An interactive possibilistic programming approach for multiple objective supply chain master planning

Fuzzy Sets and Systems 159 (2008) 193–214www.elsevier.com/locate/fss

An interactive possibilistic programming approach for multipleobjective supply chain master planning

S.A. Torabia,∗,1, E. Hassinib

aDepartment of Industrial Engineering, Faculty of Engineering, University of Tehran, Tehran, IranbDeGroote School of Business, McMaster University, 1280 Main St. W., Hamilton, ON, Canada L8S 4M4

Received 20 March 2007; received in revised form 27 June 2007; accepted 25 August 2007Available online 31 August 2007

Abstract

Providing an efficient production plan that integrates the procurement and distribution plans into a unified framework is critical toachieving competitive advantage. In this paper, we consider a supply chain master planning model consisting of multiple suppliers, onemanufacturer and multiple distribution centers. We first propose a new multi-objective possibilistic mixed integer linear programmingmodel (MOPMILP) for integrating procurement, production and distribution planning considering various conflicting objectivessimultaneously as well as the imprecise nature of some critical parameters such as market demands, cost/time coefficients and capacitylevels. Then, after applying appropriate strategies for converting this possibilistic model into an auxiliary crisp multi-objective linearmodel (MOLP), we propose a novel interactive fuzzy approach to solve this MOLP and finding a preferred compromise solution.The proposed model and solution method are validated through numerical tests. Computational results indicate that the proposedfuzzy method outperforms other fuzzy approaches and is very promising not only for solving our problem, but also for any MOLPmodel especially multi-objective mixed integer models.© 2007 Elsevier B.V. All rights reserved.

Keywords: Possibilistic programming; Supply chain master planning; Mixed-integer linear programs; Compromise solution

1. Introduction

The main focus of supply chain management (SCM) is the control of material flow among suppliers, plants, ware-houses and customers efficiently such that the total cost in the supply chain can be minimized [38]. A major thrust ofrecent research in this area is the development of optimization models that integrate different functions (e.g. purchasing,production and distribution) in the supply chain. The basic idea behind this approach is to simultaneously optimizedecision variables of different functions that have traditionally been optimized sequentially [26]. In this regard, one ofthe main issues facing with supply chain managers is the supply chain master planning (SCMP) problem. The majortask of SCMP is the determination of procurement, production and distribution quantities for facilities in differentechelons of a supply chain on a medium term basis [27]. Traditionally, these activities were conducted either indepen-dently or sequentially resulting in large inventories and very poor overall performance. But in the presence of emerging

∗ Corresponding author. Tel.: +9821 88021067; fax: +9821 88013102.E-mail addresses: [email protected] (S.A. Torabi), [email protected] (E. Hassini).

1 Currently a Post Doctoral Fellow at DeGroote School of Business, McMaster University, Hamilton, ON, Canada.

0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2007.08.010

http://www.elsevier.com/locate/fss

mailto:[email protected]

mailto:[email protected]

194 S.A. Torabi, E. Hassini / Fuzzy Sets and Systems 159 (2008) 193–214

computing power and increasing global competitive pressures, coordinating the procurement, production and distribu-tion plans in an integrated and centralized framework is critical to the success of supply chains.

In such environments, where decisions involve resources and data that are owned by different entities within thesupply chain, there are two paramount characteristics of the problems that a decision maker will be faced with: (1)Conflicting objectives that may arise from the nature of operations (e.g., minimize cost and at the same time increasecustomer service) and the structure of the supply chain where it is often difficult to align the goals of the differentparties within the supply chain. (2) Lack of precise data (e.g., cost and lead time data) and/or presence of uncertaintywith imprecise parameters (e.g., demand fuzziness). Thus, it is important that models addressing problems in this areashould be designed to handle the foregoing two complexities.

In this paper we present a novel SCMP model consisting of multiple suppliers, one manufacturer and multipledistribution centers which integrates the procurement, production and distribution plans considering various conflict-ing objectives simultaneously as well as the imprecise nature of some critical parameters such as market demands,cost/time coefficients and capacity levels. We consider two important objective functions: the total cost of logisticsand the total value of purchasing. For constructing the former objective, we have used the modern concepts from thecosting literature known as the total cost of ownership (TCO) and activity-based costing (ABC) in order to have acomprehensive total cost function. The paper has two important applied and theoretical contributions. First, it presentsa comprehensive and practical, but tractable, optimization model for supply chain master planning. The need for suchmodel by practitioners, for example for incorporation in advanced planning systems (APS), has been highlighted byTempelmeier [37]. And second, it introduces a novel solution procedure for finding an efficient compromise solutionto a fuzzy multi-objective mixed-integer program. In our literature survey we have felt a lack of studies in this field,something that is understandable, given that mixed integer programming is known to be complex even when all datais certain and precise.

The remainder of this paper is organized as follows. The relevant literature is reported in Section 2. In Section 3 wedefine our notation, state our assumptions and propose a new multi-objective possibilistic mixed integer linear program(MOPMILP) for the proposed SCMP problem. After applying appropriate strategies for converting the possibilisticmodel into an auxiliary crisp multi-objective linear model (MOLP), we propose a novel interactive fuzzy approach tosolve this MOLP and find an efficient compromise solution in Section 4. The proposed model and solution methodare validated through numerical tests in Section 5. The data for these numerical computations have been inspired by areal life industrial case as well as randomly generated data. Conclusion remarks about our computational results andfurther research directions are the subject of Section 6.

2. Literature review

The considered SCMP problem deals with a medium term procurement, production and distribution planning prob-lem in a three-echelon supply chain involving multiple suppliers, one manufacturer and multiple distribution cen-ters. Based on characteristics of the problem which are explained in more details in the next section, we review themost relevant and recent literature in three different but somewhat close streams: supplier selection and order lot-sizing models, dynamic supply chain planning models, and application of fuzzy modeling in supply chain planningproblems.

2.1. Supplier selection and order lot-sizing

Ghodsypour and O’Brien [9] developed an integrated analytical hierarchy process (AHP) and linear programmingapproach for solving a multi-objective, single-item, single-period capacitated supplier selection and order lot-sizingproblem. Xia and Wu [41] introduced a new model that integrates an improved AHP by rough sets theory witha multi-objective mixed integer program to support supplier selection decisions in total business volume discountsenvironments. Hassini [10] presented a supplier selection and order lot-sizing model for a single item, multiple period,multiple capacitated suppliers offering lead time-dependent capacity reservation and unit price discounts. Sirias andMehra [33] studied quantity-dependent discounts versus lead time-dependent discounts in supply chains through asimulation study. They concluded that the lead time-dependent discount systems could be more promising for supplychains, especially for the manufacturing side. For additional studies in this area, the interested reader is referred to acomprehensive review provided by Aissaoui et al. [2].

S.A. Torabi, E. Hassini / Fuzzy Sets and Systems 159 (2008) 193–214 195

2.2. Dynamic supply chain planning

Literature review of the SCMP area reveals that there are considerable amount of papers integrating production anddistribution decisions (see, for example, [3,8,21,26]). But, there are very limited papers integrating procurement anddistribution stages with production. Kanyalkar and Adil [14] considered a single supplier, multi-plant, multi-selling lo-cation and multi-dock problem and presented a comprehensive monolithic crisp mixed integer linear goal programmingmodel to obtain a time and capacity aggregated production plan, a detailed production plan, a detailed procurementplan and a detailed distribution plan simultaneously to overcome the drawbacks of the hierarchical/sequential planningapproaches of not yielding a feasible and/or an optimal plan. Noorul Haq and Kannan [25] developed an integratedsupplier selection and multi-echelon distribution inventory model in a built-to-order supply chain involving singleselected supplier, multiple plants, multiple distributors, multiple wholesalers and multiple retailers. However, in thesupply side of the above integrated models, authors only considered the single supplier and/or single-item cases withoutany discount structure. Sucky [34] stated that the coordination of order and production policies between buyers andsuppliers in supply chains is of particular interest and a coordinated order and production policy can reduce total costsignificantly.

2.3. Applications of fuzzy modeling in supply chain planning

Hsu and Wang [11] provided a possibilistic linear programming model to determine appropriate strategies regardingthe safety stock levels for assembly materials, regulating dealers’ forecast demands and numbers of key machines.Wang and Liang [39] presented an interactive possibilistic linear programming approach for solving the multi-productaggregate production planning problem with imprecise forecast demand, related operating costs and capacity. Kumaret al. [15] proposed a fuzzy programming model with flexibility in some constraints for a single-item, single periodvendor selection problem. Mula et al. [23] provided a new linear programming model for the medium term productionplanning in a capacitated multi-product, multi-level and multi-period MRP system, and transformed it into the threedifferent auxiliary crisp models with flexibility in the objective function, market demand and available capacity resourcesusing different fuzzy aggregation operators. Liang [20] developed an interactive multi-objective linear programmingmodel for solving the fuzzy multi-objective transportation problems with piecewise linear membership function. Inanother work, Liang [21] proposed an interactive fuzzy multi-objective linear programming model for solving anintegrated production-transportation planning problem in supply chains. Selim and Ozkarahan [32] developed aninteractive fuzzy goal programming for the supply chain distribution network design. Chen and Chang [5] developedan approach for deriving the membership function of the fuzzy minimum total cost in a multi-product, multi-echelon,multi-period supply chain with fuzzy parameters using �-cut representation and the fuzzy extension principle. However,in most of the aforementioned works [11,15,20,21,39], the authors have applied the max–min approach of Zimmermann[42] to solve the auxiliary single-objective model. But, it is well-known that the solution yielded by max–min operatormight not be unique nor efficient [17–19].

