Measuring the perceived container leasing prices in liner shipping network design with empty container repositioning

Transportation Research Part E 94 (2016) 123–140

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

Measuring the perceived container leasing prices in liner shipping network design with empty container repositioning Jianfeng Zheng a,b,⇑, Zhuo Sun a,b,⇑, Fangjun Zhang c a

Collaborative Innovation Center for Transport Studies @ Dalian Maritime University, Dalian 116026, China Transportation Management College, Dalian Maritime University, Dalian 116026, China c Wuhan Metro Operation Company, Wuhan 430000, China b

a r t i c l e

i n f o

Article history: Received 10 April 2016 Received in revised form 6 July 2016 Accepted 2 August 2016

Keywords: Perceived container leasing prices Liner shipping network design Empty container repositioning Two-stage optimization method

a b s t r a c t This paper aims to measure the perceived container leasing prices at different ports by presenting a two-stage optimization method. In stage I, we propose a practical liner shipping network design problem with empty container repositioning. The proposed problem further considers the use of foldable containers and allows the mutual substitution between empty containers to decrease the number of empty containers to be repositioned. In stage II, the inverse optimization technique is used to determine the perceived container leasing prices at different ports, based on the solution obtained in stage I. Based on a set of candidate liner shipping service routes, a mixed-integer nonlinear programming model is built for the proposed problem in stage I. The nonlinear terms are linearized by introducing the auxiliary variables. Numerical experiments based on a realistic Asia-Europe-Oceania liner shipping network are carried out to account for the effectiveness of our two-stage optimization method. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction In liner shipping industry, different types of containers (i.e., multi-type containers) are shipped by the liner shipping companies on the regularly scheduled shipping service routes (i.e., liner shipping network). When the liner shipping network is designed, it will be operated over a seasonal planning horizon. For every seasonal planning horizon, the liner shipping company will alter its current liner shipping network, according to the container shipment demand forecasted for the next seasonal planning horizon. Based on the liner shipping network, the liner shipping company transports both laden containers and empty containers. When the liner shipping company cannot maintain a balance between supplies and demands of empty containers for certain ports, container leasing offers an alternative way for container management (Shen and Khoong, 1995; Moon et al., 2010; Dong and Song, 2012; Liu et al., 2013; Wu and Lin, 2015; Jiao et al., 2016). Evidently, different container leasing contracts lead to different leasing terms and prices (Dong and Song, 2012; Liu et al., 2013; Jiao et al., 2016). In this paper, we do not consider realistic container leasing when satisfying supplies and/or demands of empty containers at different ports. Alternatively, we aim to measure the perceived container leasing prices at different ports. Different from realistic container leasing prices in practice, the perceived container leasing prices are calculated based on the costs on repositioning empty containers via our liner shipping network. According to our perceived container leasing prices, it is helpful for the liner shipping company to make decisions on container leasing strategies at different ports. ⇑ Corresponding authors at: Transportation Management College, Dalian Maritime University, Dalian 116026, China. E-mail addresses: [email protected] (J. Zheng), [email protected] (Z. Sun). http://dx.doi.org/10.1016/j.tre.2016.08.001 1366-5545/Ó 2016 Elsevier Ltd. All rights reserved.

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Actually, if the perceived container leasing price at any particular port is larger than the realistic container leasing price in the container leasing market, it is costly to reposition their own empty containers for this port; otherwise it is economic for the liner shipping company to reposition their own empty containers, as compared with that adopts container leasing for this port. In order to determine the perceived container leasing prices, this paper presents a two-stage optimization method. In stage I, we propose a practical liner shipping network design (LSND) problem with empty container repositioning. Based on the solution obtained in stage I, the perceived container leasing prices are determined in stage II. Generally, the formulation of the LSND problem can be classified into two categories, according to whether the set of liner shipping service routes is given or not. For the first category, the LSND problem is often formulated as a multi-commodity flow assignment model or a container flow assignment model (Agarwal and Ergun, 2008; Meng and Wang, 2011; Wang and Meng, 2013; Brouer et al., 2014a, 2014b; Zheng et al., 2015b; Huang et al., 2015; Wang et al., 2015). The set of liner shipping service routes in the first category can be given in advance or generated by using some algorithms such as the column generation method as shown in Agarwal and Ergun (2008). For the latter one, the LSND is generally formulated as a vehicle routing problem (Fagerholt, 1999, 2004; Sambracos et al., 2004; Karlaftis et al., 2009; Zheng et al., 2014) or a hub location problem (Gelareh et al., 2010; Gelareh and Pisinger, 2011). This paper considers the first one. In practice, the liner shipping company will not completely redesign its liner shipping network from scratch after every seasonal planning horizon. In other words, the current liner shipping network operated by the liner shipping company can be regarded as an efficient portion of the set of candidate shipping service routes. Due to the imbalance of imports and exports, liner shipping companies have to transport laden containers and reposition empty containers. Meng and Wang (2011) demonstrated the potential cost-savings by incorporating the empty container repositioning issue in the LSND problem. In order to reduce the cost for repositioning empty containers, this paper mainly considers two strategies, i.e., the mutual substitution between empty container containers and the use of foldable containers. Both of these two strategies have been considered by the liner shipping company to reduce the repositioning cost (Chang et al., 2008; Moon et al., 2013). Note that high production cost of foldable containers have made liner shipping companies hesitant to adopt these units (Moon and Hong, 2016). By considering the mutual substitution between empty containers, the number of empty containers to be repositioned can be decreased (Chang et al., 2008). Generally, empty containers with different sizes can be easily substituted each other. However, it is basically not allowed to substitute between the general empty containers and reefer empty containers. Based on the use of foldable containers, the ship capacity occupied by empty containers can be decreased (Moon et al., 2013). Hence, the ship storage space can be saved efficiently. However, it incurs the extra handling cost for folding and unfolding empty containers. In practice, several foldable container designs have been developed, leading to different folding principles, such as 4:1, 5:1 and 6:1. For any folding principle, e.g., 4:1, four folded empty containers can be regarded as one standard empty container from the occupied capacity point of view. Furthermore, it brings us a new problem on where to fold and/or unfold empty containers and how many empty containers to be folded or unfolded at different ports. 1.1. Literature review In the aspect of empty container repositioning, there have been many recent studies in the literature, e.g., Cheung and Chen (1998), Imai and Rivera (2001), Li et al. (2007), Lam et al. (2007), Feng and Chang (2008), Dong and Song (2009), Francesco et al. (2009), Moon et al. (2010), Song and Dong (2011, 2012), Moon and Hong (2016). The majority of literature assumed that empty containers should be transported only when the laden container load of a ship is not full. Shintani et al. (2010), Moon et al. (2013) and Moon and Hong (2016) studied how the use of foldable containers could reduce the repositioning costs. Recently, Chen et al. (2016) investigated pricing and competition in a shipping market with carriers providing services between two locations, as well as considering the repositioning of empty containers. There have been some studies on container leasing. Shen and Khoong (1995), Moon et al. (2010) studied empty container repositioning, as well as considering the leasing of empty containers. Dong and Song (2012) focused on leasing term optimization in container shipping systems. Wu and Lin (2015) investigated the selection between owned and leased containers. Moon and Hong (2016) studied the reposition of empty containers using both standard and foldable containers, as well as considering container leasing. Recently, Liu et al. (2013) and Jiao et al. (2016) focused on investigating container leasing contracts. Network design, ship fleet deployment, ship scheduling and container routing for the liner shipping industry have attracted much attention, following the five review papers: Ronen (1983, 1993), Christiansen et al. (2004, 2013) and Meng et al. (2014). As mentioned before, the existing studies on the LSND problem can be classified into two categories – with and without the set of shipping service routes. Based on the set of shipping service routes, Agarwal and Ergun (2008) proposed a multi-commodity based time-space network model for the LSND problem with cargo routing, which is formulated as a mixed-integer linear programing model. In order to generate the shipping service routes, three different methods (i.e., a greedy heuristic, a column generation method and a benders decomposition method) were proposed. Agarwal and Ergun (2010) and Zheng et al. (2015a) studied the LSND problem considering liner alliances, as well as determining capacity exchange cost for sharing ship capacity among liners in an alliance. Meng and Wang (2011) introduced the concept of segment in order to formulate a LSND problem with empty container repositioning as a mixed-integer linear programing model. They assumed that the candidate ship fleet deployment plans (ship size and number of ships) and the candidate shipping service routes were given in advance.

