38] This method not only determines effect of each factoron response variable but also considers multiple effectssimultaneously DOE is extended technique that includesseveral designs such as full factorial fractional factorial andnested design Specific design varies for each problem andshould be selected based on problem condition Our problemconsists of tree effective factors namely inventory policyreorder point and lead time these factors have nonlineareffect on objective function values As the aim of DOE inthis research is producing decision space with few factors (in

6 Modelling and Simulation in Engineering

Demand arrival

Sufficientinventory isavailable

True

False

Reorderproducts

True

False

Demand satisfaction

Make an order

Passing of lead time

Demand dispose

Lost customer

Is there anyorder in way

True

False

Updating of inventory level

Demand satisfaction

(a) FQ policy

Demand arrival

Sufficientinventory isavailable

Demand satisfaction Lost customer

FalseTrue

Demand dispose

System start

Make order

Make an order

Passing of lead time

TrueFalse

Passing of reviewing period

Updating of inventory level

(b) FI policy

Demand arrival

Sufficientinventory isavailable

Lost customer

System start

Make orderFalse True

Estimatedquantity ispossible

True

False

True False

Update of estimation

Update of information

Order maximum quantity

Order estimated quantity

Passing of lead time

Updating of inventory level

Demand satisfaction

Passing of reviewing period

Demand dispose

(c) DF policy

Figure 1

Modelling and Simulation in Engineering 7

Table 2 DOE configuration

ProductsFactors

IP 119877119905

119871

minus 0 + minus 0 + minus 0 +119860 FI FQ DF 10 20 30 3 5 7119861 FI FQ DF 15 30 45 3 5 7119862 FI FQ DF 15 30 45 3 5 7119863 FI FQ DF 15 30 45 3 5 7

Table 3 Experimental design

Number of trial 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27119877119901

minus minus minus minus minus minus minus minus minus 0 0 0 0 0 0 0 0 0 + + + + + + + + +119871 minus minus minus 0 0 0 + + + minus minus minus 0 0 0 + + + minus minus minus 0 0 0 + + +IP minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 +

this research there are 3 factors) using full factorial design ispreferable On the other hand because nonlinear behaviourof response variables (ie objective functions) centre pointis considered Each factor has high level ldquo+rdquo low level ldquominusrdquoand centre point ldquo0rdquo Factors and levels configurations areshown in Table 2 while Table 3 shows experimental designIn both of these tables minus sign is representative of lowlevel of related factors while plus sign is indicator of high levelof factors and zero determines centre point For examplein the case of lead time (119871) 3 is low level 7 is high leveland 5 is centre point Lead time is considered with daysas measuring unit and begins from time order is organizeduntil the arrival of inventory Reorder point is labelled as 119877

119901

and is intended to satisfy product demand during lead time(119871) Eventually inventory policy is labelled as IP and refersto inventory policies that were described in Section 412The mentioned design includes 27 runs and there are fourresponse variables cost service level average of inventoryposition and robustness In fact first three response variablesare expected values of defined objective functions (ie resp(1) (2) and (3)) but robustness is a defined criterion whichguarantees that optimal solution remains valid in situationwhere demand variation and intensification increase Forevaluation of robustness demands are multiplied by randomvariable with normal distribution (120583 = 1 120590 = 15)This random variable is also restricted to positive values toprevent negative demands Expected value of 4000 generatednumbers with mentioned constraint is equal to 148 thisresult implies that multiplied random number increases bothdemand variation and intensification simultaneously Thenthe percent of decrease in service level is considered asrobustness for all 27 scenarios produced byDOE It is obviousthat a lower decrease in service level is desirable

422 Weight Assignment Based on PCA Method PCA isthe abbreviation of ldquoprincipal component analysisrdquo and isknown as powerful data-driven method successfully appliedin several areas [2] PCA analysis can determine key variablesthat govern most of the variability This technique helps toremove less important variables to reduce dimension of data

sets PCA considers two criteria for each variable in datasets the first is variability and the second is correlation withother variables So input for this method consists of variancecovariance matrix [39 40] Although other methods likeentropy is applicable in multiple attribute decision makingproblem PCA advantage is considerable because variablescorrelations are not ignored As our problem is constructedof highly correlated variables PCA is the preferable choiceFor correlation justification remember that improvementof inventory position criterion causes negative influence onshortage cost or improvement of service level criterion causesnegative influence on inventory in hand and holding cost Inthese circumstances there is no reason for applying variationbased methods like entropy so in this paper PCA is appliedfor weight assignment These weights are used by MADMtechnique (described in Section 423) to improve qualityof ranking Because of relative utility of scenarios linearscaleless method employed to scaleless data obtained fromsimulation

(1) Calculate scaleless matrix of 119863 = (1198891 1198892 119889

119901)119899times119901

which 119889119894119895is scaleless value of 119894th scenario in the view

point of 119895th criterion

(2) Calculate sample mean vector 119889 = (1198891 1198892 119889

119901)119901

and covariance matrix based on (7) and (8) respec-tively

119889119896=1

119899

119899

sum

119894=1

119889119894119895 (7)

119878 = (119904119896119902)119901times119901

=1

119899 minus 1(119863 minus 119889)

119879

(119863 minus 119889) (8)

(3) 1198621radic119904119896119896 is diagonal 119901 times 119901 matrix 119896th diagonal

element is 1radic119904119896119896 and sample correlation matrix (119877)is calculated as mentioned in the following equation

119877 =1198621

radic119904119896119896

sdot 119878 sdot1198621

radic119904119896119896

(9)

8 Modelling and Simulation in Engineering

(4) Solve the following equation where 119868119901is a 119901 times 119901

identity matrix10038161003816100381610038161003816119877 minus 120582119868

119901

10038161003816100381610038161003816= 0 (10)

Solving (10) results in 119875 individual eigenvaluesregarding 120582

1ge 1205822ge sdot sdot sdot ge 120582

119901and sum119901

119896=1120582119896=

119875 These eigenvalues have 119875-specific eigenvectors(119897119896

1 119897119896

2 119897

119896

119901) and 119896 = 1 119901 These vectors create

principal components as expressed in the followingequation

PC119896=

119901

sum

119902=1

119897119896

119902119889119895

119902 (11)

(5) Select sufficient number of principal components Inthis problem first119872 components whose cumulativepercentage of their eigenvalues (119862

119872) is greater than

95 are selected Cumulative percentage is calculatedbased on the following equation

119862119898=sum119898

119896=1120582119896

sum119901

119896=1120582119896

=sum119898

119896=1120582119896

119875 (12)

(6) Finally weights vector that is indicated by119885 is derivedfrom the mathematical relation in (13) while 119882

119896is

equal to 120582119896119875 if the entire elements of PC

119896are

positive otherwise minus120582119896119875 if entire component are

negative

119885 =

119872

sum

119896=1

119882119896PC119896 (13)

In other cases positive or negative 119882119896should be defined

in the order that final weight vector is positive For moredetailed discussion about 119882

119896 refer to [41] Finally for each

product there are four 119885 parameters that determine relativeimportance for the following criteria cost service levelaverage of inventory position and robustness each elementof 119885 vector is divided by the sum of vector elements fornormalization

423 VIKOR Method This method was developed by Opri-covic and is the abbreviation of expression that in Serbianmeans ldquomultiple criteria optimization and compromisingsolutionsrdquo [42] VIKOR method is based on compromisingsolutions which is the result of Yu [43] and Zeleny [44]studies This approach considers closeness to ideal solutionIn comparison with other MADM methods VIKOR is ableto present compromising solutions and substitutes thesesolutions with best one in the case of necessity [45] VIKORmethod is applicable for problems with multiple discretecriteria Generally this method can be considered as rankingtool for scenarios which are generated by DOE Ranking isbased on L-P metric function according to (14) with defini-tion of 119891

119894119895as the value of 119894th criterion for 119895th scenario In

construction of L-P metric function 119891lowast119894indicates best value

of all scenarios in the view point of 119894th criterion and 119891minus119894

represents the worst value of all scenarios with considerationof 119894th criterion 119908

119894is the weight parameter for 119894th criterion

and is derived by PCA method Consider

119871119901=

119899

sum

119894=1

[

119908119894(119891lowast

119894minus 119891119894119895)

(119891lowast

119894minus 119891minus

119894)]

119901

1119901

1 ≪ 119875 ≪ infin 119895 = 1 2 3 119869

119871infin= max

119894[

119908119894(119891lowast

119894minus 119891119894119895)

(119891lowast

119894minus 119891minus

119894)]

(14)

VIKOR method only involves 1198711and 119871

infinwhich are known

as 119878119895and 119877

119895 respectively 119878 and 119877 index refer to scenarios

so both of these functions are regarded for each scenarioVIKOR steps are as follows

(1) Calculate 119891minus119894and 119891lowast

119894for entire of criteria

(2) Calculate 119878119895and 119877

119895for entire of scenarios

(3) With the assumption that 119878lowast and 119877lowast are the smallestvalues among all of 119878

119895and 119877

119895 respectively and also

119878minus and 119877minus are the greatest values in comparison withall of 119878

119895and 119877

119895 respectively 119876

119895is calculated for 119895th

scenario based on (15) In this equation V is strategicweight that determines the importance of individualcriterion (119878

119895) against group importance of criteria

(119877119895) for each scenario Consider

119876119895=

V (119878119895minus 119878lowast

)

119878minus minus 119878lowast+ (1 minus V)

(119877119895minus 119877lowast

)

119877minus minus 119877lowast (15)

(4) Scenarios should be sorted in descending order basedon 119878119877 and119876 specifically that result in three differentranking lists

(5) According to 119876 the scenario which is indicated by 1198861015840is the best scenario if two conditions namely (C1)and (C2) are satisfied

(C1) Acceptable Advantage This condition implies that thesecond best solution (11988610158401015840) should be far enough from thebest solution (1198861015840) to be accepted as unique solution Themathematical equation is defined in (16) considering 119869 as thetotal number of scenarios Consider

119876(11988610158401015840

) minus 119876 (1198861015840

) ≫1

(119869 minus 1) (16)

(C2) Acceptable Stability This condition validates 1198861015840 as thebest stable solution if 1198861015840 is also the best solution based on 119878 or119877

In case that one or both of the mentioned conditions arenot satisfied compromised solutions are involved VIKORmethod suggests that in such case compromising solutionshave equal value to the best solution and could be consideredinterchangeably by decision makers

Modelling and Simulation in Engineering 9

If (C1) is not satisfied scenarios 1198861015840 11988610158401015840 119886119872 are consid-ered 1198861015840 is the best solution and 11988610158401015840 119886119872 are compromisingsolutions with priority of 119876 on the other hand 1198861015840 11988610158401015840are considered if (C2) is not satisfied 119872 superscript isdetermined by (17)

119876(119886119872

) minus 119876 (1198861015840

) ≫1

(119869 minus 1) (17)

5 Numerical Result

Simulation process was executed on laptop with 18 GH CPUand 4GB of RAM Running of each replication with anima-tion and maximum speed takes 7 minutes and 35 secondsAs simulation of each scenario contains 10 replications andthere are 27 scenarios the total simulation time will be 2046minutes and 36 seconds

51 Multiresolution Result for Demand Modelling Result ofdemand modelling is presented for product 119861 because thisanalysis for other products is similar For this problempolynomial functions are applied for yearly and monthlydemand rates Equations (18) and (19) are estimated bynonlinear regression model In these functions 119905

119910and 119905119898are

time variables

119891 (119905119910) = 0008942 times 119905

3

119910minus 003884 times 119905

2

119910

+ 01867 times 119905119910minus 008018

(18)

and 95 confidence interval is presented in Table 4 for eachcoefficient of (18)

In this table coefficients of (18) are visible in the firstcolumn while lower bund and upper bund of each coefficientare presented in the second and third columns respectivelyThere is no interval including zero so it is concluded thatcoefficient estimation is valid

Estimation error for 119891(119905119910) is reported based on the sum

of square errors (SSE) which is 1504 times 10minus6 Consider

119891 (119905119898) = 4768 times 10

minus5

times 1199055

119898minus 0001692 times 119905

4

119898+ 00214

times 1199053

119898minus 01159 times 119905

2

119898+ 03474 times 119905

119898minus 02286

(19)

Monthly demand rate function that is indicated by 119891(119905119898)

and its coefficients confidence interval are presented inTable 5 SEE is equal to 00005825 Based on local businessinformation total demand of product 119861 is estimated to beabout 6822 units in five years According to the informationabout product 119861 Figure 2 shows final multiresolution resultfor this product

52 Simulation Parameters and Configuration For simula-tion of problem it is necessary to configure inventory policyparameters and variables We have three different policiesnamely FQ FI and DF Variables of FQ policy are reorderpoint (119877

119901) lead time (119871) and economic quantity orders (119876

119895119905)

First two variables are configured based on Table 2 and thirdvariable is derived fromWilson formula according to average

Table 4 95 confidence interval for estimated coefficient of 119891(119905119910)

Coefficients Lower bund Upper bund0008942 0004835 001305minus003884 minus007604 minus000165201867 008654 0287minus008018 minus01568 minus0003523

Table 5 95 confidence interval for estimated coefficient of 119891(119905119898)

Coefficients Lower bund Upper bund4768 times 10minus5 4451 times 10minus5 5086 times 10minus5

minus0001692 minus0001795 minus00015900214 00202 00226minus01159 minus01221 minus0109703474 0334 03608minus02286 minus02371 minus022

Table 6 Estimation of distribution functions of demand and reor-ders quantity

Product type Distribution function 119875 value Order quantity119860 278 lowast BETA(113 328) gt015 100119861 GAMMA(678 118) gt015 92119862 EXPO(818) gt015 100119863 446 lowast BETA(0957 321) gt015 115

of demand holding cost (119862ℎ) and ordering cost (119862

119903) The

calculated values are 57 71 72 and 72 for product 119860 119861 119862and119863 respectively

Orders quantity is unknown for FI policy and is cal-culated based on probability distribution function whichis obtained by demand simulation of each product for 20daysrsquo time interval Distribution fitting is performed by inputanalyser of Arena software and graphical result for product 119861is shown in Figure 3 Results for other products are reportedin Table 6

In this table fitted probability distribution function foreach product is presented in second column while 119875 valueof fitting which is greater than 015 for fitted distributiongrantees goodness of fitting in 95 confidence level Finallyaccording to fitted distribution functions for FI policyproper ordering amount of each product is presented in thefourth column

DF policy has two unknown parameters namely 120572 and120573 The first parameter is weight of current real demand in (4)and second one is the weight of current trend in (5) Theseparameters are adjusted by simulation of five years demandDemand for the entire four products simultaneously is con-sidered and an average of lost sale percentage is consideredas response variable Different values of response variables arereported in Table 7 According to the different values of 120572 and120573 the best values for 120572 and 120573 are 09 and 01 respectivelywhich results in minimum lost percentage in average

10 Modelling and Simulation in Engineering

Table 7 Average of lost sale percentage for different values of 120572 and 120573

120572 and 120573 values 01 02 03 04 05 06 07 08 09Response to 120572 0380 0356 0334 0320 0306 0292 0284 0276 0268Response to 120573 0296 0302 0306 0306 0306 0304 0305 0304 0302

50

100

150

200

250

300

350Demand fit for each month

Dem

and

Fitted valueReal demand

00 10 20 30 40 50 60

Month

Figure 2 Comparison between real and estimated demand

100 200 300 400 500 600 700

50

100

150

200

Distribution summary

Distribution gammaExpression minus0001 + GAMM(00)Square error 0006825

Chi-square testNumber of intervals = 5Degrees of freedom = 2Test statistic = 148

Kolmogorov-Smirnov testTest statistic = 00733

Corresponding P value = 0483

Corresponding P value gt 015

Figure 3 Distribution function of customer demand for reorderinterval of product 119861

53Weight Assignment Asmentioned before PCA is appliedforweight assignment For this purposeDOE result is neededin the form of decision matrix that is shown in Table 8 Inthis table four criteria cost average of inventory positionservice level and robustness are evaluated for each scenarioof product 119861 Decision matrix should be scaleless that isperformed by linear method Then the result of eigenvalues

Table 8 Decision matrix for product 119861

Scenarionumber Cost Average of

inventory positionServicelevel

Robustness

1 32984000 4352 76 62 29214710 397514 79 33 37114636 339623 69 24 34840720 410948 71 55 36820570 344738 72 66 39191837 331091 67 27 37037180 391012 68 48 40994680 298712 64 69 41407045 319678 63 110 28282930 524462 83 711 26422145 489243 85 512 31023829 416049 79 313 30395190 487146 79 614 35393680 395812 74 515 32838055 395646 77 316 33285615 463341 76 617 39652020 335668 66 718 35269609 383944 73 219 26227680 625612 87 720 25565820 586688 88 621 28711069 49449 85 422 27301580 585472 85 723 34016415 470961 78 724 30601546 463975 82 325 29752815 560421 82 726 41024410 382994 68 627 31604741 443053 78 3

is calculated and shown in Table 9 Also result of principalcomponents is presented in Table 10

As it is visible in Table 9 cumulative proportion ofthe first and second components is 0966 which is greaterthan 095 so the first two components are considered asprincipal components Their relative weights are 078 and0186 respectively Final weights are normalized and resultsare reported in Table 11

54 Ranking and Selection of Scenarios Weights of criteriaobtained by PCA method in addition to decision matrix areinput of VIKOR method 119878 119876 and 119877 values are calculatedbased on (14) and (15) while strategic weight indicated byV is equal to 05 so individual and group utility have same

Modelling and Simulation in Engineering 11

Table 9 Result of eigenvalues

Eigenvalues 31204 07428 01111 00257Proportion 0780 0186 0028 0006Cumulative 0780 0966 0994 1000

Table 10 Result of principle components

Variables PC1 PC2 PC3 PC4Inventory 0543 0089 minus0818 0167Cost minus0538 minus0277 minus0511 minus0610Service level minus0551 minus0210 minus0230 0774Robustness 0335 minus0933 0126 0023

Table 11 Weights of criteria for different products

Criteria Products119860 119861 119862 119863

Average of inventory position 01012 00590 01942 00462Cost 02852 01836 01742 03081Service level 01195 01392 00966 01206Robustness 04941 06182 05350 05251

importance Calculated values are reported in Table 12 forentire scenario as VIKOR output

Product 119861 According to Table 12 values of 119877 119878 and 119876 arezero for scenario 18 Regarding (C1) (Acceptable Advantage)the difference between the best scenario and second rankedscenario should be more than 00038 The second rankedscenario regarding 119876 value is scenario 3 and its value is0051044 that satisfies both (C1) and (C2) Final result forproduct 119861 is unique scenario that is number 18 and consistsof the following configuration Inventory policy is DFwith119877

119901

being equal to 30 products and 7 days lead time According todecision matrix which is shown in Table 8 the performanceof this scenario for product 119861 results in 383944 products asaverage of inventory position and expected cost is equal to35269609 for product life cycle 73 of customer demandwillbe satisfied and in the case of increasing demand intensityand variation 2 decrease is expected in service level so itwill change into 71 Analyses for other products are similarso only final results are presented in this paper

Product 119860 In this case VIKORmethod ranks scenario num-ber 15 as the best one This scenario consists of DF inventorypolicy 20 products for 119877

119901and 5 days for lead time Perfor-

mance of this scenario results in 374778 products as averageof inventory position Expected cost is 31603294 and servicelevel will be 66 This scenario is robust against demandintensification and variation because changes in demandaffect service level as small as 1 But scenario number 15 doesnot satisfy (C1) so compromising solutions are consideredAlthough the best scenario is number 15 VIKOR methodimplies that compromising solutions have the same values

Table 12 VIKOR output for product 119861

Run number 119878 119877 119876

1 0634311 0796269 0715292 0003184 0185077 0094133 0059804 0042283 00510444 0536946 0592539 05647425 0718862 0796269 07575656 0116278 0089887 01030827 0428814 0388808 04088118 086078 0796269 08285259 002043 0140653 008054110 0677386 1 083869311 0266254 0592539 042939612 0043962 0185077 01145213 0571695 0796269 068398214 0515097 0592539 055381815 009173 0185077 013840416 0648693 0796269 072248117 1 1 118 0 0 019 0630698 1 081534920 04239 0796269 061008421 0139527 0388808 026416722 06581 1 08290523 0819338 1 090966924 0022245 0185077 010366125 0726183 1 086309126 0849516 0796269 082289327 0072768 0185077 0128922

So in real situation compromising solution can be substi-tuted This solution regarding their priority is as follows

Scenario number 24 inventory policy is DF 119877119901is

equal to 30 products and lead time is 5 daysScenario number 18 inventory policy is DF 119877

119901is

equal to 20 products and lead time is 7 daysScenario number 12 inventory policy isDF119877

119901is equal

to 20 products and lead time is 3 days

Product 119862 For this product scenario number 1 is ranked asthe best one This scenario consists of FI inventory policy 119877

119901

is 15 products and lead time is 3 days Result of this scenario is4387 products as the average of inventory position Expectedcost is 66143000 and service level will be 61 with robustnessof 1 Compromising solutions are as follows

Scenario number 6 inventory policy is DF 119877119901is 15

products and lead time is 5 daysScenario number 3 inventory policy is DF 119877

119901is 15

products and lead time is 3 daysScenario number 7 inventory policy is FI 119877

119901is 15

products and lead time is 7 days

12 Modelling and Simulation in Engineering

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 4 Interaction plot for product 119862 average of inventory position

Scenario number 4 inventory policy is FI 119877119901is 15

products and lead time is 5 days

Product119863 Scenario number 3 has the best rank for this prod-uct This scenario consists of DF inventory policy 119877

119901is 15

products and lead time is 3 days This scenario leads to

363896 products as average of inventory position with62648337 expected cost and 61 service level with 1robustness Compromising solutions are as follows

Scenario number 6 this scenario is mentioned beforeas a compromising solution for product C

Modelling and Simulation in Engineering 13

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 5 Interaction plot for product 119862 cost

14 Modelling and Simulation in Engineering

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

3

5

7

L

3

5

7

L

3

5

7

L

L

(b)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 6 Interaction plot for product 119862 service level

Scenario number 17 inventory policy is FI119877119901is equal

to 15 products and lead time is 7 days

55 Sensitivity Analysis As mentioned in the previous sec-tion all scenarios related to product 119860 prefer DF inventory

policy so it is concluded that DF policy is unique optimalpolicy while lead time includes values of 3 5 and 7 in selectedscenarios This situation implies that product 119860 is insensitiveto the delivery time It can be wise to decide lead time about 5days but there is no need for strict control on lead time while

Modelling and Simulation in Engineering 15

119877119901should be controlled strictly to prevent inventory in hand

from reaching lower than 20 productsIn the case of product 119861 (C1) is satisfied so there is a

unique scenario while there are four compromising solutionfor product 119862 In this situation based on VIKOR methodcompromising solutions have same value while interactionplot can be applied for investigation of existing differencebetween compromising solutions As mentioned before thebest scenario suggests FI inventory policy for this product butin compromising solutions there are both FI and DF policiesLead time varies from 3 to 7 days So this confusing situationis resolved by deeper analysis and applying interaction plotFigure 4 shows interaction plot for DOE factors and averageof inventory position as response variables

Figure 4 shows that with either DF or FI policy there isno significant difference among different lead times in theview point of inventory position but DF leads to less averageof inventory position With 119877

119901equaling 15 there is less

sensitivity to different lead times So configuring 119877119901which

is equal to 15 products the average of inventory positioncriterion will be robust against lead time variations

Figure 5 regards cost criterion and shows that FI andDF policies are equal and make least sensitivity related tolead time On the other hand three plots of 119877

119901against lead

time are parallel The parallel plots are interpreted as lackof relation between 119877

119901and lead time With 119877

119901equaling 15

increase in inventory cost could be mitigated when lead timesets to 3 days

In Figure 6 with 3 days for lead time there is no signifi-cant difference between inventory policies in the view pointof service level119877

119901and lead time are independent and the best

service level is related to FI policy which is also very sensitiveto 119877119901 As 119877

119901increases service level will grow According to

the interaction plot analyses optimal decision for product119862 is FI inventory policy 3 days lead time and 15 or moreproducts for 119877

119901

Decision about product 119863 is simple Based on the in-formation obtained from solutions it is recommended toemploy DF inventory policy with 119877

119901being equal to 15

products and 3 days for lead time

6 Conclusion

In this paper a simulation optimization framework is pro-posed for robust optimization of retailer inventory systemThis framework is less time consuming in comparison withmetaheuristic or SPSA approach it also provides robustsolution for multiple objective functions In this research alsotrend and cyclic change of customers demand are consideredfor more realistic view of problem Proposed framework con-sists of discrete event simulation and surrogate model In thisframework full factorial design of experiment is employedfor producing decision space Based on the three decisionfactors 27 different scenarios are produced and simulated Insimulation modelling multiresolution method is applied forsimulation of demand with highly dynamic pattern Becausethere are multiple criteria for inventory system performanceVIKORmethod is employed asmulticriteria decisionmaking

technique In addition robustness is considered as perfor-mance criterion and PCA is used for improving performanceof VIKOR method Finally developed framework gave thebest ranked and compromising solutions so interaction plotapplied for more investigation and sensitivity analysis ofobtained solutions

Due to the novelty of the proposed framework it isrecommended that future studies encompass optimization ofinventory system of integrated supply chain Also improve-ment of developed framework can be considered as futurestudy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers for theirsignificant roles in improvement of this research quality

References

[1] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[2] S Yin S XDingAHaghaniHHao andP Zhang ldquoA compar-ison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[3] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[4] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[5] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall International Series in Management New YorkNY USA 1963

[6] E Naddor Inventory Systems John Wiley amp Sons New YorkNY USA 1966

[7] R Peterson and E A Silver Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 2nd edition 1985

[8] T C E Cheng ldquoEconomic order quantity model with demand-dependent unit production cost and imperfect productionprocessesrdquo IIE Transactions vol 23 no 1 pp 23ndash28 1991

[9] S H Chen C CWang and A Ramer ldquoBackorder fuzzy inven-tory model under function principlerdquo Information Sciences vol95 no 1-2 pp 71ndash79 1996

[10] X Zhao J Xie and J C Wei ldquoThe value of early order com-mitment in a two-level supply chainrdquo European Journal of Op-erational Research vol 180 no 1 pp 194ndash214 2007

[11] R S M Lau J Xie and X Zhao ldquoEffects of inventory policyon supply chain performance a simulation study of critical

16 Modelling and Simulation in Engineering

decision parametersrdquo Computers and Industrial Engineeringvol 55 no 3 pp 620ndash633 2008

[12] J Xu and L Zhao ldquoA multi-objective decision-making modelwith fuzzy rough coefficients and its application to the inventoryproblemrdquo Information Sciences vol 180 no 5 pp 679ndash6962010

[13] F Hnaien X Delorme and A Dolgui ldquoMulti-objective opti-mization for inventory control in two-level assembly systemsunder uncertainty of lead timesrdquo Computers amp OperationsResearch vol 37 no 11 pp 1835ndash1843 2010

[14] H D Purnomo H M Wee and Y Praharsi ldquoTwo inventoryreview policies on supply chain configuration problemrdquo Com-puters and Industrial Engineering vol 63 no 2 pp 448ndash4552012

[15] B L Nelson ldquo50th anniversary article stochastic simulationresearch in management sciencerdquoManagement Science vol 50no 7 pp 855ndash868 2004

[16] J Boesel R O Bowden Jr F Glover J P Kelly and E WestwigldquoFuture of simulation optimizationrdquo inProceedings of theWinterSimulation Conference pp 1466ndash1469 Arlington Va USADecember 2001

[17] M C Fu ldquoOptimization for simulation theory vs practicerdquoINFORMS Journal on Computing vol 14 no 3 pp 192ndash2152002

[18] L Wang ldquoA hybrid genetic algorithm-neural network strategyfor simulation optimizationrdquoApplied Mathematics and Compu-tation vol 170 no 2 pp 1329ndash1343 2005

[19] B B Keskin S H Melouk and I L Meyer ldquoA simulation-optimization approach for integrated sourcing and inventorydecisionsrdquo Computers and Operations Research vol 37 no 9pp 1648ndash1661 2010

[20] E Mazhari J Zhao N Celik S Lee Y Son and L HeadldquoHybrid simulation and optimization-based design and oper-ation of integrated photovoltaic generation storage units andgridrdquo Simulation Modelling Practice and Theory vol 19 no 1pp 463ndash481 2011

[21] Q Duan and T W Liao ldquoOptimization of replenishment poli-cies for decentralized and centralized capacitated supply chainsunder various demandsrdquo International Journal of ProductionEconomics vol 142 no 1 pp 194ndash204 2013

[22] J C Spall ldquoMultivariate stochastic approximation usinga simultaneous perturbation gradient approximationrdquo IEEETransactions on Automatic Control vol 37 no 3 pp 332ndash3411992

[23] J C Spall ldquoImplementation of the simultaneous perturbationalgorithm for stochastic optimizationrdquo IEEE Transactions onAerospace and Electronic Systems vol 34 no 3 pp 817ndash8231998

[24] J C Spall ldquoStochastic optimization and the simultaneousperturbation methodrdquo in Proceedings of the Winter SimulationConference (WSC rsquo99) pp 101ndash109 December 1999

[25] J D Schwartz W Wang and D E Rivera ldquoSimulation-based optimization of process control policies for inventorymanagement in supply chainsrdquo Automatica vol 42 no 8 pp1311ndash1320 2006

