Simultaneous Optimization of Berth Allocation, Quay Crane Assignment and Quay Crane Scheduling Problems in Container Terminals

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1 Smultaneous Optmzaton of Berth Allocaton, Quay Crane Assgnment and Quay Crane Schedulng Problems n Contaner Termnals Necat Aras, Yavuz Türkoğulları, Z. Caner Taşkın, Kuban Altınel Abstract In ths work, we focus on the ntegrated plannng of the followng problems faced wthn the context of seasde operatons at contaner termnals: berth allocaton, quay crane assgnment, and quay crane schedulng. Frst, we formulate a new bnary nteger lnear program for the ntegrated soluton of the berth allocaton and quay crane assgnment problems called BACAP. Then we extend t by ncorporatng the crane schedulng problem as well, whch s named BACASP. Although the model for BACAP s very effcent and even large nstances up to 60 vessels can be solved to optmalty, only small nstances for BACASP can be solved optmally. To be able to solve large nstances, we present a necessary and suffcent condton for generatng an optmal soluton of BACASP from an optmal soluton of BA- CAP usng a postprocessng algorthm. We also develop a cuttng plane algorthm for the case where ths condton s not satsfed. Ths algorthm solves BACAP repeatedly by addng cuts generated from the optmal solutons at each tral untl the aforementoned condton holds. 1 Introducton There has been a consderable growth n the share of contanerzed trade n the world s total dry cargo durng the last 30 years. Therefore, the effcent management of seaport contaner termnals has become a crucal ssue [2]. In ths work, we concentrate on the ntegrated plannng of seasde operatons, whch ncludes the berth allocaton problem (BAP), quay crane assgnment problem (CAP) and quay crane schedulng problem (CSP). Generally, BAP deals wth the determnaton of the optmal berthng tmes and postons of vessels n contaner termnals. The focus of CSP, on the other hand, s manly on the problem of determnng an optmal Necat Aras, Yavuz Türkoğulları, Caner Taşkın, Kuban Altınel Boğazç Unversty, İstanbul, Turkey e-mal: {arasn, turkogullar, caner.taskn, altnel}@boun.edu.tr 1

2 2 Aras et al. handlng sequence of vessels for the avalable cranes at the termnal. However, as can be realzed, the assgnment of the cranes to vessels has a drect effect on the processng tmes of the vessels. As a result, crane assgnment decsons can be embedded wthn ether BAP or CSP models. In ths work we formulate two new MILP formulatons ntegratng frst BAP and CAP (BACAP), and then BAP, CAP, and CSP (BACASP). Both of them consder a contnuous berth layout where vessels can berth at arbtrary postons wthn the range of the quay and dynamc vessel arrvals where vessels cannot berth before the expected arrval tme. The crane schedule found by solvng the BACASP formulaton determnes the specfc crane allocaton to vessels for every tme perod. These MILP models are the frst models solved exactly rather than heurstcally n the lterature for relatvely large nstances. 2 Model Formulaton The underlyng assumptons of our models are gven as follows. The plannng horzon s dvded nto equal-szed tme perods. The berth s dvded nto equal-szed berth sectons. Each berth secton s occuped by no more than one vessel n each tme perod. Each quay crane can be assgned to at most one vessel per tme perod. Each vessel has a mnmum and maxmum number of quay cranes that can be assgned to t. The servce of a vessel by quay cranes begns upon that vessel s berthng at the termnal, and t s not dsrupted untl the vessel departs. The number of quay cranes assgned to a vessel does not change durng ts stay at the berth, whch s referred to as a tme-nvarant assgnment [1]. Furthermore, the set of specfc cranes assgned to a vessel s kept the same. By lettng the ndex of vessels, g the ndex of crane groups, j the ndex of berth sectons, k the ndex of number of cranes, t the ndex of tme perods, c g l the ndex of the leftmost crane n group g, c g r the ndex of the rghtmost crane n group g, and C(g) the ndex set of cranes n group g, we defne the followng parameters: B= the number of berth sectons, G= the number of crane groups, N= the number of avalable quay cranes, T = the number of tme perods n the plannng horzon, V = the number of vessels, d = due tme of vessel, e = arrval tme of vessel, = lower bound on the number of cranes that can be assgned to vessel, = upper bound on the number of cranes that can be assgned to vessel, l = the length of vessel measured n terms of the number of berth sectons occuped, p = processng tme of vessel f k cranes are assgned to t, s = desred berth secton of vessel, φ 1 = cost of one unt devaton from the desred berth secton for vessel, φ 2 = cost of berthng one perod later than the arrval tme for vessel, φ 3 = cost of departng one perod later than the due tme for vessel. Let us defne a bnary varable X k jt, whch s equal to one f vessel starts berthng at secton j n tme perod t, and k quay cranes are assgned to t, and zero otherwse. Constrant (1) ensures that each vessel berths at a unque secton and tme perod, and the number of quay cranes assgned to t les between the mnmum and maxmum allowed quanttes.

