A Comparison of Three Real-Time Shortest Path Models in Dynamic Interval Graph

Abstract

The Dynamic Interval (DI) graph models the updating uncertainty of the arc cost in the graph, which shows great application prospects in unstable-road transportation planning and management. This paper studies the Real-time Shortest Path (RTSP) problems in the DI graph. First, the RTSP problem is defined in mathematical equations. Second, three models for RTSP are proposed, which are the Dynamic Robust Shortest Path (DRSP) model, the Dynamic Greedy Robust Shortest Path (DGRSP) model and the Dynamic Mean Shortest Path (DMSP) model. Then, three solution methods are designed. Finally, a numerical study is conducted to compare the efficiency of the models and corresponding solution methods. It shows that the DGRSP model and DMSP model generally present better results than the others. In the real road network test, they have the minimum average-regret-ratio of DGSP 7.8% and DMSP 7.1%; while in the generated network test, they both have a minimum average-regret-ratio of 0.5%.

1. Introduction

With the fast development of Global Positioning System (GPS) technologies and the wide use of mobile phones, innovative services have emerged, such as electronic maps, ride-sharing and ride-hailing. Among these services, static and dynamic Vehicle Routing (VR) functions are frequently used, especially in unstable-road transportation. It mainly helps the traveler find the Real-time Shortest Path (RTSP) from the starting position (node) to the destination position (node).

There are six kinds of Shortest Paths for VR functions corresponding to six kinds of graphs (see

Table 1. Six kinds of shortest path problems in six graphs.

Graphs	Problems	Arc Cost	Search
SD	SP	exact value	static
SS	SSP	random variable	static
SI	RSP	interval value	static
DD	DSP	exact value depending on time	dynamic
DS	DSSP	random variable depending on time	dynamic
DI	DRSP	interval value depending on time	dynamic

and Figure 1).

The Static and Deterministic (SD) graph and Shortest Path (SP) problem (Figure 1a). In the SD graph, the cost of each arc is directly known. Dijkstra [1] first studies the SP problem in the SD graph, and designs a famous dynamic programming algorithm, named the Dijkstra algorithm. Then, the Floyd–Warshall algorithm and Bellman–Ford algorithm are proposed for the SP problem by Floyd [2], Bellman [3] and Ford and Fulkerson [4]. Himmich et al. [5] study an extension of the SP problem. In addition to the deterministic arc cost, the resource constraints are considered in an R-dimensional resource consumption vector. A multi-phase dynamic programming algorithm is designed to solve the model. Bahel et al. [6] study an application of the SP in the area of agents shipping their respective demands, that is, more than one traveler (visitor) should find their SPs in the model.

The Dynamic and Deterministic (DD) graph and Dynamic Shortest Path (DSP) problem (Figure 1d). In the DD graph, the cost of each arc changes over time (or stage) t. Previous work on algorithms for DSP problems in DD graph includes Murchland [7], Dionne [8], and Goto and Sangiovanni–Vincentelli [9]. Then, Ramalingam and Reps [10] develop a widely-used algorithm, named RR. It presents good performance in most situations. By observing that the RR algorithm updates, in the data structure aspect, the shortest path graph instead of the shortest path tree, Demetrescu, et al. [11] propose a specialization of the RR algorithm for updating a shortest path tree. Then, Buriol, et al. [12] introduce a new generic technique to reduce the heap (priority queue) size.

The Static and Stochastic (SS) graph and Stochastic Shortest Path (SSP) problem (Figure 1b). The cost of each arc in the graph is modeled by an independent random variable. Martin [13] studies the shortest path in stochastic networks, which is also directed and acyclic. Then, Frank [14] and Hassin and Zemel [15] study the stochastic graph where all the arcs’ cost is known before entering into the network and the goal is to find the shortest path in this implementation. Mirchandani and Soroush [16,17] assume that the actual cost of all arcs is unknown before the path is selected, and the goal is to choose a path that minimizes the expected total cost. Beigy and Meybodi [18] study the SSP with the positive-valued random variable arc cost sampled from a distribution, and an iterative stochastic algorithm using a distributed learning automata to find the SSP by taking a sufficient number of samples. Ketkov et al. [19] study the SSP with the arc cost subject to distributional uncertainty, and the decision-maker minimizes the worst-case expected loss over an ambiguity set (or a family) of candidate distributions. It is the combination of stochastic uncertainty and interval uncertainty. Lee et al. [20] study the SSP to maximize the probability of arriving at the destination within a given time budget with the arc cost (travel time) following an independent normal distribution and develop pseudo-polynomial time exact algorithms. Song and Cheng [21] concentrate on the SSP problem with the mean and standard deviation of arc cost (travel time) directly known. They transform the problem into a mixed-integer conic quadratic program and design a generalized Bender’s decomposition solution approach.

