Abstract
The service network design problem is one of the key problems in the field of railroad intermodal transportation of hazardous materials (hazmat). It lies in determining the frequency of the trains and the allocation of both hazardous and ordinary shipments to these trains, as well as the railroad connecting plan of all shipments based on the established train formation plan. Based on a unified method, we build a multiobjective mixed-integer linear-programming model that comprehensively considers both hazardous and ordinary materials, the compatibility of goods and services, the capacity of the transfer station, and the delivery time limit of each shipment. An augmented ε-constraint algorithm is customized to solve the model. Computational results of a practical example in China show that an increase of 8.4 million yuan in the total cost could reduce nearly 24541 the total risk. Moreover, the algorithms comparison and sensitivity analysis indicate that the augmented ε-constraint algorithm is a more suitable approach to handle the model because of its advantage in the uniqueness of distinct nondominated solutions. Finally, the sensitivity analysis shows that the total cost and risk could simultaneously be reduced if the delivery time limit is relaxed or if the transfer time is shortened.
1. Introduction
Hazardous materials (hazmat), such as combustibles, explosives, poisons, and radioactive substances, are the basic components of our modern industrialized life. Due to the different production and consumption locations of most hazmat, there are a large number of hazmat transportation in real life. With the increase in transportation volume, the number of transportation accidents with hazmat is also increasing. Different from general transportation accidents, transportation accidents with hazmat usually derive more serious consequences such as combustion, explosion, and leakage, resulting in casualties, property losses, and environmental and ecological damage. Railroad intermodal transportation is a transportation mode with both cost advantages and risk reduction for long-distance inland transportation of hazmat, which has attracted more and more attention from scholars and hazmat carriers.
The service network design optimization problem is a key and difficult point in the operation and management of railroad intermodal transportation of hazmat. Railroad intermodal transportation involves three links, rail-haul, inbound and outbound drayage, and transfer operation. Meanwhile, considering the particularity of hazmat, the service network design optimization problem of hazmat is more complex than that of ordinary materials.
In China, railway freight transport services take hazmat and ordinary materials into consideration together. Except that some services such as special trains for postal and, other trains can carry both hazmat and ordinary materials at the same time. Generally speaking, after arriving at the freight station, hazmat and ordinary materials are loaded into railway vehicles, and railway vehicles loaded with various materials are sent to the technical station. Various vehicle flows are grouped into train flows according to the freight train marshaling plan, and the train flows are grouped online according to the freight train diagram, and they are then finally transported to their destination station through various processes.
The current transportation organization mode in China is based on the annual planned train operation technical documents, namely the train formation plan and train operation diagram, and takes the daily dispatching and command work as the core of train operation organization. After China Railway Corporation issues the transportation plan, each railway bureau will report the prepared traffic flow plan according to the economic investigation and other procedures. Then, China Railway Corporation adjusts the plan after a comprehensive balance. Based on this, each railway bureau will finally formulate a traffic flow plan.
In the actual work, in a short time (usually one month), Each railway bureau can finally determine the train type, frequency, origins and destinations, and the distribution scheme of freight flow according to the actual prediction of freight flow of hazardous and ordinary materials. This could be used as a specific train work plan in the transportation scheme to ensure transportation, reduce costs and risks, and improve the quality of railway freight transportation service.
To sum up, this paper studies the railroad intermodal transportation service network design problem with comprehensive consideration of both hazardous and ordinary materials, as well as the actual operation organization in China. We can summarize the main contributions of this paper as follows.
Firstly, we simplified the railroad intermodal transportation service network into a network with different types of transportation services and transfer services by skillfully using the unified method.
Secondly, we constructed a multiobjective mixed-integer linear-programming model that considers both hazardous and ordinary goods, taking fully into account the characteristics of the organization forms of various transportation modes in railroad intermodal transportation, multitypes, and multicommodity flow, as well as the cost, risk, and time consumption of the transfer process.
Thirdly, the modeling method based on the service network makes the model compact and clear, which can effectively reduce the problem scale and adapt to the large-scale characteristics of practical problems. Thus, we customized an augmented ε-constraint algorithm to solve the model and provide abundant representative nondominated frontiers for decision-makers with different preferences in different situations.
The rest of this article is organized as follows. Section 2 gives the literature reviews related to the railroad intermodal transportation service network design problem of hazmat. The problem description and the mathematical formulation are given in Section 3, and an augmented ε-constraint algorithm is given in Section 4. Furthermore, Section 5 presents the empirical studies. The final section presents the conclusions and future research.