[26] H G Neddermeijer G J V Oortmarssen and N P R DekkerldquoA framework for response surface methodology for simulationoptimizationrdquo in Proceedings of the Winter Simulation Confer-ence vol 1 Orlando Fla USA December 1999

[27] B Can and C Heavey ldquoA comparison of genetic programmingand artificial neural networks in metamodeling of discrete-event simulation modelsrdquo Computers amp Operations Researchvol 39 no 2 pp 424ndash436 2012

[28] A Azadeh Z S Faiz S M Asadzadeh and R Tavakkoli-Moghaddam ldquoAn integrated artificial neural network-comput-er simulation for optimization of complex tandem queuesystemsrdquoMathematics and Computers in Simulation vol 82 no4 pp 666ndash678 2011

[29] R Bornatico J Hussy A Witzig and L Guzzella ldquoSurrogatemodeling for the fast optimization of energy systemsrdquo Energyvol 57 pp 653ndash662 2013

[30] X Wan J F Pekny and G V Reklaitis ldquoSimulation-basedoptimizationwith surrogatemodels application to supply chainmanagementrdquoComputers and Chemical Engineering vol 29 no6 pp 1317ndash1328 2005

[31] W Shi Z Liu J Shang and Y Cui ldquoMulti-criteria robust designof a JIT-based cross-docking distribution center for an autoparts supply chainrdquo European Journal of Operational Researchvol 229 no 3 pp 695ndash706 2013

[32] G Dellino J P C Kleijnen and C Meloni ldquoRobust optimiza-tion in simulation taguchi and Response Surface Methodol-ogyrdquo International Journal of Production Economics vol 125 no1 pp 52ndash59 2010

[33] T Yang Y Wen and F Wang ldquoEvaluation of robustness of sup-ply chain information-sharing strategies using a hybrid Taguchiand multiple criteria decision-making methodrdquo InternationalJournal of Production Economics vol 134 no 2 pp 458ndash4662011

[34] M E Kuhl and J R Wilson ldquoModeling and simulating Poissonprocesses having trends or nontrigonometric cyclic effectsrdquoEuropean Journal of Operational Research vol 133 no 3 pp566ndash582 2001

[35] P Narayan and J Subramanian Inventory ManagementmdashPrinciples and Practices Excel Books New Delhi India 2008

[36] E A Silver and R Peterson Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 1985

[37] R G Brown Smoothing Forecasting and Prediction of DiscreteTime Series Courier 2004

[38] D C Montgomery Design and analysis of experiments JohnWiley amp Sons New York NY USA 6th edition 2005

[39] J Zhu ldquoData envelopment analysis vs principal componentanalysis an illustrative study of economic performance ofChinese citiesrdquo European Journal of Operational Research vol111 no 1 pp 50ndash61 1998

[40] I M Premachandra ldquoA note on DEA vs principal componentanalysis an improvement to Joe Zhursquos approachrdquo EuropeanJournal of Operational Research vol 132 no 3 pp 553ndash5602001

[41] D Slottje G W Scully G J Hirschberg and K Hayes Mea-suring the Quality of Life across Countries A MultidimensionalAnalysis Westview Press Boulder Colo USA 1991

[42] S Opricovic Multicriteria optimization of civil engineeringsystems [PhD thesis] Faculty of Civil Engineering BelgradeSerbia 1998

[43] P L Yu ldquoA class of solutions for group decision problemsrdquoManagement Science vol 19 no 8 pp 936ndash946 1973

Modelling and Simulation in Engineering 17

[44] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[45] S Opricovic and G H Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

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Page 6: Research Article Robust Inventory System Optimization ...downloads.hindawi.com/journals/mse/2014/305120.pdf · Robust Inventory System Optimization Based on Simulation and Multiple

6 Modelling and Simulation in Engineering

Demand arrival

Sufficientinventory isavailable

True

False

Reorderproducts

True

False

Demand satisfaction

Make an order

Passing of lead time

Demand dispose

Lost customer

Is there anyorder in way

True

False

Updating of inventory level

Demand satisfaction

(a) FQ policy

Demand arrival

Sufficientinventory isavailable

Demand satisfaction Lost customer

FalseTrue

Demand dispose

System start

Make order

Make an order

Passing of lead time

TrueFalse

Passing of reviewing period

Updating of inventory level

(b) FI policy

Demand arrival

Sufficientinventory isavailable

Lost customer

System start

Make orderFalse True

Estimatedquantity ispossible

True

False

True False

Update of estimation

Update of information

Order maximum quantity

Order estimated quantity

Passing of lead time

Updating of inventory level

Demand satisfaction

Passing of reviewing period

Demand dispose

(c) DF policy

Figure 1

Modelling and Simulation in Engineering 7

Table 2 DOE configuration

ProductsFactors

IP 119877119905

119871

minus 0 + minus 0 + minus 0 +119860 FI FQ DF 10 20 30 3 5 7119861 FI FQ DF 15 30 45 3 5 7119862 FI FQ DF 15 30 45 3 5 7119863 FI FQ DF 15 30 45 3 5 7

Table 3 Experimental design

Number of trial 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27119877119901

minus minus minus minus minus minus minus minus minus 0 0 0 0 0 0 0 0 0 + + + + + + + + +119871 minus minus minus 0 0 0 + + + minus minus minus 0 0 0 + + + minus minus minus 0 0 0 + + +IP minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 +

this research there are 3 factors) using full factorial design ispreferable On the other hand because nonlinear behaviourof response variables (ie objective functions) centre pointis considered Each factor has high level ldquo+rdquo low level ldquominusrdquoand centre point ldquo0rdquo Factors and levels configurations areshown in Table 2 while Table 3 shows experimental designIn both of these tables minus sign is representative of lowlevel of related factors while plus sign is indicator of high levelof factors and zero determines centre point For examplein the case of lead time (119871) 3 is low level 7 is high leveland 5 is centre point Lead time is considered with daysas measuring unit and begins from time order is organizeduntil the arrival of inventory Reorder point is labelled as 119877

119901

and is intended to satisfy product demand during lead time(119871) Eventually inventory policy is labelled as IP and refersto inventory policies that were described in Section 412The mentioned design includes 27 runs and there are fourresponse variables cost service level average of inventoryposition and robustness In fact first three response variablesare expected values of defined objective functions (ie resp(1) (2) and (3)) but robustness is a defined criterion whichguarantees that optimal solution remains valid in situationwhere demand variation and intensification increase Forevaluation of robustness demands are multiplied by randomvariable with normal distribution (120583 = 1 120590 = 15)This random variable is also restricted to positive values toprevent negative demands Expected value of 4000 generatednumbers with mentioned constraint is equal to 148 thisresult implies that multiplied random number increases bothdemand variation and intensification simultaneously Thenthe percent of decrease in service level is considered asrobustness for all 27 scenarios produced byDOE It is obviousthat a lower decrease in service level is desirable

422 Weight Assignment Based on PCA Method PCA isthe abbreviation of ldquoprincipal component analysisrdquo and isknown as powerful data-driven method successfully appliedin several areas [2] PCA analysis can determine key variablesthat govern most of the variability This technique helps toremove less important variables to reduce dimension of data

sets PCA considers two criteria for each variable in datasets the first is variability and the second is correlation withother variables So input for this method consists of variancecovariance matrix [39 40] Although other methods likeentropy is applicable in multiple attribute decision makingproblem PCA advantage is considerable because variablescorrelations are not ignored As our problem is constructedof highly correlated variables PCA is the preferable choiceFor correlation justification remember that improvementof inventory position criterion causes negative influence onshortage cost or improvement of service level criterion causesnegative influence on inventory in hand and holding cost Inthese circumstances there is no reason for applying variationbased methods like entropy so in this paper PCA is appliedfor weight assignment These weights are used by MADMtechnique (described in Section 423) to improve qualityof ranking Because of relative utility of scenarios linearscaleless method employed to scaleless data obtained fromsimulation

(1) Calculate scaleless matrix of 119863 = (1198891 1198892 119889

119901)119899times119901

which 119889119894119895is scaleless value of 119894th scenario in the view

point of 119895th criterion

(2) Calculate sample mean vector 119889 = (1198891 1198892 119889

119901)119901

and covariance matrix based on (7) and (8) respec-tively

119889119896=1

119899

119899

sum

119894=1

119889119894119895 (7)

119878 = (119904119896119902)119901times119901

=1

119899 minus 1(119863 minus 119889)

119879

(119863 minus 119889) (8)

(3) 1198621radic119904119896119896 is diagonal 119901 times 119901 matrix 119896th diagonal

element is 1radic119904119896119896 and sample correlation matrix (119877)is calculated as mentioned in the following equation

119877 =1198621

radic119904119896119896

sdot 119878 sdot1198621

radic119904119896119896

(9)

8 Modelling and Simulation in Engineering

(4) Solve the following equation where 119868119901is a 119901 times 119901

identity matrix10038161003816100381610038161003816119877 minus 120582119868

119901

10038161003816100381610038161003816= 0 (10)

Solving (10) results in 119875 individual eigenvaluesregarding 120582

1ge 1205822ge sdot sdot sdot ge 120582

119901and sum119901

119896=1120582119896=

119875 These eigenvalues have 119875-specific eigenvectors(119897119896

1 119897119896

2 119897

119896

119901) and 119896 = 1 119901 These vectors create

principal components as expressed in the followingequation

PC119896=

119901

sum

119902=1

119897119896

119902119889119895

119902 (11)

(5) Select sufficient number of principal components Inthis problem first119872 components whose cumulativepercentage of their eigenvalues (119862

119872) is greater than

95 are selected Cumulative percentage is calculatedbased on the following equation

119862119898=sum119898

119896=1120582119896

sum119901

119896=1120582119896

=sum119898

119896=1120582119896

119875 (12)

(6) Finally weights vector that is indicated by119885 is derivedfrom the mathematical relation in (13) while 119882

119896is

equal to 120582119896119875 if the entire elements of PC

119896are

positive otherwise minus120582119896119875 if entire component are

negative

119885 =

119872

sum

119896=1

119882119896PC119896 (13)

In other cases positive or negative 119882119896should be defined

in the order that final weight vector is positive For moredetailed discussion about 119882

119896 refer to [41] Finally for each

product there are four 119885 parameters that determine relativeimportance for the following criteria cost service levelaverage of inventory position and robustness each elementof 119885 vector is divided by the sum of vector elements fornormalization

423 VIKOR Method This method was developed by Opri-covic and is the abbreviation of expression that in Serbianmeans ldquomultiple criteria optimization and compromisingsolutionsrdquo [42] VIKOR method is based on compromisingsolutions which is the result of Yu [43] and Zeleny [44]studies This approach considers closeness to ideal solutionIn comparison with other MADM methods VIKOR is ableto present compromising solutions and substitutes thesesolutions with best one in the case of necessity [45] VIKORmethod is applicable for problems with multiple discretecriteria Generally this method can be considered as rankingtool for scenarios which are generated by DOE Ranking isbased on L-P metric function according to (14) with defini-tion of 119891

119894119895as the value of 119894th criterion for 119895th scenario In

construction of L-P metric function 119891lowast119894indicates best value

of all scenarios in the view point of 119894th criterion and 119891minus119894

represents the worst value of all scenarios with considerationof 119894th criterion 119908

119894is the weight parameter for 119894th criterion

and is derived by PCA method Consider

119871119901=

119899

sum

119894=1

[

119908119894(119891lowast

119894minus 119891119894119895)

(119891lowast

119894minus 119891minus

119894)]

119901

1119901

1 ≪ 119875 ≪ infin 119895 = 1 2 3 119869

119871infin= max

119894[

119908119894(119891lowast

119894minus 119891119894119895)

(119891lowast

119894minus 119891minus

119894)]

(14)

VIKOR method only involves 1198711and 119871

infinwhich are known

as 119878119895and 119877

119895 respectively 119878 and 119877 index refer to scenarios

so both of these functions are regarded for each scenarioVIKOR steps are as follows

(1) Calculate 119891minus119894and 119891lowast

119894for entire of criteria

(2) Calculate 119878119895and 119877

119895for entire of scenarios

(3) With the assumption that 119878lowast and 119877lowast are the smallestvalues among all of 119878

119895and 119877

119895 respectively and also

119878minus and 119877minus are the greatest values in comparison withall of 119878

119895and 119877

119895 respectively 119876

119895is calculated for 119895th

scenario based on (15) In this equation V is strategicweight that determines the importance of individualcriterion (119878

119895) against group importance of criteria

(119877119895) for each scenario Consider

119876119895=

V (119878119895minus 119878lowast

)

119878minus minus 119878lowast+ (1 minus V)

(119877119895minus 119877lowast

)

119877minus minus 119877lowast (15)

(4) Scenarios should be sorted in descending order basedon 119878119877 and119876 specifically that result in three differentranking lists

(5) According to 119876 the scenario which is indicated by 1198861015840is the best scenario if two conditions namely (C1)and (C2) are satisfied

(C1) Acceptable Advantage This condition implies that thesecond best solution (11988610158401015840) should be far enough from thebest solution (1198861015840) to be accepted as unique solution Themathematical equation is defined in (16) considering 119869 as thetotal number of scenarios Consider

119876(11988610158401015840

) minus 119876 (1198861015840

) ≫1

(119869 minus 1) (16)

(C2) Acceptable Stability This condition validates 1198861015840 as thebest stable solution if 1198861015840 is also the best solution based on 119878 or119877

In case that one or both of the mentioned conditions arenot satisfied compromised solutions are involved VIKORmethod suggests that in such case compromising solutionshave equal value to the best solution and could be consideredinterchangeably by decision makers

Modelling and Simulation in Engineering 9

If (C1) is not satisfied scenarios 1198861015840 11988610158401015840 119886119872 are consid-ered 1198861015840 is the best solution and 11988610158401015840 119886119872 are compromisingsolutions with priority of 119876 on the other hand 1198861015840 11988610158401015840are considered if (C2) is not satisfied 119872 superscript isdetermined by (17)

119876(119886119872

) minus 119876 (1198861015840

) ≫1

(119869 minus 1) (17)

5 Numerical Result

Simulation process was executed on laptop with 18 GH CPUand 4GB of RAM Running of each replication with anima-tion and maximum speed takes 7 minutes and 35 secondsAs simulation of each scenario contains 10 replications andthere are 27 scenarios the total simulation time will be 2046minutes and 36 seconds

51 Multiresolution Result for Demand Modelling Result ofdemand modelling is presented for product 119861 because thisanalysis for other products is similar For this problempolynomial functions are applied for yearly and monthlydemand rates Equations (18) and (19) are estimated bynonlinear regression model In these functions 119905

119910and 119905119898are

time variables

119891 (119905119910) = 0008942 times 119905

3

119910minus 003884 times 119905

2

119910

+ 01867 times 119905119910minus 008018

(18)

and 95 confidence interval is presented in Table 4 for eachcoefficient of (18)

In this table coefficients of (18) are visible in the firstcolumn while lower bund and upper bund of each coefficientare presented in the second and third columns respectivelyThere is no interval including zero so it is concluded thatcoefficient estimation is valid

Estimation error for 119891(119905119910) is reported based on the sum

of square errors (SSE) which is 1504 times 10minus6 Consider

119891 (119905119898) = 4768 times 10

minus5

times 1199055

119898minus 0001692 times 119905

4

119898+ 00214

times 1199053

119898minus 01159 times 119905

2

119898+ 03474 times 119905

119898minus 02286

(19)

Monthly demand rate function that is indicated by 119891(119905119898)

and its coefficients confidence interval are presented inTable 5 SEE is equal to 00005825 Based on local businessinformation total demand of product 119861 is estimated to beabout 6822 units in five years According to the informationabout product 119861 Figure 2 shows final multiresolution resultfor this product

52 Simulation Parameters and Configuration For simula-tion of problem it is necessary to configure inventory policyparameters and variables We have three different policiesnamely FQ FI and DF Variables of FQ policy are reorderpoint (119877

119901) lead time (119871) and economic quantity orders (119876

119895119905)

First two variables are configured based on Table 2 and thirdvariable is derived fromWilson formula according to average

Table 4 95 confidence interval for estimated coefficient of 119891(119905119910)

Coefficients Lower bund Upper bund0008942 0004835 001305minus003884 minus007604 minus000165201867 008654 0287minus008018 minus01568 minus0003523

Table 5 95 confidence interval for estimated coefficient of 119891(119905119898)

Coefficients Lower bund Upper bund4768 times 10minus5 4451 times 10minus5 5086 times 10minus5

minus0001692 minus0001795 minus00015900214 00202 00226minus01159 minus01221 minus0109703474 0334 03608minus02286 minus02371 minus022

Table 6 Estimation of distribution functions of demand and reor-ders quantity

Product type Distribution function 119875 value Order quantity119860 278 lowast BETA(113 328) gt015 100119861 GAMMA(678 118) gt015 92119862 EXPO(818) gt015 100119863 446 lowast BETA(0957 321) gt015 115

of demand holding cost (119862ℎ) and ordering cost (119862

119903) The

calculated values are 57 71 72 and 72 for product 119860 119861 119862and119863 respectively

Orders quantity is unknown for FI policy and is cal-culated based on probability distribution function whichis obtained by demand simulation of each product for 20daysrsquo time interval Distribution fitting is performed by inputanalyser of Arena software and graphical result for product 119861is shown in Figure 3 Results for other products are reportedin Table 6

In this table fitted probability distribution function foreach product is presented in second column while 119875 valueof fitting which is greater than 015 for fitted distributiongrantees goodness of fitting in 95 confidence level Finallyaccording to fitted distribution functions for FI policyproper ordering amount of each product is presented in thefourth column

DF policy has two unknown parameters namely 120572 and120573 The first parameter is weight of current real demand in (4)and second one is the weight of current trend in (5) Theseparameters are adjusted by simulation of five years demandDemand for the entire four products simultaneously is con-sidered and an average of lost sale percentage is consideredas response variable Different values of response variables arereported in Table 7 According to the different values of 120572 and120573 the best values for 120572 and 120573 are 09 and 01 respectivelywhich results in minimum lost percentage in average

10 Modelling and Simulation in Engineering

Table 7 Average of lost sale percentage for different values of 120572 and 120573

120572 and 120573 values 01 02 03 04 05 06 07 08 09Response to 120572 0380 0356 0334 0320 0306 0292 0284 0276 0268Response to 120573 0296 0302 0306 0306 0306 0304 0305 0304 0302

50

100

150

200

250

300

350Demand fit for each month

Dem

and

Fitted valueReal demand

00 10 20 30 40 50 60

Month

Figure 2 Comparison between real and estimated demand

100 200 300 400 500 600 700

50

100

150

200

Distribution summary

Distribution gammaExpression minus0001 + GAMM(00)Square error 0006825

Chi-square testNumber of intervals = 5Degrees of freedom = 2Test statistic = 148

Kolmogorov-Smirnov testTest statistic = 00733

Corresponding P value = 0483

Corresponding P value gt 015

Figure 3 Distribution function of customer demand for reorderinterval of product 119861

53Weight Assignment Asmentioned before PCA is appliedforweight assignment For this purposeDOE result is neededin the form of decision matrix that is shown in Table 8 Inthis table four criteria cost average of inventory positionservice level and robustness are evaluated for each scenarioof product 119861 Decision matrix should be scaleless that isperformed by linear method Then the result of eigenvalues

Table 8 Decision matrix for product 119861

Scenarionumber Cost Average of

inventory positionServicelevel

Robustness

1 32984000 4352 76 62 29214710 397514 79 33 37114636 339623 69 24 34840720 410948 71 55 36820570 344738 72 66 39191837 331091 67 27 37037180 391012 68 48 40994680 298712 64 69 41407045 319678 63 110 28282930 524462 83 711 26422145 489243 85 512 31023829 416049 79 313 30395190 487146 79 614 35393680 395812 74 515 32838055 395646 77 316 33285615 463341 76 617 39652020 335668 66 718 35269609 383944 73 219 26227680 625612 87 720 25565820 586688 88 621 28711069 49449 85 422 27301580 585472 85 723 34016415 470961 78 724 30601546 463975 82 325 29752815 560421 82 726 41024410 382994 68 627 31604741 443053 78 3

is calculated and shown in Table 9 Also result of principalcomponents is presented in Table 10

As it is visible in Table 9 cumulative proportion ofthe first and second components is 0966 which is greaterthan 095 so the first two components are considered asprincipal components Their relative weights are 078 and0186 respectively Final weights are normalized and resultsare reported in Table 11

54 Ranking and Selection of Scenarios Weights of criteriaobtained by PCA method in addition to decision matrix areinput of VIKOR method 119878 119876 and 119877 values are calculatedbased on (14) and (15) while strategic weight indicated byV is equal to 05 so individual and group utility have same

Modelling and Simulation in Engineering 11

Table 9 Result of eigenvalues

Eigenvalues 31204 07428 01111 00257Proportion 0780 0186 0028 0006Cumulative 0780 0966 0994 1000

Table 10 Result of principle components

Variables PC1 PC2 PC3 PC4Inventory 0543 0089 minus0818 0167Cost minus0538 minus0277 minus0511 minus0610Service level minus0551 minus0210 minus0230 0774Robustness 0335 minus0933 0126 0023

Table 11 Weights of criteria for different products

Criteria Products119860 119861 119862 119863

Average of inventory position 01012 00590 01942 00462Cost 02852 01836 01742 03081Service level 01195 01392 00966 01206Robustness 04941 06182 05350 05251

importance Calculated values are reported in Table 12 forentire scenario as VIKOR output

Product 119861 According to Table 12 values of 119877 119878 and 119876 arezero for scenario 18 Regarding (C1) (Acceptable Advantage)the difference between the best scenario and second rankedscenario should be more than 00038 The second rankedscenario regarding 119876 value is scenario 3 and its value is0051044 that satisfies both (C1) and (C2) Final result forproduct 119861 is unique scenario that is number 18 and consistsof the following configuration Inventory policy is DFwith119877

119901

being equal to 30 products and 7 days lead time According todecision matrix which is shown in Table 8 the performanceof this scenario for product 119861 results in 383944 products asaverage of inventory position and expected cost is equal to35269609 for product life cycle 73 of customer demandwillbe satisfied and in the case of increasing demand intensityand variation 2 decrease is expected in service level so itwill change into 71 Analyses for other products are similarso only final results are presented in this paper

Product 119860 In this case VIKORmethod ranks scenario num-ber 15 as the best one This scenario consists of DF inventorypolicy 20 products for 119877

119901and 5 days for lead time Perfor-

mance of this scenario results in 374778 products as averageof inventory position Expected cost is 31603294 and servicelevel will be 66 This scenario is robust against demandintensification and variation because changes in demandaffect service level as small as 1 But scenario number 15 doesnot satisfy (C1) so compromising solutions are consideredAlthough the best scenario is number 15 VIKOR methodimplies that compromising solutions have the same values

Table 12 VIKOR output for product 119861

Run number 119878 119877 119876

1 0634311 0796269 0715292 0003184 0185077 0094133 0059804 0042283 00510444 0536946 0592539 05647425 0718862 0796269 07575656 0116278 0089887 01030827 0428814 0388808 04088118 086078 0796269 08285259 002043 0140653 008054110 0677386 1 083869311 0266254 0592539 042939612 0043962 0185077 01145213 0571695 0796269 068398214 0515097 0592539 055381815 009173 0185077 013840416 0648693 0796269 072248117 1 1 118 0 0 019 0630698 1 081534920 04239 0796269 061008421 0139527 0388808 026416722 06581 1 08290523 0819338 1 090966924 0022245 0185077 010366125 0726183 1 086309126 0849516 0796269 082289327 0072768 0185077 0128922

So in real situation compromising solution can be substi-tuted This solution regarding their priority is as follows

Scenario number 24 inventory policy is DF 119877119901is

equal to 30 products and lead time is 5 daysScenario number 18 inventory policy is DF 119877

119901is

equal to 20 products and lead time is 7 daysScenario number 12 inventory policy isDF119877

119901is equal

to 20 products and lead time is 3 days

Product 119862 For this product scenario number 1 is ranked asthe best one This scenario consists of FI inventory policy 119877

119901

is 15 products and lead time is 3 days Result of this scenario is4387 products as the average of inventory position Expectedcost is 66143000 and service level will be 61 with robustnessof 1 Compromising solutions are as follows

Scenario number 6 inventory policy is DF 119877119901is 15

products and lead time is 5 daysScenario number 3 inventory policy is DF 119877

119901is 15

products and lead time is 3 daysScenario number 7 inventory policy is FI 119877

119901is 15

products and lead time is 7 days

12 Modelling and Simulation in Engineering

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 4 Interaction plot for product 119862 average of inventory position

Scenario number 4 inventory policy is FI 119877119901is 15

products and lead time is 5 days

Product119863 Scenario number 3 has the best rank for this prod-uct This scenario consists of DF inventory policy 119877

119901is 15

products and lead time is 3 days This scenario leads to

363896 products as average of inventory position with62648337 expected cost and 61 service level with 1robustness Compromising solutions are as follows

Scenario number 6 this scenario is mentioned beforeas a compromising solution for product C

Modelling and Simulation in Engineering 13

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 5 Interaction plot for product 119862 cost

14 Modelling and Simulation in Engineering

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

3

5

7

L

3

5

7

L

3

5

7

L

L

(b)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 6 Interaction plot for product 119862 service level

Scenario number 17 inventory policy is FI119877119901is equal

to 15 products and lead time is 7 days

55 Sensitivity Analysis As mentioned in the previous sec-tion all scenarios related to product 119860 prefer DF inventory

policy so it is concluded that DF policy is unique optimalpolicy while lead time includes values of 3 5 and 7 in selectedscenarios This situation implies that product 119860 is insensitiveto the delivery time It can be wise to decide lead time about 5days but there is no need for strict control on lead time while

Modelling and Simulation in Engineering 15

119877119901should be controlled strictly to prevent inventory in hand

from reaching lower than 20 productsIn the case of product 119861 (C1) is satisfied so there is a

unique scenario while there are four compromising solutionfor product 119862 In this situation based on VIKOR methodcompromising solutions have same value while interactionplot can be applied for investigation of existing differencebetween compromising solutions As mentioned before thebest scenario suggests FI inventory policy for this product butin compromising solutions there are both FI and DF policiesLead time varies from 3 to 7 days So this confusing situationis resolved by deeper analysis and applying interaction plotFigure 4 shows interaction plot for DOE factors and averageof inventory position as response variables

Figure 4 shows that with either DF or FI policy there isno significant difference among different lead times in theview point of inventory position but DF leads to less averageof inventory position With 119877

119901equaling 15 there is less

sensitivity to different lead times So configuring 119877119901which

is equal to 15 products the average of inventory positioncriterion will be robust against lead time variations

Figure 5 regards cost criterion and shows that FI andDF policies are equal and make least sensitivity related tolead time On the other hand three plots of 119877

119901against lead

time are parallel The parallel plots are interpreted as lackof relation between 119877

119901and lead time With 119877

119901equaling 15

increase in inventory cost could be mitigated when lead timesets to 3 days

In Figure 6 with 3 days for lead time there is no signifi-cant difference between inventory policies in the view pointof service level119877

119901and lead time are independent and the best

service level is related to FI policy which is also very sensitiveto 119877119901 As 119877

119901increases service level will grow According to

the interaction plot analyses optimal decision for product119862 is FI inventory policy 3 days lead time and 15 or moreproducts for 119877

119901

Decision about product 119863 is simple Based on the in-formation obtained from solutions it is recommended toemploy DF inventory policy with 119877

119901being equal to 15

products and 3 days for lead time

6 Conclusion

In this paper a simulation optimization framework is pro-posed for robust optimization of retailer inventory systemThis framework is less time consuming in comparison withmetaheuristic or SPSA approach it also provides robustsolution for multiple objective functions In this research alsotrend and cyclic change of customers demand are consideredfor more realistic view of problem Proposed framework con-sists of discrete event simulation and surrogate model In thisframework full factorial design of experiment is employedfor producing decision space Based on the three decisionfactors 27 different scenarios are produced and simulated Insimulation modelling multiresolution method is applied forsimulation of demand with highly dynamic pattern Becausethere are multiple criteria for inventory system performanceVIKORmethod is employed asmulticriteria decisionmaking

technique In addition robustness is considered as perfor-mance criterion and PCA is used for improving performanceof VIKOR method Finally developed framework gave thebest ranked and compromising solutions so interaction plotapplied for more investigation and sensitivity analysis ofobtained solutions

Due to the novelty of the proposed framework it isrecommended that future studies encompass optimization ofinventory system of integrated supply chain Also improve-ment of developed framework can be considered as futurestudy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers for theirsignificant roles in improvement of this research quality

References

[1] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[2] S Yin S XDingAHaghaniHHao andP Zhang ldquoA compar-ison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[3] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[4] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[5] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall International Series in Management New YorkNY USA 1963

[6] E Naddor Inventory Systems John Wiley amp Sons New YorkNY USA 1966

[7] R Peterson and E A Silver Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 2nd edition 1985

[8] T C E Cheng ldquoEconomic order quantity model with demand-dependent unit production cost and imperfect productionprocessesrdquo IIE Transactions vol 23 no 1 pp 23ndash28 1991

[9] S H Chen C CWang and A Ramer ldquoBackorder fuzzy inven-tory model under function principlerdquo Information Sciences vol95 no 1-2 pp 71ndash79 1996