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Wang and Meng (2013) investigated a special sub-problem of LSND by focusing on the choice of reversing port rotation in a given liner shipping network. Song and Dong (2013) designed a single long-haul liner shipping service route by incorporating ship fleet deployment and empty container repositioning. Brouer et al. (2014a) presented an integer programming model and a benchmark suite for the LSND problem, as well as providing easy access to the data sources. Later, Brouer et al. (2014b) proposed an integer programming based heuristic (a matheuristic) for LSND. The matheuristic is composed of four main algorithmic components: a construction heuristic, an improvement heuristic, a reinsertion heuristic and a perturbation heuristic. Recently, Wang et al. (2015) explored the segment-based network alteration in order to optimize the liner shipping network. Without the set of shipping service routes, the feeder shipping service route design problem with one hub port and many feeder ports was investigated in Fagerholt (1999, 2004), Sambracos et al. (2004), and Karlaftis et al. (2009). Imai et al. (2006, 2009) studied and compared hub-and-spoke (H&S) strategy by mega-ship with multi-port-calling (MPC) strategy by conventional ship size between two regions, and Imai et al. (2009) also considered the impact of empty container repositioning. Shintani et al. (2007) aimed to design a single shipping service route considering empty container repositioning. They employed the genetic algorithm to solve their problem. Gelareh et al. (2010) proposed a competitive hub location problem for designing liner shipping networks. Gelareh and Pisinger (2011) analyzed the impact of ship fleet deployment on the H&S network design by using a pre-determined discount factor to reflect economies of scale in ship size. Zheng et al. (2014) proposed a two-phase mathematical programing model for the liner H&S network design problem, where the phase I determines hub location and feeder allocation, and ship routing and fleet deployment in phase II is formulated as a vehicle routing problem. As we know, Shintani et al. (2007), Imai et al. (2009), Meng and Wang (2011), Song and Dong (2013) and Huang et al. (2015) are the five papers found in the literature that explicitly consider the empty container repositioning issue together with the LSND problem. However, Shintani et al. (2007) and Song and Dong (2013) focus on designing a single shipping service route. Container routing is not an issue in Imai et al. (2009). Meng and Wang (2011) and Huang et al. (2015) formulated the LSND problem with empty container repositioning as a mixed-integer linear programming model. Meng and Wang (2011) assumed that the candidate ship fleet deployment plans were given in advance. Huang et al. (2015) assumed that ship scheduling was given for each shipping service route, in order to check the feasible laden container shipment plans. Different from Shintani et al. (2007), Imai et al. (2009) and Song and Dong (2013), this paper considers multiple shipping service routes, as well as solving the container routing problem. Different from Meng and Wang (2011), the ship fleet deployment problem, the shipping service route selection problem and the container routing problem are solved together in this paper, without providing the candidate ship fleet deployment plans for each shipping service route. Different from Huang et al. (2015), this paper does not consider the ship scheduling problem. In addition, this paper further considers the use of foldable containers and the mutual substitution between empty containers, which are not considered in these five papers. Moreover, based on the results of empty container repositioning in our liner shipping network, this paper determines the perceived container leasing prices, which are helpful for the liner shipping company to make decisions on container leasing strategies at different ports. 1.2. Contributions The contributions of this paper are two-fold. Firstly, a two-stage optimization method is presented to measure the perceived container leasing prices at different ports. In stage I, a practical liner shipping network design problem with empty container repositioning is proposed. Based on the solution obtained in stage I, the perceived container leasing prices at different ports are determined in stage II. Secondly, the proposed problem in stage I considers two strategies (i.e., the use of foldable containers and the mutual substitution between empty containers), in order to reduce the repositioning costs. The rest of this paper is organized as follows. Section 2 gives notation and assumptions. Section 3 presents a two-stage optimization method. Section 4 carries out the numerical experiments based on the Asia-Europe-Oceania liner shipping services. Finally, a summary is given in Section 5. 2. Notation and assumptions 2.1. Notation The main notation used in this paper is shown as follows. P: P1: P2: R: V:

Set Set Set Set Set

of of of of of

ports; hub ports; feeder ports; predetermined candidate shipping service routes; ship types; (continued on next page)

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A: W: Daod : sv : Capv : tav ;p : ^tav ;p : ~ta : v ;p cfix v : cbunker : v cbunker;port : v cberth v ;p : cload a;p : ~cload a;p : cfold a;p : cinv a :

Set of container types; Set of Origin-Destination (OD) port pairs; Weekly number of laden containers with type a 2 A transported for OD port pair ðo; dÞ 2 W over a seasonal planning horizon; Average speed for ship v when sailing at sea; Capacity of a ship with type v ; Time spent for loading or discharging one laden container with type a for type v ship at port p; Time spent for loading or discharging one folded empty container with type a for type v ship at port p; Time spent for loading or discharging one unfolded empty container with type a for type v ship at port p; Fixed weekly operating cost for type v ship, which comprises the ship maintenance cost, crew payment and insurance cost; Bunker cost per hour for type v ship when sailing at sea; Bunker cost per hour for type v ship when berthing at port; Berth charge per hour for type v ship when berthing at port p; Cost for loading or unloading one unfolded (laden or empty) container with type a at port p; Cost for loading or unloading one folded empty container with type a at port p; Cost for folding or unfolding one empty container with type a at port p; Inventory cost for transporting one folded or unfolded empty container with type a per hour.

2.2. Shipping service routes Generally, the port calling sequence for a particular shipping service route r 2 R can be expressed as follows:

pr1 ! pr2 !    ! prNr ! prðNr þ1Þ :¼ pr1

ð1Þ

where N r is the number of ports served by shipping service route r. Let Irp denote the set of coding indices of port p on shipping service route r. Without loss of generality, a number of ships (i.e., a ship fleet) are deployed on each shipping service route in set R to maintain a weekly service. If the round-trip time for any shipping service route r 1 2 R deployed by ships with type v 1 2 V is 168  2 h (i.e., two weeks), a ship fleet composed of two type v 1 2 V ships is deployed to maintain a weekly service. We further assume that more than one ship fleet can be deployed on any shipping service route. For instance, we can deploy four type v 1 2 V ships (i.e., two ship fleets) on shipping service route r 1 2 R, and then the ship capacity becomes twice. 2.3. Routing laden and empty containers A route passed through by laden containers from their origin port to their destination port is referred to as a laden container route, which may be a combination of some relevant shipping service routes. A laden container route does not involve any container transshipment operation is referred to as a direct shipping service. Following Meng and Wang (2011), Song and Dong (2012) and Zheng et al. (2014, 2015a, 2015b), this paper assumes that laden containers cannot be transshipped for more than two times from their origin ports to their destination ports. To reflect empty container repositioning, let Eap denote the difference between the incoming and outgoing container flow with container type a 2 A at port p:

Eap ¼

X

8q2P

Daqp 

X

Dapq ;

8a 2 A;

8p 2 P

ð2Þ

8q2P

Empty containers are transported from the surplus ports to the deficit ports. In order to reduce the costs on repositioning empty containers, this paper considers two strategies: the use of foldable containers, and the mutual substitution between empty containers. 2.4. Time structure and cost structure For any shipping service route r 2 R deployed by ships with type v 2 V , the round-trip time mainly includes two terms: sailing time for type v ship at sea and berthing time for handling containers at each port called by this shipping service route. Note that the time spent for folding or unfolding empty containers at each port is not considered in this paper, since the folding and unfolding operations can be completed whether the ship is sailing at sea or berthing at the port. For any shipping service route r 2 R deployed by ships with type v 2 V , the total weekly operating cost mainly consists of the following four terms: fixed weekly operating cost, bunker cost, berth charge and container handling cost at each called port. Actually, liner shipping companies concern the time for repositioning empty containers, which can be described by ‘‘per diem” in liner shipping industry. Alternatively, we consider inventory cost for transporting empty containers. In practice, the

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time spent when a ship stays at any port depends on the time window coordinated between the liner shipping company and port operator. Hence, this paper mainly considers the inventory cost for transporting empty containers when sailing at sea. Note that the inventory cost for transporting laden containers is not considered in this paper. Obviously, each cargo has a preferential container type when transporting from origin to destination. This paper assumes that the mutual substitution between empty containers brings a substitution cost. Let csub a;b be the substitution cost for one empty container with type a substituted by container type b. Evidently, the production cost of foldable containers is higher than that of standard containers. In order to integrate the cost of purchasing or possessing foldable containers in our problem, this paper considers a holding cost when using foldable containers. Let chold denote the holding cost per week for one foldable empty container with a type a. For simplicity, the holding cost of any standard container is not considered in this paper. 3. Two-stage optimization method Here, we propose a two-stage optimization method. In stage I, a practical liner shipping network design problem with empty container repositioning is proposed, as shown in Section 3.1. In stage II, we aim to determine the perceived container leasing prices at different ports based on the solution obtained in stage I, as shown in Section 3.2. 3.1. Liner shipping network design with empty container repositioning 3.1.1. Decision variables The decision variables for the liner shipping network design problem with empty container repositioning in stage I can be defined as follows: xr v : nr v : mr v : nfold a : t port v ;ri : za;o ri : ua;o : ri a;o

f ri : ^zari : ~zari : ^ ari : u ~ ari : u g^ap : g~ap : epa;b : ^f a : ri ~f a : ri

A binary variable which takes value 1 if shipping service route r 2 R is served by ships with type v 2 V, and 0 otherwise; Number of ships with type v 2 V deployed on shipping service route r 2 R; Number of ship fleets with type v 2 V deployed on shipping service route r 2 R; Number of foldable containers with type a 2 A over a seasonal planning horizon; Time spent for type v 2 V ship when berthing at the ith port of shipping service route r 2 R; Number of laden containers with type a 2 A originated from port o 2 P and loaded at the ith port of shipping service route r 2 R; Number of laden containers with type a 2 A originated from port o 2 P and discharged at the ith port of shipping service route r 2 R; Number of laden containers with type a 2 A originated from port o 2 P and stowed on board of a ship deployed on shipping service route r 2 R and sailing on the ith leg of shipping service route r 2 R; Number of folded empty containers with type a 2 A loaded at the ith port of shipping service route r 2 R; Number of unfolded empty containers with type a 2 A loaded at the ith port of shipping service route r 2 R; Number of folded empty containers with type a 2 A discharged at the ith port of shipping service route r 2 R; Number of unfolded empty containers with type a 2 A discharged at the ith port of shipping service route r 2 R; Number of unfolded empty containers with type a 2 A folded at port p 2 P; Number of folded empty containers with type a 2 A unfolded at port p 2 P; Number of empty containers with type a 2 A substituted by container type b 2 A at port p 2 P; Number of folded empty containers with type a 2 A stowed on board of a ship deployed on shipping service route r 2 R and sailing on the ith leg of shipping service route r 2 R; Number of unfolded empty containers with type a 2 A stowed on board of a ship deployed on shipping service route r 2 R and sailing on the ith leg of shipping service route r 2 R.

