Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 406218, 21 pages

http://dx.doi.org/10.1155/2015/406218

## Modeling the Multicommodity Multimodal Routing Problem with Schedule-Based Services and Carbon Dioxide Emission Costs

^{1}School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China^{2}MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology, Beijing Jiaotong University, Beijing 100044, China

Received 7 September 2015; Revised 22 November 2015; Accepted 23 November 2015

Academic Editor: Young Hae Lee

Copyright © 2015 Yan Sun and Maoxiang Lang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We explore a freight routing problem wherein the aim is to assign optimal routes to move commodities through a multimodal transportation network. This problem belongs to the operational level of service network planning. The following formulation characteristics will be comprehensively considered: (1) multicommodity flow routing; (2) a capacitated multimodal transportation network with schedule-based rail services and time-flexible road services; (3) carbon dioxide emissions consideration; and (4) a generalized costs optimum oriented to customer demands. The specific planning of freight routing is thus defined as a capacitated time-sensitive multicommodity multimodal generalized shortest path problem. To solve this problem systematically, we first establish a node-arc-based mixed integer nonlinear programming model that combines the above formulation characteristics in a comprehensive manner. Then, we develop a linearization method to transform the proposed model into a linear one. Finally, a computational experiment from the Chinese inland container export business is presented to demonstrate the feasibility of the model and linearization method. The computational results indicate that implementing the proposed model and linearization method in the mathematical programming software Lingo can effectively solve the large-scale practical multicommodity multimodal transportation routing problem.

#### 1. Introduction

##### 1.1. Multimodal Transportation

Multimodal transportation utilizes more than one transportation service (rail, road, air, and maritime transportation) on the routes that serve to move commodities from their origins to their destinations [1–3]. Multimodal transportation differs significantly from traditional unimodal transportation which employs single transportation service provided by one transportation carrier. Due to the integrative combination of different advantages of various transportation services, multimodal transportation shows superiorities in lowering transportation costs and abating environmental pollution when compared with unimodal transportation in a long-haul transportation setting. Moreover, containers are widely used to carry commodities in multimodal transportation. They can simplify the packaging of product, economize on the package materials, and ensure transportation safety [4]. The standardization of containers in the physical structure of the transportation system also benefits the mechanization of the loading and unloading operations and can help improve the efficiency of these operations at terminals [5].

Several empirical comparative studies have demonstrated the superiorities of multimodal transportation in costs and environmental protection. Janic [6] formulated the full costs (linear summation of the internal costs and external costs) of a multimodal and road freight transportation network. By using the simplified European multimodal transportation network (truck-rail combination) and its equivalent road transportation network as a case, that study summarized the trends of variation of the full costs of multimodal transportation and road transportation in relation to the transportation distances. These trends showed that, compared with road transportation, multimodal transportation decreases the average full costs more rapidly as the distance increases and shows fewer full costs if the transportation distance exceeds approximately 1000 km (the critical distance when the rail service frequency is 5 trains/week). Moreover, the critical distance decreases significantly as the rail service frequency increases. Liao et al. [7] compared the carbon dioxide emissions of the long-haul truck transportation and the multimodal transportation (truck-ocean combination) in a case study of container transportation between Kaohsiung and Keelung in Taiwan. The activity-based method was adopted to estimate the carbon dioxide emissions in the transportation process, and the computational results of the case indicated that the carbon dioxide emissions fell from 566,525 tonnes to 163,830 tonnes, corresponding to a 71% decrease, when truck transportation was replaced by multimodal transportation. These results supported the view that the use of multimodal transportation as an alternative to long-haul truck transportation can markedly decrease carbon dioxide emissions from the transportation sector.

