Selecting a Multicriteria Inventory Classification Model

Research ArticleSelecting a Multicriteria Inventory Classification Model toImprove Customer Order Fill Rate

Qamar Iqbal1 DonMalzahn1 and Lawrence E Whitman2

1 Industrial Systems and Manufacturing Engineering Department Wichita State University 1845 Fairmount StWichita KS 67260 USA2College of Engineering and Information Technology University of Arkansas at Little Rock 2801 S University AveLittle Rock AR 72204 USA

Correspondence should be addressed to Qamar Iqbal qxiqbalshockerswichitaedu

Received 13 January 2017 Accepted 5 March 2017 Published 5 April 2017

Academic Editor Panos Pardalos

Copyright copy 2017 Qamar Iqbal et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Multicriteria models have been proposed for inventory classification in previous studies However it is important to make adecision when a particular multicriteria inventory classification model should be preferred over other models and also if thehighest performingmodel remains the highest performing at all times Companies always look for ways to improve customer orderfulfillment process This paper shows how better inventory classification can improve customer order fill rate in variable settingsThemethod to compare the inventory classificationmodelswith regard to improving customer order fill rate is proposedThe cut-offpoint is calculated which indicates when a model currently in use should be dropped in favor of another model to increase revenueby filling more orders Sensitivity analysis is also performed to determine how holding cost and demand uncertainty affect theperformance metric Finally regression analysis and hypothesis testing inform the decision-maker of how a modelrsquos performancediffers from other models at various values of holding cost and standard deviation of demand

1 Introduction

In order to efficiently manage inventory companies use anABC classification by assigning items to one of three classesso that specific inventory control policies can be applied[1] Class A is considered very important class B is seen asmoderately important and class C is considered the leastimportant [2] The traditional ABC classification is based onPareto analysis and the criterion typically used is annualdollar value usage [3] This criterion is most frequently usedin practice because managers pay more attention to theinventory that has a high dollar value [4]

The traditional ABC method is easy to use becauseit only considers dollar usage The following studies havesuggested that using other criteria in combination makesinventory classification more effective The criteria that areoften employed in multicriteria methods include lead timecommonality obsolescence and criticality [2 5ndash8] The ana-lytic hierarchy process (AHP) distance-based modeling andneural network techniques have also been used in different

studies [6 9ndash11] In later studies optimization models areused to classify inventoryThese are discussed in the literaturereview The study focuses on optimization models

Since the single-criteria method is easy to use it is mostwidely employed in classifying inventory The multicriterialiterature claims that more than one criterion should beused to enhance the inventory classification This argumentcannot be validated unless the results of the single-criteriaand multicriteria models are compared quantitatively with acommon metric

Inventory classification directly affects the ability ofinventory to satisfy customer orders Poor inventory clas-sification on one hand will result in inventory buildup ofthose items that are not needed to meet customer demandon the other hand this may cause inventory shortages ofitems when they are needed the most The metric that isusually used in the industry is customer order fill rate whichis defined as the probability of filling an entire customerorder within a specified period [12] A comparison of orderfill rates obtained by inventory classification from different

HindawiAdvances in Decision SciencesVolume 2017 Article ID 5028919 11 pageshttpsdoiorg10115520175028919

2 Advances in Decision Sciences

Table 1 Methods and techniques used in ABC classification [13]

Method Techniques ResearchSingle-criteria Traditional ABC [14ndash18]

Multicriteria

Analytic hierarchy process(AHP) [19 20]

Cluster analysis [21 22]Decision tree [23 24]

Distance modeling [25]Genetic algorithm [6]Graphical matrix [26ndash28]Neural network [2 29 30]

Optimization models [1 3 31ndash37]

optimizationmodels will highlight the shortcomings of usingonemodel over othermodelsThis will help a decision-makerin selecting the right model for the data This has not beenaddressed in previous studies

This study fills these gapsThe contribution of this paper isas follows (1) The performances of the models are comparedusing order fill rate (2) Since inventory holding cost andstandard deviation of demand may vary with time this canaffect the decision of model selection Therefore assessmentof modelsrsquo behavior at varying levels of holding cost andstandard deviation of demand is presented using sensitivityanalysis (3) Hypothesis testing is included to understand ifthe difference in modelsrsquo behavior is significant at varyinglevels of inventory holding cost and standard deviation ofdemand (4) Finally a cut-off point method is introducedto understand the revenue impact when the highest perfor-mance shifts from one model to another

2 Literature Review

Both single-criteria and multicriteria methods are used toclassify inventory Several different techniques are used inmulticriteria classification In general inventory classificationtechniques can be divided into nonoptimization techniquesand optimization techniques Studies that use these methodsare listed in Table 1 In optimization methods linear ornonlinear optimization models are used In these modelsitems receive an aggregate optimal inventory score basedon the objective function [3] Scores of inventory items aremaximized and later they are classified into A B or Ccategories Table 1 summarizes these models

Techniques that do not use optimization models mayinclude the analytic hierarchy process cluster analysis deci-sion tree distance modeling graphical matrix and neuralnetwork AHP is widely used in earlier studies This processis based on a pairwise comparison of all criteria where theuser must define the direction and degree of preferenceThe method is criticized for its limitation which involvessubjective judgment when making pairwise comparisons [3]

Since this research focuses on using optimization modelsin classifying inventory the literature review discusses onlythese models

21 Multicriteria Analysis Using Weighted Linear Optimiza-tion Ramanathan [3] proposed an approach called theweighted linear optimizationmodel which uses multiple cri-teria The Ramanathan model (R model) employs a weightedadditive function to aggregate the performance of eachinventory item into a single score referred to as the optimalinventory score The model is a maximization objectivefunctionWeights are automatically assignedwhen themodelis solved The output also gives the optimal inventory scoreSeveral modifications to this model have been suggested

Ng [1] presented a modified version of the R modelreferred to as the Ng model He ranked the criteria indescending order and used normalized weights He alsoproposed a transformation technique to solve the modelwithout the need of linear optimization software

Zhou and Fan [31] pointed out shortcomings of theR model whereby an item can get classified as A whenit scores high in less favorable criteria This leads to aninappropriate classification of ABC items Their proposedmodel (ZFmodel) uses themost favorable and least favorableweights for each item

Chen [34] further improved the ZF model by introduc-ing a peer-estimation approach This method selects twocommon sets of criteria weights namely the most favorableand least favorable weights The resulting scores are thenaggregated for each item without any subjectivity

Hadi-Vencheh [32] observed that in the Ng model thefinal scores of each item do not depend on the weights ofeach criterion obtained from themodel hence the itemsmaybe inappropriately classified He modified the Ng model andproposed a nonlinear optimization model (HV model)

Torabi et al [35] proposed a model to incorporate bothquantitative and qualitative criteria This is the first modelof its kind that uses both quantitative and qualitative criteriato classify inventory into A B or C categories The authorsrefer to it as a modified linear programming model insteadof an optimization model Jeddou [36] used the Ng model toclassify vehicle spare parts items

Park et al [37] developed a cross-evaluation-basedweighted linear optimization (CE-WLO) model using cross-efficiency evaluations in weighted linear optimization toprovide a classification of inventory items Their model (PBBmodel) performs in a peer-evaluation mode

Iqbal and Malzahn [38] extended the ZF model andincluded the descending ranking criteria constraint in themodel They named it modified ZF model They also pro-posed a model fitness test The test evaluates the modelrsquosability to classify items without resulting in infeasibility Theuser can compare different models by comparing infeasibilityarising from each model before making a choice in modelselection for inventory classification The test results suggestthat models that use descending ranking criteria constraintresult in low or no classification infeasibility

It will be useful to know if the classification of itemsfrommodels using descending ranking criteria also performsany better than other models In this study we evaluated theperformance ofmodels in fulfilling customer ordersWe usedorder fill rate as a measure of comparison

Advances in Decision Sciences 3

Table 2 Summary of calculations of overall order fill rate for modified ZF model (first dataset)

Leadtime Item

Class ofmodified

ZF

Avg unitcost

Annualdemand(119863119894)

Orderingqty (119876119894)

Holdingcost unit CSL Safety

factor (119870119894) 119866(119870119894) Fr119894Satisfieddemand (Fr119894 times 119863119894)

