A MATHEMATICAL MODEL FOR TWO DIMENSIONAL LOADING PROBLEM IN CROSS-DOCKING NETWORK DESIGN. İlker Küçükoğlu, Aslı Aksoy, Seval Ene and Nursel Öztürk

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1 Mathematial and Computational Appliation Vol. 18 No. 3 pp A MATHEMATICAL MODEL FOR TWO DIMENSIONAL LOADING PROBLEM IN CROSS-DOCKING NETWORK DESIGN İlker Küçükoğlu Alı Akoy Seval Ene and Nurel Öztürk Department of Indutrial Engineering Uludag Univerity Gorukle Bura Turkey ikuukoglu@uludag.edu.tr aliakoy@uludag.edu.tr evalene@uludag.edu.tr nurel@uludag.edu.tr Abtrat- In thi paper the tranportation problem of ro-doking network i taken into aount. Cro-doking enter differ from ditribution enter or warehoue in that the load are tranported from origin to detination through ro-doking enter without toring them for a long time. Good are unloaded from ioming truk and reloaded immediately onto outgoing truk in thee ro-doking enter. Thi problem i formulated uing a mixed integer programming with two-dimenional loading ontraint. The two-dimenional hape are applied for both truk and load in order to find exat apaity of eah truk in bai of eah produt. The illutrative problem are olved and reult how that the propoed model find pratial olution with twodimenional loading ontraint. Key Word- Cro-doking tranportation problem mathematial modeling 1. INTRODUCTION Cro-doking i a logiti trategy reently ued by many ompanie in order to redue inventory and improve utomer atifation. Produt are tranferred from ioming truk to outgoing truk without toring them for a long time (generally le than 24 hour) in thee enter. Thi trategy provide different advantage ompared with traditional ditribution enter: the onolidation of hipment a horter delivery lead time the redution of ot improved utomer ervie fewer overtok et. [1]. A a reult of thee advantage ro-doking ha beome an intereting logiti trategy that an give ompanie important ompetitive benefit. Coniderable reearh on ro-doking ha been invetigated in the literature. Thee tudie an be laified baed on the problem type: loation of ro-dok rodoking layout deign ro-doking network vehile routing dok door aignment truk heduling torage and other iue. Some of thee problem are more oerned about long term deiion (trategi or tatial) while other deal with hort-term deiion (operational). Cro-doking network problem onit of upplier utomer and ro-doking faility et. Eah et ontain one or more loation and the aim i to determine the flow of good from upplier to utomer through ro-dok in order to redue the total tranportation ot. Lim et al. [2] tudied the ro-doking network problem by extending the traditional tranhipment problem. The tranhipment problem onit of a number of upplier tranhipment and demand node with the apaitated ar.

2 274 İ. Küçükoğlu A. Akoy S. Ene and N. Öztürk Moreover upplier and utomer time window are onidered to determine inventory holding ot in thi tudy. Chen et al. [3] tudied a imilar problem in ro-doking network deign. The notieable differee between thee tudie are that upplie and demand are not allowed to plit and different produt an be onidered. An integer programming formulation of the problem i provided and three heuriti algorithm are propoed a a olution approah. Mua et al. [4] evaluate the total ot a ditit from other tudie by onidering vehile tranportation ot. They formulated the problem with integer programming and propoed an ant olony optimization meta-heuriti algorithm to olve the problem. Ma et al. [5] tudied imilar problem by onidering only one type of produt and formulated the problem with time ot and truk etup ot. The author propoed a olution approah whih onit of two tage heuriti. Alpan et al. [6] tudied the tranhipment heduling problem in a multiple inbound and outbound dok onfiguration. Diret hipping and inventory holding trategie are allowed in the problem. The objetive i to find bet hedule of tranhipment operation. They propoed everal heuriti algorithm to attain the olution. Miao et al. [7] onidered the tranhipment problem with oft and hard time window ontraint. They propoed two type of meta-heuriti algorithm: tabu earh and geneti algorithm. On the other hand there are oniderable number of tudy related with the truk loading or pallet loading in the literature. Chen et al. [8] propoed a binary mathematial model for two-dimenional pallet paking problem for non-uniform box ize and multiple pallet. Zahariadi et al. [9] developed a tabu earh meta-heuriti algorithm for vehile routing problem with two-dimenional loading ontraint and teted their algorithm on everal behmark intae. They ahieved to everal new bet olution. Fuellerer et al. [10] onidered the vehile routing problem with threedimenional loading ontraint. They preented an ant olony optimization algorithm a a olution approah. Zahariadi et al. [11] introdued a new tranportation problem alled the pallet-paking vehile routing problem and ued tabu earh baed heuriti algorithm to olve problem. Finally Leung et al. [12] ued a imulated annealing metaheuriti algorithm for heterogeneou fleet vehile routing problem with twodimenional loading ontraint and teted with behmark intae derived from the two-dimenional loading vehile routing problem. However only a paper whih onider truk loading in ro-doking network problem ould be found to the bet of our knowledge. Charkhgard & Tabar [13] onidered three dimenional produt and truk hape to find exat apaity of the truk. But only one type of truk and ubi produt are aumed in their tudy. Moreover there i no deiion variable for truk loading plan. In thi tudy we onider the two-dimenional loading problem in ro-doking network deign. The two-dimenional hape are applied for both truk and load in order to find exat apaity of eah truk in bai of eah produt. The objetive i to find the bet network flow route and truk loading plan deiion that minimize the total tranportation ot.

