Optimization of loading sequence and rehandling strategy for multi-quay crane operations in container terminals

Transportation Research Part E 80 (2015) 1–19

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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

Optimization of loading sequence and rehandling strategy for multi-quay crane operations in container terminals Mingjun Ji ⇑, Wenwen Guo, Huiling Zhu, Yongzhi Yang Transportation Management College, Dalian Maritime University, Dalian 116026, China

a r t i c l e

i n f o

Article history: Received 10 November 2014 Received in revised form 8 April 2015 Accepted 6 May 2015

Keywords: Container Loading sequence Rehandling strategy Genetic algorithm

a b s t r a c t In this paper, we consider the optimization of loading sequence and rehandling strategy in the terminal operation. We present an optimization strategy to minimize the number of rehandles, and establish a mathematical model to integrate the loading sequence and the rehandling strategy under the parallel operation of multi-quay cranes. Furthermore, we give an improved genetic algorithm to solve the model. We show the efficiency of the optimization strategy and algorithm by comparing them with previous strategies and heuristics. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Container terminals play an important role in global manufacturing and international business by serving as multi-modal interfaces, usually between sea and land transport (Zhang et al., 2002). Terminal yard is one of the most important constituent part of container terminals. During the loading process, containers will be picked up from the yard to the vessel considering the stowage plan and yard stacking status. The stowage plan, illustrated in Fig. 1(a), shows a side view of the ship-bays, where the ship-bay is described with the two-dimensional coordinates of stack and tier. The yard stacking status, illustrated in Fig. 1(b), are described by the three-dimensional coordinates of bay, stack and tier. A bay consists of multiple stacks and a stack consists of multiple tiers. Since the containers are stacked in the vertical direction, rehandling must be performed for retrieving a container, called as a target container, which is not on the top tier. Furthermore, moving an obstructed container to another stack may lead to additional rehandling moves of containers during the retrieval process (Sauri and Martin, 2011). The obstructed container may be moved to the nearest, lowest or optimization stack in the same yard bay according to the nearest stack, lowest stack and optimization strategy. Different strategies result in different additional rehandling. Thus, there are two factors affecting the rehandling operation, one is the rehandling strategy (Petering and Hussein, 2013), and another is the loading sequence. Rehandling is time-consuming, because it is an unproductive move in container terminals. The number of rehandles is one of the most important factors affecting the operational efficiency in container terminals, and it is correlated with the loading sequence and the rehandling strategy. To improve the operational efficiency of container terminals, ports should focus on reducing the number of rehandles in the yard, including additional rehandling.

⇑ Corresponding author. E-mail addresses: [email protected] (M. Ji), [email protected] (W. Guo), [email protected] (H. Zhu), [email protected] (Y. Yang). http://dx.doi.org/10.1016/j.tre.2015.05.004 1366-5545/Ó 2015 Elsevier Ltd. All rights reserved.

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M. Ji et al. / Transportation Research Part E 80 (2015) 1–19

10 08 06 04 02 01 03 05 07 09

08 06 04 02 01 03 05 07

Fig. 1. Ship-bays and the corresponding yard stacking areas.

In this paper, we study the multi-quay crane parallel operations during the loading process without considering the number of vessels. Based on the given containership stowage plan and yard storage characteristics, we develop a mathematical model to minimize the number of rehandles. Moreover, we design a genetic algorithm integrated loading sequence and rehandling strategy (GA-ILSRS) for the lowest stack, the nearest stack and the optimization strategy. Finally, we verify the robustness of the algorithm compared with other heuristic. This paper is organized as follows: Section 2 reviews the literature. A detailed description of the problem is given in Section 3. Section 4 presents mathematical models for different strategies. Section 5 outlines the design of GA-ILSRS to solve the model. Section 6 verifies the effectiveness of the GA-ILSRS compared with other heuristics, and shows the results for three rehandling strategies and different quay cranes. Additionally, the t-test and lower bound are showed. Lastly, this paper presents final conclusions.

2. Literature review The rehandling problem is usually related to the stowage plan for container ships and yards. Some literatures considered rehandling problem in the perspective of the containership stowage plan or the yard storage characteristic. They focused on the rehandling strategy under a certain loading sequence without regarding for the effect of loading sequence on rehandling. Kim (1997) proposed an algorithm to calculate the expected number of unproductive moves, where the height and width of a bay in the container stack were important decision variables in designing the storage configuration. Differing from the previous research by Kim (1997), Avriel et al. (1998) presented an integer linear formulation to find the optimal solution for stowage in a single bay of a ship, where containers’ origin and destination ports are known in advance. Similar to the previous work by Kim (1997), Kim et al. (2000), Kim and Hong (2006) and Imai et al. (2006) developed mathematical models and solution methods to determine the overall amount of rehandles. Later, Sauri and Martin (2011) described three stacking strategies, which take into account the containers’ arrival and departure rates and the storage yard characteristics, and developed a model to estimate the number of rehandles. Woo and Kim (2011) introduced a method for allocating storage space to groups of outbound containers, and discussed the impacts of various space-reservation strategies on the productivity of the loading operation for outbound containers. Zhu et al. (2012) investigated iterative deepening A⁄ algorithms using new lower bound measures and heuristics, and examined a more difficult variant of the problem that had previously been ignored in the literature. Lee et al. (2013) discussed a novel approach that integrated yard truck scheduling and storage allocation, and designed a hybrid insertion algorithm for the problem. Regarding the container yard operations, the rehandling process was usually described as a two-dimensional problem concentrating on the research of yard-stacks and tiers within a yard-bay. Based on some assumptions taken from Kim and Hong (2006), Caserta et al. (2011) described a dynamic programming formulation and determined a two-dimensional corridor with width and height parameters. The block relocation problem (BRP) is encountered in the maritime container shipping industry and other industries where containers are stored in stacks. Some papers discussed the BRP from a single yard-bay. After surveying the work done on the BRP, Petering and Hussein (2013) considered the containers in a single yard-bay, introduced a new mathematical formulation for the BRP, and expressed a new look-ahead algorithm to obtain superior solutions for the problem. Jin et al. (2014) extended the previous work (Jin et al., 2013), and presented an improved greedy look-ahead heuristic to find an optimized operation plan for the crane with the fewest number of container relocations. Moreover, considering the automated stacking problem, Gharehgozli et al. (2014) studied a yard crane scheduling problem to carry out a set of container storage and retrieval requests in a single container block.

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The existing approaches to rectify rehandling inefficiencies considered different restricted variants of the block relocation problem. The restricted variant was studied by Caserta et al. (2009), who used a dynamic programming formulation with an exponential number of states. Wan et al. (2009) introduced an integer programming model with one extra restriction that each container above the target can be relocated exactly once during any retrieval. In addition, Several variants of the rehandling problem, with different assumptions made about the containers, have been investigated by Zhang et al. (2010), Kang et al. (2006) and Park and Kim (2010). Caserta et al. (2012) developed a simple heuristic based upon a set of relocation rules, and made realistic assumptions to solve the relocation problem. Besides minimizing the number of relocations, other ways to model the costs of container retrieval operations have been considered. For example, Zhu et al. (2010) measured the distance of traverse travel and hoist moves. Lee and Lee (2010) allowed the relocation of containers across bays. Jang et al. (2013) addressed the optimization of a block stacking storage system and showed how the number of relocations can be reduced. The algorithms for solving the relocation problems are diverse, and the genetic algorithm is usually adopted to compute the experiments. Bazzazia et al. (2009) presented an efficient genetic algorithm to solve an extended storage space allocation problem in a container terminal. He et al. (2010) employed heuristic rules and parallel genetic algorithm to resolve the yard crane scheduling problem. At automated container terminals, genetic algorithm is also employed to improve container handling operations. Skinner et al. (2013) focused on scheduling container transfers and solved the problem using a genetic algorithm based on the optimization approach. Homayouni et al. (2014) integrated the scheduling of quay cranes, the automated guided vehicles and the handling platforms, and proposed a genetic algorithm to solve the problem. In contrast to the previous researches, this paper investigates the optimization of rehandling strategies. Considering the stowage plan for ships and yards, we propose a mathematical model to integrate the loading sequence and the rehandling strategy. Moreover, the two-dimensional rehandling problem is extended to a three-dimensional problem including yard-bays, stacks and tiers, which is more challenging.