7 1 A 7121 0483 0260 14242 099 2326 0003 1 04829 048297 5 A 865 1200 0372 173 099 2326 0003 0999 11992 119926 8 B 5168 2000 0622 10336 095 1645 0021 0996 19917 199174 9 B 1466 48000 5722 2932 095 1645 0021 0991 475772 4757725 10 B 72 3000 0645 144 095 1645 0021 0995 29836 298364 2 C 5845 8000 1170 1169 09 1282 0047 0984 78714 787143 3 C 4082 4000 0990 8164 09 1282 0047 0992 39671 396712 4 C 198 4004 1422 396 09 1282 0047 0995 39853 398534 6 C 712 12000 1298 1424 09 1282 0047 0978 117393 1173934 7 C 784 4000 0714 1568 09 1282 0047 0987 39474 39474

Sum =866871

Sum =857453

3 Method

The model uses three criteria lead time average annualdemand and average unit cost These criteria have consis-tently been used in other studies [1 3 31] We consideredlead time as the first criterion because the higher the leadtime of an item is the more time it takes to replenish itwhen stockout occurs which can affect a companyrsquos abilityto satisfy customer demand We included Ng model ZFmodel HV model PBB model and modified ZF model inour analysis However we excluded models that result inclassification infeasibility in Iqbal and Malzahn [38] Theexclusions include single-criteria model and R model Thereason to include modified ZF model is to understand if thedescending ranking criteria improve the performance of themodel

31 Order Fill Rate To determine how suitable an inventoryclassification model is we calculated the order fill rate foreachmodelThen the order fill rates ofmodelswere comparedto see which model results in the highest order fill rate Themethod to calculate order fill rate is explained in the work ofBabai et al [39] The formulae used in the calculations are asfollows

FR119894 = 1 minus 120590119894radic119871 119894119866 (119896119894)119876119894

FR119879 = sum119873119894=1 FR119894119863119894sum119873119894=1119863119894

(1)

where 119873 is number of inventory items in the inventorysystem ℎ119894 is unit inventory holding cost of item 119894 119896119894 is safetyfactor for item 119894 against the customer service level (CSL) 119863119894is mean annual demand of item 119894 120590119894 is standard deviation ofannual demand for item 119894 119871 119894 is lead time of item 119894119876119894 is orderquantity of item 119894 FR119894 is fill rate of item 119894 FR119879 is overall fillrate of the inventory system and 119866(119909) is loss function of thestandard normal distribution

311 Order Fill Rate for Sample Dataset 1 The order fill ratehas been calculated for several multicriteria models Firstsample dataset 1 which contains ten items is used Thisdataset is extracted from Park et al [37] Holding cost andstandard deviation of demand are not known To calculatethe order fill rate we assumed annual holding cost of 20of the average unit cost as used in Park et al [37] and astandard deviation of demand at 25 In the sensitivityanalysis section we will use multiple values of the holdingcost and standard deviation and we will evaluate how theseaffect the final result A summary of calculations of theoverall order fill rate for the modified ZF model is shown inTable 2

FR119879 = sum119894=10119894=1 (Fr119894 times 119863119894)sum119894=10119894=1 (119863119894) = 857453

866871 = 09891 (2)

A summary of the overall order fill rate of all othermulticriteria models is shown in Table 3 We excluded thesingle-criteria method the R model and the ZF modelbecause they all resulted in classification infeasibility [38]But once the descending ranking criteria constraint is addedto the ZF model (ie modified ZF model) the classificationinfeasibility does not exist However the PBB model resultedin the highest order fill rate

We also notice that resulting order fill rates from themodified ZF model and Ng model are identical This isbecause both models classify items into the same classes indataset 1 which may not be the case for other datasets wheresample size is large In the next section we test multicriteriamodels using 47-item dataset

312 Order Fill Rate for Sample Dataset 2 The seconddataset contains 47 inventory items This dataset is used inRamanathan [3] We calculated the order fill rate for themulticriteria models We used a holding cost of 20 of theaverage unit cost and a standard deviation of 25 of demandThe ABC classification of items used in these calculations is


Table 3 Summary of overall order fill rate for all other multicriteriamodels from dataset 1

Method Model Order fill rate

Multicriteria models

HV 09905Modified ZF 09891

PBB 09949Ng 09891

Table 4 Summary of overall order fill rate for multicriteria modelsfrom dataset 2



HV 09813ZF 09822

Modified ZF 09886PBB 09799Ng 09762

shown in Appendix We set a 99 service level for class Aitems 95 service level for class B items and 90 servicelevel for class C items Results are shown in Table 4

We observed fromTable 4 that results aremore distinctivecompared to dataset 1 We found that the order fill rate of themodified ZFmodel is better than that of the ZFmodel whichshows that adding a descending-order constraint improvesthe performance of the ZFmodelThese results are consistentwith the results of sample dataset 1 In this example themodified ZF model results in the highest order fill rate

We notice that in dataset 1 modified ZFmodel improvedthe classification feasibility And in dataset 2 apart fromimproving classification feasibility it also improved the orderfill rate This infers that using criteria in descending order(some criteria more important some less) provides an equalor better order fill rate compared to the case when thisconstraint is not considered in the model

32 Sensitivity Analysis In calculating the order fill rate forexample dataset 1 a standard deviation of 25 of demandand a holding cost 20 of average unit cost are used Thesevalues can differ whichmay change the results of the findingsSensitivity analysis was done to investigate how the resultschange at different levels of demand uncertainty and holdingcost

321 Sensitivity Analysis for Sample Dataset 1 We used fourmodels for comparison HV modified ZF PBB and NgWe did not include the R model and ZF model becauseinventory classification is infeasible if we use them [38] Wecalculated the order fill rate at five different values of standarddeviation of demand (25 1 05 025 and 01) andfor each value of standard deviation we used different valuesof holding cost ranging from 01 to 50 of the average costWe used 99 service level for class A items 95 service levelfor class B items and 90 service level for class C items Someof these plots are shown in Figure 1

0001 0005 001 005 01 02 03 04 05Holding cost

0001 0005 001 005 01 02 03 04 05Holding cost

Standard deviation 25 of demand


HVModified ZF

PBBNg

HVModified ZF

PBBNg

0988

099

0992

0994

0996

0998

1

Fill

rate

097

0975

098

0985

099

0995

1

Fill

rate

Figure 1 Overall order fill rate from multiple models for dataset 1

005 004 003 002 001 0001Standard deviation of demand

HVModified ZF

PBBNg

096

097

098

099

1

101

Fill

rate

Holding ＝ＩＭＮ = 02 lowast verage unit cost

Figure 2 Order fill rates at different values of standard deviation ofdemand with constant holding cost

Graphs at standard deviation 05 025 and 01 ofdemand are not shown because they show a similar trendas we have seen at 25 and 1 It is interesting to seehow models behave when the standard deviation of demandchanges at a constant value of the holding cost We used anannual holding cost of 20 of the average unit cost and ranthe models at different values of standard deviation Resultsare shown in Figure 2

There is a little difference among these four models atlower values of holding cost The PBB model scores thehighest overall fill rate at all values of standard deviation and


holding cost The Ng model and modified ZF model providethe same model value which is why their lines overlap Twomore important results can be drawn from Figures 1 and2 First the difference in fill rate among models is reducedwhen holding cost is decreased at any given value of demanduncertainty At a standard deviation of 025 of demandor below when the holding cost reaches 0001 or 01 ofthe average cost the difference between the order fill ratesof the models becomes nearly zero Therefore it becomesirrelevant which model is used for inventory classificationwhen the standard deviation drops below 025 and theholding cost is 01 or below The analysis shows that atlower values of holding cost it does notmakemuchdifferencewhich model should be used for inventory classification Butcare should be taken when the inventory holding cost ishigh Second we find an increasing trend in the fill ratewhen the standard deviation is decreasing This means wecannot ignore the effect of the standard deviation of demandwhen comparing model results It is always advantageousto perform a sensitivity analysis to determine which modelresults in the highest fill rate at any given value of standarddeviation and holding cost If a model results in the highestorder fill rate at a given value of standard deviation andholding cost then it is wrong to assume that it will alwaysresult in the highest order fill rate at any value of standarddeviation and holding cost

