Iterative heuristic for simultaneous allocations of berths, quay cranes, and yards under practical situations

Transportation Research Part E xxx (xxxx) xxx–xxx

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

Iterative heuristic for simultaneous allocations of berths, quay cranes, and yards under practical situations Changchun Liu Department of Industrial Engineering, Tsinghua University, Beijing 100084, China

ARTICLE INFO

ABSTRACT

Keywords: Discrete berth allocation problem Quay crane assignment problem Yard assignment problem Transshipment hubs Crane productivity

This study investigates a problem regarding simultaneous allocations of berths, quay cranes, and yards in transshipment hubs. Transshipment and import/export activities, general quay crane assignment decisions, and productivity losses incurred by quay crane interference are considered. A mixed-integer programming model is proposed to minimize the total cost, including speed-up cost, delay cost, penalty cost of vessels, and operational cost on container loading and unloading. An iterative heuristic is developed to solve the proposed model. Extensive numerical experiments are conducted to validate the effectiveness of the proposed model and the efficiency of the algorithm.

1. Introduction Container terminals are the most representative logistics facilities that handle inbound, outbound, and transshipment containers from and to seas and lands. Recently, the increasing number of containers and vessels poses new challenges to port management and resource scheduling due to scarce land, high labor cost, and limited technical equipment. Therefore, high operational efficiency must be sought via certain managerial tools. Accordingly, an increasing number of practitioners and scholars have studied widely and deeply on the manner in which efficiency and productivity can be improved using limited resources in terminals. Among these limited resources, berths, quay cranes (QCs), and yards are the most important and have attracted considerable research efforts. For surveys, we refer to the review works of Vis and Koster (2003), Steenken et al. (2004), Stahlbock and Voß (2008), Bierwirth and Meisel (2010, 2015). In this paper, we study the simultaneous allocations of berths, QCs, and yards under some practical situations. Generally, berth allocation problems (BAPs) concern assigning the berthing positions and times for arriving vessels. Imai et al. (1997) studied a BAP which can be reduced to a classical assignment problem. Lim (1998) transformed the BAP into a two-dimensional packing problem and proved that BAP is NP-Complete. Legato and Mazza (2001) developed a simulation model for a discrete BAP. Cordeau et al. (2005) presented a tabu search heuristic and Monaco and Sammarra (2007) developed a Lagrangean approach. Wang and Lim (2007) transformed the problem into a multiple stage decision making procedure and proposed a new multiple stage search method. Imai et al. (2008b) considered a variation of BAP at multi-user terminals. Hansen and Oguzb (2008) proposed a variable neighborhood search method and Golias et al. (2009) developed a genetic algorithm based heuristic. Lee et al. (2010) studied a continuous and dynamic BAP, and Giallombardo et al. (2010) integrated a tactical BAP. Buhrkal et al. (2011) reviewed and described three models of BAP. Zhen et al. (2011b) and Zhen (2015) studied operational and tactical decision models for BAP under uncertainty, respectively. Xiang et al. (2017) and Liu et al. (2017) also studied a robust BAP. Lalla-Ruiz et al. (2016) presented a set-partitioning-based model and Lalla-Ruiz et al. (2017) developed a popmusic-based approach. Imai et al. (2005) divided the BAP into three types: discrete, continuous, and hybrid. In the present study, we focus on the discrete berth type.

E-mail addresses: [email protected], [email protected]. https://doi.org/10.1016/j.tre.2019.11.008 Received 2 October 2018; Received in revised form 8 October 2019; Accepted 16 November 2019 1366-5545/ © 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Changchun Liu, Transportation Research Part E, https://doi.org/10.1016/j.tre.2019.11.008

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Recently, the integration of BAP and QC assignment problem (QCAP) has been widely studied by researchers. Handling times of vessels are influenced by assignment decisions for QCs. On this basis, advanced berth models have been integrated with QC-to-vessel assignments. Park and Kim (2003) presented an integration model for continuous BAP and QCAP that jointly determined berthing times, positions, and QC assignment decisions of each vessel. Imai et al. (2008a) considered the constraints that QCs cannot pass one another. Meisel and Bierwirth (2009) proposed three costs for vessels that should be considered, that is, speed-up, delay, and penalty costs. Liang et al. (2009) determined vessels’ berthing positions and times and the number of QCs allocated to each vessel simultaneously. Zhang et al. (2010) considered the coverage ranges of QCs and allowed for limited adjustments of QCs during loading and discharging. Giallombardo et al. (2010) considered discrete BAP and QCAP and proposed the concept of QC profiles to facilitate the combination of BAP and QCAP. Meisel and Bierwirth (2013) presented a framework for integrating BAP, QCAP, and QC scheduling. Vacca et al. (2013) proposed an exact branch-and-price algorithm that can solve instances with up to 20 vessels and 5 berths. Iris et al. (2015) extended existing integrated BAP and QCAP models by proposing a novel set of partitioning formulations. Li et al. (2015) solved an integrated model of the continuous BAP and specific QCAP with QC coverage range. Liu et al. (2016) focused on a rescheduling problem about the disruptions caused when QCs breakdown unexpectedly in the middle of the execution of a planned schedule. Iris et al. (2017) considered the decrease in the marginal productivity of QCs and the increase in handling times due to deviation from the desired position. In this paper, we consider general QC assignment decisions, which will assign specific QCs to each vessel at each time segment, and the productivity losses incurred by QC interferences. Yard allocation problem (YAP) aims to assign yard storage locations (sub-blocks) in yards to each vessel. YAP also aims to minimize transport distances or total transportation costs of moving containers (Chen and Lu, 2012; Zhen et al., 2016) between berths and the storage locations. Various studies have also investigated YAP. Kim and Hong (1999) considered the manner in which storage spaces can be allocated for import containers. Kim and Hong (2002) also studied problems regarding assignments of storage spaces and transfer cranes for handling import containers. Kim et al. (2000) determined storage locations of arriving export containers by considering their weights. Zhang et al. (2003) studied problems regarding storage space allocation by considering inbound and outbound containers. Kozan and Preston (2007) presented an iterative search algorithm to determine optimal locations and corresponding handling schedules. Chen and Lu (2012) considered problems regarding storage location assignments of outbound containers. Jin et al. (2016) studied problems regarding daily management of storage yards and proposed the concept of “yard crane profiles.” In the current study, we apply a consignment strategy, which was studied by Dekker et al. (2006, 2011a, 2016) in yard management. This strategy attempts to store containers together in particular dedicated storage areas for the same destination vessel. Transshipments are becoming increasingly popular and important in ports worldwide (Cordeau et al., 2007; Moccia et al., 2009; Nishimura et al., 2009; Bell et al., 2013; Fransoo and Lee, 2013; Wang et al., 2015). Besides, Lee and Chen (2009, 2008, 2013, 2016) studied yard templates that primarily concern problems regarding assignments of storage areas for transshipment hubs. In reality, BAP, QCAP, and YAP are constantly interrelated. The berth positions determined by BAP influence the assignment of sub-blocks to vessels to minimize transport distances, thereby reducing transportation cost. Meanwhile, the assignment of sub-blocks in YAP has a great influence on the vessels’ preferred berth positions, thereby affecting the BAP. Moreover, the start of berthing and departure time of vessels, which are determined by BAP, affect the workload distribution of yard cranes (YCs), thereby influencing the efficiency of YC and the traffic congestion of prime movers. Zhen et al. (2011a) studied integrated berth and yard template planning problems in transshipment hubs. They developed a heuristic algorithm for solving problems in large-scale realistic environments. Wang et al. (2018a) investigated an integrated optimization problem on the three major types of resources used in container terminals, namely, berth, QC, and yard storage spaces. They developed a column generation based heuristic procedure that could provide a lower bound for the integrated problem to solve the model on large-scale instances, in which they designed an exact pseudopolynomial algorithm for pricing problems. In the present study, we investigate an integration of BAP, QCAP, and YAP. We consider transshipment and import/export activities, general QC assignment decisions, and productivity losses incurred by QC interferences. For a clearer picture, the classification of the aforementioned and other related literature in terms of the addressed problems and the adopted methods is shown in Table 1. Zhen et al. (2011a), Wang et al. (2018a) studied an integration model for BAP, QCAP, and YAP. In comparison with the two studies, the present work will investigate a more practical problem.

