Container fleet sizing and empty repositioning in liner shipping systems

Transportation Research Part E 45 (2009) 860–877

Contents lists available at ScienceDirect

Transportation Research Part E

journal homepage: www.elsevier .com/locate / t re

Container fleet sizing and empty repositioning in liner shipping systems

Jing-Xin Dong, Dong-Ping Song *

International Shipping and Logistics Group, Business School, University of Plymouth, Cookworthy Building, Drake Circus, Plymouth PL4 8AA, UK

a r t i c l e i n f o

Article history:Received 10 November 2008Received in revised form 12 March 2009Accepted 7 May 2009

Keywords:Fleet sizingEmpty container repositioningThreshold control policyGenetic AlgorithmsEvolutionary StrategyStochasticSimulation

1366-5545/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.tre.2009.05.001

* Corresponding author. Tel.: +44 01752 585630;E-mail addresses: [email protected] (

a b s t r a c t

This paper considers the joint container fleet sizing and empty container repositioningproblem in multi-vessel, multi-port and multi-voyage shipping systems with dynamic,uncertain and imbalanced customer demands. The objective is to minimize the expectedtotal costs including inventory-holding costs, lifting-on/lifting-off costs, transportationcosts, repositioning costs, and lost-sale penalty costs. A simulation-based optimization toolis developed to optimize the container fleet size and the parameterized empty reposition-ing policy simultaneously. The optimization procedure is based on Genetic Algorithms andEvolutionary Strategy combined with an adjustment mechanism. Case studies are given todemonstrate the results.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In the last few decades, ever expanding global economy leads to rapid growth of container shipping. Container vessel car-rying capacity and world container traffic both have maintained two digits annual growth rates (www.ci-online.co.uk). Theworld container fleet size also keeps growing around 10% pa according to the Institute of International Lessor (www.iicl.org).Liner shipping is a special sector of industries that involve a huge amount of investment in assets such as vessels, containers,chassis, etc.

Shipping lines operate a number of vessels that are deployed on specific shipping routes. A shipping route may be definedas a sequenced list of ports. A schedule is the timetable of when each vessel will call at each port in the shipping route. Forexample, a Europe–Asia route may consists of a few ports in Europe and a few ports in Asia. Usually eight to ten vessels willbe deployed in such route to provide weekly service. Vessels will continuously run round-trips according to the publishedschedules. One round trip is often termed as a voyage. It should be pointed out that in one round trip some ports may becalled at several times (e.g. hub ports) and different ports may be called at different weekdays. Laden containers are gener-ated according to customer demands and will be carried by vessels from origin ports to destination ports. After laden con-tainers have reached destinations, they will be unpacked and become empty. Empty containers may be stored in ports forfuture reuse or repositioned to other ports to meet demands there. Therefore, a liner shipping system provides a regulartransport service for a specific shipping network, which involves multiple vessels, multiple ports and multiple voyages (Songet al., 2005).

Container availability is essential for shipping lines to meet customer demands. However, the management of containerfleets is complicated and requires intelligent trade-off among many activities. For example, smaller container fleet sizes

. All rights reserved.

fax: +44 01752 585713.J.-X. Dong), [email protected] (D.-P. Song).

http://www.ci-online.co.uk

http://www.iicl.org

mailto:[email protected]

mailto:[email protected]

http://www.sciencedirect.com/science/journal/13665545

http://www.elsevier.com/locate/tre

J.-X. Dong, D.-P. Song / Transportation Research Part E 45 (2009) 860–877 861

would save capital investment and inventory costs, but this may increase risk of losing customer demands due to unavail-ability. Repositioning empty containers from one port to another is an essential mechanism to overcome the trade imbalancearising from most of the geographical locations. It was reported that empty containers have accounted for at least 20% ofglobal port handling activity ever since 1998 (Drewry, 2006). The imbalance of trade demands has become even more prom-inent in recent years (Song and Carter, 2009). Repositioning may reduce container waiting times and increase their utiliza-tion, but it incurs additional transportation and handling costs and occupies precious vessel spaces, especially when theywere repositioned to a port where demands fall or too many empties have already arrived.

The fleet sizing problem and the empty repositioning problem are highly related. On the one hand, large fleet sizes maydecrease the requirements of empty repositioning. On the other hand, efficient empty repositioning could improve con-tainer’s utilization and therefore equivalently increase the fleet capacity. In the broad context of general vehicle fleet sizingand empty vehicle allocation, a rich literature exists (e.g. Dejax and Crainic 1987; Cheung and Powell, 1996; Crainic, 2000).However, as pointed out by Kochel et al. (2003), only a few papers considered the joint fleet sizing and empty allocationproblem. For example, Beaujon and Turnquist (1991) presented a non-linear network optimization model to optimize deci-sions on sizing a vehicle fleet and utilizing the fleet simultaneously. Du and Hall (1997) applied inventory and queuing the-ory and proposed a threshold policy to redistribute empties in a hub-and-spoke system. A decomposition approach was thendeveloped to determine the fleet size and control parameters. Kochel et al. (2003) combined the simulation with GeneticAlgorithms to deal with the fleet sizing and vehicle allocation problem. Song and Earl (2008) showed that the optimal emptyvehicle repositioning policy is of the threshold control type for a two-depot service system and the optimal fleet size and theoptimal threshold values can be derived analytically.

The above few studies on fleet sizing and empty repositioning mainly concentrated on vehicle fleet management in inlandtransportation systems, which is significantly different from container fleet management in liner shipping industry. Linershipping systems have fixed and regular schedules for vessels while inland vehicles are usually not. Containers have to becarried by vessels and their movements are subject to constraints such as vessels’ pre-specified routes, timetables and car-rying capacities. The empty container repositioning is further constrained by dynamic customer demands and vessel sparecapacities since laden containers have priority to be shipped. In addition, because one port in a shipping route may be calledseveral times in one round-trip, containers designated to different ports should be distinguished and appropriately selectedto be loaded onto a vessel when it calls.

In the line of empty container repositioning problem, much of the work focused on deterministic systems (e.g. White,1972; Shen and Khoong, 1995; Bourbeau et al., 2000; Choong et al., 2002; Erera et al., 2005; Olivo et al., 2005; Cheangand Lim, 2005; Jula et al., 2006; Shintani et al., 2007; Song and Carter, 2009). The stochastic nature of the problem has at-tracted attention since 1990s and a few mathematical models have been developed, e.g. a two-stage model and mix integerprogram for inland empty container allocation among ports, depots and customers (Crainic et al. 1993a,b); simulation model(Lai et al., 1995), two-stage stochastic network model (Cheung and Chen, 1998), dynamic programming models (Li et al.,2004; Song, 2007; Lam et al., 2007), inventory-based heuristic policies (Li et al., 2007; Song and Dong, 2008) for empty con-tainer repositioning between sea ports. However, little research has been reported on the joint optimization of containerfleet sizing and empty container repositioning in liner shipping systems, to which we attempt to contribute.

The rest of this paper is organized as follows: in next section, we formulate the joint fleet sizing and empty containerrepositioning problem mathematically, and explain the evolution of the system in an event-driven mode. In Section 3, solu-tion methods are addressed. The decision-making process is first explained and a parameterized rule-based repositioningpolicy is then presented. A simple heuristic algorithm is proposed to determine the parameters in the rule-based policy.The structure of the simulation model is briefly described. In Section 4, a simulation-based optimization procedure is devel-oped based on Genetic Algorithm and Evolutionary Strategy to optimize the container fleet size and control parameters inthe rule-based policy. In Section 5, a range of numerical examples are given to demonstrate the effectiveness of the proposedapproach by comparing the ‘optimized’ policy with the heuristic policy and the non-repositioning policy. Finally, conclusionsare made in Section 6.

2. Problem formulation

Consider a shipping service system that is composed of a fleet of vessels, a fleet of containers and a set of ports in order tomeet external customer demands, which represent the requirements of empty containers to generate laden containers andthe requirements of moving laden containers from original ports to destination ports. Vessels make regular round trips alongthe ports based on pre-specified route and timetable. When a vessel arrives at a port, it will unload laden/empty containersfirst, and then collect laden/empty containers from the current port and continue its journey.

An example of liner service routes is shown in Fig. 1. This is a Europe–Asia service route, in which eight vessels aredeployed to call at ports: Southampton (SOU), Hamburg (HAM), Rotterdam (ROT), Port Klang (PKL), Singapore (SIN), She-kou (SK), Hong Kong (HK), Ningbo (NB), Shanghai (SH), Yantian (YT), Hong Kong (HK), Singapore (SIN), in sequence and thenback to Southampton. The transit time in days from Southampton to other ports and the days of calling at each port areshown in Fig. 1. The round-trip takes 56 days and eight vessels provide a weekly service for the shipping route, e.g. on everyWednesday there will be a vessel calls at Southampton. However, eight vessels may have slightly different carryingcapacities.

SOU

HAM

ROT

PKL

SIN

SK

YT

HK

NB

SHPort Trans. time Day of callSOU 0 WedHAM 4 SunROT 6 TuePKL 23 FriSIN 24 SatSK 27 TueHK 28 WedNB 30 FriSH 31 SatYT 35 WedHK 36 ThuSIN 40 MonSOU 56 Wed

Fig. 1. A liner shipping service route.