We did a similar analysis at other service levels as wellIn the second comparison we used paired service levelsof 95 90 and 85 with classes A B and C itemsrespectively In the third comparison we used the pairingof service levels at 90 85 and 80 service for classesA B and C items respectively The PBB model results inthe highest fill rate This result is shown in Appendix Weobserve the same trend as shown in the first pair of servicelevels That is the difference in fill rate is decreasing amongmodels when the holding cost is decreasing Also the fill rateimproves for all models as the standard deviation of demanddrops

322 Sensitivity Analysis for Sample Dataset 2 Sensitivityanalysis of sample dataset 2 was done by varying the holdingcost at different values standard deviation of demand whilecalculating the order fill rate To be consistent we used thesame values for the standard deviation of demand at 251 05 025 and 01 and the same values of holding costranging from 01 to 50 of average cost as those we used inthe sensitivity analysis of dataset 1 In the first comparisonservice levels of 99 95 and 90 are paired with classesA B and C items respectively The fill rate is calculatedfrom four models HV modified ZF PBB and Ng Resultsare shown in Figures 3 and 4 Graphs at 1 05 025 and01 standard deviation of demand are not shown becausethey show a similar trend

Here we see a similar trend as shown for dataset 1 With adecreased standard deviation the overall fill rate is improvedAlso the overall fill rate shows an increasing trend withdecreasing holding cost The difference between the resultsof the four models is reduced when holding cost is reduced

0001 0005 001 005 01 02 03 04 05Holding cost


HVModified ZF

PBBNg

094

095

096

097

098

099

1

Fill

rate



092093094095096097098099

1101

Fill

rate

HVModified ZF

PBBNg



However themodified ZFmodel results in the highest overallfill rate in this case at all values of standard deviation andholding cost used in the calculations exceptwhen the holdingcost becomes 01 of the average cost or drops below thisvalue at the standard deviation of demand at 025 or belowThis means that the standard deviation and holding costshould be given consideration when selecting the modelSensitivity analysis will highlight which model results in thehighest overall fill rate

We also calculated the overall fill rate at other values ofcustomer service levels The second pairing of service levelwas 95 90 and 85 for class A class B and class C itemsThe third pairing of service level was 90 85 and 80 forclass A class B and class C items Results for these are thesame as shown in the case of pair 1 service level

323 Regression Analysis of Difference in Slope In Figures 2and 4 we see an increasing trend in fill rate at lower values ofstandard deviationHowever the slope of eachmodel appearsdifferent This difference indicates that the performanceamongmodels varies with the standard deviation of demandIt is useful in decision-making to analyze the difference in


Table 5 Some output of regression lines

HV Modified ZF PBB Ng

Sample size 6 6 6 6Slope minus03773 minus04345 minus02 minus04345SE coefficient 0000702 0000727 0000001 0000727Degrees offreedom 8 8 8 8

slope of the models We selected case study 1 (Figure 2)to compare the slopes of different models The regressionanalysis to fit the individual model lines shown in Figure 2was obtained using the software Minitab Some of thesestatistics are shown in Table 5

Hypothesis test is given as follows

Ho 1198871 = 1198872 (no difference in slope)versusH1 1198871 = 1198872 (difference in slope exists)

We used a 119905-test (since 119899 lt 25) and the rejection region wasset as follows

10038161003816100381610038161199051198881003816100381610038161003816 gt 119905(120572df) where df = 1198991 + 1198992 minus 4 (3)

We find 119905(0958) = 18595119879statistic is found from the formula

119905119888 = 1198871 minus 1198872radicSE21198871 + SE21198872

(4)

Results of the hypothesis test are shown in Table 6 Thecomparison is made at 95 confidence interval We alreadyknow from Figure 2 that the PBBmodel results in the highestfill rate and the difference between the PBB model andothermodels shrinks when the standard deviation of demanddecreases Table 6 provides insight that this difference issignificant Users should be careful in selecting amodel whenthe variation in demand is high A model can result in betterperformance at one value of standard deviation but at othervalues another model might outperform the first model

4 Discussion

In this study we present a method for comparing the per-formances of the models The sensitivity analysis regressionanalysis and hypothesis test are used to understand theeffect of variability in inventory holding cost and standarddeviation of demand on the decision of model selection Theanalysis also determines if descending ranking criteria haveany significance in improving the results and performance ofthe model We used two datasets for the analysis

We evaluated the performance of the model in fulfillingcustomer orders The metric used in the industry is ldquooverallfill raterdquo Inventory classification that is received from eachmodel is used in calculating the order fill rate Results were

then compared to determine where each model stands interms of satisfying customer orders In the first dataset wefound that the PBB model leads other models In the seconddataset we found that the modified ZF model shows thebest performanceTherefore selection of a model to performinventory classification depends on the dataset

In a comparative analysis it was also revealed that afteradding descending-order criteria to the ZFmodel the result-ing order fill rate was improved This indicates that givingunequal weights to the criteria enhances the performance ofthe model

Sensitivity analysis showed that holding cost and standarddeviation play very important roles in determining whichmodel results in the highest order fill rate By varying thesevalues we analyzed the response of different models whencompared We discovered that the difference in the order fillrate of models is reduced as the holding cost and standarddeviation of demand decrease At one point the difference isnegligible Below this point it becomes insignificant whichmodel is used for inventory classification A given modelcannot outperform other models at all values of holding costand standard deviation of demandwhen comparing the orderfill rates Sensitivity analysis helps in making an informeddecision aboutmodel selection at a given value of holding costand standard deviation of demand

Results from regression analysis and hypothesis testingshowed that the difference between slopes of the models issignificant thus indicating that the best performing modelmay not be superior to other models when the standarddeviation of demand is reduced Selecting amodel for the bestperformance depends on the standard deviation of demandand holding cost Users should take this into considerationwhen doing a comparative analysis for model selection

If we further extend this analysis it is useful to find thecut-off point where the performance of originally superiormodel becomes equal to anothermodelWhenwe know thesecut-off points it becomes easier to select a model which givesthe highest order fill rate

5 Conclusion

This paper compares the performances of various modelsusing the order fill rate The study presents an analysis toshow that the best performing model does not remain thebest at all times A method to calculate a model cut-off pointis also shown where the highest performing model losesto another model Two different datasets are used in thisstudy Sensitivity analysis regression analysis and hypothesistest are also included to understand the usefulness andsignificance of the results

The study shows that the inclusion of descending criteriaconstraint improves the performance of the model Thismeans that when criteria are set in order of importancethe resulting classification of inventory items produces betterorder fill rate As was seen in the case of the ZF modelwhen the modified ZFmodel which includes the descendingranking criteria is used the order FRT of the model isimproved


Table 6 Test results of slopes comparison (for dataset 1)

ParameterHV model

versus modifiedZF Model

HV modelversus PBBmodel

HV modelversus Ng model

Modified ZFmodel versusPBB model

Modified ZFmodel versus Ng

model

PBB modelversus Ng model

119905119888 5696 minus25325 5696 minus32566 0 32566119875 value 0 0 0 0 1 0

Inference

Slope ofmodified ZFmodel is

significantlyhigher than thatof HV model

Slope of HVmodel is

significantlyhigher than thatof PBB model

Slope of Ngmodel is




Slopes of bothmodels are notsignificantlydifferent

Slope of Ngmodel is


We also found an important relationship between theresults of each model when the holding cost and standarddeviation of demand vary Regression analysis shows thatthe slopes of models are significantly different Sensitivityanalysis confirms that the end result of models comparisondoes not change at other customer service levels Howeverthe lead of the highest scoring model shrinks when theholding cost and standard deviation of demand decreaseThestudy also shows that selection of a model depends on thedataset and values of the holding cost and standard deviationof demand

The method explained in this study tends to improvethe order fill rate if the results of comparative evaluation ofmodels are used in selecting the model For example oncea cut-off point of a model is determined and achieved theuser can switch to new highest performing model By doingthis order fill rate is improved which results in an increase inrevenue in a specified period

For future research it would be interesting to comparethe results when a particular product has a variable lead timeespecially when a product is purchased from more than onesupplier as each suppliermay have different lead times for thesame product

Appendix

A Mathematical Form of Multicriteria Models

A1 R Model

max119869sum119895=1

V119898119895119910119898119895

st 119869sum119895=1

V119898119895119910119899119895 le 1 119899 = 1 2 119873V119898119895 ge 0 119895 = 1 2 119869

