The Multi-Objective Shortest Path Problem with Multimodal Transportation for Emergency Logistics

Abstract

The optimization of emergency logistical transportation is crucial for the timely dispatch of aid and support to affected areas. By incorporating practical constraints into emergency logistics, this study establishes a multi-objective shortest path mixed-integer programming model based on a multimodal transportation network. To solve multi-objective shortest path problems with multimodal transportation, we design an ideal point method and propose a procedure for constructing the complete Pareto frontier based on the k-shortest path multi-objective algorithm. We use modified Dijkstra and Floyd multimodal transportation shortest path algorithms to build a k-shortest path multi-objective algorithm. The effectiveness of the proposed multimodal transportation shortest path algorithm is verified using empirical experiments carried out on test sets of different scales and a comparison of the runtime using a commercial solver. The results show that the modified Dijkstra algorithm has a runtime that is 100 times faster on average than the modified Floyd algorithm, which highlights its greater applicability in large-scale multimodal transportation networks, demonstrating that the proposed method both has practical significance and can generate satisfactory solutions to the multi-objective shortest path problem with multimodal transportation in the context of emergency logistics.

1. Introduction

As many logistical sectors (e.g., packaging, transportation, warehousing, distribution, reverse logistics) must respond to major natural disasters and public security events, emergency logistics is a crucial element of effective emergency management. Emergency logistics refers to activities carried out in response to severe natural disasters, sudden public health events, public security events, military conflicts, and other emergencies; such activities aim to maximize time benefits and minimize disaster losses [1]. Providing access to rescue teams and their accompanying supplies promptly is a critical aspect of this response. In this study, we consider a multi-objective shortest path problem with multimodal transportation (MOSPP-MT) to improve the efficiency of emergency rescue and mitigate losses.

Emergency logistics plays a crucial role in minimizing the impact of disasters by ensuring the timely and correct provision of rescue materials and conducting rescue operations. A critical challenge in emergency logistics is the effective utilization of limited transportation resources while considering various practical constraints. Therefore, optimizing the allocation and utilization of transportation resources is crucial. During a disaster, transportation resources may be limited and must be exploited to deliver rescue teams and materials. Karatas et al. [2] have investigated the most prominent decision-related problems observed during epidemics. Careful planning and coordination are required to ensure that resources are utilized to their full extent. Selecting appropriate routes is essential to ensure the swift and efficient transportation of rescue teams and materials. Factors such as road conditions, traffic congestion, and accessibility must be considered when determining optimal routes. By fully utilizing limited transportation resources and selecting appropriate routes, rescue teams and materials can be transported in the shortest possible time while accounting for various practical constraints. There may be limited access (e.g., restricted time, manpower, and transportation capacity) to roads or airports, making it difficult to deliver supplies via a single mode of transport. Emergency logistics is an important field within emergency management, and multimodal transportation in emergency logistics can help to overcome these constraints. For instance, we can use advanced transportation modes like drones for aerial dispatch to reach isolated areas [3,4]. Similarly, multimodal transportation can help improve routing and inventory management to ensure the efficient distribution of resources. Multimodal transportation, which involves multiple modes of transportation (such as air, rail, and road) in delivering goods and people, has attracted significant attention in recent years due to its potential to improve efficiency, reduce costs, and minimize environmental impact. General emergency logistics problems may involve multiple demand locations and supply centers to which commodities need to be efficiently transported from multiple starting points. In such problems, recommending transportation routes between each pair of the supply and demand locations while considering different transportation modes is of great significance within logistical decision making, in which multiple criteria may be involved. Considering potential difficulties encountered due to restrictions on rescue facilities and capacity, the MOSPP-MT algorithm may play an essential role in emergency rescue; it therefore requires further investigation.

Significant advances have been made in tackling the shortest path problem (SPP), which is a classic combinatorial optimization problem. It has been widely investigated in graph theory and network optimization. The Dijkstra algorithm [5], Bellman algorithm [6] and A* algorithm [7] were developed to solve the SPP. In addition, some scholars have conducted extensive studies on the fuzzy shortest circuit problem, considering the uncertainties of the network (e.g., line interruptions and equipment maintenance) [8,9]. However, in real-world scenarios, there are often multiple objectives or criteria that need to be considered when seeking an optimal path. This has led to the development of the multi-objective shortest path problem (MOSPP), in which the goal is to find a set of paths that optimize multiple objectives simultaneously, for example, minimizing both distance and travel time. Serafini [10] proved that the multi-objective shortest path problem is NP-complete. Casas et al. [11] proposed a multi-objective Dijkstra algorithm to solve shortest path problems and compared it with the classic MOSPP algorithm proposed by Martins [12]. Hiroki et al. [13] proposed a method based on constructing ideal points to solve multi-objective problems by determining the feasible solution closest to an ideally constructed point that satisfies all constraints via the final solution. José et al. [14] presented an overview of the MOSPP and explored a multi-objective evolutionary algorithm applied to the MOSPP. Salzman et al. [15] presented an overview of the field with an emphasis on results obtained in recent years using heuristic search techniques among other approaches (including evolutionary algorithms, integer programming, and reinforcement-learning algorithms). Siddiqi et al. [16,17] proposed a stochastic evolution-based algorithm for solving the MOSPP that works on a single solution and proposed two new evolutionary algorithms to overcome the shortcomings of existing evolutionary algorithms. To compare different approaches to MOSPP on multigraphs, Beke et al. [18] extended popular genetic representations to a multigraph case and compared qualities of the proposed solution. Zheng et al. [19] developed a novel method based on probability theory for solving the MOSPP. Dhodiya et al. [20] suggested that the aspiration level-based non-dominated sorting genetic algorithm II and the aspiration level-based non-dominated sorting genetic algorithm III can handle the fuzzy multi-objective shortest path problem effectively and efficiently, thereby providing optimal outputs. Ren et al. [21] have developed a fast A*-based algorithm that can find an optimal solution while satisfying multiple resource constraints.

