Aalborg Universitet. Positioning empty containers under dependent demand process. Dang, Vinh Quang; Yun, Won-Young; Kopfer, Herbert

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1 Downloaded from vbn.aau.dk on: marts 20, 2019 Aalborg Unverstet Postonng empty contaners under dependent demand process Dang, Vnh Quang; Yun, Won-Young; Kopfer, Herbert Publshed n: Computers & Industral Engneerng DOI (lnk to publcaton from Publsher): do: /j.ce Publcaton date: 2012 Document Verson Accepted author manuscrpt, peer revewed verson Lnk to publcaton from Aalborg Unversty Ctaton for publshed verson (APA): Dang, V. Q., Yun, W-Y., & Kopfer, H. (2012). Postonng empty contaners under dependent demand process. Computers & Industral Engneerng, 62(3), General rghts Copyrght and moral rghts for the publcatons made accessble n the publc portal are retaned by the authors and/or other copyrght owners and t s a condton of accessng publcatons that users recognse and abde by the legal requrements assocated wth these rghts.? Users may download and prnt one copy of any publcaton from the publc portal for the purpose of prvate study or research.? You may not further dstrbute the materal or use t for any proft-makng actvty or commercal gan? You may freely dstrbute the URL dentfyng the publcaton n the publc portal? Take down polcy If you beleve that ths document breaches copyrght please contact us at vbn@aub.aau.dk provdng detals, and we wll remove access to the work mmedately and nvestgate your clam.

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3 Computers & Industral Engneerng 62 (2012) Contents lsts avalable at ScVerse ScenceDrect Computers & Industral Engneerng journal homepage: Postonng empty contaners under dependent demand process Quang-Vnh Dang a, Won-Young Yun b,, Herbert Kopfer c a Department of Mechancal and Manufacturng Engneerng, Aalborg Unversty, Aalborg East, DK 9220, Denmark b Department of Industral Engneerng, Pusan Natonal Unversty, Busan, South Korea c Faculty of Busness Studes and Economcs, Unversty of Bremen, Bremen, Germany artcle nfo abstract Artcle hstory: Avalable onlne 4 December 2011 Keywords: Empty contaners Inventory polces Genetc algorthm Owng to trade mbalance, shppng companes poston empty contaners between ports or depots perodcally. The most dffcult problem for postonng s that t s not possble to know the exact amounts of empty contaners requred n the future. The paper deals wth the problem of postonng empty contaners n a port area wth multple depots. Customer demands and returnng contaners n depots per unt tme perod are assumed to be serally-correlated and dependent random varables. Three optons are consdered to prepare the requred extent of postonng: postonng from other overseas ports, nland postonng between depots, and leasng. The polces for empty-contaner management consst of three parts as follows: a coordnated, (s, S) nventory polcy for postonng from other ports, (r, R ) polcy for nland postonng between depots; and a smple leasng polcy wth zero lead-tme. For nland postonng polcy, four heurstc methods are proposed to reposton empty contaners between depots. The objectve s to obtan optmal polces correspondng to dfferent methods of nland postonng n order to mnmze the expected total costs. A genetc-based optmzaton procedure s developed to fnd the optmal parameters (s, S) and (r, R ). Some numercal examples and senstvty analyses are gven to demonstrate the results. Ó 2011 Elsever Ltd. All rghts reserved. 1. Introducton Empty contaners are mportant logstcal resources n the lght of changes n the nternatonal logstcs envronment; shppng companes wsh to manage and operate them effcently. Owng to the trade mbalance, empty contaners must be postoned from surplus areas to shortage areas perodcally and shppng companes have nventory polces to allocate empty contaners. They reposton empty contaners among hub areas, ports and depots. Hence, the effcent management of empty contaners becomes a source of compettve advantage for shppng companes to mprove ther customer-servce levels and productvty. In empty-contaner allocaton problems, some researchers pad attenton to determnstc systems (e.g. Choong, Cole, & Kutanoglu, 2002; D Francesco, Manca, & Zuddas, 2006; Moon, Do Ngoc, & Hur, 2010; Olvo, Zuddas, Francesco, & Manca, 2005; Shntan, Ima, Nshmura, & Papadmtrou, 2007). Stochastc problems have been studed snce late 1990s. Cheung and Chen (1998) consdered a two-stage stochastc network model for the dynamc empty contaner allocaton problem. L, Lu, Leung, and La (2004) and L, Wu, and Lu (2007) developed a new (u,d) polcy for the empty contaner Correspondng author. Address: Department of Industral Engneerng, Pusan Natonal Unversty, 30-san, Jangjeon-dong, Geumjong-gu, Busan , South Korea. Tel.: ; fax: E-mal address: wonyun@pusan.ac.kr (W.-Y. Yun). allocaton problem between ports. Lam, Lee, and Tang (2007) proposed a dynamc stochastc model for a smple, two-port twovoyage system to prove the effectveness of the approxmately optmal results from a two-port two-voyage model that utlzed lnear approxmaton archtecture. Song and Dong (2008) dealt wth an empty-contaner management problem n a cyclc route to seek the optmal repostonng polcy n a dynamc and stochastc stuaton. Ths study was then extended by Dong and Song (2009) n case of the jont contaner fleet szng and empty contaner repostonng problem n mult-vessel, mult-port and mult-voyage shppng systems wth dynamc, uncertan and mbalanced customer demands. Hwang (2008) establshed a smulaton model by Arena that took account of postonng empty contaners between mult-ports under uncertan factors such as shppers demand and navgaton tme of vessels. Che (2009) also developed an Arena smulaton model to analyze empty contaner management consderng uncertan demands and supples n a two-depot system. Yun, Lee, and Cho (2011) consdered an nventory control problem of empty contaners n an nland transportaton system wth a smple polcy to reposton contaners from other hubs. Most of those research studed empty contaner repostonng between seaports; however, lttle research has been reported on the coordnated optmzaton of emptycontaner postonng from overseas ports and postonng between depots n an nland mult-depot system under dependent demand process, to whch we attempt to contrbute. Furthermore, shortterm leasng of empty contaners s also taken nto consderaton /$ - see front matter Ó 2011 Elsever Ltd. All rghts reserved. do: /j.ce

4 Q.-V. Dang et al. / Computers & Industral Engneerng 62 (2012) Nomenclature t I L o L r L d L e C f C o C h C l C r j A t B t D t dscrete tme-perod set of depots ( 2 I: ndex of depots) lead-tme for overseas postonng lead-tme for nland re-postonng from one depot to another lead-tme for dstrbuton from termnal to each depot leasng perod fxed cost per replenshment for overseas postonng varable cost per unt for overseas postonng holdng cost per unt per perod at depot leasng cost per unt per perod at depot cost of nland re-postonng per unt per perod from depot I to depot j set of depots that can poston out empty contaners n tme-perod t set of depots that can poston n empty contaners n tme-perod t customer demand at depot n tme-perod t V t number of contaners returned from customers to depot n tme-perod t W expected net demand at depot (W ¼ l D l V wthl D ; l V : means of customer demand and number of contaners returned at depot ). I t estmated nventory poston at depot at the begnnng of tme-perod t RO t number of empty contaners that can be postoned out from depot n tme-perod t RI t number of empty contaners that can be postoned n depot n tme-perod t The decson varables S order-up-to level for overseas postonng s reorder pont for overseas postonng R nland-postonng-out level at depot ( 2 I) r nland-postonng-n level at depot ( 2 I) The rest of ths paper s organzed as follows: n the next secton, we address the model descrpton and propose replenshment polces for empty contaners n the nland mult-depot system. In Secton 3 and 4, a soluton method to obtan optmal polces s dscussed. The structure of the smulaton model s extensvely descrbed. A smulaton-based optmzaton procedure s then developed based on genetc algorthm (GA) to fnd the optmal parameters (s, S) and (r, R ). In Secton V, numercal examples and senstvty analyses are gven to demonstrate the