3. Problem description A major goal of containership’s operation is to finish the loading process efficiently. Dispatchers usually arrange the loading sequence according to personal experience during practical operations. For the description of the container yard, we assume that the stacking status of the containers in the container yard is known and that the container yard receives the ship’s stowage plan. In addition, in order to ensure operational safety, containers with the same size in the yard are allowed to be stacked within a bay, and assumed that every obstructed container is moved to a different slot in the same bay. Based on these assumptions, we focus on the parallel operation of multi-quay cranes, aiming to determine a reasonable loading sequence to minimize the number of rehandles. We introduce the problem by an example of double quay cranes for three rehandling strategies. Most container terminals adopt the lowest or the nearest stack strategy, which only takes into account the rehandling location. In this paper, we present the optimization strategy considering both the loading sequence and the rehandling location to decrease the number of rehandles. To facilitate the description of the problem, we define the parameters related to the rehandling problem. m represents the number of containers on ship-bay A, and n represents the number of containers on ship-bay B. Am and Bn show the serial number of container on ship-bay A and B, respectively. Coordinate ði; j; kÞ defines the container on yard-bay i, yard-stack j and yard-tier k. Coordinate ðo; p; qÞ represents the position in the ship for container ði; j; kÞ, which is on ship-bay o, ship-stack p and ship-tier q. We denote the loading container for the s-th time in the loading sequence by ls . r represents the serial number of the quay crane, and the set of all quay cranes is denoted by X ¼ f1; . . . ; r; . . . ; Rg. Since the quay cranes operations are parallel, each quay crane serves a single ship-bay. In other words, quay cranes can be used to express l

s ship-bays. In addition, hijs expresses the height of the yard-bay i and yard-stack j before loading target container ls . xlrijk equals

to 1 if the container ði; j; kÞ is loaded onto the ship-bay in the s-th time by quay crane r, and equals to 0 otherwise. For ease of analysis, this work assigns the values of m and n as 9, and the value of r is 2 for ship-bays A and B. Bays on the ship and the corresponding stacking status are simplified as shown in Fig. 2, where (a) and (b) represent the container

A2 A5

A6

A7

A8

A2

A3

A4

A9

B5

B6

B7

B8

B2

B3

B4

A1

B1

(a) ship-bay A

(b) ship-bay B

B9

B9

A1

B1

B3

B8

A8

B2 A7

B4

A6

A4

A3

A5

B5

A9

B6

B7

(c) corresponding yard stacking status

Fig. 2. Ship stowage plan and yard storage characteristics.

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stowage on two nonadjacent bays (A and B), and (c) represents the corresponding stacking status. Next, the major focus of the paper is to determine a reasonable loading sequence. We assume that quay crane A serves ship-bay A, quay crane B serves ship-bay B, and quay cranes A; B have the same efficiency of loading operations. Fig. 2 shows that the yard may produce numerous loading sequences based on the known stowage plan, which lead to different numbers of rehandles. For example, bay A may be loaded according to sequences such as A1 ¼ ðA1; A2; A3; A4; A5; A6; A7; A8; A9Þ, and A2 ¼ ðA1; A2; A5; A6; A3; A7; A4; A8; A9Þ. Similarly, bay B also have different loading sequences, for instance B1 ¼ ðB1; B2; B3; B4; B5; B6; B7; B8; B9Þ or B2 ¼ ðB1; B2; B6; B5; B3; B4; B7; B8; B9Þ. So, it is necessary to ascertain an efficient loading sequence to minimize the number of rehandles. During the loading process, rehandling will occur when the target container is not on the top of a stack. Containers above the target one are named the obstructed containers. The location of obstructed containers can affect the next loading of container in the same or different ship-bay. Moving the obstructed container to an unreasonable stack may lead to additional rehandling. In order to minimize the number of rehandles, three strategies are presented as follows. 3.1. Nearest stack strategy A certain loading sequence is assumed, where bay A is loaded according to A1 ¼ ðA1; A2; A3; A4; A5; A6; A7; A8; A9Þ, and bay B is loaded according to B1 ¼ ðB1; B2; B3; B4; B5; B6; B7; B8; B9Þ. Due to the same efficiency of double quay cranes, the loading sequence is as follows. First, container A1 is loaded, which is on the top of the stack and cannot incur any rehandling. Next, container B1 is loaded, which results in the rehandling operation of the obstructed container A2, and the height of the target l

s ¼ 3  1 ¼ 2 . The number of obstructed containers is calculated by stack for container B1 is calculated by hijs ¼ k  xlrijk   ls h P P P ls ij xls ¼ 4  3  0 ¼ 1. According to the nearest stack strategy, we should move A2 to the r hij  k  r 0 2Xnfrg k0 ¼kþ1 r 0 ijk0

nearest stack, and the yard stacking status becomes shown in Fig. 3. Meanwhile, the height of the located stack is the sum of its original height and the obstructed containers, which is calculated by   P P Phlijs ls ls 1 ls xls ¼ 2 þ ð4  3  0Þ ¼ 3. If the nearest stack is also the lowest stack, the hiðj1Þ ¼ hiðj1Þ þ r hij  k  r0 2Xnfrg k0 ¼kþ1 r 0 ijk0 obstructed container is relocated to the lowest stack. According to the equation for calculating the number of rehandles,   P P PPP P Phlijs l s xls . For this strategy and loading sequence, the number of rehandles  xlrijk Z ¼ ls k j i r hijs  k  r0 2Xnfrg k0 ¼kþ1 r 0 ijk0 is 6. If the loading sequence changes to A2 ¼ ðA1; A2; A5; A6; A3; A7; A4; A8; A9Þ for bay A and B2 ¼ ðB1; B2; B5; B6; B3; B7; B4; B8; B9Þ for bay B, the number of rehandles for this strategy and loading sequence is 10. This result shows that different loading sequences can lead to different numbers of rehandles. 3.2. Lowest stack strategy A certain loading sequence is assumed, that ship-bay A must be loaded according to A1 and B1 for ship-bay B, in accordance with the nearest stack strategy. Container A1 is loaded, which is on the top of the stack, and cannot incur rehandling. While container B1 is loaded, there exists the obstructed container A2, and the height of the target stack for container B1 is l

s ¼31¼2 . The number of obstructed containers is calculated by calculated by hijs ¼ k  xlrijk   ls P Phij P ls ls ¼ 4  3  0 ¼ 1. According to the lowest stack strategy, container A2 is moved to the x r hij  k  r 0 2Xnfrg k0 ¼kþ1 r 0 ijk0

l

l

l

l

lowest stack J. The height of the lowest stack J is obtained by hiJs1 ¼ minfhi1s1 ; hi2s1 ; . . . ; hijs1 ; . . .g ¼ minf2; 3; 2; 3; 4g ¼ 2 ,   P P Phls l l l and changes by hiJs ¼ hiJs1 þ r hijs  k  r0 2Xnfrg k0ij¼kþ1 xlrs0 ijk0 ¼ 2 þ ð4  3  0Þ ¼ 3 . Moreover, the yard stacking status is shown in Fig. 4. If the lowest stack is also the nearest stack, the height of located stack changes on the basis of the nearest stack. The number of rehandles for this strategy and loading sequence is 6.

B9 A2

B1

B3

B8

A8

B2 A7

B4

A6

A4

A3

A5

B5

A9

B6

B7

Fig. 3. Yard stacking status based on the nearest stack strategy.

M. Ji et al. / Transportation Research Part E 80 (2015) 1–19

5

B9

B8

B1

B3

A2

A6

A4

A8

B2

A3

A5

B5

A7

A9

B6

B7

B4

Fig. 4. Yard stacking status based on the lowest stack strategy.

B8 B4

A8 A7

A6

A4

B9

A5

B5

A9

B6

B7

Fig. 5. Yard stacking status based on the optimization strategy.