(A1)

Notation

119895 = criteria119898 = inventory items

V119898119895 = weight of119898th inventory items under criteria 119895119910119898119895 = score of119898th inventory items under criteria 119895

A2 Ng Model

max 119878119894 =119869sum119895=1

119908119894119895119910119894119895

st 119869sum119895=1

119908119894119895 = 1119908119894119895 minus 119908119894(119895+1) ge 0 119895 = 1 2 (119869 minus 1)119908119894119895 ge 0 119895 = 1 2 119869

(A2)

Notation

119895 = criteria119894 = inventory items119908119894119895 = weight of the 119894th inventory item under criteria 119895119910119894119895 = score of 119894th inventory item under criteria 119895

A3 ZF Model

119887119868119894 = min119873sum119899=1

119908119887119894119899119910119894119899

st 119873sum119899=1

119908119887119894119899119910119898119899 ge 1 119898 = 1 2 119872

119908119887119894119899 ge 0

(A3)

Notation

119894 = inventory items119899 = criteria119910119894119899 = score of 119894th inventory item under criteria 119899119908119887119894119899 = weight of 119894th inventory item under criteria 119899


Table 7

Items Lead time Annual demand Average unit Single-criteria R model HV model ZF model PBB model Ng model(weeks) (unit) cost ($) Class Score Class Score Class Score Class Score Class Score Class

13 7 1200 8650 A 10000 A 10494 A 08517 A 09135 A 1 A29 7 200 13434 A 10000 A 10947 A 06411 A 09453 A 1 A34 7 2700 707 A 10000 A 10072 A 05000 B 08486 A 1 A45 7 100 3440 A 10000 A 10048 A 05000 B 08486 A 1 A9 6 3300 7344 Infeasible 09055 B 08967 A 07632 A 08081 A 083 A18 6 1200 4950 Infeasible 08571 B 08516 B 06423 A 07619 B 083 A28 6 400 7840 Infeasible 08571 B 08702 B 06114 B 07783 B 083 A40 6 200 5168 Infeasible 08571 B 08547 B 05301 B 07501 B 083 B2 5 2700 21000 B 10000 A 10373 A 10000 A 08355 A 067 B12 5 5000 2087 B 07863 B 07128 B 04466 B 06762 B 067 B14 5 800 11040 B 07588 B 07711 B 06080 B 07037 B 067 B19 5 1200 4750 B 07158 B 06947 B 05246 B 06440 B 067 B31 5 300 7200 B 07142 Infeasible 07118 B 04621 B 06504 B 067 B33 5 400 4948 B 07142 Infeasible 06894 B 04496 B 06344 B 067 B37 5 500 3000 B 07142 Infeasible 06772 B 04093 C 06172 B 067 B39 5 200 5960 B 07142 Infeasible 06966 B 04268 B 06398 B 067 B43 5 200 2989 B 07142 Infeasible 06753 C 03617 C 06130 C 067 B47 5 300 846 B 07142 Infeasible 06703 C 03209 C 05940 C 067 B3 4 21200 2376 B 10000 A 10644 A 06345 A 08129 A 075 B10 4 1500 16050 C 07812 B 07710 B 06965 A 06511 B 05 C8 4 4800 5500 C 06621 C 05964 C 04761 B 05911 C 05 C17 4 4800 1466 C 06454 C 05485 C 03065 C 05512 C 05 C23 4 500 8650 C 06000 C 05815 C 04126 C 05564 C 05 C21 4 1900 2440 C 05872 C 05159 C 03093 C 05161 C 05 C22 4 700 6500 C 05787 C 05488 C 03955 C 05291 C 05 C20 4 800 5845 C 05779 C 05404 C 03977 C 05326 C 05 C7 3 10000 2820 C 06280 C 05762 C 03509 C 05331 C 04 C6 3 9400 3124 C 06165 C 05597 C 03502 C 05226 C 0385 C5 3 6000 5798 C 05784 C 05048 C 03882 C 04983 C 033 C15 3 1200 7120 C 04830 C 04211 C 03280 C 04368 C 033 C16 3 1800 4500 C 04597 C 03812 C 02733 C 04191 C 033 C38 3 200 6740 C 04531 C 03923 C 01927 C 04174 C 033 C24 3 1200 3320 C 04416 C 03563 C 02310 C 03982 C 033 C35 3 300 6060 C 04398 C 03848 C 02196 C 04116 C 033 C26 3 1000 3384 C 04376 C 03536 C 02298 C 04099 C 033 C36 3 400 4082 C 04285 C 03536 C 02020 C 03937 C 033 C44 3 100 4830 C 04285 C 03618 C 01420 C 03964 C 033 C46 3 100 2880 C 04285 C 03407 C 01420 C 03775 C 033 C11 2 21000 512 C 09905 A 08202 B 04940 B 05612 C 058 C1 2 11700 4992 C 07070 C 05546 C 04123 C 04684 C 036 C32 2 400 5302 C 03227 C 02402 C 01123 C 02908 C 017 C42 2 200 3770 C 02884 C 02042 C 00542 C 02716 C 017 C41 2 400 1980 C 02857 C 01791 C 00525 C 07474 B 017 C4 1 17200 2773 C 08469 B 05832 C 04041 C 04227 C 0405 C27 1 400 8403 C 04001 C 02309 C 01242 C 02146 C 01333 C30 1 400 5600 C 02666 C 01501 C 00405 C 01815 C 00866 C25 1 1000 3705 C 02019 C 01154 C 00000 C 01690 C 00666 C


New ZF


0005 001 005 01 02 03 04 05Holding cost

HV PBBNg

09975

0998

09985

0999

09995

1

Fill

rate

0001 0005 001 005 01 02 03 04 05Holding cost


HVNew ZF

PBBNg

095

096

097

098

099

1Fi

ll ra

teStandard deviation 1 of demand

New ZF

0001 0005 001 005 01 02 03 04 05Holding cost

HV PBBNg

098

0985

099

0995

1

Fill

rate

New ZF


0001 0005 001 005 01 02 03 04 05Holding cost

HV PBBNg

0988099

0992099409960998

1

Fill

rate

New ZF


0001 0005 001 005 01 02 03 04 05Holding cost

HV PBBNg

099409950996099709980999

1

Fill

rate

Figure 5

119887119868119894 = least favorable aggregate score of 119894th inventoryitem

119892119868119894 = max119873sum119899=1

119908119892119894119899119884119894119899st 119873sum

119899=1

119908119892119894119899119910119898119899 le 1 119898 = 1 2 119872119908119892119894119899 ge 0

(A4)

119892119868119894 is most favorable aggregate score of 119894th inventory itemFurthermore the two indexes are combined to produce a

ldquocomposite indexrdquo using formula (A5)

119899119868119894 (120582) = (120582 (119892119868119894 minus 119892119868119894)(119892119868lowast minus 119892119868119894) ) + ((1 minus 120582) (119887119868119894 minus 119887119868119894)(119887119868lowast minus 119887119868119894) ) (A5)

where 119892119868lowast is maximum value of good index 119887119868lowast is minimumvalue of bad index 120582 is control parameter that is specified byuser and 119899119868119894 is composite index for an item

A4 HV Model

max 119878119894 =119869sum119895=1

119910119894119895119908119895

st 119869sum119895=1

1199082119895 = 1

119908119895 ge 119908119895+1 ge 0 119895 = 1 2 (119869 minus 1)119908119895 ge 0 119895 = 1 2 119869

(A6)

Notation

119908119895 = relative importance weight attached to 119895thcriteria119910119894119895 = score of 119894th inventory item under criteria 119895


A5 PBB Model

119862119901119896 = min119904sum119903=1

119906119903119896119910119903119901st 119904sum

119903=1

119906119903119896119910119903119896 = 119897lowast119896119904sum119903=1

119906119903119896119910119903119895 le 1 119895 = 1 119899 119895 = 119896119906119903119896 ge 0 forall119903

(A7)

Notation

119906119903119896 = weight given to the 119903th criterion of the 119896th item(119910119903119896)119910119903119896 = score on 119896 item under criterion 119903119897lowast119896 = optimal performance of inventory score asreceived from R model119862119901119896 = the optimal cross-efficiency score for item 119901evaluated by item 119896