results. Fnally, conclusons are made n Secton VI. 2. Inventory control model 2.1. Model descrpton Fg. 1 shows an nland transportaton network. Shppng companes have nland depots to store empty contaners and to provde them for transportaton of freghts across/to termnals, depots, and customer locatons. Due to the mbalance n trade, some ports accumulate a large number of empty contaners, whle other ports are often faced wth a shortage of empty contaners. The mbalance problem occurs across depots n an nland transport system. To solve the mbalance problem, three optons for replenshment are taken nto account. Frstly, shppng companes may regularly place orders for empty contaners from overseas ports. After repostonng from overseas and arrval at the termnal, the empty contaners wll be dstrbuted among the nland depots. Secondly, empty contaners can be repostoned between depots. Though both the overseas replenshment and nland repostonng of empty contaners are undertaken, shortage may stll occur. In that case, shppng companes can lease empty contaners to make up the shortfall at once, but the leased empty contaners must be returned to the leasng companes after a specfed perod. A large number of empty contaners can be repostoned from overseas ports wth moderate prces; however, t requres a long replenshment lead-tme and may overstock the depots. Otherwse, repostonng empty contaners between depots wth shorter lead-tme s more flexble to cope wth the fluctuaton of demands, but ths plan has hgher transportaton cost. Therefore, to reduce expendture and to be more responsve to customer demands, the challenge for shppng companes s to successfully allocate empty contaners between mult-depots and to lease a mnmum number of empty contaners from leasng companes Assumptons In ths paper, an nventory control problem for empty contaners s studed under the followng assumptons: There are n nland depots. Sngle commodty, 40 feet contaners, s consdered. The demand and supply of empty contaners per unt tme are serally-correlated and dependent varables. The lead-tmes for overseas orders are ndependent and dentcally dstrbuted random varables. All lead-tmes for nland postonng across multple depots are dentcal and constant. The lead-tmes for dstrbutng empty contaners from the termnal to depots are constant. Short-term lease wth zero lead-tmes s avalable Empty-contaner replenshment polces Coordnated overseas postonng polcy A dscrete-tme (s, S) polcy (refer to Slver, Pyke, & Peterson, 1998) s consdered to decde on orderng tme and quantty of empty contaners based on the estmated total nventory poston of all depots n the future, as shown n Fg. 2 below. At the startng pont of each tme-perod t, we estmate the nventory poston I tþe½lošþd of all depots followng a tme-lapse of ðe½l o ŠþdÞ, whch s the sum of expected average lead-tme for overseas postonng, denoted by E½L o Š, and the smallest value among the lead-tmes for dstrbuton from termnal to the depots, denoted by d. If the estmated nventory poston s less than s, we order empty contaners up to S n advance; otherwse, we do not place any order of empty contaners from overseas ports. An autoregressve model (AR(1)) s used to forecast the customer demands and the amount of returnng contaners durng the expected lead-tme for overseas postonng. The AR(1) s defned as: D t ¼ n D þ h D D t 1 þ e D ð1þ V t ¼ n V þ h V V t 1 þ e V ð2þ where h D s the parameter of the model, the constant term, n D, equals l D ð1 h D Þ and the error term, e D, s assumed to be a

5 710 Q.-V. Dang et al. / Computers & Industral Engneerng 62 (2012) Leasng Company Leasng Company Customer Depot Inland poston Depot Inland poston Return Lease Return Lease Demand Demand Customer Supply Supply Inland poston Customer Demand Depot Inland poston Depot Demand Customer Supply Supply Dstrbuton Dstrbuton Overseas Poston Termnal Overseas Poston Fg. 1. Inland transportaton system for empty contaners. Fg. 2. A coordnated (s, S) oversea postonng polcy. normally dstrbuted varable wth zero mean and constant varance (wth l D, r 2 D are mean and varance of customer demands of depot ). Analogcal defntons are used for h V, n V and e V n case of returnng