[10] X Zhao J Xie and J C Wei ldquoThe value of early order com-mitment in a two-level supply chainrdquo European Journal of Op-erational Research vol 180 no 1 pp 194ndash214 2007

[11] R S M Lau J Xie and X Zhao ldquoEffects of inventory policyon supply chain performance a simulation study of critical

16 Modelling and Simulation in Engineering

decision parametersrdquo Computers and Industrial Engineeringvol 55 no 3 pp 620ndash633 2008

[12] J Xu and L Zhao ldquoA multi-objective decision-making modelwith fuzzy rough coefficients and its application to the inventoryproblemrdquo Information Sciences vol 180 no 5 pp 679ndash6962010

[13] F Hnaien X Delorme and A Dolgui ldquoMulti-objective opti-mization for inventory control in two-level assembly systemsunder uncertainty of lead timesrdquo Computers amp OperationsResearch vol 37 no 11 pp 1835ndash1843 2010

[14] H D Purnomo H M Wee and Y Praharsi ldquoTwo inventoryreview policies on supply chain configuration problemrdquo Com-puters and Industrial Engineering vol 63 no 2 pp 448ndash4552012

[15] B L Nelson ldquo50th anniversary article stochastic simulationresearch in management sciencerdquoManagement Science vol 50no 7 pp 855ndash868 2004

[16] J Boesel R O Bowden Jr F Glover J P Kelly and E WestwigldquoFuture of simulation optimizationrdquo inProceedings of theWinterSimulation Conference pp 1466ndash1469 Arlington Va USADecember 2001

[17] M C Fu ldquoOptimization for simulation theory vs practicerdquoINFORMS Journal on Computing vol 14 no 3 pp 192ndash2152002

[18] L Wang ldquoA hybrid genetic algorithm-neural network strategyfor simulation optimizationrdquoApplied Mathematics and Compu-tation vol 170 no 2 pp 1329ndash1343 2005

[19] B B Keskin S H Melouk and I L Meyer ldquoA simulation-optimization approach for integrated sourcing and inventorydecisionsrdquo Computers and Operations Research vol 37 no 9pp 1648ndash1661 2010

[20] E Mazhari J Zhao N Celik S Lee Y Son and L HeadldquoHybrid simulation and optimization-based design and oper-ation of integrated photovoltaic generation storage units andgridrdquo Simulation Modelling Practice and Theory vol 19 no 1pp 463ndash481 2011

[21] Q Duan and T W Liao ldquoOptimization of replenishment poli-cies for decentralized and centralized capacitated supply chainsunder various demandsrdquo International Journal of ProductionEconomics vol 142 no 1 pp 194ndash204 2013

[22] J C Spall ldquoMultivariate stochastic approximation usinga simultaneous perturbation gradient approximationrdquo IEEETransactions on Automatic Control vol 37 no 3 pp 332ndash3411992

[23] J C Spall ldquoImplementation of the simultaneous perturbationalgorithm for stochastic optimizationrdquo IEEE Transactions onAerospace and Electronic Systems vol 34 no 3 pp 817ndash8231998

[24] J C Spall ldquoStochastic optimization and the simultaneousperturbation methodrdquo in Proceedings of the Winter SimulationConference (WSC rsquo99) pp 101ndash109 December 1999

[25] J D Schwartz W Wang and D E Rivera ldquoSimulation-based optimization of process control policies for inventorymanagement in supply chainsrdquo Automatica vol 42 no 8 pp1311ndash1320 2006

[26] H G Neddermeijer G J V Oortmarssen and N P R DekkerldquoA framework for response surface methodology for simulationoptimizationrdquo in Proceedings of the Winter Simulation Confer-ence vol 1 Orlando Fla USA December 1999

[27] B Can and C Heavey ldquoA comparison of genetic programmingand artificial neural networks in metamodeling of discrete-event simulation modelsrdquo Computers amp Operations Researchvol 39 no 2 pp 424ndash436 2012

[28] A Azadeh Z S Faiz S M Asadzadeh and R Tavakkoli-Moghaddam ldquoAn integrated artificial neural network-comput-er simulation for optimization of complex tandem queuesystemsrdquoMathematics and Computers in Simulation vol 82 no4 pp 666ndash678 2011

[29] R Bornatico J Hussy A Witzig and L Guzzella ldquoSurrogatemodeling for the fast optimization of energy systemsrdquo Energyvol 57 pp 653ndash662 2013

[30] X Wan J F Pekny and G V Reklaitis ldquoSimulation-basedoptimizationwith surrogatemodels application to supply chainmanagementrdquoComputers and Chemical Engineering vol 29 no6 pp 1317ndash1328 2005

[31] W Shi Z Liu J Shang and Y Cui ldquoMulti-criteria robust designof a JIT-based cross-docking distribution center for an autoparts supply chainrdquo European Journal of Operational Researchvol 229 no 3 pp 695ndash706 2013

[32] G Dellino J P C Kleijnen and C Meloni ldquoRobust optimiza-tion in simulation taguchi and Response Surface Methodol-ogyrdquo International Journal of Production Economics vol 125 no1 pp 52ndash59 2010

[33] T Yang Y Wen and F Wang ldquoEvaluation of robustness of sup-ply chain information-sharing strategies using a hybrid Taguchiand multiple criteria decision-making methodrdquo InternationalJournal of Production Economics vol 134 no 2 pp 458ndash4662011

[34] M E Kuhl and J R Wilson ldquoModeling and simulating Poissonprocesses having trends or nontrigonometric cyclic effectsrdquoEuropean Journal of Operational Research vol 133 no 3 pp566ndash582 2001

[35] P Narayan and J Subramanian Inventory ManagementmdashPrinciples and Practices Excel Books New Delhi India 2008

[36] E A Silver and R Peterson Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 1985

[37] R G Brown Smoothing Forecasting and Prediction of DiscreteTime Series Courier 2004

[38] D C Montgomery Design and analysis of experiments JohnWiley amp Sons New York NY USA 6th edition 2005

[39] J Zhu ldquoData envelopment analysis vs principal componentanalysis an illustrative study of economic performance ofChinese citiesrdquo European Journal of Operational Research vol111 no 1 pp 50ndash61 1998

[40] I M Premachandra ldquoA note on DEA vs principal componentanalysis an improvement to Joe Zhursquos approachrdquo EuropeanJournal of Operational Research vol 132 no 3 pp 553ndash5602001

[41] D Slottje G W Scully G J Hirschberg and K Hayes Mea-suring the Quality of Life across Countries A MultidimensionalAnalysis Westview Press Boulder Colo USA 1991

[42] S Opricovic Multicriteria optimization of civil engineeringsystems [PhD thesis] Faculty of Civil Engineering BelgradeSerbia 1998

[43] P L Yu ldquoA class of solutions for group decision problemsrdquoManagement Science vol 19 no 8 pp 936ndash946 1973

Modelling and Simulation in Engineering 17

[44] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[45] S Opricovic and G H Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

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Page 7: Research Article Robust Inventory System Optimization ...downloads.hindawi.com/journals/mse/2014/305120.pdf · Robust Inventory System Optimization Based on Simulation and Multiple

Modelling and Simulation in Engineering 7

Table 2 DOE configuration

ProductsFactors

IP 119877119905

119871

minus 0 + minus 0 + minus 0 +119860 FI FQ DF 10 20 30 3 5 7119861 FI FQ DF 15 30 45 3 5 7119862 FI FQ DF 15 30 45 3 5 7119863 FI FQ DF 15 30 45 3 5 7

Table 3 Experimental design

Number of trial 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27119877119901

minus minus minus minus minus minus minus minus minus 0 0 0 0 0 0 0 0 0 + + + + + + + + +119871 minus minus minus 0 0 0 + + + minus minus minus 0 0 0 + + + minus minus minus 0 0 0 + + +IP minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 + minus 0 +

this research there are 3 factors) using full factorial design ispreferable On the other hand because nonlinear behaviourof response variables (ie objective functions) centre pointis considered Each factor has high level ldquo+rdquo low level ldquominusrdquoand centre point ldquo0rdquo Factors and levels configurations areshown in Table 2 while Table 3 shows experimental designIn both of these tables minus sign is representative of lowlevel of related factors while plus sign is indicator of high levelof factors and zero determines centre point For examplein the case of lead time (119871) 3 is low level 7 is high leveland 5 is centre point Lead time is considered with daysas measuring unit and begins from time order is organizeduntil the arrival of inventory Reorder point is labelled as 119877

119901

and is intended to satisfy product demand during lead time(119871) Eventually inventory policy is labelled as IP and refersto inventory policies that were described in Section 412The mentioned design includes 27 runs and there are fourresponse variables cost service level average of inventoryposition and robustness In fact first three response variablesare expected values of defined objective functions (ie resp(1) (2) and (3)) but robustness is a defined criterion whichguarantees that optimal solution remains valid in situationwhere demand variation and intensification increase Forevaluation of robustness demands are multiplied by randomvariable with normal distribution (120583 = 1 120590 = 15)This random variable is also restricted to positive values toprevent negative demands Expected value of 4000 generatednumbers with mentioned constraint is equal to 148 thisresult implies that multiplied random number increases bothdemand variation and intensification simultaneously Thenthe percent of decrease in service level is considered asrobustness for all 27 scenarios produced byDOE It is obviousthat a lower decrease in service level is desirable

422 Weight Assignment Based on PCA Method PCA isthe abbreviation of ldquoprincipal component analysisrdquo and isknown as powerful data-driven method successfully appliedin several areas [2] PCA analysis can determine key variablesthat govern most of the variability This technique helps toremove less important variables to reduce dimension of data

sets PCA considers two criteria for each variable in datasets the first is variability and the second is correlation withother variables So input for this method consists of variancecovariance matrix [39 40] Although other methods likeentropy is applicable in multiple attribute decision makingproblem PCA advantage is considerable because variablescorrelations are not ignored As our problem is constructedof highly correlated variables PCA is the preferable choiceFor correlation justification remember that improvementof inventory position criterion causes negative influence onshortage cost or improvement of service level criterion causesnegative influence on inventory in hand and holding cost Inthese circumstances there is no reason for applying variationbased methods like entropy so in this paper PCA is appliedfor weight assignment These weights are used by MADMtechnique (described in Section 423) to improve qualityof ranking Because of relative utility of scenarios linearscaleless method employed to scaleless data obtained fromsimulation

(1) Calculate scaleless matrix of 119863 = (1198891 1198892 119889

119901)119899times119901

which 119889119894119895is scaleless value of 119894th scenario in the view

point of 119895th criterion

(2) Calculate sample mean vector 119889 = (1198891 1198892 119889

119901)119901

and covariance matrix based on (7) and (8) respec-tively

119889119896=1

119899

119899

sum

119894=1

119889119894119895 (7)

119878 = (119904119896119902)119901times119901

=1

119899 minus 1(119863 minus 119889)

119879

(119863 minus 119889) (8)

(3) 1198621radic119904119896119896 is diagonal 119901 times 119901 matrix 119896th diagonal

element is 1radic119904119896119896 and sample correlation matrix (119877)is calculated as mentioned in the following equation

119877 =1198621

radic119904119896119896

sdot 119878 sdot1198621

radic119904119896119896

(9)

8 Modelling and Simulation in Engineering

(4) Solve the following equation where 119868119901is a 119901 times 119901

identity matrix10038161003816100381610038161003816119877 minus 120582119868

119901

10038161003816100381610038161003816= 0 (10)

Solving (10) results in 119875 individual eigenvaluesregarding 120582

1ge 1205822ge sdot sdot sdot ge 120582

119901and sum119901

119896=1120582119896=

119875 These eigenvalues have 119875-specific eigenvectors(119897119896

1 119897119896

2 119897

119896

119901) and 119896 = 1 119901 These vectors create

principal components as expressed in the followingequation

PC119896=

119901

sum

119902=1

119897119896

119902119889119895

119902 (11)

(5) Select sufficient number of principal components Inthis problem first119872 components whose cumulativepercentage of their eigenvalues (119862

119872) is greater than

95 are selected Cumulative percentage is calculatedbased on the following equation

119862119898=sum119898

119896=1120582119896

sum119901

119896=1120582119896

=sum119898

119896=1120582119896

119875 (12)

(6) Finally weights vector that is indicated by119885 is derivedfrom the mathematical relation in (13) while 119882

119896is

equal to 120582119896119875 if the entire elements of PC

119896are

positive otherwise minus120582119896119875 if entire component are

negative

119885 =

119872

sum

119896=1

119882119896PC119896 (13)

In other cases positive or negative 119882119896should be defined

in the order that final weight vector is positive For moredetailed discussion about 119882

119896 refer to [41] Finally for each

product there are four 119885 parameters that determine relativeimportance for the following criteria cost service levelaverage of inventory position and robustness each elementof 119885 vector is divided by the sum of vector elements fornormalization

423 VIKOR Method This method was developed by Opri-covic and is the abbreviation of expression that in Serbianmeans ldquomultiple criteria optimization and compromisingsolutionsrdquo [42] VIKOR method is based on compromisingsolutions which is the result of Yu [43] and Zeleny [44]studies This approach considers closeness to ideal solutionIn comparison with other MADM methods VIKOR is ableto present compromising solutions and substitutes thesesolutions with best one in the case of necessity [45] VIKORmethod is applicable for problems with multiple discretecriteria Generally this method can be considered as rankingtool for scenarios which are generated by DOE Ranking isbased on L-P metric function according to (14) with defini-tion of 119891

119894119895as the value of 119894th criterion for 119895th scenario In

construction of L-P metric function 119891lowast119894indicates best value

of all scenarios in the view point of 119894th criterion and 119891minus119894

represents the worst value of all scenarios with considerationof 119894th criterion 119908

119894is the weight parameter for 119894th criterion

and is derived by PCA method Consider

119871119901=

119899

sum

119894=1

[

119908119894(119891lowast

119894minus 119891119894119895)

(119891lowast

119894minus 119891minus

119894)]

119901

1119901

1 ≪ 119875 ≪ infin 119895 = 1 2 3 119869

119871infin= max

119894[

119908119894(119891lowast

119894minus 119891119894119895)

(119891lowast

119894minus 119891minus

119894)]

(14)

VIKOR method only involves 1198711and 119871

infinwhich are known

as 119878119895and 119877

119895 respectively 119878 and 119877 index refer to scenarios

so both of these functions are regarded for each scenarioVIKOR steps are as follows

(1) Calculate 119891minus119894and 119891lowast

119894for entire of criteria

(2) Calculate 119878119895and 119877

119895for entire of scenarios

(3) With the assumption that 119878lowast and 119877lowast are the smallestvalues among all of 119878

119895and 119877

119895 respectively and also

119878minus and 119877minus are the greatest values in comparison withall of 119878

119895and 119877

119895 respectively 119876

119895is calculated for 119895th

scenario based on (15) In this equation V is strategicweight that determines the importance of individualcriterion (119878

119895) against group importance of criteria

(119877119895) for each scenario Consider

119876119895=

V (119878119895minus 119878lowast

)

119878minus minus 119878lowast+ (1 minus V)

(119877119895minus 119877lowast

)

119877minus minus 119877lowast (15)

(4) Scenarios should be sorted in descending order basedon 119878119877 and119876 specifically that result in three differentranking lists

(5) According to 119876 the scenario which is indicated by 1198861015840is the best scenario if two conditions namely (C1)and (C2) are satisfied

(C1) Acceptable Advantage This condition implies that thesecond best solution (11988610158401015840) should be far enough from thebest solution (1198861015840) to be accepted as unique solution Themathematical equation is defined in (16) considering 119869 as thetotal number of scenarios Consider

119876(11988610158401015840

) minus 119876 (1198861015840

) ≫1

(119869 minus 1) (16)

(C2) Acceptable Stability This condition validates 1198861015840 as thebest stable solution if 1198861015840 is also the best solution based on 119878 or119877

In case that one or both of the mentioned conditions arenot satisfied compromised solutions are involved VIKORmethod suggests that in such case compromising solutionshave equal value to the best solution and could be consideredinterchangeably by decision makers

Modelling and Simulation in Engineering 9

If (C1) is not satisfied scenarios 1198861015840 11988610158401015840 119886119872 are consid-ered 1198861015840 is the best solution and 11988610158401015840 119886119872 are compromisingsolutions with priority of 119876 on the other hand 1198861015840 11988610158401015840are considered if (C2) is not satisfied 119872 superscript isdetermined by (17)

119876(119886119872

) minus 119876 (1198861015840

) ≫1

(119869 minus 1) (17)

5 Numerical Result

Simulation process was executed on laptop with 18 GH CPUand 4GB of RAM Running of each replication with anima-tion and maximum speed takes 7 minutes and 35 secondsAs simulation of each scenario contains 10 replications andthere are 27 scenarios the total simulation time will be 2046minutes and 36 seconds

51 Multiresolution Result for Demand Modelling Result ofdemand modelling is presented for product 119861 because thisanalysis for other products is similar For this problempolynomial functions are applied for yearly and monthlydemand rates Equations (18) and (19) are estimated bynonlinear regression model In these functions 119905

119910and 119905119898are

time variables

119891 (119905119910) = 0008942 times 119905

3

119910minus 003884 times 119905

2

119910

+ 01867 times 119905119910minus 008018

(18)

and 95 confidence interval is presented in Table 4 for eachcoefficient of (18)

In this table coefficients of (18) are visible in the firstcolumn while lower bund and upper bund of each coefficientare presented in the second and third columns respectivelyThere is no interval including zero so it is concluded thatcoefficient estimation is valid

Estimation error for 119891(119905119910) is reported based on the sum

of square errors (SSE) which is 1504 times 10minus6 Consider

119891 (119905119898) = 4768 times 10

minus5

times 1199055

119898minus 0001692 times 119905

4

119898+ 00214

times 1199053

119898minus 01159 times 119905

2

119898+ 03474 times 119905

119898minus 02286

(19)

Monthly demand rate function that is indicated by 119891(119905119898)

and its coefficients confidence interval are presented inTable 5 SEE is equal to 00005825 Based on local businessinformation total demand of product 119861 is estimated to beabout 6822 units in five years According to the informationabout product 119861 Figure 2 shows final multiresolution resultfor this product

52 Simulation Parameters and Configuration For simula-tion of problem it is necessary to configure inventory policyparameters and variables We have three different policiesnamely FQ FI and DF Variables of FQ policy are reorderpoint (119877

119901) lead time (119871) and economic quantity orders (119876

119895119905)

First two variables are configured based on Table 2 and thirdvariable is derived fromWilson formula according to average

Table 4 95 confidence interval for estimated coefficient of 119891(119905119910)

Coefficients Lower bund Upper bund0008942 0004835 001305minus003884 minus007604 minus000165201867 008654 0287minus008018 minus01568 minus0003523

Table 5 95 confidence interval for estimated coefficient of 119891(119905119898)

Coefficients Lower bund Upper bund4768 times 10minus5 4451 times 10minus5 5086 times 10minus5

minus0001692 minus0001795 minus00015900214 00202 00226minus01159 minus01221 minus0109703474 0334 03608minus02286 minus02371 minus022

Table 6 Estimation of distribution functions of demand and reor-ders quantity

Product type Distribution function 119875 value Order quantity119860 278 lowast BETA(113 328) gt015 100119861 GAMMA(678 118) gt015 92119862 EXPO(818) gt015 100119863 446 lowast BETA(0957 321) gt015 115

of demand holding cost (119862ℎ) and ordering cost (119862

119903) The

calculated values are 57 71 72 and 72 for product 119860 119861 119862and119863 respectively

Orders quantity is unknown for FI policy and is cal-culated based on probability distribution function whichis obtained by demand simulation of each product for 20daysrsquo time interval Distribution fitting is performed by inputanalyser of Arena software and graphical result for product 119861is shown in Figure 3 Results for other products are reportedin Table 6

In this table fitted probability distribution function foreach product is presented in second column while 119875 valueof fitting which is greater than 015 for fitted distributiongrantees goodness of fitting in 95 confidence level Finallyaccording to fitted distribution functions for FI policyproper ordering amount of each product is presented in thefourth column

DF policy has two unknown parameters namely 120572 and120573 The first parameter is weight of current real demand in (4)and second one is the weight of current trend in (5) Theseparameters are adjusted by simulation of five years demandDemand for the entire four products simultaneously is con-sidered and an average of lost sale percentage is consideredas response variable Different values of response variables arereported in Table 7 According to the different values of 120572 and120573 the best values for 120572 and 120573 are 09 and 01 respectivelywhich results in minimum lost percentage in average

10 Modelling and Simulation in Engineering

Table 7 Average of lost sale percentage for different values of 120572 and 120573

120572 and 120573 values 01 02 03 04 05 06 07 08 09Response to 120572 0380 0356 0334 0320 0306 0292 0284 0276 0268Response to 120573 0296 0302 0306 0306 0306 0304 0305 0304 0302

50

100

150

200

250

300

350Demand fit for each month

Dem

and

Fitted valueReal demand

00 10 20 30 40 50 60

Month

Figure 2 Comparison between real and estimated demand

100 200 300 400 500 600 700

50

100

150

200

Distribution summary

Distribution gammaExpression minus0001 + GAMM(00)Square error 0006825

Chi-square testNumber of intervals = 5Degrees of freedom = 2Test statistic = 148

Kolmogorov-Smirnov testTest statistic = 00733

Corresponding P value = 0483

Corresponding P value gt 015

Figure 3 Distribution function of customer demand for reorderinterval of product 119861

53Weight Assignment Asmentioned before PCA is appliedforweight assignment For this purposeDOE result is neededin the form of decision matrix that is shown in Table 8 Inthis table four criteria cost average of inventory positionservice level and robustness are evaluated for each scenarioof product 119861 Decision matrix should be scaleless that isperformed by linear method Then the result of eigenvalues

Table 8 Decision matrix for product 119861

Scenarionumber Cost Average of

inventory positionServicelevel

Robustness

1 32984000 4352 76 62 29214710 397514 79 33 37114636 339623 69 24 34840720 410948 71 55 36820570 344738 72 66 39191837 331091 67 27 37037180 391012 68 48 40994680 298712 64 69 41407045 319678 63 110 28282930 524462 83 711 26422145 489243 85 512 31023829 416049 79 313 30395190 487146 79 614 35393680 395812 74 515 32838055 395646 77 316 33285615 463341 76 617 39652020 335668 66 718 35269609 383944 73 219 26227680 625612 87 720 25565820 586688 88 621 28711069 49449 85 422 27301580 585472 85 723 34016415 470961 78 724 30601546 463975 82 325 29752815 560421 82 726 41024410 382994 68 627 31604741 443053 78 3

is calculated and shown in Table 9 Also result of principalcomponents is presented in Table 10

As it is visible in Table 9 cumulative proportion ofthe first and second components is 0966 which is greaterthan 095 so the first two components are considered asprincipal components Their relative weights are 078 and0186 respectively Final weights are normalized and resultsare reported in Table 11

54 Ranking and Selection of Scenarios Weights of criteriaobtained by PCA method in addition to decision matrix areinput of VIKOR method 119878 119876 and 119877 values are calculatedbased on (14) and (15) while strategic weight indicated byV is equal to 05 so individual and group utility have same

Modelling and Simulation in Engineering 11

Table 9 Result of eigenvalues

Eigenvalues 31204 07428 01111 00257Proportion 0780 0186 0028 0006Cumulative 0780 0966 0994 1000

Table 10 Result of principle components

Variables PC1 PC2 PC3 PC4Inventory 0543 0089 minus0818 0167Cost minus0538 minus0277 minus0511 minus0610Service level minus0551 minus0210 minus0230 0774Robustness 0335 minus0933 0126 0023

Table 11 Weights of criteria for different products

Criteria Products119860 119861 119862 119863

Average of inventory position 01012 00590 01942 00462Cost 02852 01836 01742 03081Service level 01195 01392 00966 01206Robustness 04941 06182 05350 05251

importance Calculated values are reported in Table 12 forentire scenario as VIKOR output

Product 119861 According to Table 12 values of 119877 119878 and 119876 arezero for scenario 18 Regarding (C1) (Acceptable Advantage)the difference between the best scenario and second rankedscenario should be more than 00038 The second rankedscenario regarding 119876 value is scenario 3 and its value is0051044 that satisfies both (C1) and (C2) Final result forproduct 119861 is unique scenario that is number 18 and consistsof the following configuration Inventory policy is DFwith119877

119901

being equal to 30 products and 7 days lead time According todecision matrix which is shown in Table 8 the performanceof this scenario for product 119861 results in 383944 products asaverage of inventory position and expected cost is equal to35269609 for product life cycle 73 of customer demandwillbe satisfied and in the case of increasing demand intensityand variation 2 decrease is expected in service level so itwill change into 71 Analyses for other products are similarso only final results are presented in this paper

Product 119860 In this case VIKORmethod ranks scenario num-ber 15 as the best one This scenario consists of DF inventorypolicy 20 products for 119877

119901and 5 days for lead time Perfor-

mance of this scenario results in 374778 products as averageof inventory position Expected cost is 31603294 and servicelevel will be 66 This scenario is robust against demandintensification and variation because changes in demandaffect service level as small as 1 But scenario number 15 doesnot satisfy (C1) so compromising solutions are consideredAlthough the best scenario is number 15 VIKOR methodimplies that compromising solutions have the same values

Table 12 VIKOR output for product 119861

Run number 119878 119877 119876

1 0634311 0796269 0715292 0003184 0185077 0094133 0059804 0042283 00510444 0536946 0592539 05647425 0718862 0796269 07575656 0116278 0089887 01030827 0428814 0388808 04088118 086078 0796269 08285259 002043 0140653 008054110 0677386 1 083869311 0266254 0592539 042939612 0043962 0185077 01145213 0571695 0796269 068398214 0515097 0592539 055381815 009173 0185077 013840416 0648693 0796269 072248117 1 1 118 0 0 019 0630698 1 081534920 04239 0796269 061008421 0139527 0388808 026416722 06581 1 08290523 0819338 1 090966924 0022245 0185077 010366125 0726183 1 086309126 0849516 0796269 082289327 0072768 0185077 0128922

So in real situation compromising solution can be substi-tuted This solution regarding their priority is as follows

Scenario number 24 inventory policy is DF 119877119901is

equal to 30 products and lead time is 5 daysScenario number 18 inventory policy is DF 119877

119901is

equal to 20 products and lead time is 7 daysScenario number 12 inventory policy isDF119877

119901is equal

to 20 products and lead time is 3 days

Product 119862 For this product scenario number 1 is ranked asthe best one This scenario consists of FI inventory policy 119877

119901

is 15 products and lead time is 3 days Result of this scenario is4387 products as the average of inventory position Expectedcost is 66143000 and service level will be 61 with robustnessof 1 Compromising solutions are as follows

Scenario number 6 inventory policy is DF 119877119901is 15

products and lead time is 5 daysScenario number 3 inventory policy is DF 119877

119901is 15

products and lead time is 3 daysScenario number 7 inventory policy is FI 119877

119901is 15

products and lead time is 7 days

12 Modelling and Simulation in Engineering

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 4 Interaction plot for product 119862 average of inventory position

Scenario number 4 inventory policy is FI 119877119901is 15

products and lead time is 5 days

Product119863 Scenario number 3 has the best rank for this prod-uct This scenario consists of DF inventory policy 119877

119901is 15

products and lead time is 3 days This scenario leads to

363896 products as average of inventory position with62648337 expected cost and 61 service level with 1robustness Compromising solutions are as follows

Scenario number 6 this scenario is mentioned beforeas a compromising solution for product C

Modelling and Simulation in Engineering 13

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 5 Interaction plot for product 119862 cost

14 Modelling and Simulation in Engineering

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

3

5

7

L

3

5

7

L

3

5

7

L

L

(b)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 6 Interaction plot for product 119862 service level

Scenario number 17 inventory policy is FI119877119901is equal

to 15 products and lead time is 7 days

55 Sensitivity Analysis As mentioned in the previous sec-tion all scenarios related to product 119860 prefer DF inventory

policy so it is concluded that DF policy is unique optimalpolicy while lead time includes values of 3 5 and 7 in selectedscenarios This situation implies that product 119860 is insensitiveto the delivery time It can be wise to decide lead time about 5days but there is no need for strict control on lead time while

Modelling and Simulation in Engineering 15

119877119901should be controlled strictly to prevent inventory in hand

from reaching lower than 20 productsIn the case of product 119861 (C1) is satisfied so there is a

unique scenario while there are four compromising solutionfor product 119862 In this situation based on VIKOR methodcompromising solutions have same value while interactionplot can be applied for investigation of existing differencebetween compromising solutions As mentioned before thebest scenario suggests FI inventory policy for this product butin compromising solutions there are both FI and DF policiesLead time varies from 3 to 7 days So this confusing situationis resolved by deeper analysis and applying interaction plotFigure 4 shows interaction plot for DOE factors and averageof inventory position as response variables

Figure 4 shows that with either DF or FI policy there isno significant difference among different lead times in theview point of inventory position but DF leads to less averageof inventory position With 119877

119901equaling 15 there is less

sensitivity to different lead times So configuring 119877119901which

is equal to 15 products the average of inventory positioncriterion will be robust against lead time variations

Figure 5 regards cost criterion and shows that FI andDF policies are equal and make least sensitivity related tolead time On the other hand three plots of 119877