The Dynamic and Stochastic (DS) graph and Dynamic Stochastic Shortest Path (DSSP) problem (Figure 1e). The cost of each arc in the graph is modeled by a random variable depending on time. Hall [22] studies the stochastic network with travel times that are both random and time-dependent. The goal is to find the least expected travel time path. It is concluded that an adaptive decision rule is better than a simple path. Bertsimas and Van Ryzin [23,24] study the stochastic and dynamic vehicle routing problem, where vehicles traveling at constant velocities must serve stochastic demands. Psaraftis and Tsitsiklis [25] examine the SP problems, where the arc costs are the known functions of certain environmental variables following independent Markov dynamic processes. For more relevant research see Azaron and Kianfar [26], Pattanamekar et al. [27], and Ojeda Rios et al. [28].

The Static and Interval (SI) graph Robust Shortest Path (RSP) problem (Figure 1c). The cost of each arc in the graph is uncertain and modeled by an interval value. Dias and Climaco [29] first study the Robust Shortest Path (RSP) problem in an interval graph. Averbakh and Lebedev [30] and Zielinski [31] simultaneously prove the NP-hardness of the RSP problem. Karasan et al. [32] introduce a mixed integer programming (MIP) formulation for the RSP problem. Then, Montemanni et al. [33] present a branch and bound algorithm for RSP, while Montemanni and Gambardella [34] give an exact algorithm based on Benders decomposition. The approximate algorithm for RSP is designed by Kasperski and Zielinski [35]. Catanzaro et al. [36] reduce the search domain by giving sufficient conditions for nodes and arcs which are always (or never) in an optimal solution. Chassein et al. [37] consider RSP problems to find a path that optimizes the worst-case performance over an uncertainty set for arc costs where one of the uncertainty measures is the interval uncertainty. The data-driven robust optimization is conducted based on real-world traffic measurements from Chicago City. Davoodi and Ghaffari [38] model the uncertain networks with interval arc. To find the SP, they propose a practical approach with two phases, that is, a preprocessing phase and a query phase. Zhang [39] studies the unique shortest path routing problem, which aims to find an optimal weight set subject to four sets of constraints. Interval arc cost appears as a set constraint of link capacity.

The Dynamic and Interval (DI) graph Dynamic Robust Shortest Path (DRSP) problem (Figure 1f). The cost of each arc in the graph is uncertain and modeled by an interval value depending on time. Compared to the SI graph, the DI graph makes full use of the updated information and shows great application prospects in the transportation application. Xu and Zhou [40] study the Dynamic Robust Shortest Path (DRSP) problem in the DI graph. The Nested Dijkstra (ND) algorithm is given as a heuristic algorithm.

This paper concentrates on the DI graph. First, a general formula of the RTSP problem in the DI graph is given. Then, instead of the DRSP model, two models are proposed, which are the Dynamic Greedy Robust Shortest Path (DGRSP) model and the Dynamic Mean Shortest Path (DMSP) model. The DRSP model considers the uncertainty of all arcs equally; the DGRSP model considers more about the uncertainty of the nearby arcs; the DMSP model uses the interval mean value to approximately replace the uncertainty. Then, three solution methods are designed for the corresponding three models, which are the CPLEX optimizer, ND algorithm and Dijkstra algorithm. Finally, a numerical study is conducted in both real road networks and generated networks to validate the efficiency of the models and algorithms. It shows that the DGRSP model and ND algorithm generally return the best solution.

This paper is organized as follows: The SP problem in the DI graph is formally described in Section 2. The DRSP model, DGRSP model and MSP model, as well as their corresponding algorithms and solution methods, are the main body of this paper. They are described in Section 3, Section 4 and Section 5, respectively. In Section 6, the numerical study is conducted, which validates the efficiency of the models and algorithms. The main conclusions are summarized in Section 7. The abbreviations used in this paper are in

Table 2. Most used abbreviations in this paper.

