Study on Berth and Quay-crane Allocation under Stochastic Environments in Container Terminal

Transcription

1 Systems Engineering Theory & Practice Volume 28, Issue 1, January 2008 Online English edition of the Chinese language journal Cite this article as: SETP, 2008, 28(1): Study on Berth and Quay-crane Allocation under Stochastic Environments in Container Terminal ZHOU Peng-fei, KANG Hai-gui State key Lab. of Coastal and Offshore Engineering, Dalian University of Technolgoy, Dalian , China Abstract: Effective berth and quay-crane allocation improves service level of container terminal. For an efficient assignment, stochastic characteristic of containership arrival time and handling time is a key factor. A berth & quay-crane allocation model under stochastic environments is proposed in this article, so as to minimize the average waiting time of containership in terminal. However, it is solved in polynomial CPU time. So to obtain a good solution with considerably small computational effort, a genetic algorithm is developed with a reduced search set on its property. Numerical experiments show that the proposed model is capable of efficiently and dynamically allocating berths and quay-cranes to calling containerships in real stochastic environments and reflects the risk preference of decisionmaker. And the solutions of GA are stable and satisfactory in acceptable CPU time. Key Words: container terminal; berth & quay-crane allocation; stochastic programming; genetic algorithm 1 Introduction Multi-User container Terminal (MUT) concept, which has been employed by many major container hub ports such as Singapore port, HongKong port, etc., could save container operating costs and improve the utilization of terminal resources. Meanwhile, most container terminals in china are managed as the MUT, as the limited terminal space has to be efficiently utilized to meet huge container traffic. One of the issues that affect the efficiency of MUT operations is berth and quay-crane allocation for calling vessels to determine their berthing times and positions as well as loading/discharging cranes. Researches on this issue mainly fall into two categories: one is to employ simulation to evaluate and optimize the assignment strategy of the resources (such as References [1 6]); another is to use mathematical programming to schedule the berths and quay-cranes. This article discusses the resource allocation problem in MUT with a stochastic mathematical programming model. Recently, based on mathematical programming method, some studies related to the berth & quay-crane allocation problems have been performed. Considering Fist Comes Fist Serves (FCFS) allocation strategy, Lai and Shih [7] proposed a heuristics algorithm to assign berths to calling containerships, and evaluated the methods with different criteria (such as ship minimum average waiting time and maximum average berth utilization). Brown, et al. [8,9] treated ship berthing in naval ports. They identified the optimal set of ship-to-berth assignments that maximized the sum of the benefits for ships in terminal. Berth shifting and overwhelmingly assigning for newly arriving ship is important and different from one in commence ports. Imai, et al. [10,11] proposed solution techniques for the static and dynamic berth allocation to minimize ship waiting time. Nishimura, et al. [12] further expended the dynamic vision of the berth allocation problem with different water-depth configurations, and designed a genetic algorithm to solve the model. References [13,14] developed their berth allocation models at various service priorities of containerships and proposed a genetic algorithm to solve the non-linear problem. Considering simultaneous handling in two sides of indented container berth to improve the operating efficiency, Imai, et al. [15] suggested a mixed integrity programming model. Wang, et al. [16] designed a random beam search algorithm to solve berth allocation problems. Li, et al. [17] proposed a hybrid optimization strategy that could increase the population diversity of genetic algorithm and accelerate the convergence process. In the above studies, the entire terminal space is partitioned into several parts (or berths) and the allocation is based on the divided berth space. Another class of the BAP is with a continuous location. Ships are allowed to gain service wherever the empty spaces are available to physically accommodate the ships via a continuous location system. This class of problem resembles more or less the cuttingstock problem. On the basis of the graph theory, Lim [18] transformed the berthing problem to a restricted form of the two-dimensional packing problem, using a graph theoretical representation to capture the problem succinctly, and propose an effective heuristic for the problem. Considering a continuous berth line, Kim, et al. [19] suggested a mixed-integer-linear-programming (MIP) model by simulated annealing algorithm, which is formulated for the berthscheduling problem to determine berth times and positions of vessels. Imai, at al. [20] classified the solution process into two stages: First, discrete berth allocation model was solved Received date: December 1, 2005 Corresponding author: Tel: ; pfzhou@dlut.edu.cn Foundation item: Supported by the National Nature Science Foundation (No ) Copyright c 2008, Systems Engineering Society of China. Published by Elsevier BV. All rights reserved.