3 A Mathematial Model for Two Dimenional Loading Problem 275 The ret of the paper i organized a follow. In Setion 2 problem definition and mixed integer mathematial model i preented. In Setion 3 the illutrative problem and their olution are given. Finally Setion 4 olude the tudy. 2. PROBLEM DESCRIPTION In thi paper ro-doking network i onidered for S upplier (origin) D utomer (detination) and C ro-doking failitie. The produt flow from origin to detination through ro-dok aording to utomer demand. Eah produt of upplier i loaded into ioming and outgoing truk by onidering it detination. Thu the objetive i to find the bet tranhipment plan regarding the two dimenional truk loading operation in order to minimize total tranportation ot. Mua et al. [4] and Charkhgard & Tabar [8] onidered ome aumption in their mathematial model. In thi paper we have aepted everal of them and enhaed our model with the following new ondition: Diretly hipping i not allowed from upplier to detination. Truk apaity i taken into aount with dimenional ontraint on the ontrary of weight or amount of load. Load and truk are onidered a retangular hape. The truk may have different ize. Thu loading plan are affeted by the truk hoie. Moreover eah produt to be ent from different origin to different detination may have different ize. The tranportation ot are related with only travelled ditae among the loation. The oept of the two-dimenional loading problem in ro-doking network i depited in Figure 1 whih illutrate an example of two upplier two ro-dok failitie and three utomer. Supplier I S111 S112 S131 S132 S121 S111 S211 S112 S212 Cutomer I S121 Cro-Dok I Supplier II S211 S212 S231 Cro-Dok II S : The produt i at upplier moved to detinadion (utomer) d Figure 1. An illutrative example of deribed problem S131 S132 S231 Cutomer II Cutomer III

4 276 İ. Küçükoğlu A. Akoy S. Ene and N. Öztürk Aording to aumption deribed above the problem an be formulated a mixed integer programming uing the following notation: S D C S : The et of origin : The et of detination : The et of ro-dok V _ : Number of truk at origin where S V _ : Number of truk at ro-dok where C d C F : Number of boxe flow at origin to detination d where S d D L_ : Length of truk m at origin where S S m L_ : Length of truk n at ro-dok where C C n W _ : Width of truk m at origin where S S m W _ : Width of truk n at ro-dok where C C n m 1... V _ S n 1... V _ m 1... V _ S C n 1... V _ p_ : Cot of a truk from origin to ro-dok where S C p_ : Cot of a truk from ro-dok to detination d where C d D d l : Length of box i at origin for detination d where S d D i 1...F d w : Width of box i at origin for detination d where S d D M i 1...F d : Arbitrarily large ontant The variable are: x_ : x oordinate of the outh wet orner of box i in truk m from origin to ro-dok related to detination d where S d D C i 1...F d m 1... x_ : x oordinate of the outh wet orner of box i in truk n from rodok to detination d related to origin where S d D C i 1...F d n 1... V _ C y_ : y oordinate of the outh wet orner of box i in truk m from origin to ro-dok related to detination d where S d D C i 1...F d m 1... C