Many researchers have studied the empirical applications of multimodal transportation in specific cases; for example, Bookbinder and Fox [8] discussed multimodal transportation shortest time and minimal costs route selection for five commodity flows that need to be routed from Canada to Mexico in the North American Free Trade Agreement area. Banomyong and Beresford [9] explored multimodal transportation route selection for exporting commodities from Laos to the European Union and proposed a cost model for multimodal transportation to help exporters make decisions. Cho et al. [10] presented a weighted constrained shortest path problem model and a label setting algorithm to select multimodal transportation routes from Busan to Rotterdam. Meethom and Kengpol [11] combined the AHP (Analytic Hierarchy Process) method with 0-1 goal programming to design a decision support system for selecting an optimal multimodal transportation route from Bangkok to Danang.

Currently, with the rapid development of economic globalization, international trade and the accompanying global circulation of commodities have grown remarkably. Consequently, transportation networks have expanded widely and commodity distribution routes have extended significantly. These changes represent challenges to transportation performance in terms of, for example, costs, timeliness, and environmental concerns [2, 12]. In response to these challenges, increasing numbers of companies have adopted multimodal transportation schemes to transport their products or raw materials. According to the relevant statistics, logistics costs represent 30–50% of the product cost [2]. For this reason, logistics cost is one of the most effective targets for companies to lower product cost, raise profits, and maintain competitiveness in the international trade market. Therefore, freight routing for commodity transportation has been given great importance by both the management decision makers and multimodal transport operators.

##### 1.2. Formulation Characteristics of the Study

The practical demand for lowering logistics costs motivates research interests in the freight routing problem in multimodal transportation networks. The essence of this problem is to select optimal routes to move commodities from their origins to their destinations through a multimodal transportation network. In our study, we consider following formulation characteristics to make the formulation of this problem correspond more closely to actual practice.

*(**1) Multicommodity Flow Routing*. Freight routing aims at satisfying customer demands at minimal costs. Each transportation demand can be represented by a commodity flow that is characterized by five attributes: origin, destination, volume, release time, and due date. From the perspective of the entire multimodal transportation network, more than one transportation demand within the planning horizon needs to be satisfied. Various transportation demands differ from each other in at least one of the five above-mentioned attributions. Thus, multiple commodity flows must be routed to satisfy the various transportation demands. Therefore, the freight routing problem explored in this study will be developed as a multicommodity problem.

*(**2) Schedule-Based and Time-Flexible Transportation Services*. In a multimodal transportation network, we can combine different transportation services to move commodities along routes. The most obvious distinction among the transportation services is in terms of their operating modes. In this study, we define two categories for classifying transportation services. The first category is that of schedule-based services, that is, operations controlled by schedules. The second category is that of time-flexible services, that is, operations not controlled by schedules. This distinction is similar to that drawn in Moccia et al.’s study [13]. Schedule-based services (e.g., rail services) are operated based on schedules that are specified in advance. Conversely, time-flexible services (e.g., road services) are not constrained by time and can travel through the network flexibly. The operations of the two types of transportation services and the transshipments among them will be emphasized in Section 2.

*(**3) Consideration of Carbon Dioxide Emissions*. With the evolution of society and improvement in the awareness of sustainable development, environmental issues have drawn increasing attention from both the government and the public, especially in developing countries, for example, China, a rapidly industrializing country. One of the most pressing environmental issues is global warming caused by greenhouse gas emissions. Carbon dioxide represents approximately 80% of the total greenhouse gas emissions [7], and the reduction of these emissions is acknowledged to be a highly challenging problem worldwide. In the transportation sector, various activities have been shown to represent up to 19% of the global energy consumption [14] and to produce large amounts of greenhouse gases (e.g., carbon dioxide, carbon monoxide, and methane) and air pollutants (e.g., sulfur dioxide and oxynitride) [15]. For example, according to Li et al.’s evaluation [16], carbon dioxide emissions from the Chinese transportation sector represent 8.37% (53.96%) of the total carbon dioxide emissions from fossil fuel (liquid fuel) consumption in 2007. Therefore, transportation is considered a key target in reducing carbon dioxide emissions. For this reason, it is necessary for decision makers to seriously consider the carbon dioxide emissions when planning multimodal transportation routes.

