Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 246578 13 pageshttpdxdoiorg1011552013246578

Research ArticleOrder Batching in Warehouses by Minimizing TotalTardiness A Hybrid Approach of Weighted Association RuleMining and Genetic Algorithms

Amir Hossein Azadnia1 Shahrooz Taheri2 Pezhman Ghadimi3

Muhamad Zameri Mat Saman1 and Kuan Yew Wong1

1 Department of Manufacturing and Industrial Engineering Faculty of Mechanical Engineering Universiti Teknologi MalaysiaJohor Bahru 81310 UTM Skudai Malaysia

2 Department of Computer Science Faculty of Computer Science Universiti Teknologi Malaysia Johor Bahru81310 UTM Skudai Malaysia

3 Enterprise Research Centre University of Limerick Limerick Ireland

Correspondence should be addressed to Amir Hossein Azadnia azadniaiegmailcom

Received 17 April 2013 Accepted 23 May 2013

Academic Editors Y-P Huang and P Melin

Copyright copy 2013 Amir Hossein Azadnia et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

One of the cost-intensive issues in managing warehouses is the order picking problem which deals with the retrieval of items fromtheir storage locations in order to meet customer requests Many solution approaches have been proposed in order to minimizetraveling distance in the process of order picking However in practice customer orders have to be completed by certain due datesin order to avoid tardiness which is neglected in most of the related scientific papers Consequently we proposed a novel solutionapproach in order tominimize tardiness which consists of four phases First of all weighted association rulemining has been used tocalculate associations between orders with respect to their due date Next a batching model based on binary integer programminghas been formulated to maximize the associations between orders within each batch Subsequently the order picking phase willcome up which used a Genetic Algorithm integrated with the Traveling Salesman Problem in order to identify the most suitabletravel path Finally the Genetic Algorithm has been applied for sequencing the constructed batches in order to minimize tardinessIllustrative examples and comparisons are presented to demonstrate the proficiency and solution quality of the proposed approach

1 Introduction

Based on ELAAT Kearney [1] about twenty percent of thelogistic costs of the surveyed companies were incurred dueto warehousing in 2003 A vital part of the logistics systemof a company is involved with its warehouses Optimizationof operations within every facility must be considered as animportant part of its policies in order to promptly delivergoods or services to its customers at the least cost [2] Respon-siveness is a critical success factor in a warehousing systemProduct movement within a warehouse can be facilitatedeffectively by consolidating orders into batches which can bedone before picking customer orders [3]

In order picking systems certain due dates are assignedto customer orders which should not be violated In order toavoid production delays item retrieval from the warehouseneeds to be done at its appropriate time In these casesthe tardiness of customer orders should be involved withorder batching instead of using the total processing timeas a measure for the solution quality [4] Henn et al [5]defined the tardiness of a customer order as the positivevalue between the completion time of a customer orderwith its due date The time that an order picker finishesherhis tour of gathering all required items and comes backto the starting point is called completion time Obviouslyone of the factors that can influence the completion time is

2 The Scientific World Journal

the processing time of the orders According to the aboveexplanation reducing the processing time and travel timefor constructing a batch is an important fact in reducing thecosts and minimizing delays of the customer responses Onthe other hand not considering the orderrsquos due date in theprocess of order batching can cause a huge dissatisfaction incustomer expectations regarding the efficient responsivenessof the company For explanation purposes assume that wehave some orders with different due dates considering justprocessing time or traveling time in the process of orderbatching regardless of their due dates may consolidate orderswith different due dates in a same batch So this issue maycause a problem for meeting customersrsquo demand in a certaindue date Consequently it is highlighted that order duedates should be considered as an important factor for orderbatching

Based on the above illuminated problem this researchpaper attempts to develop a novel model to overcome theaforementioned issue in the multiple parallel aisles withmanual order picking system In our proposed model itwas assumed that orders with similar items are groupedtogetherTherefore batching orders with higher resemblancecan reduce the order pickersrsquo traveled distance Also forconsidering the due dates in our model the due date of eachorder is defined as a weight parameter in calculating the asso-ciation between orders in terms of support The associationinformation integrates the bathing process with various dataregarding each order For calculating the association betweenorders in terms of support mining association rules withweighted items (MINWAL) were utilized Then a clusteringapproach based on a binary integer programming modelhas been used in order to maximize the customer demandassociation which is called support Next the Genetic Algo-rithm (GA) has been utilized to solve the Traveling SalesmanProblem (TSP) to minimize the travelling time for collectingall items in a constructed batch This process was followedby a batch sequencing process which utilized a GA in orderto minimize the average tardiness of all orders This papercopes with integrating batch ordering picker routing andthe batch sequencing problem regarding ordersrsquo due datein order to minimize the average tardiness of all ordersWe believe that the proposed approach namely ATGH isnot considered in any existing published paper In the nextsection a comprehensive literature survey is presented inorder to review the existing research activities related to thecurrent research paper

2 Literature Review

21 Order Batching Problem Order picking is an actionwhich takes place to fulfill a customer demand (inter-nalexternal) that is normally done by retrieving items frombuffer locations inside the warehouse [6ndash8] In practice thereare two types of systems for order picking systems whichare totally manual and involved with human order pickersand the systems that are entirely computerized in case ofretrieving items from the warehouse Picker-to-parts andparts-to-picker systems can be categorized as the systemsthat belong to the first group of warehousing systems [5 9]

Picker-to-parts systems are themost commonly used systemsin warehousing in which picking the items is done by anoperator that drives or walks along the aisles [10 11] Thisscenario is different in parts-to-picker systems Order pickersdo not move in this system and are located in the depotPallets or bins (unit load) from the warehouse are retrievedby automated storage and retrieval systems (ASRS) anddelivered to them at the depot After that the requested itemsare detached by the operators stationed in the depot and thepallets or bins are returned to its spot in the warehouse by theASRS [5]

In a large amount of orders a single order picking policymight be applied in which one order can be picked ineach picking tour However in a small amount of orderspicking loads in a single picking tour can lead to a decreasein travel times [12] Order batching is classified as a NP-hard problem which can be done to improve warehouseefficiency by reducing operational costs [12 13] Thus manyheuristic algorithms are proposed in the literature which helpto solve this problem Savings and seed algorithms are themost commonly used order batching heuristics in the case ofmanual picking systems Seed algorithms were introduced by[14] in which a seed order would be selected for each batchWith respect to the picking device capacity other orderswould be assigned to the seed orders Clarke andWright [15]first developed the savings algorithms inspired by a vehiclerouting algorithmThis algorithm works based on the savingthat would be achieved in travel distance or travel time Afterthat a pair of batches would be selected iteratively by thealgorithm and combined with each other until a capacityconstraint threshold is reached that stops combining of thebatches

In the current literature review it was tried to reviewall existing papers which are involved in order batchingproblems considering the minimization of due dates totaltravel distance and processing time First Vinod [16] sug-gested that integer programming can be used as a flexiblemethod for grouping any kind of objects Armstrong et al[17] also presented an integer programming model based onpredetermined batch sizes considering proximity batching

They tried to minimize the total processing time of allbatches Ratliff and Rosenthal [18] tried to minimize traveldistance based on a procedure which helps to find the bestpicking tour traveled Bymeans of a distinguished TSP a tourwas formed in a 50 aisles warehouse Kusiak et al [19] useda quadratic integer programming model to obtain batchesfrom eight orders In their proposed model it was tried tominimize the total sum of distances

One of their limitations was related to their model rangeof applicability which is not well suited for large numbers oforders Gibson and Sharp [20] presented an order batchingprocedure integrated with a computer simulation whichresulted in minimization of travel distance for each tourRosenwein [21] utilized Centre of Gravity (COG) and Min-imumAdditional Aisle (MAA) for order batching in order tomeasure travel distance It was shown that the MAA metricperformed better than the COG metric regarding travel tourgeneration Gademann et al [22] targeted to minimize thetotal travel time for all batches using a branch-and-bound

The Scientific World Journal 3

algorithm They tested their model by several test sets with amaximumorder number of 32whichwere compared togetherregarding their CPU times Hsu et al [3] applied GeneticAlgorithms (GAs) to deal with order batching problemswith any kind of configuration In their proposed approachthey endeavored to minimize the total travel distance Inanother research activity done by Chen and Wu [23] datamining and integer programming were combined to forma solution approach regarding the order batching problemThey have tried to maximize the similarity between orders ina batch in terms of support without considering ordersrsquo duedates Based on the presented results which was attemptedto compare first-come first-served (FCFS) MAA COGand Gibson and Sharprsquos method (GSM) the authors statedthat the proposed approach is effective in solving the orderbatching problems in terms of reducing traveling distancesBased on the analysis done by Gademann and van de Velde[24] the authors stated that the order batching problemis NP-hard to a great extent provided that no batch hasmore than two orders They developed a branch-and-pricealgorithm to formulate the problem in order to minimizethe total travel distance Three years later a mixed-integerprogrammingmodel was developed by Bozer andKile [25] inorder to minimize the length of trips or travel distance Theystated that acquiring minimum travel distance for a largenumber of orders requires substantial computational timeKulak et al [26] proposed a solution approach to reduce themathematical burden of order batching and picker routingproblems They proved the efficiency of their cluster-basedtabu search approaches by running several test examplesThey have focused onminimizing the travel distance betweenlocations

In practice certain completion due dates are assigned tocustomer orders For instance in distribution warehousesthe scheduled departure of trucks has to be done by certaindue dates to ensure the on time delivery of the requesteditems to the customers [22] Inmaterial warehouses avoidingproduction delays can be guaranteed if retrieving the itemsfrom thewarehouse is done based on their schedules In thesecases assessing the tardiness of the customer orders needs tobe focused in the process of batching customer orders intopicking orders [4] The tardiness is the difference betweenthe order finish time and the due date if this differenceis positive So it is very important to consider the ordersrsquodue dates when batching processes are being done To ourknowledge there are few papers that consider due dates orminimize tardiness in their model Elsayed et al [4] andElsayed and Lee [27] focused on developing minimizationmodels of order tardiness and their incurred costs Thesolution approaches proposed by these two research papersare of limited applicability tomanual picker-to-parts systemsSingle processing times are used in both solution approachesin order to determine a sequence of batches Therefore theywill not yield competitive results in the situation describedin our paper Won and Olafsson [28] endeavored to integrateorder batching and picker routing problems together in orderto minimize the travel time They compared their proposedheuristics with other existing algorithms such as FCFS Itcan be perceived that their simulation results improvement is

reduced when the numbers of items for each order increaseTsai et al [29] attempted to solve a batch picking modelconsidering earliness and tardiness penalty using a multiple-GA method for obtaining the best possible batch pickingplans They tried to develop a flexible model to cope withthe current dynamic environment in terms of responsivenessUpon studying their proposed method it was determinedthat the orders were split which causes items in an order tobe collected in different tours Also their proposed modelallocates penalty to batches overweight (when the batchvolume exceeds the picker capacity) which is not applicablein practice

22 Picker Routing Problem The picker routing problem isdefined as the process of identifying the minimum distancewhich would be traveled by the order picker in a warehouseupon identifying which order should be picked first [30 31]There are some existing heuristics for picker routing suchas return s-shape largest gap and combined routing DeKoster et al [32] tried to solve order batching problems byevaluating two groups of heuristic methods such as seedalgorithms and time savings algorithms by means of s-shapeand largest gap strategies as two different routing strategiesAccording to the obtained results integrating the s-shapestrategy and an outsized capacity pick device together withseed algorithms can yield better performance In contrast thelargest gap strategy using a small capacity pick device givesbest performance when it integrated with time savings algo-rithms Ho and Tseng [33] considered the largest gap routingstrategy together with a simulated annealing optimizationmethod not only to optimize the total travel distance but alsoto find alternative routes that are better than ones found byjust the largest gap strategy In a research activity done byTheys et al [34] s-shape largest gap and some other routingheuristics were compared with the Lin-Kernighan-HelsgaunTSP heuristic Based on their results this TSP heuristic helpsto reduce travel distance up to the rate of 48 on average Inthe current research activity a GA TSP model has been usedin order to solve the picker routing problem

23 Batch Sequencing Problem The problem of batch seq-uencing can be defined as finding the orders of constructedbatches to be processed further Henn and Schmid [35]used metaheuristics to solve their proposed model of orderbatching and sequencing to minimize the ordersrsquo tardinessIterated Local Search and Attribute-Based Hill Climber arethe two employed metaheuristics in their model Accordingto different test problems the efficiency of their proposedmethods was illustrated Provided solutions can be improvedby 46 on average in comparison with the ones obtainedby standard constructive heuristics such as an application ofthe Earliest Due Date rule They conducted their proposedmethod for a maximum of 80 items

Other sections of this paper are as follows First theproposed model is described in Section 3 which covers allthe aspects that are involved with it After that Section 4presents the research methodology of this paper This isfollowed by Section 5 which encompasses the numericalexamples results and discussion of this research activity

4 The Scientific World Journal

Finally Section 6 gives an end to the paper with a briefconclusion

3 Model Description

The assumptions of this research are described as follows

(1) A manual picker to part system in a parallel aislewarehouse is considered in this research

(2) There is only one location for storing each type of itemand vice versa

(3) Multiple picker devices are not allowedTherefore themaximum number of pickers is only one

(4) The volume for each order should not exceed thecapacity of the picker

(5) Batch size does not exceed the capacity of picker(6) The storage size in each location is identical(7) All customer orders are known in advance

31 Weighted Association Rule Mining Binary Integer Cluster-ing Model In this section an association based clusteringmodel developed by Chen and Wu [23] was modified inorder to create the batch structure regarding ordersrsquo duedate First the association between customersrsquo demand willbe calculated based on the similarity of items in orderswith respect to their due date using MINWAL Ordersrsquodue dates are considered as the weight of association rulemining Although batching orders with higher associationsgenerated by similarity between their items can decrease thedistance travelled by order pickers not considering the duedates would cause the orders to be batched inappropriatelywith different due dates Some problems may occur by theaforementioned issue such as shortage or excess inventorycosts In order to solve this problem the order due date hasbeen considered as a weight of each order for calculatingthe support between orders Relationship maximization ofcustomer orders based on similar items within each orderand due dates of each order can be done with the weightedassociation rules Travel distance would be reduced by takinginto account batching of orders with higher relationshipsOther than that higher associations can batch orders withsimilar due dates in order to minimize tardiness This isfollowed by utilizing a binary integer programming modelintegrated with a clustering algorithm which assembles theorders into their respective batches

311 Weighted Association Mining for Determining CustomerOrder Relationships Association rule mining can be definedas a data mining method which tries to distinguish interrela-tions of variables where large databases exist [36] Some rulesare involved with the association rule model where there isan association between some set of items with another set ofitems [37] In order to calculate rule interestingness supportand confidence can be considered as the twomost appropriatemeasures in association ruleminingThe support for an item-set is the percentage of transactions that contain the itemset

Table 1 Order-item data

Order Items Due date1198741

A(3) B(4) 10

1198742

A(2) C(3) 30

1198743

A(1) B(3) 5

1198744

A(2) B(4) C(2) 50

Table 2 Item-order data

Item OrderA 119874

1 1198742 1198743 1198744

B 1198741 1198743 1198744

C 1198742 1198744

in the database A probability shows how frequently the rulehead occurs among all the groups containing the rule bodywhich can be defined as the confidence of an association rule(body rArr head) [38 39]

Based on the definition by Agrawal et al [36] theassociation rule mining problem is addressed as followsitems are described as 119868 = 119894

1 1198942 119894

119899 which is a set

of n binary attributes The database is represented as 119863 =

1199051 1199052 119905

119898 which is a set of transactions An exclusive

transaction 119868119863 belongs to each transaction in119863 together witha subset of the items in 119868 A rule is defined as an implicationof the form 119883 rArr 119884 where 119883119884 sube 119868 and 119883⋂119884 = 120601 Theassociation rule 119883 rArr 119884 [support = 119904 confidence = 119888]holds in the item-order data 119863 with confidence 119888 if 119888 ofOrder Identifiers in 119863 contain orderset 119883 and also containsorderset 119884 The rule 119883 rArr 119884 has support 119904 if 119904 of the item-order 119863 contains both 119883 and 119884 For instance upon batchingthe orders by taking into account the customer order patternsassociation rule mining involves finding out the amountof support between demands of customers from the orderdatabase In order batching Table 1 which shows the order-item data is transposed to the item-order data form shownin Table 2 since the order associations are demanded and theproduct item associations is not necessary In this exampleassociation rule mining can perform its job to define theassociations between customer orders that can be perceivedfrom information presented in Table 2

In the area of order batching association rule miningwas first used by Chen and Wu [23] In their researchpaper association rule mining was used to calculate thecorrelations between customersrsquo demands in terms of sup-port They pointed out that orders with similar items shouldbe consolidated together in order to minimize the totaltravel distance traversed by the picker An Apriori algorithmhas been used in their proposed method to calculate theassociation between orders It should be highlighted that theyonly considered the relationships between orders in termsof similarities between items However in batch processingeach order has a due date which can be considered as theweight but this weight is not considered by any other existingmethod in the literature

The classical Apriori algorithm [40] extracts binary asso-ciation rules based on the downward closure property which

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Table 3 Weight of each order

Order Weight1198741

05827

1198742

01979

1198743

07634

1198744

00672

proves that subsets of a frequent itemset are also frequent [41]However the Apriori algorithm cannot be applied because itcannot handle the weighted case Therefore in this researchactivity theMINWAL algorithm is applied in order to extractthe weighted supports among the pair of orders to fulfill thetask of considering both the support and the weights factors

In order to use MINWAL [42] for mining weighted asso-ciation rules the weight should be in the range of the 0 to1 interval Based on the aforementioned fact larger weightsshould be assigned to orders with smaller due dates in acomparison with other orders that have lots of time to bedelivered in order to minimize the average tardiness of allorders Therefore in this study (1) has been used to calculatethe appropriate weights for each order as following

Weight (119874119894) = 119890(minus01lowast120573lowast119889119874119894

Maximume Due Date) (1)

where 119874119894denotes an order and 119889

119874119894refers to the due date

of order 119874119894 120573 is a constant value that is used to normalize

the due dates which is considered as 30 (30 minute) in thisstudy Also the maximum due date of the orders was usedin order to provide an appropriate interval for the weightsFor example the weight of each order is tabulated in Table 3which is calculated by (1)

Given a set of orders 119874 = 1199001 1199002 119900

119898 a weight 119908

119895for

each order 119900119895 within the values of 0 to 1 is assigned by (1) via

its due date where 119895 = 1 2 119898 shows the importance ofit The weighted support for the weighted association ordersof 1198741and 119874

2as 1198741rArr 1198742is defined if the weighted support

of such a pair of orders is no less than theminimumweightedsupport (minsup) threshold In this study the minimumsupport threshold is considered as 0 in order to identify all theassociations among orders Equation (2) shows the support ofan association for any pair of orders as following

(sum sum

119908119895isin(1198741cup1198742)

119908119895) lowast Support (119874

1cup 1198742) (2)

Algorithm 1 shows the MINWAL steps in order to mineweighted association rules This algorithm consists of 6subroutines such as Search Counting Join Prune Checkingand Rules The search subroutine provides the size of thelargest itemsets in database Counting is responsible forcounting every 1-itemset inside the database The Join step isresponsible for generating the 119896-itemsets (119862119896) from (119896 minus 1)-itemsets (119862119896 minus 1) The Prune step removes inappropriateitemsets which do not exist The Checking step updatesthe count of 119896-itemsets inside the transaction and prunesitemsets which do not meet the minimum support thresholdIn the Rules step every association rule will be extractedTable 4 shows the notations of Algorithm 1

(1) Main Algorithm (wminsup mincon 119891119863 119908)(2) size = Search (119863)(3) 119871 = 0(4) for (119894 = 1 119894 le size 119894++)(5) 119862

119894= 119871119894= 0

(6) for each transaction do(7) (SC 119862

1 size) =Counting (119863 119908)

(8) 119896 = 1(9) while (|119862119896| ge 119896)(10) 119896++(11) 119862

119896= Join (119862

119896minus1)

(12) 119862119896=Prune (119862

119896)

(13) (119862119896 119871119896) =Checking (119862

119896119863)

(14) 119871 = 119871 cup 119871119896

(15)Rules (SC 119871)(16) ends

Algorithm 1 MINWAL algorithm [42]

Table 4 Notations of Algorithm 1

119863 The database119908 The set of item weights119871119896

Set of large 119896-itemsets

119862119896

Set of 119896-itemsets which may be 119896-subsets of large119895-itemsets for 119895 ge 119896

SC (119883) No of transactions containing itemset119883wminsup Weighted support thresholdminconf Confidence thresholdSize Maximum possible large weighted itemsets

312 Association Based Binary Integer Programming forOrders Batching In order to create the batch structure abinary integer clustering model which has been developed byChen andWu [23] was appliedThey developed a binary inte-ger programming model for maximizing similarity betweenorders in terms of support in order to minimize the totaltraveling distance In this research activity support betweenorders was calculated based on weighted association miningusing MINWAL which was described in Section 311 Thebinary integer model for order batching is described asfollows

Parameters

119878119894119895Support between order 119894 and 119895 determined by MIN-WAL

119862V Capacity of picking vehicle

119881119894Weight of order 119894

119870 Number of batches

1198831198941198951 if order 119894 is assigned to batch 119895

0 otherwise

6 The Scientific World Journal

1198841198941 if order 119895 is chosen as a batch median0 otherwise

Maximize119873

sum

119894=1

119873

sum

119895=1

119878119894119895lowast 119883119894119895 (3)

Subject to119873

sum

119895=1

119883119894119895= 1 for 119894 119895 = 1 2 119873

(4)

119883119894119895le 119884119895

for 119894 119895 = 1 2 119873 (5)

119873

sum

119895=1

119884119895= 119870 (6)

119873

sum

119894=1

119881119894119883119894119895le 119862V for 119895 = 1 2 119873 (7)

119883119894119895= 0 1 for 119894 119895 = 1 2 119873 (8)

119884119895= 0 1 for 119895 = 1 2 119873 (9)

In the current model the objective function (3) maxi-mizes the sum of support from all orders to their relevantbatch medians as 119870 orders are selected as batch mediansThis model will maximize the support between orders in thebatches Membership of each order in just one batch wouldbe guaranteed by constraint (4) The number of batches with119870 is limited using constraints (5) and (6) The total quantityin a tour is also represented as a limitation in constraint (7) Itcannot exceed the capacity of order picker 119862V For the orderbatchingmodel to have a binary solutionwould be ensured byconstraints (8) and (9) For calculating the initial numbers ofbatches (10) will be used where sum119881

119894represents total volume

of all orders as following

119870initial = lceilsum119881119894

119862V

rceil (10)

32 Picker Routing

321 Genetic Algorithms In this research GAs which wereintroduced by Holland [43] have been used in order tosolve the picker routing and batch sequencing problem Thebiological evolution process is used as a base fact in a GA forproblem solving to a great extentThemain steps of a GA canbe described as follows [44 45]

(i) populating a series of solutions(ii) solutions assessment(iii) picking the most appropriate solutions(iv) generating new solutions based on some genetic

operation

In step (iii) selection forms the main part of a GA Thebad solutions are removed by selection and the good onesremain Random generation over the feasible or infeasible

solution space of a problem is used as a basis for creatingthe population Each solution is evaluated using the fitnessfunction Selection would be done according to certain selec-tions of fitness together with the probability that is assignedto it Generating a fresh population and a better solutionare involved with recombination of the individual solutionCrossover and mutation are the most used GA operators forgenerating new offspring from the old generation of parentsParameters of the GA are originated from the probabilitiesof mutation and crossover Exploration of the sample spaceis diversified by these operators In this research the GA hasbeenutilized in two steps First theGAhas beenused in orderto solve the picker routing problem Consider that in thewarehouse a set of items should be picked by a picker in onetour Therefore the picker routing problem in the warehousecould be categorized as a TSP The goal in the picker routingproblem as a TSP in the warehouse is to find an optimal pathfor minimizing the travel distance traversed by the pickingmachine where each node is visited at least once In thisresearch for a given number of itemset in a batch GA TSPhas been used to find the minimum travel distance whichis traversed by the picking machine Second a GA has beenutilized in order to sequence all batches namely GA sequencingwhichwould be further processedThe batches are sequencedbased on a fitness function which minimizes the tardiness ofall ordersThedetailed procedures ofGA TSP andGA sequencingare illustrated in Sections 323 and 324

322 Warehouse Layout Thewarehouse discussed through-out this research is assumed to have a 3D multi-parallel-aisle warehouse layout The picker will start the tour fromthe depot which is located in front of the leftmost aisle andreturns back to it A picker leaves the depot After that ittravels to a certain buffer spot where it picks the requestedorder(s) and puts the order(s) into a certain picker Then itcontinue on its way to another spot These procedures wouldbe repeated again and again so that all the requested ordersare picked Then it is time for the picker to return to thedepot

The graphic layout of the warehouse zone involved in thisresearch activity is pictorially displayed in Figure 1 It showsthe 119909 and 119910 dimensions of the 3D warehouse layout Eachitem in the warehouse has a specific location For examplethe location of item 119894 can be addressed by 119871

119894= (119909119894 119910119894 119911119894)

Consider two locations as 119871119886= (119909119886 119910119886 119911119886) and 119871

119887= (119909119887

119910119887 119911119887) If two locations are located in the same sub-aisles

of the same block the distance between these two itemslocations (119863

119886119887) will be calculated using

119863119886119887

=1003816100381610038161003816119909119886 minus 119909119887

1003816100381610038161003816 +1003816100381610038161003816119910119886 minus 119910119887

1003816100381610038161003816 +1003816100381610038161003816119911119886 minus 119911119887

1003816100381610038161003816 (11)

Equation (11) is not applicable if the two storage locationsbelong to different subaisles that are from the same block [34]In order to solve this problem distances of all possible pathsbetween two locations have been calculated and the shortestone was considered

323 GA TSP for the Picker Routing Problem In order todetermine the optimal route of a picker related to a given

The Scientific World Journal 7

Lb=(xbybzb)

La=(xayaza)

Li=(xiyizi)

Depot

Figure 1 Warehouse layout

1 2 3 4 5 6 7 8 9L0 L1 L8 L11 L9

L13 L12 L10 L0

Figure 2 A feasible chromosome encoding in GA TSP

set of items in a batch a GA has been used to solve theTSP problem to minimize travelling time In a GA eachsolution is considered as a chromosome The steps of GA TSPare described as follows

(1) for each batch establishing the structure of a chromo-somes

(2) generate initial feasible chromosomes(3) evaluate each chromosome based on the GA TSP fit-

ness function(4) apply the Selection operation(5) apply the Crossover operation(6) apply the Mutation operation(7) evaluate each offspring based on the fitness function(8) go to step 4 if the termination criterion is not satis-

fied otherwise terminate and return the best solutionas the optimal path

At the first step of GA TSP the structure of the chro-mosome should be established In the GA TSP the locationof an item that should be visited is presented by the valueof a gene Accordingly the sequence of visiting the itemlocation is denoted by the order of a gene in a chromosomeTherefore the total number of items locations in a specificbatch is considered equal to the length of a chromosome Forexample if the length of a chromosome is nine it means nineplaces should be visited by the picker as shown in Figure 2If 119871119894represents the item location in the warehouse 119894 = 1

119899 it shows that the picker leaves the depot (1198710denotes

the location of a depot) goes to 1198711 1198718 11987111 1198719 11987113 11987112 11987110

and then comes back to the depotIn the second step the initial number of feasible chro-

mosomes should be generated randomly for each batch

The number of initial chromosomes could be different basedon the complexity of the problem It should be consideredthat the first and the last gene of each chromosome should beassigned individually to the location of a depot In step threeeach chromosome will be evaluated based on the GA TSPfitness function which is formulated as follows

fitness = Min119899minus1

sum

119895=1

119863119895119895+1

(12)

where 119863119895119895+1

denotes the distance between two subsequentgenes in a chromosome 119895 = 1 119899 minus 1 represents the orderof the genes and 119899 is number of genes

Subsequently step four is involved with the selectionoperation In each iteration the selection operator usesfitness values to select the parents of the next generationIn this research the roulette wheel selection policy hasbeen used to guarantee that the most appropriate pairs ofchromosomes have been selected to generate offspring Itmeans that a higher probability of being selected to createa new population will be assigned to the chromosomes withhigher fitness values

Steps five and six are involved with conducting thecrossover and mutation operation The construction of theoffspring is realized by the crossover and the diversity of theindividuals is maintained by the mutation operator Blocksof genes between chromosomes are traded by means of thecrossover operation inwhich the exploitation of a particularlyprofitable portion of the parameter space is allowed to theblock Here permutation order based crossover (POP) as avariation of the distinguished order crossover has been usedAs shown in Figure 3 the original idea is about choosing twoparents and a point for cutting The first portion of offspring2 contains the first portion of parent 1 up to the cut pointTo form the second portion of offspring 2 consider all genesin parent 2 except those genes that already exist in the firstportion of the offspring 2

As shown in Figure 4 two positions are randomlyselected in a chromosome and exchanged with each othernamely the order based mutation approach SWAP isadopted for solving GA TSP Step six is followed by step sevenwhich includes the evaluation of each offspring that has beengenerated in the previous step based on the GA TSP fitnessfunction In step eight termination criteria will be checkedIf the termination condition is met then the process will beterminated After that the best chromosome will be selectedas the optimal solution Otherwise the selection process is tobe conducted again to generate a new population

324 GA for Sequencing the Batches Sequencing the batchesshould be conducted in order to minimize the averagetardiness of all orders The problem of batch sequencing canbe defined as finding the orders of batches which should beprocessed further For this purpose the GA sequencing methodhas been utilized In this research all of the batches whichwere constructed using binary integer programming andGA TSP will be sequenced by using the GA The populationgeneration selectionmutation and stopping threshold in theGA sequencing are identical to those in the GA TSP However

8 The Scientific World Journal

Parent 1

Parent 2

Offspring 2

Offspring 1

L1 L8 L9

L1 L1L8 L8L9

L11 L13 L12 L10

L1 L8 L9 L11L13 L12L10

L13 L9 L11 L11 L13 L12

L12

L10

L10L1L8L12

L13 L9 L11 L10

L1 L8 L12L13 L9 L11 L10

Figure 3 Crossover operation

Parent L1

L1

L8

L8

L9

L9

L11

L11

L13

L13

L12

L12

L10

L10Offspring

Figure 4 Mutation operation

1 2 3 4 5b1 b4 b3 b5 b2

Figure 5 Chromosome structure

the fitness function and chromosome structure are differentthan that of GA TSP In contrast with GA TSP chromosomesstructure batch number is presented by the value of a geneAccordingly the sequence of batch is denoted by the orderof a gene in a chromosome For instance Figure 5 shows thechromosome structure If 119887

119894isin 119861 denotes a set of batches

Figure 5 shows that 1198871 1198874 1198873 1198875 1198872should be processed

respectively Therefore the length of a chromosome is equalto the number of the batches The GA sequencing fitness func-tion for chromosome 119910 is described in

fitness119910=

1

119873

119899

sum

119896=1

sum

119894isin119887119896

119862119879 (119887119896) minus 119863119906 (119900

119894) (13)

119862119879 (119887119896) = 119862119879 (119887

119896minus1) + Pt (119887119896) (14)

Pt (119887119896) =Opt (119887

119896)

119881119878 (15)

where119873 denotes for the total number of orders 119896 = 1 119899

represents the batch sequence in which 119899 is the total numberof batches 119900

119894stands for the order 119894 in which 119894 isin 119887

119896 119887119896

denotes the batch in 119896th position of a sequence for specificchromosome 119862119879(119887

119896) stands for the completion time of a

batch in the 119896th position of a sequence 119863119906(119900119894) is the symbol

for due date of order 119894which belongs to 119887119896 Pt(119887119896) is the process

time of 119887119896 Opt(119887

119896) is the optimal path solution for 119887

119896 Finally

119881119878 is the moving speed of the picker

4 Research Methodology

In this research the proposed methodology consists of fivesteps These steps are described as follows and shown pic-torially in Figure 6

Step 1 It is involved with utilizing theMINWAL algorithm inorder to determine the associations between orders in termsof support considering their due dates In this step an orderdue date will be considered as the weight of an order It meansorders that are similar in items and due date should havemoresupport for being batched together

Step 2 It encompasses determining the initial number ofbatches In this step the initial number of batches will becalculated by (10)

Step 3 It includes the problem of modeling and solving firstthe binary integer programming model for order clusteringand batching will be modeled Next this model will be solvedin order to determine the structures of batches

Step 4 It is about the picker routing problem The optimalpath for a picker within each batch will be determined bysolving a TSP using GA

Step 5 It is about sequencing the batches based on the fitnessfunction which minimizes the average tardiness of all ordersIn this step GA algorithmwill be utilized in order to solve thebatch sequencing problem regarding the fitness function

5 Computational Experiments and Results

51 Test Problems and Parameter Setting In this sectionfive numerical test problems with different settings andparameters have been carried out in order to illustrate theproficiency of the proposed method with respect to averagetardiness of all orders Table 5 summarizes the descriptionof these test problems This table includes information withregard to number of items number of orders capacity of thepicker total weight of all the items in an order andminimumnumber of batches

In these test problems the quantity of items quantityof orders capacity of the picker device (119862V) and amount ofeach item are predetermined Each order contains a certain

The Scientific World Journal 9

Start

Input order informationwarehouse setting picker

setting

Compute the supportbetween customersrsquo

orders using MINWAL

Calculating initialnumber of batches (K) to

assign the orders

Solving the binaryinteger programmingclustering model for

constructing the batches

Feasiblesolution

Yes

No

Solving picker routing

Solving batch sequencingproblem using

End

K = K + 1

problem using GA TSP

GA sequencing

Figure 6 Methodology framework

quantity of items and also each item has a certain quantityThese quantities are produced randomly The distributionfunction type for the number of items in each order is Normal[5 10] Also the distribution function type for each itemquantity in an order is uniform [1 10] The picker movingspeed is set as 119881119878 = 2 (ms) The warehouse has 3000 itemlocation The GA control parameters for solving the TSP andsequencing problem are tabulated in Table 6

All of the proposed approaches are coded usingMATLABversion 790 and Java 15 programming language Also theIBM ILOG CPLEX optimization studio version 124 softwarewas utilized jointly with MATLAB in order to facilitate theprocess of solving the binary integer programming problemExamples are run by means of a computer featured by

4 Gigabytes random access memory (RAM) and a 220Gigahertz Intel processor (T6600)

52 Computational Results and Discussion In order to revealthe competency of the proposed solution approach it wascompared with the other existing methods in the literatureOne of the latest published research activities in the literaturewhich deals with minimization of tardiness has been con-ducted by Henn and schmid [35] They utilized the EarliestDueDate (EDD) approach as their constructive algorithm forcomparing with their proposed approaches They proposedtwo heuristics Iterated Local Search andAttribute-BasedHillClimber are the two employed metaheuristics in their modelAccording to different test problems the efficiency of their

10 The Scientific World Journal

Table 5 Test problems description

Example 1 Example 2 Example 3 Example 4 Example 5Number of orders 40 60 80 100 150

Number of items 80 120 160 200 200

Capacity of picker 200 200 300 400 550

Total weight 2212 3668 4393 6110 8369

Minimum number of batch 11 19 15 16 16

Due dates range 60 90 120 180 240

Table 6 GA control parameters

GA TSP GA sequencing

Population size 500 200

Cross over strategyPermutation orderbased crossover

(POP)

Permutation orderbased crossover

(POP)Cross over probability 08 08

Mutation strategy SWAP SWAPMutation probability 02 02

Iteration number 10000 10000

Parent selection method Roulette wheel Roulette wheel

proposed methods was illustrated Their obtained resultsfrom their proposed approaches showed a 46 improvementon tardiness compared to EDD They used an s-shape andlargest gap routing strategy for their proposed method Inorder to make a reasonable comparison with their proposedmethod EDD was integrated with the s-shape routing strat-egy and GA sequencing Then based on improvement on EDDwe can compare our proposed method with their proposedmethod In the EDD approach due dates play an importantrole in sorting all orders Based on an ascending sequenceassigning the orders to batches ensures that no capacityviolation can happen for the picking device

For the warehouse condition which is described inSection 323 our proposed approach EDD integrated withGA TSP and GA sequencing and EDD integrated with s-shapealgorithm and GA sequencing have been evaluated in termsof average tardiness for all orders (119879) which is calculatedbased on (13) number of batches (119870) total process time(Pt) and average process time for each batch (119860Pt) All ofthe GA setting for the TSP and sequencing problems areconsidered the same for all methods Also the settings whichhave been mentioned in Section 51 were considered for all ofthe compared methods

The results for all of the test problems for the threeapproaches are tabulated in Tables 7 8 and 9 As it isillustrated in Table 10 EDD integrated with GA TSP andGA sequencing has a 3176 improvement on average in termsof average tardiness of all orders than EDD integrated withthe s-shape algorithm and GA sequencing It can be perceivedthat GA TSP performs better than s-shape Also the ATGHmethod has a 6811 improvement in average tardinessoff all orders than EDD integrated with s-shape algorithm

and GA sequencing Consequently the ATGH method achievesmuch improvement against Henn and Schmidrsquos [35] pro-posedmethod According to Tables 7 and 8 it is also worth tomention thatGA TSP performs better than the s-shape routingstrategy in terms of process time The improvements whichhave been achieved in this research could be justified basedon the following reasons

The problem of order batching deals with several sig-nificant drivers in which the number of batches can beconsidered as a key factor [23] As a justification to thismatter it is worth to mention that total travel distance maynot decrease directly due to a decline in the number ofbatches But a decline in the number of batches will increasebatch sizes which may lead to an increase in the distancetravelled by order picker However an advanced exploitationof picker capacity to load efficiently may reduce the numberof constructed batches together reducing the total traveldistance Improving the total travel distance traversed by thepicker can minimize the average tardiness of all orders Interms of number of the batches according to Tables 7 8 and9 it can be perceived that our proposed method assigns theorders to the batches optimally The ATGH method assignsthe orders to batches based on their item similarity andtheir due dates and it uses the picker capacity efficiently Butthe EDD method allocates the orders to batches based onan ascending sequence of their due date and it should beensured that no capacity violation will happen for the pickingdevice which may lead to an inefficient order batching Tohighlight the problem an example can be made based on theresults tabulated in Tables 7 and 8 The number of batchesfor example 1 in our method is 11 in comparison with 13obtained fromEDDmethod 13 batches can be justified basedon this scenario that two subsequent orders say the firstorder is 100 kg and the other is 150 kg cannot be batchedtogether due to violated batch capacity which is set to be200 kg The number of batches would increase due to theaforementioned issue which affects the total process time tohave an increasing manner This situation will rarely happenin the ATGH method because of the weighted associationbased binary integer programming utilizedwhich batches theorders in order to maximize the association between themwhich is calculated based on the item similarity of ordersand the weight of their corresponding due dates The ATGHmethod uses the picker capacity efficiently

Obviously one of the factors that can influence thecompletion time and the average tardiness of all orders is theprocessing time of the orders Reducing the processing time

The Scientific World Journal 11

Table 7 Computational results of ATGH

Example 1 Example 2 Example 3 Example 4 Example 5119879 56204 29948 90202 163 58898

Pt 605833 1217833 105833 121033 1339

119870 11 19 15 16 16

119860Pt 55076 6409 7055 7564 8368

Table 8 Computational results of EDD integrated with GA TSP and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 124550 42976 18222 5981 1547

Pt 667333 133516 116083 13185 141916

119870 13 21 16 17 16

119860Pt 51333 6357 7255 7775 8869

Table 9 Computational results of EDD integrated with s-shape algorithm and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 1651 62677 2825 13365 17593

Pt 75266 16636 13323 14333 14926

119870 13 21 16 17 16

119860Pt 5789 892 833 8431 932

Table 10 Comparison results

EDD + s-shape + GA sequencing EDD + GA TSP + GA sequencing ATGHTardiness (min) Tardiness (min) Improvement () Tardiness (min) Improvement ()

(1) 1651 124550 2456 56204 6596(2) 62677 42976 3143 29948 5222(3) 2825 18222 3550 90202 6807(4) 13365 5981 5525 1627 8783(5) 17593 1547 1206 58898 6652Average improvement 3176 6811

and travel time for constructing a batch is an outstanding factin reducing the costs and minimizing delays of the customerresponses In our proposed model orders with similar itemsare consolidated together Therefore batching orders withmore similarity can reduce the distance traversed by thepicker facility Also the due date of each order is definedas a weight parameter in calculating the association betweenorders in terms of support which was done for consideringthe due dates in our model Thereupon the model considersdue date and the process time in an integrated manner whichsimultaneously affected the average tardiness of all ordersThis spoken issue is different in the EDD approach in whichthe ordersrsquo due dates are just considered Example 5 canbe considered as evidence of this matter in which the totaland average process time for ATGH is lower than EDDintegrated with GA TSP and GA sequencing with respect to thefact that number of constructed batches for both approachesare happened to be equal after obtaining the results

6 Conclusion

One of the most important concerns of warehouse managersis involvedwith finding the best optimal and cost efficientwayto pick orders placed by customers in order to be known as aresponsible company with satisfied customers in terms of ontime delivery In the current research activity order batchingpicker routing and batch sequencing problems of warehouseprocesses were addressed which are to be solved jointly asthey are mixed up with many manufacturing industriesMathematical solutions of these NP-hard problems might beavailable for problems that are small in nature But thesekinds of solutions may not be applicable to be consideredby warehouse managers Hence we presented novel solutionapproach namely ATGH to solve this issue in accordancewith minimizing the average tardiness of all customersrsquoorders which can be considered as a major objective ofthis research activity A weighted association rule mining

12 The Scientific World Journal

algorithm namely MINWAL was utilized to determine thecustomersrsquo orders association regarding similarity betweenitems and ordersrsquo due dates Actually in our proposedmethodthe due date of each order is defined as a weight parameterin calculating the association between orders So orders withsimilar items and due dates have more chance to be in asame batch This issue is not considered in most of theresearch activities in the literature related to our work andconsidering this issue in the order batching problem is themain contribution of our research activity The previous stepwas followed by a binary integer clustering model in orderto solve the order batching problem which maximizes thesimilarity between orders in terms of support After that aGA was applied to solve a TSP in order to find the optimalrouting path

Finally a batch sequencing problem was addressed inconjunction with a GA

The other research objective of this research addresseswhether the proposed ATGH algorithm can further improvethe solution quality for a multiple parallel aisle warehousewith manual picking system Hence comparisons were madeconsideringATGHEDD integratedGA TSP andGA sequencingand EDD integrated s-shape and GA sequencing algorithms Asdemonstrated by the obtained results ATGH comes out tobe the most striking solution approach in terms of tardinessIn spite of the fact that considering all customer orders asdiscussed in this paper would lead to have a static case beinginvolved with dynamic cases in which different points of timeare assigned to customerrsquos orders arrival would be beneficialSo there might be some potential room of research to extendour discussed approach with respect to dynamic orders

Conflict of Interests

The IBM ILOG CPLEX optimization studio version 124software is not used for financial gain and there is no conflictof interests with any party as it is only a student version

References

[1] ELAAT Kearney Excellence in Logistics ELA Brussels Bel-gium 2004

[2] Y-C Ho T-S Su and Z-B Shi ldquoOrder-batching methods foran order-picking warehouse with two cross aislesrdquo Computersamp Industrial Engineering vol 55 no 2 pp 321ndash347 2008

[3] C-M Hsu K-Y Chen and M-C Chen ldquoBatching ordersin warehouses by minimizing travel distance with geneticalgorithmsrdquo Computers in Industry vol 56 no 2 pp 169ndash1782005

[4] E A Elsayed M-K Lee S Kim and E Scherer ldquoSequencingand batching procedures forminimizing earliness and tardinesspenalty of order retrievalsrdquoThe International Journal of Produc-tion Research vol 31 no 3 pp 727ndash738 1993

[5] S Henn S Koch and W Gerhard Order Batching in OrderPicking Warehouses A Survey of Solution Approaches SpringerLondon UK 2012

[6] J Drury Towards More Efficient Order Picking IMM Mono-graph no 1 The Institute of Materials Management CranfieldUK 1988

[7] M Goetschalckx and J Ashayeri ldquoClassification and design oforder picking systemsrdquo Logistics World pp 99ndash106 1989

[8] J A Tompkins J A White Y A Bozer E H Frazelle and J MA Tanchoco Facilities Planning John Wiley amp Sons HobokenNJ USA 2003

[9] S Henn and G Wascher ldquoTabu search heuristics for the orderbatching problem in manual order picking systemsrdquo EuropeanJournal of Operational Research vol 222 no 3 pp 484ndash4942012

[10] S Henn ldquoAlgorithms for on-line order batching in an orderpicking warehouserdquo Computers and Operations Research vol39 no 11 pp 2549ndash2563 2012

