Abstract

This study explores an operational-level container routing problem in the road-rail multimodal service network. In response to the demand for an environmentally friendly transportation, we extend the problem into a green version by using both emission charging method and bi-objective optimization to optimize the CO₂ emissions in the routing. Two uncertain factors, including capacity uncertainty of rail services and travel time uncertainty of road services, are formulated in order to improve the reliability of the routes. By using the triangular fuzzy numbers and time-dependent travel time to separately model the capacity uncertainty and travel time uncertainty, we establish a fuzzy chance-constrained mixed integer nonlinear programming model. A linearization-based exact solution strategy is designed, so that the problem can be effectively solved by any exact solution algorithm on any mathematical programming software. An empirical case is presented to demonstrate the feasibility of the proposed methods. In the case discussion, sensitivity analysis and bi-objective optimization analysis are used to find that the bi-objective optimization method is more effective than the emission charging method in lowering the CO₂ emissions for the given case. Then, we combine sensitivity analysis and fuzzy simulation to identify the best confidence value in the fuzzy chance constraint. All the discussion will help decision makers to better organize the green multimodal transportation.

1. Introduction

With the rapid development of globalization, companies are seeking for specialized partners all over the world to outsource their businesses and also for new markets to make more profit. International trade and global commodity circulation thus get significantly motivated during the last decades [1]. Supported by the transportation industry, this prosperity results in a remarkable expansion of the transportation network from a limited region to the entire world. The expanded transportation network extends the distribution channels and enhances the difficulty of transportation organization from the viewpoint of economy, efficiency, reliability, and environmental protection. The traditional unimodal transportation is no longer suitable for this situation. Meanwhile, an advanced transportation mode, namely, the multimodal transportation, emerges in the transportation industry. It combines different transportation modes to generate origin-to-destination routes and integrates their respective advantages in the transportation process. Furthermore, by using standardized containers, multimodal transportation can be highly mechanized and the efficiency of the operations at terminals is greatly improved [2]. Many empirical case studies, for example, Bookbinder and Fox [3], Janic [4], and Liao et al. [5], have demonstrated the superiority of the multimodal transportation in the long-haul transportation setting. Nowadays, multimodal transportation is getting more and more popular and plays a crucial role in international trade. Since the transportation chains in this area are usually long combining different actors, effective solutions for their coordination are necessary in order to ensure that the goods are delivered according to the customer’s requirements. In this context, the multimodal routing problem has gained interest of both researchers and practitioners [6, 7]. This operational problem aims at selecting the best routes to move containers from their origins to destinations through the multimodal service network according to customer demands [2]. In order to achieve this objective, a number of different factors have to be considered.

First of all, multimodal transportation combines different modes with different operational constraints that have to be represented in the mathematical model. In general, it can be distinguished between time-flexible modes (e.g., road) that can be used whenever they are needed and schedule-based modes (e.g., rail) that operate according to fixed schedules planned in advance. These schedules regulate the route as well as the (un-)loading, departure, and arrival times of the transportation service. Since these schedules cannot be easily changed, they should be considered when planning the routes of the containers and their transshipment between different transportation services; otherwise, the multimodal plans might be infeasible in reality.

Secondly, the actors and stakeholders involved in multimodal transportation systems have usually different interests which are often in conflict. Whereas transportation operators usually tend to minimize only the transportation costs, customers are also interested in the reliability of the routes in order to avoid situations where a plan becomes infeasible due to late arrival of a service or insufficient capacity. Since the multimodal routing is done before the start of transportation, when the information about the transportation services (e.g., traffic situation, travel time, and capacity) is not completely known, possible sources of uncertainty should be considered already during the planning [8]. As an example, the scheduled rail services have usually limited capacity due to the limited length of the tracks in the railway stations and limited locomotive power [9]. However, the available capacities of rail services are uncertain since they are also dependent on other transportation plans that are difficult to predict [10]. If only deterministic capacities are considered in the model, two consequences may happen in practice: the planned routes are either infeasible if the capacities are estimated to be too large, or the planned routes are not the actual best ones if the capacity estimation is too small. Therefore, capacity uncertainty is a factor that should be considered in the decision-making process. In addition to that, road services frequently suffer from various disruptions, such as natural disasters (e.g., bad weather) and manmade disruptions (e.g., traffic accidents and congestion) [11]. Among these disruptions, traffic congestion is very common due to the rapidly increasing traffic volume in recent years [12]. Traffic congestion results in the travel time uncertainty and delayed arrivals of the containers at the nodes, which disrupts the transshipment if the containers have to take another scheduled transportation service at the same node and also possibly violates the due date constraint at the destination. Thus, congestion is another factor that should be also considered in the decision-making process.