3.1.2. Model development Before the mathematical programming model is proposed, we provide some descriptions on multi-type foldable containers. Generally, all folded empty containers should be finally unfolded when they are delivered to their destination ports. Hence, for any type of container at any particular port, the inflow of the folded empty containers (i.e., folded empty containers to be discharged and unfolded empty containers to be folded) is equal to the outflow of the folded empty containers (i.e., folded empty containers to be loaded and folded empty containers to be unfolded). Namely,

g^ap þ

XX XX ^zari ; ^ ari ¼ g~ap þ u

r2Rp i2I rp

8a 2 A;

8p 2 P

ð3Þ

r2Rp i2I rp

where Rp is the set of candidate shipping service routes which call at port p. Furthermore, Eq. (3) also implies that, for any type of container at any particular port, the number of folded empty containers to be unfolded should not be larger than the number of folded empty containers to be discharged. Namely,

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J. Zheng et al. / Transportation Research Part E 94 (2016) 123–140

g~ap 6

XX ^ ari ; u

8a 2 A;

8p 2 P

ð4Þ

r2Rp i2I rp

Due to the use of foldable empty containers, the ship capacity constraint on each leg of any shipping service route can be described as follows:

XX

a;o 

ba  f ri

þ

o2P a2A

Xh

 i X ba  a  ^f ari þ ~f ari 6 ðCapv  mrv Þ; v 2V

a2A

8i ¼ 1; 2; . . . ; Nr ;

8r 2 R

ð5Þ

where ba is a parameter used to convert one unit of container type a into a number of TEUs, and a is used to describe the folding principle. For any container type, empty containers cannot be substituted by the same container type at any single port. Namely,

8a 2 A;

epa;a ¼ 0;

8p 2 P

ð6Þ

Clearly, one can obtain Eq. (6) when the parameter csub a;a is set to be a positive number, so as to minimize the substitution cost. Moreover, empty containers should be balanced after the reposition of empty containers by considering mutual substitution between empty containers, as well as folding and unfolding of empty containers. Namely,

2 ba 

4Ea p

3  X XX ~ ari  ~zari Þ5 ¼ þ ðg~ap  g^ap Þ þ ðu bb  epb;a  ba  epa;b ; r2Rp i2I rp

8a 2 A;

8p 2 P

ð7Þ

b2A

According to Eq. (7), one can find that there are empty containers with type a 2 A to substitute other empty containers with P P P p a ~a ^a ~ a ~a different types at any particular port p 2 P, i.e., b2A eb;a > 0 if Ep þ ðg p  g p Þ þ r2Rp i2I rp ðuri  zri Þ > 0 is satisfied. If P P P p p a a a a a ~ ri  ~zri Þ 6 0 is satisfied, we can have eb;a ¼ 0 (8b 2 A) and b2A ea;b P 0. Ep þ ðg~p  g^p Þ þ r2Rp i2I rp ðu By considering the use of multi-type foldable containers, the proposed liner shipping network design problem with empty container repositioning can be formulated as the following mixed-integer nonlinear programming model (denoted by model A):

" # " # Nr   Nr X XX XX X XLri Lri fix bunker inv a a ^ ~ þ cv  nrv þ mrv  cv  ca  ðf ri þ f ri Þ   xrv sv r2R v 2V r2R i¼1 a2A v 2V sv i¼1

min þ

Nr Xh Nr XXh i X i XX XX  a   a  a;o a;o ~zri þ u ^ ~ ari  cload ^a ~load þ cload ðchold  nfold a;ri þ zri þ uri  ca;ri a;ri  ðzri þ uri Þ þ a a Þ r2R i¼1 a2A

þ

XXh p2P a2A

r2R i¼1 a2A o2P

i

a2A

Nr Xh i XXX XX p bunker;port ^a ~a cfold ðcsub t port þ cberth a;p  ðg p þ g p Þ þ a;b  ea;b Þ þ v ;ri Þ v ;ri  ðcv r2R i¼1 v 2V

p2P a2A b2A

ð8Þ

subject to (3), (5), (7),

X xrv 6 1;

8r 2 R;

nrv 6 Mxrv ;

8r 2 R;

8v 2 V;

ð10Þ

mrv 6 Mxrv ;

8r 2 R;

8v 2 V;

ð11Þ

v 2V

mrv 6 nrv ;

ð9Þ

8v 2 V;

8r 2 R;

XX mrv P 1;

r2Rp v 2V

8p 2 P;

X a XX a;o zri ¼ Dod ; r2Ro i2Iro

ua;o ri ¼ 0;

ð12Þ ð13Þ

8a 2 A;

8o 2 P;

ð14Þ

ðo;dÞ2W

8i 2 Iro ;

8r 2 Ro ;

8a 2 A;

XX a;o a ðzri  ua;o ri Þ ¼ Doj ; j–o;

8o 2 P;

8o; j 2 P;

ð15Þ

8a 2 A;

8ðo; jÞ 2 W ;

ð16Þ

r2Rj i2Irj

XX a;o ðzri  ua;o ri Þ ¼ 0; j–o; r2Rj i2Irj

8o; j 2 P;

8a 2 A;

8ðo; jÞ R W ;

ð17Þ

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J. Zheng et al. / Transportation Research Part E 94 (2016) 123–140 a;o

a;o

8i ¼ 1; 2; . . . ; N r ;

a;o f ri þ za;o rðiþ1Þ ¼ f rðiþ1Þ þ urðiþ1Þ ;

8o 2 P;

8a 2 A;

8r 2 R;

ð18Þ

^a ^f a þ ^za ^a ri rðiþ1Þ ¼ f rðiþ1Þ þ urðiþ1Þ ;

8i ¼ 1; 2; . . . ; Nr ;

8a 2 A;

8r 2 R;

ð19Þ

~a ~f a þ ~za ~a ri rðiþ1Þ ¼ f rðiþ1Þ þ urðiþ1Þ ;

8i ¼ 1; 2; . . . ; Nr ;

8a 2 A;

8r 2 R;

ð20Þ

168  nrv þ M  ð1  xrv Þ P

Nr   N r XXh X X  i Lri a;o  mrv þ tav ;ri  za;o ri þ uri sv i¼1 i¼1 a2A o2P

þ

N r Xh X    i ^t av ;ri  ^zari þ u ^ ari þ ~tav ;ri  ~zari þ u ~ ari ;

8r

i¼1 a2A

8v 2 V;

2 R; tport v ;ri þ M  ð1  xrv Þ P

X a2A

^ ari Þ þ ~tav ;ri  ð~zari þ u ~ari Þ þ ½^t av ;ri  ð^zari þ u

¼ 1; 2; . . . ; Nr ; nfold  a

X g^ap P 0;

ð21Þ

8r 2 R;

8v 2 V;

8a 2 A;

XX a;o ½t av ;ri  ðza;o ri þ uri Þ;

8i

a2A o2P

ð22Þ ð23Þ

p2P

xrv 2 f0; 1g; a;o

8r 2 R;

8v 2 V;

ð24Þ

f ri P 0;

8i ¼ 1; 2; . . . ; Nr ;

8r 2 R;

8o 2 P;

8a 2 A;

ð25Þ

za;o ri P 0;

8i ¼ 1; 2; . . . ; Nr ;

8r 2 R;

8o 2 P;

8a 2 A;

ð26Þ

ua;o ri P 0;

8i ¼ 1; 2; . . . ; Nr ;

8r 2 R;

8o 2 P;

8a 2 A;

ð27Þ

^f a P 0; ri

8i ¼ 1; 2; . . . ; Nr ;

8r 2 R;

8a 2 A;

ð28Þ

~f a P 0; ri

8i ¼ 1; 2; . . . ; Nr ;

8r 2 R;

8a 2 A;

ð29Þ

^zari P 0;

8i ¼ 1; 2; . . . ; Nr ;

8r 2 R;

8a 2 A;

ð30Þ

~zari P 0;

8i ¼ 1; 2; . . . ; Nr ;

8r 2 R;

8a 2 A;

ð31Þ

^ ari P 0; u

8i ¼ 1; 2; . . . ; Nr ;

8r 2 R;

8a 2 A;

ð32Þ

~ ari P 0; u

8i ¼ 1; 2; . . . ; Nr ;

