Solving the Robust Shortest Path Problem with Multimodal Transportation

Abstract

This paper explores the challenges of finding robust shortest paths in multimodal transportation networks. With the increasing complexity and uncertainties in modern transportation systems, developing efficient and reliable routing strategies that can adapt to various disruptions and modal changes is essential. By incorporating practical constraints in parameter uncertainty, this paper establishes a robust shortest path mixed-integer programming model based on a multimodal transportation network under transportation time uncertainty. To solve robust shortest path problems with multimodal transportation, we propose a modified Dijkstra algorithm that integrates parameter uncertainty with multimodal transportation. The effectiveness of the proposed multimodal transportation shortest path algorithm is verified using empirical experiments on test sets of different scales and a comparison of the runtime using a commercial solver. The experimental results on the multimodal transportation networks demonstrate the effectiveness of our approach in providing robust and efficient routing solutions. The results demonstrate that the proposed method can generate optimal solutions to the robust shortest path problem in multimodal transportation under time uncertainty and has practical significance.

1. Introduction

In the realm of urban transportation, the use of multiple modes of transportation has become a common practice for individuals seeking to navigate through bustling cities efficiently. Mulitimodal transportation includes multiple transportation modes, such as airway, rail, and road. However, the inherent uncertainties and variabilities present in each mode of transportation pose challenges to ensuring a seamless and reliable travel experience. Factors such as traffic congestion, service disruptions, and unforeseen events can significantly impact the travel time and overall comfort of commuters. To address these challenges, researchers and transportation planners have turned their attention to the robust shortest path problem within multimodal transportation networks. This complex issue involves finding optimal routes that are resilient to uncertainties and fluctuations in the transportation system, ultimately aiming to enhance the reliability and efficiency of urban travel.

In the robust shortest path problem in multimodal transportation (RSPP-MT), the study seeks to develop innovative solutions and algorithms that can navigate individuals through urban landscapes with confidence and reliability. Through an exploration of the interplay between different modes of transportation and uncertainties, this research endeavors to revolutionize urban mobility and improve the overall quality of life for city dwellers.

To solve the RSPP-MT in the strong sense for arbitrary uncertainty sets, we focus on the budgeted uncertainty sets introduced by Bertsimas and Sim [1] and their extension to variable uncertainty by Pugliese et al. [2]. In this context, robustness refers to the ability of the transportation system to handle disruptions, shown as the fluctuations of the edge weight (i.e., transportation time).

The RSPP-MT appears to be a challenging issue in transportation planning and optimization. The problem involves finding the most efficient and reliable path for passengers or freight to travel through a network with multimodal transportation. The path selection with multimodal transportation is essentially a shortest path problem (SPP) in network optimization. The SPP is a classic problem in the field of combinatorial optimization, which has attracted extensive attention in the fields of graph theory and complex networks. It has been motivated by many real-world applications such as network optimization. The earliest researched and widely used classical algorithms for SPP were proposed by Dijkstra [3] and Bellman [4]. To solve the linear optimization problems with uncertain data, Bertsimas and Sim [1] proposed a robust approach to solve linear programming (LP) and mixed-integer programming (MIP) with data uncertainties. Sivakumar et al. [5] studied the variance-constrained shortest path problem, which can be used to model any application in which the travel costs on a link are not deterministic but follow a distribution that has a possible correlation with the travel costs on other links. Stefan et al. [6] proposed a classification and a generic formulation and briefly discussed complex modeling issues involving resources. Various aspects of robust optimization have been investigated by different researchers. Jose et al. [7] studied binary linear programming problems in the presence of uncertainties that may cause solution values to change during implementation. Weimin et al. [8] proposed a tractable robust counterpart for solving the uncertain linear optimization problem with correlated uncertainties. Bertsimas and Thiele [9] proposed a general methodology based on robust optimization to address the problem of optimally controlling a supply chain subject to stochastic demand in discrete time. In addition, Bertsimas et al. [10] surveyed the known landscape of the theory and applications of robust optimization. Considering the uncertainty of the network itself, such as traffic congestion, delays, and service interruptions, Pugliese et al· [2] focused on a resource and modeled its variability through the budget of uncertainty set. Diah et al. [11] investigated the robust shortest path problem using the robust linear optimization methodology and discussed two types of uncertainty, namely box uncertainty and ellipsoidal uncertainty. Endra et al. [12] considered the application of robust optimization to model predictive control by using a box uncertainty set. Studies have shown that the use of uncertainty sets can lead to more robust and flexible decision-making strategies, allowing decision-makers to hedge against adverse outcomes and adapt to changing conditions.