To the best of our knowledge, there is no research work in the supply chain master planning literature integratingprocurement, production and distribution planning activities in a fuzzy environment. Hence, in this study we developa comprehensive multi-objective SCMP framework for a three-echelon supply chain involving multiple suppliers,one manufacturer and multiple distribution centers in a fuzzy environment to determine the coordinated purchasing,production and distribution quantities over a given multi-period decision horizon. The main contributions of this papercan be summarized as follows:

• Introducing a novel SCMP model in a fuzzy environment for integrating different activities in a multi-echelon, multi-product and multi-period supply chain network, where we consider a simultaneous quantity and lead time-dependentdiscount policy for the multi-item, multi-period supplier selection and order lot-sizing sub-problem.

• Proposing a new fuzzy programming approach for solving the auxiliary crisp multi-objective linear programmingmodel to provide a preferred compromise solution for the decision maker.

It should be noted that unlike previous studies that assume dynamic deterministic demand (e.g., [6,10]), criticalparameters (such as market demands and capacity levels) are imprecise (fuzzy) in nature due to incompleteness and/orunavailability of required data over the mid-term decision horizon. In such conditions, the retailer/distributor knows itsdemand requirements almost certainly, but quotes it in an imprecise manner, e.g., 100±10 units. Therefore, we have to


estimate the problem parameters subjectively based on current insufficient data and the decision maker’s experience.That is why in this paper we chose to apply a fuzzy modeling approach. For a recent review of different approachesfor dealing with uncertainty in production planning problems the reader is advised to consult Mula et al. [24].

The proposed SCMP model could be used by every manufacturer dealing with different suppliers offering lead time-dependent discounts as well as multiple customer zones dispersed in different areas for distributing its final products.This seems to be especially relevant for global chemical processing companies, as has been mentioned to one of theauthors by a chemical engineer that attended a conference presentation of an earlier version of this study. Furthermore,the proposed multi-objective framework enables the manufacturer to consider the various criteria in developing anintegrated procurement, production and distribution plan in order to reach a compromise solution considering differentsupply parties’ concerns (i.e., implementing the win–win strategy).

3. Problem description and formulation

A manufacturer produces different types of products using a common set of input items (i.e., raw materials and/or sub-assembly parts) which are supplied from a pre-determined set of qualified suppliers. The final products are then deliveredto different distribution centers (warehouses/wholesalers) in order to satisfy their associated dynamic demands. Thisproblem in fact integrates three planning sub-problems: (1) supplier selection and order lot-sizing in the first echelon,(2) production planning at the manufacturer in the second echelon, and (3) distribution planning in the last echelonof the supply chain. Our objective is to find the best planning decisions over a multi-period mid-term horizon, in anintegrated and coordinated manner, for the following issues:

• Procurement plan: The purchase quantity for each item from each supplier in each period.• Production plan: The production quantity for each final product in each period.• Distribution plan: The number of each final product to be delivered to each distribution center in each period.

In order to develop the supply chain master plan, we consider two quantitative objectives: minimizing the total costof logistics and maximizing the total value of purchasing which are the most important objectives in SCMP problem(e.g., [14,25,41]).

Decision making in such a complex supply chain network requires considering conflicting objectives as well asdifferent constraints imposed by the suppliers, manufacturers and distributors. Moreover, in practical situations, mostof the parameters embedded in SCMP problem are frequently fuzzy in nature because of incompleteness and/orunavailability of required data over the mid-term horizon, and can be just obtained subjectively [5,39]. For example,in a real decision problem, market demands, cost/time coefficients and amount of available resources are usuallyimprecise over the planning horizon, and therefore assigning a set of crisp values for such ambiguous parameters is notappropriate. We rely on possibility theory to model this fuzziness. This theory uses possibility distributions to handlethis inherent ambiguous phenomenon in the problem parameters [11,16,20,39].

3.1. Problem assumptions and notation

Below are the main characteristics and assumptions used in the problem formulation:

• The final products have deterministic dynamic demand at each distribution center over a given finite planning horizon(which is usually between 3 and 6 months).

• A pool of pre-determined qualified suppliers is given.• The production system at the manufacturer and each supplier is aggregated into a capacitated single stage system

(i.e., by focusing on the bottleneck stage).• Production capacities at the manufacturer and suppliers are estimated by considering a rough estimate of various

contingencies (such as set-ups and machine break downs) and possible capacity expansions (using overtime and/orsubcontracting).

• The customers do not wait to receive their orders in a future period. Therefore, inventory shortages (stockouts) arenot allowed at each echelon.

• It is assumed that the different supply parties have been located in close proximity, so the procurement and distributionlead times are negligible (in contrast to the length of each planning period).


Fig. 1. The triangular possibility distribution of fuzzy parameter n.

• Due to competitive environments in bidding, there is some flexibility in the replenishment scheme of each supplier. Inparticular, they offer lead time-dependent capacity reservation and unit price discounts in their discount schedule byconsidering possible capacity expansions in the future periods. The practical situations of such lead time-dependentdiscount schemes can be found in several industries such as the biologics and semiconductor industries [10,28] wherethe suppliers and their customers engage in flexible quantity contracts that envision different pricing and capacityavailabilities depending on lead times.

• Each supplier may indicate a minimum acceptable utilized capacity (MAUC) in each period based on a correspondingminimum acceptable utilization rate (MAUR) of its reserved capacity. That is, each supplier only accepts orderswhich for the utilized capacity would be equal or greater than an economic pre-defined amount. Considering suchcapacity constraints in the model can improve the win–win strategy between supply chain partners and enforces themanufacturer to select fewer suppliers in each period.

• Inventory holding costs are linear and non-decreasing in succeeding echelons.• Due to incompleteness and/or unavailability of required data over the mid-term decision horizon, critical parameters

(such as market demands and capacity levels) are assumed to be imprecise (fuzzy) in nature. Furthermore, the patternof triangular fuzzy number is adopted to represent each fuzzy parameter. The triangular possibility distribution isthe most common tool for modeling the imprecise nature of the ambiguous parameters due to its computationalefficiency and simplicity in data acquisition [20,42]. Generally, a possibility distribution can be stated as the degreeof occurrence of an event with imprecise data. Fig. 1 presents the triangular possibility distribution of fuzzy numbern = (np, nm, no), where np, nm and no are the most pessimistic value, the most possible value, and the most optimisticvalue of n estimated by a decision maker.

The indices, parameters and variables used to formulate the problem mathematically are described below. We use thesuperscripts 1, 2 and 3 to denote the different echelons of the supply chain, i.e., the suppliers’ level, the manufacturer’sand the distributors’ level, respectively.

Indices:

i index of items (i = 1, . . ., I )

j index of suppliers (j = 1, . . ., J )

k index of final products (k = 1, . . ., K)

� index of distribution centers (� = 1, . . ., L)

t index of time periods (t = 1, . . ., T )

r index of discount intervals (which are supplier-dependent)

Parameters:

dk�t demand of final product k at the distribution center � in period tsi set of qualified suppliers offering item i (i.e., si ⊆ {1, . . . , J })pj set of items offered by supplier j (i.e., pj ⊆ {1, . . . , I })DIj number of discount intervals offered by supplier j, (DIj �T )

SDjr {Ljr , . . ., Ujr }; the mutually exclusive sub-intervals from {1, . . ., T } representing the rth discount interval proposed by supplierj · Ljr and Ujr are the shortest and longest periods within this interval, respectively

urjr minimum acceptable capacity utilization rate for supplier j at each period of rth discount interval (i.e., urj t = urjr , ∀t ∈ SDjr )

cap1jr reserved production capacity of supplier j at each period of rth discount interval (i.e., cap1

j t = cap1jr , ∀t ∈ SDjr )

a1ij unit capacity requirement at supplier j for item i


cap2t production capacity of manufacturer at period t

a2k unit capacity requirement at manufacturer for final product k

bik quantity of item i required to produce one unit of final product kvri unit storage volume required for item iW 2

r storage capacity (in volume) of receiving warehouse at the manufacturervfk unit storage volume required for final product kW 2

f storage capacity (in volume) of shipping warehouse at the manufacturerW 3

� storage capacity (in volume) at the distribution center �

TCO total cost of ownership (i.e., the total purchasing costs)TCP total production costsTCD total distribution costsTC total costs of logistics systemslc total supplier level costs over the planning horizonslcj total costs associated with supplier j over the planning horizonolc total order level costs over the planning horizonolcj t total costs incurred by placing an order to supplier j in period t (such as ordering, transportation and receiving costs per order)ulc total unit level costs over the planning horizoncijr unit price of item i charged by supplier j at each period of rth discount interval of this supplier’s discount schedule (i.e., cij t = cijr ,