862 J.-X. Dong, D.-P. Song / Transportation Research Part E 45 (2009) 860–877

In practice, the operation of the shipping system differs in many details. We make the following assumptions:

1. The container fleet is a decision variable but fixed in one experiment. No short-term leasing is considered.2. Only one type of containers, 20-foot equivalent unit (TEU) is considered.3. Vessels have finite capacities and sail according to pre-specified service routes and schedules. Every vessel will repeat the

same route pattern.4. Meeting customer demands and repositioning empty containers are also decision variables that are made and executed

when a vessel departs from a port.5. Unloaded laden/empty containers from a vessel can be used to meet customer demands in the current port when the ves-

sel departs from the port. This implies that after the laden container arrived at its destination, they become immediatelyavailable.

6. Only the demands that received 1 day before the vessel departure will be satisfied. Unmet demands due to unavailabilityof empty containers or insufficient vessel capacity will be lost and incur lost-sale costs. The demands received in the lastday will be satisfied by next vessel.

The validity of the above assumptions is explained below. Fleet sizing is a, relatively, long-term decision. It is reasonableto assume it to be fixed in one experiment. Because short-term leasing is often expensive, shipping lines prefer to lease con-tainers in long-term. The long-term leased containers can be treated as owned containers (e.g. Cheung and Chen, 1998). Inpractice, 40-foot containers are often used, which could be treated as two TEUs (e.g. Li et al., 2004, 2007; Choong et al., 2002).As for assumption 3, shipping companies often publish their service schedules 6 months in advance and it is a common prac-tice to repeat the route pattern and maintain the regularity. As for assumption 4, due to the dynamic and stochastic nature ofthe underlying systems, it is reasonable to assume the use of decentralized decision-making mechanism in an event-drivenmode at the operational level. Assumption 5 is another common assumption in the literature (e.g. Li et al., 2004, 2007;Choong et al., 2002). Because our focus is on the seaborne container transportation and inland transportation time is relativeshort compared with seaborne travel time, we assume unloaded containers are immediately available for reuse. As forassumption 6, the 1 day delay reflects the fact that a certain period of time is required in port to prepare for loading; thelost sale assumption may be justified by the facts that container shipping is a very competitive industry, many similar ser-vices are provided by competitors, and customers may be impatient.

The problem can be formulated in an event-driven way. The state of the system is only updated when a vessel arrival ordeparture event occurs. These events naturally divide the system evolution process into a series of stages.

2.1. Notation

To describe the system, we first introduce the following notation.

k
the kth event, which could be either an arrival event or departure event of a vessel tk the kth event occurring time N the number of ports in the system. Ports are coded by number 1, 2, . . ., N nij(tk) the customer demands (in TEUs) received by the shipping company on day tk from port i to port j, which follows a
random variable nij


x
a sample process of the customer demands in the planning horizon. For a given sample, the demands {nij(tk), for any i,j, k} are fixed
Chi
the inventory holding cost per TEU per day at port i
Cpi
the penalty cost for unmet demands per TEU at port i
Coi
the lifting-on cost per TEU at port i
Cfi
the lifting-off cost per TEU at port i
Cij
the transportation cost per TEU (for both laden and empty) from port i to port j Nc the container fleet size in TEUs, which is a decision variable rv the carrying capacity in TEUs of vessel v dij(k) the cumulative demands from port i to port j, which are accumulated from the time that the last vessel (in the same
direction) departed from the port i to the occurring epoch of event k
Pv(k,i) the set of ports in the route that vessel v will call at during the period from the time it departs from port i at tk to the
time the same vessel returns to the port i. For the example in Fig. 1, if a vessel from Europe to Asia calls at Singaporeat tk, then Pv(k,SIN) = {SK, HK, NB, SH, YT}; on the other hand, if a vessel from Asia to Europe calls at Singapore at tk,then Pv(k,SIN) = {SOU, HAM, ROT, PKL}

si(k)
the inventory level of empty containers at port i just after event k, and si(0) represents the initial number of emptycontainers at port i
xvijðkÞ
the number of empty containers on vessel v from port i to port j just after event k. It is a decision variable if vessel v
departs from port i at tk; otherwise, it is a state variable
yv
ijðkÞ
the number of laden containers on vessel v from port i to port j just after event k. It is a decision variable if vessel vdeparts from port i at tk; otherwise, it is a state variable
p
a control policy which determines how to satisfy customer demands and how to reposition empty containers in theplanning horizon. It may be represented by fyv
lj ðkÞ; xvlj ðkÞ : anyl; j; v ; and kg

In order to facilitate the formulation of the system in an event-driven way, the following identification functions are intro-duced to describe the events in the system.

p(v,k,A) = i
if vessel v arrives at port i at time epoch tk; otherwise takes �1 p(v,k,D) = i if vessel v departs from port i at time epoch tk; otherwise takes �1 I{condition} the indicator function. It takes 1 if the condition is true, otherwise 0
In the container shipping system under consideration, there are three types of decisions: container fleet size Nc; laden con-tainers to be loaded on a vessel when the vessel departs from a port, i.e. yv

ijðkÞ when p(v,k,D) = i; empty containers to beloaded on a vessel to others when the vessel departs from a port, i.e. xv

ijðkÞ when p(v,k,D) = i. It should be pointed out thatif p(v,k,D) – i, then both yv

ijðkÞ and xvijðkÞ are not decision variables, instead, they are regarded as state variables representing

the laden and empty containers on vessel v at time tk.

2.2. System evolution for an arrival event

Suppose a vessel v in the shipping service arrives at port i at time tk. The following activities can be observed: (i) the cus-tomer demands for each port-pair are accumulated; (ii) laden and empty containers on board of vessel v that are designatedto port i are unloaded; (iii) the states of inventory levels at ports and states of containers on vessel v are then updated.Mathematically,

dljðkÞ ¼ dljðk� 1Þ þ nljðtk�1 þ 1Þ þ � � � þ nljðtkÞ for 8l; j 2 f1;2; . . . ;Ng; ð1Þ

siðkÞ ¼ siðk� 1Þ þX

l½xv

li ðk� 1Þ þ yvli ðk� 1Þ�; ð2Þ

sjðkÞ ¼ sjðk� 1Þ for 8j 2 f1;2; . . . ;Ng n fig; ð3Þxv

li ðkÞ ¼ 0 for 8l 2 f1;2; . . . ;Ng; ð4Þyv

li ðkÞ ¼ 0 for 8l 2 f1;2; . . . ;Ng; ð5Þxv

lj ðkÞ ¼ xvljðk� 1Þ for 8l 2 f1;2; . . . ;Ng; 8j 2 f1;2; . . . ;Ng n fig; ð6Þ

yvlj ðkÞ ¼ yv

lj ðk� 1Þ for 8l 2 f1;2; . . . ;Ng; 8j 2 f1;2; . . . ;Ng n fig; ð7Þ

xuljðkÞ ¼ xu

ljðk� 1Þ for 8u – v ; 8l; j 2 f1;2; . . . ;Ng; ð8Þ

yuljðkÞ ¼ yu

ljðk� 1Þ for 8u – v ; 8l; j 2 f1;2; . . . ;Ng: ð9Þ


Eq. (1) represents the activity to accumulate customer demands from the previous event to the current event. Eq. (2) rep-resents that laden and empty containers are unloaded from vessel v to port i and the inventory of empty containers at porti is updated. The inventory of empty containers at other ports remains the same as shown in (3), where f1;2; . . . ;Ng n figdenotes the set of all ports excluding port i. Eqs. (4) and (5) reset the numbers of empty and laden containers on vessel vdesignated to port i to be 0, because they have just been unloaded. Eqs. (6) and (7) indicate that all the containers on vesselv that are not designated to port i remain on board. Finally, Eqs. (8) and (9) represent that the state variables for any othervessel remain the same.