Due to the uncertainties regarding transportation, the study of the SPP in an uncertainty network is much closer to the actual situation. Several research studies have addressed the robust shortest path problem with multimodal transportation using various mathematical models and optimization techniques. One approach is to develop robust optimization models that consider the uncertainty and variability in travel times and service availability. Jonathan et al. [13] extended two mathematical formulations to represent the vehicle routing problem with time windows and multiple deliverymen under uncertainty using the robust optimization paradigm with budgeted uncertainty sets and developed effective exact solution methods for solving each of them. Virginie et al. [14] considered the application of a recent criterion for the shortest path problem with uncertain arc lengths, proposing a formulation in terms of a large-scale integer linear program. As multimodal transportation has developed, Asyia et al. [15] optimized a path for transportation networks with multiple intermediate sinks between an initial source and a final sink by developing a mathematical model using forward and backward recursive computations with the objective to explore different modes of transportation for minimizing the per unit cost to deliver raw material. Rahim et al. [16] proposed an adapted evolutionary algorithm in which chromosomes with variable lengths and particularly defined evolutionary stages were used to solve the problem of time-dependent shortest multimodal paths in complex and large urban areas. Following the idea of data-driven robust optimization, André et al. [17] considered a range of uncertainty sets from the current literature based on real-world traffic measurements and then compared the performance of the resulting robust paths within and outside the sample, which allowed us to draw conclusions on the suitability of uncertainty sets. Wang et al. [18] proposed a data-driven distributionally robust shortest path model where the distribution of the travel time is only observable through a finite training dataset. Yuli et al. [19] presented a robust shortest path (RSP) model that only requires partial distribution information of travel times, including the support set, mean, variance, and correlation matrix. Wang et al. [20] proposed a data-driven distributionally robust shortest path model where the distribution of the travel time in the transportation network can only be partially observed through a finite number of samples. Ketkov et al. [21] formulated a distributionally robust version of the shortest path problem, where the arc costs were governed by some probability distribution, which was itself subject to uncertainty. However, there are a few studies on the SPP with multimodal transportation. Another approach is to use advanced algorithms (heuristic algorithms and exact algorithms), such as evolutionary algorithms, genetic algorithms, and branch and bound algorithms, to solve the robust shortest path problem. These algorithms can efficiently search for optimal or satisfactory solutions in large and complex transportation networks, considering the trade-offs between travel time, reliability, and other factors. Montemanni et al. [22] presented a branch and bound algorithm for robust shortest path problems on interval digraphs, where intervals represent uncertainty regarding real costs and a robust path is not too far from the shortest path for each possible configuration of the arc costs. Artur et al. [23] introduced new robust labels that yielded dynamic programming algorithms for the problems with time windows and capacity constraints. Mehrdad et al. [24] reformulated the SPP as a conic quadratic program and developed an outer-approximation algorithm based on this formulation.