If the loading sequence is changed to A2 and B2 , as previously given, the number of rehandles for the lowest stack strategy is 8. 3.3. Optimization strategy A certain loading sequence is assumed, in accordance with the above strategies. The optimization strategy is to select the optimal stack for an obstructed container. A selectable stack is checked to see whether it has the next loading container. If the stack does not contain the next loading container, it is selected and its height changes according to   P P Phls l l l s hiJs ¼ hiJs1 þ r hijs  k  r0 2Xnfrg k0ij¼kþ1 xlrs0 ijk0  xlrijk . Otherwise, the stack will not be selected as it may cause additional rehandling. As a result, this obstructed container is moved to the lowest stack J whose origin height is l

l

l

l

. Then, the height of the located stack J changes to hiJs1 ¼ minfhi1s1 ; hi2s1 ; . . . ; hijs1 ; . . .g   ls h P P P l l l s . In this example, we directly load the container A1; A2; A3 and hiJs ¼ hiJs1 þ r hijs  k  r0 2Xnfrg k0ij¼kþ1 xlrs0 ijk0  xlrijk B1; B2; B3 without obstructed containers. When loading the container A4, there exists an obstructed container B9. If moving the obstructed container B9 above the container A7, it leads to secondary rehandling. It should be moved above the container A9, based on the optimization strategy. The status of the yard in this case is shown in Fig. 5. According to the optimization strategy and the loading sequence, the number of rehandles is 5. If the loading sequence changes to A2 and B2 , the number of rehandles is 7. By analysis of the simultaneous operation of double quay cranes, a variety of loading sequences can lead to different numbers of rehandles, and the selected location of an obstructed container may also lead to different numbers of rehandles. Thus we can reduce the number of rehandles by integrating the loading sequence and the rehandling strategy. 4. Modeling 4.1. Model assumptions The model is established using the following assumptions: (1) (2) (3) (4)

This paper studies the ship-bays and yard-bays without considering the number of vessels. The stowage plan of the ship and the status of the container yard are known. The obstructed containers are moved to different stacks of the same bay, which have enough rehandling locations. The number of trucks is not considered to be a restrictive factor, and the loading process will not be delayed due to waiting for a truck.

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4.2. Model symbols and decision variables

Notations i Bay of containers in the yard i0 A second bay of containers in the yard I The total number of yard bays j Stack of containers in the yard j0 A second stack of containers in the yard J Located stack of rehandling containers in the yard k Tier of containers in the yard 0 k A second tier of containers in the yard o Bay of containers in the ship corresponding to the location of ði; j; kÞ in the yard 0 o0 Bay of containers in the ship corresponding to the location of ði0 ; j0 ; k Þ in the yard p Stack of containers in the ship corresponding to the location of ði; j; kÞ in the yard 0 p0 Stack of containers in the ship corresponding to the location of ði0 ; j0 ; k Þ in the yard q Tier of containers in the ship corresponding to the location of ði; j; kÞ in the yard 0 q0 Tier of containers in the ship corresponding to the location of ði0 ; j0 ; k Þ in the yard N The total number of containers l The loading sequence in the yard, l ¼ fl1 ; l2 ; . . . ; ls ; . . . ; lN g s The loading number of containers in the loading sequence l s0 The second loading number of containers in the loading sequence l ls The loading container for the s-th time in the loading sequence r The serial number of quay crane X The set of all quay cranes, X ¼ f1; . . . ; r; . . . ; Rg r0 The second serial number of quay crane other than quay crane r; r0 2 X n frg M ij The highest tier of stockpiling in bay i and stack j Decision variable s xlrijk Equals to 1 if the container ði; j; kÞ is loaded onto the ship-bay in the s-th time by quay crane r, and 0 otherwise Dependent variable l hijs The height of the yard-bay i and yard-stack j before loading target container ls

4.3. Model establishment In this section, we establish mathematical models to integrate the loading sequence and the rehandling strategy for multi-quay crane operations based on the lowest stack, the nearest stack and the optimization strategy. The objective function of the models aims to minimize the number of rehandles considering both loading sequence and rehandling location. According to the models, we randomly generate a loading sequence and calculate the number of rehandles by P Phls l hijs  k  r0 2Xnfrg k0ij¼kþ1 xlrs0 ijk0 while the target container ði; j; kÞ is loaded. Then, select the located stack J for three rehandling l

strategies. And on this basis, we update the dependent variable hijs , which is the height of target stack or located stack for three rehandling strategies. Since all containers are loaded, we obtain the total number of rehandles. Meanwhile, we generate another loading sequence again and do the same calculation. Finally, we obtain the optimal loading sequence to minimize the number of rehandles. The detailed descriptions of the mathematical models are shown as follows:

0 X X X X XB l X min Z ¼ @hijs  k  ls

k

j

i

r

h ls

ij X

r0 2Xnfrg k0 ¼kþ1

1

C s xlrs0 ijk0 A  xlrijk

ð1Þ

In which: The height of the target stack for loaded container ði; j; kÞ is calculated by Eq. (2).

  l l s s hijs ¼ k  xlrijk þ 1  xlrijk  hijs

ð2Þ

The height of the located stack J for loaded container ði; j; kÞ is calculated by Eqs. (3)–(5) under three rehandling strategies. During the computational process, we firstly determine the located stack J for three strategies. For the lowest stack strategy, J is the lowest stack in the yard-bay i.

M. Ji et al. / Transportation Research Part E 80 (2015) 1–19

7

n o l l l l hiJs1 ¼ min hi1s1 ; hi2s1 ; . . . ; hijs1 ; . . .

l

l

hiJs ¼ hiJs1

0 XB l X þ @hijs  k 

r0 2Xnfrg k0 ¼kþ1

r

1

h ls

ij X

C s xlrs0 ijk0 A  xlrijk

ð3Þ

For the nearest stack strategy, J is the neighboring stack of the yard stack j.

J ¼ j þ 1 or J ¼ j  1 When i ¼ 1; J is on the right side of the target stack j.

0 X X l ls1 B ls hiJs ¼ hiðjþ1Þ þ @hij  k  r

1

hls

ij X

C s xlrs0 ijk0 A  xlrijk

0

r 0 2Xnfrg

k ¼kþ1

When i ¼ I; I expresses the last stack. J is on the left side of the target stack j.

0 X X B ls l ls1 hiJs ¼ hiðj1Þ þ @hij  k 

1

hls

ij X

r 0 2Xnfrg k0 ¼kþ1

r

C s xlrs0 ijk0 A  xlrijk

When 1 < i < I; J is on the left or right side of the target stack j l hiJs

¼

ls1 hiðjþ1Þ

0 XB l X þ @hijs  k  r

1

hls

ij X

C xlrs0 ijk0 A

0

r 0 2Xnfrg

s  xlrijk

k ¼kþ1

Or

0 X X B ls l ls1 hiJs ¼ hiðj1Þ þ @hij  k 

1

hls

ij X

r 0 2Xnfrg k0 ¼kþ1

r

C s xlrs0 ijk0 A  xlrijk

ð4Þ

For the optimization strategy, J is the optimal stack in the yard bay i. When the located stack J cannot contain the container with the same ship-stack and lower ship-tier compared with the relocated container, l

l

hiJs ¼ hiJs1

0 XB l X þ @hijs  k 

h ls

ij X

r0 2Xnfrg k0 ¼kþ1

r

1 C s xlrs0 ijk0 A  xlrijk

Otherwise, the lowest stack would be the optimization stack,

n o l l l l hiJs1 ¼ min hi1s1 ; hi2s1 ; . . . ; hijs1 ; . . .

l

l

hiJs ¼ hiJs1

0 XB l X þ @hijs  k  r

r0 2Xnfrg

h ls

ij X 0

1 C s xlrs0 ijk0 A  xlrijk

ð5Þ

k ¼kþ1

Subject to

XXXXX ls

j

k

i

s xlrijk ¼N

ð6Þ

r

0

ðq0  xlsropq0  q  xlsropq Þ  ðs0  sÞ P 0

ð7Þ

l hijs

ð8Þ

6 M ij

s xlrijk

2 f0; 1g

ð9Þ

Notes: Eq. (1) shows the objective function for minimizing the number of rehandles. Eq. (2) represents the height of target stack. l

Eq. (3) expresses the value of hijs for the lowest stack strategy, and refers to the height of the lowest stack plus the number of l

containers relocated to the stack. Eq. (4) calculates the value of hijs for the nearest stack strategy, and refers to the height of

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M. Ji et al. / Transportation Research Part E 80 (2015) 1–19 l

the nearest stack plus the number of containers relocated to the stack. Eq. (5) expresses the value of hijs for the optimization strategy, and means that if the next target container is on the stack, the relocation stack will be changed to the lowest stack. Constraint (6) indicates that all containers will be extracted. Constraint (7) requires that the container cannot be suspended on board which implies that when containers are in the same bay and stack, the subjacent containers have to be put down before the above containers can be positioned. Constraint (8) considers the restrictions of stack height. Constraint (9) imposes the binary requirements on the flow variables.