∀t ∈ SDjr )

aulcij t additional unit level costs of item i bought from supplier j at period tulcij t total unit level costs of item i bought from supplier j at period tpckt unit variable production cost of final product k at period t (except material cost)t ck�t unit shipping cost of final product k to distribution center � at period t

hr2it unit holding cost of item i at period t (based on the average price in period t)

hf2kt unit holding cost of final product k at the manufacturer in period t

hf3k�t unit holding cost of final product k at the distribution center � in period t

ssk�t safety stock level of final product k at the distribution center � in period tqij average defective rate of item i supplied by supplier jQi manufacturer’s acceptable defective rate for incoming shipments of item islj average service level (the percentage of on-time deliveries) of supplier jSL manufacturer’s acceptable service level per periodRj overall score (weight) of supplier j considering qualitative performance factorsTVP total value of purchasingubxijt an upper bound on xijt value

Decision variables:

xijt purchasing quantity of item i from supplier j in period tpkt production quantity of final product k in period tsk�t shipping quantity of final product k to distribution center � in period tIr2

it ending inventory level of item i at the manufacturer in period tIf 2

kt ending inventory level of final product k at the manufacturer in period tIf 3

k�t ending inventory level of final product k at the distribution center � in period tyjt 1, if an order is placed with supplier j in period t, 0, otherwisezj 1, if an order is placed with supplier j over the decision horizon, 0, otherwise

Note that the lead time-dependent capacity reservations and unit price discounts offered by suppliers indicate thatwe would have: cij,r+1 < cijr , and cap1

j,r+1 > cap1j,r .

3.2. Problem formulation

Generally, fuzzy mathematical programming is classified into two following major classes [13]:

• Fuzzy mathematical programming with vagueness when there is flexibility in the given target values of objectivefunctions and the elasticity of constraints. This class is referred to as flexible programming [15,21,23,32].

• Fuzzy mathematical programming with ambiguous coefficients in objective functions and constraints which is calledpossibilistic programming [11,16,17,20,39].


In flexible programming models, the membership functions of fuzzy objectives and constraints are generallypreference-based and determined by the decision maker subjectively. Possibilistic programming is based on the objec-tive degree of event occurrence for each imprecise data and hence the related possibility distributions are determinedobjectively relying on some available historical data with an analogous to the probability distributions. Our proposedfuzzy programming model lies in the second group because some parameters in the total cost objective function, andsome technological coefficients and right-hand sides in some constraints are ambiguous in nature.

3.2.1. Objective functionsWe consider two important and conflicting objectives in our SCMP problem: the total cost of logistics (TC) and the to-

tal value of purchasing (TVP). Several other recent studies have also considered similar objectives [e.g., 14,21,25,32,41].

3.2.1.1. Objective 1: minimizing the total cost of logistics. The total costs of logistics is the most practical decisionobjective which is usually used in the supply chain master planning models [14,21,25,27]. The total costs of logisticsinclude the total costs of purchasing, production and distribution activities and could be calculated by the followingequation:

Min TC = TCO + TCP + TCD.

In this work, in order to construct a comprehensive total cost function, we have adopted the activity-based costing(ABC) and the total cost of ownership (TCO) concepts. The ABC is an effective costing tool which is being increasinglyapplied in different industries [30]. The ABC enables the cost analyzer to categorize and analyze all the related activitiesand the associated costs of a specific process (e.g. purchasing process) on the basis of their hierarchical structure. Itshould be noted that although this concept provides a more descriptive rather than a normative tool for cost management,it provides a framework to understand cost behavior and translate the hierarchical cost structure of a given process intoa mathematical programming model to make optimal decisions [7].

On the other hand, the TCO is a comprehensive financial estimate approach which reflects all the resources consumedin performing the purchasing-related activities and measures all the costs and benefits associated with these activitieswithin the company’s value chain (i.e. the life cycle costs of the purchased items/services). In other words, TCO couldhelp a company to assess their direct and indirect costs related to the purchase of any capital investment (includinginitial purchase price, repairs, maintenance, and personnel training, among other expenses).

Degraeve et al. [6,7] presented a typical hierarchical structure of purchasing activities consisting of three levels: (1)the supplier level activities, (2) the order level activities and (3) the unit level activities. The first hierarchical leveldescribes costs incurred and conditions imposed whenever the purchasing company actually uses the supplier over thedecision horizon. Examples of costs on the supplier level include a quality audit cost incurred by the buyer for theevaluation of a supplier, and additional research and development costs due to using a particular supplier. The orderlevel parameters indicate costs incurred and conditions imposed each time an order is placed with a particular supplierand include, amongst others, costs associated with reception, invoicing, transportation, ordering and receiving creditnotes. At the unit level we find costs incurred and conditions imposed related to the units of the products for whichthe procurement decision has to be made, for example, price, internal failure (e.g. due to quality problems), externalfailure, and inventory holding costs. It is important to make this classification of activities into separate levels since theoverall cost driver (i.e., number of suppliers, number of orders, number of units procured) for each level of activity isindependent of the activities in other levels.

Applying the ABC and TCO concepts, the total purchasing cost consists of the total supplier level costs, the totalorder level costs and the total unit level costs which could be estimated as follows:

TCO = slc + olc + ulc,

such that:

slc =J∑

j=1

slcj · zj , olc =T∑

t=1

J∑j=1

olcj t · yjt , ulc =T∑

t=1

J∑j=1

∑i∈pj

(cij t+aulcij t ) · xijt+T∑

t=1

I∑i=1

hr2it · Ir2

it .

The unit prices quoted by suppliers are assumed to be crisp, and hence the total unit level costs of item i bought fromsupplier j at period t is estimated by ulcij t = cij t + aulcij t .


The total production cost is equal to the sum of variable production costs (except material costs) and inventoryholding costs of final products at the manufacturer:

TCP =T∑

t=1

K∑k=1

(pckt · pkt + hf2kt · If 2

kt ).

The total distribution cost is equal to the sum of transportation costs and inventory holding costs of final products atthe distributors:

TCD =T∑

t=1

L∑�=1

K∑k=1

(tck�t · sk�t + hf3k�t · If 3

k�t ).

Thus, the objective of minimizing the total cost of logistics can be stated as follows:

min TC =J∑

j=1

slcj · zj +T∑

t=1

J∑j=1

olcj t · yjt +T∑

t=1

J∑j=1

∑i∈pj

(cij t + aulcij t ) · xijt +T∑

t=1

I∑i=1

hr2it ·Ir2

it

+T∑

t=1

K∑k=1

(pckt · pkt + hf2kt · If 2

kt ) +T∑

t=1

L∑�=1

K∑k=1

(tck�t · sk�t + hf3k�t · If 3

k�t ). (1)

3.2.1.2. Objective 2: maximizing the total value of purchasing. The total value of purchasing [9,41] considers theimpact of qualitative (intangible) performance criteria in purchasing decisions (such as after sale services, businessstructure and technical capabilities of the suppliers), and can be estimated as follows:

max TVP =J∑

j=1

Rj

T∑t=1

∑i∈pj

·xijt , (2)

where Rj is the overall score (weight) of supplier j. These global weights could be generated using, for example, thefuzzy analytical hierarchy process (FAHP) [4,25]. Other important quantitative factors affecting the supplier selectionand order lot-sizing decision (i.e., the quality and on-time delivery criteria) will be incorporated in the model as softconstraints.

3.2.2. Model constraints3.2.2.1. Inventory level constraints. All relevant inventory balancing constraints at the manufacturer and distributioncenters are summarized as follows:

Ir2i,t−1 +

∑j∈si

xij t − Ir2it =

∑k

bik · pkt ∀i, t, (3)

If 2k,t−1 + pkt − If 2

kt =∑

�

sk�t ∀k, t, (4)

If 3k�,t−1 + sk�t − If 3

k�t = dk�t ∀k, �, t, (5)

If 3k�t � ssk�t ∀k, �, t. (6)

Constraints (3) and (4) are inventory balancing equations for components and final products at the manufacturer’swarehouses, respectively. The right-hand sides in constraints (3) represent the dependent demands of components basedon the production plan of final products. Moreover, constraints (5) and (6) indicate the inventory equations for finalproducts and the safety stock levels at the distribution centers, respectively. Also, ssk�t = �kdk�,t+1, where �k is theforward inventory coverage policy factor for product k [14]. It is noted that the safety stock level for the last period Tis calculated based on the first period’s demand.


