# Modeling a Carbon-Efficient Road–Rail Intermodal Routing Problem with Soft Time Windows in a Time-Dependent and Fuzzy Environment by Chance-Constrained Programming

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## Abstract

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_{2}emissions is considered by the routing. Soft time windows are incorporated into the routing to optimize the timeliness of the first-mile pickup and last-mile delivery services in intermodal transportation. The routing is further modeled in a time-dependent and fuzzy environment where the average truck speeds of the road depend on the truck departure times and are simultaneously considered fuzzy along with rail capacities. The fuzzy truck speed leads to the fuzziness of three aspects, including speed-dependent CO

_{2}emissions of the road, a timetable-constrained transfer process from road to rail, and delivery time window violation. This study formulates the routing problem under the above considerations and carbon tax regulation as a combination of transportation path planning problem and truck departure time and speed matching problem. A fuzzy nonlinear optimization model is then established for the proposed routing problem. Furthermore, chance-constrained programming with general fuzzy measure is used to conduct the defuzzification of the model to make the problem solvable, and linearization techniques are adopted to linearize the model to enhance the efficiency of problem-solving. Finally, this study presents an empirical case to demonstrate the effectiveness of the designed approach. This case study evaluates the performance of carbon tax regulation by comparing it with multi-objective optimization. It also focuses on sensitivity analysis to discuss the influence of the optimistic–pessimistic parameter and confidence level on the optimization results. Several managerial insights are revealed based on the case study.

## 1. Introduction

_{2}emissions that considerably harm environmental quality and sustainability [1,2]. To achieve energy conservation and environmental protection in the transportation sector, China is promoting a modal shift policy to motivate the change of long-haul goods transportation from road to rail [3]. The modal shift policy aims to increase the proportion of railways in long-haul goods transportation and establish a reasonable transportation structure that can effectively combine road and rail transportation. To implement the policy and achieve efficient and green transportation, developing road–rail intermodal transportation (RRIT) is being attached to great importance by the transportation industry. RRIT uses road transportation that has the advantages of high flexibility and mobility and simplified maintenance for short/medium-distance pickup and delivery activities and adopts rail transportation, which is a cost-effective and energy-saving means for long-haul transportation [4,5]. Coordinating road and rail to realize intermodality can integrate their respective advantages and make RRIT a more excellent competitor to road transportation.

_{2}emissions and other objectives, which can enable the RRIT to reach its full potential as an environmentally friendly means of transportation and help to enhance the environmental goal of the road-to-rail modal shift policy.

_{2}emissions, and its performance should be estimated by comparing it with multi-objective optimization. The timeliness of pickup and delivery services expected by shippers and receivers are represented by soft time windows such that the time window violation is allowed and penalized to improve customer service flexibility that could be beneficial to other objectives. The proposed RRIRP aims to minimize the total costs, including transportation costs, emission costs, and penalty costs. The cost objective reflects a combination of economy, environmental sustainability (carbon efficiency), and timeliness of transportation. The minimization of the cost objective of the RRIRP shows a balance among these considerations.

_{2}emissions of the road mainly depend on speed, the time dependency of truck speeds leads to the time-dependent emission rates of the road [11].

## 2. Literature Review

_{2}emissions. Resat and Turkay [25] also formulated truck speed optimization to avoid traffic congestion, but their work on the optimization of the intermodal transportation network did not take emission reduction into account. In addition, some studies simply focus on speed optimization for intermodal transportation, e.g., Kontovas and Psaraftis [26] and Fan et al. [27].

_{2}emissions and (or) fuel consumptions and better satisfy soft or hard time window constraints. Compared with the VRP with time-dependent speeds, studies on intermodal routing considering such a characteristic are pretty limited. Braekers et al. [31] modeled a time-dependent routing of drayage operations in the service area of intermodal terminals, in which the time-dependent speeds are converted into time-dependent travel times. Although discussed in an intermodal transportation setting, their work is still a VRP.

- (1)
- Although carbon tax regulation has been widely used in carbon-efficient intermodal routing studies, the carbon tax rate is usually a constant in these studies, and the performance and feasibility of carbon tax regulation are rarely estimated in their case analysis.
- (2)
- Moreover, under carbon tax regulation, emission reduction is simply driven by transportation path modification in the current research, which limits the effective improvement of the carbon efficiency of the intermodal routing.
- (3)
- Time dependency and uncertainty of truck speeds are the natural characteristics of the road but are not fully modeled in the current studies. Ignoring the time-dependent truck speeds limits the diversity of approaches that can drive the emission reduction of the routing.
- (4)
- Although capacity uncertainty has been explored, the combination of speed uncertainty and capacity uncertainty that can achieve improved reliability of routing is not considered in the current studies.
- (5)
- The uncertainty of the transfer from road to rail caused by truck speed uncertainty is rarely modeled, which affects the reliability of the intermodal transfer of the routing optimization. Moreover, soft time windows in an uncertain environment resulting from truck speed uncertainty are not well established by existing studies.

- (1)
- A carbon-efficient RRIRP is modeled in a time-dependent and fuzzy environment in which the time-dependent truck speeds of the road enable a truck departure time and speed matching to further strengthen the economy, carbon efficiency, timeliness, and reliability of the optimization.
- (2)
- Under carbon tax regulation, the carbon emission reduction in our model is driven by transportation path modification and truck departure time and speed matching. The feasibility of carbon tax regulation is verified by comparing it with multi-objective optimization in the empirical case study.
- (3)
- Multiple sources of uncertainty are incorporated into the RRIRP, including truck speeds and rail capacities. The induced system uncertainty is comprehensively formulated in the RRIRP, including speed-dependent emission rates of roads, transfer from road to rail, and violation of soft delivery time windows.
- (4)
- A CCP model with general fuzzy measure is employed to deal with the proposed RRIRP and provides decision-makers with the optimum solutions with reference to their attitudes on the objective and constraints.

