OPTIMISING YARD OPERATIONS IN PORT CONTAINER TERMINALS

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1 Advaced OR ad AI Methods i Trasportatio OPTIMISING YARD OPERATIONS IN PORT CONTAINER TERMINALS Louise K TRANBERG Abstract. This paper deals with the problem of positioig cotaiers i a yard block of a port cotaier termial. The objective of the cotaier positioig problem (CPP) is to miimise the total hadlig time i the block, i.e. the time required for storage ad reshufflig of cotaiers. Oe of the costrait types, cocerig the last-i first-out (LIFO) priciple, implies major modellig challeges. A mixed-iteger liear programmig model for the geeral CPP is formulated, implemeted i the modellig tool Mosel, ad validated by the solutio of a test case usig the Xpress-MP solver. 1. Itroductio As a result of globalisatio ad ecoomic growth, the eed for cotaier trasportatio has become very sigificat, ad cosequetly so has the competitio betwee ports. Thus, the importace of efficiet hadlig i cotaier termials is costatly icreasig. A very importat issue whe optimisig port efficiecy cocers storage ad movemet i the yard. The operatios i yard storage blocks offer may differet problem aspects ad cosiderable challeges cocerig modellig ad solutio approaches. The purpose of this paper is to cotribute to the hadlig of yard operatio problems. No operatios research approaches to the cotaier positioig problem (CPP) have bee foud i the literature. Therefore, the mai cotributio of this paper will be the itroductio of a mathematical formulatio of the CPP. The CPP occurs i all port cotaier termials. However, the parameters - especially the dimesios ad the equipmet - vary. For iformatio about the differet types of port desig ad equipmet, see [2]. I Figure 1 a example of a port layout, characteristic for a automated termial, is show. I geeral, three types of operatios are carried out i port cotaier termials: 1) quayside, 2) yard, ad 3) ladside operatios. For import cotaiers, the followig equipmet types ad operatios ca be idetified (the sequece of operatios is reversed for export cotaiers). Cetre for Traffic ad Trasport, Techical Uiversity of Demark, Buildig 115, DK-2800 Lygby. lkt@ctt.dtu.dk

2 Optimisig yard operatios i port cotaier termials 387 QC PP RMG DP Railway Storage block Trai Truck/chassis Cotaier vessel AGV 1) Quayside operatios 2) Yard operatios 3) Ladside operatios Figure 1. Pricipal layout of a automated port cotaier termial (ot true to size), ispired by [2]. After a vessel is assiged to a berth, Quay Craes (QCs) uload the cotaiers ad Automated Guided Vehicles (AGVs) trasport each cotaier to oe of the pick-up poits (PPs) by the storage blocks. Due to the lack of accurate data before the arrival of vessels i the port, the plaig of the cotaier positios i the yard caot begi before the uloadig starts [2]. At the PP, cotaiers are picked up by Rail Mouted Gatry Craes (RMGs) ad stored at some predetermied positio. Each cotaier might be reshuffled a umber of times before beig moved to the drop-off poit (DP). At the DP, cotaiers are loaded oto trucks or chassis by which they are trasported ito the hiterlad either by road or by rail. 2. Formulatig ad modellig the CPP The focus of this paper is to formulate ad solve the problem cocerig yard operatios at oe storage block, the CPP. The objective of the CPP is to determie the optimal positios of cotaiers c i oe block by miimisig the total umber of reshuffles. A storage block cosists of a umber of positios p = (1,..., P ) i the horizotal plae where p = 1 correspods to the PP ad p= P correspods to the DP. Figure 2 provides a overview of the pricipal structure of a storage block iclusive PP ad DP ad how the positios are umbered. I the CPP model, the biary variable x cp is equal to 1 if cotaier c is placed at positio p after reshuffles, ad 0 otherwise. At each positio, a umber of cotaiers ca be stacked o top of each other, the maximum level depedig o the stackig equipmet. Whe movig a cotaier placed udereath other cotaier(s) at a positio, this or these must be removed first. This last-i first-out (LIFO) priciple restricts the arrival, storage, ad movemet times, idicated by the cotiuous decisio variables t a cp, t s cp, ad t m cp. Figure 3 shows the LIFO priciple for two cotaiers, c ad c, at positio p. The followig must be assured: if cotaier c ad c both arrive at positio p, ad if cotaier c arrives before cotaier c ad departs from the positio after the arrival