Thirdly, the environmental aspects of transportation have drawn considerable attention from the public and the government [13]. Among the various environmental issues, global warming caused by greenhouse gas emissions has been considered as a tremendous threat to human sustainable development. CO₂ accounts for approximately 80% of the entire greenhouse gas emissions [5]. During recent decades, large numbers of countries have proposed regulations to ease the global warming by reducing CO₂ emissions. For example, in 2012, the National Development and Reform Commission of China proposed a pilot project on trading carbon emissions [2]. Since the transportation sector is one of the biggest contributors to the CO₂ emissions [14, 15], the transportation industry should accept this responsibility and consider controlling and optimizing the CO₂ emissions during the transportation planning stage to promote the development of an environmentally friendly transportation. In order to respond to these environmental requirements, in this study, we extend the multimodal routing problem into a green version by considering the CO₂ emissions.

The inclusion of uncertain factors and the combination of multiple objectives (i.e., economic and environmental) in multimodal transport planning is an emerging field which has been studied in recent years [13–15]. However, the authors usually consider only one source of uncertainty or model the CO₂ emissions in form of charges which limits their influence on the resulting routes. Therefore, we present a multimodal routing planning approach which deals with the following question:

How can the sustainability and reliability of multimodal routing be improved by considering environmental factors and uncertainty?

The contributions of the paper are fivefold: (1)A mixed integer nonlinear optimization model for multimodal routing including multiple objectives (i.e., economic and environmental) and multiple uncertainty factors (i.e., road traffic congestion and rail service capacity) is defined.(2)A fuzzy logic is proposed to model the capacity uncertainty, and a piecewise linear function is used to represent the road traffic congestion.(3)The influence of CO₂ emissions on the routes is analyzed comparing the CO₂ emissions charging method and a bi-objective approach in which minimizing CO₂ emissions is set as an independent objective.(4)Since the proposed optimization model is nonlinear, a linearization approach is proposed in order to reduce the complexity of the model.(5)The planning approach is applied to a real-world case study from China with specific network design and parameters. In this case, the network is limited to road and rail services; however, other modes with fixed schedules (e.g., inland waterway and deep sea shipping) could be also treated in a similar way as it is the case of rail.

The paper is organized as follows: In Section 2, we review the current literature on multimodal transportation routing problem. In Section 3, we present a detailed modelling methodology. The improvements of our settings compared to the existing multimodal routing literature are indicated by comparing the two sections. In Section 4, based on the proposed modelling framework, we establish a fuzzy chance-constrained mixed-integer nonlinear programming model to mathematically describe the multimodal routing problem explored in this study. Since the proposed model is nonlinear, in Section 5 we design a linearization-based exact solution strategy to solve the problem. In Section 6, an empirical case study based on the Chinese scenario is given to demonstrate the feasibility of the methods in dealing with the practical problem. Also in the case study section, sensitivity analysis, bi-objective optimization scenario analysis on CO₂ emissions, and fuzzy simulation are adopted to draw conclusions which can further help the decision makers to better organize the multimodal transportation. Finally, conclusions of this study are drawn in Section 7.

2. Literature Review

The multimodal routing problem is a combinatorial optimization problem which deals with finding the optimal routes for customer demands respecting the constraints of the network [16]. Customer demands are represented by transportation orders characterized by their origin, destination, volume of containers, release time at origin, and due date at destination. Whereas the release time is usually considered as fixed, the due date can be either considered as a hard constraint (see, e.g., Verma et al. [17], Sun and Lang [2, 18], and Sun et al. [19]) or delayed deliveries can be accepted causing additional penalty costs (see, e.g., Demir et al. [20], Hrušovský et al. [14], and Zhang et al. [21].

However, considering only late arrivals might not be sufficient since early deliveries can also cause problems with the storage of the goods until they are needed. Therefore, various studies consider due date time windows instead of time points [8, 22], which is also the case in this study. Moreover, modelling a soft time window is more suitable than a hard one, because customers allow the violation to the time window (i.e., early and delayed deliveries) within a certain degree.

