Inventory-Based Empty Container Repositioning in a Multi-Port System

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1 Invenory-Based Emy Conainer Reosiioning in a Muli-Por ysem Loo Hay Lee Ek Peng Chew Yi Luo earmen of Indusrial & ysems Engineering Naional Universiy of ingaore ingaore s: {iseleelh isece iseluoy}@nus.edu.sg Absrac he urose of his sudy is o develo an invenory-based conrol olicy o reosiion emy conainer in a muli-or sysem wih uncerain cusomer demand. A singlelevel olicy wih reosiioning rule in erms of minimizing he reosiioning cos is roosed o manage he emy conainer wih eriodical review. he objecive is o oimize he arameers of he olicy o minimize he execed oal cos er eriod incurred by reosiioning emy conainer holding unused emy conainer and leasing emy conainer. he roblem is solved by alying non-linear rogramming and a gradien search aroach wih Infiniesimal Perurbing Analysis (IPA esimaor. Numerical examles are given o demonsrae he effeciveness of he roosed olicy. Keywords- emy conainer reosiioning; invenory conrol; simulaion; Infiniesimal Perurbaion Analysis (IPA. I. INRUCIN In he las few decades he conainerizaion of cargo ransoraion has been he fases growing secor of he mariime indusries. he growh of conainerized shiing has resened challenges ineviably in aricular o he managemen of emy conainers arising from he highly imbalanced rade beween counries. I is reored ha emy conainer movemens consiue aroximaely 20% of he world ors handing aciviy ever since 1998 [1]. ong [2] reors ha he cos of reosiioning emy conainer is jus under $15 billions which is 27% of he oal world flee running cos based on he daa for If he cos of reosiioning emy conainer can be reduced he shiing comany could increase rofi and imrove comeiiveness. herefore how o effecively and efficienly manage ECs is a very imoran issue for shiing comany and i is known as emy conainer reosiioning (ECR roblem. Much aenion abou ECR roblem has been focused on uilizing mahemaic models o solve his issue [3-7]. Mahemaic models can ofen caure he naure of he roblem while give rise o concerns such as requiremen of a re-secified lanning horizon sensiiviy of he decisions o daa accuracy and variabiliy and imlemenaion of he decisions in he sochasic sysems [8 9]. Recenly several auhors urn o exlore he invenory-based mechanism in addressing he ECR roblem in he sochasic sysems [ ]. hese sudies demonsrae ha he oimal reosiioning olicies are of he hreshold conrol ye characerized by some arameers and rules in some siuaions such as oneor and wo-or sysems. Researchers exend he above works o more general sysems and focus on he imlemenaion of hreshold-ye conrol olicies [9 12~14]. In his aer we consider he ECR roblem in a mulior sysem which comrises a se of ors conneced o each oher and a flee of owned conainers are used o mee he sochasic cusomer demands. A single-level hreshold olicy wih reosiioning rule in erms of minimizing he reosiioning cos is roosed o manage he EC wih eriodical review. he objecive of he aer is o minimize he execed oal cos er eriod including ransoraion cos and holding and leasing cos by oimizing he arameers of he single-level olicy. he aer is organized as follows. ecion II resens he formulaion for our roblem. ecion III describes he Infiniesimal Perurbaion Analysis (IPA-based gradien echnique o solve he roblem. ecion IV illusraes he numerical sudies. Conclusions are rovided in he las secion. II. PRBLEM FRMULAIN We consider a muli-or sysem consising of ors conneced wih each oher. A flee of owned ECs mees exogenous cusomer demands which are defined as he requiremens for ransforming ECs o laden conainers and hen ransoring hese laden conainers from original ors o desinaion ors. A single-level hreshold olicy wih eriodical review is emloyed o manage he ECs. A he beginning of a eriod he ECR decisions are made for each or involving wheher o reosiion ECs o/from which ors and in wha quaniy. hen when he cusomer demands occur in he eriod we can use hose conainers ha are currenly sored a he or and hose ECs ha are reosiioned o he or in he eriod o saisfy. If i is no enough we need o lease addiional ECs immediaely from vendors. We make he following assumions: he owned conainer flee is fixed. hor-erm leasing is considered and he quaniy of he leased ECs is always available in he or a any ime. he leased ECs are no disinguished from owned conainer. weny-foo equivalen uni (EU is used o reresen a conainer. he ravel ime for each - air (from or o or is less han one eriod lengh. When he reosiioned ECs arrive a he desinaion ors hey will become available immediaely. 80