Abbr	Full Terms	Abbr	Full Terms
DD	Dynamic and Deterministic	RC	Robust Cost
DGRSP	Dynamic Greedy Robust Shortest Path	RD	Robust Deviation
DI	Dynamic Interval	RSP	Robust Shortest Path
DMSP	Dynamic Mean Shortest Path	RTSP	Real-time Shortest Path
DRSP	Dynamic Robust Shortest Path	SD	Static and Deterministic
DS	Dynamic and Stochastic	SI	Static and Interval
DSP	Dynamic Shortest Path	SP	Shortest Path
DSSP	Dynamic Stochastic Shortest Path	SS	Static and Stochastic
DVR	Dynamic Vehicle Routing	SSP	Stochastic Shortest Path
MIP	Mixed Integer Programming	VR	Vehicle Routing
ND	Nested Dijkstra

.

2. Problem Definition

The DI graph aims to model the real-time variation of the graph. The RTSP problem in the DI graph is proposed for two reasons. First, in unstable areas such as earthquake rescue areas and war rescue zones, it tries to reduce the timely uncertainty and helps more people escape from difficulties; second, in daily transportation, it timely reduces the uncertainty and improves the scheduling efficiency. The real-time tracking system including GPS makes this operation possible. The RTSP model returns the real-time solution path with the next visiting node which forms the final routing path with minimized cost for the traveler. Compared to the traditional SP, it dynamically changes routes with time and controls the uncertainty by interval cost and minimized regret.

Suppose $G=(V,A)$ is a directed graph with nodes set V and arcs set A, $\left|V\right|=n$, $\left|A\right|=m$. Node s is the starting (origin) node, and e is the end (destination) node, $s,e\in V$. Suppose p is a path from s to e, and P is a path set containing all paths from s to e. At least one path p, from s to e, exists in the graph G. We have $p\in P$ and $P\ne \varnothing$. When the visitor in stage t stays at a node $s\left(t\right)$ in the graph, there are k frozen nodes determined to visit for the next k stages, which are $\left\{s\right(t+1),s(t+2),\cdots ,s(t+k\left)\right\}$. Generally, the number of frozen nodes is often 1 or 2 in the transportation model, that is, $k=1,2$. Too many frozen nodes may reduce the usage of the updated information. We optimize the path from $s(t+k)$ to e.

At the beginning of initialization, SP finds an optimized path from $s\left(1\right)$ to e to initiate the frozen nodes set (see in Figure 2a). In the first stage, the visitor stays at node s, and $s\left(1\right)=s$. The frozen nodes are $s\left(2\right),s\left(3\right),\cdots ,s(1+k)$. It visits node $s\left(2\right)$ in the path directly connected to $s\left(1\right)$, finds an optimized path from $s(1+k)$ to e, and adds node $s(2+k)$ to the frozen node set. At the 2nd stage, the visitor stays at node $s\left(2\right)$. The frozen nodes become $s\left(3\right),s\left(4\right),\cdots ,s(2+k)$. It visits node $s\left(3\right)$, finds an optimized path from $s(2+k)$ to e, and adds node $s(3+k)$ to the frozen node set. So again, at the tth stage, the frozen nodes are $s(t+1),s(t+2),\cdots ,s(t+k)$. The visitor visits node $s(t+1)$, finds an optimized path from $s(t+k)$ to e, and adds node $s(t+k+1)$ to the frozen node set, $t=1,2,3,\cdots ,n_p-k-1$. Finally, suppose at the $n_p$th stage, the visitor reaches the destination node e, and finishes the travel with $s\left(n_p\right)=e$. So the whole path is $s\left(1\right),s\left(2\right),s\left(3\right),\cdots ,s\left(t\right),\cdots ,s\left(n_p\right)$.

The possible cost $c_{ij}\left(t\right)$ of arc $(i,j)$ in stage t is in the interval $[{\underline{c}}_{ij}\left(t\right),{\overline{c}}_{ij}\left(t\right)]$, that is, ${\underline{c}}_{ij}\left(t\right)\le c_{ij}\left(t\right)\le {\overline{c}}_{ij}\left(t\right)$ (see in Figure 1f). $c_{ij}\left(t\right)$ is known in stage t, and thus in the final stage $t=n$ the problem is a DSP problem (offline problem), and the optimal path can be calculated by dynamic programming. However, before stage $t=n$, it is an online problem, only God knows the optimal path. In the real-world application scenario, the road may face extreme variations such as sudden closures. It can be modeled by setting ${\overline{c}}_{ij}\left(t\right)$ with a large number if the closures happen in stage t.