The containers from transportation orders need to be transported in the multimodal service network by various transportation services with their specific characteristics. One of these characteristics is the usually fixed schedule of the high-capacity and environmentally friendly transportation modes (e.g., railway or inland waterway) in contrast to the time-flexible (nonscheduled) road service. However, the majority of the related articles neglects the schedules and considers rail services to be also time-flexible. This facilitates the modelling since only transshipment times between services in terminals need to be considered (Sun et al. [23], Jiang and Lu [24], Cai et al. [25], Zhang et al. [21], and Xiong and Wang [26]). However, this does not correspond to reality where railway services are usually planned well in advance [2, 19]. Therefore, the schedules are usually fixed in the tactical network design phase (see, e.g., Wieberneit [27], Andersen et al. [28], and SteadieSeifi et al. [29] for this area) and have to be accepted in the operative multimodal routing considered in this paper (Liu et al. [30], Lin [31], Demir et al. [20], and Hrušovský et al. [14]).

The routes can be optimized according to different criteria. Traditionally, the optimization models concentrate on costs or time [32]. However, various surveys show that multiple criteria, among others also sustainability and reliability, are important for transportation planners [33, 34]. In case of sustainability, vast majority of available literature only considers economic and environmental criteria whereas social aspects of transportation are neglected [35, 36]. As a result, the comparison of transportation costs and CO₂ emissions is decisive for finding the optimal routes.

The CO₂ emissions can be either integrated into the cost objective function as emission charges or modelled as an independent objective in multiobjective optimization. The majority of the articles on optimizing CO₂ emissions in transportation adopts the first approach (e.g., Chang et al. [37], Zhang et al. [21], Wen et al. [38], Sun and Lang [2], Chen et al. [39], Demir et al. [20], and Hrušovský et al. [14]), whereas only a few studies used the second approach (e.g., Sun and Chen [40] and Qu et al. [41]). However, it is difficult to decide which method is better to use since the charging method is easier to implement but might not always be applicable, especially in cases where the emission charges are too low in comparison to transportation costs and therefore do not have any influence on the resulting routes [22].

Another important decision is the choice of the emission calculation method since most of the models require a lot of parameters (e.g., fuel type, vehicle and route characteristics, and traffic condition) that cannot be always estimated in advance [20, 42–45]. As an alternative, the activity-based method that multiplies activity intensity (unit: TEU-km) with the CO₂ emission factor (unit: g/TEU-km) to calculate the total CO₂ emissions [5] yields a better feasibility in transportation practice and has been already applied in various studies [2, 5, 37, 39, 41, 46].

The reliability of transportation is influenced by various sources of uncertainty (e.g., weather, accidents, congestion, and technical issues) that can cause disruptions in transportation processes. Despite this fact, most of the existing models consider all factors as deterministic and do not take into account uncertainty [47]. In order to improve the reliability of transportation plans, the uncertainty should be considered during the planning phase. Despite the variety of possible reasons for disruption, their effects influence either the demand, the travel time, or the supply (capacity) of transportation. Whereas extensive literature about demand uncertainty in multimodal routing is available (see, e.g., Lium et al. [48] and Ho et al. [49]), the consideration of travel time and capacity uncertainty is rather limited [50, 51]. Moreover, usually the models deal with only one source of uncertainty. However, the consideration of multiple uncertainties at the same time might lead to improved routes and therefore in our study the travel time uncertainty for road services and capacity uncertainty for rail services should be combined.

Road traffic congestion leads to travel time uncertainty of road services. Only a few studies, for example, Demir et al. [20], Sun and Chen [40], and Uddin and Huynh [11], adopted stochastic programming to formulate the multimodal service network problem under disruptions. However, in practical planning, due to the lack of reliable historical data [52], it is quite difficult to fit the probability distributions to the travel times of all road services in the multimodal service network. Such disadvantage of stochastic programming in addressing the uncertainty is emphasized by Zarandi et al. [52] in a location-routing problem. Furthermore, the fuzzy programming that uses fuzzy variables (usually triangular or trapezoidal numbers) to evaluate the uncertainty fails to represent the variation of the travel time of a road service en route with respect to the time of the day [53]. Therefore, it is worthwhile to find an applicable approach (e.g., time-dependent travel time that is widely discussed in the vehicle routing problems [53, 54]) to formulate the road traffic congestion and introduce it to the multimodal routing modelling.

