Reducing hinterland transportation costs through container sharing

Transcription

1 Flex Serv Manuf J DOI /s y Reducing hinterland transportation costs through container sharing Sebastian Sterzik Herbert Kopfer Won-Young Yun Ó Springer Science+Business Media New York 2012 Abstract Based on two scenarios which describe the movement of full and empty containers within seaport hinterland transportation, this contribution determines possible benefits of exchanging empty containers between cooperating trucking companies. In the first scenario, trucking companies only have access to their own containers. In the second scenario, empty containers are permitted to be interchanged among several owners. Descriptions of both scenarios are given by comprehensive mathematical formulations considering vehicle routing, simultaneously with empty container repositioning. Realistic-sized instances are solved by applying an efficient tabu search heuristic. The results show that container sharing between trucking companies leads to remarkable cost savings. Keywords Hinterland container truck transportation Container sharing Vehicle routing and scheduling Empty container repositioning Integrated routing Pickup and delivery 1 Introduction Cost saving possibilities for maritime transport chains have mainly been investigated by focusing on acquisitions and strategic alliances for sea transport only. Since the remaining potential for the reduction of expenses at sea declines, the S. Sterzik H. Kopfer (&) Department of Economics and Business Studies, University of Bremen, Wilhelm-Herbst-Str. 5, Bremen, Germany kopfer@uni-bremen.de S. Sterzik sterzik@uni-bremen.de W.-Y. Yun Department of Industrial Engineering, Pusan National University, Pusan , Korea

2 S. Sterzik et al. pressure to decrease costs in areas of intermodal door-to-door services is constantly rising. Remarkably, the research on coordination in the hinterland of seaports has received limited attention within the last decades although the portion of inland costs in the total costs of container shipping ranges from 40 to 80 % (Notteboom and Rodrigue 2005). Thereby, empty container repositioning constitutes a major cost driver for operating trucking companies since most of the routes in the hinterland are actually pendulum tours among terminals, trucking companies depots and customers (receivers/shippers). Solutions to reduce the amount of empty container movements mainly focus on the movement of available empty containers from particular customer locations directly to places where they will be needed next (Veenstra 2005). Hence, recently emptied containers at a receiver s location can be integrated into a successional transport chain by transporting them either to the depot, the seaport terminal or to a shipper. The comprehensive integration of container movements into overall transport chains is a promising approach to reduce container repositioning. Based on this approach, this contribution investigates the additional possibilities to reposition empty containers if trucking companies in the seaport hinterland cooperate with each other. Thereby, companies share their information on empty container location so that cooperating companies can agree to assign containers to transportation tasks which seem most appropriate for cost reduction. In detail, we present, analyze and compare two scenarios. While the first scenario describes a hinterland region in which trucking companies only use their own empty containers for their underlying transportation requests (see Fig. 1a), the second scenario permits the exchange of empty containers between all operating trucking companies in the hinterland region (see Fig. 1b). Both scenarios are defined formally by an integrated mathematical model simultaneously considering empty container repositioning and vehicle routing. Test cases known from literature are modified to represent several Fig. 1 Empty container repositioning in a seaport hinterland

3 Reducing hinterland transportation costs cooperating or non-cooperating partners in the scenario. These cases are solved by adopting an efficient tabu search heuristic which has already shown to be very efficient to solve original test cases for container hinterland transportation problems; i.e. unchanged instances from literature which have not been extended by introducing several partners to the underlying transportation scenario (Sterzik and Kopfer 2012). To analyze the potential of container sharing for different problem settings, we also determine the impact of tight and wide time windows at customer locations. Contrary to tight time windows, wide time windows offer the possibility to stack empty containers at customer locations before and after completing the containers loading or unloading. The obtained results give evidence that container sharing between cooperating trucking companies leads to remarkable cost savings. The paper is organized as follows: Section 2 reviews the relevant literature. A detailed problem description and the exact mathematical formulation of the integrated routing problems are given for scenarios without cooperation and with cooperation in Sects. 3 and 4, respectively. While Sect. 5 refers to the presentation of the solution heuristic, the obtained results are discussed in Sect. 6. Finally, Sect. 7 contains the conclusions and future research opportunities. 2 Literature review The literature directly related to the discussed topic comprises the field of empty container repositioning and the methodological approaches to solve truck transportation problems in the hinterland of seaports. Early descriptions dealing with the optimization of empty container allocation can be found in Crainic et al. (1993) who describe a dynamic deterministic scenario which handles the allocation of empty containers according to current and future customer demands. Thereby, they propose a general modelling framework reflecting the operational and planning complexity of hinterland transportation. Based on this framework Abrache et al. (1999) propose a decomposition algorithm for the deterministic multi-commodity model. Cheung and Chen (1998) try to improve the repositioning of empty containers for liner operators by modelling the dynamic container allocation problem as a two-stage stochastic model. They aim to determine the number of leased containers needed to meet customers demands over time. Another solution approach to reduce empty container repositioning costs is delivered by Choong et al. (2002) who define an integer formulation for a broader hinterland in which empty containers can be moved by barges at very low costs. More recently, Jula et al. (2006) analyze the impact of applying two empty container reuse strategies ( depot direct and street turn ) on the reduction of the number and the cost of truck trips in the Los Angeles port area. The authors assume the maritime terminal as the only container-depot in the underlying region so that empty containers cannot be stacked at a company s depot. The objective is to reduce congestions at the port area. The depot direct strategy benefits from constituting further off-dock container depots, which are intended to reduce the required number of empty trips. According to the street turn -strategy, empty containers are directly moved from receiver to shipper customers. By implementing a two-phase optimization technique