## 3. Background Information

#### 3.1. Establishing the Transportation System

#### 3.2. Modeling the Time-Dependent and Fuzzy Environment

#### 3.2.1. Modeling of the Time-Dependent and Fuzzy Truck Speed

#### 3.2.2. Modeling of the Induced System Fuzziness Resulting from Fuzzy Truck Speed

#### Modeling of the Speed-Dependent and Fuzzy CO_{2} Emissions of Road

_{2}emissions from roads are significantly influenced by truck speed [23]. Modeling the speed-dependent emissions enables the truck departure time and speed matching problem to consider the improvement of carbon efficiency as its objective. We adopt the methodology for calculating transport emissions and energy consumption (MEET) [60] to determine speed-dependent emission rates. The feasibility of MEET in carbon-efficient transportation planning has already been verified by various research articles, e.g., [6,23,61]. Based on MEET, we have Equation (1) for a deterministic emission rate in kg/TEU/km for heavy-duty trucks loaded with TEU (twenty-foot equivalent unit) containers when departing from the node in the time range $p$:

_{2}emissions of roads. When the road service yields fuzzy truck speed $\left({v}_{1}^{p},{v}_{2}^{p},{v}_{3}^{p},{v}_{4}^{p}\right)$ in time range $p$, the prominent points of its corresponding fuzzy emission rate are based on fuzzy arithmetic operations as in Equation (2):

#### Modeling of the Fuzziness of the Transfer Process from Road to Rail

#### Modeling of the Soft Delivery Time Window in a Fuzzy Environment

- (1)
- When ${t}_{d}^{4}<{b}^{-}$, it is definite that the entire fuzzy delivery accomplishment time violates the lower bound of the time window, and the resulting fuzzy time window violation can be represented by $\left({b}^{-}-{t}_{d}^{4},{b}^{-}{t}_{d}^{3},{b}^{-}{t}_{d}^{2},{b}^{-}-{t}_{d}^{1}\right)$.
- (2)
- When ${t}_{d}^{1}>{b}^{-}$, the fuzzy delivery accomplishment time satisfies the lower bound, and the fuzzy time window violation is $\left(0,0,0,0\right)$.
- (3)
- When the fuzzy delivery accomplishment time is between the above two situations (i.e., ${t}_{d}^{1}\le {b}^{-}$ and ${t}_{d}^{4}\ge {b}^{-}$), part of the fuzzy delivery accomplishment time violates the lower bound, and the corresponding fuzzy time window violation can be formulated as $\stackrel{~}{\eta}=\left(\mathrm{max}\left\{{b}^{-}-{t}_{d}^{4},0\right\},\mathrm{max}\left\{{b}^{-}-{t}_{d}^{3},0\right\},\mathrm{max}\left\{{b}^{-}-{t}_{d}^{2},0\right\},\mathrm{max}\left\{{b}^{-}-{t}_{d}^{1},0\right\}\right)$, which also includes the first two situations.

#### 3.3. Problem Formulation

- (1)
- Enhancing the transfer efficiency between road and rail by reducing the in-transit inventory.
- (2)
- Lowering the CO
_{2}emissions of road services selected for transportation orders to improve the carbon efficiency of the road sector. - (3)
- Improving the timeliness of transportation by minimizing the time window violation of both pickup and delivery services.
- (4)
- Ensuring the smooth transfer from road to rail in a fuzzy environment to achieve reliable transportation.

## 4. Fuzzy Nonlinear Optimization Model

#### 4.1. Optimization Objective

_{2}emission costs under the carbon tax policy in the RRIRP (denoted by ${\stackrel{~}{F}}_{4}$). In this item, the emission rate of a road service in a time range (i.e., ${\stackrel{~}{e}}_{ijsp}$) is a fuzzy parameter since the truck speed of the road service is considered fuzzy. Based on the MEET and fuzzy arithmetic operations, ${\stackrel{~}{e}}_{ijsp}$ is determined by Equations (4) and (5).

#### 4.2. Constraint Set

## 5. Solution Approach

#### 5.1. Chance-Constrained Programming with General Fuzzy Measure

#### 5.1.1. Establishing the Expected Objective

#### 5.1.2. Establishing the Chance Constraints

#### 5.2. Model Linearization

**Proof:**

**Proof:**

## 6. Empirical Case Study

#### 6.1. Case Description

#### 6.2. Case Analysis

#### 6.2.1. Sensitivity of the Optimization Results Concerning the Carbon Tax Rate

_{2}emissions reflect the carbon efficiency (i.e., environmental objective) of the proposed RRIRP. The time window violation degrees determine its timeliness objective. As we can see from Table 5, when the carbon tax rate varies in an actually feasible range of [0.01, 0.10] CNY/kg, the following conclusions can be drawn:

- (1)
- The increase of the carbon tax rate leads to a constant rise of the total costs of the RRIRP. However, the travel costs and loading/unloading operation costs of the RRIRP remain unchanged, which clarifies that the transportation paths for the transportation orders are insensitive to the carbon tax rate.
- (2)
- The increase of the carbon tax rate cannot guarantee lowering the CO
_{2}emissions of the RRIRP. In some cases, increasing the carbon tax rate (e.g., from 0.04 to 0.05 or from 0.06 to 0.07) leads to increased costs and emissions. - (3)
- The decrease (increase) of the CO
_{2}emissions increases (decreases) the time window violation degrees. The carbon efficiency and timeliness of the RRIRP cannot reach their respective optimum simultaneously under a feasible carbon tax rate range. - (4)
- The change of CO
_{2}emissions concerning the carbon tax rate is caused by the truck departure time and speed matching that makes tradeoffs among lowering costs, reducing emissions, and improving timeliness, which is reflected by the change of the in-transit inventory costs (i.e., $E\left[{\stackrel{~}{F}}_{3}\right]$) and the time window violation degrees.