5. GA-ILSRS In this paper, we assume the number of retrieval containers is N in the terminal yard. The loading sequence of N containers is diversiform, and the number of loading sequence is N!. For the each certain loading sequence, we should obtain the minimum number of rehandles, which increases the complexity of the problem and difficulty of the solution. Moreover, due to the require of the stowage plan, unreasonable loading sequence is eliminated. As a result, N! is an estimated value for the complexity of this problem. For the genetic algorithm (GA), it can be used to solve more complex problems, and the parallelism of GA improves the computational efficiency. The general GA has a high degree of randomness, which randomly generates the initial population, cross-bit and mutation points. Based on an improvement to the general GA, we propose the GA-ILSRS and employ the algorithm to reduce the iterations and randomness using two cross-bit method to produce the cross points and mutation points. The main improvement of the GA-ILSRS is to avoid the ineffective crossover and mutation operation, and to exclude unfeasible and repeated solutions. During the process of coding, crossover and mutation, the GA-ILSRS considers both loading sequence and rehandling location. Meanwhile, the GA-ILSRS integrates the loading sequence and the rehandling strategy to calculate fitness, which is more challenging. The operating flow path of the GA-ILSRS is shown in Fig. 6. Detailed descriptions of the GA-ILSRS are as follows.

Fig. 6. The operating flow path of the GA-ILSRS.

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5.1. Set initial population In this paper, we use L to denote the solution vector of the model. The serial of vector L is a loading sequence. For example: L ¼ ð 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Þ. The corresponding loading sequence is 1–2–3–4–5–6–7–8–9–1 0–11–12–13–14. The matrix Eðs; o; p; q; i; j; kÞ denotes the information of containers in the yard and ship, where s represents the serial number of container, o denotes the ship’s bay with container s; p is the stack of container s in bay o; q refers to the tier of container s; i is the bay of container s in the yard, j is the stack of container s in the yard, k represents the tier container s in the yard. In the matrix E, there is a one to one correspondence between the position in the ship-bay and the position in the yard-bay. For example,

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14

B1 B B B6 B B E ¼ B1 B B1 B B @1

2 1 2 1 2 1 2 1

2

1

2

1

6 5 5 6 6 7 7 4

4

5

5

6

1 2 2 2 2 2 2 3

3

3

3

3

1 2 3 1 4 3 2 5

2

1

2

1

3 3 3 3 2 2 2 3

4

5

5

6

2 2 1 2 3 2 3 3 2

3

3

3

3

1

2 C C C 6 C C C 3 C C 2 C C C 6 A 3

In this example, there are 14 containers to be loaded. The vector L is a chromosome. The matrix E contains the information of containers in vector L. During the process of generating the initial population, the matrix E ensures the feasible loading sequence considering both the positions of containers in the ship-bay and yard-bay, which reduces the randomness of the algorithm. 5.2. Crossover We use the two cross-bit method to cross. Before the crossover, we choose two chromosomes from parents. Then, we randomly generate two intersections. For the genes before the first cross-bit and after the second cross-bit, the two offsprings inherit their parents’ genes. For the genes between the first cross-bit and the second cross-bit, offspring 1 inherits the genes’ sequence, which is the same with parent 2, and offspring 2 inherits the genes’ sequence, which is the same with parent 1. Finally, according to the information of containers in the matrix E, we judge the solution. Excluding unfeasible and repeated solutions, we get M solutions as new populations by repeating the above operation. The judgment process gets rid of those solutions that can lead to rehandling in the ship. It reflects the coordination of the ship-bay and yard-bay. For example, we randomly produce two intersections and carry out the crossover operation. The crossover process of chromosomes is shown in Fig. 7. 5.3. Mutation This study applies the two cross-bit method to produce the mutation points. The number of containers is N. We produce the two mutation points in the vector L where the mutation points are exchanged. One mutation point is produced from 1 to N , and another is from N2 þ 1 to N. The method to produce the mutation points reduces the randomness of mutation and 2 improves the running efficiency of the algorithm. During the mutation operation, the container loading sequence cannot be exchanged in some cases. For example, considering the previous vector L and matrix E, container 1 and 5 cannot be exchanged because container 5 is above container 1, which exists in the same bay and same stack on ship. As a result, we must first load container 1 before container 5. To avoid unrealistic situations, the following procedures are utilized. First, a function l is designed and assigned an initial value of 0. Then, the loading sequence of any container C 1 and C 2 is exchanged, where the loading sequence of container C 1 is above the container C 2 . After that, the loading containers between containers C 1 and C 2 are judged. If there exists a container whose tier is higher than that of container C 1 or lower than that of the container C 2 in the same stack and same bay, then l ¼ l þ 1, and the container C 1 and C 2 are not exchanged. Then continue to produce two new mutation points until the

Parent 1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

1

2

3

4

5

6

9

8

7

10

11

12

13

14

Offspring 1

Parent 2

3

1

5

13

4

12

2

6

9

11

8

10

7

14

3

1

5

13

4

2

6

9

12

11

8

10

7

14

Offspring 2

Fig. 7. Chromosome cross.

10

M. Ji et al. / Transportation Research Part E 80 (2015) 1–19

exchange requirements of l ¼ 0 are satisfied. The process aims to coordinate the ship-bay and yard-bay as well. Therefore, we exchange the order of the loading sequence to generate a new feasible solution. 5.4. Calculate fitness The termination rule is set as the maximum number of generation G. For three rehandling strategies, we firstly calculate l

hijs , which is the height of the located stack and the target stack based on the Eqs. (2)–(5) respectively. Then we obtain the value of the objective function under Eq. (1). In this approach, the objective function is the fitness function, which integrates the loading sequence and the rehandling strategy. 5.5. Selection This work adopts roulette wheel selection, and the steps are as follows: (1) Define the one-dimensional array wð2MÞ for storing the score value of the parent and offspring individuals. The number of individuals is 2M. (2) For each individual Ai in the population 2M, we randomly select q individuals from the population, and compare the fitness of Li with the fitness of q individuals. The number of wðiÞ is the number of q individuals which have higher fitness than Li . In this way we get the array of wð2MÞ. (3) Select the maximum fitness value of M individuals as a new parent. This approach ensures that the best individual in each generation can survive, and excellent individuals within the parent and offspring populations are most probably preserved. Meanwhile, due to the random selection of q individuals, the algorithm maintains the diversity of the population and avoids a premature convergence. This approach increases the possibility of obtaining the global optimal solution. In summary, the algorithm employs the roulette wheel selection, two cross-bit method and mutation over an exact range to generate new populations. This method integrates the loading sequence and the rehandling strategy based on a general GA, and reduces the randomness of the algorithm, which has more merits. 6. Computational experiments In this paper, we compare the results of the computational experiments in the perspective of algorithm, rehandling strategy and multi-quay crane operations. The results of the experiments are obtained using a computer with 2 gigabytes of RAM, Windows XP Professional operating system and an Intel Core 2 Duo with 3.00 gigahertz cores. Through the comparisons of experiments, we verify the robustness and universality of algorithm and obtain the corresponding conclusions. 6.1. Comparisons for algorithm In order to verify the effectiveness of the algorithm, we compare the performance of the GA-ILSRS with the branch-and-bound and heuristic algorithm (Kim and Hong, 2006). For the branch-and-bound and heuristic algorithm, the loading sequence is certain before loading operation. Moreover, they assume that rehandling occurs only at the moment when a target container is to be picked up and the obstructed containers are relocated to other stacks in the same bay. As a result, for the GA with certain loading sequence (GA-CLS), the assumptions on the problem including loading sequence are same with that of the branch-and-bound and heuristic algorithm. For the GA-ILSRS, the assumptions on the problem are same with that of the branch-and-bound and heuristic algorithm, except that the loading sequence is uncertain. Tables 1 and 2 show the number of rehandles and running time obtained for the different algorithms. Tables 1 and 2 show the average total number of rehandles for the heuristic algorithm, the branch-and-bound and the HA NP LP OP ; B&B , B&B ; B&B and OP , the value is either bigger than 1 or smaller than 1. If it is bigger than GA-ILSRS. For the index of B&B HA 1, the number of rehandles for the algorithm in the numerator of fraction is more than that in the denominator of fraction. Otherwise, the number of rehandles in the numerator is less. By the comparisons of the number of rehandles, the following conclusions can be drawn: (1) Table 1 shows that the number of rehandles for the GA-CLS is nearly equal to the number for B&B, and less than the number for HA. What’s more, the average running time for GA-CLS is shortest, which implies that GA-CLS is more effective when applied to the large instances. (2) Tables 1 and 2 show that the number of rehandles for the GA-ILSRS is superior to that of the B&B; HA and the GA-CLS, which implies that the loading sequence and the rehandling strategy have an important effect on the number of rehandles. The integration of loading sequence and rehandling strategy is more important to minimize the number of rehandles.