5 A Mathematial Model for Two Dimenional Loading Problem 277 y_ : y oordinate of the outh wet orner of box i in truk n from rodok to detination d related to origin where S d D C i 1...F d n 1... V _ C v_ : i a binary variable and equal to 1 if truk m i ued for ro-dok at origin otherwie it i equal to 0 where S C m 1... V _ S nd v _ : i a binary variable and equal to 1 if truk n i ued for detination d at ro-dok otherwie it i equal to 0 where d D C n 1... V _ C z _ : i a binary variable and equal to 1 if box i of origin related to detination d i hipped from origin to ro-dok with truk m otherwie it i equal to 0 where S d D C i 1...F d m 1... z_ : i a binary variable and equal to 1 if box i of origin related to detination d i hipped from ro-dok to detination d with truk n otherwie it i equal to 0 where S d D C i 1...F d n 1... V _ C _ doif : i a binary variable and equal to 1 if box i belong to detination d i on the left ide of box f belong to detination o in truk m whih flow from origin to ro-dok otherwie it i equal to 0 where S d od C i 1...F d f 1...F o m 1... _ doif : i a binary variable and equal to 1 if box i belong to detination d i on the right ide of box f belong to detination o in truk m whih flow from origin to ro-dok otherwie it i equal to 0 where S d od C i 1...F d f 1...F o m 1... _ doif : i a binary variable and equal to 1 if box i belong to detination d i on the bottom ide of box f belong to detination o in truk m whih flow from origin to ro-dok otherwie it i equal to 0 where S d od C i 1...F d f 1...F o m 1... _ doif : i a binary variable and equal to 1 if box i belong to detination d i on the top ide of box f belong to detination o in truk m whih flow from origin to ro-dok otherwie it i equal to 0 where S d od C i 1...F d f 1...F o m 1... _ udif : i a binary variable and equal to 1 if box i loaded from origin i on the left ide of box f loaded from origin u in truk n whih flow from rodok to detination d otherwie it i equal to 0 where us d D C i 1...F d f 1...F ud n 1... V _ C

6 278 İ. Küçükoğlu A. Akoy S. Ene and N. Öztürk _ udif : i a binary variable and equal to 1 if box i loaded from origin i on the right ide of box f loaded from origin u in truk n whih flow from ro-dok to detination d otherwie it i equal to 0 where us d D C i 1...F d f 1...F ud n 1... V _ C _ udif : i a binary variable and equal to 1 if box i loaded from origin i on the bottom ide of box f loaded from origin u in truk n whih flow from ro-dok to detination d otherwie it i equal to 0 where us d D C i 1...F d f 1...F ud n 1... V _ C _ udif : i a binary variable and equal to 1 if box i loaded from origin i on the top ide of box f loaded from origin u in truk n whih flow from rodok to detination d otherwie it i equal to 0 where us d D C i 1...F d f 1...F ud n 1... V _ C The mathematial model an be formulated a follow: V _ S V _ C nd _ d _ S C m1 C dd n1 Min p v p v (1).t. C m1 z _ 1 S d D i 1... F ( 2) v _ 1 S m 1... V _ S (3) C F d dd i1 V _ C C n1 z _ M v _ S C m 1... V _ S (4) z _ 1 S d D i 1... F ( 5 ) v _ nd 1 C n 1... V _ C ( 6 ) dd F d dd i1 nd z _ M v _ d D C n 1... V _ C ( 7 ) Fd V _ S Fd V _ C S dd i1 m1 S dd i1 n1 z _ z _ C x _ l x _ M 1 _ of doif S d od m... V _ S C i... F f... F x _ l x _ M 1 _ of of doif d o S d od m... V _ S C i... F f... F d o d d ( 8 ) ( 9 ) ( 10 )

