Designing a Genetic Algorithm to Optimize Fulfilled Orders in Order Picking Planning Problem with Probabilistic Demand

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1 In. J. Research n Indusral Engneerng pp Volume 1 Number Inernaonal Journal of Research n Indusral Engneerng ournal homepage: esgnng a Genec Algorhm o Opmze Fulflled Orders n Order Pcng Plannng Problem wh Probablsc emand M. Seyedrezae 1* S.E. Naaf 1 A. Aghaan 2 H. Bagherzadeh Valam 3 1 eparmen of Indusral Engneerng Scence and Research Branch Islamc Azad Unversy ehran Iran 2 eparmen of Indusral Engneerng Mazandaran Unversy of Scence and echnology Babol Iran 3 eparmen of Appled Mahemacs Shahr-e-Rey Branch Islamc Azad Unversy ehran Iran A R I C L E I N F O Arcle hsory : Receved: May Revsed: Augus Acceped: Sepember Keywords : Supply chan Order pcng problem mahemacal model dsrbuon cener fulflled order genec algorhm. A B S R A C srbuon ceners Cs play an mporan ey role n supply chan. elverng he rgh ems o he rgh cusomers a he rgh me a he rgh cos s a crcal msson of he Cs. oday cusomer sasfacon s an mporan facor for suppler companes n order o gan more profs. Opmzng he number of fulflled orders An order ha he requred quany of all ems n ha order are avalable from he nvenory and can be send o he cusomer n a me perod may lead o delay some maor orders; and consequenly lead o dssasfacon of hese cusomers ulmaely loss hem and lead o lower profs. In addon some nvenory may reman n he warehouse n a me-perod and over he me become corrup. I also leads o reduce he benef of suppler companes n he supply chan. herefore n hs paper we wll presen a dynamc mahemacal model o flow process /sorage process of goods for order pcng plannng problem OPP n Cs. And we wll opmze he number of fulflled orders n hs problem wh regard o a he coeffcen of each cusomer b o mee each cusomer's needs n he leas me c probablsc demand of cusomers and d ang nvenory o send o cusomers a he earles opporuny o preven her decay. Afer presenng he mahemacal model we use Lngo sofware o solve small sze problems. Complexy of he mahemacal model wll nensfy by ncreasng he numbers of cusomers and producs n dsrbuon cener herefore Lngo sofware wll no able o solve hese problems n a reasonable me. herefore we wll develop and use a genec algorhm GA for solvng hese problems. 1. Inroducon Order pcng s a process by whch producs are rereved from specfed sorage locaons wh respec o cusomer orders. Order pcng s a labor-nensve as n warehousng mprovng he performance of order pcng generally can lead o a large amoun of savngs n warehousng coss [1]. he effcency of order pcng s dependable on facors such as he sorage racs C layou and conrol mechansms. he overall logscs servce level also can * Correspondng auhor E-mal address: seyedreza@yahoo.com

2 41 esgnng a Genec Algorhm o Opmze Fulflled Orders n Order Pcng be mproved hrough effcen warehousng operaons. In an order pcng operaon order pcers may pc one order a he me sngle order pcng. Warehouse or C managers are neresed n fndng he mos economcal way of pcng orders whch opmze he coss nvolved n erms of cusomer sasfacons or me. In addon many developed researches consdered demand n deermnsc manner whereas n he real world s no deermnsc because of changng n requremens and shorer produc lfe cycles. herefore n hs sudy we esmae he demand of each em ypes n each perod accordng o he prevous perods and probably dsrbuon funcon and develop a mahemacal model for OPP as he decson process o opmze fulflled orders n a C wh respec o he pershable goods. Alhough hs sep s very mporan for large Cs bu developng dynamc mahemacal models o mul me-perod has no receved aenon from he academa and praconers so far. 2. Leraure revew In he followng wo subsecons a bref descrpon of he wor whch has been done o address he desgn models for a dsrbuon cener researches have been developed n Order pcng sysem desgn models would be brefly dscussed. 2.1.esgnng models for a dsrbuon cener warehouse esgnng a C or a warehouse s a complex as nvolvng a number of desgn parameers. A desgner s face wh he challenge of how bes o address all he desgnng ssues whou ncreasng he complexy of problem. Several models have been presened n he leraure o desgn a C. Ashayer and Gelders [2] revew varous analycal- and smulaon-based approaches proposed for warehouse desgnng problems. hey conclude ha n general neher a pure analycal approach nor a pure smulaon approach wll lead o a praccal desgnng mehod. As a poenal soluon hey sugges a wo-sep procedure for warehouse desgnng. As a frs sep of hs procedure analycal models could be used o qucly compare dfferen desgnng alernaves and o selec only a few feasble alernaves hus reducng he search space. As a second sep an elaborae smulaon model can be bul o nclude dynamcs ha could no be modeled usng analycal approaches. Gray e al. [3] propose a mul-sage herarchcal decson approach o model he compose desgn and operang problems for a ypcal order-consoldaon warehouse. Annualzed ncremenal nal coss plus he warehouse operang coss are mnmzed as an obecve funcon. Van den Berg [4] presens a leraure survey on mehods and echnques for he plannng and conrol of warehousng sysems. Plannng-relaed decsons nclude nvenory managemen and sorage locaon assgnmen whle conrol-relaed decsons nclude pcerroung sequencng schedulng and order bachng. Rouwenhors e al. [5] presen a reference framewor and a classfcaon of warehouse desgn and conrol problems. hey defne a warehouse desgn as a srucured approach of decson mang a a sraegc accal and operaonal level n an aemp o mee a number of well-defned performance crera.