#### 6.2.2. Multi-Objective Optimization Analysis Considering the Minimization of Emissions

_{2}emissions as an independent objective can be an alternative for the intermodal operator to balance the economy and carbon efficiency of the RRIRP. In multi-objective optimization, the two objectives are Equations (76) and (77). Then, we adopt the interactive fuzzy programming approach with the bounded objective function [37] to obtain the Pareto solutions to the problem.

- (1)
- When the satisfaction degrees of the environmental objective are low (i.e., $0\le \u2206\le 0.6$), the transportation path planning does not contribute to the emission reduction resulting from the increase of $\u2206$ since the unchanged travel costs (see Figure 9) of the RRIRP indicates the transportation paths remain unchanged during the emission reduction. In this case, reducing the CO
_{2}emissions depends on the truck departure time and speed matching reflected by the changing in-transit inventory costs and time window violation degrees (see Figure 10 and Figure 11). The truck departure time and speed matching contributes to reducing CO_{2}emissions by 25.56% when compared with the maximum emissions that are 569,546 kg. - (2)
- In the cases where there are lower satisfaction degrees of the environmental objective, reducing the CO
_{2}emissions makes the time window violations increase (see Figure 11), which means the carbon efficiency and timeliness of the RRIRP are in conflict with each other in this case. - (3)
- When the satisfaction degrees of the environmental objective are high (i.e., $0.6\le \u2206\le 1.0$), the transportation path planning and truck departure time and speed matching both contribute to the emission reduction, which is reflected by the significant change of travel costs, in-transit inventory costs, and time window violation degrees (see Figure 9, Figure 10 and Figure 11). The combination of transportation path planning and truck departure time and speed matching by RRIRP efficiently decreases CO
_{2}emissions by 44.26% compared with the maximum emissions. - (4)
- In the cases where there are higher satisfaction degrees of the environmental objective, the time window violation degrees tend to decrease when the CO
_{2}emissions are reduced, except for the variation of $\u2206$ from 0.9 to 1.0 that increases the violations (see Figure 11).

_{2}emissions of the RRIRP at the expense of scarifying the economy.

#### 6.2.3. Sensitivity of the Optimization Results Concerning $\lambda $ and $\alpha $

- (1)
- For each value of $\lambda $, the increase of $\alpha $ leads to higher total costs that are a combinational reflection of transportation costs, carbon efficiency, and timeliness of the RRIRP since increasing $\alpha $ compresses the feasible solution space of the RRIRP and worsens the solutions found in a smaller space.
- (2)
- Confidence level $\alpha $ reflects the reliability that the optimization results of the RRIRP are applicable in the actual transportation by selecting the rail services with sufficient capacities and arranging a smooth transfer from road to rail for the transportation orders. Figure 12 demonstrates that the reliability of the RRIRP is conflicting with its total costs. Customers should pay more total costs to the intermodal operator for their transportation orders if they are very cautious on the capacity constraint and fixed operation time window constraint and prefer reliable transportation,
- (3)
- With the increase of $\lambda $, the sensitivity of the total costs of the RRIRP concerning $\alpha $ becomes more significant. Especially, if the customers hold cautious attitudes on the constraints and prefer a higher confidence level to ensure reliable transportation of their containers (i.e., $\alpha \ge 0.6$), the increase of $\lambda $ raises the total costs when $\lambda <\alpha $.

_{2}emissions (carbon efficiency), and time window violation degrees (timeliness) in the total costs concerning $\lambda $ and $\alpha $.

- (1)
- With the increase of $\alpha $, the transportation costs of the RRIRP tend to decrease. However, the CO
_{2}emissions and the time window violation degrees tend to increase. The increase of the carbon tax and penalty costs are more significant than the decrease of the transportation costs, which leads to the increase of the total costs illustrated by Figure 12. - (2)
- The optimization of the RRIRP balances the three components included in the total costs that are treated as the RRIRP objective. As $\alpha $ increases to improve reliability, transportation costs are saved at the expense of scarifying the carbon efficiency and timeliness of the RRIRP. In this case, improving the economy and reliability of the RRIRP worsens its timeliness and carbon efficiency.
- (3)
- The change of $\lambda $ fluctuates the transportation costs, CO
_{2}emissions, and time window violation degrees of the RRIRP. As $\lambda $ increases, it is more sensitive to $\alpha $.

- (1)
- The increase of $\alpha $ results in the change of the travel costs, loading/unloading operation costs, and in-transit inventory costs, which shows that the transportation path is modified and the rematch of truck departure times and speeds occurs.
- (2)
- With the increase of $\alpha $, the decrease of the loading/unloading operation costs proves that more transportation orders are accomplished by truck-only transportation where the intermodal transfer that causes loading/unloading operation costs does not exist.
- (3)
- (4)
- However, truck-only transportation can avoid the risk associated with the fuzzy constraints that the capacities are insufficient and disrupt the intermodal transfer. Therefore, truck-only transportation enhances the reliability of the RRIRP and is realized by transportation path planning. Consequently, the RRIT system established in our study (see Figure 1) is more reliable than the hub-and-spoke system.
- (5)
- When high confidence levels are needed (i.e., $\alpha \ge 0.8$), the increasing use of truck-only transportation should reduce the in-transit inventory costs that only exist in the intermodal transfer. However, as indicated by Figure 18, the in-transit inventory costs still increase, which means that truck departure times and speeds are rematched to enable an early arrival of trucks at the intermodal terminals to ensure that the transfer from road to rail can be accomplished smoothly.