3 388 L. K. Traberg z PP, p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 p = 7 p = Stack of cotaiers DP, p = P x y Figure 2. Pricipal structure of a storage block cosistig of a umber of positios p i the horizotal plae. At each positio, a umber of cotaiers c ca be stacked. The positios are umbered from 1 to P : p=1correspods to the pick-up poit PP, p=2,..., P 1 correspods to positios i the block, ad p= P correspods to the drop-off poit DP. Positio p: ~ c Storage(c) c ~ Storage(c) Arrival(c) Arrival(c) ~ Departure(c) ~ Departure(c) Time Figure 3. The LIFO priciple: if cotaier c arrives before ad departs after cotaier c arrives at positio p, the c must also depart after c leaves p. of cotaier c (i.e. there is a time overlap betwee the storage times for the two cotaiers at the positio), the cotaier c must depart after cotaier c leaves positio p. This ca be formulated as follows (where N): x cp = x cp = 1 t a cp+µ t ã cp t a cp+ t s cp+µ t ã cp } t a cp+ t s cp t ã cp + t s cp +µ These relatios are liearised i the formulatio of the CPP by the itroductio of two sets of biary variables,α c cp adδ c cp, esurig the LIFO coditios. Parameters i the problem are the pick-up ad drop-off times for a cotaier c, PT c ad DT c, the time required to move a cotaier betwee two positios, p ad p, deoted by TT p p, the legth of the plaig periodτ, a small time itervalµ, ad a small umber ǫ 1. Figure 4, showig a sequece of positios for a cotaier c, provides a overview of the structure of the CPP preseted.

4 Optimisig yard operatios i port cotaier termials 389 Positio DP p +1 p xcp 2 2 p 2 x cpp1 PP a tcpp1 tcp a 2 2 Cotaier c x cp x cp xcdp + 2 tcp a t a cp+1+ 1 t a cdp + 2 Time t m cpp1 t s cp 2 2 t s cp t m cp t s cp+1+ 1 t m cp+1+ 1 Figure 4. The CPP structure: overview of a sequece of positios for oe cotaier c Notatio The otatio for the CPP model is as follows. C={1,..., C } is the set of cotaiers. The set of positiosp={1,..., P } icludes the PP ad DP.N={1,..., N } is the set of umbers of reshuffles where =1correspods to the PP (p=1) ad 2correspods to actual positios i the block. However, for the DP (p= P ), 3, sice there is at least oe actual positio i the block. The setz=c P N is itroduced, where z=(c, p, ) Z. Furthermore, let q=(π,φ)=(q π, q φ ) Q=Π Φ, the x(q)= q Q x q adq π ={q=(π,φ) Q π =π}, i.e. x(q π )= φ Φ x (π,φ), π. Likewise, the macros t a (Q), t s (Q) ad t m (Q) are used. Table 1 provides a overview of the otatio for the CPP model. Sets C P N Parameters PT c DT c TT p p Cotaiers Positios icludig pick-up ad drop-off poits (PP ad DP) Numbers of reshuffles Pick-up times for cotaier c Drop-off time for cotaier c Trasportatio time betwee positio p ad positio p Time horizo τ µ Small time iterval ǫ Small umber ( 1) Decisio variables x z B Z Equals 1 iff cotaier c is placed at positio p after reshuffles tz a R Z + Arrival time for cotaier c at positio p after reshuffles tz s R Z + Storage time for cotaier c at positio p after reshuffles tz m R Z + Movemet time for cotaier c after positio p ad reshuffles α c cp B C 2 P Equals 1 iff cotaier c arrives before cotaier c at positio p δ c cp B C 2 P Equals 1 iff cotaier c leaves before cotaier c arrives at positio p Table 1. Overview of the otatio for the CPP model