Establishing multi-commodity and multi-mode transportation or material distribution models can aid in solving emergency logistics problems in multimodal traffic networks. Based on the space–time network, a multi-commodity, multi-supply node, multi-mode transportation model for emergency supply distribution was presented by Haghani et al. [22]. Barbarosoglu et al. [23] proposed a scenario-based two-stage stochastic programming model to plan disaster relief transportation and developed a multi-commodity and multi-mode network flow formula to describe logistics in urban transportation networks. Many scholars have designed and improved optimization algorithms to solve complex multimodal transport problems. Yi et al. [24] proposed an ant colony optimization algorithm for the emergency logistics multimodal transportation problem and decomposed the original transportation problem into a vehicle routing optimization problem and a multi-commodity scheduling problem in order to solve it. Wang et al. [25] studied routing and congestion problems in a multimodal transportation network and demonstrated how we might increase network capacity as much as possible while saving both time and money. Yang et al. [26] proposed a novel travel-mode recommendation system for a multimodal transportation system. Their model focused on scenarios in which multimodal transportation recommendations may be implemented.

To the best of our knowledge, very few studies have been conducted on the MOSPP-MT, and efficient algorithms that can provide a satisfactory solution within a reasonable amount of computation time are still lacking. The multiple objectives involved in this problem need to be carefully balanced. In most cases, the optimal solution for one specific objective is not optimal for the other objectives; therefore, a satisfactory solution should be a Pareto solution [27], such that the current solution cannot be improved for any single objective without worsening the other objectives. To obtain the Pareto frontier, Yu et al. [28] presented a ripple-spreading algorithm, which determined the complete Pareto frontier by performing a ripple relay race to obtain the set of Pareto optimal path solutions. Considering the practical difficulties in emergency logistics, multiple objectives must be considered, such as the total shipping time, transportation distance, and shipping cost. MOSPP-MT has broad applications in transportation, communication, project scheduling, and other research areas. It is crucial to apply multimodal transportation while considering the characteristics of emergency logistics (e.g., limited road capacity and high material demands). However, it is still difficult to determine a balanced solution for MOSPP-MT.

While there has been extensive research on multi-objective optimization in various domains, including engineering, logistics, and operations research, there is a noticeable gap in the literature regarding multimodal transportation in the context of multi-objective optimization. Specifically, existing studies often focus on optimizing a single mode of transportation, such as road or rail, without considering the interactions between and interdependence of different modes. Research is needed to develop optimization models and algorithms that can effectively integrate multiple modes of transportation while taking into account the characteristics of each transportation mode and the connections between them. While some studies have considered multimodal transportation in the context of single-objective optimization, there is a lack of research that explicitly addresses the multi-objective nature of the problem. In the context of emergency logistics, the recommendation of optimal paths between specific locations is of great importance. Additionally, the trade-offs involved in optimizing various objectives—such as time, cost, and the impact of the distances within a multimodal network—remain underexplored. Most methodologies do not account for real-time data and flexible routing, which are increasingly relevant to today’s dynamic transportation landscape. We must therefore develop multi-objective optimization approaches that can effectively balance conflicting objectives in the context of multimodal transportation.

Motivated by the above limitations and requirements, in this paper, we develop a multi-objective shortest path model based on a multimodal transportation network. We propose an ideal point method and a procedure through which to obtain the Pareto frontier based on the k-shortest multi-objective algorithm to solve the model. In addition, we modify Dijkstra and Floyd’s classical shortest circuit algorithm to solve the MOSPP-MT. Aiming to achieve minimal shipping time, transportation distance, and total shipping cost, the algorithm for each individual objective is exact, and a Pareto solution can be obtained with multiple objectives. After the optimal solution for each objective is obtained, the ideal point method based on the k-shortest multi-objective algorithm can obtain the Pareto solution to the multi-objective problem. The effectiveness of the proposed method is proven herein through extensive computational tests and a comparison with the commercial solver CPLEX.

The remainder of this paper is organized as follows. In Section 2, we describe the problem in detail and present a multi-objective shortest path mixed-integer programming model. In Section 3, we devise improved multimodal transportation algorithms for the single-objective shortest path problem. In Section 4, we present an ideal point method and a procedure through which to obtain the Pareto frontier based on the k-shortest multi-objective algorithm. Computational experiments are presented and discussed in Section 5, which are followed by conclusions and suggestions for future research in Section 6.

2. A Description of the Problem and Mathematical Formulation

When a disaster occurs, the emergency materials within the affected area may be used up rapidly. To minimize the impact of disasters as soon as possible, it is critical that we provide rescue materials and perform timely rescue operations via emergency logistics. To meet the needs of the disaster area, it is necessary to transport materials from other areas to supplement shortages rapidly, and multimodal transportation should be considered for this task. The higher the priority of a given transportation mode, the shorter its shipping time. To reduce the total shipping time, we should use high-priority transportation modes as much as possible; that said, high-priority transportation modes also come at a great cost. Limited transportation resources should be fully utilized, and the multimodal transportation route needs to be planned so that rescue teams and materials may be transported in as little time as possible, all while considering various practical constraints.

Multimodal transportation networks and the details of the task at hand should be considered when modeling the optimization of emergency logistics plans. A dispatch task normally has a specific starting point and a destination, and the total amount of supplies to be delivered (i.e., the total number of batches of commodities in the emergency rescue task) is also provided. A multimodal transport network comprises nodes and edges. For each node, the loading and unloading capacity information accompanying different modes of transport (i.e., the number of batches that can be loaded and unloaded within a time unit) is provided. We must consider the traffic information between nodes (i.e., the number of batches that can be transported within each unit of time) when assessing the edges.

Based on these requirements, a multi-objective shortest path mixed-integer programming model is established in this section. Several relevant assumptions are presented initially:

•

Three transportation modes are considered—air, railway, and road—in order of decreasing priority.

•

The transportation mode can only be transferred from a higher priority mode to lower priority modes. We denote $f_1$ < $f_2$ if transportation mode $f_1$ has higher priority than transportation mode $f_2$.

•

The transportation task must be continuously delivered both spatially and temporally.

•

Transfer between different transportation modes occurs only at nodes in the transport network.

•

The transportation task can be split into batches. Each batch in the task cannot be split during transportation; in other words, a batch is the smallest possible unit for transportation.