_{2}emissions, and time window violation degrees concerning $\lambda $. Furthermore, it is difficult to determine which value of $\lambda $ is the best for RRIRP since it refers to the optimistic–pessimistic attitudes of decision-makers (intermodal operator and customers) that are subjective and might be changed in different decision-making conditions.

## 7. Conclusions

_{2}emissions is driven by the synchronous optimization of the two subproblems in the optimization problem. The truck speed fuzziness and rail capacity fuzziness are simultaneously modeled in the proposed RRIRP to improve the reliability of capacity assignment and intermodal transfer arrangement in the RRIRP to provide customers with reliable transportation in which the induced fuzziness of emission rates of roads, the transfer from road to rail, and the violation of the soft delivery time windows are comprehensively modeled. Based on the above considerations, this study proposes a CCP model with general fuzzy measure to deal with the proposed RRIRP.

- (1)
- When the carbon tax rate is in a reasonable range, carbon tax regulation depends on the truck departure time and speed matching to drive the reduction of CO
_{2}emissions. Furthermore, carbon tax regulation depends on a relatively high tax rate to realize the emission reduction of the RRIRP, which might cause its infeasibility in some cases. - (2)
- Multi-objective optimization provides an effective option for the intermodal operator to balance the economy and carbon efficiency of the RRIRP. It also reveals that lowering the transportation and penalty costs are conflicting with reducing CO
_{2}emissions. The intermodal operator can make effective tradeoffs between them using the Pareto solutions. - (3)
- Transportation path planning dominates the RRIRP when improving carbon efficiency is paid less attention, in which transportation services with lower travel costs are selected to form the transportation paths. However, a significant improvement in carbon efficiency resulting from the requirement for environmental protection requires cooperation between transportation path planning and truck departure time and speed matching.
- (4)
- Customers’ attitudes toward the objective and constraints considerably influence the RRIRP. Especially when customers are cautious about the constraints and prefer reliable transportation, they need to pay more costs for their transportation orders. In this case, the cost objective that combines transportation costs, CO
_{2}emissions, and violation of soft time windows reaches a balanced state where transportation costs are saved at the expense of scarifying the carbon efficiency and timeliness of the RRIRP. - (5)
- The RRIT system indicated by Figure 1 is more reliable than the hub-and-spoke system and should be attached to great importance in practical transportation. Both transportation path planning that motivates the use of truck-only transportation and truck departure time and speed matching that ensures the early arrival of trucks at intermodal terminals contribute to improving the reliability of the RRIRP.
- (6)
- Using the proposed CCP model, the intermodal operator can always find an optimum solution as the transportation scheme to organize the RRIT to meet the customers’ preferences and accomplish the transportation orders with the best service.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Relationship among different fuzzy measures [50].

**Figure 6.**The distributions of $Me\left\{\stackrel{~}{\phi}\le \beta \right\}$ and $Me\left\{\stackrel{~}{\phi}\ge \beta \right\}$ concerning $\beta $.

**Figure 17.**Sensitivity of the loading/unloading operation costs concerning $\lambda $ and $\alpha $.

Sets, Indices, and Parameters of the Transportation Orders | |
---|---|

$K$ | Set of transportation orders served by the intermodal operator. |

$k$ | $\mathrm{Index}\mathrm{of}\mathrm{a}\mathrm{transportation}\mathrm{order},\mathrm{and}k\in K$. |

${q}_{k}$ | $\mathrm{Demand}\mathrm{in}\mathrm{TEU}\mathrm{of}\mathrm{the}\mathrm{containers}\mathrm{of}\mathrm{transportation}\mathrm{order}k$. |

$\left({\tau}_{k}^{-},{\tau}_{k}^{+}\right)$ | $\mathrm{Origin}-\mathrm{to}-\mathrm{destination}\mathrm{pair}\mathrm{of}\mathrm{transportation}\mathrm{order}k$$\mathrm{where}{\tau}_{k}^{-}$$\mathrm{and}{\tau}_{k}^{+}$ are the origin and destination nodes, respectively. |

$\left[{a}_{k}^{-},{a}_{k}^{+}\right]$ | $\mathrm{Soft}\mathrm{pickup}\mathrm{time}\mathrm{window}\mathrm{of}\mathrm{the}\mathrm{shipper}\mathrm{of}\mathrm{transportation}\mathrm{order}k$. |

$\left[{b}_{k}^{-},{b}_{k}^{+}\right]$ | $\mathrm{Soft}\mathrm{delivery}\mathrm{time}\mathrm{window}\mathrm{of}\mathrm{the}\mathrm{receiver}\mathrm{of}\mathrm{transportation}\mathrm{order}k$. |

Sets and indices of the transportation network | |

$\left(N,A,S\right)$ | $\mathrm{A}\mathrm{RRIT}\mathrm{network},\mathrm{where}N$$\mathrm{is}\mathrm{the}\mathrm{node}\mathrm{set},A$$\mathrm{is}\mathrm{the}\mathrm{directed}\mathrm{arc}\mathrm{set},\mathrm{and}S$ is the transportation service set. |

$h,i,\mathrm{a}\mathrm{n}\mathrm{d}j$ | $\mathrm{Indices}\mathrm{of}\mathrm{nodes},\mathrm{and}h,i,\mathrm{a}\mathrm{n}\mathrm{d}j\in N$. |