[11] R de Koster ldquoHow to assess a warehouse operation in a singletourrdquo Tech Rep RSM Erasmus University Rotterdam TheNetherlands 2004

[12] R de Koster T Le-Duc and K J Roodbergen ldquoDesign andcontrol of warehouse order picking a literature reviewrdquo Euro-pean Journal of Operational Research vol 182 no 2 pp 481ndash5012007

[13] R A Ruben and F R Jacobs ldquoBatch construction heuristics andstorage assignment strategies for walkride and pick systemsrdquoManagement Science vol 45 no 4 pp 575ndash596 1999

[14] E A Elsayed ldquoAlgorithms for optimal material handling inautomatic warehousing systemsrdquo International Journal of Pro-duction Research vol 19 no 5 pp 525ndash535 1981

[15] G Clarke and J W Wright ldquoScheduling of vehicles froma central depot to a number of delivery pointsrdquo OperationsResearch vol 12 no 4 pp 568ndash581 1964

[16] H D Vinod ldquoInteger programming and the theory of group-ingrdquo Journal of the American Statistical Association vol 64 no326 pp 506ndash519 1969

[17] R D Armstrong W D Cook and A L Saipe ldquoOptimalbatching in a semi-automated order picking systemrdquo Journal ofthe Operational Research Society vol 30 no 8 pp 711ndash720 1979

[18] H D Ratliff and A S Rosenthal ldquoOrder-picking in a rect-angular warehouse a solvable case of the traveling salesmanproblemrdquo Operations Research vol 31 no 3 pp 507ndash521 1983

[19] A Kusiak A Vannelli and R K Kumar ldquoClustering analysismodels and algorithmsrdquo Cybernetics vol 15 pp 139ndash154 1986

[20] D R Gibson and G P Sharp ldquoOrder batching proceduresrdquoEuropean Journal of Operational Research vol 58 no 1 pp 57ndash67 1992

[21] M B Rosenwein ldquoA comparison of heuristics for the problemofbatching orders for warehouse selectionrdquo International Journalof Production Research vol 34 no 3 pp 657ndash664 1996

[22] A J RMGademann J P van denBerg andHH van derHoffldquoAn order batching algorithm for wave picking in a parallel-aislewarehouserdquo IIE Transactions vol 33 no 5 pp 385ndash398 2001

[23] M-C Chen and H-P Wu ldquoAn association-based clusteringapproach to order batching considering customer demandpatternsrdquo Omega vol 33 no 4 pp 333ndash343 2005

[24] NGademann and S van deVelde ldquoOrder batching tominimizetotal travel time in a parallel-aisle warehouserdquo IIE Transactionsvol 37 no 1 pp 63ndash75 2005

[25] Y A Bozer and J W Kile ldquoOrder batching in walk-and-pick order picking systemsrdquo International Journal of ProductionResearch vol 46 no 7 pp 1887ndash1909 2008

[26] O Kulak Y Sahin and M E Taner ldquoJoint order batching andpicker routing in single and multiple-cross-aisle warehousesusing cluster-based tabu search algorithmsrdquo Flexible Servicesand Manufacturing Journal vol 24 no 1 pp 52ndash80 2012

The Scientific World Journal 13

[27] E A Elsayed and M-K Lee ldquoOrder processing in automatedstorageretrieval systems with due datesrdquo IIE Transactions vol28 no 7 pp 567ndash577 1996

[28] J Won and S Olafsson ldquoJoint order batching and order pickingin warehouse operationsrdquo International Journal of ProductionResearch vol 43 no 7 pp 1427ndash1442 2005

[29] C-Y Tsai J J H Liou and T-M Huang ldquoUsing a multiple-GA method to solve the batch picking problem consideringtravel distance and order due timerdquo International Journal of Pro-duction Research vol 46 no 22 pp 6533ndash6555 2008

[30] Y KaoM H Chen and Y T Huang ldquoA hybrid algorithm basedon ACO and PSO for capacitated vehicle routing problemsrdquoMathematical Problems in Engineering vol 2012 Article ID726564 17 pages 2012

[31] X Gan Y Wang S Li and B Niu ldquoVehicle routing problemwith time windows and simultaneous delivery and pick-up ser-vice based on MCPSOrdquoMathematical Problems in Engineeringvol 2012 Article ID 104279 11 pages 2012

[32] M B M de Koster E S van der Poort and M WoltersldquoEfficient orderbatching methods in warehousesrdquo InternationalJournal of ProductionResearch vol 37 no 7 pp 1479ndash1504 1999

[33] Y-C Ho and Y-Y Tseng ldquoA study on order-batching methodsof order-picking in a distribution centre with two cross-aislesrdquoInternational Journal of Production Research vol 44 no 17 pp3391ndash3417 2006

[34] C Theys O Braysy W Dullaert and B Raa ldquoUsing a TSPheuristic for routing order pickers in warehousesrdquo EuropeanJournal of Operational Research vol 200 no 3 pp 755ndash7632010

[35] S Henn and V Schmid Metaheuristics for Order Batching andSequencing in Manual Order Picking Systems Univ Fak furWirtschaftswiss 2011

[36] M Hahsler B Grun and K Hornik ldquoArulesmdasha computationalenvironment for mining association rules and frequent itemsetsrdquo Journal of Statistical Software vol 14 2005

[37] R Agrawal T Imielinski and A Swami ldquoMining associationrules between sets of items in large databasesrdquo ACM SIGMODRecord vol 22 no 2 pp 207ndash216 1993

[38] S Encheva Y Kondratenko M Z Solesvik and S TuminldquoDecision support systems in logisticsrdquo in Proceeding of Inter-national electronic conference on computer science pp 254ndash2562007

[39] M-C Chen ldquoConfiguration of cellular manufacturing systemsusing association rule inductionrdquo International Journal of Pro-duction Research vol 41 no 2 pp 381ndash395 2003

[40] R Srikant and R Agrawal ldquoMining generalized associationrulesrdquo Future Generation Computer Systems vol 13 no 2-3 pp161ndash180 1997

[41] J Wu and XM Li ldquoAn effective mining algorithm for weightedassociation rules in communication networksrdquo Journal of Com-puters vol 3 no 10 pp 20ndash27 2008

[42] C H Cai A W C Fu C H Cheng and W W KwongldquoMining association rules with weighted itemsrdquo in Proceedingsof the International Symposium on Database Engineering ampApplications pp 68ndash77 Cardiff UK 1998

[43] J HollandAdaptation in Natural and Artificial Systems Univer-sity of Michigan Press Ann Arbor Mich USA 1975

[44] A A Taleizadeh F Barzinpour and H-M Wee ldquoMeta-heuristic algorithms for solving a fuzzy single-period problemrdquoMathematical and Computer Modelling vol 54 no 5-6 pp1273ndash1285 2011

[45] H Yang J Yi J Zhao and Zh Dong ldquoExtreme learning ma-chine based genetic algorithm and its application in powersystem economic dispatchrdquo Neurocomputing vol 102 pp 154ndash162 2013

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

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Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

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2 The Scientific World Journal

the processing time of the orders According to the aboveexplanation reducing the processing time and travel timefor constructing a batch is an important fact in reducing thecosts and minimizing delays of the customer responses Onthe other hand not considering the orderrsquos due date in theprocess of order batching can cause a huge dissatisfaction incustomer expectations regarding the efficient responsivenessof the company For explanation purposes assume that wehave some orders with different due dates considering justprocessing time or traveling time in the process of orderbatching regardless of their due dates may consolidate orderswith different due dates in a same batch So this issue maycause a problem for meeting customersrsquo demand in a certaindue date Consequently it is highlighted that order duedates should be considered as an important factor for orderbatching

Based on the above illuminated problem this researchpaper attempts to develop a novel model to overcome theaforementioned issue in the multiple parallel aisles withmanual order picking system In our proposed model itwas assumed that orders with similar items are groupedtogetherTherefore batching orders with higher resemblancecan reduce the order pickersrsquo traveled distance Also forconsidering the due dates in our model the due date of eachorder is defined as a weight parameter in calculating the asso-ciation between orders in terms of support The associationinformation integrates the bathing process with various dataregarding each order For calculating the association betweenorders in terms of support mining association rules withweighted items (MINWAL) were utilized Then a clusteringapproach based on a binary integer programming modelhas been used in order to maximize the customer demandassociation which is called support Next the Genetic Algo-rithm (GA) has been utilized to solve the Traveling SalesmanProblem (TSP) to minimize the travelling time for collectingall items in a constructed batch This process was followedby a batch sequencing process which utilized a GA in orderto minimize the average tardiness of all orders This papercopes with integrating batch ordering picker routing andthe batch sequencing problem regarding ordersrsquo due datein order to minimize the average tardiness of all ordersWe believe that the proposed approach namely ATGH isnot considered in any existing published paper In the nextsection a comprehensive literature survey is presented inorder to review the existing research activities related to thecurrent research paper

2 Literature Review

21 Order Batching Problem Order picking is an actionwhich takes place to fulfill a customer demand (inter-nalexternal) that is normally done by retrieving items frombuffer locations inside the warehouse [6ndash8] In practice thereare two types of systems for order picking systems whichare totally manual and involved with human order pickersand the systems that are entirely computerized in case ofretrieving items from the warehouse Picker-to-parts andparts-to-picker systems can be categorized as the systemsthat belong to the first group of warehousing systems [5 9]

Picker-to-parts systems are themost commonly used systemsin warehousing in which picking the items is done by anoperator that drives or walks along the aisles [10 11] Thisscenario is different in parts-to-picker systems Order pickersdo not move in this system and are located in the depotPallets or bins (unit load) from the warehouse are retrievedby automated storage and retrieval systems (ASRS) anddelivered to them at the depot After that the requested itemsare detached by the operators stationed in the depot and thepallets or bins are returned to its spot in the warehouse by theASRS [5]

In a large amount of orders a single order picking policymight be applied in which one order can be picked ineach picking tour However in a small amount of orderspicking loads in a single picking tour can lead to a decreasein travel times [12] Order batching is classified as a NP-hard problem which can be done to improve warehouseefficiency by reducing operational costs [12 13] Thus manyheuristic algorithms are proposed in the literature which helpto solve this problem Savings and seed algorithms are themost commonly used order batching heuristics in the case ofmanual picking systems Seed algorithms were introduced by[14] in which a seed order would be selected for each batchWith respect to the picking device capacity other orderswould be assigned to the seed orders Clarke andWright [15]first developed the savings algorithms inspired by a vehiclerouting algorithmThis algorithm works based on the savingthat would be achieved in travel distance or travel time Afterthat a pair of batches would be selected iteratively by thealgorithm and combined with each other until a capacityconstraint threshold is reached that stops combining of thebatches

In the current literature review it was tried to reviewall existing papers which are involved in order batchingproblems considering the minimization of due dates totaltravel distance and processing time First Vinod [16] sug-gested that integer programming can be used as a flexiblemethod for grouping any kind of objects Armstrong et al[17] also presented an integer programming model based onpredetermined batch sizes considering proximity batching

They tried to minimize the total processing time of allbatches Ratliff and Rosenthal [18] tried to minimize traveldistance based on a procedure which helps to find the bestpicking tour traveled Bymeans of a distinguished TSP a tourwas formed in a 50 aisles warehouse Kusiak et al [19] useda quadratic integer programming model to obtain batchesfrom eight orders In their proposed model it was tried tominimize the total sum of distances

One of their limitations was related to their model rangeof applicability which is not well suited for large numbers oforders Gibson and Sharp [20] presented an order batchingprocedure integrated with a computer simulation whichresulted in minimization of travel distance for each tourRosenwein [21] utilized Centre of Gravity (COG) and Min-imumAdditional Aisle (MAA) for order batching in order tomeasure travel distance It was shown that the MAA metricperformed better than the COG metric regarding travel tourgeneration Gademann et al [22] targeted to minimize thetotal travel time for all batches using a branch-and-bound

The Scientific World Journal 3

algorithm They tested their model by several test sets with amaximumorder number of 32whichwere compared togetherregarding their CPU times Hsu et al [3] applied GeneticAlgorithms (GAs) to deal with order batching problemswith any kind of configuration In their proposed approachthey endeavored to minimize the total travel distance Inanother research activity done by Chen and Wu [23] datamining and integer programming were combined to forma solution approach regarding the order batching problemThey have tried to maximize the similarity between orders ina batch in terms of support without considering ordersrsquo duedates Based on the presented results which was attemptedto compare first-come first-served (FCFS) MAA COGand Gibson and Sharprsquos method (GSM) the authors statedthat the proposed approach is effective in solving the orderbatching problems in terms of reducing traveling distancesBased on the analysis done by Gademann and van de Velde[24] the authors stated that the order batching problemis NP-hard to a great extent provided that no batch hasmore than two orders They developed a branch-and-pricealgorithm to formulate the problem in order to minimizethe total travel distance Three years later a mixed-integerprogrammingmodel was developed by Bozer andKile [25] inorder to minimize the length of trips or travel distance Theystated that acquiring minimum travel distance for a largenumber of orders requires substantial computational timeKulak et al [26] proposed a solution approach to reduce themathematical burden of order batching and picker routingproblems They proved the efficiency of their cluster-basedtabu search approaches by running several test examplesThey have focused onminimizing the travel distance betweenlocations

In practice certain completion due dates are assigned tocustomer orders For instance in distribution warehousesthe scheduled departure of trucks has to be done by certaindue dates to ensure the on time delivery of the requesteditems to the customers [22] Inmaterial warehouses avoidingproduction delays can be guaranteed if retrieving the itemsfrom thewarehouse is done based on their schedules In thesecases assessing the tardiness of the customer orders needs tobe focused in the process of batching customer orders intopicking orders [4] The tardiness is the difference betweenthe order finish time and the due date if this differenceis positive So it is very important to consider the ordersrsquodue dates when batching processes are being done To ourknowledge there are few papers that consider due dates orminimize tardiness in their model Elsayed et al [4] andElsayed and Lee [27] focused on developing minimizationmodels of order tardiness and their incurred costs Thesolution approaches proposed by these two research papersare of limited applicability tomanual picker-to-parts systemsSingle processing times are used in both solution approachesin order to determine a sequence of batches Therefore theywill not yield competitive results in the situation describedin our paper Won and Olafsson [28] endeavored to integrateorder batching and picker routing problems together in orderto minimize the travel time They compared their proposedheuristics with other existing algorithms such as FCFS Itcan be perceived that their simulation results improvement is

reduced when the numbers of items for each order increaseTsai et al [29] attempted to solve a batch picking modelconsidering earliness and tardiness penalty using a multiple-GA method for obtaining the best possible batch pickingplans They tried to develop a flexible model to cope withthe current dynamic environment in terms of responsivenessUpon studying their proposed method it was determinedthat the orders were split which causes items in an order tobe collected in different tours Also their proposed modelallocates penalty to batches overweight (when the batchvolume exceeds the picker capacity) which is not applicablein practice

22 Picker Routing Problem The picker routing problem isdefined as the process of identifying the minimum distancewhich would be traveled by the order picker in a warehouseupon identifying which order should be picked first [30 31]There are some existing heuristics for picker routing suchas return s-shape largest gap and combined routing DeKoster et al [32] tried to solve order batching problems byevaluating two groups of heuristic methods such as seedalgorithms and time savings algorithms by means of s-shapeand largest gap strategies as two different routing strategiesAccording to the obtained results integrating the s-shapestrategy and an outsized capacity pick device together withseed algorithms can yield better performance In contrast thelargest gap strategy using a small capacity pick device givesbest performance when it integrated with time savings algo-rithms Ho and Tseng [33] considered the largest gap routingstrategy together with a simulated annealing optimizationmethod not only to optimize the total travel distance but alsoto find alternative routes that are better than ones found byjust the largest gap strategy In a research activity done byTheys et al [34] s-shape largest gap and some other routingheuristics were compared with the Lin-Kernighan-HelsgaunTSP heuristic Based on their results this TSP heuristic helpsto reduce travel distance up to the rate of 48 on average Inthe current research activity a GA TSP model has been usedin order to solve the picker routing problem

23 Batch Sequencing Problem The problem of batch seq-uencing can be defined as finding the orders of constructedbatches to be processed further Henn and Schmid [35]used metaheuristics to solve their proposed model of orderbatching and sequencing to minimize the ordersrsquo tardinessIterated Local Search and Attribute-Based Hill Climber arethe two employed metaheuristics in their model Accordingto different test problems the efficiency of their proposedmethods was illustrated Provided solutions can be improvedby 46 on average in comparison with the ones obtainedby standard constructive heuristics such as an application ofthe Earliest Due Date rule They conducted their proposedmethod for a maximum of 80 items

Other sections of this paper are as follows First theproposed model is described in Section 3 which covers allthe aspects that are involved with it After that Section 4presents the research methodology of this paper This isfollowed by Section 5 which encompasses the numericalexamples results and discussion of this research activity

4 The Scientific World Journal

Finally Section 6 gives an end to the paper with a briefconclusion

3 Model Description

The assumptions of this research are described as follows

(1) A manual picker to part system in a parallel aislewarehouse is considered in this research

(2) There is only one location for storing each type of itemand vice versa

(3) Multiple picker devices are not allowedTherefore themaximum number of pickers is only one

(4) The volume for each order should not exceed thecapacity of the picker

(5) Batch size does not exceed the capacity of picker(6) The storage size in each location is identical(7) All customer orders are known in advance

31 Weighted Association Rule Mining Binary Integer Cluster-ing Model In this section an association based clusteringmodel developed by Chen and Wu [23] was modified inorder to create the batch structure regarding ordersrsquo duedate First the association between customersrsquo demand willbe calculated based on the similarity of items in orderswith respect to their due date using MINWAL Ordersrsquodue dates are considered as the weight of association rulemining Although batching orders with higher associationsgenerated by similarity between their items can decrease thedistance travelled by order pickers not considering the duedates would cause the orders to be batched inappropriatelywith different due dates Some problems may occur by theaforementioned issue such as shortage or excess inventorycosts In order to solve this problem the order due date hasbeen considered as a weight of each order for calculatingthe support between orders Relationship maximization ofcustomer orders based on similar items within each orderand due dates of each order can be done with the weightedassociation rules Travel distance would be reduced by takinginto account batching of orders with higher relationshipsOther than that higher associations can batch orders withsimilar due dates in order to minimize tardiness This isfollowed by utilizing a binary integer programming modelintegrated with a clustering algorithm which assembles theorders into their respective batches

311 Weighted Association Mining for Determining CustomerOrder Relationships Association rule mining can be definedas a data mining method which tries to distinguish interrela-tions of variables where large databases exist [36] Some rulesare involved with the association rule model where there isan association between some set of items with another set ofitems [37] In order to calculate rule interestingness supportand confidence can be considered as the twomost appropriatemeasures in association ruleminingThe support for an item-set is the percentage of transactions that contain the itemset

Table 1 Order-item data

Order Items Due date1198741

A(3) B(4) 10

1198742

A(2) C(3) 30

1198743

A(1) B(3) 5

1198744

A(2) B(4) C(2) 50

Table 2 Item-order data

Item OrderA 119874

1 1198742 1198743 1198744

B 1198741 1198743 1198744

C 1198742 1198744

in the database A probability shows how frequently the rulehead occurs among all the groups containing the rule bodywhich can be defined as the confidence of an association rule(body rArr head) [38 39]

Based on the definition by Agrawal et al [36] theassociation rule mining problem is addressed as followsitems are described as 119868 = 119894

1 1198942 119894

119899 which is a set

of n binary attributes The database is represented as 119863 =

1199051 1199052 119905

119898 which is a set of transactions An exclusive

transaction 119868119863 belongs to each transaction in119863 together witha subset of the items in 119868 A rule is defined as an implicationof the form 119883 rArr 119884 where 119883119884 sube 119868 and 119883⋂119884 = 120601 Theassociation rule 119883 rArr 119884 [support = 119904 confidence = 119888]holds in the item-order data 119863 with confidence 119888 if 119888 ofOrder Identifiers in 119863 contain orderset 119883 and also containsorderset 119884 The rule 119883 rArr 119884 has support 119904 if 119904 of the item-order 119863 contains both 119883 and 119884 For instance upon batchingthe orders by taking into account the customer order patternsassociation rule mining involves finding out the amountof support between demands of customers from the orderdatabase In order batching Table 1 which shows the order-item data is transposed to the item-order data form shownin Table 2 since the order associations are demanded and theproduct item associations is not necessary In this exampleassociation rule mining can perform its job to define theassociations between customer orders that can be perceivedfrom information presented in Table 2

In the area of order batching association rule miningwas first used by Chen and Wu [23] In their researchpaper association rule mining was used to calculate thecorrelations between customersrsquo demands in terms of sup-port They pointed out that orders with similar items shouldbe consolidated together in order to minimize the totaltravel distance traversed by the picker An Apriori algorithmhas been used in their proposed method to calculate theassociation between orders It should be highlighted that theyonly considered the relationships between orders in termsof similarities between items However in batch processingeach order has a due date which can be considered as theweight but this weight is not considered by any other existingmethod in the literature

The classical Apriori algorithm [40] extracts binary asso-ciation rules based on the downward closure property which

The Scientific World Journal 5

Table 3 Weight of each order

Order Weight1198741

05827

1198742

01979

1198743

07634

1198744

00672

proves that subsets of a frequent itemset are also frequent [41]However the Apriori algorithm cannot be applied because itcannot handle the weighted case Therefore in this researchactivity theMINWAL algorithm is applied in order to extractthe weighted supports among the pair of orders to fulfill thetask of considering both the support and the weights factors

In order to use MINWAL [42] for mining weighted asso-ciation rules the weight should be in the range of the 0 to1 interval Based on the aforementioned fact larger weightsshould be assigned to orders with smaller due dates in acomparison with other orders that have lots of time to bedelivered in order to minimize the average tardiness of allorders Therefore in this study (1) has been used to calculatethe appropriate weights for each order as following

Weight (119874119894) = 119890(minus01lowast120573lowast119889119874119894

Maximume Due Date) (1)

where 119874119894denotes an order and 119889

119874119894refers to the due date

of order 119874119894 120573 is a constant value that is used to normalize

the due dates which is considered as 30 (30 minute) in thisstudy Also the maximum due date of the orders was usedin order to provide an appropriate interval for the weightsFor example the weight of each order is tabulated in Table 3which is calculated by (1)

Given a set of orders 119874 = 1199001 1199002 119900

119898 a weight 119908

119895for

each order 119900119895 within the values of 0 to 1 is assigned by (1) via

its due date where 119895 = 1 2 119898 shows the importance ofit The weighted support for the weighted association ordersof 1198741and 119874

2as 1198741rArr 1198742is defined if the weighted support

of such a pair of orders is no less than theminimumweightedsupport (minsup) threshold In this study the minimumsupport threshold is considered as 0 in order to identify all theassociations among orders Equation (2) shows the support ofan association for any pair of orders as following

(sum sum

119908119895isin(1198741cup1198742)

119908119895) lowast Support (119874

1cup 1198742) (2)

Algorithm 1 shows the MINWAL steps in order to mineweighted association rules This algorithm consists of 6subroutines such as Search Counting Join Prune Checkingand Rules The search subroutine provides the size of thelargest itemsets in database Counting is responsible forcounting every 1-itemset inside the database The Join step isresponsible for generating the 119896-itemsets (119862119896) from (119896 minus 1)-itemsets (119862119896 minus 1) The Prune step removes inappropriateitemsets which do not exist The Checking step updatesthe count of 119896-itemsets inside the transaction and prunesitemsets which do not meet the minimum support thresholdIn the Rules step every association rule will be extractedTable 4 shows the notations of Algorithm 1

(1) Main Algorithm (wminsup mincon 119891119863 119908)(2) size = Search (119863)(3) 119871 = 0(4) for (119894 = 1 119894 le size 119894++)(5) 119862

119894= 119871119894= 0

(6) for each transaction do(7) (SC 119862

1 size) =Counting (119863 119908)

(8) 119896 = 1(9) while (|119862119896| ge 119896)(10) 119896++(11) 119862

119896= Join (119862

119896minus1)

(12) 119862119896=Prune (119862

119896)

(13) (119862119896 119871119896) =Checking (119862

119896119863)

(14) 119871 = 119871 cup 119871119896

(15)Rules (SC 119871)(16) ends

Algorithm 1 MINWAL algorithm [42]

Table 4 Notations of Algorithm 1

119863 The database119908 The set of item weights119871119896

Set of large 119896-itemsets

119862119896

Set of 119896-itemsets which may be 119896-subsets of large119895-itemsets for 119895 ge 119896

SC (119883) No of transactions containing itemset119883wminsup Weighted support thresholdminconf Confidence thresholdSize Maximum possible large weighted itemsets

312 Association Based Binary Integer Programming forOrders Batching In order to create the batch structure abinary integer clustering model which has been developed byChen andWu [23] was appliedThey developed a binary inte-ger programming model for maximizing similarity betweenorders in terms of support in order to minimize the totaltraveling distance In this research activity support betweenorders was calculated based on weighted association miningusing MINWAL which was described in Section 311 Thebinary integer model for order batching is described asfollows

Parameters

119878119894119895Support between order 119894 and 119895 determined by MIN-WAL

119862V Capacity of picking vehicle

119881119894Weight of order 119894

119870 Number of batches

1198831198941198951 if order 119894 is assigned to batch 119895

0 otherwise

6 The Scientific World Journal

1198841198941 if order 119895 is chosen as a batch median0 otherwise

Maximize119873

sum

119894=1

119873

sum

119895=1

119878119894119895lowast 119883119894119895 (3)

Subject to119873

sum

119895=1

119883119894119895= 1 for 119894 119895 = 1 2 119873

(4)

119883119894119895le 119884119895

for 119894 119895 = 1 2 119873 (5)

119873

sum

119895=1

119884119895= 119870 (6)

119873

sum

119894=1

119881119894119883119894119895le 119862V for 119895 = 1 2 119873 (7)

119883119894119895= 0 1 for 119894 119895 = 1 2 119873 (8)

119884119895= 0 1 for 119895 = 1 2 119873 (9)

In the current model the objective function (3) maxi-mizes the sum of support from all orders to their relevantbatch medians as 119870 orders are selected as batch mediansThis model will maximize the support between orders in thebatches Membership of each order in just one batch wouldbe guaranteed by constraint (4) The number of batches with119870 is limited using constraints (5) and (6) The total quantityin a tour is also represented as a limitation in constraint (7) Itcannot exceed the capacity of order picker 119862V For the orderbatchingmodel to have a binary solutionwould be ensured byconstraints (8) and (9) For calculating the initial numbers ofbatches (10) will be used where sum119881

119894represents total volume

of all orders as following

119870initial = lceilsum119881119894

119862V

rceil (10)

32 Picker Routing

321 Genetic Algorithms In this research GAs which wereintroduced by Holland [43] have been used in order tosolve the picker routing and batch sequencing problem Thebiological evolution process is used as a base fact in a GA forproblem solving to a great extentThemain steps of a GA canbe described as follows [44 45]

(i) populating a series of solutions(ii) solutions assessment(iii) picking the most appropriate solutions(iv) generating new solutions based on some genetic

operation

In step (iii) selection forms the main part of a GA Thebad solutions are removed by selection and the good onesremain Random generation over the feasible or infeasible

solution space of a problem is used as a basis for creatingthe population Each solution is evaluated using the fitnessfunction Selection would be done according to certain selec-tions of fitness together with the probability that is assignedto it Generating a fresh population and a better solutionare involved with recombination of the individual solutionCrossover and mutation are the most used GA operators forgenerating new offspring from the old generation of parentsParameters of the GA are originated from the probabilitiesof mutation and crossover Exploration of the sample spaceis diversified by these operators In this research the GA hasbeenutilized in two steps First theGAhas beenused in orderto solve the picker routing problem Consider that in thewarehouse a set of items should be picked by a picker in onetour Therefore the picker routing problem in the warehousecould be categorized as a TSP The goal in the picker routingproblem as a TSP in the warehouse is to find an optimal pathfor minimizing the travel distance traversed by the pickingmachine where each node is visited at least once In thisresearch for a given number of itemset in a batch GA TSPhas been used to find the minimum travel distance whichis traversed by the picking machine Second a GA has beenutilized in order to sequence all batches namely GA sequencingwhichwould be further processedThe batches are sequencedbased on a fitness function which minimizes the tardiness ofall ordersThedetailed procedures ofGA TSP andGA sequencingare illustrated in Sections 323 and 324

322 Warehouse Layout Thewarehouse discussed through-out this research is assumed to have a 3D multi-parallel-aisle warehouse layout The picker will start the tour fromthe depot which is located in front of the leftmost aisle andreturns back to it A picker leaves the depot After that ittravels to a certain buffer spot where it picks the requestedorder(s) and puts the order(s) into a certain picker Then itcontinue on its way to another spot These procedures wouldbe repeated again and again so that all the requested ordersare picked Then it is time for the picker to return to thedepot

The graphic layout of the warehouse zone involved in thisresearch activity is pictorially displayed in Figure 1 It showsthe 119909 and 119910 dimensions of the 3D warehouse layout Eachitem in the warehouse has a specific location For examplethe location of item 119894 can be addressed by 119871

119894= (119909119894 119910119894 119911119894)

Consider two locations as 119871119886= (119909119886 119910119886 119911119886) and 119871

119887= (119909119887

119910119887 119911119887) If two locations are located in the same sub-aisles

of the same block the distance between these two itemslocations (119863

119886119887) will be calculated using

119863119886119887

=1003816100381610038161003816119909119886 minus 119909119887

1003816100381610038161003816 +1003816100381610038161003816119910119886 minus 119910119887

1003816100381610038161003816 +1003816100381610038161003816119911119886 minus 119911119887

1003816100381610038161003816 (11)

Equation (11) is not applicable if the two storage locationsbelong to different subaisles that are from the same block [34]In order to solve this problem distances of all possible pathsbetween two locations have been calculated and the shortestone was considered

323 GA TSP for the Picker Routing Problem In order todetermine the optimal route of a picker related to a given

The Scientific World Journal 7

Lb=(xbybzb)

La=(xayaza)

Li=(xiyizi)

Depot

Figure 1 Warehouse layout

1 2 3 4 5 6 7 8 9L0 L1 L8 L11 L9

L13 L12 L10 L0

Figure 2 A feasible chromosome encoding in GA TSP

set of items in a batch a GA has been used to solve theTSP problem to minimize travelling time In a GA eachsolution is considered as a chromosome The steps of GA TSPare described as follows

(1) for each batch establishing the structure of a chromo-somes

(2) generate initial feasible chromosomes(3) evaluate each chromosome based on the GA TSP fit-

ness function(4) apply the Selection operation(5) apply the Crossover operation(6) apply the Mutation operation(7) evaluate each offspring based on the fitness function(8) go to step 4 if the termination criterion is not satis-

fied otherwise terminate and return the best solutionas the optimal path

At the first step of GA TSP the structure of the chro-mosome should be established In the GA TSP the locationof an item that should be visited is presented by the valueof a gene Accordingly the sequence of visiting the itemlocation is denoted by the order of a gene in a chromosomeTherefore the total number of items locations in a specificbatch is considered equal to the length of a chromosome Forexample if the length of a chromosome is nine it means nineplaces should be visited by the picker as shown in Figure 2If 119871119894represents the item location in the warehouse 119894 = 1

119899 it shows that the picker leaves the depot (1198710denotes

the location of a depot) goes to 1198711 1198718 11987111 1198719 11987113 11987112 11987110

and then comes back to the depotIn the second step the initial number of feasible chro-

mosomes should be generated randomly for each batch

The number of initial chromosomes could be different basedon the complexity of the problem It should be consideredthat the first and the last gene of each chromosome should beassigned individually to the location of a depot In step threeeach chromosome will be evaluated based on the GA TSPfitness function which is formulated as follows

fitness = Min119899minus1

sum

119895=1

119863119895119895+1

(12)

where 119863119895119895+1

denotes the distance between two subsequentgenes in a chromosome 119895 = 1 119899 minus 1 represents the orderof the genes and 119899 is number of genes

Subsequently step four is involved with the selectionoperation In each iteration the selection operator usesfitness values to select the parents of the next generationIn this research the roulette wheel selection policy hasbeen used to guarantee that the most appropriate pairs ofchromosomes have been selected to generate offspring Itmeans that a higher probability of being selected to createa new population will be assigned to the chromosomes withhigher fitness values

Steps five and six are involved with conducting thecrossover and mutation operation The construction of theoffspring is realized by the crossover and the diversity of theindividuals is maintained by the mutation operator Blocksof genes between chromosomes are traded by means of thecrossover operation inwhich the exploitation of a particularlyprofitable portion of the parameter space is allowed to theblock Here permutation order based crossover (POP) as avariation of the distinguished order crossover has been usedAs shown in Figure 3 the original idea is about choosing twoparents and a point for cutting The first portion of offspring2 contains the first portion of parent 1 up to the cut pointTo form the second portion of offspring 2 consider all genesin parent 2 except those genes that already exist in the firstportion of the offspring 2

As shown in Figure 4 two positions are randomlyselected in a chromosome and exchanged with each othernamely the order based mutation approach SWAP isadopted for solving GA TSP Step six is followed by step sevenwhich includes the evaluation of each offspring that has beengenerated in the previous step based on the GA TSP fitnessfunction In step eight termination criteria will be checkedIf the termination condition is met then the process will beterminated After that the best chromosome will be selectedas the optimal solution Otherwise the selection process is tobe conducted again to generate a new population

324 GA for Sequencing the Batches Sequencing the batchesshould be conducted in order to minimize the averagetardiness of all orders The problem of batch sequencing canbe defined as finding the orders of batches which should beprocessed further For this purpose the GA sequencing methodhas been utilized In this research all of the batches whichwere constructed using binary integer programming andGA TSP will be sequenced by using the GA The populationgeneration selectionmutation and stopping threshold in theGA sequencing are identical to those in the GA TSP However

8 The Scientific World Journal

Parent 1

Parent 2

Offspring 2

Offspring 1

L1 L8 L9

L1 L1L8 L8L9

L11 L13 L12 L10

L1 L8 L9 L11L13 L12L10

L13 L9 L11 L11 L13 L12

L12

L10

L10L1L8L12

L13 L9 L11 L10

L1 L8 L12L13 L9 L11 L10

Figure 3 Crossover operation

Parent L1

L1

L8

L8

L9

L9

L11

L11

L13

L13

L12

L12

L10

L10Offspring

Figure 4 Mutation operation

1 2 3 4 5b1 b4 b3 b5 b2

Figure 5 Chromosome structure

the fitness function and chromosome structure are differentthan that of GA TSP In contrast with GA TSP chromosomesstructure batch number is presented by the value of a geneAccordingly the sequence of batch is denoted by the orderof a gene in a chromosome For instance Figure 5 shows thechromosome structure If 119887

119894isin 119861 denotes a set of batches

Figure 5 shows that 1198871 1198874 1198873 1198875 1198872should be processed

respectively Therefore the length of a chromosome is equalto the number of the batches The GA sequencing fitness func-tion for chromosome 119910 is described in

fitness119910=

1

119873

119899

sum

119896=1

sum

119894isin119887119896

119862119879 (119887119896) minus 119863119906 (119900

119894) (13)

119862119879 (119887119896) = 119862119879 (119887

119896minus1) + Pt (119887119896) (14)

Pt (119887119896) =Opt (119887

119896)

119881119878 (15)

where119873 denotes for the total number of orders 119896 = 1 119899

represents the batch sequence in which 119899 is the total numberof batches 119900

119894stands for the order 119894 in which 119894 isin 119887

119896 119887119896

denotes the batch in 119896th position of a sequence for specificchromosome 119862119879(119887

119896) stands for the completion time of a

batch in the 119896th position of a sequence 119863119906(119900119894) is the symbol

for due date of order 119894which belongs to 119887119896 Pt(119887119896) is the process

time of 119887119896 Opt(119887

119896) is the optimal path solution for 119887

119896 Finally

119881119878 is the moving speed of the picker

4 Research Methodology

In this research the proposed methodology consists of fivesteps These steps are described as follows and shown pic-torially in Figure 6

Step 1 It is involved with utilizing theMINWAL algorithm inorder to determine the associations between orders in termsof support considering their due dates In this step an orderdue date will be considered as the weight of an order It meansorders that are similar in items and due date should havemoresupport for being batched together

Step 2 It encompasses determining the initial number ofbatches In this step the initial number of batches will becalculated by (10)

Step 3 It includes the problem of modeling and solving firstthe binary integer programming model for order clusteringand batching will be modeled Next this model will be solvedin order to determine the structures of batches

Step 4 It is about the picker routing problem The optimalpath for a picker within each batch will be determined bysolving a TSP using GA

Step 5 It is about sequencing the batches based on the fitnessfunction which minimizes the average tardiness of all ordersIn this step GA algorithmwill be utilized in order to solve thebatch sequencing problem regarding the fitness function

5 Computational Experiments and Results

51 Test Problems and Parameter Setting In this sectionfive numerical test problems with different settings andparameters have been carried out in order to illustrate theproficiency of the proposed method with respect to averagetardiness of all orders Table 5 summarizes the descriptionof these test problems This table includes information withregard to number of items number of orders capacity of thepicker total weight of all the items in an order andminimumnumber of batches

In these test problems the quantity of items quantityof orders capacity of the picker device (119862V) and amount ofeach item are predetermined Each order contains a certain

The Scientific World Journal 9

Start

Input order informationwarehouse setting picker

setting

Compute the supportbetween customersrsquo

orders using MINWAL

Calculating initialnumber of batches (K) to

assign the orders

Solving the binaryinteger programmingclustering model for

constructing the batches

Feasiblesolution

Yes

No

Solving picker routing

Solving batch sequencingproblem using

End

K = K + 1

problem using GA TSP

GA sequencing

Figure 6 Methodology framework

quantity of items and also each item has a certain quantityThese quantities are produced randomly The distributionfunction type for the number of items in each order is Normal[5 10] Also the distribution function type for each itemquantity in an order is uniform [1 10] The picker movingspeed is set as 119881119878 = 2 (ms) The warehouse has 3000 itemlocation The GA control parameters for solving the TSP andsequencing problem are tabulated in Table 6

All of the proposed approaches are coded usingMATLABversion 790 and Java 15 programming language Also theIBM ILOG CPLEX optimization studio version 124 softwarewas utilized jointly with MATLAB in order to facilitate theprocess of solving the binary integer programming problemExamples are run by means of a computer featured by

4 Gigabytes random access memory (RAM) and a 220Gigahertz Intel processor (T6600)

52 Computational Results and Discussion In order to revealthe competency of the proposed solution approach it wascompared with the other existing methods in the literatureOne of the latest published research activities in the literaturewhich deals with minimization of tardiness has been con-ducted by Henn and schmid [35] They utilized the EarliestDueDate (EDD) approach as their constructive algorithm forcomparing with their proposed approaches They proposedtwo heuristics Iterated Local Search andAttribute-BasedHillClimber are the two employed metaheuristics in their modelAccording to different test problems the efficiency of their

10 The Scientific World Journal

Table 5 Test problems description

Example 1 Example 2 Example 3 Example 4 Example 5Number of orders 40 60 80 100 150

Number of items 80 120 160 200 200

Capacity of picker 200 200 300 400 550

Total weight 2212 3668 4393 6110 8369

Minimum number of batch 11 19 15 16 16

Due dates range 60 90 120 180 240

Table 6 GA control parameters

GA TSP GA sequencing

Population size 500 200

Cross over strategyPermutation orderbased crossover

(POP)

Permutation orderbased crossover

(POP)Cross over probability 08 08

Mutation strategy SWAP SWAPMutation probability 02 02

Iteration number 10000 10000

Parent selection method Roulette wheel Roulette wheel

proposed methods was illustrated Their obtained resultsfrom their proposed approaches showed a 46 improvementon tardiness compared to EDD They used an s-shape andlargest gap routing strategy for their proposed method Inorder to make a reasonable comparison with their proposedmethod EDD was integrated with the s-shape routing strat-egy and GA sequencing Then based on improvement on EDDwe can compare our proposed method with their proposedmethod In the EDD approach due dates play an importantrole in sorting all orders Based on an ascending sequenceassigning the orders to batches ensures that no capacityviolation can happen for the picking device

For the warehouse condition which is described inSection 323 our proposed approach EDD integrated withGA TSP and GA sequencing and EDD integrated with s-shapealgorithm and GA sequencing have been evaluated in termsof average tardiness for all orders (119879) which is calculatedbased on (13) number of batches (119870) total process time(Pt) and average process time for each batch (119860Pt) All ofthe GA setting for the TSP and sequencing problems areconsidered the same for all methods Also the settings whichhave been mentioned in Section 51 were considered for all ofthe compared methods

The results for all of the test problems for the threeapproaches are tabulated in Tables 7 8 and 9 As it isillustrated in Table 10 EDD integrated with GA TSP andGA sequencing has a 3176 improvement on average in termsof average tardiness of all orders than EDD integrated withthe s-shape algorithm and GA sequencing It can be perceivedthat GA TSP performs better than s-shape Also the ATGHmethod has a 6811 improvement in average tardinessoff all orders than EDD integrated with s-shape algorithm

and GA sequencing Consequently the ATGH method achievesmuch improvement against Henn and Schmidrsquos [35] pro-posedmethod According to Tables 7 and 8 it is also worth tomention thatGA TSP performs better than the s-shape routingstrategy in terms of process time The improvements whichhave been achieved in this research could be justified basedon the following reasons

The problem of order batching deals with several sig-nificant drivers in which the number of batches can beconsidered as a key factor [23] As a justification to thismatter it is worth to mention that total travel distance maynot decrease directly due to a decline in the number ofbatches But a decline in the number of batches will increasebatch sizes which may lead to an increase in the distancetravelled by order picker However an advanced exploitationof picker capacity to load efficiently may reduce the numberof constructed batches together reducing the total traveldistance Improving the total travel distance traversed by thepicker can minimize the average tardiness of all orders Interms of number of the batches according to Tables 7 8 and9 it can be perceived that our proposed method assigns theorders to the batches optimally The ATGH method assignsthe orders to batches based on their item similarity andtheir due dates and it uses the picker capacity efficiently Butthe EDD method allocates the orders to batches based onan ascending sequence of their due date and it should beensured that no capacity violation will happen for the pickingdevice which may lead to an inefficient order batching Tohighlight the problem an example can be made based on theresults tabulated in Tables 7 and 8 The number of batchesfor example 1 in our method is 11 in comparison with 13obtained fromEDDmethod 13 batches can be justified basedon this scenario that two subsequent orders say the firstorder is 100 kg and the other is 150 kg cannot be batchedtogether due to violated batch capacity which is set to be200 kg The number of batches would increase due to theaforementioned issue which affects the total process time tohave an increasing manner This situation will rarely happenin the ATGH method because of the weighted associationbased binary integer programming utilizedwhich batches theorders in order to maximize the association between themwhich is calculated based on the item similarity of ordersand the weight of their corresponding due dates The ATGHmethod uses the picker capacity efficiently

Obviously one of the factors that can influence thecompletion time and the average tardiness of all orders is theprocessing time of the orders Reducing the processing time

The Scientific World Journal 11

Table 7 Computational results of ATGH

Example 1 Example 2 Example 3 Example 4 Example 5119879 56204 29948 90202 163 58898

Pt 605833 1217833 105833 121033 1339

119870 11 19 15 16 16

119860Pt 55076 6409 7055 7564 8368

Table 8 Computational results of EDD integrated with GA TSP and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 124550 42976 18222 5981 1547

Pt 667333 133516 116083 13185 141916

119870 13 21 16 17 16

119860Pt 51333 6357 7255 7775 8869

Table 9 Computational results of EDD integrated with s-shape algorithm and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 1651 62677 2825 13365 17593

Pt 75266 16636 13323 14333 14926

119870 13 21 16 17 16

119860Pt 5789 892 833 8431 932

Table 10 Comparison results

EDD + s-shape + GA sequencing EDD + GA TSP + GA sequencing ATGHTardiness (min) Tardiness (min) Improvement () Tardiness (min) Improvement ()

(1) 1651 124550 2456 56204 6596(2) 62677 42976 3143 29948 5222(3) 2825 18222 3550 90202 6807(4) 13365 5981 5525 1627 8783(5) 17593 1547 1206 58898 6652Average improvement 3176 6811

and travel time for constructing a batch is an outstanding factin reducing the costs and minimizing delays of the customerresponses In our proposed model orders with similar itemsare consolidated together Therefore batching orders withmore similarity can reduce the distance traversed by thepicker facility Also the due date of each order is definedas a weight parameter in calculating the association betweenorders in terms of support which was done for consideringthe due dates in our model Thereupon the model considersdue date and the process time in an integrated manner whichsimultaneously affected the average tardiness of all ordersThis spoken issue is different in the EDD approach in whichthe ordersrsquo due dates are just considered Example 5 canbe considered as evidence of this matter in which the totaland average process time for ATGH is lower than EDDintegrated with GA TSP and GA sequencing with respect to thefact that number of constructed batches for both approachesare happened to be equal after obtaining the results