• Zhen et al. (2011a), Wang et al. (2018a) only considered the transshipment containers that occupied 80% of containers utilized in the ports of Singapore. By contrast, the present study considers transshipment and import/export containers. • We propose an adaptive QC profile strategy to generate general QC assignment decisions. On one hand, we generate QC profiles in • •

each iteration, which can help facilitate the combination of BAP and QCAP. On the other hand, the mechanism of changing QC profiles in each iteration can enlarge search space, thus improving the solution quality compared with the traditional fixed QC profile strategy. In each iteration, the general QC assignment decision can be obtained on the basis of the selected QC profile. We also consider the productivity losses incurred by QC interferences. Because using the average productivity rate of QCs is imprecise and frequently causes overestimation or underestimation of handling times (Meisel and Bierwirth, 2013), we describe the productivity loss using an interference exponent (Schonfeld and Sharafeldien, 1985). We permit the violation of the latest end times and add large penalty costs into the objective if the departure of the vessels is beyond the latest end times to ensure that the problem is permanently feasible, even in extreme cases where a large number of vessels arrive simultaneously.

The structure of this article is as follows. Section 2 formulates the model. Section 3 proposes an iterative heuristic algorithm. Section 4 conducts numerical experiments and reports the results. Finally, Section 5 draws the conclusions. 2

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Table 1 The classification of the referred literature. Literature

BAP (1)

QCAP (2)

(3)

(4)

YAP

Solution Approach

(5)

Lim (1998)

–

Lee et al. (2010)

Greedy randomized adaptive search

Zhen et al. (2011b)

Meta-heuristic

Liu et al. (2017)

Relax-and-fix

Xiang et al. (2017)

Adaptive grey wolf optimizer

Cordeau et al. (2005)

Tabu search

Imai et al. (1997)

–

Legato and Mazza (2001)

Simulation

Monaco and Sammarra (2007)

Lagrangean approach

Imai et al. (2008b)

Genetic algorithm

Hansen and Oguzb (2008)

Variable neighborhood search

Golias et al. (2009)

Genetic algorithm

Buhrkal et al. (2011)

Set-partitioning

Lalla-Ruiz et al. (2016)

Set Partitioning

Zhen (2015)

Meta-heuristic

Lalla-Ruiz et al. (2017)

Popmusic-based approach

Meisel and Bierwirth (2009)

Lagrangian relaxation

Iris et al. (2015)

Set partitioning

Iris et al. (2017)

Two-stage heuristic

IIris and Lam (2019)

Two-stage heuristic

Wang et al. (2018b)

Branch-and-bound

Liang et al. (2009)

Genetic algorithm

Park and Kim (2003)

Lagrangian relaxation

Liu et al. (2016)

MIP-based relax-and-x

Imai et al. (2008a)

Genetic algorithm

Zhang et al. (2010)

Lagrangian relaxation

Li et al. (2015)

Spatiotemporal conict-based heuristic

Agra and Oliveira (2018)

Rolling horizon heuristic

Xiang et al. (2018)

Rolling horizon heuristic

Giallombardo et al. (2010)

Tabu search based heuristic

Vacca et al. (2013)

Branch-and-price

Xie et al. (2019)

Branch-and-price

Dekker et al. (2006)

–

Jin et al. (2016)

Divide-and-conquer

Zhen et al. (2016)

Heuristic

Zhen et al. (2011a)

Heuristic

Wang et al. (2018a)

Column-generation based heuristic

This study

Iterative heuristic

(1) Continuous; (2) Discrete; (3) Assign number of QCs; (4) Assign specific QCs; (5) QC profile.

2. Model formulation We formulate a mixed-integer linear programming (MILP) model in this section. We first describe the problem, then introduce the notations, objective function, and constraints. 2.1. Problem description Before formulating the integrated model, we first describe the background briefly. If readers want to know some details, they could reader some references, i.e., Zhen et al. (2011a), Wang et al. (2018a). BAP aims to assign the berthing position and time for each arriving vessel. For the detailed description of BAP’s objective, please refer to Meisel and Bierwirth (2009). We also consider the 1) to characterize the productivity loss. productivity losses incurred by QC interferences. We use an interference exponent , (0 < In practical applications, should be gathered from historical data observed for the terminals under consideration. For example, for terminals of the port of Kaohsiung (Taiwan), Chu and Huang (2002) reported that the values of ranges from 0.806 to 0.996. On this 3