contaners Dstrbuton polcy A dstrbuton polcy s proposed for delverng empty contaners from the termnal to the depots. Because the lead-tme for overseas postonng s a source of randomness, the nland mult-depot system checks for each tme-perod t whether or not overseas orders placed n the prevous perods wll arrve. As soon as empty contaners that are postoned from overseas ports arrve at the termnal, they are dvded nto lots n proporton to net demands of the depots. These lots are then transported to the correspondng depots n the system Inland postonng polcy Inland postonng s an alternatve approach for replenshng empty contaners wth shorter lead-tmes. Hence, at the startng ponts of all tme perods, we also make another mportant decson about the nland postonng of empty contaners across multple depots. The nventory poston, I tþlr of depot after the lead-tme for nland postonng, L r, s estmated. The AR(1) model s also appled to

6 Q.-V. Dang et al. / Computers & Industral Engneerng 62 (2012) forecast the customer demands and the supply by returnng contaners durng the lead-tme for nland postonng. Next, we classfy depots nto three sets. The frst set ncludes depots whch can poston out empty contaners. n A t ¼ I tþlr RO t ¼ðI tþlr ðd tþlr ðd tþlr V tþlr Þ > R o V tþlr ÞÞ R and RI t ¼ 0 The second set ncludes depots whch can poston n empty contaners. n o B t ¼ I tþlr ðd tþlr V tþlr Þ < r RO t ¼ 0 and RI t ¼ r ði tþlr ðd tþlr V tþlr ÞÞ The fnal set ncludes depots whch nether poston out nor poston n empty contaners. Beneft ¼ðC h þ C l j Cr j ÞmnðROt ; RIt j Þ Empty contaners are progressvely repostoned one-by-one from an orgn depot n set A t to a destnaton depot n set B t based on one of four methods of nland postonng polcy. After postonng one empty contaner from the orgn to the destnaton, the costs used to determne the flow of nland postonng are updated. The nland postonng procedure contnues untl ether the sum of all RO t ( 2 A t ) or the sum of all RI t ( 2 B t ) equals to zero Leasng polcy Short-term leases wth a zero lead-tme are employed to meet the shortage n a depot at once. Leased empty contaners must be returned to the leasng companes after a leasng perod L e. The leasng perods of empty contaners at all depots are the same. 3. Smulaton model The flowchart of the smulaton model n Fg. 3 descrbes the sequence of replenshment polces proposed. r < I tþlr ðd tþlr RO t ¼ 0 and RI t ¼ 0 V tþlr Þ < R for any To defne an orgn depot n set A t and a destnaton depot n set B t, n other words, to defne the flow of repostonng empty contaners, four methods of nland postonng polcy are proposed usng cost factors whch are C h ; Cl ; Cr j. Holdng costs for ROt contaners are consdered at orgn depots n set A t whle leasng costs for RI t contaners are taken nto account at destnaton depots n set B t. In addton, nland postonng costs for mnðro t ; RIt jþ contaners are consdered from orgn depots n set A t to destnaton depots n set B t. Four heurstc methods of nland postonng polcy are descrbed as follows: (1) Method 1 Empty contaners are repostoned from that depot n set A t whch has the hghest holdng cost, C h RO t, to depot j n set Bt whch has the hghest leasng cost, C l j RIt j. If there s merely one depot n set A t (B t ), the leasng cost of B t (the holdng cost of A t ) s consdered only. (2) Method 2 Empty contaners are repostoned from depot n set A t whch has the hghest holdng cost, C h RO t, to depot j n set Bt whch has the lowest cost of nland postonng, C r j mnðrot ; RIt jþ. In case that there s merely one depot n set A t (B t ), the cost of nland postonng (the holdng cost) s consdered only. (3) Method 3 Empty contaners are repostoned to that depot j n set B t whch has the hghest leasng cost, C l j RIt j, from depot n set At whch has the lowest cost of nland postonng, C r j mnðrot ; RIt jþ. In case that there s one depot n set A t (B t ), the leasng cost (the cost of nland postonng) s consdered only. (4) Method 4 Empty contaner repostonng from depot n set A t to depot j n set B t s performed f the tuple (,j) obtans the most beneft among all tuples of depots. The beneft s defned as follows: Fg. 3. Flowchart of the smulaton model.