6 Conclusion

One of the most important concerns of warehouse managersis involvedwith finding the best optimal and cost efficientwayto pick orders placed by customers in order to be known as aresponsible company with satisfied customers in terms of ontime delivery In the current research activity order batchingpicker routing and batch sequencing problems of warehouseprocesses were addressed which are to be solved jointly asthey are mixed up with many manufacturing industriesMathematical solutions of these NP-hard problems might beavailable for problems that are small in nature But thesekinds of solutions may not be applicable to be consideredby warehouse managers Hence we presented novel solutionapproach namely ATGH to solve this issue in accordancewith minimizing the average tardiness of all customersrsquoorders which can be considered as a major objective ofthis research activity A weighted association rule mining

12 The Scientific World Journal

algorithm namely MINWAL was utilized to determine thecustomersrsquo orders association regarding similarity betweenitems and ordersrsquo due dates Actually in our proposedmethodthe due date of each order is defined as a weight parameterin calculating the association between orders So orders withsimilar items and due dates have more chance to be in asame batch This issue is not considered in most of theresearch activities in the literature related to our work andconsidering this issue in the order batching problem is themain contribution of our research activity The previous stepwas followed by a binary integer clustering model in orderto solve the order batching problem which maximizes thesimilarity between orders in terms of support After that aGA was applied to solve a TSP in order to find the optimalrouting path

Finally a batch sequencing problem was addressed inconjunction with a GA

The other research objective of this research addresseswhether the proposed ATGH algorithm can further improvethe solution quality for a multiple parallel aisle warehousewith manual picking system Hence comparisons were madeconsideringATGHEDD integratedGA TSP andGA sequencingand EDD integrated s-shape and GA sequencing algorithms Asdemonstrated by the obtained results ATGH comes out tobe the most striking solution approach in terms of tardinessIn spite of the fact that considering all customer orders asdiscussed in this paper would lead to have a static case beinginvolved with dynamic cases in which different points of timeare assigned to customerrsquos orders arrival would be beneficialSo there might be some potential room of research to extendour discussed approach with respect to dynamic orders

Conflict of Interests

The IBM ILOG CPLEX optimization studio version 124software is not used for financial gain and there is no conflictof interests with any party as it is only a student version

References

[1] ELAAT Kearney Excellence in Logistics ELA Brussels Bel-gium 2004

[2] Y-C Ho T-S Su and Z-B Shi ldquoOrder-batching methods foran order-picking warehouse with two cross aislesrdquo Computersamp Industrial Engineering vol 55 no 2 pp 321ndash347 2008

[3] C-M Hsu K-Y Chen and M-C Chen ldquoBatching ordersin warehouses by minimizing travel distance with geneticalgorithmsrdquo Computers in Industry vol 56 no 2 pp 169ndash1782005

[4] E A Elsayed M-K Lee S Kim and E Scherer ldquoSequencingand batching procedures forminimizing earliness and tardinesspenalty of order retrievalsrdquoThe International Journal of Produc-tion Research vol 31 no 3 pp 727ndash738 1993

[5] S Henn S Koch and W Gerhard Order Batching in OrderPicking Warehouses A Survey of Solution Approaches SpringerLondon UK 2012

[6] J Drury Towards More Efficient Order Picking IMM Mono-graph no 1 The Institute of Materials Management CranfieldUK 1988

[7] M Goetschalckx and J Ashayeri ldquoClassification and design oforder picking systemsrdquo Logistics World pp 99ndash106 1989

[8] J A Tompkins J A White Y A Bozer E H Frazelle and J MA Tanchoco Facilities Planning John Wiley amp Sons HobokenNJ USA 2003

[9] S Henn and G Wascher ldquoTabu search heuristics for the orderbatching problem in manual order picking systemsrdquo EuropeanJournal of Operational Research vol 222 no 3 pp 484ndash4942012

[10] S Henn ldquoAlgorithms for on-line order batching in an orderpicking warehouserdquo Computers and Operations Research vol39 no 11 pp 2549ndash2563 2012

[11] R de Koster ldquoHow to assess a warehouse operation in a singletourrdquo Tech Rep RSM Erasmus University Rotterdam TheNetherlands 2004

[12] R de Koster T Le-Duc and K J Roodbergen ldquoDesign andcontrol of warehouse order picking a literature reviewrdquo Euro-pean Journal of Operational Research vol 182 no 2 pp 481ndash5012007

[13] R A Ruben and F R Jacobs ldquoBatch construction heuristics andstorage assignment strategies for walkride and pick systemsrdquoManagement Science vol 45 no 4 pp 575ndash596 1999

[14] E A Elsayed ldquoAlgorithms for optimal material handling inautomatic warehousing systemsrdquo International Journal of Pro-duction Research vol 19 no 5 pp 525ndash535 1981

[15] G Clarke and J W Wright ldquoScheduling of vehicles froma central depot to a number of delivery pointsrdquo OperationsResearch vol 12 no 4 pp 568ndash581 1964

[16] H D Vinod ldquoInteger programming and the theory of group-ingrdquo Journal of the American Statistical Association vol 64 no326 pp 506ndash519 1969

[17] R D Armstrong W D Cook and A L Saipe ldquoOptimalbatching in a semi-automated order picking systemrdquo Journal ofthe Operational Research Society vol 30 no 8 pp 711ndash720 1979

[18] H D Ratliff and A S Rosenthal ldquoOrder-picking in a rect-angular warehouse a solvable case of the traveling salesmanproblemrdquo Operations Research vol 31 no 3 pp 507ndash521 1983

[19] A Kusiak A Vannelli and R K Kumar ldquoClustering analysismodels and algorithmsrdquo Cybernetics vol 15 pp 139ndash154 1986

[20] D R Gibson and G P Sharp ldquoOrder batching proceduresrdquoEuropean Journal of Operational Research vol 58 no 1 pp 57ndash67 1992

[21] M B Rosenwein ldquoA comparison of heuristics for the problemofbatching orders for warehouse selectionrdquo International Journalof Production Research vol 34 no 3 pp 657ndash664 1996

[22] A J RMGademann J P van denBerg andHH van derHoffldquoAn order batching algorithm for wave picking in a parallel-aislewarehouserdquo IIE Transactions vol 33 no 5 pp 385ndash398 2001

[23] M-C Chen and H-P Wu ldquoAn association-based clusteringapproach to order batching considering customer demandpatternsrdquo Omega vol 33 no 4 pp 333ndash343 2005

[24] NGademann and S van deVelde ldquoOrder batching tominimizetotal travel time in a parallel-aisle warehouserdquo IIE Transactionsvol 37 no 1 pp 63ndash75 2005

[25] Y A Bozer and J W Kile ldquoOrder batching in walk-and-pick order picking systemsrdquo International Journal of ProductionResearch vol 46 no 7 pp 1887ndash1909 2008

[26] O Kulak Y Sahin and M E Taner ldquoJoint order batching andpicker routing in single and multiple-cross-aisle warehousesusing cluster-based tabu search algorithmsrdquo Flexible Servicesand Manufacturing Journal vol 24 no 1 pp 52ndash80 2012

The Scientific World Journal 13

[27] E A Elsayed and M-K Lee ldquoOrder processing in automatedstorageretrieval systems with due datesrdquo IIE Transactions vol28 no 7 pp 567ndash577 1996

[28] J Won and S Olafsson ldquoJoint order batching and order pickingin warehouse operationsrdquo International Journal of ProductionResearch vol 43 no 7 pp 1427ndash1442 2005

[29] C-Y Tsai J J H Liou and T-M Huang ldquoUsing a multiple-GA method to solve the batch picking problem consideringtravel distance and order due timerdquo International Journal of Pro-duction Research vol 46 no 22 pp 6533ndash6555 2008

[30] Y KaoM H Chen and Y T Huang ldquoA hybrid algorithm basedon ACO and PSO for capacitated vehicle routing problemsrdquoMathematical Problems in Engineering vol 2012 Article ID726564 17 pages 2012

[31] X Gan Y Wang S Li and B Niu ldquoVehicle routing problemwith time windows and simultaneous delivery and pick-up ser-vice based on MCPSOrdquoMathematical Problems in Engineeringvol 2012 Article ID 104279 11 pages 2012

[32] M B M de Koster E S van der Poort and M WoltersldquoEfficient orderbatching methods in warehousesrdquo InternationalJournal of ProductionResearch vol 37 no 7 pp 1479ndash1504 1999

[33] Y-C Ho and Y-Y Tseng ldquoA study on order-batching methodsof order-picking in a distribution centre with two cross-aislesrdquoInternational Journal of Production Research vol 44 no 17 pp3391ndash3417 2006

[34] C Theys O Braysy W Dullaert and B Raa ldquoUsing a TSPheuristic for routing order pickers in warehousesrdquo EuropeanJournal of Operational Research vol 200 no 3 pp 755ndash7632010

[35] S Henn and V Schmid Metaheuristics for Order Batching andSequencing in Manual Order Picking Systems Univ Fak furWirtschaftswiss 2011

[36] M Hahsler B Grun and K Hornik ldquoArulesmdasha computationalenvironment for mining association rules and frequent itemsetsrdquo Journal of Statistical Software vol 14 2005

[37] R Agrawal T Imielinski and A Swami ldquoMining associationrules between sets of items in large databasesrdquo ACM SIGMODRecord vol 22 no 2 pp 207ndash216 1993

[38] S Encheva Y Kondratenko M Z Solesvik and S TuminldquoDecision support systems in logisticsrdquo in Proceeding of Inter-national electronic conference on computer science pp 254ndash2562007

[39] M-C Chen ldquoConfiguration of cellular manufacturing systemsusing association rule inductionrdquo International Journal of Pro-duction Research vol 41 no 2 pp 381ndash395 2003

[40] R Srikant and R Agrawal ldquoMining generalized associationrulesrdquo Future Generation Computer Systems vol 13 no 2-3 pp161ndash180 1997

[41] J Wu and XM Li ldquoAn effective mining algorithm for weightedassociation rules in communication networksrdquo Journal of Com-puters vol 3 no 10 pp 20ndash27 2008

[42] C H Cai A W C Fu C H Cheng and W W KwongldquoMining association rules with weighted itemsrdquo in Proceedingsof the International Symposium on Database Engineering ampApplications pp 68ndash77 Cardiff UK 1998

[43] J HollandAdaptation in Natural and Artificial Systems Univer-sity of Michigan Press Ann Arbor Mich USA 1975

[44] A A Taleizadeh F Barzinpour and H-M Wee ldquoMeta-heuristic algorithms for solving a fuzzy single-period problemrdquoMathematical and Computer Modelling vol 54 no 5-6 pp1273ndash1285 2011

[45] H Yang J Yi J Zhao and Zh Dong ldquoExtreme learning ma-chine based genetic algorithm and its application in powersystem economic dispatchrdquo Neurocomputing vol 102 pp 154ndash162 2013

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

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The Scientific World Journal 3

algorithm They tested their model by several test sets with amaximumorder number of 32whichwere compared togetherregarding their CPU times Hsu et al [3] applied GeneticAlgorithms (GAs) to deal with order batching problemswith any kind of configuration In their proposed approachthey endeavored to minimize the total travel distance Inanother research activity done by Chen and Wu [23] datamining and integer programming were combined to forma solution approach regarding the order batching problemThey have tried to maximize the similarity between orders ina batch in terms of support without considering ordersrsquo duedates Based on the presented results which was attemptedto compare first-come first-served (FCFS) MAA COGand Gibson and Sharprsquos method (GSM) the authors statedthat the proposed approach is effective in solving the orderbatching problems in terms of reducing traveling distancesBased on the analysis done by Gademann and van de Velde[24] the authors stated that the order batching problemis NP-hard to a great extent provided that no batch hasmore than two orders They developed a branch-and-pricealgorithm to formulate the problem in order to minimizethe total travel distance Three years later a mixed-integerprogrammingmodel was developed by Bozer andKile [25] inorder to minimize the length of trips or travel distance Theystated that acquiring minimum travel distance for a largenumber of orders requires substantial computational timeKulak et al [26] proposed a solution approach to reduce themathematical burden of order batching and picker routingproblems They proved the efficiency of their cluster-basedtabu search approaches by running several test examplesThey have focused onminimizing the travel distance betweenlocations

In practice certain completion due dates are assigned tocustomer orders For instance in distribution warehousesthe scheduled departure of trucks has to be done by certaindue dates to ensure the on time delivery of the requesteditems to the customers [22] Inmaterial warehouses avoidingproduction delays can be guaranteed if retrieving the itemsfrom thewarehouse is done based on their schedules In thesecases assessing the tardiness of the customer orders needs tobe focused in the process of batching customer orders intopicking orders [4] The tardiness is the difference betweenthe order finish time and the due date if this differenceis positive So it is very important to consider the ordersrsquodue dates when batching processes are being done To ourknowledge there are few papers that consider due dates orminimize tardiness in their model Elsayed et al [4] andElsayed and Lee [27] focused on developing minimizationmodels of order tardiness and their incurred costs Thesolution approaches proposed by these two research papersare of limited applicability tomanual picker-to-parts systemsSingle processing times are used in both solution approachesin order to determine a sequence of batches Therefore theywill not yield competitive results in the situation describedin our paper Won and Olafsson [28] endeavored to integrateorder batching and picker routing problems together in orderto minimize the travel time They compared their proposedheuristics with other existing algorithms such as FCFS Itcan be perceived that their simulation results improvement is

reduced when the numbers of items for each order increaseTsai et al [29] attempted to solve a batch picking modelconsidering earliness and tardiness penalty using a multiple-GA method for obtaining the best possible batch pickingplans They tried to develop a flexible model to cope withthe current dynamic environment in terms of responsivenessUpon studying their proposed method it was determinedthat the orders were split which causes items in an order tobe collected in different tours Also their proposed modelallocates penalty to batches overweight (when the batchvolume exceeds the picker capacity) which is not applicablein practice

22 Picker Routing Problem The picker routing problem isdefined as the process of identifying the minimum distancewhich would be traveled by the order picker in a warehouseupon identifying which order should be picked first [30 31]There are some existing heuristics for picker routing suchas return s-shape largest gap and combined routing DeKoster et al [32] tried to solve order batching problems byevaluating two groups of heuristic methods such as seedalgorithms and time savings algorithms by means of s-shapeand largest gap strategies as two different routing strategiesAccording to the obtained results integrating the s-shapestrategy and an outsized capacity pick device together withseed algorithms can yield better performance In contrast thelargest gap strategy using a small capacity pick device givesbest performance when it integrated with time savings algo-rithms Ho and Tseng [33] considered the largest gap routingstrategy together with a simulated annealing optimizationmethod not only to optimize the total travel distance but alsoto find alternative routes that are better than ones found byjust the largest gap strategy In a research activity done byTheys et al [34] s-shape largest gap and some other routingheuristics were compared with the Lin-Kernighan-HelsgaunTSP heuristic Based on their results this TSP heuristic helpsto reduce travel distance up to the rate of 48 on average Inthe current research activity a GA TSP model has been usedin order to solve the picker routing problem

23 Batch Sequencing Problem The problem of batch seq-uencing can be defined as finding the orders of constructedbatches to be processed further Henn and Schmid [35]used metaheuristics to solve their proposed model of orderbatching and sequencing to minimize the ordersrsquo tardinessIterated Local Search and Attribute-Based Hill Climber arethe two employed metaheuristics in their model Accordingto different test problems the efficiency of their proposedmethods was illustrated Provided solutions can be improvedby 46 on average in comparison with the ones obtainedby standard constructive heuristics such as an application ofthe Earliest Due Date rule They conducted their proposedmethod for a maximum of 80 items

Other sections of this paper are as follows First theproposed model is described in Section 3 which covers allthe aspects that are involved with it After that Section 4presents the research methodology of this paper This isfollowed by Section 5 which encompasses the numericalexamples results and discussion of this research activity

4 The Scientific World Journal

Finally Section 6 gives an end to the paper with a briefconclusion

3 Model Description

The assumptions of this research are described as follows

(1) A manual picker to part system in a parallel aislewarehouse is considered in this research

(2) There is only one location for storing each type of itemand vice versa

(3) Multiple picker devices are not allowedTherefore themaximum number of pickers is only one

(4) The volume for each order should not exceed thecapacity of the picker

(5) Batch size does not exceed the capacity of picker(6) The storage size in each location is identical(7) All customer orders are known in advance

31 Weighted Association Rule Mining Binary Integer Cluster-ing Model In this section an association based clusteringmodel developed by Chen and Wu [23] was modified inorder to create the batch structure regarding ordersrsquo duedate First the association between customersrsquo demand willbe calculated based on the similarity of items in orderswith respect to their due date using MINWAL Ordersrsquodue dates are considered as the weight of association rulemining Although batching orders with higher associationsgenerated by similarity between their items can decrease thedistance travelled by order pickers not considering the duedates would cause the orders to be batched inappropriatelywith different due dates Some problems may occur by theaforementioned issue such as shortage or excess inventorycosts In order to solve this problem the order due date hasbeen considered as a weight of each order for calculatingthe support between orders Relationship maximization ofcustomer orders based on similar items within each orderand due dates of each order can be done with the weightedassociation rules Travel distance would be reduced by takinginto account batching of orders with higher relationshipsOther than that higher associations can batch orders withsimilar due dates in order to minimize tardiness This isfollowed by utilizing a binary integer programming modelintegrated with a clustering algorithm which assembles theorders into their respective batches

311 Weighted Association Mining for Determining CustomerOrder Relationships Association rule mining can be definedas a data mining method which tries to distinguish interrela-tions of variables where large databases exist [36] Some rulesare involved with the association rule model where there isan association between some set of items with another set ofitems [37] In order to calculate rule interestingness supportand confidence can be considered as the twomost appropriatemeasures in association ruleminingThe support for an item-set is the percentage of transactions that contain the itemset

Table 1 Order-item data

Order Items Due date1198741

A(3) B(4) 10

1198742

A(2) C(3) 30

1198743

A(1) B(3) 5

1198744

A(2) B(4) C(2) 50

Table 2 Item-order data

Item OrderA 119874

1 1198742 1198743 1198744

B 1198741 1198743 1198744

C 1198742 1198744

in the database A probability shows how frequently the rulehead occurs among all the groups containing the rule bodywhich can be defined as the confidence of an association rule(body rArr head) [38 39]

Based on the definition by Agrawal et al [36] theassociation rule mining problem is addressed as followsitems are described as 119868 = 119894

1 1198942 119894

119899 which is a set

of n binary attributes The database is represented as 119863 =

1199051 1199052 119905

119898 which is a set of transactions An exclusive

transaction 119868119863 belongs to each transaction in119863 together witha subset of the items in 119868 A rule is defined as an implicationof the form 119883 rArr 119884 where 119883119884 sube 119868 and 119883⋂119884 = 120601 Theassociation rule 119883 rArr 119884 [support = 119904 confidence = 119888]holds in the item-order data 119863 with confidence 119888 if 119888 ofOrder Identifiers in 119863 contain orderset 119883 and also containsorderset 119884 The rule 119883 rArr 119884 has support 119904 if 119904 of the item-order 119863 contains both 119883 and 119884 For instance upon batchingthe orders by taking into account the customer order patternsassociation rule mining involves finding out the amountof support between demands of customers from the orderdatabase In order batching Table 1 which shows the order-item data is transposed to the item-order data form shownin Table 2 since the order associations are demanded and theproduct item associations is not necessary In this exampleassociation rule mining can perform its job to define theassociations between customer orders that can be perceivedfrom information presented in Table 2

In the area of order batching association rule miningwas first used by Chen and Wu [23] In their researchpaper association rule mining was used to calculate thecorrelations between customersrsquo demands in terms of sup-port They pointed out that orders with similar items shouldbe consolidated together in order to minimize the totaltravel distance traversed by the picker An Apriori algorithmhas been used in their proposed method to calculate theassociation between orders It should be highlighted that theyonly considered the relationships between orders in termsof similarities between items However in batch processingeach order has a due date which can be considered as theweight but this weight is not considered by any other existingmethod in the literature

The classical Apriori algorithm [40] extracts binary asso-ciation rules based on the downward closure property which

The Scientific World Journal 5

Table 3 Weight of each order

Order Weight1198741

05827

1198742

01979

1198743

07634

1198744

00672

proves that subsets of a frequent itemset are also frequent [41]However the Apriori algorithm cannot be applied because itcannot handle the weighted case Therefore in this researchactivity theMINWAL algorithm is applied in order to extractthe weighted supports among the pair of orders to fulfill thetask of considering both the support and the weights factors

In order to use MINWAL [42] for mining weighted asso-ciation rules the weight should be in the range of the 0 to1 interval Based on the aforementioned fact larger weightsshould be assigned to orders with smaller due dates in acomparison with other orders that have lots of time to bedelivered in order to minimize the average tardiness of allorders Therefore in this study (1) has been used to calculatethe appropriate weights for each order as following

Weight (119874119894) = 119890(minus01lowast120573lowast119889119874119894

Maximume Due Date) (1)

where 119874119894denotes an order and 119889

119874119894refers to the due date

of order 119874119894 120573 is a constant value that is used to normalize

the due dates which is considered as 30 (30 minute) in thisstudy Also the maximum due date of the orders was usedin order to provide an appropriate interval for the weightsFor example the weight of each order is tabulated in Table 3which is calculated by (1)

Given a set of orders 119874 = 1199001 1199002 119900

119898 a weight 119908

119895for

each order 119900119895 within the values of 0 to 1 is assigned by (1) via

its due date where 119895 = 1 2 119898 shows the importance ofit The weighted support for the weighted association ordersof 1198741and 119874

2as 1198741rArr 1198742is defined if the weighted support

of such a pair of orders is no less than theminimumweightedsupport (minsup) threshold In this study the minimumsupport threshold is considered as 0 in order to identify all theassociations among orders Equation (2) shows the support ofan association for any pair of orders as following

(sum sum

119908119895isin(1198741cup1198742)

119908119895) lowast Support (119874

1cup 1198742) (2)

Algorithm 1 shows the MINWAL steps in order to mineweighted association rules This algorithm consists of 6subroutines such as Search Counting Join Prune Checkingand Rules The search subroutine provides the size of thelargest itemsets in database Counting is responsible forcounting every 1-itemset inside the database The Join step isresponsible for generating the 119896-itemsets (119862119896) from (119896 minus 1)-itemsets (119862119896 minus 1) The Prune step removes inappropriateitemsets which do not exist The Checking step updatesthe count of 119896-itemsets inside the transaction and prunesitemsets which do not meet the minimum support thresholdIn the Rules step every association rule will be extractedTable 4 shows the notations of Algorithm 1

(1) Main Algorithm (wminsup mincon 119891119863 119908)(2) size = Search (119863)(3) 119871 = 0(4) for (119894 = 1 119894 le size 119894++)(5) 119862

119894= 119871119894= 0

(6) for each transaction do(7) (SC 119862

1 size) =Counting (119863 119908)

(8) 119896 = 1(9) while (|119862119896| ge 119896)(10) 119896++(11) 119862

119896= Join (119862

119896minus1)

(12) 119862119896=Prune (119862

119896)

(13) (119862119896 119871119896) =Checking (119862

119896119863)

(14) 119871 = 119871 cup 119871119896

(15)Rules (SC 119871)(16) ends

Algorithm 1 MINWAL algorithm [42]

Table 4 Notations of Algorithm 1

119863 The database119908 The set of item weights119871119896

Set of large 119896-itemsets

119862119896

Set of 119896-itemsets which may be 119896-subsets of large119895-itemsets for 119895 ge 119896

SC (119883) No of transactions containing itemset119883wminsup Weighted support thresholdminconf Confidence thresholdSize Maximum possible large weighted itemsets

312 Association Based Binary Integer Programming forOrders Batching In order to create the batch structure abinary integer clustering model which has been developed byChen andWu [23] was appliedThey developed a binary inte-ger programming model for maximizing similarity betweenorders in terms of support in order to minimize the totaltraveling distance In this research activity support betweenorders was calculated based on weighted association miningusing MINWAL which was described in Section 311 Thebinary integer model for order batching is described asfollows

Parameters

119878119894119895Support between order 119894 and 119895 determined by MIN-WAL

119862V Capacity of picking vehicle

119881119894Weight of order 119894

119870 Number of batches

1198831198941198951 if order 119894 is assigned to batch 119895

0 otherwise

6 The Scientific World Journal

1198841198941 if order 119895 is chosen as a batch median0 otherwise

Maximize119873

sum

119894=1

119873

sum

119895=1

119878119894119895lowast 119883119894119895 (3)

Subject to119873

sum

119895=1

119883119894119895= 1 for 119894 119895 = 1 2 119873

(4)

119883119894119895le 119884119895

for 119894 119895 = 1 2 119873 (5)

119873

sum

119895=1

119884119895= 119870 (6)

119873

sum

119894=1

119881119894119883119894119895le 119862V for 119895 = 1 2 119873 (7)

119883119894119895= 0 1 for 119894 119895 = 1 2 119873 (8)

119884119895= 0 1 for 119895 = 1 2 119873 (9)

In the current model the objective function (3) maxi-mizes the sum of support from all orders to their relevantbatch medians as 119870 orders are selected as batch mediansThis model will maximize the support between orders in thebatches Membership of each order in just one batch wouldbe guaranteed by constraint (4) The number of batches with119870 is limited using constraints (5) and (6) The total quantityin a tour is also represented as a limitation in constraint (7) Itcannot exceed the capacity of order picker 119862V For the orderbatchingmodel to have a binary solutionwould be ensured byconstraints (8) and (9) For calculating the initial numbers ofbatches (10) will be used where sum119881

119894represents total volume

of all orders as following

119870initial = lceilsum119881119894

119862V

rceil (10)

32 Picker Routing

321 Genetic Algorithms In this research GAs which wereintroduced by Holland [43] have been used in order tosolve the picker routing and batch sequencing problem Thebiological evolution process is used as a base fact in a GA forproblem solving to a great extentThemain steps of a GA canbe described as follows [44 45]

(i) populating a series of solutions(ii) solutions assessment(iii) picking the most appropriate solutions(iv) generating new solutions based on some genetic

operation

In step (iii) selection forms the main part of a GA Thebad solutions are removed by selection and the good onesremain Random generation over the feasible or infeasible

solution space of a problem is used as a basis for creatingthe population Each solution is evaluated using the fitnessfunction Selection would be done according to certain selec-tions of fitness together with the probability that is assignedto it Generating a fresh population and a better solutionare involved with recombination of the individual solutionCrossover and mutation are the most used GA operators forgenerating new offspring from the old generation of parentsParameters of the GA are originated from the probabilitiesof mutation and crossover Exploration of the sample spaceis diversified by these operators In this research the GA hasbeenutilized in two steps First theGAhas beenused in orderto solve the picker routing problem Consider that in thewarehouse a set of items should be picked by a picker in onetour Therefore the picker routing problem in the warehousecould be categorized as a TSP The goal in the picker routingproblem as a TSP in the warehouse is to find an optimal pathfor minimizing the travel distance traversed by the pickingmachine where each node is visited at least once In thisresearch for a given number of itemset in a batch GA TSPhas been used to find the minimum travel distance whichis traversed by the picking machine Second a GA has beenutilized in order to sequence all batches namely GA sequencingwhichwould be further processedThe batches are sequencedbased on a fitness function which minimizes the tardiness ofall ordersThedetailed procedures ofGA TSP andGA sequencingare illustrated in Sections 323 and 324

322 Warehouse Layout Thewarehouse discussed through-out this research is assumed to have a 3D multi-parallel-aisle warehouse layout The picker will start the tour fromthe depot which is located in front of the leftmost aisle andreturns back to it A picker leaves the depot After that ittravels to a certain buffer spot where it picks the requestedorder(s) and puts the order(s) into a certain picker Then itcontinue on its way to another spot These procedures wouldbe repeated again and again so that all the requested ordersare picked Then it is time for the picker to return to thedepot

The graphic layout of the warehouse zone involved in thisresearch activity is pictorially displayed in Figure 1 It showsthe 119909 and 119910 dimensions of the 3D warehouse layout Eachitem in the warehouse has a specific location For examplethe location of item 119894 can be addressed by 119871

119894= (119909119894 119910119894 119911119894)

Consider two locations as 119871119886= (119909119886 119910119886 119911119886) and 119871

119887= (119909119887

119910119887 119911119887) If two locations are located in the same sub-aisles

of the same block the distance between these two itemslocations (119863

119886119887) will be calculated using

119863119886119887

=1003816100381610038161003816119909119886 minus 119909119887

1003816100381610038161003816 +1003816100381610038161003816119910119886 minus 119910119887

1003816100381610038161003816 +1003816100381610038161003816119911119886 minus 119911119887

1003816100381610038161003816 (11)

Equation (11) is not applicable if the two storage locationsbelong to different subaisles that are from the same block [34]In order to solve this problem distances of all possible pathsbetween two locations have been calculated and the shortestone was considered

323 GA TSP for the Picker Routing Problem In order todetermine the optimal route of a picker related to a given

The Scientific World Journal 7

Lb=(xbybzb)

La=(xayaza)

Li=(xiyizi)

Depot

Figure 1 Warehouse layout

1 2 3 4 5 6 7 8 9L0 L1 L8 L11 L9

L13 L12 L10 L0

Figure 2 A feasible chromosome encoding in GA TSP

set of items in a batch a GA has been used to solve theTSP problem to minimize travelling time In a GA eachsolution is considered as a chromosome The steps of GA TSPare described as follows

(1) for each batch establishing the structure of a chromo-somes

(2) generate initial feasible chromosomes(3) evaluate each chromosome based on the GA TSP fit-

ness function(4) apply the Selection operation(5) apply the Crossover operation(6) apply the Mutation operation(7) evaluate each offspring based on the fitness function(8) go to step 4 if the termination criterion is not satis-

fied otherwise terminate and return the best solutionas the optimal path

At the first step of GA TSP the structure of the chro-mosome should be established In the GA TSP the locationof an item that should be visited is presented by the valueof a gene Accordingly the sequence of visiting the itemlocation is denoted by the order of a gene in a chromosomeTherefore the total number of items locations in a specificbatch is considered equal to the length of a chromosome Forexample if the length of a chromosome is nine it means nineplaces should be visited by the picker as shown in Figure 2If 119871119894represents the item location in the warehouse 119894 = 1

119899 it shows that the picker leaves the depot (1198710denotes

the location of a depot) goes to 1198711 1198718 11987111 1198719 11987113 11987112 11987110

and then comes back to the depotIn the second step the initial number of feasible chro-

mosomes should be generated randomly for each batch

The number of initial chromosomes could be different basedon the complexity of the problem It should be consideredthat the first and the last gene of each chromosome should beassigned individually to the location of a depot In step threeeach chromosome will be evaluated based on the GA TSPfitness function which is formulated as follows

fitness = Min119899minus1

sum

119895=1

119863119895119895+1

(12)

where 119863119895119895+1

denotes the distance between two subsequentgenes in a chromosome 119895 = 1 119899 minus 1 represents the orderof the genes and 119899 is number of genes

Subsequently step four is involved with the selectionoperation In each iteration the selection operator usesfitness values to select the parents of the next generationIn this research the roulette wheel selection policy hasbeen used to guarantee that the most appropriate pairs ofchromosomes have been selected to generate offspring Itmeans that a higher probability of being selected to createa new population will be assigned to the chromosomes withhigher fitness values

Steps five and six are involved with conducting thecrossover and mutation operation The construction of theoffspring is realized by the crossover and the diversity of theindividuals is maintained by the mutation operator Blocksof genes between chromosomes are traded by means of thecrossover operation inwhich the exploitation of a particularlyprofitable portion of the parameter space is allowed to theblock Here permutation order based crossover (POP) as avariation of the distinguished order crossover has been usedAs shown in Figure 3 the original idea is about choosing twoparents and a point for cutting The first portion of offspring2 contains the first portion of parent 1 up to the cut pointTo form the second portion of offspring 2 consider all genesin parent 2 except those genes that already exist in the firstportion of the offspring 2

As shown in Figure 4 two positions are randomlyselected in a chromosome and exchanged with each othernamely the order based mutation approach SWAP isadopted for solving GA TSP Step six is followed by step sevenwhich includes the evaluation of each offspring that has beengenerated in the previous step based on the GA TSP fitnessfunction In step eight termination criteria will be checkedIf the termination condition is met then the process will beterminated After that the best chromosome will be selectedas the optimal solution Otherwise the selection process is tobe conducted again to generate a new population

324 GA for Sequencing the Batches Sequencing the batchesshould be conducted in order to minimize the averagetardiness of all orders The problem of batch sequencing canbe defined as finding the orders of batches which should beprocessed further For this purpose the GA sequencing methodhas been utilized In this research all of the batches whichwere constructed using binary integer programming andGA TSP will be sequenced by using the GA The populationgeneration selectionmutation and stopping threshold in theGA sequencing are identical to those in the GA TSP However

8 The Scientific World Journal

Parent 1

Parent 2

Offspring 2

Offspring 1

L1 L8 L9

L1 L1L8 L8L9

L11 L13 L12 L10

L1 L8 L9 L11L13 L12L10

L13 L9 L11 L11 L13 L12

L12

L10

L10L1L8L12

L13 L9 L11 L10

L1 L8 L12L13 L9 L11 L10

Figure 3 Crossover operation

Parent L1

L1

L8

L8

L9

L9

L11

L11

L13

L13

L12

L12

L10

L10Offspring

Figure 4 Mutation operation

1 2 3 4 5b1 b4 b3 b5 b2

Figure 5 Chromosome structure

the fitness function and chromosome structure are differentthan that of GA TSP In contrast with GA TSP chromosomesstructure batch number is presented by the value of a geneAccordingly the sequence of batch is denoted by the orderof a gene in a chromosome For instance Figure 5 shows thechromosome structure If 119887

119894isin 119861 denotes a set of batches

Figure 5 shows that 1198871 1198874 1198873 1198875 1198872should be processed

respectively Therefore the length of a chromosome is equalto the number of the batches The GA sequencing fitness func-tion for chromosome 119910 is described in

fitness119910=

1

119873

119899

sum

119896=1

sum

119894isin119887119896

119862119879 (119887119896) minus 119863119906 (119900

119894) (13)

119862119879 (119887119896) = 119862119879 (119887

119896minus1) + Pt (119887119896) (14)

Pt (119887119896) =Opt (119887

119896)

119881119878 (15)

where119873 denotes for the total number of orders 119896 = 1 119899

represents the batch sequence in which 119899 is the total numberof batches 119900

119894stands for the order 119894 in which 119894 isin 119887

119896 119887119896

denotes the batch in 119896th position of a sequence for specificchromosome 119862119879(119887

119896) stands for the completion time of a

batch in the 119896th position of a sequence 119863119906(119900119894) is the symbol

for due date of order 119894which belongs to 119887119896 Pt(119887119896) is the process

time of 119887119896 Opt(119887

119896) is the optimal path solution for 119887

119896 Finally

119881119878 is the moving speed of the picker

4 Research Methodology

In this research the proposed methodology consists of fivesteps These steps are described as follows and shown pic-torially in Figure 6

Step 1 It is involved with utilizing theMINWAL algorithm inorder to determine the associations between orders in termsof support considering their due dates In this step an orderdue date will be considered as the weight of an order It meansorders that are similar in items and due date should havemoresupport for being batched together

Step 2 It encompasses determining the initial number ofbatches In this step the initial number of batches will becalculated by (10)

Step 3 It includes the problem of modeling and solving firstthe binary integer programming model for order clusteringand batching will be modeled Next this model will be solvedin order to determine the structures of batches

Step 4 It is about the picker routing problem The optimalpath for a picker within each batch will be determined bysolving a TSP using GA

Step 5 It is about sequencing the batches based on the fitnessfunction which minimizes the average tardiness of all ordersIn this step GA algorithmwill be utilized in order to solve thebatch sequencing problem regarding the fitness function

5 Computational Experiments and Results

51 Test Problems and Parameter Setting In this sectionfive numerical test problems with different settings andparameters have been carried out in order to illustrate theproficiency of the proposed method with respect to averagetardiness of all orders Table 5 summarizes the descriptionof these test problems This table includes information withregard to number of items number of orders capacity of thepicker total weight of all the items in an order andminimumnumber of batches

In these test problems the quantity of items quantityof orders capacity of the picker device (119862V) and amount ofeach item are predetermined Each order contains a certain

The Scientific World Journal 9

Start

Input order informationwarehouse setting picker

setting

Compute the supportbetween customersrsquo

orders using MINWAL

Calculating initialnumber of batches (K) to

assign the orders

Solving the binaryinteger programmingclustering model for

constructing the batches

Feasiblesolution

Yes

No

Solving picker routing

Solving batch sequencingproblem using

End

K = K + 1

problem using GA TSP

GA sequencing

Figure 6 Methodology framework

quantity of items and also each item has a certain quantityThese quantities are produced randomly The distributionfunction type for the number of items in each order is Normal[5 10] Also the distribution function type for each itemquantity in an order is uniform [1 10] The picker movingspeed is set as 119881119878 = 2 (ms) The warehouse has 3000 itemlocation The GA control parameters for solving the TSP andsequencing problem are tabulated in Table 6

All of the proposed approaches are coded usingMATLABversion 790 and Java 15 programming language Also theIBM ILOG CPLEX optimization studio version 124 softwarewas utilized jointly with MATLAB in order to facilitate theprocess of solving the binary integer programming problemExamples are run by means of a computer featured by

4 Gigabytes random access memory (RAM) and a 220Gigahertz Intel processor (T6600)

52 Computational Results and Discussion In order to revealthe competency of the proposed solution approach it wascompared with the other existing methods in the literatureOne of the latest published research activities in the literaturewhich deals with minimization of tardiness has been con-ducted by Henn and schmid [35] They utilized the EarliestDueDate (EDD) approach as their constructive algorithm forcomparing with their proposed approaches They proposedtwo heuristics Iterated Local Search andAttribute-BasedHillClimber are the two employed metaheuristics in their modelAccording to different test problems the efficiency of their

10 The Scientific World Journal

Table 5 Test problems description

Example 1 Example 2 Example 3 Example 4 Example 5Number of orders 40 60 80 100 150

Number of items 80 120 160 200 200

Capacity of picker 200 200 300 400 550

Total weight 2212 3668 4393 6110 8369

Minimum number of batch 11 19 15 16 16

Due dates range 60 90 120 180 240

Table 6 GA control parameters

GA TSP GA sequencing

Population size 500 200

Cross over strategyPermutation orderbased crossover

(POP)

Permutation orderbased crossover

(POP)Cross over probability 08 08

Mutation strategy SWAP SWAPMutation probability 02 02

Iteration number 10000 10000

Parent selection method Roulette wheel Roulette wheel

proposed methods was illustrated Their obtained resultsfrom their proposed approaches showed a 46 improvementon tardiness compared to EDD They used an s-shape andlargest gap routing strategy for their proposed method Inorder to make a reasonable comparison with their proposedmethod EDD was integrated with the s-shape routing strat-egy and GA sequencing Then based on improvement on EDDwe can compare our proposed method with their proposedmethod In the EDD approach due dates play an importantrole in sorting all orders Based on an ascending sequenceassigning the orders to batches ensures that no capacityviolation can happen for the picking device

For the warehouse condition which is described inSection 323 our proposed approach EDD integrated withGA TSP and GA sequencing and EDD integrated with s-shapealgorithm and GA sequencing have been evaluated in termsof average tardiness for all orders (119879) which is calculatedbased on (13) number of batches (119870) total process time(Pt) and average process time for each batch (119860Pt) All ofthe GA setting for the TSP and sequencing problems areconsidered the same for all methods Also the settings whichhave been mentioned in Section 51 were considered for all ofthe compared methods

The results for all of the test problems for the threeapproaches are tabulated in Tables 7 8 and 9 As it isillustrated in Table 10 EDD integrated with GA TSP andGA sequencing has a 3176 improvement on average in termsof average tardiness of all orders than EDD integrated withthe s-shape algorithm and GA sequencing It can be perceivedthat GA TSP performs better than s-shape Also the ATGHmethod has a 6811 improvement in average tardinessoff all orders than EDD integrated with s-shape algorithm

and GA sequencing Consequently the ATGH method achievesmuch improvement against Henn and Schmidrsquos [35] pro-posedmethod According to Tables 7 and 8 it is also worth tomention thatGA TSP performs better than the s-shape routingstrategy in terms of process time The improvements whichhave been achieved in this research could be justified basedon the following reasons

The problem of order batching deals with several sig-nificant drivers in which the number of batches can beconsidered as a key factor [23] As a justification to thismatter it is worth to mention that total travel distance maynot decrease directly due to a decline in the number ofbatches But a decline in the number of batches will increasebatch sizes which may lead to an increase in the distancetravelled by order picker However an advanced exploitationof picker capacity to load efficiently may reduce the numberof constructed batches together reducing the total traveldistance Improving the total travel distance traversed by thepicker can minimize the average tardiness of all orders Interms of number of the batches according to Tables 7 8 and9 it can be perceived that our proposed method assigns theorders to the batches optimally The ATGH method assignsthe orders to batches based on their item similarity andtheir due dates and it uses the picker capacity efficiently Butthe EDD method allocates the orders to batches based onan ascending sequence of their due date and it should beensured that no capacity violation will happen for the pickingdevice which may lead to an inefficient order batching Tohighlight the problem an example can be made based on theresults tabulated in Tables 7 and 8 The number of batchesfor example 1 in our method is 11 in comparison with 13obtained fromEDDmethod 13 batches can be justified basedon this scenario that two subsequent orders say the firstorder is 100 kg and the other is 150 kg cannot be batchedtogether due to violated batch capacity which is set to be200 kg The number of batches would increase due to theaforementioned issue which affects the total process time tohave an increasing manner This situation will rarely happenin the ATGH method because of the weighted associationbased binary integer programming utilizedwhich batches theorders in order to maximize the association between themwhich is calculated based on the item similarity of ordersand the weight of their corresponding due dates The ATGHmethod uses the picker capacity efficiently

Obviously one of the factors that can influence thecompletion time and the average tardiness of all orders is theprocessing time of the orders Reducing the processing time

The Scientific World Journal 11

Table 7 Computational results of ATGH

Example 1 Example 2 Example 3 Example 4 Example 5119879 56204 29948 90202 163 58898

Pt 605833 1217833 105833 121033 1339

119870 11 19 15 16 16

119860Pt 55076 6409 7055 7564 8368

Table 8 Computational results of EDD integrated with GA TSP and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 124550 42976 18222 5981 1547

Pt 667333 133516 116083 13185 141916

119870 13 21 16 17 16

119860Pt 51333 6357 7255 7775 8869

Table 9 Computational results of EDD integrated with s-shape algorithm and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 1651 62677 2825 13365 17593

Pt 75266 16636 13323 14333 14926

119870 13 21 16 17 16

119860Pt 5789 892 833 8431 932

Table 10 Comparison results

EDD + s-shape + GA sequencing EDD + GA TSP + GA sequencing ATGHTardiness (min) Tardiness (min) Improvement () Tardiness (min) Improvement ()

(1) 1651 124550 2456 56204 6596(2) 62677 42976 3143 29948 5222(3) 2825 18222 3550 90202 6807(4) 13365 5981 5525 1627 8783(5) 17593 1547 1206 58898 6652Average improvement 3176 6811

and travel time for constructing a batch is an outstanding factin reducing the costs and minimizing delays of the customerresponses In our proposed model orders with similar itemsare consolidated together Therefore batching orders withmore similarity can reduce the distance traversed by thepicker facility Also the due date of each order is definedas a weight parameter in calculating the association betweenorders in terms of support which was done for consideringthe due dates in our model Thereupon the model considersdue date and the process time in an integrated manner whichsimultaneously affected the average tardiness of all ordersThis spoken issue is different in the EDD approach in whichthe ordersrsquo due dates are just considered Example 5 canbe considered as evidence of this matter in which the totaland average process time for ATGH is lower than EDDintegrated with GA TSP and GA sequencing with respect to thefact that number of constructed batches for both approachesare happened to be equal after obtaining the results

6 Conclusion

One of the most important concerns of warehouse managersis involvedwith finding the best optimal and cost efficientwayto pick orders placed by customers in order to be known as aresponsible company with satisfied customers in terms of ontime delivery In the current research activity order batchingpicker routing and batch sequencing problems of warehouseprocesses were addressed which are to be solved jointly asthey are mixed up with many manufacturing industriesMathematical solutions of these NP-hard problems might beavailable for problems that are small in nature But thesekinds of solutions may not be applicable to be consideredby warehouse managers Hence we presented novel solutionapproach namely ATGH to solve this issue in accordancewith minimizing the average tardiness of all customersrsquoorders which can be considered as a major objective ofthis research activity A weighted association rule mining