8r 2 R;

8a 2 A;

ð33Þ

g^ap P 0;

8p 2 P;

8a 2 A

ð34Þ

g~ap P 0;

8p 2 P;

8a 2 A;

ð35Þ

epa;b P 0;

8p 2 P;

8a; b 2 A;

ð36Þ

mrv 2 Zþ [ f0g;

8r 2 R;

8v 2 V;

ð37Þ

nrv 2 Zþ [ f0g;

8r 2 R;

8v 2 V;

ð38Þ

tport v ;ri P 0;

8i ¼ 1; 2; . . . ; Nr ;

P 0; nfold a

8a 2 A:

8r 2 R;

8v 2 V;

ð39Þ ð40Þ

where M is a big positive constant. 168 in constraints (21) is the number of hours in one week. The objective function (8) aims to minimize the total weekly operating cost, which includes eight terms: (i) the fixed operating cost and the bunker cost when sailing at sea; (ii) the inventory cost for empty containers; (iii) the cost for loading and discharging laden containers; (iv) the cost for loading and discharging empty containers; (v) the holding cost of foldable

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J. Zheng et al. / Transportation Research Part E 94 (2016) 123–140

containers; (vi) the cost for folding and unfolding empty containers; (vii) the substitution cost for empty containers; and (viii) the berth charge and the bunker cost when berthing at the port. Constraints (9) show that only one type of ship (i.e., ships with the same type) will be deployed on the opened shipping service route. Constraints (10) and (11) describe that a number of ships with any particular type will be deployed on a single shipping service route when this ship type is determined to serve this shipping service route. Constraints (12) mean that there is more than one ship for each ship fleet. Constraints (13) enforce that each port is served by at least one ship fleet. Constraints (14)–(20) are flow conservation constraints. Constraints (21) mean that a number of ships with a suitable ship type are deployed on each single ship route so as to maintain a weekly service. Constraints (22) are used to determine the time spent at the port for handling containers. Constraints (23) aim to determine the number of foldable containers in order to calculate the holding cost of foldable containers. Finally, constraints (24)–(40) define the domain of the decision variables. As shown in Appendix A, we also provide three comparable models (denoted by models B, C and D), in order to analyze the effect of two strategies (i.e., the use of foldable containers and the mutual substitution between empty container containers) on our problem. Only the use of foldable containers is considered in model B, and only the mutual substitution between empty container containers is considered in model C. None of these two strategies is considered in model D. Obviously, these four models (i.e., models A, B, C and D) are nonlinear because of f^f a  xrv g and/or f~f a  xrv g in their objective ri

ri

functions. In order to linearize our models, these two nonlinear terms can be simply replaced by two non-negative auxiliary ^ariv g and fq ~ariv g, respectively. Furthermore, the following constraints should be satisfied: variables fq

^ariv P ^f ari  Mð1  xrv Þ; q

8a 2 A;

8i ¼ 1; 2; . . . ; N r ;

8r 2 R;

8v 2 V

ð41Þ

~ariv P ~f ari  Mð1  xrv Þ; q

8a 2 A;

8i ¼ 1; 2; . . . ; N r ;

8r 2 R;

8v 2 V

ð42Þ

3.2. Calculation of the perceived container leasing prices Based on the solution obtained by solving model A in stage I, we aim to determine the perceived container leasing prices at different ports in stage II, by making use of the inverse optimization technique (Ahuja and Orlin, 2001). Firstly, we investigate a reduced model of our problem proposed in stage I, denoted by model RA, where only the empty container repositioning issue is studied. Similar to Benders decomposition (Benders, 1962), by considering variables fxrv g, a;o a;o fnrv g, fmrv g, fza;o ri g, furi g and ff ri g the complicating ones, which are used to determine the shipping service route selection, ship fleet deployment and laden container routing, it is possible to reformulate model A to obtain an equivalent problem. To achieve this, the other variables of model A are projected out through the parameterization of variables fxrv g, fnrv g, fmrv g, a;o a;o fza;o ri g, furi g and ff ri g, which results in the following primal linear model RA: N r Xh  XX i XX emp;a ^a ^ a ~ ari Þ  cload ~load cout;p  yaout;p  cemp;a  yain;p  ð~zari þ u a;ri þ ðzri þ uri Þ  c a;ri in;p

max

a2A p2P



XXh

i

^a ~a cfold a;p  ðg p þ g p Þ 

r2R i¼1 a2A

N r X XX

p2P a2A

cinv a 

r2R i¼1 a2A

XXX Lri p  xrv r  ð^f ari þ ~f ari Þ  ðcsub a;b  ea;b Þ sv r p2P a2A b2A

ð43Þ

subject to (28)–(36),

g^ap  g~ap þ

XX ^ ari  ^zari Þ ¼ 0; ðu

8a 2 A;

8p 2 P;

ð44Þ

r2Rp i2I rp

2 ba 

4ðya out;p



yain;p Þ

3   XX  X  a a a a ~zri  u ~ ri 5 þ þ g^p  g~p þ bb  epb;a  ba  epa;b ¼ ba  Eap ; r2Rp i2I rp

8p 2 P;

8a 2 A;

ð45Þ

b2A

^f a  ^f a ^a ^a ri rðiþ1Þ þ zrðiþ1Þ  urðiþ1Þ ¼ 0;

8i ¼ 1; 2; . . . ; N r ;

8a 2 A;

8r 2 R;

ð46Þ

~f a  ~f a ~a ~a ri rðiþ1Þ þ zrðiþ1Þ  urðiþ1Þ ¼ 0;

8i ¼ 1; 2; . . . ; N r ;

8a 2 A;

8r 2 R;

ð47Þ

XX X  rv r  ba  ða  ^f ari þ ~f ari Þ 6 Capv r  m ðba  f a;o ri Þ; a2A

8i ¼ 1; 2; . . . ; Nr ;

8r 2 R;

ð48Þ

o2P a2A

 N r Xh Nr  X X    i Lri a a ~tav ;ri  ~zari þ u ^tav ;ri  ^zari þ u  rv r ^ ~   m þ M  ð1  x Þ  þ 6 168  n r v r v r r ri ri r r sv r i¼1 a2A i¼1 

N r XXh X i¼1 o2P a2A

i  a;o t av r ;ri  ðza;o ri þ uri Þ ;

8r 2 R;

ð49Þ

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J. Zheng et al. / Transportation Research Part E 94 (2016) 123–140

yain;p P 0; yaout;p P 0;

8p 2 P; 8p 2 P;

8a 2 A;

ð50Þ

8a 2 A:

ð51Þ

a;o  rv r g, fm  rv r g, fza;o  a;o where f xrv r g, fn ri g, furi g and ff ri g are fixed vectors obtained by solving model A in stage I, and v r is the type of ship deployed on shipping service route r 2 R. cemp;a and cemp;a out;p are the perceived prices for weekly leasing in and off one in;p emp;a empty container with type a (8a 2 A) at port p 2 P, respectively. Generally, cemp;a is satisfied. yain;p and yaout;p are the out;p 6 c in;p

weekly number of empty containers with type a (8a 2 A) leased in and off at port p 2 P, respectively. It seems that we consider the leasing of empty containers in model RA. This paper aims to determine the parameters of model RA (i.e., fcemp;a in;p g and fcemp;a out;p g), in order that the projection of the optimal solution of model A is the optimal solution of model RA. Next, we make use of the dual of model RA to determine the perceived container leasing prices at different ports (i.e., emp;a fcemp;a in;p g and fcout;p g), following the inverse optimization technique (Ahuja and Orlin, 2001). k ¼ f^ ka ; i ¼ 1; 2; . . . ; N r ; a 2 A; r 2 Rg, ~ k ¼ f~ ka ; i ¼ 1; 2; . . . ; N r ; a 2 A; r 2 Rg, Let r ¼ fra ; a 2 A; p 2 Pg, s ¼ fsa ; p 2 P; a 2 Ag, ^ p

p

ri

ri

p ¼ fpri : pri P 0; i ¼ 1; 2; . . . ; Nr ; r 2 Rg and l ¼ flr : lr P 0; r 2 Rg be the dual variables associated with constraints (44)–

P r Lri P r P P a a;o  a;o  rv r  Ni¼1  rv r þ M  ð1   m ¼ 168  n xrv r Þ  Ni¼1 (49), respectively. Let t emp r o2P a2A ½t v r ;ri  ðzri þ uri Þ (8r 2 R) and sv r P P  rv r  o2P a2A ðba  f a;o Capemp ¼ Capv r  m 8r 2 R). The dual of model RA (denoted by model DRA) can ri Þ (8i ¼ 1; 2; . . . ; N r ; ri be described as follows:

min

Nr XX XX X ðba  Eap  sap Þ þ ðCapemp  pri Þ þ ðtemp  lr Þ r ri a2A p2P

r2R i¼1

ð52Þ

r2R

subject to

ba  sap P cemp;a out;p ;

8p 2 P;

ba  sap P cemp;a in;p ;

8a 2 A;

8p 2 P;

ð53Þ

8a 2 A;

load ~a ~a ba  sap þ k rði1Þ þ t v r ;ri  lr P ca;ri ;

8a 2 A;

load ~a ~a ba  sap  k rði1Þ þ t v r ;ri  lr P c a;ri ;