*(**4) Generalized Costs Optimum Oriented to Customer Demands*. Transportation is a service industry, and transportation operation oriented to customer demands has been promoted vigorously with the development of traditional transportation practice into modern logistics. In freight routing, the goal of the shippers is to pay minimal costs for the transportation of their commodities. As service providers, the multimodal transport operators and the third party logistics companies should base their planning on customer demands, which means that lowering costs should be chosen as their planning objective for the transportation process. In this study, we use generalized costs to evaluate how much money the shippers should pay for transportation. The generalized costs cover not only the transportation costs but also the inventory costs, operation costs at terminals, carbon dioxide emissions costs, and additional service costs.

##### 1.3. Literature Review

Because multimodal transportation routing has attracted substantial interest [17, 18], many relevant studies have been conducted in previous decades. Among the early studies (studies conducted before 2005), Min [2] developed a chance-constrained goal program model to provide the distribution manager with decision support in choosing international multimodal transportation routes. Minimizing the transportation costs and risk and satisfying the requirements of on-time service are formulated in the proposed model. Barnhart and Ratliff [19] discussed the minimal costs multimodal transportation routing problem and introduced two solution approaches that separately involved a shortest path algorithm and a matching algorithm. The two approaches were adopted to solve the routing problems with rail service costs expressed per trailer and per flatcar, respectively. Boardman et al. [20] designed a -shortest path double-swap method to solve a multimodal transportation routing problem and incorporated this method with the a database and user interface to build a decision support system for the real-time routing of shipments through a multimodal transportation network. Ziliaskopoulos and Wardell [21] systematically presented a time-dependent multimodal optimum path algorithm for multimodal transportation networks. In their algorithm, many time parameters, including schedules, dynamic arc travel times, and transshipment delays, were comprehensively considered. This approach enhanced the feasibility of the algorithm in addressing the practical problem. Lozano and Storchi [22] defined the shortest viable path within an origin-destination pair in the multimodal transportation network and proposed an ad hoc modification of the chronological algorithm to solve the multimodal shortest path problem by obtaining a solution set of the problem. Lam and Srikanthan [18] improved the computational efficiency of the -shortest algorithm by using clustering technique and applied this algorithm to multimodal transportation routing. Boussedjra et al. [23] addressed the multimodal shortest path problem and proposed a shortest path algorithm involving label correcting method to solve the single origin-destination pair problem. The efficiency of the algorithm was verified by comparing its performance with that of the branch-and-bound method.

Essentially, in these previous studies, with the exception of Min’s study, researchers have emphasized the use of algorithms for selecting optimal multimodal transportation routes for commodities, and the development of optimization models attracted limited attention. However, it is necessary to construct optimization models, because it is difficult to find a universal optimization model to solve all types of multimodal transportation routing problems considering different formulation characteristics. Moreover, optimization models can provide an exact benchmark for systematically testing various solution algorithms.