2 Figure 1. Procedure for container vessel handling in terminal and then adjusted the discrete solutions to acquire a continuous distribution solution, which is a compromise solution for the continuous berth allocation. As it is hard to solve the continuous berth allocation, few related researches have been reported. In addition, Li, et al. [21] allocate vessels (jobs) to a berth with multiple quay cranes (processors) to minimize vessel handling time in port, where multiple consecutive cranes may process a vessel simultaneously. Similarly, Guan, et al. [22] proposed processor assignment model to minimize the total weighted completion time. Otherwise, Daganzo [23] divided all the loading/unloading container tasks into many operation sections, and use integrity programming model to solve the static quay-crane assignment. Peterkofsky and Daganzo [24] also regarded quay-crane programming as a open-job planning problem, and solved it using branch and bound algorithm. Zeng, et al. [25] proposed a mixed integrity programming model for quay-crane allocation, and a heuristic algorithm was developed based on genetic algorithm. In addition, the flexible container terminal handling operation model [26] also included the optimal scheduling of quay-cranes. In the above-mentioned studies, the instances are all considered in the hard environments. However, uncertainty characters always exist widely and impact decision-making, sometime even definitively. Considering operation fuzzy factors, Zhou, et al. [27] used the fuzzy theory to study container berth allocation and obtained satisfactory solutions. Stochastic factors also prevail in berth allocation. Therefore, with the development of stochastic programming theory and demand for effective resource programming, it is necessary to strengthen the study under stochastic environment. In addition, the previous berth allocation models mostly simplified quay-crane allocation that directly related to berth allocation effect, or separately allocate quay-crane so that not all quay resources are always adequately utilized. Therefore, we develop a novel dynamic berth allocation model to minimize total waiting time of calling vessels, in which arrival times and loading/discharging times of vessels are considered as stochastic parameters and the entire terminal is regarded as a discrete berth set. Dynamic means that vessels can arrive after others have started to be handled. 2 Problem description Nowadays, usually before or when containerships arrive in port, berths and quay-cranes are allocated to them by the terminal dispatcher according to relevant ship and terminal information and scheduling strategy. Figure 1 illustrates a brief process of containership including arrival, berthing, loading, unloading, and departure. Berth & quay-crane allocation refers to allocating an appropriate berth to a calling ship and assigning reasonable quay-cranes to minimize the ship staying time in terminal, to enhance terminal resource utilization and ship line satisfaction, and to reduce their operation costs. Imai, et al. [10] illustrated that for a quick turnaround in MUT system, optimal ship-to-berth assignments should be found without considering the FCFS basis. However, this may result in some ships dissatisfaction with order of service and waiting time. Therefore, authors ignore FCFS to decrease average waiting time of ships in port. Besides, a special acceptable waiting time for every calling ship is introduced to guarantee each ship satisfaction based on stochastic probability. The model of berth & quay-crane allocation is based on the following assumptions: 1) arrival times of ships are stochastic with some certainty distributions because of the random characteristics of weather, machine, and others; 2) the loading/unloading ship time is decided by the berths, number of cranes, and other factors, which are also random variables; 3) different maximum acceptable waiting times are chosen for ships on their different time demands and importance ranks; 4) ships must be serviced at any berth with acceptable physical conditions such as water depth and quay length; 5) Taking no account of ships berth shift, each ship has only one berthing opportunity; 6) The ship length should meet the requirements of assigned quay-crane s operation, namely: the number of allocated quay-cranes is not more than that of quay-cranes allowed by the ship at the same time; 7) When more than one quay-cranes are handling a ship at the same time, inevitably they will mutually interfere with each other, and it will decrease the quay-crane efficiency. By actual statistics, the crane efficiency discount can be treated with the gap between the maximum allowable quay-crane numbers and the actual crane number, namely: the corresponding crane efficiency discounts are 0.9, 0.95, 1.0 corresponding to gap 0, 1, 2 and the more, respectively. 3 Berth & quay-crane allocation model 3.1 Related notation Tv, Tb, Tc The numbers of arrival ships, berths and quay-cranes considered, respectively; V, B, C The set of arrival ships, berths, and quay-