2. Literature Review
In the theoretical research field of multimodal transport, railroad intermodal transportation has attracted extensive attention of researchers; detailed information could be seen in Crainic’s review [1]. However, most of the research mainly focuses on the transportation of ordinary goods. Erkut et al. [2] summarized a large number of literature on hazmat logistics and pointed out that multimodal transport of dangerous goods had not been studied since 2007. Since then, scholars began to study this problem. This section mainly introduces the research related to the transportation network design problem of hazmat and the multimodal transportation service network design of ordinary goods.
2.1. Transportation Network Design Problem of Hazmat
Compared with the urgent demand for research results in this field, there is a lack of research on the service network design problem of railroad intermodal transportation of hazmat at home and abroad. In the existing research, more attention is paid to the physical network design problem for hazmat road transportation. This problem is mainly from the perspective of government departments to decide which links are open to hazmat transportation in the existing road networks to avoid densely populated areas. The differences in the research mainly lie in the different models (bilevel programming model [3, 4], single-objective model [5], etc.) and different objectives (considering risk equity [4], time factor [5], or conditional value at risk [6], etc.). In the research field of railroad transportation of hazmat, more attention is paid to how to reduce the risk [7] and optimize the railroad route plan [8].
But in the research field of service network design problems for hazmat multimodal transportation, at present, there were six documents out of which three from the same research team have discussed some of the contents of this problem. Verma et al. jointly designed the number and route of intermodal trains with different levels of services for hazmat and ordinary materials from the perspective of railway intermodal transport companies providing door-to-door transportation services [9–11]. Among them, papers [9, 10] assume that at least one intermodal transportation service and the route can meet the delivery time demand set by the customer. Literature [11] allows the delay of delivery time, but the transporter needs to pay a certain penalty cost in the case of delivery delay. All three works of literature ignore the waiting time of each node, such as the transfer station of intermodal transportation, and assume that the multimodal transport network has sufficient capacity and equipment to complete all the shipments of transportation. Based on these assumptions, all three works of literature have established a biobjective mixed-integer linear-programming model based on a link model with the objectives of minimizing total cost and total risk of the rail-haul process, as well as the inbound and outbound processes. Then they changed the biobjective problem into a single-objective problem using the linear weighted-sum algorithm. According to the characteristics of the single-objective problem, different solutions are designed. Assadipour et al. successively studied the intermodal transportation problem for hazmat. They first jointly designed the number and route of various trains for both hazmat and regular containers; they proposed a biobjective optimization model that not only considers congestion at intermodal yards but also determines the appropriate equipment capacity. Then they proposed a bilevel biobjective model to identify the terminals that have the most impact on system risks and determine the tolls that should be assigned to those terminals [12, 13]. Abuobidalla et al. presented a nonlinear model for railway hazmat transportation planning problems with blocking decisions, which is to minimize the total costs with the constraint that the total population exposure and environmental risks should be lower than the given thresholds [14].
2.2. Multimodal Transportation Service Network Design of Ordinary Goods
Network design and service network design are the most common problems studied, and detailed information could be seen in Archetti’s survey [15]. Specifically, Crainic gave the first model of multimodal transport service network design, which considered multimodal and multicommodity cargo transportation. They built a model to minimize the total system cost and then proposed a solution algorithm based on decomposition and column generation technology [16]. Newman and Yano studied the application of multimodal transport service network design involving railways. They compared the differences between decentralized and centralized planning methods composed of hub-and-spoke combined with direct connections, established an integer programming model to minimize the total cost, and designed a heuristic algorithm to solve the model [17]. Pedersen and Crainic established a mixed-integer programming model based on the time-space network to optimize the train timetable in the multimodal transportation network. In addition to the operation cost and delay cost of the train, the model included the transferring time cost and it could be solved directly by commercial optimization software [18]. Sharypova et al. proposed a new model that accurately evaluates the occurrence time of transportation events and the number of containers transferred between vehicles. The model aimed to minimize the service network design cost, which could be solved directly by commercial optimization software [19]. Zhou et al. addressed distribution planning and network flow optimization under multimodal transportation taking into consideration both economic and environmental performance [20]. Duan et al. presented a new multimodal service network design model aiming at solving the service frequency and allocation scheme, with the consideration of transshipments, capacity constraints, and heterogeneous users [21]. Inghels et al. presented a dynamic tactical planning model that minimized the sum of transportation costs, as well as external environmental and societal costs. Their model offered an allocation plan to transport modes and determine transportation frequencies over a planning horizon [22]. Fontaine et al. proposed a deterministic mixed-integer linear-programming model for the scheduled service network design problem and developed an efficient Benders decomposition algorithm that included a tailored partial decomposition technique [23].