B Scores and Classification ofItems in Dataset 2

See Table 7

C Overall Order Fill Rate of Dataset 1at 95 90 and 85 Service Level forClass A Class B and Class C Items

See Figure 5

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] W L Ng ldquoA simple classifier formultiple criteria ABC analysisrdquoEuropean Journal of Operational Research vol 177 no 1 pp344ndash353 2007

[2] F Y Partovi and M Anandarajan ldquoClassifying inventory usingan artificial neural network approachrdquo Computers amp IndustrialEngineering vol 41 no 4 pp 389ndash404 389

[3] R Ramanathan ldquoABC inventory classification with multiple-criteria using weighted linear optimizationrdquo Computers andOperations Research vol 33 no 3 pp 695ndash700 2006

[4] R E Stanford and W Martin ldquoTowards a normative model forinventory cost management in a generalized ABC classificationsystemrdquo Journal of the Operational Research Society vol 58 no7 pp 922ndash928 2007

[5] P Hautaniemi and T Pirttila ldquoThe choice of replenishmentpolicies in an MRP environmentrdquo International Journal ofProduction Economics vol 59 no 1 pp 85ndash92 1999

[6] HAGuvenir andE Erel ldquoMulticriteria inventory classificationusing a genetic algorithmrdquo European Journal of OperationalResearch vol 105 no 1 pp 29ndash37 1998

[7] B E Flores D L Olson and CWolfe ldquoJudgmental adjustmentof forecasts a comparison of methodsrdquo International Journal ofForecasting vol 7 no 4 pp 421ndash433 1992

[8] H Jamshidi and A Jain ldquoMulti-criteria ABC inventory classifi-cation with exponential smoothing weightsrdquo Journal of GlobalBusiness Issues vol 2 no 1 2008

[9] F Y Partovi and J Burton ldquoUsing the analytic hierarchyprocess for ABC analysisrdquo International Journal of Operationsamp Production Management vol 13 no 9 pp 29ndash44 1993

[10] P P Gajpal L S Ganesh and C Rajendran ldquoCriticalityanalysis of spare parts using the analytic hierarchy processrdquoInternational Journal of Production Economics vol 35 no 1ndash3pp 293ndash297 1994

[11] A Bhattacharya B Sarkar and S K Mukherjee ldquoDistance-based consensus method for ABC analysisrdquo International Jour-nal of Production Research vol 45 no 15 pp 3405ndash3420 2007

[12] J-S Song ldquoOn the order fill rate in a multi-item base-stockinventory systemrdquo Operations Research vol 46 no 6 pp 831ndash845 1998

[13] T J Van Kampen R Akkerman and D P Van Donk ldquoSKUclassification a literature review and conceptual frameworkrdquoInternational Journal of Operations amp Production Managementvol 32 no 7 pp 850ndash876 2012

[14] A G Canen and R D Galvao ldquoAn application of ABC analysisto control imported materialrdquo Interfaces vol 10 no 4 pp 22ndash24 1980

[15] J J Chrisman ldquoBasic production techniques for small manu-facturers ii inventory control methods and MRPrdquo Productionamp Inventory Management vol 26 no 3 pp 48ndash64 1985

[16] E S Gardner Jr ldquoEvaluating forecast performance in aninventory control systemrdquo Management Science vol 36 no 4pp 490ndash499 1990

[17] V Portougal ldquoDemand forecast for a catalog retailing companyrdquoProduction and Inventory Management Journal vol 43 no 1-2pp 29ndash34 2002

[18] G C Onwubolu and B C Dube ldquoImplementing an improvedinventory control system in a small company a case studyrdquoProduction Planning and Control vol 17 no 1 pp 67ndash76 2006

[19] C U I Nan-fang and L U O Xue ldquoABC classification basedon AHP in servicing spare partrdquo Industrial Engineering andManagement vol 6 article 008 2004

[20] AMolenaers H Baets L Pintelon andGWaeyenbergh ldquoCrit-icality classification of spare parts a case studyrdquo InternationalJournal of Production Economics vol 140 no 2 pp 570ndash5782012

[21] R Ernst and M A Cohen ldquoOperations related groups (ORGs)a clustering procedure for productioninventory systemsrdquo Jour-nal of Operations Management vol 9 no 4 pp 574ndash598 1990

[22] L Canetta N Cheikhrouhou and R Glardon ldquoApplyingtwo-stage SOM-based clustering approaches to industrial dataanalysisrdquo Production Planning and Control vol 16 no 8 pp774ndash784 2005

[23] J Huiskonen ldquoMaintenance spare parts logistics special char-acteristics and strategic choicesrdquo International Journal of Pro-duction Economics vol 71 no 1ndash3 pp 125ndash133 2001

[24] J E Boylan AA Syntetos andGCKarakostas ldquoClassificationfor forecasting and stock control a case studyrdquo Journal of theOperational Research Society vol 59 no 4 pp 473ndash481 2008


[25] Y Chen K W Li D Marc Kilgour and K W Hipel ldquoAcase-based distance model for multiple criteria ABC analysisrdquoComputers and Operations Research vol 35 no 3 pp 776ndash7962008

[26] A J DrsquoAlessandro and A Baveja ldquoDivide and conquer rohmand Haasrsquo response to a changing specialty chemicals marketrdquoInterfaces vol 30 no 6 pp 1ndash16 2000

[27] A A Ghobbar and C H Friend ldquoSources of intermittentdemand for aircraft spare parts within airline operationsrdquoJournal of Air Transport Management vol 8 no 4 pp 221ndash2312002

[28] A A Syntetos J E Boylan and J D Croston ldquoOn thecategorization of demand patternsrdquo Journal of the OperationalResearch Society vol 56 no 5 pp 495ndash503 2005

[29] M-C Yu ldquoMulti-criteria ABC analysis using artificial-intel-ligence-based classification techniquesrdquo Expert Systems withApplications vol 38 no 4 pp 3416ndash3421 2011

[30] G Kabir and M A A Hasin ldquoMulti-criteria inventory classi-fication through integration of fuzzy analytic hierarchy processand artificial neural networkrdquo International Journal of Industrialand Systems Engineering vol 14 no 1 pp 74ndash103 2013

[31] P Zhou and L Fan ldquoA note on multi-criteria ABC inventoryclassification using weighted linear optimizationrdquo EuropeanJournal of Operational Research vol 182 no 3 pp 1488ndash14912007

[32] A Hadi-Vencheh ldquoAn improvement to multiple criteria ABCinventory classificationrdquo European Journal of OperationalResearch vol 201 no 3 pp 962ndash965 2010

[33] S M Hatefi and S A Torabi ldquoA common weight MCDA-DEA approach to construct composite indicatorsrdquo EcologicalEconomics vol 70 no 1 pp 114ndash120 2010

[34] J-X Chen ldquoPeer-estimation for multiple criteria ABC inven-tory classificationrdquo Computers and Operations Research vol 38no 12 pp 1784ndash1791 2011

[35] S A Torabi S M Hatefi and B S Pay ldquoABC inventoryclassification in the presence of both quantitative and qualitativecriteriardquo Computers amp Industrial Engineering vol 63 no 2 pp530ndash537 2012

[36] M B Jeddou ldquoMulti-criteria ABC inventory classification-a case of vehicles spare parts itemsrdquo Journal of AdvancedManagement Science vol 2 no 3 pp 181ndash185 2014

[37] J Park H Bae and J Bae ldquoCross-evaluation-based weightedlinear optimization for multi-criteria ABC inventory classifica-tionrdquo Computers and Industrial Engineering vol 76 no 1 pp40ndash48 2014

[38] Q Iqbal and D Malzahn ldquoEvaluating discriminating powerof single-criteria and multi-criteria models towards inventoryclassificationrdquo Computers amp Industrial Engineering vol 104 pp219ndash223 2017

[39] M Z Babai T Ladhari and I Lajili ldquoOn the inventoryperformance of multi-criteria classification methods empiricalinvestigationrdquo International Journal of Production Research vol53 no 1 pp 279ndash290 2015

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Table 1 Methods and techniques used in ABC classification [13]

Method Techniques ResearchSingle-criteria Traditional ABC [14ndash18]

Multicriteria

Analytic hierarchy process(AHP) [19 20]

Cluster analysis [21 22]Decision tree [23 24]

Distance modeling [25]Genetic algorithm [6]Graphical matrix [26ndash28]Neural network [2 29 30]