7 712 Q.-V. Dang et al. / Computers & Industral Engneerng 62 (2012) Genetc algorthm Ths secton presents a smulaton-based GA to seek the optmal polces mnmzng the long-run expected total cost per unt tme n an nland mult-depot system. The man dea of the optmzaton algorthm s ntroduced and we wll show how we apply the GA to optmze the parameters (.e. the values of the decson varables S, sr, r wth e I) for all nland postonng methods presented n Secton 2. The overall strategy s explaned as follows: Gene representaton Intalzaton Evaluaton va smulaton Selecton Recombnaton, mutaton and adjustment 4.1. Gene representaton The proper representaton of a soluton plays a key role n the development of a GA. For the system under consderaton, a soluton can be represented by a chromosome of non-negatve ntegers ðs; s; R 1 ; r 1 ;...; R n ; r n Þ where S and s are the order-up-to level and reorder pont for overseas postonng, R and r are nland-postonng-out and n levels of depot for the nland postonng polcy. A vald chromosome should satsfy two mandatory condtons. Frst, the value of S s not less than that of s ðs > sþ. Second, the value of R s not less than that of r for each depot ðr > r Þ Intalzaton For the ntal generaton, each gene of an ndvdual s unformly generated wthn a predetermned range of value. After ntalzaton, for the par of (s, S) and each par of (r, R ) for any, f S < s or R < r, then postons of genes n that par are swapped wth each other to satsfy two mandatory condtons. The ntal ndvduals are evaluated through the smulaton model Evaluaton va smulaton Each ndvdual n the populaton represents a potental strategy for solvng the nland mult-depot system and, of course, a soluton when applyng ths strategy. The elements n the chromosome of the ndvdual are used for decson makng of overseas postonng and nland postonng empty contaner durng the smulaton. The smulaton s then run to evaluate the expected total cost per unt tme for each ndvdual and to gve a feedback on ts performance whch s used as a ftness value of the ndvdual n the GA. real-value chromosome. In ths paper, we produce offsprng wth unform crossover whch s descrbed as follows. After havng selected two parents for applyng the crossover operator, an offsprng s generated by copyng the alleles of the genes of the parent chromosomes randomly from one of the two chosen parents. However, as shown n Fg. 4, a par of elements correspondng to the same depot s coped nto the offsprng together as a block because of ther close relatonshp. The unform crossover acts wth probablty P c. Whenever an offsprng s produced, mutaton s appled wth probablty P m. The operaton of mutaton changes a randomly chosen gene on a chromosome by assgnng to ts allele a new number unformly generated n a predetermned range. If, after mutaton, S < sor R < r for any, then we swap the postons of S and s or R and r respectvely, n order to ensure that the two above mentoned mandatory condtons are always satsfed. 5. Numercal example and senstvty analyss In ths secton, a numercal example s explored to llustrate the procedure of the smulaton-based genetc algorthm. To obtan best possble nventory polces based on the four proposed methods for nland postonng, the decson varables ðs; s; R 1 ; r 1 ;...; R n ; r n Þmust be specfed n order to aspre the goal of mnmzng the expected total cost per unt tme whch s used as the optmzaton crteron. More numercal computatons are then conducted to examne the senstvty of the results wth respect to the system parameters such as lead-tme and unt costs. The data for ths example whch takes account of four dentcal depots are gven n Tables 1 4. The smulaton length and warm-up perod for the model are 730 and 120 days, respectvely Numercal example For ths example, to determne the approprate values of GA parameters, several experments were conducted. Three dfferent populaton sze (10, 20, and 50), three dfferent P c (0.4, 0.6, and 0.8), and three dfferent P m (0.05, 0.1, and 0.15) are checked The populaton sze of 50, P c of 0.4 and P m of 0.1 gave the best performance wth the lowest total cost. Hence, the populaton sze, P c, and P m are set to 50, 0.4, and 0.1, respectvely. The termnaton rule s to stop when reachng the maxmum of 200 generatons. The best nventory polces wth respect to the four heurstc methods (M)of nland postonng under dependent demands are shown n Table 5 and the relevant costs are smlarly shown n Table 6 and Fg Selecton and populaton management Varous evolutonary methods can be appled to ths problem. We use (l + k) selecton for selectng ndvduals for reproducton. Under ths method, l parents and k offsprng compete for survval and the l best out of the set of offsprng and old parents, n other words, the l lowest n terms of total cost, are selected as parents of the next generaton Crossover, mutaton and adjustment In the crossover operaton, two ndvduals have to be selected from the parent populaton to generate an offsprng. The Roulette-wheel selecton s used n our algorthm, whch probablstcally selects ndvduals based on ther ftness values. There are many dfferent crossover methods that can be performed on the Fg. 4. Chromosome and recombnaton operaton. Table 1 Dstrbuton of demand and supply. Depot Demand Supply 1 NORM (100, 20 2 ) NORM (100, 20 2 ) 2 NORM (120, 20 2 ) NORM (110, 20 2 ) 3 NORM (260, 30 2 ) NORM (200, 30 2 ) 4 NORM (300, 30 2 ) NORM (220, 30 2 )