${N}_{i}^{-}$ | $\mathrm{Set}\mathrm{of}\mathrm{the}\mathrm{predecessor}\mathrm{nodes}\mathrm{to}\mathrm{node}i$$,\mathrm{and}{N}_{i}^{-}\subseteq N$. |

${N}_{i}^{+}$ | $\mathrm{Set}\mathrm{of}\mathrm{the}\mathrm{successor}\mathrm{nodes}\mathrm{to}\mathrm{node}i$$,\mathrm{and}{N}_{i}^{+}\subseteq N$. |

$\left(i,j\right)$ | $\mathrm{A}\mathrm{directed}\mathrm{arc}\mathrm{from}\mathrm{node}i$$\mathrm{to}\mathrm{node}j$$,\mathrm{and}\left(i,j\right)\in A$. |

$r\mathrm{a}\mathrm{n}\mathrm{d}s$ | $\mathrm{Indices}\mathrm{of}\mathrm{transportation}\mathrm{services},\mathrm{and}r\mathrm{a}\mathrm{n}\mathrm{d}s\in S$. |

${S}_{ij}={\mathsf{\Pi}}_{ij}\bigcup {\mathsf{\Gamma}}_{ij}$ | $\mathrm{Set}\mathrm{of}\mathrm{transportation}\mathrm{services}\mathrm{on}\mathrm{arc}\left(i,j\right)$$,\mathrm{and}{\mathsf{\Pi}}_{ij}$$\mathrm{and}{\mathsf{\Gamma}}_{ij}$ are the sets of rail services and road services on the arc, respectively. |

${P}_{ijs}$ | $\mathrm{Set}\mathrm{of}\mathrm{continuous}\mathrm{time}\mathrm{ranges}\mathrm{for}\mathrm{road}\mathrm{service}s$$\mathrm{on}\mathrm{arc}\left(i,j\right)$. |

$p$ | $\mathrm{Index}\mathrm{of}\mathrm{a}\mathrm{time}\mathrm{range},\mathrm{and}p\in {P}_{ijs}$. |

Parameters of the transportation services | |

${d}_{ijs}$ | $\mathrm{Travel}\mathrm{distance}\mathrm{in}\mathrm{km}\mathrm{of}\mathrm{transportation}\mathrm{service}s$$\mathrm{running}\mathrm{from}\mathrm{node}i$$\mathrm{to}\mathrm{node}j$. |

${t}_{i}^{s}$ | $\mathrm{Loading}/\mathrm{unloading}\mathrm{operation}\mathrm{time}\mathrm{in}\mathrm{hr}/\mathrm{TEU}\mathrm{of}\mathrm{transportation}\mathrm{service}s$$\mathrm{at}\mathrm{node}i$. |

$[{l}_{i}^{{s}^{-}},{l}_{i}^{s+}]$ | $\mathrm{Fixed}\mathrm{loading}/\mathrm{unloading}\mathrm{operation}\mathrm{time}\mathrm{window}\mathrm{of}\mathrm{rail}\mathrm{service}s$$\mathrm{at}\mathrm{node}i$. |

${\stackrel{~}{\varpi}}_{ijs}$ | $\mathrm{Trapezoidal}\mathrm{fuzzy}\mathrm{capacity}\mathrm{in}\mathrm{TEU}\mathrm{of}\mathrm{rail}\mathrm{service}s$$\mathrm{on}\mathrm{arc}\left(i,j\right)$$,\mathrm{and}{\stackrel{~}{\varpi}}_{ijs}=\left({\varpi}_{ijs}^{1},{\varpi}_{ijs}^{2},{\varpi}_{ijs}^{3},{\varpi}_{ijs}^{4}\right)$. |

$\left[\left.{o}_{ijs}^{p-},{o}_{ijs}^{p+}\right)\right.$ | $\mathrm{Interval}\mathrm{of}\mathrm{time}\mathrm{range}p$$\mathrm{of}\mathrm{road}\mathrm{service}s$$\mathrm{on}\mathrm{arc}\left(i,j\right)$. |

${\stackrel{~}{v}}_{ijsp}$ | $\mathrm{Trapezoidal}\mathrm{fuzzy}\mathrm{average}\mathrm{speed}\mathrm{in}\mathrm{km}/\mathrm{hr}\mathrm{of}\mathrm{road}\mathrm{service}s$$\mathrm{running}\mathrm{from}\mathrm{node}i$$\mathrm{to}\mathrm{node}j$$\mathrm{when}\mathrm{its}\mathrm{trucks}\mathrm{depart}\mathrm{from}\mathrm{node}i$$\mathrm{in}\mathrm{time}\mathrm{range}p$$,\mathrm{and}{\stackrel{~}{v}}_{ijsp}=\left({v}_{ijsp}^{1},{v}_{ijsp}^{2},{v}_{ijsp}^{3},{v}_{ijsp}^{4}\right)$. |

${e}_{ijs}$ | ${\mathrm{CO}}_{2}\mathrm{emission}\mathrm{rate}\mathrm{in}\mathrm{kg}/\mathrm{TEU}/\mathrm{km}\mathrm{of}\mathrm{rail}\mathrm{service}s$$\mathrm{on}\mathrm{arc}\left(i,j\right)$. |

${\stackrel{~}{e}}_{ijsp}$ | $\mathrm{Speed}-\mathrm{dependent}\mathrm{and}\mathrm{trapezoidal}\mathrm{fuzzy}{\mathrm{CO}}_{2}\mathrm{emission}\mathrm{rate}\mathrm{in}\mathrm{kg}/\mathrm{TEU}/\mathrm{km}\mathrm{of}\mathrm{road}\mathrm{service}s$$\mathrm{on}\mathrm{arc}\left(i,j\right)$$\mathrm{in}\mathrm{time}\mathrm{range}p$$,\mathrm{and}{\stackrel{~}{e}}_{ijsp}=\left({e}_{ijsp}^{1},{e}_{ijsp}^{2},{e}_{ijsp}^{3},{e}_{ijsp}^{4}\right)$. |

Parameters of costs | |

${c}_{\mathrm{r}\mathrm{o}\mathrm{a}\mathrm{d}}^{1}$ | Travel cost rate in CNY (Chinese Yuan)/TEU/km of road services carrying out pickup and delivery services in the intermodal transportation. |

${c}_{\mathrm{r}\mathrm{o}\mathrm{a}\mathrm{d}}^{2}$ | Travel cost rate in CNY/TEU/km of truck-only transportation service. |

${c}_{\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{l}}^{1}\mathrm{a}\mathrm{n}\mathrm{d}{c}_{\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{l}}^{2}$ | $\mathrm{Travel}\mathrm{cos}\mathrm{t}\mathrm{rates}\mathrm{of}\mathrm{rail}\mathrm{services}\mathrm{where}{c}_{\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{l}}^{1}$$\mathrm{is}\mathrm{the}\mathrm{rate}\mathrm{in}\mathrm{CNY}/\mathrm{TEU}\mathrm{and}{c}_{\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{l}}^{2}$$\mathrm{is}\mathrm{the}\mathrm{rate}\mathrm{in}\mathrm{CNY}/\mathrm{TEU}/\mathrm{km}.\mathrm{The}\mathrm{combination}\mathrm{of}{c}_{\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{l}}^{1}$$\mathrm{and}{c}_{\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{l}}^{2}$ determines the travel costs of rail services. |

${c}^{s}$ | $\mathrm{Loading}/\mathrm{unloading}\mathrm{operation}\mathrm{cos}\mathrm{t}\mathrm{rate}\mathrm{in}\mathrm{CNY}/\mathrm{TEU}\mathrm{of}\mathrm{transportation}\mathrm{service}s$. |

${c}_{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{e}}$ | Inventory cost rate in CNY/TEU/hr of intermodal terminals. |

$\pi $ | An inventory period in hr that is free of charge at intermodal terminals. |

${c}_{\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{y}}$ | Penalty cost rate in CNY/TEU/hr for violating the soft pickup or delivery time windows. |

${c}_{\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}$ | Carbon tax rate in CNY/kg. |

Auxiliary parameter and index | |

$\xi $ | A sufficient big number. |

$\theta $ | $\mathrm{Index}\mathrm{of}\mathrm{a}\mathrm{prominent}\mathrm{point}\mathrm{of}\mathrm{a}\mathrm{trapezoidal}\mathrm{fuzzy}\mathrm{number},\mathrm{and}\theta \in \left\{1,2,3,4\right\}.$ |

Deterministic variables | |

${x}_{ijs}^{k}$ | $0\u20131\mathrm{binary}\mathrm{variable}\mathrm{representing}\mathrm{the}\mathrm{use}\mathrm{of}\mathrm{a}\mathrm{specific}\mathrm{transportation}\mathrm{service}\mathrm{by}\mathrm{a}\mathrm{transportation}\mathrm{order}.{x}_{ijs}^{k}=1$$\mathrm{means}\mathrm{that}\mathrm{transportation}\mathrm{service}s$$\mathrm{on}\mathrm{arc}\left(i,j\right)$$\mathrm{is}\mathrm{used}\mathrm{to}\mathrm{move}\mathrm{the}\mathrm{containers}\mathrm{of}\mathrm{transportation}\mathrm{order}k$$;{x}_{ijs}^{k}=0$ otherwise. |

${u}_{ijsp}^{k}$ | $0\u20131\mathrm{binary}\mathrm{variable}\mathrm{representing}\mathrm{the}\mathrm{use}\mathrm{of}\mathrm{a}\mathrm{specific}\mathrm{road}\mathrm{service}\mathrm{in}\mathrm{a}\mathrm{specific}\mathrm{time}\mathrm{range}\mathrm{by}\mathrm{a}\mathrm{transportation}\mathrm{order}.{u}_{ijsp}^{k}=1$$\mathrm{means}\mathrm{that}\mathrm{road}\mathrm{service}$$\mathrm{on}\mathrm{arc}\left(i,j\right)$$\mathrm{in}\mathrm{time}\mathrm{range}p$$\mathrm{is}\mathrm{used}\mathrm{to}\mathrm{move}\mathrm{the}\mathrm{containers}\mathrm{of}\mathrm{transportation}\mathrm{order}$$k$$;{u}_{ijsp}^{k}=0$ otherwise. |

${w}_{k}$ | $\mathrm{Non}-\mathrm{negative}\mathrm{variable}\mathrm{representing}\mathrm{the}\mathrm{pickup}\mathrm{start}\mathrm{time}(\mathrm{i}.\mathrm{e}.,\mathrm{the}\mathrm{time}\mathrm{when}\mathrm{the}\mathrm{containers}\mathrm{start}\mathrm{to}\mathrm{be}\mathrm{loaded})\mathrm{of}\mathrm{transportation}\mathrm{order}k$$\mathrm{at}\mathrm{its}\mathrm{origin}\mathrm{node}{\tau}_{k}^{-}$. |

${g}_{i}^{k}$ | $\mathrm{Non}-\mathrm{negative}\mathrm{variable}\mathrm{representing}\mathrm{the}\mathrm{start}\mathrm{time}\mathrm{of}\mathrm{loading}\mathrm{containers}\mathrm{of}\mathrm{transportation}\mathrm{order}k$$\mathrm{on}\mathrm{trucks}\mathrm{at}\mathrm{node}i$. |

${n}_{ijs}^{k}$ | $\mathrm{Non}-\mathrm{negative}\mathrm{integer}\mathrm{variable}\mathrm{representing}\mathrm{the}\mathrm{day}\mathrm{that}\mathrm{road}\mathrm{service}s$$\mathrm{departs}\mathrm{from}\mathrm{node}i$$\mathrm{when}\mathrm{moving}\mathrm{the}\mathrm{containers}\mathrm{of}\mathrm{transportation}\mathrm{order}k$$\mathrm{on}\mathrm{arc}\left(i,j\right)$. |

${\vartheta}_{ijsk}$ | $\mathrm{Non}-\mathrm{negative}\mathrm{variable}\mathrm{representing}\mathrm{the}\mathrm{charged}\mathrm{in}-\mathrm{transit}\mathrm{inventory}\mathrm{period}\mathrm{in}\mathrm{hr}\mathrm{of}\mathrm{the}\mathrm{containers}\mathrm{of}\mathrm{transportation}\mathrm{order}k$$\mathrm{at}\mathrm{node}i$$\mathrm{before}\mathrm{being}\mathrm{moved}\mathrm{being}\mathrm{moved}\mathrm{from}\mathrm{node}i$$\mathrm{to}\mathrm{node}j$$\mathrm{by}\mathrm{transportation}\mathrm{service}s$. |

${\delta}_{k}$ | $\mathrm{Non}-\mathrm{negative}\mathrm{variable}\mathrm{representing}\mathrm{the}\mathrm{violation}\mathrm{in}\mathrm{hr}\mathrm{of}\mathrm{pickup}\mathrm{start}\mathrm{time}{w}_{k}$$\mathrm{of}\mathrm{transportation}\mathrm{order}k$$\mathrm{regarding}\left[{a}_{k}^{-},{a}_{k}^{+}\right]$. |

Trapezoidal fuzzy variables | |

${\stackrel{~}{y}}_{ik}$ | $\mathrm{Non}-\mathrm{negative}\mathrm{trapezoidal}\mathrm{fuzzy}\mathrm{variable}\mathrm{representing}\mathrm{the}\mathrm{time}\mathrm{when}\mathrm{the}\mathrm{containers}\mathrm{of}\mathrm{transportation}\mathrm{order}k$$\mathrm{arrive}\mathrm{and}\mathrm{get}\mathrm{unloaded}\mathrm{at}\mathrm{node}i$$\mathrm{using}\mathrm{road},\mathrm{and}{\stackrel{~}{y}}_{ik}=\left({y}_{ik}^{1},{y}_{ik}^{2},{y}_{ik}^{3},{y}_{ik}^{4}\right)$. |

${\stackrel{~}{z}}_{ijsk}$ | $\mathrm{Non}-\mathrm{negative}\mathrm{trapezoidal}\mathrm{fuzzy}\mathrm{variable}\mathrm{representing}\mathrm{the}\mathrm{in}-\mathrm{transit}\mathrm{inventory}\mathrm{period}\mathrm{in}\mathrm{hr}\mathrm{of}\mathrm{the}\mathrm{containers}\mathrm{of}\mathrm{transportation}\mathrm{order}k$$\mathrm{at}\mathrm{node}i$$\mathrm{before}\mathrm{being}\mathrm{moved}\mathrm{from}\mathrm{node}i$$\mathrm{to}\mathrm{node}j$$\mathrm{by}\mathrm{rail}\mathrm{service}s$$,\mathrm{and}{\stackrel{~}{z}}_{ijsk}=\left({z}_{ijsk}^{1},{z}_{ijsk}^{2},{z}_{ijsk}^{3},{z}_{ijsk}^{4}\right)$ |

${\stackrel{~}{m}}_{ijsk}$ | $\mathrm{Non}-\mathrm{negative}\mathrm{trapezoidal}\mathrm{fuzzy}\mathrm{variable}\mathrm{representing}\mathrm{the}\mathrm{charged}\mathrm{in}-\mathrm{transit}\mathrm{inventory}\mathrm{period}\mathrm{in}\mathrm{hr}\mathrm{of}\mathrm{the}\mathrm{containers}\mathrm{of}\mathrm{transportation}\mathrm{order}k$$\mathrm{at}\mathrm{node}i$$\mathrm{before}\mathrm{being}\mathrm{moved}\mathrm{from}\mathrm{node}i$$\mathrm{to}\mathrm{node}j$$\mathrm{by}\mathrm{rail}\mathrm{service}s$$,\mathrm{and}{\stackrel{~}{m}}_{ijsk}=\left({m}_{ijsk}^{1},{m}_{ijsk}^{2},{m}_{ijsk}^{3},{m}_{ijsk}^{4}\right)$ |

${\stackrel{~}{\eta}}_{k}$ | $\mathrm{Non}-\mathrm{negative}\mathrm{trapezoidal}\mathrm{fuzzy}\mathrm{variable}\mathrm{representing}\mathrm{the}\mathrm{violation}\mathrm{in}\mathrm{hr}\mathrm{of}\mathrm{the}\mathrm{trapezoidal}\mathrm{fuzzy}\mathrm{delivery}\mathrm{accomplishment}\mathrm{time}\mathrm{of}\mathrm{transportation}\mathrm{order}k$$\mathrm{regarding}{b}_{k}^{-}$$,\mathrm{and}{\stackrel{~}{\eta}}_{k}=\left({\eta}_{k}^{1},{\eta}_{k}^{2},{\eta}_{k}^{3},{\eta}_{k}^{4}\right)$. |