^a ^a ~load rap þ k rði1Þ þ t v r ;ri  lr P ca;ri ;

ð54Þ

8i 2 Irp ;

8a 2 A;

8a 2 A;

8r 2 Rp ;

8i 2 Irp ;

8i 2 Irp ;

8p 2 P;

8r 2 Rp ;

8r 2 Rp ;

ð55Þ

8p 2 P;

8p 2 P;

ð56Þ ð57Þ

rap  k^arði1Þ þ ^tav r ;ri  lr P ~cload 8a 2 A; 8i 2 Irp ; 8r 2 Rp ; 8p 2 P; a;ri ;

ð58Þ

rap þ ba  sap P cfold 8p 2 P; 8a 2 A; a;p ;

ð59Þ

rap  ba  sap P cfold a;p ;

ð60Þ

8p 2 P;

8a 2 A;

 cinv a  Lri  xr v r ^a ^a  k ; k ri rði1Þ þ ba  a  pri P  sv r  cinv a  Lri  xr v r ~a ~a  k ; k ri rði1Þ þ ba  pri P  sv r ba  ðsbp  sap Þ P csub a;b ;

8p 2 P;

8i ¼ 1; 2; . . . ; Nr ;

8i ¼ 1; 2; . . . ; Nr ;

8a; b 2 A;

8a 2 A;

8a 2 A;

8r 2 R;

8r 2 R;

ð61Þ

ð62Þ ð63Þ

pri P 0; 8i ¼ 1; 2; . . . ; Nr ; 8r 2 R;

ð64Þ

lr P 0; 8r 2 R:

ð65Þ

a ~ a ~ ~a  ^a  a ^ a p ~a ^ a;o  a;o  ^a   rv ; n  rv ; f a;o Let the vector ð xr v ; m ri ; zri ; uri ; f ri ; f ri ; zri ; zri ; uri ; uri ; g p ; g p ; ea;b Þ denote the optimal solution of model A. Following the inverse optimization technique (Ahuja and Orlin, 2001), we aim to determine the parameters cemp;a (8p 2 P; 8a 2 A) and cemp;a out;p in;p (8p 2 P; 8a 2 A) of model RA, in order that the optimal solution of model RA follows the optimal solution of model A. Namely, (8p 2 P; 8a 2 A) and cemp;a (8p 2 P; 8a 2 A), the vector based on the proper values of cemp;a out;p in;p

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J. Zheng et al. / Transportation Research Part E 94 (2016) 123–140

  ^a    ^ ari ; u ~ ari ; g ^ap ; g ~ap ;  ain;p ; y aout;p Þ can become the optimal solution of model RA, where y ain;p ¼ 0 (8p 2 P; 8a 2 A) and ð^f ari ; ~f ari ;  zri ; ~zari ; u epa;b ; y   p   ~zari ; u ^zari ;  ^ ari ; u ~ ari ; g ^ap ; g ~ap ;  ain;p ; y aout;p Þ is a feasible solution of model RA. aout;p ¼ 0 (8p 2 P; 8a 2 A). Obviously, the vector ð^f ari ; ~f ari ;  ea;b ; y y   a a a a ^ ~  p; s p ; kri ; kri ; p  ri ; l  r Þ represent a feasible solution of model DRA. One form of the linear programming optimalLet the vector ðr   ^a    ^ ari ; u ~ ari ; g ^ap ; g ~ap ;  ain;p ; y aout;p Þ and the dual solution ity conditions states that the primal solution ð^f ari ; ~f ari ;  zri ; ~zari ; u epa;b ; y  a    ap ; s ap ; ^ ðr kari ; ~ kri ; pri ; lr Þ are optimal for their problems if these two solutions are feasible, and together they satisfy the following primal-dual complementary slackness conditions:   P    ri ¼ 0. (1) If a2A ba  a  ^f ari þ ~f ari < Capemp (8i ¼ 1; 2; . . . ; N r ; 8r 2 R), then p ri a  a i PN r P h a emp   a a a  r ¼ 0. ^ ri Þ þ ~tv r ;ri  ~zri þ u ~ ri < t r (8r 2 R), then l (2) If i¼1 a2A ^t v r ;ri  ð^zri þ u     cinv a Lri xr v r ^a  k ^a  (3) If ^f ari > 0 (8i ¼ 1; 2; . . . ; N r ; 8a 2 A; 8r 2 R), then k . ri rði1Þ þ ba  a  pri ¼  sv r inv L   c x   rv r ri a ~a  k ~a  (4) If ~f ari > 0 (8i ¼ 1; 2; . . . ; N r ; 8a 2 A; 8r 2 R), then k . ri rði1Þ þ ba  pri ¼  sv r  ^a ^a   ap þ k ^zari > 0 (8a 2 A; 8i 2 Irp ; 8r 2 Rp ; 8p 2 P), then r ~load (5) If  rði1Þ þ t v r ;ri  lr ¼ ca;ri .  a a a a  ~ p þ krði1Þ þ ~tv ;ri  l  r ¼ cload (6) If ~zri > 0 (8a 2 A; 8i 2 Irp ; 8r 2 Rp ; 8p 2 P), then ba  s a;ri . r   a a a a load ^ ^  p  krði1Þ þ tv ;ri  l  r ¼ ~ca;ri . ^ ri > 0 (8a 2 A; 8i 2 Irp ; 8r 2 Rp ; 8p 2 P), then r (7) If u r   ~a ~ta  l ap  k  r ¼ cload ~ ari > 0 (8a 2 A; 8i 2 Irp ; 8r 2 Rp ; 8p 2 P), then ba  s (8) If u þ rði1Þ a;ri . v r ;ri a a a fold   þb s  ¼ c . (9) If g^ > 0 (8p 2 P; 8a 2 A), then r p

p

a

p

a;p

 ap  ba  s ap ¼ cfold ~ap > 0 (8p 2 P; 8a 2 A), then r (10) If g a;p . p bp  s ap Þ ¼ csub (11) If  ea;b > 0 (8p 2 P; 8a; b 2 A), then ba  ðs a;b .

n o P  ~a ^a   emp Let Xcap a2A ba  ða  f ri þ f ri Þ < Capri ; 8i ¼ 1; 2; . . . ; N r ; 8r 2 R , ri ¼ ði; rÞj n P P h i o r   ^a ~a emp ; 8r 2 R , ^a  ~a  ^a ~a Xtime ¼ rj Ni¼1 r a2A t v r ;ri  ðzri þ uri Þ þ t v r ;ri  ðzri þ uri Þ < t r n o  Xfoldlegflow ¼ ða; i; rÞj^f ari > 0; 8i ¼ 1; 2; . . . ; N r ; 8a 2 A; 8r 2 R , ria n o  Xunfoldlegflow ¼ ða; i; rÞj~f ari > 0; 8i ¼ 1; 2; . . . ; N r ; 8a 2 A; 8r 2 R , ria n o ^zari > 0; 8a 2 A; 8i 2 Irp ; 8r 2 Rp ; 8p 2 P , Xfoldload ¼ ða; i; rÞj ria

za > 0; 8a 2 A; 8i 2 Irp ; 8r 2 Rp ; 8p 2 P , Xunfoldload ¼ ða; i; rÞj~ ria ri n o  ^ a > 0; 8a 2 A; 8i 2 I ; 8r 2 R ; 8p 2 P , Xfolddis ¼ ða; i; rÞju ria

ri

rp

p

  ~ ari > 0; 8a 2 A; 8i 2 Irp ; 8r 2 Rp ; 8p 2 P , X ¼ ða; i; rÞju n o ^ap > 0; 8a 2 A; 8p 2 P , Xfoldnum ¼ ða; pÞjg pa n o ~ap > 0; 8a 2 A; 8p 2 P , Xunfoldnum ¼ ða; pÞjg pa n o Xsubstinum ¼ ða; b; pÞjepa;b > 0; 8a; b 2 A; 8p 2 P . pab unfolddis ria

Now, we further discuss an inverse optimization problem (denoted by IOP), which aims to find a feasible dual solution and the proper values of parameters cemp;a (8p 2 P; 8a 2 A) and cemp;a out;p (8p 2 P; 8a 2 A) satisfying the primal-dual complemenin;p tary slackness conditions. The following theorem shows the non-uniqueness of the solution of IOP.     satisfies all constraints of IOP, then cemp;a Theorem 1. If a vector cemp;a ; cemp;a þ c; cemp;a out;p out;p  c is also a feasible vector for IOP, in;p in;p where c is an arbitrary nonnegative number. ^a  ~a   emp;a  a a  emp;a emp;a Proof. For the vector ðcemp;a in;p ; cout;p Þ, we assume that a dual solution ðrp ; sp ; kri ; kri ; pri ; lr Þ and the vector ðcin;p ; c out;p Þ satisfies all constraints of IOP (i.e., all constraints of model DRA and the primal-dual complementary slackness conditions). For this þ c; cemp;a feasible dual solution, we can also find a feasible vector ðcemp;a out;p  cÞ, where c is an arbitrary nonnegative number, in;p since all constraints of IOP can be satisfied. h Based on Theorem 1, we can define the objective function of IOP as the minimization of the difference between the perceived leasing-in price and the perceived off-leasing price at each port. Hence, IOP can be formulated as follows:

min

XX emp;a ðcin;p  cemp;a out;p Þ a2A p2P

subject to (64), (65)