119901against lead

time are parallel The parallel plots are interpreted as lackof relation between 119877

119901and lead time With 119877

119901equaling 15

increase in inventory cost could be mitigated when lead timesets to 3 days

In Figure 6 with 3 days for lead time there is no signifi-cant difference between inventory policies in the view pointof service level119877

119901and lead time are independent and the best

service level is related to FI policy which is also very sensitiveto 119877119901 As 119877

119901increases service level will grow According to

the interaction plot analyses optimal decision for product119862 is FI inventory policy 3 days lead time and 15 or moreproducts for 119877

119901

Decision about product 119863 is simple Based on the in-formation obtained from solutions it is recommended toemploy DF inventory policy with 119877

119901being equal to 15

products and 3 days for lead time

6 Conclusion

In this paper a simulation optimization framework is pro-posed for robust optimization of retailer inventory systemThis framework is less time consuming in comparison withmetaheuristic or SPSA approach it also provides robustsolution for multiple objective functions In this research alsotrend and cyclic change of customers demand are consideredfor more realistic view of problem Proposed framework con-sists of discrete event simulation and surrogate model In thisframework full factorial design of experiment is employedfor producing decision space Based on the three decisionfactors 27 different scenarios are produced and simulated Insimulation modelling multiresolution method is applied forsimulation of demand with highly dynamic pattern Becausethere are multiple criteria for inventory system performanceVIKORmethod is employed asmulticriteria decisionmaking

technique In addition robustness is considered as perfor-mance criterion and PCA is used for improving performanceof VIKOR method Finally developed framework gave thebest ranked and compromising solutions so interaction plotapplied for more investigation and sensitivity analysis ofobtained solutions

Due to the novelty of the proposed framework it isrecommended that future studies encompass optimization ofinventory system of integrated supply chain Also improve-ment of developed framework can be considered as futurestudy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers for theirsignificant roles in improvement of this research quality

References

[1] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[2] S Yin S XDingAHaghaniHHao andP Zhang ldquoA compar-ison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[3] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[4] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[5] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall International Series in Management New YorkNY USA 1963

[6] E Naddor Inventory Systems John Wiley amp Sons New YorkNY USA 1966

[7] R Peterson and E A Silver Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 2nd edition 1985

[8] T C E Cheng ldquoEconomic order quantity model with demand-dependent unit production cost and imperfect productionprocessesrdquo IIE Transactions vol 23 no 1 pp 23ndash28 1991

[9] S H Chen C CWang and A Ramer ldquoBackorder fuzzy inven-tory model under function principlerdquo Information Sciences vol95 no 1-2 pp 71ndash79 1996

[10] X Zhao J Xie and J C Wei ldquoThe value of early order com-mitment in a two-level supply chainrdquo European Journal of Op-erational Research vol 180 no 1 pp 194ndash214 2007

[11] R S M Lau J Xie and X Zhao ldquoEffects of inventory policyon supply chain performance a simulation study of critical

16 Modelling and Simulation in Engineering

decision parametersrdquo Computers and Industrial Engineeringvol 55 no 3 pp 620ndash633 2008

[12] J Xu and L Zhao ldquoA multi-objective decision-making modelwith fuzzy rough coefficients and its application to the inventoryproblemrdquo Information Sciences vol 180 no 5 pp 679ndash6962010

[13] F Hnaien X Delorme and A Dolgui ldquoMulti-objective opti-mization for inventory control in two-level assembly systemsunder uncertainty of lead timesrdquo Computers amp OperationsResearch vol 37 no 11 pp 1835ndash1843 2010

[14] H D Purnomo H M Wee and Y Praharsi ldquoTwo inventoryreview policies on supply chain configuration problemrdquo Com-puters and Industrial Engineering vol 63 no 2 pp 448ndash4552012

[15] B L Nelson ldquo50th anniversary article stochastic simulationresearch in management sciencerdquoManagement Science vol 50no 7 pp 855ndash868 2004

[16] J Boesel R O Bowden Jr F Glover J P Kelly and E WestwigldquoFuture of simulation optimizationrdquo inProceedings of theWinterSimulation Conference pp 1466ndash1469 Arlington Va USADecember 2001

[17] M C Fu ldquoOptimization for simulation theory vs practicerdquoINFORMS Journal on Computing vol 14 no 3 pp 192ndash2152002

[18] L Wang ldquoA hybrid genetic algorithm-neural network strategyfor simulation optimizationrdquoApplied Mathematics and Compu-tation vol 170 no 2 pp 1329ndash1343 2005

[19] B B Keskin S H Melouk and I L Meyer ldquoA simulation-optimization approach for integrated sourcing and inventorydecisionsrdquo Computers and Operations Research vol 37 no 9pp 1648ndash1661 2010

[20] E Mazhari J Zhao N Celik S Lee Y Son and L HeadldquoHybrid simulation and optimization-based design and oper-ation of integrated photovoltaic generation storage units andgridrdquo Simulation Modelling Practice and Theory vol 19 no 1pp 463ndash481 2011

[21] Q Duan and T W Liao ldquoOptimization of replenishment poli-cies for decentralized and centralized capacitated supply chainsunder various demandsrdquo International Journal of ProductionEconomics vol 142 no 1 pp 194ndash204 2013

[22] J C Spall ldquoMultivariate stochastic approximation usinga simultaneous perturbation gradient approximationrdquo IEEETransactions on Automatic Control vol 37 no 3 pp 332ndash3411992

[23] J C Spall ldquoImplementation of the simultaneous perturbationalgorithm for stochastic optimizationrdquo IEEE Transactions onAerospace and Electronic Systems vol 34 no 3 pp 817ndash8231998

[24] J C Spall ldquoStochastic optimization and the simultaneousperturbation methodrdquo in Proceedings of the Winter SimulationConference (WSC rsquo99) pp 101ndash109 December 1999

[25] J D Schwartz W Wang and D E Rivera ldquoSimulation-based optimization of process control policies for inventorymanagement in supply chainsrdquo Automatica vol 42 no 8 pp1311ndash1320 2006

[26] H G Neddermeijer G J V Oortmarssen and N P R DekkerldquoA framework for response surface methodology for simulationoptimizationrdquo in Proceedings of the Winter Simulation Confer-ence vol 1 Orlando Fla USA December 1999

[27] B Can and C Heavey ldquoA comparison of genetic programmingand artificial neural networks in metamodeling of discrete-event simulation modelsrdquo Computers amp Operations Researchvol 39 no 2 pp 424ndash436 2012

[28] A Azadeh Z S Faiz S M Asadzadeh and R Tavakkoli-Moghaddam ldquoAn integrated artificial neural network-comput-er simulation for optimization of complex tandem queuesystemsrdquoMathematics and Computers in Simulation vol 82 no4 pp 666ndash678 2011

[29] R Bornatico J Hussy A Witzig and L Guzzella ldquoSurrogatemodeling for the fast optimization of energy systemsrdquo Energyvol 57 pp 653ndash662 2013

[30] X Wan J F Pekny and G V Reklaitis ldquoSimulation-basedoptimizationwith surrogatemodels application to supply chainmanagementrdquoComputers and Chemical Engineering vol 29 no6 pp 1317ndash1328 2005

[31] W Shi Z Liu J Shang and Y Cui ldquoMulti-criteria robust designof a JIT-based cross-docking distribution center for an autoparts supply chainrdquo European Journal of Operational Researchvol 229 no 3 pp 695ndash706 2013

[32] G Dellino J P C Kleijnen and C Meloni ldquoRobust optimiza-tion in simulation taguchi and Response Surface Methodol-ogyrdquo International Journal of Production Economics vol 125 no1 pp 52ndash59 2010

[33] T Yang Y Wen and F Wang ldquoEvaluation of robustness of sup-ply chain information-sharing strategies using a hybrid Taguchiand multiple criteria decision-making methodrdquo InternationalJournal of Production Economics vol 134 no 2 pp 458ndash4662011

[34] M E Kuhl and J R Wilson ldquoModeling and simulating Poissonprocesses having trends or nontrigonometric cyclic effectsrdquoEuropean Journal of Operational Research vol 133 no 3 pp566ndash582 2001

[35] P Narayan and J Subramanian Inventory ManagementmdashPrinciples and Practices Excel Books New Delhi India 2008

[36] E A Silver and R Peterson Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 1985

[37] R G Brown Smoothing Forecasting and Prediction of DiscreteTime Series Courier 2004

[38] D C Montgomery Design and analysis of experiments JohnWiley amp Sons New York NY USA 6th edition 2005

[39] J Zhu ldquoData envelopment analysis vs principal componentanalysis an illustrative study of economic performance ofChinese citiesrdquo European Journal of Operational Research vol111 no 1 pp 50ndash61 1998

[40] I M Premachandra ldquoA note on DEA vs principal componentanalysis an improvement to Joe Zhursquos approachrdquo EuropeanJournal of Operational Research vol 132 no 3 pp 553ndash5602001

[41] D Slottje G W Scully G J Hirschberg and K Hayes Mea-suring the Quality of Life across Countries A MultidimensionalAnalysis Westview Press Boulder Colo USA 1991

[42] S Opricovic Multicriteria optimization of civil engineeringsystems [PhD thesis] Faculty of Civil Engineering BelgradeSerbia 1998

[43] P L Yu ldquoA class of solutions for group decision problemsrdquoManagement Science vol 19 no 8 pp 936ndash946 1973

Modelling and Simulation in Engineering 17

[44] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[45] S Opricovic and G H Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

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Page 8: Research Article Robust Inventory System Optimization ...downloads.hindawi.com/journals/mse/2014/305120.pdf · Robust Inventory System Optimization Based on Simulation and Multiple

8 Modelling and Simulation in Engineering

(4) Solve the following equation where 119868119901is a 119901 times 119901

identity matrix10038161003816100381610038161003816119877 minus 120582119868

119901

10038161003816100381610038161003816= 0 (10)

Solving (10) results in 119875 individual eigenvaluesregarding 120582

1ge 1205822ge sdot sdot sdot ge 120582

119901and sum119901

119896=1120582119896=

119875 These eigenvalues have 119875-specific eigenvectors(119897119896

1 119897119896

2 119897

119896

119901) and 119896 = 1 119901 These vectors create

principal components as expressed in the followingequation

PC119896=

119901

sum

119902=1

119897119896

119902119889119895

119902 (11)

(5) Select sufficient number of principal components Inthis problem first119872 components whose cumulativepercentage of their eigenvalues (119862

119872) is greater than

95 are selected Cumulative percentage is calculatedbased on the following equation

119862119898=sum119898

119896=1120582119896

sum119901

119896=1120582119896

=sum119898

119896=1120582119896

119875 (12)

(6) Finally weights vector that is indicated by119885 is derivedfrom the mathematical relation in (13) while 119882

119896is

equal to 120582119896119875 if the entire elements of PC

119896are

positive otherwise minus120582119896119875 if entire component are

negative

119885 =

119872

sum

119896=1

119882119896PC119896 (13)

In other cases positive or negative 119882119896should be defined

in the order that final weight vector is positive For moredetailed discussion about 119882

119896 refer to [41] Finally for each

product there are four 119885 parameters that determine relativeimportance for the following criteria cost service levelaverage of inventory position and robustness each elementof 119885 vector is divided by the sum of vector elements fornormalization

423 VIKOR Method This method was developed by Opri-covic and is the abbreviation of expression that in Serbianmeans ldquomultiple criteria optimization and compromisingsolutionsrdquo [42] VIKOR method is based on compromisingsolutions which is the result of Yu [43] and Zeleny [44]studies This approach considers closeness to ideal solutionIn comparison with other MADM methods VIKOR is ableto present compromising solutions and substitutes thesesolutions with best one in the case of necessity [45] VIKORmethod is applicable for problems with multiple discretecriteria Generally this method can be considered as rankingtool for scenarios which are generated by DOE Ranking isbased on L-P metric function according to (14) with defini-tion of 119891

119894119895as the value of 119894th criterion for 119895th scenario In

construction of L-P metric function 119891lowast119894indicates best value

of all scenarios in the view point of 119894th criterion and 119891minus119894

represents the worst value of all scenarios with considerationof 119894th criterion 119908

119894is the weight parameter for 119894th criterion

and is derived by PCA method Consider

119871119901=

119899

sum

119894=1

[

119908119894(119891lowast

119894minus 119891119894119895)

(119891lowast

119894minus 119891minus

119894)]

119901

1119901

1 ≪ 119875 ≪ infin 119895 = 1 2 3 119869

119871infin= max

119894[

119908119894(119891lowast

119894minus 119891119894119895)

(119891lowast

119894minus 119891minus

119894)]

(14)

VIKOR method only involves 1198711and 119871

infinwhich are known

as 119878119895and 119877

119895 respectively 119878 and 119877 index refer to scenarios

so both of these functions are regarded for each scenarioVIKOR steps are as follows

(1) Calculate 119891minus119894and 119891lowast

119894for entire of criteria

(2) Calculate 119878119895and 119877

119895for entire of scenarios

(3) With the assumption that 119878lowast and 119877lowast are the smallestvalues among all of 119878

119895and 119877

119895 respectively and also

119878minus and 119877minus are the greatest values in comparison withall of 119878

119895and 119877

119895 respectively 119876

119895is calculated for 119895th

scenario based on (15) In this equation V is strategicweight that determines the importance of individualcriterion (119878

119895) against group importance of criteria

(119877119895) for each scenario Consider

119876119895=

V (119878119895minus 119878lowast

)

119878minus minus 119878lowast+ (1 minus V)

(119877119895minus 119877lowast

)

119877minus minus 119877lowast (15)

(4) Scenarios should be sorted in descending order basedon 119878119877 and119876 specifically that result in three differentranking lists

(5) According to 119876 the scenario which is indicated by 1198861015840is the best scenario if two conditions namely (C1)and (C2) are satisfied

(C1) Acceptable Advantage This condition implies that thesecond best solution (11988610158401015840) should be far enough from thebest solution (1198861015840) to be accepted as unique solution Themathematical equation is defined in (16) considering 119869 as thetotal number of scenarios Consider

119876(11988610158401015840

) minus 119876 (1198861015840

) ≫1

(119869 minus 1) (16)

(C2) Acceptable Stability This condition validates 1198861015840 as thebest stable solution if 1198861015840 is also the best solution based on 119878 or119877

In case that one or both of the mentioned conditions arenot satisfied compromised solutions are involved VIKORmethod suggests that in such case compromising solutionshave equal value to the best solution and could be consideredinterchangeably by decision makers

Modelling and Simulation in Engineering 9

If (C1) is not satisfied scenarios 1198861015840 11988610158401015840 119886119872 are consid-ered 1198861015840 is the best solution and 11988610158401015840 119886119872 are compromisingsolutions with priority of 119876 on the other hand 1198861015840 11988610158401015840are considered if (C2) is not satisfied 119872 superscript isdetermined by (17)

119876(119886119872

) minus 119876 (1198861015840

) ≫1

(119869 minus 1) (17)

5 Numerical Result

Simulation process was executed on laptop with 18 GH CPUand 4GB of RAM Running of each replication with anima-tion and maximum speed takes 7 minutes and 35 secondsAs simulation of each scenario contains 10 replications andthere are 27 scenarios the total simulation time will be 2046minutes and 36 seconds

51 Multiresolution Result for Demand Modelling Result ofdemand modelling is presented for product 119861 because thisanalysis for other products is similar For this problempolynomial functions are applied for yearly and monthlydemand rates Equations (18) and (19) are estimated bynonlinear regression model In these functions 119905

119910and 119905119898are

time variables

119891 (119905119910) = 0008942 times 119905

3

119910minus 003884 times 119905

2

119910

+ 01867 times 119905119910minus 008018

(18)

and 95 confidence interval is presented in Table 4 for eachcoefficient of (18)

In this table coefficients of (18) are visible in the firstcolumn while lower bund and upper bund of each coefficientare presented in the second and third columns respectivelyThere is no interval including zero so it is concluded thatcoefficient estimation is valid

Estimation error for 119891(119905119910) is reported based on the sum

of square errors (SSE) which is 1504 times 10minus6 Consider

119891 (119905119898) = 4768 times 10

minus5

times 1199055

119898minus 0001692 times 119905

4

119898+ 00214

times 1199053

119898minus 01159 times 119905

2

119898+ 03474 times 119905

119898minus 02286

(19)

Monthly demand rate function that is indicated by 119891(119905119898)

and its coefficients confidence interval are presented inTable 5 SEE is equal to 00005825 Based on local businessinformation total demand of product 119861 is estimated to beabout 6822 units in five years According to the informationabout product 119861 Figure 2 shows final multiresolution resultfor this product

52 Simulation Parameters and Configuration For simula-tion of problem it is necessary to configure inventory policyparameters and variables We have three different policiesnamely FQ FI and DF Variables of FQ policy are reorderpoint (119877

119901) lead time (119871) and economic quantity orders (119876

119895119905)

First two variables are configured based on Table 2 and thirdvariable is derived fromWilson formula according to average

Table 4 95 confidence interval for estimated coefficient of 119891(119905119910)

Coefficients Lower bund Upper bund0008942 0004835 001305minus003884 minus007604 minus000165201867 008654 0287minus008018 minus01568 minus0003523

Table 5 95 confidence interval for estimated coefficient of 119891(119905119898)

Coefficients Lower bund Upper bund4768 times 10minus5 4451 times 10minus5 5086 times 10minus5

minus0001692 minus0001795 minus00015900214 00202 00226minus01159 minus01221 minus0109703474 0334 03608minus02286 minus02371 minus022

Table 6 Estimation of distribution functions of demand and reor-ders quantity

Product type Distribution function 119875 value Order quantity119860 278 lowast BETA(113 328) gt015 100119861 GAMMA(678 118) gt015 92119862 EXPO(818) gt015 100119863 446 lowast BETA(0957 321) gt015 115

of demand holding cost (119862ℎ) and ordering cost (119862

119903) The

calculated values are 57 71 72 and 72 for product 119860 119861 119862and119863 respectively

Orders quantity is unknown for FI policy and is cal-culated based on probability distribution function whichis obtained by demand simulation of each product for 20daysrsquo time interval Distribution fitting is performed by inputanalyser of Arena software and graphical result for product 119861is shown in Figure 3 Results for other products are reportedin Table 6

In this table fitted probability distribution function foreach product is presented in second column while 119875 valueof fitting which is greater than 015 for fitted distributiongrantees goodness of fitting in 95 confidence level Finallyaccording to fitted distribution functions for FI policyproper ordering amount of each product is presented in thefourth column

DF policy has two unknown parameters namely 120572 and120573 The first parameter is weight of current real demand in (4)and second one is the weight of current trend in (5) Theseparameters are adjusted by simulation of five years demandDemand for the entire four products simultaneously is con-sidered and an average of lost sale percentage is consideredas response variable Different values of response variables arereported in Table 7 According to the different values of 120572 and120573 the best values for 120572 and 120573 are 09 and 01 respectivelywhich results in minimum lost percentage in average

10 Modelling and Simulation in Engineering

Table 7 Average of lost sale percentage for different values of 120572 and 120573

120572 and 120573 values 01 02 03 04 05 06 07 08 09Response to 120572 0380 0356 0334 0320 0306 0292 0284 0276 0268Response to 120573 0296 0302 0306 0306 0306 0304 0305 0304 0302

50

100

150

200

250

300

350Demand fit for each month

Dem

and

Fitted valueReal demand

00 10 20 30 40 50 60

Month

Figure 2 Comparison between real and estimated demand

100 200 300 400 500 600 700

50

100

150

200

Distribution summary

Distribution gammaExpression minus0001 + GAMM(00)Square error 0006825

Chi-square testNumber of intervals = 5Degrees of freedom = 2Test statistic = 148

Kolmogorov-Smirnov testTest statistic = 00733

Corresponding P value = 0483

Corresponding P value gt 015

Figure 3 Distribution function of customer demand for reorderinterval of product 119861

53Weight Assignment Asmentioned before PCA is appliedforweight assignment For this purposeDOE result is neededin the form of decision matrix that is shown in Table 8 Inthis table four criteria cost average of inventory positionservice level and robustness are evaluated for each scenarioof product 119861 Decision matrix should be scaleless that isperformed by linear method Then the result of eigenvalues

Table 8 Decision matrix for product 119861

Scenarionumber Cost Average of

inventory positionServicelevel

Robustness

1 32984000 4352 76 62 29214710 397514 79 33 37114636 339623 69 24 34840720 410948 71 55 36820570 344738 72 66 39191837 331091 67 27 37037180 391012 68 48 40994680 298712 64 69 41407045 319678 63 110 28282930 524462 83 711 26422145 489243 85 512 31023829 416049 79 313 30395190 487146 79 614 35393680 395812 74 515 32838055 395646 77 316 33285615 463341 76 617 39652020 335668 66 718 35269609 383944 73 219 26227680 625612 87 720 25565820 586688 88 621 28711069 49449 85 422 27301580 585472 85 723 34016415 470961 78 724 30601546 463975 82 325 29752815 560421 82 726 41024410 382994 68 627 31604741 443053 78 3

is calculated and shown in Table 9 Also result of principalcomponents is presented in Table 10

As it is visible in Table 9 cumulative proportion ofthe first and second components is 0966 which is greaterthan 095 so the first two components are considered asprincipal components Their relative weights are 078 and0186 respectively Final weights are normalized and resultsare reported in Table 11

54 Ranking and Selection of Scenarios Weights of criteriaobtained by PCA method in addition to decision matrix areinput of VIKOR method 119878 119876 and 119877 values are calculatedbased on (14) and (15) while strategic weight indicated byV is equal to 05 so individual and group utility have same

Modelling and Simulation in Engineering 11

Table 9 Result of eigenvalues

Eigenvalues 31204 07428 01111 00257Proportion 0780 0186 0028 0006Cumulative 0780 0966 0994 1000

Table 10 Result of principle components

Variables PC1 PC2 PC3 PC4Inventory 0543 0089 minus0818 0167Cost minus0538 minus0277 minus0511 minus0610Service level minus0551 minus0210 minus0230 0774Robustness 0335 minus0933 0126 0023

Table 11 Weights of criteria for different products

Criteria Products119860 119861 119862 119863

Average of inventory position 01012 00590 01942 00462Cost 02852 01836 01742 03081Service level 01195 01392 00966 01206Robustness 04941 06182 05350 05251

importance Calculated values are reported in Table 12 forentire scenario as VIKOR output

Product 119861 According to Table 12 values of 119877 119878 and 119876 arezero for scenario 18 Regarding (C1) (Acceptable Advantage)the difference between the best scenario and second rankedscenario should be more than 00038 The second rankedscenario regarding 119876 value is scenario 3 and its value is0051044 that satisfies both (C1) and (C2) Final result forproduct 119861 is unique scenario that is number 18 and consistsof the following configuration Inventory policy is DFwith119877

119901

being equal to 30 products and 7 days lead time According todecision matrix which is shown in Table 8 the performanceof this scenario for product 119861 results in 383944 products asaverage of inventory position and expected cost is equal to35269609 for product life cycle 73 of customer demandwillbe satisfied and in the case of increasing demand intensityand variation 2 decrease is expected in service level so itwill change into 71 Analyses for other products are similarso only final results are presented in this paper

Product 119860 In this case VIKORmethod ranks scenario num-ber 15 as the best one This scenario consists of DF inventorypolicy 20 products for 119877

119901and 5 days for lead time Perfor-

mance of this scenario results in 374778 products as averageof inventory position Expected cost is 31603294 and servicelevel will be 66 This scenario is robust against demandintensification and variation because changes in demandaffect service level as small as 1 But scenario number 15 doesnot satisfy (C1) so compromising solutions are consideredAlthough the best scenario is number 15 VIKOR methodimplies that compromising solutions have the same values

Table 12 VIKOR output for product 119861

Run number 119878 119877 119876

1 0634311 0796269 0715292 0003184 0185077 0094133 0059804 0042283 00510444 0536946 0592539 05647425 0718862 0796269 07575656 0116278 0089887 01030827 0428814 0388808 04088118 086078 0796269 08285259 002043 0140653 008054110 0677386 1 083869311 0266254 0592539 042939612 0043962 0185077 01145213 0571695 0796269 068398214 0515097 0592539 055381815 009173 0185077 013840416 0648693 0796269 072248117 1 1 118 0 0 019 0630698 1 081534920 04239 0796269 061008421 0139527 0388808 026416722 06581 1 08290523 0819338 1 090966924 0022245 0185077 010366125 0726183 1 086309126 0849516 0796269 082289327 0072768 0185077 0128922

So in real situation compromising solution can be substi-tuted This solution regarding their priority is as follows

Scenario number 24 inventory policy is DF 119877119901is

equal to 30 products and lead time is 5 daysScenario number 18 inventory policy is DF 119877

119901is

equal to 20 products and lead time is 7 daysScenario number 12 inventory policy isDF119877

119901is equal

to 20 products and lead time is 3 days

Product 119862 For this product scenario number 1 is ranked asthe best one This scenario consists of FI inventory policy 119877

119901

is 15 products and lead time is 3 days Result of this scenario is4387 products as the average of inventory position Expectedcost is 66143000 and service level will be 61 with robustnessof 1 Compromising solutions are as follows

Scenario number 6 inventory policy is DF 119877119901is 15

products and lead time is 5 daysScenario number 3 inventory policy is DF 119877

119901is 15

products and lead time is 3 daysScenario number 7 inventory policy is FI 119877

119901is 15

products and lead time is 7 days

12 Modelling and Simulation in Engineering

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 4 Interaction plot for product 119862 average of inventory position

Scenario number 4 inventory policy is FI 119877119901is 15

products and lead time is 5 days

Product119863 Scenario number 3 has the best rank for this prod-uct This scenario consists of DF inventory policy 119877

119901is 15

products and lead time is 3 days This scenario leads to

363896 products as average of inventory position with62648337 expected cost and 61 service level with 1robustness Compromising solutions are as follows

Scenario number 6 this scenario is mentioned beforeas a compromising solution for product C

Modelling and Simulation in Engineering 13

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 5 Interaction plot for product 119862 cost

14 Modelling and Simulation in Engineering

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

3

5

7

L

3

5

7

L

3

5

7

L

L

(b)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 6 Interaction plot for product 119862 service level

Scenario number 17 inventory policy is FI119877119901is equal

to 15 products and lead time is 7 days

55 Sensitivity Analysis As mentioned in the previous sec-tion all scenarios related to product 119860 prefer DF inventory

policy so it is concluded that DF policy is unique optimalpolicy while lead time includes values of 3 5 and 7 in selectedscenarios This situation implies that product 119860 is insensitiveto the delivery time It can be wise to decide lead time about 5days but there is no need for strict control on lead time while

Modelling and Simulation in Engineering 15

119877119901should be controlled strictly to prevent inventory in hand

from reaching lower than 20 productsIn the case of product 119861 (C1) is satisfied so there is a

unique scenario while there are four compromising solutionfor product 119862 In this situation based on VIKOR methodcompromising solutions have same value while interactionplot can be applied for investigation of existing differencebetween compromising solutions As mentioned before thebest scenario suggests FI inventory policy for this product butin compromising solutions there are both FI and DF policiesLead time varies from 3 to 7 days So this confusing situationis resolved by deeper analysis and applying interaction plotFigure 4 shows interaction plot for DOE factors and averageof inventory position as response variables

Figure 4 shows that with either DF or FI policy there isno significant difference among different lead times in theview point of inventory position but DF leads to less averageof inventory position With 119877

119901equaling 15 there is less

sensitivity to different lead times So configuring 119877119901which

is equal to 15 products the average of inventory positioncriterion will be robust against lead time variations

Figure 5 regards cost criterion and shows that FI andDF policies are equal and make least sensitivity related tolead time On the other hand three plots of 119877

119901against lead

time are parallel The parallel plots are interpreted as lackof relation between 119877

119901and lead time With 119877

119901equaling 15

increase in inventory cost could be mitigated when lead timesets to 3 days

In Figure 6 with 3 days for lead time there is no signifi-cant difference between inventory policies in the view pointof service level119877

119901and lead time are independent and the best

service level is related to FI policy which is also very sensitiveto 119877119901 As 119877

119901increases service level will grow According to

the interaction plot analyses optimal decision for product119862 is FI inventory policy 3 days lead time and 15 or moreproducts for 119877

119901

Decision about product 119863 is simple Based on the in-formation obtained from solutions it is recommended toemploy DF inventory policy with 119877

119901being equal to 15

products and 3 days for lead time

6 Conclusion

In this paper a simulation optimization framework is pro-posed for robust optimization of retailer inventory systemThis framework is less time consuming in comparison withmetaheuristic or SPSA approach it also provides robustsolution for multiple objective functions In this research alsotrend and cyclic change of customers demand are consideredfor more realistic view of problem Proposed framework con-sists of discrete event simulation and surrogate model In thisframework full factorial design of experiment is employedfor producing decision space Based on the three decisionfactors 27 different scenarios are produced and simulated Insimulation modelling multiresolution method is applied forsimulation of demand with highly dynamic pattern Becausethere are multiple criteria for inventory system performanceVIKORmethod is employed asmulticriteria decisionmaking

technique In addition robustness is considered as perfor-mance criterion and PCA is used for improving performanceof VIKOR method Finally developed framework gave thebest ranked and compromising solutions so interaction plotapplied for more investigation and sensitivity analysis ofobtained solutions