3 42 M. Seyedrezae e al. For effcen order pcng plannng researchers have elaboraed on warehouse layou sorage polcy and pcng polcy [7]. Warehouse layou problem addresses he shape of warehouse shelf locaon number of asles locaon and number of sarng/endng pons of pcng operaon [8 9]. Sorage polcy ncludes random sorage dedcaed sorage classbased sorage and volume-based sorage [7]. 2.2.Order pcng sysem desgnng models Order pcng has been denfed as a maor acvy n a C and he prme conrbuor o overall C operang expenses. Several research conrbuons propose models for he desgn and selecon of an approprae orders pcng sysem OPS - for a gven applcaon. Yoon and Sharp [10] presen a srucured procedure for he analyss and desgn of order pcng sysems ha consders nerdependen relaonshps beween dfferen funconal areas e.g. recevng pcng sorng ec. of OPS. Ho and seng [11] have suded he order-bachng mehods for a C wh wo cross-asles; and conduc an exensve expermen over he combnaons of nne seed-order selecon rules and en accompanyng-order selecon rules along wh wo roue-plannng mehods and wo pcng frequency dsrbuons. Pcer roung problem s o deermne how many pcers o ravel along wha sequence n order o mnmze he oal ravel dsance. Many researchers have suded hs problem. Among hose Hseh and sa [12] conduc a smulaon over a se of varables such as he number of cross-asles sorage polcy order combnaon and pcng polcy n order o fnd ou an effcen desgn of he warehouse. Peersen and Aase [7] conduc a smulaon sudy o compare he mpac of pcng polcy sorage polcy and roung polcy on he pcng effcency; and concluded ha he oal pcng me can be reduced he mos by order combnaon followed by sorage polcy and roung polcy. A recen revew paper by Gu Goeschalcx and McGnns [13] denfes order pcng as one area of warehouse operaons. hey classfy order pcng no bachng sequencng and roung and sorng all of whch belong o operaonal level no plannng level. Chen and Wu [14] descrbe he developmen of an order-bachng approach based on daa mnng and neger programmng; and presen an order-cluserng model based on 0 1 neger programmng o maxmze he assocaons beween orders whn each bach. anels Rummel and Schanz [15] address he problem n whch ems may be sored n mulple locaons so ha order pcng requres choosng a subse of he locaons ha sore he requred em o collec he requred quany. hey formulae a model for smulaneously deermnng he assgnmen and sequencng decsons and compare o prevous models for order pcng. e Koser van der Poor and Wolers [16] propose wo groups of heursc algorhms for effcen order bachng he Seed algorhms and he me Savngs algorhms and evaluae he performance of he algorhms usng wo dfferen roung sraeges: he so-called S-shape and larges gap sraeges. Won and Olafsson [17] formulae he bachng and order-pcng problem only as a combnaoral opmzaon problem; and evaluae he benefs of addressng he on problem and propose smple bu effecve heurscs for s soluon. Gademann and van de Velde [18] address problem of bachng