3 Smultaneous Optmzaton n Contaner Termnals 3 k= T p +1, X k jt = 1 = 1,...,V. (1) t=e Constrant set (2) guarantees that each berth secton s occuped by at most one vessel n each tme perod. To put t dfferently, there should not be any overlap among the rectangles representng vessels n the two-dmensonal tme-berth secton space, whch are located between max ( e,t p + 1) and mn ( T p + 1,t) on the tme dmenson, and between max(1, j l + 1) and mn(b l + 1, j) on the berth secton dmenson. V =1 mn(, j) j =max(1, j l +1) k= mn(t p k+1,t) X k t =max(e,t p k+1) j t 1 j = 1,...,B;t = 1,...,T (2) We next dscuss how quay crane avalablty can be handled n the BACAP model. Let us denote the number of avalable quay cranes by N. Constrant set (3) ensures that n each tme perod the number of actve quay cranes s less than or equal to the avalable number of cranes: V =1 k= mn(t p +1,t) kx t =max(e,t p k+1) k jt N t = 1,...,T (3) The objectve functon (4) of our model mnmzes the total cost, whose components for each vessel are: ) the cost of devaton from the desred berth secton, ) the cost of berthng later than the arrval tme, and ) the cost of departng later than the due tme. Our nteger programmng formulaton for BACAP can be summarzed as follows: mn V =1 k= subject to constrants (1),(2),(3) T p +1 ) } + {φ 1 j s + φ 2 (t e ) + φ 3 (t + p k 1 d X k jt t=e (4) X k jt {0,1} = 1,...,V ; j = 1,...,B l + 1;k =,..., ;t = e,...,t p + 1. Recall that although the avalablty of quay cranes s consdered n constrant set (3) n BACAP, a schedule s not generated for each quay crane. To develop a mathematcal programmng formulaton for BACASP we extend the formulaton for BACAP by ncludng the constrant sets (1) (3) and defnng new varables and constrants so that feasble schedules are obtaned for quay cranes, whch do not ncur setup due to the change n the relatve order of cranes. We should remark that f quay cranes 1 and + 1 are assgned to a vessel n a tme perod, then quay crane has to be assgned to the same vessel as well snce quay cranes are located along the berth on a sngle ralway. Hence, we defne a crane group as a set of adjacent quay cranes and

4 4 Aras et al. let the bnary varable Y g t denote whether crane group g assgned to vessel starts servce n tme perod t. Constrant set (5) relates the X and Y-varables. It ensures that f k quay cranes are assgned to vessel, t must be served by a crane group g that s formed by C(g) = k cranes, where C(g) s the ndex set of cranes n group g and denotes the cardnalty of a set. Moreover, G s the total number of crane groups. X k jt G g=1 C(g) =k Y g t = 0 = 1,...,V ;k =,..., ;t = e,...,t p + 1 (5) It should be emphaszed that each crane can be a member of multple crane groups. However, each crane can operate as a member of at most one group n each tme perod. The next set of constrants (6) guarantees that ths condton holds: V G mn(t p k+1,t) Y g =1 g=1 t=max(e,t p k+1) c C(g) t 1 c = 1,...,N;t = 1,...,T (6) Even though constrants (5) and (6) make sure that each quay crane s assgned to at most one vessel n any tme perod, they do not guarantee that quay cranes are assgned to vessels n the correct sequence. In partcular, let the quay cranes be ndexed n such a way that a crane postoned closer to the begnnng of the berth has a lower ndex. Snce all cranes perform ther duty along a ral at the berth, they cannot pass each other or stated dfferently ther order cannot be changed. The next four constrant sets help to ensure preservng the crane orderng. Here, Z ct denotes the poston of crane c n tme perod t. Z ct Z (c+1)t c = 1,...,N 1;t = 1,...,T (7) Z Nt B t = 1,...,T (8) g Z g c l t + B(1 Y t ) jt k= jx k Z g cr t ( j + l 1)X k jt + B(1 Y g k= = 1,...,V ;g = 1,...,G; t = e,...,t p + 1;t t t + p 1 (9) t ) = 1,...,V ;g = 1,...,G; t = e,...,t p + 1;t t t + p 1 (10) Constrant set (7) smply states that the postons of the cranes (n terms of berth sectons) are respected by the ndex of the cranes. Ths means that the poston of crane c s always less than or equal to the poston of crane c+1 durng the plannng horzon. Constrant set (8) makes sure that the last crane (crane N) s postoned wthn the berth. By defnng c g l and c g r as the ndex of the crane that s, respectvely,