2.3. System evolution for a departure event

Suppose a vessel v in the shipping service departs from port i at time tk. The occurring activities will be: (i) the customerdemands for each port-pair are accumulated; (ii) the customer demands that arrived 1 day before the vessel departure aresatisfied as much as possible, whereas unmet demands will be lost; (iii) make decisions on empty container repositioning, i.e.how many to reposition out and where to go; (iv) load laden and empty containers onto vessel v; (v) update states of inven-tory levels at ports and states of containers on vessel v. Mathematically,

d0ljðkÞ ¼ dljðk� 1Þ þ nljðtk�1 þ 1Þ þ � � � þ nljðtkÞ for 8l; j 2 f1;2; . . . ;Ng; ð10ÞXj

yvijðkÞ ¼minfsiðk� 1Þ; rv �

Xl

Xj

½xvlj ðk� 1Þ þ yv

lj ðk� 1Þ�;X

j

ðd0ijðkÞ � nijðtkÞÞg; ð11Þ

0 6X

j

xvijðkÞ 6minfsiðk� 1Þ �

Xj

yvijðkÞ; rv �

Xl

Xj


lj ðk� 1Þ� �X

j

yvijðkÞg; ð12Þ

siðkÞ ¼ siðk� 1Þ �X

j

½yvijðkÞ þ xv

ijðkÞ�; ð13Þ

sjðkÞ ¼ sjðk� 1Þ for 8j 2 f1;2; . . . ;Ng n fig; ð14Þ

yvlj ðkÞ ¼ yv

ljðk� 1Þ for 8l 2 f1;2; . . . ;Ng n fig; 8j 2 f1;2; . . . ;Ng; ð15Þ

xvljðkÞ ¼ xv

lj ðk� 1Þ for 8l 2 f1;2; . . . ;Ng n fig; j 2 f1;2; . . . ;Ng; ð16Þ

dijðkÞ ¼ nijðtkÞ for any j 2 Pvðk; iÞ; ð17Þ

dijðkÞ ¼ d0ijðkÞ for any j =2 Pvðk; iÞ; ð18Þ

dljðkÞ ¼ d0ljðkÞ for 8l 2 f1;2; . . . ;Ng n fig; 8j 2 f1;2; . . . ;Ng; ð19Þ

xuljðkÞ ¼ xu

ljðk� 1Þ for 8u – v; 8l; j 2 f1;2; . . . ;Ng; ð20Þ

yuljðkÞ ¼ yu

ljðk� 1Þ for 8u – v ; 8l; j 2 f1;2; . . . ;Ng: ð21Þ

Different from the vessel arrival event, here the cumulative demands are updated in two steps. The first step is to receive thecustomer demands and the second step is to update the demand states after meeting demands. To avoid the confusion, weuse d0ijðkÞ to represent the state of demands in the first step (i.e. after receiving customer demands), and dij(k) to represent thestate of demands in the second step (i.e. after meeting demands).

Eq. (10) represents the activity to accumulate customer demands from the previous event to the current event. Eqs. (11)and (12) represent the activities to meet customer demands and reposition empty containers following the decisions subjectto the constraints of container inventories and vessel capacity. Here the decisions on meeting demands and repositioningempties are based on the company’s managerial policy. Eqs. (13) and (14) update the inventory of empty containers at allports. Eqs. (15) and (16) indicate that all the containers on vessel v that are not originated from port i remain on board.Eq. (17) represents that unmet demands from current port i to every port in Pv(k, i) will be lost except those received inthe last day. Eq. (18) represents that customer demands from current port i to every port that does not belong to Pv(k, i) willremain at the port. It should be pointed out that for any j R Pv(k, i), we should have yv

ijðkÞ ¼ 0 and xvijðkÞ ¼ 0 since the vessel

will not visit those ports before it calls at the current port i again. Eq. (19) indicates that for any other port the cumulativedemands remain the same. Finally, Eqs. (20) and (21) represent that the state variables for any other vessel remain the same.

2.4. Objective function

The problem is to find the optimal fleet size and the optimal control policy that minimize the following expected totalcost,

JðNc;pÞ ¼ EJðNc;p;xÞ ¼ limn!1

1n

Xn

i¼1

JðNc;p;xiÞ; ð22Þ


where J(Nc, p, xi) is a sample cost for all events in the planning horizon, and xi is an individual sample process, which is arealization of the stochastic process of the customer demands. The sample cost consists of inventory-holding costs, lost-salepenalty costs, lifting-on costs, lifting-off costs, and laden/empty container transportation costs. More specifically,

JðNc;p;xÞ ¼X

k

Xi

Chi � siðk� 1Þ � ðtk � tk�1Þ þ

Xk

Xi

Cpi �X

u

Xv

Xj

½dij � nijðtkÞ � yvijðkÞ� � Ifj 2 Pvðk; iÞg

� Ifpðv; k;DÞ ¼ ig þX

k

Xi

Coi �X

u

Xv

Xj

½xvijðkÞ þ yv

ijðkÞ� � Ifpðv; k;DÞ ¼ ig

þX

k

Xi

Cfi �X

u

Xv

Xj

½xvjiðk� 1Þ þ yv

jiðk� 1Þ� � Ifpðv; k;AÞ ¼ ig

þX

k

Xi

Xu

Xv

Ci�1;i �X

j

Xl

½xvjl ðk� 1Þ þ yv

jl ðk� 1Þ� � Ifpðv ; k;AÞ ¼ ig: ð23Þ

The first term on the right hand side of (23) represents the empty container holding costs at ports, which is related to theinventory level and the storage time. The second term represents the penalty costs incurred by losing unmet customer de-mands due to insufficient vessel capacity or unavailable empty containers. The third term represents the lifting-on costs forboth laden and empty containers. The fourth term represents the lifting-off costs for laden and empty containers. Finally, thefifth term represents the container transportation costs, where Ci–1,i represents the transportation costs per TEU moved fromthe previous port in the service route u to the current port i. Here we assume that the transportation cost is addictive toocean legs. This is reasonable since distance is the main factor for shipping operational costs. The transportation costs areonly calculated for the vessel arrival events. In other words, when a vessel docks at a port, no transportation cost isconsidered.

3. Solution methods

This section tries to solve the optimization problem in (22). However, for service routes with more than two ports andmultiple deployed vessels, it is extremely difficult to find the optimal fleet size Nc and the optimal policy p analytically.In this section, we first discuss the decision-making process for meeting customer demands and repositioning empty con-tainers, and simplify the problem; then present heuristics for a rule-based policy; finally describe a simulator that will beused later to optimize the rule-based policy.

3.1. Decision-making: value-based policy and parameterized rule-based policy

From the definition, a control policy p determines how to satisfy customer demands and how to reposition empty con-tainers. We may classify control policies into two groups: value-based policies and parameterised rule-based policies.

The policy in the first group can be represented by a series of values that describe the numbers of laden and empty con-tainers moved by each vessel for each port-pair at each event, i.e. p ¼ fyv

lj ðkÞ; xvlj ðkÞ : any l; j;v ; and kg. The decision-making

process is to design fyvljðkÞ; xv

ljðkÞ : any l; j; v; and kg in advance. This type of policies is often used in classic mathematicalprogramming. It is a natural and intuitive definition for the deterministic systems. However, in the situations with dynamicoperations and stochastic nature, it is difficult to find good value-based policies and even more difficult to execute the pol-icies, because the values specified in yv

lj ðkÞ and xvljðkÞ may be feasible for one sample process, but infeasible for other sample

processes. For example, if the actual available empty containers at port l at time tk are smaller than xvljðkÞ in the designed

policy due to uncertainty in customer demands, then the operator cannot execute the policy as planned. Therefore, addi-tional mechanisms are required in order to apply such value-based policies to stochastic systems.

The policy in the second group is characterized by a set of rules and a set of parameters. The decision-making process is toapply the rules and utilize the parameters to determine fyv

lj ðkÞ; xvljðkÞ : any l; j;v; and kg dynamically. The key difference is

that the numbers of containers moved by each vessel are not designed in advance under the policy in the second group. In-stead, the rules and the parameters should be designed in advance. In other words, the rules and parameters should be thesame for different sample processes, while fyv

lj ðkÞ; xvlj ðkÞ : any l; j;v ; and kg are probabilistic functions depending on the

realization of the stochastic process. In practice, this type of policies appears to be more appropriate since they provide flex-ibility to cope with dynamic changes and unexpected fluctuations. Many inventory-based policies belong to this type (e.g. Liet al., 2004, 2007; Song, 2007; Song and Dong, 2008).

We restrict ourselves to a class of simple structured control policies that belong to the second group. It is assumed thatcustomer demands should be satisfied as much as possible in order to maximize the revenue and also assume that the cus-tomers with the largest volume will have priority to be satisfied. These two rules determine the decision variables fyv

ijðkÞgwhen p(v, k, D) = i, in respond to dynamic system state and uncertain customer demands. However, if j R Pv(k, i), thenyv

ijðkÞ ¼ 0 because those demands should not be met by the current vessel since it will not call at the destination ports beforeit visits the current port again.

The empty repositioning decisions are determined by threshold control parameters. We assume that each port i has a pairof threshold parameters (Di, Ui) representing the lower and upper levels of empty container inventories. The empty reposi-tioning requirements can be described as follows: if the empty container inventory is greater than the upper bound Ui, then


empty containers should be moved out of port i and bring the inventory level back to Ui; if it is less than the lower bound Di,then empty containers should be moved into port i from other ports and bring the inventory level up to Di; otherwise, norepositioning is required for port i.

Under the above threshold repositioning policies, the empty repositioning is essentially implemented in two steps foreach departure event as described below. Suppose k is a departure event of vessel v from port i, i.e. p(v, k, D) = i;

Step 1: Compute the estimated number of empty containers to be repositioned out or repositioned in for each port(denoted as EOv

j ðkÞ or EIvj ðkÞ) based on inventory of empty containers on hand and the threshold parameters:

s0iðkÞ :¼ siðk� 1Þ �X

j

yvijðkÞ;

s0jðkÞ :¼ sjðk� 1Þ for j – i;

EOvj ðkÞ ¼ s0jðkÞ � Uj; and EIvj ðkÞ ¼ 0; if s0jðkÞ > Uj for any j;

EOvj ðkÞ ¼ 0; and EIvj ðkÞ ¼ Dj � s0jðkÞ; if s0jðkÞ < Dj for any j;

EOvj ðkÞ ¼ 0; and EIvj ðkÞ ¼ 0; if Dj 6 s0jðkÞ 6 Uj for any j;

where s0iðkÞ represents the inventory of empty containers at port i after meeting customer demands at event k; and s0jðkÞ forj – i represents the inventory of empty containers at port j at event k.

