1. Introduction
As an effective alternative to the road transportation that is the most typical representative of traditional monomodal transportation, road–rail multimodal transportation combines the good mobility of road transportation in short- or medium-distance pickup and delivery and the mass capacity and outstanding reliability of rail transportation in long-distance transportation, which can provide the customers with seamless door-to-door services [
1,
2]. Furthermore, by using the standard containers to carry the goods, the mechanization of transportation can be significantly improved, which leads to a remarkable enhancement of the transportation efficiency [
3]. Nierat [
4,
5] points out that transportation distance is an important factor that influences the economic competition between road–rail multimodal transportation and the monomodal road transportation. As for the long-haul transportation setting, empirical studies conducted by Janic [
6], Bookbinder and Fox [
7], Du et al. [
8] and Göçmen and Erol [
9] show that multimodal transportation is more competitive than the monomodal transportation in improving not only the transportation economy but also efficiency and sustainability, especially in an extensive global commodity circulation network. The freight industry in Europe is mainly enabled by road transportation [
6]. In Northern America (including the United States, Canada, and Mexico) and Asia, the focus is mainly shifting transportation to more cost-efficient and environmental-friendly modes. There, multimodal transportation is more economically and environmentally viable and shows better competition than road transportation, and thereby gets wide utilization in these areas [
6,
7,
8]. Especially in China, since the 1980s, road–rail multimodal transportation was widely applied in the container transportation and now plays an important role in China’s inland transportation system for both international and domestic business. The transportation industry attached great importance to promote the road–rail multimodal transportation in practice, which motivates its optimization research. Among relative optimization problems (e.g., network design problem [
10,
11,
12] and location problem [
13,
14]), the road–rail multimodal routing problem, an operational-level planning problem, drew attention from both researchers and practitioners and has become one of the forefronts in the transportation optimization field [
15,
16].
The road–rail multimodal routing problem is related to the optimal utilization of the existing transportation facilities and equipment in the multimodal transportation network. It aims at planning the best origin-to-destination routes to move the containers through the road–rail multimodal transportation network based on the transportation orders proposed by customers [
3,
17,
18]. As a service-oriented planning, the road–rail multimodal routing always take satisfying the customer demand by providing the best transportation scheme as its goal.
First of all, the customers would like to pay least expenditure to accomplish their transportation orders, so that they cut down on the logistics cost to reduce the entire product cost in order to make more profit [
19]. Consequently, minimizing the customers’ expenditure used to accomplish their transportation orders is always set as an essential optimization objective of the multimodal routing modelling, regardless of single-objective modelling (e.g., Sun and Lang [
17], Sun et al. [
3], Ayar and Yaman [
20], and Hrušovský et al. [
21]) or multi-objective modelling (e.g., Chang [
19], Xiong and Wang [
22], Xiong and Dong [
23], and Sun and Chen [
24]). A state-of-the-art representation that can be used to formulate the expenditure paid to accomplish the transportation orders is “generalized costs” [
25]. Generalized costs covers all the expenditure created in the whole transportation process, including the transportation costs created in the pre-haul, long-haul and end-haul processes, handling costs (mainly refer to the loading/unloading operation costs) and inventory costs created in the transshipment process [
20,
26]. Moreover, generalized costs also contains penalty costs when there are soft due dates and on-time delivery is considered [
21] and emission costs (e.g., carbon emission costs) when sustainability issues are considered [
17]. Therefore in this study, minimizing the generalized costs for accomplishing multiple transportation orders will be the optimization objective of our road–rail multimodal routing model.