ð66Þ

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J. Zheng et al. / Transportation Research Part E 94 (2016) 123–140

ba  sap  cemp;a out;p P 0; 8p 2 P; ba  sap þ cemp;a P 0; in;p

8a 2 A;

8p 2 P;

ð67Þ

8a 2 A;

load ~a ~a ba  sap þ k rði1Þ þ t v r ;ri  lr P ca;ri ;

8a 2 A;

load ~a ~a ba  sap  k rði1Þ þ t v r ;ri  lr P c a;ri ;

^a ^a ~load rap þ k rði1Þ þ t v r ;ri  lr P ca;ri ;

ð68Þ

8i 2 Irp ;

8a 2 A;

8a 2 A;

8r 2 Rp ;

8i 2 Irp ;

8i 2 Irp ;

8p 2 P;

8r 2 Rp ;

8r 2 Rp ;

8ða; i; rÞ R Xunfoldload ; ria

8p 2 P;

8p 2 P;

8ða; i; rÞ R Xunfolddis ; ria

8ða; i; rÞ R Xfoldload ; ria

ð69Þ ð70Þ ð71Þ

rap  k^arði1Þ þ ^tav r ;ri  lr P ~cload 8a 2 A; 8i 2 Irp ; 8r 2 Rp ; 8p 2 P; 8ða; i; rÞ R Xfolddis ; a;ri ; ria

ð72Þ

rap þ ba  sap P cfold 8p 2 P; 8a 2 A; 8ða; pÞ R Xfoldnum ; a;p ; pa

ð73Þ

rap  ba  sap P cfold a;p ;

ð74Þ

8p 2 P;

8a 2 A;

8ða; pÞ R Xunfoldnum ; pa

 cinv a  Lri  xr v r ^a ^a  k ; k ri rði1Þ þ ba  a  pri P  sv r  cinv a  Lri  xr v r ~a  k ~a k ; ri rði1Þ þ ba  pri P  sv r ba  ðsbp  sap Þ P csub a;b ;

8p 2 P;

8i ¼ 1; 2; . . . ; Nr ;

8i ¼ 1; 2; . . . ; Nr ;

8a; b 2 A;

8a 2 A;

8a 2 A;

8ða; b; pÞ R Xsubstinum ; pab

8r 2 R;

8r 2 R;

8ða; i; rÞ R Xfoldlegflow ; ria

8ða; i; rÞ R Xunfoldlegflow ; ria

ð75Þ

ð76Þ ð77Þ

pri ¼ 0; 8ði; rÞ 2 Xcap ri ;

ð78Þ

lr ¼ 0; 8r 2 Xtime ; r

ð79Þ

 cinv a  Lri  xrv r ^a  k ^a k ; ri rði1Þ þ ba  a  pri ¼  sv r  cinv a  Lri  xrv r ~a  k ~a k ; ri rði1Þ þ ba  pri ¼  sv r ^a ^a ~load rap þ k rði1Þ þ t v r ;ri  lr ¼ c a;ri ;

8ða; i; rÞ 2 Xfoldlegflow ; ria

8ða; i; rÞ 2 Xunfoldlegflow ; ria

8ða; i; rÞ 2 Xfoldload ; ria

load ~a ~a ba  sap þ k rði1Þ þ t v r ;ri  lr ¼ c a;ri ;

8ða; i; rÞ 2 Xunfoldload ; ria

ð80Þ

ð81Þ ð82Þ ð83Þ

rap  k^arði1Þ þ ^tav r ;ri  lr ¼ ~cload 8ða; i; rÞ 2 Xfolddis ; ria a;ri ;

ð84Þ

load ~a ~a ba  sap  k rði1Þ þ t v r ;ri  lr ¼ c a;ri ;

ð85Þ

8ða; i; rÞ 2 Xunfolddis ; ria

rap þ ba  sap ¼ cfold 8ða; pÞ 2 Xfoldnum ; a;p ; pa

ð86Þ

rap  ba  sap ¼ cfold a;p ;

8ða; pÞ 2 Xunfoldnum ; pa

ð87Þ

ba  ðsbp  sap Þ ¼ csub a;b ;

8ða; b; pÞ 2 Xsubstinum ; pab

ð88Þ

cemp;a out;p P 0;

8p 2 P;

8a 2 A;

ð89Þ

P 0; cemp;a in;p

8p 2 P;

8a 2 A:

ð90Þ

Constraints (67)–(77) follow constraints (53)–(63) of model DRA. Constraints (78)–(88) represent the primal-dual complementary slackness conditions. On the optimal solution of IOP, the perceived leasing-in price and the perceived off-leasing price for any container type at any particular port are identical, as shown in the following theorem.

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J. Zheng et al. / Transportation Research Part E 94 (2016) 123–140

  satisfies the optimal solution of IOP, then cemp;a Theorem 2. If a vector cemp;a ; cemp;a ¼ cemp;a out;p out;p (8p 2 P; in;p in;p

8a 2 A).

emp;a Proof. We show a proof by contradiction. Let the vector ðcemp;a in;p ; c out;p Þ satisfy the optimal solution of IOP. Without loss of gener







  emp;a emp;a > cemp;a ality, we assume cemp;a out;p , according to constraints (67) and in;p –cout;p for any p 2 P and a 2 A. Then we can have cin;p emp;a

(68). Then the objective value of IOP is larger than 0. Let ~cin;p  cemp;a in;p

 cemp;a out;p

emp;a

ap , where the decision variable fs ap g satisfies ¼ ~cout;p ¼ ba  s  ~cemp;a in;p





  and are replaced by and ~cemp;a the optimal solution of IOP. When out;p for p 2 P and a 2 A in the vector   emp;a , the objective value of IOP can be further reduced, and all constraints of IOP are not violated. It means that the vector cemp;a in;p ; c out;p   emp;a cannot satisfy the optimal solution of IOP. h cemp;a in;p ; c out;p

4. Numerical experiments In this section, we provide the numerical results for a realistic Asia-Europe-Oceania shipping service network with 46 ports, as shown in Fig. 1. The set of candidate shipping service routes can be found on the websites provided by different liner shipping companies (e.g., OOCL, http://www.oocl.com; Maersk, http://www.maerskline.com/appmanager). In Appendix B, we show the set of candidate shipping service routes used in our numerical experiments. Following Meng and Wang (2011), we assume that laden containers can only be transshipped at hub ports and empty containers can be transshipped at any port. From the practical point of view, the 10 ports, i.e., Rotterdam, Sokhna, Salalah, Colombo, Singapore, Hong Kong, Kaohsiung, Shanghai, Pusan and Yokohama are selected as hub ports. In order to meet the transshipment requirements for laden containers, the following constraints should be satisfied for models A, B, C and D:

XX a;o zri ¼ 0; j–o;

8j 2 P n P 1 ;

8o 2 P;

8a 2 A

ð91Þ

r2Rj i2Irj

For heterogeneous ships, we consider four different ship types, and the ship type related parameters are shown in Table 1. As for the set of container types, two container types, i.e., TEU and FEU are considered. As for the space or capacity occupied, one FEU is regarded as two TEUs. Namely, bTEU ¼ 1 and bFEU ¼ 2. The OD container demand is derived from the liner shipping company and we assume that about 70% of the container flows are assumed to be FEUs. In our numerical experiments, the 4:1 folding principle is considered, i.e., a ¼ 0:25. For the container handling time, we assume that tav ;p ¼ ~t av ;p for ship v handling containers with type a at any port p 2 P. Since folded empty containers are handled (loaded or discharged) as a batch, we assume that tav ;p ¼ 4^t av ;p . We further assume that each port has the same port productivity. Hence, the container handling

Fig. 1. Ports in an Asia-Europe-Oceania shipping service network.