In recent years, with the constant improvement of the design of algorithms for solving problems in this area of study, increasing numbers of multimodal transportation routing models have been developed. In these studies, a few researchers have concentrated on the development of GIS-based models for the multimodal transportation routing problem. For example, Winebrake et al. [24] constructed a geospatial multimodal transportation routing model to select minimal costs, minimal time, and minimal carbon dioxide emissions routes for an origin-destination pair. The construction and solution of them model were conducted with ArcGIS software. Other researchers have continued to pursue studies on goal programming models and solution algorithms. Zhang and Guo [25] and Zhang et al. [26] separately presented the foundational frameworks of integer programming models for the multimodal transportation routing problem. Sun et al. [27] studied the basic uncapacitated single-commodity multimodal transportation routing problem without a demanded delivery time constraint and used the label correcting algorithm to solve it. A similar study was also conducted by Sun and Chen [28], whereas this study addressed an uncertain transportation case and considered biobjective optimization, including the minimization of total transportation costs and total carbon dioxide emissions. Using these studies as a springboard, many studies have highlighted transportation due date constraints and capacity constraints. In the formulation of transportation due date constraints, several studies (e.g., Liu et al. [29]) have treated the transportation due date as a hard constraint, which the total transportation time should not exceed. Furthermore, others have considered the transportation due date as an index for charging penalty costs, which means that if the transportation due date is violated, penalty costs must be paid to compensate for the loss, as in, for example, Zeng et al. [30], Wang et al. [31], Fan and Le [32], Li et al. [33], and Tang and Huo [34]. Regarding the capacity constraint, vehicle carrying capacity has been widely considered, for example, by Kang et al. [35], Wang et al. [31], Liu et al. [29], Li et al. [33], Çakır [36], Verma et al. [37], Cai et al. [38], Tang and Huo [34], and Lei et al. [39]. A few studies (e.g., Chang et al. [40]) have also defined terminal operating capacity and vehicle carrying capacity as capacity constraints. As for the scheduling issues, current studies, for example, Cai et al. [38], Xiong and Wang [41], and Lei et al. [39], have viewed schedules as “terminal schedule-based service time windows” to formulate a multimodal transportation routing problem with time windows. In these studies, the terminal scheduled service time windows are only related to the terminals, and different transportation services share one schedule-based service time window at the same terminal. A terminal cannot be covered in the multimodal transportation route if the arrival time of the commodity at this terminal is out of the range of the schedule-based service time window, and this consideration is also formulated as a hard constraint in the optimization models [13, 38, 39, 41, 42].

In terms of comparisons with our study, first, apart from Çakır [36], Verma et al. [37], and Chang et al. [40], all other previous studies cited above preferred to optimize the single-commodity routing problem in a multimodal transportation network [27–35, 38, 39, 41], and this type of optimization cannot guarantee an optimum result for the overall performance of the entire multimodal transportation network. Second, in many studies, all types of transportation services were assumed to adopt a time-flexible service mode, and the operations at terminals were simplified as a continuous “arrival transshipment departure” process [27–41], which cannot be expected to match the practical reality that some transportation services, for example, rail, ocean, and air services, are operated according to schedules. The actual transshipments among the schedule-based and time-flexible transportation services are much more complicated than the “arrival transshipment departure” process, as will be explained in Section 2. Although some researchers used time windows to address the scheduling issues [38, 39, 41], the consideration of terminal schedule-based service time windows is not expected to represent the schedules in the model formulation exactly, because schedules regulate not only the transportation service times but also the transportation service routes. Additionally, different schedules regulate different service times even at the same terminal. Moreover, formulation of terminal schedule-based service time windows as a hard constraint itself [13, 38, 39, 41, 42] does not reflect the actual practice, because if their arrival times are not within the time windows, the commodities can wait until the lower bound of the current time window or that of the next time window and then be transshipped. Consequently, the transportation schemes designed by the studies above are less supportive of decision-making. Additionally, the network deformation method is widely used by researchers to convert actual multimodal transportation networks into a standard graph with one link between two conjoint nodes [27–29, 34, 38]. This method can simplify model formulation but will result in a substantial expansion of the network scale. Thus, this procedure may be feasible for a small-scale multimodal transportation networks, but it will be unfeasible for large scale ones in practice.

##### 1.4. Similar Works and Problem

Among existing studies, those most similar to ours are those of Chang [42], Moccia et al. [13], and Ayar and Yaman [5]. Chang [42] addressed the problem of selecting the best routes to move commodities through international multimodal transportation networks. In his study, he considered multiobjective optimization and schedule-based transportation services and demanded delivery times and transportation economies of scale, and he formulated the route selection problem as a multiobjective multimodal multicommodity flow problem with time windows and concave costs. In his study, time window constraints were used to represent the restriction of schedules and demanded delivery times relative to best route selection in empirical cases. A concave piecewise linear function was adopted to measure the transportation costs of all commodity flows. Each commodity flow was considered to be splittable. In the proposed model, minimizing the total transportation costs and total transportation time were set as the objectives, and the weighted summation method was used to address the multicriteria-based optimization. The problem was decomposed into a series of more easily solved single-commodity flow subproblems by using the Lagrangian relaxation technique to relax the capacity constraint of the model. The subgradient optimization algorithm was then used to obtain the solutions to the subproblems. Finally, a reoptimization technique was designed to modify the solutions of subproblems to construct a feasible solution of the initial problem.