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basis, we set = 0.9 in this paper. Traffic congestion commonly occurs in YAP. As previously mentioned, we use the consignment strategy. Thus, loading in the subblocks must be completed in a limited time, thereby resulting in the high workload of the YCs and prime movers near the sub-blocks. To avoid traffic congestion, two practical restrictions are given: (i) neighbor sub-blocks should not be reserved for the same vessel, and (ii) two or more sub-blocks in each block should not be reserved for one vessel considering the limited YCs deployed to each block. 2.2. Notation Input Parameters: V Set of vessels, which is indexed by i , j, V = {1, …, i, …, V } ; B Set of berths, which is indexed by b, B = {1, …, b, …, B } ; K Set of available sub-blocks, which is indexed by k, K = {1, …, k , …, K } ; H Total number of time segments in planning horizon; HE Extended planning horizon with consideration of periodicity; T Set of time segments, T = {1, …, H + HE } , which is indexed by t; P Total number of QCs, which is indexed by p; Ai Total operation time of QCs for vessel i, which is expressed as the number of crane-time segments, required to unload and load all the containers; MAi , MIi Maximum and minimum number of QCs that can be assigned to vessel i, respectively; [ai f , bi f ] Feasible service time segments for vessel i V ; [aie , bie ] Expected service time segments for vessel i V ; S Set of neighborhood pairs, which is indexed by s, S = {1, …, s, …, S } ; g Set of sub-blocks in the same block; G Set of all the blocks, g G ; ri Requirement of sub-blocks reserved for vessel i V , which is determined by the number of containers loaded onto vessel i V ; vij Number of containers transshipped from vessel i to vessel j; viI , viE Number of import and export containers for vessel i V , respectively; L U Dkb , Dkb Distance between berth segment b B and sub-block k K for loading and unloading operations, respectively; U DIb Distance between berth segment b B and the area reserved for import container for unloading operation. Here the area U contains multiple sub-blocks and DIb denotes the average distance; L DkE Distance between land and sub-block k K for loading operation. Here “E” denotes the gate location for export containers entering the terminal; ci1, ci2, ci3 Cost rate for speed-up, delay, and penalty of vessel i V , respectively; c 4 Cost rate for transporting a container in the yard; M Sufficiently large real number. Decision variables: Start berthing and departure time of vessel i, respectively; Start and finish time considering the periodicity for berth b; µi = 1, if the finishing time of vessel i exceeds bi f , that is, i > bi f , and 0 otherwise; it = 1, if vessel i starts loading/unloading in time segment t, and 0 otherwise; it = 1, if vessel i is handling in time segment t, and 0 otherwise; Xit Number of QCs assigned to vessel i at time segment t; ib = 1, if vessel i moors at berth b, and 0 otherwise; ijb = 1, if vessels i and j moor at berth b and vessel i berths before vessel j, and 0 otherwise; ik = 1 if sub-block k is assigned to vessel i, and 0 otherwise; itk = 1, if ik and it equal to 1, and 0 otherwise; Zipt = 1, if QC p is assigned to vessel i at time segment t, and 0 otherwise; Lipt = 1, if Zipt Zi, p 1, t = 1, and 0 otherwise, which denotes the left-most QC assigned to vessel i; Ript = 1, if Zipt Zi, p + 1, t = 1, and 0 otherwise, which denotes the right-most QC assigned to vessel i; a+ a + aie + i , respectively, which are used to linearize the item (aie Equal to aie i and i ) in the objective; i , i e e e + b+ b Equal to i bi and i + bi , respectively, which are used to linearize the item ( i bi ) in the objective; i , i jkib, jkb are binary variables, which equal to jk ib and jk jb , respectively. i,

i

b, b

2.3. Mathematical model We first show how to calculate the distances from berths to the sub-blocks. Yards are utilized as temporary container storages for shipping liners by using the consignment strategy (Han et al., 2008; Jiang et al., 2012). Further details about the calculation of the distances from berths to the sub-blocks are as follows. As shown in Fig. 1, four types of distances (namely, loading and unloading 4

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Fig. 1. An example for distance calculation.

distances for transshipment containers, unloading distance for import containers, and loading distance for export containers) should L U L U , Dkb , DIb , and DkE , respectively, whose definition are presented in Section 2.2. In practice, there may be calculated on the basis of Dkb exist more than one gates for import and export. For example in Fig. 1, gates 1, 2, and 3 are used to import and gates 4, 5, and 6 L U represent export gates. The distance DIb , and DkE are calculated by average values: U DIb =

D IU1 b + D IU2 b + D IU3 b 3

,

L DkE =

L L L D kE + D kE + D kE 4 5 6

3

As shown in Fig. 1, vessel j berths at berth 3, and its containers are stored at sub-blocks K24, K27, and K35. Therefore, r j = 3, where r j denotes the number of sub-blocks reserved for vessel j. The four types of distances are calculated as follows:

• Distance for unloading import containers of vessel j. This distance can be calculated as follows: U DIb

Dis1j =

jb

b B

• Distance for unloading transshipment containers of vessel j from vessel i. For the example in Fig. 1, the unloading average distance is

U + D U + DU D24,1 27,1 35,1

3

Dis2ij =

k K

, which can be calculated as:

b B

U Dkb

jk

ib

rj

=

k K

U Dkb

b B

jkib

rj

• Distance for loading export containers of vessel j. This distance contains two parts: the distance between berth and sub-blocks and the distance between sub-blocks and export gate location, which be calculated as follows:

Dis3j =

k K

L Dkb

b B

jk

jb

rj

+

L DkE

k K

jk

rj

k K

=

b B

L Dkb

jkb

+

k K

L DkE

jk

rj

• Distance for loading transshipment containers of vessel j from vessel i. For the example in Fig. 1, the loading average distance is

L +DL +DL D24,3 27,3 35,3

Dis4 ij =

3 k K

, which can be calculated as:

b B

rj

L Dkb

jk

jb

=

k K

b B

L Dkb

jkb

rj

5

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The objective function of the proposed model is given as follows.

[ci1 (aie

minZ =

+ ci2 (

+ i)

bie)+ + ci3 µi ] + c 4

i

i V

i V U viI DIb

+ c4

ib

[ci1

=

a+ i

L viE (DkE ik +

+ c4

i V b B

+ ci2

b+ i

L Dkb ikb )

+ ci3 µi ] + c 4

vij i V

U viI DIb

c4

b B

j V ,j i b B k K

ib

+

i V b B

j V ,j i b B k K

L viE (DkE ik +

c4

U L Dkb jkib + Dkb jkb rj

ri

i V k K

i V

+

vij

L b B Dkb ikb )

ri

i V k K

U L Dkb jkib + Dkb jkb rj

(1)

Objective function (1) aims to minimize the total cost, which includes the penalty costs for BAP and the operation costs for YAP. For BAP cost, we employ the objective function proposed by Meisel and Bierwirth (2009), which includes three kinds of costs: (i) total speed-up cost which is caused by berthing earlier than expected arrival time aie , (ii) total delay cost in departure later than bie , and (iii) total penalty cost which is caused by departing later than bi f . The YAP cost includes four kinds of transportation costs: (i) loading cost for transshipment containers, (ii) unloading cost for transshipment containers, (iii) unloading cost for import containers, and (iv) loading cost for export containers. The constraints are given as follows. (I) Berth-related constraints.

= 1,

ib

i

V

(2)

b B

H

= 1,

it

i

V

(3)

t=1 H i

=

it ,

i

V

(4)

t=1

H i

=

+

i

it

1,

i

j

+ M (1

V

(5)

t=1

H i

+

it

ijb ),

i, j

V, i

j, b

B

(6)

t=1

ijb

+

jib

0.5(

ijb

+

jib

ib

ib

+

+

jb ),

i, j

V, i

j, b

B

(7)

jb

1,

i, j

V, i

j, b

B

(8)

i

ai f ,

i

bi f

Mµi ,

i

i

bi f

M ( µi

1),

b

i

i

+ M (1

b

i

M (1

b

b

H

(9)

V

(10)

V

ib ), ib ),

1,

b

(11)

i

V

i

V, b

B

(12)

i

V, b

B

(13) (14)

B

aie

i

=

a+ i

a i

,

i

V

(15)

i

bie =

b+ i

b i

,

i

V

(16)