4 43 esgnng a Genec Algorhm o Opmze Fulflled Orders n Order Pcng orders n a parallel-asle warehouse o mnmze he oal ravelng me needed o pc all ems. hey model he problem as a generalzed se paronng problem and presen a column generaon algorhm o solve s lnear programmng relaxaon. For furher research resuls on order pcng he neresed readers may refer o [13]. Parh [6] desgned an order pcng sysems for Cs. In hs paper he focused on he followng wo eys n orders pcng sysem-desgnng ssues: confguraon of he sorage sysem and selecon beween bach and zone order pcng sraeges. he overall goal of hs research s o develop a se of analycal models o help he desgner n desgnng orderpcng sysems n a dsrbuon cener. In dsrbuon ceners some orders may no be fulflled due o he shorage of nvenory. Rm and Par [19] consdered he case where an order s pced and shpped only when all ems n he order s avalable n he nvenory; and he unfulflled orders are ransferred o he nex day wh hgher prory. Gven daly nvenory and a se of orders arrved durng a day he problem was o assgn he nvenory o he orders so as he order fll rae OFR o maxmze. hey formulaed hs problem as a lnear programmng problem. hey also consder he weghed OFR o reflec he mporance of each order. hs problem do no dale wh n a dynamc manner. hs s a praccal model bu does no ae no consderaon ceran ssues ha are addressed whn hs presen paper: forecasng he demand of each cusomer accordng o he demand funcon capacy lmaon of warehouse n C and consderng pershable goods n plannng horzon of mulple me-perods. he proposed model belongs o he class of NP-complee problems. herefore we use genec algorhm o solve hs model. 3. emand Forecasng Mehods Forecasng s an aemp o deermne n advance he mos lely oucome of an unceran varable. Plannng and conrollng logscs sysems need predcons for he level of fuure economc acves because of he me lag n machng supply o demand. hroughou he leraure revew and mahemacal models developmen emphass has been pu upon he deermnsc demand of ems and sable producon whereas n he real world demand s no deermnsc and mus be esmaed accordng o he Probably and sasc mehods. In hs research we supposed demand of ems n orders are dscree and have Bnomal dsrbuon. he essenal parameers o esmae he demand of ems n an order are: P Probably of demand em n order a me perod S Sandard devaon of demand em n order a me perod E Mahemacal expecaon of demand em n order a me perod accordng o he demand funcon n Quany of demand em n order cusomer a pervous me perods 1 α Confdence nerval Z 1- α Is a pon s. P z > Z α =α 2 Z s he normal dsrbuon 1-2 2

5 44 M. Seyedrezae e al. he followng relaons calculae he mean and sandard devaon of demand funcon E = n P I J 1a S = n P 1 P I J 1b Accordng o he Cenral lm heorem we can esmae he bnomal dsrbuon by normal dsrbuon. We suppose confdence nerval of em demand s 95% so Inequaly 1c ensures a 95% Confdence nerval for demand of ems n each order. E Z α S E + Z α S I J 1c Mahemacal model Suppose here s a C whch s suppled by Producon ceners and supples many ems o many real sores. he cusomers realers send orders o he C once n every me-perod. he orders-whose quany s avalable from he curren nvenory for all ems-are pced and shpped o he cusomers on he curren me-perod. he orders; whch are no repled due o he shorage of a leas one em n ha order are carred forward o he nex me-perod called bac order. Assumpons he assumpons of he mahemacal model are as followng: 1. emand of ems n cusomer order s deermned accordng o bnomal dsrbuon n each me-perod. 2. Confdence nerval for demand of ems n cusomer order should be predeermned 95%. 3. Bac-orders are allowed. 4. he order of real ceners s ndependen. 5. he produc's lfe s specfc. 6. vdng of ems n a cusomer order s no allowed. 7. he capacy of C s lm and specfc. 8. mporance of each cusomer's order for he C s defne and specfc Indexes Index of orders cusomers =12 I Index of em ypes =1 2 J q Index of me perods q =12 W Wegh of order cusomer S Sandard devaon of demand funcon of produc for cusomer n perod

6 45 esgnng a Genec Algorhm o Opmze Fulflled Orders n Order Pcng E Mahemacal expecaon of demand em n order a me perod accordng o he demand funcon Parameers L Longevy of produc n erms of me perod V he space ha occuped by each un of produc n erms of m 3 M A large posve number capacy V Capacy of dsrbuon cener Model decson varables: R Invenory remanng of dsrbuon cener for em a he end of me perod S Quany of em mus be suppled from manufacurng ceners o meeng he demand of cusomers a me perod Esmaed quany of ems for demand of cusomer a me perod Y 1 If demand of cusomer for produc n perod s mee perod ; or 0 oherwse F 1 f order ha requesed n me-perod s fulflled n me-perod ; or 0 oherwse Mahemacal model F W W Maxmze Z I I + = 2 Subec o J I Y 1 3 J I M Y F J R S F I J F R S R I + = 1 6 J q F R I L q q V V R S capacy J J I Z S E 2 1- α 9 J I Z + S E 2 1- α 10