12 The Scientific World Journal

algorithm namely MINWAL was utilized to determine thecustomersrsquo orders association regarding similarity betweenitems and ordersrsquo due dates Actually in our proposedmethodthe due date of each order is defined as a weight parameterin calculating the association between orders So orders withsimilar items and due dates have more chance to be in asame batch This issue is not considered in most of theresearch activities in the literature related to our work andconsidering this issue in the order batching problem is themain contribution of our research activity The previous stepwas followed by a binary integer clustering model in orderto solve the order batching problem which maximizes thesimilarity between orders in terms of support After that aGA was applied to solve a TSP in order to find the optimalrouting path

Finally a batch sequencing problem was addressed inconjunction with a GA

The other research objective of this research addresseswhether the proposed ATGH algorithm can further improvethe solution quality for a multiple parallel aisle warehousewith manual picking system Hence comparisons were madeconsideringATGHEDD integratedGA TSP andGA sequencingand EDD integrated s-shape and GA sequencing algorithms Asdemonstrated by the obtained results ATGH comes out tobe the most striking solution approach in terms of tardinessIn spite of the fact that considering all customer orders asdiscussed in this paper would lead to have a static case beinginvolved with dynamic cases in which different points of timeare assigned to customerrsquos orders arrival would be beneficialSo there might be some potential room of research to extendour discussed approach with respect to dynamic orders

Conflict of Interests

The IBM ILOG CPLEX optimization studio version 124software is not used for financial gain and there is no conflictof interests with any party as it is only a student version

References

[1] ELAAT Kearney Excellence in Logistics ELA Brussels Bel-gium 2004

[2] Y-C Ho T-S Su and Z-B Shi ldquoOrder-batching methods foran order-picking warehouse with two cross aislesrdquo Computersamp Industrial Engineering vol 55 no 2 pp 321ndash347 2008

[3] C-M Hsu K-Y Chen and M-C Chen ldquoBatching ordersin warehouses by minimizing travel distance with geneticalgorithmsrdquo Computers in Industry vol 56 no 2 pp 169ndash1782005

[4] E A Elsayed M-K Lee S Kim and E Scherer ldquoSequencingand batching procedures forminimizing earliness and tardinesspenalty of order retrievalsrdquoThe International Journal of Produc-tion Research vol 31 no 3 pp 727ndash738 1993

[5] S Henn S Koch and W Gerhard Order Batching in OrderPicking Warehouses A Survey of Solution Approaches SpringerLondon UK 2012

[6] J Drury Towards More Efficient Order Picking IMM Mono-graph no 1 The Institute of Materials Management CranfieldUK 1988

[7] M Goetschalckx and J Ashayeri ldquoClassification and design oforder picking systemsrdquo Logistics World pp 99ndash106 1989

[8] J A Tompkins J A White Y A Bozer E H Frazelle and J MA Tanchoco Facilities Planning John Wiley amp Sons HobokenNJ USA 2003

[9] S Henn and G Wascher ldquoTabu search heuristics for the orderbatching problem in manual order picking systemsrdquo EuropeanJournal of Operational Research vol 222 no 3 pp 484ndash4942012

[10] S Henn ldquoAlgorithms for on-line order batching in an orderpicking warehouserdquo Computers and Operations Research vol39 no 11 pp 2549ndash2563 2012

[11] R de Koster ldquoHow to assess a warehouse operation in a singletourrdquo Tech Rep RSM Erasmus University Rotterdam TheNetherlands 2004

[12] R de Koster T Le-Duc and K J Roodbergen ldquoDesign andcontrol of warehouse order picking a literature reviewrdquo Euro-pean Journal of Operational Research vol 182 no 2 pp 481ndash5012007

[13] R A Ruben and F R Jacobs ldquoBatch construction heuristics andstorage assignment strategies for walkride and pick systemsrdquoManagement Science vol 45 no 4 pp 575ndash596 1999

[14] E A Elsayed ldquoAlgorithms for optimal material handling inautomatic warehousing systemsrdquo International Journal of Pro-duction Research vol 19 no 5 pp 525ndash535 1981

[15] G Clarke and J W Wright ldquoScheduling of vehicles froma central depot to a number of delivery pointsrdquo OperationsResearch vol 12 no 4 pp 568ndash581 1964

[16] H D Vinod ldquoInteger programming and the theory of group-ingrdquo Journal of the American Statistical Association vol 64 no326 pp 506ndash519 1969

[17] R D Armstrong W D Cook and A L Saipe ldquoOptimalbatching in a semi-automated order picking systemrdquo Journal ofthe Operational Research Society vol 30 no 8 pp 711ndash720 1979

[18] H D Ratliff and A S Rosenthal ldquoOrder-picking in a rect-angular warehouse a solvable case of the traveling salesmanproblemrdquo Operations Research vol 31 no 3 pp 507ndash521 1983

[19] A Kusiak A Vannelli and R K Kumar ldquoClustering analysismodels and algorithmsrdquo Cybernetics vol 15 pp 139ndash154 1986

[20] D R Gibson and G P Sharp ldquoOrder batching proceduresrdquoEuropean Journal of Operational Research vol 58 no 1 pp 57ndash67 1992

[21] M B Rosenwein ldquoA comparison of heuristics for the problemofbatching orders for warehouse selectionrdquo International Journalof Production Research vol 34 no 3 pp 657ndash664 1996

[22] A J RMGademann J P van denBerg andHH van derHoffldquoAn order batching algorithm for wave picking in a parallel-aislewarehouserdquo IIE Transactions vol 33 no 5 pp 385ndash398 2001

[23] M-C Chen and H-P Wu ldquoAn association-based clusteringapproach to order batching considering customer demandpatternsrdquo Omega vol 33 no 4 pp 333ndash343 2005

[24] NGademann and S van deVelde ldquoOrder batching tominimizetotal travel time in a parallel-aisle warehouserdquo IIE Transactionsvol 37 no 1 pp 63ndash75 2005

[25] Y A Bozer and J W Kile ldquoOrder batching in walk-and-pick order picking systemsrdquo International Journal of ProductionResearch vol 46 no 7 pp 1887ndash1909 2008

[26] O Kulak Y Sahin and M E Taner ldquoJoint order batching andpicker routing in single and multiple-cross-aisle warehousesusing cluster-based tabu search algorithmsrdquo Flexible Servicesand Manufacturing Journal vol 24 no 1 pp 52ndash80 2012

The Scientific World Journal 13

[27] E A Elsayed and M-K Lee ldquoOrder processing in automatedstorageretrieval systems with due datesrdquo IIE Transactions vol28 no 7 pp 567ndash577 1996

[28] J Won and S Olafsson ldquoJoint order batching and order pickingin warehouse operationsrdquo International Journal of ProductionResearch vol 43 no 7 pp 1427ndash1442 2005

[29] C-Y Tsai J J H Liou and T-M Huang ldquoUsing a multiple-GA method to solve the batch picking problem consideringtravel distance and order due timerdquo International Journal of Pro-duction Research vol 46 no 22 pp 6533ndash6555 2008

[30] Y KaoM H Chen and Y T Huang ldquoA hybrid algorithm basedon ACO and PSO for capacitated vehicle routing problemsrdquoMathematical Problems in Engineering vol 2012 Article ID726564 17 pages 2012

[31] X Gan Y Wang S Li and B Niu ldquoVehicle routing problemwith time windows and simultaneous delivery and pick-up ser-vice based on MCPSOrdquoMathematical Problems in Engineeringvol 2012 Article ID 104279 11 pages 2012

[32] M B M de Koster E S van der Poort and M WoltersldquoEfficient orderbatching methods in warehousesrdquo InternationalJournal of ProductionResearch vol 37 no 7 pp 1479ndash1504 1999

[33] Y-C Ho and Y-Y Tseng ldquoA study on order-batching methodsof order-picking in a distribution centre with two cross-aislesrdquoInternational Journal of Production Research vol 44 no 17 pp3391ndash3417 2006

[34] C Theys O Braysy W Dullaert and B Raa ldquoUsing a TSPheuristic for routing order pickers in warehousesrdquo EuropeanJournal of Operational Research vol 200 no 3 pp 755ndash7632010

[35] S Henn and V Schmid Metaheuristics for Order Batching andSequencing in Manual Order Picking Systems Univ Fak furWirtschaftswiss 2011

[36] M Hahsler B Grun and K Hornik ldquoArulesmdasha computationalenvironment for mining association rules and frequent itemsetsrdquo Journal of Statistical Software vol 14 2005

[37] R Agrawal T Imielinski and A Swami ldquoMining associationrules between sets of items in large databasesrdquo ACM SIGMODRecord vol 22 no 2 pp 207ndash216 1993

[38] S Encheva Y Kondratenko M Z Solesvik and S TuminldquoDecision support systems in logisticsrdquo in Proceeding of Inter-national electronic conference on computer science pp 254ndash2562007

[39] M-C Chen ldquoConfiguration of cellular manufacturing systemsusing association rule inductionrdquo International Journal of Pro-duction Research vol 41 no 2 pp 381ndash395 2003

[40] R Srikant and R Agrawal ldquoMining generalized associationrulesrdquo Future Generation Computer Systems vol 13 no 2-3 pp161ndash180 1997

[41] J Wu and XM Li ldquoAn effective mining algorithm for weightedassociation rules in communication networksrdquo Journal of Com-puters vol 3 no 10 pp 20ndash27 2008

[42] C H Cai A W C Fu C H Cheng and W W KwongldquoMining association rules with weighted itemsrdquo in Proceedingsof the International Symposium on Database Engineering ampApplications pp 68ndash77 Cardiff UK 1998

[43] J HollandAdaptation in Natural and Artificial Systems Univer-sity of Michigan Press Ann Arbor Mich USA 1975

[44] A A Taleizadeh F Barzinpour and H-M Wee ldquoMeta-heuristic algorithms for solving a fuzzy single-period problemrdquoMathematical and Computer Modelling vol 54 no 5-6 pp1273ndash1285 2011

[45] H Yang J Yi J Zhao and Zh Dong ldquoExtreme learning ma-chine based genetic algorithm and its application in powersystem economic dispatchrdquo Neurocomputing vol 102 pp 154ndash162 2013

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4 The Scientific World Journal

Finally Section 6 gives an end to the paper with a briefconclusion

3 Model Description

The assumptions of this research are described as follows

(1) A manual picker to part system in a parallel aislewarehouse is considered in this research

(2) There is only one location for storing each type of itemand vice versa

(3) Multiple picker devices are not allowedTherefore themaximum number of pickers is only one

(4) The volume for each order should not exceed thecapacity of the picker

(5) Batch size does not exceed the capacity of picker(6) The storage size in each location is identical(7) All customer orders are known in advance

31 Weighted Association Rule Mining Binary Integer Cluster-ing Model In this section an association based clusteringmodel developed by Chen and Wu [23] was modified inorder to create the batch structure regarding ordersrsquo duedate First the association between customersrsquo demand willbe calculated based on the similarity of items in orderswith respect to their due date using MINWAL Ordersrsquodue dates are considered as the weight of association rulemining Although batching orders with higher associationsgenerated by similarity between their items can decrease thedistance travelled by order pickers not considering the duedates would cause the orders to be batched inappropriatelywith different due dates Some problems may occur by theaforementioned issue such as shortage or excess inventorycosts In order to solve this problem the order due date hasbeen considered as a weight of each order for calculatingthe support between orders Relationship maximization ofcustomer orders based on similar items within each orderand due dates of each order can be done with the weightedassociation rules Travel distance would be reduced by takinginto account batching of orders with higher relationshipsOther than that higher associations can batch orders withsimilar due dates in order to minimize tardiness This isfollowed by utilizing a binary integer programming modelintegrated with a clustering algorithm which assembles theorders into their respective batches

311 Weighted Association Mining for Determining CustomerOrder Relationships Association rule mining can be definedas a data mining method which tries to distinguish interrela-tions of variables where large databases exist [36] Some rulesare involved with the association rule model where there isan association between some set of items with another set ofitems [37] In order to calculate rule interestingness supportand confidence can be considered as the twomost appropriatemeasures in association ruleminingThe support for an item-set is the percentage of transactions that contain the itemset

Table 1 Order-item data

Order Items Due date1198741

A(3) B(4) 10

1198742

A(2) C(3) 30

1198743

A(1) B(3) 5

1198744

A(2) B(4) C(2) 50

Table 2 Item-order data

Item OrderA 119874

1 1198742 1198743 1198744

B 1198741 1198743 1198744

C 1198742 1198744

in the database A probability shows how frequently the rulehead occurs among all the groups containing the rule bodywhich can be defined as the confidence of an association rule(body rArr head) [38 39]

Based on the definition by Agrawal et al [36] theassociation rule mining problem is addressed as followsitems are described as 119868 = 119894

1 1198942 119894

119899 which is a set

of n binary attributes The database is represented as 119863 =

1199051 1199052 119905

119898 which is a set of transactions An exclusive

transaction 119868119863 belongs to each transaction in119863 together witha subset of the items in 119868 A rule is defined as an implicationof the form 119883 rArr 119884 where 119883119884 sube 119868 and 119883⋂119884 = 120601 Theassociation rule 119883 rArr 119884 [support = 119904 confidence = 119888]holds in the item-order data 119863 with confidence 119888 if 119888 ofOrder Identifiers in 119863 contain orderset 119883 and also containsorderset 119884 The rule 119883 rArr 119884 has support 119904 if 119904 of the item-order 119863 contains both 119883 and 119884 For instance upon batchingthe orders by taking into account the customer order patternsassociation rule mining involves finding out the amountof support between demands of customers from the orderdatabase In order batching Table 1 which shows the order-item data is transposed to the item-order data form shownin Table 2 since the order associations are demanded and theproduct item associations is not necessary In this exampleassociation rule mining can perform its job to define theassociations between customer orders that can be perceivedfrom information presented in Table 2

In the area of order batching association rule miningwas first used by Chen and Wu [23] In their researchpaper association rule mining was used to calculate thecorrelations between customersrsquo demands in terms of sup-port They pointed out that orders with similar items shouldbe consolidated together in order to minimize the totaltravel distance traversed by the picker An Apriori algorithmhas been used in their proposed method to calculate theassociation between orders It should be highlighted that theyonly considered the relationships between orders in termsof similarities between items However in batch processingeach order has a due date which can be considered as theweight but this weight is not considered by any other existingmethod in the literature

The classical Apriori algorithm [40] extracts binary asso-ciation rules based on the downward closure property which

The Scientific World Journal 5

Table 3 Weight of each order

Order Weight1198741

05827

1198742

01979

1198743

07634

1198744

00672

proves that subsets of a frequent itemset are also frequent [41]However the Apriori algorithm cannot be applied because itcannot handle the weighted case Therefore in this researchactivity theMINWAL algorithm is applied in order to extractthe weighted supports among the pair of orders to fulfill thetask of considering both the support and the weights factors

In order to use MINWAL [42] for mining weighted asso-ciation rules the weight should be in the range of the 0 to1 interval Based on the aforementioned fact larger weightsshould be assigned to orders with smaller due dates in acomparison with other orders that have lots of time to bedelivered in order to minimize the average tardiness of allorders Therefore in this study (1) has been used to calculatethe appropriate weights for each order as following

Weight (119874119894) = 119890(minus01lowast120573lowast119889119874119894

Maximume Due Date) (1)

where 119874119894denotes an order and 119889

119874119894refers to the due date

of order 119874119894 120573 is a constant value that is used to normalize

the due dates which is considered as 30 (30 minute) in thisstudy Also the maximum due date of the orders was usedin order to provide an appropriate interval for the weightsFor example the weight of each order is tabulated in Table 3which is calculated by (1)

Given a set of orders 119874 = 1199001 1199002 119900

119898 a weight 119908

119895for

each order 119900119895 within the values of 0 to 1 is assigned by (1) via

its due date where 119895 = 1 2 119898 shows the importance ofit The weighted support for the weighted association ordersof 1198741and 119874

2as 1198741rArr 1198742is defined if the weighted support

of such a pair of orders is no less than theminimumweightedsupport (minsup) threshold In this study the minimumsupport threshold is considered as 0 in order to identify all theassociations among orders Equation (2) shows the support ofan association for any pair of orders as following

(sum sum

119908119895isin(1198741cup1198742)

119908119895) lowast Support (119874

1cup 1198742) (2)

Algorithm 1 shows the MINWAL steps in order to mineweighted association rules This algorithm consists of 6subroutines such as Search Counting Join Prune Checkingand Rules The search subroutine provides the size of thelargest itemsets in database Counting is responsible forcounting every 1-itemset inside the database The Join step isresponsible for generating the 119896-itemsets (119862119896) from (119896 minus 1)-itemsets (119862119896 minus 1) The Prune step removes inappropriateitemsets which do not exist The Checking step updatesthe count of 119896-itemsets inside the transaction and prunesitemsets which do not meet the minimum support thresholdIn the Rules step every association rule will be extractedTable 4 shows the notations of Algorithm 1

(1) Main Algorithm (wminsup mincon 119891119863 119908)(2) size = Search (119863)(3) 119871 = 0(4) for (119894 = 1 119894 le size 119894++)(5) 119862

119894= 119871119894= 0

(6) for each transaction do(7) (SC 119862

1 size) =Counting (119863 119908)

(8) 119896 = 1(9) while (|119862119896| ge 119896)(10) 119896++(11) 119862

119896= Join (119862

119896minus1)

(12) 119862119896=Prune (119862

119896)

(13) (119862119896 119871119896) =Checking (119862

119896119863)

(14) 119871 = 119871 cup 119871119896

(15)Rules (SC 119871)(16) ends

Algorithm 1 MINWAL algorithm [42]

Table 4 Notations of Algorithm 1

119863 The database119908 The set of item weights119871119896

Set of large 119896-itemsets

119862119896

Set of 119896-itemsets which may be 119896-subsets of large119895-itemsets for 119895 ge 119896

SC (119883) No of transactions containing itemset119883wminsup Weighted support thresholdminconf Confidence thresholdSize Maximum possible large weighted itemsets

312 Association Based Binary Integer Programming forOrders Batching In order to create the batch structure abinary integer clustering model which has been developed byChen andWu [23] was appliedThey developed a binary inte-ger programming model for maximizing similarity betweenorders in terms of support in order to minimize the totaltraveling distance In this research activity support betweenorders was calculated based on weighted association miningusing MINWAL which was described in Section 311 Thebinary integer model for order batching is described asfollows

Parameters

119878119894119895Support between order 119894 and 119895 determined by MIN-WAL

119862V Capacity of picking vehicle

119881119894Weight of order 119894

119870 Number of batches

1198831198941198951 if order 119894 is assigned to batch 119895

0 otherwise

6 The Scientific World Journal

1198841198941 if order 119895 is chosen as a batch median0 otherwise

Maximize119873

sum

119894=1

119873

sum

119895=1

119878119894119895lowast 119883119894119895 (3)

Subject to119873

sum

119895=1

119883119894119895= 1 for 119894 119895 = 1 2 119873

(4)

119883119894119895le 119884119895

for 119894 119895 = 1 2 119873 (5)

119873

sum

119895=1

119884119895= 119870 (6)

119873

sum

119894=1

119881119894119883119894119895le 119862V for 119895 = 1 2 119873 (7)

119883119894119895= 0 1 for 119894 119895 = 1 2 119873 (8)

119884119895= 0 1 for 119895 = 1 2 119873 (9)

In the current model the objective function (3) maxi-mizes the sum of support from all orders to their relevantbatch medians as 119870 orders are selected as batch mediansThis model will maximize the support between orders in thebatches Membership of each order in just one batch wouldbe guaranteed by constraint (4) The number of batches with119870 is limited using constraints (5) and (6) The total quantityin a tour is also represented as a limitation in constraint (7) Itcannot exceed the capacity of order picker 119862V For the orderbatchingmodel to have a binary solutionwould be ensured byconstraints (8) and (9) For calculating the initial numbers ofbatches (10) will be used where sum119881

119894represents total volume

of all orders as following

119870initial = lceilsum119881119894

119862V

rceil (10)

32 Picker Routing

321 Genetic Algorithms In this research GAs which wereintroduced by Holland [43] have been used in order tosolve the picker routing and batch sequencing problem Thebiological evolution process is used as a base fact in a GA forproblem solving to a great extentThemain steps of a GA canbe described as follows [44 45]

(i) populating a series of solutions(ii) solutions assessment(iii) picking the most appropriate solutions(iv) generating new solutions based on some genetic

operation

In step (iii) selection forms the main part of a GA Thebad solutions are removed by selection and the good onesremain Random generation over the feasible or infeasible

solution space of a problem is used as a basis for creatingthe population Each solution is evaluated using the fitnessfunction Selection would be done according to certain selec-tions of fitness together with the probability that is assignedto it Generating a fresh population and a better solutionare involved with recombination of the individual solutionCrossover and mutation are the most used GA operators forgenerating new offspring from the old generation of parentsParameters of the GA are originated from the probabilitiesof mutation and crossover Exploration of the sample spaceis diversified by these operators In this research the GA hasbeenutilized in two steps First theGAhas beenused in orderto solve the picker routing problem Consider that in thewarehouse a set of items should be picked by a picker in onetour Therefore the picker routing problem in the warehousecould be categorized as a TSP The goal in the picker routingproblem as a TSP in the warehouse is to find an optimal pathfor minimizing the travel distance traversed by the pickingmachine where each node is visited at least once In thisresearch for a given number of itemset in a batch GA TSPhas been used to find the minimum travel distance whichis traversed by the picking machine Second a GA has beenutilized in order to sequence all batches namely GA sequencingwhichwould be further processedThe batches are sequencedbased on a fitness function which minimizes the tardiness ofall ordersThedetailed procedures ofGA TSP andGA sequencingare illustrated in Sections 323 and 324

322 Warehouse Layout Thewarehouse discussed through-out this research is assumed to have a 3D multi-parallel-aisle warehouse layout The picker will start the tour fromthe depot which is located in front of the leftmost aisle andreturns back to it A picker leaves the depot After that ittravels to a certain buffer spot where it picks the requestedorder(s) and puts the order(s) into a certain picker Then itcontinue on its way to another spot These procedures wouldbe repeated again and again so that all the requested ordersare picked Then it is time for the picker to return to thedepot

The graphic layout of the warehouse zone involved in thisresearch activity is pictorially displayed in Figure 1 It showsthe 119909 and 119910 dimensions of the 3D warehouse layout Eachitem in the warehouse has a specific location For examplethe location of item 119894 can be addressed by 119871

119894= (119909119894 119910119894 119911119894)

Consider two locations as 119871119886= (119909119886 119910119886 119911119886) and 119871

119887= (119909119887

119910119887 119911119887) If two locations are located in the same sub-aisles

of the same block the distance between these two itemslocations (119863

119886119887) will be calculated using

119863119886119887

=1003816100381610038161003816119909119886 minus 119909119887

1003816100381610038161003816 +1003816100381610038161003816119910119886 minus 119910119887

1003816100381610038161003816 +1003816100381610038161003816119911119886 minus 119911119887

1003816100381610038161003816 (11)

Equation (11) is not applicable if the two storage locationsbelong to different subaisles that are from the same block [34]In order to solve this problem distances of all possible pathsbetween two locations have been calculated and the shortestone was considered

323 GA TSP for the Picker Routing Problem In order todetermine the optimal route of a picker related to a given

The Scientific World Journal 7

Lb=(xbybzb)

La=(xayaza)

Li=(xiyizi)

Depot

Figure 1 Warehouse layout

1 2 3 4 5 6 7 8 9L0 L1 L8 L11 L9

L13 L12 L10 L0

Figure 2 A feasible chromosome encoding in GA TSP

set of items in a batch a GA has been used to solve theTSP problem to minimize travelling time In a GA eachsolution is considered as a chromosome The steps of GA TSPare described as follows

(1) for each batch establishing the structure of a chromo-somes

(2) generate initial feasible chromosomes(3) evaluate each chromosome based on the GA TSP fit-

ness function(4) apply the Selection operation(5) apply the Crossover operation(6) apply the Mutation operation(7) evaluate each offspring based on the fitness function(8) go to step 4 if the termination criterion is not satis-

fied otherwise terminate and return the best solutionas the optimal path

At the first step of GA TSP the structure of the chro-mosome should be established In the GA TSP the locationof an item that should be visited is presented by the valueof a gene Accordingly the sequence of visiting the itemlocation is denoted by the order of a gene in a chromosomeTherefore the total number of items locations in a specificbatch is considered equal to the length of a chromosome Forexample if the length of a chromosome is nine it means nineplaces should be visited by the picker as shown in Figure 2If 119871119894represents the item location in the warehouse 119894 = 1

119899 it shows that the picker leaves the depot (1198710denotes

the location of a depot) goes to 1198711 1198718 11987111 1198719 11987113 11987112 11987110

and then comes back to the depotIn the second step the initial number of feasible chro-

mosomes should be generated randomly for each batch

The number of initial chromosomes could be different basedon the complexity of the problem It should be consideredthat the first and the last gene of each chromosome should beassigned individually to the location of a depot In step threeeach chromosome will be evaluated based on the GA TSPfitness function which is formulated as follows

fitness = Min119899minus1

sum

119895=1

119863119895119895+1

(12)

where 119863119895119895+1

denotes the distance between two subsequentgenes in a chromosome 119895 = 1 119899 minus 1 represents the orderof the genes and 119899 is number of genes

Subsequently step four is involved with the selectionoperation In each iteration the selection operator usesfitness values to select the parents of the next generationIn this research the roulette wheel selection policy hasbeen used to guarantee that the most appropriate pairs ofchromosomes have been selected to generate offspring Itmeans that a higher probability of being selected to createa new population will be assigned to the chromosomes withhigher fitness values

Steps five and six are involved with conducting thecrossover and mutation operation The construction of theoffspring is realized by the crossover and the diversity of theindividuals is maintained by the mutation operator Blocksof genes between chromosomes are traded by means of thecrossover operation inwhich the exploitation of a particularlyprofitable portion of the parameter space is allowed to theblock Here permutation order based crossover (POP) as avariation of the distinguished order crossover has been usedAs shown in Figure 3 the original idea is about choosing twoparents and a point for cutting The first portion of offspring2 contains the first portion of parent 1 up to the cut pointTo form the second portion of offspring 2 consider all genesin parent 2 except those genes that already exist in the firstportion of the offspring 2

As shown in Figure 4 two positions are randomlyselected in a chromosome and exchanged with each othernamely the order based mutation approach SWAP isadopted for solving GA TSP Step six is followed by step sevenwhich includes the evaluation of each offspring that has beengenerated in the previous step based on the GA TSP fitnessfunction In step eight termination criteria will be checkedIf the termination condition is met then the process will beterminated After that the best chromosome will be selectedas the optimal solution Otherwise the selection process is tobe conducted again to generate a new population

324 GA for Sequencing the Batches Sequencing the batchesshould be conducted in order to minimize the averagetardiness of all orders The problem of batch sequencing canbe defined as finding the orders of batches which should beprocessed further For this purpose the GA sequencing methodhas been utilized In this research all of the batches whichwere constructed using binary integer programming andGA TSP will be sequenced by using the GA The populationgeneration selectionmutation and stopping threshold in theGA sequencing are identical to those in the GA TSP However

8 The Scientific World Journal

Parent 1

Parent 2

Offspring 2

Offspring 1

L1 L8 L9

L1 L1L8 L8L9

L11 L13 L12 L10

L1 L8 L9 L11L13 L12L10

L13 L9 L11 L11 L13 L12

L12

L10

L10L1L8L12

L13 L9 L11 L10

L1 L8 L12L13 L9 L11 L10

Figure 3 Crossover operation

Parent L1

L1

L8

L8

L9

L9

L11

L11

L13

L13

L12

L12

L10

L10Offspring

Figure 4 Mutation operation

1 2 3 4 5b1 b4 b3 b5 b2

Figure 5 Chromosome structure

the fitness function and chromosome structure are differentthan that of GA TSP In contrast with GA TSP chromosomesstructure batch number is presented by the value of a geneAccordingly the sequence of batch is denoted by the orderof a gene in a chromosome For instance Figure 5 shows thechromosome structure If 119887

119894isin 119861 denotes a set of batches

Figure 5 shows that 1198871 1198874 1198873 1198875 1198872should be processed

respectively Therefore the length of a chromosome is equalto the number of the batches The GA sequencing fitness func-tion for chromosome 119910 is described in

fitness119910=

1

119873

119899

sum

119896=1

sum

119894isin119887119896

119862119879 (119887119896) minus 119863119906 (119900

119894) (13)

119862119879 (119887119896) = 119862119879 (119887

119896minus1) + Pt (119887119896) (14)

Pt (119887119896) =Opt (119887

119896)

119881119878 (15)

where119873 denotes for the total number of orders 119896 = 1 119899

represents the batch sequence in which 119899 is the total numberof batches 119900

119894stands for the order 119894 in which 119894 isin 119887

119896 119887119896

denotes the batch in 119896th position of a sequence for specificchromosome 119862119879(119887

119896) stands for the completion time of a

batch in the 119896th position of a sequence 119863119906(119900119894) is the symbol

for due date of order 119894which belongs to 119887119896 Pt(119887119896) is the process

time of 119887119896 Opt(119887

119896) is the optimal path solution for 119887

119896 Finally

119881119878 is the moving speed of the picker

4 Research Methodology

In this research the proposed methodology consists of fivesteps These steps are described as follows and shown pic-torially in Figure 6

Step 1 It is involved with utilizing theMINWAL algorithm inorder to determine the associations between orders in termsof support considering their due dates In this step an orderdue date will be considered as the weight of an order It meansorders that are similar in items and due date should havemoresupport for being batched together

Step 2 It encompasses determining the initial number ofbatches In this step the initial number of batches will becalculated by (10)

Step 3 It includes the problem of modeling and solving firstthe binary integer programming model for order clusteringand batching will be modeled Next this model will be solvedin order to determine the structures of batches

Step 4 It is about the picker routing problem The optimalpath for a picker within each batch will be determined bysolving a TSP using GA

Step 5 It is about sequencing the batches based on the fitnessfunction which minimizes the average tardiness of all ordersIn this step GA algorithmwill be utilized in order to solve thebatch sequencing problem regarding the fitness function

5 Computational Experiments and Results

51 Test Problems and Parameter Setting In this sectionfive numerical test problems with different settings andparameters have been carried out in order to illustrate theproficiency of the proposed method with respect to averagetardiness of all orders Table 5 summarizes the descriptionof these test problems This table includes information withregard to number of items number of orders capacity of thepicker total weight of all the items in an order andminimumnumber of batches

In these test problems the quantity of items quantityof orders capacity of the picker device (119862V) and amount ofeach item are predetermined Each order contains a certain

The Scientific World Journal 9

Start

Input order informationwarehouse setting picker

setting

Compute the supportbetween customersrsquo

orders using MINWAL

Calculating initialnumber of batches (K) to

assign the orders

Solving the binaryinteger programmingclustering model for

constructing the batches

Feasiblesolution

Yes

No

Solving picker routing

Solving batch sequencingproblem using

End

K = K + 1

problem using GA TSP

GA sequencing

Figure 6 Methodology framework

quantity of items and also each item has a certain quantityThese quantities are produced randomly The distributionfunction type for the number of items in each order is Normal[5 10] Also the distribution function type for each itemquantity in an order is uniform [1 10] The picker movingspeed is set as 119881119878 = 2 (ms) The warehouse has 3000 itemlocation The GA control parameters for solving the TSP andsequencing problem are tabulated in Table 6

All of the proposed approaches are coded usingMATLABversion 790 and Java 15 programming language Also theIBM ILOG CPLEX optimization studio version 124 softwarewas utilized jointly with MATLAB in order to facilitate theprocess of solving the binary integer programming problemExamples are run by means of a computer featured by

4 Gigabytes random access memory (RAM) and a 220Gigahertz Intel processor (T6600)

52 Computational Results and Discussion In order to revealthe competency of the proposed solution approach it wascompared with the other existing methods in the literatureOne of the latest published research activities in the literaturewhich deals with minimization of tardiness has been con-ducted by Henn and schmid [35] They utilized the EarliestDueDate (EDD) approach as their constructive algorithm forcomparing with their proposed approaches They proposedtwo heuristics Iterated Local Search andAttribute-BasedHillClimber are the two employed metaheuristics in their modelAccording to different test problems the efficiency of their

10 The Scientific World Journal

Table 5 Test problems description

Example 1 Example 2 Example 3 Example 4 Example 5Number of orders 40 60 80 100 150

Number of items 80 120 160 200 200

Capacity of picker 200 200 300 400 550

Total weight 2212 3668 4393 6110 8369

Minimum number of batch 11 19 15 16 16

Due dates range 60 90 120 180 240

Table 6 GA control parameters

GA TSP GA sequencing

Population size 500 200

Cross over strategyPermutation orderbased crossover

(POP)

Permutation orderbased crossover

(POP)Cross over probability 08 08

Mutation strategy SWAP SWAPMutation probability 02 02

Iteration number 10000 10000

Parent selection method Roulette wheel Roulette wheel

proposed methods was illustrated Their obtained resultsfrom their proposed approaches showed a 46 improvementon tardiness compared to EDD They used an s-shape andlargest gap routing strategy for their proposed method Inorder to make a reasonable comparison with their proposedmethod EDD was integrated with the s-shape routing strat-egy and GA sequencing Then based on improvement on EDDwe can compare our proposed method with their proposedmethod In the EDD approach due dates play an importantrole in sorting all orders Based on an ascending sequenceassigning the orders to batches ensures that no capacityviolation can happen for the picking device

For the warehouse condition which is described inSection 323 our proposed approach EDD integrated withGA TSP and GA sequencing and EDD integrated with s-shapealgorithm and GA sequencing have been evaluated in termsof average tardiness for all orders (119879) which is calculatedbased on (13) number of batches (119870) total process time(Pt) and average process time for each batch (119860Pt) All ofthe GA setting for the TSP and sequencing problems areconsidered the same for all methods Also the settings whichhave been mentioned in Section 51 were considered for all ofthe compared methods

The results for all of the test problems for the threeapproaches are tabulated in Tables 7 8 and 9 As it isillustrated in Table 10 EDD integrated with GA TSP andGA sequencing has a 3176 improvement on average in termsof average tardiness of all orders than EDD integrated withthe s-shape algorithm and GA sequencing It can be perceivedthat GA TSP performs better than s-shape Also the ATGHmethod has a 6811 improvement in average tardinessoff all orders than EDD integrated with s-shape algorithm

and GA sequencing Consequently the ATGH method achievesmuch improvement against Henn and Schmidrsquos [35] pro-posedmethod According to Tables 7 and 8 it is also worth tomention thatGA TSP performs better than the s-shape routingstrategy in terms of process time The improvements whichhave been achieved in this research could be justified basedon the following reasons

The problem of order batching deals with several sig-nificant drivers in which the number of batches can beconsidered as a key factor [23] As a justification to thismatter it is worth to mention that total travel distance maynot decrease directly due to a decline in the number ofbatches But a decline in the number of batches will increasebatch sizes which may lead to an increase in the distancetravelled by order picker However an advanced exploitationof picker capacity to load efficiently may reduce the numberof constructed batches together reducing the total traveldistance Improving the total travel distance traversed by thepicker can minimize the average tardiness of all orders Interms of number of the batches according to Tables 7 8 and9 it can be perceived that our proposed method assigns theorders to the batches optimally The ATGH method assignsthe orders to batches based on their item similarity andtheir due dates and it uses the picker capacity efficiently Butthe EDD method allocates the orders to batches based onan ascending sequence of their due date and it should beensured that no capacity violation will happen for the pickingdevice which may lead to an inefficient order batching Tohighlight the problem an example can be made based on theresults tabulated in Tables 7 and 8 The number of batchesfor example 1 in our method is 11 in comparison with 13obtained fromEDDmethod 13 batches can be justified basedon this scenario that two subsequent orders say the firstorder is 100 kg and the other is 150 kg cannot be batchedtogether due to violated batch capacity which is set to be200 kg The number of batches would increase due to theaforementioned issue which affects the total process time tohave an increasing manner This situation will rarely happenin the ATGH method because of the weighted associationbased binary integer programming utilizedwhich batches theorders in order to maximize the association between themwhich is calculated based on the item similarity of ordersand the weight of their corresponding due dates The ATGHmethod uses the picker capacity efficiently

Obviously one of the factors that can influence thecompletion time and the average tardiness of all orders is theprocessing time of the orders Reducing the processing time

The Scientific World Journal 11

Table 7 Computational results of ATGH

Example 1 Example 2 Example 3 Example 4 Example 5119879 56204 29948 90202 163 58898

Pt 605833 1217833 105833 121033 1339

119870 11 19 15 16 16

119860Pt 55076 6409 7055 7564 8368

Table 8 Computational results of EDD integrated with GA TSP and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 124550 42976 18222 5981 1547

Pt 667333 133516 116083 13185 141916

119870 13 21 16 17 16

119860Pt 51333 6357 7255 7775 8869

Table 9 Computational results of EDD integrated with s-shape algorithm and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 1651 62677 2825 13365 17593

Pt 75266 16636 13323 14333 14926

119870 13 21 16 17 16

119860Pt 5789 892 833 8431 932

Table 10 Comparison results

EDD + s-shape + GA sequencing EDD + GA TSP + GA sequencing ATGHTardiness (min) Tardiness (min) Improvement () Tardiness (min) Improvement ()

(1) 1651 124550 2456 56204 6596(2) 62677 42976 3143 29948 5222(3) 2825 18222 3550 90202 6807(4) 13365 5981 5525 1627 8783(5) 17593 1547 1206 58898 6652Average improvement 3176 6811

and travel time for constructing a batch is an outstanding factin reducing the costs and minimizing delays of the customerresponses In our proposed model orders with similar itemsare consolidated together Therefore batching orders withmore similarity can reduce the distance traversed by thepicker facility Also the due date of each order is definedas a weight parameter in calculating the association betweenorders in terms of support which was done for consideringthe due dates in our model Thereupon the model considersdue date and the process time in an integrated manner whichsimultaneously affected the average tardiness of all ordersThis spoken issue is different in the EDD approach in whichthe ordersrsquo due dates are just considered Example 5 canbe considered as evidence of this matter in which the totaland average process time for ATGH is lower than EDDintegrated with GA TSP and GA sequencing with respect to thefact that number of constructed batches for both approachesare happened to be equal after obtaining the results

6 Conclusion

One of the most important concerns of warehouse managersis involvedwith finding the best optimal and cost efficientwayto pick orders placed by customers in order to be known as aresponsible company with satisfied customers in terms of ontime delivery In the current research activity order batchingpicker routing and batch sequencing problems of warehouseprocesses were addressed which are to be solved jointly asthey are mixed up with many manufacturing industriesMathematical solutions of these NP-hard problems might beavailable for problems that are small in nature But thesekinds of solutions may not be applicable to be consideredby warehouse managers Hence we presented novel solutionapproach namely ATGH to solve this issue in accordancewith minimizing the average tardiness of all customersrsquoorders which can be considered as a major objective ofthis research activity A weighted association rule mining

12 The Scientific World Journal

algorithm namely MINWAL was utilized to determine thecustomersrsquo orders association regarding similarity betweenitems and ordersrsquo due dates Actually in our proposedmethodthe due date of each order is defined as a weight parameterin calculating the association between orders So orders withsimilar items and due dates have more chance to be in asame batch This issue is not considered in most of theresearch activities in the literature related to our work andconsidering this issue in the order batching problem is themain contribution of our research activity The previous stepwas followed by a binary integer clustering model in orderto solve the order batching problem which maximizes thesimilarity between orders in terms of support After that aGA was applied to solve a TSP in order to find the optimalrouting path

Finally a batch sequencing problem was addressed inconjunction with a GA

The other research objective of this research addresseswhether the proposed ATGH algorithm can further improvethe solution quality for a multiple parallel aisle warehousewith manual picking system Hence comparisons were madeconsideringATGHEDD integratedGA TSP andGA sequencingand EDD integrated s-shape and GA sequencing algorithms Asdemonstrated by the obtained results ATGH comes out tobe the most striking solution approach in terms of tardinessIn spite of the fact that considering all customer orders asdiscussed in this paper would lead to have a static case beinginvolved with dynamic cases in which different points of timeare assigned to customerrsquos orders arrival would be beneficialSo there might be some potential room of research to extendour discussed approach with respect to dynamic orders

Conflict of Interests

The IBM ILOG CPLEX optimization studio version 124software is not used for financial gain and there is no conflictof interests with any party as it is only a student version

References

[1] ELAAT Kearney Excellence in Logistics ELA Brussels Bel-gium 2004

[2] Y-C Ho T-S Su and Z-B Shi ldquoOrder-batching methods foran order-picking warehouse with two cross aislesrdquo Computersamp Industrial Engineering vol 55 no 2 pp 321ndash347 2008

[3] C-M Hsu K-Y Chen and M-C Chen ldquoBatching ordersin warehouses by minimizing travel distance with geneticalgorithmsrdquo Computers in Industry vol 56 no 2 pp 169ndash1782005

[4] E A Elsayed M-K Lee S Kim and E Scherer ldquoSequencingand batching procedures forminimizing earliness and tardinesspenalty of order retrievalsrdquoThe International Journal of Produc-tion Research vol 31 no 3 pp 727ndash738 1993

[5] S Henn S Koch and W Gerhard Order Batching in OrderPicking Warehouses A Survey of Solution Approaches SpringerLondon UK 2012

[6] J Drury Towards More Efficient Order Picking IMM Mono-graph no 1 The Institute of Materials Management CranfieldUK 1988

[7] M Goetschalckx and J Ashayeri ldquoClassification and design oforder picking systemsrdquo Logistics World pp 99ndash106 1989

[8] J A Tompkins J A White Y A Bozer E H Frazelle and J MA Tanchoco Facilities Planning John Wiley amp Sons HobokenNJ USA 2003

[9] S Henn and G Wascher ldquoTabu search heuristics for the orderbatching problem in manual order picking systemsrdquo EuropeanJournal of Operational Research vol 222 no 3 pp 484ndash4942012

[10] S Henn ldquoAlgorithms for on-line order batching in an orderpicking warehouserdquo Computers and Operations Research vol39 no 11 pp 2549ndash2563 2012

[11] R de Koster ldquoHow to assess a warehouse operation in a singletourrdquo Tech Rep RSM Erasmus University Rotterdam TheNetherlands 2004

[12] R de Koster T Le-Duc and K J Roodbergen ldquoDesign andcontrol of warehouse order picking a literature reviewrdquo Euro-pean Journal of Operational Research vol 182 no 2 pp 481ndash5012007

[13] R A Ruben and F R Jacobs ldquoBatch construction heuristics andstorage assignment strategies for walkride and pick systemsrdquoManagement Science vol 45 no 4 pp 575ndash596 1999

[14] E A Elsayed ldquoAlgorithms for optimal material handling inautomatic warehousing systemsrdquo International Journal of Pro-duction Research vol 19 no 5 pp 525ndash535 1981

[15] G Clarke and J W Wright ldquoScheduling of vehicles froma central depot to a number of delivery pointsrdquo OperationsResearch vol 12 no 4 pp 568ndash581 1964

[16] H D Vinod ldquoInteger programming and the theory of group-ingrdquo Journal of the American Statistical Association vol 64 no326 pp 506ndash519 1969

[17] R D Armstrong W D Cook and A L Saipe ldquoOptimalbatching in a semi-automated order picking systemrdquo Journal ofthe Operational Research Society vol 30 no 8 pp 711ndash720 1979

[18] H D Ratliff and A S Rosenthal ldquoOrder-picking in a rect-angular warehouse a solvable case of the traveling salesmanproblemrdquo Operations Research vol 31 no 3 pp 507ndash521 1983

[19] A Kusiak A Vannelli and R K Kumar ldquoClustering analysismodels and algorithmsrdquo Cybernetics vol 15 pp 139ndash154 1986

[20] D R Gibson and G P Sharp ldquoOrder batching proceduresrdquoEuropean Journal of Operational Research vol 58 no 1 pp 57ndash67 1992

[21] M B Rosenwein ldquoA comparison of heuristics for the problemofbatching orders for warehouse selectionrdquo International Journalof Production Research vol 34 no 3 pp 657ndash664 1996

[22] A J RMGademann J P van denBerg andHH van derHoffldquoAn order batching algorithm for wave picking in a parallel-aislewarehouserdquo IIE Transactions vol 33 no 5 pp 385ndash398 2001

[23] M-C Chen and H-P Wu ldquoAn association-based clusteringapproach to order batching considering customer demandpatternsrdquo Omega vol 33 no 4 pp 333ndash343 2005