11

M. Ji et al. / Transportation Research Part E 80 (2015) 1–19 Table 1 The comparison for the GA-CLS. Scale

Tier  Stack

9 12 15 18 21 24 16 20 24 28 20 25 30

33 34 35 3 6 37 38 44 45 46 47 54 55 56

Branch-andbound

Heuristic algorithm

GA-CLS

B&B

ACT

HA

ACT

NP

LP

OP

ACT

3.45 5.10 7.23 8.35 9.88 11.33 9.53 11.85 13.75 16.70 12.63 15.78 21.00

7.3 8.0 13.2 14.5 45.5 142.6 14.8 33.4 152.4 226.8 49.5 223.3 2657.5

3.55 5.25 7.43 8.65 10.35 11.8 10.33 12.63 14.78 18.28 14.13 18.03 24.38

1 1 1 1 1 1 1 1 2 2 1 2 2

3.67 4.75 7.67 9.25 11.47 12.21 11.15 12.78 15.01 18.21 14.25 18.23 23.87

3.40 4.52 7.45 8.63 10.68 11.60 10.45 12.50 14.67 17.56 13.67 17.68 23.33

3.14 4.25 7.25 8.33 10.33 11.60 10.33 12.33 14.33 17.33 13.67 17.33 22.67

<1 <1 <1 <1 <1 <1 <1 <1 <1 1 1 1 1

HA B&B

NP B&B

LP B&B

OP B&B

OP HA

1.03 1.03 1.03 1.04 1.05 1.04 1.08 1.07 1.08 1.10 1.12 1.14 1.16

1.06 0.93 1.06 1.11 1.16 1.08 1.17 1.08 1.09 1.09 1.12 1.16 1.13

0.99 0.89 1.03 1.03 1.08 1.02 1.09 1.05 1.07 1.05 1.08 1.12 1.11

0.91 0.83 1.00 0.99 1.05 1.02 1.08 1.04 1.04 1.04 1.08 1.09 1.08

0.88 0.81 0.97 0.96 0.99 0.98 1.00 0.98 0.97 0.95 0.97 0.96 0.93

Table 2 The comparison for the GA-ILSRS. Scale

Tier  Stack

9 12 15 18 21 24 16 20 24 28 20 25 30

3 3 3 3 3 3 4 4 4 4 5 5 5

            

3 4 5 6 7 8 4 5 6 7 4 5 6

Branch-andbound

Heuristic algorithm

GA-ILSRS

B&B

ACT

HA

ACT

NP

LP

OP

ACT

3.45 5.10 7.23 8.35 9.88 11.33 9.53 11.85 13.75 16.70 12.63 15.78 21.00

7.3 8.0 13.2 14.5 45.5 142.6 14.8 33.4 152.4 226.8 49.5 223.3 2657.5

3.55 5.25 7.43 8.65 10.35 11.8 10.33 12.63 14.78 18.28 14.13 18.03 24.38

1 1 1 1 1 1 1 1 2 2 1 2 2

2.67 3.67 7.28 7.78 9.86 10.67 10.23 11.01 13.63 15.77 11.25 17.01 20.44

2.01 3.13 6.67 7.25 9.13 9.67 9.33 10.33 12.25 14.14 10.33 15.13 19.13

1.67 2.75 6.15 6.33 8.75 9.25 8.13 9.67 11.87 13.25 9.67 14.45 18.67

<1 <1 <1 <1 <1 <1 <1 1 1 1 1 1 2

NP B&B

LP B0 &B

OP B&B

OP HA

0.77 0.72 1.00 0.93 0.99 0.94 1.07 0.93 0.99 0.94 0.89 1.08 0.97

0.58 0.61 0.92 0.87 0.92 0.85 0.98 0.87 0.89 0.85 0.82 0.96 0.91

0.48 0.54 0.85 0.76 0.89 0.82 0.85 0.82 0.86 0.79 0.77 0.92 0.89

0.47 0.52 0.83 0.73 0.85 0.78 0.79 0.77 0.80 0.72 0.68 0.80 0.77

Notes: HA represents the heuristic algorithm. B&B represents the branch-and-bound algorithm. GA-CLS represents the GA with certain loading sequence. GA-ILSRS represents the GA integrated loading sequence and rehandling strategy. ACT represents the average running time(s). N  P represents the nearest stack strategy, L  P represents the lowest stack strategy, and O  P represents the optimization strategy.

6.2. Comparisons for rehandling strategy 6.2.1. Comparisons of significance test In order to further validate the significance of the model and the effectiveness of the algorithm, we randomly generate 13 experiments for small, medium and large scale. The container scales are 36, 46, 52, 60, 82, 90, 96, 106, 112, 124, 244, 436 and 980. These containers are stockpiling in different yard-bays, yard-stacks and yard-tiers, which indicates the problem is considered in a three-dimensional angle. Generally, for the largest ship of 18,000 TEU (twenty-foot equivalent unit), the container scale in a ship-bay reaches 300 TEU. When the experiment scale for double quay cranes reaches 980, the container scale in a ship-bay reaches 490 TEU. In reality, it is impossible to stockpile 490 containers in a ship-bay. Thus, the instance only verifies the validity of the algorithm. Meanwhile, the t-test is used to analyze repeated experiments to show the significant differences between two rehandling strategies. The t-test formula (John, 2007) is as follows,

X1  X2 t 12 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 S1 þ S22  2r 12 S1 S2 X2  X3 t 23 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 S2 þ S23  2r 23 S2 S3 X1  X3 t 13 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 S1 þ S23  2r 13 S1 S3 In which:

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M. Ji et al. / Transportation Research Part E 80 (2015) 1–19

X 1 ; S1 : mean and standard deviation of 13 trials under the nearest stack strategy; X 2 ; S2 : mean and standard deviation of 13 trials under the lowest stack strategy; X 3 ; S3 : mean and standard deviation of 13 trials under the optimization strategy; r12 ; t 12 : correlation coefficient and significance t-value between the nearest and the lowest strategy; r23 ; t 23 : correlation coefficient and significance t-value between the lowest and the optimization strategy; r13 ; t 13 : correlation coefficient and significance t-value between the nearest and the optimization strategy. Table 3 lists the minimum and maximum number of rehandles by repeated test, and the corresponding t-test values for the 13 trials based on three rehandling strategies. The t-test values show a significant difference among the three strategies. Table 3 The number of rehandles and t value for different container scales. Scales