Due to the novelty of the proposed framework it isrecommended that future studies encompass optimization ofinventory system of integrated supply chain Also improve-ment of developed framework can be considered as futurestudy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers for theirsignificant roles in improvement of this research quality

References

[1] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[2] S Yin S XDingAHaghaniHHao andP Zhang ldquoA compar-ison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[3] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[4] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[5] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall International Series in Management New YorkNY USA 1963

[6] E Naddor Inventory Systems John Wiley amp Sons New YorkNY USA 1966

[7] R Peterson and E A Silver Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 2nd edition 1985

[8] T C E Cheng ldquoEconomic order quantity model with demand-dependent unit production cost and imperfect productionprocessesrdquo IIE Transactions vol 23 no 1 pp 23ndash28 1991

[9] S H Chen C CWang and A Ramer ldquoBackorder fuzzy inven-tory model under function principlerdquo Information Sciences vol95 no 1-2 pp 71ndash79 1996

[10] X Zhao J Xie and J C Wei ldquoThe value of early order com-mitment in a two-level supply chainrdquo European Journal of Op-erational Research vol 180 no 1 pp 194ndash214 2007

[11] R S M Lau J Xie and X Zhao ldquoEffects of inventory policyon supply chain performance a simulation study of critical

16 Modelling and Simulation in Engineering

decision parametersrdquo Computers and Industrial Engineeringvol 55 no 3 pp 620ndash633 2008

[12] J Xu and L Zhao ldquoA multi-objective decision-making modelwith fuzzy rough coefficients and its application to the inventoryproblemrdquo Information Sciences vol 180 no 5 pp 679ndash6962010

[13] F Hnaien X Delorme and A Dolgui ldquoMulti-objective opti-mization for inventory control in two-level assembly systemsunder uncertainty of lead timesrdquo Computers amp OperationsResearch vol 37 no 11 pp 1835ndash1843 2010

[14] H D Purnomo H M Wee and Y Praharsi ldquoTwo inventoryreview policies on supply chain configuration problemrdquo Com-puters and Industrial Engineering vol 63 no 2 pp 448ndash4552012

[15] B L Nelson ldquo50th anniversary article stochastic simulationresearch in management sciencerdquoManagement Science vol 50no 7 pp 855ndash868 2004

[16] J Boesel R O Bowden Jr F Glover J P Kelly and E WestwigldquoFuture of simulation optimizationrdquo inProceedings of theWinterSimulation Conference pp 1466ndash1469 Arlington Va USADecember 2001

[17] M C Fu ldquoOptimization for simulation theory vs practicerdquoINFORMS Journal on Computing vol 14 no 3 pp 192ndash2152002

[18] L Wang ldquoA hybrid genetic algorithm-neural network strategyfor simulation optimizationrdquoApplied Mathematics and Compu-tation vol 170 no 2 pp 1329ndash1343 2005

[19] B B Keskin S H Melouk and I L Meyer ldquoA simulation-optimization approach for integrated sourcing and inventorydecisionsrdquo Computers and Operations Research vol 37 no 9pp 1648ndash1661 2010

[20] E Mazhari J Zhao N Celik S Lee Y Son and L HeadldquoHybrid simulation and optimization-based design and oper-ation of integrated photovoltaic generation storage units andgridrdquo Simulation Modelling Practice and Theory vol 19 no 1pp 463ndash481 2011

[21] Q Duan and T W Liao ldquoOptimization of replenishment poli-cies for decentralized and centralized capacitated supply chainsunder various demandsrdquo International Journal of ProductionEconomics vol 142 no 1 pp 194ndash204 2013

[22] J C Spall ldquoMultivariate stochastic approximation usinga simultaneous perturbation gradient approximationrdquo IEEETransactions on Automatic Control vol 37 no 3 pp 332ndash3411992

[23] J C Spall ldquoImplementation of the simultaneous perturbationalgorithm for stochastic optimizationrdquo IEEE Transactions onAerospace and Electronic Systems vol 34 no 3 pp 817ndash8231998

[24] J C Spall ldquoStochastic optimization and the simultaneousperturbation methodrdquo in Proceedings of the Winter SimulationConference (WSC rsquo99) pp 101ndash109 December 1999

[25] J D Schwartz W Wang and D E Rivera ldquoSimulation-based optimization of process control policies for inventorymanagement in supply chainsrdquo Automatica vol 42 no 8 pp1311ndash1320 2006

[26] H G Neddermeijer G J V Oortmarssen and N P R DekkerldquoA framework for response surface methodology for simulationoptimizationrdquo in Proceedings of the Winter Simulation Confer-ence vol 1 Orlando Fla USA December 1999

[27] B Can and C Heavey ldquoA comparison of genetic programmingand artificial neural networks in metamodeling of discrete-event simulation modelsrdquo Computers amp Operations Researchvol 39 no 2 pp 424ndash436 2012

[28] A Azadeh Z S Faiz S M Asadzadeh and R Tavakkoli-Moghaddam ldquoAn integrated artificial neural network-comput-er simulation for optimization of complex tandem queuesystemsrdquoMathematics and Computers in Simulation vol 82 no4 pp 666ndash678 2011

[29] R Bornatico J Hussy A Witzig and L Guzzella ldquoSurrogatemodeling for the fast optimization of energy systemsrdquo Energyvol 57 pp 653ndash662 2013

[30] X Wan J F Pekny and G V Reklaitis ldquoSimulation-basedoptimizationwith surrogatemodels application to supply chainmanagementrdquoComputers and Chemical Engineering vol 29 no6 pp 1317ndash1328 2005

[31] W Shi Z Liu J Shang and Y Cui ldquoMulti-criteria robust designof a JIT-based cross-docking distribution center for an autoparts supply chainrdquo European Journal of Operational Researchvol 229 no 3 pp 695ndash706 2013

[32] G Dellino J P C Kleijnen and C Meloni ldquoRobust optimiza-tion in simulation taguchi and Response Surface Methodol-ogyrdquo International Journal of Production Economics vol 125 no1 pp 52ndash59 2010

[33] T Yang Y Wen and F Wang ldquoEvaluation of robustness of sup-ply chain information-sharing strategies using a hybrid Taguchiand multiple criteria decision-making methodrdquo InternationalJournal of Production Economics vol 134 no 2 pp 458ndash4662011

[34] M E Kuhl and J R Wilson ldquoModeling and simulating Poissonprocesses having trends or nontrigonometric cyclic effectsrdquoEuropean Journal of Operational Research vol 133 no 3 pp566ndash582 2001

[35] P Narayan and J Subramanian Inventory ManagementmdashPrinciples and Practices Excel Books New Delhi India 2008

[36] E A Silver and R Peterson Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 1985

[37] R G Brown Smoothing Forecasting and Prediction of DiscreteTime Series Courier 2004

[38] D C Montgomery Design and analysis of experiments JohnWiley amp Sons New York NY USA 6th edition 2005

[39] J Zhu ldquoData envelopment analysis vs principal componentanalysis an illustrative study of economic performance ofChinese citiesrdquo European Journal of Operational Research vol111 no 1 pp 50ndash61 1998

[40] I M Premachandra ldquoA note on DEA vs principal componentanalysis an improvement to Joe Zhursquos approachrdquo EuropeanJournal of Operational Research vol 132 no 3 pp 553ndash5602001

[41] D Slottje G W Scully G J Hirschberg and K Hayes Mea-suring the Quality of Life across Countries A MultidimensionalAnalysis Westview Press Boulder Colo USA 1991

[42] S Opricovic Multicriteria optimization of civil engineeringsystems [PhD thesis] Faculty of Civil Engineering BelgradeSerbia 1998

[43] P L Yu ldquoA class of solutions for group decision problemsrdquoManagement Science vol 19 no 8 pp 936ndash946 1973

Modelling and Simulation in Engineering 17

[44] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[45] S Opricovic and G H Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

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Page 9: Research Article Robust Inventory System Optimization ...downloads.hindawi.com/journals/mse/2014/305120.pdf · Robust Inventory System Optimization Based on Simulation and Multiple

Modelling and Simulation in Engineering 9

If (C1) is not satisfied scenarios 1198861015840 11988610158401015840 119886119872 are consid-ered 1198861015840 is the best solution and 11988610158401015840 119886119872 are compromisingsolutions with priority of 119876 on the other hand 1198861015840 11988610158401015840are considered if (C2) is not satisfied 119872 superscript isdetermined by (17)

119876(119886119872

) minus 119876 (1198861015840

) ≫1

(119869 minus 1) (17)

5 Numerical Result

Simulation process was executed on laptop with 18 GH CPUand 4GB of RAM Running of each replication with anima-tion and maximum speed takes 7 minutes and 35 secondsAs simulation of each scenario contains 10 replications andthere are 27 scenarios the total simulation time will be 2046minutes and 36 seconds

51 Multiresolution Result for Demand Modelling Result ofdemand modelling is presented for product 119861 because thisanalysis for other products is similar For this problempolynomial functions are applied for yearly and monthlydemand rates Equations (18) and (19) are estimated bynonlinear regression model In these functions 119905

119910and 119905119898are

time variables

119891 (119905119910) = 0008942 times 119905

3

119910minus 003884 times 119905

2

119910

+ 01867 times 119905119910minus 008018

(18)

and 95 confidence interval is presented in Table 4 for eachcoefficient of (18)

In this table coefficients of (18) are visible in the firstcolumn while lower bund and upper bund of each coefficientare presented in the second and third columns respectivelyThere is no interval including zero so it is concluded thatcoefficient estimation is valid

Estimation error for 119891(119905119910) is reported based on the sum

of square errors (SSE) which is 1504 times 10minus6 Consider

119891 (119905119898) = 4768 times 10

minus5

times 1199055

119898minus 0001692 times 119905

4

119898+ 00214

times 1199053

119898minus 01159 times 119905

2

119898+ 03474 times 119905

119898minus 02286

(19)

Monthly demand rate function that is indicated by 119891(119905119898)

and its coefficients confidence interval are presented inTable 5 SEE is equal to 00005825 Based on local businessinformation total demand of product 119861 is estimated to beabout 6822 units in five years According to the informationabout product 119861 Figure 2 shows final multiresolution resultfor this product

52 Simulation Parameters and Configuration For simula-tion of problem it is necessary to configure inventory policyparameters and variables We have three different policiesnamely FQ FI and DF Variables of FQ policy are reorderpoint (119877

119901) lead time (119871) and economic quantity orders (119876

119895119905)

First two variables are configured based on Table 2 and thirdvariable is derived fromWilson formula according to average

Table 4 95 confidence interval for estimated coefficient of 119891(119905119910)

Coefficients Lower bund Upper bund0008942 0004835 001305minus003884 minus007604 minus000165201867 008654 0287minus008018 minus01568 minus0003523

Table 5 95 confidence interval for estimated coefficient of 119891(119905119898)

Coefficients Lower bund Upper bund4768 times 10minus5 4451 times 10minus5 5086 times 10minus5

minus0001692 minus0001795 minus00015900214 00202 00226minus01159 minus01221 minus0109703474 0334 03608minus02286 minus02371 minus022

Table 6 Estimation of distribution functions of demand and reor-ders quantity

Product type Distribution function 119875 value Order quantity119860 278 lowast BETA(113 328) gt015 100119861 GAMMA(678 118) gt015 92119862 EXPO(818) gt015 100119863 446 lowast BETA(0957 321) gt015 115

of demand holding cost (119862ℎ) and ordering cost (119862

119903) The

calculated values are 57 71 72 and 72 for product 119860 119861 119862and119863 respectively

Orders quantity is unknown for FI policy and is cal-culated based on probability distribution function whichis obtained by demand simulation of each product for 20daysrsquo time interval Distribution fitting is performed by inputanalyser of Arena software and graphical result for product 119861is shown in Figure 3 Results for other products are reportedin Table 6

In this table fitted probability distribution function foreach product is presented in second column while 119875 valueof fitting which is greater than 015 for fitted distributiongrantees goodness of fitting in 95 confidence level Finallyaccording to fitted distribution functions for FI policyproper ordering amount of each product is presented in thefourth column

DF policy has two unknown parameters namely 120572 and120573 The first parameter is weight of current real demand in (4)and second one is the weight of current trend in (5) Theseparameters are adjusted by simulation of five years demandDemand for the entire four products simultaneously is con-sidered and an average of lost sale percentage is consideredas response variable Different values of response variables arereported in Table 7 According to the different values of 120572 and120573 the best values for 120572 and 120573 are 09 and 01 respectivelywhich results in minimum lost percentage in average

10 Modelling and Simulation in Engineering

Table 7 Average of lost sale percentage for different values of 120572 and 120573

120572 and 120573 values 01 02 03 04 05 06 07 08 09Response to 120572 0380 0356 0334 0320 0306 0292 0284 0276 0268Response to 120573 0296 0302 0306 0306 0306 0304 0305 0304 0302

50

100

150

200

250

300

350Demand fit for each month

Dem

and

Fitted valueReal demand

00 10 20 30 40 50 60

Month

Figure 2 Comparison between real and estimated demand

100 200 300 400 500 600 700

50

100

150

200

Distribution summary

Distribution gammaExpression minus0001 + GAMM(00)Square error 0006825

Chi-square testNumber of intervals = 5Degrees of freedom = 2Test statistic = 148

Kolmogorov-Smirnov testTest statistic = 00733

Corresponding P value = 0483

Corresponding P value gt 015

Figure 3 Distribution function of customer demand for reorderinterval of product 119861

53Weight Assignment Asmentioned before PCA is appliedforweight assignment For this purposeDOE result is neededin the form of decision matrix that is shown in Table 8 Inthis table four criteria cost average of inventory positionservice level and robustness are evaluated for each scenarioof product 119861 Decision matrix should be scaleless that isperformed by linear method Then the result of eigenvalues

Table 8 Decision matrix for product 119861

Scenarionumber Cost Average of

inventory positionServicelevel

Robustness

1 32984000 4352 76 62 29214710 397514 79 33 37114636 339623 69 24 34840720 410948 71 55 36820570 344738 72 66 39191837 331091 67 27 37037180 391012 68 48 40994680 298712 64 69 41407045 319678 63 110 28282930 524462 83 711 26422145 489243 85 512 31023829 416049 79 313 30395190 487146 79 614 35393680 395812 74 515 32838055 395646 77 316 33285615 463341 76 617 39652020 335668 66 718 35269609 383944 73 219 26227680 625612 87 720 25565820 586688 88 621 28711069 49449 85 422 27301580 585472 85 723 34016415 470961 78 724 30601546 463975 82 325 29752815 560421 82 726 41024410 382994 68 627 31604741 443053 78 3

is calculated and shown in Table 9 Also result of principalcomponents is presented in Table 10

As it is visible in Table 9 cumulative proportion ofthe first and second components is 0966 which is greaterthan 095 so the first two components are considered asprincipal components Their relative weights are 078 and0186 respectively Final weights are normalized and resultsare reported in Table 11

54 Ranking and Selection of Scenarios Weights of criteriaobtained by PCA method in addition to decision matrix areinput of VIKOR method 119878 119876 and 119877 values are calculatedbased on (14) and (15) while strategic weight indicated byV is equal to 05 so individual and group utility have same

Modelling and Simulation in Engineering 11

Table 9 Result of eigenvalues

Eigenvalues 31204 07428 01111 00257Proportion 0780 0186 0028 0006Cumulative 0780 0966 0994 1000

Table 10 Result of principle components

Variables PC1 PC2 PC3 PC4Inventory 0543 0089 minus0818 0167Cost minus0538 minus0277 minus0511 minus0610Service level minus0551 minus0210 minus0230 0774Robustness 0335 minus0933 0126 0023

Table 11 Weights of criteria for different products

Criteria Products119860 119861 119862 119863

Average of inventory position 01012 00590 01942 00462Cost 02852 01836 01742 03081Service level 01195 01392 00966 01206Robustness 04941 06182 05350 05251

importance Calculated values are reported in Table 12 forentire scenario as VIKOR output

Product 119861 According to Table 12 values of 119877 119878 and 119876 arezero for scenario 18 Regarding (C1) (Acceptable Advantage)the difference between the best scenario and second rankedscenario should be more than 00038 The second rankedscenario regarding 119876 value is scenario 3 and its value is0051044 that satisfies both (C1) and (C2) Final result forproduct 119861 is unique scenario that is number 18 and consistsof the following configuration Inventory policy is DFwith119877

119901

being equal to 30 products and 7 days lead time According todecision matrix which is shown in Table 8 the performanceof this scenario for product 119861 results in 383944 products asaverage of inventory position and expected cost is equal to35269609 for product life cycle 73 of customer demandwillbe satisfied and in the case of increasing demand intensityand variation 2 decrease is expected in service level so itwill change into 71 Analyses for other products are similarso only final results are presented in this paper

Product 119860 In this case VIKORmethod ranks scenario num-ber 15 as the best one This scenario consists of DF inventorypolicy 20 products for 119877

119901and 5 days for lead time Perfor-

mance of this scenario results in 374778 products as averageof inventory position Expected cost is 31603294 and servicelevel will be 66 This scenario is robust against demandintensification and variation because changes in demandaffect service level as small as 1 But scenario number 15 doesnot satisfy (C1) so compromising solutions are consideredAlthough the best scenario is number 15 VIKOR methodimplies that compromising solutions have the same values

Table 12 VIKOR output for product 119861

Run number 119878 119877 119876

1 0634311 0796269 0715292 0003184 0185077 0094133 0059804 0042283 00510444 0536946 0592539 05647425 0718862 0796269 07575656 0116278 0089887 01030827 0428814 0388808 04088118 086078 0796269 08285259 002043 0140653 008054110 0677386 1 083869311 0266254 0592539 042939612 0043962 0185077 01145213 0571695 0796269 068398214 0515097 0592539 055381815 009173 0185077 013840416 0648693 0796269 072248117 1 1 118 0 0 019 0630698 1 081534920 04239 0796269 061008421 0139527 0388808 026416722 06581 1 08290523 0819338 1 090966924 0022245 0185077 010366125 0726183 1 086309126 0849516 0796269 082289327 0072768 0185077 0128922

So in real situation compromising solution can be substi-tuted This solution regarding their priority is as follows

Scenario number 24 inventory policy is DF 119877119901is

equal to 30 products and lead time is 5 daysScenario number 18 inventory policy is DF 119877

119901is

equal to 20 products and lead time is 7 daysScenario number 12 inventory policy isDF119877

119901is equal

to 20 products and lead time is 3 days

Product 119862 For this product scenario number 1 is ranked asthe best one This scenario consists of FI inventory policy 119877

119901

is 15 products and lead time is 3 days Result of this scenario is4387 products as the average of inventory position Expectedcost is 66143000 and service level will be 61 with robustnessof 1 Compromising solutions are as follows

Scenario number 6 inventory policy is DF 119877119901is 15

products and lead time is 5 daysScenario number 3 inventory policy is DF 119877

119901is 15

products and lead time is 3 daysScenario number 7 inventory policy is FI 119877

119901is 15

products and lead time is 7 days

12 Modelling and Simulation in Engineering

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 4 Interaction plot for product 119862 average of inventory position

Scenario number 4 inventory policy is FI 119877119901is 15

products and lead time is 5 days

Product119863 Scenario number 3 has the best rank for this prod-uct This scenario consists of DF inventory policy 119877

119901is 15

products and lead time is 3 days This scenario leads to

363896 products as average of inventory position with62648337 expected cost and 61 service level with 1robustness Compromising solutions are as follows

Scenario number 6 this scenario is mentioned beforeas a compromising solution for product C

Modelling and Simulation in Engineering 13

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 5 Interaction plot for product 119862 cost

14 Modelling and Simulation in Engineering

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

3

5

7

L

3

5

7

L

3

5

7

L

L

(b)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 6 Interaction plot for product 119862 service level

Scenario number 17 inventory policy is FI119877119901is equal

to 15 products and lead time is 7 days

55 Sensitivity Analysis As mentioned in the previous sec-tion all scenarios related to product 119860 prefer DF inventory

policy so it is concluded that DF policy is unique optimalpolicy while lead time includes values of 3 5 and 7 in selectedscenarios This situation implies that product 119860 is insensitiveto the delivery time It can be wise to decide lead time about 5days but there is no need for strict control on lead time while

Modelling and Simulation in Engineering 15

119877119901should be controlled strictly to prevent inventory in hand

from reaching lower than 20 productsIn the case of product 119861 (C1) is satisfied so there is a

unique scenario while there are four compromising solutionfor product 119862 In this situation based on VIKOR methodcompromising solutions have same value while interactionplot can be applied for investigation of existing differencebetween compromising solutions As mentioned before thebest scenario suggests FI inventory policy for this product butin compromising solutions there are both FI and DF policiesLead time varies from 3 to 7 days So this confusing situationis resolved by deeper analysis and applying interaction plotFigure 4 shows interaction plot for DOE factors and averageof inventory position as response variables

Figure 4 shows that with either DF or FI policy there isno significant difference among different lead times in theview point of inventory position but DF leads to less averageof inventory position With 119877

119901equaling 15 there is less

sensitivity to different lead times So configuring 119877119901which

is equal to 15 products the average of inventory positioncriterion will be robust against lead time variations

Figure 5 regards cost criterion and shows that FI andDF policies are equal and make least sensitivity related tolead time On the other hand three plots of 119877

119901against lead

time are parallel The parallel plots are interpreted as lackof relation between 119877

119901and lead time With 119877

119901equaling 15

increase in inventory cost could be mitigated when lead timesets to 3 days

In Figure 6 with 3 days for lead time there is no signifi-cant difference between inventory policies in the view pointof service level119877

119901and lead time are independent and the best

service level is related to FI policy which is also very sensitiveto 119877119901 As 119877

119901increases service level will grow According to

the interaction plot analyses optimal decision for product119862 is FI inventory policy 3 days lead time and 15 or moreproducts for 119877

119901

Decision about product 119863 is simple Based on the in-formation obtained from solutions it is recommended toemploy DF inventory policy with 119877

119901being equal to 15

products and 3 days for lead time

6 Conclusion

In this paper a simulation optimization framework is pro-posed for robust optimization of retailer inventory systemThis framework is less time consuming in comparison withmetaheuristic or SPSA approach it also provides robustsolution for multiple objective functions In this research alsotrend and cyclic change of customers demand are consideredfor more realistic view of problem Proposed framework con-sists of discrete event simulation and surrogate model In thisframework full factorial design of experiment is employedfor producing decision space Based on the three decisionfactors 27 different scenarios are produced and simulated Insimulation modelling multiresolution method is applied forsimulation of demand with highly dynamic pattern Becausethere are multiple criteria for inventory system performanceVIKORmethod is employed asmulticriteria decisionmaking

technique In addition robustness is considered as perfor-mance criterion and PCA is used for improving performanceof VIKOR method Finally developed framework gave thebest ranked and compromising solutions so interaction plotapplied for more investigation and sensitivity analysis ofobtained solutions

Due to the novelty of the proposed framework it isrecommended that future studies encompass optimization ofinventory system of integrated supply chain Also improve-ment of developed framework can be considered as futurestudy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers for theirsignificant roles in improvement of this research quality

References

[1] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[2] S Yin S XDingAHaghaniHHao andP Zhang ldquoA compar-ison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[3] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[4] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[5] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall International Series in Management New YorkNY USA 1963

[6] E Naddor Inventory Systems John Wiley amp Sons New YorkNY USA 1966

[7] R Peterson and E A Silver Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 2nd edition 1985

[8] T C E Cheng ldquoEconomic order quantity model with demand-dependent unit production cost and imperfect productionprocessesrdquo IIE Transactions vol 23 no 1 pp 23ndash28 1991

[9] S H Chen C CWang and A Ramer ldquoBackorder fuzzy inven-tory model under function principlerdquo Information Sciences vol95 no 1-2 pp 71ndash79 1996

[10] X Zhao J Xie and J C Wei ldquoThe value of early order com-mitment in a two-level supply chainrdquo European Journal of Op-erational Research vol 180 no 1 pp 194ndash214 2007

[11] R S M Lau J Xie and X Zhao ldquoEffects of inventory policyon supply chain performance a simulation study of critical

16 Modelling and Simulation in Engineering

decision parametersrdquo Computers and Industrial Engineeringvol 55 no 3 pp 620ndash633 2008

[12] J Xu and L Zhao ldquoA multi-objective decision-making modelwith fuzzy rough coefficients and its application to the inventoryproblemrdquo Information Sciences vol 180 no 5 pp 679ndash6962010

[13] F Hnaien X Delorme and A Dolgui ldquoMulti-objective opti-mization for inventory control in two-level assembly systemsunder uncertainty of lead timesrdquo Computers amp OperationsResearch vol 37 no 11 pp 1835ndash1843 2010

[14] H D Purnomo H M Wee and Y Praharsi ldquoTwo inventoryreview policies on supply chain configuration problemrdquo Com-puters and Industrial Engineering vol 63 no 2 pp 448ndash4552012

[15] B L Nelson ldquo50th anniversary article stochastic simulationresearch in management sciencerdquoManagement Science vol 50no 7 pp 855ndash868 2004

[16] J Boesel R O Bowden Jr F Glover J P Kelly and E WestwigldquoFuture of simulation optimizationrdquo inProceedings of theWinterSimulation Conference pp 1466ndash1469 Arlington Va USADecember 2001

[17] M C Fu ldquoOptimization for simulation theory vs practicerdquoINFORMS Journal on Computing vol 14 no 3 pp 192ndash2152002

[18] L Wang ldquoA hybrid genetic algorithm-neural network strategyfor simulation optimizationrdquoApplied Mathematics and Compu-tation vol 170 no 2 pp 1329ndash1343 2005

[19] B B Keskin S H Melouk and I L Meyer ldquoA simulation-optimization approach for integrated sourcing and inventorydecisionsrdquo Computers and Operations Research vol 37 no 9pp 1648ndash1661 2010

[20] E Mazhari J Zhao N Celik S Lee Y Son and L HeadldquoHybrid simulation and optimization-based design and oper-ation of integrated photovoltaic generation storage units andgridrdquo Simulation Modelling Practice and Theory vol 19 no 1pp 463ndash481 2011

[21] Q Duan and T W Liao ldquoOptimization of replenishment poli-cies for decentralized and centralized capacitated supply chainsunder various demandsrdquo International Journal of ProductionEconomics vol 142 no 1 pp 194ndash204 2013

[22] J C Spall ldquoMultivariate stochastic approximation usinga simultaneous perturbation gradient approximationrdquo IEEETransactions on Automatic Control vol 37 no 3 pp 332ndash3411992

[23] J C Spall ldquoImplementation of the simultaneous perturbationalgorithm for stochastic optimizationrdquo IEEE Transactions onAerospace and Electronic Systems vol 34 no 3 pp 817ndash8231998

[24] J C Spall ldquoStochastic optimization and the simultaneousperturbation methodrdquo in Proceedings of the Winter SimulationConference (WSC rsquo99) pp 101ndash109 December 1999

[25] J D Schwartz W Wang and D E Rivera ldquoSimulation-based optimization of process control policies for inventorymanagement in supply chainsrdquo Automatica vol 42 no 8 pp1311ndash1320 2006

[26] H G Neddermeijer G J V Oortmarssen and N P R DekkerldquoA framework for response surface methodology for simulationoptimizationrdquo in Proceedings of the Winter Simulation Confer-ence vol 1 Orlando Fla USA December 1999

[27] B Can and C Heavey ldquoA comparison of genetic programmingand artificial neural networks in metamodeling of discrete-event simulation modelsrdquo Computers amp Operations Researchvol 39 no 2 pp 424ndash436 2012

[28] A Azadeh Z S Faiz S M Asadzadeh and R Tavakkoli-Moghaddam ldquoAn integrated artificial neural network-comput-er simulation for optimization of complex tandem queuesystemsrdquoMathematics and Computers in Simulation vol 82 no4 pp 666ndash678 2011

[29] R Bornatico J Hussy A Witzig and L Guzzella ldquoSurrogatemodeling for the fast optimization of energy systemsrdquo Energyvol 57 pp 653ndash662 2013

[30] X Wan J F Pekny and G V Reklaitis ldquoSimulation-basedoptimizationwith surrogatemodels application to supply chainmanagementrdquoComputers and Chemical Engineering vol 29 no6 pp 1317ndash1328 2005

[31] W Shi Z Liu J Shang and Y Cui ldquoMulti-criteria robust designof a JIT-based cross-docking distribution center for an autoparts supply chainrdquo European Journal of Operational Researchvol 229 no 3 pp 695ndash706 2013

[32] G Dellino J P C Kleijnen and C Meloni ldquoRobust optimiza-tion in simulation taguchi and Response Surface Methodol-ogyrdquo International Journal of Production Economics vol 125 no1 pp 52ndash59 2010

[33] T Yang Y Wen and F Wang ldquoEvaluation of robustness of sup-ply chain information-sharing strategies using a hybrid Taguchiand multiple criteria decision-making methodrdquo InternationalJournal of Production Economics vol 134 no 2 pp 458ndash4662011

[34] M E Kuhl and J R Wilson ldquoModeling and simulating Poissonprocesses having trends or nontrigonometric cyclic effectsrdquoEuropean Journal of Operational Research vol 133 no 3 pp566ndash582 2001

[35] P Narayan and J Subramanian Inventory ManagementmdashPrinciples and Practices Excel Books New Delhi India 2008

[36] E A Silver and R Peterson Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 1985

[37] R G Brown Smoothing Forecasting and Prediction of DiscreteTime Series Courier 2004