Secondly, besides transportation economy, customers also show considerable concern if their transportation orders can be accomplished punctually, especially when the ‘just-in-time’ production strategy gets more and more popular and is employed by various industries [
27,
28]. As a result, improving the transportation efficiency is another important goal demanded by customers’ due dates of accomplishing the transportation order shows the customer requirement on transportation efficiency. Majority of the existing literature formulates the due dates as time points and constructs a hard constraint that the accomplishment of a transportation order should not be later than its due date [
3,
17,
26,
29,
30,
31,
32]. Zhang et al. [
33] formulate the due dates as time windows and also propose a hard constraint that the accomplishment of a transportation order must falls into its time window. There are also studies that consider the due dates represented by time points are soft and could be violated to a certain degree [
21,
34]. To make the setting of the due date better match the customer requirement on transportation efficiency, we consider a soft time window more suitable than the above forms, because (1) compared with time point, it can better describe the customer requirement on on-time transportation that the transportation orders should be accomplished neither too early nor too late, and (2) compared with hard due date, it can reflect the fact that customer can tolerate the early or delayed accomplishment of a transportation order with some extent. In this study, we use a positive parameter named penalty costs per TEU per hour to try to avoid the early or delayed accomplishment of all the transportation orders. The entire penalty costs of all the transportation orders are integrated into the generalized costs. The minimization of the generalized costs helps to lower the penalty costs, which further helps to optimize the timeliness of accomplishing the transportation orders. The setting of penalty costs has already been employed by Hrušovský [
21] and Zhao et al. [
34] in improving the transportation efficiency of the multimodal routing that uses time points to represent the due dates. However, these two studies only use penalty costs to avoid delayed accomplishment.
Last but not least, growing attention is paid to the reliability of transportation plans that is influenced by uncertainty [
35]. The uncertainty can disrupt the transportation process, which will influence whether the transportation orders can be accomplished successfully [
3]. Thus, improving the reliability of road–rail multimodal routing by considering uncertainty is vitally necessary. The uncertain source from the side of customers is their demands [
36]. The road–rail multimodal routing decision making is an advance work that should be done earlier than the actual transportation starts [
18,
37]. During the decision-making process, the rapidly changed market makes customers difficult to determine their demands ahead of time, and the lack of effective information communication between customers and planners will also lead to the imprecision of demand data [
38]. The customers can define their demands only when the actual transportation starts [
35]. Therefore, demand uncertainty exists in the road–rail multimodal routing decision-making process.
Majority of the current studies on the multimodal routing neglects the demand uncertainty and use deterministic numbers to represent the demands [
2,
3,
17,
19,
20,
21,
22,
23,
24,
26,
29,
30,
31,
32,
34]. When using deterministic data to estimate the uncertain parameters, underestimation and overestimation might exist [
18]. The multimodal transportation network is capacitated [
18]. Underestimation on the demand might result in the infeasibility of the planned road–rail multimodal route since the actual demand might exceed the capacities of some transportation services on the route. Overestimation might mean that the actual best road–rail multimodal route is identified to be infeasible since the estimated demand exceeds the capacities of some transportation services on the actual best route, while the actual demand satisfies the capacity constraint. When the decision makers need to serve multiple transportation orders, underestimation and overestimation of the demands might exist simultaneously, which makes the situation more complicated. Consequently, using deterministic demands in the road–rail multimodal routing problem is clearly infeasible. The demand uncertainty hence should be integrated into the problem in order to improve the transportation reliability.
Demand uncertainty can be formulated by fuzzy programming and stochastic programming. Currently, there are limited studies on the road–rail multimodal routing problem that consider the demand uncertainty. Existing literature addresses the demand uncertainty by fuzzy programming. Fazayeli et al. [
36], Sun et al. [
37], and Yu et al. [
39] utilize triangular fuzzy numbers to represent the uncertain demands when optimizing the multimodal (location-) routing problem and employ fuzzy possibility measure to model the objective and constraints with fuzzy demands. To the best of our knowledge, there is no literature that adopts stochastic programming to deal with the demand uncertainty in the multimodal routing problem. However, stochastic programming gains wide application in the vehicle routing problem with demand uncertainty, an optimization problem that also belongs to the operational level of transportation planning [
40]. Various studies on the vehicle routing problem with stochastic demands can be found, e.g., Gaur et al. [
41], Gutierrez et al. [
42], Mendoza et al. [
43], and Bianchi et al. [
44].