Bay  Stack  Tier

Small instances 36 364

Strategies

Rehandling and t value Min

Max

NP LP OP

13 9 9

t a ð18Þ ¼ 2:10 Mean

t value

Significance

20 15 14

17 12.9 11.9

t12 ¼ 7:01 t23 ¼ 0:94 t13 ¼ 3:81

Y N Y

46

364

NP LP OP

23 20 18

30 24 23

27.3 21.4 20

t12 ¼ 3:94 t23 ¼ 3:06 t13 ¼ 4:33

Y Y Y

52

364

NP LP OP

25 22 20

32 25 23

28.5 23.4 21.8

t12 ¼ 5:62 t23 ¼ 5:50 t13 ¼ 7:58

Y Y Y

60

364

NP LP OP

28 25 25

38 34 30

33.7 29 26.3

t12 ¼ 1:64 t23 ¼ 0:96 t13 ¼ 3:4

N N Y

82

664

NP LP OP

37 32 30

46 37 37

40.5 34.8 32.4

t12 ¼ 2:11 t23 ¼ 2:44 t13 ¼ 3:35

Y Y Y

90

664

NP LP OP

50 37 34

58 47 39

53.3 41.8 36.4

t12 ¼ 3:83 t23 ¼ 1:74 t13 ¼ 14:16

Y N Y

NP LP OP

49 43 42

56 50 46

52.7 47.7 43.3

t12 ¼ 2:39 t23 ¼ 3:44 t13 ¼ 5:91

Y Y Y

Medium instances 96 664

106

664

NP LP OP

53 46 44

59 50 47

55.3 47.4 44.9

t12 ¼ 7:20 t23 ¼ 3:81 t13 ¼ 11:13

Y Y Y

112

664

NP LP OP

60 47 45

68 53 49

64 49.7 46.9

t12 ¼ 9:84 t23 ¼ 2:93 t13 ¼ 9:24

Y Y Y

124

664

NP LP OP

73 61 58

78 63 61

75.4 62.4 59.1

t12 ¼ 19:09 t23 ¼ 7:83 t13 ¼ 24:96

Y Y Y

Large instances 244

12  6  4

NP LP OP

172 131 119

180 137 122

176.2 134.3 119.9

t12 ¼ 11:64 t23 ¼ 4:24 t13 ¼ 9:71

Y Y Y

436

24  6  4

NP LP OP

287 223 212

290 225 215

288.9 224.4 213.1

t12 ¼ 32:25 t23 ¼ 12:56 t13 ¼ 33:84

Y Y Y

980

48  6  4

NP LP OP

619 486 469

623 489 471

620.6 487.5 469.8

t12 ¼ 53:24 t23 ¼ 19:67 t13 ¼ 75:41

Y Y Y

Notes: N  P represents the nearest stack strategy, L  P represents the lowest stack strategy, and O  P represents the optimization strategy. Significant level a ¼ 0:05.

13

M. Ji et al. / Transportation Research Part E 80 (2015) 1–19

The performance of the optimization strategy is the best, and the lowest stack strategy is better than the nearest stack strategy. Table 3 shows the number of rehandles corresponding to three rehandling strategies is different. And the t value indicates that the rehandling strategy passes the test of significance, which shows the difference among the three strategies. From Table 3, we can see that the scales of the experiments are large enough and suitable for any vessels. For the small instances, the value of t13 is the biggest compared with t 12 and t 23 , which indicates the difference between the nearest and optimization strategy is the most distinctive. For the medium and large instances, the value of t 23 is the smallest, which indicates the difference between the lowest and optimization strategy is insignificant, and reduces with the increase of container scales. In addition, the experiments show the model and the algorithm are suitable for the three-dimensional problem, which is described by the yard-bays, stacks and tiers.

Table 4 The minimum number of rehandles for different container scales. Scales

Bay  Stack  Tier

Small instances 36 364

Strategy

Min

LB

M  LB

MLS (%)

NLP

NLPN (%)

LOP

LOPL (%)

NP L-P OP

13 9 9

12 9 9

1 0 0

2.78 0.00 0.00

4

30.77

0

0.00

46

364

NP LP OP

23 20 18

18 16 15

5 4 3

10.87 8.70 6.52

3

13.04

2

10.00

52

364

NP LP OP

25 22 20

23 20 19

2 2 1

3.85 3.85 1.92

3

12.00

2

9.09

60

364

NP LP OP

28 25 25

26 23 23

2 2 2

3.33 3.33 3.33

3

10.71

0

0.00

82

664

NP LP OP

37 32 30

32 27 26

5 5 4

6.10 6.10 4.88

5

13.51

2

6.25

90

664

NP LP OP

50 37 34

41 30 29

9 7 5

10.00 7.78 5.56

13

26.00

3

8.11

NP LP OP

49 43 42

41 35 35

8 8 7

8.33 8.33 7.29

6

12.24

1

2.32

Medium instances 96 664

106

664

NP LP OP

53 46 44

41 36 35

12 10 9

11.32 9.43 8.49

7

13.21

2

4.35

112

664

NP LP OP

60 47 45

48 38 37

12 9 8

10.71 8.04 7.14

13

21.67

2

4.26

124

664

NP LP OP

73 61 58

56 46 45

17 15 13

13.71 12.10 10.48

12

21.67

2

4.26

NP LP OP

172 131 119

73 68 63

99 63 56

40.57 25.82 22.95

41

23.84

12

9.16

Large instances 244 12  6  4

436

24  6  4

NP LP OP

287 223 212

168 153 146

119 70 66

27.29 16.06 15.14

64

22.30

11

4.93

980

48  6  4

NP LP OP

619 486 469

318 304 292

301 182 177

30.71 18.57 18.06

133

21.49

17

3.50

14

M. Ji et al. / Transportation Research Part E 80 (2015) 1–19

6.2.2. Comparisons of lower bound To evaluate the quality of the model, it is necessary to obtain a tight lower bound. Through analysis, we realize that the rehandling contains both the direct rehandling, which occurs due to loading sequence, and the indirect rehandling which occurs due to the relocated positions. The direct rehandling is inevitable, so the minimum total number of rehandles is obtained when the number of indirect rehandling is 0. Thus, the number of direct rehandling is as an estimator of the lower bound. It is determined by the number of obstructed containers above the target container, where the obstructed containers are the containers after the target container in the loading sequence. The steps of solving the lower bound are as follows: (1) Based on the ship stowage plan and the stockpiling state in the yard, we first determine the loading sequence. (2) Based on the loading sequence, we calculate the minimum number of rehandles, which is denoted by LB. The equation of the LB is as follows:

LB ¼

XXXXX ls

k

j

i

l

s ðhijs  kÞ  xlrijk

r

(3) Determining the other loading sequence through the GA-ILSRS, we then obtain a different minimum number of rehandles LB1 . (4) Compare the number LB with LB1 and select the smaller one as LB. Return to step (3), and repeat to obtain the true LB. Table 4 lists the minimum number of rehandles and the lower bounds for the 13 experiments. In Table 4, LB represents the lower bound of rehandling. M  LB refers to the difference between the minimum number of rehandles and the lower bounds, which is an estimator of the indirect rehandling. N is the number of container scales. MLS means the percentage of the indirect rehandling in terms of the total container scale, which is calculated by MLB . NLP and LOP N are the differences among the three strategies, i.e., NLP ¼ ðN  PÞ  ðL  PÞ , LOP ¼ ðL  PÞ  ðO  PÞ. In addition, . NLPN means the increased range of the lowest stack strategy, which is calculated according to NLPN ¼ ðNPÞðLPÞ ðNPÞ . LOPL is the increased range of the optimization strategy, LOPL ¼ ðLPÞðOPÞ ðLPÞ

Fig. 8. The number of MLS in different scales.

15

M. Ji et al. / Transportation Research Part E 80 (2015) 1–19

In order to facilitate the comparisons, a line chart comparing the percentage of the number of indirect rehandling, in terms of the container scale, for different strategies is shown in Fig. 8. The lateral axis shows the container scale and the vertical axis shows the number of MLS, which means the percentage of the indirect rehandling. From the comparisons of the lower bound in Table 4 and Fig. 8, the following conclusions can be drawn: (1) NLP and LOP are not negative, which suggest that the lowest stack strategy is better than the nearest stack strategy, and the optimization strategy is better than the lowest. They reflect the differences among the three rehandling strategies. (2) NLPN and LOPL reflect the increased range for the three rehandling strategies. The row of NLPN shows that the lowest stack strategy reduces the number of rehandles by 10% to 30% compared with the nearest stack strategy. LOPL indicates that the optimization strategy reduces the number of rehandles by 10% compared with the lowest stack strategy. Table 5 The minimum number of rehandles for single quay crane operation. Scales

Bay  Stack  Tier

Small instances 34 364

Strategy

Min

Max

Mean

LB

M  LB

NP LP OP

17 15 12

25 21 18

22.1 17.9 15.2

7 5 4

t value

Significance

NLPN (%)

LOPL (%)