The transportation network is denoted $N=(V,E)$. We denote the starting point and destination of the tasks s and q, respectively, and $s,q\in V$. The subnetwork of transportation mode f can be abstracted: $N_f=(V_f,E_f)$, where the index of a node is $i\in V_f$, and ${(i,j)}_f\in E_f$ represents the undirected edge between i and j in transportation mode f. If ${(i,j)}_f\in E_f$ exists, we assume that we can transport batches between i and j in both directions in transportation mode f. The node set is $V=\{V_{f_1}\cup V_{f_2}\cup \dots \cup V_{f_{\left|F\right|}}\}$, and the edge set is $E=\{E_{f_1}\cup E_{f_2}\cup \dots \cup E_{f_{\left|F\right|}}\}$. In this study, we consider three modes of transportation. Figure 1 presents an example of a multimodal transportation network. Figure 1 shows the actual transportation network, which has been abstracted and modeled into a layered network. In Figure 1, each subnetwork stores the nodes and edges that relate to a particular mode. The directed edges between the subnetworks represent possible transfers between different transportation modes. The directed edges between subnetworks (i.e., the transfer edges) do not take any shipping time, but they have limited transfer capacities (i.e., the minimum value between the uploading capacity on the unloading end and the loading capacity on the other). We also consider transfer costs between the subnetworks. Specifically, at nodes s and q, the transfer edges do not trigger any cost and have unlimited transfer capacity, as the transfer will not be executed at the supply and demand locations. The commodities may not be dispatched all at once due to capacity constraints. Therefore, the total shipping time includes both the transportation time along the selected route and the dispatching time due to the limited capacity along the route, which can be determined by the smallest edge capacity along the route.

To describe the model clearly,

Table 1. Notation of the related parameters.

Notation	Definition
F	Set of transportation modes
f	Index of transportation modes and $f\in F$
$N_f$	Subnetwork of transportation mode f
$l_i^f$	Periodic loading capacity of node i under transportation mode f
$u{l}_i^f$	Periodic unloading capacity of node i under transportation mode f
$r_{ij}^f$	Periodic traffic capacity of edge ${(i,j)}_f$ under transportation mode f
$l{r}_{ij}^f$	Length of edge ${(i,j)}_f$ under transportation mode f
$t{r}_{ij}^f$	Transportation time of the edge ${(i,j)}_f$ under transportation mode f
u	Number of batches in the task
$s{w}^{f_1,f_2}$	Whether the task can transfer from transportation mode $f_1$ to $f_2$
	($f_1$ has a higher priority than $f_2$)
s	Starting point of the task
q	Destination of the task
$C^f$	Unit cost of transportation mode f
$tc$	Unit cost of the transfer
T	Time horizon of transportation planning
t	Index of time period and $t\in T$

presents the notations of the related parameters used in the models, and

Table 2. Notation of the decision variables.

Notation	Definition
$x_{ij}^f$	Binary variable indicating whether the task selects
	the edge ${(i,j)}_f$ to transport under the transportation mode f
$y_{ijt}^f$	Number of batches in transportation in
	the edge ${(i,j)}_f$ in period t under the transportation mode f
$swi{t}_i^{f_1,f_2}$	Binary variable indicating whether the task transfers
	from transportation mode $f_1$ to $f_2$ at node i
	($f_1$ has a higher priority than $f_2$)
$X_t$	Binary variable indicating whether there are any batches
	in the task delivered from starting point s in period t

presents the notations of the decision variables.

We model the problem as follows:

$$\begin{array}{cc}\hfill & min{z}_1=\sum _{f\in F}\sum _{{(i,j)}_f\in E_f}t{r}_{ij}^f·x_{ij}^f+\sum _{t\in T}{X}_t;\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & min{z}_2=\sum _{f\in F}\sum _{{(i,j)}_f\in E_f}l{r}_{ij}^f·x_{ij}^f;\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & min{z}_3=\sum _{f\in F}\sum _{{(i,j)}_f\in E_f}{C}^f·l{r}_{ij}^f·x_{ij}^f+\sum _{f_1\in F}\sum _{f_2\in F:f_1<f_2}\sum _{i\in V}swi{t}_i^{f_1,f_2}·tc;\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & \mathrm{s}.\mathrm{t}.\sum _{f\in F}\sum _{{(s,j)}_f\in E_f}{x}_{s,j}^f-\sum _{f\in F}\sum _{{(j,s)}_f\in E_f}{x}_{j,s}^f=1;\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & \sum _{f\in F}\sum _{{(q,j)}_f\in E_f}{x}_{q,j}^f-\sum _{{(j,q)}_f\in E_f}\sum _{f\in F}{x}_{j,q}^f=-1;\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & \sum _{f\in F}\sum _{{(i,j)}_f\in E_f}{x}_{ij}^f-\sum _{f\in F}\sum _{{(j,i)}_f\in E_f}{x}_{ji}^f=0,\forall i\in V\backslash \{s,q\};\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & \sum _{j:{(j,i)}_{f_1}\in E_{f_1}}{x}_{ji}^{f_1}\ge swi{t}_i^{f_1,f_2},\forall i\in V\backslash \{s,q\},\forall f_1,f_2\in F;\hfill \end{array}$$

(7)

$$\sum _{j:{(i,j)}_{f_2}\in E_{f_2}}{x}_{ij}^{f_2}\ge swi{t}_i^{f_1,f_2},\forall i\in V\backslash \{s,q\},\forall f_1,f_2\in F;$$

(8)

$$\sum _{f_2\in F\backslash f_1}swi{t}_i^{f_1,f_2}+\sum _{j:{(j,i)}_{f_2}\in E_{f_2}}{x}_{ji}^{f_2}=\sum _{k:{(i,k)}_{f_2}\in E_{f_2}}{x}_{ik}^{f_2}+\sum _{f_3\in F\backslash \{f_1,f_2\}}swi{t}_i^{f_2,f_3},$$