3 cranes considered, V = {1, 2,, Tv }, B={1, 2,, Tb}, C={1, 2,, Tc}; O The ship sequence set according to the ship arriving time, 1 corresponds to the first ship, O={1, 2,, Tv}; Tvh j The loading/unloading amount of ship j, j V ; ξj tv Containership j arrival time, j V : ξj tc Loading/unloading time of ship j, j V ; Ls j Ship j s length (including horizontal safety length), j V ; Lb i Berthi s length, i B; Ds j Ship j s draft (including vertical safety depth), j V ; Db i Berthi s depth of water, i B; mt j Ship j s maximum acceptable waiting time, j V ; BerthingT Ship s berthing and departing time. Decision variable: x ijk 1,ifShipj gets service at the kth service point at berth i; 0, otherwise, i B, j V, k O; Dependent variables: ov ik The loading/unloading ship at berth i while assigning quay-cranes to the kth loading/unloading ship, i B, k O; Completed berthing time of loading/unloading ship at berth i while assigning quay-cranes to the kth loading/unloading ship, i B, k O; vb k The berth allocated for the kth loading/unloading ship, k O; vc k The total number of quay-cranes allocated for the kth loading/unloading ship, k O; ξik tbc ξk tb ξik tbs Loading/unloading time of the kth ship, k O; The starting time of the kth service available at berth i, i B, k O; ξgk tcs The starting time of the gth quay-crane service available while allocating quay-cranes to the kth loading/unloading ship, g C, k O; bc gk The berth at which the gth quay-crane is located while allocating quay-cranes to the kth loading/unloading ship, g C, k O; ξgk tcso The time of the gth quay-crane arrival ship when allocating quay-cranes to the kth loading/unloading ship, g C, k O; ξk tch The starting time of the kth loading/unloading ship, k O; wb j, wc j Ship j s waiting berth time and waiting quay-crane time, j V. 3.2 Dependent variable calculation To facilitate the description, we introduce temporary variables curb i, curv, i B. Step 1 k =1. ξik tbs =0, i B. ξtcs gk =0, g C. initialize bc gk. Step 2 Assume curb i =1,if j V x ij,curb i =0, curb i = curb i +1, ov ik = j V (j x ij,curb i ), ξik tbc = ξov tv ik + BerthingT, i B. Choosing the corresponding smallest berth ξik tbc as vb k. Assume curv = ov vbk,k. Step 3 (Selecting quay-cranes) Assume Lef tcrane, RightCrane. By the current state of quay-cranes (ξik tbs and ) and the principle that quay-cranes cannot cross, decide ξ tcs gk each quay-crane s ξgk tcso. Choosing the quay-cranes whose ξgk tcso is smaller than ξvb tbc k,k. Iftheyexists,ξtch k = ξvb tbc k,k ; Otherwise choose the first quay-crane available, ξk tch = ξgk tcso. Make LeftCrane and RightCrane equal to the minimum and maximum crane number in the chosen quaycrane set, respectively. If the total number of quay-cranes is larger than the number of ones ship curv allows, then Lef tcrane +1and RightCrane 1 until RightCrane LeftCrane +1equals to the number of quay-cranes that the ship allows. Otherwise, continue. Step 4 vc k = RightCrane LeftCrane+1. Decide ξ tb k according to the assumption (7), vc k and Tvh curv. Step 5 if k Tv, then k = k +1,renewξgk tcs and bc gk of quay-cranes, renew ξ tbs ; Otherwise end. ik Step 6 While { j V x ij,curb i =0and curb vbk Tv}, {curb vbk = curb vbk +1}. If curb vbk Tv, Let ξik tbc equals to a large enough number; Otherwise, ov ik = j V (j x ij,curb i ), ξik tbc = max(ξov tv ik,ξik tbs)+ BerthingT, i B. Choose the corresponding smallest berth ξik tbc as vb k. Assume curv = ov vbk,k. Return Step Stochastic 0-1 programming model The dynamic berth & quay-crane allocation model can be described as follow: First, allocate arriving ships to berths by decision variable x ijk, and arrange the berthing sequence for ships; And then, according to current berth state, choose the next handling ship, and according to quay-cranes state and sequential ships state, assign quay-cranes to the current ship. Ship s waiting time contains two parts: the waiting time for berth and the waiting time for quay-cranes. The latter is generally smaller. This problem can be showed by the following model: Obj. min. k O(wb k + wc k ) (1) s.t. x ijk =0 j V (2) i B k O x ijk 1 i B, k O (3) j V P (ξ tbs vb k,k ξ tv ov vbk,k <wb k) α k O (4) P (ξk tch ξvb tbc k,k <wc k ) β k O (5) (x ijk Db i ) Ds j j V (6) i B k O (x ijk Lb i ) Ls j j V (7) i B k O wb k + wc k mt ovvbk,k k O (8) wb k 0, wc k 0 k O (9) x ijk {0, 1} i B, j V, k O (10) The model is a 0-1 nonlinear programming with stochastic coefficients. The objective (1) minimizes the sum of ships waiting time (including waiting berth time wb k and waiting quay-crane time wc k ). Equation (2) states that each