Constraint (2) indicates that each vessel can be allocated to one berth. Constraint (3) states that each vessel can only start at one time segment. Constraints (4) and (5) define the start of berthing and departure time of each vessel. Constraint (6) ensures that no conflict occurs for two vessels mooring at the same berth. Constraints (7) and (8) indicate a time sequence for two vessels berthing at the same berth. Constraint (9) guarantees that each vessel should berth after its earliest arrival time. Constraints (10) and (11) use the binary variable µi to determine whether the departure time for a vessel exceeds its latest departure time bi f . If µi = 1, a penalty cost is incurred. Constraints (12) and (13) define the start and finish times for berth b. Constraint (14) states that b b does not exceed a+ H 1. Constraint (15) relates aie and ia +. Constraint (16) defines the relationship between i bie and ib+, ib . i with i (II) Berth-QC-related constraints. 6

Transportation Research Part E xxx (xxxx) xxx–xxx

C. Liu i, t 1,

it

it

it

Ziqt ,

Xit

i

(17)

V , t = 1, …, H

(18)

V , p = 1, …, P , t = 1, …, H

M it ,

Xit

i

i

P,

(19)

V , t = 1, …, H

t

{HE + 1, …, H }

(20)

i V

Xit +

Xi, t + H

i V

P,

t

{1, …, HE }

(21)

i V

H

(Xit )

Ai ,

i

V

(22)

t=1

Zipt

1,

p = 1, …, P , t = 1, …, H

(23)

i V

1

(Zipt

(Zipt 1

Zi, p

Zi, p (Zipt

(Zipt

1, t )

1, t )

M (1

MLipt

Zi, p + 1, t )

Zi, p + 1, t )

0,

M (1

MRipt

0,

Lipt )

0,

i Ript )

i

i

V , p = 1, …, P , t = 1, …, H

(24)

V , p = 1, …, P , t = 1, …, H

(25)

0,

(26)

i

V , p = 1, …, P , t = 1, …, H

V , p = 1, …, P , t = 1, …, H

(27)

P

Lipt =

it ,

i

V , t = 1, …, H

Ript =

it ,

i

V , t = 1, …, H

Zipt = Xit ,

i

V , t = 1, …, H

(28)

p=1 P

(29)

p=1

P

(30)

p=1

Constraints (17) and (18) relate variables it and it , it and Ziqt , respectively. Constraint (19) relates Xit and it . Constraints (20) and (21) ensure that the number of QCs used in each time segment cannot exceed the QC capacity for the cyclical berth planning. Constraint (22) ensures the completion of the operation of vessel i. Here is a known parameter to characterize the productivity loss. Constraint (23) ensures that each QC can be assigned to no more than one vessel per time segment. Constraints (24) and (25) and (26) and (27) define Lipt and Ript , respectively. Constraints (28)–(30) ensure that all the QCs between the leftmost and rightmost QCs are assigned to a vessel. If it = 1, then there exists one Lipt = 1 and one Ript = 1. If there is one QC between the leftmost and rightmost QCs, which is not assigned to the vessel, there must have more than one Lipt and Ript equaling to 1. A contradiction occurs since it is a binary variable. 1), which is used to charIn Constraint (22), there exists a nonlinear item (Xit ) . Here is an interference exponent, (0 < acterize the productivity loss. It is a known parameter and we set = 0.9 in this paper. Evidently, Xit {0, MIi, …, MAi} , which has MAi MIi + 2 possible values. We let (Xit ) = 0Wit0 + (MIi ) WitMIi + …+ (MAi) WitMAi, i V , t = 1, …, H , then Constraint (22) can be replaced by Constraints (31) and (33).

Wit0 + WitMIi + …+ WitMAi = 1, Xit = H

0Wit0

+

MIi WitMIi

+

i

(31)

V , t = 1, …, H

…+ MAi WitMAi,

[0Wit0 + (MIi ) WitMIi+…+ (MAi ) WitMAi]

i

Ai ,

V , t = 1, …, H

i

V

(32) (33)

t=1

Here, are 0–1 binary variables. Constraint (31) ensures that Xit must be located in one and only one piece in {0, MIi, …, MAi } . Constraint (32) relates Xit and Wit0, WitMIi, …, WitMAi and Constraint (33) ensures the completion of the operation of vessel i. (III) Yard-related constraints.

WitMIi,

…, WitMAi

ik

1,

k

K

ik

= ri,

i

V

Wit0,

(34)

i V

(35)

k K

7

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C. Liu

itk

+

ik

1,

it

1,

itk

i

s

V, t

S, t

T, k

(36)

K

{HE + 1, …, H }

(37)

k s i V itk

+

k s i V

1,

i (t + H ) k

s

S, t

{1, …, HE }

(38)

k s i V

1,

itk

g

G, t

{HE + 1, …, H }

(39)

k g i V

itk

+

k g i V

1,

i (t + H ) k

g

G, t

{1, …, HE }

(40)

k g i V

jkib

jk

+

ib

1,

i, j

jkb

jk

+

jb

1,

j

V, i

j, k

V, k

K, b

K, b

B

(41)

B

(42)

Constraint (34) ensures that each sub-block can only be assigned to at most one vessel. Constraint (35) guarantees that the assigned sub-blocks should satisfy the requirement of vessels. Constraint (36) implies that itk = 1 if ik = 1 and it = 1. Constraints (37) and (38) ensure that two neighbor sub-blocks should not be assigned for the same vessel. Constraints (39) and (40) state that two or more sub-blocks in each block should not be reserved for one vessel in each block. Constraints (41) and (42) define jkib and jkb . (IV) Variables’ domains. i,

a+ i ,

i,

b, b

µi it ,

Xit

a i

0, {0, 1},

it ,

,

b+ i ,

b

B i

b i

0,

i

i

(45)

V i

V , t = 1, …, H

(47) (48)

{0, 1},

i

ijb

{0, 1},

i, j

ik

{0, 1},

i

V, k

itk

{0, 1},

i

V , t = 1, …, H , k

jkib,

jkb

(46)

V , t = 1, …, H

ib

Zipt , Lipt , Ript

(43)

V

(44)

{0, 1},

it

0,

V, b

V, i

{0, 1},

{0, 1},

B

j, b

(50)

K

i

i, j

(49)

B

(51)

K

(52)

V , p = 1, …, P , t = 1, …, H

V, k

K, i

j, b

(53)

B

3. Solution approach The proposed model is complex and commercial software, e.g., CPLEX and LINDO, can only solve small-scale problems. Therefore, an effective iterative heuristic is proposed to solve large-scale problems in this paper. In addition, we employ the framework of algorithm proposed by Zhen et al. (2011a) with some modifications. 3.1. General framework Through analyzing the problem structure, we divide the variables into three types: berth-related, QC-related, and yard-related variables. In each type, we divide it into major variables, associated variables, and auxiliary variables. The specific classification is shown in Table 2. Here the values of associated variables can be obtained once the major variables are determined. For example, once we determine the values of it and ib , we can obtain i by Constraints (4) and (17), and i through Constraints (4), (5) and (17). Table 2 Specific classification of decision variables. Berth-related variables Major