4 S. Sterzik et al. which seeks to find the best match between supply and demand of empty containers over a number of periods, it is concluded that the two introduced reuse strategies can reduce the traffic around the ports significantly. Since the focus is on avoiding congestion, the effects on the companies transportation costs are not considered. Veenstra (2005) reports about the Boxsharing concept at the Rotterdam port. This concept is based on a database system where the members of a coalition of participating trucking companies can put their own empty container surpluses in. In return, the members of the initiative can search for empty containers that they might want to use. Similar to the container sharing idea, companies can integrate empty containers into their routes. Launched in November 2002 the system contained only 300 empty containers stationed all over Europe in It is assumed that companies will be eager to participate if possible benefits of sharing empty containers with others are uncovered. A recent publication analyzing the idea of container sharing can be found in Kopfer and Sterzik (2012). In this, the authors describe a hinterland setting which is similar to the underlying setting in this contribution, but is restricted to consider solely one terminal. By defining an integrated mixed integer model which includes container allocation as well as vehicle routing, small test instances are solved with the solver software CPLE. The generated results indicate the large potential of container sharing. A more detailed overview of planning models concerning empty container repositioning in the hinterland of seaports can be found in Braekers et al. (2011b). Methodological approaches for solving truck transportation problems in hinterland regions have been becoming a research field with growing attention in the last few years. Erera et al. (2005) were the first authors to describe a single model which comprises the integration of vehicle routing and empty container repositioning. It is shown that integrated models are substantially larger than repositioning models but can represent the existing problems comprehensively. A full truckload problem described as vehicle routing problem with full containers (VRPFC) is addressed by Imai et al. (2007). They could identify near optimum solutions by using a subgradient heuristic based on Lagrangian relaxation. Jula et al. (2005) propose an asymmetric multi-traveling salesman problem with time windows (m-tsptw) to model the container movement by trucks in the hinterland of seaports. The authors apply a two-phase exact algorithm based on dynamic programming as well as a modified genetic algorithm to solve the problem. Similarly, Zhang et al. (2009) model a container truck transportation problem with multiple depots, two types of customers and one terminal as a m-tsptw. The authors develop a cluster method and a reactive tabu search (RTS) to solve the problem. Zhang et al. (2010) extended the setting of Zhang et al. (2009) by considering more than one terminal. In the following, this problem is defined as the inland container transportation problem (ICT). Inspired by Wang and Regan (2002) the authors apply a windowpartition based method (WPB method) to solve realistic-sized instances. 3 Problem description The underlying setting for our scenarios is based on the comprehensive ICT by Zhang et al. (2010). Though this problem considers a setting with multiple depots,

5 Reducing hinterland transportation costs only one trucking company is in charge to serve the underlying transportation requests in the regarded hinterland region. Therefore, the ICT has to be modified in order to be suitable for the measurement of the benefit of container sharing among a number of cooperating companies. In the following, we firstly present the Basic Setting that describes the modified ICT. Subsequently, the Multi-Company ICT (MCICT) characterizing the non-cooperative scenario and the cooperative scenario of the MCICT with Container Sharing (MCICT CS) are presented. 3.1 Basic setting In a local region full and empty containers have to be moved between different locations by at least two trucking companies. In detail, we consider a hinterland of at least two terminals, a number of customers and several depots each belonging to a trucking company. At every depot, an arbitrary number of vehicles can be parked and, moreover, the depots are defined as repositories for an unlimited number of empty containers. Regarding the terminals, not only seaports can be enclosed within the hinterland region. Further terminal types can be provided by rail yards or river ports. As can be seen in Fig. 2 we distinguish four transportation request types: inbound full (IF), inbound empty (IE), outbound full (OF) and outbound empty (OE). These requests can be separated into those requiring the transportation of inbound containers and those referring to outbound containers. Incoming containers located at a terminal that need to be moved to their destinations in the hinterland are called inbound containers. Reversely, containers located in the hinterland that need to be delivered to a terminal are called outbound containers. The defined container terms derive from the well-known research field of inbound and outbound logistics. We distinguish two types of customers. On the one hand shippers offer freight which is to be transported to a foreign region via a terminal. The flow of a full container from a shipper to a terminal is defined as OF request. As stated, this transportation request is defined as outbound full since a full container needs to be moved from the hinterland to a terminal. On the other hand, receivers require the transport of their goods from an outside region via a terminal. The full container which has to be transported from a terminal to a receiver is called IF container. For Fig. 2 Modified ICT with one terminal