[38] D C Montgomery Design and analysis of experiments JohnWiley amp Sons New York NY USA 6th edition 2005

[39] J Zhu ldquoData envelopment analysis vs principal componentanalysis an illustrative study of economic performance ofChinese citiesrdquo European Journal of Operational Research vol111 no 1 pp 50ndash61 1998

[40] I M Premachandra ldquoA note on DEA vs principal componentanalysis an improvement to Joe Zhursquos approachrdquo EuropeanJournal of Operational Research vol 132 no 3 pp 553ndash5602001

[41] D Slottje G W Scully G J Hirschberg and K Hayes Mea-suring the Quality of Life across Countries A MultidimensionalAnalysis Westview Press Boulder Colo USA 1991

[42] S Opricovic Multicriteria optimization of civil engineeringsystems [PhD thesis] Faculty of Civil Engineering BelgradeSerbia 1998

[43] P L Yu ldquoA class of solutions for group decision problemsrdquoManagement Science vol 19 no 8 pp 936ndash946 1973

Modelling and Simulation in Engineering 17

[44] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[45] S Opricovic and G H Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

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Page 10: Research Article Robust Inventory System Optimization ...downloads.hindawi.com/journals/mse/2014/305120.pdf · Robust Inventory System Optimization Based on Simulation and Multiple

10 Modelling and Simulation in Engineering

Table 7 Average of lost sale percentage for different values of 120572 and 120573

120572 and 120573 values 01 02 03 04 05 06 07 08 09Response to 120572 0380 0356 0334 0320 0306 0292 0284 0276 0268Response to 120573 0296 0302 0306 0306 0306 0304 0305 0304 0302

50

100

150

200

250

300

350Demand fit for each month

Dem

and

Fitted valueReal demand

00 10 20 30 40 50 60

Month

Figure 2 Comparison between real and estimated demand

100 200 300 400 500 600 700

50

100

150

200

Distribution summary

Distribution gammaExpression minus0001 + GAMM(00)Square error 0006825

Chi-square testNumber of intervals = 5Degrees of freedom = 2Test statistic = 148

Kolmogorov-Smirnov testTest statistic = 00733

Corresponding P value = 0483

Corresponding P value gt 015

Figure 3 Distribution function of customer demand for reorderinterval of product 119861

53Weight Assignment Asmentioned before PCA is appliedforweight assignment For this purposeDOE result is neededin the form of decision matrix that is shown in Table 8 Inthis table four criteria cost average of inventory positionservice level and robustness are evaluated for each scenarioof product 119861 Decision matrix should be scaleless that isperformed by linear method Then the result of eigenvalues

Table 8 Decision matrix for product 119861

Scenarionumber Cost Average of

inventory positionServicelevel

Robustness

1 32984000 4352 76 62 29214710 397514 79 33 37114636 339623 69 24 34840720 410948 71 55 36820570 344738 72 66 39191837 331091 67 27 37037180 391012 68 48 40994680 298712 64 69 41407045 319678 63 110 28282930 524462 83 711 26422145 489243 85 512 31023829 416049 79 313 30395190 487146 79 614 35393680 395812 74 515 32838055 395646 77 316 33285615 463341 76 617 39652020 335668 66 718 35269609 383944 73 219 26227680 625612 87 720 25565820 586688 88 621 28711069 49449 85 422 27301580 585472 85 723 34016415 470961 78 724 30601546 463975 82 325 29752815 560421 82 726 41024410 382994 68 627 31604741 443053 78 3

is calculated and shown in Table 9 Also result of principalcomponents is presented in Table 10

As it is visible in Table 9 cumulative proportion ofthe first and second components is 0966 which is greaterthan 095 so the first two components are considered asprincipal components Their relative weights are 078 and0186 respectively Final weights are normalized and resultsare reported in Table 11

54 Ranking and Selection of Scenarios Weights of criteriaobtained by PCA method in addition to decision matrix areinput of VIKOR method 119878 119876 and 119877 values are calculatedbased on (14) and (15) while strategic weight indicated byV is equal to 05 so individual and group utility have same

Modelling and Simulation in Engineering 11

Table 9 Result of eigenvalues

Eigenvalues 31204 07428 01111 00257Proportion 0780 0186 0028 0006Cumulative 0780 0966 0994 1000

Table 10 Result of principle components

Variables PC1 PC2 PC3 PC4Inventory 0543 0089 minus0818 0167Cost minus0538 minus0277 minus0511 minus0610Service level minus0551 minus0210 minus0230 0774Robustness 0335 minus0933 0126 0023

Table 11 Weights of criteria for different products

Criteria Products119860 119861 119862 119863

Average of inventory position 01012 00590 01942 00462Cost 02852 01836 01742 03081Service level 01195 01392 00966 01206Robustness 04941 06182 05350 05251

importance Calculated values are reported in Table 12 forentire scenario as VIKOR output

Product 119861 According to Table 12 values of 119877 119878 and 119876 arezero for scenario 18 Regarding (C1) (Acceptable Advantage)the difference between the best scenario and second rankedscenario should be more than 00038 The second rankedscenario regarding 119876 value is scenario 3 and its value is0051044 that satisfies both (C1) and (C2) Final result forproduct 119861 is unique scenario that is number 18 and consistsof the following configuration Inventory policy is DFwith119877

119901

being equal to 30 products and 7 days lead time According todecision matrix which is shown in Table 8 the performanceof this scenario for product 119861 results in 383944 products asaverage of inventory position and expected cost is equal to35269609 for product life cycle 73 of customer demandwillbe satisfied and in the case of increasing demand intensityand variation 2 decrease is expected in service level so itwill change into 71 Analyses for other products are similarso only final results are presented in this paper

Product 119860 In this case VIKORmethod ranks scenario num-ber 15 as the best one This scenario consists of DF inventorypolicy 20 products for 119877

119901and 5 days for lead time Perfor-

mance of this scenario results in 374778 products as averageof inventory position Expected cost is 31603294 and servicelevel will be 66 This scenario is robust against demandintensification and variation because changes in demandaffect service level as small as 1 But scenario number 15 doesnot satisfy (C1) so compromising solutions are consideredAlthough the best scenario is number 15 VIKOR methodimplies that compromising solutions have the same values

Table 12 VIKOR output for product 119861

Run number 119878 119877 119876

1 0634311 0796269 0715292 0003184 0185077 0094133 0059804 0042283 00510444 0536946 0592539 05647425 0718862 0796269 07575656 0116278 0089887 01030827 0428814 0388808 04088118 086078 0796269 08285259 002043 0140653 008054110 0677386 1 083869311 0266254 0592539 042939612 0043962 0185077 01145213 0571695 0796269 068398214 0515097 0592539 055381815 009173 0185077 013840416 0648693 0796269 072248117 1 1 118 0 0 019 0630698 1 081534920 04239 0796269 061008421 0139527 0388808 026416722 06581 1 08290523 0819338 1 090966924 0022245 0185077 010366125 0726183 1 086309126 0849516 0796269 082289327 0072768 0185077 0128922

So in real situation compromising solution can be substi-tuted This solution regarding their priority is as follows

Scenario number 24 inventory policy is DF 119877119901is

equal to 30 products and lead time is 5 daysScenario number 18 inventory policy is DF 119877

119901is

equal to 20 products and lead time is 7 daysScenario number 12 inventory policy isDF119877

119901is equal

to 20 products and lead time is 3 days

Product 119862 For this product scenario number 1 is ranked asthe best one This scenario consists of FI inventory policy 119877

119901

is 15 products and lead time is 3 days Result of this scenario is4387 products as the average of inventory position Expectedcost is 66143000 and service level will be 61 with robustnessof 1 Compromising solutions are as follows

Scenario number 6 inventory policy is DF 119877119901is 15

products and lead time is 5 daysScenario number 3 inventory policy is DF 119877

119901is 15

products and lead time is 3 daysScenario number 7 inventory policy is FI 119877

119901is 15

products and lead time is 7 days

12 Modelling and Simulation in Engineering

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 4 Interaction plot for product 119862 average of inventory position

Scenario number 4 inventory policy is FI 119877119901is 15

products and lead time is 5 days

Product119863 Scenario number 3 has the best rank for this prod-uct This scenario consists of DF inventory policy 119877

119901is 15

products and lead time is 3 days This scenario leads to

363896 products as average of inventory position with62648337 expected cost and 61 service level with 1robustness Compromising solutions are as follows

Scenario number 6 this scenario is mentioned beforeas a compromising solution for product C

Modelling and Simulation in Engineering 13

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 5 Interaction plot for product 119862 cost

14 Modelling and Simulation in Engineering

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

3

5

7

L

3

5

7

L

3

5

7

L

L

(b)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 6 Interaction plot for product 119862 service level

Scenario number 17 inventory policy is FI119877119901is equal

to 15 products and lead time is 7 days

55 Sensitivity Analysis As mentioned in the previous sec-tion all scenarios related to product 119860 prefer DF inventory

policy so it is concluded that DF policy is unique optimalpolicy while lead time includes values of 3 5 and 7 in selectedscenarios This situation implies that product 119860 is insensitiveto the delivery time It can be wise to decide lead time about 5days but there is no need for strict control on lead time while

Modelling and Simulation in Engineering 15

119877119901should be controlled strictly to prevent inventory in hand

from reaching lower than 20 productsIn the case of product 119861 (C1) is satisfied so there is a

unique scenario while there are four compromising solutionfor product 119862 In this situation based on VIKOR methodcompromising solutions have same value while interactionplot can be applied for investigation of existing differencebetween compromising solutions As mentioned before thebest scenario suggests FI inventory policy for this product butin compromising solutions there are both FI and DF policiesLead time varies from 3 to 7 days So this confusing situationis resolved by deeper analysis and applying interaction plotFigure 4 shows interaction plot for DOE factors and averageof inventory position as response variables

Figure 4 shows that with either DF or FI policy there isno significant difference among different lead times in theview point of inventory position but DF leads to less averageof inventory position With 119877

119901equaling 15 there is less

sensitivity to different lead times So configuring 119877119901which

is equal to 15 products the average of inventory positioncriterion will be robust against lead time variations

Figure 5 regards cost criterion and shows that FI andDF policies are equal and make least sensitivity related tolead time On the other hand three plots of 119877

119901against lead

time are parallel The parallel plots are interpreted as lackof relation between 119877

119901and lead time With 119877

119901equaling 15

increase in inventory cost could be mitigated when lead timesets to 3 days

In Figure 6 with 3 days for lead time there is no signifi-cant difference between inventory policies in the view pointof service level119877

119901and lead time are independent and the best

service level is related to FI policy which is also very sensitiveto 119877119901 As 119877

119901increases service level will grow According to

the interaction plot analyses optimal decision for product119862 is FI inventory policy 3 days lead time and 15 or moreproducts for 119877

119901

Decision about product 119863 is simple Based on the in-formation obtained from solutions it is recommended toemploy DF inventory policy with 119877

119901being equal to 15

products and 3 days for lead time

6 Conclusion

In this paper a simulation optimization framework is pro-posed for robust optimization of retailer inventory systemThis framework is less time consuming in comparison withmetaheuristic or SPSA approach it also provides robustsolution for multiple objective functions In this research alsotrend and cyclic change of customers demand are consideredfor more realistic view of problem Proposed framework con-sists of discrete event simulation and surrogate model In thisframework full factorial design of experiment is employedfor producing decision space Based on the three decisionfactors 27 different scenarios are produced and simulated Insimulation modelling multiresolution method is applied forsimulation of demand with highly dynamic pattern Becausethere are multiple criteria for inventory system performanceVIKORmethod is employed asmulticriteria decisionmaking

technique In addition robustness is considered as perfor-mance criterion and PCA is used for improving performanceof VIKOR method Finally developed framework gave thebest ranked and compromising solutions so interaction plotapplied for more investigation and sensitivity analysis ofobtained solutions

Due to the novelty of the proposed framework it isrecommended that future studies encompass optimization ofinventory system of integrated supply chain Also improve-ment of developed framework can be considered as futurestudy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers for theirsignificant roles in improvement of this research quality

References

[1] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[2] S Yin S XDingAHaghaniHHao andP Zhang ldquoA compar-ison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[3] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[4] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[5] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall International Series in Management New YorkNY USA 1963

[6] E Naddor Inventory Systems John Wiley amp Sons New YorkNY USA 1966

[7] R Peterson and E A Silver Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 2nd edition 1985

[8] T C E Cheng ldquoEconomic order quantity model with demand-dependent unit production cost and imperfect productionprocessesrdquo IIE Transactions vol 23 no 1 pp 23ndash28 1991

[9] S H Chen C CWang and A Ramer ldquoBackorder fuzzy inven-tory model under function principlerdquo Information Sciences vol95 no 1-2 pp 71ndash79 1996

[10] X Zhao J Xie and J C Wei ldquoThe value of early order com-mitment in a two-level supply chainrdquo European Journal of Op-erational Research vol 180 no 1 pp 194ndash214 2007

[11] R S M Lau J Xie and X Zhao ldquoEffects of inventory policyon supply chain performance a simulation study of critical

16 Modelling and Simulation in Engineering

decision parametersrdquo Computers and Industrial Engineeringvol 55 no 3 pp 620ndash633 2008

[12] J Xu and L Zhao ldquoA multi-objective decision-making modelwith fuzzy rough coefficients and its application to the inventoryproblemrdquo Information Sciences vol 180 no 5 pp 679ndash6962010

[13] F Hnaien X Delorme and A Dolgui ldquoMulti-objective opti-mization for inventory control in two-level assembly systemsunder uncertainty of lead timesrdquo Computers amp OperationsResearch vol 37 no 11 pp 1835ndash1843 2010

[14] H D Purnomo H M Wee and Y Praharsi ldquoTwo inventoryreview policies on supply chain configuration problemrdquo Com-puters and Industrial Engineering vol 63 no 2 pp 448ndash4552012

[15] B L Nelson ldquo50th anniversary article stochastic simulationresearch in management sciencerdquoManagement Science vol 50no 7 pp 855ndash868 2004

[16] J Boesel R O Bowden Jr F Glover J P Kelly and E WestwigldquoFuture of simulation optimizationrdquo inProceedings of theWinterSimulation Conference pp 1466ndash1469 Arlington Va USADecember 2001

[17] M C Fu ldquoOptimization for simulation theory vs practicerdquoINFORMS Journal on Computing vol 14 no 3 pp 192ndash2152002

[18] L Wang ldquoA hybrid genetic algorithm-neural network strategyfor simulation optimizationrdquoApplied Mathematics and Compu-tation vol 170 no 2 pp 1329ndash1343 2005

[19] B B Keskin S H Melouk and I L Meyer ldquoA simulation-optimization approach for integrated sourcing and inventorydecisionsrdquo Computers and Operations Research vol 37 no 9pp 1648ndash1661 2010

[20] E Mazhari J Zhao N Celik S Lee Y Son and L HeadldquoHybrid simulation and optimization-based design and oper-ation of integrated photovoltaic generation storage units andgridrdquo Simulation Modelling Practice and Theory vol 19 no 1pp 463ndash481 2011

[21] Q Duan and T W Liao ldquoOptimization of replenishment poli-cies for decentralized and centralized capacitated supply chainsunder various demandsrdquo International Journal of ProductionEconomics vol 142 no 1 pp 194ndash204 2013

[22] J C Spall ldquoMultivariate stochastic approximation usinga simultaneous perturbation gradient approximationrdquo IEEETransactions on Automatic Control vol 37 no 3 pp 332ndash3411992

[23] J C Spall ldquoImplementation of the simultaneous perturbationalgorithm for stochastic optimizationrdquo IEEE Transactions onAerospace and Electronic Systems vol 34 no 3 pp 817ndash8231998

[24] J C Spall ldquoStochastic optimization and the simultaneousperturbation methodrdquo in Proceedings of the Winter SimulationConference (WSC rsquo99) pp 101ndash109 December 1999

[25] J D Schwartz W Wang and D E Rivera ldquoSimulation-based optimization of process control policies for inventorymanagement in supply chainsrdquo Automatica vol 42 no 8 pp1311ndash1320 2006

[26] H G Neddermeijer G J V Oortmarssen and N P R DekkerldquoA framework for response surface methodology for simulationoptimizationrdquo in Proceedings of the Winter Simulation Confer-ence vol 1 Orlando Fla USA December 1999

[27] B Can and C Heavey ldquoA comparison of genetic programmingand artificial neural networks in metamodeling of discrete-event simulation modelsrdquo Computers amp Operations Researchvol 39 no 2 pp 424ndash436 2012

[28] A Azadeh Z S Faiz S M Asadzadeh and R Tavakkoli-Moghaddam ldquoAn integrated artificial neural network-comput-er simulation for optimization of complex tandem queuesystemsrdquoMathematics and Computers in Simulation vol 82 no4 pp 666ndash678 2011

[29] R Bornatico J Hussy A Witzig and L Guzzella ldquoSurrogatemodeling for the fast optimization of energy systemsrdquo Energyvol 57 pp 653ndash662 2013

[30] X Wan J F Pekny and G V Reklaitis ldquoSimulation-basedoptimizationwith surrogatemodels application to supply chainmanagementrdquoComputers and Chemical Engineering vol 29 no6 pp 1317ndash1328 2005

[31] W Shi Z Liu J Shang and Y Cui ldquoMulti-criteria robust designof a JIT-based cross-docking distribution center for an autoparts supply chainrdquo European Journal of Operational Researchvol 229 no 3 pp 695ndash706 2013

[32] G Dellino J P C Kleijnen and C Meloni ldquoRobust optimiza-tion in simulation taguchi and Response Surface Methodol-ogyrdquo International Journal of Production Economics vol 125 no1 pp 52ndash59 2010

[33] T Yang Y Wen and F Wang ldquoEvaluation of robustness of sup-ply chain information-sharing strategies using a hybrid Taguchiand multiple criteria decision-making methodrdquo InternationalJournal of Production Economics vol 134 no 2 pp 458ndash4662011

[34] M E Kuhl and J R Wilson ldquoModeling and simulating Poissonprocesses having trends or nontrigonometric cyclic effectsrdquoEuropean Journal of Operational Research vol 133 no 3 pp566ndash582 2001

[35] P Narayan and J Subramanian Inventory ManagementmdashPrinciples and Practices Excel Books New Delhi India 2008

[36] E A Silver and R Peterson Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 1985

[37] R G Brown Smoothing Forecasting and Prediction of DiscreteTime Series Courier 2004

[38] D C Montgomery Design and analysis of experiments JohnWiley amp Sons New York NY USA 6th edition 2005

[39] J Zhu ldquoData envelopment analysis vs principal componentanalysis an illustrative study of economic performance ofChinese citiesrdquo European Journal of Operational Research vol111 no 1 pp 50ndash61 1998

[40] I M Premachandra ldquoA note on DEA vs principal componentanalysis an improvement to Joe Zhursquos approachrdquo EuropeanJournal of Operational Research vol 132 no 3 pp 553ndash5602001

[41] D Slottje G W Scully G J Hirschberg and K Hayes Mea-suring the Quality of Life across Countries A MultidimensionalAnalysis Westview Press Boulder Colo USA 1991

[42] S Opricovic Multicriteria optimization of civil engineeringsystems [PhD thesis] Faculty of Civil Engineering BelgradeSerbia 1998

[43] P L Yu ldquoA class of solutions for group decision problemsrdquoManagement Science vol 19 no 8 pp 936ndash946 1973

Modelling and Simulation in Engineering 17

[44] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[45] S Opricovic and G H Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

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Page 11: Research Article Robust Inventory System Optimization ...downloads.hindawi.com/journals/mse/2014/305120.pdf · Robust Inventory System Optimization Based on Simulation and Multiple

Modelling and Simulation in Engineering 11

Table 9 Result of eigenvalues

Eigenvalues 31204 07428 01111 00257Proportion 0780 0186 0028 0006Cumulative 0780 0966 0994 1000

Table 10 Result of principle components

Variables PC1 PC2 PC3 PC4Inventory 0543 0089 minus0818 0167Cost minus0538 minus0277 minus0511 minus0610Service level minus0551 minus0210 minus0230 0774Robustness 0335 minus0933 0126 0023

Table 11 Weights of criteria for different products

Criteria Products119860 119861 119862 119863

Average of inventory position 01012 00590 01942 00462Cost 02852 01836 01742 03081Service level 01195 01392 00966 01206Robustness 04941 06182 05350 05251

importance Calculated values are reported in Table 12 forentire scenario as VIKOR output

Product 119861 According to Table 12 values of 119877 119878 and 119876 arezero for scenario 18 Regarding (C1) (Acceptable Advantage)the difference between the best scenario and second rankedscenario should be more than 00038 The second rankedscenario regarding 119876 value is scenario 3 and its value is0051044 that satisfies both (C1) and (C2) Final result forproduct 119861 is unique scenario that is number 18 and consistsof the following configuration Inventory policy is DFwith119877

119901

being equal to 30 products and 7 days lead time According todecision matrix which is shown in Table 8 the performanceof this scenario for product 119861 results in 383944 products asaverage of inventory position and expected cost is equal to35269609 for product life cycle 73 of customer demandwillbe satisfied and in the case of increasing demand intensityand variation 2 decrease is expected in service level so itwill change into 71 Analyses for other products are similarso only final results are presented in this paper

Product 119860 In this case VIKORmethod ranks scenario num-ber 15 as the best one This scenario consists of DF inventorypolicy 20 products for 119877

119901and 5 days for lead time Perfor-

mance of this scenario results in 374778 products as averageof inventory position Expected cost is 31603294 and servicelevel will be 66 This scenario is robust against demandintensification and variation because changes in demandaffect service level as small as 1 But scenario number 15 doesnot satisfy (C1) so compromising solutions are consideredAlthough the best scenario is number 15 VIKOR methodimplies that compromising solutions have the same values

Table 12 VIKOR output for product 119861

Run number 119878 119877 119876

1 0634311 0796269 0715292 0003184 0185077 0094133 0059804 0042283 00510444 0536946 0592539 05647425 0718862 0796269 07575656 0116278 0089887 01030827 0428814 0388808 04088118 086078 0796269 08285259 002043 0140653 008054110 0677386 1 083869311 0266254 0592539 042939612 0043962 0185077 01145213 0571695 0796269 068398214 0515097 0592539 055381815 009173 0185077 013840416 0648693 0796269 072248117 1 1 118 0 0 019 0630698 1 081534920 04239 0796269 061008421 0139527 0388808 026416722 06581 1 08290523 0819338 1 090966924 0022245 0185077 010366125 0726183 1 086309126 0849516 0796269 082289327 0072768 0185077 0128922

So in real situation compromising solution can be substi-tuted This solution regarding their priority is as follows

Scenario number 24 inventory policy is DF 119877119901is

equal to 30 products and lead time is 5 daysScenario number 18 inventory policy is DF 119877

119901is

equal to 20 products and lead time is 7 daysScenario number 12 inventory policy isDF119877

119901is equal

to 20 products and lead time is 3 days

Product 119862 For this product scenario number 1 is ranked asthe best one This scenario consists of FI inventory policy 119877

119901

is 15 products and lead time is 3 days Result of this scenario is4387 products as the average of inventory position Expectedcost is 66143000 and service level will be 61 with robustnessof 1 Compromising solutions are as follows

Scenario number 6 inventory policy is DF 119877119901is 15

products and lead time is 5 daysScenario number 3 inventory policy is DF 119877

119901is 15

products and lead time is 3 daysScenario number 7 inventory policy is FI 119877

119901is 15

products and lead time is 7 days

12 Modelling and Simulation in Engineering

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 4 Interaction plot for product 119862 average of inventory position

Scenario number 4 inventory policy is FI 119877119901is 15

products and lead time is 5 days

Product119863 Scenario number 3 has the best rank for this prod-uct This scenario consists of DF inventory policy 119877

119901is 15

products and lead time is 3 days This scenario leads to

363896 products as average of inventory position with62648337 expected cost and 61 service level with 1robustness Compromising solutions are as follows

Scenario number 6 this scenario is mentioned beforeas a compromising solution for product C

Modelling and Simulation in Engineering 13

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 5 Interaction plot for product 119862 cost

14 Modelling and Simulation in Engineering

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

3

5

7

L

3

5

7

L

3

5

7

L

L

(b)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 6 Interaction plot for product 119862 service level

Scenario number 17 inventory policy is FI119877119901is equal

to 15 products and lead time is 7 days

55 Sensitivity Analysis As mentioned in the previous sec-tion all scenarios related to product 119860 prefer DF inventory

policy so it is concluded that DF policy is unique optimalpolicy while lead time includes values of 3 5 and 7 in selectedscenarios This situation implies that product 119860 is insensitiveto the delivery time It can be wise to decide lead time about 5days but there is no need for strict control on lead time while

Modelling and Simulation in Engineering 15

119877119901should be controlled strictly to prevent inventory in hand

from reaching lower than 20 productsIn the case of product 119861 (C1) is satisfied so there is a

unique scenario while there are four compromising solutionfor product 119862 In this situation based on VIKOR methodcompromising solutions have same value while interactionplot can be applied for investigation of existing differencebetween compromising solutions As mentioned before thebest scenario suggests FI inventory policy for this product butin compromising solutions there are both FI and DF policiesLead time varies from 3 to 7 days So this confusing situationis resolved by deeper analysis and applying interaction plotFigure 4 shows interaction plot for DOE factors and averageof inventory position as response variables

Figure 4 shows that with either DF or FI policy there isno significant difference among different lead times in theview point of inventory position but DF leads to less averageof inventory position With 119877

119901equaling 15 there is less

sensitivity to different lead times So configuring 119877119901which

is equal to 15 products the average of inventory positioncriterion will be robust against lead time variations

Figure 5 regards cost criterion and shows that FI andDF policies are equal and make least sensitivity related tolead time On the other hand three plots of 119877

119901against lead

time are parallel The parallel plots are interpreted as lackof relation between 119877

119901and lead time With 119877

119901equaling 15

increase in inventory cost could be mitigated when lead timesets to 3 days

In Figure 6 with 3 days for lead time there is no signifi-cant difference between inventory policies in the view pointof service level119877

119901and lead time are independent and the best

service level is related to FI policy which is also very sensitiveto 119877119901 As 119877

119901increases service level will grow According to

the interaction plot analyses optimal decision for product119862 is FI inventory policy 3 days lead time and 15 or moreproducts for 119877

119901

Decision about product 119863 is simple Based on the in-formation obtained from solutions it is recommended toemploy DF inventory policy with 119877

119901being equal to 15

products and 3 days for lead time

6 Conclusion

In this paper a simulation optimization framework is pro-posed for robust optimization of retailer inventory systemThis framework is less time consuming in comparison withmetaheuristic or SPSA approach it also provides robustsolution for multiple objective functions In this research alsotrend and cyclic change of customers demand are consideredfor more realistic view of problem Proposed framework con-sists of discrete event simulation and surrogate model In thisframework full factorial design of experiment is employedfor producing decision space Based on the three decisionfactors 27 different scenarios are produced and simulated Insimulation modelling multiresolution method is applied forsimulation of demand with highly dynamic pattern Becausethere are multiple criteria for inventory system performanceVIKORmethod is employed asmulticriteria decisionmaking

technique In addition robustness is considered as perfor-mance criterion and PCA is used for improving performanceof VIKOR method Finally developed framework gave thebest ranked and compromising solutions so interaction plotapplied for more investigation and sensitivity analysis ofobtained solutions

Due to the novelty of the proposed framework it isrecommended that future studies encompass optimization ofinventory system of integrated supply chain Also improve-ment of developed framework can be considered as futurestudy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers for theirsignificant roles in improvement of this research quality

References

[1] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[2] S Yin S XDingAHaghaniHHao andP Zhang ldquoA compar-ison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[3] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[4] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[5] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall International Series in Management New YorkNY USA 1963

[6] E Naddor Inventory Systems John Wiley amp Sons New YorkNY USA 1966

[7] R Peterson and E A Silver Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 2nd edition 1985

[8] T C E Cheng ldquoEconomic order quantity model with demand-dependent unit production cost and imperfect productionprocessesrdquo IIE Transactions vol 23 no 1 pp 23ndash28 1991

[9] S H Chen C CWang and A Ramer ldquoBackorder fuzzy inven-tory model under function principlerdquo Information Sciences vol95 no 1-2 pp 71ndash79 1996

[10] X Zhao J Xie and J C Wei ldquoThe value of early order com-mitment in a two-level supply chainrdquo European Journal of Op-erational Research vol 180 no 1 pp 194ndash214 2007

[11] R S M Lau J Xie and X Zhao ldquoEffects of inventory policyon supply chain performance a simulation study of critical

16 Modelling and Simulation in Engineering

decision parametersrdquo Computers and Industrial Engineeringvol 55 no 3 pp 620ndash633 2008

[12] J Xu and L Zhao ldquoA multi-objective decision-making modelwith fuzzy rough coefficients and its application to the inventoryproblemrdquo Information Sciences vol 180 no 5 pp 679ndash6962010

[13] F Hnaien X Delorme and A Dolgui ldquoMulti-objective opti-mization for inventory control in two-level assembly systemsunder uncertainty of lead timesrdquo Computers amp OperationsResearch vol 37 no 11 pp 1835ndash1843 2010

[14] H D Purnomo H M Wee and Y Praharsi ldquoTwo inventoryreview policies on supply chain configuration problemrdquo Com-puters and Industrial Engineering vol 63 no 2 pp 448ndash4552012