8 Q.-V. Dang et al. / Computers & Industral Engneerng 62 (2012) Table 2 Cost factors. Depot Holdng cost Leasng cost Fxed overseas cost Varable overseas cost Table 6 Relevant costs. M Holdng cost Overseas cost Inland cost Leasng cost Total cost Table 3 Cost of nland postonng (per unt per day). Depot Table 7 Near-optmal polces under changng L R. L r S s R 1 r 1 R 2 r 2 R 3 r 3 R 4 r Table 8 Relevant costs under changng L R. Table 4 Lead-tme factors (Day). Depot Dstrbuton Overseas Inland Leasng UNIFORM Olvo et al., 2005; 1 28 Urban, L r Holdng cost Overseas cost Inland cost Leasng cost Total cost Table 5 Near-optmal polces. M S s R 1 r 1 R 2 r 2 R 3 r 3 R 4 r Fg. 6. Trends of costs wth varyng lead-tmes for nland postonng. Fg. 5. Comparson of relevant costs of numercal example. Table 6 shows that holdng costs as well as overseas postonng costs of the solutons are almost the same for all methods M. Its consstent wth the stabltes of S and s n Table 5. Ths demonstrates that the dfferent heurstc methods of nland postonng do not affect the coordnated overseas postonng polcy. Moreover, t s nterestng to observe that for all methods M the values of R and r of depots 3 and 4 are greater than those of depots 1 and 2. Ths can be explaned by the fact that depots 3 and 4 have much more shortage of empty contaners than depots 1 and 2, so they would rather to receve than poston out empty contaners. The hgher the value of R s, the less empty contaners are postoned out by depots 3 and 4 and the hgher r s, the more empty contaners are receved by depots 3 and 4 from the other depots (an opposte explanaton s appled to the case of depots 1 and 2).

9 714 Q.-V. Dang et al. / Computers & Industral Engneerng 62 (2012) It can be observed from Table 6 and Fg. 5 that method 4 has the lowest expected total cost, n other words ths method s the most effectve approach to manage and reposton empty contaners between mult-depots (hence, method 4 wll be used n all cases of senstvty analyss). Ths can be explaned by the fact that method 4 consdered holdng cost at surplus depots, leasng cost at shortage depots and cost for nland postonng from surplus to shortage depots Senstvty analyss Lead-tme parameter In ths analyss we vary the lead-tme for postonng empty contaners from one depot to another. The best found polces for method 4 and the relevant costs under four dfferent lead-tmes L r for nland postonng are shown n Tables 7 and 8, whle trends of relevant costs are dsplayed n Fg. 6. From the results, we can fnd that when lead-tme for nland postonng ncreases, the values of S and s become larger and larger n order to hold more empty contaners. Ths leads to the ncrement of holdng cost whle overseas postonng cost almost does not change. In addton, the values R and r have upward trends whle cost of nland postonng and leasng cost ncrease. As a result, the total cost ncreases Unt cost parameters Leasng cost per unt per day. We ncrease the leasng cost per unt per day of all depots n steps of 20%. The parameters of the found polces and the relevant costs are shown n Tables 9 and 10, whle trends of costs are dsplayed n Fg. 7 below. From the results, t can be seen that when unt leasng cost ncreases under the same other unt costs, the values of S and s steadly ncrease so as to hold more empty contaners, whle r follows an upward trend. The growng values of r lead to a slght ncrease of holdng cost and cost of nland postonng. In addton, cost for overseas postonng almost does not change and cost for leasng substantally rses due to the upward trend of ts unt cost. As a result, the total cost goes up Cost of nland postonng per unt per day. We ncrease the cost for nland postonng per unt per day of all depots n steps of 20%. The best found polces and the relevant costs are shown n Tables 11 and 12, whle trends of costs are dsplayed n Fg. 8 below. Table 9 Near-optmal polces under changng C l. % C l S s R 1 r 1 R 2 r 2 R 3 r 3 R 4 r Table 11 Near-optmal polces under changng C r j. % C r j S s R 1 r 1 R 2 r 2 R 3 r 3 R 4 r Table 10 Relevant costs under changng unt leasng cost C l. % C l Holdng cost Overseas cost Inland cost Leasng cost Total cost Table 12 Relevant costs under changng C r j. % C r j Holdng cost Overseas cost Inland cost Leasng cost Total cost Fg. 7. Trends of costs wth change of unt leasng cost. Fg. 8. Trends of costs wth change of unt nland postonng cost.