A comparative analysis of some aforementioned literature is shown in Table 1. From the table, we find that there is a lack of research on the service network design problem of railroad intermodal transportation of hazmat. The difference in the existing research mainly lies in the objective types of the model, while their solution method is mostly a linear weighted-sum algorithm. In addition, the existing research does not consider the characteristics of the organization forms of various transportation modes in railroad intermodal transportation, nor the situation of multiple types of hazmat, along with the cost, risk, and time consumption of the transfer process, even to the extent that no linear optimization model suitable for the large-scale characteristics of practical problems is built.
Hence, we try to develop theoretically rigorous and practically efficient multiobjective optimization approaches for the railroad service network design problem for both hazardous and ordinary materials. Many practical situations such as multiple types and multicommodity of demand, compatibility between goods and various services, and delivery time limit are considered in the problem. We first formulate the problem as a complete and compact biobjective mixed-integer linear-programming model. Then we customize the augmented ε-constraint algorithm to find the representative nondominated solutions and compare them with the weighted-sum algorithm. A large-scale realistic-based case is used to test the proposed multiobjective optimization model and algorithm.
3. Problem Description and Model Formulation
3.1. Problem Description
The key process of railroad intermodal transportation service network design is to design the railway transportation service, and its specific process is shown in Figure 1. The railroad intermodal transportation service studied in this paper includes the inbound process from the starting nodes to the appropriate railroad intermodal transportation stations, the transfer process at the stations, the rail-haul process which may include the transfer process between railway services, the transfer process at the terminals of railway services, and the outbound process from the terminals to the destination of goods, as shown in Figure 2.
Thus, the railroad intermodal transportation service network design problem (O) comprehensively considering hazmat and ordinary materials is defined as follows.
Given the railroad intermodal transportation network information (intermodal transportation network topology, the distance of various links, station capacity, etc.), freight flow information (category, starting node, destination, freight volume, etc.), and the existing freight train marshaling plan (including train classification, starting node, ending node, train route and Stop Schedule Plan of each train), the task of this problem is to determine the following parts.(1)The highway and railway connection scheme of various freight flows.(2)The allocation scheme of freight flows in various trains.(3)The number of various trains.
Thus, we could transport each shipment of goods from the starting node to the destination according to the customer's service requirements under the objectives of minimizing the total transportation risk and the total cost of the operator, while the relevant operational and capacity constraints (such as the compatibility between goods types and services, railway station capacity, train capacity, and the arrival time limit required by customers) will not be violated.
3.2. Assumptions
Due to the complexity of the actual problem, the problem is appropriately simplified and assumed in the modeling process, which is summarized as follows:(1)Assume that each shipment of goods cannot be transported separately.(2)The transportation volume of various goods is measured by the number of vehicles (1 vehicle equals 40 feet container), and we assume that the transportation volume is 60 tons per container.(3)Assume that the cost includes fixed cost and variable cost of operation, in which the fixed cost is only related to the number of trains, while the variable cost is related to the number of vehicles and the transport distance.(4)The difference in unit variable transportation cost of different types of goods is expressed by the cost discount coefficient of goods. And the difference in unit transportation risk of various services/different types of goods is expressed by the service risk discount coefficient/risk discount coefficient of goods.(5)Assume that the cost and risk of the transfer process are related to the type of goods and the types of transportation services before and after transit. To simplify the model, we ignore the loading and unloading time, costs, and risks of goods at the origin and destination.(6)Assume that each service runs directly between its origin and destination, and the section capacity constraints are ignored.
3.3. Construction of Railroad Intermodal Transportation Service Network
To solve the problem that is difficult to describe in the process of freight transportation, two auxiliary networks are defined first, and then we could transform the service network design problem of railroad intermodal transportation of hazmat and ordinary materials into a special multicommodity network flow problem. The first auxiliary network is the physical network of railroad intermodal transportation, which is recorded as an undirected graph , where represents the set of nodes in the network, that is the set composed of the starting and ending nodes of each shipment as well as the railway stations. is the set of undirected arcs, which represents railway and highway links. Based on the physical network , each node and link may provide various types of service. Based on this, a second auxiliary network, namely the railroad intermodal transportation service network, is constructed, which is recorded as a directed graph , where is the set of service nodes and is the set of service arcs, including highway transportation service arcs, railway transportation service arcs, transshipment service arcs, and transfer service arcs. To further simplify the problem and model, the highway transportation service and railway transportation service are treated indiscriminately, and they are distinguished only by service types. At the same time, the transshipment service arc is regarded as a special transferring service arc by parameters setting of different transferring and transshipment services. The schematic diagram of the two auxiliary networks is shown in Figure 3. The upper figure shows the railroad intermodal transportation physical network from a to f, where a and f are respectively the starting node and ending node of transportation and b, c, d, and e are railway stations. The lower figure shows the railroad intermodal transportation service network from a to f. Each transportation service has a service starting node and service ending node. When there are several different services at a physical node, there would be multiple service nodes at this node. The arcs between different service nodes at the same physical node are called transfer service arcs, and the arcs between different service nodes of different physical nodes are called transportation service arcs.