[24] NGademann and S van deVelde ldquoOrder batching tominimizetotal travel time in a parallel-aisle warehouserdquo IIE Transactionsvol 37 no 1 pp 63ndash75 2005

[25] Y A Bozer and J W Kile ldquoOrder batching in walk-and-pick order picking systemsrdquo International Journal of ProductionResearch vol 46 no 7 pp 1887ndash1909 2008

[26] O Kulak Y Sahin and M E Taner ldquoJoint order batching andpicker routing in single and multiple-cross-aisle warehousesusing cluster-based tabu search algorithmsrdquo Flexible Servicesand Manufacturing Journal vol 24 no 1 pp 52ndash80 2012

The Scientific World Journal 13

[27] E A Elsayed and M-K Lee ldquoOrder processing in automatedstorageretrieval systems with due datesrdquo IIE Transactions vol28 no 7 pp 567ndash577 1996

[28] J Won and S Olafsson ldquoJoint order batching and order pickingin warehouse operationsrdquo International Journal of ProductionResearch vol 43 no 7 pp 1427ndash1442 2005

[29] C-Y Tsai J J H Liou and T-M Huang ldquoUsing a multiple-GA method to solve the batch picking problem consideringtravel distance and order due timerdquo International Journal of Pro-duction Research vol 46 no 22 pp 6533ndash6555 2008

[30] Y KaoM H Chen and Y T Huang ldquoA hybrid algorithm basedon ACO and PSO for capacitated vehicle routing problemsrdquoMathematical Problems in Engineering vol 2012 Article ID726564 17 pages 2012

[31] X Gan Y Wang S Li and B Niu ldquoVehicle routing problemwith time windows and simultaneous delivery and pick-up ser-vice based on MCPSOrdquoMathematical Problems in Engineeringvol 2012 Article ID 104279 11 pages 2012

[32] M B M de Koster E S van der Poort and M WoltersldquoEfficient orderbatching methods in warehousesrdquo InternationalJournal of ProductionResearch vol 37 no 7 pp 1479ndash1504 1999

[33] Y-C Ho and Y-Y Tseng ldquoA study on order-batching methodsof order-picking in a distribution centre with two cross-aislesrdquoInternational Journal of Production Research vol 44 no 17 pp3391ndash3417 2006

[34] C Theys O Braysy W Dullaert and B Raa ldquoUsing a TSPheuristic for routing order pickers in warehousesrdquo EuropeanJournal of Operational Research vol 200 no 3 pp 755ndash7632010

[35] S Henn and V Schmid Metaheuristics for Order Batching andSequencing in Manual Order Picking Systems Univ Fak furWirtschaftswiss 2011

[36] M Hahsler B Grun and K Hornik ldquoArulesmdasha computationalenvironment for mining association rules and frequent itemsetsrdquo Journal of Statistical Software vol 14 2005

[37] R Agrawal T Imielinski and A Swami ldquoMining associationrules between sets of items in large databasesrdquo ACM SIGMODRecord vol 22 no 2 pp 207ndash216 1993

[38] S Encheva Y Kondratenko M Z Solesvik and S TuminldquoDecision support systems in logisticsrdquo in Proceeding of Inter-national electronic conference on computer science pp 254ndash2562007

[39] M-C Chen ldquoConfiguration of cellular manufacturing systemsusing association rule inductionrdquo International Journal of Pro-duction Research vol 41 no 2 pp 381ndash395 2003

[40] R Srikant and R Agrawal ldquoMining generalized associationrulesrdquo Future Generation Computer Systems vol 13 no 2-3 pp161ndash180 1997

[41] J Wu and XM Li ldquoAn effective mining algorithm for weightedassociation rules in communication networksrdquo Journal of Com-puters vol 3 no 10 pp 20ndash27 2008

[42] C H Cai A W C Fu C H Cheng and W W KwongldquoMining association rules with weighted itemsrdquo in Proceedingsof the International Symposium on Database Engineering ampApplications pp 68ndash77 Cardiff UK 1998

[43] J HollandAdaptation in Natural and Artificial Systems Univer-sity of Michigan Press Ann Arbor Mich USA 1975

[44] A A Taleizadeh F Barzinpour and H-M Wee ldquoMeta-heuristic algorithms for solving a fuzzy single-period problemrdquoMathematical and Computer Modelling vol 54 no 5-6 pp1273ndash1285 2011

[45] H Yang J Yi J Zhao and Zh Dong ldquoExtreme learning ma-chine based genetic algorithm and its application in powersystem economic dispatchrdquo Neurocomputing vol 102 pp 154ndash162 2013

Submit your manuscripts athttpwwwhindawicom

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ArtificialNeural Systems

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RoboticsJournal of

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Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 5

Table 3 Weight of each order

Order Weight1198741

05827

1198742

01979

1198743

07634

1198744

00672

proves that subsets of a frequent itemset are also frequent [41]However the Apriori algorithm cannot be applied because itcannot handle the weighted case Therefore in this researchactivity theMINWAL algorithm is applied in order to extractthe weighted supports among the pair of orders to fulfill thetask of considering both the support and the weights factors

In order to use MINWAL [42] for mining weighted asso-ciation rules the weight should be in the range of the 0 to1 interval Based on the aforementioned fact larger weightsshould be assigned to orders with smaller due dates in acomparison with other orders that have lots of time to bedelivered in order to minimize the average tardiness of allorders Therefore in this study (1) has been used to calculatethe appropriate weights for each order as following

Weight (119874119894) = 119890(minus01lowast120573lowast119889119874119894

Maximume Due Date) (1)

where 119874119894denotes an order and 119889

119874119894refers to the due date

of order 119874119894 120573 is a constant value that is used to normalize

the due dates which is considered as 30 (30 minute) in thisstudy Also the maximum due date of the orders was usedin order to provide an appropriate interval for the weightsFor example the weight of each order is tabulated in Table 3which is calculated by (1)

Given a set of orders 119874 = 1199001 1199002 119900

119898 a weight 119908

119895for

each order 119900119895 within the values of 0 to 1 is assigned by (1) via

its due date where 119895 = 1 2 119898 shows the importance ofit The weighted support for the weighted association ordersof 1198741and 119874

2as 1198741rArr 1198742is defined if the weighted support

of such a pair of orders is no less than theminimumweightedsupport (minsup) threshold In this study the minimumsupport threshold is considered as 0 in order to identify all theassociations among orders Equation (2) shows the support ofan association for any pair of orders as following

(sum sum

119908119895isin(1198741cup1198742)

119908119895) lowast Support (119874

1cup 1198742) (2)

Algorithm 1 shows the MINWAL steps in order to mineweighted association rules This algorithm consists of 6subroutines such as Search Counting Join Prune Checkingand Rules The search subroutine provides the size of thelargest itemsets in database Counting is responsible forcounting every 1-itemset inside the database The Join step isresponsible for generating the 119896-itemsets (119862119896) from (119896 minus 1)-itemsets (119862119896 minus 1) The Prune step removes inappropriateitemsets which do not exist The Checking step updatesthe count of 119896-itemsets inside the transaction and prunesitemsets which do not meet the minimum support thresholdIn the Rules step every association rule will be extractedTable 4 shows the notations of Algorithm 1

(1) Main Algorithm (wminsup mincon 119891119863 119908)(2) size = Search (119863)(3) 119871 = 0(4) for (119894 = 1 119894 le size 119894++)(5) 119862

119894= 119871119894= 0

(6) for each transaction do(7) (SC 119862

1 size) =Counting (119863 119908)

(8) 119896 = 1(9) while (|119862119896| ge 119896)(10) 119896++(11) 119862

119896= Join (119862

119896minus1)

(12) 119862119896=Prune (119862

119896)

(13) (119862119896 119871119896) =Checking (119862

119896119863)

(14) 119871 = 119871 cup 119871119896

(15)Rules (SC 119871)(16) ends

Algorithm 1 MINWAL algorithm [42]

Table 4 Notations of Algorithm 1

119863 The database119908 The set of item weights119871119896

Set of large 119896-itemsets

119862119896

Set of 119896-itemsets which may be 119896-subsets of large119895-itemsets for 119895 ge 119896

SC (119883) No of transactions containing itemset119883wminsup Weighted support thresholdminconf Confidence thresholdSize Maximum possible large weighted itemsets

312 Association Based Binary Integer Programming forOrders Batching In order to create the batch structure abinary integer clustering model which has been developed byChen andWu [23] was appliedThey developed a binary inte-ger programming model for maximizing similarity betweenorders in terms of support in order to minimize the totaltraveling distance In this research activity support betweenorders was calculated based on weighted association miningusing MINWAL which was described in Section 311 Thebinary integer model for order batching is described asfollows

Parameters

119878119894119895Support between order 119894 and 119895 determined by MIN-WAL

119862V Capacity of picking vehicle

119881119894Weight of order 119894

119870 Number of batches

1198831198941198951 if order 119894 is assigned to batch 119895

0 otherwise

6 The Scientific World Journal

1198841198941 if order 119895 is chosen as a batch median0 otherwise

Maximize119873

sum

119894=1

119873

sum

119895=1

119878119894119895lowast 119883119894119895 (3)

Subject to119873

sum

119895=1

119883119894119895= 1 for 119894 119895 = 1 2 119873

(4)

119883119894119895le 119884119895

for 119894 119895 = 1 2 119873 (5)

119873

sum

119895=1

119884119895= 119870 (6)

119873

sum

119894=1

119881119894119883119894119895le 119862V for 119895 = 1 2 119873 (7)

119883119894119895= 0 1 for 119894 119895 = 1 2 119873 (8)

119884119895= 0 1 for 119895 = 1 2 119873 (9)

In the current model the objective function (3) maxi-mizes the sum of support from all orders to their relevantbatch medians as 119870 orders are selected as batch mediansThis model will maximize the support between orders in thebatches Membership of each order in just one batch wouldbe guaranteed by constraint (4) The number of batches with119870 is limited using constraints (5) and (6) The total quantityin a tour is also represented as a limitation in constraint (7) Itcannot exceed the capacity of order picker 119862V For the orderbatchingmodel to have a binary solutionwould be ensured byconstraints (8) and (9) For calculating the initial numbers ofbatches (10) will be used where sum119881

119894represents total volume

of all orders as following

119870initial = lceilsum119881119894

119862V

rceil (10)

32 Picker Routing

321 Genetic Algorithms In this research GAs which wereintroduced by Holland [43] have been used in order tosolve the picker routing and batch sequencing problem Thebiological evolution process is used as a base fact in a GA forproblem solving to a great extentThemain steps of a GA canbe described as follows [44 45]

(i) populating a series of solutions(ii) solutions assessment(iii) picking the most appropriate solutions(iv) generating new solutions based on some genetic

operation

In step (iii) selection forms the main part of a GA Thebad solutions are removed by selection and the good onesremain Random generation over the feasible or infeasible

solution space of a problem is used as a basis for creatingthe population Each solution is evaluated using the fitnessfunction Selection would be done according to certain selec-tions of fitness together with the probability that is assignedto it Generating a fresh population and a better solutionare involved with recombination of the individual solutionCrossover and mutation are the most used GA operators forgenerating new offspring from the old generation of parentsParameters of the GA are originated from the probabilitiesof mutation and crossover Exploration of the sample spaceis diversified by these operators In this research the GA hasbeenutilized in two steps First theGAhas beenused in orderto solve the picker routing problem Consider that in thewarehouse a set of items should be picked by a picker in onetour Therefore the picker routing problem in the warehousecould be categorized as a TSP The goal in the picker routingproblem as a TSP in the warehouse is to find an optimal pathfor minimizing the travel distance traversed by the pickingmachine where each node is visited at least once In thisresearch for a given number of itemset in a batch GA TSPhas been used to find the minimum travel distance whichis traversed by the picking machine Second a GA has beenutilized in order to sequence all batches namely GA sequencingwhichwould be further processedThe batches are sequencedbased on a fitness function which minimizes the tardiness ofall ordersThedetailed procedures ofGA TSP andGA sequencingare illustrated in Sections 323 and 324

322 Warehouse Layout Thewarehouse discussed through-out this research is assumed to have a 3D multi-parallel-aisle warehouse layout The picker will start the tour fromthe depot which is located in front of the leftmost aisle andreturns back to it A picker leaves the depot After that ittravels to a certain buffer spot where it picks the requestedorder(s) and puts the order(s) into a certain picker Then itcontinue on its way to another spot These procedures wouldbe repeated again and again so that all the requested ordersare picked Then it is time for the picker to return to thedepot

The graphic layout of the warehouse zone involved in thisresearch activity is pictorially displayed in Figure 1 It showsthe 119909 and 119910 dimensions of the 3D warehouse layout Eachitem in the warehouse has a specific location For examplethe location of item 119894 can be addressed by 119871

119894= (119909119894 119910119894 119911119894)

Consider two locations as 119871119886= (119909119886 119910119886 119911119886) and 119871

119887= (119909119887

119910119887 119911119887) If two locations are located in the same sub-aisles

of the same block the distance between these two itemslocations (119863

119886119887) will be calculated using

119863119886119887

=1003816100381610038161003816119909119886 minus 119909119887

1003816100381610038161003816 +1003816100381610038161003816119910119886 minus 119910119887

1003816100381610038161003816 +1003816100381610038161003816119911119886 minus 119911119887

1003816100381610038161003816 (11)

Equation (11) is not applicable if the two storage locationsbelong to different subaisles that are from the same block [34]In order to solve this problem distances of all possible pathsbetween two locations have been calculated and the shortestone was considered

323 GA TSP for the Picker Routing Problem In order todetermine the optimal route of a picker related to a given

The Scientific World Journal 7

Lb=(xbybzb)

La=(xayaza)

Li=(xiyizi)

Depot

Figure 1 Warehouse layout

1 2 3 4 5 6 7 8 9L0 L1 L8 L11 L9

L13 L12 L10 L0

Figure 2 A feasible chromosome encoding in GA TSP

set of items in a batch a GA has been used to solve theTSP problem to minimize travelling time In a GA eachsolution is considered as a chromosome The steps of GA TSPare described as follows

(1) for each batch establishing the structure of a chromo-somes

(2) generate initial feasible chromosomes(3) evaluate each chromosome based on the GA TSP fit-

ness function(4) apply the Selection operation(5) apply the Crossover operation(6) apply the Mutation operation(7) evaluate each offspring based on the fitness function(8) go to step 4 if the termination criterion is not satis-

fied otherwise terminate and return the best solutionas the optimal path

At the first step of GA TSP the structure of the chro-mosome should be established In the GA TSP the locationof an item that should be visited is presented by the valueof a gene Accordingly the sequence of visiting the itemlocation is denoted by the order of a gene in a chromosomeTherefore the total number of items locations in a specificbatch is considered equal to the length of a chromosome Forexample if the length of a chromosome is nine it means nineplaces should be visited by the picker as shown in Figure 2If 119871119894represents the item location in the warehouse 119894 = 1

119899 it shows that the picker leaves the depot (1198710denotes

the location of a depot) goes to 1198711 1198718 11987111 1198719 11987113 11987112 11987110

and then comes back to the depotIn the second step the initial number of feasible chro-

mosomes should be generated randomly for each batch

The number of initial chromosomes could be different basedon the complexity of the problem It should be consideredthat the first and the last gene of each chromosome should beassigned individually to the location of a depot In step threeeach chromosome will be evaluated based on the GA TSPfitness function which is formulated as follows

fitness = Min119899minus1

sum

119895=1

119863119895119895+1

(12)

where 119863119895119895+1

denotes the distance between two subsequentgenes in a chromosome 119895 = 1 119899 minus 1 represents the orderof the genes and 119899 is number of genes

Subsequently step four is involved with the selectionoperation In each iteration the selection operator usesfitness values to select the parents of the next generationIn this research the roulette wheel selection policy hasbeen used to guarantee that the most appropriate pairs ofchromosomes have been selected to generate offspring Itmeans that a higher probability of being selected to createa new population will be assigned to the chromosomes withhigher fitness values

Steps five and six are involved with conducting thecrossover and mutation operation The construction of theoffspring is realized by the crossover and the diversity of theindividuals is maintained by the mutation operator Blocksof genes between chromosomes are traded by means of thecrossover operation inwhich the exploitation of a particularlyprofitable portion of the parameter space is allowed to theblock Here permutation order based crossover (POP) as avariation of the distinguished order crossover has been usedAs shown in Figure 3 the original idea is about choosing twoparents and a point for cutting The first portion of offspring2 contains the first portion of parent 1 up to the cut pointTo form the second portion of offspring 2 consider all genesin parent 2 except those genes that already exist in the firstportion of the offspring 2

As shown in Figure 4 two positions are randomlyselected in a chromosome and exchanged with each othernamely the order based mutation approach SWAP isadopted for solving GA TSP Step six is followed by step sevenwhich includes the evaluation of each offspring that has beengenerated in the previous step based on the GA TSP fitnessfunction In step eight termination criteria will be checkedIf the termination condition is met then the process will beterminated After that the best chromosome will be selectedas the optimal solution Otherwise the selection process is tobe conducted again to generate a new population

324 GA for Sequencing the Batches Sequencing the batchesshould be conducted in order to minimize the averagetardiness of all orders The problem of batch sequencing canbe defined as finding the orders of batches which should beprocessed further For this purpose the GA sequencing methodhas been utilized In this research all of the batches whichwere constructed using binary integer programming andGA TSP will be sequenced by using the GA The populationgeneration selectionmutation and stopping threshold in theGA sequencing are identical to those in the GA TSP However

8 The Scientific World Journal

Parent 1

Parent 2

Offspring 2

Offspring 1

L1 L8 L9

L1 L1L8 L8L9

L11 L13 L12 L10

L1 L8 L9 L11L13 L12L10

L13 L9 L11 L11 L13 L12

L12

L10

L10L1L8L12

L13 L9 L11 L10

L1 L8 L12L13 L9 L11 L10

Figure 3 Crossover operation

Parent L1

L1

L8

L8

L9

L9

L11

L11

L13

L13

L12

L12

L10

L10Offspring

Figure 4 Mutation operation

1 2 3 4 5b1 b4 b3 b5 b2

Figure 5 Chromosome structure

the fitness function and chromosome structure are differentthan that of GA TSP In contrast with GA TSP chromosomesstructure batch number is presented by the value of a geneAccordingly the sequence of batch is denoted by the orderof a gene in a chromosome For instance Figure 5 shows thechromosome structure If 119887

119894isin 119861 denotes a set of batches

Figure 5 shows that 1198871 1198874 1198873 1198875 1198872should be processed

respectively Therefore the length of a chromosome is equalto the number of the batches The GA sequencing fitness func-tion for chromosome 119910 is described in

fitness119910=

1

119873

119899

sum

119896=1

sum

119894isin119887119896

119862119879 (119887119896) minus 119863119906 (119900

119894) (13)

119862119879 (119887119896) = 119862119879 (119887

119896minus1) + Pt (119887119896) (14)

Pt (119887119896) =Opt (119887

119896)

119881119878 (15)

where119873 denotes for the total number of orders 119896 = 1 119899

represents the batch sequence in which 119899 is the total numberof batches 119900

119894stands for the order 119894 in which 119894 isin 119887

119896 119887119896

denotes the batch in 119896th position of a sequence for specificchromosome 119862119879(119887

119896) stands for the completion time of a

batch in the 119896th position of a sequence 119863119906(119900119894) is the symbol

for due date of order 119894which belongs to 119887119896 Pt(119887119896) is the process

time of 119887119896 Opt(119887

119896) is the optimal path solution for 119887

119896 Finally

119881119878 is the moving speed of the picker

4 Research Methodology

In this research the proposed methodology consists of fivesteps These steps are described as follows and shown pic-torially in Figure 6

Step 1 It is involved with utilizing theMINWAL algorithm inorder to determine the associations between orders in termsof support considering their due dates In this step an orderdue date will be considered as the weight of an order It meansorders that are similar in items and due date should havemoresupport for being batched together

Step 2 It encompasses determining the initial number ofbatches In this step the initial number of batches will becalculated by (10)

Step 3 It includes the problem of modeling and solving firstthe binary integer programming model for order clusteringand batching will be modeled Next this model will be solvedin order to determine the structures of batches

Step 4 It is about the picker routing problem The optimalpath for a picker within each batch will be determined bysolving a TSP using GA

Step 5 It is about sequencing the batches based on the fitnessfunction which minimizes the average tardiness of all ordersIn this step GA algorithmwill be utilized in order to solve thebatch sequencing problem regarding the fitness function

5 Computational Experiments and Results

51 Test Problems and Parameter Setting In this sectionfive numerical test problems with different settings andparameters have been carried out in order to illustrate theproficiency of the proposed method with respect to averagetardiness of all orders Table 5 summarizes the descriptionof these test problems This table includes information withregard to number of items number of orders capacity of thepicker total weight of all the items in an order andminimumnumber of batches

In these test problems the quantity of items quantityof orders capacity of the picker device (119862V) and amount ofeach item are predetermined Each order contains a certain

The Scientific World Journal 9

Start

Input order informationwarehouse setting picker

setting

Compute the supportbetween customersrsquo

orders using MINWAL

Calculating initialnumber of batches (K) to

assign the orders

Solving the binaryinteger programmingclustering model for

constructing the batches

Feasiblesolution

Yes

No

Solving picker routing

Solving batch sequencingproblem using

End

K = K + 1

problem using GA TSP

GA sequencing

Figure 6 Methodology framework

quantity of items and also each item has a certain quantityThese quantities are produced randomly The distributionfunction type for the number of items in each order is Normal[5 10] Also the distribution function type for each itemquantity in an order is uniform [1 10] The picker movingspeed is set as 119881119878 = 2 (ms) The warehouse has 3000 itemlocation The GA control parameters for solving the TSP andsequencing problem are tabulated in Table 6

All of the proposed approaches are coded usingMATLABversion 790 and Java 15 programming language Also theIBM ILOG CPLEX optimization studio version 124 softwarewas utilized jointly with MATLAB in order to facilitate theprocess of solving the binary integer programming problemExamples are run by means of a computer featured by

4 Gigabytes random access memory (RAM) and a 220Gigahertz Intel processor (T6600)

52 Computational Results and Discussion In order to revealthe competency of the proposed solution approach it wascompared with the other existing methods in the literatureOne of the latest published research activities in the literaturewhich deals with minimization of tardiness has been con-ducted by Henn and schmid [35] They utilized the EarliestDueDate (EDD) approach as their constructive algorithm forcomparing with their proposed approaches They proposedtwo heuristics Iterated Local Search andAttribute-BasedHillClimber are the two employed metaheuristics in their modelAccording to different test problems the efficiency of their

10 The Scientific World Journal

Table 5 Test problems description

Example 1 Example 2 Example 3 Example 4 Example 5Number of orders 40 60 80 100 150

Number of items 80 120 160 200 200

Capacity of picker 200 200 300 400 550

Total weight 2212 3668 4393 6110 8369

Minimum number of batch 11 19 15 16 16

Due dates range 60 90 120 180 240

Table 6 GA control parameters

GA TSP GA sequencing

Population size 500 200

Cross over strategyPermutation orderbased crossover

(POP)

Permutation orderbased crossover

(POP)Cross over probability 08 08

Mutation strategy SWAP SWAPMutation probability 02 02

Iteration number 10000 10000

Parent selection method Roulette wheel Roulette wheel

proposed methods was illustrated Their obtained resultsfrom their proposed approaches showed a 46 improvementon tardiness compared to EDD They used an s-shape andlargest gap routing strategy for their proposed method Inorder to make a reasonable comparison with their proposedmethod EDD was integrated with the s-shape routing strat-egy and GA sequencing Then based on improvement on EDDwe can compare our proposed method with their proposedmethod In the EDD approach due dates play an importantrole in sorting all orders Based on an ascending sequenceassigning the orders to batches ensures that no capacityviolation can happen for the picking device

For the warehouse condition which is described inSection 323 our proposed approach EDD integrated withGA TSP and GA sequencing and EDD integrated with s-shapealgorithm and GA sequencing have been evaluated in termsof average tardiness for all orders (119879) which is calculatedbased on (13) number of batches (119870) total process time(Pt) and average process time for each batch (119860Pt) All ofthe GA setting for the TSP and sequencing problems areconsidered the same for all methods Also the settings whichhave been mentioned in Section 51 were considered for all ofthe compared methods

The results for all of the test problems for the threeapproaches are tabulated in Tables 7 8 and 9 As it isillustrated in Table 10 EDD integrated with GA TSP andGA sequencing has a 3176 improvement on average in termsof average tardiness of all orders than EDD integrated withthe s-shape algorithm and GA sequencing It can be perceivedthat GA TSP performs better than s-shape Also the ATGHmethod has a 6811 improvement in average tardinessoff all orders than EDD integrated with s-shape algorithm

and GA sequencing Consequently the ATGH method achievesmuch improvement against Henn and Schmidrsquos [35] pro-posedmethod According to Tables 7 and 8 it is also worth tomention thatGA TSP performs better than the s-shape routingstrategy in terms of process time The improvements whichhave been achieved in this research could be justified basedon the following reasons

The problem of order batching deals with several sig-nificant drivers in which the number of batches can beconsidered as a key factor [23] As a justification to thismatter it is worth to mention that total travel distance maynot decrease directly due to a decline in the number ofbatches But a decline in the number of batches will increasebatch sizes which may lead to an increase in the distancetravelled by order picker However an advanced exploitationof picker capacity to load efficiently may reduce the numberof constructed batches together reducing the total traveldistance Improving the total travel distance traversed by thepicker can minimize the average tardiness of all orders Interms of number of the batches according to Tables 7 8 and9 it can be perceived that our proposed method assigns theorders to the batches optimally The ATGH method assignsthe orders to batches based on their item similarity andtheir due dates and it uses the picker capacity efficiently Butthe EDD method allocates the orders to batches based onan ascending sequence of their due date and it should beensured that no capacity violation will happen for the pickingdevice which may lead to an inefficient order batching Tohighlight the problem an example can be made based on theresults tabulated in Tables 7 and 8 The number of batchesfor example 1 in our method is 11 in comparison with 13obtained fromEDDmethod 13 batches can be justified basedon this scenario that two subsequent orders say the firstorder is 100 kg and the other is 150 kg cannot be batchedtogether due to violated batch capacity which is set to be200 kg The number of batches would increase due to theaforementioned issue which affects the total process time tohave an increasing manner This situation will rarely happenin the ATGH method because of the weighted associationbased binary integer programming utilizedwhich batches theorders in order to maximize the association between themwhich is calculated based on the item similarity of ordersand the weight of their corresponding due dates The ATGHmethod uses the picker capacity efficiently

Obviously one of the factors that can influence thecompletion time and the average tardiness of all orders is theprocessing time of the orders Reducing the processing time

The Scientific World Journal 11

Table 7 Computational results of ATGH

Example 1 Example 2 Example 3 Example 4 Example 5119879 56204 29948 90202 163 58898

Pt 605833 1217833 105833 121033 1339

119870 11 19 15 16 16

119860Pt 55076 6409 7055 7564 8368

Table 8 Computational results of EDD integrated with GA TSP and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 124550 42976 18222 5981 1547

Pt 667333 133516 116083 13185 141916

119870 13 21 16 17 16

119860Pt 51333 6357 7255 7775 8869

Table 9 Computational results of EDD integrated with s-shape algorithm and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 1651 62677 2825 13365 17593

Pt 75266 16636 13323 14333 14926

119870 13 21 16 17 16

119860Pt 5789 892 833 8431 932

Table 10 Comparison results

EDD + s-shape + GA sequencing EDD + GA TSP + GA sequencing ATGHTardiness (min) Tardiness (min) Improvement () Tardiness (min) Improvement ()

(1) 1651 124550 2456 56204 6596(2) 62677 42976 3143 29948 5222(3) 2825 18222 3550 90202 6807(4) 13365 5981 5525 1627 8783(5) 17593 1547 1206 58898 6652Average improvement 3176 6811

and travel time for constructing a batch is an outstanding factin reducing the costs and minimizing delays of the customerresponses In our proposed model orders with similar itemsare consolidated together Therefore batching orders withmore similarity can reduce the distance traversed by thepicker facility Also the due date of each order is definedas a weight parameter in calculating the association betweenorders in terms of support which was done for consideringthe due dates in our model Thereupon the model considersdue date and the process time in an integrated manner whichsimultaneously affected the average tardiness of all ordersThis spoken issue is different in the EDD approach in whichthe ordersrsquo due dates are just considered Example 5 canbe considered as evidence of this matter in which the totaland average process time for ATGH is lower than EDDintegrated with GA TSP and GA sequencing with respect to thefact that number of constructed batches for both approachesare happened to be equal after obtaining the results

6 Conclusion

One of the most important concerns of warehouse managersis involvedwith finding the best optimal and cost efficientwayto pick orders placed by customers in order to be known as aresponsible company with satisfied customers in terms of ontime delivery In the current research activity order batchingpicker routing and batch sequencing problems of warehouseprocesses were addressed which are to be solved jointly asthey are mixed up with many manufacturing industriesMathematical solutions of these NP-hard problems might beavailable for problems that are small in nature But thesekinds of solutions may not be applicable to be consideredby warehouse managers Hence we presented novel solutionapproach namely ATGH to solve this issue in accordancewith minimizing the average tardiness of all customersrsquoorders which can be considered as a major objective ofthis research activity A weighted association rule mining

12 The Scientific World Journal

algorithm namely MINWAL was utilized to determine thecustomersrsquo orders association regarding similarity betweenitems and ordersrsquo due dates Actually in our proposedmethodthe due date of each order is defined as a weight parameterin calculating the association between orders So orders withsimilar items and due dates have more chance to be in asame batch This issue is not considered in most of theresearch activities in the literature related to our work andconsidering this issue in the order batching problem is themain contribution of our research activity The previous stepwas followed by a binary integer clustering model in orderto solve the order batching problem which maximizes thesimilarity between orders in terms of support After that aGA was applied to solve a TSP in order to find the optimalrouting path

Finally a batch sequencing problem was addressed inconjunction with a GA

The other research objective of this research addresseswhether the proposed ATGH algorithm can further improvethe solution quality for a multiple parallel aisle warehousewith manual picking system Hence comparisons were madeconsideringATGHEDD integratedGA TSP andGA sequencingand EDD integrated s-shape and GA sequencing algorithms Asdemonstrated by the obtained results ATGH comes out tobe the most striking solution approach in terms of tardinessIn spite of the fact that considering all customer orders asdiscussed in this paper would lead to have a static case beinginvolved with dynamic cases in which different points of timeare assigned to customerrsquos orders arrival would be beneficialSo there might be some potential room of research to extendour discussed approach with respect to dynamic orders

Conflict of Interests

The IBM ILOG CPLEX optimization studio version 124software is not used for financial gain and there is no conflictof interests with any party as it is only a student version

References

[1] ELAAT Kearney Excellence in Logistics ELA Brussels Bel-gium 2004

[2] Y-C Ho T-S Su and Z-B Shi ldquoOrder-batching methods foran order-picking warehouse with two cross aislesrdquo Computersamp Industrial Engineering vol 55 no 2 pp 321ndash347 2008

[3] C-M Hsu K-Y Chen and M-C Chen ldquoBatching ordersin warehouses by minimizing travel distance with geneticalgorithmsrdquo Computers in Industry vol 56 no 2 pp 169ndash1782005

[4] E A Elsayed M-K Lee S Kim and E Scherer ldquoSequencingand batching procedures forminimizing earliness and tardinesspenalty of order retrievalsrdquoThe International Journal of Produc-tion Research vol 31 no 3 pp 727ndash738 1993

[5] S Henn S Koch and W Gerhard Order Batching in OrderPicking Warehouses A Survey of Solution Approaches SpringerLondon UK 2012

[6] J Drury Towards More Efficient Order Picking IMM Mono-graph no 1 The Institute of Materials Management CranfieldUK 1988

[7] M Goetschalckx and J Ashayeri ldquoClassification and design oforder picking systemsrdquo Logistics World pp 99ndash106 1989

[8] J A Tompkins J A White Y A Bozer E H Frazelle and J MA Tanchoco Facilities Planning John Wiley amp Sons HobokenNJ USA 2003

[9] S Henn and G Wascher ldquoTabu search heuristics for the orderbatching problem in manual order picking systemsrdquo EuropeanJournal of Operational Research vol 222 no 3 pp 484ndash4942012

[10] S Henn ldquoAlgorithms for on-line order batching in an orderpicking warehouserdquo Computers and Operations Research vol39 no 11 pp 2549ndash2563 2012

[11] R de Koster ldquoHow to assess a warehouse operation in a singletourrdquo Tech Rep RSM Erasmus University Rotterdam TheNetherlands 2004

[12] R de Koster T Le-Duc and K J Roodbergen ldquoDesign andcontrol of warehouse order picking a literature reviewrdquo Euro-pean Journal of Operational Research vol 182 no 2 pp 481ndash5012007

[13] R A Ruben and F R Jacobs ldquoBatch construction heuristics andstorage assignment strategies for walkride and pick systemsrdquoManagement Science vol 45 no 4 pp 575ndash596 1999

[14] E A Elsayed ldquoAlgorithms for optimal material handling inautomatic warehousing systemsrdquo International Journal of Pro-duction Research vol 19 no 5 pp 525ndash535 1981

[15] G Clarke and J W Wright ldquoScheduling of vehicles froma central depot to a number of delivery pointsrdquo OperationsResearch vol 12 no 4 pp 568ndash581 1964

[16] H D Vinod ldquoInteger programming and the theory of group-ingrdquo Journal of the American Statistical Association vol 64 no326 pp 506ndash519 1969

[17] R D Armstrong W D Cook and A L Saipe ldquoOptimalbatching in a semi-automated order picking systemrdquo Journal ofthe Operational Research Society vol 30 no 8 pp 711ndash720 1979

[18] H D Ratliff and A S Rosenthal ldquoOrder-picking in a rect-angular warehouse a solvable case of the traveling salesmanproblemrdquo Operations Research vol 31 no 3 pp 507ndash521 1983

[19] A Kusiak A Vannelli and R K Kumar ldquoClustering analysismodels and algorithmsrdquo Cybernetics vol 15 pp 139ndash154 1986

[20] D R Gibson and G P Sharp ldquoOrder batching proceduresrdquoEuropean Journal of Operational Research vol 58 no 1 pp 57ndash67 1992

[21] M B Rosenwein ldquoA comparison of heuristics for the problemofbatching orders for warehouse selectionrdquo International Journalof Production Research vol 34 no 3 pp 657ndash664 1996

[22] A J RMGademann J P van denBerg andHH van derHoffldquoAn order batching algorithm for wave picking in a parallel-aislewarehouserdquo IIE Transactions vol 33 no 5 pp 385ndash398 2001

[23] M-C Chen and H-P Wu ldquoAn association-based clusteringapproach to order batching considering customer demandpatternsrdquo Omega vol 33 no 4 pp 333ndash343 2005

[24] NGademann and S van deVelde ldquoOrder batching tominimizetotal travel time in a parallel-aisle warehouserdquo IIE Transactionsvol 37 no 1 pp 63ndash75 2005

[25] Y A Bozer and J W Kile ldquoOrder batching in walk-and-pick order picking systemsrdquo International Journal of ProductionResearch vol 46 no 7 pp 1887ndash1909 2008

[26] O Kulak Y Sahin and M E Taner ldquoJoint order batching andpicker routing in single and multiple-cross-aisle warehousesusing cluster-based tabu search algorithmsrdquo Flexible Servicesand Manufacturing Journal vol 24 no 1 pp 52ndash80 2012

The Scientific World Journal 13

[27] E A Elsayed and M-K Lee ldquoOrder processing in automatedstorageretrieval systems with due datesrdquo IIE Transactions vol28 no 7 pp 567ndash577 1996

[28] J Won and S Olafsson ldquoJoint order batching and order pickingin warehouse operationsrdquo International Journal of ProductionResearch vol 43 no 7 pp 1427ndash1442 2005

[29] C-Y Tsai J J H Liou and T-M Huang ldquoUsing a multiple-GA method to solve the batch picking problem consideringtravel distance and order due timerdquo International Journal of Pro-duction Research vol 46 no 22 pp 6533ndash6555 2008

[30] Y KaoM H Chen and Y T Huang ldquoA hybrid algorithm basedon ACO and PSO for capacitated vehicle routing problemsrdquoMathematical Problems in Engineering vol 2012 Article ID726564 17 pages 2012

[31] X Gan Y Wang S Li and B Niu ldquoVehicle routing problemwith time windows and simultaneous delivery and pick-up ser-vice based on MCPSOrdquoMathematical Problems in Engineeringvol 2012 Article ID 104279 11 pages 2012

[32] M B M de Koster E S van der Poort and M WoltersldquoEfficient orderbatching methods in warehousesrdquo InternationalJournal of ProductionResearch vol 37 no 7 pp 1479ndash1504 1999

[33] Y-C Ho and Y-Y Tseng ldquoA study on order-batching methodsof order-picking in a distribution centre with two cross-aislesrdquoInternational Journal of Production Research vol 44 no 17 pp3391ndash3417 2006

[34] C Theys O Braysy W Dullaert and B Raa ldquoUsing a TSPheuristic for routing order pickers in warehousesrdquo EuropeanJournal of Operational Research vol 200 no 3 pp 755ndash7632010

[35] S Henn and V Schmid Metaheuristics for Order Batching andSequencing in Manual Order Picking Systems Univ Fak furWirtschaftswiss 2011

[36] M Hahsler B Grun and K Hornik ldquoArulesmdasha computationalenvironment for mining association rules and frequent itemsetsrdquo Journal of Statistical Software vol 14 2005

[37] R Agrawal T Imielinski and A Swami ldquoMining associationrules between sets of items in large databasesrdquo ACM SIGMODRecord vol 22 no 2 pp 207ndash216 1993

[38] S Encheva Y Kondratenko M Z Solesvik and S TuminldquoDecision support systems in logisticsrdquo in Proceeding of Inter-national electronic conference on computer science pp 254ndash2562007

[39] M-C Chen ldquoConfiguration of cellular manufacturing systemsusing association rule inductionrdquo International Journal of Pro-duction Research vol 41 no 2 pp 381ndash395 2003

[40] R Srikant and R Agrawal ldquoMining generalized associationrulesrdquo Future Generation Computer Systems vol 13 no 2-3 pp161ndash180 1997

[41] J Wu and XM Li ldquoAn effective mining algorithm for weightedassociation rules in communication networksrdquo Journal of Com-puters vol 3 no 10 pp 20ndash27 2008

[42] C H Cai A W C Fu C H Cheng and W W KwongldquoMining association rules with weighted itemsrdquo in Proceedingsof the International Symposium on Database Engineering ampApplications pp 68ndash77 Cardiff UK 1998

[43] J HollandAdaptation in Natural and Artificial Systems Univer-sity of Michigan Press Ann Arbor Mich USA 1975

[44] A A Taleizadeh F Barzinpour and H-M Wee ldquoMeta-heuristic algorithms for solving a fuzzy single-period problemrdquoMathematical and Computer Modelling vol 54 no 5-6 pp1273ndash1285 2011

[45] H Yang J Yi J Zhao and Zh Dong ldquoExtreme learning ma-chine based genetic algorithm and its application in powersystem economic dispatchrdquo Neurocomputing vol 102 pp 154ndash162 2013

Submit your manuscripts athttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

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Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

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ArtificialNeural Systems

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RoboticsJournal of

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Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 The Scientific World Journal

1198841198941 if order 119895 is chosen as a batch median0 otherwise

Maximize119873

sum

119894=1

119873

sum

119895=1

119878119894119895lowast 119883119894119895 (3)

Subject to119873

sum

119895=1

119883119894119895= 1 for 119894 119895 = 1 2 119873

(4)

119883119894119895le 119884119895

for 119894 119895 = 1 2 119873 (5)

119873

sum

119895=1

119884119895= 119870 (6)

119873

sum

119894=1

119881119894119883119894119895le 119862V for 119895 = 1 2 119873 (7)

119883119894119895= 0 1 for 119894 119895 = 1 2 119873 (8)

119884119895= 0 1 for 119895 = 1 2 119873 (9)

In the current model the objective function (3) maxi-mizes the sum of support from all orders to their relevantbatch medians as 119870 orders are selected as batch mediansThis model will maximize the support between orders in thebatches Membership of each order in just one batch wouldbe guaranteed by constraint (4) The number of batches with119870 is limited using constraints (5) and (6) The total quantityin a tour is also represented as a limitation in constraint (7) Itcannot exceed the capacity of order picker 119862V For the orderbatchingmodel to have a binary solutionwould be ensured byconstraints (8) and (9) For calculating the initial numbers ofbatches (10) will be used where sum119881

119894represents total volume

of all orders as following

119870initial = lceilsum119881119894

119862V

rceil (10)

32 Picker Routing

321 Genetic Algorithms In this research GAs which wereintroduced by Holland [43] have been used in order tosolve the picker routing and batch sequencing problem Thebiological evolution process is used as a base fact in a GA forproblem solving to a great extentThemain steps of a GA canbe described as follows [44 45]

(i) populating a series of solutions(ii) solutions assessment(iii) picking the most appropriate solutions(iv) generating new solutions based on some genetic

operation

In step (iii) selection forms the main part of a GA Thebad solutions are removed by selection and the good onesremain Random generation over the feasible or infeasible

solution space of a problem is used as a basis for creatingthe population Each solution is evaluated using the fitnessfunction Selection would be done according to certain selec-tions of fitness together with the probability that is assignedto it Generating a fresh population and a better solutionare involved with recombination of the individual solutionCrossover and mutation are the most used GA operators forgenerating new offspring from the old generation of parentsParameters of the GA are originated from the probabilitiesof mutation and crossover Exploration of the sample spaceis diversified by these operators In this research the GA hasbeenutilized in two steps First theGAhas beenused in orderto solve the picker routing problem Consider that in thewarehouse a set of items should be picked by a picker in onetour Therefore the picker routing problem in the warehousecould be categorized as a TSP The goal in the picker routingproblem as a TSP in the warehouse is to find an optimal pathfor minimizing the travel distance traversed by the pickingmachine where each node is visited at least once In thisresearch for a given number of itemset in a batch GA TSPhas been used to find the minimum travel distance whichis traversed by the picking machine Second a GA has beenutilized in order to sequence all batches namely GA sequencingwhichwould be further processedThe batches are sequencedbased on a fitness function which minimizes the tardiness ofall ordersThedetailed procedures ofGA TSP andGA sequencingare illustrated in Sections 323 and 324

322 Warehouse Layout Thewarehouse discussed through-out this research is assumed to have a 3D multi-parallel-aisle warehouse layout The picker will start the tour fromthe depot which is located in front of the leftmost aisle andreturns back to it A picker leaves the depot After that ittravels to a certain buffer spot where it picks the requestedorder(s) and puts the order(s) into a certain picker Then itcontinue on its way to another spot These procedures wouldbe repeated again and again so that all the requested ordersare picked Then it is time for the picker to return to thedepot

The graphic layout of the warehouse zone involved in thisresearch activity is pictorially displayed in Figure 1 It showsthe 119909 and 119910 dimensions of the 3D warehouse layout Eachitem in the warehouse has a specific location For examplethe location of item 119894 can be addressed by 119871

119894= (119909119894 119910119894 119911119894)

Consider two locations as 119871119886= (119909119886 119910119886 119911119886) and 119871

119887= (119909119887

119910119887 119911119887) If two locations are located in the same sub-aisles

of the same block the distance between these two itemslocations (119863

119886119887) will be calculated using

119863119886119887

=1003816100381610038161003816119909119886 minus 119909119887

1003816100381610038161003816 +1003816100381610038161003816119910119886 minus 119910119887

1003816100381610038161003816 +1003816100381610038161003816119911119886 minus 119911119887

1003816100381610038161003816 (11)

Equation (11) is not applicable if the two storage locationsbelong to different subaisles that are from the same block [34]In order to solve this problem distances of all possible pathsbetween two locations have been calculated and the shortestone was considered

323 GA TSP for the Picker Routing Problem In order todetermine the optimal route of a picker related to a given

The Scientific World Journal 7

Lb=(xbybzb)

La=(xayaza)

Li=(xiyizi)

Depot

Figure 1 Warehouse layout

1 2 3 4 5 6 7 8 9L0 L1 L8 L11 L9

L13 L12 L10 L0

Figure 2 A feasible chromosome encoding in GA TSP

set of items in a batch a GA has been used to solve theTSP problem to minimize travelling time In a GA eachsolution is considered as a chromosome The steps of GA TSPare described as follows