10 10 8

t 12 ¼ 3:52 t 23 ¼ 4:38 t 13 ¼ 8:4

Y Y Y

11.7

13.33

44

364

NP LP OP

2 2 1

6 4 3

4.5 3 1.8

2 1 1

0 1 0

t 12 ¼ 4:09 t 23 ¼ 8:67 t 13 ¼ 6:55

Y Y Y

0.00

50.00

58

464

NP LP OP

22 18 14

26 22 20

23.7 20.1 17.6

10 7 6

12 11 8

t 12 ¼ 7:62 t 23 ¼ 2:83 t 13 ¼ 5:88

Y Y Y

18.18

22.22

66

464

NP LP OP

3 3 3

8 6 5

6.2 4.9 3.7

2 1 1

1 2 2

t 12 ¼ 3:39 t 23 ¼ 3:57 t 13 ¼ 3:71

Y Y Y

0.00

0.00

76

464

NP LP OP

10 8 7

15 12 10

12 10.1 8.3

7 6 6

3 2 1

t 12 ¼ 2:95 t 23 ¼ 3:94 t 13 ¼ 4:48

Y Y Y

20.00

12.50

88

664

NP LP OP

2 2 2

7 6 4

5.3 4 2.7

2 1 1

0 1 1

t 12 ¼ 3:09 t 23 ¼ 4:02 t 13 ¼ 4:15

Y Y Y

0.00

0.00

NP LP OP

59 48 44

70 53 50

64.2 50.9 47.1

31 28 27

28 20 17

t 12 ¼ 4:15 t 23 ¼ 4:64 t 13 ¼ 4:81

Y Y Y

18.64

8.33

Medium instances 100 664

106

664

NP LP OP

61 44 40

68 49 48

65.1 47 43.9

32 30 29

29 14 11

t 12 ¼ 9:95 t 23 ¼ 2:49 t 13 ¼ 16:3

Y Y Y

27.87

9.09

118

664

NP LP OP

4 3 3

8 7 5

6.4 4.9 3.6

3 2 2

1 1 1

t 12 ¼ 7:94 t 23 ¼ 4:84 t 13 ¼ 8:03

Y Y Y

25.00

0.00

132

664

NP LP OP

76 56 54

85 65 59

82.1 60.1 56.4

48 47 45

28 9 9

t 12 ¼ 6:22 t 23 ¼ 2:38 t 13 ¼ 8:61

Y Y Y

26.32

3.57

NP LP OP

138 119 110

142 122 112

139.4 120.9 111.2

98 92 89

40 27 21

t 12 ¼ 10:1 t 23 ¼ 6:7 t 13 ¼ 9:5

Y Y Y

13.77

7.56

Large instances 218 12  6  4

303

18  6  4

NP LP OP

201 171 151

208 174 154

205.2 172.6 152.3

113 113 109

58 58 42

t 12 ¼ 13:5 t 23 ¼ 13:4 t 13 ¼ 16:5

Y Y Y

14.93

11.70

490

24  6  4

NP LP OP

342 296 265

348 299 269

344.9 297.3 267.4

171 158 146

171 138 119

t 12 ¼ 13:9 t 23 ¼ 12:1 t 13 ¼ 19:3

Y Y Y

13.45

10.47

16

M. Ji et al. / Transportation Research Part E 80 (2015) 1–19

(3) The minimum number of rehandles is not less than the lower bound for the three rehandling strategies, which verify the availability of the established model and algorithm. And Table 4 shows that the container scale in a ship bay reaches 490 TEU, which proves that the model and the algorithm are available for any vessels. (4) Fig. 8 shows that MLS for the nearest and the lowest stack strategy is approximately equal for small and medium instances, yet greater than the optimization strategy. Meanwhile, for large instances, due to the increasing of indirect rehandling, the MLS for the nearest strategy gets significantly larger than that for the lowest and the optimization strategy. This indicates that the number of rehandles for the optimization strategy decreases with the increase of container scales. Thus, the lowest stack is selected as the optimal stack, which makes the MLS for the lowest and optimization strategy approximately equal. In addition, for the nearest and the lowest stack strategy, only the loading sequence is taken into account, but for the optimization strategy, both the loading sequence and the rehandling location are considered which lead to the minimum number of rehandles.

Table 6 The minimum number of rehandles for triple quay crane operations. Scales

Bay  Stack  Tier

Small instances 54 364

Strategy

Min

Max

Mean

LB

NP LP OP

25 21 19

29 24 22

26.7 22.5 20.4

18 14 11

M  LB

t value

Significance

NLPN (%)

LOPL (%)

7 7 8

t12 ¼ 2:19 t23 ¼ 7:63 t13 ¼ 12:5

Y Y Y

16.00

9.52

69

664

NP LP OP

37 29 26

43 32 29

28.3 30.5 27.5

22 19 18

15 10 8

t12 ¼ 8:05 t23 ¼ 6:57 t13 ¼ 12:3

Y Y Y

21.62

10.34

78

664

NP LP OP

36 31 27

39 34 31

37.4 32.3 29.2

23 21 19

13 10 8

t12 ¼ 10:5 t23 ¼ 8:05 t13 ¼ 14:4

Y Y Y

13.89

12.91

90

664

NP LP OP

65 48 44

71 51 48

67.7 49.4 45.4

37 35 32

28 13 12

t12 ¼ 13:5 t23 ¼ 8:11 t13 ¼ 14:9

Y Y Y

26.15

8.33

123

664

NP LP OP

68 58 50

74 61 53

70.4 59.5 51.3

49 43 38

19 15 12

t12 ¼ 8:05 t23 ¼ 15:7 t13 ¼ 15:6

Y Y Y

14.71

13.79

135

12  6  4

NP LP OP

96 83 75

103 88 79

100.1 85.7 76.5

71 62 58

25 21 17

t12 ¼ 7:91 t23 ¼ 8:6 t13 ¼ 13:3

Y Y Y

13.54

9.64

NP LP OP

107 91 82

110 94 85

108.2 92.2 83.5

79 68 62

28 23 20

t12 ¼ 10:8 t23 ¼ 17:2 t13 ¼ 13:2

Y Y Y

14.95

9.89

Medium instances 144 12  6  4

159

12  6  4

NP LP OP

93 85 77

99 93 79

96.1 88.8 78

69 63 56

24 22 21

t12 ¼ 3:46 t23 ¼ 4:5 t13 ¼ 4:7

Y Y Y

8.61

9.41

168

12  6  4

NP LP OP

90 77 74

97 82 79

94.1 80 77.1

61 59 57

29 18 17

t12 ¼ 7:36 t23 ¼ 5:38 t13 ¼ 9:09

Y Y Y

14.44

3.90

186

12  6  4

NP LP OP

98 79 76

106 83 79

102.4 80.9 77.3

64 58 57

34 21 19

t12 ¼ 3:73 t23 ¼ 9:80 t13 ¼ 2:79

Y Y Y

19.39

3.80

NP LP OP

159 127 108

164 130 113

162.3 128.5 111

93 78 72

66 49 36

t12 ¼ 10:9 t23 ¼ 6:25 t13 ¼ 21:2

Y Y Y

20.13

14.96

Large instances 366 18  6  4

654

36  6  4

NP LP OP

265 203 197

269 208 202

266.9 205.5 199.6

192 173 167

73 30 30

t12 ¼ 20:4 t23 ¼ 2:56 t13 ¼ 22:4

Y Y Y

23.40

2.96

981

48  6  4

NP LP OP

587 467 446

591 472 450

589.2 469 447.9

383 358 345

204 109 101

t12 ¼ 21:6 t23 ¼ 7:82 t13 ¼ 25:4

Y Y Y

20.44

4.50

M. Ji et al. / Transportation Research Part E 80 (2015) 1–19

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Fig. 9. The difference between the minimum number of rehandles and the lower bound for single quay crane.

Fig. 10. The difference between the minimum number of rehandles and the lower bound for triple quay cranes.