4 calling ship must be served only once in any service order at berth. Constraint (3) ensures that not more than one ship is served in any service order at berth. Constraint (4) and (5) indicate that the probabilities of minimum waiting time of each ship (wb k and wc k ) are not less than α and β, respectively. Constraint (6) and (7) require that the allocated berth should meet the physical conditions of ships (water depth and length). Constraint (8) states that each ship s waiting time should be smaller than the maximum acceptable waiting time (mt ovvbk ). Constraint (9) requires waiting time,k for ships to be non-negative. Constraint (10) says decision variable x ijk is binary. 3.4 Stochastic variable treatment According to statistics, the deviations of actual arriving time and loading/discharging time of ships can be regarded as normal distribution. Then the arriving time and loading/discharging time of ships can be expressed by ξ(= a + ξ d ), is a planned or estimated time (real number), and represents the deviation (Normal distributing stochastic variables which expectation values are 0). The stochastic variables in the model include arrival time of ship (ξj tv ), loading/discharging time of container (ξj tc ) and their dependent variables. The stochastic variables mainly involve in addition calculation and comparison. Two stochastic variables with normal distribution are denoted as ξ 1 (μ 1,σ1) 2 and ξ 2 (μ 2,σ2), 2 respectively. Then ξ 1 +ξ 2 = ξ(μ 1 +μ 2,σ1 2 +σ2). 2 It is supposed that ξ 1 >ξ 2 if and only if μ 1 >μ 2. 4 Solution based on genetic algorithm 4.1 Representation of individuals The symbolic string (genome) for a solution consists of two sub-strings (sub-genomes). One of these determines the berths assigned to calling ships by the order of their arrival. Another shows service orders for each ship at each berth. Before encoding individuals, calling ships are ordered by their arrival time, and an Order Limit Number (OLN) is selected to construct service orders at berths, and berths should be identified with their indices. Then genomes can be formed. In Figure 2, we propose only a part of the solution with four berths and OLN (=3). The symbols in substring 1 are the identifications of berth No, ones in sub-string 2reflect service order at berths, and the total of the symbols in both sub-string 1 and 2 is equal to the calling ship number. Under ship 1, the symbol 1 in sub-string 1 and 1 in sub-string 2 show together that berth 1 serves ship 1 and the first place of service order at berth 1 is assigned to ship 1. Under ship 5, 1 in substring 1 and 3 in sub-string 2 say that ship 5 is also served at berth 1 and the third place of the remainder of service order at berth 1, that is, ship 5 is served at the forth service location of berth 1. The service order at berth 1 is ship 1, 8, 11, 5, 15 and so on. Thereby considering constraint set (2), (3), and (10), if OLN is appropriate, each feasible solution in the feasible solution set can be mapped onto an encoded solution, and each encoded solution can be mapped onto a feasible solution. Generally, to avoid long waiting time of ships, OLN is provided with a small number such as 3. In this manner, we can get a reduced searching space, which is showed in the next section. By this coding method, we can get optimal solution to constraints Figure 2. Illustration of feasible solution encoding (2), (3), and (10). Constraints (6) and (7) are considered in population initialization and mutation operation, and constraints (4), (5), (8), and (9) will be reflected in the fitness value. 