5 390 L. K. Traberg 2.2. The CPP model The objective of the CPP is to miimise the total umber of reshuffles i the storage block, subject to costraits of the followig three types: (1) time ad flow restrictios, (2) LIFO costraits, ad (3) logical coectios betwee decisio variables ad parameters. The CPP model ca be stated as follows: mi x(z) (1) s.t. t a c11 = PT c, c C (2) x(z cp p= P, 3)=DT c, c C (3) x(z c ) x(z c 1 ), c, C N\{1} (4) t m cp TT p p (x cp + x c p+1 1), c, p, p, C P P N\{ N } (5) t m cp TT p p (x cp + x c p+1 1)+τ(2 x cp x c p+1 ), t a cp+ t s cp+ t m cp t a c p+1 +τ(2 x cp x c p+1 ), t a cp+ t s cp+ t m cp t a c p+1 τ(2 x cp x c p+1 ), c, p, p, C P P N\{ N } (6) c, p, p, C P P N\{ N } (7) c, p, p, C P P N\{ N } (8) t a cp2 = PT c+ t m c11, c C (9) α c cp x(z cp )+ x(z cp )+ǫ(t a (Z cp ) δ c cp x(z cp )+ x(z cp )+ǫ(t a (Z cp )+t s (Z cp )+µ t a (Z cp )+t s (Z cp ) t a (Z cp )+t s (Z cp )+µ (t a (Z cp )+µ)) 2, c, c, p C P (10) t a (Z cp )) 2, c, c, p C P (11) τ(2 α c cp δ c cp ), c, c, p C P (12) t a z+ t s z+ t m z 3τx z, z Z (13) x(z c ) 1, c, C N (14) x(z cp ) 1, c, p C P (15) x z B Z, t a z, t s z, t m z R Z +,α c cp,δ c cp B C 2 P (16) The objective fuctio (1) miimises the total umber of reshuffles i the block. Alteratively, the total movemet time t m (Z) could be miimised or several terms could be weighted accordig to importace i a multi-objective fuctio. The costraits (2) - (9) cocer the time ad flow restrictios. Equatios (2) set the first arrival time equal to the pick-up time for every cotaier c. Correspodigly, equatios (3)

6 Optimisig yard operatios i port cotaier termials 391 impose that the drop-off times are met by settig the arrival time at the DP. The iequalities (4) esure the cotiuity of a sequece of positios by forcig x(z c 1 )=1 if x(z c )=1. The movemet times betwee actual pairs of positios are determied by (5) ad (6), settig the lower ad the upper boud o tcp, m respectively. The same techique is used i iequalities (7) ad (8), esurig the balace betwee the arrival, storage ad movemet times. As for the costrait pair (5) ad (6), either both iequalities (7) ad (8) are bidig (i.e. x cp = x c p+1 = 1) or both are o-bidig. I equatios (9) the storage time at the PP is set equal to 0 for all cotaiers. The costraits (10) - (12) cocer the LIFO restrictios. Iequalities (10) setα c cp = 1 if cotaier c arrives at positio p before cotaier c. Iequalities (11) setδ c cp = 1 if cotaier c departs from positio p after cotaier c arrives at p. The iequalities (12) coect the two above costraits by esurig that cotaier c departs from positio p after cotaier c if bothα c cp adδ c cp equal 1. The costraits (13) - (16) represet logical restrictios ad variable domais for the CPP. I (13) the coectio betwee the biary x z ad the cotiuous tz a, tz s ad tz m is formulated, iequalities (14) esure that cotaier c is placed at maximum oe positio p at a time (i.e. per umber ), ad the iequalities (15) imply that cotaier c caot be placed at positio p more tha oce. The latter costraits are icluded for modellig reasos. Fially, the decisio variables domais are stated i (16). 3. Computatioal results ad perspectives The model is implemeted i the modellig tool Mosel ad validated by the solutio of a small test case usig the Xpress-MP solver [1]. The optimal solutio for the test case was foud withi two miutes usig a Petium 2.00 GHz 1.00 GB RAM. Future work with the CPP will iclude solutio of larger test cases i order to evaluate the model s potetial relevace to real-live problem istaces. The proposed mixed-iteger liear model presets a comprehesive formulatio of the CPP, providig a overview of the complexity of the problem. Further work withi the developmet of alterative solutio approaches, such as heuristics, decompositio, etc., for solvig larger problem istaces will be a importat cotributio to the hadlig of cotaier storage problems i marie termials. Refereces [1] Dash Optimizatio. Xpress-mosel user guide. Release 1.2. Aditioal iformatio at [2] D. Steeke, S. Voß, ad R. Stahlbock. Cotaier termial operatio ad operatios research - a classificatio ad literature review. I H.-O. Güther ad K. H. Kim, editors, Cotaier Termials ad Automated Trasport Systems, chapter 8, pages Spriger, 2005.