$$\begin{array}{cc}\hfill & \forall i\in V\backslash \{s,q\},\forall f_1,f_2,f_3\in F;\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & \sum _{f\in F}\sum _{{(j,i)}_f\in E_f}{y}_{j,i,t-t{r}_{ji}^f}^f=\sum _{f\in F}\sum _{{(i,k)}_f\in E_f}{y}_{i,k,t}^f,\forall t\in T,\forall i\in V\backslash \{s,q\};\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & \sum _{t\in T}\sum _{f\in F}\sum _{{(s,j)}_f\in E_f}{y}_{sjt}^f=\sum _{t\in T}\sum _{f\in F}\sum _{{(i,q)}_f\in E_f}{y}_{iqt}^f=u;\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & y_{ijt}^f\le x_{ij}^f·r_{ij}^f,\forall {(i,j)}_f\in E_f,\forall f\in F,\forall t\in T;\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & \sum _{{(s,j)}_f\in E_f}{y}_{s,j,t}^f\le l_s^f,\forall t\in T,\forall f\in F;\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & \sum _{{(i,q)}_f\in E_f}{y}_{i,q,t}^f\le u{l}_q^f,\forall t\in T,\forall f\in F;\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & \sum _{j:{(i,j)}_{f_1}\in E_{f_1}}{y}_{i,j,t}^{f_1}\le swi{t}_j^{f_1,f_2}·u{l}_j^{f_1},\forall t\in T,\forall f_1,f_2\in F;\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & \sum _{i:{(i,j)}_{f_2}\in E_{f_2}}{y}_{i,j,t}^{f_2}\le swi{t}_i^{f_1,f_2}·l_i^{f_2},\forall t\in T,\forall f_1,f_2\in F;\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & swi{t}_i^{f_1,f_2}\le s{w}^{f_1,f_2},\forall i\in V,\forall f_1,f_2\in F;\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & X_t·M\ge \sum _{f\in F}\sum _{{(s,j)}_f\in E_f}{y}_{sjt}^f,\forall t\in T;\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & X_t\in \{0,1\},\forall t\in T;\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & x_{ij}^f\in \{0,1\},\forall f\in F,\forall {(i,j)}_f\in E_f;\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & swi{t}_i^{f_1,f_2}\in \{0,1\},\forall i\in V,\forall f_1,f_2\in F;\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & y_{ijt}^f\ge 0,\forall f\in F,\forall {(i,j)}_f\in E_f,\forall t\in T.\hfill \end{array}$$

(22)

Objective function (1) minimizes the total shipping time (i.e., the arrival time of the last batch) with the time taken to dispatch all commodity batches also considered. Objective function (2) minimizes the total transportation distance of the multimodal transport paths (i.e., the sum of the edge lengths). Objective function (3) minimizes the total shipping cost (i.e., the transportation and transfer costs generated by different transportation modes).

Constraints (4)–(6) ensure that the transportation route of a task is geographically continuous. This means that all segments of the route must be connected without any interruptions. Constraints (7)–(9) ensure connectivity between transfer node i and the selected route. When $swi{t}_i^{f_1,f_2}=1$, the transfer from transportation mode $f_1$ to $f_2$ is executed at node i. This is crucial for facilitating the change in transportation methods at specific points along the route. Constraints (10) and (11) ensure that the number of batches transported on the selected routes is continuous. Constraints (12)–(14) determine the loading and unloading capacities and the traffic capacity limits of the transportation route. Constraints (15) and (16) ensure that the unloading and loading capacity of the node are respected and that transfer occurs when the number of batches before and after transfer remains consistent. Constraint (17) ensures that transfer can only be executed at nodes on which it is allowed. Constraint (18) is used to label the periods required to complete the dispatch of all batches. Constraints (19)–(21) stipulate that the decision variables $X_t$, $x_{ij}^f$, and $swi{t}_i^{f_1,f_2}$ are binary variables, and Constraint (22) requires the decision variable $y_{ijt}^f$ to be non-negative.

3. Multimodal Shortest Path Algorithms with Single Objective Functions

In order to address the MOSPP-MT, we develop an ideal point framework (as presented in Section 4) based on the k-shortest path multi-objective algorithm. When using the ideal point method, it is necessary to obtain optimal values of each objective function; thus, we must first devise multimodal shortest path algorithms with single-objective functions to obtain the optimal values of each objective function. In this section, we introduce multimodal transportation shortest path algorithms, based on the Dijkstra and Floyd algorithms, which can effectively handle single-objective problems and can be used in the subsequent ideal point framework.

3.1. Modified Dijkstra Algorithm

The Dijkstra algorithm was proposed by Dutch computer scientist Dijkstra in 1959 [5]. It obtains the shortest path from a node (called the “source node’’) to all other nodes in a graph by adopting a greedy strategy.

Table 3. Notations related to the modified Dijkstra algorithm.

Notations	Definition
$N_{f_l}$	Subnetwork of priority level l under transportation mode f
$s_l$	Starting point of the subnetwork of priority level l
$q_l$	Endpoint of the subnetwork of priority level l
$f^i$	Transportation mode f reaching node i
$v_j^{f_j}$	Node j on a partial path reached by transportation mode f

lists the related parameters used to describe the modified Dijkstra algorithm.

The modified Dijkstra algorithm for the multimodal transport shortest path problem proceeds as follows:

•

The subnetwork of the highest priority $N_{f_{high}}$ of the starting point s is determined, and the priority level is denoted as a, starting from node $s_a$.

•

Sets P and H are introduced. Set P records the node whose minimum objective value has been obtained; set H records the node that has not been visited. Initially, set P contains only node $s_a$.

•

Set H is traversed to determine the node with the minimum objective value; then, the node is added to set P, and the objective value and path of the node in set H are updated.

•

The above steps are repeated until the endpoint $q_p(p\ge a)$ is visited, and the objective values of different endpoints $q_p(p\ge a)$ are compared to determine the minimum value.

The pseudocodes for the modified Dijkstra algorithm addressing the multimodal transport shortest path problem are presented in Algorithms A1–A3 in Appendix A [29]. Algorithms A1–A3 have a time complexity of $O\left(\left(\right|V|+|E\left|\right)·log\left(\right|V\left|\right)\right)$ [30]. The modified algorithm for objectives $z_1$ is an approximately exact algorithm, and the modified algorithm for objective $z_2$ and $z_3$ is an exact algorithm.

3.2. Modified Floyd Algorithm

The Floyd algorithm [31] is also a shortest path algorithm based on dynamic programming. It is widely used to resolve various practical problems.

The procedure of the modified Floyd algorithm for the multimodal transport shortest path problem is described as follows:

•

The adjacency matrix S of the multimodal transportation network is constructed. The entry of $S\left[v\right]\left[u\right]$ indicates the time spent on the path from node v to node u.

•

The subnetwork of the highest priority $N_{f_{high}}$ with the starting point s is determined, and the priority level is denoted a, starting from node $s_a$.

•

The matrix D is introduced in order to record the shortest path. The entry of $D\left[v\right]\left[u\right]$ means that the shortest path is that running directly from node v to node u.