6.3. Comparisons for multi-quay crane operations In practical operation, there are different numbers of quay cranes operating simultaneously in a container terminal. In this work, we represent the number of quay cranes as r. In order to verify the versatility of the GA-ILSRS, we carry out the comparisons for single quay crane operation and triple quay crane operations. Thirteen experiments involving different container scales are enumerated. Tables 5 and 6 display the results on the minimum number of rehandles and the lower bound. When r ¼ 1. Fig. 9 shows the number of the indirect rehandling under three rehandling strategies concerning different scales of containers. When r = 3 Fig. 10 shows the number of the indirect rehandling under the three rehandling strategies considering different scales of containers for triple quay crane operations. Tables 5 and 6 show the same conclusions with the double quay cranes. Moreover, the amplitude change of NLPN and LOPL for the single quay crane is relatively larger than that for the triple quay cranes. The number of rehandles decreases with the number of quay cranes. Through the comparison between the minimum number of rehandles for the single and triple quay crane operations, we can see that the optimization strategy is superior to the nearest and the lowest stack strategy. The method for double quay cranes also applies to single and triple quay cranes, and is more suitable for multi-quay cranes. The results show the versatility of the algorithm and present a theoretical basis for dispatchers to reasonably arrange the loading sequence.

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M. Ji et al. / Transportation Research Part E 80 (2015) 1–19

7. Conclusions Based on the nearest stack and the lowest stack strategy, we developed the optimization strategy and established the mathematical models to integrate the loading sequence and the rehandling strategy. Under the preconditions of a known stowage plan and multi-quay crane operations, this paper obtained the optimal loading sequence and minimized the number of rehandles for three different strategies using the GA-ILSRS. Through the comparison of specific examples, the three strategies have significant differences which indicate the necessity of integrating the loading sequence and the rehandling strategy. Moreover, the results of experiments show that the optimization strategy reduces the number of rehandles by 10% compared with the lowest stack strategy, and the lowest stack strategy reduces the number of rehandles by 10–30% compared with the nearest stack strategy. In addition, similar studies were carried out on single quay crane and triple quay crane operations in order to prove the versatility of the model and algorithm. Experiments indicate that the model and the algorithm are suitable for any vessels and available for three-dimensional problem. Acknowledgements This research is supported by the Key Project of National Social Science Fund (No. 14ZDB131) and the National Nature Science Foundation of China (No. 71072081). References Avriel, M., Penn, M., Shpirer, N., Witteboon, S., 1998. Stowage planning for container ships to reduce the number of shifts. Ann. Oper. Res. 76, 55–71. Bazzazia, M., Safaei, N., Javadian, N., 2009. A genetic algorithm to solve the storage space allocation problem in a container terminal. Comput. Ind. Eng. 56, 44–52. Caserta, M., Voß, S., Sniedovich, M., 2009. Applying the corridor method to a blocks relocation problem. J. Oper. Res. Spectrum 33 (4), 915–929. Caserta, M., Schwarze, S., Voß, S., 2012. A mathematical formulation and complexity considerations for the blocks relocation problem. Eur. J. Oper. Res. 219 (1), 96–104. Caserta, M., Voß, S., Sniedovich, M., 2011. Applying the corridor method to a blocks relocation problem. Oper. Res. Spectrum 33, 915–929. Gharehgozli, A.H., Yu, Y., Koster, R., Udding, J.T., 2014. An exact method for scheduling a yard crane. Eur. J. Oper. Res. 235, 431–447. He, J.L., Chang, D.F., Mi, W.J., Yan, W., 2010. A hybrid parallel genetic algorithm for yard crane scheduling. Transp. Res. Part E: Logist. Transp. Rev. 46, 136– 155. Homayouni, S.M., Tang, S.H., Motlagh, O., 2014. A genetic algorithm for optimization of integrated scheduling of cranes, vehicles, and storage platforms at automated container terminals. J. Comput. Appl. Math. 270, 545–556. Imai, A., Sasaki, K., Nishimura, E., Papadimitriou, S., 2006. Multi-objective simultaneous stowage and load planning for a container ship with container rehandle in yard stacks. Eur. J. Oper. Res. 171, 373–389. Jang, D.W., Kim, S.W., Kim, K.H., 2013. The optimization of mixed block stacking requiring relocations. Int. J. Prod. Econ. 143 (2), 256–262. Jin, B., Lim, A., Zhu, W., 2013. A greedy look-ahead heuristic for the container relocation problem. Recent Trends Appl. Artif. Intell. 7906, 181–190. Jin, B., Zhu, W.B., Lim, A., 2014. Solving the container relocation problem by an improved greedy look-ahead heuristic. Eur. J. Oper. Res., in press. 10.1016/j. ejor.2014.07.038. John, A.R., 2007. Mathematical Statistics and Data Analysis, third ed. Thomson Brooks Cole. Kang, J., Ryu, K.R., Kim, K.H., 2006. Deriving stacking strategies for export containers with uncertain weight information. J. Intell. Manuf. 17 (4), 399–410. Kim, K.H., 1997. Evaluation of the number of rehandles in container yards. Comput. Ind. Eng. 32 (4), 701–702. Kim, K.H., Hong, G.P., 2006. A heuristic rule for relocating blocks. Comput. Oper. Res. 33 (4), 940–954. Kim, K.H., Park, Y.M., Ryu, K.R., 2000. Deriving decision rules to locate export containers in container yards. Eur. J. Oper. Res. 124, 89–101. Lee, Y., Lee, Y.J., 2010. A heuristic for retrieving containers from a yard. Comput. Oper. Res. 37 (6), 1139–1147. Lee, D.H., Cao, J.X., Shi, Q.X., Chen, J.H., 2013. A heuristic algorithm for yard truck scheduling and storage allocation problems. Transp. Res. Part E: Logist. Transp. Rev. 45 (5), 810–820. Park, T.K., Kim, K.H., 2010. Comparing handling and space costs for various types of stacking methods. Comput. Ind. Eng. 58 (3), 501–508. Petering, M.E.H., Hussein, M.I., 2013. A new mixed integer program and extended look-ahead heuristic algorithm for the block relocation problem. Eur. J. Oper. Res. 231, 120–130. Sauri, S., Martin, E., 2011. Space allocating strategies for improving import yard performance at marine terminals. Transp. Res. Part E: Logist. Transp. Rev. 47 (6), 1038–1067. Skinner, B., Yuan, S., Huang, S.D., Liu, D., Cai, B., Dissanayake, G., Lau, H., Bott, A., Pagac, D., 2013. Optimisation for job scheduling at automated container terminals using genetic algorithm. Comput. Ind. Eng. 64, 511–523. Wan, Y.W., Liu, J., Tsai, P.C., 2009. The assignment of storage locations to containers for a container stack. Naval Res. Logist. 56 (8), 699–713. Woo, J., Kim, K.H., 2011. Estimating the space requirement for outbound container inventories in port container terminals. Int. J. Prod. Econ. 133, 293–301. Zhang, C., Chen, W., Zheng, L., 2010. A note on deriving decision rules to locate export containers in container yards. Eur. J. Oper. Res. 205 (2), 483–485. Zhang, C.Q., Wan, Y.W., Liu, J.Y., Richard, J.L., 2002. Dynamic crane deployment in container storage yards. Transp. Res. Part B 36, 537–555. Zhu, M., Fan, X., He, W., 2010. A heuristic approach for transportation planning optimization in container yard. In: 2010 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM), pp. 1766–1770. Zhu, W.B., Hu, Q., Andrew, L., Zhang, H.D., 2012. Iterative deepening A⁄ algorithms for the container relocation problem. IEEE Trans. Autom. Sci. Eng. 9 (4), 710–722. Dr. Mingjun Ji received the Master degree in Applied Mathematics from Inner Mongolia University, China, in 2001, and the Ph.D. degree in Operational Research from Dalian University of Technology, China, in 2004. He had been University of Cambridge to do post-doctorial research from 2005 to 2006. Current, he is a professor in the Transportation Management College, Dalian Maritime University. His main research interests include supply chain management, shipping management, global optimization and heuristic algorithms. He has published about 30 research papers. Miss Wenwen Guo received the Bachelor’s degree in Logistics Engineering and Management from Dalian Maritime University, China, in 2013. Now she is a postgraduate student in Transportation Management College, Dalian Maritime University, Dalian, China. Her main research interests are containership transportation and container terminal operation.

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Miss Huiling Zhu received the Bachelor’s degree in Logistics Engineering and Management from Dalian Maritime University, China, in 2012. Now she is a PhD student in Transportation Management College, Dalian Maritime University, Dalian, China. Her main research interests are containership stowage and container terminal operation. Mr. Yongzhi Yang is an associate professor in Transportation Management College, Dalian Maritime University, Dalian, China. His research interest is mainly container multimodal transport.