7 46 M. Seyedrezae e al. F 1 I 11 F = 0 I J > 12 Y = 0 I J > 13 F {01} I J 14 Y {01} I J 15 R 0 and n eger J 16 0 and n eger I J 17 S 0 and n eger J 18 he obecve funcon 2 aemped o maxmze he number of fulflled orders wh respec o he mporance of cusomers and bac-order coeffcen. erm W W shows mporance of cusomer o he oher cusomers. hs erm ensures meeng demand of mporan cusomers n he hgher prory. Coeffcen + guaranees assgnng prory of bac-orders. By opmzng of he obecve funcon sasfacon of cusomers wll be ncreased because of cusomer mporance and bac-order coeffcen. Consran 3 shows ems of an order n a me-perod are suppled only once n plannng horzon of mulple me-perods. Consran 4 forces F o be zero f any one em of he order n meperod s no fulflled by he nvenory of me-perod. Consran 5 means ha he oal assgnmen for em n me-perod canno exceed he nvenory of he em n begnnng of hs me-perod he nvenory of each me-perod for each em s he nvenory carred over from he prevous perod plus suppled amoun of curren perod. Consran 6 calculae remanng amoun of em a he end of each meperod. Consran 7 guarany remanng ems n he end of each me-perod n C mus be used before pershng. hs consran ensures assgnng he ems carred forward from prevous perods o avod pershng. herefore here s no pershed em n C n plannng horzon of mulple me-perods and we do no allow ems n C o be pershed. Consran 8 specfes he C capacy lmaon. Consran 9 and 10 ensure confdence nerval for demand of ems n each order. Consran 11 ensures ha an order of a cusomer n a me-perod s suppled only once n plannng horzon of mulple me-perods. Consran 12 guarany he orders of me perod can be fulflled on he nex me-perods no on he pervous me-perodsfor example: he demanded order n me perod 3 can' be fulflled n me-perod 1 or 2 hs order can be fulflled n perod 3 and fuure me-perods 45. Consran 13 resrcs llogcal meeng of ems n an order. Consrans are he logcal bnary and non-negavy neger requremens on he decson varables. 5. Genec algorhm In spe of oher sochasc search mehods GA searches he feasbly space by seng of feasbly soluons smulaneously n order o fnd opmal or near-opmal soluons. he I

8 47 esgnng a Genec Algorhm o Opmze Fulflled Orders n Order Pcng procedure s carred ou by he use of genec operaons. GA was developed nally by Holland [20]. Goldberg [21] gave an neresng survey of some praccal wors carred ou n hs area. In he nex secon we descrbe an els genec algorhm o solvng our proposed mahemacal model. 6. Els genec algorhm for solvng he OPP In hs secon an els genec algorhm for solvng he OPP s nroduced. For desgnng he GA some prncple facors are descrbed as follows: 6.1.Soluon codng chromosome srucure he srucure of he soluon comprses of four marxes fgure 5 explaned below. 1. Marx [ F ] I supplyng me marx ndcaes supply me of orders n plannng horzon of mulple me- perods. Each elemen of marx [F] s deermned by equaon 19 whch s generaed from he unform dsrbuon of U+1 fgure 1. f = + 1 I 19 f = means he demand of cusomer n me-perod s suppled n me perod 1. Fgure 1.Componen of chromosome supplyng me marx 2. Marx [] s a complex marx and consss of genes relang o he forecased amoun of ems for cusomers. Elemen d n marx ] I J [ sgnfes forecased demand of em by cusomer n me-perod. each elemen of hese marxes are deermned by equaons 9 and 10. Fgure 2 shows he srucure of marx []. 1 2 = [[ ] I J [ ] I JK [ ] I J K[ ] I J ]