The capacity uncertainty has only been considered by a few studies in different transportation planning problems, for example, multiobjective solid minimal cost flow problem by Wang [55], multiobjective multi-item solid transportation problem by Kaur and Kumar [56], and resource allocation problem for containerized cargo transportation networks by Kundu et al. [57]. Whereas Wang [55] used stochastic programming to address this issue, Kaur and Kumar [56] and Kundu et al. [57] formulated the capacity uncertainty from a fuzzy programming viewpoint. The fuzzy programming approach has been also used in the context of vehicle routing problem [52, 58] and shortest path problem [59], since it can avoid the infeasibility of the stochastic programming in case of lack of historical data [58]. However, to the best of our knowledge, there exists no study which would consider capacity uncertainty in multimodal routing problem.

The considered aspects of the multimodal routing approach increase the complexity of the problem. The modelling of constraints for schedule-based services leads to the nonlinearity of the model, which can be seen in the models proposed by Liu et al. [30], Lin [31], and Chang [60]. Moreover, the modelling of uncertainty sources might also lead to nonlinearity of the model. As for the problem formulated by a nonlinear model, Xie et al. [61] pointed out that the nonlinearity makes the problem very difficult to solve, while using linearization technique to get the equivalent linear model can enable the problem to be relatively easily solved. The linearization technique should be designed according to the specific nonlinearity of the nonlinear model. Its main idea is to introduce new linear forms to replace the nonlinear components in the model equivalently.

As the literature review shows, the interest of researchers in multimodal routing has been increasing during the last years, considering different aspects of the studied planning problem. However, there still exists potential for improvement especially in the area of sustainability and reliability of transportation. Therefore, our paper contributes to these aspects of the multimodal routing since it introduces and compares two different methods for modelling CO₂ emissions. Moreover, two different sources of uncertainty (travel time and capacity) are considered within the planning process which should reduce the influence of potential disruptions on created transportation plans and improve the reliability. In case of rail service capacity, the fuzzy programming approach is used in order to better represent the possible capacity variations. Although the application of the model to the Chinese case study requires some specific assumptions, the general model can be used for any multimodal service network.

In order to give an overview of the current state of the literature, the most important articles with the covered aspects are shown in Table 1.

3. Modelling Methodology

As already mentioned, the objective of the multimodal routing problem in this study is to select the best green routes to move containers from multiple transportation orders through the multimodal service network with road traffic congestion and rail service capacity uncertainty. The multimodal service network is composed of nodes and arcs, where nodes represent the terminals serving as origins and destinations of transportation orders as well as transshipment points between rail and road services. The arcs represent the services connecting the nodes. Related information on road and rail services, such as schedules, transportation distance, and capacity, is regarded as the parameters of the nodes and arcs.

In the studied network, rail services are the backbone for multimodal container transportation, while road services play an important role in origin pickup and destination delivery. In some cases, road services also take direct origin-to-destination transportation. Figure 1 shows the basic topological structure of the multimodal service network. Overall, the network structure is similar to a hub-and-spoke network with multiple hubs where origins and destinations represent spokes while railway container stations represent hubs.

Figure 1 

Basic topological structure of the multimodal service network.

3.1. Modelling Transportation Services

Service characteristics are dependent on the chosen transportation mode. In case of road service, the trucks are assumed to be fully flexible which means that they are available every time they are needed and it is possible to assign as many trucks as necessary to carry the containers. As a result, the road services are uncapacitated and their departure time can be adjusted according to the customer requirements and the current state of the network. This means that the departure time can be either immediately after the containers are loaded on the vehicle, but can be also postponed in order to avoid traffic congestion. However, it has to be ensured that the truck arrives to its destination before the loading cutoff time of the next planned service in order to avoid infeasibility of the transportation plan. Besides that, too early arrivals should be also avoided as inventory costs can be charged for containers waiting for the next service at a terminal. Therefore, it is important to determine the planned departure time of a truck from the current node.