Based on the above analysis, the second auxiliary network constructed above can be simplified, which is recorded as a directed graph , where is a set of service nodes, and each vertex in represents that a specific transportation service passes through a specific node. Let and represent its corresponding service and station respectively. is a set of service arcs, which includes transportation arcs and transfer arcs . Arc connects two adjacent vertices in the physical network and represents the operation process of the service between those two nodes. Let and represent the corresponding service and road section, if , then , . Arc indicates the transfer process of goods between different services at the same station, if , then , .
3.4. Definitions of Other Symbols
To facilitate modeling, other important symbols are defined and described (Table 2).
3.5. Mathematical Model
To sum up, the optimization model [P] of railroad intermodal transportation service network design problem considering hazmat and ordinary materials is built as follows:
In model [P], (1) and (2) are objective functions minimizing system cost and system risk respectively. Equation (1) includes the variable cost of transportation service, fixed cost of transportation service, and transfer service cost. Equation (2) only includes the risk of transportation service and the risk of transfer service.
Equations (3) to (11) are constraints. Among them, equations (3) and (4) are the assignment constraints, and equation (5) is flow conservation constraint of service nodes, which ensure that each shipment of goods will be transported from the starting node to the destination. Equation (6) indicates that only when a certain type of goods is allowed to be handled in the service arc, this type of goods can be transported through the arc. Equation (7) represents the relationship between variables and constraints . Equation (8) represents the capacity restriction constraint of railroad intermodal transportation stations. It should be pointed out that (8) also has two equivalent alternative formulas, namely and , respectively, representing that the flow volume arriving at the node could not exceed the capacity of node and the transfer volume at node could not exceed the capacity of node . Equation (9) represents the arrival time limit requirements of each shipment of goods. Equation (10) is the 0-1 integer constraint of the variables. Equation (11) is the positive integer constraint of the variables.
4. Solution Methodology
The service network design problem of railroad intermodal transportation of hazmat is a biobjective mixed-integer programming problem. For this kind of problem, there are no efficient algorithms in the literature now which can find all nondominated solutions in a polynomial time. Nevertheless, there exist several simple and easy-to-implement multiobjective algorithms which could be used to find a required number of representative nondominated solutions within a reasonable time provided that the corresponding single-objective model is well formulated. Model [P] has variables and linear constraints. Fortunately, preliminary computational results show that our model [P] exhibits a compact structure and enables its single-objective models to be solved optimality in a reasonable time for large-scale problems using commercial optimization software. The service network design problem of railroad intermodal transportation of hazmat may consider the two objectives of minimizing the total cost and total risk at the same time. For such kind of problem, it is often difficult for decision-makers to identify the relative importance of cost and risk objectives. They are more willing to obtain several uniformly distributed nondominated solutions and then choose the most satisfactory solution based on their judgment. Based on this, we customized an augmented ε-constraint algorithm to solve the proposed railroad intermodal transportation service network design problem of hazmat.
The ε-constraint algorithm was first proposed by Haimes et al. [24] in 1971. It is one of the most attractive multiobjective optimization algorithms in theory and calculation. As a generation method, the ε-constraint method has a lot of benefits [25, 26]. However, the original ε-constraint algorithm only optimizes one objective when constructing the auxiliary model and restricts other objectives than could not be inferior to a specific value within its variation range with inequality constraints. By approximately adjusting the level of the value of the objective constraint, all or a group of representative nondominated solutions can be identified. When the inequality constraints of all constrained objectives worked, the original ε-constraint algorithm can correctly identify the nondominated solution, but when the inequality constraint of some constrained objectives does not work, the original ε-constraint algorithm may output weak nondominated solutions.