6 S. Sterzik et al. both full transportation types the pickup and delivery location are always given in advance. Obviously, these transportation tasks lead to an empty container positioning or repositioning problem. Firstly, before an OF task can be processed, a shipper requires an empty container to be loaded with his freight. The origin of this empty container must be determined during the solution process. Secondly, the receiver of an IF task obtains an empty container after the container is unloaded. The determination of the container s destination also requires a decision for allocating empty containers. Due to the imbalance between import- and export-dominated areas, we also need to consider OE or IE containers which either have to be moved to a terminal or originate from it. The origin of an OE container within the hinterland (i.e. which container to take for the OE process) and vice versa the destination of an IE container is not given in advance and thus has to be determined during the solution process. Considering an import-dominated area, a surplus of empty containers is available in the hinterland related to this area. Therefore, these additional empty transportation resources must be moved to export-dominated regions as OE containers via the terminals. The possible origins of these containers are the locations at which empty containers accrue. Within the underlying setting these locations are the depot and the receiver locations after an IF container is unloaded. In an export-dominated area, a lack of empty transportation resources arises and leads to necessary transportations of empty containers from different regions via the terminals to the hinterland. Therefore, the trucking company needs to move empty containers from the terminals to locations at which empty containers are required. If there is no shipper node which needs an empty container, there is the possibility to store the containers temporarily at a depot. The predefined pickup and delivery nodes of IF and OF transportation requests, as well as the repositioning problem for empty container movements can be seen in Tables 1 and 2. Due to the intransparency of local container flows in hinterland areas and global flows between hinterland areas, it is possible that there are OE containers as well as IE containers at the same time and for the same hinterland area. Table 1 Predefined locations of full container movements IF OF Origin Terminal Shipper Destination Receiver Terminal Table 2 Repositioning problem of empty containers IE OE Empty container For a shipper From a receiver Origin Terminal Receiver or depot Receiver, terminal Receiver or depot Destination Shipper or depot Terminal Shipper Shipper, terminal or depot

7 Reducing hinterland transportation costs To complete the problem description, it has to be noted that each trucking company considered in the modified ICT serves its requests using a homogeneous fleet of vehicles. Since our analysis is restricted to 40-feet-containers, a vehicle can only move one container at a time. Each vehicle starts and ends its tour at the depot of its trucking company. While time windows at these nodes do not have to be considered, the time windows at the customer nodes and at the terminal vertices have to be kept. Containers are made available at customer locations for predefined time-intervals. During these time-intervals the containers can be loaded or unloaded by the customers. Since a truck needs not to stay at the customer location during its container s predefined time interval, it can perform some other transportation tasks before the container will be picked up. The flexibility of vehicle routing and scheduling is even increased further by the fact that it is not required that the delivery and the pickup of a certain container is performed by the same truck. The predefined time-interval for a container at a customer location is determined by two surrounding time windows at each customer location. During the first time window the full/empty container has to be delivered to the receiver/shipper location. After the container is unloaded/loaded, it can be picked up by a vehicle during the second time window. The assumption differs from the ICT of Zhang et al. (2010) who solely define one time window at a customer location. In this case, an operating vehicle which moves a container to a customer location has to wait at the location until the container is dispatched. Changing this assumption by introducing a second time window allows vehicles to skip the container s time interval in between the customers time windows. Skipping a container s service time in the hinterland is for example a typical proceeding for the port of Rotterdam. As Veenstra (2005) stated the transportation tasks for a container s delivery and pickup are usually not done by the same vehicle since loading/unloading a container needs considerable time and at locations where containers are delivered regularly, a truck driver could pick up empty containers delivered the day before. By knowing all transportation tasks in advance for a given period the objective is to minimize the total fulfilment costs. These fulfilment costs consist of fixed costs and variable costs. Hence, in a first step the number of used vehicles should be minimized while in the second step the optimization of the operating time symbolizing the transportation costs should be pursued (Toth and Vigo 2002). 3.2 Multi-company inland container transportation problem (MCICT) The MCICT defines a scenario in which empty containers can only be switched between locations belonging to a specified trucking company. Figure 3 gives an example for this common situation in the hinterland of terminals. Hereby, trucking company 1 has to serve an OE and an OF transportation request. Trucking company 2 is in charge of an IE and an IF request (see Fig. 3a). As can be seen in Fig. 3b the opportunities to allocate empty containers within this example are very restrictive for the non-cooperating case since the required empty containers can only derive either from the depot (trucking company 1) or have to be moved to the depot (trucking company 2).