Optimization models [1 3 31ndash37]

optimizationmodels will highlight the shortcomings of usingonemodel over othermodelsThis will help a decision-makerin selecting the right model for the data This has not beenaddressed in previous studies

This study fills these gapsThe contribution of this paper isas follows (1) The performances of the models are comparedusing order fill rate (2) Since inventory holding cost andstandard deviation of demand may vary with time this canaffect the decision of model selection Therefore assessmentof modelsrsquo behavior at varying levels of holding cost andstandard deviation of demand is presented using sensitivityanalysis (3) Hypothesis testing is included to understand ifthe difference in modelsrsquo behavior is significant at varyinglevels of inventory holding cost and standard deviation ofdemand (4) Finally a cut-off point method is introducedto understand the revenue impact when the highest perfor-mance shifts from one model to another

2 Literature Review

Both single-criteria and multicriteria methods are used toclassify inventory Several different techniques are used inmulticriteria classification In general inventory classificationtechniques can be divided into nonoptimization techniquesand optimization techniques Studies that use these methodsare listed in Table 1 In optimization methods linear ornonlinear optimization models are used In these modelsitems receive an aggregate optimal inventory score basedon the objective function [3] Scores of inventory items aremaximized and later they are classified into A B or Ccategories Table 1 summarizes these models

Techniques that do not use optimization models mayinclude the analytic hierarchy process cluster analysis deci-sion tree distance modeling graphical matrix and neuralnetwork AHP is widely used in earlier studies This processis based on a pairwise comparison of all criteria where theuser must define the direction and degree of preferenceThe method is criticized for its limitation which involvessubjective judgment when making pairwise comparisons [3]

Since this research focuses on using optimization modelsin classifying inventory the literature review discusses onlythese models

21 Multicriteria Analysis Using Weighted Linear Optimiza-tion Ramanathan [3] proposed an approach called theweighted linear optimizationmodel which uses multiple cri-teria The Ramanathan model (R model) employs a weightedadditive function to aggregate the performance of eachinventory item into a single score referred to as the optimalinventory score The model is a maximization objectivefunctionWeights are automatically assignedwhen themodelis solved The output also gives the optimal inventory scoreSeveral modifications to this model have been suggested

Ng [1] presented a modified version of the R modelreferred to as the Ng model He ranked the criteria indescending order and used normalized weights He alsoproposed a transformation technique to solve the modelwithout the need of linear optimization software

Zhou and Fan [31] pointed out shortcomings of theR model whereby an item can get classified as A whenit scores high in less favorable criteria This leads to aninappropriate classification of ABC items Their proposedmodel (ZFmodel) uses themost favorable and least favorableweights for each item

Chen [34] further improved the ZF model by introduc-ing a peer-estimation approach This method selects twocommon sets of criteria weights namely the most favorableand least favorable weights The resulting scores are thenaggregated for each item without any subjectivity

Hadi-Vencheh [32] observed that in the Ng model thefinal scores of each item do not depend on the weights ofeach criterion obtained from themodel hence the itemsmaybe inappropriately classified He modified the Ng model andproposed a nonlinear optimization model (HV model)

Torabi et al [35] proposed a model to incorporate bothquantitative and qualitative criteria This is the first modelof its kind that uses both quantitative and qualitative criteriato classify inventory into A B or C categories The authorsrefer to it as a modified linear programming model insteadof an optimization model Jeddou [36] used the Ng model toclassify vehicle spare parts items

Park et al [37] developed a cross-evaluation-basedweighted linear optimization (CE-WLO) model using cross-efficiency evaluations in weighted linear optimization toprovide a classification of inventory items Their model (PBBmodel) performs in a peer-evaluation mode

Iqbal and Malzahn [38] extended the ZF model andincluded the descending ranking criteria constraint in themodel They named it modified ZF model They also pro-posed a model fitness test The test evaluates the modelrsquosability to classify items without resulting in infeasibility Theuser can compare different models by comparing infeasibilityarising from each model before making a choice in modelselection for inventory classification The test results suggestthat models that use descending ranking criteria constraintresult in low or no classification infeasibility

It will be useful to know if the classification of itemsfrommodels using descending ranking criteria also performsany better than other models In this study we evaluated theperformance ofmodels in fulfilling customer ordersWe usedorder fill rate as a measure of comparison



Leadtime Item

Class ofmodified

ZF

Avg unitcost





7 1 A 7121 0483 0260 14242 099 2326 0003 1 04829 048297 5 A 865 1200 0372 173 099 2326 0003 0999 11992 119926 8 B 5168 2000 0622 10336 095 1645 0021 0996 19917 199174 9 B 1466 48000 5722 2932 095 1645 0021 0991 475772 4757725 10 B 72 3000 0645 144 095 1645 0021 0995 29836 298364 2 C 5845 8000 1170 1169 09 1282 0047 0984 78714 787143 3 C 4082 4000 0990 8164 09 1282 0047 0992 39671 396712 4 C 198 4004 1422 396 09 1282 0047 0995 39853 398534 6 C 712 12000 1298 1424 09 1282 0047 0978 117393 1173934 7 C 784 4000 0714 1568 09 1282 0047 0987 39474 39474

Sum =866871

Sum =857453

3 Method



FR119894 = 1 minus 120590119894radic119871 119894119866 (119896119894)119876119894

FR119879 = sum119873119894=1 FR119894119863119894sum119873119894=1119863119894

(1)




866871 = 09891 (2)









PBB 09949Ng 09891




HV 09813ZF 09822







0001 0005 001 005 01 02 03 04 05Holding cost

0001 0005 001 005 01 02 03 04 05Holding cost



HVModified ZF

PBBNg

HVModified ZF

PBBNg

0988

099

0992

0994

0996

0998

1

Fill

rate

097

0975

098

0985

099

0995

1

Fill

rate



HVModified ZF

PBBNg

096

097

098

099

1

101

Fill

rate










0001 0005 001 005 01 02 03 04 05Holding cost


HVModified ZF

PBBNg

094

095

096

097

098

099

1

Fill

rate



092093094095096097098099

1101

Fill

rate

HVModified ZF

PBBNg
















119905119888 = 1198871 minus 1198872radicSE21198871 + SE21198872

(4)


4 Discussion








5 Conclusion





ParameterHV model






model



Inference



Slope of HVmodel is


Slope of Ngmodel is





Slope of Ngmodel is





Appendix


A1 R Model

max119869sum119895=1

V119898119895119910119898119895

st 119869sum119895=1

V119898119895119910119899119895 le 1 119899 = 1 2 119873V119898119895 ge 0 119895 = 1 2 119869

(A1)

Notation



A2 Ng Model

max 119878119894 =119869sum119895=1

119908119894119895119910119894119895

st 119869sum119895=1

119908119894119895 = 1119908119894119895 minus 119908119894(119895+1) ge 0 119895 = 1 2 (119869 minus 1)119908119894119895 ge 0 119895 = 1 2 119869

(A2)

Notation


A3 ZF Model

119887119868119894 = min119873sum119899=1

119908119887119894119899119910119894119899

st 119873sum119899=1

119908119887119894119899119910119898119899 ge 1 119898 = 1 2 119872

119908119887119894119899 ge 0

(A3)

Notation



Table 7




New ZF


0005 001 005 01 02 03 04 05Holding cost

HV PBBNg

09975

0998

09985

0999

09995

1

Fill

rate

0001 0005 001 005 01 02 03 04 05Holding cost


HVNew ZF

PBBNg

095

096

097

098

099

1Fi

ll ra


New ZF

0001 0005 001 005 01 02 03 04 05Holding cost

HV PBBNg

098

0985

099

0995

1

Fill

rate

New ZF


0001 0005 001 005 01 02 03 04 05Holding cost

HV PBBNg

0988099

0992099409960998

1

Fill

rate

New ZF


0001 0005 001 005 01 02 03 04 05Holding cost

HV PBBNg

099409950996099709980999

1

Fill

rate

Figure 5


119892119868119894 = max119873sum119899=1

119908119892119894119899119884119894119899st 119873sum

119899=1

119908119892119894119899119910119898119899 le 1 119898 = 1 2 119872119908119892119894119899 ge 0

(A4)