In most cases, are not enough data for decision makers to fit the probability distributions of demands [
10,
45,
46], which reduces the feasibility of stochastic programming in dealing with the uncertainty. However, it is easier for decision makers to give their own estimations on the uncertain parameter. The estimation might be pessimistic, optimistic, and moderately optimistic [
47]. Therefore, it is worthwhile to use fuzzy logic to address the uncertainty according to the expert experiences and opinions. There are three forms of fuzzy numbers that are most commonly used in the fuzzy programming model, i.e., interval numbers, triangular fuzzy numbers and trapezoidal fuzzy numbers [
48]. Compared with triangular fuzzy numbers that have already been used in modeling the road–rail multimodal routing problem with demand uncertainty [
36,
37,
39], trapezoidal fuzzy numbers is more flexible since it allows that different decision makers hold different opinions on the most likely values of the demand and hence better match the practical decision-making situation [
18]. Consequently, in this study, we select trapezoidal fuzzy numbers to model the uncertain demand.
Additionally, the existing studies [
36,
37,
39] use fuzzy possibility measure to build the fuzzy chance constraint(s). As stressed by Zarandi et al. [
45] and Zheng and Liu [
46], fuzzy possibility measure is not self-dual, i.e., a fuzzy event might fail even its possibility is 1. However, fuzzy credibility measure is self-dual, i.e., a fuzzy event must hold when its credibility is 1 and must fail when its credibility is 0. Therefore, fuzzy credibility measure is more suitable to build fuzzy chance constraint(s) and will be applied in this study.
Above all, as for the road–rail multimodal routing problem, there is still research potential to improve its efficiency and reliability by considering soft due date time windows and demand uncertainty, respectively. Furthermore, as demonstrated in various vehicle routing studies [
45,
46,
49], the reliability of routing related to demand uncertainty or travel time uncertainty influences its economy and efficiency. Therefore, it is meaningful to explore the effect of demand uncertainty on the capacitated road–rail multimodal routing problem with time windows, which will be the main contribution of this study.
The remaining sections of this study are organized as follows. In
Section 2, we formulate demand uncertainty by trapezoidal fuzzy numbers. In
Section 3, we describe a multimodal routing scenario with a specific network topology and a schedule-constrained transportation process. In
Section 4, a FMINLP model is established for the capacitated road–rail multimodal routing problem with fuzzy demands and soft due date time windows under the given scenario, in which the objective is to minimize the generalized costs for accomplishing multiple transportation orders. In
Section 5, we present a fuzzy programming method to address the fuzzy problem by using fuzzy expected value model and fuzzy chance constraint based on fuzzy credibility measure to separately deal with the fuzzy objective and fuzzy constraint. Linearization technique is then used in the same section to generate an equivalent MILP model so that the problem can be solved by standard mathematical programming software. In
Section 6, a numerical case is given to demonstrate the feasibility of the proposed method. We quantify the effect of demand uncertainty on the economy and efficiency of the road–rail multimodal routing by sensitivity analysis, and proposing a framework based on fuzzy simulation to help decision makers to make effective tradeoff among transportation economy, efficiency, and reliability. Finally, the conclusions of this study are drawn in
Section 7.
2. Modeling Demand Fuzziness
Considering the infeasibility of formulating stochastic demand that has been stated in
Section 1, in this study, we employ fuzzy set theory to model demand uncertainty by using trapezoidal fuzzy numbers. Trapezoidal fuzzy numbers use four prominent points to describe the imprecise parameters, which can be seen in
Figure 1 [
45]. For a trapezoidal fuzzy number
that measures a fuzzy demand [
50], the definition of its prominent points are as follows.
(1) The value is the most pessimistic estimation, which is unlikely in practice but still slightly possible in the real world if the case turns out badly.
(2) The interval is the most likely estimation and corresponds to the most realistic case. It matches the fact that different decision makers might hold different opinions on the most likely values of a demand.
(3) The value is the most optimistic estimation, which is as unlikely as in the real world but might be possible if the case turns out well.