3.2.2.2. Capacity constraints. The following constraints are related to the production capacity levels at the suppliersand the manufacturer:∑

i∈pj

a1ij · xijt � cap1

j t ∀j, t, (7)

∑i∈pj

a1ij · xijt � urj t · cap1

j t · yjt ∀j, t, (8)

∑k

a2k · pkt � cap2

t ∀t. (9)

Constraints (7) and (8) represent the maximum and minimum level of utilized capacities at the suppliers in each period,respectively. Constraints (8) ensure that the manufacturer to take into account the suppliers’ MAUC requirements. Thesetypes of constraints would be active whenever a supplier j is selected in period t. Constraints (9) state the manufacturer’sproduction capacity in each period. Moreover, the warehouses’ space limitations at the manufacturer and the distributorsare as follows:∑

i

vri · Ir2it �W 2

r ∀t, (10)∑k

vfk · If 2kt �W 2

f ∀t, (11)∑k

vfk · If 3k�t �W 3

� ∀�, t. (12)

Constraints (10) and (11) indicate the limited storage space at the receiving and shipping warehouses at the manu-facturer, respectively. The storage capacities at the distributors are imposed by constraints (12).

3.2.2.3. Quality and on-time delivery constraints. Two other important quantitative factors imposed by the manu-facturer on the supplier selection and order lot-sizing sub-problem are the minimum acceptable levels of quality andon time delivery (or service level) provided by each supplier, respectively [9,10,15,41]. These requirements could beimposed by the following constraints:∑

j∈si

qij · xijt �Qi

∑j∈si

xij t ∀i, t, (13)

∑i

∑j∈si

slj xij t � SL∑

i

∑j∈si

xij t ∀t. (14)

3.2.2.4. Constraints on variables. The integrality and non-negativity constraints are as follows:

xijt �ubxijt · yjt ∀i, j ∈ si, t, (15)

zj �∑

t

yj t ∀j, (16)

yjt �zj ∀j, t, (17)

yjt , zj ∈ {0, 1} ∀j, t, (18)

xijt , pkt , sk�t , If 3k�t , If 2

kt , Ir2it �0 ∀i, j ∈ si, k, �, t. (19)

Constraints (15) ensure that if there will be an order from supplier j at period t (i.e., yjt= 1), then the amount of orderfor each item say i (xijt ) will be limited to its upper bound which can be calculated by the following equation:

ubxijt = min

⎛⎝cap1oj t

a1pij

,

T∑h=t

K∑k=1

L∑�=1

bikdok�h

⎞⎠ ∀i, j ∈ si, t,


where cap1oj t and a

1pij are the most optimistic and the most pessimistic values of capacity level and consumption rate,

respectively. Also, dok�h denotes the most optimistic value of demand. When yjt= 1, the corresponding order level

cost (olcj t ) will be charged by the total cost function (1). Constraints (16) and (17) are integrality constraints. Usingconstraint (16), the decision variable zj will be equal to zero, if the model suggests not to buy from the supplierj over the planning horizon (i.e.,

∑t yj t = 0), while constraint (17) forces zj to be equal to 1, if during some

time period, an order is placed with supplier j. In this case, an appropriate supplier level cost (slcj ) will be chargedthrough the total cost function (1). Finally, constraints (18) and (19) are non-negativity constraints of continuousvariables.

4. Solution methodology

Recalling above-mentioned objective functions and constraints, we are dealing with a multiple objective possibilisticmixed integer linear programming model (MOPMILP). To solve this problem, we apply a two-phase approach. In thefirst phase, the original problem is converted into an equivalent auxiliary crisp multiple objective mixed integer linearmodel. Then, in the second phase, a novel interactive fuzzy programming approach is proposed for finding a preferredcompromise solution through an interaction between the decision maker and model analyzer.

4.1. An auxiliary multi-objective mixed integer linear model

We apply an extended version of a well-known approach proposed by Lai and Hwang [16,18] to transform theMOPMILP model into an auxiliary crisp multiple objective mixed integer linear programming model. To do so, weshould apply appropriate strategies for converting the fuzzy total cost objective function as well as some soft constraintsinto the equivalent crisp equations.

4.1.1. Treating the imprecise total cost objective functionGiven the imprecise coefficients in the objective function, in general, one cannot guarantee an ideal solution to problem

(1)–(19). There have been several approaches for obtaining compromise solutions in the literature [16,22,31,35,36].As stated in Hsu and Hwang [11], the last four approaches [22,31,35,36] have restrictive assumptions and are oftendifficult to implement in practice, thus, we chose to implement that of Lai and Hwang [16].

Since some of the parameters in the fuzzy total cost of logistics TC have triangular possibility distributions, theTC objective function would have a triangular possibility distribution as well. Geometrically, this fuzzy objective canbe fully defined by the three prominent points (TCp, 0), (TCm, 1) and (TCo, 0). So, this imprecise objective can beminimized by pushing the three points towards the left. Consequently, minimizing the imprecise objective functionTC requires minimizing TCp, TCm and TCo simultaneously. However, there may exist a conflict in the simultaneousminimization of these crisp objectives. So, using the Lai and Hwang’s approach [16] which is also adopted by otherresearchers [11,39], we minimize TCm, maximize (TCm − TCp), and minimize (TCo − TCm) instead of minimizingTCp, TCm and TCo simultaneously. These three objectives still serve the purpose of pushing the three objective pointsto the left. In this manner, the original fuzzy total cost of logistics (1) is replaced by the following three crisp objectivesto obtain a compromise solution:

min Z1 = TCm =J∑

j=1

slcmj · zj +

T∑t=1

J∑j=1

olcmj t · yjt +

T∑t=1

J∑j=1

∑i∈pj

(cij t + aulcmij t ) · xijt

+T∑

t=1

I∑i=1

hr2mit · Ir2

it +T∑

t=1

K∑k=1

(pcmkt · pkt + hf 2m

kt · If 2kt )

+T∑

t=1

L∑�=1

K∑k=1

(tcmk�t · sk�t + hf 3m

k�t · If 3k�t ), (20)

max Z2 = (TCm − TCp) =J∑

j=1

(slcmj − slc

pj ) · zj +

T∑t=1

J∑j=1

(olcmj t − olc

pj t ) · yjt


+T∑

t=1

J∑j=1

∑i∈pj

(aulcmij t − aulc

pij t ) · xijt+

T∑t=1

I∑i=1

(hr2mit − hr

2pit ) · Ir2

it +T∑

t=1

K∑k=1

(pcmkt − pc

pkt ) · pkt

+T∑

t=1

K∑k=1

(hf 2mkt − hf

2pkt ) · If 2

kt +T∑

t=1

L∑�=1

K∑k=1

[(tcmk�t − tc

pk�t ) · sk�t + (hf 3m

k�t − hf3pk�t ) · If 3

k�t ], (21)

min Z3 = (TCo − TCm) =J∑

j=1

(slcoj − slcm

j ) · zj +T∑

t=1

J∑j=1

(olcoj t − olcm

j t ) · yjt

+T∑

t=1

J∑j=1

∑i∈pj

(aulcoij t − aulcm

ij t ) · xijt+T∑

t=1

I∑i=1

(hr2oit − hr2m

it ) · Ir2it +

T∑t=1

K∑k=1

(pcokt − pcm

kt ) · pkt

+T∑

t=1

K∑k=1

(hf 2okt − hf 2m

kt ) · If 2kt +

T∑t=1

L∑�=1

K∑k=1

[(tcok�t − tcm

k�t ) · sk�t + (hf 3ok�t − hf 3m

k�t ) · If 3k�t ]. (22)

4.1.2. Treating the soft constraintsTo resolve the imprecise demands in the right-hand sides of constraints (5) and (6), the weighted average method

[16,20,39] is used for the defuzzification process and converting the dk�t parameter into a crisp number. So, if theminimum acceptable degree of feasibility (i.e., minimum acceptable possibility), �, is given, then the equivalentauxiliary crisp constraints can be represented as follows:

If 3k�,t−1 + sk�t − If 3

k�t = w1dpk�t,� + w2d

mk�t,� + w3d

ok�t,� ∀k, �, t, (23)

If 3k�t ��k[w1d

pk�,t+1,� + w2d

mk�,t+1,� + w3d

ok�,t+1,�] ∀k, � and t = 1, . . . , T − 1, (24)

where w1 + w2 + w3 = 1, and w1,w2 and w3denote the weights of the most pessimistic, the most possible and themost optimistic value of the fuzzy demand, respectively. The suitable values for these weights as well as � usually aredetermined subjectively by the experience and knowledge of the decision maker. However, based on the concept ofthe most likely values proposed by Lai and Hwang [16] and considering several relevant works [20,39], we set theseparameters as: w2 = 4/6, w1 = w3 = 1/6 and � = 0.5. Furthermore, as it is mentioned earlier, the safety stock levelfor the last period T is calculated based on the first period’s demand.