In contrast to that, rail services are modelled as block container trains driving between two nodes according to fixed schedules with the following characteristics: (1) Operation time window at the loading/unloading organization station that regulates the operation start time and cutoff time regarding loading/unloading containers on/off the train(2) Scheduled departure/arrival time from/at the loading/unloading organization station(3) Operation period. For the convenience of modelling, same rail services in different operation periods are considered as different services.

Since the loading/unloading operations of both road and rail services are highly mechanized, the time needed for loading/unloading is relatively low. Consequently, we assume that these times can be neglected in the multimodal routing model. Besides that, the capacities of rail services are limited. Taking into account these assumptions, the origin-to-destination transportation process combining road and rail services is depicted in Figure 2 and can be described as follows: (1) The containers will first depart from the origin (node 1) at the planned departure time that is not earlier than the earliest release time defined by the customer, and then arrive at the loading organization station (node 2).(2) In order to be able to load the containers on the train operated from node 2 to node 3, their arrival time at node 2 should be no later than the operation start time. If the arrival time is earlier than the operation start time, there will be an inventory period from the arrival time to the operation start time at node 2, and the containers should wait until the operation start time and then get loaded. Moreover, there exists a free-of-charge period for inventory costs [64]. Only when the inventory period is longer than this period will inventory costs be created.(3) When loaded on the train, the containers will wait until the scheduled departure time and then leave node 2 along with the train.(4) The containers will arrive at the unloading organization station (node 3) at the scheduled arrival time of the train. However, they should wait until the operation start time and then get unloaded. Therefore, the effective arrival time of the containers at node 3 is the operation start time instead of the scheduled arrival time.(5) After unloading from the train, the transportation organizer can select a planned departure time that is no earlier than the operation start time to move containers to their destination (node 4). In case of an early arrival to the destination, the containers have to be stored again until the planned delivery date in a warehouse provided by the third-party inventory companies that can have different conditions in comparison to warehouses used in intermodal terminals [65].

Figure 2 

Diagram of a simple road-rail multimodal route.

3.2. Modelling Uncertainty

Considering the uncertainty factors, traffic congestion is an important factor in road transportation that influences the travel times. Congestion can occur either regularly due to insufficient road capacity or unexpectedly due to, for example, an accident. Since the latter one is hard to predict, only regularly occurring congestion is considered in this study. However, this congestion can vary with the time of the day or the section considered, therefore piecewise linear functions can be used to model such situations [53, 54]. By combining limited historical data and experiences from the experts, such travel time functions can be established. Figure 3 shows an example of the time-dependent travel time function. There are two peak hours from 9 am to 10 am and from 6 pm to 8 pm on the arc served by road services.

Figure 3 

Time-dependent travel time formulated by piecewise linear function.

Besides the road congestion, the available capacity of a rail service can be also uncertain due to various reasons (e.g., order cancellation, vehicle breakdown, and lack of available vehicles). Since the reasons are manifold and the historical data about capacity variation might not be available, using the stochastic programming for capacity uncertainty might not be feasible. As a consequence, capacity uncertainty can be effectively expressed in a fuzzy way by using the triangular fuzzy variable represented by the minimum of the possible capacities, most possible capacity and maximum of the possible capacities. Similar to the traffic congestion issue, fuzzy capacity can be attained by limited historical data and expert experiences.

3.3. Modelling and Optimizing the CO₂ Emissions

As stated in Section 2, there are two approaches to optimize the CO₂ emissions, including the CO₂ emission charging method and the bi-objective optimization method. The optimization model is firstly constructed in Section 4 using the charging method, and the bi-objective method is then introduced in Section 6 where both methods are compared. For calculating the CO₂ emissions, the activity-based method is used. This method calculates the total CO₂ emissions for a transportation service using the formula () where EM is the CO₂ emission factor of the service (unit: g/TEU-km), is the number of the containers carried by the transportation service (unit: TEU), and is the corresponding transportation distance (unit: km).

3.4. Modelling the Customer Demands

Transportation orders are considered to be unsplittable which means that all containers of an order have to use the same route. The release times of orders are fixed whereas the delivery can be done within a due date time window limited by a lower and upper bound. For each transportation order, it is best if the arrival time of the containers falls within the time window. In this case, there will be no extra costs at the destination. However, if the containers arrive at the destination earlier than the lower bound, inventory costs will be created, because the goods need to be stored before being processed at the destination. If the arrival time is later than the upper bound, penalty costs should be charged for the delayed delivery.