135

J. Zheng et al. / Transportation Research Part E 94 (2016) 123–140 Table 1 Ship type related parameters. Ship type Ship capacity (TEUs) Ship speed (knots) Bunker cost at sea (USD/h) Fixed cost (USD/d)

0 1500 16.2 956 10,000

1 3000 18.9 1361 15,000

2 5000 22.5 4000 20,000

3 10,000 23 5000 35,000

time only depends on ship type. The number of standard containers to be handled per hour at each port for four ship types shown in Table 1 are 40, 70, 95 and 120, respectively. For simplicity, we assume that each port has an identical berth charge per hour for the same ship type. Following Wang and Meng (2012), the hourly berth charges for four ship types shown in Table 1 are 500 USD, 1000 USD, 1666 USD and 3333 USD, respectively. Following Brouer et al. (2014a), the hourly bunker cost when berthing at port for these four ship types are 52 USD, 94 USD, 115 USD and 167 USD, respectively. For the inveninv tory cost, cinv TEU ¼ c FEU ¼ 0:5 is adopted, following Wang and Meng (2015). For the substitution cost, we assume that sub hold hold cTEU;FEU ¼ 30 (USD) and csub FEU;TEU ¼ 50 (USD). For the holding cost of foldable containers, we assume c TEU ¼ cFEU ¼ 20 (USD per week), following Moon and Hong (2016). The proposed models are solved by CPLEX, which runs on a 3.2 GHz Dual Core desktop PC with the Windows 7 operating system and 4 GB of RAM. Our models can be solved in a reasonable time, according to the CPU times used for solving different models, as shown in Table 2. Table 2 shows a comparison among four different models. The column ‘‘Ship cost” in Table 2 represents the total ship related cost (i.e., the fixed operating cost, the bunker cost and the berth charge) for all ships deployed on each shipping service routes, and the column ‘‘Container cost” means the total container related cost (i.e., the container handling cost, the substitution cost, the inventory cost and the holding cost). Compared with other three models, model D has the highest total cost. As compared with model D, the total cost is reduced by 4.51%, 3.57% and 3.91% for models A, B and C, respectively. Evidently, by comparing models B with C, the ship related cost is reduced, however the container related cost is increased, when the use of foldable containers is considered in model B. It means that the use of foldable containers has an effective impact on ship fleet deployment. Furthermore, Table 2 shows that only FEU empty containers are folded and unfolded for both models A and B. This is because, 70% of the container flows are assumed to be FEUs, and the cost for folding or unfolding one FEU empty container is less than that for two TEU empty containers. When the mutual substitution between empty containers is allowed, the number of empty containers to be repositioned is reduced, and the container related cost can be decreased. As shown in Table 2, the number of TEU empty containers to be substituted is twice as large as the number of FEU empty containers to be substituted for both models A and C. This is because the number of empty containers to be substituted between TEU and FEU should be balanced. On the optimal solution of IOP, let cemp;a ¼ cemp;a ¼ cemp;a p out;p denote the perceived container leasing price for container type in;p a 2 A at port p 2 P. From the liner shipping company perspective, let cemp;a and cemp;a out;p denote the realistic prices for weekly in;p leasing in and off one empty container with type a 2 A at port p 2 P, respectively. Generally, cemp;a > cemp;a out;p is satisfied for any in;p with cemp;a and cemp;a container type a 2 A at port p 2 P. By comparing cemp;a p out;p , it is helpful for decision making on container in;p leasing at any particular port p 2 P. If cemp;a P cemp;a is satisfied, it is an economic way to lease in and/or off empty containers p in;p . If with type a 2 A at port p 2 P, because the repositioning cost for port p 2 P is very high, leading to a high value of cemp;a p cemp;a < cemp;a p out;p is satisfied, it is an economic way to reposition empty containers with type a 2 A for port p 2 P, rather than emp;a < cemp;a the leasing of empty containers at port p 2 P. If cemp;a is satisfied, we consider two cases. In the first case, out;p < cp in;p port p 2 P has a surplus of empty containers with type a 2 A, and it is better to lease off empty containers with type a 2 A at port p 2 P. In the second case, port p 2 P has a deficit of empty containers with type a 2 A, and it is a better strategy by repositioning empty containers with type a 2 A for port p 2 P, because the container leasing-in price is high. Here we do not consider the comparison between the realistic container leasing prices and the perceived container leasing prices at different ports.

Table 2 Comparison among different models. CPU time (s)

Model Model Model Model

D C B A

126.81 112.51 211.87 186.89

The cost (107 USD)

Number of empty containers folded

Number of empty containers substituted

Ship cost

Container cost

Total cost

TEUs

FEUs

TEUs

FEUs

1.752 1.71 1.683 1.662

0.907 0.845 0.881 0.877

2.659 2.555 2.564 2.539

– – 0 0

– – 69 448

– 2104 – 2100

– 1052 – 1050

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J. Zheng et al. / Transportation Research Part E 94 (2016) 123–140

Table 3 The perceived container leasing prices at different ports. Port

Brisbane Fremantle Adelaide Melbourne Sydney Chittagong Zeebrugge Antwerp Sokhna Lehavre Hamburg Bremerhaven Hongkong Cochin Chennai Nhavasheva Jakarta Tokyo Nagoya Yokohama Kobe Aqabah Pusan

The perceived container leasing prices TEU

FEU

21.1442 90.041 26.6993 15.2672 77.832 56.4603 2.46825 65.7077 107.421 50.4943 6.92989 31.6508 217.71 110.062 116.312 47.0648 0 255.681 173.938 315.272 223.544 77.4255 252.309

92.2884 120.082 103.399 80.5344 95.664 162.921 54.9365 71.4153 264.841 40.9885 63.8598 3.30159 375.421 160.123 172.623 144.13 50 561.362 287.876 570.544 387.087 204.851 444.617

Port

Kwangyang Portklang Rotterdam Dalian Xingang Qingdao Ningbo Shanghai Salalah Karachi Manila Jeddah Singapore Yantian Xiamen Chiwan Colombo Kaohsiung Leamchabang Jebelali Thamesport Southampton Hochiminh

The perceived container leasing prices TEU

FEU

258.287 176.995 5.71561 242.938 272.704 248.608 290.988 239.152 103.491 103.8 153.179 21.791 169.513 225.706 254.841 239.665 154.497 258.996 254.658 119.238 62.3042 6.42989 247.964

456.574 396.008 61.4312 425.877 485.407 437.216 521.975 418.304 256.981 257.601 356.359 93.582 389.026 391.413 449.682 419.331 248.995 457.992 449.317 288.476 64.6085 62.8598 435.928

In Table 3, we show the results of the perceived container leasing prices for different container types at each port. For the port of Bremerhaven in Europe, its perceived container leasing price for FEU container type is happened to be smaller than that for TEU container type. This is because this port becomes a deficit port on certain container type when randomly distinguishing the container demands between TEU and FEU during data processing. The demand of empty containers for any port in Europe can be easily satisfied, since many empty containers are accumulated in Europe. For any other port, the corresponding perceived container leasing price for FEU container type is expected to be larger than that for TEU container type. This is because one FEU empty container is regarded as two TEU empty containers from occupied ship capacity point of view, however the marginal cost on repositioning one FEU empty container is usually less than that on repositioning two TEU empty containers. As shown in Table 3, there is one port (i.e., Jakarta), where the perceived container leasing price is 0 and 50 for TEU container type and FEU container type, respectively. This is because only the substitution cost is spent on repositioning empty containers for this port. Except for this port, European ports have the comparatively low perceived container leasing prices, while Asian ports have the comparatively high perceived container leasing prices. This is due to the imbalance of imports and exports between Asia and Europe, many empty containers are accumulated in Europe, while ports in Asia have a deficit. Moreover, Yokohama has the highest perceived container leasing prices for both TEU container type and FEU container type. As located in Northeast Asia shown in Fig. 1, the demand of empty containers at Yokohama is satisfied by repositioning empty containers from the surplus ports which are far away from Yokohama. Hence the repositioning cost is quite high, leading to a high perceived container leasing price. Note that the perceived container leasing prices are quite different for some adjacent ports (e.g., Tokyo and Nagoya), however the realistic container leasing prices might not be different too much in the same country or region. One can further consider the container transportation issue in the hinterland, in order to reduce such kind of differences on the perceived container leasing prices. For simplicity, the hinterland transportation is not considered in this paper. In Fig. 2, we show the results of the perceived container leasing prices for FEU container type at different ports. Clearly, ports can be classified into four clusters identified by three rectangles, according to different geographical locations. European ports form a single cluster, where the perceived container leasing prices are quite low. Ports between Suez Canal and India are grouped into the second cluster, where the perceived container leasing prices are at a medium level. Jakarta and ports in Australia are in the third cluster, where the perceived container leasing prices are very low, similar to those at European ports. As located in Asia (i.e., Southeast, East and Northeast Asia), the rest ports belong to the fourth cluster, where the perceived container leasing prices are quite high. As mentioned before, high production cost of foldable containers have made liner shipping companies hesitant to adopt these units (Moon and Hong, 2016). In practice, the production cost of foldable container becomes lower, in order that foldable containers can be popularized in liner shipping industry. It would be interesting to investigate our problem by considering the impact of different production costs of foldable containers. Alternatively, this paper considers different holding costs of foldable containers, as shown in Fig. 3. Evidently, the number of foldable containers decreases dramatically, as

J. Zheng et al. / Transportation Research Part E 94 (2016) 123–140

137

Fig. 2. The perceived container leasing prices for FEU container type at different ports.

Fig. 3. The results for different holding costs.

the holding cost increases. However, both the ship related cost and the container related cost almost keep unchanged for different holding costs of foldable containers. In other words, the ship related cost and the container related cost are not sensitive to the holding cost of foldable containers.