A4 HV Model

max 119878119894 =119869sum119895=1

119910119894119895119908119895

st 119869sum119895=1

1199082119895 = 1

119908119895 ge 119908119895+1 ge 0 119895 = 1 2 (119869 minus 1)119908119895 ge 0 119895 = 1 2 119869

(A6)

Notation



A5 PBB Model

119862119901119896 = min119904sum119903=1

119906119903119896119910119903119901st 119904sum

119903=1

119906119903119896119910119903119896 = 119897lowast119896119904sum119903=1

119906119903119896119910119903119895 le 1 119895 = 1 119899 119895 = 119896119906119903119896 ge 0 forall119903

(A7)

Notation



See Table 7


See Figure 5



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






Leadtime Item

Class ofmodified

ZF

Avg unitcost





7 1 A 7121 0483 0260 14242 099 2326 0003 1 04829 048297 5 A 865 1200 0372 173 099 2326 0003 0999 11992 119926 8 B 5168 2000 0622 10336 095 1645 0021 0996 19917 199174 9 B 1466 48000 5722 2932 095 1645 0021 0991 475772 4757725 10 B 72 3000 0645 144 095 1645 0021 0995 29836 298364 2 C 5845 8000 1170 1169 09 1282 0047 0984 78714 787143 3 C 4082 4000 0990 8164 09 1282 0047 0992 39671 396712 4 C 198 4004 1422 396 09 1282 0047 0995 39853 398534 6 C 712 12000 1298 1424 09 1282 0047 0978 117393 1173934 7 C 784 4000 0714 1568 09 1282 0047 0987 39474 39474

Sum =866871

Sum =857453

3 Method



FR119894 = 1 minus 120590119894radic119871 119894119866 (119896119894)119876119894

FR119879 = sum119873119894=1 FR119894119863119894sum119873119894=1119863119894

(1)




866871 = 09891 (2)









PBB 09949Ng 09891




HV 09813ZF 09822







0001 0005 001 005 01 02 03 04 05Holding cost

0001 0005 001 005 01 02 03 04 05Holding cost



HVModified ZF

PBBNg

HVModified ZF

PBBNg

0988

099

0992

0994

0996

0998

1

Fill

rate

097

0975

098

0985

099

0995

1

Fill

rate



HVModified ZF

PBBNg

096

097

098

099

1

101

Fill

rate










0001 0005 001 005 01 02 03 04 05Holding cost


HVModified ZF

PBBNg

094

095

096

097

098

099

1

Fill

rate



092093094095096097098099

1101

Fill

rate

HVModified ZF

PBBNg
















119905119888 = 1198871 minus 1198872radicSE21198871 + SE21198872

(4)


4 Discussion








5 Conclusion





ParameterHV model






model



Inference



Slope of HVmodel is


Slope of Ngmodel is





Slope of Ngmodel is





Appendix


A1 R Model

max119869sum119895=1

V119898119895119910119898119895

st 119869sum119895=1

V119898119895119910119899119895 le 1 119899 = 1 2 119873V119898119895 ge 0 119895 = 1 2 119869

(A1)

Notation



A2 Ng Model

max 119878119894 =119869sum119895=1

119908119894119895119910119894119895

st 119869sum119895=1

119908119894119895 = 1119908119894119895 minus 119908119894(119895+1) ge 0 119895 = 1 2 (119869 minus 1)119908119894119895 ge 0 119895 = 1 2 119869

(A2)

Notation


A3 ZF Model

119887119868119894 = min119873sum119899=1

119908119887119894119899119910119894119899

st 119873sum119899=1

119908119887119894119899119910119898119899 ge 1 119898 = 1 2 119872

119908119887119894119899 ge 0

(A3)

Notation



Table 7




New ZF


0005 001 005 01 02 03 04 05Holding cost

HV PBBNg

09975

0998

09985

0999

09995

1

Fill

rate

0001 0005 001 005 01 02 03 04 05Holding cost


HVNew ZF

PBBNg

095

096

097

098

099

1Fi

ll ra


New ZF

0001 0005 001 005 01 02 03 04 05Holding cost

HV PBBNg

098

0985

099

0995

1

Fill

rate

New ZF


0001 0005 001 005 01 02 03 04 05Holding cost

HV PBBNg

0988099

0992099409960998

1

Fill

rate

New ZF


0001 0005 001 005 01 02 03 04 05Holding cost

HV PBBNg

099409950996099709980999

1

Fill

rate

Figure 5


119892119868119894 = max119873sum119899=1

119908119892119894119899119884119894119899st 119873sum

119899=1

119908119892119894119899119910119898119899 le 1 119898 = 1 2 119872119908119892119894119899 ge 0

(A4)





A4 HV Model

max 119878119894 =119869sum119895=1

119910119894119895119908119895

st 119869sum119895=1

1199082119895 = 1

119908119895 ge 119908119895+1 ge 0 119895 = 1 2 (119869 minus 1)119908119895 ge 0 119895 = 1 2 119869

(A6)

Notation



A5 PBB Model

119862119901119896 = min119904sum119903=1

119906119903119896119910119903119901st 119904sum

119903=1

119906119903119896119910119903119896 = 119897lowast119896119904sum119903=1

119906119903119896119910119903119895 le 1 119895 = 1 119899 119895 = 119896119906119903119896 ge 0 forall119903

(A7)

Notation



See Table 7


See Figure 5



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of









PBB 09949Ng 09891




HV 09813ZF 09822







0001 0005 001 005 01 02 03 04 05Holding cost

0001 0005 001 005 01 02 03 04 05Holding cost



HVModified ZF

PBBNg

HVModified ZF

PBBNg

0988

099

0992

0994

0996

0998

1

Fill

rate

097

0975

098

0985

099

0995

1

Fill

rate



HVModified ZF

PBBNg

096

097

098

099

1

101

Fill

rate










0001 0005 001 005 01 02 03 04 05Holding cost


HVModified ZF

PBBNg

094

095

096

097

098

099

1

Fill

rate



092093094095096097098099

1101

Fill

rate

HVModified ZF

PBBNg
















119905119888 = 1198871 minus 1198872radicSE21198871 + SE21198872

(4)


4 Discussion








5 Conclusion





ParameterHV model






model



Inference



Slope of HVmodel is


Slope of Ngmodel is





Slope of Ngmodel is





Appendix


A1 R Model

max119869sum119895=1

V119898119895119910119898119895

st 119869sum119895=1

V119898119895119910119899119895 le 1 119899 = 1 2 119873V119898119895 ge 0 119895 = 1 2 119869

(A1)

Notation



A2 Ng Model

max 119878119894 =119869sum119895=1

119908119894119895119910119894119895

st 119869sum119895=1

119908119894119895 = 1119908119894119895 minus 119908119894(119895+1) ge 0 119895 = 1 2 (119869 minus 1)119908119894119895 ge 0 119895 = 1 2 119869

(A2)

Notation


A3 ZF Model

119887119868119894 = min119873sum119899=1

119908119887119894119899119910119894119899

st 119873sum119899=1

119908119887119894119899119910119898119899 ge 1 119898 = 1 2 119872

119908119887119894119899 ge 0

(A3)

Notation



Table 7




New ZF


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Figure 5


119892119868119894 = max119873sum119899=1

119908119892119894119899119884119894119899st 119873sum

119899=1

119908119892119894119899119910119898119899 le 1 119898 = 1 2 119872119908119892119894119899 ge 0

(A4)





A4 HV Model

max 119878119894 =119869sum119895=1

119910119894119895119908119895

st 119869sum119895=1

1199082119895 = 1

119908119895 ge 119908119895+1 ge 0 119895 = 1 2 (119869 minus 1)119908119895 ge 0 119895 = 1 2 119869

(A6)

Notation



A5 PBB Model

119862119901119896 = min119904sum119903=1

119906119903119896119910119903119901st 119904sum

119903=1

119906119903119896119910119903119896 = 119897lowast119896119904sum119903=1

119906119903119896119910119903119895 le 1 119895 = 1 119899 119895 = 119896119906119903119896 ge 0 forall119903

(A7)

Notation



See Table 7


See Figure 5



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of









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092093094095096097098099

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119905119888 = 1198871 minus 1198872radicSE21198871 + SE21198872

(4)