Moreover, regarding the other soft constraints (i.e., capacity levels, quality and service level constraints) which haveimprecise parameters both in the left-hand side and right-hand side, we can use the fuzzy ranking concept [16,29,39],and replace each imprecise constraint with three equivalent auxiliary inequality constraints. In this manner, we canobtain the following auxiliary capacity constraints:∑

i∈pj

a1pij,� · xijt �cap

1pj t,� ∀j, t, (25)

∑i∈pj

a1mij,� · xijt �cap1m

j t,� ∀j, t, (26)

∑i∈pj

a1oij,� · xijt �cap1o

j t,� ∀j, t, (27)

∑i∈pj

a1pij,� · xijt �ur

pj t,� · cap

1pj t,� · yjt ∀j, t, (28)

∑i∈pj

a1mij,� · xijt �urm

j t,� · cap1mj t,� · yjt ∀j, t, (29)

∑i∈pj

a1oij,� · xijt �uro

j t,� · cap1oj t,� · yjt ∀j, t, (30)

∑k

a2pk,� · pkt �cap

2pt,� ∀t, (31)

204 S.A. Torabi, E. Hassini / Fuzzy Sets and Systems 159 (2008) 193–214∑k

a2mk,� · pkt �cap2m

t,� ∀t, (32)∑k

a2ok,� · pkt �cap2o

t,� ∀t. (33)

In a similar way we can have the following auxiliary constraints for the fuzzy quality and service level constraints:∑j∈si

qpij,� · xijt �Q

pi,�

∑j∈si

xij t ∀i, t, (34)

∑j∈si

qmij,� · xijt �Qm

i,�

∑j∈si

xij t ∀i, t, (35)

∑j∈si

qoij,� · xijt �Qo

i,�

∑j∈si

xij t ∀i, t, (36)

∑i

∑j∈si

slpj,�xijt �SL

p�

∑i

∑j∈si

xij t ∀t, (37)

∑i

∑j∈si

slmj,�xijt �SLm

�

∑i

∑j∈si

xij t ∀t, (38)

∑i

∑j∈si

sloj,�xijt �SLo

�

∑i

∑j∈si

xij t ∀t. (39)

Consequently, we would have an auxiliary crisp multi-objective mixed integer linear programming model (MOMILP)as follows:

MOMILP:

min Z = [Z1, −Z2, Z3, −Z4],Z1 = TCm, Z2 = TCm − TCp,

Z3 = TC0 − TCm, Z4 = TVP

s.t. v ∈ F(v), (40)

where v denotes a feasible solution vector involving all of the continuous and binary variables in the original problem.Also, F(v) denotes the feasible region involving crisp constraints (3)–(4), (10)–(12), (15)–(19) and (23) up to (39).

4.2. Proposed interactive fuzzy programming solution approach

There are several methods in the literature for solving multi-objective linear programming (MOLP) models, amongthem; the fuzzy programming approaches are being increasingly applied. The main advantage of fuzzy approaches isthat they are capable to measure the satisfaction degree of each objective function explicitly. This issue can help thedecision maker to make her/his final decision by choosing a preferred efficient solution according to the satisfactiondegree and preference (relative importance) of each objective function.

Zimmermann developed the first fuzzy approach for solving a MOLP called max–min approach [42], but it is well-known that the solution yielded by max–min operator might not be unique nor efficient [17–19]. Therefore, after thatseveral methods were proposed to remove this deficiency. Of particular interest, Lai and Hwang [17] developed theaugmented max–min approach (hereafter the LH method), Selim and Ozkarahan [32] presented a modified version ofWerners’ approach [40] (hereafter the MW method), and Li et al. [19] proposed a two-phase fuzzy approach (hereafterthe LZL method). A brief discussion of these three approaches has been presented in Appendix A.

In our initial numerical tests for applying these approaches to solve the problem, we have observed some deficiencies.Among the single-phase methods (i.e. the LH and MW) which solve the original model directly by just one auxiliarycrisp model, the LH method sometimes generates inefficient solutions dominated by the solution of LZL method, and theMW method usually yields an efficient but unbalanced and poorly compromised solution so that the satisfaction degreesof objectives have considerable differences, which is often not acceptable by the decision maker. On the other hand,


although LZL always produces an efficient solution, it applies a two-phase method which needs more computationalefforts than the single-phase methods, especially for solving the multi-objective mixed integer linear models.

Consequently, we tried to develop a new single-phase approach removing above deficiencies which led to a newfuzzy approach. The proposed approach (hereafter the TH method) is actually a hybridization of the LH and MWmethods. Of particular interest, we were able to prove the efficiency of this method using a similar technique to thatused by Li et al. [19]; which has been represented in Appendix B.

In summary, our proposed interactive solution procedure to solve the original MOPMILP model is as follows:Step 1: Determine appropriate triangular possibility distributions for the imprecise parameters and formulate the

original MOPMILP model for the SCMP problem.Step 2: Convert the original fuzzy total cost of logistics TC into the three equivalent crisp objectives (20)–(22).Step 3: Given the minimum acceptable possibility level for imprecise parameters, �, convert the fuzzy constraints

into the corresponding crisp ones, and formulate the auxiliary crisp MOMILP model.Step 4: Determine the positive ideal solution (PIS) and negative ideal solution (NIS) for each objective function by

solving the corresponding MILP model as follows [1,12,15–21,32,39,42]:

ZPIS1 = min TCm, ZNIS

1 = max TCm,

ZPIS2 = max(TCm − TCp), ZNIS

2 = min(TCm − TCp),

ZPIS3 = min(TCo − TCm), ZNIS

3 = max(TCo − TCm),

ZPIS4 = max TVP, ZNIS

4 = min TVP.

s.t. v ∈ F(v).

It should be noted that determining the above ideal solutions requires solving eight mixed integer linear programwhich could be computationally very cumbersome especially in large-sized problem instances. In order to alleviate thecomputational complexity, we apply the following heuristic rules:

• Obtaining an approximate positive ideal solution for each objective function by solving the corresponding MILPheuristically to obtain a satisfactory feasible integer solution. To do so, the MIP solver is run until reaching pre-specified termination criteria, e.g., CPU time and/or optimality gap [14].

• Instead of solving a separate MILP for determining each NIS, we can estimate them using the positive ideal solutions.Let v∗

h and Zh(v∗h) denote the decision vector associated with the PIS of hth objective function and the corresponding

value of hth objective function, respectively. So, the related NIS could be estimated as follows:

ZNISh = max

k=1,...,4{Zh(v

∗k )}; h = 1, 3; ZNIS

h = mink=1,...,4

{Zh(v∗k )}; h = 2, 4.

Step 5: Specify a linear membership function for each objective function as follows:

�1(v) =

⎧⎪⎪⎨⎪⎪⎩1 if Z1 < ZPIS

1 ,

ZNIS1 − Z1

ZNIS1 − ZPIS

1

if ZPIS1 �Z1 �ZNIS

1 ,

0 if Z1 > ZNIS1 ,

(41)

�2(v) =

⎧⎪⎪⎨⎪⎪⎩1 if Z1 > ZPIS

2 ,

Z2 − ZNIS2

ZPIS2 − ZNIS

2

if ZNIS2 �Z2 �ZPIS

2 ,

0 if Z2 < ZNIS2 ,

(42)

�3(v) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩1 if Z3 < ZPIS

3 ,

ZNIS3 − Z3

ZNIS3 − ZPIS

3

if ZPIS3 �Z3 �ZNIS

3 ,

0 if Z3 > ZNIS3 ,

(43)


Fig. 2. Linear membership function for Z1 (Z3).

Fig. 3. Linear membership function for Z2 (Z4).

�4(v) =

⎧⎪⎪⎨⎪⎪⎩1 if Z4 > ZPIS

4 ,

Z4 − ZNIS4

ZPIS4 − ZNIS

4

if ZNIS4 �Z4 �ZPIS

4 ,

0 if Z4 < ZNIS4 .

(44)

In fact, �h(v) denote the satisfaction degree of hth objective function for the given solution vector v. Figs. 2 and 3represent the graphs of these membership functions.

Step 6: Convert the auxiliary MOMILP model into an equivalent single-objective MILP using the following newauxiliary crisp formulation (45).

Auxiliary MILP:

max �(v) = ��0 + (1 − �)∑h

�h�h(v)

s.t. �0 ��h(v), h = 1, . . . , 4,

v ∈ F(v), �0 and � ∈ [0, 1], (45)

where�h(v) and�0 = minh{�h(v)}denote the satisfaction degree of hth objective function and the minimum satisfactiondegree of objectives, respectively. This formulation has a new achievement function defined as a convex combination ofthe lower bound for satisfaction degree of objectives (�0), and the weighted sum of these achievement degrees (�h(v))

to ensure yielding an adjustably balanced compromise solution. Moreover, �h and � indicate the relative importance ofthe hth objective function and the coefficient of compensation, respectively. The �h parameters are determined by thedecision maker based on her/his preferences such that

∑h �h = 1, �h > 0. Also, � controls the minimum satisfaction

level of objectives as well as the compromise degree among the objectives implicitly. That is, the proposed formulationis capable of yielding both unbalanced and balanced compromised solutions for a given problem instance based on thedecision maker’s preferences through adjusting the value of parameter �.

In this regard, a higher value for � means more attention is paid to obtain a higher lower bound for the satisfactiondegree of objectives (�0) and accordingly more balanced compromise solutions. On the contrary, the lower value for� means more attention is paid to obtain a solution with high satisfaction degree for some objectives with higherrelative importance without any attention paid to the satisfaction degree of other objectives (i.e., yielding unbalancedcompromise solutions).