9 48 M. Seyedrezae e al. Fgure 2. Componen of chromosome emand marx 3. Marx [ S ] J consss of genes whch ndcae he quany of ems for meeng he demand of cusomers n plannng horzon of me-perods. Each elemen of marx specfed by equaon 20. Fgure 3 shows he srucure of marx [S]. [ ] J S s I Fgure 3. Componen of chromosome C demand marx R S J s f = Marx [ R ] J consss of genes Fgure 4 demonsrang he res of ems a he end of each me-perod. Each elemen of hs marx s deermned accordng o he equaons 8 and 21. R Fgure 4. Componen of chromosome res of ems n C = Max{ 0 S + R 1 } J s f 21. = I Fgure 5. Chromosome srucure

10 49 esgnng a Genec Algorhm o Opmze Fulflled Orders n Order Pcng 6.2.Improved GA operaors In hs paper he chromosome srucure s formed as a complex marx. When he GA operaors are appled offsprng chromosome may no be feasble soluon so hese soluons are correced o be feasble Crossover operaor he crossover operaor produces chldren by exchangng nformaon conaned n he parens. In hs paper we adop he sandard one-pon crossover whch randomly generaes one-crossover pons along he lengh of array. hs crossover pon dvdes each paren no wo segmens; subsequenly he offsprngs are creaed by exchangng he places of he second segmens. he crossover process s llusraed n fgure 6a and 6b. Paren1 Paren2 Fgure 6a. Crossover operaor parens

11 50 M. Seyedrezae e al. Offsprng1 Offsprng Muaon operaor Fgure 6b. offsprng -Crossover operaor For carryng ou he muaon hree dfferen operaors are developed: Sngle muaon o mplemen hs ype of muaon a cusomer n a sngle me perod s randomly chosen and hen he muaon operaor selecs a dfferen me-perod whch demand of hs cusomer can be suppled llusraed n fgure 7.

12 51 esgnng a Genec Algorhm o Opmze Fulflled Orders n Order Pcng Mul row muaon Fgure 7. Sngle muaon he mul muaon performs he sngle muaon on all he orders of a pced cusomer n plannng horzon of mulple me-perods llusraed n fgure 8. Fgure 8. Mul row muaon Mul column muaon hs ype of muaon performs he sngle muaon on all he orders of a pced me-perod llusraed n fgure 9. Fgure 9. Mul column muaon 6.3.Mang pool selecon sraegy For creang he new generaon s necessary o selec some chromosomes mang pool wh he laes fness n he curren generaon for recombnng or creang chromosomes relaed o he new generaon. In hs case ournamen selecon and Els selecon are used n whch he fness of curren generaon chromosomes s calculaed accordng o obecve funcon. Fgure 10 shows he procedure of mang pool selecon sraegy.

13 52 M. Seyedrezae e al. Fgure 10. Mang pool selecon sraegy 6.4.Soppng crera he algorhm ermnaes f he number of generaons exceeds he specfc number Els GA procedure Fgure 11 shows he flowchar of GA. he seps need o mplemen GA are as follow: Sep 1: Creae a random feasble paren populaon P0 of populaon sze Se =0. Sep 2: Assgn a fness value o each soluon n P0 accordng o he obecve funcon Sep 3: Use ournamen selecon and Els selecon wh her raes o selecng chromosomes from P and creang mang pool members of populaon sze. Sep 4: generae a new populaon of populaon sze as fallow: 4-1: chose ow soluons from mang pool wh predefned crossover rae o generae offsprng and add hem o populaon of nex generaon P : mue each soluon n mang pool wh predefned muaon rae and add o populaon of nex generaon P : selec he res of populaon sze form mang pool accordng o he bes obecve funcon fness value and copy hem n o he populaon of nex generaon P + 1. Sep 5: If he soppng creron s sasfed sop and reurn he bes soluon of he generaon. Sep 6: Se =+1 and go o Sep 3.

14 53 esgnng a Genec Algorhm o Opmze Fulflled Orders n Order Pcng Fgure 11. Flowchar of els GA 7. Compuaonal resuls In order o shows he effcency of developed GA; we solve 10 es problems by hs algorhm and Lngo 9 on a PC Penum IV 3.06 GHz processor 1024 MB RAM and wndows XP Professonal Operang Sysem. he GA was developed usng Malab7.0. he npu parameers used n GA for all problems are showed n able 1. he obaned resuls are shown n able 2. Ieraons 100 able 1. Parameers used n GA Populaon sze Elsm Muaon rae Crossover rae 0.88