2 When he laden conainers arrive a he desinaion ors hey will become emy and be available a he beginning of nex eriod. A. Noaion o formulae he roblem following noaions are inroduced firsly. he flee of owned emy conainers he se of ors he discree ime decision eriod he surlus or subse in eriod he defici or subse in eriod he balanced or subse in eriod he beginning on-hand invenory of or in eriod he invenory osiion of or in eriod afer making he ECR decisions he amoun of ECs reosiioned from surlus or o he or in eriod he random cusomer demand for he - air in eriod he vecor of he beginning on-hand invenory in eriod he vecor of he invenory osiion in eriod he array of reosiioned quaniies for all ors he amoun of esimaed EC suly for surlus or in eriod he amoun of esimaed EC demand for defici or in eriod he sochasic cusomer demands in eriod which is he array of a realizaion of he cusomer demands he cos of reosiioning an EC from or o or he cos of holding an EC a or er eriod he cos of leasing an EC a or er eriod he hreshold of or Vecor of he hresholds o simlify he narraive he following noaions are inroduced. he sum of ECs reosiioned ou from or in eriod he sum of ECs reosiioned ino or in eriod he sum of exored laden conainers of or in eriod he sum of imored laden conainers of or in eriod he cumulaive disribuion funcion for he amoun of he difference beween he laden conainer inbound and oubound of or in eriod I should be oined ou ha is a given sae variable i.e. he given iniial on-handing invenory; while is a decision variable for. he ECR decisions are made a he beginning of eriod firsly. hen he invenory osiion can be obained by I y x u u P (1 Afer cusomer demands are realized and he laden conainers become available he beginning on-hand invenory for he nex eriod can be udaed by x 1 y P (2 Nex we resen he single-level hreshold olicy o deermine he reosiioned quaniies in eriod. B. A ingle-level hreshold Policy o make he ECR decisions a single-level hreshold olicy is develoed which ries o mainain he invenory osiion a a arge hreshold value. More secifically or has a arge hreshold namely ; in each eriod such as in eriod if he beginning on-handing invenory of or namely is greaer han is hreshold value i.e. hen i is a surlus or and he quaniy excess of can be reosiioned ou o oher ors ha may need i o ry o bring he invenory osiion down o ; if is less han hen i is a defici or and ECs should be reosiioned ino his or from surlus ors o ry o bring he invenory osiion u o ; if is equal o hen i is a balanced or and nohing is done. Wihou loss of generaliy we consider he ECR decisions in eriod. According o he hreshold olicy hree subses i.e. surlus or subse defici or subse and balanced or subse can be obained as follows: ; ;. When eiher he surlus or subse or he defici or subse is emy we do nohing. ha is no ECs are reosiioned and we can have. However when and are nonemy we can comue he amouns of EC sulies of surlus ors and EC demands of defici ors by: a x i P (3 i i i a j j xj j P (4 hen he roblem is abou moving ECs from surlus ors o defici ors in he righ quaniy a he leas movemen cos. A ransoraion model is formulaed o solve his roblem as follows: R min C z ip jp i j i j (5 s. z i j a i P jp i (6 z i j a j P ip j (7 z min( a a ip i j i j jp ip jp zi j 0 i P j P (9 Consrains (6 and (7 are resource consrains. Consrain (8 imlies ha he amoun of oal exored ECs from he surlus ors is caaciaed by he amoun of oal demands of all defici ors; hus we can ry o bring he invenory osiion of each or back o is hreshold level. Consrains (9 are he non-negaive reosiioned EC quaniy consrains. olving he ransoraion model we can obain he reosiioned quaniies from he surlus ors o he defici (8 81