•

All nodes are traversed. If $S\left[v\right]\left[u\right]>S\left[v\right]\left[r\right]+S\left[r\right]\left[u\right]$ on node r, $S\left[v\right]\left[u\right]=S\left[v\right]\left[r\right]+S\left[r\right]\left[u\right]$ is set, and the entry is updated from $D\left[v\right]\left[u\right]$ to $D\left[v\right]\left[r\right]$.

•

The minimum value is determined by comparing the objective values from node $s_a$ to a different destination $q_j(j\ge a)$.

When the objective is to minimize the path time, the value in the adjacency matrix $S\left[v\right]\left[u\right]$ represents the transfer time. If nodes v and u are adjacent nodes in the same network, the entry in the adjacency matrix $S\left[v\right]\left[u\right]$ represents the path time, and the other entries are set to infinity. The pseudocodes for the modified Floyd algorithm addressing the multimodal transport shortest path problem are presented in Algorithms A4 and A5 in Appendix B [29]. Algorithms A4 and A5 have a time complexity of $O\left({\left|V\right|}^3\right)$ [31].

4. The Ideal Point Method and Procedure for Building the Pareto Frontier Based on the $\mathit{K}$-Shortest Path Multi-Objective Algorithm

Building upon the preceding multimodal shortest path algorithms with single-objective functions, in this section, we propose the ideal point method based on the k-shortest path multi-objective algorithm to address MOSPP-MT. A basic ideal point method based on the multi-objective shortest path problem was proposed by Coutinho-Rodrigues et al. [32] and Current et al. [33]. MOSPP-MT has P objectives. The value of objective p of the transport path is $z_p\left(path\right)(p=1,2,3)$. We refer to the shortest path of objective p as $pat{h}_p^{min}(p=1,2,3)$, which has an optimal value of $z_p\left(pat{h}_p^{min}\right)(p=1,2,3)$ for objective p. Therefore, $(z_1\left(pat{h}_1^{min}\right),z_2\left(pat{h}_2^{min}\right),z_3\left(pat{h}_3^{min}\right))$ is the ideal point for MOSPP-MT.

The k-shortest paths multi-objective algorithm produces the k-shortest paths of three objectives. We refer to the wth-shortest path of objective r as $pat{h}_r^w(w=1,2,\dots ,k;r=1,2,3)$ and its value of objective p as $z_p\left(pat{h}_r^w\right)(w=1,2,\dots ,k;r=1,2,3;p=1,2,3)$. For each objective P, there must be a minimum and a maximum value of $z_p\left(pat{h}_r^w\right)(w=1,2,\dots ,k;r=1,2,3)$; therefore, the range of $z_p\left(pat{h}_r^w\right)$ is $[z_p^{min},z_p^{max}](p=1,2,3)$.

We normalize all objective values to the value interval [0, 1]. For each $pat{h}_r^w$, the objective values can be normalized as follows:

$$\begin{array}{c}\hfill {\overline{z}}_p\left(pat{h}_r^w\right)={\displaystyle \frac{z_p\left(pat{h}_r^w\right)-z_p^{min}}{z_p^{max}-z_p^{min}}},(p=1,2,3).\end{array}$$

(23)

After normalization, the value of the ideal point is $(0,0,0)$. We use the normalized path point $({\overline{z}}_1\left(pat{h}_r^w\right),{\overline{z}}_2\left(pat{h}_r^w\right),{\overline{z}}_3\left(pat{h}_r^w\right))$ to calculate the distance $d_w$ to the ideal point.

$$\begin{array}{c}\hfill d_w={[\sum _{p=1}^3({\lambda }_p·{\left|{\overline{z}}_p\left(pat{h}_r^w\right)\right|}^2)]}^{1/2}\end{array}$$

(24)

where ${\lambda }_p$ is the weight of the $pth$ objective value, which reflects the decision maker’s preference and the importance of each objective in emergency logistics.

Owing to the complex multimodal transport network in practical problems, enumerating the paths from the starting point to the destination is impossible; therefore, the shortest path problem with single objectives should be solved first. First, the shortest multimodal transport path $pat{h}_p^{min}$ with p individual objectives and the minimum value $z_p^{min}$ is obtained. Subsequently, the upper limit of p individual objectives $z_p^{max}$ and the shortest path set $\mathsf{\Omega }$ of multimodal transport are obtained using the k-shortest path algorithm. It has been proven that the optimal path for a multi-objective problem must be obtained in the k-shortest path routing of a particular objective [34].

Based on the above description, we present the ideal point method based on the k-shortest multi-objective algorithm. The main procedure of the algorithm is described as follows:

•

Step 1: Based on the multimodal transportation shortest path algorithms, the shortest path $pat{h}_p^{min}$ and minimum value $z_p^{min}$ of p individual objectives are obtained.

•

Step 2: If $pat{h}_1^{min}=pat{h}_2^{min}= \dots =pat{h}_P^{min}$, then the shortest path optimal solution of multi-objective multimodal transport is obtained and terminated. Otherwise, the ideal point is determined by the minimum value $z_p^{min}$, and we set $k=2$ before moving on to Step 3.

•

Step 3: The k-shortest multimodal transport path is solved for each individual objective using the k-shortest path multi-objective algorithm.

•

Step 4: The multimodal shortest path set from the starting point to the endpoint is determined via each objective’s k-shortest multimodal transport path. The value $z_p^w$ corresponding to each $pat{h}_r^w$ from the set is then calculated, and the value interval $[z_p^{min},z_p^{max}]$ corresponding to each objective is determined.

•

Step 5: The value and ideal point of the $pat{h}_r^w$ from the set are normalized.

•

Step 6: The weighted Euclidean distance $d_w$ between the objective value and ideal point value is calculated after normalization, and the path corresponding to the minimum distance $d^{min}$ is determined.

•

Step 7: The previously obtained path is used for comparison. If the obtained path is the same, the path can be called a satisfactory solution of MOSPP-MT; otherwise, we set $k=k+1$ before moving on to Step 3.

The time complexity of the ideal point method based on the k-shortest multi-objective algorithm depends on the number of iterations k. We cannot guarantee a certain time complexity based on the input information. According to our experimental data, the algorithm will converge after three to four iterations. The ideal point method based on the k-shortest multi-objective algorithm can also determine a Pareto solution for the MOSPP-MT.

The Pareto frontier is an important concept in multi-objective optimization problems; it represents a set of solutions among which no single solution dominates. Each solution in the Pareto frontier is non-dominant, meaning that no other solution can achieve better results for all objectives at once.