5. Summary This paper aims to determine the perceived container leasing prices for different container types at different ports by presenting a two-stage optimization method. In stage I, we propose a practical liner shipping network design problem with empty container repositioning. A mixed-integer nonlinear programming model is developed for our proposed problem. The nonlinear terms are linearized by introducing the auxiliary variables. Based on the solution obtained in stage I, the perceived container leasing prices for different container types at each port are determined in stage II, by making use of the inverse optimization technique. Two strategies (i.e., the use of foldable containers and the mutual substitution between empty containers) are taken into account in order to reduce the repositioning costs. Results show that the total cost for our proposed problem in stage I can be reduced by using these two strategies. When the use of foldable containers is

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considered, the ship related cost can be reduced. When the mutual substitution between empty containers is considered, the container related cost can be obviously reduced. Furthermore, with respect to the perceived container leasing prices at different ports in an Asia-Europe-Oceania shipping service network, ports are partitioned into several clusters. To simply our problem, this study has made several assumptions, for example candidate shipping service routes are fixed and given. The future work could extend our problem from the following perspectives: (1) consider the generation of shipping service routes when solving the liner shipping network design problem; (2) investigate the impact of hinterland transportation on the perceived container leasing prices; (3) consider the realistic container leasing prices and analyze the competition between container leasing and empty container repositioning; and (4) investigate the routing of laden and empty containers in a global liner shipping network and design an efficient decomposition method to solve it. Acknowledgements We would like to thank anonymous referees for their useful comments, which significantly improve the presentation of this paper. This research is partly supported by the National Basic Research Program of China (2012CB725400), the National Natural Science Foundation of China (71501021, 61304179, 71431001), and the Special Program for the Fundamental Research Funds for the Central Universities (20110116201). Appendix A. As for comparing with model A, we also provide some comparable models. In order to explore the effect of a single strategy (i.e., the use of foldable containers or the mutual substitution between empty containers) on the liner shipping network design problem with empty container repositioning, we further provide two similar models (denoted by models B and C). In model B, only the use of foldable containers is considered. Without considering the mutual substitution between empty containers, model B can be described as follows:

" # Nr   N r XXh XX XX X  a;o i Lri a;o fix bunker þ cv  nrv þ mrv  cv  cload a;ri  zri þ uri s

min

r2R v 2V

i¼1

v

r2R i¼1 a2A o2P

Nr Xh i XXh  i X XX  a   a  ~zri þ u ^ ~ ari  cload ^a ^a ~a þ ~load þ cfold ðchold  nfold þ a;ri þ zri þ uri  ca;ri a;p  g p þ g p a a Þ r2R i¼1 a2A

p2P a2A

a2A

" # XX Nr X N r Xh   XL  i XX ri inv a a bunker;port ^ ~ þ ca  f ri þ f ri   xr v t port þ cberth þ v ;ri v ;ri  c v r2R i¼1 a2A v 2V sv r2R i¼1 v 2V

ðA1Þ

subject to (3), (5), (9)–(35), (37)–(40),

ðg^ap  g~ap Þ þ

XX ~ ari Þ ¼ Eap ; ð~zari  u

8a 2 A;

8p 2 P:

ðA2Þ

r2Rp i2I rp

In model C, only the mutual substitution between empty containers is taken into account. Without considering the use of foldable containers, model C can be described as follows:

" # Nr   N r XX XX XX X XXX Lri p a;o a;o fix bunker þ min cv  nrv þ mrv  cv  ½cload ðcsub a;ri  ðzri þ uri Þ þ a;b  ea;b Þ s v r2R v 2V r2R i¼1 a2A o2P p2P a2A b2A i¼1 " # Nr Xh Nr X i XX XX XLri  a  inv ~f a  ~zri þ u ~ ari  cload þ c   x þ rv a;ri a ri r2R i¼1 a2A r2R i¼1 a2A v 2V sv þ

Nr Xh XX r2R i¼1 v 2V

 i bunker;port t port þ cberth v ;ri v ;ri  cv

ðA3Þ

subject to (9)–(18), (20), (24)–(27), (29), (31), (33), (36)–(39),

X X XX a;o ðba  f ri Þ þ ðba  ~f ari Þ 6 ðCapv  mrv Þ; o2P a2A

ba 

v 2V

a2A

8i ¼ 1; 2; . . . ; Nr ;

XX X  ~zari  u ~ ari ¼ ba  Eap þ ðba  epa;b  bb  epb;a Þ; r2Rp i2I rp

8a 2 A;

8r 2 R;

ðA4Þ

8p 2 P;

ðA5Þ

b2A

168  nrv þ M  ð1  xrv Þ P

Nr   Nr XX Nr Xh i X X X Lri a;o ~t a  ð~za þ u ~ ari Þ ;  mrv þ ½t av ;ri  ðza;o ri v ;ri ri þ uri Þ þ s v i¼1 i¼1 a2A o2P i¼1 a2A

2 R;

8v 2 V;

8r ðA6Þ

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J. Zheng et al. / Transportation Research Part E 94 (2016) 123–140

tport v ;ri þ M  ð1  xrv Þ P

X a2A

~ ari Þ þ ½~t av ;ri  ð~zari þ u

XX a;o ½tav ;ri  ðza;o ri þ uri Þ;

8i ¼ 1; 2; . . . ; Nr ;

8r 2 R;

8v 2 V:

ðA7Þ

a2A o2P

In order for comparison, we also provide a general model (denoted by model D) without considering both the use of foldable containers and the mutual substitution between empty containers. Model D can be described as follows:

" # Nr   N r XXh Nr X XX X XX  a;o i X X Lri a;o bunker ~ ari Þ cfix  n þ m  c  cload ½ð~zari þ u þ þ r v r v a;ri  zri þ uri v v s v r2R v 2V r2R a2A o2P r2R a2A i¼1 i¼1 i¼1 " # XX Nr X Nr X XX XLri bunker;port ~a cinv  xr v ½t port þ cberth þ  cload v ;ri Þ a;ri  þ a  f ri  v ;ri  ðcv r2R i¼1 a2A v 2V sv r2R i¼1 v 2V

min

ðA8Þ

subject to (9)–(18), (20), (24)–(27), (29), (31), (33), (37)–(39), (A4), (A6), (A7),

XX ~ ari Þ ¼ Eap ; ð~zari  u

8a 2 A;

8p 2 P:

ðA9Þ

r2Rp i2I rp

Appendix B. The candidate shipping service routes used in our numerical experiments are as follows: No.

Port calling sequence of each candidate shipping service route

1

Rotterdam ? Hamburg ? Southampton ? Lehavre ? Colombo ? Singapore ? Hongkong ? Kobe ? Nagoya ? Tokyo ? Hongkong ? Singapore ? Jeddah ? Lehavre ? Southampton ? Hamburg ? Rotterdam Lehavre ? Southampton ? Hamburg ? Rotterdam ? Jeddah ? Singapore ? Yantian ? Ningbo ? Shanghai ? Yantian ? Singapore ? Southampton ? Hamburg ? Rotterdam ? Lehavre Rotterdam ? Hamburg ? Southampton ? Colombo ? Singapore ? Kwangyang ? Pusan ? Shanghai ? Ningbo ? Yantian ? Chiwan ? Singapore ? Colombo ? Southampton ? Hamburg ? Rotterdam Southampton ? Antwerp ? Hamburg ? Rotterdam ? Jebelali ? Singapore ? Chiwan ? Kaohsiung ? Xiamen ? Chiwan ? Hongkong ? Singapore ? Colombo ? Rotterdam ? Hamburg ? Antwerp ? Southampton Rotterdam ? Hamburg ? Southampton ? Salalah ? Colombo ? Singapore ? Hongkong ? Yantian ? Qingdao ? Shanghai ? Hongkong ? Yantian ? Singapore ? Salalah ? Southampton ? Hamburg ? Rotterdam Jebelali ? Nhavasheva ? Jeddah ? Sokhna ? Southampton ? Rotterdam ? Hamburg ? Antwerp ? Lehavre ? Sokhna ? Jebelali Sokhna ? Singapore ? Ningbo ? Shanghai ? Chiwan ? Hongkong ? Singapore ? Sokhna Salalah ? Singapore ? Hongkong ? Pusan ? Shanghai ? Ningbo ? Chiwan ? Hongkong ? Singapore ? Salalah Singapore ? Brisbane ? Sydney ? Melbourne ? Adelaide ? Fremantle ? Singapore Xiamen ? Chiwan ? Hongkong ? Singapore ? Portklang ? Salalah ? Jeddah ? Aqabah ? Salalah ? Singapore ? Xiamen Hochiminh ? Leamchabang ? Singapore ? Portklang ? Hochiminh Shanghai ? Yokohama ? Tokyo ? Nagoya ? Kobe ? Shanghai Singapore ? Laemchabang ? Hongkong ? Yantian ? Tokyo ? Yokohama ? Nagoya ? Kaohsiung ? Singapore Fremantle ? Melbourne ? Sydney ? Brisbane ? Yokohama ? Pusan ? Qingdao ? Shanghai ? Ningbo ? Brisbane ? Sydney ? Melbourne ? Fremantle Singapore ? Jakarta ? Fremantle ? Adelaide ? Melbourne ? Sydney ? Brisbane ? Singapore Manila ? Kaohsiung ? Xiamen ? Hongkong ? Yantian ? Chiwan ? Hongkong ? Manila Dalian ? Xingang ? Qingdao ? Shanghai ? Ningbo ? Shanghai ? Kwangyang ? Pusan ? Dalian Chittagong ? Chennai ? Colombo ? Cochin ? Nhavasheva ? Cochin ? Colombo ? Chennai ? Chittagong Sokhna ? Aqabah ? Jeddah ? Salalah ? Karachi ? Jebelali ? Salalah ? Sokhna Southampton ? Thamesport ? Hamburg ? Bremerhaven ? Rotterdam ? Antwerp ? Zeebrugge ? Lehavre ? Southampton

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

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