15 54 M. Seyedrezae e al. able 2. Comparng resul of GA and Lngo Problem Lngo GA Ex. n me S No. fulflled orders Ob me S No. fulflled orders Ob Problems 9 and 10 have been feasble afer and seconds respecvely. We nerruped runnng of hese problems before obanng global opmal soluons. 7.1.Comparson mercs o compare he resul obaned by wo solvng mehod we defne he hree followng mercs and compare he soluons accordng o he mercs. 1- Elapsed CPU me: hs merc shows he elapsed me o solve he problems. By consderng he resul of able 2 and hs merc he developed GA performs beer han Lngo sofware. 2- Number of fulflled orders: hs merc demonsrae he number of fulflled orders n plannng horzon of mulple me-perods. Accordng hs merc f number of fulflled orders n a procedure s more anoher he procedure performs beer. able 3 and Fgure 8 compare he resul obaned by GA and Lngo. 3- Obecve fness funcon: hs merc represens fness of each soluon. In addon hs merc shows he prmacy of soluon procedure

16 55 esgnng a Genec Algorhm o Opmze Fulflled Orders n Order Pcng Accordng o he Comparson mercs and consderng he resuls of able 2 when he complexy of problems ncrease because of excessve number of varables and consrans lngo can no solve hese problems n reasonable me. However he elapsed me o solve hese problems by GA s very noceable and qualy of hese soluons approxmaely s as same as Lngo procedure. Fgures 12 and 13 show qualy of soluons ha have produced by GA and Lngo sofware. hese resul shows effcency of developed GA. Fgure 12. Resuled number of fulflled orders n GA and Lngo Fgure 13. Qualy obecve funcon of obaned soluons accordng o he GA and Lngo 8. Conclusons In hs arcle we have presened a new dynamc mahemacal model for OPP wh consderng produc lfe cusomer mporance probablsc demand and bac-order sraegy

17 56 M. Seyedrezae e al. o meeng demand of realers n a C. In addon we have desgned prncples of GA o solve he proposed model because of s complexy. Fnally we have solved numerous examples o compare he resul of GA and lngo. Numercal examples showed ha he proposed GA s effcen and effecve n searchng for near opmal soluons. Accordng o he comparng resul and model's properes we can dscover usng mporance of hs model; so usng he facors consdered n hs paper helps o he C manager sysem n geng sasfacon of cusomers n he compeve mare. Mnmzng em-handlng cos n hs model and usng he oher mea-heurscs o solve hs problem are nvesgaed as he nex researches. References [1] ompns J.A. Whe J.A. Bozer Y.A. Frazelle E.H. anchoco J.M.A. and revno J Facles Plannng 2nd ed. Wley New Yor. [2] Ashayer J. and Gelders L.F Warehouse desgn opmzaon European Journal of Operaon Research Vol. 21 No. 3 pp [3] Gray A.E. Karmarar U.S. and Sedmann A esgn and operaon of an order-consoldaon warehouse models and applcaon European Journal of Operaonal Research Vol. 58 No. 1 pp [4] Van den Berg J.P A leraure survey on plannng and conrol of warehousng sysems IIE ransacons Vol. 31 No. 8 pp [5] Rouwenhors B. Reuer B. Socrahm V. van Houum G.J. Manel R.J. and Zm W.H.M Warehouse desgn and conrol: Framewor and leraure revew European Journal of Operaonal Research Vol. 122 No. 3 pp [6] Parh P.J esgnng Order Pcng Sysems for srbuon Ceners Faculy of he Vrgna Polyechnc Insue and Sae Unversy n paral fulfllmen of he requremens for he degree of ocor of Phlosophy n Indusral and Sysems Engneerng. [7] Peersen C.G. and Aase G A comparson of pcng sorage and roung polces n manual order pcng Inernaonal Journal of Producon Economcs Vol. 92 No. 1 pp [8] Ralff H.. and Rosenhal S Order pcng n a recangular warehouse: A solvable case of he ravelng salesman problem Operaon Research Vol. 31 No. 3 pp [9] Vaughan.S. and Peersen C.G he effec of cross asles on order pcng effcency Inernaonal Journal of Producon Research Vol. 37 No. 4 pp [10] Yoon C.S. and Sharp G.P Example applcaon of he cognve desgn procedure for an order pc sysem: Case sudy European Journal of Operaonal Research Vol. 87 No. 2 pp [11] Ho Y.C. and seng Y.Y A sudy on order-bachng mehods of orderpcng n a dsrbuon cenre wh wo cross-asle Inernaonal Journal of Producon Research Vol. 44 No. 17 pp

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