Step 2: Determine the actual number of empty containers to be repositioned out from the current port i (denoted asXv

i ðkÞ) and split it over destination ports in proportion to those ports’ requirements,

Xvi ðkÞ :¼min EOv

i ðkÞ;X

j

EIvj ðkÞ; rv �X

l

Xj


lj ðk� 1Þ� �X

j

yvijðkÞ

( );

xvijðkÞ ¼ Xv

i ðkÞ � EIvj ðkÞ=X

j

EIvj ðkÞ; if j 2 Pvðk; iÞ;

xvijðkÞ ¼ 0; if j =2 Pvðk; iÞ:

In step 2, the min operation imposes the constraints to the estimated exporting empties from the current port by the totalrequirements from other ports and the vessel spare capacity. We apply the linear rationing rule to split the empty containeramount among destination ports. This rule reflects the fact that the company may tend to bring the container inventory levelback to the target stock level more evenly across the route. In practice, many other rules could be used, e.g. the nearest portfirst, the most demanding port first. Nevertheless, the simulation-based evolutionary optimization tool developed in Section4 can be applied in the same way. The reason for setting xv

ijðkÞ ¼ 0 for j R Pv(k, i) in step 2 is that the allocated empty con-tainers to those ports should not be loaded onto the current vessel since it will not call at the destination ports before it visitsthe current port again.

From now on, we focus on the above threshold repositioning policies characterized by a set of parameters {(Di, Ui):1 6 i 6 N}. In other words, the control policy p is essentially determined by {(Di, Ui): 1 6 i 6 N} since the rules to meet cus-tomer demands are fixed. The problem is therefore simplified as to find the optimal fleet size Nc and the optimal parameters{(Di, Ui): 1 6 i 6 N} that minimize the expected total cost. However, due to the complexity of the underlying system, it is stilldifficult to derive the analytical solutions. We present two alternatives: the first is based on heuristics and the second isbased on a simulator. The simulator will then be used as an evaluation tool in a simulation-based evolutionary optimizationapproach proposed in Section 4.

3.2. Heuristics

By utilizing the statistics of customer demands and the frequency of the shipping service, the threshold parameters thatcharacterize the empty container repositioning may be determined heuristically. Let lij and rij denote the mean and thestandard deviation of the daily demand nij. Let Ts be the time interval of consecutive vessel calls in the shipping service(determined by the frequency of the service), e.g. for a weekly service we have Ts = 7. In fact, most shipping services inthe current container shipping industry are weekly services.

We use the following heuristics to determine the threshold parameters. The upper level for port i is determined by theweekly total exports plus its standard deviation. Here the weekly total exports are estimated by summarizing the daily aver-age export demands over one week. Taking port i as an example, its weekly total exports are given by

Pjlij � 7. The lower

level for port i is determined by the weekly net exports and the weekly total export’s standard deviation. A port’s dailynet exports are defined as the differences between total daily export demands and total daily import demands, e.g.P

jlij �P

ilij for port i. The reason to use weekly net exports instead of weekly total exports in determining the lower levelis to encourage the movements of empty containers from surplus ports to deficit ports and minimize the movements of emp-ties in the opposite direction. More specially, the threshold parameters are given by


Ui ¼X

j

lij � Ts þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX

j

ðrijÞ2 � Ts

sfor any i;

Di ¼ max 0;X

j

lij �X

i

lij

!� Ts

( )þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXj

ðrijÞ2 � Ts

sfor any i:

We call the above heuristics the heuristic repositioning policy (HRP). The advantage of HRP is that the parameters {(Di, Ui):1 6 i 6 N} can be easily determined using the statistics of the underlying stochastic systems (i.e. the average and standard devi-ation of customer demands). Now the remaining question is how to determine the container fleet size. This is a single parameteroptimization problem. Note that the trade-off relationship between inventory costs and lost-sale costs, it is reasonable to as-sume that the optimal container fleet size cannot be too small or too large, and the cost function could have a U-shape withrespect to the fleet size. An efficient approach to optimize a U-shape function over an interval without using derivatives isthe golden section search method (Gerald and Wheatley, 2004), which can be used together with HRP to solve (22).

The main concern for the heuristic repositioning policy is that the threshold parameters are not optimal, and there is noindication on how good HRP is. In that sense, a simulation-based optimization approach may be more appropriate. We de-scribe the simulator that we developed in next section and present the simulation-based evolutionary algorithm in Section 4.

3.3. Simulator

Simulation has been very successful in modelling complex systems and solving optimization problems (Fu, 1994). Thehigh-level structure of the simulation model for our systems is described in Fig. 2. The purpose of the simulation modelis to evaluate the performance for any given scenario under a set of control parameters representing the fleet size andthreshold values.

The simulation consists of two important modules: the simulator module and the decision-making module. The simulatormodule takes a set of input data including: customer demands, vessels with attributes (e.g. capacities), ports with attributes(e.g. cost coefficients), shipping network with distances between ports, shipping routes with sequenced port lists, voyages(specifying the timetable of each vessel during the simulation). The simulator module handles the occurring vessel arrivalor departure events and drives the evolution of the system states. The simulator interacts with the decision-making module,executes the decisions such as meeting customer demands and repositioning empty containers, and handles the laden andempty lifting-on/off activities.

The decision-making module takes a set of control parameters as inputs. It generates the container fleet according to thecontrol parameter. The generated container fleet is then used as a fixed asset in the simulator. The decision-making modulealso determines the quantities of laden and empty containers to be loaded on vessel, and splits empty containers over des-tination ports. This decision-making process is performed dynamically in event-driven mode based on the input thresholdvalues, customer demands and dynamic information obtained from the simulator module.

The simulation completes its current run when neither arrival events nor departure events are found in the system. Thecumulative performance measures such as total costs and individual cost components are outputted from the simulator.

4. Simulation-based evolutionary optimization

This section presents a simulation-based optimization tool to seek the optimal fleet size and the optimal thresholdparameters simultaneously. Combined with the simulation, we developed an evolutionary optimization algorithm basedon Genetic Algorithms (GA) and Evolutionary Strategy (ES). Both GA and ES are random search methods that take overthe principle of biological evolution for the optimization of technical systems (Goldberg, 1989; Back, 1996). ES uses real

Decision-making

Vessel, port,attributesContainers

Network,route, voyage

Output and feedback systemperformance

Fleet sizing

Customerdemands

Simulator

Handle vessel arrival anddeparture events

Handle laden and emptycontainers

Emptyrepositioning

rule

Dynamicinformation

Decisions

Controlparameters

Fig. 2. A high-level structure of the simulation model.

Initialization AdjustmentEvaluation viasimulation

Termination

Reduce mutationdeviation

Mutation

Recombination

Selection

Best solutionYes

No

Fig. 3. The simulation-based evolutionary algorithm.


variable to represent a solution and is more suitable for parameter optimization (Song et al., 2006). The output of the sim-ulation-based optimization approach is termed as an evolutionary algorithm-based policy (EAP).

For the system under consideration, a solution can be represented by a vector of non-negative integers, s: = (Nc,D1, U1, . . ., DN, UN), where (Di, Ui) are the lower and upper threshold parameters for port i and N is the number of ports inthe system. The expected total cost can be rewritten as J(s). Our evolutionary algorithm is a modified version based on ESand GA and combined with an adjustment mechanism by considering the structure of the underlying problem. The proce-dure of the simulation-based evolutionary algorithm is illustrated in Fig. 3.

The procedure consists of several components as depicted in Fig. 3 and also involves a few control parameters to executethe operations, which are introduced below.

4.1. Initialization

In each generation, a set of individuals (solutions) will be manipulated. Those individuals are called the parent genera-tion; its size is denoted as NP. Initially, all parents are the same, in which the container fleet size is initialized and uniformlydistributed among all ports; the threshold parameters for all ports are set as zero. This implies that there is no empty con-tainer repositioning in the system under the initial solution. The initial solution is evaluated through the simulation.

4.2. Selection and recombination

The evolutionary algorithm is performed generation by generation. The parent generation will produce the same numberof the offspring generation. The parent generation is updated according to the following mechanism: if the offspring gener-ation includes the best solution up to now, then the whole offspring generation will be selected and copied into the nextparent generation; otherwise, the worst individual in the offspring generation is replaced by the best solution up to now,and then the offspring generation is copied into the next parent generation. Such selection mechanism is a modified elitistselection and guarantees that the best solution up to now is always in the parent generation.