(1) for each batch establishing the structure of a chromo-somes

(2) generate initial feasible chromosomes(3) evaluate each chromosome based on the GA TSP fit-

ness function(4) apply the Selection operation(5) apply the Crossover operation(6) apply the Mutation operation(7) evaluate each offspring based on the fitness function(8) go to step 4 if the termination criterion is not satis-

fied otherwise terminate and return the best solutionas the optimal path

At the first step of GA TSP the structure of the chro-mosome should be established In the GA TSP the locationof an item that should be visited is presented by the valueof a gene Accordingly the sequence of visiting the itemlocation is denoted by the order of a gene in a chromosomeTherefore the total number of items locations in a specificbatch is considered equal to the length of a chromosome Forexample if the length of a chromosome is nine it means nineplaces should be visited by the picker as shown in Figure 2If 119871119894represents the item location in the warehouse 119894 = 1

119899 it shows that the picker leaves the depot (1198710denotes

the location of a depot) goes to 1198711 1198718 11987111 1198719 11987113 11987112 11987110

and then comes back to the depotIn the second step the initial number of feasible chro-

mosomes should be generated randomly for each batch

The number of initial chromosomes could be different basedon the complexity of the problem It should be consideredthat the first and the last gene of each chromosome should beassigned individually to the location of a depot In step threeeach chromosome will be evaluated based on the GA TSPfitness function which is formulated as follows

fitness = Min119899minus1

sum

119895=1

119863119895119895+1

(12)

where 119863119895119895+1

denotes the distance between two subsequentgenes in a chromosome 119895 = 1 119899 minus 1 represents the orderof the genes and 119899 is number of genes

Subsequently step four is involved with the selectionoperation In each iteration the selection operator usesfitness values to select the parents of the next generationIn this research the roulette wheel selection policy hasbeen used to guarantee that the most appropriate pairs ofchromosomes have been selected to generate offspring Itmeans that a higher probability of being selected to createa new population will be assigned to the chromosomes withhigher fitness values

Steps five and six are involved with conducting thecrossover and mutation operation The construction of theoffspring is realized by the crossover and the diversity of theindividuals is maintained by the mutation operator Blocksof genes between chromosomes are traded by means of thecrossover operation inwhich the exploitation of a particularlyprofitable portion of the parameter space is allowed to theblock Here permutation order based crossover (POP) as avariation of the distinguished order crossover has been usedAs shown in Figure 3 the original idea is about choosing twoparents and a point for cutting The first portion of offspring2 contains the first portion of parent 1 up to the cut pointTo form the second portion of offspring 2 consider all genesin parent 2 except those genes that already exist in the firstportion of the offspring 2

As shown in Figure 4 two positions are randomlyselected in a chromosome and exchanged with each othernamely the order based mutation approach SWAP isadopted for solving GA TSP Step six is followed by step sevenwhich includes the evaluation of each offspring that has beengenerated in the previous step based on the GA TSP fitnessfunction In step eight termination criteria will be checkedIf the termination condition is met then the process will beterminated After that the best chromosome will be selectedas the optimal solution Otherwise the selection process is tobe conducted again to generate a new population

324 GA for Sequencing the Batches Sequencing the batchesshould be conducted in order to minimize the averagetardiness of all orders The problem of batch sequencing canbe defined as finding the orders of batches which should beprocessed further For this purpose the GA sequencing methodhas been utilized In this research all of the batches whichwere constructed using binary integer programming andGA TSP will be sequenced by using the GA The populationgeneration selectionmutation and stopping threshold in theGA sequencing are identical to those in the GA TSP However

8 The Scientific World Journal

Parent 1

Parent 2

Offspring 2

Offspring 1

L1 L8 L9

L1 L1L8 L8L9

L11 L13 L12 L10

L1 L8 L9 L11L13 L12L10

L13 L9 L11 L11 L13 L12

L12

L10

L10L1L8L12

L13 L9 L11 L10

L1 L8 L12L13 L9 L11 L10

Figure 3 Crossover operation

Parent L1

L1

L8

L8

L9

L9

L11

L11

L13

L13

L12

L12

L10

L10Offspring

Figure 4 Mutation operation

1 2 3 4 5b1 b4 b3 b5 b2

Figure 5 Chromosome structure

the fitness function and chromosome structure are differentthan that of GA TSP In contrast with GA TSP chromosomesstructure batch number is presented by the value of a geneAccordingly the sequence of batch is denoted by the orderof a gene in a chromosome For instance Figure 5 shows thechromosome structure If 119887

119894isin 119861 denotes a set of batches

Figure 5 shows that 1198871 1198874 1198873 1198875 1198872should be processed

respectively Therefore the length of a chromosome is equalto the number of the batches The GA sequencing fitness func-tion for chromosome 119910 is described in

fitness119910=

1

119873

119899

sum

119896=1

sum

119894isin119887119896

119862119879 (119887119896) minus 119863119906 (119900

119894) (13)

119862119879 (119887119896) = 119862119879 (119887

119896minus1) + Pt (119887119896) (14)

Pt (119887119896) =Opt (119887

119896)

119881119878 (15)

where119873 denotes for the total number of orders 119896 = 1 119899

represents the batch sequence in which 119899 is the total numberof batches 119900

119894stands for the order 119894 in which 119894 isin 119887

119896 119887119896

denotes the batch in 119896th position of a sequence for specificchromosome 119862119879(119887

119896) stands for the completion time of a

batch in the 119896th position of a sequence 119863119906(119900119894) is the symbol

for due date of order 119894which belongs to 119887119896 Pt(119887119896) is the process

time of 119887119896 Opt(119887

119896) is the optimal path solution for 119887

119896 Finally

119881119878 is the moving speed of the picker

4 Research Methodology

In this research the proposed methodology consists of fivesteps These steps are described as follows and shown pic-torially in Figure 6

Step 1 It is involved with utilizing theMINWAL algorithm inorder to determine the associations between orders in termsof support considering their due dates In this step an orderdue date will be considered as the weight of an order It meansorders that are similar in items and due date should havemoresupport for being batched together

Step 2 It encompasses determining the initial number ofbatches In this step the initial number of batches will becalculated by (10)

Step 3 It includes the problem of modeling and solving firstthe binary integer programming model for order clusteringand batching will be modeled Next this model will be solvedin order to determine the structures of batches

Step 4 It is about the picker routing problem The optimalpath for a picker within each batch will be determined bysolving a TSP using GA

Step 5 It is about sequencing the batches based on the fitnessfunction which minimizes the average tardiness of all ordersIn this step GA algorithmwill be utilized in order to solve thebatch sequencing problem regarding the fitness function

5 Computational Experiments and Results

51 Test Problems and Parameter Setting In this sectionfive numerical test problems with different settings andparameters have been carried out in order to illustrate theproficiency of the proposed method with respect to averagetardiness of all orders Table 5 summarizes the descriptionof these test problems This table includes information withregard to number of items number of orders capacity of thepicker total weight of all the items in an order andminimumnumber of batches

In these test problems the quantity of items quantityof orders capacity of the picker device (119862V) and amount ofeach item are predetermined Each order contains a certain

The Scientific World Journal 9

Start

Input order informationwarehouse setting picker

setting

Compute the supportbetween customersrsquo

orders using MINWAL

Calculating initialnumber of batches (K) to

assign the orders

Solving the binaryinteger programmingclustering model for

constructing the batches

Feasiblesolution

Yes

No

Solving picker routing

Solving batch sequencingproblem using

End

K = K + 1

problem using GA TSP

GA sequencing

Figure 6 Methodology framework

quantity of items and also each item has a certain quantityThese quantities are produced randomly The distributionfunction type for the number of items in each order is Normal[5 10] Also the distribution function type for each itemquantity in an order is uniform [1 10] The picker movingspeed is set as 119881119878 = 2 (ms) The warehouse has 3000 itemlocation The GA control parameters for solving the TSP andsequencing problem are tabulated in Table 6

All of the proposed approaches are coded usingMATLABversion 790 and Java 15 programming language Also theIBM ILOG CPLEX optimization studio version 124 softwarewas utilized jointly with MATLAB in order to facilitate theprocess of solving the binary integer programming problemExamples are run by means of a computer featured by

4 Gigabytes random access memory (RAM) and a 220Gigahertz Intel processor (T6600)

52 Computational Results and Discussion In order to revealthe competency of the proposed solution approach it wascompared with the other existing methods in the literatureOne of the latest published research activities in the literaturewhich deals with minimization of tardiness has been con-ducted by Henn and schmid [35] They utilized the EarliestDueDate (EDD) approach as their constructive algorithm forcomparing with their proposed approaches They proposedtwo heuristics Iterated Local Search andAttribute-BasedHillClimber are the two employed metaheuristics in their modelAccording to different test problems the efficiency of their

10 The Scientific World Journal

Table 5 Test problems description

Example 1 Example 2 Example 3 Example 4 Example 5Number of orders 40 60 80 100 150

Number of items 80 120 160 200 200

Capacity of picker 200 200 300 400 550

Total weight 2212 3668 4393 6110 8369

Minimum number of batch 11 19 15 16 16

Due dates range 60 90 120 180 240

Table 6 GA control parameters

GA TSP GA sequencing

Population size 500 200

Cross over strategyPermutation orderbased crossover

(POP)

Permutation orderbased crossover

(POP)Cross over probability 08 08

Mutation strategy SWAP SWAPMutation probability 02 02

Iteration number 10000 10000

Parent selection method Roulette wheel Roulette wheel

proposed methods was illustrated Their obtained resultsfrom their proposed approaches showed a 46 improvementon tardiness compared to EDD They used an s-shape andlargest gap routing strategy for their proposed method Inorder to make a reasonable comparison with their proposedmethod EDD was integrated with the s-shape routing strat-egy and GA sequencing Then based on improvement on EDDwe can compare our proposed method with their proposedmethod In the EDD approach due dates play an importantrole in sorting all orders Based on an ascending sequenceassigning the orders to batches ensures that no capacityviolation can happen for the picking device

For the warehouse condition which is described inSection 323 our proposed approach EDD integrated withGA TSP and GA sequencing and EDD integrated with s-shapealgorithm and GA sequencing have been evaluated in termsof average tardiness for all orders (119879) which is calculatedbased on (13) number of batches (119870) total process time(Pt) and average process time for each batch (119860Pt) All ofthe GA setting for the TSP and sequencing problems areconsidered the same for all methods Also the settings whichhave been mentioned in Section 51 were considered for all ofthe compared methods

The results for all of the test problems for the threeapproaches are tabulated in Tables 7 8 and 9 As it isillustrated in Table 10 EDD integrated with GA TSP andGA sequencing has a 3176 improvement on average in termsof average tardiness of all orders than EDD integrated withthe s-shape algorithm and GA sequencing It can be perceivedthat GA TSP performs better than s-shape Also the ATGHmethod has a 6811 improvement in average tardinessoff all orders than EDD integrated with s-shape algorithm

and GA sequencing Consequently the ATGH method achievesmuch improvement against Henn and Schmidrsquos [35] pro-posedmethod According to Tables 7 and 8 it is also worth tomention thatGA TSP performs better than the s-shape routingstrategy in terms of process time The improvements whichhave been achieved in this research could be justified basedon the following reasons

The problem of order batching deals with several sig-nificant drivers in which the number of batches can beconsidered as a key factor [23] As a justification to thismatter it is worth to mention that total travel distance maynot decrease directly due to a decline in the number ofbatches But a decline in the number of batches will increasebatch sizes which may lead to an increase in the distancetravelled by order picker However an advanced exploitationof picker capacity to load efficiently may reduce the numberof constructed batches together reducing the total traveldistance Improving the total travel distance traversed by thepicker can minimize the average tardiness of all orders Interms of number of the batches according to Tables 7 8 and9 it can be perceived that our proposed method assigns theorders to the batches optimally The ATGH method assignsthe orders to batches based on their item similarity andtheir due dates and it uses the picker capacity efficiently Butthe EDD method allocates the orders to batches based onan ascending sequence of their due date and it should beensured that no capacity violation will happen for the pickingdevice which may lead to an inefficient order batching Tohighlight the problem an example can be made based on theresults tabulated in Tables 7 and 8 The number of batchesfor example 1 in our method is 11 in comparison with 13obtained fromEDDmethod 13 batches can be justified basedon this scenario that two subsequent orders say the firstorder is 100 kg and the other is 150 kg cannot be batchedtogether due to violated batch capacity which is set to be200 kg The number of batches would increase due to theaforementioned issue which affects the total process time tohave an increasing manner This situation will rarely happenin the ATGH method because of the weighted associationbased binary integer programming utilizedwhich batches theorders in order to maximize the association between themwhich is calculated based on the item similarity of ordersand the weight of their corresponding due dates The ATGHmethod uses the picker capacity efficiently

Obviously one of the factors that can influence thecompletion time and the average tardiness of all orders is theprocessing time of the orders Reducing the processing time

The Scientific World Journal 11

Table 7 Computational results of ATGH

Example 1 Example 2 Example 3 Example 4 Example 5119879 56204 29948 90202 163 58898

Pt 605833 1217833 105833 121033 1339

119870 11 19 15 16 16

119860Pt 55076 6409 7055 7564 8368

Table 8 Computational results of EDD integrated with GA TSP and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 124550 42976 18222 5981 1547

Pt 667333 133516 116083 13185 141916

119870 13 21 16 17 16

119860Pt 51333 6357 7255 7775 8869

Table 9 Computational results of EDD integrated with s-shape algorithm and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 1651 62677 2825 13365 17593

Pt 75266 16636 13323 14333 14926

119870 13 21 16 17 16

119860Pt 5789 892 833 8431 932

Table 10 Comparison results

EDD + s-shape + GA sequencing EDD + GA TSP + GA sequencing ATGHTardiness (min) Tardiness (min) Improvement () Tardiness (min) Improvement ()

(1) 1651 124550 2456 56204 6596(2) 62677 42976 3143 29948 5222(3) 2825 18222 3550 90202 6807(4) 13365 5981 5525 1627 8783(5) 17593 1547 1206 58898 6652Average improvement 3176 6811

and travel time for constructing a batch is an outstanding factin reducing the costs and minimizing delays of the customerresponses In our proposed model orders with similar itemsare consolidated together Therefore batching orders withmore similarity can reduce the distance traversed by thepicker facility Also the due date of each order is definedas a weight parameter in calculating the association betweenorders in terms of support which was done for consideringthe due dates in our model Thereupon the model considersdue date and the process time in an integrated manner whichsimultaneously affected the average tardiness of all ordersThis spoken issue is different in the EDD approach in whichthe ordersrsquo due dates are just considered Example 5 canbe considered as evidence of this matter in which the totaland average process time for ATGH is lower than EDDintegrated with GA TSP and GA sequencing with respect to thefact that number of constructed batches for both approachesare happened to be equal after obtaining the results

6 Conclusion

One of the most important concerns of warehouse managersis involvedwith finding the best optimal and cost efficientwayto pick orders placed by customers in order to be known as aresponsible company with satisfied customers in terms of ontime delivery In the current research activity order batchingpicker routing and batch sequencing problems of warehouseprocesses were addressed which are to be solved jointly asthey are mixed up with many manufacturing industriesMathematical solutions of these NP-hard problems might beavailable for problems that are small in nature But thesekinds of solutions may not be applicable to be consideredby warehouse managers Hence we presented novel solutionapproach namely ATGH to solve this issue in accordancewith minimizing the average tardiness of all customersrsquoorders which can be considered as a major objective ofthis research activity A weighted association rule mining

12 The Scientific World Journal

algorithm namely MINWAL was utilized to determine thecustomersrsquo orders association regarding similarity betweenitems and ordersrsquo due dates Actually in our proposedmethodthe due date of each order is defined as a weight parameterin calculating the association between orders So orders withsimilar items and due dates have more chance to be in asame batch This issue is not considered in most of theresearch activities in the literature related to our work andconsidering this issue in the order batching problem is themain contribution of our research activity The previous stepwas followed by a binary integer clustering model in orderto solve the order batching problem which maximizes thesimilarity between orders in terms of support After that aGA was applied to solve a TSP in order to find the optimalrouting path

Finally a batch sequencing problem was addressed inconjunction with a GA

The other research objective of this research addresseswhether the proposed ATGH algorithm can further improvethe solution quality for a multiple parallel aisle warehousewith manual picking system Hence comparisons were madeconsideringATGHEDD integratedGA TSP andGA sequencingand EDD integrated s-shape and GA sequencing algorithms Asdemonstrated by the obtained results ATGH comes out tobe the most striking solution approach in terms of tardinessIn spite of the fact that considering all customer orders asdiscussed in this paper would lead to have a static case beinginvolved with dynamic cases in which different points of timeare assigned to customerrsquos orders arrival would be beneficialSo there might be some potential room of research to extendour discussed approach with respect to dynamic orders

Conflict of Interests

The IBM ILOG CPLEX optimization studio version 124software is not used for financial gain and there is no conflictof interests with any party as it is only a student version

References

[1] ELAAT Kearney Excellence in Logistics ELA Brussels Bel-gium 2004

[2] Y-C Ho T-S Su and Z-B Shi ldquoOrder-batching methods foran order-picking warehouse with two cross aislesrdquo Computersamp Industrial Engineering vol 55 no 2 pp 321ndash347 2008

[3] C-M Hsu K-Y Chen and M-C Chen ldquoBatching ordersin warehouses by minimizing travel distance with geneticalgorithmsrdquo Computers in Industry vol 56 no 2 pp 169ndash1782005

[4] E A Elsayed M-K Lee S Kim and E Scherer ldquoSequencingand batching procedures forminimizing earliness and tardinesspenalty of order retrievalsrdquoThe International Journal of Produc-tion Research vol 31 no 3 pp 727ndash738 1993

[5] S Henn S Koch and W Gerhard Order Batching in OrderPicking Warehouses A Survey of Solution Approaches SpringerLondon UK 2012

[6] J Drury Towards More Efficient Order Picking IMM Mono-graph no 1 The Institute of Materials Management CranfieldUK 1988

[7] M Goetschalckx and J Ashayeri ldquoClassification and design oforder picking systemsrdquo Logistics World pp 99ndash106 1989

[8] J A Tompkins J A White Y A Bozer E H Frazelle and J MA Tanchoco Facilities Planning John Wiley amp Sons HobokenNJ USA 2003

[9] S Henn and G Wascher ldquoTabu search heuristics for the orderbatching problem in manual order picking systemsrdquo EuropeanJournal of Operational Research vol 222 no 3 pp 484ndash4942012

[10] S Henn ldquoAlgorithms for on-line order batching in an orderpicking warehouserdquo Computers and Operations Research vol39 no 11 pp 2549ndash2563 2012

[11] R de Koster ldquoHow to assess a warehouse operation in a singletourrdquo Tech Rep RSM Erasmus University Rotterdam TheNetherlands 2004

[12] R de Koster T Le-Duc and K J Roodbergen ldquoDesign andcontrol of warehouse order picking a literature reviewrdquo Euro-pean Journal of Operational Research vol 182 no 2 pp 481ndash5012007

[13] R A Ruben and F R Jacobs ldquoBatch construction heuristics andstorage assignment strategies for walkride and pick systemsrdquoManagement Science vol 45 no 4 pp 575ndash596 1999

[14] E A Elsayed ldquoAlgorithms for optimal material handling inautomatic warehousing systemsrdquo International Journal of Pro-duction Research vol 19 no 5 pp 525ndash535 1981

[15] G Clarke and J W Wright ldquoScheduling of vehicles froma central depot to a number of delivery pointsrdquo OperationsResearch vol 12 no 4 pp 568ndash581 1964

[16] H D Vinod ldquoInteger programming and the theory of group-ingrdquo Journal of the American Statistical Association vol 64 no326 pp 506ndash519 1969

[17] R D Armstrong W D Cook and A L Saipe ldquoOptimalbatching in a semi-automated order picking systemrdquo Journal ofthe Operational Research Society vol 30 no 8 pp 711ndash720 1979

[18] H D Ratliff and A S Rosenthal ldquoOrder-picking in a rect-angular warehouse a solvable case of the traveling salesmanproblemrdquo Operations Research vol 31 no 3 pp 507ndash521 1983

[19] A Kusiak A Vannelli and R K Kumar ldquoClustering analysismodels and algorithmsrdquo Cybernetics vol 15 pp 139ndash154 1986

[20] D R Gibson and G P Sharp ldquoOrder batching proceduresrdquoEuropean Journal of Operational Research vol 58 no 1 pp 57ndash67 1992

[21] M B Rosenwein ldquoA comparison of heuristics for the problemofbatching orders for warehouse selectionrdquo International Journalof Production Research vol 34 no 3 pp 657ndash664 1996

[22] A J RMGademann J P van denBerg andHH van derHoffldquoAn order batching algorithm for wave picking in a parallel-aislewarehouserdquo IIE Transactions vol 33 no 5 pp 385ndash398 2001

[23] M-C Chen and H-P Wu ldquoAn association-based clusteringapproach to order batching considering customer demandpatternsrdquo Omega vol 33 no 4 pp 333ndash343 2005

[24] NGademann and S van deVelde ldquoOrder batching tominimizetotal travel time in a parallel-aisle warehouserdquo IIE Transactionsvol 37 no 1 pp 63ndash75 2005

[25] Y A Bozer and J W Kile ldquoOrder batching in walk-and-pick order picking systemsrdquo International Journal of ProductionResearch vol 46 no 7 pp 1887ndash1909 2008

[26] O Kulak Y Sahin and M E Taner ldquoJoint order batching andpicker routing in single and multiple-cross-aisle warehousesusing cluster-based tabu search algorithmsrdquo Flexible Servicesand Manufacturing Journal vol 24 no 1 pp 52ndash80 2012

The Scientific World Journal 13

[27] E A Elsayed and M-K Lee ldquoOrder processing in automatedstorageretrieval systems with due datesrdquo IIE Transactions vol28 no 7 pp 567ndash577 1996

[28] J Won and S Olafsson ldquoJoint order batching and order pickingin warehouse operationsrdquo International Journal of ProductionResearch vol 43 no 7 pp 1427ndash1442 2005

[29] C-Y Tsai J J H Liou and T-M Huang ldquoUsing a multiple-GA method to solve the batch picking problem consideringtravel distance and order due timerdquo International Journal of Pro-duction Research vol 46 no 22 pp 6533ndash6555 2008

[30] Y KaoM H Chen and Y T Huang ldquoA hybrid algorithm basedon ACO and PSO for capacitated vehicle routing problemsrdquoMathematical Problems in Engineering vol 2012 Article ID726564 17 pages 2012

[31] X Gan Y Wang S Li and B Niu ldquoVehicle routing problemwith time windows and simultaneous delivery and pick-up ser-vice based on MCPSOrdquoMathematical Problems in Engineeringvol 2012 Article ID 104279 11 pages 2012

[32] M B M de Koster E S van der Poort and M WoltersldquoEfficient orderbatching methods in warehousesrdquo InternationalJournal of ProductionResearch vol 37 no 7 pp 1479ndash1504 1999

[33] Y-C Ho and Y-Y Tseng ldquoA study on order-batching methodsof order-picking in a distribution centre with two cross-aislesrdquoInternational Journal of Production Research vol 44 no 17 pp3391ndash3417 2006

[34] C Theys O Braysy W Dullaert and B Raa ldquoUsing a TSPheuristic for routing order pickers in warehousesrdquo EuropeanJournal of Operational Research vol 200 no 3 pp 755ndash7632010

[35] S Henn and V Schmid Metaheuristics for Order Batching andSequencing in Manual Order Picking Systems Univ Fak furWirtschaftswiss 2011

[36] M Hahsler B Grun and K Hornik ldquoArulesmdasha computationalenvironment for mining association rules and frequent itemsetsrdquo Journal of Statistical Software vol 14 2005

[37] R Agrawal T Imielinski and A Swami ldquoMining associationrules between sets of items in large databasesrdquo ACM SIGMODRecord vol 22 no 2 pp 207ndash216 1993

[38] S Encheva Y Kondratenko M Z Solesvik and S TuminldquoDecision support systems in logisticsrdquo in Proceeding of Inter-national electronic conference on computer science pp 254ndash2562007

[39] M-C Chen ldquoConfiguration of cellular manufacturing systemsusing association rule inductionrdquo International Journal of Pro-duction Research vol 41 no 2 pp 381ndash395 2003

[40] R Srikant and R Agrawal ldquoMining generalized associationrulesrdquo Future Generation Computer Systems vol 13 no 2-3 pp161ndash180 1997

[41] J Wu and XM Li ldquoAn effective mining algorithm for weightedassociation rules in communication networksrdquo Journal of Com-puters vol 3 no 10 pp 20ndash27 2008

[42] C H Cai A W C Fu C H Cheng and W W KwongldquoMining association rules with weighted itemsrdquo in Proceedingsof the International Symposium on Database Engineering ampApplications pp 68ndash77 Cardiff UK 1998

[43] J HollandAdaptation in Natural and Artificial Systems Univer-sity of Michigan Press Ann Arbor Mich USA 1975

[44] A A Taleizadeh F Barzinpour and H-M Wee ldquoMeta-heuristic algorithms for solving a fuzzy single-period problemrdquoMathematical and Computer Modelling vol 54 no 5-6 pp1273ndash1285 2011

[45] H Yang J Yi J Zhao and Zh Dong ldquoExtreme learning ma-chine based genetic algorithm and its application in powersystem economic dispatchrdquo Neurocomputing vol 102 pp 154ndash162 2013

Submit your manuscripts athttpwwwhindawicom

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Electrical and Computer Engineering

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ArtificialNeural Systems

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RoboticsJournal of

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Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 7

Lb=(xbybzb)

La=(xayaza)

Li=(xiyizi)

Depot

Figure 1 Warehouse layout

1 2 3 4 5 6 7 8 9L0 L1 L8 L11 L9

L13 L12 L10 L0

Figure 2 A feasible chromosome encoding in GA TSP

set of items in a batch a GA has been used to solve theTSP problem to minimize travelling time In a GA eachsolution is considered as a chromosome The steps of GA TSPare described as follows

(1) for each batch establishing the structure of a chromo-somes

(2) generate initial feasible chromosomes(3) evaluate each chromosome based on the GA TSP fit-

ness function(4) apply the Selection operation(5) apply the Crossover operation(6) apply the Mutation operation(7) evaluate each offspring based on the fitness function(8) go to step 4 if the termination criterion is not satis-

fied otherwise terminate and return the best solutionas the optimal path

At the first step of GA TSP the structure of the chro-mosome should be established In the GA TSP the locationof an item that should be visited is presented by the valueof a gene Accordingly the sequence of visiting the itemlocation is denoted by the order of a gene in a chromosomeTherefore the total number of items locations in a specificbatch is considered equal to the length of a chromosome Forexample if the length of a chromosome is nine it means nineplaces should be visited by the picker as shown in Figure 2If 119871119894represents the item location in the warehouse 119894 = 1

119899 it shows that the picker leaves the depot (1198710denotes

the location of a depot) goes to 1198711 1198718 11987111 1198719 11987113 11987112 11987110

and then comes back to the depotIn the second step the initial number of feasible chro-

mosomes should be generated randomly for each batch

The number of initial chromosomes could be different basedon the complexity of the problem It should be consideredthat the first and the last gene of each chromosome should beassigned individually to the location of a depot In step threeeach chromosome will be evaluated based on the GA TSPfitness function which is formulated as follows

fitness = Min119899minus1

sum

119895=1

119863119895119895+1

(12)

where 119863119895119895+1

denotes the distance between two subsequentgenes in a chromosome 119895 = 1 119899 minus 1 represents the orderof the genes and 119899 is number of genes

Subsequently step four is involved with the selectionoperation In each iteration the selection operator usesfitness values to select the parents of the next generationIn this research the roulette wheel selection policy hasbeen used to guarantee that the most appropriate pairs ofchromosomes have been selected to generate offspring Itmeans that a higher probability of being selected to createa new population will be assigned to the chromosomes withhigher fitness values

Steps five and six are involved with conducting thecrossover and mutation operation The construction of theoffspring is realized by the crossover and the diversity of theindividuals is maintained by the mutation operator Blocksof genes between chromosomes are traded by means of thecrossover operation inwhich the exploitation of a particularlyprofitable portion of the parameter space is allowed to theblock Here permutation order based crossover (POP) as avariation of the distinguished order crossover has been usedAs shown in Figure 3 the original idea is about choosing twoparents and a point for cutting The first portion of offspring2 contains the first portion of parent 1 up to the cut pointTo form the second portion of offspring 2 consider all genesin parent 2 except those genes that already exist in the firstportion of the offspring 2

As shown in Figure 4 two positions are randomlyselected in a chromosome and exchanged with each othernamely the order based mutation approach SWAP isadopted for solving GA TSP Step six is followed by step sevenwhich includes the evaluation of each offspring that has beengenerated in the previous step based on the GA TSP fitnessfunction In step eight termination criteria will be checkedIf the termination condition is met then the process will beterminated After that the best chromosome will be selectedas the optimal solution Otherwise the selection process is tobe conducted again to generate a new population

324 GA for Sequencing the Batches Sequencing the batchesshould be conducted in order to minimize the averagetardiness of all orders The problem of batch sequencing canbe defined as finding the orders of batches which should beprocessed further For this purpose the GA sequencing methodhas been utilized In this research all of the batches whichwere constructed using binary integer programming andGA TSP will be sequenced by using the GA The populationgeneration selectionmutation and stopping threshold in theGA sequencing are identical to those in the GA TSP However

8 The Scientific World Journal

Parent 1

Parent 2

Offspring 2

Offspring 1

L1 L8 L9

L1 L1L8 L8L9

L11 L13 L12 L10

L1 L8 L9 L11L13 L12L10

L13 L9 L11 L11 L13 L12

L12

L10

L10L1L8L12

L13 L9 L11 L10

L1 L8 L12L13 L9 L11 L10

Figure 3 Crossover operation

Parent L1

L1

L8

L8

L9

L9

L11

L11

L13

L13

L12

L12

L10

L10Offspring

Figure 4 Mutation operation

1 2 3 4 5b1 b4 b3 b5 b2

Figure 5 Chromosome structure

the fitness function and chromosome structure are differentthan that of GA TSP In contrast with GA TSP chromosomesstructure batch number is presented by the value of a geneAccordingly the sequence of batch is denoted by the orderof a gene in a chromosome For instance Figure 5 shows thechromosome structure If 119887

119894isin 119861 denotes a set of batches

Figure 5 shows that 1198871 1198874 1198873 1198875 1198872should be processed

respectively Therefore the length of a chromosome is equalto the number of the batches The GA sequencing fitness func-tion for chromosome 119910 is described in

fitness119910=

1

119873

119899

sum

119896=1

sum

119894isin119887119896

119862119879 (119887119896) minus 119863119906 (119900

119894) (13)

119862119879 (119887119896) = 119862119879 (119887

119896minus1) + Pt (119887119896) (14)

Pt (119887119896) =Opt (119887

119896)

119881119878 (15)

where119873 denotes for the total number of orders 119896 = 1 119899

represents the batch sequence in which 119899 is the total numberof batches 119900

119894stands for the order 119894 in which 119894 isin 119887

119896 119887119896

denotes the batch in 119896th position of a sequence for specificchromosome 119862119879(119887

119896) stands for the completion time of a

batch in the 119896th position of a sequence 119863119906(119900119894) is the symbol

for due date of order 119894which belongs to 119887119896 Pt(119887119896) is the process

time of 119887119896 Opt(119887

119896) is the optimal path solution for 119887

119896 Finally

119881119878 is the moving speed of the picker

4 Research Methodology

In this research the proposed methodology consists of fivesteps These steps are described as follows and shown pic-torially in Figure 6

Step 1 It is involved with utilizing theMINWAL algorithm inorder to determine the associations between orders in termsof support considering their due dates In this step an orderdue date will be considered as the weight of an order It meansorders that are similar in items and due date should havemoresupport for being batched together

Step 2 It encompasses determining the initial number ofbatches In this step the initial number of batches will becalculated by (10)

Step 3 It includes the problem of modeling and solving firstthe binary integer programming model for order clusteringand batching will be modeled Next this model will be solvedin order to determine the structures of batches

Step 4 It is about the picker routing problem The optimalpath for a picker within each batch will be determined bysolving a TSP using GA

Step 5 It is about sequencing the batches based on the fitnessfunction which minimizes the average tardiness of all ordersIn this step GA algorithmwill be utilized in order to solve thebatch sequencing problem regarding the fitness function

5 Computational Experiments and Results

51 Test Problems and Parameter Setting In this sectionfive numerical test problems with different settings andparameters have been carried out in order to illustrate theproficiency of the proposed method with respect to averagetardiness of all orders Table 5 summarizes the descriptionof these test problems This table includes information withregard to number of items number of orders capacity of thepicker total weight of all the items in an order andminimumnumber of batches

In these test problems the quantity of items quantityof orders capacity of the picker device (119862V) and amount ofeach item are predetermined Each order contains a certain

The Scientific World Journal 9

Start

Input order informationwarehouse setting picker

setting

Compute the supportbetween customersrsquo

orders using MINWAL

Calculating initialnumber of batches (K) to

assign the orders

Solving the binaryinteger programmingclustering model for

constructing the batches

Feasiblesolution

Yes

No

Solving picker routing

Solving batch sequencingproblem using

End

K = K + 1

problem using GA TSP

GA sequencing

Figure 6 Methodology framework

quantity of items and also each item has a certain quantityThese quantities are produced randomly The distributionfunction type for the number of items in each order is Normal[5 10] Also the distribution function type for each itemquantity in an order is uniform [1 10] The picker movingspeed is set as 119881119878 = 2 (ms) The warehouse has 3000 itemlocation The GA control parameters for solving the TSP andsequencing problem are tabulated in Table 6

All of the proposed approaches are coded usingMATLABversion 790 and Java 15 programming language Also theIBM ILOG CPLEX optimization studio version 124 softwarewas utilized jointly with MATLAB in order to facilitate theprocess of solving the binary integer programming problemExamples are run by means of a computer featured by

4 Gigabytes random access memory (RAM) and a 220Gigahertz Intel processor (T6600)

52 Computational Results and Discussion In order to revealthe competency of the proposed solution approach it wascompared with the other existing methods in the literatureOne of the latest published research activities in the literaturewhich deals with minimization of tardiness has been con-ducted by Henn and schmid [35] They utilized the EarliestDueDate (EDD) approach as their constructive algorithm forcomparing with their proposed approaches They proposedtwo heuristics Iterated Local Search andAttribute-BasedHillClimber are the two employed metaheuristics in their modelAccording to different test problems the efficiency of their

10 The Scientific World Journal

Table 5 Test problems description

Example 1 Example 2 Example 3 Example 4 Example 5Number of orders 40 60 80 100 150

Number of items 80 120 160 200 200

Capacity of picker 200 200 300 400 550

Total weight 2212 3668 4393 6110 8369

Minimum number of batch 11 19 15 16 16

Due dates range 60 90 120 180 240

Table 6 GA control parameters

GA TSP GA sequencing

Population size 500 200

Cross over strategyPermutation orderbased crossover

(POP)

Permutation orderbased crossover

(POP)Cross over probability 08 08

Mutation strategy SWAP SWAPMutation probability 02 02

Iteration number 10000 10000

Parent selection method Roulette wheel Roulette wheel

proposed methods was illustrated Their obtained resultsfrom their proposed approaches showed a 46 improvementon tardiness compared to EDD They used an s-shape andlargest gap routing strategy for their proposed method Inorder to make a reasonable comparison with their proposedmethod EDD was integrated with the s-shape routing strat-egy and GA sequencing Then based on improvement on EDDwe can compare our proposed method with their proposedmethod In the EDD approach due dates play an importantrole in sorting all orders Based on an ascending sequenceassigning the orders to batches ensures that no capacityviolation can happen for the picking device

For the warehouse condition which is described inSection 323 our proposed approach EDD integrated withGA TSP and GA sequencing and EDD integrated with s-shapealgorithm and GA sequencing have been evaluated in termsof average tardiness for all orders (119879) which is calculatedbased on (13) number of batches (119870) total process time(Pt) and average process time for each batch (119860Pt) All ofthe GA setting for the TSP and sequencing problems areconsidered the same for all methods Also the settings whichhave been mentioned in Section 51 were considered for all ofthe compared methods

The results for all of the test problems for the threeapproaches are tabulated in Tables 7 8 and 9 As it isillustrated in Table 10 EDD integrated with GA TSP andGA sequencing has a 3176 improvement on average in termsof average tardiness of all orders than EDD integrated withthe s-shape algorithm and GA sequencing It can be perceivedthat GA TSP performs better than s-shape Also the ATGHmethod has a 6811 improvement in average tardinessoff all orders than EDD integrated with s-shape algorithm

and GA sequencing Consequently the ATGH method achievesmuch improvement against Henn and Schmidrsquos [35] pro-posedmethod According to Tables 7 and 8 it is also worth tomention thatGA TSP performs better than the s-shape routingstrategy in terms of process time The improvements whichhave been achieved in this research could be justified basedon the following reasons

The problem of order batching deals with several sig-nificant drivers in which the number of batches can beconsidered as a key factor [23] As a justification to thismatter it is worth to mention that total travel distance maynot decrease directly due to a decline in the number ofbatches But a decline in the number of batches will increasebatch sizes which may lead to an increase in the distancetravelled by order picker However an advanced exploitationof picker capacity to load efficiently may reduce the numberof constructed batches together reducing the total traveldistance Improving the total travel distance traversed by thepicker can minimize the average tardiness of all orders Interms of number of the batches according to Tables 7 8 and9 it can be perceived that our proposed method assigns theorders to the batches optimally The ATGH method assignsthe orders to batches based on their item similarity andtheir due dates and it uses the picker capacity efficiently Butthe EDD method allocates the orders to batches based onan ascending sequence of their due date and it should beensured that no capacity violation will happen for the pickingdevice which may lead to an inefficient order batching Tohighlight the problem an example can be made based on theresults tabulated in Tables 7 and 8 The number of batchesfor example 1 in our method is 11 in comparison with 13obtained fromEDDmethod 13 batches can be justified basedon this scenario that two subsequent orders say the firstorder is 100 kg and the other is 150 kg cannot be batchedtogether due to violated batch capacity which is set to be200 kg The number of batches would increase due to theaforementioned issue which affects the total process time tohave an increasing manner This situation will rarely happenin the ATGH method because of the weighted associationbased binary integer programming utilizedwhich batches theorders in order to maximize the association between themwhich is calculated based on the item similarity of ordersand the weight of their corresponding due dates The ATGHmethod uses the picker capacity efficiently

Obviously one of the factors that can influence thecompletion time and the average tardiness of all orders is theprocessing time of the orders Reducing the processing time

The Scientific World Journal 11

Table 7 Computational results of ATGH

Example 1 Example 2 Example 3 Example 4 Example 5119879 56204 29948 90202 163 58898

Pt 605833 1217833 105833 121033 1339

119870 11 19 15 16 16

119860Pt 55076 6409 7055 7564 8368

Table 8 Computational results of EDD integrated with GA TSP and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 124550 42976 18222 5981 1547

Pt 667333 133516 116083 13185 141916

119870 13 21 16 17 16

119860Pt 51333 6357 7255 7775 8869

Table 9 Computational results of EDD integrated with s-shape algorithm and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 1651 62677 2825 13365 17593

Pt 75266 16636 13323 14333 14926

119870 13 21 16 17 16

119860Pt 5789 892 833 8431 932

Table 10 Comparison results

EDD + s-shape + GA sequencing EDD + GA TSP + GA sequencing ATGHTardiness (min) Tardiness (min) Improvement () Tardiness (min) Improvement ()

(1) 1651 124550 2456 56204 6596(2) 62677 42976 3143 29948 5222(3) 2825 18222 3550 90202 6807(4) 13365 5981 5525 1627 8783(5) 17593 1547 1206 58898 6652Average improvement 3176 6811

and travel time for constructing a batch is an outstanding factin reducing the costs and minimizing delays of the customerresponses In our proposed model orders with similar itemsare consolidated together Therefore batching orders withmore similarity can reduce the distance traversed by thepicker facility Also the due date of each order is definedas a weight parameter in calculating the association betweenorders in terms of support which was done for consideringthe due dates in our model Thereupon the model considersdue date and the process time in an integrated manner whichsimultaneously affected the average tardiness of all ordersThis spoken issue is different in the EDD approach in whichthe ordersrsquo due dates are just considered Example 5 canbe considered as evidence of this matter in which the totaland average process time for ATGH is lower than EDDintegrated with GA TSP and GA sequencing with respect to thefact that number of constructed batches for both approachesare happened to be equal after obtaining the results

6 Conclusion

One of the most important concerns of warehouse managersis involvedwith finding the best optimal and cost efficientwayto pick orders placed by customers in order to be known as aresponsible company with satisfied customers in terms of ontime delivery In the current research activity order batchingpicker routing and batch sequencing problems of warehouseprocesses were addressed which are to be solved jointly asthey are mixed up with many manufacturing industriesMathematical solutions of these NP-hard problems might beavailable for problems that are small in nature But thesekinds of solutions may not be applicable to be consideredby warehouse managers Hence we presented novel solutionapproach namely ATGH to solve this issue in accordancewith minimizing the average tardiness of all customersrsquoorders which can be considered as a major objective ofthis research activity A weighted association rule mining

12 The Scientific World Journal

algorithm namely MINWAL was utilized to determine thecustomersrsquo orders association regarding similarity betweenitems and ordersrsquo due dates Actually in our proposedmethodthe due date of each order is defined as a weight parameterin calculating the association between orders So orders withsimilar items and due dates have more chance to be in asame batch This issue is not considered in most of theresearch activities in the literature related to our work andconsidering this issue in the order batching problem is themain contribution of our research activity The previous stepwas followed by a binary integer clustering model in orderto solve the order batching problem which maximizes thesimilarity between orders in terms of support After that aGA was applied to solve a TSP in order to find the optimalrouting path

Finally a batch sequencing problem was addressed inconjunction with a GA

The other research objective of this research addresseswhether the proposed ATGH algorithm can further improvethe solution quality for a multiple parallel aisle warehousewith manual picking system Hence comparisons were madeconsideringATGHEDD integratedGA TSP andGA sequencingand EDD integrated s-shape and GA sequencing algorithms Asdemonstrated by the obtained results ATGH comes out tobe the most striking solution approach in terms of tardinessIn spite of the fact that considering all customer orders asdiscussed in this paper would lead to have a static case beinginvolved with dynamic cases in which different points of timeare assigned to customerrsquos orders arrival would be beneficialSo there might be some potential room of research to extendour discussed approach with respect to dynamic orders

Conflict of Interests

The IBM ILOG CPLEX optimization studio version 124software is not used for financial gain and there is no conflictof interests with any party as it is only a student version

References

[1] ELAAT Kearney Excellence in Logistics ELA Brussels Bel-gium 2004

[2] Y-C Ho T-S Su and Z-B Shi ldquoOrder-batching methods foran order-picking warehouse with two cross aislesrdquo Computersamp Industrial Engineering vol 55 no 2 pp 321ndash347 2008

[3] C-M Hsu K-Y Chen and M-C Chen ldquoBatching ordersin warehouses by minimizing travel distance with geneticalgorithmsrdquo Computers in Industry vol 56 no 2 pp 169ndash1782005

[4] E A Elsayed M-K Lee S Kim and E Scherer ldquoSequencingand batching procedures forminimizing earliness and tardinesspenalty of order retrievalsrdquoThe International Journal of Produc-tion Research vol 31 no 3 pp 727ndash738 1993

[5] S Henn S Koch and W Gerhard Order Batching in OrderPicking Warehouses A Survey of Solution Approaches SpringerLondon UK 2012

[6] J Drury Towards More Efficient Order Picking IMM Mono-graph no 1 The Institute of Materials Management CranfieldUK 1988

[7] M Goetschalckx and J Ashayeri ldquoClassification and design oforder picking systemsrdquo Logistics World pp 99ndash106 1989

[8] J A Tompkins J A White Y A Bozer E H Frazelle and J MA Tanchoco Facilities Planning John Wiley amp Sons HobokenNJ USA 2003

[9] S Henn and G Wascher ldquoTabu search heuristics for the orderbatching problem in manual order picking systemsrdquo EuropeanJournal of Operational Research vol 222 no 3 pp 484ndash4942012

[10] S Henn ldquoAlgorithms for on-line order batching in an orderpicking warehouserdquo Computers and Operations Research vol39 no 11 pp 2549ndash2563 2012

[11] R de Koster ldquoHow to assess a warehouse operation in a singletourrdquo Tech Rep RSM Erasmus University Rotterdam TheNetherlands 2004

[12] R de Koster T Le-Duc and K J Roodbergen ldquoDesign andcontrol of warehouse order picking a literature reviewrdquo Euro-pean Journal of Operational Research vol 182 no 2 pp 481ndash5012007

[13] R A Ruben and F R Jacobs ldquoBatch construction heuristics andstorage assignment strategies for walkride and pick systemsrdquoManagement Science vol 45 no 4 pp 575ndash596 1999

[14] E A Elsayed ldquoAlgorithms for optimal material handling inautomatic warehousing systemsrdquo International Journal of Pro-duction Research vol 19 no 5 pp 525ndash535 1981