[15] B L Nelson ldquo50th anniversary article stochastic simulationresearch in management sciencerdquoManagement Science vol 50no 7 pp 855ndash868 2004

[16] J Boesel R O Bowden Jr F Glover J P Kelly and E WestwigldquoFuture of simulation optimizationrdquo inProceedings of theWinterSimulation Conference pp 1466ndash1469 Arlington Va USADecember 2001

[17] M C Fu ldquoOptimization for simulation theory vs practicerdquoINFORMS Journal on Computing vol 14 no 3 pp 192ndash2152002

[18] L Wang ldquoA hybrid genetic algorithm-neural network strategyfor simulation optimizationrdquoApplied Mathematics and Compu-tation vol 170 no 2 pp 1329ndash1343 2005

[19] B B Keskin S H Melouk and I L Meyer ldquoA simulation-optimization approach for integrated sourcing and inventorydecisionsrdquo Computers and Operations Research vol 37 no 9pp 1648ndash1661 2010

[20] E Mazhari J Zhao N Celik S Lee Y Son and L HeadldquoHybrid simulation and optimization-based design and oper-ation of integrated photovoltaic generation storage units andgridrdquo Simulation Modelling Practice and Theory vol 19 no 1pp 463ndash481 2011

[21] Q Duan and T W Liao ldquoOptimization of replenishment poli-cies for decentralized and centralized capacitated supply chainsunder various demandsrdquo International Journal of ProductionEconomics vol 142 no 1 pp 194ndash204 2013

[22] J C Spall ldquoMultivariate stochastic approximation usinga simultaneous perturbation gradient approximationrdquo IEEETransactions on Automatic Control vol 37 no 3 pp 332ndash3411992

[23] J C Spall ldquoImplementation of the simultaneous perturbationalgorithm for stochastic optimizationrdquo IEEE Transactions onAerospace and Electronic Systems vol 34 no 3 pp 817ndash8231998

[24] J C Spall ldquoStochastic optimization and the simultaneousperturbation methodrdquo in Proceedings of the Winter SimulationConference (WSC rsquo99) pp 101ndash109 December 1999

[25] J D Schwartz W Wang and D E Rivera ldquoSimulation-based optimization of process control policies for inventorymanagement in supply chainsrdquo Automatica vol 42 no 8 pp1311ndash1320 2006

[26] H G Neddermeijer G J V Oortmarssen and N P R DekkerldquoA framework for response surface methodology for simulationoptimizationrdquo in Proceedings of the Winter Simulation Confer-ence vol 1 Orlando Fla USA December 1999

[27] B Can and C Heavey ldquoA comparison of genetic programmingand artificial neural networks in metamodeling of discrete-event simulation modelsrdquo Computers amp Operations Researchvol 39 no 2 pp 424ndash436 2012

[28] A Azadeh Z S Faiz S M Asadzadeh and R Tavakkoli-Moghaddam ldquoAn integrated artificial neural network-comput-er simulation for optimization of complex tandem queuesystemsrdquoMathematics and Computers in Simulation vol 82 no4 pp 666ndash678 2011

[29] R Bornatico J Hussy A Witzig and L Guzzella ldquoSurrogatemodeling for the fast optimization of energy systemsrdquo Energyvol 57 pp 653ndash662 2013

[30] X Wan J F Pekny and G V Reklaitis ldquoSimulation-basedoptimizationwith surrogatemodels application to supply chainmanagementrdquoComputers and Chemical Engineering vol 29 no6 pp 1317ndash1328 2005

[31] W Shi Z Liu J Shang and Y Cui ldquoMulti-criteria robust designof a JIT-based cross-docking distribution center for an autoparts supply chainrdquo European Journal of Operational Researchvol 229 no 3 pp 695ndash706 2013

[32] G Dellino J P C Kleijnen and C Meloni ldquoRobust optimiza-tion in simulation taguchi and Response Surface Methodol-ogyrdquo International Journal of Production Economics vol 125 no1 pp 52ndash59 2010

[33] T Yang Y Wen and F Wang ldquoEvaluation of robustness of sup-ply chain information-sharing strategies using a hybrid Taguchiand multiple criteria decision-making methodrdquo InternationalJournal of Production Economics vol 134 no 2 pp 458ndash4662011

[34] M E Kuhl and J R Wilson ldquoModeling and simulating Poissonprocesses having trends or nontrigonometric cyclic effectsrdquoEuropean Journal of Operational Research vol 133 no 3 pp566ndash582 2001

[35] P Narayan and J Subramanian Inventory ManagementmdashPrinciples and Practices Excel Books New Delhi India 2008

[36] E A Silver and R Peterson Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 1985

[37] R G Brown Smoothing Forecasting and Prediction of DiscreteTime Series Courier 2004

[38] D C Montgomery Design and analysis of experiments JohnWiley amp Sons New York NY USA 6th edition 2005

[39] J Zhu ldquoData envelopment analysis vs principal componentanalysis an illustrative study of economic performance ofChinese citiesrdquo European Journal of Operational Research vol111 no 1 pp 50ndash61 1998

[40] I M Premachandra ldquoA note on DEA vs principal componentanalysis an improvement to Joe Zhursquos approachrdquo EuropeanJournal of Operational Research vol 132 no 3 pp 553ndash5602001

[41] D Slottje G W Scully G J Hirschberg and K Hayes Mea-suring the Quality of Life across Countries A MultidimensionalAnalysis Westview Press Boulder Colo USA 1991

[42] S Opricovic Multicriteria optimization of civil engineeringsystems [PhD thesis] Faculty of Civil Engineering BelgradeSerbia 1998

[43] P L Yu ldquoA class of solutions for group decision problemsrdquoManagement Science vol 19 no 8 pp 936ndash946 1973

Modelling and Simulation in Engineering 17

[44] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[45] S Opricovic and G H Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 12: Research Article Robust Inventory System Optimization ...downloads.hindawi.com/journals/mse/2014/305120.pdf · Robust Inventory System Optimization Based on Simulation and Multiple

12 Modelling and Simulation in Engineering

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

60

45

30

60

45

30

60

45

30

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 4 Interaction plot for product 119862 average of inventory position

Scenario number 4 inventory policy is FI 119877119901is 15

products and lead time is 5 days

Product119863 Scenario number 3 has the best rank for this prod-uct This scenario consists of DF inventory policy 119877

119901is 15

products and lead time is 3 days This scenario leads to

363896 products as average of inventory position with62648337 expected cost and 61 service level with 1robustness Compromising solutions are as follows

Scenario number 6 this scenario is mentioned beforeas a compromising solution for product C

Modelling and Simulation in Engineering 13

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 5 Interaction plot for product 119862 cost

14 Modelling and Simulation in Engineering

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

3

5

7

L

3

5

7

L

3

5

7

L

L

(b)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 6 Interaction plot for product 119862 service level

Scenario number 17 inventory policy is FI119877119901is equal

to 15 products and lead time is 7 days

55 Sensitivity Analysis As mentioned in the previous sec-tion all scenarios related to product 119860 prefer DF inventory

policy so it is concluded that DF policy is unique optimalpolicy while lead time includes values of 3 5 and 7 in selectedscenarios This situation implies that product 119860 is insensitiveto the delivery time It can be wise to decide lead time about 5days but there is no need for strict control on lead time while

Modelling and Simulation in Engineering 15

119877119901should be controlled strictly to prevent inventory in hand

from reaching lower than 20 productsIn the case of product 119861 (C1) is satisfied so there is a

unique scenario while there are four compromising solutionfor product 119862 In this situation based on VIKOR methodcompromising solutions have same value while interactionplot can be applied for investigation of existing differencebetween compromising solutions As mentioned before thebest scenario suggests FI inventory policy for this product butin compromising solutions there are both FI and DF policiesLead time varies from 3 to 7 days So this confusing situationis resolved by deeper analysis and applying interaction plotFigure 4 shows interaction plot for DOE factors and averageof inventory position as response variables

Figure 4 shows that with either DF or FI policy there isno significant difference among different lead times in theview point of inventory position but DF leads to less averageof inventory position With 119877

119901equaling 15 there is less

sensitivity to different lead times So configuring 119877119901which

is equal to 15 products the average of inventory positioncriterion will be robust against lead time variations

Figure 5 regards cost criterion and shows that FI andDF policies are equal and make least sensitivity related tolead time On the other hand three plots of 119877

119901against lead

time are parallel The parallel plots are interpreted as lackof relation between 119877

119901and lead time With 119877

119901equaling 15

increase in inventory cost could be mitigated when lead timesets to 3 days

In Figure 6 with 3 days for lead time there is no signifi-cant difference between inventory policies in the view pointof service level119877

119901and lead time are independent and the best

service level is related to FI policy which is also very sensitiveto 119877119901 As 119877

119901increases service level will grow According to

the interaction plot analyses optimal decision for product119862 is FI inventory policy 3 days lead time and 15 or moreproducts for 119877

119901

Decision about product 119863 is simple Based on the in-formation obtained from solutions it is recommended toemploy DF inventory policy with 119877

119901being equal to 15

products and 3 days for lead time

6 Conclusion

In this paper a simulation optimization framework is pro-posed for robust optimization of retailer inventory systemThis framework is less time consuming in comparison withmetaheuristic or SPSA approach it also provides robustsolution for multiple objective functions In this research alsotrend and cyclic change of customers demand are consideredfor more realistic view of problem Proposed framework con-sists of discrete event simulation and surrogate model In thisframework full factorial design of experiment is employedfor producing decision space Based on the three decisionfactors 27 different scenarios are produced and simulated Insimulation modelling multiresolution method is applied forsimulation of demand with highly dynamic pattern Becausethere are multiple criteria for inventory system performanceVIKORmethod is employed asmulticriteria decisionmaking

technique In addition robustness is considered as perfor-mance criterion and PCA is used for improving performanceof VIKOR method Finally developed framework gave thebest ranked and compromising solutions so interaction plotapplied for more investigation and sensitivity analysis ofobtained solutions

Due to the novelty of the proposed framework it isrecommended that future studies encompass optimization ofinventory system of integrated supply chain Also improve-ment of developed framework can be considered as futurestudy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers for theirsignificant roles in improvement of this research quality

References

[1] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[2] S Yin S XDingAHaghaniHHao andP Zhang ldquoA compar-ison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[3] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[4] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[5] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall International Series in Management New YorkNY USA 1963

[6] E Naddor Inventory Systems John Wiley amp Sons New YorkNY USA 1966

[7] R Peterson and E A Silver Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 2nd edition 1985

[8] T C E Cheng ldquoEconomic order quantity model with demand-dependent unit production cost and imperfect productionprocessesrdquo IIE Transactions vol 23 no 1 pp 23ndash28 1991

[9] S H Chen C CWang and A Ramer ldquoBackorder fuzzy inven-tory model under function principlerdquo Information Sciences vol95 no 1-2 pp 71ndash79 1996

[10] X Zhao J Xie and J C Wei ldquoThe value of early order com-mitment in a two-level supply chainrdquo European Journal of Op-erational Research vol 180 no 1 pp 194ndash214 2007

[11] R S M Lau J Xie and X Zhao ldquoEffects of inventory policyon supply chain performance a simulation study of critical

16 Modelling and Simulation in Engineering

decision parametersrdquo Computers and Industrial Engineeringvol 55 no 3 pp 620ndash633 2008

[12] J Xu and L Zhao ldquoA multi-objective decision-making modelwith fuzzy rough coefficients and its application to the inventoryproblemrdquo Information Sciences vol 180 no 5 pp 679ndash6962010

[13] F Hnaien X Delorme and A Dolgui ldquoMulti-objective opti-mization for inventory control in two-level assembly systemsunder uncertainty of lead timesrdquo Computers amp OperationsResearch vol 37 no 11 pp 1835ndash1843 2010

[14] H D Purnomo H M Wee and Y Praharsi ldquoTwo inventoryreview policies on supply chain configuration problemrdquo Com-puters and Industrial Engineering vol 63 no 2 pp 448ndash4552012

[15] B L Nelson ldquo50th anniversary article stochastic simulationresearch in management sciencerdquoManagement Science vol 50no 7 pp 855ndash868 2004

[16] J Boesel R O Bowden Jr F Glover J P Kelly and E WestwigldquoFuture of simulation optimizationrdquo inProceedings of theWinterSimulation Conference pp 1466ndash1469 Arlington Va USADecember 2001

[17] M C Fu ldquoOptimization for simulation theory vs practicerdquoINFORMS Journal on Computing vol 14 no 3 pp 192ndash2152002

[18] L Wang ldquoA hybrid genetic algorithm-neural network strategyfor simulation optimizationrdquoApplied Mathematics and Compu-tation vol 170 no 2 pp 1329ndash1343 2005

[19] B B Keskin S H Melouk and I L Meyer ldquoA simulation-optimization approach for integrated sourcing and inventorydecisionsrdquo Computers and Operations Research vol 37 no 9pp 1648ndash1661 2010

[20] E Mazhari J Zhao N Celik S Lee Y Son and L HeadldquoHybrid simulation and optimization-based design and oper-ation of integrated photovoltaic generation storage units andgridrdquo Simulation Modelling Practice and Theory vol 19 no 1pp 463ndash481 2011

[21] Q Duan and T W Liao ldquoOptimization of replenishment poli-cies for decentralized and centralized capacitated supply chainsunder various demandsrdquo International Journal of ProductionEconomics vol 142 no 1 pp 194ndash204 2013

[22] J C Spall ldquoMultivariate stochastic approximation usinga simultaneous perturbation gradient approximationrdquo IEEETransactions on Automatic Control vol 37 no 3 pp 332ndash3411992

[23] J C Spall ldquoImplementation of the simultaneous perturbationalgorithm for stochastic optimizationrdquo IEEE Transactions onAerospace and Electronic Systems vol 34 no 3 pp 817ndash8231998

[24] J C Spall ldquoStochastic optimization and the simultaneousperturbation methodrdquo in Proceedings of the Winter SimulationConference (WSC rsquo99) pp 101ndash109 December 1999

[25] J D Schwartz W Wang and D E Rivera ldquoSimulation-based optimization of process control policies for inventorymanagement in supply chainsrdquo Automatica vol 42 no 8 pp1311ndash1320 2006

[26] H G Neddermeijer G J V Oortmarssen and N P R DekkerldquoA framework for response surface methodology for simulationoptimizationrdquo in Proceedings of the Winter Simulation Confer-ence vol 1 Orlando Fla USA December 1999

[27] B Can and C Heavey ldquoA comparison of genetic programmingand artificial neural networks in metamodeling of discrete-event simulation modelsrdquo Computers amp Operations Researchvol 39 no 2 pp 424ndash436 2012

[28] A Azadeh Z S Faiz S M Asadzadeh and R Tavakkoli-Moghaddam ldquoAn integrated artificial neural network-comput-er simulation for optimization of complex tandem queuesystemsrdquoMathematics and Computers in Simulation vol 82 no4 pp 666ndash678 2011

[29] R Bornatico J Hussy A Witzig and L Guzzella ldquoSurrogatemodeling for the fast optimization of energy systemsrdquo Energyvol 57 pp 653ndash662 2013

[30] X Wan J F Pekny and G V Reklaitis ldquoSimulation-basedoptimizationwith surrogatemodels application to supply chainmanagementrdquoComputers and Chemical Engineering vol 29 no6 pp 1317ndash1328 2005

[31] W Shi Z Liu J Shang and Y Cui ldquoMulti-criteria robust designof a JIT-based cross-docking distribution center for an autoparts supply chainrdquo European Journal of Operational Researchvol 229 no 3 pp 695ndash706 2013

[32] G Dellino J P C Kleijnen and C Meloni ldquoRobust optimiza-tion in simulation taguchi and Response Surface Methodol-ogyrdquo International Journal of Production Economics vol 125 no1 pp 52ndash59 2010

[33] T Yang Y Wen and F Wang ldquoEvaluation of robustness of sup-ply chain information-sharing strategies using a hybrid Taguchiand multiple criteria decision-making methodrdquo InternationalJournal of Production Economics vol 134 no 2 pp 458ndash4662011

[34] M E Kuhl and J R Wilson ldquoModeling and simulating Poissonprocesses having trends or nontrigonometric cyclic effectsrdquoEuropean Journal of Operational Research vol 133 no 3 pp566ndash582 2001

[35] P Narayan and J Subramanian Inventory ManagementmdashPrinciples and Practices Excel Books New Delhi India 2008

[36] E A Silver and R Peterson Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 1985

[37] R G Brown Smoothing Forecasting and Prediction of DiscreteTime Series Courier 2004

[38] D C Montgomery Design and analysis of experiments JohnWiley amp Sons New York NY USA 6th edition 2005

[39] J Zhu ldquoData envelopment analysis vs principal componentanalysis an illustrative study of economic performance ofChinese citiesrdquo European Journal of Operational Research vol111 no 1 pp 50ndash61 1998

[40] I M Premachandra ldquoA note on DEA vs principal componentanalysis an improvement to Joe Zhursquos approachrdquo EuropeanJournal of Operational Research vol 132 no 3 pp 553ndash5602001

[41] D Slottje G W Scully G J Hirschberg and K Hayes Mea-suring the Quality of Life across Countries A MultidimensionalAnalysis Westview Press Boulder Colo USA 1991

[42] S Opricovic Multicriteria optimization of civil engineeringsystems [PhD thesis] Faculty of Civil Engineering BelgradeSerbia 1998

[43] P L Yu ldquoA class of solutions for group decision problemsrdquoManagement Science vol 19 no 8 pp 936ndash946 1973

Modelling and Simulation in Engineering 17

[44] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[45] S Opricovic and G H Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 13: Research Article Robust Inventory System Optimization ...downloads.hindawi.com/journals/mse/2014/305120.pdf · Robust Inventory System Optimization Based on Simulation and Multiple

Modelling and Simulation in Engineering 13

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

L

3

5

7

L

3

5

7

L

3

5

7

L

(b)

80

60

40

times106

80

60

40

times106

80

60

40

times106

15 30 45 3 5 7 Fixedinterval

Fixedorder

Demandforecasting

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 5 Interaction plot for product 119862 cost

14 Modelling and Simulation in Engineering

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

3

5

7

L

3

5

7

L

3

5

7

L

L

(b)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 6 Interaction plot for product 119862 service level

Scenario number 17 inventory policy is FI119877119901is equal

to 15 products and lead time is 7 days

55 Sensitivity Analysis As mentioned in the previous sec-tion all scenarios related to product 119860 prefer DF inventory

policy so it is concluded that DF policy is unique optimalpolicy while lead time includes values of 3 5 and 7 in selectedscenarios This situation implies that product 119860 is insensitiveto the delivery time It can be wise to decide lead time about 5days but there is no need for strict control on lead time while

Modelling and Simulation in Engineering 15

119877119901should be controlled strictly to prevent inventory in hand

from reaching lower than 20 productsIn the case of product 119861 (C1) is satisfied so there is a

unique scenario while there are four compromising solutionfor product 119862 In this situation based on VIKOR methodcompromising solutions have same value while interactionplot can be applied for investigation of existing differencebetween compromising solutions As mentioned before thebest scenario suggests FI inventory policy for this product butin compromising solutions there are both FI and DF policiesLead time varies from 3 to 7 days So this confusing situationis resolved by deeper analysis and applying interaction plotFigure 4 shows interaction plot for DOE factors and averageof inventory position as response variables

Figure 4 shows that with either DF or FI policy there isno significant difference among different lead times in theview point of inventory position but DF leads to less averageof inventory position With 119877

119901equaling 15 there is less

sensitivity to different lead times So configuring 119877119901which

is equal to 15 products the average of inventory positioncriterion will be robust against lead time variations

Figure 5 regards cost criterion and shows that FI andDF policies are equal and make least sensitivity related tolead time On the other hand three plots of 119877

119901against lead

time are parallel The parallel plots are interpreted as lackof relation between 119877

119901and lead time With 119877

119901equaling 15

increase in inventory cost could be mitigated when lead timesets to 3 days

In Figure 6 with 3 days for lead time there is no signifi-cant difference between inventory policies in the view pointof service level119877

119901and lead time are independent and the best

service level is related to FI policy which is also very sensitiveto 119877119901 As 119877

119901increases service level will grow According to

the interaction plot analyses optimal decision for product119862 is FI inventory policy 3 days lead time and 15 or moreproducts for 119877

119901

Decision about product 119863 is simple Based on the in-formation obtained from solutions it is recommended toemploy DF inventory policy with 119877

119901being equal to 15

products and 3 days for lead time

6 Conclusion

In this paper a simulation optimization framework is pro-posed for robust optimization of retailer inventory systemThis framework is less time consuming in comparison withmetaheuristic or SPSA approach it also provides robustsolution for multiple objective functions In this research alsotrend and cyclic change of customers demand are consideredfor more realistic view of problem Proposed framework con-sists of discrete event simulation and surrogate model In thisframework full factorial design of experiment is employedfor producing decision space Based on the three decisionfactors 27 different scenarios are produced and simulated Insimulation modelling multiresolution method is applied forsimulation of demand with highly dynamic pattern Becausethere are multiple criteria for inventory system performanceVIKORmethod is employed asmulticriteria decisionmaking

technique In addition robustness is considered as perfor-mance criterion and PCA is used for improving performanceof VIKOR method Finally developed framework gave thebest ranked and compromising solutions so interaction plotapplied for more investigation and sensitivity analysis ofobtained solutions

Due to the novelty of the proposed framework it isrecommended that future studies encompass optimization ofinventory system of integrated supply chain Also improve-ment of developed framework can be considered as futurestudy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers for theirsignificant roles in improvement of this research quality

References

[1] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[2] S Yin S XDingAHaghaniHHao andP Zhang ldquoA compar-ison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[3] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[4] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[5] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall International Series in Management New YorkNY USA 1963

[6] E Naddor Inventory Systems John Wiley amp Sons New YorkNY USA 1966

[7] R Peterson and E A Silver Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 2nd edition 1985

[8] T C E Cheng ldquoEconomic order quantity model with demand-dependent unit production cost and imperfect productionprocessesrdquo IIE Transactions vol 23 no 1 pp 23ndash28 1991

[9] S H Chen C CWang and A Ramer ldquoBackorder fuzzy inven-tory model under function principlerdquo Information Sciences vol95 no 1-2 pp 71ndash79 1996

[10] X Zhao J Xie and J C Wei ldquoThe value of early order com-mitment in a two-level supply chainrdquo European Journal of Op-erational Research vol 180 no 1 pp 194ndash214 2007

[11] R S M Lau J Xie and X Zhao ldquoEffects of inventory policyon supply chain performance a simulation study of critical

16 Modelling and Simulation in Engineering

decision parametersrdquo Computers and Industrial Engineeringvol 55 no 3 pp 620ndash633 2008

[12] J Xu and L Zhao ldquoA multi-objective decision-making modelwith fuzzy rough coefficients and its application to the inventoryproblemrdquo Information Sciences vol 180 no 5 pp 679ndash6962010

[13] F Hnaien X Delorme and A Dolgui ldquoMulti-objective opti-mization for inventory control in two-level assembly systemsunder uncertainty of lead timesrdquo Computers amp OperationsResearch vol 37 no 11 pp 1835ndash1843 2010

[14] H D Purnomo H M Wee and Y Praharsi ldquoTwo inventoryreview policies on supply chain configuration problemrdquo Com-puters and Industrial Engineering vol 63 no 2 pp 448ndash4552012

[15] B L Nelson ldquo50th anniversary article stochastic simulationresearch in management sciencerdquoManagement Science vol 50no 7 pp 855ndash868 2004

[16] J Boesel R O Bowden Jr F Glover J P Kelly and E WestwigldquoFuture of simulation optimizationrdquo inProceedings of theWinterSimulation Conference pp 1466ndash1469 Arlington Va USADecember 2001

[17] M C Fu ldquoOptimization for simulation theory vs practicerdquoINFORMS Journal on Computing vol 14 no 3 pp 192ndash2152002

[18] L Wang ldquoA hybrid genetic algorithm-neural network strategyfor simulation optimizationrdquoApplied Mathematics and Compu-tation vol 170 no 2 pp 1329ndash1343 2005

[19] B B Keskin S H Melouk and I L Meyer ldquoA simulation-optimization approach for integrated sourcing and inventorydecisionsrdquo Computers and Operations Research vol 37 no 9pp 1648ndash1661 2010

[20] E Mazhari J Zhao N Celik S Lee Y Son and L HeadldquoHybrid simulation and optimization-based design and oper-ation of integrated photovoltaic generation storage units andgridrdquo Simulation Modelling Practice and Theory vol 19 no 1pp 463ndash481 2011

[21] Q Duan and T W Liao ldquoOptimization of replenishment poli-cies for decentralized and centralized capacitated supply chainsunder various demandsrdquo International Journal of ProductionEconomics vol 142 no 1 pp 194ndash204 2013

[22] J C Spall ldquoMultivariate stochastic approximation usinga simultaneous perturbation gradient approximationrdquo IEEETransactions on Automatic Control vol 37 no 3 pp 332ndash3411992

[23] J C Spall ldquoImplementation of the simultaneous perturbationalgorithm for stochastic optimizationrdquo IEEE Transactions onAerospace and Electronic Systems vol 34 no 3 pp 817ndash8231998

[24] J C Spall ldquoStochastic optimization and the simultaneousperturbation methodrdquo in Proceedings of the Winter SimulationConference (WSC rsquo99) pp 101ndash109 December 1999

[25] J D Schwartz W Wang and D E Rivera ldquoSimulation-based optimization of process control policies for inventorymanagement in supply chainsrdquo Automatica vol 42 no 8 pp1311ndash1320 2006

[26] H G Neddermeijer G J V Oortmarssen and N P R DekkerldquoA framework for response surface methodology for simulationoptimizationrdquo in Proceedings of the Winter Simulation Confer-ence vol 1 Orlando Fla USA December 1999

[27] B Can and C Heavey ldquoA comparison of genetic programmingand artificial neural networks in metamodeling of discrete-event simulation modelsrdquo Computers amp Operations Researchvol 39 no 2 pp 424ndash436 2012

[28] A Azadeh Z S Faiz S M Asadzadeh and R Tavakkoli-Moghaddam ldquoAn integrated artificial neural network-comput-er simulation for optimization of complex tandem queuesystemsrdquoMathematics and Computers in Simulation vol 82 no4 pp 666ndash678 2011

[29] R Bornatico J Hussy A Witzig and L Guzzella ldquoSurrogatemodeling for the fast optimization of energy systemsrdquo Energyvol 57 pp 653ndash662 2013

[30] X Wan J F Pekny and G V Reklaitis ldquoSimulation-basedoptimizationwith surrogatemodels application to supply chainmanagementrdquoComputers and Chemical Engineering vol 29 no6 pp 1317ndash1328 2005

[31] W Shi Z Liu J Shang and Y Cui ldquoMulti-criteria robust designof a JIT-based cross-docking distribution center for an autoparts supply chainrdquo European Journal of Operational Researchvol 229 no 3 pp 695ndash706 2013

[32] G Dellino J P C Kleijnen and C Meloni ldquoRobust optimiza-tion in simulation taguchi and Response Surface Methodol-ogyrdquo International Journal of Production Economics vol 125 no1 pp 52ndash59 2010

[33] T Yang Y Wen and F Wang ldquoEvaluation of robustness of sup-ply chain information-sharing strategies using a hybrid Taguchiand multiple criteria decision-making methodrdquo InternationalJournal of Production Economics vol 134 no 2 pp 458ndash4662011

[34] M E Kuhl and J R Wilson ldquoModeling and simulating Poissonprocesses having trends or nontrigonometric cyclic effectsrdquoEuropean Journal of Operational Research vol 133 no 3 pp566ndash582 2001

[35] P Narayan and J Subramanian Inventory ManagementmdashPrinciples and Practices Excel Books New Delhi India 2008

[36] E A Silver and R Peterson Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 1985

[37] R G Brown Smoothing Forecasting and Prediction of DiscreteTime Series Courier 2004

[38] D C Montgomery Design and analysis of experiments JohnWiley amp Sons New York NY USA 6th edition 2005

[39] J Zhu ldquoData envelopment analysis vs principal componentanalysis an illustrative study of economic performance ofChinese citiesrdquo European Journal of Operational Research vol111 no 1 pp 50ndash61 1998

[40] I M Premachandra ldquoA note on DEA vs principal componentanalysis an improvement to Joe Zhursquos approachrdquo EuropeanJournal of Operational Research vol 132 no 3 pp 553ndash5602001

[41] D Slottje G W Scully G J Hirschberg and K Hayes Mea-suring the Quality of Life across Countries A MultidimensionalAnalysis Westview Press Boulder Colo USA 1991

[42] S Opricovic Multicriteria optimization of civil engineeringsystems [PhD thesis] Faculty of Civil Engineering BelgradeSerbia 1998

[43] P L Yu ldquoA class of solutions for group decision problemsrdquoManagement Science vol 19 no 8 pp 936ndash946 1973

Modelling and Simulation in Engineering 17

[44] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[45] S Opricovic and G H Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 14: Research Article Robust Inventory System Optimization ...downloads.hindawi.com/journals/mse/2014/305120.pdf · Robust Inventory System Optimization Based on Simulation and Multiple

14 Modelling and Simulation in Engineering

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

30

45

15

30

45

15

30

45

15

Rp

RpRpRp

(a)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

3

5

7

L

3

5

7

L

3

5

7

L

L

(b)