It is noteworthy that there exists a correlation between � and the range of �h values (i.e. maxh{�h} − minh{�h}) sothat there will be a limited reasonable interval of � in which it could be selected for a given � vector. For example, forthe considerably large values of this range, corresponding � should be selected as a small value (e.g. smaller than 0.3)because of explicit preference of the decision maker for getting an unbalanced compromise solution in this case.

Step 7: Given the coefficient of compensation � and relative importance of the fuzzy goals (� vector), solve theproposed auxiliary crisp model (45) by the MIP solver. If the decision maker is satisfied with this current efficientcompromise solution, stop. Otherwise, provide another efficient solution by changing the value of some controllableparameters say � and �, and then go back to Step 3.

5. Computational experiments

5.1. Experimental design

To demonstrate the validity and practicality of the proposed model and solution method, an industrial case scenarioinspired form a home appliances manufacturer is presented. This supply chain involves four suppliers, one manufacturerand three distribution centers located in different customer zones. The factory produces three types of products usingten common purchased items. The supplier-item matrix representing the pj and si sets is given in Table 1 where pair(j, i) is 1 if the supplier j offers item i, and it is zero, otherwise.

These eligible suppliers have been selected through an initial screening process performed by the quality controldepartment. Especially, the fuzzy AHP process [25] was used to determine the overall scores of suppliers, and finallythe above four suppliers having the highest overall scores Rj were selected. The overall global weights vector afternormalization was adjusted to Rj = (0.32, 0.26, 0.18, 0.24).

The planning horizon is 3 months consisting of 12 weekly periods. The longest material acquisition lead time is 2days (less than one week) and indicates that the lead times can be ignored in contrast with the length of each period.

Because of confidentiality as well as the lack of some required data, we decided to generate most of the requiredparameters randomly. However, the generation of random data was done in such a way that they will be close to thereal data available in the company. Without loss of generality and just to simplify the generation of fuzzy parameters,we applied symmetrical triangular possibility distribution for our numerical test. So, the most possible value of eachimprecise parameter was first generated with an appropriate probability distribution, and then the corresponding mostpessimistic and optimistic values were determined by multiplying the most possible value with 0.8 and 1.2, respectively[32]. Moreover, for lead time-dependent capacity reservation and unit price discounts, the number of discount intervalsoffered by each supplier (DIj ) was first generated randomly between 1 and 3, and then the associated sub-intervals(SDjr ) were considered with equal periods. Table 2 summarizes the information about the source of random datageneration.

Other input data have been shown in Tables 3 and 4. Table 3 represents the bill of material data (bik) where the entryof (k, i) denotes the number of items i required to produce one unit of product k. Also, the forward inventory coveragepolicy factors for determination of safety stock levels were considered as �k = 5% for all final products.

Regarding the manufacturer capacity, we could check the following necessary feasibility conditions:

t∑h=1

L∑�=1

K∑k=1

a2mk dm

k�h �t∑

h=1

cap2mh , t = 1, . . . , T . (46)

Table 1The supplier–item matrix

Supplier (j) Item (i)

1 2 3 4 5 6 7 8 9 10

1 1 1 1 1 1 1 0 0 1 12 1 1 1 1 1 1 1 1 0 03 1 1 1 1 0 0 0 0 1 14 1 1 1 1 0 0 1 1 0 0


Table 2The sources of random generation of data set

Parameter Corresponding random distribution

dm1�t N(150, 10)

dm2�t N(400, 20)

dm3�t N(250, 15)

a2mk U(3, 5)

cap2mt

∑t

∑k

∑�

(a2mk dm

k�t )/T · U(1.1, 1.3)

a1mij U(1, 3)

cap1mj1

∑t

∑i∈pj

∑k

∑�

(a1mij bikd

mk�t )/T · U(0.5, 0.8)

cap1mjr , (r > 1) U(1.1, 1.2) · capj ·r−1

urmjr U(0.2, 0.3)

cij1 U(4, 8)

cijr (r > 1) U(0.8, 1) · cij ·r−1

aulcmijr U(0.1, 0.2) · cijr

pcmkt , (t = 1) U(1, 3)

pcmkt , (t > 1) pcm

kt = pcmk,t−1 · U(1, 1.1)

tcmk�t , (t = 1) U(0.2, 0.4)

tcmk�t , (t > 1) tcm

k�t = tcmk�,t−1 · U(1, 1.05)

hr2mit averagej∈si {cij t } · U(0.005, 0.01)

hf 2mkt

∑i|bik �=0

(∑j∈si

cij t /|si |)

· bik + pcmkt · U(0.005, 0.01)

hf 3mk�t hf 2m

kt · U(1.05, 1.10)

slcmj U(1000, 1500)

olcmj t U(100, 200)

Ir2i0

∑k

∑� bikd

Pk�1 · U(0.8, 1.0)

qmij U(0.01, 0.03)

Qmi U(0.03, 0.05)

slmj U(0.90, 0.95)

SLm U(0.90, 0.95)

Table 3Bill of material matrix (bik values)

Product (k) Item (i)

1 2 3 4 5 6 7 8 9 10

1 2 1 0 1 0 2 1 0 1 12 1 3 1 0 1 2 0 1 2 23 1 0 2 1 1 1 3 0 1 1

These necessary feasibility conditions require the existence a sufficient cumulative capacity in each period based onthe most possible situation. Moreover, in the supply side there exists enough capacity provided by the suppliers andtherefore no need to check the capacity feasibility.

Furthermore, the decision maker provided the relative importance of objectives linguistically as: �1 > �4 > �2 = �3,and based on this relationships we set the objectives weight vector as: � = (0.5, 0.15, 0.15, 0.2). After some initialexperiments, the stopping criteria for solving each MIP as well as controllable parameters were set as: CPU time = 300seconds, optimality gap = 0.05, � = 0.01 and � = 0.4, respectively. It is noted that the reason for selecting � = 0.4is that the Z1 is the most important objective and also Z2 and Z3 are actually relative measures from Z1. Thus thesomewhat unbalanced compromise solution with highest satisfaction degree for Z1 is of particular interest. In thisrespect, our initial experiments show that any value of � between 0.3 and 0.8 could be appropriate for obtaining acompromise solution with �1 > �4 > �2 = �3. However, it seems that � = 0.4 is more appropriate. It should be noted


Table 4Storage capacity data

Unit storage volume required for item i (vri ) (3, 1, 2, 1, 1, 3, 2, 1, 2, 1)

Unit storage volume required for product k (vfk) (5, 8, 6)

Storage capacity of receiving warehouse at the manufacturer (18500)Storage capacity of shipping warehouse at the manufacturer (1800)Storage capacity (in volume) at the distribution center � (W 3

� ) (1300, 900, 1200)

that although the size of problem instances are considerable and acceptable in real-size scale (including about 1550constraints, 800 continuous variables and 52 binary variables), the CPU time was not an issue in our experiments andfortunately all of the above approaches lead to “good” feasible solution (with mean optimality gap less than 2%) withinjust a few seconds.

5.2. Performance analysis

To evaluate the performance of the aforementioned fuzzy approaches (i.e. the LZL, LH, MW and proposed THmethods), 50 problem instances were randomly generated. These approaches were coded in GAMS and the OSL solverfrom IBM was used for solving the MIP models on a Pentium 4 with a 1.8 GHz CPU processor and 256 MB of RAM.Due to space limitations, the details of the solutions found by the different approaches are not presented here, but canbe made available upon request. In summary, we make the following observations based on our numerical experiments:

• The solutions found by the Zimmermann max–min method are always inefficient and dominated by the solutions ofLZL and/or TH methods.

• In the 27 cases out of 50 problem instances (i.e., in about 54%), the solutions found by the LH method were equalto the solution of LZL and/or TH method. Also, in the 10 cases out of 50 problem instances (i.e., in about 20%), itssolutions were inefficient and dominated by the solutions of LZL and/or TH method. However, it should be noted thatthese solutions were close to the corresponding efficient solutions, and may be the main source of this inefficiencyis related to the computational errors.

• The solutions found by the MW method are usually efficient but at the same time unbalanced and poorly compromisedwhich would be often unacceptable by the decision maker. In other words, the minimum satisfaction level �0 is verysmall and the most attention is just paid to objectives with higher weights without any attention paid to the satisfactiondegree of other objectives.

• The solutions found by the TH method are always efficient. Also in the 25 and 28 cases out of 50 problem instances(i.e., in about 50% and 56%), the solutions found by the proposed TH method were equal to the solution of LH andLZL methods, respectively.