Associated Auxiliary

it , i,

QC-related variables

ib

i , b , b, µi , it , ijb a+ a b+ b i , i , i , i

Xit , Zipt

ik

Lipt , Ript

itk

Wit0, WitMIi, …, WitMAi

8

Yard-related variables

jkib, jkb

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The basic idea of the iterative heuristic is as follows: We first solve the integrated BAP and QCAP model, which aims to minimize the BAP cost with Constraints (2)–(21), (23)–(33), and obtain the values of it , ib , Xit , and Zipt . Then we fix a proportion ( ) of the berth- and QC-related decision variables (that is, it , ib , Xit , Zipt ) to optimize the YAP model, which minimizes the YAP cost with Constraints (34)–(42), and obtain the value of ik . In order to improve the solution quality, the procedure enters the iteration process. That is, we solve the integrated model with objective (1) and Constraints (2)–(21)(23)–(42) by fixing a proportion ( ) of the yardrelated decision variables, and then solve YAP model by fixing a proportion ( ) of the berth- and QC-related decision variables. The iterative process is repeated until the termination condition is satisfied. The framework of the algorithm is shown in Algorithm 1. Algorithm 1. Framework of iterative algorithm

3.2. Solving BAP and QCAP When the number of vessels is large, the integration of BAP and QCAP without consideration of the yard-related variables is still intractable (Agra and Oliveira, 2018). On the basis of the results of some preliminary experiments, the solver cannot find any feasible solution quickly for some instances. Therefore, to address this difficulty, we propose an adaptive QC profile strategy based modified rolling horizon heuristic (MRHH) to solve the integration of BAP and QCAP. In the adaptive QC profile strategy, QC profiles in each iteration is changeable. In other words, different QC profiles are generated in each iteration. This operation can not only facilitate the combination of BAP and QCAP, but also increase search space and improve the solution quality. In each iteration, the general QC assignment decisions ( Xit and Zipt ,) can be obtained according the selected QC profile. The main idea of the original rolling horizon heuristic (RHH) is to divide the planning horizon into smaller sub-horizon and then solve each subproblem repeatedly. On this basis, we propose an MRHH to solve the integration of BAP and QCAP. In comparison with the existing RHH, which has a fixed number of vessels in each iteration, the number of vessels in each iteration in the proposed MRHH is variable. In the following subsections, we will present some details used in the MRHH, e.g., QC profile generation method, vessel sequence generation rules, vessels determination method, enhancement technique, and formulation for subproblems. 3.2.1. QC profile generation The problems in the general QC assignment decisions are complex. Therefore, we proposed an adaptive QC profiles strategy to solve the QCAP. By comparing previous studies of Zhen et al. (2011a), Wang et al. (2018a), which generated QC profiles only once, we generate the set of possible QC profiles in each iteration. Such an operation can increase the diversity of QC profiles. For vessel i, its QC profile has three important parameters: (i) hiq , the operation time of vessel i by using QC profile q, (ii) qiqm , the number of QCs assigned to vessel i in the mth time segment for QC profile q, and (iii) liqm , =1, if qiqm > 0 , and 0, otherwise. himax himin + 1 Certain rules should be obeyed when generating the set of possible QC profiles. Here, we assume that Pi (where Pi denotes the set of possible QC profiles for vessel i, and himax and himin denote the maximum and minimum handling time of vessel i, respectively), which can guarantee that a QC profile exists in set Pi , whose handling time can be equal to the values between himin and himax . The algorithm for generating the QC profile set is shown in Algorithm 2.

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Algorithm 2. Algorithm for generating QC profile set

3.2.2. Sequence generation rules This subsection proposes specific rules to determine the sequence of vessels. Some rules are given as follows: time b ; • R1: The increasing order of the expected end • R2: The decreasing order of the cost rates c , c , and c , which reflect the weights of vessels; . • R3: The decreasing order of minq Pi hiq bi

f

1 i

e i

2 i

3 i

f

ai + 1

We use R1, R2, and R3 as the primary, secondary, and tertiary criteria to generate the vessels’ sequence, respectively. 3.2.3. Vessel determination in each subproblem Let {v1, v2, …, vn, …, vN } denote the sequence of vessels. We solve N vessels sequentially. In the nth iteration, we determine the vessels contained in subproblem VnB and the fixed vessel set VnF as follows:

• M1: V • M2: V • M3: V • M4: V

B n B n B n B n

= = = =

{vn {vn {vn {vn

anf anf ane ane

n} and VnF = {vn n} and VnF = {vn n} and VnF = {vn n} and VnF = {vn

bnf , n bne , n bnf , n bne , n

n n n n

< n} ; < n} ; < n} ; < n} .

An example is given as follows. A data set with 15 vessels is given in Table 3. The results are shown in Table 4. Evidently, VnB of Table 3 Data set. No. of vessels

aie

bie

ai f

bi f

No. of vessels

aie

bie

ai f

bi f

No. of vessels

aie

bie

ai f

bi f

1 2 3 4 5

0 0 2 5 10

4 5 6 8 15

0 0 0 0 0

11 12 14 14 21

6 7 8 9 10

13 14 20 24 27

17 17 23 27 30

5 8 14 18 21

25 23 29 33 36

11 12 13 14 15

28 30 33 35 37

31 34 36 40 40

22 22 27 25 31

37 42 42 47 46

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Table 4 Vessels contained in the subproblem (VnB ) for different methods. Iteration

M1

M2

M3

M4

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

{1, {2, {3, {4, {5, {6, {7, {8, {9,

{1, 2, 3, 4, 5} {2, 3, 4, 5, 6} {3, 4, 5, 6} {4, 5, 6, 7} {5, 6, 7, 8} {6, 7, 8} {7, 8} {8, 9, 10, 11, 12} {9, 10, 11, 12, 13, 14} {10, 11, 12, 13, 14} {11, 12, 13, 14, 15}

{1, 2, 3, 4, 5} {2, 3, 4, 5} {3, 4, 5, 6, 7} {4, 5, 6, 7} {5, 6, 7, 8} {6, 7, 8, 9} {7, 8} {8, 9, 10, 11} {9, 10, 11, 12, 13} {10, 11, 12, 13, 14} {11, 12, 13, 14, 15}

{1, 2, 3} {2, 3, 4} {3, 4} {4} {5, 6, 7} {6, 7} {7} {8} {9, 10} {10, 11, 12} {11, 12} {12, 13} {13, 14} {14, 15}

2, 3, 4, 5, 6, 7} 3, 4, 5, 6, 7} 4, 5, 6, 7, 8} 5, 6, 7, 8} 6, 7, 8, 9, 10} 7, 8, 9, 10, 11, 12} 8, 9, 10, 11, 12} 9, 10, 11, 12, 13, 14} 10, 11, 12, 13, 14, 15}

M1 is larger than that of M4. We also perform some experiments to find the best methods for determining the vessels in each subproblem in Section 4.2.2. In addition, when vN VnB , we only solve the remaining vessels once. 3.2.4. Enhancement technique for reducing computational time Some subproblems with the same vessel set will be resolved. For example, in Table 4, if we use M1, then the first, third, and seventh iterations will solve the same problem and obtain the same solution because the vessel set is the same. Thus, we conduct the following judgment before solving the subproblem at the nth iteration to reduce the computational time.