4.2 Discussion of the searching space Theorem 4.1 Considering constraint (2), (3), and (10), the size of feasible search space of decision variables is PTb Tv Tv, in which Tv and Tbrepresents the number of ships and berths, respectively. Proof. If we consider only the constraints (2), (3), and (10), the solving of model is similar to the distribution of Tv balls to Tb Tv different boxes. Apparently, the size of the search space is PTb Tv Tv. Theorem 4.2 The size of the search space in proposed genetic algorithm is (Tb OLN), Tv, Tb, and OLN represents the number of ships, berths, and the most delay service times, respectively. Proof. The size of the search space in sub-string 1 is Tb Tb Tb, the size of the search space }{{} in sub-string 2 is OLN OLN OLN. There- }{{} fore, the size of the search space of the genetic algorithm is Tb Tb Tb OLN OLN OLN = }{{}}{{} (Tb OLN) Tv. Usually OLN Tv,andthen(Tb OLN) Tv PTb Tv Tv. 4.3 Initialization Because of large search space, population is initialized by randomly selecting berths available with uniform distribution considering (6) and (7), and by randomly choosing order numbers from {1, 2,, OLN} with non-uniform distribution. As little order number maybe generally related to little waiting time, it is assigned to have large frequency. In detail, Substring 1 is formed by choosing berths continuously and randomly from berths available sets; Sub-string 2 is constructed by randomly selecting order numbers from {1, 2,, OLN} with given distributions. 4.4 Crossover and mutation operation A modified operation is proposed based on partially match crossover, which allows the crossover between parallel sections in two parents. Figure 3 illustrates how the crossover operation works. First, two cut points are randomly chosen, and two sub-genomes are equally parted into two groups. Then, genes parted by two cut point and groups are exchanged so that each of the two genomes possesses new partial genetic information from the opposite to produce two new genomes (such as: O1 and O2 in Figure 3). The

5 Figure 3. Illustration of crossover operation Figure 4. Illustration of a mutation new genome O1 possesses sideward genetic information of P1 and the middle of P2. Thus, two new offsprings are obtained. As both P1 and P2 meet (6) and (7), the two new offsprings from them would fulfill the constraints. The mutation objective is to disrupt current chromosome slight by inserting a new gene. In this research, we use the following mutation based on displacement mutation: we randomly select a position and then changes genes just as initialization at random, which must fulfill (6) and (7) (Figure 4). 4.5 Individual evaluation In the processes of individuals coding, initializing, crossover, and mutation operation, all constraints have been considered to accept for constraint (8) (constraints 4, 5 and 9 are used to calculate the objective value). So the individual fitness value is composed of the objective function value and the sufficient punishment value if the individual cannot meet constraint (8). 4.6 Realization of genetic algorithm The process of genetic algorithm is illustrated in Figure 5. To adapt them to fitness landscape by the objective function, the algorithm works on the principle of evolving a population of trial solution, over iterations. 5 Computational experiments and analysis 5.1 Test problems Considering the distribution of containership s arrival time, loading/unloading quay-crane technical parameters, samples of a container terminal with 4 berths and 12 quaycranes are generated systemically to evaluate the proposed model and algorithm. The arrival times of ships are calculated by forming their expected value from an exponential interval distribution with an expected value (λ) and selecting randomly their variance from {3 2, 4 2,, 9 2 }. Three types Figure 5. Procedure of generic algorithm of ships (I, II, III) are generated randomly with uniform distribution, whose draft and length are based on their ship type. The expected loading/unloading quantity of each ship is stochastically selected based on its type (I: ; II: ; III: ). The productivity of Quay-crane depends on crane type, the weather, and other factors. And the consuming time of loading/discharging a container is chosen randomly within 2 3 minutes. A maximum acceptable waiting time for each ship is generated within 3 10 hours at random. Twelve groups of samples are introduced with different λ values (3, 4 and 5 hours) and calling ships (25, 50, 75 to 100). For each group, 4 sample problems are generated randomly with different seed sets. The berths available and maximum allowable quay-cranes for I, II, and III types of ships are 0-2 # berth and 5 quay-cranes; 0-3 # berth and 4 quay-cranes; 0-4 # berth and 3 quay-cranes, respectively.