In order to analyze and compare the performance of these fuzzy approaches, we have used two performance measures:(1) the well-known distance measure, and (2) a new proposed balancing measure. The distance measure is used fordetermining the degree of closeness of each solution to the corresponding ideal solution. In this regard, we define thefollowing family of distance functions [1,18]:

dp(v) =[∑

h

�ph(1 − �h(v))p

]1/p

; p�1 and integer. (47)

Since the satisfaction degree of each objective is defined as the relative closeness of the solution to the ideal point orthe relative remoteness to the anti-ideal point, they are used explicitly in Eq. (47). The power p represents a distanceparameter and especially p = 1, 2 and ∞ are operationally important so that d1 (the Manhattan distance) and d2 (theEuclidean distance) are the longest and shortest distances in the geometrical sense; and d∞ (the Tchebycheff distance)is the shortest distance in the numerical sense. Generally speaking, when p increases, the amount of distance dp andalso the credibility of the distance function dp decreases [18]. It is noted that based on the definition of dp, the fuzzyapproach with minimum dp (especially for p = 1), would be preferred to the other methods.


Table 5Performance comparison of fuzzy approaches for � = (0.5, 0.15, 0.15, 0.2), � = 0.01, and � = 0.4

Fuzzy approaches Distance measures Dispersion measured1 d2 d∞ RSD

LZL 0.357 0.204 0.168 0.317LH 0.360 0.203 0.165 0.304MW 0.310 0.181 0.133 0.771TH 0.353 0.199 0.159 0.311

Table 6Sensitivity analysis on � value in the MW and TH methods

�-Value TH method MW method�Z1

(v) �Z2(v) �Z3

(v) �Z4(v) �Z1

(v) �Z2(v) �Z3

(v) �Z4(v)

0 0.925 0.061 0.939 0.292 0.925 0.061 0.939 0.2920.1 0.944 0.116 0.884 0.219 0.909 0.074 0.926 0.3080.2 0.714 0.467 0.533 0.546 0.929 0.056 0.944 0.2890.3 0.685 0.5 0.5 0.573 0.917 0.062 0.938 0.3120.4 0.685 0.5 0.5 0.573 0.925 0.061 0.939 0.2920.5 0.685 0.5 0.5 0.573 0.943 0.042 0.958 0.2710.6–0.9 0.685 0.5 0.5 0.573 0.685 0.5 0.5 0.5731 0.678 0.5 0.5 0.557 0.678 0.5 0.5 0.557

Corresponding max–min solution = (0.5, 0.5, 0.5, 0.713).Corresponding LZL solution = (0.619, 0.5, 0.5, 0.713).Corresponding LH solution = (0.697, 0.5, 0.5, 0.557).

We also propose a new measure, i.e., the range of satisfaction degrees (RSD) which is a dispersion index and iscomputed as follows:

RSD(v) = maxh

(�h(v)) − minh

(�h(v)). (48)

In fact, this index measures the balancing amount of a compromise solution via calculating the maximum differencebetween the satisfaction degrees of objectives. It also indicates the level of consistency between the priority vector �quoted by the decision maker and the satisfaction degrees vector �. For example, given strong differences in �h values,considerable differences in �h values, i.e. the higher RSD will be more desirable. In this case, parameter � should beset as a small number (less than 0.3) indicating less attention is paid to the minimum satisfaction level �0. Table 5summarizes the numerical results of the four fuzzy approaches in terms of above-mentioned performance indices.

Furthermore, to analyze the impact of changing parameter � on the final solution of MW and proposed TH method,we solved some problem instances with different values of �. As an example, for a specific instance (with seed number27 in GAMS), the corresponding solutions have been provided in Table 6.

From the above comparison and sensitivity analysis, we can derive the following information:

• From Table 5, the MW method seems better than other methods in terms of distance measures, but at the same timehas the largest RSD value because the related solutions are usually unbalanced and poorly compromised. Generally,such compromise solutions are often unacceptable by the decision maker.

• From Table 6 we can observe that MW method is sensitive to parameter � so that it produces different unbalancedsolutions for � values less than 0.6. On the other hand, the proposed TH method is not very sensitive to � value so thatit produces appropriately unique balanced solution for � values between 0.3 and 0.9. However, the MW and proposedTH methods have very similar unbalanced solutions for small values of � (especially for ��0.1) and very similarbalanced solutions for large values of � (especially for ��0.6). Therefore, we can control the degree of compromiseamong the objectives by the proposed TH method, and a spectrum of unbalanced and balanced solutions based onthe decision maker preferences, can be obtained for a given relative importance vector �.


• Among LZL, LH and the proposed TH method which generally produce the balanced compromised solutions,surprisingly the proposed TH method outperforms the others in terms of distance measure.

• Regarding the RSD measure, the TH and LZL methods have higher values than the LH method. Through moreinvestigation it can be realized that the main reason is related to the solutions characteristic so that the satisfactiondegree of objectives in the TH and LZL’s solutions are compatible with the decision maker’s preferences (i.e.�1 > �4 > �2 = �3), that is why these methods have higher RSD than the LH method.

Based on the above information, we can conclude that the proposed TH is the most appropriate method among theconsidered fuzzy approaches because: (1) it is more robust and reliable than the LH and MW approach as it alwaysgenerates efficient solutions and is able to produce both unbalanced and balanced solutions based on the decision maker’spreferences, (2) its solutions are consistent with the decision maker’s preferences (i.e., the consistency between weightvector � and satisfaction vector �), (3) it is more flexible than LZL and LH approaches because it is able to find differentefficient solutions for a specific problem instance with a given weight vector � through changing the � value, and finallyfrom the computational stand point (4) the proposed TH method, due to its single-phase characteristic, is more suitablethan the LZL method especially for solving multi-objective mixed integer linear models. In summary, the proposed THmethod contains all the advantages of existing methods and at the same time, it overcomes their shortcomings.

6. Conclusion remarks

This study proposes a novel multi-objective possibilistic programming model to formulate a supply chain masterplanning problem integrating procurement, production and distribution planning in a multi-echelon, multi-product andmulti-period supply chain network. A two-phase interactive fuzzy programming procedure has been developed. In thefirst phase, the possibilistic programming model is converted into an auxiliary crisp MOMILP by applying appropriatestrategies. Then, a novel fuzzy approach (called TH method) is applied to find an efficient compromise solution.

The numerical experiments indicate that the proposed TH method is very promising fuzzy approach which can pro-duce both unbalanced and balanced efficient solutions based on the decision maker’s preferences along with offeringappropriate flexibility to provide different solutions to help the decision maker in selecting the final preferred compro-mise solution. This approach can also be used for solving other practical MOLP models due to it’s computationallyadvantages.

Finally, there are some possible directions for further research. Among them, is to test the proposed approach on acomplete real life problem, such as a chemical processing supply chain where the use of lead-time dependent discountsis common. In addition, extending the proposed model to more general supply chain networks with multiple plantsin parallel and also considering other important discount policies for example, business volume discounts [41] are ofparticular interest. Moreover, as it mentioned earlier, the CPU time was not an issue in our numerical experiments. Thatis why we used a limit of 300 s of CPU time in our numerical tests. However, in other large-scaled practical problems itmight be an issue. Therefore, developing an efficient metaheuristic algorithm to solve the corresponding MILP modelsshould be helpful in reaching efficient solutions.

Appendix A

In this appendix we provide an abstract version of three previously developed approaches (i.e., the LZL, LH andMW methods).

A.1. Li et al. (LZL) two-phase method [19]

LZL model:

max �(v) =∑h

�h�h(v)

s.t. �0h ��h(v), h = 1, . . . , 4,

v ∈ F(v), �0h, �h(v) ∈ [0, 1]. (A.1)


In the above formulation, �0h denotes the minimum satisfaction degree of hth objective function which is found by

solving the Zimmermann’s max–min approach [42] as follows:max–min model:

max �

s.t. �h(v)��, h = 1, . . . , 4,

v ∈ F(v), � ∈ [0, 1]. (A.2)

A.2. Lai and Hwang (LH) augmented max–min method [17]

LH model:

max �(v) = �0 + �∑h

�h�h(v)

s.t. �0 ��h(v), h = 1, . . . , 4,

v ∈ F(v), �0 ∈ [0, 1]. (A.3)

Here, �0 denotes the minimum satisfaction degree of objectives which is determined along with the variables �h(v) viasolving the LH model directly in a single phase. Also, � is a sufficiently small positive number which is usually set to0.01 [17,18].

A.3. Selim and Ozkarahan extended Werners (MW) method [32]

MW model:

max �(v) = ��0 + (1 − �)∑h

�h�h

s.t. �h(v)��0 + �h, h = 1, . . . , 4,

v ∈ F(v), �, �0 and �h ∈ [0, 1]. (A.4)

In this model, �0 and �h(v) denote the minimum satisfaction degree of objectives and satisfaction degree of objectiveh, respectively, which simultaneously are determined through solving the MW model. Moreover, � is the coefficient ofcompensation [32], and we have set it to 0.4 based on our initial tests.

Appendix B

In this appendix we proof a theorem to establish the efficiency of solutions produced by the proposed TH method.We start by defining a fuzzy-efficient solution to (45).

Definition. A vector v∗ is an optimal solution to auxiliary MILP model (45) or an efficient solution to MOMILPmodel (40), iff there does not exist any v ∈ F(v) such that �h(v)��h(v

∗) for all h and �s(v) > �s(v∗) for at least one

s ∈ {1, . . . , 4}.