In the recombination operation, two individuals have to be selected from the parent population to generate a descendant.The Roulette-wheel selection is used in our algorithm, which probabilistically selects individuals based on their fitness val-ues (Goldberg, 1989). Let J(sp) denote the expected total cost under the solution represented by parent sp. The fitness value ofeach parent is defined as: Fp: = max{J(sp): 1 6 p 6 NP} – J(sp). The individuals’ expected probabilities to be selected are givenby Fp/RpFp. Individuals are then mapped one-to-one into contiguous intervals in the range [0, 1]. The size of each individualinterval corresponds to the fitness value of the associated individual. Clearly, the fittest individual occupies the largest inter-val; whereas the less fit have correspondingly smaller intervals within the roulette wheel. To select an individual, a randomnumber is generated in the interval [0, 1] and the individual whose segment spans the random number is selected.

There are many different recombination (or crossover) operators that can be performed on real-values chromosome,e.g. simple crossover, two-point crossover, uniform crossover, arithmetical crossover, BLX crossover (Herrera et al.,2003). Although the crossover operators may affect the efficiency of the search process, the quality of solutions is oftenreasonably close. In our experiment, we will use the uniform crossover which is described below. After selecting the par-ents, a descendant is generated by randomly copying the elements from two selected parents. However, the pair of ele-ments (threshold values) corresponding to the same port is copied into the descendant together due to their closerelationship. An example is shown in Fig. 4, in which the shaded elements are copied from parent 2 while other elementsare copied from parent 1.

4.3. Mutation

The mutation operation enables a descendant’s genotype to differ from that of its parents. The deviations refer to indi-vidual genes, and are random and independent of each other. For example, for a descendant s(n) at the nth generation, a uni-form random vector z is generated such that its first element follows a uniform distribution z1 � U(�cn, cn), while otherelements follow another uniform distribution zi � U(�cn/N, cn/N) for 1 < i 6 2N + 1. It is reasonable to deviate the first

Nc

Nc

Parent 1

Parent 2

Descendant

Fleet size and threshold parameters

Nc

U1D1 U2D2 UNDN... ...

U1D1 U2D2 UNDN... ...

U1D1 U2D2... ... UNDN

Fig. 4. Illustration of the recombination operation.


element in a larger degree than other elements since the first element represents the total fleet size while the others are forinventory levels at individual ports. Here cn indicates the degree of deviation for mutation at the nth generation. Then, themutation operator is done by setting s(n) = s(n) + z.

4.4. Adjustment

After mutation, each descendant s(n) is adjusted by taking into account the knowledge of the underlying system. Firstly,the fleet size Nc will be checked and adjusted if necessary to make sure it falls within a specified interval, e.g. between 0 andthree times of the total vessel capacity. Secondly, if a port is a surplus port (i.e. its total import exceeds its total export), itscorresponding lower threshold value is set to be 0, while its upper threshold value is ensured to be non-negative. The reasonis that surplus ports do not require empty containers to be repositioned in. For example, European ports and American portsare surplus ports and there is no need to reposition empty containers to those ports. Such knowledge is often available toshipping companies in advance based on historical data. Thirdly, if a port is a deficit port (i.e. its total import is less thanits total export), then its corresponding threshold values will be adjusted as follows: (i) if a threshold value is less than 0,then it is set as the average daily net exports (here the net exports are defined the difference between exports and imports)multiplied by the service frequency (e.g. multiplied by seven for weekly services); (ii) check each pair of threshold valuescorresponding to the same port, if Di > Ui, then swap their position to make sure that the upper level is not less than the lowerlevel. The above adjustment may be justified by the fact that those deficit ports require empty containers to be repositionedin and should maintain a certain level of inventory if possible.

4.5. Evaluation via simulation

For each fixed descendant s(n), its first element is extracted and used to generate container fleet in the simulation. Thecontainer fleet is then uniformly distributed among ports. Other elements in the descendant are used to make empty con-tainer repositioning decisions dynamically during the simulation. The simulation is then run to evaluate the total costcorresponding to the descendant s(n) and feedback the performance measure to the optimization procedure. In otherwords, the ‘Control parameters’ box and the ‘Output and feedback system performance’ box in Fig. 2 are the interfacebetween the simulation model and the optimization procedure. It should be pointed out that the total number of runsrequired in the evolutionary optimization procedure could be NP � NG (i.e. the population multiplied by the generations),where NG is the maximum number of generations. Ideally, for each solution, multiple samples should be run to averagethe sample costs in order to obtain the accurate expected total cost. However, this will dramatically increase the compu-tation effort.

4.6. Termination criteria and mutation deviation reduction

Let nc denote the number of consecutive generations in which no improvement is achieved for the cost function. It will beset to be 0 whenever an improvement is achieved. The deviation for mutation cn is reduced by a constant factor a if nc isgreater than a predetermined number (denoted by Ns). In other words, Ns is a threshold value to trigger the reduction ofthe deviation cn. The mutation deviation reduction is a common operation in Evolutionary Strategy (Back, 1996). It bearsthe similarity to the temperature cooling mechanism in the simulated annealing method. By reducing the mutation devia-tion, it essentially adjusts the search step sizes. The justification is that we may search the best parameters using larger stepsizes in earlier stages, while using smaller step sizes in later stages.

In this paper, it is assumed that the optimization procedure is terminated if cn < 1 or n > NG. The best solution up to now,denoted by s*, is then returned as the ‘‘optimal” solution. This solution provides liner operators with the information aboutthe container fleet size and the threshold control parameters for each port in the shipping system. However, it should bepointed out that multiple high-quality local optima might exist due to the combinatorial nature of the problem.

5. Numerical examples

We have presented a heuristics repositioning policy (HRP) in Section 3, and an evolutionary algorithm-based policy (EAP)in Section 4. To facilitate the comparison, the non-repositioning policy (NRP), is also introduced in numerical examples.


Under the NRP policy, only laden containers will be loaded onto and unloaded from vessels, and no empty containers will berepositioned explicitly. This policy will be used as a reference point to quantify the scale of benefits achieved by other twopolicies through better repositioning.

A range of numerical examples is experimented in this section. In principle, the expected total cost should be estimatedby averaging over multiple sample processes. However, it is very time-consuming to use multiple sample processes to eval-uate every new solution, particularly in EAP. Therefore, we first apply NRP, HRP, and EAP to two case studies based on a sin-gle sample process, contrast their relative differences, and demonstrate the convergence of the evolutionary algorithm.Secondly, the obtained results are evaluated through 100 random samples to perform the statistical analysis on whetherthe results are valid generally. Thirdly, more numerical computations are conducted to examine the sensitivity of the resultswith respect to system parameters.

In obtaining EAP, the simulation-based evolutionary algorithm includes a number of control parameters such as NG (par-ent population size), c0 (initial deviation for mutation), a (reducing factor of the mutation deviation), Ns (the threshold valueto trigger deviation reduction), and NG (the maximum number of generations). The performance of evolutionary algorithmsmay be affected by the control parameters, which are often chosen based on pilot runs. In this paper, the control parametersare set as follows: NP = 10, which is about one to three times of the number of ports in the shipping route; c0 = 2000 for case 1and c0 = 1000 for case 2, which is about the average vessel capacity; a = 0.9; Ns = 5; and NG = 100. The initial fleet size in theevolutionary algorithm is 9000 TEUs for both cases, which is close to the total vessel capacity. The initial threshold values inthe evolutionary algorithm are set to be 0 for all ports. The termination criteria are set by either NG � NP = 1000 (i.e. the prod-uct of generations and parent population equals 1000) or cn < 1 (i.e. the mutation deviation is less than 1).

For NRP and HRP, meeting customer demands and empty container repositioning are implicit, but there is no mechanismto determine the container fleet size. As discussed in Section 3.2, the golden section search method (Gerald and Wheatley,2004) can be used to find the best fleet size for NRP and HRP. The interval of the fleet size is set to be [5000, 50,000] in allscenarios. This interval should include the optimal fleet size since the lower bound is much less than the total vessel carryingcapacity, while the upper bound is more than five times of the total vessel carrying capacity. The termination criterion of thegolden section search is that the searching interval of the fleet size bounds becomes less than 100.

5.1. Case study 1

The first case is a trans-pacific shipping service (CLX) provided by a Chinese shipping line (www.ci-online.co.uk). Theshipping route consists of three ports: Ningbo (NB), Shanghai (SH), and Long Beach (LB) as shown in Fig. 5. The transit timesbetween ports (including berth time at ports) are also shown in Fig. 5.

There are four vessels deployed in this shipping route and the journey time on a round trip for each vessel is 28 days. Theyprovide a weekly service for all three ports. The capacity of each vessel is 2700, 2700, 2824 and 2824 TEUs, respectively. Inour experiment, each vessel will run consecutively 20 round-trips starting from Ningbo, which covers a period about22 months. The timetable of the first round-trip (i.e. voyage) for each vessel is shown in Table 1 (assuming the first vesselstarts on October 1st). The unit costs of inventory, lifting-on/lifting-off, and lost sales penalty are given in Table 2. The trans-portation cost per TEU between two ports is assumed to the transit time multiplied by a constant £2.