Moccia et al. [13] explored the multimodal multicommodity flow problem with pickup and delivery time windows, and their study was similar to Chang’s [42]. The problem addressed in their study was oriented to a multimodal transportation network with time-flexible road services and schedule-based rail services. To address this problem, the authors first presented a virtual network representation to convert the physical multimodal transportation network into a detailed representation of operations by adding nodes to represent the pickup and delivery time windows and the scheduled departure times. Then, two mixed integer programming models, including an arc-node-based model and a path-based model, were formulated. The two models were both single-objective ones wherein the aim was to minimize the total transportation costs of all commodity flows. Nonconvex piecewise linear costs, time windows, and side constraints were all considered and formulated in the optimization models, and each commodity flow was unsplittable. Finally, a column generation algorithm was developed to achieve the lower bound of the problem, and it was embedded within heuristics to obtain feasible solutions of the problem.

Ayar and Yaman’s study [5] was a special case of the multimodal multicommodity flow problem previously formulated by Moccia et al. [13]. In their study, the release times and due dates of commodities replaced the pickup time and delivery time windows, respectively; that is, the transportation of a certain commodity was defined to begin at or after its release time at its origin and was to be achieved before its due date at its destination. Additionally, in contrast to Chang’s [42] and Moccia et al.’s [13] approaches, to make the costs calculations more accessible and to make them correspond better to the real-world cases, the total transportation costs were evaluated by generalized costs covering transportation costs en route, operation costs, and inventory costs at terminals. Then, the authors proposed two mixed integer programming models and valid inequalities to solve the multicommodity routing problem in a truck-ocean transportation network where truck services were considered to be time flexible and uncapacitated, whereas maritime services were capacitated and operated according to schedules. The solution algorithm designed in this study was somewhat similar to Chang’s [42] in that a Lagrangian relaxation technique was used to relax the capacity constraint.

Note that the freight routing problem that this study focuses on is extremely similar to the multicommodity multimodal transportation network design problem. Both of these problems relate to planning optimal routes to move multiple commodities through the transportation network by using multiple transportation services rationally. They also show obvious similarities in model formulation and solution algorithm design. The multicommodity multimodal transportation network design problem has always been a focus of transportation planning, and many studies have already been conducted to address this problem. Some representative studies are reviewed here. Crainic and Rousseau [43] presented a foundational modeling and algorithmic framework to solve the multicommodity multimodal transportation network design problem. An integer nonlinear programming model was constructed, and a solution algorithm involving decomposition and column generation was developed in this study. Their work provided a solid foundation for future related studies. Daeki et al. [44] explored the truck-aircraft service network design problem for express package delivery. The authors addressed this problem as a multimodal transportation network design problem with time windows, separately formulated an approximate model and an exact one to describe this problem, and designed a linear programming relaxation method optimized by valid inequalities to attain the optimal solution of the large-scale problem. Zhang et al. [45] developed a bilevel programming model to optimize the multicommodity multimodal transportation design problem with carbon dioxide emissions and economies of terminal scale. The upper level of the model adopted a genetic algorithm to design the optimal topology of the terminal network, while the lower level served to distribute the multicommodity flow through the multimodal transportation network. Qu et al. [46] built an integer nonlinear programming model considering transfer costs and carbon dioxide emission costs for the multicommodity multimodal transportation network design problem. Using linearization technique, the proposed model was transformed into a linear one that could be solved easily by mathematical programming software.