We can also apply the k-shortest multi-objective algorithm to obtain the Pareto frontier, the main procedure of which may be described as follows:

•

Step 1: Based on the algorithms, the shortest path $pat{h}_p^{min}$ and the minimum value $z_p^{min}$ of each individual objective are obtained.

•

Step 2: The shortest path $pat{h}_p^{min}$ and the objectives’ values are placed in the non-dominant solution set $\mathsf{\Omega }$, and $k=2$ is set before moving on to Step 3.

•

Step 3: The k-shortest path multimodal transport path is solved for each individual objective using the k-shortest multi-objective algorithm.

•

Step 4: The set $\mathsf{\Omega }$ from the starting point to the endpoint is determined via the k-shortest multimodal transport path of each objective. The set $\mathsf{\Omega }$ is updated according to the Pareto domination principle.

•

Step 5: If the elements in set $\mathsf{\Omega }$ have not been updated and the incumbent set $\mathsf{\Omega }$ contains the Pareto frontier, the procedure is stopped; otherwise, $k=k+1$ is set before moving on to Step 3.

The procedure for building the Pareto frontier based on the k-shortest multi-objective algorithm can determine the Pareto frontier for the MOSPP-MT.

5. Computational Results

The computational experiments were performed on a PC equipped with an AMD Ryzen 7 5800H CPU with Radeon Graphics running at 3.20 GHz and 16 GB of RAM on a Windows 10 64-bit OS.

Formulae were solved using CPLEX 12.6.3. The CPLEX solver was set to use only one thread with all other parameters set to their default values. The algorithms were implemented in Python 3.8.

5.1. Test Instances

To the best of our knowledge, there are no benchmark datasets for the MOSPP-MT, so we have generated our own instance sets.

For each network of the test set, node information contains the periodic loading and unloading capacity of each node in different transportation modes. Networks are generated with the following node numbers: n = 20, 40, 60, 80 and 100. Periodic loading and unloading capacities are in the range of [1, 10], [5, 15], and [5, 10] for air, rail, and road transport, respectively.

Edge data describe the structure of different traffic networks, periodic transport capacity, time consumption, and distance to each edge. We primarily consider air, rail, and road transport the modes of transportation in each network. We generate the number of edges $\left|E\right|$ in the air, rail, and road modes using 20%, 30%, and 50% connection probabilities, respectively. The distance of each edge ranges from [200 to 400 km], [150 to 300 km], and [50 to 200 km] and the periodic transport capacity of each edge ranges from [1 to 5], [5 to 20], and [4 to 15] for air, rail, and road transport, respectively.

A total of 10 networks are generated for each node number. For each network, we randomly select three different starting points and destinations, which gives a total number of instances equal to $5\times 10\times 3=150$. The modified Dijkstra, modified Floyd, and CPLEX solver are thoroughly tested and compared. The runtime limit for each method on each instance is set at 40,000 milliseconds in comparison experiments involving the single-objective shortest path problem. The modified Dijkstra and Floyd methods are tested and compared for the MOSPP-MT.

5.2. Computational Results

5.2.1. Results of the Single-Objective Shortest Path Problem

We obtain the shortest path for single-objective multimodal transport by considering three objectives: total shipping time, total transportation distance, and total shipping cost. In this section, we test the generated instances. We randomly select three starting points and destinations within each network to generate three instances. We report the average runtime of the three instances based on each transportation network and each method.

The comparative results include the runtimes of the three objectives using the modified Dijkstra and Floyd algorithms and the CPLEX solver.

Table 4. Results of instances with $n=20$.

Runtime (ms)
Network Number	Time		Distance			Cost
	Dijkstra	Floyd	Dijkstra	Floyd	CPLEX	Dijkstra	Floyd	CPLEX
1	1.526	307.890	1.164	137.606	2010	1.120	137.475	1790
2	1.768	275.436	1.368	123.896	2000	1.108	125.149	2230
3	1.923	320.109	1.279	142.395	2200	1.186	146.075	2400
4	1.850	307.852	1.140	136.122	1360	1.203	138.782	2110
5	2.104	320.884	0.724	145.947	2450	0.905	144.727	2260
6	2.113	313.636	1.198	144.244	2110	1.203	143.978	1790
7	2.072	319.563	0.888	143.051	2000	1.309	143.779	2230
8	1.534	319.649	1.297	143.424	2200	0.945	144.944	2400
9	1.818	320.990	1.191	143.913	1360	0.983	147.005	2110
10	1.629	318.767	1.080	142.678	2450	0.903	144.687	2360

,

Table 5. Results of instances with $n=40$.

Runtime (ms)
Network Number	Time		Distance			Cost
	Dijkstra	Floyd	Dijkstra	Floyd	CPLEX	Dijkstra	Floyd	CPLEX
1	4.063	2517.492	2.105	1126.335	9400	2.808	1118.806	9100
2	5.880	2481.849	3.247	1110.388	9100	2.728	1121.214	8500
3	6.099	2509.503	3.566	1102.087	8500	2.773	1123.159	7800
4	6.066	2541.424	3.422	1106.818	10,000	2.955	1121.096	10100
5	6.279	2481.422	3.226	1103.656	9700	2.934	1118.578	10,000
6	5.687	2494.197	3.275	1111.882	9400	2.593	1111.512	9500
7	5.534	2514.802	2.515	1118.317	7700	2.540	1111.975	8000
8	5.111	2486.996	2.298	1118.430	7800	2.476	1118.745	7600
9	5.533	2456.301	2.917	1127.519	10,000	2.969	1107.332	10100
10	5.077	2471.517	2.591	1123.270	10,000	2.648	1123.721	10,000

,

Table 6. Results of instances with $n=60$.

Runtime (ms)
Network Number	Time		Distance		Cost
	Dijkstra	Floyd	Dijkstra	Floyd	Dijkstra	Floyd
1	8.013	8799.1752	4.852	5135.983	4.707	5034.457
2	11.487	8853.004	5.996	4269.060	6.184	4321.070
3	14.504	8909.514	6.646	4284.304	5.696	4138.266
4	14.050	8826.988	7.396	4316.894	6.271	4076.871
5	20.095	9007.987	7.824	4259.850	6.470	4077.893
6	10.417	8775.713	7.073	4252.811	5.870	4078.573
7	12.512	8780.538	6.418	4283.324	6.567	4133.936
8	13.451	9048.343	4.647	4182.861	6.435	4152.424
9	11.907	8791.527	8.081	4321.064	6.762	4113.640
10	12.879	8732.004	5.911	4212.571	6.313	4112.017

,

Table 7. Results of instances with $n=80$.