8 S. Sterzik et al. Fig. 3 Example for an MCICT 3.3 Multi-company inland container transportation problem with container sharing (MCICT CS) In the cooperative setting the exchange of empty containers among cooperating partners is permitted. That is why benefits arise through the emerging additional flexibility to allocate empty containers to a vehicle s tour. In detail, this scenario allows companies to use foreign empty containers which are obtained at a terminal or customer location. Companies have access to IE containers and can use obtained empty containers at receiver locations of cooperating companies. However, empty containers stacked at an external depot are excluded to be used in a cooperation. It should be noted that the MCICT CS does not reduce to a single company version of the MCICT and that it is different to the ICT since the participating companies only exchange available empty containers within the hinterland and do not exchange hinterland transportation requests; i.e. every company still has to serve its own transportation requests (OF, IF, OE). As in the MCICT a truck has to start and end its tour at the depot of the operating trucking company. To optimize the scenario we disregard the interests of a single company and measure the companies total benefits from a central point of view; i.e. the benefit of a particular company is subordinated for the sake of the global optimum of the scenario. The impact of container sharing in the MCICT CS can be best seen in the stated example of Fig. 3. Beside the possibility to use the depot as the origin of an empty container, trucking company 1 can now integrate either the IE container or the obtained empty container at the receiver location of trucking company 2 in its route to serve its requests (see Fig. 4a). Obviously, it has to be assumed that these time windows are consistent with the shipper s or the terminal s time window for the OE request. As can be seen in Fig. 4b the possibilities for company 2 to reposition empty containers did not change compared to the non-cooperative case. Nevertheless, company 2 will still profit by container sharing and, thus, reduce transportation costs if company 1 handles its IE request or the empty container at the receiver location.

9 Reducing hinterland transportation costs Fig. 4 Benefits of container sharing If further customers and trucking companies are included, the benefit of container sharing is assumed to grow tremendously through the rising flexibility to allocate empty containers; i.e. the more empty containers are shared with other cooperating companies, the higher the probability to save travelled distances that are induced by transportation requests. The reduction of container carriage distances, obviously, goes along with the reduction of the trucks travelling distances since a container needs to be moved from one location to the other. Due to the interdependency of the transportation resource and the means of transport, the emerging additional flexibility to allocate empty containers will consequently cause a minimization of the trucks transportation costs. 4 Model formulation The resulting problem leads to an integrated model which does not only consider vehicle routing and scheduling but also simultaneously the allocation of empty containers. By considering the containers as scarce transportation resources which have to be routed and scheduled in order to fulfil the given freight requests, it is possible to determine within the mathematical model (a) at which location empty containers should be picked up for OF and OE transportation requests, (b) where IE containers and empty containers obtained at receiver locations should be delivered, and (c) in which order and by which vehicle the containers transportation tasks should be carried to. The integrated model is based on the directed graph G = {V, A} whereas V describes the node set and A ¼fði; jþji; j 2 Vg denotes the arc set. V contains the customer node set V C, terminal node set V T and depot node set V D. There are two

10 S. Sterzik et al. types of customers (V C ¼ V S [ V R ), the node sets V S ¼ V S i [ V S o and V R ¼ V R i [ V R o describe the shipper and the receiver node sets. V S i and V R i refer to the first time window of the shipper/receiver, in which an empty/full container has to be made available. After the container has been completely loaded or unloaded, container c 2 C can be picked up by a vehicle k 2 K during the second time window of the shipper/receiver (V S o and V R o). The terminal node set V T refers to the transportation types, i.e. V T ¼ V T IE [ V T IF [ V T OE [ V T OF. The number of all customer and terminal nodes is defined by v. Since for each IF and OF transportation request, the pickup and delivery node are explicitly given by the input data, every customer has its corresponding terminal node. In case of an OF transportation request this means that, after a shipper i 2 V S o has been served by a vehicle, the full container has to be moved to terminal node ði þ nþ 2V T OF. Thereby, n defines the number of customers to be served. In case of an IF transportation request, a full container has to be moved from terminal node i 2 V T IF to its corresponding receiver location ði 2nÞ 2V R i. Depot set V D is subdivided into the start and end depot node sets V D s and V D e. Each depot corresponds to one of the d trucking companies. The company owning and the company assigned to are specified by d veh k and d cus i. E.g. if vehicle 3 belongs to trucking company 1, d veh 3 gets value 1. Similarly, if customer 14 should be served by trucking company 5 d cus 14 gets value 5. Since a company s depot constitutes the start and end location for a vehicle k 2 K, we doubled all depot vertices so that nodes ðv þ dk veh Þ2V D s and ðv þ d þ dk veh Þ2V D e describe the same depot. Furthermore, a large number of a empty containers can be stacked at the depots. During a route, node i 2 V C [ V T has to be reached during its time window, determined by the interval [b i /e i ]. Thus, a vehicle has to arrive at location i before time b i. However, arrival before a i is allowed and leads to waiting time for the vehicle. For each two distinct stop locations, t ij represents the travel time from location i to location j. Picking up or dropping off a container at node i 2 V C [ V T takes a service time of s i. While the binary decision variables y ijc and x ijk define whether container c/vehicle k traverses the arc from location i to j, L ic and T ik specify the arrival time of a container/vehicle at a location. The following model refers to Kopfer and Sterzik (2012) who have introduced a mixed-integer programming formulation for the MCICT and the MCICT CS with one terminal. min z ¼ ðvþdþd vehþk T k ðvþd vehþkþ ð1þ k k2kðt y ijc ¼ 1 8i C [ V T IF [ V T IE ð2þ j2v c2c y ijc ¼ a i2v D s j2v c2c i2v j2v T OF [V T OE [V D e ð3þ y ijc ¼ 1 8c 2 C ð4þ

11 Reducing hinterland transportation costs j2v S i [V D e c2c i2v R o [V D s c2c y ijc ¼ 1 8i 2 V T IE ð5þ y ijc ¼ 1 8j 2 V T OE ð6þ y iði 2nÞc ¼ 1 8i 2 V T IF ð7þ c2c y iðiþnþc ¼ 1 8i 2 V S [ V R i ð8þ c2c y jic y ijc ¼ 0 8i 2 V C ; c 2 C ð9þ j2v j2v L jc L ic þ t ij þ s i Mð1 y ijc Þ 8i; j 2 V; c 2 C ð10þ x ijk ¼ 1 8i 2 V C [ V T ð11þ j2v k2k x ðvþd vehþjk ¼ 1 8k 2 K ð12þ k j2v x iðvþdþd vehþk ¼ 1 8k 2 K ð13þ k i2v x jik x ijk ¼ 0 8i 2 V C [ V T ; k 2 K ð14þ j2v j2v T jk T ik þ t ij þ s i Mð1 x ijk Þ 8i; j 2 V; k 2 K ð15þ b i T ik e i 8i 2 V C [ V T ; k 2 K ð16þ x ijk dk veh ¼ x ijk di cus 8i 2 V C [ V T ; j 2 V; k 2 K ð17þ x ijk y ijc 8i 2 V S o [ V R o [ V T ; j 2 V; c 2 C ð18þ k2k x ijk y ijc 8i 2 V C [ V T ; j 2 V S i [ V R i [ V T [ V D e; c 2 C ð19þ k2k T ik ¼ L ic 8i 2 V C [ V T ; k 2 K; c 2 C ð20þ x ijk ; y ijc 2f0; 1g 8i; j 2 V; k 2 K; c 2 C ð21þ T ik ; L ic : real variables 8i 2 V; k 2 K; c 2 C ð22þ By considering empty container repositioning on the one hand (2 10) and vehicle routing and scheduling (11 17) on the other hand, the presented model pursues the minimization of the vehicles total operating time. Thereby, the operating time of a vehicle comprises the travel time from one node to the other as well as the service and waiting times at a location. The minimization of fixed costs is achieved by raising the number of operating vehicles until a feasible solution is found. The main component of the integrated model is given through equations (18 21) which assure the interlinking of the transportation resource and the means of transport.

12 S. Sterzik et al. Restrictions (2 3) define that every customer node is visited once and that a container flow begins either at the terminal as an inbound container or at the depot of a trucking company. A container s final destination is given by the terminal or the depot, stated by (4). While the possible origins/destinations of empty containers are defined by (5 6), restrictions (7 8) assure the defined locations of a full container transportation task. Besides, (8) also states that a container has to pass the loading/ unloading process at a shipper/receiver node. Equations (9) and (10) ensure the route and time continuity. Thereby, M marks a sufficiently large constant. Further, a vehicle starts and ends its tour at the depot of its trucking company (12 13). It also has to be assured that a customer and a terminal location is visited exactly once by a truck (11). A vehicle s route continuity as well as the time restrictions are defined by (14 16). Since a container always has to be moved by a vehicle, it has to be ensured that the vehicles cover the containers flows. We achieved this by interlinking the containers flows and the vehicles routes with each other (18 19); i.e. the flows of the containers are covered but the vehicles have the possibility to interrupt these flows and e.g. use different untraveled arcs. Obviously, if a vehicle moves a container, both have to leave a node at the same time (20). Restriction (17) defines the MCICT in which container sharing is not permitted. If the exchange of empty containers is allowed (MCICT CS), the equation changes to: x ijk d veh k ¼ x ijk d cus i 8i 2 V S [ V R i [ V T OF [ V T OE; j 2 V; k 2 K ð23þ 5 Solution procedure The proposed scenarios are based on the ICT which is classified as an extension of the NP-hard vehicle routing problem with time windows (VRPTW) (Zhang et al. 2010). Hence, only relatively small instances for the MCICT and MCICT CS can be solved to optimality with e.g. the help of the illustrated integrated routing model. In order to handle real-world instances we present a heuristic for both scenarios. The heuristic is based on Sterzik and Kopfer (2012) who propose a tabu search algorithm for the ICT and compare it to the WPB method of Zhang et al. (2010). The WPB method for the ICT determines an upper and a lower bound. Regarding the upper bound the authors define an over-constrained model whereas the latest limit e i of each time window is considered as the arrival time of a vehicle at a certain node i. The ratio of the lower bound to the feasible solution of the upper bound indicates the quality of the obtained solution. By decomposing the time windows better solutions can be obtained. The observed computational results show that the tabu search heuristic outperforms the WPB method in 19 of 20 test instances with respect to effectiveness and efficiency. The general outline of our solution methodology can be described as follows. For constructing an initial solution for the MCICT a modified Clark & Wright-savings algorithm is used. We then apply the tabu search heuristic to generate a final solution. For the solution process, arcs ði; jþ : i; j 2 V where d i cus = d j cus are penalized, so that t ij = M. Due to the fact that the solution space of the MCICT CS comprises all feasible solutions of the MCICT for the same data set, the final solution of the MCICT