7 A Mathematial Model for Two Dimenional Loading Problem 279 y _ w y _ M 1 _ of doif S d o D m... V _ S C i... F f... F y _ w y _ M 1 _ ( 12 ) of of doif d o S d od m... V _ S C i... F f... F d o _ doif _ doif _ doif _ doif z _ z _ of 1 ( 13 ) S d od m 1... V _ S C i 1... Fd f 1... Fo x _ l L _ Sm S d D m 1... V _ S i 1... Fd ( 14 ) C y _ w W _ Sm S d D m 1... V _ S i 1... Fd ( 15 ) C (11) 1 1 (16) x _ y _ M z _ S d D C m... V _ S i... F d x _ l x _ M 1 _ udf udif u S d D n... V _ C C i... F f... F x _ l x _ M 1 _ udf udf udif d ud u S d D n... V _ C C i... F f... F y _ w y _ M 1 _ udf udif d ud u S d D n... V _ C C i... F f... F y _ w y _ M 1 _ udf udf udif d ud u S d D n... V _ C C i... F f... F d ud z _ z _ 1 udif udif udif ıudif of 1 1 d 1 ud u S d D n... V _ C C i... F f... F ( 17 ) ( 18 ) ( 19 ) ( 20 ) x _ l L _ C S d D n 1... V _ C C i 1... F ( 22 ) n d y _ w W _ C S d D n 1... V _ C C i 1... F ( 23 ) n d x _ y _ M z _ S d D n 1... V _ C C i 1... F ( 24 ) v_ x _ d v_ y_ nd z_ x_ z_ y 0 doif _ doif _ doif _ doif _ udif _ udif _ udif _ udif ( 21 ) Binary The objetive (1) i to minimize the total tranportation ot. The ontraint onit of two part: ro-doking onolidation (ontraint (2) to (8)) and two dimenional truk loading (ontraint (9) to (24)) deiion. For the ro-doking onolidation deiion ontraint (2) and (5) aign the produt to ioming and outgoing truk and enure that eah produt an be aigned only one ioming and outgoing truk repetively. Contraint (3) provide that eah truk at origin an be ent at mot one ro-doking faility and alo ontraint (6) maintain thi ondition at ro-doking truk that eah truk at ro-doking failitie an be

8 280 İ. Küçükoğlu A. Akoy S. Ene and N. Öztürk ent at mot one detination. Contraint (4) and (7) define that eah produt an be aigned to ioming or outgoing truk if the deignated truk i ued. Contraint (8) fore the produt ontinuity at ro-doking failitie. For the truk loading deiion ontraint (9) to (12) and ontraint (17) to (20) provide that produt do not overlap eah other if a pair of produt i in the ame truk and it i determined by ontraint (13) and ontraint (21) aording to ioming and outgoing tranport repetively. Contraint (14) and (15) enure that all the produt loaded in an ioming truk fit within the dimenion of the truk. Likewie ontraint (22) and (23) enure the ame ondition for outgoing truk. If a truk i not ued then there hould not be any produt loaded in it. Thi requirement i handled by ontraint (16) for ioming truk and ontraint (24) for outgoing truk. 3. COMPUTATIONAL EXPERIMENTS Beaue ro-doking network deign and two-dimenional truk loading problem have been not onidered jointly before there i no behmark tet et available. Therefore we generated our own data et randomly ilude different enario to hek the validity of the model. The tet problem are deribed with four bai parameter (S/C/D/F max ): the number of upplier S the number of ro-dok failitie C the number of detination D and maximum flow amount in network F max. Table 1 preent the ret of the parameter range ued for randomly generated intae. Table 1. Parameter range Tranportation ot range (p_ p_ d ) : U[50 150] Truk width range (W_S m W_C n ) : U[50 150] Produt dimenion range (w l ) : U[5 25] Truk length range (L_S m L_C n : U[ ] Flow range (F d ) : U[0 F max ] Truk number (V S V C ) : U[0 8] We performed 10 different problem ategorie and eah ategory ontain five randomly generated problem intae. Intae are olved with CPLEX on a Pentium 2.2GHz CPU with 2.0 GB memory. Eah output of the model i examined to verify olution. To illutrate the experiment Table 2 ilude a data et belong to problem ategory 3/2/4/4 and Table 3 repreent the optimal olution (objetive fution value = 944.0) of the illutrative example. Eah row deribe the integrated produt flow plan of a produt in a network deign and ilude the produt-vehile aignment and produt loation (x and y oordinate) for ioming and outgoing truk repetively. Alo aignment of the ioming truk to ro-doking enter and aignment of the outgoing truk to utomer an be obtained by the table. Table 2. A data et for 3/2/4/4 problem type V_S = {3 2 4} V_C = {3 2} L_S m = { } L_C n = { } W_S m = { } W_C n = { } p_ = { } p_ d = { } F d = { } l = { } w = { }