In this regard, Mavrotas [27] proposed an improved auxiliary model in his augmented ε-constraint algorithm, by introducing slack or surplus variables to standardize the inequality constraints of each restricted objective into equality constraints. At the same time, the slack or surplus variables corresponding to all restricted objectives are dimensionless and then extended to the objective function of the original auxiliary model with a sufficiently small weight . Mavrotas has proved that the improved auxiliary model can ensure that only nondominated solutions could be searched. For the original problem (O), without losing generality, the total cost objective is minimized, and the total risk objective is constrained. Let be the change range of the total risk objective. The nondominated solutions of the original problem (O) can be obtained by solving the following criteria ε-constraint model:
According to the research of Mavrotas [27], let be the relaxation variable of the constraint of the total risk objective. In addition, to avoid the scaling problem, the introduced relaxation variable is further divided by the difference of the total risk objective . Let be the current constraint level of the total risk objective. Criteria ε-constraint model (12) for finding nondominated solutions of the original problem (O) can be strengthened as
Based on the constructed single-objective auxiliary model (13), the ε-constraint algorithm searches for different nondominated solutions by dynamically adjusting the value of the coefficient on the right of the constraint of each restricted objective (i.e., dynamically adjusting the constraint level of each restricted objective). When then ε-constraint algorithm is used to solve multiobjective optimization problems, many strategies can be used to adjust the constraint level of restricted objectives. The basic principle of Mavrotas [27] bisection method is to first use a given number of lattice points to divide the value range of each restricted objective into several intervals, and each lattice point of the restricted objective represents its one bisection constraint level; then, they are sorted in a descending order according to the quality of the value of objectives. Finally, all lattice points of constrained objectives are combined to generate nondominated solutions which are to be explored. For the original problem (O), let be the number of equal partitions of the total risk objective. That is to say, the length of the equal partitions is , and the number of equal grid points is . Since the original problem (O) has only two objectives, it means that the number of nondominated solutions to be explored is also . Let be the index of the equal grid points of the total risk objective, and is numbered from 0. For each equal grid point , let the corresponding constraint level be , and then solve model (13). If there is an optimal solution , the nondominated solution is obtained.
The standard augmented ε-constraint algorithm still could be improved further. Mavrotas [27, 28] developed two acceleration mechanisms, namely early departure mechanism and jump mechanism. The former can end the exploration of invalid grids of some constraint objectives in advance, and the latter can safely jump redundant grids of some constraint objectives. Both mechanisms are expected to speed up the solution efficiency of the algorithm. For the original problem (O), when the augmented ε-constraint algorithm is starting, only the most relaxed limit () is imposed on the total risk objective , and then the constraint level is gradually strengthened. If there is no feasible solution at a grid point , it means that strengthening the value level of the total risk objective at the following grid points will also result in infeasible solutions. In such a case, it is obvious that continuing the algorithm is unnecessary. Hence, the algorithm is terminated.
On the other hand, if there is an optimal solution at the current grid point , and if , it means that the same nondominated solution will be obtained at the next one or a few grid point(s), and the only difference is the value of the slack variable . Let be the bypass coefficient of the total risk objective, and . If , we can bypass from grid point to grid point . Note that for the original problem (O), preset bisection intervals for the total risk objective may not exactly obtain nondominated solutions. If the number of nondominated solutions obtained is lower than the demand, we can increase the value of the parameter according to the missing quantity of nondominated solutions.
5. Computational Experiments
5.1. Overview of the Network
We take the railway transportation network under the management of Chengdu Railway Bureau (hereinafter referred to CRB) and the highway transportation network in Chongqing, Guizhou, and Sichuan province as the background to construct the physical network of the test example. Based on the freight train formation plan of CRB in 2019, the railroad intermodal transportation service network is designed for both hazmat and ordinary materials. Assume that some administrative cities are the origins and destinations of railroad intermodal transportation of hazmat and ordinary materials. 20 administrative cities with an annual total industrial output value of more than 40 billion yuan are selected as the origins and destinations of hazmat, which are nodes 1 to 20, as shown in Figure 4. 34 administrative cities contained in the test area can be set as the origins and destinations of ordinary materials, which are nodes 1 to 34, as shown in Figure 4. The railway freight stations are transfer nodes of railroad intermodal transportation. Stipulate that if the goods are transported by road, it shall be carried out according to the shortest route of the road, and then the physical network of railroad intermodal transportation in the study area is constructed, as shown in Figure 4. The intensity of colour of each city represents the population density of the region, and with greater population density comes darker colour. The data in the network mainly include the length and the population density of the highway from the cargo origin and destination to the nearby railway stations, as well as the length and the population density of the railway between the railway stations. Among them, the length of highway links is taken as the shortest path in the highway network, and the length of railway links is taken as the actual length of the railway section. For simplicity, if the link passes through administrative cities A and B, its population density is calculated by dividing the sum of the population of A and B by the sum of the areas of the two cities.