13 Reducing hinterland transportation costs Fig. 5 Classified container movements is used as the initial solution for the MCICT CS. Obviously, the distance matrix for the cooperative case has to be adapted in order to permit the exchange of empty containers among different trucking companies. Subsequently, the initial solution is improved by the proposed tabu search heuristic which is also applied for the MCICT. Due to a better comprehension of the heuristics we will mainly talk about container movements which comprise one or two locations as can be seen in Fig. 5. Two basic types are distinguished: while the first type describes the full container transportation requests which always comprise an origin and a destination location, the second basic transportation type describes container movements which require the allocation of empty containers and, thus, are only defined through one location (origin or destination location). This is an important difference since the localsearch operators as well as operators of the construction heuristic deal with container movements being part of routes. However, considering the distance matrix, we do not merge the nodes of a full container transportation request into one node as can be seen in Zhang et al. (2010). For both scenarios we adapt the distance matrices so that container movements that cannot succeed each other are excluded. For example, an operating vehicle which just served an IF transportation request is not permitted to serve an OF or OE request since it does not carry an empty container. In consideration of the container movements and according to the travel distances between requests as well as the underlying time windows, we therefore define a distance matrix for each container movement r 2 R. Certainly, through the additional consideration of external containers for a particular company these matrices are mainly bigger for the MCICT CS. By adapting the distance matrices, the solution space can be enormously reduced and, in consequence, so can the computational time. In what follows, we first describe the construction heuristic for the MCICT. We then give a detailed description of the tabu search heuristic. 5.1 Modified Clark & Wright-savings algorithm The savings algorithm of Clarke and Wright (1964) was introduced for the vehicle routing problem (VRP). To adapt the algorithm for the MCICT we additionally consider multiple depots and terminals, different customer types as well as time constraints. In detail, we initially assign each container movement to its corresponding trucking company. Thereby, one vehicle serves exactly one movement r 2 R in a pendulum tour. Thus, considering container movement type 2, empty containers originate from the depot or have to be delivered to the depot. Subsequently, we compute the savings sav ij for each r of a depot as follows:

14 S. Sterzik et al. sav ij ¼ t iðvþdþd cus i Þ þ t ðvþd cus j Þj t ij 8i; j 2 V C [ V T ð24þ The container movements of each trucking company are sorted in descending order of the savings. Beginning at the top of this list the routes are then merged into one. The established routes have to be feasible and must not delete/interfere with a previously defined connection between two container movements. 5.2 Tabu search heuristic Tabu search is a local-search meta heuristic that is based on an iterative process for finding the best solution in the neighbourhood N(s) of a given solution s. By using a memory structure, cycling, i.e. revisiting a solution again and again in a loop of the search trajectory, can be banned from the solution space for H iterations (Glover 1986). Our tabu search heuristic comprises an initial phase and a main phase. In the initial phase the algorithm seeks to reduce the number of required vehicles at depot i 2 V D s to the defined amount m i. The aim is to diminish the solution space to a great extent. An adequate value for m i that is not too large or too small depends highly on the investigated hinterland region and the considered time windows at the customer and terminal locations. The risk of excluding qualitatively good solutions if m i is defined too small for a particular depot needs to be minimized by applying numerical experiments for a data set type. Every additional vehicle which exceeds the truck limit m i is penalized with the additional costs cost pen. Thereby, p(s) determines the summation of all penalty costs which have to be added to the objective value f(s). The initial phase ends if p(s) = 0. During the main phase enduring iter max 1 iterations the excess of the defined truck limit is forbidden. While the first phase is mainly characterized by the Operator Selection component which seeks rapidly to find a solution that does not include penalty costs, the second phase specially emphasizes in the Intensification Strategy as can be seen in the outline of Algorithm 1. Further general criteria that affect the search process of the tabu search heuristic are determined by the calculation of the objective function, the consideration of diversification elements as well as the tabu tenure and aspiration criteria. Algorithm 1 Framework of the Tabu Search Heuristic 1: H number of tabu iterations; 2: Solution of Savings-Algorithm is used as s; 3: while p(s) [ 0 do 4: Operator Selection is applied; 5: end while 6: while iter 1 \ iter max 1 do 7: Operator Selection is applied; 8: Intensification Strategy is applied; 9: iter 1 = iter 1? 1; 10: end while