The daily customer demands from port i to port j are assumed to follow uniform distribution U(0, 2 � lij) or normal dis-tribution N(lij, (0.2 � lij)2). The mean of daily demands (in TEUs) for each port-pair is given in Table 3. This implies that thevessel utilization from China to USA is about 0.8870 and from USA to China is about 0.5069. There are no demands betweenNingbo and Shanghai since they are usually met by local short-sea (or feeder) shipping services instead of deep-sea services.The ratio of the demands from China to USA to that from USA to China is 1.75, which indicates the degree of trade imbalance.

Table 4 gives the total costs for uniform distributions and normal distributions of customer demands under three policies:non-repositioning policy (NRP), heuristics repositioning policy (HRP), and evolutionary algorithm-based policies (EAP). Thepercentages of the cost reduction achieved by EAP from NRP and HRP are also given in Table 4. The optimization tool isdeveloped using C++ and run on a PC with 1.86 GHz processor. For the uniform demands, the CPU time is 227 s for NRP usingthe golden section search; 240 s for HRP using the golden section search; 7995 s for EAP. The computation times for the nor-mal demand scenarios are similar.

It can be seen that the evolutionary algorithm reduces the costs significantly, e.g. by 16.76% (18.68%) from the heuristicsrepositioning policy, and by 60.09% (65.23%) from the non-repositioning policy under uniform (normal) demand distribu-tions. This reveals that NRP is obviously not appropriate; HRP is reasonably good, about 40% better than NRP; but there

NB

SH LB

13

13

2

Fig. 5. A trans-pacific shipping service.


Table 1The first voyage of four vessels in case 1.

Vessel 1 2 3 4

Ningbo ETD 01-October 08-October 15-October 22-OctoberShanghai ETA 03-October 11-October 17-October 24-October

ETD 03-October 11-October 17-October 24-OctoberLong Beach ETA 16-October 23-October 30-October 06-November

ETD 16-October 23-October 30-October 06-NovemberNingbo ETA 29-October 05-November 12-November 19-November

Table 2Unit costs of inventory and lifting costs in £ at ports.

Port Inventory cost per day Lifting-on or off cost Lost sales penalty

Ningbo 1.0 30 1000Shanghai 1.0 30 1000Long Beach 1.0 30 1000

Table 3Average daily demands in TEUs in case 1.

Ningbo Shanghai Long Beach

Ningbo 0 0 200Shanghai 0 0 150Long Beach 100 100 0

Table 4Costs in million £ under different policies in case 1.

NRP % Reduction HRP % Reduction EAP

Uniform 120.76 60.09 57.90 16.76 48.20Normal 117.23 65.23 50.12 18.68 40.76


are still significant room to improve by optimizing the fleet size and the threshold values. The costs under normal demands issignificantly lower than those under uniform demands, which reveals that higher variation in customer demands incurshigher logistics cost.

The evolution of performance (the minimum cost and the maximum cost at each generation) in the evolutionary algo-rithm is shown in Figs. 6 and 7 for uniform demands and normal demands, respectively. It demonstrates that not onlythe best performance is converging, but also the range of performance difference between the maximum cost and the min-imum cost at each generation is decreasing to 0.

The best solutions under three policies are given in Table 5. The EAP requires fewer fleet sizes than the HRP because ofbetter setting the threshold values in three ports. The optimal fleet size under EAP is around the total vessel capacity. Thismay reflect the assumption that containers unloaded at a vessel’s arrival event epoch become re-usable when the vessel de-parts. Ningbo has higher threshold values than Shanghai because both ports have the similar importing demands whileNingbo has more exporting demands than Shanghai. It is interesting to observe that there is little inventory of empty

Fig. 6. Evolution of solutions by EAP with uniform demands in case 1. (Square line: maximum cost in each generation; diamond line: minimum cost in eachgeneration.)

Fig. 7. Evolution of solutions by EAP with normal demands in case 1. (Square line: maximum cost in each generation; diamond line: minimum cost in eachgeneration.)

Table 5The best solutions corresponding to three policies in case 1.

Policy Fleet size Ningbo Shanghai Long Beach

D1 U1 D2 U2 D3 U3

Uniform NRP 8610 0 0 0 0 0 0HRP 11,934 992 1685 602 1288 219 1598EAP 11,208 3300 4657 1950 3138 0 57

Normal NRP 7792 0 0 0 0 0 0HRP 11,365 787 1494 413 1120 75 1489EAP 10,777 2037 2247 1169 1605 0 16


containers required at Long Beach under EAP. This can be explained by the fact that Long Beach has much more importingdemands than exporting demands. It is consistent with the practice adopted by some shipping lines that they are reposition-ing empty containers back to Asia as soon as possible. However, the contribution of using EAP is that it provides specificupper and lower inventory levels in Asian ports (e.g. Shanghai and Ningbo in this case), in other words, not only the directionof empty container movements, but also the quantity split over deficit ports. The fleet size under NRP is lower than thoseunder HRP and EAP, this is due to the trade-off between holding inventory and lost-sale penalty. Clearly under NRP, theinventory cost accumulated in surplus ports could dominate the total expected cost because of no empty repositioning. An-other observation is that the fleet size and threshold values in normal demand scenarios are lower than those in uniformdemand scenarios. This reflects the intuition that reliable customer demands (with less variation) can reduce the fleet sizeand the inventory levels at ports.

5.2. Case study 2

The second case is a Europe–Asia shipping service (EU3) provided by the Grand Alliance (www.ci-online.co.uk), which ismore complicated. The shipping route consists of twelve port-of-calls with ten different ports: Southampton (SOU), Ham-burg (HAM), Rotterdam (ROT), Port Klang (PKL), Singapore (SIN), Shekou (SK), Hong Kong (HK), Ningbo (NB), Shanghai(SH), Yantian (YT), Hong Kong (HK), and Singapore (SIN), as shown in Fig. 1. The transit times between ports (including berthtime at ports) are also shown in Fig. 1.

Eight vessels are deployed in this shipping route and the total journey time for one voyage (a round trip) for each vessel is56 days, which provides a weekly service. The capacity of each vessel is 8749, 8750, 8400, 8100, 8652, 8652, 8400, and 8750TEUs, respectively. However, in order to save the computational time, the vessel capacity is reduced by dividing the actualdeployed vessel capacity by 10 in our simulation experiments. Each vessel will run consecutively 10 round-trips startingfrom Southampton, which covers about 20 months. The timetable of the first round-trip (i.e. voyage) for each vessel is shownin Table 6 (assuming the first vessel starts on October 1st). The unit costs of inventory, lifting-on/lifting-off, lost sales penalty,and the transportation cost per TEU between two ports are set in the same way as case 1.

The daily customer demands from port i to port j are assumed to follow uniform distribution U(0, 2 � lij) or normal dis-tribution N(lij, (0.2 � lij)2). The mean of daily demands (in TEUs) for each port-pair is given in Table 7. There are no demandsfrom Hamburg or Rotterdam to Southampton since the vessels in this service sail to Asia first before calling at Southampton.Table 6 implies that the vessel utilization from Asia to Europe is about 0.9081 and from Europe to Asia is about 0.5645. Theratio of the demands from Asia to Europe to that from Europe to Asia is 1.61.

The total costs for uniform and normal demands under three policies are given in Table 8. The ‘‘% reduction” columns inTable 8 indicate the percentage of the cost reduction achieved by EAP compared to NRP and HRP. For the uniform demands,the CPU time is 1264 s for NRP with golden section search; 546 s for HRP with golden section search; 22,820 s for EAP.


Table 6The first voyage of eight vessels in case 2.

Vessel 1 2 3 4 5 6 7 8

SOU 01-October 08-October 15-October 22-October 29-October 05-November 12-November 19-NovemberHAM 05-October 12-October 19-October 26-October 02-November 09-November 16-November 23-NovemberROT 07-October 14-October 21-October 28-October 04-November 11-November 18-November 25-NovemberPKL 24-October 31-October 07-November 14-November 21-November 28-November 05-December 12-DecemberSIN 25-October 01-November 08-November 15-November 22-November 29-November 06-December 13-DecemberSK 28-October 04-November 11-November 18-November 25-November 02-December 09-December 16-DecemberHK 29-October 05-November 12-November 19-November 26-November 03-December 10-December 17-DecemberNB 31-October 07-November 14-November 21-November 28-November 05-December 12-December 19-DecemberSH 01-November 08-November 15-November 22-November 29-November 06-December 13-December 20-DecemberYT 05-November 12-November 19-November 26-November 03-December 10-December 17-December 24-DecemberHK 06-November 13-November 20-November 27-November 04-December 11-December 18-December 25-DecemberSIN 10-November 17-November 24-November 01-December 08-December 15-December 22-December 29-DecemberSOU 26-November 03-December 10-December 17-December 24-December 31-December 07-January 14-January

Table 7Average daily demands in TEUs in case 2.