15 30 45 3 5 7

80

70

60

80

70

60

Fixedinterval

Fixedorder

Demandforecasting

80

70

60

IP

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

IPFixed intervalFixed orderDemand forecasting

(c)

Figure 6 Interaction plot for product 119862 service level

Scenario number 17 inventory policy is FI119877119901is equal

to 15 products and lead time is 7 days

55 Sensitivity Analysis As mentioned in the previous sec-tion all scenarios related to product 119860 prefer DF inventory

policy so it is concluded that DF policy is unique optimalpolicy while lead time includes values of 3 5 and 7 in selectedscenarios This situation implies that product 119860 is insensitiveto the delivery time It can be wise to decide lead time about 5days but there is no need for strict control on lead time while

Modelling and Simulation in Engineering 15

119877119901should be controlled strictly to prevent inventory in hand

from reaching lower than 20 productsIn the case of product 119861 (C1) is satisfied so there is a

unique scenario while there are four compromising solutionfor product 119862 In this situation based on VIKOR methodcompromising solutions have same value while interactionplot can be applied for investigation of existing differencebetween compromising solutions As mentioned before thebest scenario suggests FI inventory policy for this product butin compromising solutions there are both FI and DF policiesLead time varies from 3 to 7 days So this confusing situationis resolved by deeper analysis and applying interaction plotFigure 4 shows interaction plot for DOE factors and averageof inventory position as response variables

Figure 4 shows that with either DF or FI policy there isno significant difference among different lead times in theview point of inventory position but DF leads to less averageof inventory position With 119877

119901equaling 15 there is less

sensitivity to different lead times So configuring 119877119901which

is equal to 15 products the average of inventory positioncriterion will be robust against lead time variations

Figure 5 regards cost criterion and shows that FI andDF policies are equal and make least sensitivity related tolead time On the other hand three plots of 119877

119901against lead

time are parallel The parallel plots are interpreted as lackof relation between 119877

119901and lead time With 119877

119901equaling 15

increase in inventory cost could be mitigated when lead timesets to 3 days

In Figure 6 with 3 days for lead time there is no signifi-cant difference between inventory policies in the view pointof service level119877

119901and lead time are independent and the best

service level is related to FI policy which is also very sensitiveto 119877119901 As 119877

119901increases service level will grow According to

the interaction plot analyses optimal decision for product119862 is FI inventory policy 3 days lead time and 15 or moreproducts for 119877

119901

Decision about product 119863 is simple Based on the in-formation obtained from solutions it is recommended toemploy DF inventory policy with 119877

119901being equal to 15

products and 3 days for lead time

6 Conclusion

In this paper a simulation optimization framework is pro-posed for robust optimization of retailer inventory systemThis framework is less time consuming in comparison withmetaheuristic or SPSA approach it also provides robustsolution for multiple objective functions In this research alsotrend and cyclic change of customers demand are consideredfor more realistic view of problem Proposed framework con-sists of discrete event simulation and surrogate model In thisframework full factorial design of experiment is employedfor producing decision space Based on the three decisionfactors 27 different scenarios are produced and simulated Insimulation modelling multiresolution method is applied forsimulation of demand with highly dynamic pattern Becausethere are multiple criteria for inventory system performanceVIKORmethod is employed asmulticriteria decisionmaking

technique In addition robustness is considered as perfor-mance criterion and PCA is used for improving performanceof VIKOR method Finally developed framework gave thebest ranked and compromising solutions so interaction plotapplied for more investigation and sensitivity analysis ofobtained solutions

Due to the novelty of the proposed framework it isrecommended that future studies encompass optimization ofinventory system of integrated supply chain Also improve-ment of developed framework can be considered as futurestudy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers for theirsignificant roles in improvement of this research quality

References

[1] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[2] S Yin S XDingAHaghaniHHao andP Zhang ldquoA compar-ison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[3] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[4] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[5] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall International Series in Management New YorkNY USA 1963

[6] E Naddor Inventory Systems John Wiley amp Sons New YorkNY USA 1966

[7] R Peterson and E A Silver Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 2nd edition 1985

[8] T C E Cheng ldquoEconomic order quantity model with demand-dependent unit production cost and imperfect productionprocessesrdquo IIE Transactions vol 23 no 1 pp 23ndash28 1991

[9] S H Chen C CWang and A Ramer ldquoBackorder fuzzy inven-tory model under function principlerdquo Information Sciences vol95 no 1-2 pp 71ndash79 1996

[10] X Zhao J Xie and J C Wei ldquoThe value of early order com-mitment in a two-level supply chainrdquo European Journal of Op-erational Research vol 180 no 1 pp 194ndash214 2007

[11] R S M Lau J Xie and X Zhao ldquoEffects of inventory policyon supply chain performance a simulation study of critical

16 Modelling and Simulation in Engineering

decision parametersrdquo Computers and Industrial Engineeringvol 55 no 3 pp 620ndash633 2008

[12] J Xu and L Zhao ldquoA multi-objective decision-making modelwith fuzzy rough coefficients and its application to the inventoryproblemrdquo Information Sciences vol 180 no 5 pp 679ndash6962010

[13] F Hnaien X Delorme and A Dolgui ldquoMulti-objective opti-mization for inventory control in two-level assembly systemsunder uncertainty of lead timesrdquo Computers amp OperationsResearch vol 37 no 11 pp 1835ndash1843 2010

[14] H D Purnomo H M Wee and Y Praharsi ldquoTwo inventoryreview policies on supply chain configuration problemrdquo Com-puters and Industrial Engineering vol 63 no 2 pp 448ndash4552012

[15] B L Nelson ldquo50th anniversary article stochastic simulationresearch in management sciencerdquoManagement Science vol 50no 7 pp 855ndash868 2004

[16] J Boesel R O Bowden Jr F Glover J P Kelly and E WestwigldquoFuture of simulation optimizationrdquo inProceedings of theWinterSimulation Conference pp 1466ndash1469 Arlington Va USADecember 2001

[17] M C Fu ldquoOptimization for simulation theory vs practicerdquoINFORMS Journal on Computing vol 14 no 3 pp 192ndash2152002

[18] L Wang ldquoA hybrid genetic algorithm-neural network strategyfor simulation optimizationrdquoApplied Mathematics and Compu-tation vol 170 no 2 pp 1329ndash1343 2005

[19] B B Keskin S H Melouk and I L Meyer ldquoA simulation-optimization approach for integrated sourcing and inventorydecisionsrdquo Computers and Operations Research vol 37 no 9pp 1648ndash1661 2010

[20] E Mazhari J Zhao N Celik S Lee Y Son and L HeadldquoHybrid simulation and optimization-based design and oper-ation of integrated photovoltaic generation storage units andgridrdquo Simulation Modelling Practice and Theory vol 19 no 1pp 463ndash481 2011

[21] Q Duan and T W Liao ldquoOptimization of replenishment poli-cies for decentralized and centralized capacitated supply chainsunder various demandsrdquo International Journal of ProductionEconomics vol 142 no 1 pp 194ndash204 2013

[22] J C Spall ldquoMultivariate stochastic approximation usinga simultaneous perturbation gradient approximationrdquo IEEETransactions on Automatic Control vol 37 no 3 pp 332ndash3411992

[23] J C Spall ldquoImplementation of the simultaneous perturbationalgorithm for stochastic optimizationrdquo IEEE Transactions onAerospace and Electronic Systems vol 34 no 3 pp 817ndash8231998

[24] J C Spall ldquoStochastic optimization and the simultaneousperturbation methodrdquo in Proceedings of the Winter SimulationConference (WSC rsquo99) pp 101ndash109 December 1999

[25] J D Schwartz W Wang and D E Rivera ldquoSimulation-based optimization of process control policies for inventorymanagement in supply chainsrdquo Automatica vol 42 no 8 pp1311ndash1320 2006

[26] H G Neddermeijer G J V Oortmarssen and N P R DekkerldquoA framework for response surface methodology for simulationoptimizationrdquo in Proceedings of the Winter Simulation Confer-ence vol 1 Orlando Fla USA December 1999

[27] B Can and C Heavey ldquoA comparison of genetic programmingand artificial neural networks in metamodeling of discrete-event simulation modelsrdquo Computers amp Operations Researchvol 39 no 2 pp 424ndash436 2012

[28] A Azadeh Z S Faiz S M Asadzadeh and R Tavakkoli-Moghaddam ldquoAn integrated artificial neural network-comput-er simulation for optimization of complex tandem queuesystemsrdquoMathematics and Computers in Simulation vol 82 no4 pp 666ndash678 2011

[29] R Bornatico J Hussy A Witzig and L Guzzella ldquoSurrogatemodeling for the fast optimization of energy systemsrdquo Energyvol 57 pp 653ndash662 2013

[30] X Wan J F Pekny and G V Reklaitis ldquoSimulation-basedoptimizationwith surrogatemodels application to supply chainmanagementrdquoComputers and Chemical Engineering vol 29 no6 pp 1317ndash1328 2005

[31] W Shi Z Liu J Shang and Y Cui ldquoMulti-criteria robust designof a JIT-based cross-docking distribution center for an autoparts supply chainrdquo European Journal of Operational Researchvol 229 no 3 pp 695ndash706 2013

[32] G Dellino J P C Kleijnen and C Meloni ldquoRobust optimiza-tion in simulation taguchi and Response Surface Methodol-ogyrdquo International Journal of Production Economics vol 125 no1 pp 52ndash59 2010

[33] T Yang Y Wen and F Wang ldquoEvaluation of robustness of sup-ply chain information-sharing strategies using a hybrid Taguchiand multiple criteria decision-making methodrdquo InternationalJournal of Production Economics vol 134 no 2 pp 458ndash4662011

[34] M E Kuhl and J R Wilson ldquoModeling and simulating Poissonprocesses having trends or nontrigonometric cyclic effectsrdquoEuropean Journal of Operational Research vol 133 no 3 pp566ndash582 2001

[35] P Narayan and J Subramanian Inventory ManagementmdashPrinciples and Practices Excel Books New Delhi India 2008

[36] E A Silver and R Peterson Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 1985

[37] R G Brown Smoothing Forecasting and Prediction of DiscreteTime Series Courier 2004

[38] D C Montgomery Design and analysis of experiments JohnWiley amp Sons New York NY USA 6th edition 2005

[39] J Zhu ldquoData envelopment analysis vs principal componentanalysis an illustrative study of economic performance ofChinese citiesrdquo European Journal of Operational Research vol111 no 1 pp 50ndash61 1998

[40] I M Premachandra ldquoA note on DEA vs principal componentanalysis an improvement to Joe Zhursquos approachrdquo EuropeanJournal of Operational Research vol 132 no 3 pp 553ndash5602001

[41] D Slottje G W Scully G J Hirschberg and K Hayes Mea-suring the Quality of Life across Countries A MultidimensionalAnalysis Westview Press Boulder Colo USA 1991

[42] S Opricovic Multicriteria optimization of civil engineeringsystems [PhD thesis] Faculty of Civil Engineering BelgradeSerbia 1998

[43] P L Yu ldquoA class of solutions for group decision problemsrdquoManagement Science vol 19 no 8 pp 936ndash946 1973

Modelling and Simulation in Engineering 17

[44] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[45] S Opricovic and G H Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 15: Research Article Robust Inventory System Optimization ...downloads.hindawi.com/journals/mse/2014/305120.pdf · Robust Inventory System Optimization Based on Simulation and Multiple

Modelling and Simulation in Engineering 15

119877119901should be controlled strictly to prevent inventory in hand

from reaching lower than 20 productsIn the case of product 119861 (C1) is satisfied so there is a

unique scenario while there are four compromising solutionfor product 119862 In this situation based on VIKOR methodcompromising solutions have same value while interactionplot can be applied for investigation of existing differencebetween compromising solutions As mentioned before thebest scenario suggests FI inventory policy for this product butin compromising solutions there are both FI and DF policiesLead time varies from 3 to 7 days So this confusing situationis resolved by deeper analysis and applying interaction plotFigure 4 shows interaction plot for DOE factors and averageof inventory position as response variables

Figure 4 shows that with either DF or FI policy there isno significant difference among different lead times in theview point of inventory position but DF leads to less averageof inventory position With 119877

119901equaling 15 there is less

sensitivity to different lead times So configuring 119877119901which

is equal to 15 products the average of inventory positioncriterion will be robust against lead time variations

Figure 5 regards cost criterion and shows that FI andDF policies are equal and make least sensitivity related tolead time On the other hand three plots of 119877

119901against lead

time are parallel The parallel plots are interpreted as lackof relation between 119877

119901and lead time With 119877

119901equaling 15

increase in inventory cost could be mitigated when lead timesets to 3 days

In Figure 6 with 3 days for lead time there is no signifi-cant difference between inventory policies in the view pointof service level119877

119901and lead time are independent and the best

service level is related to FI policy which is also very sensitiveto 119877119901 As 119877

119901increases service level will grow According to

the interaction plot analyses optimal decision for product119862 is FI inventory policy 3 days lead time and 15 or moreproducts for 119877

119901

Decision about product 119863 is simple Based on the in-formation obtained from solutions it is recommended toemploy DF inventory policy with 119877

119901being equal to 15

products and 3 days for lead time

6 Conclusion

In this paper a simulation optimization framework is pro-posed for robust optimization of retailer inventory systemThis framework is less time consuming in comparison withmetaheuristic or SPSA approach it also provides robustsolution for multiple objective functions In this research alsotrend and cyclic change of customers demand are consideredfor more realistic view of problem Proposed framework con-sists of discrete event simulation and surrogate model In thisframework full factorial design of experiment is employedfor producing decision space Based on the three decisionfactors 27 different scenarios are produced and simulated Insimulation modelling multiresolution method is applied forsimulation of demand with highly dynamic pattern Becausethere are multiple criteria for inventory system performanceVIKORmethod is employed asmulticriteria decisionmaking

technique In addition robustness is considered as perfor-mance criterion and PCA is used for improving performanceof VIKOR method Finally developed framework gave thebest ranked and compromising solutions so interaction plotapplied for more investigation and sensitivity analysis ofobtained solutions

Due to the novelty of the proposed framework it isrecommended that future studies encompass optimization ofinventory system of integrated supply chain Also improve-ment of developed framework can be considered as futurestudy

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers for theirsignificant roles in improvement of this research quality

References

[1] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[2] S Yin S XDingAHaghaniHHao andP Zhang ldquoA compar-ison study of basic data-driven fault diagnosis and processmonitoring methods on the benchmark Tennessee Eastmanprocessrdquo Journal of Process Control vol 22 no 9 pp 1567ndash15812012

[3] S Yin S X Ding A H A Sari and H Hao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[4] S Yin G Wang and H R Karimi ldquoData-driven design ofrobust fault detection system for wind turbinesrdquo Mechatronicsvol 24 no 4 pp 298ndash306 2014

[5] G Hadley and T M Whitin Analysis of Inventory SystemsPrentice-Hall International Series in Management New YorkNY USA 1963

[6] E Naddor Inventory Systems John Wiley amp Sons New YorkNY USA 1966

[7] R Peterson and E A Silver Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 2nd edition 1985

[8] T C E Cheng ldquoEconomic order quantity model with demand-dependent unit production cost and imperfect productionprocessesrdquo IIE Transactions vol 23 no 1 pp 23ndash28 1991

[9] S H Chen C CWang and A Ramer ldquoBackorder fuzzy inven-tory model under function principlerdquo Information Sciences vol95 no 1-2 pp 71ndash79 1996

[10] X Zhao J Xie and J C Wei ldquoThe value of early order com-mitment in a two-level supply chainrdquo European Journal of Op-erational Research vol 180 no 1 pp 194ndash214 2007

[11] R S M Lau J Xie and X Zhao ldquoEffects of inventory policyon supply chain performance a simulation study of critical

16 Modelling and Simulation in Engineering

decision parametersrdquo Computers and Industrial Engineeringvol 55 no 3 pp 620ndash633 2008

[12] J Xu and L Zhao ldquoA multi-objective decision-making modelwith fuzzy rough coefficients and its application to the inventoryproblemrdquo Information Sciences vol 180 no 5 pp 679ndash6962010

[13] F Hnaien X Delorme and A Dolgui ldquoMulti-objective opti-mization for inventory control in two-level assembly systemsunder uncertainty of lead timesrdquo Computers amp OperationsResearch vol 37 no 11 pp 1835ndash1843 2010

[14] H D Purnomo H M Wee and Y Praharsi ldquoTwo inventoryreview policies on supply chain configuration problemrdquo Com-puters and Industrial Engineering vol 63 no 2 pp 448ndash4552012

[15] B L Nelson ldquo50th anniversary article stochastic simulationresearch in management sciencerdquoManagement Science vol 50no 7 pp 855ndash868 2004

[16] J Boesel R O Bowden Jr F Glover J P Kelly and E WestwigldquoFuture of simulation optimizationrdquo inProceedings of theWinterSimulation Conference pp 1466ndash1469 Arlington Va USADecember 2001

[17] M C Fu ldquoOptimization for simulation theory vs practicerdquoINFORMS Journal on Computing vol 14 no 3 pp 192ndash2152002

[18] L Wang ldquoA hybrid genetic algorithm-neural network strategyfor simulation optimizationrdquoApplied Mathematics and Compu-tation vol 170 no 2 pp 1329ndash1343 2005

[19] B B Keskin S H Melouk and I L Meyer ldquoA simulation-optimization approach for integrated sourcing and inventorydecisionsrdquo Computers and Operations Research vol 37 no 9pp 1648ndash1661 2010

[20] E Mazhari J Zhao N Celik S Lee Y Son and L HeadldquoHybrid simulation and optimization-based design and oper-ation of integrated photovoltaic generation storage units andgridrdquo Simulation Modelling Practice and Theory vol 19 no 1pp 463ndash481 2011

[21] Q Duan and T W Liao ldquoOptimization of replenishment poli-cies for decentralized and centralized capacitated supply chainsunder various demandsrdquo International Journal of ProductionEconomics vol 142 no 1 pp 194ndash204 2013

[22] J C Spall ldquoMultivariate stochastic approximation usinga simultaneous perturbation gradient approximationrdquo IEEETransactions on Automatic Control vol 37 no 3 pp 332ndash3411992

[23] J C Spall ldquoImplementation of the simultaneous perturbationalgorithm for stochastic optimizationrdquo IEEE Transactions onAerospace and Electronic Systems vol 34 no 3 pp 817ndash8231998

[24] J C Spall ldquoStochastic optimization and the simultaneousperturbation methodrdquo in Proceedings of the Winter SimulationConference (WSC rsquo99) pp 101ndash109 December 1999

[25] J D Schwartz W Wang and D E Rivera ldquoSimulation-based optimization of process control policies for inventorymanagement in supply chainsrdquo Automatica vol 42 no 8 pp1311ndash1320 2006

[26] H G Neddermeijer G J V Oortmarssen and N P R DekkerldquoA framework for response surface methodology for simulationoptimizationrdquo in Proceedings of the Winter Simulation Confer-ence vol 1 Orlando Fla USA December 1999

[27] B Can and C Heavey ldquoA comparison of genetic programmingand artificial neural networks in metamodeling of discrete-event simulation modelsrdquo Computers amp Operations Researchvol 39 no 2 pp 424ndash436 2012

[28] A Azadeh Z S Faiz S M Asadzadeh and R Tavakkoli-Moghaddam ldquoAn integrated artificial neural network-comput-er simulation for optimization of complex tandem queuesystemsrdquoMathematics and Computers in Simulation vol 82 no4 pp 666ndash678 2011

[29] R Bornatico J Hussy A Witzig and L Guzzella ldquoSurrogatemodeling for the fast optimization of energy systemsrdquo Energyvol 57 pp 653ndash662 2013

[30] X Wan J F Pekny and G V Reklaitis ldquoSimulation-basedoptimizationwith surrogatemodels application to supply chainmanagementrdquoComputers and Chemical Engineering vol 29 no6 pp 1317ndash1328 2005

[31] W Shi Z Liu J Shang and Y Cui ldquoMulti-criteria robust designof a JIT-based cross-docking distribution center for an autoparts supply chainrdquo European Journal of Operational Researchvol 229 no 3 pp 695ndash706 2013

[32] G Dellino J P C Kleijnen and C Meloni ldquoRobust optimiza-tion in simulation taguchi and Response Surface Methodol-ogyrdquo International Journal of Production Economics vol 125 no1 pp 52ndash59 2010

[33] T Yang Y Wen and F Wang ldquoEvaluation of robustness of sup-ply chain information-sharing strategies using a hybrid Taguchiand multiple criteria decision-making methodrdquo InternationalJournal of Production Economics vol 134 no 2 pp 458ndash4662011

[34] M E Kuhl and J R Wilson ldquoModeling and simulating Poissonprocesses having trends or nontrigonometric cyclic effectsrdquoEuropean Journal of Operational Research vol 133 no 3 pp566ndash582 2001

[35] P Narayan and J Subramanian Inventory ManagementmdashPrinciples and Practices Excel Books New Delhi India 2008

[36] E A Silver and R Peterson Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 1985

[37] R G Brown Smoothing Forecasting and Prediction of DiscreteTime Series Courier 2004

[38] D C Montgomery Design and analysis of experiments JohnWiley amp Sons New York NY USA 6th edition 2005

[39] J Zhu ldquoData envelopment analysis vs principal componentanalysis an illustrative study of economic performance ofChinese citiesrdquo European Journal of Operational Research vol111 no 1 pp 50ndash61 1998

[40] I M Premachandra ldquoA note on DEA vs principal componentanalysis an improvement to Joe Zhursquos approachrdquo EuropeanJournal of Operational Research vol 132 no 3 pp 553ndash5602001

[41] D Slottje G W Scully G J Hirschberg and K Hayes Mea-suring the Quality of Life across Countries A MultidimensionalAnalysis Westview Press Boulder Colo USA 1991

[42] S Opricovic Multicriteria optimization of civil engineeringsystems [PhD thesis] Faculty of Civil Engineering BelgradeSerbia 1998

[43] P L Yu ldquoA class of solutions for group decision problemsrdquoManagement Science vol 19 no 8 pp 936ndash946 1973

Modelling and Simulation in Engineering 17

[44] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[45] S Opricovic and G H Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 16: Research Article Robust Inventory System Optimization ...downloads.hindawi.com/journals/mse/2014/305120.pdf · Robust Inventory System Optimization Based on Simulation and Multiple

16 Modelling and Simulation in Engineering

decision parametersrdquo Computers and Industrial Engineeringvol 55 no 3 pp 620ndash633 2008

[12] J Xu and L Zhao ldquoA multi-objective decision-making modelwith fuzzy rough coefficients and its application to the inventoryproblemrdquo Information Sciences vol 180 no 5 pp 679ndash6962010

[13] F Hnaien X Delorme and A Dolgui ldquoMulti-objective opti-mization for inventory control in two-level assembly systemsunder uncertainty of lead timesrdquo Computers amp OperationsResearch vol 37 no 11 pp 1835ndash1843 2010

[14] H D Purnomo H M Wee and Y Praharsi ldquoTwo inventoryreview policies on supply chain configuration problemrdquo Com-puters and Industrial Engineering vol 63 no 2 pp 448ndash4552012

[15] B L Nelson ldquo50th anniversary article stochastic simulationresearch in management sciencerdquoManagement Science vol 50no 7 pp 855ndash868 2004

[16] J Boesel R O Bowden Jr F Glover J P Kelly and E WestwigldquoFuture of simulation optimizationrdquo inProceedings of theWinterSimulation Conference pp 1466ndash1469 Arlington Va USADecember 2001

[17] M C Fu ldquoOptimization for simulation theory vs practicerdquoINFORMS Journal on Computing vol 14 no 3 pp 192ndash2152002

[18] L Wang ldquoA hybrid genetic algorithm-neural network strategyfor simulation optimizationrdquoApplied Mathematics and Compu-tation vol 170 no 2 pp 1329ndash1343 2005

[19] B B Keskin S H Melouk and I L Meyer ldquoA simulation-optimization approach for integrated sourcing and inventorydecisionsrdquo Computers and Operations Research vol 37 no 9pp 1648ndash1661 2010

[20] E Mazhari J Zhao N Celik S Lee Y Son and L HeadldquoHybrid simulation and optimization-based design and oper-ation of integrated photovoltaic generation storage units andgridrdquo Simulation Modelling Practice and Theory vol 19 no 1pp 463ndash481 2011

[21] Q Duan and T W Liao ldquoOptimization of replenishment poli-cies for decentralized and centralized capacitated supply chainsunder various demandsrdquo International Journal of ProductionEconomics vol 142 no 1 pp 194ndash204 2013

[22] J C Spall ldquoMultivariate stochastic approximation usinga simultaneous perturbation gradient approximationrdquo IEEETransactions on Automatic Control vol 37 no 3 pp 332ndash3411992

[23] J C Spall ldquoImplementation of the simultaneous perturbationalgorithm for stochastic optimizationrdquo IEEE Transactions onAerospace and Electronic Systems vol 34 no 3 pp 817ndash8231998

[24] J C Spall ldquoStochastic optimization and the simultaneousperturbation methodrdquo in Proceedings of the Winter SimulationConference (WSC rsquo99) pp 101ndash109 December 1999

[25] J D Schwartz W Wang and D E Rivera ldquoSimulation-based optimization of process control policies for inventorymanagement in supply chainsrdquo Automatica vol 42 no 8 pp1311ndash1320 2006

[26] H G Neddermeijer G J V Oortmarssen and N P R DekkerldquoA framework for response surface methodology for simulationoptimizationrdquo in Proceedings of the Winter Simulation Confer-ence vol 1 Orlando Fla USA December 1999

[27] B Can and C Heavey ldquoA comparison of genetic programmingand artificial neural networks in metamodeling of discrete-event simulation modelsrdquo Computers amp Operations Researchvol 39 no 2 pp 424ndash436 2012

[28] A Azadeh Z S Faiz S M Asadzadeh and R Tavakkoli-Moghaddam ldquoAn integrated artificial neural network-comput-er simulation for optimization of complex tandem queuesystemsrdquoMathematics and Computers in Simulation vol 82 no4 pp 666ndash678 2011

[29] R Bornatico J Hussy A Witzig and L Guzzella ldquoSurrogatemodeling for the fast optimization of energy systemsrdquo Energyvol 57 pp 653ndash662 2013

[30] X Wan J F Pekny and G V Reklaitis ldquoSimulation-basedoptimizationwith surrogatemodels application to supply chainmanagementrdquoComputers and Chemical Engineering vol 29 no6 pp 1317ndash1328 2005

[31] W Shi Z Liu J Shang and Y Cui ldquoMulti-criteria robust designof a JIT-based cross-docking distribution center for an autoparts supply chainrdquo European Journal of Operational Researchvol 229 no 3 pp 695ndash706 2013

[32] G Dellino J P C Kleijnen and C Meloni ldquoRobust optimiza-tion in simulation taguchi and Response Surface Methodol-ogyrdquo International Journal of Production Economics vol 125 no1 pp 52ndash59 2010

[33] T Yang Y Wen and F Wang ldquoEvaluation of robustness of sup-ply chain information-sharing strategies using a hybrid Taguchiand multiple criteria decision-making methodrdquo InternationalJournal of Production Economics vol 134 no 2 pp 458ndash4662011

[34] M E Kuhl and J R Wilson ldquoModeling and simulating Poissonprocesses having trends or nontrigonometric cyclic effectsrdquoEuropean Journal of Operational Research vol 133 no 3 pp566ndash582 2001

[35] P Narayan and J Subramanian Inventory ManagementmdashPrinciples and Practices Excel Books New Delhi India 2008

[36] E A Silver and R Peterson Decision Systems for InventoryManagement and Production Planning JohnWiley amp Sons NewYork NY USA 1985

[37] R G Brown Smoothing Forecasting and Prediction of DiscreteTime Series Courier 2004

[38] D C Montgomery Design and analysis of experiments JohnWiley amp Sons New York NY USA 6th edition 2005

[39] J Zhu ldquoData envelopment analysis vs principal componentanalysis an illustrative study of economic performance ofChinese citiesrdquo European Journal of Operational Research vol111 no 1 pp 50ndash61 1998

[40] I M Premachandra ldquoA note on DEA vs principal componentanalysis an improvement to Joe Zhursquos approachrdquo EuropeanJournal of Operational Research vol 132 no 3 pp 553ndash5602001

[41] D Slottje G W Scully G J Hirschberg and K Hayes Mea-suring the Quality of Life across Countries A MultidimensionalAnalysis Westview Press Boulder Colo USA 1991

[42] S Opricovic Multicriteria optimization of civil engineeringsystems [PhD thesis] Faculty of Civil Engineering BelgradeSerbia 1998

[43] P L Yu ldquoA class of solutions for group decision problemsrdquoManagement Science vol 19 no 8 pp 936ndash946 1973

Modelling and Simulation in Engineering 17

[44] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[45] S Opricovic and G H Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 17: Research Article Robust Inventory System Optimization ...downloads.hindawi.com/journals/mse/2014/305120.pdf · Robust Inventory System Optimization Based on Simulation and Multiple

Modelling and Simulation in Engineering 17

[44] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[45] S Opricovic and G H Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Page 18: Research Article Robust Inventory System Optimization ...downloads.hindawi.com/journals/mse/2014/305120.pdf · Robust Inventory System Optimization Based on Simulation and Multiple

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of