15 Reducing hinterland transportation costs Objective function The calculation of the values of the objective function during the solution procedure is a problem of its own. As mentioned before, the objective function seeks to minimize the trucks total operating time. The degree of freedom for determining the vehicles arrival times for a given route is relatively large since it depends on the number and character of container movement types in a vehicle s route, the travel times between these locations, and mainly on the time windows amplitude at the corresponding customer and terminal vertices. We propose a heuristic approach to reduce waiting times and thus to estimate the best start and end times of vehicle k 2 K. First of all, we check if a route is feasible. Thereby, we allow a surplus of waiting time since T ik should always be defined as the minimal arrival time, i.e. a vehicle that traverses arc (i, j) arrives at node j at time a j where applicable. Otherwise T jk is defined by T ik? s i? t ij if a j \ T ik? s i? t ij B e j. Assuming that the route of vehicle k is feasible we then try to reduce unnecessary waiting times. The determined arrival time at the last customer on the route of vehicle k is used to recursively improve the arrival times of the prior customers. Since multiple deployments of trucks are not permitted, vehicle k 2 K is used as a synonym for the route of k in the following. If k l determines the node that marks the last position of k, T kl 1 k is then defined as T kl k s kl 1 t kl 1 k 1 or as e kl 1 if T k1 k s kl 1 t kl 1 k 1 e kl 1. The heuristic then uses the determined arrival time successively to calculate the remaining arrival times until T ðvþd veh k Þk is defined. The number of required routes is also minimized by introducing a further penalty parameter cost rout that is added to the travel time of every required route during the search process. Necessarily, these costs are deducted from the best solution s best that is determined when the algorithm terminates Operator selection The neighbourhood of a current solution s is composed of all solutions that can be reached by applying one of the local-search operators. Three types of moves are used in the given tabu search approach: The insertion-operator removes a randomly selected container movement r* from its route and inserts it in another route or at another place in its current route. The cross-operator swaps a randomly selected container movement r* from its route and exchanges it with container movement r 2 R nfrg. The combine-operator tries to reduce the number of routes by reinserting the elements of each short route into another route. Thereby, a short route is defined as a route which comprises less than y container movements. Algorithm 2 Operator Selection 1: N random number in the interval [0,1]; 2: a probability value; 3: if N\a then 4: Combine-Operator is applied;

16 S. Sterzik et al. Algorithm 2 continued 5: s s; 6: Tabu list T is updated; 7: end if 8: Cross-Operator is applied; 9: s s; 10: T is updated; 11: Insertion-Operator is applied; 12: s s; 13: T; N are updated; As it can be seen in the pseudocode of Algorithm 2, the insertion- and crossoperator are applied in each iteration while the usage of a combine-operator depends on the probability value a. Due to the fact that applying this operator takes a lot of computational time one has to find an adequate value for a which does not impair the algorithm s efficiency. After applying an operator the best non-tabu solution s 2NðsÞ becomes the new current solution s. The tabu list T has to be updated Intensification and diversification strategies The usual search process can be interrupted for an intensification strategy that is defined in Algorithm 3. The frequency of interruption depends on the probability value b and the quality of s which is related to (1? c)*f(s best ) whereas s best determines the best known solution so far and c is a constant parameter in the interval [0, 1]. Thereby, s is modified R times by using each r in s once for the cross- and insertion-operator. Both solutions are compared and the best solution according to the objective value is chosen. Based on this modified solution the operator selection algorithm is applied for iter 2 max iterations. For an efficient tabu search algorithm this iteration limit value should be defined well since the execution of the operator selection algorithm is applied for R *iter 2 max iterations. The intensification strategy is an important component of the Tabu Search Heuristic and is, therefrom, defined as an autonomous algorithm within this framework. Hence, the tabu list is restarted each time the intensification strategy is applied. Furthermore, H can be adapted according to the modified framework within this component. Algorithm 3 Intensification Strategy 1: b probability value; 2: c number in the interval [0,1]; 3: if ðn\bþ^ðfðsþ\ð1þcþfðs best ÞÞ then 4: T = {}; 5: H is updated; 6: for r 2 R do 7: r is used for cross-operator and leads to solution s CO ;

17 Reducing hinterland transportation costs Algorithm 3 continued 8: r is used for insertion-operator and leads to solution s IO ; 9: if f(s CO ) \ f(s IO ) then 10: s s CO ; 11: else 12: s s IO ; 13: end if 14: while iter 2 \ iter max 2 do 15: Operator Selection is applied; 16: iter 2 = iter 2? 1; 17: end while 18: end for 19: H is updated; 20: end if To diversify the search, we implement a mechanism which penalizes any neighbourhood solution s N 2 NðsÞ by a factor that is proportional to the addition frequency of its attributes and a scaling factor. In detail, q rk describes the number of times container movement r has been added to route k during the search process. The intensity of the diversification process can be adjusted by parameter k. Thus, unless f(s N ) \ f(s best ) penalty term k*q rk is added to the total solution costs f(s N ). The illustrated diversification strategy is a modification of the mechanism used in Taillard (1993) Tabu tenure and aspiration criteria The tabu list T is constituted as a deterministic list which records each container movement r that is removed from its routes k. After the removal r is not allowed to be served by vehicle k for H iterations. An exception of applying the tabu status can be marked for the combine-operator. Hereby, we want to grab any chance to always get the best neighbourhood solution even if r is tabu for k. The risk to get caught in a cycle is not given since the combine-operator is not applied in every iteration and, moreover, the other two operators would be applied before returning to this move type. However, in general a tabu status is overruled if the algorithm finds a solution which is better than any solution known so far. 6 Computational results In this section we analyze the benefit of container sharing by comparing the MCICT with the MCICT CS for realistic-sized instances. Our computational results are based on the data sets for the ICT (Zhang et al. 2010) which include five depots, three terminals and 75 transportation requests. Since the ICT solely considers one trucking company and one time window at each customer location, we had to modify these sets. In detail, we assigned the transportation requests equally to the