4. Mathematical Model for the Multimodal Routing Problem

4.1. Notations

In this study, represents a road-rail multimodal service network in form of a directed graph, where , , and separately denote the node set, arc set, and transportation service set in the network. For each transportation order ( where is the transportation order set), its five attributes are represented as origin , destination , volume (unit: TEU), earliest release time at origin, and due date where and are separately the lower bound and upper bound of the due date time window, respectively. The rest of the symbols in the mathematical model and their respective representations are summarized as follows in Table 2.

4.2. A Fuzzy Chance-Constrained MINLP Model

The framework of the mathematical model is derived from the classical arc-node-based formulation that has been widely used in the routing problems of various research fields, for example, telecommunication systems, manufacturing systems, and internet service systems [1]. The mathematical model for the multimodal routing problem explored in this study is presented as follows. The proposed model is a combination of the fuzzy chance-constrained programming and the mixed integer nonlinear programming (MINLP).

Objective function:

Eqs. (1), (2), (3), (4), and (5) are successively the transportation costs en route, loading and unloading costs at the nodes, inventory costs, penalty costs caused by delayed deliveries at destinations, and CO₂ emission costs. Their linear summation represents the generalized costs. The objective function minimizes the generalized costs for multiple transportation orders using multimodal transportation. This setting reflects the objective of satisfying customer demands and minimizing monetary expenditures. Therefore, the multimodal routing in this study is a kind of service-oriented planning.

Subject to

Eq. (6) is the flow conservation constraint which ensures the continuity of the nodes and arcs on the planned routes, which is a common constraint in the node-arc-based formulation [1]. Eq. (7) means that no more than one transportation service on one arc can be utilized to serve a transportation order. The combination of the two equations above ensures that containers in a transportation order are unsplittable, which is claimed in Section 3.4.

Eq. (8) is the fuzzy chance constraint of the rail service capacity based on the fuzzy credibility measure [66]. In this equation, represents the credibility of the event in . Eq. (8) ensures that the credibility of the event that the entire loaded container volume on a rail service does not exceed its available capacity should not be lower than a predetermined confidence . The value of indicates the decision makers’ preference for addressing the corresponding decision-making.

Eq. (9) assumes that the arrival time of the containers of transportation order at the origin equals its earliest release time.

Eqs. (10), (11), (12), and (13) compute the arrival times of the containers of transportation order at the nodes on the route by road services. First, we determine the planned departure time of the containers at the current node by (10). Then, we normalize such departure time into range [0, 24] by using the floor integral function (11) and normalization formula , so that we can further generate the corresponding travel time of the containers on arc by road service according to the piecewise linear function (12). Finally, we can gain the arrival time of container flow at the following node by (13).

Eq. (14) computes the arrival times of the containers of transportation order at the nodes on the route by rail services. As claimed in Section 3.1, is the effective arrival time of containers at node . Because we assume that loading/unloading time can be neglected, is also the earliest departure time of containers at transshipping node.

Eq. (15) is the rail service operation cutoff time constraint. It ensures that the arrival time of the containers of transportation order at node should not exceed the operation cutoff time of rail service at the same node, if this rail service has to be used in order to move these containers on arc .

Eqs. (16) and (17) compute the charged inventory time of the containers of transportation order at node before being moved on arc by a rail service and a road service, respectively. In (17), especially if and , .

Eqs. (18), (19), (20), (21), (22), and (23) are the variable domain constraints.

The above mathematical model for the multimodal routing problem is summarized in Figure 4. Figure 4 classifies all equations (excluding variable domain constraints) into three categories, including “Modelling the network” which realizes the methodology proposed in Section 3, “Setting the objective” which reflects that satisfying customer demands is the core of the routing problem, and “Basic assumptions” which are the basic conditions of the model. This should help the readers to better understand the functions and roles of various equations in the mathematical model.

Figure 4 

Framework of the mathematical model.

By using the solution strategy designed in the next section, the proposed model can be used to optimize any green multimodal routing problems that are composed of scheduled rail services with uncertain capacities and time-flexible road services with uncertain travel times. It can plan the best multimodal routes to accomplish multiple transportation orders with soft due date time windows.