SOU HAM ROT KPL SIN SK HK NB SH YT

SOU 0 2 2 2 6 2 5 1 5 0HAM 0 0 2 2 6 2 6 2 6 0ROT 0 0 0 2 6 2 6 2 6 0PKL 0 0 0 0 2 0 2 0 2 0SIN 10 10 10 0 0 1 4 1 4 0SK 0 0 0 0 0 0 2 2 2 0HK 10 10 10 0 1 0 0 1 4 0NB 5 5 5 0 1 0 1 0 0 1SH 10 10 10 0 1 0 4 0 0 1YT 2 2 2 0 1 0 2 0 0 0

Table 8Costs in million £ under different policies in case 2.

NRP % Reduction HRP % Reduction EAP

Uniform 38.33 58.78 18.74 15.71 15.80Normal 37.89 58.89 19.34 19.45 15.58

Fig. 8. Evolution of solutions by EAP with uniform demands in case 2. (Square line: maximum cost in each generation; diamond line: minimum cost in eachgeneration.)


Similar to the first case, it can be observed that the evolutionary algorithm significantly reduces the costs, e.g. by 15.71%(19.45%) from the heuristics repositioning policy, and by 58.78% (58.89%) from the non-repositioning policy under uniform(normal) demand distributions. This confirms the effectiveness of the evolutionary algorithm. The evolution of performance(the minimum cost and the maximum cost at each generation) in the evolutionary algorithm is shown in Figs. 8 and 9 foruniform demands and normal demands, respectively. It shows the convergence of the best performance and the differencebetween the maximum cost and the minimum cost over generations.

The best solutions under three policies are given in Table 9. Similar to case 1, the optimal fleet size under EAP is aroundthe total vessel capacity. Although EAP and HRP require comparable fleet sizes, EAP has much higher inventory levels at def-icit ports in Asia such as Singapore, Hong Kong, Ningbo, Shanghai and Yantian, and lower inventory levels at surplus ports in

Fig. 9. Evolution of solutions by EAP with normal demands in case 2. (Square line: maximum cost in each generation; diamond line: minimum cost in eachgeneration.)

Table 9The best solutions corresponding to three policies in case 2.

Fleet size SOU HAM ROT PKL SIN SK HK NB SH YTD1, U1 D2, U2 D3, U3 D4, U4 D5, U5 D6, U6 D7, U7 D8, U8 D9, U9 D10, U10

UniNRP 33,687 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0HRP 7533 15, 204 16, 205 15, 190 4, 46 131, 313 4, 46 47, 285 75, 152 75, 292 54, 75EAP 6788 0, 8 0, 11 0, 69 71, 123 388, 533 0, 37 250, 276 274, 777 238, 535 234, 498

NormNRP 34,351 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0HRP 7511 5,173 6,188 6,174 2, 44 114, 289 1, 36 37, 254 67, 130 58, 261 44, 58EAP 6733 0, 4 0, 39 0,241 43, 381 433, 522 0, 0 264, 343 227, 387 224, 366 215, 293


Europe such as Southampton, Hamburg and Rotterdam. The NRP requires a huge fleet size due to the lack of active reposi-tioning mechanism. Comparing the uniform demand scenarios with the normal demand scenarios, it can be observed thatthe fleet size and threshold values in normal demand scenarios are generally lower than those in uniform demand scenarios.This again confirms the intuition that reliable customer demands can reduce the fleet size and the inventory levels at ports.

5.3. Statistical analysis over multiple samples

The results in Sections 5.1 and 5.2 are based on a single sample process (i.e. x) of customer demands in the planning hori-zon. In this section, we evaluate the solutions obtained in the above two sub-sections using 100 random samples. The sta-tistics for two cases (CLX and EU3) with uniform and normal demand distributions are given in Table 10. The statisticsinclude the mean, the standard deviation, the minimum value, and the maximum value of the samples costs (in million

Table 10The statistics of 100 sample costs under the obtained solutions.

NRP (% reduction) HRP (% reduction) EAP

CLX Mean 119.57 (59.76%) 57.25 (15.95%) 48.12Uniform Std. dev 3.68 2.19 1.86

Min 110.63 53.18 44.33Max 130.30 63.30 52.96

CLX Mean 119.48 (65.39%) 50.26 (17.71%) 41.36Normal Std. dev 1.51 0.83 0.42

Min 115.96 48.52 40.54Max 122.31 52.75 42.50

EU3 Mean 37.67 (57.72%) 19.06 (16.44%) 15.93Uniform Std. dev 0.52 0.35 0.19

Min 36.41 17.96 15.53Max 39.06 19.86 16.36

EU3 Mean 37.81 (58.76%) 19.33 (19.31%) 15.59Normal Std. dev 0.19 0.24 0.06

Min 37.32 18.80 15.42Max 38.33 19.97 15.76

Table 11Costs in million £ under different policies with varying cost parameters (CLX).

Inv. cost Lifting cost Penalty NRP (% reduction) HRP (% reduction) EAP

1 30 1000 117.23 (65.23%) 50.12 (18.68%) 40.765 30 1000 120.51 (63.22%) 57.53 (22.95%) 44.33

10 30 1000 123.13 (60.77%) 65.08 (25.79%) 48.3020 30 1000 128.28 (57.21%) 80.88 (32.14%) 54.89

1 60 1000 133.11 (49.08%) 75.77 (10.55%) 67.781 90 1000 148.99 (36.36%) 101.42 (6.51%) 94.811 120 1000 164.87 (26.13%) 127.06 (4.15%) 121.791 30 500 70.98 (42.63%) 44.92 (9.37%) 40.721 30 2000 184.13 (77.86%) 60.74 (32.88%) 40.771 30 3000 248.37 (83.58%) 71.19 (42.73%) 40.77

Table 12Costs in million £ under different policies with varying cost parameters (EU3).

Inv. cost Lifting cost Penalty NRP (% reduction) HRP (% reduction) EAP

1 30 1000 37.89 (58.89%) 19.34 (19.45%) 15.585 30 1000 45.25 (59.17%) 23.09 (20.00%) 18.47

10 30 1000 50.99 (58.75%) 27.98 (24.84%) 21.0320 30 1000 62.47 (56.90%) 36.77 (26.77%) 26.92

1 60 1000 44.39 (46.27%) 27.50 (13.28%) 23.851 90 1000 49.84 (35.34%) 35.66 (9.63%) 32.231 120 1000 55.33 (26.73%) 43.83 (7.49%) 40.541 30 500 25.30 (39.47%) 16.91 (9.41%) 15.321 30 2000 45.29 (65.65%) 23.13 (32.73%) 15.561 30 3000 50.75 (69.15%) 25.43 (38.45%) 15.65


£) under three policies, respectively. The percentages of cost reduction from NRP or HRP to EAP (for the average costs) arealso given in Table 10.

It can be seen from Table 10 that: (i) the sample costs have very small standard deviations in all scenarios (in particularrelative to the mean). The minimum and maximum values of the sample costs are pretty close, especially for scenarios withnormal demand distributions. These observations indicate that the sample cost is a good representative for the expectedcost. This may be explained in two aspects. Firstly, the planning horizon is around 20 months in both case studies, the sto-chastic effect (caused by random daily demands) over a, relatively, long period of time may compensate the effect of lackingsamples (over space). Secondly, because both CLX and EU3 are weekly services, the customer demands that a vessel is re-quired to meet are accumulated over 7 days. This pooling operation reduces the effect of the randomness in daily demandsaccording to the law of large numbers in probability theory; (ii) comparing the cost reduction percentages based on 100 sam-ples in Table 10 with those based on a single sample in Tables 4 and 8, they are fairly close. Therefore, the results about therelative differences between three policies are generally valid in the situations with multiple sample processes.

5.4. More numerical computations and sensitivity analysis

In this section, more numerical computations are conducted to examine the sensitivity of the results with respect to sys-tem parameters. We will only focus on the cases with normal demand distribution because we have observed that the resultsin two demand distributions have similar pattern.

Three types of parameters are considered: inventory cost, lifting-on and lifting-off cost, and lost-sale penalty cost. Wevary the above parameters one at a time. Again the container fleet sizes in NRP and HRP have been optimized using the gold-en section search method. A single sample process is used in evaluation to save the computation time. The results are givenin Tables 11 and 12 for services CLX and EU3, respectively.

It can be observed from Tables 11 and 12 that: (i) as each of cost parameters increases, the total cost under a given policyis increasing; (ii) as the inventory cost increases, the cost saving from NRP to EAP remains similar; while the cost saving fromHRP to EAP is increasing. The latter may be explained by the fact that higher inventory cost encourages more frequentlyrepositioning; (iii) as the lifting cost increases, the cost savings from both NRP and HRP to EAP are decreasing. This reflectsthat fact that higher lifting costs discourage the repositioning of empty containers; (iv) as the lost-sale penalty cost increases,the cost savings from both NRP and HRP to EAP are increasing.

6. Conclusions

This paper considers the joint container fleet sizing and empty container repositioning problem in liner shipping systems.Focusing on a type of parameterized rule-based empty container repositioning policies, a simulation-based optimization tool


is developed to optimize simultaneously the container fleet size and the parameters used in the repositioning policy by min-imizing the expected total costs consisting of inventory-holding costs, lifting-on/off costs, transportation costs, repositioningcosts, and lost demand penalty costs. The optimization procedure is based on Genetic Algorithms (GA) and EvolutionaryStrategy (ES) combined with adjustments by taking into account the knowledge of the underlying problems. The algorithmcan generate significantly better results compared with a heuristic policy and the non-repositioning policy.