The membership function of trapezoidal fuzzy number
is as Equation (1) [
18].
The α-cut level of trapezoidal fuzzy number
is
where
. As illustrated by
Figure 1,
is the confidence interval of
at α level where
and
are the lower bound and upper bound of the interval, respectively.
6. Computational Experiment
6.1. Numerical Case
In this section, we present a numerical case to demonstrate the feasibility of the proposed method and further discuss the effect of demand uncertainty on the capacitated road–rail multimodal routing problem with time windows. The multimodal transportation network in the case is shown as
Figure 5.
The schedules of all the rail services in the multimodal transportation network are given in
Table 2. The numbers listed along the road transportation arcs are their travel times in hour, transportation costs in CNY per TEU and transportation capacities in TEU, successively.
The loading/unloading operation costs per TEU of rail service and road service are separately 195 CNY and 20 CNY. The inventory costs per TEU per hour are 3 CNY. The penalty costs per TEU per hour are 50 CNY. It should be noted that the unit penalty costs is set by the decision makers and reflect relative importance of transportation efficiency in the multimodal routing. In practical decision making, multiple attribute decision-making methods, e.g., classical analytic hierarchy process (AHP) method, can be used to help decision makers to set the value in a more precise way [
10]. The transportation orders that need to be accomplished in the numerical case is presented in
Table 3. Note that all the numbers regarding release time and due date time window are all discretized.
6.2. Illustration of the Best Road–Rail Multimodal Routes
We use the standard Branch-and-Bound Algorithm to solve the problem formulated by the MILP model. The algorithm is run by Lingo 12 developed by LINDO Systems Inc., Chicago, IL, USA [
66]. All the computation is carried out on a ThinkPad Laptop with Intel Core i5-5200U 2.20 GHz CPU 8 GB RAM. When the confidence level
is set to 0.9 by the decision makers, the best road–rail multimodal routes are planned as shown in
Table 4.
6.3. Sensivity Analysis of the Capacitated Road–Rail Multimodal Routing with Respect to the Confidence Level
In the fuzzy chance constraint, the confidence level is set by decision makers manually, which might influence the optimization results [
3,
18,
67]. Therefore, it is necessary to explore whether and how demand uncertainty influences the capacitated road–rail multimodal routing via confidence level
. In this study, we use sensitivity analysis to test if the routing results change with the variation of confidence level
. We increase confidence level
from 0.1 to 1.0 with a step size of 0.1 and gain the best road–rail multimodal routes and corresponding minimal generalized costs, which can be seen in
Figure 6.
As we can see from
Figure 6, following conclusions can be drawn for the case presented in
Section 6.1:
(1) The capacitated road–rail multimodal routing is sensitive to the confidence level in the fuzzy chance constraint regarding fuzzy demand. The sensitivity is significant when the confidence level exceeds 0.5.
(2) Increasing confidence level will result in the increase of the minimal generalized costs corresponding to the best road–rail multimodal routes, and the increase is stepwise.
(3) It is impossible for decision makers to plan the best road–rail multimodal routes that can achieve the best transportation economy and best transportation reliability simultaneously.
We also calculate the total hours that lead to penalty costs under different confidence level, which can be seen in
Figure 7. As we can see from
Figure 7, as for the case presented in
Section 6.1, the confidence level also influence the transportation efficiency of the capacitated road–rail multimodal routing.
6.4. Fuzzy Simulation to Determine the Best Confidence Level
As shown in
Section 6.3, the confidence level has significant effect on the capacitated multimodal routing with soft due date time windows. Therefore, the decision makers have to decide which confidence level is optimal if they would like to gain a crisp solution to the problem [
63].
In the practical decision-making process, the decision makers prefer reliable plans on the road–rail multimodal routes, so that all the transportation orders can be successfully accomplished. Since the transportation efficiency is modeled by penalty costs that are caused by violation of the due date time windows, the generalized costs can reflect a combination of transportation economy and transportation efficiency. Then under acceptable reliability, the decision makers prefer plans with lower generalized costs.