• If V V , then the subproblem at the nth iteration should no longer be solved, and the schedule of vessel n can be fixed as the solution obtained at the n 1th iteration; • Otherwise, we solve the subproblem at the nth iteration and obtain the solution of vessel n. B n 1

B n

Given this operation, we can reduce the number of subproblems. Formerly, the instance in Table 3 is solved with 9, 11, 11, and 14 iterations using the four methods. However, if we add the preceding judgment, then the number of iterations will be 6, 7, 9, and 9, respectively. 3.2.5. Formulation for subproblems We introduce two sets VnB, VnF in the nth iteration and formulate the model for the nth iteration as follows.

• Objective function. minZnB =

i VnB

[ci1

a+ i

+ ci2

b+ i

+ ci3 µi ]

(54)

• Fix the decision variables of the vessels in V . That is, the values of , n 1th iteration. The constraints in the subproblem contain two parts: • Constraints. – Constraints (2)–(21) and (23)–(33) by replacing vessel set V with set V . F n

it

ib ,

Xit and Zipt are equal to the values obtained in the

B n

– Some new constraints are added to relate the QC profile with QC assignment decisions Xit and Zipt . Some notations are defined: [ iq ] =1, if vessel i selects profile q; 0, otherwise; [ iqt ] =1, if both iq and it equal to 1; 0, otherwise. Some constraints are added as follows: iq

= 1,

i

V

(55)

q Pi

iqt

it

+

iq

1,

liqm

i (t m + 1) q Pi

i

iqt ,

V, q

i

Pi, t

V, t

T

T, m

(56)

{1, …, max {hiq}} p Pi

11

(57)

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qiq (t

m + 1) · iqm

= Xit ,

i

V , m = max{1, t

hiq + 1}, …, hiq

(58)

q Pi

Constraint (55) ensures that each vessel can only use one QC profile. Constraint (56) links three decision variables it , iqt . Constraint (57) links decision variables it and iqt . Constraint (58) defines the relationship between iqm and Xit .

iq ,

and

By solving the subproblem, we can select a QC profile ( iq ) for each vessel in each iteration. Then based on the value of iq , we can obtain the value of Xit through Constraints (56)–(58), and obtain the value of Zipt by Constraints (19), (23)–(30) based on the value of Xit . 3.2.6. Procedure of MRHH The adaptive QC profile strategy based MRHH is presented in Algorithm 3. If Algorithm 3 is used to solve the integrated model in Line 13 in Algorithm 3, then the QC profile used in the optimal solution will be reserved for the newly generated Pi . Algorithm 3. Procedure of adaptive QC profile strategy based MRHH

3.3. Solving YAP On the basis of the outputs of BAP and QCAP (that is, berth assignment ib and loading time yard-related decision variables, that is, ik . The YAP model is formulated as follows:

• Objective function. minZY =c 4

vij i V

+

j V ,j i b B k K L viE (DkE ik +

c4 i V k K

U L Dkb jkib + Dkb jkb rj

b B

U viI DIb

+ c4

it ),

we can solve the YAP and obtain

ib

i V b B

L Dkb ikb )

ri

(59)

• Constraints.

Constraints (34)–(42).

4. Computational results In this section, we conduct extensive numerical experiments to validate the effectiveness of the proposed solution approach. In addition, we run all codes on a 64-bit server machine (Configuration: 64-bit, 2.80 GHz, Intel(R) Core(TM) i7-7600U CPU). 4.1. Instance generation For the instance data, we refer to Zhen et al. (2011a), Wang et al. (2018a). We set the planning horizon as one week and H = 42 . 12

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Table 5 Scale of instances groups in experiments. Group ID

No. of vessels V

No. of berths B

No. of sub-blocks K

No. of QCs P

No. of time segments H

G1 G2 G3 G4 G5 G6 G7

15 20 30 35 45 50 60

2 3 4 5 6 7 8

80 120 160 200 240 280 320

5 7 11 12 16 18 20

42 42 42 42 42 42 42

Thus, each time segment contains 4 h. On the basis of the number of vessels, the instance can be divided into seven groups as shown in Table 5. Vessels are categorized into three classes: feeder, medium, and jumbo (Meisel and Bierwirth, 2009). vij is randomly generated in the ranges of [192, 480], [576, 1344], and [1440, 1920] for three vessel classes, respectively. viI , viO are randomly generated in the ranges of [24, 60], [72, 168], and [180, 240]. For the yard configurations, we refer to the model of Zhen et al. (2011a). We set c 4 = 2.5 × 10 6 in the following experiments. ri is randomly generated in the ranges of [2, 3], [4, 7], [8, 10] for three vessel classes. We randomly distribute aie , ai f , bie , and bi f along the planning horizon. For each vessel, bie aie is approximately the same length as the average length ai f is approximately five times as long as the average length of its handling time. of its handling time, and bi f 4.2. Parameter and rule determination for MRHH In this section, three experiments are conducted.

• Investigate the effectiveness of vessel sequence generation rules in Section 4.2.1. • Select a vessel determination method from methods M1, M2, M3, and M4 in Section 4.2.2. • Determine the best probability for fixing the decision variables in Section 4.2.3. 4.2.1. Determination of vessel sequence generation methods In this subsection, we investigate the influences of different vessel sequence generation methods on the solution quality. Three rules have been introduced in Section 3.2.2. By adjusting the priority of these different rules, we obtain six sequence generation methods. We adopt M1 as the vessel determination method. We solve the integrated BAP and QCAP (Line 5 in Algorithm 1) and the average BAP cost and CPU time in seconds are reported in Table 6. As shown in Table 6, using R1 as the primary criterion performs better than using R2 or R3 as the primary criterion in terms of cost and CPU time. This result is caused by the last few vessels in the sequence, which have a small solution space because the resources are occupied by high weighted vessels, thereby resulting in a large objective value and long CPU time. Moreover, using R2 as the secondary criterion has a better performance than using R3 by comparing the results in column R1,R2,R3 and R1,R3,R2. In conclusion, R1,R2,R3 is the best vessel sequence generation method among the six combinations and will be used in the following experiments. 4.2.2. Influence of vessel determination method in each subproblem In this subsection, we investigate the influence of vessel determination method in each subproblem by using the MRHH on the solution quality. We solve the same problem in Section 4.2.1 and the average BAP cost and CPU time in seconds in the first iteration are reported in Table 7. From the table, the quality of solution decreases from M1 to M4. Therefore, M1 has the best performance among the four Table 6 Average BAP cost and CPU time (second) under different sequence generation methods. Group

G1 G2 G3 G4 G5 G6 G7 Avg.