5.2. Parameters’ Setting
Determining the service network design scheme of railroad intermodal transportation of hazmat requires inputting a lot of data. Due to the lack of practical data, in the test example, the data such as train formation plan information, network information, types of hazmat, cost, and risk of railway and highway are derived from real statistical data, while the various types of OD flow matrix and its delivery time limit are derived from randomly generated data.
In the test case, we consider three types of hazmat with the largest transportation volume, namely 1st type (flammable liquid), 2nd type (toxic and infectious materials), and 3rd type (corrosive materials). Different types of hazmat have slightly different cost and risk coefficients. According to the transportation price rules in Chinese railway freight transportation, the cost coefficient of the three types of hazmat is taken as 1.5, 2, and 1.5, respectively. Meanwhile, based on a comprehensive judgment of the harmfulness to human health, flammability, and reactivity characteristics, the risk coefficient of the three types of hazmat is valued as 1, 1.1, and 1.2, respectively.
This paper only considers the marshaling plan of freight heavy trains that the origins and destinations are both in the jurisdiction. According to the 2016 basic train diagram and freight train formation plan of CRB, there are about 215 trains under the jurisdiction of CRB, including all kinds of direct trains, through train, section trains, pick-up trains, small running trains, and express trains. Deduct the repeated services of pick-up trains and section trains with the same origin and destination; ignoring small running trains, a total of 82 alternative train services are obtained. According to the origins and destinations of train services, the transfer stations of railroad intermodal are from nodes 35 to 73 as shown in Figure 5. The capacity parameters of transfer stations include marshaling stations (such as Chengdu North and Xinglong yard), section stations (such as Chuanshuibei and Nangongshan), and intermediate stations (such as Tuanjiecun and Luohuang) which are set according to the type of transfer station, namely 60000, 15000, and 3000 vehicles/month.
(a)
(b)
Taking one month as the test cycle, OD data are randomly generated by Excel. The generation rules are as follows.
Select the cities with a distance of more than 400 km as OD nodes of ordinary materials, and cities with a distance of more than 400 km and an annual industrial output value of more than 40 billion yuan as OD points of hazmat. Assume that the total monthly traffic volume of ordinary materials in the bureau is 20000 vehicles, several effective OD pairs are randomly generated in the collection of ordinary materials OD nodes, and the traffic volume is distributed according to the GDP of the cities of origin and destination. The total monthly traffic volume of hazmat in the bureau is 2000 vehicles, several effective OD pairs are randomly generated in the OD nodes set of hazmat, and the traffic volume is distributed according to the GDP of the cities of origin and destination. The shipment transportation time is limited to 5 days (120 hours) in case that the farthest possible cargo could be transported in time.
According to the analysis of the train formation plan within the administration of CRA and the actual situation of highway transportation, the transportation services are divided into six categories. According to the actual situation in China, except that the express train could not handle hazmat, other services can serve all four categories of goods. The average speed of each category of services is shown in Table 1 according to reference [11] and the actual situation in China. The value of the cost parameters is determined by referring to the freight rate. The railway freight rate refers to the provisions of the freight rate in China's railway 95306 network, taking the average value of the whole vehicle freight rate; then, we obtain the base operating price of 16.64 yuan per ton and the base departure arrival price of 0.1032 yuan per ton per km [29]. Taking the base operating price as the fixed cost rate for train operation and the base departure arrival price as the variable cost rate, we could get that the fixed freight rate is 49920 yuan per train, and the variable freight rate is 6.192 yuan per vehicle per km. The highway freight cost rate also refers to the provisions on receiving and delivery fees and storage fees in China Railway 95306 network. The minimum rate of highway transportation of a vehicle is 15 yuan per ton, and the rate is 0.8 yuan per ton-km [30]. Assuming that there are 60 tons per vehicle, the fixed rate of highway transportation service is set to be 900 yuan, and the variable rate is set to be 48 yuan per vehicle-km. As per the relationship between railway freight price and cost in reference [31], the railway transportation cost is about 82.5% of the freight price. Similarly, according to the relationship between highway freight rate and cost in reference [32], the highway transportation cost is about 94% of the freight rate. To sum up, the fixed and variable costs of various transportation services are shown in Table 3.