Furthermore, some studies have focused on the interval data problem. Kasperski and Zielinski [25] proposed a polynomial time approximation algorithm to solve the problem of minimizing the maximal regret in combinatorial optimization problems with interval data. They also studied the robust shortest path problem in edge series and parallel multidigraphs with interval costs [26]. Roberto et al. [27] proposed a Benders decomposition approach for robust shortest path problems on digraphs with interval costs, where intervals represent uncertainty regarding real costs and a robust path is not too far from the shortest path for each possible configuration of the arc costs. Paweł [28] investigated the computational complexity of the robust version of the shortest path problem, where arc lengths are specified as interval numbers. Oya et al. [29] modeled data uncertainty by treating the arc lengths as interval ranges.

However, most of the existing studies handled uncertain factors with a mono-modal mode of transportation in the network. Few studies dealt with a robust shortest path problem with multimodal transportation, incorporating budget uncertainty sets into the decision models.

Overall, the literature on the robust shortest path problem in multimodal transportation demonstrates the importance of considering both efficiency and reliability in transportation planning and optimization. The future research in this area may continue to explore new mathematical models, optimization techniques, and decision support tools to address the complex challenges of multimodal transportation and improve the overall travel experience for passengers and freight.

2. Problem Description and Mathematical Formulation

In modern cities, people often need to utilize various modes of transportation (such as airway, railway, road, etc.) to transport materials. However, each mode of transportation may be subject to uncertainties in its operation and service quality, such as traffic congestion, vehicle breakdowns, and service interruptions, which can impact the efficiency and comfort of travel. For such multimodal transportation networks, it is necessary to study methods to solve the robust shortest path problem. The robust shortest path problem takes into account uncertainties in transportation, aiming to find a path that performs well under various possible scenarios to ensure travel efficiency and reliability.

In this study, multimodal transportation networks and task information should be considered when modeling the optimization problem. A dispatch task normally has a specific origin and a destination, and the total amount of supplies to be delivered. We must consider the traffic information between nodes (i.e., the number of batches that can be transported in each time unit) for the edges and the time uncertainty of the traffic network.

Based on these demands, a robust shortest path mixed-integer programming model is established in this section. To accomplish this, several relevant assumptions are presented first:

• Three transportation modes are considered: airway, railway, and road in decreasing priority order.
• The transportation mode can only be transferred from a higher-priority mode to a lower-priority mode. 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 unit for transportation.

The transportation network is denoted as $N=(V,E)$, shown in Figure 1. We denote the origin and destination of the task as s and q, respectively, and $s,q\in V$. The sub-network of transportation mode f can be abstracted as $N_f=(V_f,E_f)$, 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 under 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 under 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 $E=\{E_{f_1}\cup E_{f_2}\cup \dots \cup E_{f_{\left|F\right|}}\}$.

To describe the model clearly,

Table 1. Notations of the related parameters.

Notations	Definition
F	Set of transportation modes
f	Index of transportation modes and $f\in F$
$N_f$	Sub-network of transportation mode f
$t{r}_{ij}^f$	Transportation time of the edge ${(i,j)}_f$ under transportation mode f
${\widehat{tr}}_{ij}^f$	Transportation time uncertainty of the edge ${(i,j)}_f$ under transportation mode f
$s{w}^{f_1,f_2}$	Whether the task can transfer from transportation mode $f_1$ to $f_2$
	($f_1$ has a higher priority than $f_2$)
s	Origin of the task
q	Destination of the task
$\Gamma$	The count number of uncertainty

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

Table 2. Notations of the decision variables.

Notations	Definition
$x_{ij}^f$	Binary variable indicating whether the task selects
	the edge ${(i,j)}_f$ to transport under transportation mode f
$y_{ij}^f$	Binary variable indicating whether the task selects
	the edge ${(i,j)}_f$ with time uncertainty under 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 $f_2$)

presents the notations of the decision variables.