R1,R2,R3

R1,R3,R2

R2,R1,R3

R2,R3,R1

R3,R1,R2

R3,R2,R1

Cost

Time

Cost

Time

Cost

Time

Cost

Time

Cost

Time

Cost

Time

13.2 10.6 3.2 11.5 25.2 5.9 6.7 10.9

12.2 7.6 13.2 28.3 99.1 85.1 74.3 45.7

13.3 10.7 3.2 11.6 25.3 5.9 6.7 11.0

12.3 7.7 13.3 28.5 99.6 85.1 74.9 45.9

21.0 17.2 5.3 19.0 41.3 9.2 10.5 17.6

30.8 19.4 38.9 78.3 264.7 236.0 219.1 126.7

21.4 17.3 5.4 19.0 41.6 9.2 11.0 17.8

30.9 19.5 40.3 80.0 268.3 242.5 226.7 129.7

23.3 18.4 5.4 20.4 42.8 10.1 11.8 18.9

36.9 25.0 39.9 87.0 321.7 258.5 234.5 143.3

23.4 18.7 5.6 20.4 44.7 10.5 12.2 19.4

36.9 25.5 40.6 90.3 322.3 263.3 242.7 145.9

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Table 7 Average BAP cost and CPU time (second) under different vessels determination method. Group

G1 G2 G3 G4 G5 G6 G7 Avg.

M1

M2

M3

M4

Cost

Time

Cost

Time

Cost

Time

Cost

Time

13.2 10.6 3.2 11.5 25.2 5.9 6.7 10.9

12.2 7.6 13.2 28.3 99.1 85.1 74.3 45.7

14.4 10.6 3.9 12.4 29.3 13.5 10.0 13.4

1.8 2.4 5.9 9.2 25.0 35.0 20.5 14.3

14.0 11.2 4.3 21.5 37.6 12.3 11.7 16.1

1.0 1.6 3.7 7.3 17.6 20.1 11.5 9.0

15.4 16.8 7.5 19.7 61.0 19.3 18.2 22.6

0.5 1.0 2.3 3.3 7.9 11.8 7.9 5.0

Note: The time is the average CPU time per subproblem.

methods. From the results of M4, a small value of VnB may lead to worst objective value. Thus, in view of the solution quality and CPU time, we select M1 to determine the vessels in each subproblem in the following experiments. 4.2.3. Influence of variable fixed probability A major difference between our algorithm and the heuristic algorithm proposed by Zhen et al. (2011a) is that we can allow a proportion 1 of yard-related decisions to be re-optimized while solving the integrated model, and vice versa. Evidently, as increases, the CPU time spent on each iteration decreases. However, a good solution can be found if we re-optimize additional decision variables (as decreases). Therefore, we need to find an appropriate to balance the CPU time and solution quality. The preliminary experiments show that the algorithm converges slowly. Thus, we vary from 0.8 to 1.0 with a step of 0.02 to run the instances in groups G1 and G3. The results are shown in Fig. 2. From the figure, the solution quality will initially increase and then remain stable as decreases. The CPU time has a rapid growth trend as decreases because it becomes increasingly difficult to be solved in each iteration. Therefore, we set = 0.92 in the following experiments in terms of solution quality and computational time. 4.3. Algorithm performance To evaluate the performance of the proposed heuristic, we compare the obtained results with the lower bound LBdecoup proposed by Zhen et al. (2011a), which is obtained by decoupling the original model and solving the independent berth and yard problems separately. We elaborate the details for calculating LBdecoup in the Appendix in Zhen et al. (2011a). The obtained LBdecoup is obviously no greater than the optimal value of the original problem. We use LBdecoup as the benchmark to evaluate the proposed heuristics. Table 8 displays the comparison of the proposed heuristics with the lower bound (LBdecoup ). The value of lower bound (LB), the computation time of obtaining LB (t1), the objective value obtained by our proposed method (Obj), the computation time of proposed Obj LB method (t2 ), and the relative gap (Gap), which is calculated by Gap = LB × 100%, are reported. The average gaps of the instances in Groups G1, G2, and G3 are 3.6%, 4.5%, and 6.7%, respectively. The experiment results imply that the proposed heuristic is effective for solving the proposed model. 4.4. System analysis In this section, we conduct some experiments to analyze the effects of yard congestion, the percentage of import/export containers, and vessels’ time windows. 4.4.1. Analysis of yard congestion In this subsection, we analyze the effect of yard congestion, which is presented by Constraints (37)–(40). We initially solve the model without considering Constraints (37)–(40), and then we resolve the model with consideration of Constraints (37)–(40). The following indicators are recorded: (a) objective values; (b) CPU time; (c) Count 1: the number of cases that two neighbor sub-blocks are loading simultaneously; and (d) Count 2 : the number of cases that each block has more than one sub-block are loading simultaneously. Table 9 reports the results of the instances with 15–60 vessels. As shown in Table 9, the problem without considering yard congestion has a smaller CPU time than that where yard congestion is considered and is thus more easily solvable. Moreover, the objective values obtained by the problem without considering yard congestion is lower than that considered yard congestion. Constraints (37)–(40) enforce that two neighbor sub-blocks cannot have loading process simultaneously at each time segment in the planning horizon, and the number of sub-blocks that are in the loading process cannot exceed one in each block. Without considering Constraints (37)–(40), the obtained schedule has a large Count 1 and Count 2 , which results in yard congestion. 4.4.2. Influence of the percentage of import and export containers To investigate whether the percentage of import/export containers affects the solutions, we vary the percentage of import/export 14

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Fig. 2. Objective values and CPU time with varying

for instances groups G1 and G3.

Table 8 The objective value and CPU time (minute) of instances G1, G2, G3 for LBdecoup and proposed algorithm. ID

1 2 3 4 5 6 7 8 9 10 Avg.

G1 (15 vessels)

G2 (20 vessels)

G3 (30 vessels)

LB

t1

Obj

t2

Gap

LB

t1

Obj

t2

Gap (%)

LB

t1

Obj

t2

Gap (%)

23.41 21.97 19.42 27.41 17.03 26.00 20.53 19.78 25.01 17.99

3.9 3.1 9.0 2.9 3.8 18.3 5.0 6.0 11.4 11.1

23.78 22.33 20.61 28.55 18.00 26.37 21.49 20.71 25.65 18.59

4.1 3.4 9.6 3.3 4.3 19.6 5.8 6.9 13.3 12.3 3.6

1.6 1.7 6.1 4.2 5.7 1.4 4.7 4.7 2.6 3.3

26.16 27.15 26.06 23.43 29.64 27.82 23.97 27.22 26.81 30.70

7.3 4.4 11.7 19.3 4.7 11.1 4.5 16.1 4.1 6.8

27.28 27.68 27.42 24.54 31.79 29.43 24.61 28.90 28.32 31.02

8.5 4.8 13.0 21.1 5.3 12.4 5.0 18.5 4.7 8.0

4.3 2.0 5.2 4.7 7.3 5.8 2.7 6.2 5.6 1.0 4.5

41.77 46.18 42.84 43.35 43.73 45.31 42.72 40.92 50.38 43.22

31.5 16.9 39.1 23.0 67.5 105.7 8.3 27.1 41.3 22.2

44.75 49.18 45.86 46.78 46.63 48.89 44.84 43.83 53.63 45.40

36.2 18.4 43.1 24.4 71.2 120.8 9.5 31.6 43.6 23.8

7.2 6.5 7.0 7.9 6.6 7.9 5.0 7.1 6.4 5.0 6.7

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Table 9 Performance analysis of yard congestion. Group

G1 G2 G3 G4 G5 G6 G7

Without consideration of yard congestion Obj

Time (min)