4 Discussion








5 Conclusion





ParameterHV model






model



Inference



Slope of HVmodel is


Slope of Ngmodel is





Slope of Ngmodel is





Appendix


A1 R Model

max119869sum119895=1

V119898119895119910119898119895

st 119869sum119895=1

V119898119895119910119899119895 le 1 119899 = 1 2 119873V119898119895 ge 0 119895 = 1 2 119869

(A1)

Notation



A2 Ng Model

max 119878119894 =119869sum119895=1

119908119894119895119910119894119895

st 119869sum119895=1

119908119894119895 = 1119908119894119895 minus 119908119894(119895+1) ge 0 119895 = 1 2 (119869 minus 1)119908119894119895 ge 0 119895 = 1 2 119869

(A2)

Notation


A3 ZF Model

119887119868119894 = min119873sum119899=1

119908119887119894119899119910119894119899

st 119873sum119899=1

119908119887119894119899119910119898119899 ge 1 119898 = 1 2 119872

119908119887119894119899 ge 0

(A3)

Notation



Table 7




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Figure 5


119892119868119894 = max119873sum119899=1

119908119892119894119899119884119894119899st 119873sum

119899=1

119908119892119894119899119910119898119899 le 1 119898 = 1 2 119872119908119892119894119899 ge 0

(A4)





A4 HV Model

max 119878119894 =119869sum119895=1

119910119894119895119908119895

st 119869sum119895=1

1199082119895 = 1

119908119895 ge 119908119895+1 ge 0 119895 = 1 2 (119869 minus 1)119908119895 ge 0 119895 = 1 2 119869

(A6)

Notation



A5 PBB Model

119862119901119896 = min119904sum119903=1

119906119903119896119910119903119901st 119904sum

119903=1

119906119903119896119910119903119896 = 119897lowast119896119904sum119903=1

119906119903119896119910119903119895 le 1 119895 = 1 119899 119895 = 119896119906119903119896 ge 0 forall119903

(A7)

Notation



See Table 7


See Figure 5



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of














119905119888 = 1198871 minus 1198872radicSE21198871 + SE21198872

(4)


4 Discussion








5 Conclusion





ParameterHV model






model



Inference



Slope of HVmodel is


Slope of Ngmodel is





Slope of Ngmodel is





Appendix


A1 R Model

max119869sum119895=1

V119898119895119910119898119895

st 119869sum119895=1

V119898119895119910119899119895 le 1 119899 = 1 2 119873V119898119895 ge 0 119895 = 1 2 119869

(A1)

Notation



A2 Ng Model

max 119878119894 =119869sum119895=1

119908119894119895119910119894119895

st 119869sum119895=1

119908119894119895 = 1119908119894119895 minus 119908119894(119895+1) ge 0 119895 = 1 2 (119869 minus 1)119908119894119895 ge 0 119895 = 1 2 119869

(A2)

Notation


A3 ZF Model

119887119868119894 = min119873sum119899=1

119908119887119894119899119910119894119899

st 119873sum119899=1

119908119887119894119899119910119898119899 ge 1 119898 = 1 2 119872

119908119887119894119899 ge 0

(A3)

Notation



Table 7




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Figure 5


119892119868119894 = max119873sum119899=1

119908119892119894119899119884119894119899st 119873sum

119899=1

119908119892119894119899119910119898119899 le 1 119898 = 1 2 119872119908119892119894119899 ge 0

(A4)





A4 HV Model

max 119878119894 =119869sum119895=1

119910119894119895119908119895

st 119869sum119895=1

1199082119895 = 1

119908119895 ge 119908119895+1 ge 0 119895 = 1 2 (119869 minus 1)119908119895 ge 0 119895 = 1 2 119869

(A6)

Notation



A5 PBB Model

119862119901119896 = min119904sum119903=1

119906119903119896119910119903119901st 119904sum

119903=1

119906119903119896119910119903119896 = 119897lowast119896119904sum119903=1

119906119903119896119910119903119895 le 1 119895 = 1 119899 119895 = 119896119906119903119896 ge 0 forall119903

(A7)

Notation



See Table 7


See Figure 5



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






ParameterHV model






model



Inference



Slope of HVmodel is


Slope of Ngmodel is





Slope of Ngmodel is





Appendix


A1 R Model

max119869sum119895=1

V119898119895119910119898119895

st 119869sum119895=1

V119898119895119910119899119895 le 1 119899 = 1 2 119873V119898119895 ge 0 119895 = 1 2 119869

(A1)

Notation



A2 Ng Model

max 119878119894 =119869sum119895=1

119908119894119895119910119894119895

st 119869sum119895=1

119908119894119895 = 1119908119894119895 minus 119908119894(119895+1) ge 0 119895 = 1 2 (119869 minus 1)119908119894119895 ge 0 119895 = 1 2 119869

(A2)

Notation


A3 ZF Model

119887119868119894 = min119873sum119899=1

119908119887119894119899119910119894119899

st 119873sum119899=1

119908119887119894119899119910119898119899 ge 1 119898 = 1 2 119872

119908119887119894119899 ge 0

(A3)

Notation



Table 7




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Figure 5


119892119868119894 = max119873sum119899=1

119908119892119894119899119884119894119899st 119873sum

119899=1

119908119892119894119899119910119898119899 le 1 119898 = 1 2 119872119908119892119894119899 ge 0

(A4)





A4 HV Model

max 119878119894 =119869sum119895=1

119910119894119895119908119895

st 119869sum119895=1

1199082119895 = 1

119908119895 ge 119908119895+1 ge 0 119895 = 1 2 (119869 minus 1)119908119895 ge 0 119895 = 1 2 119869

(A6)

Notation



A5 PBB Model

119862119901119896 = min119904sum119903=1

119906119903119896119910119903119901st 119904sum

119903=1

119906119903119896119910119903119896 = 119897lowast119896119904sum119903=1

119906119903119896119910119903119895 le 1 119895 = 1 119899 119895 = 119896119906119903119896 ge 0 forall119903

(A7)

Notation



See Table 7


See Figure 5



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Table 7




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Figure 5


119892119868119894 = max119873sum119899=1

119908119892119894119899119884119894119899st 119873sum

119899=1

119908119892119894119899119910119898119899 le 1 119898 = 1 2 119872119908119892119894119899 ge 0

(A4)





A4 HV Model

max 119878119894 =119869sum119895=1

119910119894119895119908119895

st 119869sum119895=1

1199082119895 = 1

119908119895 ge 119908119895+1 ge 0 119895 = 1 2 (119869 minus 1)119908119895 ge 0 119895 = 1 2 119869

(A6)

Notation



A5 PBB Model

119862119901119896 = min119904sum119903=1

119906119903119896119910119903119901st 119904sum

119903=1

119906119903119896119910119903119896 = 119897lowast119896119904sum119903=1

119906119903119896119910119903119895 le 1 119895 = 1 119899 119895 = 119896119906119903119896 ge 0 forall119903

(A7)

Notation



See Table 7


See Figure 5



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





New ZF


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Figure 5


119892119868119894 = max119873sum119899=1

119908119892119894119899119884119894119899st 119873sum

119899=1

119908119892119894119899119910119898119899 le 1 119898 = 1 2 119872119908119892119894119899 ge 0

(A4)





A4 HV Model

max 119878119894 =119869sum119895=1

119910119894119895119908119895

st 119869sum119895=1

1199082119895 = 1

119908119895 ge 119908119895+1 ge 0 119895 = 1 2 (119869 minus 1)119908119895 ge 0 119895 = 1 2 119869

(A6)

Notation



A5 PBB Model

119862119901119896 = min119904sum119903=1

119906119903119896119910119903119901st 119904sum

119903=1

119906119903119896119910119903119896 = 119897lowast119896119904sum119903=1

119906119903119896119910119903119895 le 1 119895 = 1 119899 119895 = 119896119906119903119896 ge 0 forall119903

(A7)

Notation



See Table 7


See Figure 5



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





A5 PBB Model

119862119901119896 = min119904sum119903=1

119906119903119896119910119903119901st 119904sum

119903=1

119906119903119896119910119903119896 = 119897lowast119896119904sum119903=1

119906119903119896119910119903119895 le 1 119895 = 1 119899 119895 = 119896119906119903119896 ge 0 forall119903

(A7)

Notation



See Table 7


See Figure 5



References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Documents

Selecting a Multicriteria Inventory Classification Model