In this section, we propose a robust linear formulation. For every selected edge, we introduce an integer parameter $\Gamma$. Let $\Gamma$ be the count number of coefficients $t{r}_{ij}^f$, which is subject to the transportation time uncertainty. $\Gamma$ takes values according to a symmetric distribution with mean equal to the transportation time of the edge $t{r}_{ij}^f$ in the interval [$t{r}_{ij}^f-{\widehat{tr}}_{ij}^f$, $t{r}_{ij}^f+{\widehat{tr}}_{ij}^f$]. As clarified below, the role of parameter $\Gamma$ is to adjust the robustness of the proposed method against the level of conservatism of the solution. Speaking intuitively, it is unlikely that all the $t{r}_{ij}^f$ values will change. Our goal is to protect against all these coefficients of $t{r}_{ij}^f$ that are allowed to change, but some of the coefficients are changed by $\Gamma$. We model the problem as follows:

$$\begin{array}{cc}\hfill & minT\hfill \end{array}$$

(1)

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

(2)

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

(3)

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

(4)

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

(5)

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

(6)

$$\begin{array}{cc}\hfill & \sum _{f_2\in F\setminus f_1}swi{t}_i^{f_1,f_2}+\sum _{j:{(j,i)}_{f_2}\in E_{f_2}}{x}_{ji}^{f_2}=\sum _{k:{(i,k)}_{f_2}\in E_{f_2}}{x}_{ik}^{f_2}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\sum _{f_3\in F\setminus \{f_1,f_2\}}swi{t}_i^{f_2,f_3},\forall i\in V\setminus \{s,q\},\forall f_1,f_2,f_3\in F;\hfill \end{array}$$

(7)

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

(8)

$$\begin{array}{cc}\hfill & \sum _{f\in F}\sum _{{(i,j)}_f\in E_f}t{r}_{ij}^f·x_{ij}^f+\underset{\{S_f\mid S_f\subseteq E_f,{\sum }_{f\in F}|S_f|=\Gamma \}}{max}\left\{\sum _{f\in F}\sum _{{(i,j)}_f\in {\mathcal{S}}_f}{\widehat{tr}}_{ij}^f{y}_{ij}^f\right\}\le T;\hfill \end{array}$$

(9)

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

(10)

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

(11)

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

(12)

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

(13)

Objective function (1) minimizes the total transportation time. Constraints (2)–(4) ensure that the transportation route of a task is geographically continuous. Constraints (5)–(7) 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. Constraint (8) ensures that transfer can only be executed on nodes where it is allowed. Constraint (9) ensures that the solution adheres to a predefined limit on the total transportation time. Constraint (10) ensures that the transportation time uncertainty of the edge must occur only after it has been selected. Constraint (11) ensures that decision variable $y_{ijt}^f$ is non-negative. Constraints (12) and (13) stipulate that the decision variables $x_{ij}^f$, and $swi{t}_i^{f_1,f_2}$ are binary variables.

By strong duality, since constraint (9) is feasible and bounded for $\Gamma$, then the dual constraint (9) is also feasible and bounded and objective values coincide. Using a proposition introduced by Bertsimas and Sim [1], we reformulate the constraint (9) as a linear constraint as follows:

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

(14)

$$\begin{array}{cc}\hfill & z+p_{ij}^f\ge {\widehat{tr}}_{ij}^f{y}_{ij}^f,\forall f\in F,\forall {(i,j)}_f\in E_f;\hfill \end{array}$$

(15)

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

(16)

$$\begin{array}{cc}\hfill & z\ge 0;\hfill \end{array}$$

(17)

Through the application of duality transformations, we derive a model that is amenable to linear optimization techniques. This transformation has enabled us to reformulate the problem into a more tractable form, facilitating the application of linear programming methods to achieve an optimal solution.

3. Robust Shortest Path Algorithm with Multimodal Transportation

In order to address the RSPP-MT, we develop a robust shortest path algorithm with multimodal transportation. In this section, we introduce the multimodal transportation shortest path algorithm based on the Dijkstra algorithm that can effectively handle the RSPP-MT.

The Dijkstra algorithm was proposed by Dutch computer scientist Dijkstra in 1959. 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. Related notations of the modified Dijkstra algorithm.