9 A Mathematial Model for Two Dimenional Loading Problem 281 Table 3. Solution of the illutrative example # Supplier Ioming Truk Outgoing Truk Cutomer Produt Cro_Dok Vehile x_oordinate y_oordinate Vehile x_oordinate y_oordinate () (d) (i) () (k) (x) (y) (k) (x) (y) Table 4 preent the reult of all intae. The firt olumn of eah intae i the objetive fution value of the optimum olution and the eond olumn i the omputational time (in eond) of the CPLEX. It i learly een from the table that an optimum olution ould be found for eah intae by the propoed model. On the other hand omputational time how that CPLEX take le than 10 minute (without the run time limitation) to reah optimum olution for eah problem. Moreover the average omputational time of the intae i 2.5 minute and generally olution ould be found in one minute. Conequently omputational reult validate the apability of the propoed model. Problem Table 4. Reult of the intae Intae I Intae II Intae III Intae IV Intae V Average OFV Time() OFV Time() OFV Time() OFV Time() OFV Time() OFV Time() 1/1/5/ /1/2/ /2/6/ /2/4/ /4/2/ /1/2/ /5/3/ /2/3/ /2/2/ /2/4/ OFV: Objetive Fution Value 4. CONCLUSION AND FUTURE WORK In thi work the two-dimenional truk loading problem wa tudied firt time for ro-doking network deign in order to minimize the total tranportation ot. Thu

10 282 İ. Küçükoğlu A. Akoy S. Ene and N. Öztürk loading and tranportation operation an be arried out more realiti and appliable in real life. Problem i formulated with the mixed integer mathematial model. Propoed model i teted with everal randomly generated intae and reult are examined to verify olution. Reult how that our model verifie it apability. A a reult thi model able to find pratial olution to ro-doking operation. For further work on thi problem i to find a powerful meta-heuriti algorithm to find effetive olution on large ale problem. Aknowledgement-Thi work wa upported by The Commiion of Sientifi Reearh Projet of Uludag Univerity Projet number KUAP(M)-2012/ REFERENCES 1. J.V. Belle P. Valkenaer D. Cattrye Cro-doking: State of the art Omega 40(6) A. Lim Z. Miao B. Rodrigue Z. Xu Tranhipment through rodok with inventory and time window Naval Reearh Logiti 52(8) P. Chen Y. Guo A. Lim B. Rodrigue Multiple rodok with inventory and time window Computer & Operation Reearh R. Mua J. Arnaout H. Jung Ant olony optimization algorithm to olve the tranportation problem of ro-doking network Computer & Indutrial Engineering H. Ma Z. Miao A. Lim B. Rodrigue Crodoking ditribution network with etup ot and time window ontraint Omega G. Alpan A. Ladier R. Larbi B. Penz Heuriti olution for tranhipment problem in a multiple door ro doking warehoue Computer & Indutrial Engineering Z. Miao F. Yang K. Fu D. Xu Tranhipment ervie through rodok with both oft and hard time window Annal of Operation Reearh C. S. Chen S. Sarin B. Ram The pallet paking problem for non-uniform box ize International Journal of Prodution Reearh 29(10) E. E. Zahariadi C.D. Tarantili C.T. Kiranoudi A guided tabu earh for the vehile routing problem with two-dimenional loading ontraint European Journal of Operational Reearh G. Fuellerer K.F. Doerner R.H. Hartl M. Iori Metaheuriti for vehile routing problem with three-dimenional loading ontraint European Journal of Operational Reearh E. E. Zahariadi C.D. Tarantili C.T. Kiranoudi The pallet-paking vehile routing problem Tranportation Siee 46(3) S. C. H. Leung Z. Zhang D. Zhang X. Hua M.K. Lim A meta-heuriti algorithm for heterogeneou fleet vehile routing problem with two-dimenional loading ontraint European Journal of Operational Reearh H. Charkhgard A.A.Y. Tabar Tranportation problem of ro-doking network with three-dimenional truk Afrian Journal of Buine Management 5(22)