The objective of online RTSP is to find the shortest path, with minimized cost, from s to e. Suppose $s\left(1\right),s\left(2\right),s\left(3\right),\cdots ,s\left(t\right),\cdots ,s\left(n_p\right)$ is a path p from s to e in path set P. Then, the objective is

$$C_{\mathrm{on}}=\underset{p\in P}{min}\sum _{t=1}^{n_p-1}{c}_{s\left(t\right)s(t+1)}$$

(1)

which compares all possible paths in P to achieve the optimal path. The variable of arc cost $c_{s\left(t\right)s(t+1)}$ is essential since it affects the final total cost. Suppose the final offline optimal path for RTSP from s to e is $\tilde{s}\left(1\right),\tilde{s}\left(2\right),\tilde{s}\left(3\right),\cdots ,\tilde{s}\left(t\right),\cdots ,\tilde{s}\left(\tilde{n}\right)$, where $\tilde{s}\left(1\right)=s$, $\tilde{s}\left(\tilde{n}\right)=e$, and $\tilde{n}\le n$. Then, the length of the optimal path is

$$C_{\mathrm{opt}}=\sum _{t=1}^{\tilde{n}-1}{c}_{\tilde{s}\left(t\right)\tilde{s}(t+1)}$$

(2)

By comparing Equation (1) and (2), the difference between $C_{\mathrm{on}}$ and $C_{\mathrm{opt}}$ is defined as the regret $R_{\mathrm{on}}=C_{\mathrm{on}}-C_{\mathrm{opt}}$. Minimizing $C_{\mathrm{on}}$ is equivalent to minimizing $R_{\mathrm{on}}$, since $C_{\mathrm{opt}}$ is a constant value.

Since it is scarcely possible to obtain the optimal solution before stage n, we can construct a dynamic model (see Figure 2a). That is, find an optimal path in each stage $t<n$, add the corresponding node in the frozen nodes set, and finally construct an acceptable path. The following are three dynamic models and the corresponding solution methods for the RTSP problem in stage t.

3. DRSP Model in DI Graph

3.1. Problem Description

The DRSP model is an extension of RSP to the dynamic version to make the full usage of the updated information. Figure 3 gives an example to show its advantage. DRSP in Figure 3 controls the realization of the maximum regret $R_{\mathrm{on}}=C_{\mathrm{on}}-C_{\mathrm{opt}}$, which is also shown in

Table 3. Maximum regret relates to the visitor’s location in DRSP.

Possible	Maximum Regret Relating to Location in				Final	Final
Paths	Node $\mathit{s}$	Node 1	Node 0	Node $\mathit{e}$	Cost	Regret
{s,e}	15	6	6	6	12	6
{s,1,e}	$\mathbf{6}$	6	6	6	12	6
{s,1,0,e}	7	$\mathbf{2}$	$\mathbf{0}$	$\mathbf{0}$	$\mathbf{6}$	0

Bold characters represent the least costs (or regret) among all paths.

.

When the visitor starts at node s (in Figure 3a), path {s,1,e} is the optimal path with the minimized maximum regret of 6 which is achieved by Definition 1–4. If we use this result without any change, it is the static RSP model with the final regret of 6. However, the updated information of arc $(s,1)$ and $(s,e)$ may be used when the visitor stays in node 1 (in Figure 3b), which results in the new optimal path of {s,1,0,e} with minimized maximum regret 2. When the visitor reaches node 0 (in Figure 3c) and the node e (in Figure 3d), path {s,1,0,e} is still optimal with the regret 0. That is to say, the static RSP model only gives the fixed path {s,1,e} with the final realization of regret 6. However, the DRSP uses the updated information to change the path in node 1 from {s,1,e} to {s,1,0,e} resulting in the final realization of regret reduced to 0. Figure 3 and Table 3 make a mini-correction of that in Xu and Zhou [40].