Notations	Definition
$N_{f_l}$	Sub-network of priority level l under transportation mode f
$s_l^m$	Origin of the sub-network of priority level l under mth state
$q_l^m$	Endpoint of the sub-network of priority level l under mth state
$f^i$	Transportation mode f reaching node i
$v_j^{f_j}$	Node j in a partial path reached by transportation mode f

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

The main concept of the modified Dijkstra algorithm for the RSPP-MT is described as follows:

• Determine the sub-network of the highest priority $N_{f_{high}}$ of the origin s and mark the priority level as a, starting from node $s_a$.
• Introduce $\Gamma$ to record the time uncertainty. Set every node that has $\Gamma +1$ states. Each state records the specific time uncertainty count; i.e., the ith state counts the nodes with i edge time uncertainties. The ith state also records the minimum objective value with $i-1$ edge time uncertainties. The nodes under ith state can only update minimum objective values by the nodes with the ith state and the nodes with the ($i-1$)th state plus one more edge time uncertainty. Initially, the 1st state counts the nodes with one edge time uncertainty and the minimum objective value without edge time uncertainties.
• Introduce sets P and H. Set P records the node with $\Gamma +1$ states whose minimum objective value has been obtained; set H records the node with $\Gamma +1$ states that has not been visited. Initially, set P contains only node $s_a$ with 1th state.
• Traverse set H to determine the node with the minimum objective values for $\Gamma +1$ states and then add the node to set P and update the objective value and path of the node in set H.
• Repeat the above steps until the endpoint $q_p(p\ge a)$ under ($\Gamma +1$)th state is reached and compare the objective value of different endpoints $q_p(p\ge a)$ under ($\Gamma +1$)th state to determine the minimum value.

The pseudocodes for the modified Dijkstra algorithm addressing the RSPP-MT are presented in Algorithm 1. The algorithm has a time complexity of $O\left(\left(\right|V|+|E\left|\right)·log\left(\right|V\left|\right)·(\Gamma +1)\right)$. The modified algorithm for the total transportation time objective T is an exact algorithm.

To verify the correctness of the exact algorithms, Proposition 1 proves that the RSPP-MT under the total transportation time objective T has an optimal substructure, where $SP(i^m,j^m,f^{i^m},f^{j^m})$ is the shortest path from i to j using transportation modes with priority levels from $f^i$ to $f^j$ under the mth state.

Algorithm 1: modified Dijkstra algorithm for time minimization

Proposition 1. 

Assume that node q under the Γ state has a set $U_q^{\Gamma }$ with $(\Gamma +1)$ largest transportation time uncertainties among the path. For all the nodes $\{k,\dots ,m,\dots ,n\}$ directly connect to node q, the robust shortest path $SP(s^{\Gamma },q^{\Gamma })$ is denoted as $SP(s^{\Gamma },q^{\Gamma })=min\{(SP(s^{\Gamma },k^{\Gamma })+t{r}_{kq}^f-U_k^{\Gamma }+U_q^{\Gamma }),\dots ,(SP(s^{\Gamma },m^{\Gamma })+t{r}_{mq}^f-U_m^{\Gamma }+U_q^{\Gamma }),\dots ,$ $(SP(s^{\Gamma },n^{\Gamma })+t{r}_{nq}^f-U_n^{\Gamma }+U_q^{\Gamma }),\dots ,$ $(SP(s^{(\Gamma -1)},k^{(\Gamma -1)})+t{r}_{kq}^f-U_k^{(\Gamma -1)}+U_q^{\Gamma }),\dots ,$ $(SP(s^{(\Gamma -1)},m^{(\Gamma -1)})+t{r}_{mq}^f-U_m^{(\Gamma -1)}+U_q^{\Gamma }),\dots ,$  $(SP(s^{(\Gamma -1)},n^{(\Gamma -1)})+t{r}_{nq}^f-U_n^{(\Gamma -1)}+U_q^{\Gamma })\}$. There must be $SP(w^{\Gamma },j^{\Gamma })$, which is the shortest path from node w to node j under the Γ state. Therefore, the multimodal shortest path has an optimal substructure.