Assume that the transfer time between railway and highway is 24 hours per shipment [33] and the transfer cost is taken as 500 yuan per vehicle, where 300 is the loading and unloading cost according to the provisions on loading and unloading expenses in the charging methods [34], plus another 200 of the costs of pick-up, delivery, and marshaling. The transfer risk coefficient is 1. The transfer time, cost, and risk between roads are not considered. The transfer time between railway services is assumed to be 8 hours per shipment, the cost is calculated as 200 yuan per vehicle, and the transfer risk coefficient is 0.5.
The expected risk theory is used to measure the transportation and transfer risk, which mainly includes accident probability and accident consequence [34]. According to reference [35], the accident probabilities of highway and railway transportation are namely and , while the transfer accident probability between railway and highway services is . The accident consequence of transportation service is , while the accident consequences of transfer service is .
5.3. Calculation Results
We test the effectiveness and efficiency of the proposed approach with the above large-scale realistic case. The augmented ε-constraint algorithm is coded in MATLAB 8.0. The underlying single-objective optimization problems are solved by invoking CPLEX 12.5. All computations are executed on a personal computer with Intel Core i5-2400 3.10 GHz CPU, 8.00 GB RAM, and Windows 7-64 bits operating system.
The summary results of 3 representative nondominated solutions for the test case are shown in Table 4. For simplicity, we only provide the cost, risk, total service time, and computation time results for each nondominated solution. The operation plan of each service under the 3 nondominated solutions is shown in Table 4. Solutions 1 and 3 represent the preference of total cost objective and total risk objective, respectively, while solution 2 represents the trade-off between the two objectives. Tables 4 and 5 show that if the decision-maker prefers to minimize the total cost, it is necessary to carry out fewer transfers. At the same time, it is necessary to minimize the transportation cost and reduce the fixed cost of services, but it may lead to an increase in total transportation risk by selecting some services with less distance but greater risk. If the decision-maker prefers to minimize the total risk, the total risk may be significantly reduced due to the use of low-risk transfer nodes and low-risk railway services, but the transfer service cost and fixed cost may be increased due to the increasing use of railway transportation services. In addition, it can be seen from Table 4 that if the decision-maker prefers to minimize the total cost, the total service time of goods may be reduced due to the increasing use of highway services and the avoidance of transfer services. In the case of minimum total risk, the increase in transfer time, as well as the total service time, is caused by the increase in transfer services.
5.4. Comparison of Algorithms
We compare the augmented ε-constraint algorithm with the weighted-sum algorithm widely used in existing works in terms of the quality of nondominated solutions and computation time. Comparison results of the two multiobjective algorithms to find 6 and 11 bounded nondominated solutions are presented in Figure 5, where the horizontal axis and vertical axis represent the total cost and the total risk, respectively. WSA represents the weighted-sum algorithm, and AECA represents the augmented ε-constraint algorithm.
In Figure 5, all solutions are nondominated solutions, and both algorithms are good at searching extreme nondominated points. Nevertheless, Figure 5 clearly shows that the WSA algorithm does not obtain uniformly distributed nondominated solutions by using uniformly distributed weights, while the AECA algorithm could obtain more distinct and uniformly distributed approximate nondominated frontiers, especially when we pay more attention to the objective of minimizing the total cost, which is more suitable with the characteristics of our problem.
In addition, in terms of the solution speed, the calculation time of the two algorithms for solving each solution is shown in Table 6. It can be seen from Table 6 that if we only focus on the risk objective, the solution time of the two algorithms is significantly reduced. However, the time reduction degree of WSA is more significant. The average computation time of WAS and AECA to find 11 bounded nondominated solutions is 2262.8 and 5425.9 seconds, respectively, which reflects that WSA has a faster computation speed.
To sum up, the two algorithms are both effective for solving the railroad intermodal transportation service network design problem considering hazmat and ordinary materials. Since the background of the problem is to determine the railroad intermodal transportation route and the operation plan of various transportation services within a month from the perspective of the railway department, this problem is from the tactical decision-making level which does not have strict requirements on the calculation time. The average solution time of the AECA algorithm is less than two hours, which is fast enough to meet the requirements of decision-makers. At the same time, the decision-makers tend to obtain more distinct representative nondominated frontiers. Thus, it can be concluded that AECE is more suitable for solving the service network design problem of railroad intermodal transportation.
5.5. Sensitivity Analysis
In this section, the sensitivity analysis method is used to evaluate the effects of the delivery time limit parameter and the transfer time parameter on the objectives of the original problem. We first analyze the influence of on the objectives. Set the benchmark of the delivery time limit to be 120 hours, and decrease or increase it by 6 hours per step to optimize the total cost and total risk objectives, respectively. The sensitivity analysis results of separately minimizing the total cost and the total risk are drawn in Figure 6.