Runtime (ms)
Network Number	Time		Distance		Cost
	Dijkstra	Floyd	Dijkstra	Floyd	Dijkstra	Floyd
1	15.953	19,913.558	8.096	8984.820	9.051	8947.401
2	26.293	20,109.992	10.923	8906.244	9.686	8774.333
3	29.172	20,063.419	11.146	8829.007	11.689	8905.774
4	26.281	20,244.182	11.692	8920.812	11.771	8888.581
5	44.265	20,331.441	10.799	8865.159	15.148	8830.332
6	26.558	20,047.608	11.505	8852.251	12.004	8859.821
7	25.592	20,124.836	12.483	8766.580	12.150	8843.509
8	25.441	20,919.458	10.704	8702.860	14.334	8979.861
9	26.128	20,989.681	10.292	8785.187	11.745	9014.375
10	26.009	20,628.163	11.304	8874.220	11.431	8898.979

and

Table 8. Results of instances with $n=100$.

Runtime (ms)
Network Number	Time		Distance		Cost
	Dijkstra	Floyd	Dijkstra	Floyd	Dijkstra	Floyd
1	33.460	39,534.733	21.826	17,342.384	12.728	17,395.374
2	50.677	39,000.124	22.125	17,178.744	28.350	17,198.035
3	44.329	39,466.146	29.243	17,047.695	26.551	17,266.670
4	61.888	39,457.899	28.127	16,904.780	28.281	17,064.228
5	31.553	39,585.347	29.506	17,120.358	28.915	17,690.199
6	41.541	39,080.722	29.305	17,726.811	25.982	18,309.474
7	51.822	39,279.392	18.221	17,177.946	24.738	17,235.529
8	51.196	40,104.247	19.273	17,288.920	26.405	17,191.714
9	50.624	39,876.867	18.493	17,061.169	28.000	17,042.998
10	43.739	39,664.945	18.910	17,091.829	18.145	17,125.862

show the average runtime of the two improved algorithms and CPLEX. The test sets are based on the aforementioned multimodal transport networks. The results show that in terms of computational efficiency, the modified Dijkstra algorithm is significantly superior to the modified Floyd algorithm and CPLEX solver. CPLEX solver uses strategies such as branch and bound with cutting planes to solve integer programming problems. CPLEX solver takes more than 40,000 milliseconds to solve an individual objective shortest-path problem in a transportation network with more than 60 nodes. In addition, the CPLEX solver cannot find the optimal solution of the objective function $z_1\left(Time\right)$ in a reasonable time even for $n=20$. We therefore do not use CPLEX solver to solve the problem within transportation networks with 60, 80, and 100 nodes, nor do we utilize it for objective function $z_1\left(Time\right)$. We can conclude that CPLEX can not solve the MOSPP-MT with a large number of nodes, $\left|V\right|$, and edges, $\left|E\right|$, in a short time.

Furthermore, Table 4, Table 5, Table 6, Table 7 and Table 8 show that the modified Dijkstra method is superior to the modified Floyd method and CPLEX solver. The modified Dijkstra and Floyd algorithms can both solve the aforementioned problem with multimodal networks, and the modified Floyd algorithm can be applied in more complicated situations, for example, when a path between each pair of locations needs to be recommended. In our study, we consider recommended paths between specific locations, and in this case, the modified Dijkstra algorithm is more efficient than the modified Floyd algorithm. The average runtime of the modified Dijkstra method is a fraction of that of the next-best method. As the multimodal transport network node number $\left|V\right|$ and directed edge number $\left|E\right|$ increase, the runtime of the modified Floyd algorithm significantly increases. This is due to the time complexity of the algorithm. Therefore, the modified Dijkstra algorithm has higher computational efficiency and is more applicable to large-scale multimodal transport networks when solving this problem. The results validate this algorithm, indicating that it can effectively solve the MOSPP-MT.

5.2.2. Results on the MOSPP-MT

We utilize the two improved k-shortest path multi-objective algorithms to solve the MOSPP-MT using the test sets. We determine that the weight on $z_2$ is 0.2, and the weights on $z_1$ and $z_3$ are set to 0.4. We randomly set the starting point, end point, and number of batches for each instance. We report the average runtime for each method.

To compare the efficiency of the modified Dijkstra and Floyd algorithms, we measure their runtime in circumstances with various numbers of nodes. We conduct experiments by running both algorithms and increasing the number of nodes incrementally. We record the runtime of each algorithm for different networks with a given number of nodes and analyze how their performance changes as the number of nodes increases.

Table 9. Results of the MOSPP-MT.

Runtime (ms)
Network Number	n  =  20		n = 40		n = 60		n = 80		n = 100
	Dijkstra	Floyd	Dijkstra	Floyd	Dijkstra	Floyd	Dijkstra	Floyd	Dijkstra	Floyd
1	11.179	588.596	44.784	4882.604	85.326	16,301.290	206.733	38,902.186	339.047	77,914.251
2	15.110	541.650	31.755	4874.433	125.709	16,714.159	210.589	38,278.791	318.762	76,877.255
3	12.801	627.474	49.628	4920.972	99.260	15,761.860	204.511	39,059.667	449.355	77,157.263
4	13.810	594.440	58.986	5044.875	84.150	16,515.763	210.435	37,124.585	309.136	75,205.317
5	11.367	618.5491	56.371	5031.441	152.132	16,456.009	229.356	37,682.677	323.428	73,454.792
6	12.744	629.288	53.592	4816.043	101.782	16,505.280	207.110	37,708.810	380.245	75,038.122
7	11.542	623.898	51.520	4958.292	173.570	16,263.829	220.948	37,676.198	374.516	74,660.072
8	9.364	613.118	54.932	4857.111	147.121	16,186.424	245.614	38,149.750	320.815	74,537.838
9	10.691	606.866	51.112	4731.408	172.031	16,274.830	243.705	36,889.729	385.712	74,555.393
10	11.225	607.886	45.855	4385.524	170.028	16,312.275	208.677	36,668.987	352.725	73,718.964

shows that the modified Dijkstra algorithm has a runtime that is 100 times faster on average than the modified Floyd algorithm. This means that for problems of a larger size, the modified Dijkstra algorithm has higher computational efficiency in solving the MOSPP-MT when a path between specific locations is recommended, meaning it is more applicable to large-scale multimodal transportation networks.