3 ors. o furher comlee he value of which involves he reosiioned quaniies for all ors we se for a balanced or for a surlus or and for a defici or which reflec he facs ha a balanced or does no reosiion in or ou ECs a surlus or does no reosiion in ECs and a defici or does no reosiion ou ECs resecively. C. he imizaion Problem Le be he execed oal cos er eriod wih he flee size and olicy arameer. he roblem which is o find he oimal arameers of he given olicy namely ha minimizes he execed oal cos er eriod can be formulaed as min J( N γ (10 γ subjec o he single-level hreshold olicy he given flee size and he invenory dynamics equaions (1 and (2. Wih a sligh misuse of he noaion we dro he subscri in he noaions of and for ease of descriion. More secifically can be formulaed as: J N γ EJ x γ E H x γ G y (11 where is he oal cos in one eriod; and are he EC reosiioning cos and he oal EC holding and leasing cos in one eriod resecively. We have H R ( x γ H ( Z C z (12 P G( y g( y P mp ( m m H L C ( y C ( y P where reresens he EC holding and leasing cos of or in one eriod;. Nex we consider he roblem under balanced scenario followed by ha under unbalanced scenario. Here balanced (unbalanced scenario is he scenario in which he oal amoun of esimaed EC suly is equal (no equal o he oal amoun of esimaed EC demand in each eriod. From (3 and (4 i is observed ha. Hence a scenario wih is a balanced scenario and a scenario wih is an unbalanced scenario. 1 Balanced cenario: Consider he roblem in he balanced scenario. We can obain he oimal soluion of (10 analyically since i only deends on he holding and leasing cos funcion. he exlanaions are as follows. From he ransoraion models we know ha he reosiioning ou (in requiremen of each surlus (defici or can be fully saisfied in each eriod in a balanced scenario. hus afer making ECR decisions he invenory osiion level of each or can be always ke a is arge hreshold level. I imlies ha he esimaed EC suly (demand of he surlus (defici or in a eriod exce he iniial eriod will be indeenden from he arameers of he flee size and he hresholds and jus deend on is cusomer demands in he revious eriod. Consequenly he EC (13 ransoraion cos and he oal EC holding and leasing cos in one eriod will be indeenden; and he execed EC ransoraion cos er eriod will be indeenden from arameers of and. Furher seaking he oimal soluion only deends on he execed oal EC holding and leasing cos funcion. he roblem (10 in he balanced scenario can be simlified o an non-linear rogramming as: min E C H ( C L ( γ P s.. P N where he value of flee size is given. Considering he convexiy of he above cos funcion and aking use of he K.K.. condiions we can obain he oimal soluion of he NLP by solving (14 and (15. H L L C C F ( C N 0 P (14 N 0 P (15 where is he Lagrange Mulilier of he balance consrain. Remark: Le be he oimal hresholds in he balanced scenario wih given flee size. Given cusomer demands we know ha can achieve he minimum holding and leasing cos for he roblem (10. However i may no achieve he minimum execed oal cos er eriod for he roblem because we can find oher hresholds achieving less ransoraion coss han ha achieved by. For examle when we se all he hresholds going o infinie so ha no ECs will be reosiioned he ransoraion cos in his scenario will be zero and less han ha in he scenario wih. hus we nex consider he unbalanced scenario. 2 Unbalanced cenario: Consider he roblem in he unbalanced scenario. By aking advanage of he srucure of he roblem we find an ineresing roery abou he ransoraion cos as follows: Proery I: In an unbalanced scenario he ransoraion cos in a eriod exce he iniial eriod could be less han or equal o ha in a balanced scenario wih same cusomer demands. Inuiively for examle if a exored-dominaed or has he robabiliy o become a surlus or i.e. i needs o reosiion ou ECs o oher ors reosiioning in less ECs han is hreshold in his or in advance when i becomes a surlus or will reduce is EC reosiioning ou quaniy. Hence less ransoraion cos in a eriod in an unbalanced scenario could be occurred. ince here is no closed-form formulaion for he comuaion of execed oal cos er eriod in he unbalanced scenario involving he reosiioned EC quaniies from he ransoraion models we ado he simulaion o esimae given values of and as shown in (16. J N 1 1 γ 1 1 J x γ H G x γ y (16 where is he oal cos in eriod ; and are he EC reosiioning cos and he oal EC 82