19.46 24.32 39.85 59.42 83.60 109.91 136.92

0.4 2.8 6.6 9.0 24.0 40.2 77.9

Count

1

117.3 172.1 260.8 329.2 431.6 507.3 623.6

With consideration of yard congestion

2

Obj

Time (min)

259.5 392.8 541.6 634.1 834.1 960.5 1085.7

22.61 28.10 46.98 72.08 100.66 126.69 160.70

8.3 10.1 42.2 36.7 52.3 56.5 98.5

Count

Count

1 0 0 0 0 0 0 0

Count

2 0 0 0 0 0 0 0

Fig. 3. Impact of the percentage of import/export containers.

containers from 0 to 1 with a step of 0.1. The results of instance groups G2 and G4 are shown in Fig. 3. From the figure, as the percentage of import/export containers increases, the objective values tend to decease for instance groups G2 and G4. Thus, the transshipment containers have a greater influence on the total cost than the import/export containers. 4.4.3. Effects of vessels’ time windows ai f When generating the instances, bie aie is approximately the same as long as the average length of its handling time and bi f is approximately five times as long as the average length of its handling time. In reality, the time windows of vessels affect the decisions. Let Z =

bi f

bie

ai f aie

. We attempt to investigate the manner how time windows affect the decisions. In these experiments, we

vary Z from 1 to 8 with a step of 0.5. The results are shown in Fig. 4. As shown in Fig. 4, the average objective values of all three instance groups tend to decrease with the increase of Z. This behavior is caused by the fact that when Z is relatively small, the penalty cost becomes high to ensure the feasibility of solutions; otherwise, when Z is relatively high, the penalty cost becomes low. When Z 5, the objective values remain stable. In these situations, the time windows have no influence on the decisions. In conclusion, the managerial implications are given as follows. First, yard congestion must be considered in the planning stage to balance the workload of yard and prime movers. Second, the transshipment containers have a greater influence on the total cost than the import/export containers. Third, the port managers need to strive for as much feasible service time as possible when negotiating with shipping companies, thereby reducing total operational cost. 5. Conclusions In this study, we model and solve the discrete integration of BAP, QCAP, and YAP in container terminals. In comparison with other scholars’ work on the related areas, the primary contributions of this study are as follows. We consider transshipment and import/export containers for integrating BAP, QCAP and YAP. We also consider general QC assignment decisions and the 16

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Fig. 4. Impact of vessels’ time windows for instance groups G1, G4 and G7. x-axis and y-axis denote the Z value and objective values, respectively.

productivity losses incurred by QC interferences. We describe the productivity losses using an interference exponent. We propose an iterative heuristic algorithm to solve this integrated model. We first solve the integrated BAP and QCAP model, and then we fix a proportion of the berth- and QC-related decision variables to optimize the YAP model. In order to improve the solution quality, the procedure enters the iteration process. That is, we solve the integrated model by fixing a proportion of the yard-related decision variables, and then solve YAP model by fixing a proportion of the berth- and QC-related decision variables. The iterative process is repeated until the termination condition is satisfied. When solving the integration of BAP and QCAP, an adaptive QC profile strategy based MRHH is proposed. On the basis of some realistic instances, we conduct extensive numerical experiments to validate the effectiveness and efficiency of the proposed algorithm. We further conduct some experiments to analyze the effects of yard congestion, percentage of import/export containers, vessels’ time windows, and integrated optimization on the system. Acknowledgements This work is supported by Beijing Municipal Science and Technology Project (No. Z181100003118001). Constructive comments from five anonymous reviewers improved the quality of the paper. References Agra, A., Oliveira, M., 2018. Mip approaches for the integrated berth allocation and quay crane assignment and scheduling problem. Eur. J. Oper. Res. 264 (1), 138–148. Bell, M.G.H., Liu, X., Angeloudis, P., Fonzone, A., Hosseinloo, S.H., 2013. A frequency-based maritime container assignment model. Transp. Res. Part B 58 (8), 58–70. Bierwirth, C., Meisel, F., 2010. A survey of berth allocation and quay crane scheduling problems in container terminals. Eur. J. Oper. Res. 202 (3), 615–627. Bierwirth, C., Meisel, F., 2015. A follow-up survey of berth allocation and quay crane scheduling problems in container terminals. Eur. J. Oper. Res. 244 (3), 675–689. Buhrkal, K., Zuglian, S., Ropke, S., Larsen, J., Lusby, R., 2011. Models for the discrete berth allocation problem: a computational comparison. Transp. Res. Part E 47 (4), 461–473. Chen, L., Lu, Z., 2012. The storage location assignment problem for outbound containers in a maritime terminal. Int. J. Prod. Econ. 135 (1), 73–80. Chu, C., Huang, W., 2002. Aggregates cranes handling capacity of container terminals: the port of kaohsiung. Marit. Policy Manage. 29 (4), 341–350. Cordeau, J.F., Gaudioso, M., Laportea, G., 2007. The service allocation problem at the gioia tauro maritime terminal. Eur. J. Oper. Res. 176 (2), 1167–1184. Cordeau, J.F., Laporte, G., Legato, P., Moccia, L., 2005. Models and tabu search heuristics for the berth-allocation problem. Transp. Sci. 39 (4), 526–538. Dekker, R., Voogd, P., Asperen, E.V., 2006. Advanced methods for container stacking. Or Spectrum 28 (4), 563–586. Fransoo, J.C., Lee, C.Y., 2013. The critical role of ocean container transport in global supply chain performance. Prod. Oper. Manage. 22 (2), 253–268. Giallombardo, G., Moccia, L., Salani, M., Vacca, I., 2010. Modeling and solving the tactical berth allocation problem. Transp. Res. Part B 44 (2), 232–245. Golias, M.M., Boile, M., Theofanis, S., 2009. Berth scheduling by customer service differentiation: a multi-objective approach. Transp. Res. Part E Logist. Transp. Rev. 45 (6), 878–892. Han, Y., Lee, L.H., Chew, E.P., Tan, K.C., 2008. A yard storage strategy for minimizing traffic congestion in a marine container transshipment hub. Or Spectrum 30 (4), 697–720. Hansen, P., Oguzb, C., 2008. Variable neighborhood search for minimum cost berth allocation. Eur. J. Oper. Res. 191 (3), 636–649. Imai, A., Chen, H.C., Nishimura, E., Papadimitriou, S., 2008a. The simultaneous berth and quay crane allocation problem. Transp. Res. Part E Logist. Transp. Rev. 44 (5), 900–920. Imai, A., Nagaiwa, K., Tat, C.W., 1997. Efficient planning of berth allocation for container terminals in asia. J. Adv. Transp. 31 (1), 75–94. Imai, A., Nishimura, E., Papadimitriou, S., 2008b. Berthing ships at a multi-user container terminal with a limited quay capacity. Transp. Res. Part E Logist. Transp. Rev. 44 (1), 136–151. Imai, A., Sun, X., Nishimura, E., Papadimitriou, S., 2005. Berth allocation in a container port: using a continuous location space approach. Transp. Res. Part B 39 (3),

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