5. An Exact Solution Strategy

5.1. Determining the Solution Strategy

The proposed mathematical model in Section 4.2 is easily understood, because it is established straightforwardly according to the settings of the specific multimodal routing problem. However, the mathematical model contains three categories of nonlinearity, including (1) fuzzy chance constraint as (8), (2) piecewise linear constraint as (12), and (3) general nonlinear components as (3), (4), (13), (14), (16), and (17). Such nonlinearity increases the difficulty of the problem to be effectively solved by the exact solution algorithms (e.g., branch-and-bound algorithm and simplex algorithm). Especially for the large-scale real-world problem, due to the nonlinearity, the solution to the problem will usually get into the local optimum, and massive computational time will be consumed to generate the solution. Therefore, if we can eliminate the nonlinearity of the model by designing the equivalent linear reformulations to the nonlinear equations in the model, then the problem can be solved with the help of the mathematical programming software (e.g., LINGO, CPLEX, and GAMS) where exact solution algorithms can be implemented efficiently.

We devote this study to developing a linearization-based exact solution strategy to address the multimodal routing problem. Although it is widely acknowledged that heuristic algorithms have better performance than the exact solution algorithms in dealing with the multimodal routing, which is known as an NP-hard problem, exact solution algorithms have two significant advantages. The first one is that they can test if the model observes the mathematical logic. The second is that they can provide an exact benchmark to verify the efficiency and accuracy of the heuristic algorithms [67]. Fazayeli et al. [68] also highlighted that the development of an exact algorithm and the comparison between the algorithms will be their future work for the location-routing problem on a multimodal transportation network.

5.2. Model Linearization Technique

5.2.1. Crisp Equivalent and Linearization of the Fuzzy Chance Constraint

From the definition of the fuzzy creditability, (24) can be obtained where is a deterministic number while is a triangular fuzzy number described by and . [69]:

According to (24), we can gain the crisp equivalency to the fuzzy chance constraint (8) by replacing with and with . However, the crisp equivalency is a piecewise linear function which is still essentially a nonlinear formula. Thus, we conduct following modifications shown as (25) and (26) to transform it into a pure linear function. After such modification, during the simulation process, we can use (25) to substitute (8) if , (26) to substitute the same equation otherwise.

5.2.2. Linearization of the Piecewise Linear Function

Similarly to (24), the piecewise linear function is also a nonlinear expression where represents for the sake of conciseness. Let represent the set of the linear segments of , the index of the segments (), and the departure time interval corresponding to the linear segment . Then, has a universal formulation shown as (27).

To linearize the piecewise linear function (27), the following procedures are conducted.

Step 1. Define a 0-1 variable . , if falls within . otherwise. And should observe the following constraint.

Step 2. Define two nonnegative variables and , and distribute them to the lower bound and upper bound of the corresponding departure time interval as their weights. Hence, we have the following equations.

Step 3. Modify the piecewise linear function as follows. where and for are all known. Finally, the pure linear equivalency to (12) is shown as (31).

5.2.3. Linearization of Other Nonlinear Components

(1) Linear Reformulation 1. Eq. (3) with nonlinear component can be linearized by (32), (33), and (34) where is a newly introduced variable.

If the containers arrive at the destination earlier than , . Consequently, . According to (32) and (33), . Minimization of (32) will then lead to . In this case, is equivalent to (32) and (33).

If the containers arrive at the destination later than , . Consequently, . According to (32) and (33), . Minimization of (32) will then lead to . In this case, is also equivalent to (32) and (33).

Above all, the equivalency above is verified. Similarly, we have the following linear reformulation.

(2) Linear Reformulation 2. Nonlinear component in (4) can be linearized by (35), (36), and (37) where is also a newly introduced variable.

(3) Linear Reformulation 3. Nonlinear constraint (12) can be reformulated by linear constraints (38) and (39).

We take as example to verify the equivalency above. According to (12), . Meanwhile, according to (45) and (46), there exists . Because is a nonnegative integer, finally . Consequently, the equivalency above is tenable.

(4) Linear Reformulation 4. By using the same linearization technique, nonlinear constraints (13) and (14) can be reformulated by linear constraints (40) and (41) and (42) and (43), respectively.

(5) Linear Reformulation 5. By using the same linearization form, nonlinear constraints (16) and (17) can be reformulated by linear constraints (44) and (45) and (46) and (47).