4 holding and leasing cos in eriod and can be obained from (12 and (13 resecively; is he amoun of he simulaion eriods. I is significan o highligh ha solving (10 in he unbalanced scenario is difficul. In order o find an oimal soluion o he roblem we need o use a search-based mehod. In nex secion we develo an oimizaion echnique namely IPA-based gradien echnique. ummarizing above discussions we can ge ha given flee size and cusomer demands he minimum execed oal cos er eriod could be achieved in eiher he balanced scenario or unbalanced scenario. he oimal soluion under balanced scenario can be obained analyically by solving (14 and (15 and under unbalanced scenario by alying he roosed IPA-based gradien echnique. Use hill climbing algorihm o udae he olicy Iniialize ( Iniializaion: e =1 olve ransoraion model o make ECR decisions Realize he cusomer demands Calculae he oal cos Esimae gradien (using he IPA Is? Yes No Comue he execed oal cos er eriod & overall gradien =+1 III. IPA-BAE GRAIEN ECHNIQUE IPA is able o esimae he gradien of he objecive funcion from one single simulaion run hus reducing he comuaional ime. Moreover i has been shown ha variance of IPA esimaor is lower comared wih many oher gradien esimaors [15]. hus we roose an IPAbased gradien echnique o search he oimal soluion in he unbalanced scenario. he overall IPA-based gradien echnique is briefly described in Fig. 1. As shown in Fig. 1 given he arameers of he flee size and he olicy wih we firs calculae he oal cos and esimae he gradien of oal cos wih resec o he hresholds in all eriods. We esimae he gradien in a eriod using he conce of erurbaion roagaion from IPA [16] he dual informaion of he LP model and he chain rule. hen we can obain he execed oal cos er eriod and he gradien of. his gradien can rovide a direcion for finding new arameers of he olicy ha may have a lower execed oal cos er eriod and hence he hill climbing algorihm is used o udae he arameers of he olicy. Finally when he erminaion crieria are saisfied he simulaion is soed. o esimae he gradien of execed oal cos er eriod we ake a arial derivaion of (16 wih resec o he hreshold of or. Wih he hel of (13 we can obain J( N γ 1 J x γ i 1 i (17 1 H ( Z E g( y y 1 i P y i where for he invenory holding and leasing cos funcion we use he execed holding and leasing cos funcion o esimae he gradien insead of using he samle ah since we are able o ge he exlici funcion o evaluae he average gradien; measures he imac of he ransoraion cos in eriod when he hreshold is changed; measures he imac of he holding and leasing cos funcion of or in eriod when he invenory osiion level is changed; measures he imac of he invenory osiion level of or in eriod when he hreshold is changed. x a i / a H Z No erminaion imal hresholds Period Period +1 u / u * I* y We define he nominal ah as he samle ah generaed by he simulaion model wih arameer and he erurbed ah as he samle ah generaed using he same model and same random seeds bu wih arameer where. Wihou loss of generaliy we only erurb he hreshold of or and kee he hresholds of he oher ors unchanged i.e. and for oher or where he value of is infiniesimally small. By sufficienly small we mean such ha he surlus or subse and defici or subse are same in he boh nominal and erurbed ahs in every eriod. fenimes we will resen he changes in various quaniies by dislaying wih argumen. We erurb in all eriods and he reresenaive erurbaion flow in eriod is shown in he Fig. 2. In our roblem wih real variables he robabiliy of having balanced or is close o 0. In oher word or should be eiher surlus or or defici or in eriod. From (3 and (4 we can derive ha and. Hence in Fig. 2 he erurbaion of will work ogeher wih he erurbaion of o affec he esimaed EC suly/demand namely for some ors. We know ha he esimaed EC suly/demand of a or is he RH of he corresonding or s consrain in he ransoraion model. I imlies ha he erurbaions of G y Figure 2. he Perurbaion Flow Yes Figure 1. he flow of he IPA-based gradien echnique x 1 i 83