Additionally, Crainic [47], Southworth and Peterson [48], Jansen et al. [49], and Yaghini and Akhavan [50] separately conducted systematic reviews of the service network design problem. All these reviews introduced this problem comprehensively from the perspectives of research content, current progress, model formulation, algorithm development, and research prospects, thus contributing to making the general research architecture more mature and ideal. Furthermore, to enhance the effectiveness of solving the network design problem, many studies discussed the solution approaches for the network design problem in detail, and a large number of solution approaches with high feasibility have been proposed, for example, a dual-ascent method by Balakrishnan et al. [51], a Lagrangian heuristic-based branch-and-bound approach by Holmberg and Yuan [52], a first multilevel cooperative tabu search algorithm by Crainic et al. [53], and a multiple choice 0-1 reformation and column-and-row generation method by Frangioni and Gendron [54]. All the studies discussed above significantly advanced the knowledge of the multicommodity multimodal transportation network design problem.

Although similarities between the two types of problems are obvious, there are still three remarkable differences between them. Considering these differences helps to define obvious distinctions between the two types of problems in terms of model formulation.

(1) In addition to the question of route selection, the network design problem also needs to solve the allocation of limited transportation resources to the network, that is, to determine the number of facilities or capacities of transportation services that should be installed on the routes, or the levels of transportation services that should be offered on the routes [47]. By contrast, the freight routing problem is based on an existing multimodal transportation network whose transportation resources have already been allocated, and it only focuses on planning origin-to-destination routes by selecting proper terminals and transportation services connecting them.

(2) The network design problem is a form of tactical planning. The due dates of the commodities are usually not considered in this problem. By contrast, the freight routing problem is a part of the operational planning and must assign great importance to customer demands. The due dates of the commodities must be formulated as a constraint to satisfy the customer demands regarding timeliness.

(3) In the network design problem, the detailed schedules of transportation services are rarely formulated by the researchers. However, in the freight routing problem, when utilizing schedule-based services to move commodities, the schedules must be followed strictly to guarantee the feasibility of routing in empirical cases. Consideration of service schedules improves the complexity of the freight routing problem in the multimodal transportation network.

##### 1.5. Organization of the Rest of the Sections

All the studies reviewed above, regardless of the problem (freight routing problem or freight network design problem) they addressed, laid a solid foundation for our study with regard to the model formulation and algorithm design. The remaining sections of our study are organized as follows.

In Section 2, we first give a detailed introduction to multiple transportation services in the multimodal transportation network, including schedule-based rail services and the time-flexible road services. We then analyze the various transshipment manners that are available in the network. All this background information is considered in formulating the model. In Section 3, we define the multicommodity multimodal transportation routing problem from the perspectives of capacity constraint and time sensitivity, and we present an example containing a single-commodity flow to illustrate the selection of time-sensitive multimodal transportation routes. In Section 4, we establish an arc-node-based mixed integer nonlinear programming model to formulate the specific freight routing problem that we explore in this study. The proposed optimization model integrates the various formulation characteristics mentioned in the abstract. Then, we develop a linearization method to linearize the proposed model to make it easily solvable by mathematical programming software. In Section 5, a computational example from the Chinese inland container export business is presented to demonstrate the feasibility of the proposed model and the use of the linearization method to address the practical problem. Finally, the conclusions of this study are presented in Section 6.

#### 2. Multiple Transportation Service Modes and Transshipments

##### 2.1. Schedule-Based and Time-Flexible Services

Rail services in the transportation network are controlled by a central authority in China. The rail services are scheduled in advance under macroscopic control. The freight trains are operated through the multimodal transportation network strictly according to the rail service schedules/timetables. A typical rail service diagram is shown in Figure 1. Some rail services show periodicity in their operating frequencies within the planning horizon. For the convenience of modeling, we consider the same rail services viewed in different operating periods as different services.