5.2.3. Case Analysis for MOSPP-MT

To better describe the procedure of solving MOSPP-MT, the application of the proposed algorithms, and the procedure for obtaining the Pareto frontier, we present a case analysis to show how we use the ideal point method based on the k-shortest multi-objective algorithm.

In this section, we only use the modified Dijkstra algorithm as an example of a k-shortest path multi-objective algorithm. In the context of emergency logistics, a major natural disaster occurs in node 19 in a multimodal transport network (as shown in the case data presented in the appendix). The numbers of nodes and directed edges of the network are $\left|V\right|=100$ and $\left|E\right|=4927$. Rescue team A, located at node 42, receives emergency instructions to support the rescue. This team needs to transport 50 batches of commodities to the endpoint, node 34, via the multimodal transportation network.

The cost per batch per kilometer is 1.5 for air transportation, 0.6 for rail transportation, and 1.1 for road transportation, and $tc=50$. For a single objective, the shortest path with the shortest total transportation distance, minimum shipping time, and minimum total shipping cost is obtained using the modified Dijkstra algorithm. The individual objective values and corresponding shortest paths are listed in

Table 10. Individual objective values and corresponding shortest paths.

Objective	Shortest Path	Value
$z_1$	Node42–A4247(rail)–Node47–A3447(rail)–Node34	7.0 h
$z_2$	Node42–A3442(road)–Node34	112.0 km
$z_3$	Node42–A3442(road)–Node34	123.2

.

As shown in Table 10, the shortest paths for each objective are determined via the ideal point method based on the k-shortest multi-objective algorithm; these are determined to be (7.0, 112.0, 123.2) based on the value of the shortest path of each objective.

The corresponding weight of objective $z_1$ is set to ${\lambda }_1$, that of objective $z_2$ to ${\lambda }_2$, and that of objective $z_3$ to ${\lambda }_3$. The weighted Euclidean distances between the objective values and the ideal points are then calculated. Paths corresponding to the different weights of each objective are listed in

Table 11. Paths corresponding to different weight values.

Weight	Euclidean Distance	Path
${\lambda }_1=0.4, {\lambda }_2=0.2, {\lambda }_3=0.4$	0.188	Node42–A1742(road)–Node17–A1734(road)–Node34
${\lambda }_1=0.2, {\lambda }_2=0.4, {\lambda }_3=0.4$	0.139	Node42–A1742(road)–Node17–A1734(road)–Node34
${\lambda }_1=0.4, {\lambda }_2=0.4, {\lambda }_3=0.2$	0.185	Node42–A1742(road)–Node17–A1734(road)–Node34
${\lambda }_1=0.05, {\lambda }_2=0.9, {\lambda }_3=0.05$	0.223	Node42–A3442(road)–Node34
${\lambda }_1=0.9, {\lambda }_2=0.05, {\lambda }_3=0.05$	0.252	Node42–A4296(road)–Node96–A3496(road)–Node34
${\lambda }_1=0.05, {\lambda }_2=0.05, {\lambda }_3=0.9$	0.097	Node42–A1742(road)–Node17–A1734(road)–Node34

. The results show that different weight values trigger different Pareto solutions; therefore, the final result can easily be influenced by the different weights assigned by decision makers in measuring each goal.

During an actual rescue mission, the decision maker may set the weight corresponding to target $z_2$ as 0.2, and the weight of $z_1$ and $z_3$ as 0.4. Therefore, the path node 42–A1742 (road)–node 17–A1734 (road)–node 34 may be a satisfactory solution for this specific case.

To show the Pareto frontier of this case, we calculate the non-dominant solution set $\mathsf{\Omega }$, as shown in

Table 12. Non-dominated solution set $\mathsf{\Omega }$.

Num	Path	Objective Value
		${\mathit{z}}_{\mathbf{1}}\mathbf{\left(}\mathit{Time}\mathbf{\right)}$	${\mathit{z}}_{\mathbf{2}}\mathbf{\left(}\mathit{Distance}\mathbf{\right)}$	${\mathit{z}}_{\mathbf{3}}\mathbf{\left(}\mathit{Cost}\mathbf{\right)}$
$pat{h}_1$	Node42–A3442(road)–Node34	15.0	112.0	123.2
$pat{h}_2$	Node42–A4247(rail)–Node47–A3447(rail)–Node34	12.0	478.8	286.8
$pat{h}_3$	Node42–A4296(road)–Node96–A3496(road)–Node34	8.0	305.0	355.5
$pat{h}_4$	Node42–A1742(road)–Node17–A1734(road)–Node34	10.0	127.0	139.7

; Table 12 lists the objective value $z_p\left(pat{h}_r^w\right)$ for each objective on each path in the set $\mathsf{\Omega }$. In this case, the four discrete non-dominant path solutions constitute the Pareto frontier.

6. Conclusions

With the aim of tackling practical transportation problems within emergency logistics, this study investigates the MOSPP-MT by establishing a mixed-integer programming model of the multi-objective shortest path with multimodal transportation. Subsequently, modified Dijkstra and Floyd algorithms are designed to solve the single-objective shortest path problem. Based on the ideal point method, we can solve the MOSPP-MT, and the obtained Pareto solution is approximately optimal. In addition, we can obtain the Pareto frontier based on the k-shortest multi-objective algorithm. This study therefore fills a significant gap in the literature concerning the MOSPP-MT (specifically its algorithmic design).

The multimodal transportation algorithm proposed in this paper provides a satisfactory solution to the MOSPP-MT. After determining the ideal points, we can utilize the ideal point method based on the k-shortest path multi-objective algorithm to produce a Pareto solution to the multi-objective problem. Finally, computational examples are used to illustrate the efficacy and results of the proposed algorithms. Our results show that the modified Dijkstra algorithm has greater computational efficiency and more robust applicability to large-scale multimodal transport networks. Intriguing topics for future research are manifold; forthcoming studies may incorporate delivery time windows for the MOSPP-MT and consider parameter uncertainty within the problem to be tackled.