The main contributions of the paper to the literature are: (i) we consider the joint optimization of container fleet sizingand empty repositioning in a stochastic dynamic liner shipping system with multi-vessels, multi-ports and multi-voyages.The literature on fleet sizing and empty repositioning mainly concentrated on vehicle fleet management in inland transpor-tation systems (Beaujon and Turnquist 1991; Du and Hall 1997; Kochel et al. 2003; Song and Earl 2008), which is signifi-cantly different from container fleet management in liner shipping industry. To the best of our knowledge, there is nopublished research on the joint optimization of container fleet sizing and empty container repositioning in liner shippingsystems. This research also extends the inventory-based repositioning policies from relative simple shipping systems(e.g. Li et al. 2004; Song 2007; Lam et al. 2007; Li et al. 2007; Song and Dong 2008) to more general systems with multi-ves-sels, multi-ports and multi-voyages; (ii) the movements of both laden and empty containers, and the constraints of vesselcapacities are explicitly modelled; vessel arrival events and departure events are treated separately. Such treatments makethe model more realistic; (iii) a simulation-based evolutionary algorithm has been developed to solve the optimization prob-lem. Its effectiveness is demonstrated using case studies in comparison with heuristics. The simulation model and the opti-mization algorithm can serve as a useful tool for liner shipping managers to design the container fleet size, set appropriatelevels of safety stocks at ports, and determine the decisions of empty container repositioning.

A range of numerical examples based on two case studies with different topological structures of the service routes anddifferent cost parameters are used to test the effectiveness of the evolutionary algorithm-based policy (EAP). It is observedthat the cost reduction from non-repositioning policy (NRP) to EAP ranges from 26% to 69%; the cost reduction from the heu-ristic repositioning policy (HRP) to EAP ranges from 4% to 38%. In most experiments the cost savings achieved by EAP fromHRP are more than 10%. A general insight is that more cost savings can be achieved by EAP if the system has higher inventorycosts, lower lifting costs, or higher lost-sale penalties. The research also confirms that fleet sizing is closely related to emptyrepositioning as the optimal fleet sizes vary when different repositioning policies are used. The EAP achieves the lowest fleetsize with better threshold levels at ports. Another observation is that the fleet size and threshold values in normal demandsituations are lower than those in uniform demand situations. This reflects the intuition that reliable customer demands(with less variation) can reduce the fleet size and the inventory levels at ports.

The main disadvantage of the evolutionary algorithm is the computational time. In our experiments, EAP requires 30–50times more CPU time than NRP or HRP. In particular, if multiple samples are used in the evaluation, EAP would require sub-stantially more CPU time. How to improve the search speed and the quality of solution is a big challenge. Another issue isabout the global optimization. Although GA is regarded as an effective algorithm to deal with many global optimizationproblems, it often leads to sub-optimal solutions due to the limitation of computational time. It should also be pointedout that even though the standard deviation of the simulation outputs over multiple samples is small (cf. Section 5.3),the proposed search algorithm may still produce a local optimal solution, which may be quite away from the optimalone. This research focuses on a specific type of rule-based repositioning policies, it is also interesting to extract and inves-tigate other types of policies from liner shipping’s practices. Moreover, the impact of inland transportation time of laden con-tainers on the performance and the optimization of more complicated shipping networks require further research.

Acknowledgements

The authors thank two referees and the editor for many valuable comments and suggestions that led to improvements ofthe paper. This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) [Grant Numbers EP/E000398/1, EP/F012918/1].

References

Back, T., 1996. Evolutionary Algorithms in Theory and Practice. Evolution Strategies, Evolutionary Programming, Genetic Algorithms. Oxford UniversityPress, Oxford.

Beaujon, G.J., Turnquist, M.A., 1991. A model for fleet sizing and vehicle allocation. Transportation Science 25 (1), 19–45.Bourbeau, B., Crainic, T.G., Gendron, B., 2000. Branch-and-bound parallelization strategies applied to a depot location and container fleet management

problem. Parallel Computing 26, 27–46.Cheang, B., Lim, A., 2005. A network flow based method for the distribution of empty containers. International Journal of Computer Applications in

Technology 22 (4), 198–204.Cheung, R.K., Chen, C.Y., 1998. A two-stage stochastic network model and solution methods for the dynamic empty container allocation problem.

Transportation Science 32 (2), 142–162.Cheung, R.K., Powell, W.B., 1996. An algorithm for multistage dynamic networks with random arc capacities, with an application to dynamic fleet

management. Operations Research 44 (6), 951–963.Choong, S.T., Cole, M.H., Kutanoglu, E., 2002. Empty container management for intermodal transportation networks. Transportation Research Part E 38 (6),

423–438.Crainic, T.G., 2000. Service network design in freight transportation. European Journal of Operational Research 122, 272–288.Crainic, T.G., Gendreau, M., Dejax, P., 1993a. Dynamic and stochastic models for the allocation of empty containers. Operations Research 41, 102–126.Crainic, T.G., Gendreau, M., Soriano, P., Toulouse, M., 1993b. A tabu search procedure for multicommodity location-allocation with balancing requirements.

Annals of Operations Research 41, 359–384.


Dejax, P., Crainic, T., 1987. A review of empty flows and fleet management models in freight transportation. Transportation Science 21 (4), 227–247.Drewry, 2006. Annual Container Market Review & Forecast 2006/07. Drewry Shipping Consultants Ltd., London.Du, Y., Hall, R., 1997. Fleet sizing and empty equipment redistribution for center-terminal transportation networks. Management Science 42 (2), 145–157.Erera, A.L., Morales, J.C., Savelsbergh, M., 2005. Global intermodal tank container management for the chemical industry. Transportation Research Part E 41

(6), 551–566.Fu, M.C., 1994. Optimization via simulation: a review. Annals of Operational Research 42, 199–247.Gerald, C.F., Wheatley, P.O., 2004. Applied Numerical Analysis, 7th ed. Addison-Wesley, New York.Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, New York, NY.Herrera, F., Lozano, M., Sanchez, A.M., 2003. A taxonomy for the crossover operator for real-coded genetic algorithms: an experimental study. International

Journal of Intelligent Systems 18, 309–338.Jula, H., Chassiakos, A., Ioannou, P., 2006. Port dynamic empty container reuse. Transportation Research Part E 42, 43–60.Kochel, P., Kunze, S., Nielander, U., 2003. Optimal control of a distributed service system with moving resources: application to the fleet sizing and allocation

problem. International Journal of Production Economics 81–82, 443–459.Lai, K.K., Lam, K., Chan, W.K., 1995. Shipping container logistics and allocation. Journal of the Operational Research Society 46, 687–697.Lam, S.W., Lee, L.H., Tang, L.C., 2007. An approximate dynamic programming approach for the empty container allocation problem. Transportation Research

Part C 15, 265–277.Li, J.A., Leung, S.C.H., Wu, Y., Liu, K., 2007. Allocation of empty containers between multi-ports. European Journal of Operational Research 182, 400–412.Li, J.A., Liu, K., Stephen, C.S., Lai, K.K., 2004. Empty container management in a port with long-run average criterion. Mathematical and Computer Modelling

40, 85–100.Olivo, A., Zuddas, P., Francesco, M.D., Manca, A., 2005. An operational model for empty container management. Maritime Economics & Logistics 7, 199–222.Shen, W.S., Khoong, C.M., 1995. A DSS for empty container distribution planning. Decision Support System 15 (1), 75–82.Shintani, K., Imai, A., Nishimura, E., Papadimitriou, S., 2007. The container shipping network design problem with empty container repositioning.

Transportation Research Part E 43, 39–59.Song, D.P., 2007. Characterizing optimal empty container reposition policy in periodic-review shuttle service systems. Journal of the Operational Research

Society 58 (1), 122–133.Song, D.P., Carter, J., 2009. Empty container repositioning in shipping industry. Maritime Policy & Management, in press.Song, D.P., Dong, J.X., 2008. Empty container management in cyclic shipping routes. Maritime Economics & Logistics 10, 335–361.Song, D.P., Earl, C.F., 2008. Optimal empty vehicle repositioning and fleet-sizing for two-depot services systems. European Journal of Operational Research

185, 760–777.Song, D.P., Hicks, C., Earl, C.F., 2006. An ordinal optimization based evolution strategy to schedule complex make-to-order products. International Journal of

Production Research 44 (22), 4877–4895.Song, D.P., Zhang, J., Carter, J., Field, T., Marshall, M., Polak, J., Schumacher, K., Sinha-Ray, P., Woods, J., 2005. On cost-efficiency of the global container

shipping network. Maritime Policy & Management 32, 15–30.White, W.W., 1972. Dynamic transhipment network: an algorithm and its application to the distribution of empty containers. Networks 2, 211–236.

Date post:	28-Nov-2023
Category:	Documents
Upload:	liverpool
View:	0 times
Download:	0 times

Container fleet sizing and empty repositioning in liner shipping systems

Documents