Proof. 

Given that the shortest path from node s to node q under the $\Gamma$ state is $SP(s^{\Gamma },q^{\Gamma })$, $SP(s^{\Gamma },q^{\Gamma })=SP(s^{\Gamma },w^{\Gamma })+SP(w^{\Gamma },j^{\Gamma })+SP(j^{\Gamma },q^{\Gamma })$. If $SP(w^{\Gamma },j^{\Gamma })$ is not the shortest path from w to j under the $\Gamma$ state, then another shortest path exists, denoted as $\overline{SP}(s^{\Gamma },q^{\Gamma })=SP(s^{\Gamma },w^{\Gamma })+\overline{SP}(w^{\Gamma },j^{\Gamma })+SP(j^{\Gamma },q^{\Gamma })$, which is the new shortest path from node s to node q with a lower transportation cost, which results in a contradiction. □

4. Computational Results

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

The formulations are solved using CPLEX 12.6.3. The CPLEX solver is set to use only one thread, with all the other parameters set to their default values. The algorithm is implemented using Python 3.8.

4.1. Test Instances

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

For each network of the test set, the node information contains the periodic loading and unloading capacities in the different transportation modes of each node. The networks are generated with node number n = 20, 40, 60, 80, and 100.

Edge data describes the structure of different traffic networks, periodic transport capacity, time consumption, and distance of each edge. We primarily consider airway, railway, and highway as three modes of transportation in each network. We generate the number of edges $\left|E\right|$ in the airway, railway, and highway modes with 20%, 30%, and 50% connection probabilities, respectively. The transport time of each edge ranges as [2, 4], [5, 6], and [7, 8] by airway, railway, and highway, respectively. The transport time uncertainty of each edge ranges as [1, 2], [1, 2], and [2, 4] by airway, railway, and highway, respectively.

Ten networks are generated for each node number. For each network, we randomly select three different origins and destinations, providing a total number of instances equal to $5\times 10\times 3=150$. The modified Dijkstra and CPLEX solver are thoroughly tested and compared for the RSPP-MT.

4.2. Computational Results

We compare the quality of the solutions by changing the uncertainty number $\Gamma$ in transportation times. We conduct our analysis on a generated dataset of three instances. $\Gamma$ takes values 1, 2, and 3 for the RSPP-MT. Based on the modified Dijkstra algorithm, we apply simulation to compare the performance of uncertainty $\Gamma$. We examine two important processing time distributions: the uniform and normal distribution. The nominal transportation times $\mu$ are drawn from the transport time of each edge $t{r}_{ij}^f$, and, for the chance-constrained models, $\sigma$ is taken from the transport time uncertainty of each edge, satisfying $\sigma \le \lfloor \mu /3\rfloor$, while the maximum deviation is taken as ${\widehat{tr}}_{ij}^f=3\sigma$ in the RSPP-MT. We generate 100 samples for each instance based on $\mu$ and $\sigma$ under each distribution and show them in Figure 2 and Figure 3.

In these figures, the x-axis represents the transportation times of the solution, while the y-axis denotes the density of the transportation times. The histogram and line chart represent the actual usage times simulated through sampling from two distributions after selecting this path via the algorithm. The red dashed line indicates the path time calculated by the algorithm. From Figure 2 and Figure 3, we see that the $\Gamma$ = 1 tends to deliver solutions with smaller sample means, variance, and percentiles compared with other $\Gamma$: RSPP-MT solutions for higher $\Gamma$ that do not protect the transportation time well for the chosen simulation setting. Larger $\Gamma$ values clearly generate overly conservative time estimates. Therefore, in this study, we prefer to use smaller $\Gamma$ values for statistical analysis as smaller $\Gamma$ values tend to yield better practical outcomes in real-world applications.