18 S. Sterzik et al. existing depots so that five trucking companies need to serve 15 requests. With regard to the time windows we want to analyze the impact of tight and wide time windows at the customer locations on the computational results. The consideration of tight time windows is motivated by the ICT-data sets which stipulate that a container can only be left at a customer location within a short time span. Therefore, the first and second customer time windows are defined tightly and are situated just before and immediately after the given service time windows for the containers. We also consider wide time windows which is common practice in many hinterland regions (e.g. Braekers et al. 2011a; Veenstra 2005). Wide time windows provide the opportunity for trucking companies to leave empty containers at customer places for a defined period before or after a container s loading or unloading is completed, respectively. For example, IF containers that are emptied at receiver locations are available after the service has finished and should be picked up before the end of the underlying time period. Empty containers for OF requests can be delivered from the beginning of the time period and should be delivered to the shipper before the service begins. For the consideration of tight and wide time windows ten data sets are defined. As shown in Sterzik and Kopfer (2012), the proposed heuristics have attested their efficiency and effectiveness for the ICT. To finally prove the performance of the adapted heuristic for the MCICT and MCICT CS, we apply the small problem instances of Kopfer and Sterzik (2012) who analyzed the MCICT and MCICT CS with only one terminal. Thereby, they applied CPLE to the corresponding mixed integer models and solved ten test instances to optimum. Each instance comprises two trucking companies that need to serve six requests. The consideration of only one terminal within the small experimental settings is extraneous for the tabu search heuristic since the terminal nodes are duplicated and can therefore be attributed with any coordinates. The algorithms have been applied in Java 6 on an Intel Core i7, 3.2 GHz PC with 12GB system memory. By using the parameters of Sterzik and Kopfer (2012) ðiter1 max ; iter2 max ; a; b; c; k; HÞ ¼ð1000; 10; 0:1; 0:12; 0:008; 1:5; 6Þ we achieved the optimal results for all instances after three runs at the most. Computational results obtained for data sets with tight time windows show that trucking companies in the hinterland of seaports benefit significantly from cooperating. Considering the variable costs, container sharing leads to a reduction of 7 % on average as can be seen in Table 3. This has a strong positive impact on the financial situation of the trucking companies since the profit margin in container trucking usually only amounts to a few percent. Regarding the fixed costs, the achieved savings are even more impressive. The experiments show that the range to reduce routes goes from 5 to 12 %. Since integrating foreign empty containers from terminal or customer locations within a vehicle s route can also lead to a saving of containers used for the transportation requests, we also analyzed whether this saving is of value. Due to the fact that the number of inbound containers cannot be reduced in any event, we only count the additional empty containers arising from the depot. Thereby, regarding tight time windows it is possible to save 8 % on average.

19 Reducing hinterland transportation costs Table 3 The impact of container sharing (tight time windows) Inst. MCICT MCICT CS Difference (in %) Vh. Cont. TT Vh. Cont. TT Vh. Con. TT , , , , , , , , , , , , , , , , , , , , Total , , Inst. instance number, Vh. operating vehicles, Cont. additional containers (from the Depot), TT total travel time Certainly, the benefit of exchanging empty containers decreases if in the noncooperative case the alternatives to integrate containers from different locations into a vehicle s route are relatively high. This is the case if wide time windows are considered (see Table 4). Thus, regarding the fixed costs for the same data sets we now determine a saving of 2 % less used vehicles and containers. Surprisingly, container sharing does not consequently cause a reduction of containers. As can be seen in test instance 7 the saving of operating vehicles in the cooperative case leads in some cases even to an increase of used containers. The vehicles total travel time for the data sets with wide time windows can be reduced by 6 % on average. By having a look on the results of a certain data set it can be seen that some companies benefit above-average by container sharing. For example, trucking company 1 in Table 5 realizes savings which are 58 and 36 % above the average values of the fixed and variable costs. Obviously, this can only be accomplished at the expense of other trucking companies like company 4 which almost does not benefit by the cooperation. 7 Concluding remarks This contribution analyzed the benefit of exchanging empty containers among trucking companies in a local hinterland region. Thereby, we presented two scenarios: while trucking companies have only access to their own containers in the MCICT, the MCICT CS allows the interchange of empty containers among several owners. For both scenarios we presented a mixed integer programming formulation that considers vehicle routing and scheduling simultaneously with empty container repositioning. Subsequently, test instances with tight and wide time windows are solved by means of an efficient tabu search heuristic. The computational results