(a)
(b)
It can be seen from Figure 6 that the parameter has a significant impact on both the total cost and total risk objectives. The total cost under the objective of minimizing total cost shows a monotonous decreasing trend with the increase of . When the change range is between −6 hours and 30 hours, the change of could not bring the change in total cost. The total risk under the objective of minimizing total cost fluctuates and decreases with the increase of . When increases by 24 hours, the total risk will no longer change. The total cost and total risk under the objective of minimizing total risk show a monotonic decreasing trend with the increase of . When the change range is between −6 hours and 30 hours, the change of could not bring the change in total cost. When increases by 12 hours, the total risk will no longer change.
Now we analyze the influence of the transfer time parameter (transfer time between railway and highway) on the calculation results. Increase or decrease in the step of 4 hours to optimize the total cost and the total risk objectives respectively. The sensitivity analysis results under the objectives of minimizing total cost and total risk are shown in Figure 7.
(a)
(b)
It can be seen from Figure 7 that the parameter has a significant impact on both total cost and total risk objective. The total cost and total risk under the objective of minimizing total cost increase with the increase of . When the change range is from −16 hours to 4 hours, the total cost remains the same. After the transfer time increases by 4 hours, the total cost begins to increase rapidly. When the change range is from −16 hours to 0 hours, the change range of total risk is small. When the transfer time increases by 8 hours, the total risk begins to increase rapidly. The total cost and total risk under the objective of minimizing total risk increase monotonically with the increase of . When the transfer time changes from −16 hours to 4 hours, the total cost remains the same. After the transfer time increases by 8 hours, the total cost begins to increase rapidly. When the transfer time is reduced by 4 hours, reducing the transfer time could not reduce the total risk, but increasing the transfer time will bring a rapid increase in the total risk.
To sum up, according to the above laws, if the freight owner of railroad intermodal transportation could appropriately relax the delivery time limit, it can significantly reduce the total cost and total risk. Therefore, the railway department can promote the freight owner to relax this limit when carrying out marketing work. In addition, if the railway department can find a scientific way to reduce the transfer time between railway and highway transportation to a certain extent, it can reduce the total cost and risk of the system.
6. Conclusions
The service network design problem is one of the key problems in the planning and management of railroad intermodal transportation of hazmat. This problem is essentially a multiobjective optimization problem. At present, most of the existing literature only focuses on the research on the multimodal transportation service network design of ordinary materials, and few works of literature have studied the service network design problem of railroad intermodal transportation considering both ordinary materials and hazmat. So far, few works of literature consider multiple types of materials at the same time when studying the service network design problem of railroad intermodal transportation and establishing a mathematical optimization model that can appropriately describe the actual situation in China. Moreover, few works of literature discuss multiobjective optimization algorithms. Firstly, we systematically analyzed the specific situation of railway freight transportation and train marshaling plan in China and then combs the service network design connotation and optimization process of railroad intermodal transportation in detail. Based on the modeling idea of a service network, we built a multiobjective mixed-integer linear-programming model that minimizes the total cost and total risk. The model included all operation and capacity constraints, especially the capacity constraints of stations and the delivery time limit of each shipment. Secondly, to find an effective nondominated frontier, we customized the classical augmented ε-constraint algorithm. Finally, the calculation results of a large-scale practical case show that an increase of 8.4 million yuan in the total cost could reduce nearly 24541 of the total risk. And the augmented ε-constraint algorithm is more suitable for solving our problem. It can not only ensure the uniform distribution of nondominated solutions but also effectively avoid redundant calculations. In addition, the sensitivity analysis shows that appropriately relaxing the delivery time limit of each shipment can reduce the total cost (by 0.16 × 108 probably) and risk (by 0.3 × 104 probably) of the system. And compressing the transfer time between railway and highway can reduce the total cost and risk of the system to a certain extent.
Future research can be expanded in two aspects. One is to design more effective algorithms to speed up the solution of multiobjective optimization problems. The second is to expand the research method of this paper to other multimodal transportation fields, such as water-road intermodal transportation and rail-water intermodal transportation.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This research was funded by the National Natural Science Foundation of China (no. 61803051), the Doctor Foundation Project of Chengdu Technological University (2019RC022), Project Plan Project of China Logistics Society, China Federation of Logistics and Purchasing (2020CSLKT3-221), Open Project Fund of Technology and Equipment Transit Operation and Maintenance Key Laboratory of Sichuan Province (2019YW003), and Key Foundation on Philosophical Social Sciences-Sichuan UAV Industry Development and Research Center (SCUAV20-A001).