We obtain the shortest path of multimodal transport with an objective of transportation time by considering the transportation time certainty. In this section, we use the generated instances for the test. We randomly select three origins and destinations of each network to generate three instances. We report the average runtime of the three instances based on each transportation network for each method.

The comparison results include the runtimes of using the modified Dijkstra, the CPLEX solver, under uncertainty $\Gamma$ = 1, 2, 3, as shown in

Table 4. Results of the RSPP-MT ($\Gamma$ = 1).

Runtime (ms)
Network Number	n = 20			n = 40			n = 60			n = 80			n = 100
	Dijkstra	CPLEX		Dijkstra	CPLEX		Dijkstra	CPLEX		Dijkstra	CPLEX		Dijkstra	CPLEX
1	7.278	423.002		110.595	2637.592		550.793	8711.174		993.246	24,327.503		2862.953	54,215.121
2	11.253	493.511		173.847	2882.893		718.668	10,433.196		2240.766	29,497.623		3964.262	66,170.130
3	24.372	441.005		157.540	2932.139		1429.670	10,783.470		2521.793	30,265.798		8196.901	70,150.986
4	18.103	430.475		233.134	2954.558		755.944	11,094.770		4505.756	30,990.257		11,078.559	71,324.390
5	15.489	510.637		159.575	2954.820		1132.525	10,962.366		4841.039	30,835.058		8426.504	72,846.445
6	19.535	463.967		128.296	3051.806		1103.992	11,132.195		2413.254	31,218.689		11,213.833	75,495.389
7	15.613	430.851		271.472	2980.387		902.963	10,999.644		4860.665	31,540.346		10,490.408	73,807.403
8	17.871	475.181		212.329	2909.462		1200.686	10,985.363		4174.486	31,643.620		10,672.481	73,901.113
9	14.301	412.058		265.682	2874.462		1162.846	10,989.971		2847.673	31,208.782		12,031.574	72,515.095
10	22.682	412.400		192.382	2960.643		1525.328	11,214.807		3497.445	31,087.398		11,068.007	71,393.307

,

Table 5. Results of the RSPP-MT ($\Gamma$ = 2).

Runtime (ms)
Network Number	n = 20			n = 40			n = 60			n = 80			n = 100
	Dijkstra	CPLEX		Dijkstra	CPLEX		Dijkstra	CPLEX		Dijkstra	CPLEX		Dijkstra	CPLEX
1	17.238	418.732		268.129	2550.418		1402.919	9462.940		4733.922	26,149.567		11,078.051	57,176.717
2	32.565	494.705		496.606	3186.625		2464.0584	10,898.998		7519.554	31,542.232		19,485.171	69,680.467
3	33.823	432.627		650.264	3264.560		2823.364	11,352.557		8517.058	33,131.995		20,796.233	72,820.373
4	45.527	510.491		680.459	3353.501		2472.157	11,629.197		8676.551	33,657.071		22,186.316	74,530.851
5	50.290	591.610		658.664	3301.744		2874.847	11,514.545		9861.291	32,806.336		24,753.942	74,546.622
6	40.464	524.168		620.006	3205.702		3118.409	11,787.018		8990.755	32,455.911		20,949.771	74,844.672
7	43.185	551.411		524.797	3145.348		2940.171	115,03.829		8922.650	32,497.091		27,031.230	74,868.860
8	51.979	506.058		634.052	3103.811		3273.129	11,674.437		9951.441	32,762.134		26,563.325	74,771.714
9	37.312	434.919		574.568	3063.391		2897.492	11622.295		9999.172	32,635.582		19,098.746	75,294.255
10	52.501	504.257		644.516	2956.802		2788.958	11,593.289		10,829.958	32,696.117		25,021.439	74,954.247

and

Table 6. Results of the RSPP-MT ($\Gamma$ = 3).