6 Figure 6. GA consumed CPU time with different vessels Figure 7. Convergence of GA 5.2 Experimental reports The following experiments are carried out on a Pentium IV (2.4 GHz) PC. By experiments, it is found that population size (P ) 100 and generation (iteration) number 2500 make a relatively stabilized objective value. The objective value is low for the same iteration when pc =0.8 and pm =0.2 (crossover probability and mutation one). So they are adopted in the following experiments. First, we examine the relationship between the calling ship number and consuming CPU time. Apparently, the string length of a genome equals to the twofold of calling vessel number. Figure 6 shows the average consuming CPU times of 4 groups of samples with different vessel numbers. The CPU time rises with the increase of total calling ship number. And the maximum time is not more than 10 minutes. Therefore, they can be acceptable completely in practice. Then we study the convergence of the genetic algorithm. Figure 7 reveals the changes in the 2 groups of average values of the total waiting times with increasing generation. Solution quality could improve a little after 1000 generation. Further, it tends to converge when the iteration exceeds To evaluate the proposed Genetic Algorithm (GA), we design a GReedy Algorithm (GRA, step-by-step optimal algorithm) that allocates the earliest berth available to the earliest coming ship and distributes most quay-cranes with restrictive operation conditions. Generally, this allocation strategy is even adopted in terminals. Table 1 gives a comparison of 6 example problems among 48 ones between GA and GRA. The objective improvement of GA from GRA is generally more than 25%. These findings are confirmed in several other experiments based on different parameter settings. Finally, the pole of probability α and β is analyzed. When α and β value equals to 0.5, constraint (4) and (5) reveal that an average value of ship waiting time is not larger than its maximum acceptable waiting time. Figure 8 illustrates the average total waiting time of several examples with different α values. On one hand, increasing α value makes the total waiting times rise. On the other hand, it leads to a significant improvement of probability in constraint (4), that is, it makes more satisfaction of calling ship than ever before. 6 Conclusions Figure 8. Illustration of effects of α values Allocating berth and quay-cranes in container terminal is related to many dynamic stochastic factors such as ship arrival time, loading/unloading time, and so on, which impact Table 1. Comparison of average waiting times by genetic algorithm and greedy algorithm (/hours) CPU time 10 mins α=0.9 P 1(3) P 2(3) P 3(3) P 4(4) P 5(4) P 6(4) Greedy algorithm 50 ships ships Genetic algorithm (G=2000) 50 ships ships Algorithm improvement by Greedy algorithm 42.2% 47.9% 41.3% 31.4% 27.1% 35.4% * P1 represents example 1; 3 represents the expected value of ship arrival time

7 the effective solution to some extent. On the basis of stochastic programming, this article proposed a berth & quay-crane allocation model in container terminal. And a genetic algorithm with reduced search space is developed based on the characteristics of the optimal solution. The experiments reveal that the stochastic programming model can effectively treat the related random factors and reflect the risk preference of decision-maker. According to different risk, decision-makers can take necessary measures to potential adverse circumstances. In acceptable time, the proposed genetic algorithm can obtain satisfactory solutions, which is significantly improved by greedy algorithm. This article has considered the random factors in the problem, but other uncertainty factors such as fuzziness as well as their combination exist widely in actual decisionmaking process. A further research of these factors will be helpful to obtain reasonable and reliable allocation solution. In addition, terminal is a continuous and cooperative logistic operation system, the berths and quay-cranes assignment are influenced by the dispatch of other terminal