[15] G Clarke and J W Wright ldquoScheduling of vehicles froma central depot to a number of delivery pointsrdquo OperationsResearch vol 12 no 4 pp 568ndash581 1964

[16] H D Vinod ldquoInteger programming and the theory of group-ingrdquo Journal of the American Statistical Association vol 64 no326 pp 506ndash519 1969

[17] R D Armstrong W D Cook and A L Saipe ldquoOptimalbatching in a semi-automated order picking systemrdquo Journal ofthe Operational Research Society vol 30 no 8 pp 711ndash720 1979

[18] H D Ratliff and A S Rosenthal ldquoOrder-picking in a rect-angular warehouse a solvable case of the traveling salesmanproblemrdquo Operations Research vol 31 no 3 pp 507ndash521 1983

[19] A Kusiak A Vannelli and R K Kumar ldquoClustering analysismodels and algorithmsrdquo Cybernetics vol 15 pp 139ndash154 1986

[20] D R Gibson and G P Sharp ldquoOrder batching proceduresrdquoEuropean Journal of Operational Research vol 58 no 1 pp 57ndash67 1992

[21] M B Rosenwein ldquoA comparison of heuristics for the problemofbatching orders for warehouse selectionrdquo International Journalof Production Research vol 34 no 3 pp 657ndash664 1996

[22] A J RMGademann J P van denBerg andHH van derHoffldquoAn order batching algorithm for wave picking in a parallel-aislewarehouserdquo IIE Transactions vol 33 no 5 pp 385ndash398 2001

[23] M-C Chen and H-P Wu ldquoAn association-based clusteringapproach to order batching considering customer demandpatternsrdquo Omega vol 33 no 4 pp 333ndash343 2005

[24] NGademann and S van deVelde ldquoOrder batching tominimizetotal travel time in a parallel-aisle warehouserdquo IIE Transactionsvol 37 no 1 pp 63ndash75 2005

[25] Y A Bozer and J W Kile ldquoOrder batching in walk-and-pick order picking systemsrdquo International Journal of ProductionResearch vol 46 no 7 pp 1887ndash1909 2008

[26] O Kulak Y Sahin and M E Taner ldquoJoint order batching andpicker routing in single and multiple-cross-aisle warehousesusing cluster-based tabu search algorithmsrdquo Flexible Servicesand Manufacturing Journal vol 24 no 1 pp 52ndash80 2012

The Scientific World Journal 13

[27] E A Elsayed and M-K Lee ldquoOrder processing in automatedstorageretrieval systems with due datesrdquo IIE Transactions vol28 no 7 pp 567ndash577 1996

[28] J Won and S Olafsson ldquoJoint order batching and order pickingin warehouse operationsrdquo International Journal of ProductionResearch vol 43 no 7 pp 1427ndash1442 2005

[29] C-Y Tsai J J H Liou and T-M Huang ldquoUsing a multiple-GA method to solve the batch picking problem consideringtravel distance and order due timerdquo International Journal of Pro-duction Research vol 46 no 22 pp 6533ndash6555 2008

[30] Y KaoM H Chen and Y T Huang ldquoA hybrid algorithm basedon ACO and PSO for capacitated vehicle routing problemsrdquoMathematical Problems in Engineering vol 2012 Article ID726564 17 pages 2012

[31] X Gan Y Wang S Li and B Niu ldquoVehicle routing problemwith time windows and simultaneous delivery and pick-up ser-vice based on MCPSOrdquoMathematical Problems in Engineeringvol 2012 Article ID 104279 11 pages 2012

[32] M B M de Koster E S van der Poort and M WoltersldquoEfficient orderbatching methods in warehousesrdquo InternationalJournal of ProductionResearch vol 37 no 7 pp 1479ndash1504 1999

[33] Y-C Ho and Y-Y Tseng ldquoA study on order-batching methodsof order-picking in a distribution centre with two cross-aislesrdquoInternational Journal of Production Research vol 44 no 17 pp3391ndash3417 2006

[34] C Theys O Braysy W Dullaert and B Raa ldquoUsing a TSPheuristic for routing order pickers in warehousesrdquo EuropeanJournal of Operational Research vol 200 no 3 pp 755ndash7632010

[35] S Henn and V Schmid Metaheuristics for Order Batching andSequencing in Manual Order Picking Systems Univ Fak furWirtschaftswiss 2011

[36] M Hahsler B Grun and K Hornik ldquoArulesmdasha computationalenvironment for mining association rules and frequent itemsetsrdquo Journal of Statistical Software vol 14 2005

[37] R Agrawal T Imielinski and A Swami ldquoMining associationrules between sets of items in large databasesrdquo ACM SIGMODRecord vol 22 no 2 pp 207ndash216 1993

[38] S Encheva Y Kondratenko M Z Solesvik and S TuminldquoDecision support systems in logisticsrdquo in Proceeding of Inter-national electronic conference on computer science pp 254ndash2562007

[39] M-C Chen ldquoConfiguration of cellular manufacturing systemsusing association rule inductionrdquo International Journal of Pro-duction Research vol 41 no 2 pp 381ndash395 2003

[40] R Srikant and R Agrawal ldquoMining generalized associationrulesrdquo Future Generation Computer Systems vol 13 no 2-3 pp161ndash180 1997

[41] J Wu and XM Li ldquoAn effective mining algorithm for weightedassociation rules in communication networksrdquo Journal of Com-puters vol 3 no 10 pp 20ndash27 2008

[42] C H Cai A W C Fu C H Cheng and W W KwongldquoMining association rules with weighted itemsrdquo in Proceedingsof the International Symposium on Database Engineering ampApplications pp 68ndash77 Cardiff UK 1998

[43] J HollandAdaptation in Natural and Artificial Systems Univer-sity of Michigan Press Ann Arbor Mich USA 1975

[44] A A Taleizadeh F Barzinpour and H-M Wee ldquoMeta-heuristic algorithms for solving a fuzzy single-period problemrdquoMathematical and Computer Modelling vol 54 no 5-6 pp1273ndash1285 2011

[45] H Yang J Yi J Zhao and Zh Dong ldquoExtreme learning ma-chine based genetic algorithm and its application in powersystem economic dispatchrdquo Neurocomputing vol 102 pp 154ndash162 2013

Submit your manuscripts athttpwwwhindawicom

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International Journal of

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Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 The Scientific World Journal

Parent 1

Parent 2

Offspring 2

Offspring 1

L1 L8 L9

L1 L1L8 L8L9

L11 L13 L12 L10

L1 L8 L9 L11L13 L12L10

L13 L9 L11 L11 L13 L12

L12

L10

L10L1L8L12

L13 L9 L11 L10

L1 L8 L12L13 L9 L11 L10

Figure 3 Crossover operation

Parent L1

L1

L8

L8

L9

L9

L11

L11

L13

L13

L12

L12

L10

L10Offspring

Figure 4 Mutation operation

1 2 3 4 5b1 b4 b3 b5 b2

Figure 5 Chromosome structure

the fitness function and chromosome structure are differentthan that of GA TSP In contrast with GA TSP chromosomesstructure batch number is presented by the value of a geneAccordingly the sequence of batch is denoted by the orderof a gene in a chromosome For instance Figure 5 shows thechromosome structure If 119887

119894isin 119861 denotes a set of batches

Figure 5 shows that 1198871 1198874 1198873 1198875 1198872should be processed

respectively Therefore the length of a chromosome is equalto the number of the batches The GA sequencing fitness func-tion for chromosome 119910 is described in

fitness119910=

1

119873

119899

sum

119896=1

sum

119894isin119887119896

119862119879 (119887119896) minus 119863119906 (119900

119894) (13)

119862119879 (119887119896) = 119862119879 (119887

119896minus1) + Pt (119887119896) (14)

Pt (119887119896) =Opt (119887

119896)

119881119878 (15)

where119873 denotes for the total number of orders 119896 = 1 119899

represents the batch sequence in which 119899 is the total numberof batches 119900

119894stands for the order 119894 in which 119894 isin 119887

119896 119887119896

denotes the batch in 119896th position of a sequence for specificchromosome 119862119879(119887

119896) stands for the completion time of a

batch in the 119896th position of a sequence 119863119906(119900119894) is the symbol

for due date of order 119894which belongs to 119887119896 Pt(119887119896) is the process

time of 119887119896 Opt(119887

119896) is the optimal path solution for 119887

119896 Finally

119881119878 is the moving speed of the picker

4 Research Methodology

In this research the proposed methodology consists of fivesteps These steps are described as follows and shown pic-torially in Figure 6

Step 1 It is involved with utilizing theMINWAL algorithm inorder to determine the associations between orders in termsof support considering their due dates In this step an orderdue date will be considered as the weight of an order It meansorders that are similar in items and due date should havemoresupport for being batched together

Step 2 It encompasses determining the initial number ofbatches In this step the initial number of batches will becalculated by (10)

Step 3 It includes the problem of modeling and solving firstthe binary integer programming model for order clusteringand batching will be modeled Next this model will be solvedin order to determine the structures of batches

Step 4 It is about the picker routing problem The optimalpath for a picker within each batch will be determined bysolving a TSP using GA

Step 5 It is about sequencing the batches based on the fitnessfunction which minimizes the average tardiness of all ordersIn this step GA algorithmwill be utilized in order to solve thebatch sequencing problem regarding the fitness function

5 Computational Experiments and Results

51 Test Problems and Parameter Setting In this sectionfive numerical test problems with different settings andparameters have been carried out in order to illustrate theproficiency of the proposed method with respect to averagetardiness of all orders Table 5 summarizes the descriptionof these test problems This table includes information withregard to number of items number of orders capacity of thepicker total weight of all the items in an order andminimumnumber of batches

In these test problems the quantity of items quantityof orders capacity of the picker device (119862V) and amount ofeach item are predetermined Each order contains a certain

The Scientific World Journal 9

Start

Input order informationwarehouse setting picker

setting

Compute the supportbetween customersrsquo

orders using MINWAL

Calculating initialnumber of batches (K) to

assign the orders

Solving the binaryinteger programmingclustering model for

constructing the batches

Feasiblesolution

Yes

No

Solving picker routing

Solving batch sequencingproblem using

End

K = K + 1

problem using GA TSP

GA sequencing

Figure 6 Methodology framework

quantity of items and also each item has a certain quantityThese quantities are produced randomly The distributionfunction type for the number of items in each order is Normal[5 10] Also the distribution function type for each itemquantity in an order is uniform [1 10] The picker movingspeed is set as 119881119878 = 2 (ms) The warehouse has 3000 itemlocation The GA control parameters for solving the TSP andsequencing problem are tabulated in Table 6

All of the proposed approaches are coded usingMATLABversion 790 and Java 15 programming language Also theIBM ILOG CPLEX optimization studio version 124 softwarewas utilized jointly with MATLAB in order to facilitate theprocess of solving the binary integer programming problemExamples are run by means of a computer featured by

4 Gigabytes random access memory (RAM) and a 220Gigahertz Intel processor (T6600)

52 Computational Results and Discussion In order to revealthe competency of the proposed solution approach it wascompared with the other existing methods in the literatureOne of the latest published research activities in the literaturewhich deals with minimization of tardiness has been con-ducted by Henn and schmid [35] They utilized the EarliestDueDate (EDD) approach as their constructive algorithm forcomparing with their proposed approaches They proposedtwo heuristics Iterated Local Search andAttribute-BasedHillClimber are the two employed metaheuristics in their modelAccording to different test problems the efficiency of their

10 The Scientific World Journal

Table 5 Test problems description

Example 1 Example 2 Example 3 Example 4 Example 5Number of orders 40 60 80 100 150

Number of items 80 120 160 200 200

Capacity of picker 200 200 300 400 550

Total weight 2212 3668 4393 6110 8369

Minimum number of batch 11 19 15 16 16

Due dates range 60 90 120 180 240

Table 6 GA control parameters

GA TSP GA sequencing

Population size 500 200

Cross over strategyPermutation orderbased crossover

(POP)

Permutation orderbased crossover

(POP)Cross over probability 08 08

Mutation strategy SWAP SWAPMutation probability 02 02

Iteration number 10000 10000

Parent selection method Roulette wheel Roulette wheel

proposed methods was illustrated Their obtained resultsfrom their proposed approaches showed a 46 improvementon tardiness compared to EDD They used an s-shape andlargest gap routing strategy for their proposed method Inorder to make a reasonable comparison with their proposedmethod EDD was integrated with the s-shape routing strat-egy and GA sequencing Then based on improvement on EDDwe can compare our proposed method with their proposedmethod In the EDD approach due dates play an importantrole in sorting all orders Based on an ascending sequenceassigning the orders to batches ensures that no capacityviolation can happen for the picking device

For the warehouse condition which is described inSection 323 our proposed approach EDD integrated withGA TSP and GA sequencing and EDD integrated with s-shapealgorithm and GA sequencing have been evaluated in termsof average tardiness for all orders (119879) which is calculatedbased on (13) number of batches (119870) total process time(Pt) and average process time for each batch (119860Pt) All ofthe GA setting for the TSP and sequencing problems areconsidered the same for all methods Also the settings whichhave been mentioned in Section 51 were considered for all ofthe compared methods

The results for all of the test problems for the threeapproaches are tabulated in Tables 7 8 and 9 As it isillustrated in Table 10 EDD integrated with GA TSP andGA sequencing has a 3176 improvement on average in termsof average tardiness of all orders than EDD integrated withthe s-shape algorithm and GA sequencing It can be perceivedthat GA TSP performs better than s-shape Also the ATGHmethod has a 6811 improvement in average tardinessoff all orders than EDD integrated with s-shape algorithm

and GA sequencing Consequently the ATGH method achievesmuch improvement against Henn and Schmidrsquos [35] pro-posedmethod According to Tables 7 and 8 it is also worth tomention thatGA TSP performs better than the s-shape routingstrategy in terms of process time The improvements whichhave been achieved in this research could be justified basedon the following reasons

The problem of order batching deals with several sig-nificant drivers in which the number of batches can beconsidered as a key factor [23] As a justification to thismatter it is worth to mention that total travel distance maynot decrease directly due to a decline in the number ofbatches But a decline in the number of batches will increasebatch sizes which may lead to an increase in the distancetravelled by order picker However an advanced exploitationof picker capacity to load efficiently may reduce the numberof constructed batches together reducing the total traveldistance Improving the total travel distance traversed by thepicker can minimize the average tardiness of all orders Interms of number of the batches according to Tables 7 8 and9 it can be perceived that our proposed method assigns theorders to the batches optimally The ATGH method assignsthe orders to batches based on their item similarity andtheir due dates and it uses the picker capacity efficiently Butthe EDD method allocates the orders to batches based onan ascending sequence of their due date and it should beensured that no capacity violation will happen for the pickingdevice which may lead to an inefficient order batching Tohighlight the problem an example can be made based on theresults tabulated in Tables 7 and 8 The number of batchesfor example 1 in our method is 11 in comparison with 13obtained fromEDDmethod 13 batches can be justified basedon this scenario that two subsequent orders say the firstorder is 100 kg and the other is 150 kg cannot be batchedtogether due to violated batch capacity which is set to be200 kg The number of batches would increase due to theaforementioned issue which affects the total process time tohave an increasing manner This situation will rarely happenin the ATGH method because of the weighted associationbased binary integer programming utilizedwhich batches theorders in order to maximize the association between themwhich is calculated based on the item similarity of ordersand the weight of their corresponding due dates The ATGHmethod uses the picker capacity efficiently

Obviously one of the factors that can influence thecompletion time and the average tardiness of all orders is theprocessing time of the orders Reducing the processing time

The Scientific World Journal 11

Table 7 Computational results of ATGH

Example 1 Example 2 Example 3 Example 4 Example 5119879 56204 29948 90202 163 58898

Pt 605833 1217833 105833 121033 1339

119870 11 19 15 16 16

119860Pt 55076 6409 7055 7564 8368

Table 8 Computational results of EDD integrated with GA TSP and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 124550 42976 18222 5981 1547

Pt 667333 133516 116083 13185 141916

119870 13 21 16 17 16

119860Pt 51333 6357 7255 7775 8869

Table 9 Computational results of EDD integrated with s-shape algorithm and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 1651 62677 2825 13365 17593

Pt 75266 16636 13323 14333 14926

119870 13 21 16 17 16

119860Pt 5789 892 833 8431 932

Table 10 Comparison results

EDD + s-shape + GA sequencing EDD + GA TSP + GA sequencing ATGHTardiness (min) Tardiness (min) Improvement () Tardiness (min) Improvement ()

(1) 1651 124550 2456 56204 6596(2) 62677 42976 3143 29948 5222(3) 2825 18222 3550 90202 6807(4) 13365 5981 5525 1627 8783(5) 17593 1547 1206 58898 6652Average improvement 3176 6811

and travel time for constructing a batch is an outstanding factin reducing the costs and minimizing delays of the customerresponses In our proposed model orders with similar itemsare consolidated together Therefore batching orders withmore similarity can reduce the distance traversed by thepicker facility Also the due date of each order is definedas a weight parameter in calculating the association betweenorders in terms of support which was done for consideringthe due dates in our model Thereupon the model considersdue date and the process time in an integrated manner whichsimultaneously affected the average tardiness of all ordersThis spoken issue is different in the EDD approach in whichthe ordersrsquo due dates are just considered Example 5 canbe considered as evidence of this matter in which the totaland average process time for ATGH is lower than EDDintegrated with GA TSP and GA sequencing with respect to thefact that number of constructed batches for both approachesare happened to be equal after obtaining the results

6 Conclusion

One of the most important concerns of warehouse managersis involvedwith finding the best optimal and cost efficientwayto pick orders placed by customers in order to be known as aresponsible company with satisfied customers in terms of ontime delivery In the current research activity order batchingpicker routing and batch sequencing problems of warehouseprocesses were addressed which are to be solved jointly asthey are mixed up with many manufacturing industriesMathematical solutions of these NP-hard problems might beavailable for problems that are small in nature But thesekinds of solutions may not be applicable to be consideredby warehouse managers Hence we presented novel solutionapproach namely ATGH to solve this issue in accordancewith minimizing the average tardiness of all customersrsquoorders which can be considered as a major objective ofthis research activity A weighted association rule mining

12 The Scientific World Journal

algorithm namely MINWAL was utilized to determine thecustomersrsquo orders association regarding similarity betweenitems and ordersrsquo due dates Actually in our proposedmethodthe due date of each order is defined as a weight parameterin calculating the association between orders So orders withsimilar items and due dates have more chance to be in asame batch This issue is not considered in most of theresearch activities in the literature related to our work andconsidering this issue in the order batching problem is themain contribution of our research activity The previous stepwas followed by a binary integer clustering model in orderto solve the order batching problem which maximizes thesimilarity between orders in terms of support After that aGA was applied to solve a TSP in order to find the optimalrouting path

Finally a batch sequencing problem was addressed inconjunction with a GA

The other research objective of this research addresseswhether the proposed ATGH algorithm can further improvethe solution quality for a multiple parallel aisle warehousewith manual picking system Hence comparisons were madeconsideringATGHEDD integratedGA TSP andGA sequencingand EDD integrated s-shape and GA sequencing algorithms Asdemonstrated by the obtained results ATGH comes out tobe the most striking solution approach in terms of tardinessIn spite of the fact that considering all customer orders asdiscussed in this paper would lead to have a static case beinginvolved with dynamic cases in which different points of timeare assigned to customerrsquos orders arrival would be beneficialSo there might be some potential room of research to extendour discussed approach with respect to dynamic orders

Conflict of Interests

The IBM ILOG CPLEX optimization studio version 124software is not used for financial gain and there is no conflictof interests with any party as it is only a student version

References

[1] ELAAT Kearney Excellence in Logistics ELA Brussels Bel-gium 2004

[2] Y-C Ho T-S Su and Z-B Shi ldquoOrder-batching methods foran order-picking warehouse with two cross aislesrdquo Computersamp Industrial Engineering vol 55 no 2 pp 321ndash347 2008

[3] C-M Hsu K-Y Chen and M-C Chen ldquoBatching ordersin warehouses by minimizing travel distance with geneticalgorithmsrdquo Computers in Industry vol 56 no 2 pp 169ndash1782005

[4] E A Elsayed M-K Lee S Kim and E Scherer ldquoSequencingand batching procedures forminimizing earliness and tardinesspenalty of order retrievalsrdquoThe International Journal of Produc-tion Research vol 31 no 3 pp 727ndash738 1993

[5] S Henn S Koch and W Gerhard Order Batching in OrderPicking Warehouses A Survey of Solution Approaches SpringerLondon UK 2012

[6] J Drury Towards More Efficient Order Picking IMM Mono-graph no 1 The Institute of Materials Management CranfieldUK 1988

[7] M Goetschalckx and J Ashayeri ldquoClassification and design oforder picking systemsrdquo Logistics World pp 99ndash106 1989

[8] J A Tompkins J A White Y A Bozer E H Frazelle and J MA Tanchoco Facilities Planning John Wiley amp Sons HobokenNJ USA 2003

[9] S Henn and G Wascher ldquoTabu search heuristics for the orderbatching problem in manual order picking systemsrdquo EuropeanJournal of Operational Research vol 222 no 3 pp 484ndash4942012

[10] S Henn ldquoAlgorithms for on-line order batching in an orderpicking warehouserdquo Computers and Operations Research vol39 no 11 pp 2549ndash2563 2012

[11] R de Koster ldquoHow to assess a warehouse operation in a singletourrdquo Tech Rep RSM Erasmus University Rotterdam TheNetherlands 2004

[12] R de Koster T Le-Duc and K J Roodbergen ldquoDesign andcontrol of warehouse order picking a literature reviewrdquo Euro-pean Journal of Operational Research vol 182 no 2 pp 481ndash5012007

[13] R A Ruben and F R Jacobs ldquoBatch construction heuristics andstorage assignment strategies for walkride and pick systemsrdquoManagement Science vol 45 no 4 pp 575ndash596 1999

[14] E A Elsayed ldquoAlgorithms for optimal material handling inautomatic warehousing systemsrdquo International Journal of Pro-duction Research vol 19 no 5 pp 525ndash535 1981

[15] G Clarke and J W Wright ldquoScheduling of vehicles froma central depot to a number of delivery pointsrdquo OperationsResearch vol 12 no 4 pp 568ndash581 1964

[16] H D Vinod ldquoInteger programming and the theory of group-ingrdquo Journal of the American Statistical Association vol 64 no326 pp 506ndash519 1969

[17] R D Armstrong W D Cook and A L Saipe ldquoOptimalbatching in a semi-automated order picking systemrdquo Journal ofthe Operational Research Society vol 30 no 8 pp 711ndash720 1979

[18] H D Ratliff and A S Rosenthal ldquoOrder-picking in a rect-angular warehouse a solvable case of the traveling salesmanproblemrdquo Operations Research vol 31 no 3 pp 507ndash521 1983

[19] A Kusiak A Vannelli and R K Kumar ldquoClustering analysismodels and algorithmsrdquo Cybernetics vol 15 pp 139ndash154 1986

[20] D R Gibson and G P Sharp ldquoOrder batching proceduresrdquoEuropean Journal of Operational Research vol 58 no 1 pp 57ndash67 1992

[21] M B Rosenwein ldquoA comparison of heuristics for the problemofbatching orders for warehouse selectionrdquo International Journalof Production Research vol 34 no 3 pp 657ndash664 1996

[22] A J RMGademann J P van denBerg andHH van derHoffldquoAn order batching algorithm for wave picking in a parallel-aislewarehouserdquo IIE Transactions vol 33 no 5 pp 385ndash398 2001

[23] M-C Chen and H-P Wu ldquoAn association-based clusteringapproach to order batching considering customer demandpatternsrdquo Omega vol 33 no 4 pp 333ndash343 2005

[24] NGademann and S van deVelde ldquoOrder batching tominimizetotal travel time in a parallel-aisle warehouserdquo IIE Transactionsvol 37 no 1 pp 63ndash75 2005

[25] Y A Bozer and J W Kile ldquoOrder batching in walk-and-pick order picking systemsrdquo International Journal of ProductionResearch vol 46 no 7 pp 1887ndash1909 2008

[26] O Kulak Y Sahin and M E Taner ldquoJoint order batching andpicker routing in single and multiple-cross-aisle warehousesusing cluster-based tabu search algorithmsrdquo Flexible Servicesand Manufacturing Journal vol 24 no 1 pp 52ndash80 2012

The Scientific World Journal 13

[27] E A Elsayed and M-K Lee ldquoOrder processing in automatedstorageretrieval systems with due datesrdquo IIE Transactions vol28 no 7 pp 567ndash577 1996

[28] J Won and S Olafsson ldquoJoint order batching and order pickingin warehouse operationsrdquo International Journal of ProductionResearch vol 43 no 7 pp 1427ndash1442 2005

[29] C-Y Tsai J J H Liou and T-M Huang ldquoUsing a multiple-GA method to solve the batch picking problem consideringtravel distance and order due timerdquo International Journal of Pro-duction Research vol 46 no 22 pp 6533ndash6555 2008

[30] Y KaoM H Chen and Y T Huang ldquoA hybrid algorithm basedon ACO and PSO for capacitated vehicle routing problemsrdquoMathematical Problems in Engineering vol 2012 Article ID726564 17 pages 2012

[31] X Gan Y Wang S Li and B Niu ldquoVehicle routing problemwith time windows and simultaneous delivery and pick-up ser-vice based on MCPSOrdquoMathematical Problems in Engineeringvol 2012 Article ID 104279 11 pages 2012

[32] M B M de Koster E S van der Poort and M WoltersldquoEfficient orderbatching methods in warehousesrdquo InternationalJournal of ProductionResearch vol 37 no 7 pp 1479ndash1504 1999

[33] Y-C Ho and Y-Y Tseng ldquoA study on order-batching methodsof order-picking in a distribution centre with two cross-aislesrdquoInternational Journal of Production Research vol 44 no 17 pp3391ndash3417 2006

[34] C Theys O Braysy W Dullaert and B Raa ldquoUsing a TSPheuristic for routing order pickers in warehousesrdquo EuropeanJournal of Operational Research vol 200 no 3 pp 755ndash7632010

[35] S Henn and V Schmid Metaheuristics for Order Batching andSequencing in Manual Order Picking Systems Univ Fak furWirtschaftswiss 2011

[36] M Hahsler B Grun and K Hornik ldquoArulesmdasha computationalenvironment for mining association rules and frequent itemsetsrdquo Journal of Statistical Software vol 14 2005

[37] R Agrawal T Imielinski and A Swami ldquoMining associationrules between sets of items in large databasesrdquo ACM SIGMODRecord vol 22 no 2 pp 207ndash216 1993

[38] S Encheva Y Kondratenko M Z Solesvik and S TuminldquoDecision support systems in logisticsrdquo in Proceeding of Inter-national electronic conference on computer science pp 254ndash2562007

[39] M-C Chen ldquoConfiguration of cellular manufacturing systemsusing association rule inductionrdquo International Journal of Pro-duction Research vol 41 no 2 pp 381ndash395 2003

[40] R Srikant and R Agrawal ldquoMining generalized associationrulesrdquo Future Generation Computer Systems vol 13 no 2-3 pp161ndash180 1997

[41] J Wu and XM Li ldquoAn effective mining algorithm for weightedassociation rules in communication networksrdquo Journal of Com-puters vol 3 no 10 pp 20ndash27 2008

[42] C H Cai A W C Fu C H Cheng and W W KwongldquoMining association rules with weighted itemsrdquo in Proceedingsof the International Symposium on Database Engineering ampApplications pp 68ndash77 Cardiff UK 1998

[43] J HollandAdaptation in Natural and Artificial Systems Univer-sity of Michigan Press Ann Arbor Mich USA 1975

[44] A A Taleizadeh F Barzinpour and H-M Wee ldquoMeta-heuristic algorithms for solving a fuzzy single-period problemrdquoMathematical and Computer Modelling vol 54 no 5-6 pp1273ndash1285 2011

[45] H Yang J Yi J Zhao and Zh Dong ldquoExtreme learning ma-chine based genetic algorithm and its application in powersystem economic dispatchrdquo Neurocomputing vol 102 pp 154ndash162 2013

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 9

Start

Input order informationwarehouse setting picker

setting

Compute the supportbetween customersrsquo

orders using MINWAL

Calculating initialnumber of batches (K) to

assign the orders

Solving the binaryinteger programmingclustering model for

constructing the batches

Feasiblesolution

Yes

No

Solving picker routing

Solving batch sequencingproblem using

End

K = K + 1

problem using GA TSP

GA sequencing

Figure 6 Methodology framework

quantity of items and also each item has a certain quantityThese quantities are produced randomly The distributionfunction type for the number of items in each order is Normal[5 10] Also the distribution function type for each itemquantity in an order is uniform [1 10] The picker movingspeed is set as 119881119878 = 2 (ms) The warehouse has 3000 itemlocation The GA control parameters for solving the TSP andsequencing problem are tabulated in Table 6

All of the proposed approaches are coded usingMATLABversion 790 and Java 15 programming language Also theIBM ILOG CPLEX optimization studio version 124 softwarewas utilized jointly with MATLAB in order to facilitate theprocess of solving the binary integer programming problemExamples are run by means of a computer featured by

4 Gigabytes random access memory (RAM) and a 220Gigahertz Intel processor (T6600)

52 Computational Results and Discussion In order to revealthe competency of the proposed solution approach it wascompared with the other existing methods in the literatureOne of the latest published research activities in the literaturewhich deals with minimization of tardiness has been con-ducted by Henn and schmid [35] They utilized the EarliestDueDate (EDD) approach as their constructive algorithm forcomparing with their proposed approaches They proposedtwo heuristics Iterated Local Search andAttribute-BasedHillClimber are the two employed metaheuristics in their modelAccording to different test problems the efficiency of their

10 The Scientific World Journal

Table 5 Test problems description

Example 1 Example 2 Example 3 Example 4 Example 5Number of orders 40 60 80 100 150

Number of items 80 120 160 200 200

Capacity of picker 200 200 300 400 550

Total weight 2212 3668 4393 6110 8369

Minimum number of batch 11 19 15 16 16

Due dates range 60 90 120 180 240

Table 6 GA control parameters

GA TSP GA sequencing

Population size 500 200

Cross over strategyPermutation orderbased crossover

(POP)

Permutation orderbased crossover

(POP)Cross over probability 08 08

Mutation strategy SWAP SWAPMutation probability 02 02

Iteration number 10000 10000

Parent selection method Roulette wheel Roulette wheel

proposed methods was illustrated Their obtained resultsfrom their proposed approaches showed a 46 improvementon tardiness compared to EDD They used an s-shape andlargest gap routing strategy for their proposed method Inorder to make a reasonable comparison with their proposedmethod EDD was integrated with the s-shape routing strat-egy and GA sequencing Then based on improvement on EDDwe can compare our proposed method with their proposedmethod In the EDD approach due dates play an importantrole in sorting all orders Based on an ascending sequenceassigning the orders to batches ensures that no capacityviolation can happen for the picking device

For the warehouse condition which is described inSection 323 our proposed approach EDD integrated withGA TSP and GA sequencing and EDD integrated with s-shapealgorithm and GA sequencing have been evaluated in termsof average tardiness for all orders (119879) which is calculatedbased on (13) number of batches (119870) total process time(Pt) and average process time for each batch (119860Pt) All ofthe GA setting for the TSP and sequencing problems areconsidered the same for all methods Also the settings whichhave been mentioned in Section 51 were considered for all ofthe compared methods

The results for all of the test problems for the threeapproaches are tabulated in Tables 7 8 and 9 As it isillustrated in Table 10 EDD integrated with GA TSP andGA sequencing has a 3176 improvement on average in termsof average tardiness of all orders than EDD integrated withthe s-shape algorithm and GA sequencing It can be perceivedthat GA TSP performs better than s-shape Also the ATGHmethod has a 6811 improvement in average tardinessoff all orders than EDD integrated with s-shape algorithm

and GA sequencing Consequently the ATGH method achievesmuch improvement against Henn and Schmidrsquos [35] pro-posedmethod According to Tables 7 and 8 it is also worth tomention thatGA TSP performs better than the s-shape routingstrategy in terms of process time The improvements whichhave been achieved in this research could be justified basedon the following reasons

The problem of order batching deals with several sig-nificant drivers in which the number of batches can beconsidered as a key factor [23] As a justification to thismatter it is worth to mention that total travel distance maynot decrease directly due to a decline in the number ofbatches But a decline in the number of batches will increasebatch sizes which may lead to an increase in the distancetravelled by order picker However an advanced exploitationof picker capacity to load efficiently may reduce the numberof constructed batches together reducing the total traveldistance Improving the total travel distance traversed by thepicker can minimize the average tardiness of all orders Interms of number of the batches according to Tables 7 8 and9 it can be perceived that our proposed method assigns theorders to the batches optimally The ATGH method assignsthe orders to batches based on their item similarity andtheir due dates and it uses the picker capacity efficiently Butthe EDD method allocates the orders to batches based onan ascending sequence of their due date and it should beensured that no capacity violation will happen for the pickingdevice which may lead to an inefficient order batching Tohighlight the problem an example can be made based on theresults tabulated in Tables 7 and 8 The number of batchesfor example 1 in our method is 11 in comparison with 13obtained fromEDDmethod 13 batches can be justified basedon this scenario that two subsequent orders say the firstorder is 100 kg and the other is 150 kg cannot be batchedtogether due to violated batch capacity which is set to be200 kg The number of batches would increase due to theaforementioned issue which affects the total process time tohave an increasing manner This situation will rarely happenin the ATGH method because of the weighted associationbased binary integer programming utilizedwhich batches theorders in order to maximize the association between themwhich is calculated based on the item similarity of ordersand the weight of their corresponding due dates The ATGHmethod uses the picker capacity efficiently

Obviously one of the factors that can influence thecompletion time and the average tardiness of all orders is theprocessing time of the orders Reducing the processing time

The Scientific World Journal 11

Table 7 Computational results of ATGH

Example 1 Example 2 Example 3 Example 4 Example 5119879 56204 29948 90202 163 58898

Pt 605833 1217833 105833 121033 1339

119870 11 19 15 16 16

119860Pt 55076 6409 7055 7564 8368

Table 8 Computational results of EDD integrated with GA TSP and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 124550 42976 18222 5981 1547

Pt 667333 133516 116083 13185 141916

119870 13 21 16 17 16

119860Pt 51333 6357 7255 7775 8869

Table 9 Computational results of EDD integrated with s-shape algorithm and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 1651 62677 2825 13365 17593

Pt 75266 16636 13323 14333 14926

119870 13 21 16 17 16

119860Pt 5789 892 833 8431 932

Table 10 Comparison results

EDD + s-shape + GA sequencing EDD + GA TSP + GA sequencing ATGHTardiness (min) Tardiness (min) Improvement () Tardiness (min) Improvement ()

(1) 1651 124550 2456 56204 6596(2) 62677 42976 3143 29948 5222(3) 2825 18222 3550 90202 6807(4) 13365 5981 5525 1627 8783(5) 17593 1547 1206 58898 6652Average improvement 3176 6811

and travel time for constructing a batch is an outstanding factin reducing the costs and minimizing delays of the customerresponses In our proposed model orders with similar itemsare consolidated together Therefore batching orders withmore similarity can reduce the distance traversed by thepicker facility Also the due date of each order is definedas a weight parameter in calculating the association betweenorders in terms of support which was done for consideringthe due dates in our model Thereupon the model considersdue date and the process time in an integrated manner whichsimultaneously affected the average tardiness of all ordersThis spoken issue is different in the EDD approach in whichthe ordersrsquo due dates are just considered Example 5 canbe considered as evidence of this matter in which the totaland average process time for ATGH is lower than EDDintegrated with GA TSP and GA sequencing with respect to thefact that number of constructed batches for both approachesare happened to be equal after obtaining the results

6 Conclusion

One of the most important concerns of warehouse managersis involvedwith finding the best optimal and cost efficientwayto pick orders placed by customers in order to be known as aresponsible company with satisfied customers in terms of ontime delivery In the current research activity order batchingpicker routing and batch sequencing problems of warehouseprocesses were addressed which are to be solved jointly asthey are mixed up with many manufacturing industriesMathematical solutions of these NP-hard problems might beavailable for problems that are small in nature But thesekinds of solutions may not be applicable to be consideredby warehouse managers Hence we presented novel solutionapproach namely ATGH to solve this issue in accordancewith minimizing the average tardiness of all customersrsquoorders which can be considered as a major objective ofthis research activity A weighted association rule mining

12 The Scientific World Journal

algorithm namely MINWAL was utilized to determine thecustomersrsquo orders association regarding similarity betweenitems and ordersrsquo due dates Actually in our proposedmethodthe due date of each order is defined as a weight parameterin calculating the association between orders So orders withsimilar items and due dates have more chance to be in asame batch This issue is not considered in most of theresearch activities in the literature related to our work andconsidering this issue in the order batching problem is themain contribution of our research activity The previous stepwas followed by a binary integer clustering model in orderto solve the order batching problem which maximizes thesimilarity between orders in terms of support After that aGA was applied to solve a TSP in order to find the optimalrouting path

Finally a batch sequencing problem was addressed inconjunction with a GA

The other research objective of this research addresseswhether the proposed ATGH algorithm can further improvethe solution quality for a multiple parallel aisle warehousewith manual picking system Hence comparisons were madeconsideringATGHEDD integratedGA TSP andGA sequencingand EDD integrated s-shape and GA sequencing algorithms Asdemonstrated by the obtained results ATGH comes out tobe the most striking solution approach in terms of tardinessIn spite of the fact that considering all customer orders asdiscussed in this paper would lead to have a static case beinginvolved with dynamic cases in which different points of timeare assigned to customerrsquos orders arrival would be beneficialSo there might be some potential room of research to extendour discussed approach with respect to dynamic orders

Conflict of Interests

The IBM ILOG CPLEX optimization studio version 124software is not used for financial gain and there is no conflictof interests with any party as it is only a student version

References

[1] ELAAT Kearney Excellence in Logistics ELA Brussels Bel-gium 2004

[2] Y-C Ho T-S Su and Z-B Shi ldquoOrder-batching methods foran order-picking warehouse with two cross aislesrdquo Computersamp Industrial Engineering vol 55 no 2 pp 321ndash347 2008

[3] C-M Hsu K-Y Chen and M-C Chen ldquoBatching ordersin warehouses by minimizing travel distance with geneticalgorithmsrdquo Computers in Industry vol 56 no 2 pp 169ndash1782005

[4] E A Elsayed M-K Lee S Kim and E Scherer ldquoSequencingand batching procedures forminimizing earliness and tardinesspenalty of order retrievalsrdquoThe International Journal of Produc-tion Research vol 31 no 3 pp 727ndash738 1993

[5] S Henn S Koch and W Gerhard Order Batching in OrderPicking Warehouses A Survey of Solution Approaches SpringerLondon UK 2012

[6] J Drury Towards More Efficient Order Picking IMM Mono-graph no 1 The Institute of Materials Management CranfieldUK 1988

[7] M Goetschalckx and J Ashayeri ldquoClassification and design oforder picking systemsrdquo Logistics World pp 99ndash106 1989

[8] J A Tompkins J A White Y A Bozer E H Frazelle and J MA Tanchoco Facilities Planning John Wiley amp Sons HobokenNJ USA 2003

[9] S Henn and G Wascher ldquoTabu search heuristics for the orderbatching problem in manual order picking systemsrdquo EuropeanJournal of Operational Research vol 222 no 3 pp 484ndash4942012

[10] S Henn ldquoAlgorithms for on-line order batching in an orderpicking warehouserdquo Computers and Operations Research vol39 no 11 pp 2549ndash2563 2012

[11] R de Koster ldquoHow to assess a warehouse operation in a singletourrdquo Tech Rep RSM Erasmus University Rotterdam TheNetherlands 2004

[12] R de Koster T Le-Duc and K J Roodbergen ldquoDesign andcontrol of warehouse order picking a literature reviewrdquo Euro-pean Journal of Operational Research vol 182 no 2 pp 481ndash5012007

[13] R A Ruben and F R Jacobs ldquoBatch construction heuristics andstorage assignment strategies for walkride and pick systemsrdquoManagement Science vol 45 no 4 pp 575ndash596 1999

[14] E A Elsayed ldquoAlgorithms for optimal material handling inautomatic warehousing systemsrdquo International Journal of Pro-duction Research vol 19 no 5 pp 525ndash535 1981

[15] G Clarke and J W Wright ldquoScheduling of vehicles froma central depot to a number of delivery pointsrdquo OperationsResearch vol 12 no 4 pp 568ndash581 1964

[16] H D Vinod ldquoInteger programming and the theory of group-ingrdquo Journal of the American Statistical Association vol 64 no326 pp 506ndash519 1969

[17] R D Armstrong W D Cook and A L Saipe ldquoOptimalbatching in a semi-automated order picking systemrdquo Journal ofthe Operational Research Society vol 30 no 8 pp 711ndash720 1979

[18] H D Ratliff and A S Rosenthal ldquoOrder-picking in a rect-angular warehouse a solvable case of the traveling salesmanproblemrdquo Operations Research vol 31 no 3 pp 507ndash521 1983

[19] A Kusiak A Vannelli and R K Kumar ldquoClustering analysismodels and algorithmsrdquo Cybernetics vol 15 pp 139ndash154 1986

[20] D R Gibson and G P Sharp ldquoOrder batching proceduresrdquoEuropean Journal of Operational Research vol 58 no 1 pp 57ndash67 1992

[21] M B Rosenwein ldquoA comparison of heuristics for the problemofbatching orders for warehouse selectionrdquo International Journalof Production Research vol 34 no 3 pp 657ndash664 1996

[22] A J RMGademann J P van denBerg andHH van derHoffldquoAn order batching algorithm for wave picking in a parallel-aislewarehouserdquo IIE Transactions vol 33 no 5 pp 385ndash398 2001

[23] M-C Chen and H-P Wu ldquoAn association-based clusteringapproach to order batching considering customer demandpatternsrdquo Omega vol 33 no 4 pp 333ndash343 2005

[24] NGademann and S van deVelde ldquoOrder batching tominimizetotal travel time in a parallel-aisle warehouserdquo IIE Transactionsvol 37 no 1 pp 63ndash75 2005

[25] Y A Bozer and J W Kile ldquoOrder batching in walk-and-pick order picking systemsrdquo International Journal of ProductionResearch vol 46 no 7 pp 1887ndash1909 2008

[26] O Kulak Y Sahin and M E Taner ldquoJoint order batching andpicker routing in single and multiple-cross-aisle warehousesusing cluster-based tabu search algorithmsrdquo Flexible Servicesand Manufacturing Journal vol 24 no 1 pp 52ndash80 2012

The Scientific World Journal 13

[27] E A Elsayed and M-K Lee ldquoOrder processing in automatedstorageretrieval systems with due datesrdquo IIE Transactions vol28 no 7 pp 567ndash577 1996

[28] J Won and S Olafsson ldquoJoint order batching and order pickingin warehouse operationsrdquo International Journal of ProductionResearch vol 43 no 7 pp 1427ndash1442 2005

[29] C-Y Tsai J J H Liou and T-M Huang ldquoUsing a multiple-GA method to solve the batch picking problem consideringtravel distance and order due timerdquo International Journal of Pro-duction Research vol 46 no 22 pp 6533ndash6555 2008

[30] Y KaoM H Chen and Y T Huang ldquoA hybrid algorithm basedon ACO and PSO for capacitated vehicle routing problemsrdquoMathematical Problems in Engineering vol 2012 Article ID726564 17 pages 2012

[31] X Gan Y Wang S Li and B Niu ldquoVehicle routing problemwith time windows and simultaneous delivery and pick-up ser-vice based on MCPSOrdquoMathematical Problems in Engineeringvol 2012 Article ID 104279 11 pages 2012

[32] M B M de Koster E S van der Poort and M WoltersldquoEfficient orderbatching methods in warehousesrdquo InternationalJournal of ProductionResearch vol 37 no 7 pp 1479ndash1504 1999

[33] Y-C Ho and Y-Y Tseng ldquoA study on order-batching methodsof order-picking in a distribution centre with two cross-aislesrdquoInternational Journal of Production Research vol 44 no 17 pp3391ndash3417 2006

[34] C Theys O Braysy W Dullaert and B Raa ldquoUsing a TSPheuristic for routing order pickers in warehousesrdquo EuropeanJournal of Operational Research vol 200 no 3 pp 755ndash7632010

[35] S Henn and V Schmid Metaheuristics for Order Batching andSequencing in Manual Order Picking Systems Univ Fak furWirtschaftswiss 2011

[36] M Hahsler B Grun and K Hornik ldquoArulesmdasha computationalenvironment for mining association rules and frequent itemsetsrdquo Journal of Statistical Software vol 14 2005

[37] R Agrawal T Imielinski and A Swami ldquoMining associationrules between sets of items in large databasesrdquo ACM SIGMODRecord vol 22 no 2 pp 207ndash216 1993

[38] S Encheva Y Kondratenko M Z Solesvik and S TuminldquoDecision support systems in logisticsrdquo in Proceeding of Inter-national electronic conference on computer science pp 254ndash2562007

[39] M-C Chen ldquoConfiguration of cellular manufacturing systemsusing association rule inductionrdquo International Journal of Pro-duction Research vol 41 no 2 pp 381ndash395 2003

[40] R Srikant and R Agrawal ldquoMining generalized associationrulesrdquo Future Generation Computer Systems vol 13 no 2-3 pp161ndash180 1997

[41] J Wu and XM Li ldquoAn effective mining algorithm for weightedassociation rules in communication networksrdquo Journal of Com-puters vol 3 no 10 pp 20ndash27 2008

[42] C H Cai A W C Fu C H Cheng and W W KwongldquoMining association rules with weighted itemsrdquo in Proceedingsof the International Symposium on Database Engineering ampApplications pp 68ndash77 Cardiff UK 1998

[43] J HollandAdaptation in Natural and Artificial Systems Univer-sity of Michigan Press Ann Arbor Mich USA 1975

[44] A A Taleizadeh F Barzinpour and H-M Wee ldquoMeta-heuristic algorithms for solving a fuzzy single-period problemrdquoMathematical and Computer Modelling vol 54 no 5-6 pp1273ndash1285 2011

[45] H Yang J Yi J Zhao and Zh Dong ldquoExtreme learning ma-chine based genetic algorithm and its application in powersystem economic dispatchrdquo Neurocomputing vol 102 pp 154ndash162 2013

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

10 The Scientific World Journal

Table 5 Test problems description

Example 1 Example 2 Example 3 Example 4 Example 5Number of orders 40 60 80 100 150

Number of items 80 120 160 200 200

Capacity of picker 200 200 300 400 550

Total weight 2212 3668 4393 6110 8369

Minimum number of batch 11 19 15 16 16

Due dates range 60 90 120 180 240

Table 6 GA control parameters

GA TSP GA sequencing

Population size 500 200

Cross over strategyPermutation orderbased crossover

(POP)

Permutation orderbased crossover

(POP)Cross over probability 08 08

Mutation strategy SWAP SWAPMutation probability 02 02

Iteration number 10000 10000

Parent selection method Roulette wheel Roulette wheel

proposed methods was illustrated Their obtained resultsfrom their proposed approaches showed a 46 improvementon tardiness compared to EDD They used an s-shape andlargest gap routing strategy for their proposed method Inorder to make a reasonable comparison with their proposedmethod EDD was integrated with the s-shape routing strat-egy and GA sequencing Then based on improvement on EDDwe can compare our proposed method with their proposedmethod In the EDD approach due dates play an importantrole in sorting all orders Based on an ascending sequenceassigning the orders to batches ensures that no capacityviolation can happen for the picking device

For the warehouse condition which is described inSection 323 our proposed approach EDD integrated withGA TSP and GA sequencing and EDD integrated with s-shapealgorithm and GA sequencing have been evaluated in termsof average tardiness for all orders (119879) which is calculatedbased on (13) number of batches (119870) total process time(Pt) and average process time for each batch (119860Pt) All ofthe GA setting for the TSP and sequencing problems areconsidered the same for all methods Also the settings whichhave been mentioned in Section 51 were considered for all ofthe compared methods

The results for all of the test problems for the threeapproaches are tabulated in Tables 7 8 and 9 As it isillustrated in Table 10 EDD integrated with GA TSP andGA sequencing has a 3176 improvement on average in termsof average tardiness of all orders than EDD integrated withthe s-shape algorithm and GA sequencing It can be perceivedthat GA TSP performs better than s-shape Also the ATGHmethod has a 6811 improvement in average tardinessoff all orders than EDD integrated with s-shape algorithm

and GA sequencing Consequently the ATGH method achievesmuch improvement against Henn and Schmidrsquos [35] pro-posedmethod According to Tables 7 and 8 it is also worth tomention thatGA TSP performs better than the s-shape routingstrategy in terms of process time The improvements whichhave been achieved in this research could be justified basedon the following reasons

The problem of order batching deals with several sig-nificant drivers in which the number of batches can beconsidered as a key factor [23] As a justification to thismatter it is worth to mention that total travel distance maynot decrease directly due to a decline in the number ofbatches But a decline in the number of batches will increasebatch sizes which may lead to an increase in the distancetravelled by order picker However an advanced exploitationof picker capacity to load efficiently may reduce the numberof constructed batches together reducing the total traveldistance Improving the total travel distance traversed by thepicker can minimize the average tardiness of all orders Interms of number of the batches according to Tables 7 8 and9 it can be perceived that our proposed method assigns theorders to the batches optimally The ATGH method assignsthe orders to batches based on their item similarity andtheir due dates and it uses the picker capacity efficiently Butthe EDD method allocates the orders to batches based onan ascending sequence of their due date and it should beensured that no capacity violation will happen for the pickingdevice which may lead to an inefficient order batching Tohighlight the problem an example can be made based on theresults tabulated in Tables 7 and 8 The number of batchesfor example 1 in our method is 11 in comparison with 13obtained fromEDDmethod 13 batches can be justified basedon this scenario that two subsequent orders say the firstorder is 100 kg and the other is 150 kg cannot be batchedtogether due to violated batch capacity which is set to be200 kg The number of batches would increase due to theaforementioned issue which affects the total process time tohave an increasing manner This situation will rarely happenin the ATGH method because of the weighted associationbased binary integer programming utilizedwhich batches theorders in order to maximize the association between themwhich is calculated based on the item similarity of ordersand the weight of their corresponding due dates The ATGHmethod uses the picker capacity efficiently

Obviously one of the factors that can influence thecompletion time and the average tardiness of all orders is theprocessing time of the orders Reducing the processing time

The Scientific World Journal 11

Table 7 Computational results of ATGH

Example 1 Example 2 Example 3 Example 4 Example 5119879 56204 29948 90202 163 58898

Pt 605833 1217833 105833 121033 1339

119870 11 19 15 16 16

119860Pt 55076 6409 7055 7564 8368

Table 8 Computational results of EDD integrated with GA TSP and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 124550 42976 18222 5981 1547

Pt 667333 133516 116083 13185 141916

119870 13 21 16 17 16

119860Pt 51333 6357 7255 7775 8869

Table 9 Computational results of EDD integrated with s-shape algorithm and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 1651 62677 2825 13365 17593

Pt 75266 16636 13323 14333 14926

119870 13 21 16 17 16

119860Pt 5789 892 833 8431 932

Table 10 Comparison results

EDD + s-shape + GA sequencing EDD + GA TSP + GA sequencing ATGHTardiness (min) Tardiness (min) Improvement () Tardiness (min) Improvement ()

(1) 1651 124550 2456 56204 6596(2) 62677 42976 3143 29948 5222(3) 2825 18222 3550 90202 6807(4) 13365 5981 5525 1627 8783(5) 17593 1547 1206 58898 6652Average improvement 3176 6811

and travel time for constructing a batch is an outstanding factin reducing the costs and minimizing delays of the customerresponses In our proposed model orders with similar itemsare consolidated together Therefore batching orders withmore similarity can reduce the distance traversed by thepicker facility Also the due date of each order is definedas a weight parameter in calculating the association betweenorders in terms of support which was done for consideringthe due dates in our model Thereupon the model considersdue date and the process time in an integrated manner whichsimultaneously affected the average tardiness of all ordersThis spoken issue is different in the EDD approach in whichthe ordersrsquo due dates are just considered Example 5 canbe considered as evidence of this matter in which the totaland average process time for ATGH is lower than EDDintegrated with GA TSP and GA sequencing with respect to thefact that number of constructed batches for both approachesare happened to be equal after obtaining the results

6 Conclusion

One of the most important concerns of warehouse managersis involvedwith finding the best optimal and cost efficientwayto pick orders placed by customers in order to be known as aresponsible company with satisfied customers in terms of ontime delivery In the current research activity order batchingpicker routing and batch sequencing problems of warehouseprocesses were addressed which are to be solved jointly asthey are mixed up with many manufacturing industriesMathematical solutions of these NP-hard problems might beavailable for problems that are small in nature But thesekinds of solutions may not be applicable to be consideredby warehouse managers Hence we presented novel solutionapproach namely ATGH to solve this issue in accordancewith minimizing the average tardiness of all customersrsquoorders which can be considered as a major objective ofthis research activity A weighted association rule mining

12 The Scientific World Journal

algorithm namely MINWAL was utilized to determine thecustomersrsquo orders association regarding similarity betweenitems and ordersrsquo due dates Actually in our proposedmethodthe due date of each order is defined as a weight parameterin calculating the association between orders So orders withsimilar items and due dates have more chance to be in asame batch This issue is not considered in most of theresearch activities in the literature related to our work andconsidering this issue in the order batching problem is themain contribution of our research activity The previous stepwas followed by a binary integer clustering model in orderto solve the order batching problem which maximizes thesimilarity between orders in terms of support After that aGA was applied to solve a TSP in order to find the optimalrouting path

Finally a batch sequencing problem was addressed inconjunction with a GA

The other research objective of this research addresseswhether the proposed ATGH algorithm can further improvethe solution quality for a multiple parallel aisle warehousewith manual picking system Hence comparisons were madeconsideringATGHEDD integratedGA TSP andGA sequencingand EDD integrated s-shape and GA sequencing algorithms Asdemonstrated by the obtained results ATGH comes out tobe the most striking solution approach in terms of tardinessIn spite of the fact that considering all customer orders asdiscussed in this paper would lead to have a static case beinginvolved with dynamic cases in which different points of timeare assigned to customerrsquos orders arrival would be beneficialSo there might be some potential room of research to extendour discussed approach with respect to dynamic orders

Conflict of Interests

The IBM ILOG CPLEX optimization studio version 124software is not used for financial gain and there is no conflictof interests with any party as it is only a student version

References

[1] ELAAT Kearney Excellence in Logistics ELA Brussels Bel-gium 2004

[2] Y-C Ho T-S Su and Z-B Shi ldquoOrder-batching methods foran order-picking warehouse with two cross aislesrdquo Computersamp Industrial Engineering vol 55 no 2 pp 321ndash347 2008

[3] C-M Hsu K-Y Chen and M-C Chen ldquoBatching ordersin warehouses by minimizing travel distance with geneticalgorithmsrdquo Computers in Industry vol 56 no 2 pp 169ndash1782005

[4] E A Elsayed M-K Lee S Kim and E Scherer ldquoSequencingand batching procedures forminimizing earliness and tardinesspenalty of order retrievalsrdquoThe International Journal of Produc-tion Research vol 31 no 3 pp 727ndash738 1993

[5] S Henn S Koch and W Gerhard Order Batching in OrderPicking Warehouses A Survey of Solution Approaches SpringerLondon UK 2012

[6] J Drury Towards More Efficient Order Picking IMM Mono-graph no 1 The Institute of Materials Management CranfieldUK 1988

[7] M Goetschalckx and J Ashayeri ldquoClassification and design oforder picking systemsrdquo Logistics World pp 99ndash106 1989

[8] J A Tompkins J A White Y A Bozer E H Frazelle and J MA Tanchoco Facilities Planning John Wiley amp Sons HobokenNJ USA 2003

[9] S Henn and G Wascher ldquoTabu search heuristics for the orderbatching problem in manual order picking systemsrdquo EuropeanJournal of Operational Research vol 222 no 3 pp 484ndash4942012

[10] S Henn ldquoAlgorithms for on-line order batching in an orderpicking warehouserdquo Computers and Operations Research vol39 no 11 pp 2549ndash2563 2012

[11] R de Koster ldquoHow to assess a warehouse operation in a singletourrdquo Tech Rep RSM Erasmus University Rotterdam TheNetherlands 2004

[12] R de Koster T Le-Duc and K J Roodbergen ldquoDesign andcontrol of warehouse order picking a literature reviewrdquo Euro-pean Journal of Operational Research vol 182 no 2 pp 481ndash5012007

[13] R A Ruben and F R Jacobs ldquoBatch construction heuristics andstorage assignment strategies for walkride and pick systemsrdquoManagement Science vol 45 no 4 pp 575ndash596 1999

[14] E A Elsayed ldquoAlgorithms for optimal material handling inautomatic warehousing systemsrdquo International Journal of Pro-duction Research vol 19 no 5 pp 525ndash535 1981

[15] G Clarke and J W Wright ldquoScheduling of vehicles froma central depot to a number of delivery pointsrdquo OperationsResearch vol 12 no 4 pp 568ndash581 1964

[16] H D Vinod ldquoInteger programming and the theory of group-ingrdquo Journal of the American Statistical Association vol 64 no326 pp 506ndash519 1969

[17] R D Armstrong W D Cook and A L Saipe ldquoOptimalbatching in a semi-automated order picking systemrdquo Journal ofthe Operational Research Society vol 30 no 8 pp 711ndash720 1979

[18] H D Ratliff and A S Rosenthal ldquoOrder-picking in a rect-angular warehouse a solvable case of the traveling salesmanproblemrdquo Operations Research vol 31 no 3 pp 507ndash521 1983

[19] A Kusiak A Vannelli and R K Kumar ldquoClustering analysismodels and algorithmsrdquo Cybernetics vol 15 pp 139ndash154 1986

[20] D R Gibson and G P Sharp ldquoOrder batching proceduresrdquoEuropean Journal of Operational Research vol 58 no 1 pp 57ndash67 1992

[21] M B Rosenwein ldquoA comparison of heuristics for the problemofbatching orders for warehouse selectionrdquo International Journalof Production Research vol 34 no 3 pp 657ndash664 1996

[22] A J RMGademann J P van denBerg andHH van derHoffldquoAn order batching algorithm for wave picking in a parallel-aislewarehouserdquo IIE Transactions vol 33 no 5 pp 385ndash398 2001

[23] M-C Chen and H-P Wu ldquoAn association-based clusteringapproach to order batching considering customer demandpatternsrdquo Omega vol 33 no 4 pp 333ndash343 2005

[24] NGademann and S van deVelde ldquoOrder batching tominimizetotal travel time in a parallel-aisle warehouserdquo IIE Transactionsvol 37 no 1 pp 63ndash75 2005

[25] Y A Bozer and J W Kile ldquoOrder batching in walk-and-pick order picking systemsrdquo International Journal of ProductionResearch vol 46 no 7 pp 1887ndash1909 2008

[26] O Kulak Y Sahin and M E Taner ldquoJoint order batching andpicker routing in single and multiple-cross-aisle warehousesusing cluster-based tabu search algorithmsrdquo Flexible Servicesand Manufacturing Journal vol 24 no 1 pp 52ndash80 2012

The Scientific World Journal 13

[27] E A Elsayed and M-K Lee ldquoOrder processing in automatedstorageretrieval systems with due datesrdquo IIE Transactions vol28 no 7 pp 567ndash577 1996

[28] J Won and S Olafsson ldquoJoint order batching and order pickingin warehouse operationsrdquo International Journal of ProductionResearch vol 43 no 7 pp 1427ndash1442 2005

[29] C-Y Tsai J J H Liou and T-M Huang ldquoUsing a multiple-GA method to solve the batch picking problem consideringtravel distance and order due timerdquo International Journal of Pro-duction Research vol 46 no 22 pp 6533ndash6555 2008

[30] Y KaoM H Chen and Y T Huang ldquoA hybrid algorithm basedon ACO and PSO for capacitated vehicle routing problemsrdquoMathematical Problems in Engineering vol 2012 Article ID726564 17 pages 2012

[31] X Gan Y Wang S Li and B Niu ldquoVehicle routing problemwith time windows and simultaneous delivery and pick-up ser-vice based on MCPSOrdquoMathematical Problems in Engineeringvol 2012 Article ID 104279 11 pages 2012

[32] M B M de Koster E S van der Poort and M WoltersldquoEfficient orderbatching methods in warehousesrdquo InternationalJournal of ProductionResearch vol 37 no 7 pp 1479ndash1504 1999

[33] Y-C Ho and Y-Y Tseng ldquoA study on order-batching methodsof order-picking in a distribution centre with two cross-aislesrdquoInternational Journal of Production Research vol 44 no 17 pp3391ndash3417 2006

[34] C Theys O Braysy W Dullaert and B Raa ldquoUsing a TSPheuristic for routing order pickers in warehousesrdquo EuropeanJournal of Operational Research vol 200 no 3 pp 755ndash7632010

[35] S Henn and V Schmid Metaheuristics for Order Batching andSequencing in Manual Order Picking Systems Univ Fak furWirtschaftswiss 2011

[36] M Hahsler B Grun and K Hornik ldquoArulesmdasha computationalenvironment for mining association rules and frequent itemsetsrdquo Journal of Statistical Software vol 14 2005

[37] R Agrawal T Imielinski and A Swami ldquoMining associationrules between sets of items in large databasesrdquo ACM SIGMODRecord vol 22 no 2 pp 207ndash216 1993

[38] S Encheva Y Kondratenko M Z Solesvik and S TuminldquoDecision support systems in logisticsrdquo in Proceeding of Inter-national electronic conference on computer science pp 254ndash2562007

[39] M-C Chen ldquoConfiguration of cellular manufacturing systemsusing association rule inductionrdquo International Journal of Pro-duction Research vol 41 no 2 pp 381ndash395 2003

[40] R Srikant and R Agrawal ldquoMining generalized associationrulesrdquo Future Generation Computer Systems vol 13 no 2-3 pp161ndash180 1997

[41] J Wu and XM Li ldquoAn effective mining algorithm for weightedassociation rules in communication networksrdquo Journal of Com-puters vol 3 no 10 pp 20ndash27 2008

[42] C H Cai A W C Fu C H Cheng and W W KwongldquoMining association rules with weighted itemsrdquo in Proceedingsof the International Symposium on Database Engineering ampApplications pp 68ndash77 Cardiff UK 1998

[43] J HollandAdaptation in Natural and Artificial Systems Univer-sity of Michigan Press Ann Arbor Mich USA 1975

[44] A A Taleizadeh F Barzinpour and H-M Wee ldquoMeta-heuristic algorithms for solving a fuzzy single-period problemrdquoMathematical and Computer Modelling vol 54 no 5-6 pp1273ndash1285 2011

[45] H Yang J Yi J Zhao and Zh Dong ldquoExtreme learning ma-chine based genetic algorithm and its application in powersystem economic dispatchrdquo Neurocomputing vol 102 pp 154ndash162 2013

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 11

Table 7 Computational results of ATGH

Example 1 Example 2 Example 3 Example 4 Example 5119879 56204 29948 90202 163 58898

Pt 605833 1217833 105833 121033 1339

119870 11 19 15 16 16

119860Pt 55076 6409 7055 7564 8368

Table 8 Computational results of EDD integrated with GA TSP and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 124550 42976 18222 5981 1547

Pt 667333 133516 116083 13185 141916

119870 13 21 16 17 16

119860Pt 51333 6357 7255 7775 8869

Table 9 Computational results of EDD integrated with s-shape algorithm and GA sequencing

Example 1 Example 2 Example 3 Example 4 Example 5119879 1651 62677 2825 13365 17593

Pt 75266 16636 13323 14333 14926

119870 13 21 16 17 16

119860Pt 5789 892 833 8431 932

Table 10 Comparison results

EDD + s-shape + GA sequencing EDD + GA TSP + GA sequencing ATGHTardiness (min) Tardiness (min) Improvement () Tardiness (min) Improvement ()

(1) 1651 124550 2456 56204 6596(2) 62677 42976 3143 29948 5222(3) 2825 18222 3550 90202 6807(4) 13365 5981 5525 1627 8783(5) 17593 1547 1206 58898 6652Average improvement 3176 6811

and travel time for constructing a batch is an outstanding factin reducing the costs and minimizing delays of the customerresponses In our proposed model orders with similar itemsare consolidated together Therefore batching orders withmore similarity can reduce the distance traversed by thepicker facility Also the due date of each order is definedas a weight parameter in calculating the association betweenorders in terms of support which was done for consideringthe due dates in our model Thereupon the model considersdue date and the process time in an integrated manner whichsimultaneously affected the average tardiness of all ordersThis spoken issue is different in the EDD approach in whichthe ordersrsquo due dates are just considered Example 5 canbe considered as evidence of this matter in which the totaland average process time for ATGH is lower than EDDintegrated with GA TSP and GA sequencing with respect to thefact that number of constructed batches for both approachesare happened to be equal after obtaining the results

6 Conclusion

One of the most important concerns of warehouse managersis involvedwith finding the best optimal and cost efficientwayto pick orders placed by customers in order to be known as aresponsible company with satisfied customers in terms of ontime delivery In the current research activity order batchingpicker routing and batch sequencing problems of warehouseprocesses were addressed which are to be solved jointly asthey are mixed up with many manufacturing industriesMathematical solutions of these NP-hard problems might beavailable for problems that are small in nature But thesekinds of solutions may not be applicable to be consideredby warehouse managers Hence we presented novel solutionapproach namely ATGH to solve this issue in accordancewith minimizing the average tardiness of all customersrsquoorders which can be considered as a major objective ofthis research activity A weighted association rule mining

12 The Scientific World Journal

algorithm namely MINWAL was utilized to determine thecustomersrsquo orders association regarding similarity betweenitems and ordersrsquo due dates Actually in our proposedmethodthe due date of each order is defined as a weight parameterin calculating the association between orders So orders withsimilar items and due dates have more chance to be in asame batch This issue is not considered in most of theresearch activities in the literature related to our work andconsidering this issue in the order batching problem is themain contribution of our research activity The previous stepwas followed by a binary integer clustering model in orderto solve the order batching problem which maximizes thesimilarity between orders in terms of support After that aGA was applied to solve a TSP in order to find the optimalrouting path

Finally a batch sequencing problem was addressed inconjunction with a GA

The other research objective of this research addresseswhether the proposed ATGH algorithm can further improvethe solution quality for a multiple parallel aisle warehousewith manual picking system Hence comparisons were madeconsideringATGHEDD integratedGA TSP andGA sequencingand EDD integrated s-shape and GA sequencing algorithms Asdemonstrated by the obtained results ATGH comes out tobe the most striking solution approach in terms of tardinessIn spite of the fact that considering all customer orders asdiscussed in this paper would lead to have a static case beinginvolved with dynamic cases in which different points of timeare assigned to customerrsquos orders arrival would be beneficialSo there might be some potential room of research to extendour discussed approach with respect to dynamic orders

Conflict of Interests

The IBM ILOG CPLEX optimization studio version 124software is not used for financial gain and there is no conflictof interests with any party as it is only a student version

References

[1] ELAAT Kearney Excellence in Logistics ELA Brussels Bel-gium 2004

[2] Y-C Ho T-S Su and Z-B Shi ldquoOrder-batching methods foran order-picking warehouse with two cross aislesrdquo Computersamp Industrial Engineering vol 55 no 2 pp 321ndash347 2008

[3] C-M Hsu K-Y Chen and M-C Chen ldquoBatching ordersin warehouses by minimizing travel distance with geneticalgorithmsrdquo Computers in Industry vol 56 no 2 pp 169ndash1782005

[4] E A Elsayed M-K Lee S Kim and E Scherer ldquoSequencingand batching procedures forminimizing earliness and tardinesspenalty of order retrievalsrdquoThe International Journal of Produc-tion Research vol 31 no 3 pp 727ndash738 1993

[5] S Henn S Koch and W Gerhard Order Batching in OrderPicking Warehouses A Survey of Solution Approaches SpringerLondon UK 2012

[6] J Drury Towards More Efficient Order Picking IMM Mono-graph no 1 The Institute of Materials Management CranfieldUK 1988

[7] M Goetschalckx and J Ashayeri ldquoClassification and design oforder picking systemsrdquo Logistics World pp 99ndash106 1989

[8] J A Tompkins J A White Y A Bozer E H Frazelle and J MA Tanchoco Facilities Planning John Wiley amp Sons HobokenNJ USA 2003

[9] S Henn and G Wascher ldquoTabu search heuristics for the orderbatching problem in manual order picking systemsrdquo EuropeanJournal of Operational Research vol 222 no 3 pp 484ndash4942012

[10] S Henn ldquoAlgorithms for on-line order batching in an orderpicking warehouserdquo Computers and Operations Research vol39 no 11 pp 2549ndash2563 2012

[11] R de Koster ldquoHow to assess a warehouse operation in a singletourrdquo Tech Rep RSM Erasmus University Rotterdam TheNetherlands 2004

[12] R de Koster T Le-Duc and K J Roodbergen ldquoDesign andcontrol of warehouse order picking a literature reviewrdquo Euro-pean Journal of Operational Research vol 182 no 2 pp 481ndash5012007

[13] R A Ruben and F R Jacobs ldquoBatch construction heuristics andstorage assignment strategies for walkride and pick systemsrdquoManagement Science vol 45 no 4 pp 575ndash596 1999

[14] E A Elsayed ldquoAlgorithms for optimal material handling inautomatic warehousing systemsrdquo International Journal of Pro-duction Research vol 19 no 5 pp 525ndash535 1981

[15] G Clarke and J W Wright ldquoScheduling of vehicles froma central depot to a number of delivery pointsrdquo OperationsResearch vol 12 no 4 pp 568ndash581 1964

[16] H D Vinod ldquoInteger programming and the theory of group-ingrdquo Journal of the American Statistical Association vol 64 no326 pp 506ndash519 1969

[17] R D Armstrong W D Cook and A L Saipe ldquoOptimalbatching in a semi-automated order picking systemrdquo Journal ofthe Operational Research Society vol 30 no 8 pp 711ndash720 1979

[18] H D Ratliff and A S Rosenthal ldquoOrder-picking in a rect-angular warehouse a solvable case of the traveling salesmanproblemrdquo Operations Research vol 31 no 3 pp 507ndash521 1983

[19] A Kusiak A Vannelli and R K Kumar ldquoClustering analysismodels and algorithmsrdquo Cybernetics vol 15 pp 139ndash154 1986

[20] D R Gibson and G P Sharp ldquoOrder batching proceduresrdquoEuropean Journal of Operational Research vol 58 no 1 pp 57ndash67 1992

[21] M B Rosenwein ldquoA comparison of heuristics for the problemofbatching orders for warehouse selectionrdquo International Journalof Production Research vol 34 no 3 pp 657ndash664 1996

[22] A J RMGademann J P van denBerg andHH van derHoffldquoAn order batching algorithm for wave picking in a parallel-aislewarehouserdquo IIE Transactions vol 33 no 5 pp 385ndash398 2001

[23] M-C Chen and H-P Wu ldquoAn association-based clusteringapproach to order batching considering customer demandpatternsrdquo Omega vol 33 no 4 pp 333ndash343 2005

[24] NGademann and S van deVelde ldquoOrder batching tominimizetotal travel time in a parallel-aisle warehouserdquo IIE Transactionsvol 37 no 1 pp 63ndash75 2005

[25] Y A Bozer and J W Kile ldquoOrder batching in walk-and-pick order picking systemsrdquo International Journal of ProductionResearch vol 46 no 7 pp 1887ndash1909 2008

[26] O Kulak Y Sahin and M E Taner ldquoJoint order batching andpicker routing in single and multiple-cross-aisle warehousesusing cluster-based tabu search algorithmsrdquo Flexible Servicesand Manufacturing Journal vol 24 no 1 pp 52ndash80 2012

The Scientific World Journal 13

[27] E A Elsayed and M-K Lee ldquoOrder processing in automatedstorageretrieval systems with due datesrdquo IIE Transactions vol28 no 7 pp 567ndash577 1996

[28] J Won and S Olafsson ldquoJoint order batching and order pickingin warehouse operationsrdquo International Journal of ProductionResearch vol 43 no 7 pp 1427ndash1442 2005

[29] C-Y Tsai J J H Liou and T-M Huang ldquoUsing a multiple-GA method to solve the batch picking problem consideringtravel distance and order due timerdquo International Journal of Pro-duction Research vol 46 no 22 pp 6533ndash6555 2008

[30] Y KaoM H Chen and Y T Huang ldquoA hybrid algorithm basedon ACO and PSO for capacitated vehicle routing problemsrdquoMathematical Problems in Engineering vol 2012 Article ID726564 17 pages 2012

[31] X Gan Y Wang S Li and B Niu ldquoVehicle routing problemwith time windows and simultaneous delivery and pick-up ser-vice based on MCPSOrdquoMathematical Problems in Engineeringvol 2012 Article ID 104279 11 pages 2012

[32] M B M de Koster E S van der Poort and M WoltersldquoEfficient orderbatching methods in warehousesrdquo InternationalJournal of ProductionResearch vol 37 no 7 pp 1479ndash1504 1999

[33] Y-C Ho and Y-Y Tseng ldquoA study on order-batching methodsof order-picking in a distribution centre with two cross-aislesrdquoInternational Journal of Production Research vol 44 no 17 pp3391ndash3417 2006

[34] C Theys O Braysy W Dullaert and B Raa ldquoUsing a TSPheuristic for routing order pickers in warehousesrdquo EuropeanJournal of Operational Research vol 200 no 3 pp 755ndash7632010

[35] S Henn and V Schmid Metaheuristics for Order Batching andSequencing in Manual Order Picking Systems Univ Fak furWirtschaftswiss 2011

[36] M Hahsler B Grun and K Hornik ldquoArulesmdasha computationalenvironment for mining association rules and frequent itemsetsrdquo Journal of Statistical Software vol 14 2005

[37] R Agrawal T Imielinski and A Swami ldquoMining associationrules between sets of items in large databasesrdquo ACM SIGMODRecord vol 22 no 2 pp 207ndash216 1993

[38] S Encheva Y Kondratenko M Z Solesvik and S TuminldquoDecision support systems in logisticsrdquo in Proceeding of Inter-national electronic conference on computer science pp 254ndash2562007

[39] M-C Chen ldquoConfiguration of cellular manufacturing systemsusing association rule inductionrdquo International Journal of Pro-duction Research vol 41 no 2 pp 381ndash395 2003

[40] R Srikant and R Agrawal ldquoMining generalized associationrulesrdquo Future Generation Computer Systems vol 13 no 2-3 pp161ndash180 1997

[41] J Wu and XM Li ldquoAn effective mining algorithm for weightedassociation rules in communication networksrdquo Journal of Com-puters vol 3 no 10 pp 20ndash27 2008

[42] C H Cai A W C Fu C H Cheng and W W KwongldquoMining association rules with weighted itemsrdquo in Proceedingsof the International Symposium on Database Engineering ampApplications pp 68ndash77 Cardiff UK 1998

[43] J HollandAdaptation in Natural and Artificial Systems Univer-sity of Michigan Press Ann Arbor Mich USA 1975

[44] A A Taleizadeh F Barzinpour and H-M Wee ldquoMeta-heuristic algorithms for solving a fuzzy single-period problemrdquoMathematical and Computer Modelling vol 54 no 5-6 pp1273ndash1285 2011

[45] H Yang J Yi J Zhao and Zh Dong ldquoExtreme learning ma-chine based genetic algorithm and its application in powersystem economic dispatchrdquo Neurocomputing vol 102 pp 154ndash162 2013

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

12 The Scientific World Journal

algorithm namely MINWAL was utilized to determine thecustomersrsquo orders association regarding similarity betweenitems and ordersrsquo due dates Actually in our proposedmethodthe due date of each order is defined as a weight parameterin calculating the association between orders So orders withsimilar items and due dates have more chance to be in asame batch This issue is not considered in most of theresearch activities in the literature related to our work andconsidering this issue in the order batching problem is themain contribution of our research activity The previous stepwas followed by a binary integer clustering model in orderto solve the order batching problem which maximizes thesimilarity between orders in terms of support After that aGA was applied to solve a TSP in order to find the optimalrouting path

Finally a batch sequencing problem was addressed inconjunction with a GA

The other research objective of this research addresseswhether the proposed ATGH algorithm can further improvethe solution quality for a multiple parallel aisle warehousewith manual picking system Hence comparisons were madeconsideringATGHEDD integratedGA TSP andGA sequencingand EDD integrated s-shape and GA sequencing algorithms Asdemonstrated by the obtained results ATGH comes out tobe the most striking solution approach in terms of tardinessIn spite of the fact that considering all customer orders asdiscussed in this paper would lead to have a static case beinginvolved with dynamic cases in which different points of timeare assigned to customerrsquos orders arrival would be beneficialSo there might be some potential room of research to extendour discussed approach with respect to dynamic orders

Conflict of Interests

The IBM ILOG CPLEX optimization studio version 124software is not used for financial gain and there is no conflictof interests with any party as it is only a student version

References

[1] ELAAT Kearney Excellence in Logistics ELA Brussels Bel-gium 2004

[2] Y-C Ho T-S Su and Z-B Shi ldquoOrder-batching methods foran order-picking warehouse with two cross aislesrdquo Computersamp Industrial Engineering vol 55 no 2 pp 321ndash347 2008

[3] C-M Hsu K-Y Chen and M-C Chen ldquoBatching ordersin warehouses by minimizing travel distance with geneticalgorithmsrdquo Computers in Industry vol 56 no 2 pp 169ndash1782005

[4] E A Elsayed M-K Lee S Kim and E Scherer ldquoSequencingand batching procedures forminimizing earliness and tardinesspenalty of order retrievalsrdquoThe International Journal of Produc-tion Research vol 31 no 3 pp 727ndash738 1993

[5] S Henn S Koch and W Gerhard Order Batching in OrderPicking Warehouses A Survey of Solution Approaches SpringerLondon UK 2012

[6] J Drury Towards More Efficient Order Picking IMM Mono-graph no 1 The Institute of Materials Management CranfieldUK 1988

[7] M Goetschalckx and J Ashayeri ldquoClassification and design oforder picking systemsrdquo Logistics World pp 99ndash106 1989

[8] J A Tompkins J A White Y A Bozer E H Frazelle and J MA Tanchoco Facilities Planning John Wiley amp Sons HobokenNJ USA 2003

[9] S Henn and G Wascher ldquoTabu search heuristics for the orderbatching problem in manual order picking systemsrdquo EuropeanJournal of Operational Research vol 222 no 3 pp 484ndash4942012

[10] S Henn ldquoAlgorithms for on-line order batching in an orderpicking warehouserdquo Computers and Operations Research vol39 no 11 pp 2549ndash2563 2012

[11] R de Koster ldquoHow to assess a warehouse operation in a singletourrdquo Tech Rep RSM Erasmus University Rotterdam TheNetherlands 2004

[12] R de Koster T Le-Duc and K J Roodbergen ldquoDesign andcontrol of warehouse order picking a literature reviewrdquo Euro-pean Journal of Operational Research vol 182 no 2 pp 481ndash5012007

[13] R A Ruben and F R Jacobs ldquoBatch construction heuristics andstorage assignment strategies for walkride and pick systemsrdquoManagement Science vol 45 no 4 pp 575ndash596 1999

[14] E A Elsayed ldquoAlgorithms for optimal material handling inautomatic warehousing systemsrdquo International Journal of Pro-duction Research vol 19 no 5 pp 525ndash535 1981

[15] G Clarke and J W Wright ldquoScheduling of vehicles froma central depot to a number of delivery pointsrdquo OperationsResearch vol 12 no 4 pp 568ndash581 1964

[16] H D Vinod ldquoInteger programming and the theory of group-ingrdquo Journal of the American Statistical Association vol 64 no326 pp 506ndash519 1969

[17] R D Armstrong W D Cook and A L Saipe ldquoOptimalbatching in a semi-automated order picking systemrdquo Journal ofthe Operational Research Society vol 30 no 8 pp 711ndash720 1979

[18] H D Ratliff and A S Rosenthal ldquoOrder-picking in a rect-angular warehouse a solvable case of the traveling salesmanproblemrdquo Operations Research vol 31 no 3 pp 507ndash521 1983

[19] A Kusiak A Vannelli and R K Kumar ldquoClustering analysismodels and algorithmsrdquo Cybernetics vol 15 pp 139ndash154 1986

[20] D R Gibson and G P Sharp ldquoOrder batching proceduresrdquoEuropean Journal of Operational Research vol 58 no 1 pp 57ndash67 1992

[21] M B Rosenwein ldquoA comparison of heuristics for the problemofbatching orders for warehouse selectionrdquo International Journalof Production Research vol 34 no 3 pp 657ndash664 1996

[22] A J RMGademann J P van denBerg andHH van derHoffldquoAn order batching algorithm for wave picking in a parallel-aislewarehouserdquo IIE Transactions vol 33 no 5 pp 385ndash398 2001

[23] M-C Chen and H-P Wu ldquoAn association-based clusteringapproach to order batching considering customer demandpatternsrdquo Omega vol 33 no 4 pp 333ndash343 2005

[24] NGademann and S van deVelde ldquoOrder batching tominimizetotal travel time in a parallel-aisle warehouserdquo IIE Transactionsvol 37 no 1 pp 63ndash75 2005

[25] Y A Bozer and J W Kile ldquoOrder batching in walk-and-pick order picking systemsrdquo International Journal of ProductionResearch vol 46 no 7 pp 1887ndash1909 2008

[26] O Kulak Y Sahin and M E Taner ldquoJoint order batching andpicker routing in single and multiple-cross-aisle warehousesusing cluster-based tabu search algorithmsrdquo Flexible Servicesand Manufacturing Journal vol 24 no 1 pp 52ndash80 2012

The Scientific World Journal 13

[27] E A Elsayed and M-K Lee ldquoOrder processing in automatedstorageretrieval systems with due datesrdquo IIE Transactions vol28 no 7 pp 567ndash577 1996

[28] J Won and S Olafsson ldquoJoint order batching and order pickingin warehouse operationsrdquo International Journal of ProductionResearch vol 43 no 7 pp 1427ndash1442 2005

[29] C-Y Tsai J J H Liou and T-M Huang ldquoUsing a multiple-GA method to solve the batch picking problem consideringtravel distance and order due timerdquo International Journal of Pro-duction Research vol 46 no 22 pp 6533ndash6555 2008

[30] Y KaoM H Chen and Y T Huang ldquoA hybrid algorithm basedon ACO and PSO for capacitated vehicle routing problemsrdquoMathematical Problems in Engineering vol 2012 Article ID726564 17 pages 2012

[31] X Gan Y Wang S Li and B Niu ldquoVehicle routing problemwith time windows and simultaneous delivery and pick-up ser-vice based on MCPSOrdquoMathematical Problems in Engineeringvol 2012 Article ID 104279 11 pages 2012

[32] M B M de Koster E S van der Poort and M WoltersldquoEfficient orderbatching methods in warehousesrdquo InternationalJournal of ProductionResearch vol 37 no 7 pp 1479ndash1504 1999

[33] Y-C Ho and Y-Y Tseng ldquoA study on order-batching methodsof order-picking in a distribution centre with two cross-aislesrdquoInternational Journal of Production Research vol 44 no 17 pp3391ndash3417 2006

[34] C Theys O Braysy W Dullaert and B Raa ldquoUsing a TSPheuristic for routing order pickers in warehousesrdquo EuropeanJournal of Operational Research vol 200 no 3 pp 755ndash7632010

[35] S Henn and V Schmid Metaheuristics for Order Batching andSequencing in Manual Order Picking Systems Univ Fak furWirtschaftswiss 2011

[36] M Hahsler B Grun and K Hornik ldquoArulesmdasha computationalenvironment for mining association rules and frequent itemsetsrdquo Journal of Statistical Software vol 14 2005

[37] R Agrawal T Imielinski and A Swami ldquoMining associationrules between sets of items in large databasesrdquo ACM SIGMODRecord vol 22 no 2 pp 207ndash216 1993

[38] S Encheva Y Kondratenko M Z Solesvik and S TuminldquoDecision support systems in logisticsrdquo in Proceeding of Inter-national electronic conference on computer science pp 254ndash2562007

[39] M-C Chen ldquoConfiguration of cellular manufacturing systemsusing association rule inductionrdquo International Journal of Pro-duction Research vol 41 no 2 pp 381ndash395 2003

[40] R Srikant and R Agrawal ldquoMining generalized associationrulesrdquo Future Generation Computer Systems vol 13 no 2-3 pp161ndash180 1997

[41] J Wu and XM Li ldquoAn effective mining algorithm for weightedassociation rules in communication networksrdquo Journal of Com-puters vol 3 no 10 pp 20ndash27 2008

[42] C H Cai A W C Fu C H Cheng and W W KwongldquoMining association rules with weighted itemsrdquo in Proceedingsof the International Symposium on Database Engineering ampApplications pp 68ndash77 Cardiff UK 1998

[43] J HollandAdaptation in Natural and Artificial Systems Univer-sity of Michigan Press Ann Arbor Mich USA 1975

[44] A A Taleizadeh F Barzinpour and H-M Wee ldquoMeta-heuristic algorithms for solving a fuzzy single-period problemrdquoMathematical and Computer Modelling vol 54 no 5-6 pp1273ndash1285 2011

[45] H Yang J Yi J Zhao and Zh Dong ldquoExtreme learning ma-chine based genetic algorithm and its application in powersystem economic dispatchrdquo Neurocomputing vol 102 pp 154ndash162 2013

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 13

[27] E A Elsayed and M-K Lee ldquoOrder processing in automatedstorageretrieval systems with due datesrdquo IIE Transactions vol28 no 7 pp 567ndash577 1996

[28] J Won and S Olafsson ldquoJoint order batching and order pickingin warehouse operationsrdquo International Journal of ProductionResearch vol 43 no 7 pp 1427ndash1442 2005

[29] C-Y Tsai J J H Liou and T-M Huang ldquoUsing a multiple-GA method to solve the batch picking problem consideringtravel distance and order due timerdquo International Journal of Pro-duction Research vol 46 no 22 pp 6533ndash6555 2008

[30] Y KaoM H Chen and Y T Huang ldquoA hybrid algorithm basedon ACO and PSO for capacitated vehicle routing problemsrdquoMathematical Problems in Engineering vol 2012 Article ID726564 17 pages 2012

[31] X Gan Y Wang S Li and B Niu ldquoVehicle routing problemwith time windows and simultaneous delivery and pick-up ser-vice based on MCPSOrdquoMathematical Problems in Engineeringvol 2012 Article ID 104279 11 pages 2012

[32] M B M de Koster E S van der Poort and M WoltersldquoEfficient orderbatching methods in warehousesrdquo InternationalJournal of ProductionResearch vol 37 no 7 pp 1479ndash1504 1999

[33] Y-C Ho and Y-Y Tseng ldquoA study on order-batching methodsof order-picking in a distribution centre with two cross-aislesrdquoInternational Journal of Production Research vol 44 no 17 pp3391ndash3417 2006

[34] C Theys O Braysy W Dullaert and B Raa ldquoUsing a TSPheuristic for routing order pickers in warehousesrdquo EuropeanJournal of Operational Research vol 200 no 3 pp 755ndash7632010

[35] S Henn and V Schmid Metaheuristics for Order Batching andSequencing in Manual Order Picking Systems Univ Fak furWirtschaftswiss 2011

[36] M Hahsler B Grun and K Hornik ldquoArulesmdasha computationalenvironment for mining association rules and frequent itemsetsrdquo Journal of Statistical Software vol 14 2005

[37] R Agrawal T Imielinski and A Swami ldquoMining associationrules between sets of items in large databasesrdquo ACM SIGMODRecord vol 22 no 2 pp 207ndash216 1993

[38] S Encheva Y Kondratenko M Z Solesvik and S TuminldquoDecision support systems in logisticsrdquo in Proceeding of Inter-national electronic conference on computer science pp 254ndash2562007

[39] M-C Chen ldquoConfiguration of cellular manufacturing systemsusing association rule inductionrdquo International Journal of Pro-duction Research vol 41 no 2 pp 381ndash395 2003

[40] R Srikant and R Agrawal ldquoMining generalized associationrulesrdquo Future Generation Computer Systems vol 13 no 2-3 pp161ndash180 1997

[41] J Wu and XM Li ldquoAn effective mining algorithm for weightedassociation rules in communication networksrdquo Journal of Com-puters vol 3 no 10 pp 20ndash27 2008

[42] C H Cai A W C Fu C H Cheng and W W KwongldquoMining association rules with weighted itemsrdquo in Proceedingsof the International Symposium on Database Engineering ampApplications pp 68ndash77 Cardiff UK 1998

[43] J HollandAdaptation in Natural and Artificial Systems Univer-sity of Michigan Press Ann Arbor Mich USA 1975

[44] A A Taleizadeh F Barzinpour and H-M Wee ldquoMeta-heuristic algorithms for solving a fuzzy single-period problemrdquoMathematical and Computer Modelling vol 54 no 5-6 pp1273ndash1285 2011

[45] H Yang J Yi J Zhao and Zh Dong ldquoExtreme learning ma-chine based genetic algorithm and its application in powersystem economic dispatchrdquo Neurocomputing vol 102 pp 154ndash162 2013

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014