Theorem. The optimal solution of auxiliary model (45) is an efficient solution to the MOMILP model (40).

Proof. Suppose that v∗ is an optimal solution of (45) which is not an efficient solution to model (40). It means thatmodel (40) must have an efficient solution say v∗∗ so that we have: �h(v

∗∗)��h(v∗); ∀h and ∃s|�s(v

∗∗) > �s(v∗).

Hence, for the minimum satisfaction level of objectives in v∗ and v∗∗ solutions, we would have �0(v∗∗)��0(v

∗), and


regarding the related objective values we would have the following inequality:

�(v∗) = ��0(v∗) + (1 − �)

∑h

�h�h(v∗) = ��0(v

∗) + (1 − �)

⎡⎣∑h�=s

�h�h(v∗) + �s�s(v

∗)

⎤⎦< ��0(v

∗∗) + (1 − �)

⎡⎣∑h�=s

�h�h(v∗∗) + �s�s(v

∗∗)

⎤⎦ = �(v∗∗).

Thus, v∗ is not the optimal solution of (45), a contradiction. �

Acknowledgments

This research is supported by both the Natural Sciences and Engineering Research Council of Canada (NSERC)and Tehran University, Iran. We are also grateful to the two anonymous reviewers for their valuable comments andconstructive criticism.

References

[1] W.F. Abd El-Wahed, S.M. Lee, Interactive fuzzy goal programming for multiobjective transportation problems, Omega: Internat. J. Manage.Sci. 34 (2006) 158–166.

[2] N. Aissaoui, M. Haouari, E. Hassini, Supplier selection and order lot sizing modeling: a review, Comput. Oper. Res. 34 (12) (2006)3516–3540.

[3] F.T.S. Chan, S.H. Chung, S. Wadhwa, A hybrid genetic algorithm for production and distribution, Omega: Internat. J. Manage. Sci. 33 (2005)345–355.

[4] D.Y. Chang, Applications of the extent analysis method on fuzzy AHP, Eur. J. Oper. Res. 95 (3) (1996) 649–655.[5] S.P. Chen, P.C. Chang, A mathematical programming approach to supply chain models with fuzzy parameters, Eng. Optim. 38 (6) (2006)

647–669.[6] Z. Degraeve, F. Roodhooft, A mathematical programming approach for procurement using activity based costing, J. Business Finance &

Accounting 27 (1–2) (2000) 69–98.[7] Z. Degraeve, F. Roodhooft, B. van Doveren, The use of total cost of ownership for strategic procurement: a company-wide management

information system, J. Oper. Res. Soc. 56 (2005) 51–59.[8] S. Elhedhli, J.L. Goffin, Efficient production-distribution system design, Manage. Sci. 51 (7) (2005) 1151–1164.[9] S.H. Ghodsypour, C. O’Brien, A decision support system for supplier selection using an integrated analytical hierarchy process and linear

programming, Internat. J. Prod. Econom. 56–57 (1998) 199–212.[10] E. Hassini, Order lot sizing with multiple capacitated suppliers offering lead time-dependent capacity reservation and unit price discounts, Prod.

Plann. Control (2006), in press.[11] H.M. Hsu, W.P. Wang, Possibilistic programming in production planning of assemble-to-order environments, Fuzzy Sets and Systems 119

(2001) 59–70.[12] C.F. Hu, C.J. Teng, S.Y. Li, A fuzzy goal programming approach to multiobjective optimization problem with priorities, Eur. J. Oper. Res. 176

(2007) 1319–1333.[13] M. Inuiguchi, J. Ramik, Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic

programming in portfolio selection problem, Fuzzy Sets and Systems 111 (2000) 3–28.[14] A.P. Kanyalkar, G.K. Adil, Aggregate and detailed production planning integrating procurement and distribution plans in a multi-site

environment. Internat. J. Prod. Res. (2006), preview article.[15] M. Kumar, P. Vrat, R. Shankar, A fuzzy programming approach for vendor selection problem in a supply chain, Internat. J. Prod. Econom. 101

(2006) 273–285.[16] Y.J. Lai, C.L. Hwang, A new approach to some possibilistic linear programming problems, Fuzzy Sets and Systems 49 (1992) 121–133.[17] Y.J. Lai, C.L. Hwang, Possibilistic linear programming for managing interest rate risk, Fuzzy Sets and Systems 54 (1993) 135–146.[18] Y.J. Lai, C.L. Hwang, Fuzzy Multiple Objective Decision Making, Methods and Applications, Springer, Berlin, 1994.[19] X.Q. Li, B. Zhang, H. Li, Computing efficient solutions to fuzzy multiple objective linear programming problems, Fuzzy Sets and Systems 157

(2006) 1328–1332.[20] T.F. Liang, Distribution planning decisions using interactive fuzzy multi-objective linear programming, Fuzzy Sets and Systems 157 (2006)

1303–1316.[21] T.F. Liang, Integrating production–transportation planning decision with fuzzy multiple goals in supply chains, Internat. J. Prod. Res. (2006),

preview article.[22] M.K. Luhandjula, Fuzzy optimization: an appraisal, Fuzzy Sets and Systems 30 (1989) 257–282.[23] J. Mula, R. Poler, J.P. Garcia, MRP with flexible constraints: a fuzzy mathematical programming approach, Fuzzy Sets and Systems 157 (2006)

74–97.


[24] J. Mula, R. Poler, J.P. Garcia-Sabater, F.C. Lario, Models for production planning under uncertainty: a review, Internat. J. Prod. Econom. 103(2006) 271–285.

[25] A. Noorul Haq, G. Kannan, Design of an integrated supplier selection and multiechelon distribution inventory model in a built-to-order supplychain environment, Internat. J. Prod. Res. 44 (10) (2006) 1963–1985.

[26] Y.B. Park, An integrated approach for production and distribution planning in supply chain management, Internat. J. Prod. Res. 43 (6) (2005)1205–1224.

[27] R. Pibernik, E. Sucky, An approach to inter-domain master planning in supply chains, Internat. J. Prod. Econom. 108 (1–2) (2007) 200–212.[28] L.P. Plambeck, T.A. Taylor, Renegotiation of supply contracts, Working Paper, Graduate School of Business, Columbia University, 2002,

Available online: 〈http://www2.gsb.columbia.edu/divisions/dro/working_papers/2005/2005-04/DRO-2005-04_Aug06.pdf〉.[29] J. Ramik, J. Rimanek, Inequality relation between fuzzy numbers and its use in fuzzy optimization, Fuzzy Sets and Systems 16 (1985)

123–138.[30] K. Rezaie, B. Ostadi, S.A. Torabi, Activity-based costing in flexible manufacturing systems with a case study in a forging industry, Internat. J.

Prod. Res. (2006), preview article.[31] M. Sakawa, H. Yano, Interactive fuzzy satisficing method for multiobjective nonlinear programming problems with fuzzy parameters, Fuzzy

Sets and Systems 30 (1989) 221–238.[32] H. Selim, I. Ozkarahan, A supply chain distribution network design model: an interactive fuzzy goal programming-based solution approach,

Internat. J. Adv. Manuf. Technol. (2007), in press.[33] D. Sirias, S. Mehra, Quantity discount versus lead time-dependent discount in an inter-organizational supply chain, Internat. J. Prod. Res. 43

(16) (2005) 3481–3496.[34] E. Sucky, Inventory management in supply chains: a bargaining problem, Internat. J. Prod. Econom. 93–94 (2005) 253–262.[35] H. Tanaka, K. Asai, Fuzzy linear programming with fuzzy numbers, Fuzzy Sets and Systems 13 (1984) 1–10.[36] H. Tanaka, H. Ichihashi, K. Asai, A formulation of linear programming problems based on comparison of fuzzy numbers, Control Cybernet.

13 (1984) 185–194.[37] H. Tempelmeier, A simple heuristic for dynamic order sizing and supplier selection with time-varying data, Prod. Oper. Manage. 11 (2002)

499–515.[38] D.J. Thomas, P.M. Griffin, Coordinated supply chain management, Eur. J. Oper. Res. 94 (1996) 1–15.[39] R.C. Wang, T.F. Liang, Applying possibilistic linear programming to aggregate production planning, Internat. J. Prod. Econom. 98 (2005)

328–341.[40] B. Werners, Aggregation models in mathematical programming, in: G. Mitra, H.J. Greenberg, F.A. Lootsma, M.J. Rijckaert, H-J. Zimmermann

(Eds.), Mathematical Models for Decision Support, Springer, Berlin, Heidelberg, New York, 1988, pp. 295–305.[41] W. Xia, Z. Wu, Supplier selection with multiple criteria in volume discount environments, Omega: Internat. J. Manage. Sci. 35 (2007)

494–504.[42] H.J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems 1 (1978) 45–55.

http://www2.gsb.columbia.edu/divisions/dro/working_papers/2005/2005-04/DRO-2005-04_Aug06.pdf

Documents

An interactive possibilistic programming approach for multiple objective supply chain master planning