10 Q.-V. Dang et al. / Computers & Industral Engneerng 62 (2012) From the results above, t can be observed that when cost of nland postonng per unt per day ncreases under unchanged other unt costs, the values of S and s are qute stable whereas R ncreases and r decreases to dmnsh nland postonng actvty. The holdng cost and overseas postonng cost have no change, whle nland postonng cost ncreases because of the upward trend of ts unt cost. Hence, the total cost goes up. 6. Concluson In ths paper, we studed a replenshment problem for empty contaners n an nland mult-depot system. For serally-correlated and dependent demands and supples of empty contaners, a coordnated ðs; SÞ orderng polcy s used to poston empty contaners from overseas ports. In addton, a new nland postonng ðr ; R Þ polcy along wth four heurstc methods to poston empty contaners between depots s proposed. Also, short-term leasng s avalable f a shortage of empty contaners occurs. A smulaton model and a genetc algorthm based heurstcs are developed to fnd optmal nventory polces correspondng to those methods of nland postonng so as to mnmze the expected total cost per unt tme. Some computatonal experments are then done to valdate the model and examne the senstvty of results wth respect to system parameters. Acknowledgments Ths work was supported by the Grant of the Korean Mnstry of Educaton, Scence and Technology (The Regonal Core Research Program/Insttute of Logstcs Informaton Technology). References Che, H. Y. (2009). A coordnated nventory control of empty contaners n two-depot system. Master Thess, Pusan Natonal Unversty, South Korea. Cheung, R. K., & Chen, C. Y. (1998). A two-stage stochastc network model and soluton methods for the dynamc empty contaner allocaton problem. Transportaton Scence, 32, Choong, S. T., Cole, M. H., & Kutanoglu, E. (2002). Empty contaner management for ntermodal transportaton networks. Transportaton Research Part E, 38, D Francesco, M., Manca, A., & Zuddas, P. (2006). Optmal management of heterogeneous fleets of empty contaners. In Proceedngs of Internatonal Conference on Informaton Systems, Logstcs and Supply Chan (pp ). Dong, J. X., & Song, D. P. (2009). Contaner fleet szng and empty repostonng n lner shppng systems. Transportaton Research Part E, 45, Hwang, H. W. (2008). A smulaton model for postonng of empty contaners. Master Thess, Pusan Natonal Unversty, South Korea. Lam, S. W., Lee, L. H., & Tang, L. C. (2007). An approxmate dynamc programmng approach for the empty contaner allocaton problem. Transportaton Research Part C, 15, L, J. A., Lu, K., Leung, S. C. H., & La, K. K. (2004). Empty contaners management n a port wth long-run average crteron. Mathematcal and Computer Modelng, 40, L, J. A., Wu, Y., & Lu, K. (2007). Allocaton of empty contaners between multdepots. European Journal of Operaton Research, 182, Moon, I. K., Do Ngoc, A. D., & Hur, Y. S. (2010). Postonng empty contaners among multple ports wth leasng and purchasng consderatons. OR Spectrum, 32, Olvo, A., Zuddas, P., Francesco, M. D., & Manca, A. (2005). An operatonal model for empty contaner management. Martme Economcs & Logstcs, 7, Shntan, K., Ima, A., Nshmura, E., & Papadmtrou, S. (2007). The contaner shppng network desgn problem wth empty contaner repostonng. Transportaton Research Part E, 43, Slver, E. A., Pyke, D. F., & Peterson, R. (1998). Inventory management and producton plannng and schedulng (3rd ed.). New York: John Wley & Sons. Song, D. P., & Dong, J. X. (2008). Empty contaner management n cyclc shppng routes. Martme Economcs & Logstcs, 10, Urban, T. L. (2005). A perodc-revew model wth serally correlated, nventory-leveldependent demand. Internatonal Journal of Producton Economcs, 95, Yun, W. Y., Lee, Y. M., & Cho, Y. S. (2011). Optmal nventory control of empty contaners n nland transportaton system. Internatonal Journal of Producton Economcs, 133,