resources including yard space and vehicles. Therefore, the cooperation research of all types of resources in container terminal is an important tread to improve the overall performance of terminal system. References [1] Legato P, Mazza R M. Berth planning and resources optimization at a container terminal via discrete event simulation. European Journal of Operational Research, 2001, 133(3): [2] Shabayek A A, Yeung W W. A simulation model for the Kwai Chung container terminals in Hong Kong. European Journal of Operational Research, 2002, 140: [3] Bielli M, Boulmakoul A, Rida M. Object oriented model for container terminal distributed simulation. European Journal of Operational Research, 2006, 175: [4] Zhang H, Jiang Z, Xu H. A simulation study of container terminal scheduling system. Journal of Shanghai Jiaotong University, 2006, 40(6): [5] Han X, Ding Y. Simulation system of container terminal charge/discharge operations. Journal of System Simulation, 2006, 18(8): [6] Cai Y, Zhang Y. Simulation optimization for berth allocation in container terminals. Chinese Journal of Construction Machinery, 2006, 4(2): [7] Lai K K, Shih K. A study of container berth allocation. Journal of Advanced Transportation, 1992, 26(1): [8] Brown G G, Lawphongpanich S, Thurman K P. Optimizing ship berthing. Naval Research Logistics, 1994, 41: [9] Brown G G, Cormican K J, Lawphongpanich S, Widdis D B. Optimizing submarine berthing with a persistence incentive, Naval Research Logistics, 1997, 44: [10] Imai A, Nagaiwa K, Tat C W. Efficient planning of berth allocation for container terminals in Asia. Journal of Advanced Transportation, 1997, 31(1): [11] Imai A, Nishimura E, Papadimitriou S. The dynamic berth allocation problem for a container port. Transportation Research Part B, 2001, 35: [12] Nishimura E, Imai A, Papadimitriou S. Berth allocation planning in the public berth system by genetic algorithms. European Journal of Operational Research, 2001, 131: [13] Imai A, Nishimura E, Papadimitriou S. Berth allocation with service priority. Transportation Research Part B, 2003, 37: [14] Zhou P, Wang K, Kang H. Study on berth allocation problem with variable service priority in multi-user container terminal. In: Proc. Sixteenth International Offshore and Polar Engineering Conference, San Francisco, CA, 2006: [15] Imai A, Nishimura E, Masahiro H. Berth allocation at indented berths for mega-containerships. European Journal of Operational Research, 2007, 179(2): [16] Wang F, Lim A. A stochastic beam search for the berth allocation problem. Decision Support Systems, 2007, 42(4): [17] Li P, Sun J, Han M. The algorithm for the berth scheduling problem by the hybrid optimization strategy GATS. Journal of Tianjin University of Technology, 2006, 22(4): [18] Lim A. The berth scheduling problem. Operations Research Letters, 1998, 22: [19] Kim K H, Moon K C. Berth scheduling by simulated annealing. Transportation Research Part B, 2003, 37: [20] Imai A, Sun X, Papadimitriou S. Berth allocation in a container port: using a continuous location space approach. Transportation Research Part B, 2005, 39: [21] Li C L, Cai X, Lee C Y. Scheduling with multiple-job-onone-processor pattern. IIE Transactions, 1998, 30: [22] Guan Y P, Xiao W Q, Cheung R K, Li C L. A multiprocessor task scheduling model for berth allocation: Heuristic and worst-case analysis. Operations Research Letters, 2002, 0: [23] Daganzo C F. The crane scheduling problem. Transportation Research B, 1989, 23 (3): [24] Peterkofsky R I, Daganzo C F. A branch and bound solution method for the crane scheduling problem. Transportation Research B, 1990, 4(3): [25] Zeng Q, Gao Y. Model and Algorithm for Quay Crane Scheduling in Container Terminals. Computer Engineering & Applications, 2006, 32: [26] Chen L, Xi L, Cai J. Integrated model and heuristic algorithm for the scheduling of container handling system in a maritime terminal. Control heory & Applications, 2006, 23(6): [27] Zhou P, Guo Z, Song X. A fuzzy vehicle-crane scheduling problem in container terminal. In: Proc. the International Conference on Sensing, computing and Automation, Chongqing, China, 2006: