Modeling a Green and Reliable Intermodal Routing Problem for Food Grain Transportation Under Carbon Tax and Trading Regulations and Multi-Source Uncertainty

Abstract

This study addresses an intermodal routing problem encountered by an intermodal transportation operator fulfilling the food grain transportation order of an agri-food company. To enhance the environmental sustainability of food logistics, carbon tax and trading regulations have been employed to reduce the carbon emissions associated with transportation. Multi-source uncertainties, including the company’s demand for food grains and various parameters related to the intermodal transportation activities, are modeled via trapezoidal fuzzy numbers to optimize the comprehensive reliability of the solution. This work incorporates wastage reduction by lowering the wastage costs and formulating a wastage threshold constraint in intermodal routing. Accordingly, a fuzzy mixed-integer nonlinear programming model for a green and reliable intermodal routing problem for food grain transportation is proposed. To overcome the model’s insolvability and the difficulty in finding the global optimum solution to a nonlinear optimization model, a two-stage solution method is developed, employing chance-constrained programming and linearization technique to reformulate the initial model. A numerical case study is given to verify the feasibility of the proposed methods. Sensitivity analysis reveals the influence of confidence levels and wastage threshold, providing insights for the agri-food company to balance economics, reliability, and wastage reduction in food grain transportation. The numerical case study also analyzes the feasibility of carbon tax and trading regulations in reducing carbon emissions, concluding that carbon tax regulations consistently achieve greater reductions and are universally feasible. In contrast, the feasibility of carbon trading regulations depends on confidence levels and wastage threshold. The findings of this work could provide strong quantitative support for intermodal transportation operators and agri-food companies seeking to implement sustainable food grain transportation.

1. Introduction

Food grains are essential for human survival and development, making their distribution critical for state and societal security [1]. As a country that is historically dominated by agriculture, China’s food production accounts for nearly 25% of global food production [2]. Its food grain yield is expected to increase to 702 million tons in 2024, with an annual growth rate of 1.15% [3]. However, spatial imbalances in the demand and supply for food grains within China have necessitated initiatives such as China‘s North-to-South food grain transportation project [4]. A high-quality food grain transportation system would play an important role in supporting the food grain distribution and food security [5].

In the context of China‘s North-to-South food grain transportation project, food grain logistics are characterized by long distances and high volumes, as highlighted in

Table 1. Food grain transportation in China.

Types of Food Grains	Grain-Producing Areas	Freight Volumes (10 Thousand Ton)	Transportation Modes	Destinations
Corn	Heilongjiang, Jilin, Liaoning, and Inner Mongolia	14,000	Road, rail, and rail–sea intermodal transportation	North China, Sichuan, Chongqing, Guangdong, Guangxi, and coastal regions
	Huang-Huai Plain	2000	Water, road, and rail transportation	Hunan, Hubei, and Hunan-Jiangxi region
Japonica rice	Northeast China	1500~1600	Rail, and rail–sea intermodal transportation	North China, East China, Southeast China, and Northwest China
	Jiangsu	600~700	Water, road, and rail transportation	Middle and lower Yangtze River reaches, and Jiangsu-Zhejiang region
Non-glutinous rice	Yangtze River basin	1600	Water, road, and rail transportation	Guangdong, Guangxi, Zhejiang, and Fujian

[4]. Similar challenges are also found in food logistics systems in countries such as the U.S. [6], Brazil [7], and Ukraine [8]. Given these circumstances, intermodal transportation is more suitable than unimodal transportation regarding food grain distribution. The combination of rail, road, and water transportation in intermodal transportation, used to convey food grains from their origins to their destinations, can fully exploit the advantages of various transportation modes to improve the economics, timeliness, and environmental sustainability of food logistics. Furthermore, the use of standardized containers for food grain transportation in intermodal systems offers advantages over traditional bag or bulk packaging, such as improved handling efficiency and reduced wastages [5]. Therefore, intermodal transportation has been widely applied for food grain distribution [9,10,11].

Intermodal transportation combines at least two transportation modes and involves various stakeholders, including shippers, receivers, intermodal transportation operators, and transportation service providers (carriers and terminal operators). Therefore, the organization of intermodal transportation presents higher complexity than that of unimodal transportation [12,13]. With regard to food grain transportation, an intermodal transportation operator should represent the customer, e.g., an agri-food company that orders food grains from a grain-producing area, seeking to select suitable transportation service providers to accomplish the transportation of food grains while considering various objectives and constraints; this is a complicated task [14]. As a result, it is an important decision-making to assist the intermodal transportation operator in selecting the transportation services and planning the transportation route for food grains. Therefore, an intermodal routing for food grain transportation would be a useful tool for analysis in order to support real-world transportation organization.

Thus far, transportation planning studies in food grain logistics have mainly focused on supply chain network design problems, e.g., [15,16,17,18], which involve tactical-level planning. A systematic literature review conducted by Taşkıner and Bilgen [19] also found that vehicle routing problems have been explored in the agri-food supply chain. However, less attention has been paid to the intermodal routing problem for food grain transportation, which involves operational-level planning. Wan et al. [5] discussed a green water–rail–road intermodal routing problem for food grain transportation under carbon tax regulation, in which the wastage costs were included in the cost objective, and a genetic algorithm was developed to efficiently find the solution. Li et al. [8] presented an intermodal grain transportation problem in the context of Ukraine, in which the disruption scenarios were considered, and they proposed a mathematical model to find the lowest-cost grain transportation scheme. Maiyar and Thakkar [20] explored a robust and sustainable intermodal transportation planning problem considering wastages and carbon tax in a hub-and-spoke network, built a mixed integer nonlinear programming model, and proposed a modified version of the particle swarm optimization algorithm to solve this problem. Maiyar and Thakkar [21] also studied a sustainable routing problem for food grain shipments in an intermodal hub-and-spoke network considering the hub location level and capacity, food grain wastage threshold, and carbon tax regulation.

Overall, the existing literature on intermodal routing for food grain transportation has considered carbon emission reduction based on carbon tax regulation, seeking to achieve sustainable transportation, and wastage reduction by minimizing wastage costs and employing a wastage threshold. However, research on intermodal transportation planning for food grain transportation is still in its infancy.

A major limitation in existing studies pertains to the use of carbon emission reduction regulations. Reducing the carbon emissions of food grain transportation to improve the environmental sustainability of food logistics has been considered in several relevant studies. These studies have mainly relied on carbon tax regulation to achieve emission reduction, while carbon trading regulation, as another popular type of carbon emission reduction regulation, has not been considered. Carbon trading regulation has been employed in intermodal routing problems for non-food-grain shipments [22,23,24,25,26]. Among these studies, Chen et al. [25] concluded that carbon tax regulation can reduce carbon emissions to a greater extent, while Wang et al. [26] reached the opposite conclusion. Therefore, it is necessary to study the green intermodal routing problem for food grain transportation while employing both carbon tax and trading regulations, compare their performance in reducing carbon emissions, and verifying their feasibility under different conditions. In this way, a feasible form of carbon emission reduction regulation can be identified and implemented to effectively coordinate green food grain transportation.

Another major weakness is related to the reliability of the intermodal routing for food grain transportation. To date, the modeling of uncertainty to achieve reliable intermodal routing for food grain transportation has received insufficient attention in relevant studies. However, process systems engineering has taken optimization under uncertainty as an active research field [27]. As highlighted in various studies [28,29,30], uncertainty is a widely discussed issue in transportation, logistics, and supply chain planning and management. Such uncertainty is caused by the fact that the intermodal routing must be conducted before the beginning of the actual transportation. Moreover, the constantly changing status of the intermodal transportation network and ineffective communication among stakeholders make it impossible to precisely determine the real-time values of the parameters involved in routing [31]. Consequently, the demand for food grains and the parameters (travel speeds, transfer times, capacities, carbon emission factors, and wastage rates) related to intermodal transportation activities cannot be set to deterministic values and can be formulated with fuzzy numbers. The modeling of this uncertainty could improve the reliability of the intermodal routing when applied in the actual food grain transportation; specifically, it could aid in avoiding the risks of the predetermined transportation route not meeting the customer’s requirements regarding timeliness and wastage threshold, exceeding the budget for food grain transportation, and causing interruptions in transportation due to insufficient capacities.

Thus far, a large number of intermodal routing studies on non-food-grain shipments have modeled the fuzziness from the perspective of the demand [32,33], capacity [34], or time [35,36]. To achieve improved reliability, the combination of various sources of uncertainty and the use of fuzzy numbers to model the uncertainty have been discussed in many studies. Some representative studies on this topic are given as follows. In Li et al.’s study on green intermodal routing [37], the fuzziness of the capacities and carbon emission factors of the intermodal transportation activities was modeled using triangular fuzzy numbers. Sun [38] formulated the uncertainty of the both demand for containers and the capacity of rail transportation using trapezoidal fuzzy numbers in an intermodal routing problem with soft time windows. Sun et al. [39] considered the uncertainty of the carbon emission factors and travel times of road transportation and the capacity of rail transportation in a green intermodal routing problem considering the improvements in the pickup and delivery services. They proposed a trapezoidal fuzzy chance-constrained linear programming model. Additionally, Zhang et al. [40] systematically formulated the uncertainty of the travel times, transfer times, and demand in a low-carbon intermodal routing with a soft time window, in which the uncertain parameters were formulated as stochastic numbers whose limitations in the modeling of uncertainty have been explained in various studies [37,38,39]. According to the above, the existing literature has not fully modeled the freight demand of the customer and the travel speeds, transfer times, capacities, carbon emission factors, and wastage rates of the intermodal transportation activities in the transportation planning for either food grains or non-food-grain shipments.

In summary, this work focuses on operational-level planning for the food grain transportation system, i.e., planning the best transportation route based on quantitative techniques to support customers and intermodal transportation operators in achieving sustainable food grain distribution. Considering the high suitability of intermodal transportation in food logistics, this work specifically studies an intermodal routing problem to optimally transport food grains from the origin warehouse to the destination warehouse. To improve the feasibility of intermodal routing optimization for food grain transportation, this work addresses critical gaps in existing research related to carbon emission reduction regulations and uncertainty modeling. The main contributions of this work are as follows:

(1) To overcome the major weakness of the existing relevant studies in the use of carbon emission reduction regulations, this work employs both carbon tax and trading regulations to reduce the carbon emissions of the food grain transportation, in which intermodal transportation activities yield uncertain carbon emission factors. Furthermore, a comparison between the two methods is performed, and their feasibility in reducing carbon emissions is analyzed in a numerical case study.

(2) To address the major limitation of the existing relevant studies in the consideration of uncertainty, and thus to improve the reliability of the intermodal routing for food grain transportation, this work fully models the multi-source uncertainty of the transportation system, including the demand for food grains and the travel speeds, transfer times, capacities, carbon emission factors, and wastage rates of intermodal transportation activities.

Additionally, the wastages during the food grain transportation are reduced by integrating the wastage costs into the objective and using a wastage threshold to control the overall wastage rate of the entire transportation process. As a result, this work proposes a novel green and reliable intermodal routing problem for food grain transportation under carbon tax and trading regulations and multi-source uncertainty. Multiple objectives are comprehensively considered, including lowering costs, ensuring timeliness, reducing wastages, reducing carbon emissions, and improving reliability.

The remaining sections of this work are organized as follows: Section 2 describes the specific routing problem in detail. Section 3 presents a fuzzy optimization model to address the green and reliable intermodal routing problem for food grain transportation. In particular, this study adopts trapezoidal fuzzy numbers, which are more flexible than triangular fuzzy numbers due to the use of an interval instead of a deterministic value to describe the most likely conditions when modeling the uncertainty. A two-stage solution method is then designed in Section 4 to achieve the global optimum solution to the problem. Section 5 demonstrates the proposed methods through a numerical case study, explores the influence of subjectively set parameters on optimization results, and discusses the performance and feasibility of carbon tax and trading regulations. Finally, the conclusions of this work are summarized in Section 6.

2. Problem Definition

2.1. Scenario Description

This work explores an intermodal routing problem for food grain transportation in a large-scale agri-food company. The company orders a batch of food grains from a farm located in a grain-producing area to produce cereal products in response to the market demand. These products are then introduced to the market. A food grain transportation order is thus generated, which is fulfilled by an intermodal transportation operator. The intermodal transportation operator represents the agri-food company, transporting the ordered food grains to the destination warehouse hired by the agri-food company.

The farm in the grain-producing area carries out pre-transportation processing of the food grains, including technical processing, delivering the food grains to the origin warehouse for pre-transportation storage and packaging the food grains to make them suitable for intermodal transportation. Based on the pre-transportation processing, the earliest release time for these food grains from the origin warehouse is determined by the farm. According to the post-transportation production schedule, the agri-food company proposes a due date for the food grain transportation order.

Based on the pre- and post-transportation processing schedules, the intermodal transportation operator needs to plan the pickup time of the food grains at the origin warehouse and select the transportation service providers, i.e., the carries that use rail, road, and water transportation to transport the food grains, as well as terminal operators who use their facilities and equipment to transfer the food grains between the three transportation modes. An intermodal transportation network is first established to connect the origin and destination warehouses. Intermodal routing optimization is then conducted by the intermodal transportation operator. An illustration of the intermodal routing problem for food grain transportation is presented in Figure 1.

2.2. Objective Formulation

When carrying out intermodal routing for food grain transportation, the intermodal transportation operator should seek to fulfill the following objectives so as to provide a high-quality transportation service when completing the food grain transportation order.

(1) Economic objective: Minimizing the total costs of the food grain transportation order is the most important requirement for the agri-food company, which seeks to reduce the total production costs to obtain a greater profit in the market. It is the key factor that influences the intermodal transportation operator’s selection of the transportation service providers [41].

(2) Timeliness objective: The intermodal transportation of food grains should be aligned with their pre- and post-transportation processing schedules, and it should take place between the earliest release time and the due date. Simultaneously, the storage of the food grains at the origin and destination warehouses should be reduced, which would in turn allow the agri-food company to employ the “Just-in-Time” strategy for food production.

(3) Wastage reduction objective: The transportation of food grains usually causes wastages, which not only worsen the quality of the intermodal transportation service but also poses a risk to the quality of the food grains themselves. It can lead to a reduction in their value and an increase in the production costs of the agri-food company [42]. Therefore, intermodal transportation should reduce the wastages by minimizing the wastage costs and controlling the overall wastage rate of the entire transportation process by setting a threshold.

(4) Reliability objective: Advanced intermodal routing optimization should avoid suffering from budget shortage or violating the due date, wastage threshold, and the capacity restriction as this leads to interruptions in transportation. The uncertainty during intermodal routing is thus modeled to enhance its reliability. The sources of uncertainty include the food grain demand of the agri-food company and the travel speeds, transfer times, capacities, carbon emission factors, and wastage rates of the intermodal transportation activities.

(5) Environmental objective: Lowering the carbon emissions within food logistics is a promising approach to achieving a sustainable food industry. When carbon emission reduction regulations are implemented, the carbon emissions are penalized and thus increase the total costs of completing the food grain transportation order. Therefore, reducing carbon emissions is not only a public goal considered by the government but also an important private goal for the agri-food company.

2.3. Multi-Source Uncertainty Modeling

This work adopts non-negative trapezoidal fuzzy numbers to describe the uncertainty of the food grain demand of the agri-food company and that of the travel speeds, transfer times, capacities, carbon emission factors, and wastage rates of the intermodal transportation activities. All possible conditions concerning these parameters in a transportation system with a constantly changing state can be covered by such modeling.

When modeling an uncertain parameter with a non-negative trapezoidal fuzzy number $\tilde{a}=\left(a^{\left(1\right)},a^{\left(2\right)},a^{\left(3\right)},a^{\left(4\right)}\right)$, $a^{\left(1\right)}$ and $a^{\left(4\right)}$ represent the minimum and maximum possible values, respectively, of the uncertain parameter in all cases. The two extreme cases, i.e., $a^{\left(1\right)}$ and $a^{\left(4\right)}$, appear in the real world with an extremely low probability, while the interval $\left[a^{\left(2\right)},a^{\left(3\right)}\right]$ shows the most likely values of the uncertain parameter, which emerge in most cases [43]. This is illustrated by the fuzzy membership degree shown in Figure 2. In this case, $\tilde{a}$ is able to include all the possible conditions that an uncertain parameter might be faced with; in this work, it is necessary to ensure that $a^{\left(4\right)}\ge a^{\left(3\right)}\ge a^{\left(2\right)}\ge a^{\left(1\right)}\ge 0$. Specifically, $\tilde{a}$ is a triangular fuzzy number when $a^{\left(2\right)}$ is equal to $a^{\left(3\right)}$.

When representing the uncertainty in the demand, $\tilde{a}$ reflects how the changing market demand leads to the fluctuations in the food grains quantities required by the agri-food company. Regarding the uncertainty in the travel speeds, transfer times, carbon emission factors, and wastage rates of the intermodal transportation activities, $a^{\left(1\right)}$ indicates that the intermodal transportation activities are being conducted in the best working conditions, while $a^{\left(4\right)}$ indicates that they occur in the worst working conditions. Regarding the uncertainty in the capacities of the intermodal transportation activities, $a^{\left(1\right)}$ refers to the situation in which these activities are performed under capacity insufficiency, while $a^{\left(4\right)}$ indicates the situation in which they are performed under sufficient capacities for the food grain transportation. $\left[a^{\left(2\right)},a^{\left(3\right)}\right]$ reflects the most likely values of these uncertain parameters.

Demand uncertainty enhances the uncertainty regarding the transfer time of the food grains at the transfer nodes. Uncertainty in the transfer time and travel speed results in uncertainty in the delivery time of the foods grains to their destination warehouse; this in turn causes the uncertainty in the storage period of the food grains at the destination warehouse. The variables regarding the delivery and storage are also non-negative trapezoidal fuzzy numbers.

3. Problem Modeling

According to the problem introduced in Section 2, this section presents a fuzzy mixed-integer nonlinear programming model to address the specific routing problem in the context of the food grain transportation.

3.1. Symbol Defination

(1) Sets and indices

$G=\left(\mathsf{\Phi },\mathsf{\Psi },\mathsf{\Omega }\right)$: an intermodal transportation network, where $\mathsf{\Phi }$, $\mathsf{\Psi }$, and $\mathsf{\Omega }$ are the sets of nodes, links, and transportation modes, respectively;

$h,i,j$: node indices, where $h,i,j\in \mathsf{\Phi }$;

$\left(i,j\right)$: link index, i.e., a link from node $i$ to node $j$, where $\left(i,j\right)\in \mathsf{\Psi }$;

$k,m$: transportation mode indices, where $k,m\in \mathsf{\Omega }$;

${\mathsf{\Phi }}^{\to i}$: node $i$’s predecessor node set, where ${\mathsf{\Phi }}^{\to i}\subseteq \mathsf{\Phi }$;

${\mathsf{\Phi }}^{i\to }$: node $i$’s successor node set, where ${\mathsf{\Phi }}^{i\to }\subseteq \mathsf{\Phi }$;

${\mathsf{\Omega }}_i$: transportation mode set that links node $i$, where ${\mathsf{\Omega }}_i\subseteq \mathsf{\Omega }$;

${\mathsf{\Omega }}_{ij}$: transportation mode set on arc $\left(i,j\right)$, where ${\mathsf{\Omega }}_{ij}\subseteq \mathsf{\Omega }$;

$o$: origin warehouse, where $o\in \mathsf{\Phi }$;

$d$: destination warehouse, where $d\in \mathsf{\Phi }$;

$f$: prominent point index for a trapezoidal fuzzy number, where $f\in \left\{1,2,3,4\right\}$;

(2) Parameters of the travel process in the intermodal transportation network

$l_{ijm}$: travel distance in km of transportation mode $m$ on arc $\left(i,j\right)$;

${\tilde{s}}_{ijm}$: trapezoidal fuzzy travel speed in km of transportation mode $m$ on arc $\left(i,j\right)$, where ${\tilde{s}}_{ijm}=\left(s_{ijm}^{\left(1\right)},s_{ijm}^{\left(2\right)},s_{ijm}^{\left(3\right)},s_{ijm}^{\left(4\right)}\right)$;

${\tilde{e}}_{ijm}$: trapezoidal fuzzy carbon emission factor in kg/(TEU·km) of transportation mode $m$ on arc $\left(i,j\right)$, where ${\tilde{e}}_{ijm}=\left(e_{ijm}^{\left(1\right)},e_{ijm}^{\left(2\right)},e_{ijm}^{\left(3\right)},e_{ijm}^{\left(4\right)}\right)$;

${\tilde{g}}_{ijm}$: trapezoidal fuzzy capacity in TEU of transportation mode $m$ on arc $\left(i,j\right)$, where ${\tilde{g}}_{ijm}=\left(g_{ijm}^{\left(1\right)},g_{ijm}^{\left(2\right)},g_{ijm}^{\left(3\right)},g_{ijm}^{\left(4\right)}\right)$;

${\tilde{w}}_{ijm}$: trapezoidal fuzzy wastage rate per km of transportation mode $m$ on arc $\left(i,j\right)$, where ${\tilde{w}}_{ijm}=\left(w_{ijm}^{\left(1\right)},w_{ijm}^{\left(2\right)},w_{ijm}^{\left(3\right)},w_{ijm}^{\left(4\right)}\right)$;

$c_{ijm}$: travel cost rate in CNY/TEU for transportation mode $m$ on arc $\left(i,j\right)$;

$c_{ijm}^{\prime }$: travel cost rate in CNY/(TEU·km) for transportation mode $m$ on arc $\left(i,j\right)$;

(3) Parameters of the transfer process in the intermodal transportation network

${\tilde{t}}_i^{km}$: trapezoidal fuzzy time in h/TEU to transfer food grains from transportation mode $k$ to transportation mode $m$ at transfer node $i$, where ${\tilde{t}}_i^{km}=\left(t_i^{km\left(1\right)},t_i^{km\left(2\right)},t_i^{km\left(3\right)},t_i^{km\left(4\right)}\right)$;

${\tilde{e}}_i^{km}$: trapezoidal fuzzy carbon emission factor in kg/TEU to transfer food grains from transportation mode $k$ to transportation mode $m$ at transfer node $i$, where ${\tilde{e}}_i^{km}=\left(e_i^{km\left(1\right)},e_i^{km\left(2\right)},e_i^{km\left(3\right)},e_i^{km\left(4\right)}\right)$;

${\tilde{g}}_i^{km}$: trapezoidal fuzzy capacity in TEU to transfer food grains from transportation mode $k$ to transportation mode $m$ at transfer node $i$, where ${\tilde{g}}_i^{km}=\left(g_i^{km\left(1\right)},g_i^{km\left(2\right)},g_i^{km\left(3\right)},g_i^{km\left(4\right)}\right)$;

${\tilde{w}}_i^{km}$: trapezoidal fuzzy wastage rate when transferring food grains from transportation mode $k$ to transportation mode $m$ at transfer node $i$, where ${\tilde{w}}_i^{km}=\left(w_i^{km\left(1\right)},w_i^{km\left(2\right)},w_i^{km\left(3\right)},w_i^{km\left(4\right)}\right)$;

$c_i^{km}$: cost rate in CNY/TEU to transfer food grains from transportation mode $k$ to transportation mode $m$ at transfer node $i$;

(4) Parameters of the carbon trading policy

$c_{\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{b}\mathrm{o}\mathrm{n}}$: carbon tax (trading price) rate in CNY/kg;

$\phi$: carbon emission limit in kg in carbon trading regulation, and carbon trading regulation can be converted into carbon tax regulation when $\phi$ is equal to 0;

(5) Parameters of the food grain transportation order

$\tilde{q}$: trapezoidal fuzzy demand in TEU of food grains, where $\tilde{q}=\left(q^{\left(1\right)},q^{\left(2\right)},q^{\left(3\right)},q^{\left(4\right)}\right)$;

$c_{\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}}$: value in CNY/TEU of food grains;

$w^{\mathrm{m}\mathrm{a}\mathrm{x}}$: wastage threshold of food grains in the entire transportation process;

$r^o$: earliest time of releasing food grains at origin destination;

$r^d$: due date of receiving food grains at destination warehouse;

$c_{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{e}}^o$: storage cost rate in CNY/(TEU·h) for origin warehouse;

$c_{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{e}}^d$: storage cost rate in CNY/(TEU·h) for destination warehouse;

(6) Variables

$x_{ijm}$: if food grains are moved by transportation mode $m$ on arc $\left(i,j\right)$, $x_{ijm}=1$; otherwise, $x_{ijm}=0$;

$y_i^{km}$: if food grains are transferred from transportation mode $k$ to transportation mode $m$ at transfer node $i$, $y_i^{km}=1$; otherwise, $y_i^{km}=0$;

$u$: pickup time of food grains at the origin warehouse, where $u\ge 0$;

$\tilde{z}$: trapezoidal fuzzy delivery time of food grains at destination warehouse, where $\tilde{z}=\left(z^{\left(1\right)},z^{\left(2\right)},z^{\left(3\right)},z^{\left(4\right)}\right)$ and $z^{\left(4\right)}\ge z^{\left(3\right)}\ge z^{\left(2\right)}\ge z^{\left(1\right)}\ge 0$;

$p^o$: storage period in h of the food grains at origin warehouse before departure;

${\tilde{p}}^d$: trapezoidal fuzzy storage period in h of food grains at destination warehouse after delivery, where ${\tilde{p}}^d=\left(p^{d\left(1\right)},p^{d\left(2\right)},p^{d\left(3\right)},p^{d\left(4\right)}\right)$ and $p^{d\left(4\right)}\ge p^{d\left(3\right)}\ge p^{d\left(2\right)}\ge p^{d\left(1\right)}\ge 0$.

3.2. Mathematical Model

$$\begin{array}{c}\begin{array}{c}\min {\displaystyle \sum _{\left(i,j\right)\in \mathsf{\Psi }}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_{ij}}}\left(c_{ijm}+c_{ijm}^{\prime }·l_{ijm}\right)·\tilde{q}·x_{ijm}+{\displaystyle \sum _{i\in \mathsf{\Phi }}}{\displaystyle \sum _{k\in {\mathsf{\Omega }}_i}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_i}}{c}_i^{km}·\tilde{q}·y_i^{km}\\ +c_{\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}}·\left({\displaystyle \sum _{\left(i,j\right)\in \mathsf{\Psi }}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_{ij}}}{\tilde{w}}_{ijm}·\tilde{q}·l_{ijm}·x_{ijm}+{\displaystyle \sum _{i\in \mathsf{\Phi }}}{\displaystyle \sum _{k\in {\mathsf{\Omega }}_i}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_i}}{\tilde{w}}_i^{km}·\tilde{q}·y_i^{km}\right)\end{array}\\ \begin{array}{c}+c_{\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{b}\mathrm{o}\mathrm{n}}·\left({\displaystyle \sum _{\left(i,j\right)\in \mathsf{\Psi }}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_{ij}}}{\tilde{e}}_{ijm}·\tilde{q}·l_{ijm}·x_{ijm}+{\displaystyle \sum _{i\in \mathsf{\Phi }}}{\displaystyle \sum _{k\in {\mathsf{\Omega }}_i}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_i}}{\tilde{e}}_i^{km}·\tilde{q}·y_i^{km}-\phi \right)\\ +c_{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{e}}^o·\tilde{q}·p^o+c_{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{e}}^d·\tilde{q}·{\tilde{p}}^d\end{array}\end{array}$$

(1)

such that

$${\displaystyle \sum _{h\in {\mathsf{\Phi }}^{\to i}}}{\displaystyle \sum _{k\in {\mathsf{\Omega }}_{hi}}}{x}_{hik}-{\displaystyle \sum _{j\in {\mathsf{\Phi }}^{i\to }}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_{ij}}}{x}_{ijm}=\left\{\begin{array}{cc}-1& i=o\\ 0& \forall i\in \mathsf{\Phi }\backslash \left\{o,d\right\}\\ 1& i=d\end{array}\right.$$

(2)

$$\begin{array}{cc}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_{ij}}}{x}_{ijm}\le 1& \forall (i,j)\in \mathsf{\Psi }\end{array}$$

(3)

$$\begin{array}{cc}{\displaystyle \sum _{k\in {\mathsf{\Omega }}_i}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_i}}{y}_i^{km}\le 1& \forall i\in \mathsf{\Phi }\backslash \left\{o,d\right\}\end{array}$$

(4)

$$\begin{array}{cc}{\displaystyle \sum _{j\in {\mathsf{\Phi }}^{i\to }}}{x}_{ijm}={\displaystyle \sum _{k\in {\mathsf{\Omega }}_i}}{y}_i^{km}& \begin{array}{cc}\forall i\in \mathsf{\Phi }\backslash \left\{o,d\right\}& \forall m\in {\mathsf{\Omega }}_i\end{array}\end{array}$$

(5)

$$\begin{array}{cc}{\displaystyle \sum _{h\in {\mathsf{\Phi }}^{\to i}}}{x}_{hik}={\displaystyle \sum _{m\in {\mathsf{\Omega }}_i}}{y}_i^{km}& \begin{array}{cc}\forall i\in \mathsf{\Phi }\backslash \left\{o,d\right\}& \forall k\in {\mathsf{\Omega }}_i\end{array}\end{array}$$

(6)

$$u\ge r^o$$

(7)

$$p^o=\max \left\{u-r^o,0\right\}$$

(8)

$$\begin{array}{cc}{z}^{\left(f\right)}=u+{\displaystyle \sum _{\left(i,j\right)\in \mathsf{\Psi }}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_{ij}}}{\displaystyle \frac{l_{ijm}}{s_{ijm}^{\left(5-f\right)}}}·x_{ijm}+{\displaystyle \sum _{i\in \mathsf{\Phi }}}{\displaystyle \sum _{k\in {\mathsf{\Omega }}_i}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_i}}{t}_i^{km\left(f\right)}·q^{\left(f\right)}·y_i^{km}& \forall f\in \left\{1,2,3,4\right\}\end{array}$$

(9)

$$\tilde{z}=\left(z^{\left(1\right)},z^{\left(2\right)},z^{\left(3\right)},z^{\left(4\right)}\right)$$

(10)

$$\begin{array}{cc}{p}^{d\left(f\right)}=\max \left\{r^d-z^{\left(5-f\right)},0\right\}& \forall f\in \left\{1,2,3,4\right\}\end{array}$$

(11)

$${\tilde{p}}^d=\left(p^{d\left(1\right)},p^{d\left(2\right)},p^{d\left(3\right)},p^{d\left(4\right)}\right)$$

(12)

$$\tilde{z}\le r^d$$

(13)

$${\displaystyle \sum _{\left(i,j\right)\in \mathsf{\Psi }}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_{ij}}}{\tilde{w}}_{ijm}·l_{ijm}·x_{ijm}+{\displaystyle \sum _{i\in \mathsf{\Phi }}}{\displaystyle \sum _{k\in {\mathsf{\Omega }}_i}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_i}}{\tilde{w}}_i^{km}·y_i^{km}\le w^{\mathrm{m}\mathrm{a}\mathrm{x}}$$

(14)

$$\begin{array}{cc}{x}_{ijm}·\tilde{q}\le {\tilde{g}}_{ijm}& \begin{array}{cc}\forall (i,j)\in \mathsf{\Psi }& \forall r\in {\mathsf{\Omega }}_{ij}\end{array}\end{array}$$

(15)

$$\begin{array}{cc}{y}_i^{km}·\tilde{q}\le {\tilde{g}}_i^{km}& \begin{array}{cc}\forall i\in \mathsf{\Phi }\backslash \left\{o,d\right\}& \begin{array}{cc}\forall k\in {\mathsf{\Omega }}_i& \forall m\in {\mathsf{\Omega }}_i\end{array}\end{array}\end{array}$$

(16)

$$\begin{array}{ccc}{x}_{ijm}\in \left\{\mathrm{0,1}\right\}& \forall \left(i,j\right)\in \mathsf{\Psi }& \forall m\in {\mathsf{\Omega }}_{ij}\end{array}$$

(17)

$$\begin{array}{ccc}{y}_i^{km}\in \left\{\mathrm{0,1}\right\}& \forall i\in \mathsf{\Phi }\backslash \left\{o,d\right\}& \begin{array}{cc}\forall k\in {\mathsf{\Omega }}_i& \forall m\in {\mathsf{\Omega }}_i\end{array}\end{array}$$

(18)

$$u\ge 0$$

(19)

$$z^{\left(4\right)}\ge z^{\left(3\right)}\ge z^{\left(2\right)}\ge z^{\left(1\right)}\ge 0$$

(20)

$$p^o\ge 0$$

(21)

$$p^{d\left(4\right)}\ge p^{d\left(3\right)}\ge p^{d\left(2\right)}\ge p^{d\left(1\right)}\ge 0$$

(22)

Equation (1) represents the optimization objective and is designed to minimize the total costs of completing the food grain transportation order. The total costs include the travel costs, transfer costs, wastage costs, carbon emission costs under carbon tax regulation ($\phi =0$) or carbon trading regulation ($\phi >0$), and pre- and post-transportation storage costs. Equations (2)–(6) represent the commonly used constraints regulating the node-to-link structure of the origin-to-destination route for a food grain transportation order [37]. Equation (2) is the flow equilibrium constraint. Equations (3) and (4) require that the food grain transportation order is unsplittable. Equations (5) and (6) ensure a smooth connection of the nodes via the transportation modes within the planned food grain transportation route. Equation (7) stipulates that the pickup of the food grain transportation order should be no earlier than its earliest release time at the origin warehouse. Equation (8) determines the pre-transportation storage period of the food grain transportation order using a piecewise linear function. Equation (9) determines the prominent points of the trapezoidal fuzzy delivery time of the food grain transportation order indicated by Equation (10) at the destination warehouse, according to the fuzzy arithmetic operations shown in Appendix A. Equations (11) and (12) show the determination of the post-transportation storage period of the food grain transportation order at the destination warehouse based on the piecewise linear function and fuzzy arithmetic operations. Equation (13) indicates that delivery of the food grain transportation order should not occur later than the due date stipulated by the agri-food company. Equation (14) indicates that the wastages of food grains in the transportation process should not exceed a given threshold set by the agri-food company. Equations (15) and (16) ensure that the capacities of the planned food grain transportation route satisfy the agri-food company’s demand for food grains. Equations (17)–(22) are the variable domain constraints.

4. A Two-Stage Solution Method

This work aims to obtain the global optimum solution to the problem to provide the agri-food company and intermodal transportation operator with quantitative routing decision support on best transportation route design. However, such a solution is not straightforwardly attainable based on the proposed model in Section 3 due to the following reasons:

(1) The proposed model contains fuzziness in both the objective and constraints. As a result, it cannot be solved straightforwardly [44], and it is impossible to obtain the best solution to support the design of food grain transportation route in the intermodal transportation network.

(2) The proposed model has nonlinear constraints, i.e., Equations (8) and (11). It is widely acknowledged that the global optimum solution is not easily attainable for a nonlinear optimization model [45], especially when such a model is used to solve a large-scale optimization problem. Although a nonlinear programming model can be solved by heuristic algorithms (e.g., genetic algorithms [46] and ant colony optimization algorithms [47]), it is difficult to determine whether a global or local optimum solution is obtained by such algorithms.

In this case, removing the fuzziness and nonlinearity is the key to obtain the global optimum solution to the problem. Therefore, it is necessary to first remove the fuzzy information in the proposed model to ensure the solvability of the optimization model. After this defuzzification processing, a model linearization processing should be conducted to finally reformulate a mixed integer linear programming model for the specific routing problem for food grain transportation. Consequently, this work develops a two-stage solution method that integrates the above processes, facilitating the attainment of the global optimum solution to the problem.

4.1. Stage I: Model Defuzzification

First of all, the well-known chance-constrained programming method is employed in this work to eliminate the fuzziness of the model proposed in Section 3.2. This method uses the fuzzy expected value operator to reformulate the fuzzy objective(s), and it then converts the fuzzy constraint(s) into chance constraint(s) [48]. An equivalent chance-constrained programming model is proposed to replace the initial fuzzy programming model for the fuzzy optimization problem.

The fuzzy expected value of a trapezoidal fuzzy number $\tilde{a}=\left(a^{\left(1\right)},a^{\left(2\right)},a^{\left(3\right)},a^{\left(4\right)}\right)$ is as given in Equation (23), where $a^{\left(4\right)}\ge a^{\left(3\right)}\ge a^{\left(2\right)}\ge a^{\left(1\right)}\ge 0$.

$$E\left[\tilde{a}\right]=\sum _{f\in \left\{1,2,3,4\right\}}{\displaystyle \frac{a^{\left(f\right)}}{4}}$$

(23)

The fuzzy expected value operator is linear [49]. Based on the fuzzy arithmetic operations shown in Appendix A, this work reformulates Equation (1) as Equation (24), in which the fuzzy information is removed. Equation (24) indicates that the routing optimization seeks to minimize the expected value of the fuzzy total costs of completing the food grain transportation order using the planned transportation route. Additionally, this equation enables the model to be easily linearized.

$$\begin{array}{c}\begin{array}{c}\min {\displaystyle \sum _{\left(i,j\right)\in \mathsf{\Psi }}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_{ij}}}{\displaystyle \sum _{f\in \left\{1,2,3,4\right\}}}\left(c_{ijm}+c_{ijm}^{\prime }·l_{ijm}\right)·{\displaystyle \frac{q^{\left(f\right)}}{4}}·x_{ijm}+{\displaystyle \sum _{i\in \mathsf{\Phi }}}{\displaystyle \sum _{k\in {\mathsf{\Omega }}_i}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_i}}{\displaystyle \sum _{f\in \left\{1,2,3,4\right\}}}{c}_i^{km}·{\displaystyle \frac{q^{\left(f\right)}}{4}}·y_i^{km}\\ +c_{\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}}·\left({\displaystyle \sum _{\left(i,j\right)\in \mathsf{\Psi }}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_{ij}}}{\displaystyle \sum _{f\in \left\{1,2,3,4\right\}}}{\displaystyle \frac{w_{ijm}^{\left(f\right)}·q^{\left(f\right)}}{4}}·l_{ijm}·x_{ijm}+{\displaystyle \sum _{i\in \mathsf{\Phi }}}{\displaystyle \sum _{k\in {\mathsf{\Omega }}_i}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_i}}{\displaystyle \sum _{f\in \left\{1,2,3,4\right\}}}{\displaystyle \frac{w_i^{km\left(f\right)}·q^{\left(f\right)}}{4}}·y_i^{km}\right)\end{array}\\ \begin{array}{c}+c_{\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{b}\mathrm{o}\mathrm{n}}·\left({\displaystyle \sum _{\left(i,j\right)\in \mathsf{\Psi }}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_{ij}}}{\displaystyle \sum _{f\in \left\{1,2,3,4\right\}}}{\displaystyle \frac{e_{ijm}^{\left(f\right)}·q^{\left(f\right)}}{4}}·l_{ijm}·x_{ijm}+{\displaystyle \sum _{i\in \mathsf{\Phi }}}{\displaystyle \sum _{k\in {\mathsf{\Omega }}_i}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_i}}{\displaystyle \sum _{f\in \left\{1,2,3,4\right\}}}{\displaystyle \frac{e_i^{km\left(f\right)}·q^{\left(f\right)}}{4}}·y_i^{km}-\phi \right)\\ +c_{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{e}}^o·{\displaystyle \sum _{f\in \left\{1,2,3,4\right\}}}{\displaystyle \frac{q^{\left(f\right)}}{4}}·p^o+c_{\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{e}}^d·{\displaystyle \sum _{f\in \left\{1,2,3,4\right\}}}{\displaystyle \frac{q^{\left(f\right)}·p^{d\left(f\right)}}{4}}\end{array}\end{array}$$

(24)

Then, this work converts Equations (13)–(16), which are the fuzzy constraints, into chance constraints. The credibility measure, which can indicate compromised decision-making attitude, is used here to build the chance constraints so as to prevent the routing decision-making from being overly optimistic or pessimistic when using possibility or necessary measures, respectively [50]. Moreover, a fuzzy event can be definitively classified as holding true or failing by setting its credibility to 0 and 1, respectively, indicating the self-dual nature of this measure [51]. Based on the credibility measure, Equations (13)–(16) are rewritten as Equations (18)–(25), respectively.

$$\mathrm{C}\mathrm{r}\left\{\tilde{z}\le r^d\right\}\ge {\lambda }_1$$

(25)

$$\mathrm{C}\mathrm{r}\left\{\sum _{\left(i,j\right)\in \mathsf{\Psi }}\sum _{m\in {\mathsf{\Omega }}_{ij}}{\tilde{w}}_{ijm}·l_{ijm}·x_{ijm}+\sum _{i\in \mathsf{\Phi }}\sum _{k\in {\mathsf{\Omega }}_i}\sum _{m\in {\mathsf{\Omega }}_i}{\tilde{w}}_i^{km}·y_i^{km}\le w^{\mathrm{m}\mathrm{a}\mathrm{x}}\right\}\ge {\lambda }_2$$

(26)

$$\begin{array}{cc}\mathrm{C}\mathrm{r}\left\{x_{ijm}·\tilde{q}\le {\tilde{g}}_{ijm}\right\}\ge {\lambda }_3& \begin{array}{cc}\forall (i,j)\in \mathsf{\Psi }& \forall r\in {\mathsf{\Omega }}_{ij}\end{array}\end{array}$$

(27)

$$\begin{array}{cc}\mathrm{C}\mathrm{r}\left\{y_i^{km}·\tilde{q}\le {\tilde{g}}_i^{km}\right\}\ge {\lambda }_3& \begin{array}{cc}\forall i\in \mathsf{\Phi }\backslash \left\{o,d\right\}& \begin{array}{cc}\forall k\in {\mathsf{\Omega }}_i& \forall m\in {\mathsf{\Omega }}_i\end{array}\end{array}\end{array}$$

(28)

In these chance constraints, $\mathrm{C}\mathrm{r}$ represents the credibility measure. ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$ denote the confidence levels, which belong to the interval [0, 1.0], and they are subjectively determined by the decision-makers. Decision-makers prefer the values of the confidence levels to be within the interval [0.5, 1.0] to prevent the decision from being unreliable. Equations (18)–(25) ensure that the credibility in satisfying a fuzzy constraint is not lower than a given confidence level.

In the above equations, ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$ reflect the degrees of emphasis of the agri-food company on whether the planned food grain transportation route can meet its requirements regarding timeliness and wastages and satisfy its demand for food grains in the actual food grain transportation process, in which the fuzzy parameters’ real-time values are eventually known. Therefore, improving the confidence levels can alleviate the risks that the use of the planned food grain transportation route will lead to the exceedance of the required due date and result in wastage and budget violations, as well as the risk that it will become infeasible due to a capacity shortage. Consequently, the reliability of the planned food grain transportation route can be strengthened by improving the confidence levels.

For a non-negative trapezoidal fuzzy number $\tilde{a}=\left(a^{\left(1\right)},a^{\left(2\right)},a^{\left(3\right)},a^{\left(4\right)}\right)$, a non-negative deterministic number $\delta$, and a confidence level $\lambda$ that belongs to the interval [0.5, 1.0], this work achieves the following deduction, whose proof is given in Appendix B. It should be noted that $\tilde{a}$ refers to the formula with fuzzy parameters and fuzzy variables on the left- or right-hand of a fuzzy inequality constraint, while $\delta$ represents the deterministic formula in the same constraint. For example, $\tilde{a}$ and $\delta$ represent $\tilde{z}$ and $r^d$, respectively, in Equation (25).

$$\mathrm{C}\mathrm{r}\left\{\tilde{a}\le \delta \right\}\ge \lambda \iff 2\left(1-\lambda \right)·a^{\left(3\right)}+\left(2\lambda -1\right)·a^{\left(4\right)}\le \delta$$

(29)

$$\mathrm{C}\mathrm{r}\left\{\tilde{a}\ge \delta \right\}\ge \lambda \iff \left(2\lambda -1\right)·a^{\left(1\right)}+2\left(1-\lambda \right)·a^{\left(2\right)}\ge \delta$$

(30)

Based on Equation (29), Equations (25) and (26) can be written as Equations (31) and (32), respectively.

$$2\left(1-{\lambda }_1\right)·z^{\left(3\right)}+\left(2{\lambda }_1-1\right)·z^{\left(4\right)}\le r^d$$

(31)

$$\begin{array}{c}2\left(1-{\lambda }_2\right)·\left({\displaystyle \sum _{\left(i,j\right)\in \mathsf{\Psi }}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_{ij}}}{w}_{ijm}^{\left(3\right)}·l_{ijm}·x_{ijm}+{\displaystyle \sum _{i\in \mathsf{\Phi }}}{\displaystyle \sum _{k\in {\mathsf{\Omega }}_i}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_i}}{w}_i^{km\left(3\right)}·y_i^{km}\right)\\ +\left(2{\lambda }_2-1\right)·\left({\displaystyle \sum _{\left(i,j\right)\in \mathsf{\Psi }}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_{ij}}}{w}_{ijm}^{\left(4\right)}·l_{ijm}·x_{ijm}+{\displaystyle \sum _{i\in \mathsf{\Phi }}}{\displaystyle \sum _{k\in {\mathsf{\Omega }}_i}}{\displaystyle \sum _{m\in {\mathsf{\Omega }}_i}}{w}_i^{km\left(4\right)}·y_i^{km}\right)\le w^{\mathrm{m}\mathrm{a}\mathrm{x}}\end{array}$$

(32)

Equation (27) should be first converted into the following equation based on the fuzzy arithmetic operations:

$$\begin{array}{c}\mathrm{C}\mathrm{r}\left\{x_{ijm}·\tilde{q}\le {\tilde{g}}_{ijm}\right\}\ge {\lambda }_3\Rightarrow \mathrm{C}\mathrm{r}\left\{{\tilde{g}}_{ijm}-x_{ijm}·\tilde{q}\ge 0\right\}\ge {\lambda }_3\Rightarrow \mathrm{C}\mathrm{r}\left\{\left(\begin{array}{c}{g}_{ijm}^{\left(1\right)}-x_{ijm}·q^{\left(4\right)},\\ g_{ijm}^{\left(2\right)}-x_{ijm}·q^{\left(3\right)},\\ \begin{array}{c}{g}_{ijm}^{\left(3\right)}-x_{ijm}·q^{\left(2\right)},\\ g_{ijm}^{\left(4\right)}-x_{ijm}·q^{\left(1\right)}\end{array}\end{array}\right)\ge 0\right\}\ge {\lambda }_3\\ \begin{array}{cc}\forall (i,j)\in \mathsf{\Psi }& \forall r\in {\mathsf{\Omega }}_{ij}\end{array}\end{array}$$

(33)

According to Equation (30), the crisp formulation of Equation (27) is Equation (34).

$$\begin{array}{cc}\left(2{\lambda }_3-1\right)·\left(g_{ijm}^{\left(1\right)}-x_{ijm}·q^{\left(4\right)}\right)+2\left(1-{\lambda }_3\right)·\left(g_{ijm}^{\left(2\right)}-x_{ijm}·q^{\left(3\right)}\right)\ge 0& \begin{array}{cc}\forall (i,j)\in \mathsf{\Psi }& \forall r\in {\mathsf{\Omega }}_{ij}\end{array}\end{array}$$

(34)

Similarly, Equation (35) is the reformation of the chance constraint in Equation (28).

$$\begin{array}{cc}\left(2{\lambda }_3-1\right)·\left(g_i^{km\left(1\right)}-y_i^{km}·q^{\left(4\right)}\right)+2\left(1-{\lambda }_3\right)·\left(g_i^{km\left(2\right)}-y_i^{km}·q^{\left(3\right)}\right)\ge 0& \begin{array}{cc}\forall i\in \mathsf{\Phi }\backslash \left\{o,d\right\}& \begin{array}{cc}\forall k\in {\mathsf{\Omega }}_i& \forall m\in {\mathsf{\Omega }}_i\end{array}\end{array}\end{array}$$

(35)

After the first-stage processing, we can obtain the chance-constrained programming formulation, which replaces the initial model presented in Section 3.2 for the specific routing problem. The crisp model has the objective indicated by Equation (24) and combines Equations (2)–(9), (11), (17)–(22), (31), (32), (34) and (35) as the constraint set.

4.2. Stage II: Model Linearization

The nonlinearity of the crisp model results from Equations (8) and (11), which employ piecewise linear functions to calculate the pre- and post-transportation storage periods. This work uses the following linear equations to restrict the lower bounds of the variables (i.e., $p^o$ and $p^{d\left(f\right)}$) together with their domain constraints (i.e., Equations (21) and (22)).

$$p^o\ge u-r^o$$

(36)

$$\begin{array}{cc}{p}^{d\left(f\right)}\ge r^d-z^{\left(5-f\right)}& \forall f\in \left\{1,2,3,4\right\}\end{array}$$

(37)

The minimization of these variables resulting from Equation (24), which minimizes the storage costs, restricts these variables’ upper bounds. The above restrictions are equivalent to Equations (8) and (11) when calculating $p^o$ and $p^{d\left(f\right)}$. Detailed proof can be found in the first author’s previous studies [38,39]. Therefore, the nonlinear Equations (8) and (11) can be replaced by the linear Equations (36) and (37).

In summary, this work obtains a linear routing optimization model. The objective is Equation (24), and the constraints are Equations (2)–(7), (9), (17)–(22), (31), (32) and (34)–(37). Mathematical programming software can be used to efficiently solve this model using the exact solution algorithms installed; the feasibility of this has already been verified in various studies [52,53,54,55]. This work utilizes the mathematical programming software Lingo to solve the problem, in which the branch-and-bound algorithm is implemented, and we obtain the global optimum solution to guide the intermodal transportation operator in routing the food grains ordered by the agri-food company in the intermodal transportation network.

5. Numerical Case Study

5.1. Numerical Case Design

This section describes a numerical case based on a commonly adopted intermodal transportation network proposed in Sun and Lang’s work [56] to verify the proposed methods and draw research conclusions. The parameters in the numerical case are set as follows.

(1) In the 35-node intermodal transportation network with rail, road, and water transportation, nodes 1 and 35 represent the origin and destination warehouses hired by the agri-food company, respectively, and the remainders are the transfer nodes.

(2) The travel distances of the three transportation modes on the links in the intermodal transportation network are set to those in Reference [56]. The parameters related to the transportation modes are given in

Table 2. Parameters related to the transportation modes in the numerical case.

Transportation Modes	${\mathit{c}}_{\mathit{i}\mathit{j}\mathit{m}}$ (CNY/TEU)	${\mathit{c}}_{\mathit{i}\mathit{j}\mathit{m}}^{\prime }$ (CNY/(TEU·km))	${\tilde{\mathit{s}}}_{\mathit{i}\mathit{j}\mathit{m}}$ (km/h)	${\tilde{\mathit{e}}}_{\mathit{i}\mathit{j}\mathit{m}}$ (kg/(TEU·km))	${\tilde{\mathit{w}}}_{\mathit{i}\mathit{j}\mathit{m}}$ (%/100 km)
Rail	500	2.03	(50, 55, 60, 65)	(0.060, 0.075, 0.085, 0.105)	(0.030, 0.040, 0.045, 0.055)
Road	15	8	(50, 60, 70, 80)	(2.150, 2.350, 2.550, 2.650)	(0.040, 0.055, 0.065, 0.080)
Water	950	0	(20, 25, 30, 40)	(0.075, 0.085, 0.105, 0.115)	(0.035, 0.045, 0.060, 0.075)

, and those related to the transfer types are listed in

Table 3. Parameters related to the transfer types in the numerical case.

Transfer Types	${\mathit{c}}_{\mathit{i}}^{\mathit{k}\mathit{m}}$ (CNY/TEU)	${\tilde{\mathit{t}}}_{\mathit{i}}^{\mathit{k}\mathit{m}}$ (min/TEU)	${\tilde{\mathit{e}}}_{\mathit{i}}^{\mathit{k}\mathit{m}}$ (kg/TEU)	${\tilde{\mathit{w}}}_{\mathit{i}}^{\mathit{k}\mathit{m}}$ (%)
Rail—Road	5	(2.0, 2.6, 3.5, 4.4)	(4.05, 4.65, 5.10, 6.05)	(0.05, 0.15, 0.25, 0.40)
Rail—Water	7	(4.0, 5.3, 6.0, 7.5)	(5.30, 5.85, 6.10, 6.60)	(0.15, 0.25, 0.40, 0.55)
Road—Water	10	(3.0, 4.5, 6.5, 8.0)	(5.20, 5.70, 6.20, 6.50)	(0.10, 0.20, 0.45, 0.60)

. In these tables, the parameters are all assumed based on the existing literature on intermodal routing problems, focusing on the Chinese scenario. $c_{ijm}$ and $c_{ijm}^{\prime }$ refers to Jiang et al.’s study [57]. $c_i^{km}$ refers to Sun et al.’s work [58], and ${\tilde{t}}_i^{km}$ is also set based on their data. ${\tilde{s}}_{ijm}$ is determined by using the data in Yuan et al. [59] as a reference. The setting of ${\tilde{e}}_{ijm}$ and ${\tilde{e}}_i^{km}$ relies on the data in Li and Lv [60] as the benchmark. Finally, ${\tilde{w}}_{ijm}$ and ${\tilde{w}}_i^{km}$ are determined with reference to Wan et al. [5].

(3) The minimum possible capacities (i.e., $g_{ijm}^{\left(1\right)}$ and $g_i^{km\left(1\right)}$) and the lower bounds (i.e., $g_{ijm}^{\left(2\right)}$ and $g_i^{km\left(2\right)}$) of the most likely capacity intervals for the intermodal transportation activities in this network are assumed to be 0.7 and 0.9 of the capacities in Sun and Lang [56], respectively.

(4) The carbon tax rate and trading price rate are set to CNY 2/kg [61], and the carbon emission limit is set to 10,000 kg in the carbon trading regulation.

(5) The food grain transportation order of the agri-food company served by the intermodal transportation operator is given in

Table 4. Food grain transportation order in the numerical case.

Parameters	$\tilde{\mathit{q}}$ (TEU)	${\mathit{c}}_{\mathbf{v}\mathbf{a}\mathbf{l}\mathbf{u}\mathbf{e}}$ (CNY/TEU)	${\mathit{r}}^{\mathit{o}}$	${\mathit{r}}^{\mathit{d}}$	${\mathit{c}}_{\mathbf{s}\mathbf{t}\mathbf{o}\mathbf{r}\mathbf{e}}^{\mathit{o}}$ (CNY/(TEU·h))	${\mathit{c}}_{\mathbf{s}\mathbf{t}\mathbf{o}\mathbf{r}\mathbf{e}}^{\mathit{d}}$ (CNY/(TEU·h))
Values	(25, 28, 31, 35)	50,000	8:00 a.m. on day 1	5:00 p.m. on day 3	30	10

.

As determined using the mathematical programming software Lingo, the computational scale of the numerical case is given in

Table 5. Computational scale and solver state of the numerical case.

Number of Variables	Number of Integer Variables	Number of Constraints	Number of Nonzeros	Solver State
1794	569	1033	10,854	Global Optimum

. In the numerical case study, the computations are conducted on a laptop equipped with a 2.1 GHz CPU and 16 GB RAM. Due to the strong suitability of Lingo for use with the branch-and-bound algorithm to solve linear optimization models, all of the computations performed in the following stages of the numerical case study are accomplished efficiently, with a short runtime, and a generator memory of 832 k is used.

5.2. Sensitivity Analysis

In the food grain transportation process, the agri-food company cannot control the operations of the intermodal transportation network or the predetermined cost rates for the transportation services and carbon emission reduction regulations. Moreover, it is difficult to freely modify the schedule for the pre- and post-transportation processing of the food grains, including the earliest release time and due date, as these schedules are determined before transportation according to the produce, supply, and sales in the market. As a result, the agri-food company can subjectively determine the confidence levels (${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$) and the wastage threshold ($w^{\mathrm{m}\mathrm{a}\mathrm{x}}$) according to its preferences. This section analyzes the influence these parameters on the intermodal routing optimization of food grain transportation.

5.2.1. Sensitivity Analysis Concerning Confidence Levels

The comprehensive reliability of the intermodal routing for food gain transportation is determined via the combination of ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$. Therefore, this work considers that these confidence levels are set to the same value. First of all, this work sets the wastage threshold to 3%, increase the confidence levels from 0.5 to 1.0 in steps of 0.1, and then calculate the optimization results under different confidence level values. Figure 1 illustrates the sensitivity of the optimization results concerning the confidence levels.

As can be observed in Figure 3, the confidence levels have a significant influence on the intermodal routing optimization of food grain transportation. The sensitivity is significant when the confidence levels are changed from 0.7 to 1.0. An improvement in the confidence levels leads to a stepwise increase in the total costs of the planned food grain transportation route under the two carbon emission reduction regulations. This is because improving the confidence levels causes the chance constraints (i.e., Equations (31), (32), (34) and (35)) to become stricter, which narrows the solution space of the optimization problem. This worsens the best solution to the problem and results in an increase in the total costs of the planned food grain transportation route.

As explained in Section 4.1, improving the confidence levels enables the reliability of the intermodal routing process for food grains to be enhanced. Therefore, it can be concluded that the economics and reliability of the planned food grain transportation route are in conflict, and the agri-food company should increase its budget for the food grain transportation order if it requires a reliable intermodal transportation. Furthermore, with the data given in Figure 3, the agri-food company can balance the economics and reliability of this transportation order by setting the confidence levels based on its preferences regarding the two objectives.

5.2.2. Sensitivity Analysis Concerning Wastage Threshold

Besides the confidence levels, the wastage threshold is also determined by the agri-food company before routing decision-making. In this section, under the premise that the confidence levels are 0.8, we increase the wastage threshold from 1% to 3% in steps of 0.5%, and calculate the optimization results under different wastage thresholds. The sensitivity of the optimization results concerning the wastage threshold is shown in Figure 4.

As can be observed in Figure 4, the wastage threshold notably influences the intermodal routing optimization of food grain transportation. The sensitivity is significant when the wastage threshold is changed from 1.0% to 2.0%. Increasing the wastage threshold helps to reduce the total costs of the planned food grain transportation route in a stepwise manner. This is because an increase in the wastage threshold relaxes the chance constraint in Equation (32) and leads to the expansion of the solution space of the optimization problem, which enables us to optimize the best solution to the problem and results in a cost reduction. Therefore, the objectives of saving budget for the transportation order and reducing the wastages of food grains during the transportation are also in conflict. In this case, the agri-food company should set a reasonable wastage threshold to balance the above two objectives, where the sensitivity illustrated in Figure 4 can be used as a reference.

Additionally, according to the sensitivity given in Figure 3 and Figure 4, the planned food grain transportation routes under the carbon trading regulation yield smaller total costs than those under the carbon tax regulation. This is because the carbon trading regulation allows a certain amount of carbon emissions to occur at no cost.

5.3. Comparison Between Carbon Tax and Trading Regulations

This work considers two widely used carbon emission reduction regulations to plan a green and reliable food grain transportation route. It is meaningful to compare the carbon tax and trading regulations in reducing carbon emissions. In intermodal routing for food grain transportation, the confidence levels and wastage threshold are subjectively determined by the agri-food company. Therefore, this work calculates the carbon emissions of the planned food grain transportation route using carbon tax and trading regulations under different values of confidence levels and wastage threshold. The results are listed in

Table 6. Carbon emissions in kg for the planned food grain transportation routes using carbon tax and trading regulations.

Wastage Thresholds		1.0%		1.5%		2.0%		2.5%		3.0%
Regulations		Tax	Trading	Tax	Trading	Tax	Trading	Tax	Trading	Tax	Trading
Confidence levels	0.5	7963	15,975	8278	13,558	8728	13,558	8728	13,558	8728	13,558
	0.6	7963	10,118	8278	13,613	8728	13,613	8728	13,613	8728	13,613
	0.7	8282	12,180	9084	13,613	9084	13,613	9084	13,613	9084	13,613
	0.8	12,776	12,776	10,373	10,373	17,607	17,607	10,706	10,706	10,706	10,750
	0.9	—	—	—	—	27,267	27,267	20,436	20,436	17,406	17,406
	1.0	—	—	—	—	—	—	—	—	28,197	28,197

, in which the “—” indicates that a feasible solution to the problem is unattainable under the corresponding settings.

As shown in Table 6, the confidence levels and wastage threshold both influence the carbon emissions of the planned food grain transportation route under the carbon tax and trading regulations. For the carbon tax regulation, improving the confidence levels results in a stepwise increase in carbon emissions of the planned food grain transportation route, while the modification of the wastage threshold leads to a fluctuating change in the carbon emissions. For the carbon trading regulation, a fluctuating change in carbon emissions is also identified when the confidence levels and wastage threshold increase.

Furthermore, this work finds that carbon tax regulation leads to the carbon emissions that are less than or equal to those when using carbon trading regulation under different values of confidence levels and wastage thresholds. This is because carbon tax regulation imposes a financial penalty for all carbon emissions, while carbon trading regulation only penalizes the company for the portion that exceeds the prescribed carbon emission limit.

Additionally, this work finds that the carbon emissions under the two carbon emission reduction methods tend to be similar with the increase in both the confidence levels and wastage threshold. In this case, carbon trading regulation might create stronger motivation for the agri-food company to carry out green transportation, since this type of regulation helps to reduce its budget for the food grain transportation order.

5.4. Verificatin of Feasibility of Carbon Tax and Trading Regulations

This section further verifies the feasibility of the two carbon emission reduction regulations in reducing the carbon emissions of the intermodal routing for food grain transportation. This work uses the minimum-emission food grain transportation routes as the benchmarks for the verification. The routes yielding the minimum carbon emissions can be obtained by solving the linear programming model, which takes Equation (38) as the objective and uses Equations (2)–(7), (9), (17)–(22), (31), (32), (34) and (35) as the constraint set.

$$\min \sum _{\left(i,j\right)\in \mathsf{\Psi }}\sum _{m\in {\mathsf{\Omega }}_{ij}}\sum _{f\in \left\{1,2,3,4\right\}}{\displaystyle \frac{e_{ijm}^{\left(f\right)}·q^{\left(f\right)}}{4}}·l_{ijm}·x_{ijm}+\sum _{i\in \mathsf{\Phi }}\sum _{k\in {\mathsf{\Omega }}_i}\sum _{m\in {\mathsf{\Omega }}_i}\sum _{f\in \left\{1,2,3,4\right\}}{\displaystyle \frac{e_i^{km\left(f\right)}·q^{\left(f\right)}}{4}}·y_i^{km}$$

(38)

The optimization results regarding intermodal routing for food grain transportation that minimizes the carbon emissions are shown in

Table 7. Carbon emissions in kg of the planned minimum-emission food grain transportation routes.

Wastage Thresholds		1.0%	1.5%	2.0%	2.5%	3.0%
Confidence levels	0.5	7963	6510	5575	5575	5575
	0.6	7963	6510	6231	6231	6231
	0.7	8282	6510	6510	6510	6510
	0.8	12,776	10,373	10,029	8892	8663
	0.9	—	—	26,178	14,978	14,749
	1.0	—	—	—	—	28,179

, in which the “—” indicates that a feasible solution to the problem is unattainable under the corresponding settings.

As shown in Table 7, improving the confidence levels and decreasing the wastage threshold result in a stepwise increase in the carbon emissions of the planned minimum-emission food grain transportation route. The factors leading to such changes are the same as those identified in Section 5.2 that lead to a change in the total costs of the planned minimum-cost food grain transportation route.

Using the carbon emissions in Table 7 as benchmarks, we can calculate the gaps in the carbon emissions between the carbon emission minimization method and the carbon emission reduction regulations according to Equation (39) [61]. In this equation, ${emmission}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and ${emmision}_{\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}$ refer to the carbon emissions of the planned food grain transportation route based on the carbon emission minimization method and carbon emission reduction regulations, respectively. The smaller the gap, the better the performance of the carbon emission reduction regulation in reducing the carbon emissions achieves [61]. The gaps in the carbon emissions are given in

Table 8. Gaps (%) in carbon emissions between carbon emission minimization method and carbon emission reduction regulations.

Wastage Thresholds		1.0%		1.5%		2.0%		2.5%		3.0%
Regulations		Tax	Trading	Tax	Trading	Tax	Trading	Tax	Trading	Tax	Trading
Confidence levels	0.5	0	50.2	21.4	52.0	36.2	58.9	36.1	58.9	36.1	58.9
	0.6	0	21.3	21.4	52.2	28.6	54.2	28.6	54.2	28.6	54.2
	0.7	0	32.0	28.3	52.2	28.3	52.2	28.3	52.2	28.3	52.2
	0.8	0	0	0	0	43.0	43.0	16.9	16.9	19.1	19.4
	0.9	—	—	—	—	4.0	4.0	26.7	26.7	15.3	15.3
	1.0	—	—	—	—	—	—	—	—	0.1	0.1

.

$$gap={\displaystyle \frac{\left|{emission}_{\mathrm{m}\mathrm{i}\mathrm{n}}-{emission}_{\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}\right|}{{emission}_{\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}}}·100\%$$

(39)

In this work, a carbon emission reduction regulation is considered feasible in reducing the carbon emissions if the gap is less than 50% [61]. Under this consideration, it can be concluded that a carbon tax policy is always feasible in the planning of green transportation route for food grains, while a carbon trading policy is not feasible in all the cases.

Table 8 indicates that the feasibility of carbon trading regulation in reducing the carbon emissions is influenced by the confidence levels and wastage threshold determined by the agri-food company. Improving the confidence levels and decreasing the wastage threshold enable the carbon trading regulation to be feasible in reducing the carbon emissions. Therefore, carbon trading regulation is suitable for the cases in which the agri-food company prefers a reliable and lower-wastage food grain transportation route for its transportation order.

Additionally, when the two carbon emission reduction regulations are both feasible in reducing the carbon emissions, carbon trading regulation is more suitable in enabling the agri-food company to realize the green transportation, as it can reduce the total costs of completing the food grain transportation order.

5.5. Managerial Implications

Based on the analysis in Section 5.2, Section 5.3 and Section 5.4, this work provides the following managerial insights, which can assist agri-food companies and intermodal transportation operators in implementing sustainable food grain transportation through intermodal strategies.

(1) The confidence levels and wastage threshold, which are subjectively determined by the agri-food company, influence the intermodal routing optimization of food grain transportation. When these parameters are set by the agri-food company, the intermodal transportation operator can use our model to plan the best food grain transportation route.

(2) Improving the reliability by increasing the confidence levels, and reducing the wastages by lowering the wastage threshold, both increase the total costs of completing the food grain transportation order. This means that the agri-food company has to allocate a larger budget for the transportation order in order to improve the quality of the transportation service.

(3) The agri-food company can balance the objectives regarding economics, reliability, and wastages by setting suitable confidence levels and wastage threshold. The sensitivity presented in this work could help the agri-food company to reasonably determine the values of these paramters.

(4) Carbon tax regulation can reduce carbon emissions to a greater extent than carbon trading regulation. It is successful in reducing the carbon emissions of the intermodal transportation of food grains, while carbon tax regulation is not always feasible. The factors influencing the feasibility of carbon trading regulation are the confidence levels and the wastage threshold.

(5) Carbon trading regulation becomes feasible in the cases where the agri-food company seeks a reliable and lower-wastage intermodal transportation for its food grain transportation order. When both regulations are feasible, carbon trading regulation is more successful in enabling the agri-food company to realize green transportation, as it can reduce the budget needed for the food grain transportation order.

6. Conclusions

This work explores a green and reliable intermodal routing problem for food grain transportation that comprehensively considers different carbon emission reduction regulations and multi-source uncertainty. It offers the following contributions and provides strong quantitative support for the agri-food company and intermodal transportation operator in implementing sustainable food grain transportation.

(1) Reducing carbon emissions in intermodal routing for food grain transportation has the potential to improve the environmental sustainability of food logistics; hence, carbon tax and trading regulations are both modeled and their performance and feasibility in reducing carbon emissions are fully analyzed.

(2) The reliability of intermodal routing for food grain transportation is effectively enhanced by modeling its multi-source uncertainty, which includes the agri-food company’s demand for food grains and the various parameters (i.e., travel speeds, transfer times, capacities, carbon emission factors, and wastage rates) related to the intermodal transportation activities. This consideration reduces the risks that the planned food grain transportation route violates the budget, due date, and wastage threshold set by the agri-food company and becomes infeasible due to a capacity insufficiency.

(3) The wastages of the food grains during the transportation process are reduced by minimizing the wastage costs as a portion of the objective and are controlled by the threshold constraint. Modeling the uncertainty in the uncertain wastage rates of the intermodal transportation activities further improves the reliability of the planned food grain transportation route on satisfying the agri-food company’s wastage requirements.

(4) Based on the use of trapezoidal fuzzy numbers to describe the uncertain parameters and resulting uncertain variables, a fuzzy mixed-integer nonlinear programming model is proposed to address the green and reliable intermodal routing problem for food grain transportation. Moreover, a two-stage solution method is designed to convert the proposed model into an equivalent chance-constrained linear programming model, ensuring the global optimum solution to the problem is attained.

However, this work focuses on a single food grain transportation order, and only formulates the earliest release time and due date to ensure the timeliness of transportation. In our future research, multiple food grain transportation orders will be taken into account. Time windows, e.g., soft time windows [62,63] and flexible time windows [64,65], will be modeled to improve the pickup and delivery services in food grain transportation to achieve timely transportation, so that agri-food companies can better adopt the “Just-in-Time” strategy. In this case, the intermodal routing problem for food grain transportation is more complicated, making a heuristic algorithm more suitable for application in the two-stage solution method than the exact solution algorithm to solve the problem. This will be considered in our future research. Furthermore, cold chain transportation is also an important part of food logistics [66,67]. Compared with food grain transportation, it yields higher wastages, and the required refrigeration incurs additional costs and carbon emissions. Our future research based on this work will also study a green and reliable intermodal routing problem for cold chain transportation, in which, besides considering wastage control, the costs and carbon emissions created by refrigeration will be reduced, and the uncertainty in the refrigeration operations will be also modeled.