Runtime (ms)
Network Number	n = 20			n = 40			n = 60			n = 80			n = 100
	Dijkstra	CPLEX		Dijkstra	CPLEX		Dijkstra	CPLEX		Dijkstra	CPLEX		Dijkstra	CPLEX
1	36.850	396.278		421.341	2592.620		2122.296	9916.470		7524.603	25,311.944		17,220.206	56,887.450
2	59.978	435.166		766.841	2854.324		3643.604	11,452.806		13,116.612	30,177.502		26,834.291	70,150.291
3	59.525	499.039		1002.466	3149.591		4057.011	11,901.889		13,966.180	31,974.920		33,399.846	73,205.453
4	65.489	436.960		1030.648	2966.882		4138.071	11,837.376		14,790.347	32,599.834		36,514.560	74,773.483
5	90.973	420.973		918.835	3030.146		5218.938	11,922.836		15,099.412	32,734.140		35,751.015	75,615.666
6	82.274	490.658		1023.838	3005.166		5001.499	12,227.741		15,279.666	32,586.722		38,369.579	74,842.386
7	83.950	451.949		1112.809	3043.065		4341.996	12,052.712		14,976.950	32,887.711		34,817.180	75,255.087
8	73.490	428.104		1038.935	3003.592		4998.427	12,078.351		15,288.732	32,947.935		36,449.765	75,578.462
9	94.633	489.919		1081.681	2968.710		4454.378	12,024.727		15,614.585	32,677.216		33,693.933	74,732.153
10	75.327	438.935		982.654	3030.528		4174.200	11,938.702		14,837.810	32,720.128		32,077.390	75,442.775

.

Table 4, Table 5 and Table 6 show the average runtime of the improved Dijkstra and CPLEX under uncertainty $\Gamma$ = 1, 2, and 3. 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 CPLEX solver. The average runtime of the modified Dijkstra method is mostly a fraction of the runner-up method. As the multimodal transport network node number $\left|V\right|$, directed edge number $\left|E\right|$, and the multimodal transport network uncertainty $\Gamma$ increase, the runtime of the CPLEX solver significantly increases. This is related to the time complexity of the CPLEX solver’s algorithm. The CPLEX solver uses strategies such as branch and bound with cutting planes to solve integer programming problems. Therefore, the algorithm based on the modified Dijkstra algorithm has a higher computational efficiency and is more capable in terms of large-scale multimodal transport networks regarding this problem. The results show that the improved Dijkstra algorithm can effectively solve the RSPP-MT and verify its validity. The modified Dijkstra algorithm can achieve the optimal solution within one percent of the time required by the CPLEX solver, greatly improving the efficiency of solving the RSPP-MT.

Table 4, Table 5 and Table 6 also show that the modified Dijkstra algorithm has a running time that is on average 10 times faster than the CPLEX solver in most cases. This means that, for larger problem sizes, the modified Dijkstra algorithm has a higher computational efficiency for the RSPP-MT with the path recommendation setting between specific locations, which has stronger applicability for large-scale multimodal transportation networks.

5. Conclusions

Aiming at practical transportation problems in parameter uncertainty, this paper investigates the RSPP-MT. We establish a mixed-integer programming model of the robust shortest path with multimodal transportation. Subsequently, the modified Dijkstra algorithm is designed to solve the RSPP-MT. The study fills the gap in the RSPP-MT, specifically regarding its algorithm design.

The multimodal transportation algorithm proposed in this paper provides an optimal solution to the RSPP-MT with the total transportation time under time uncertainty. Finally, computational instances are provided to analyze the results of the proposed algorithm. The results show that the modified Dijkstra algorithm has higher computational efficiency and more robust applicability to large-scale multimodal transport networks. The topics for future research are manifold, such as a multi-objective approach for the RSPP-MT and the consideration of multi-parameter uncertainties in the problem description.