5 Smultaneous Optmzaton n Contaner Termnals 5 the leftmost and rghtmost member of crane group g, constrant set (9) guarantees that f crane group g s assgned to vessel and vessel berths at secton j, then the poston of the leftmost member of crane group g s greater than or equal to j. Smlarly, constrant set (10) ensures that f crane group g s assgned to vessel and vessel berths at secton j, then the poston of the rghtmost member of crane group g s less than or equal to j +l 1, whch s the last secton of the berth occuped by vessel. 3 Soluton As can be observed, BACASP formulaton s sgnfcantly larger than our BACAP formulaton wth whch we can solve nstances up to 60 vessels. Hence, t should be expected that only small BACASP nstances can be solved exactly usng CPLEX Ths fact has motvated us to make use of the formulaton for BACAP n solvng larger szed BACASP nstances to optmalty. By carefully analyzng the optmal solutons of BACAP and BACASP n small szed nstances, we have fgured out that an optmal soluton of BACASP can be generated from an optmal soluton of BA- CAP provded that the condton gven n Proposton 1 s satsfed. Ths condton s based on the noton of complete sequence of vessels (wth respect to ther occuped berthng postons), whch s defned as follows. Defnton 1. A vessel sequence v 1,v 2,...,v n s complete f () v 1 s the closest vessel to the begnnng of the berth, () v n s the closest vessel to the end of the berth, () v and v +1 are two consecutve vessels wth v closer to the begnnng of the berth, and (v) two consecutve vessels n ths sequence must be at the berth durng at least one tme perod. A complete sequence s sad to be proper when the sum of the number of cranes assgned to vessels n ths sequence s less than or equal to N. Otherwse, t s called an mproper complete sequence. Proposton 1. An optmal soluton of BACASP can be obtaned from an optmal soluton of BACAP by a post-processng algorthm f and only f every complete sequence of vessels s proper. The proof of ths proposton can be found n [3]. If there s at least one mproper complete sequence of vessels n an optmal soluton of BACAP, then we cannot apply the post-processng algorthm gven as Algorthm 1 to obtan an optmal soluton of BACASP from an optmal soluton of BACAP. In Algorthm 1, V A (V NA ) denotes the set of vessels to whch cranes (no cranes) are assgned yet. Clearly, V NA V A = {1,2,...,V }. Notce that the way the vessels are pcked up from V NA and added to the set V A mples that the order of the vessels forms one or more complete sequences n the set V A. It s also ensured that these complete sequences are proper. If there exsts a complete sequence where the sum of the number of cranes assgned to vessels s larger than N, then t s possble to add the cut gven n (11)

6 6 Aras et al. Algorthm 1 Post-processng algorthm Intalzaton: Let V NA {1,2,...,V } WHILE V NA Select vessel v V NA that berths n the leftmost berth secton Fnd the vessels n V A that are n the berth wth v n at least one tme perod. Among the cranes assgned to these vessels, fnd the crane c max that s n the rghtmost berth secton. IF V A = or any vessel n V A that s at the berth wth v n at least one tme perod c max 0 ENDIF Assgn cranes ndexed from c max + 1 to c max + θ v to vessel v, where θ v s the number of cranes assgned to vessel v V NA V NA \ v ENDWHILE correspondng to an mproper complete sequence nto the formulaton of BACAP, where IS refers to an mproper complete sequence and IS s the total number of vessels nvolved n that complete sequence. Note that ths cut s used to elmnate feasble solutons that nvolve IS. X k() IS 1 (11) j()t() IS The left-hand sde of (11) conssts of the sum of the X k jt varables whch are set to one for the vessels nvolved n IS. In other words, there s only one X k jt = 1 for each vessel IS. The j,k, and t ndces for whch X k jt = 1 related to vessel are denoted as j(), k(), and t() n (11). Upon the addton of ths cut, BACAP s solved agan. The addton of these cuts s repeated untl the optmal soluton of BACAP does not contan any mproper complete sequences. At that nstant, Algorthm 1 can be called to generate an optmal soluton of BACASP from the exstng optmal soluton of BACAP. Acknowledgements We gratefully acknowledge the support of IBM through a open collaboraton research award #W granted to the frst author. References 1. Berwrth, C. and Mesel, F.: A survey of berth allocaton and quay crane schedulng problems n contaner termnals. European Journal of Operatonal Research 202, (2010) 2. Stahlbock, R. and Voß, S.: Operatons research at contaner termnals: A lterature update. OR Spectrum 30, 1 52 (2008) 3. Türokoğulları, Y.B., Aras, N., Taşkın Z.C., and Altınel, İ.K.: Smultaneous Optmzaton of Berth Allocaton, Quay Crane Assgnment and Quay Crane Schedulng Problems n Contaner Termnals, Research Paper, aras