5 of some ors could affec he oimal reosiioning quaniies of some ors which of course will affec he oal oimal reosiioned ou/in quaniies of some ors namely. From (1 we know ha. he erurbaion of will work ogeher wih he erurbaion of of some ors o affec he erurbaion on EC invenory osiions namely for some ors. Furhermore he erurbaion of will affec he ransoraion cos and he erurbaion of will affec he oal holding and leasing cos. From (2 we have which imlies ha he erurbaion on he invenory osiion will be fully roagaed o he beginning on-hand invenory of nex eriod. he flowing noaions are inroduced. he se of ors whose beginning on-hand invenory in eriod are affeced by erurbing hreshold of or he se of ors whose oal oimal reosiioned quaniies are changed by erurbing he esimaed EC suly/demand of or in eriod he corresonding dual variable for or consrain in he ransoraion model in eriod a indicaor funcion which akes 1 if he condiion is rue and oherwise 0 racing he erurbaions by following he flow in Fig. 2 we can obain ha in eriod will be eiher emy or consis of a air of ors i.e. or. imilarly will be eiher emy or consis of a air of ors i.e. or. We can obain he gradien of execed oal cos er eriod wih resec o in (17 can be aroximaed by (18. In (18 he firs erm of he RH resens he erurbaion on he ransoraion cos when ; he second erm of he RH resens he erurbaion on he ransoraion cos when ; he hird erm of he RH resens he erurbaion on he holding and leasing cos when and ; he forh erm of he RH resens he erurbaion on he holding and leasing cos when and ; he fifh erm of he RH resens he erurbaion on he holding and leasing cos when and ; can be obained by alying a roosed modified seing sone aroach wih erurbing he esimaed suly/demand of or in eriod. We know ha in he iniial eriod; and for can be obained as follows: (a when ; (b when and ; (c when and ; (d when and. J( N γ 1 i 1 where he value of J ( x γ i I ( ip I ( qi P I( Qi ( 1 i I( Qi ( 1 qi Q E g y e i i E g y i i ei I E i yi y ei 1 Qi E g y E g y q i i i qi I 1 Eq i yi y qi Qi E g y E g ye i i q q i i IEq i yi y eq i eq i i IV. in (18 is calculaed by L E g( y if 0 C y y C C F y C if 0 y H L L NUMERICAL REUL (18 (19 In his secion we aim o evaluae he erformance of he roosed single-level hreshold olicy (P. For comarison a mach back olicy (MBP is inroduced. uch olicy is widely acceed and alied in racice and is basic rincile is o mach he conainers back o he original or. Mahemaically z m 1 m (20 m he NLP in he balanced scenario is solved by Malab (version he IPA-gradien based algorihm is coded in Visual C All he numerical sudies are esed on and Inel uo Processor E GHz CPU wih 4.00 GB RAM under he Microsof Visa eraion ysem. We se he simulaion eriod wih warm-u eriod =100. For he P he erminaion crieria are ha he maximum ieraion for finding he oimal hresholds namely is achieve or he execed oal cos in he ieraion is larger han ha in he revious ieraion. We se. For he MBP since he ransoraion cos is indeenden from he arameer of flee size we se he invenory osiion in he iniial eriod be equal o he oimal invenory osiion which minimizing he execed holding and leasing cos. For a hree-or sysem we comare he erformance of boh olicies based on he execed oal cos er eriod. we give he flee size from 483 EUs o 1128 EUs o invesigae he effec of he flee size on he execed oal cos. Fig. 3 shows he resuls. I is observed ha P ouerforms MBP for all cases. he execed oal cos er eriod savings achieved by P over MBP are of he order of 12.75%~37.18%. ne ossible exlanaion is ha P makes he ECR decisions in erms of minimizing he ransoraion cos. Hence i is imoran for oeraors o use inelligen mehod in reosiioning ECs insead of resoring o simle way such as he MBP. 84