${\stackrel{~}{\mu}}_{k}$ | $\mathrm{Non}-\mathrm{negative}\mathrm{trapezoidal}\mathrm{fuzzy}\mathrm{variable}\mathrm{representing}\mathrm{the}\mathrm{violation}\mathrm{in}\mathrm{hr}\mathrm{of}\mathrm{the}\mathrm{trapezoidal}\mathrm{fuzzy}\mathrm{delivery}\mathrm{accomplishment}\mathrm{time}\mathrm{of}\mathrm{transportation}\mathrm{order}k$$\mathrm{regarding}{b}_{k}^{+}$$,\mathrm{and}{\stackrel{~}{\mu}}_{k}=\left({\mu}_{k}^{1},{\mu}_{k}^{2},{\mu}_{k}^{3},{\mu}_{k}^{4}\right)$. |

Transportation Order No. | Demands (TEU) | Soft Pickup Time Windows | Soft Delivery Time Windows |
---|---|---|---|

1 | 22 | [10, 15] | [95, 105] |

2 | 20 | [20, 28] | [140, 150] |

3 | 15 | [13, 20] | [130, 139] |

4 | 17 | [18, 25] | [85, 94] |

5 | 34 | [28, 35] | [88, 98] |

6 | 15 | [12, 17] | [110, 120] |

7 | 23 | [30, 35] | [135, 141] |

8 | 30 | [36, 41] | [142, 152] |

9 | 33 | [14, 20] | [104, 112] |

10 | 25 | [22, 30] | [131, 140] |

Cost Parameters | Values | Data Sources |
---|---|---|

${c}_{\mathrm{r}\mathrm{o}\mathrm{a}\mathrm{d}}^{1}$ | 6.0 CNY/km/TEU | Ministry of Transport of China and National Development and Reform Commission of China |

${c}_{\mathrm{r}\mathrm{o}\mathrm{a}\mathrm{d}}^{2}$ | 9.256 CNY/km/TEU | |

${c}^{s}$ | 25 CNY/TEU for road | |

195 CNY/TEU for rail | China State Railway Group Company | |

${c}_{\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{l}}^{1}$ | 440 CNY/TEU | |

${c}_{\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{l}}^{2}$ | 3.185 CNY/km/TEU | |

${c}_{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{e}}$ | 3.125 CNY/km/hr | |

${c}_{\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{y}}$ | 5 CNY/TEU/hr | Set by this study |

Time parameters | Values | Data sources |

${t}_{i}^{s}$ | 0.2 CNY/TEU for rail | Resat and Turkay [25] |

0.05 CNY/TEU for road | ||

$\pi $ | 6 hr | Set by this study |

Variables | Integer Variables | Constraints |
---|---|---|

3799 | 1251 | 8343 |

Tax Rates (CNY/t) | Total Costs (CNY) | Emissions (kg) | ${\mathit{F}}_{1}$ (CNY) | ${\mathit{F}}_{2}$ (CNY) | $\mathit{E}\left[{\stackrel{~}{\mathit{F}}}_{3}\right]$ (CNY) | Time Window violation Degrees (TEU·hr) | CPU Time (s) |
---|---|---|---|---|---|---|---|

10 | 2,499,379 | 569,546 | 2,412,655 | 37,220 | 4800 | 7802 | 67 |

20 | 2,505,075 | 569,546 | 2,412,655 | 37,220 | 4800 | 7802 | 101 |

30 | 2,510,770 | 569,546 | 2,412,655 | 37,220 | 4800 | 7802 | 111 |

40 | 2,516,994 | 565,692 | 2,412,655 | 37,220 | 5327 | 7833 | 69 |

50 | 2,522,161 | 569,546 | 2,412,655 | 37,220 | 4800 | 7802 | 298 |

60 | 2,527,741 | 547,012 | 2,412,655 | 37,220 | 5101 | 7989 | 137 |

70 | 2,533,712 | 561,249 | 2,412,655 | 37,220 | 5328 | 7845 | 83 |

80 | 2,538,630 | 534,904 | 2,412,655 | 37,220 | 5627 | 8067 | 73 |

90 | 2,544,777 | 545,388 | 2,412,655 | 37,220 | 5627 | 8038 | 40 |

100 | 2,549,347 | 521,288 | 2,412,655 | 37,220 | 5818 | 8305 | 15 |

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## Share and Cite

**MDPI and ACS Style**

Sun, Y.; Sun, G.; Huang, B.; Ge, J.
Modeling a Carbon-Efficient Road–Rail Intermodal Routing Problem with Soft Time Windows in a Time-Dependent and Fuzzy Environment by Chance-Constrained Programming. *Systems* **2023**, *11*, 403.
https://doi.org/10.3390/systems11080403

**AMA Style**

Sun Y, Sun G, Huang B, Ge J.
Modeling a Carbon-Efficient Road–Rail Intermodal Routing Problem with Soft Time Windows in a Time-Dependent and Fuzzy Environment by Chance-Constrained Programming. *Systems*. 2023; 11(8):403.
https://doi.org/10.3390/systems11080403

**Chicago/Turabian Style**

Sun, Yan, Guohua Sun, Baoliang Huang, and Jie Ge.
2023. "Modeling a Carbon-Efficient Road–Rail Intermodal Routing Problem with Soft Time Windows in a Time-Dependent and Fuzzy Environment by Chance-Constrained Programming" *Systems* 11, no. 8: 403.
https://doi.org/10.3390/systems11080403