The actual transportation situation should be known to test if the planned road–rail multimodal routes are reliable and if the planned road–rail multimodal routes can lower the generalized costs for accomplishing the transportation orders. However, as claimed in
Section 1, the actual demands of all the transportation orders cannot be known exactly during the decision-making process. Still we can simulate the actual transportation situation by using fuzzy simulation [
3,
18,
37].
Fuzzy simulation can randomly generate the actual demand of each transportation order by using the fuzzy membership function of the trapezoidal fuzzy numbers expressed by Equation (1). To better simulate the actual transportation situation, fuzzy simulation should be implemented several times (in this study, the times is set to 50) and we can do some statistical analysis on the planned road–rail multimodal routes under different confidence levels. The procedure of the fuzzy simulation is as
Figure 8. The fuzzy simulation results are given in
Appendix A.
In the fuzzy simulation ( = 1, 2, …, 50), the simulated actual demand of transportation order is . The fuzzy constraint Equation (8) can be transformed into a deterministic constraint as Equation (34).
Using these simulated actual demands, we can first analyze the reliability of the planned best road–rail multimodal routes by testing if Equation (34) is satisfied. If satisfied, the plan is successful for the
fuzzy simulation, otherwise, failed. The successful ratios of the plans under different confidence levels in the entire 50 fuzzy simulations can quantify the transportation reliability and are indicated by
Figure 9. As we can see from
Figure 9, for the sake of ensuring that the planned road–rail multimodal routes for the presented case in
Section 6.1 are exactly feasible in practice, the confidence level can be 0.6, 0.7, 0.8, 0.9, and 1.0.
Then we should determine under which confidence level, the planned best road–rail multimodal routes are closer to the actual best road–rail multimodal routes. The actual multimodal routes in the fuzzy simulation ( = 1, 2, …, 50) can be simulated by a MILP model whose optimization objective is as Equation (35) and constraints include Equations (6), (7), (10), (14)–(16), (24)–(34).
By comparing the generalized costs of transporting containers with their simulated actual demands along the planned best road–rail multimodal routes under different confidence levels with the generalized costs of the simulated actual best road–rail multimodal routes in the 50 fuzzy simulations, we can identify a crisp plan on the road–rail multimodal routs that better match the actual transportation situation. The comparison is illustrated by
Figure 10 (we only give the first 15 simulations due to the limited space).
Through
Figure 10, we can identify that the planned best road–rail multimodal routes under confidence levels of 0.6 and 0.7 are closest to the actual best situation and can lower the generalized costs compared with confidence levels of 0.8, 0.9, and 1.0. As a result, as for the case presented in
Section 6.1, the best confidence levels are 0.6 and 0.7. The road–rail multimodal routes planned under such confidence levels can be used in practice.
6.5. Comparing the Fuzzy Demands with Deterministic Demands in the Road–Rail Multimodal Routing
In the previous studies, deterministic demands are considered in the road–rail multimodal routing problem. In this study, we use the fuzzy simulation results in
Appendix A to simulate the decision makers’ behavior when estimating the demands in the multimodal routing problem with deterministic demands. As a result, we construct following four deterministic scenarios.
(1) Deterministic scenario 1 is that decision makers prefer to use the average value in the 50 fuzzy simulations to represent the deterministic demand of transportation order .
(2) Deterministic scenario 2 is that decision makers prefer to use the value that emerges more than others in the 50 fuzzy simulations to represent the deterministic demand of transportation order .
(3) Deterministic scenario 3 is that decision makers prefer to use the minimum value in the 50 fuzzy simulations to represent the deterministic demand of transportation order .
(4) Deterministic scenario 3 is that decision makers prefer to use the maximum value in the 50 fuzzy simulations to represent the deterministic demand of transportation order .
As for deterministic scenario ( = 1, 2, 3, 4), the corresponding MILP model is with optimization objective Equation (36) and constraints Equations (6), (7), (10), (14)–(16), (24)–(33), and (37), where is the deterministic demand of transportation order in deterministic scenario . For example, in deterministic scenario 1.