6 Execed oal cos P Flee size Figure 3. Execed oal cos er eriod comarison for hree-or sysem From Fig. 3 i reveals ha he oimal average oal cos aears o be convex wih resec o he flee size for each sysem. I reflecs he inuiion ha he oimal flee size is he rade-off beween he ransoraion cos and he holding and leasing cos. V. CNCLUIN AN FUURE WRK MBP In his aer he EC reosiioning roblem in a mulior sysem is considered. A single-level invenory-based olicy wih he reosiioning rule in erms of minimizing ransoraion cos is develoed o reosiion ECs eriodically by aking ino accoun demand uncerainy and dynamic oeraions. wo aroaches non-linear rogramming and IPA-based gradien echnique are develoed o solve he roblem oimizing hresholds of olicy under balanced and unbalanced scenarios resecively. he numerical resuls rovide insighs ha by reosiioning he ECs inelligenly we can significanly reduce he oal oeraion cos. he main conribuions of he sudy are as follows: (a a single-level hreshold olicy wih a reosiioning rule in erms of minimizing ransoraion cos is develoed for reosiioning ECs in a muli-or sysem. o he bes of our knowledge few works consider he reosiioning rule which is relaed o he ransoraion coss; (b by develoing he mehod o solve he difficul ECR roblem i.e. using IPA o esimae he gradien i is innovaive and rovides a oenial mehodology conribuion in his field We srongly assumed ha he ECs are disached beween each air of ors in one eriod. I may no be he righ one eriod in some general cases. Furher research is needed o relax he one-eriod assumion and consider he roblem wih differen ime dimension for he reosiioning ime. he main challenge is o rack he erurbaions along he samle ah. ransoraion ransoraion cience vol. 21(4 Nov doi: /rsc [4]. Crainic M. Gendreau and P. ejax ynamic and sochasic models for he allocaion of emy conainers eraions Research vol. 41(1 Jan. - Feb [5] W. hen and C. Khoong A for emy conainer disribuion lanning ecision uor ysems vol. 15(1 e doi: / ( [6] R. Cheung and C. Chen A wo-sage sochasic nework model and soluion mehods for he dynamic emy conainer allocaion roblem ransoraion cience vol. 32(2 May doi: /rsc [7]. W. Lam L. H. Lee and L. C. ang An aroximae dynamic rogramming aroach for he emy conainer allocaion roblem ransoraion Research Par C vol. 15(4 Aug doi: /j.rc [8]. Choong M. Cole and E. Kuanoglu Emy conainer managemen for inermodal ransoraion neworks ransoraion Research Par E vol. 38(6 Nov doi: / ( [9] J. X. ong and. P. ong Conainer flee sizing and emy reosiioning in liner shiing sysems ransoraion Research Par E vol. 45(6 Nov doi: /j.re [10]. P. ong Characerizing oimal emy conainer reosiion olicy in eriodic-review shule service sysems Journal of he eraional Research ociey vol. 58( doi: /algrave.jors [11] J. A. Li K. Liu. C. H. Leung and K. K. Lai Emy conainer managemen in a or wih long-run average crierion Mahemaical and Comuer Modeling vol. 40(1-2 July doi: /j.mcm [12]. P. ong and J. Carer imal emy vehicle redisribuion for hub-and-soke ransoraion sysems Naval Research Logisics vol. 55(2 Mar doi: /nav [13]. P. ong and J. X. ong Emy Conainer Managemen in Cyclic hiing Roues Mariime Economics & Logisics vol. 10(4 ec doi: /mel [14] J. A. Li. C. H. Leung Y. Wu and K. Liu Allocaion of emy conainers beween muli-ors Euroean Journal of eraional Research vol. 182(1 c doi: /j.ejor [15] R. uri Perurbaion analysis: he sae of he ar and research issuesexlained via he GI/G/1 queue Proceedings of he IEEE vol. 77(1 Jan doi: / [16] Y.C.Ho and X. R. Cao iscree Even ynamic ysems and Perurbaion Analysis. BosonUK Kluwer Academic Publishers REFERENCE [1] U. Naions Regional hiing and Por evelomen (Conainer raffic Forecas [2]. P. ong imal hreshold conrol of emy vehicle redisribuion in wo deo service sysems IEEE ransacions on Auomaic Conrol vol. 50(1 Jan doi: /AC ( [3] P. ejax and. Crainic urvey Paer--A Review of Emy Flows and Flee Managemen Models in Freigh 85