Then we can obtain the planned best road–rail multimodal routes and corresponding generalized costs in the above four deterministic scenarios. Furthermore, we can test their reliability in the fuzzy simulation designed in
Section 6.4. The results are given in
Table 5.
Same to the discussion around
Figure 10, we can also compare the planned best road–rail multimodal routes with fuzzy demands when confidence level is set to 0.6 or 0.7 with the planned best routes with deterministic demands, which can be seen in
Figure 11 (we only give the first 15 simulations due to the limited space). From
Figure 11, based on a statistical viewpoint, we find that:
(1) Compared with deterministic scenarios 1, 2, and 3, the road–rail multimodal routing with fuzzy demands can significantly improve the transportation reliability from 68% to 100% by approximately 47% by slightly increasing the generalized costs by averagely 2.4%.
(2) Compared with deterministic scenario 4, the road–rail multimodal routing with fuzzy demands can reduce the generalized costs by averagely 0.26% and meanwhile maintain same reliability that is 100% feasible.
Therefore, the road–rail multimodal routing with fuzzy demands performs better than the one with deterministic demands. Consequently, we can draw the conclusion that considering the demand uncertainty (fuzziness) can improve the performance of the capacitated road–rail multimodal routing problem with soft due date time windows by making effective tradeoff among transportation economy, efficiency and reliability.
6.6. Managerial Implications
Although the computational experiment is based on a numerical case, its results show important managerial implications for dealing with demand uncertainty when planning the road–rail multimodal routes.
(1) The demand uncertainty exists in practice and has significant effect on the planning. Comparing with deterministic demands, it is more meaningful to consider demand uncertainty when planning the routes, since the demand uncertainty can help decision makers make better tradeoff among different transportation goals.
(2) Increasing the confidence level in the fuzzy chance constraint will increase the generalized costs for accomplishing the targeted transportation orders; however, it will also reduce the risk that the planned road–rail multimodal routes violate the bundle capacity constraint, which is helpful to improve the transportation reliability.
(3) To obtain a crisp plan on the road–rail multimodal routes under demand certainty, it is necessary for decision makers to determine the best confidence value. By using the fuzzy simulation-based comparison, such confidence level can be effectively identified. It should be noted that the best confidence level is sensitive to the setting of the case. It might change if the case changes.
7. Conclusions
In this study, we mainly explore the effect of demand uncertainty on a capacitated road–rail multimodal routing problem with soft due date time windows. A fuzzy programming method is developed to formulate the uncertain demands and model the specific routing problem with demand fuzziness. The case study combining sensitivity analysis and fuzzy simulation indicates that the demand uncertainty influence the planning on the best road–rail multimodal routes and considering fuzzy demands can help decision makers make better tradeoff among transportation economy, efficiency and reliability when optimizing the road–rail multimodal routes. Above all, the main contributions made by this study are three fold.
(1) We improve the modelling of demand uncertainty in the road–rail multimodal routing problem by using trapezoidal fuzzy numbers to represent fuzzy demands and using fuzzy expected value model, fuzzy chance constraint and fuzzy credibility measure to deal with fuzzy routing problem.
(2) We propose soft time windows to formulate the due dates claimed by customers and integrate such formulation into the road–rail multimodal routing problem to improve the transportation efficiency.
(3) We develop a framework that combines sensitivity analysis and fuzzy simulation to quantify the effect of demand uncertainty on a capacitated road–rail multimodal routing problem with soft due date time windows, and reveal some helpful insights to help decision makers better organize the road–rail multimodal transportation.
In this study, we only consider the source of uncertainty from the customers’ side. Actually, the multimodal transportation network itself also possesses uncertainty, such as time uncertainty and demand uncertainty. Formulating multiple sources of uncertainty can further improve the performance of the road–rail multimodal routing compared with the consideration of only one source of uncertainty. In our future work, we will discuss the road–rail multimodal routing problem that contains multiple sources of uncertainty including demand, time and capacity uncertainty.