The DRSP model finds in each stage t the robust path which minimizes the robust cost (maximum deviation from the optimal shortest path) from node $s\left(T\right)$ to e, $T=t+k$. Then, it adds the nearby node $s(T+1)$ in the optimal path to the frozen nodes set. It is formally described by the following four definitions with $\forall t=1,2,\cdots ,n_p-k$.

Definition 1.

A scenario r is a realization of all arc costs, that is, in scenario r, the cost $c_{ij}^r\in [{\underline{c}}_{ij}\left(t\right),{\overline{c}}_{ij}\left(t\right)]$ is fixed, $\forall (i,j)\in A$.

Definition 2.

The robust deviation (RD), in a scenario r for path p from $s\left(T\right)$ to e, is the difference between p path cost in r and the SP cost in r from $s\left(T\right)$ to e.

Definition 3.

A robust cost (RC) for path p from $s\left(T\right)$ to e is the maximum RD for p among all possible scenarios, that is, ${\mathit{RC}}_{s\left(T\right)e}={max}_r{\mathit{RD}}_{s\left(T\right)e}$.

Definition 4.

The DRSP aims to find the path p among all paths from $s\left(T\right)$ to e at stage t with the minimized RC.

Theorem 1 is known in the literature for simplifying the RC calculation, where only the scenario induced by path p should be considered in stage t. Take Figure 3a for example, the scenario induced by path $p=\{s,1,e\}$ is depicted in Figure 4. The RC of p is $(4+8)-6=6$.

Theorem 1

(Karasan et al. [26]). The RD for path p is maximized in the scenario induced by p where the costs of all arcs in p are at upper bounds and the costs of all other arcs are at lower bounds.

3.2. Mixed Integer Programming (MIP) Formulations

The MIP formulations for the DRSP problem for path p in stage t ($t=1,2,3\cdots ,n_p-k$) are given as follows based on Theorem 1. Let $T=t+k$, then the optimization horizon in stage t is $s\left(T\right),s(T+1),\cdots ,s\left(n_p\right)$.

The objective (3) searches for a path p with minimized RC at stage t, where $\delta$ is a binary vector representing a path by constraint (5). If ${\delta }_{ij}=1$, arc $(i,j)$ is in the DRSP; 0 otherwise. Variable $x_j$ denotes the shortest total cost from node $s\left(T\right)$ to j in the scenario induced by $\delta$. Then, $x_e$ is the cost of the shortest path from $s\left(T\right)$ to e in this scenario.

In constraint (4), for a given vector $\delta$, the cost of arc $(i,j)$ is ${\underline{c}}_{ij}\left(t\right)+({\overline{c}}_{ij}\left(t\right)-{\underline{c}}_{ij}\left(t\right)){\delta }_{ij}$. It follows Theorem 1 to set the cost of arc $(i,j)$ at its upper bound on path p with ${\delta }_{ij}=1$, and all the costs of other arcs at lower bounds with ${\delta }_{ij}=0$.

Constraints (6)–(8) initialize the problem and define the domains of all the variables.

$$\left(DRSP\right)min\sum _{(i,j)\in A}{\overline{c}}_{ij}\left(t\right){\delta }_{ij}-x_e$$

(3)

$$s.t.x_j\le x_i+{\underline{c}}_{ij}\left(t\right)+({\overline{c}}_{ij}\left(t\right)-{\underline{c}}_{ij}\left(t\right)){\delta }_{ij},\forall (i,j)\in A$$

(4)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \sum _{k:(j,k)\in A}{\delta }_{jk}-\sum _{i:(i,j)\in A}{\delta }_{ij}=\left\{\begin{array}{cc}-1,\hfill & \mathrm{if}j=e\hfill \\ 1,\hfill & \mathrm{if}j=s\left(T\right)\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.\forall j\in V-\{i,k\}\end{array}\end{array}$$

(5)

$$x_{s\left(T\right)}=0$$

(6)

$${\delta }_{ij}\in \{0,1\}\forall (i,j)\in A$$

(7)

$$x_j\ge 0\forall j\in V-\left\{s\left(T\right)\right\}$$

(8)

The MIP model (3)–(8) is solved by CPLEX Optimization Studio 12.5 in the numerical study in Section 6.

4. DGRSP Model in DI Graph

In the dynamic model in Figure 2a, the recently frozen node is more important than the other far nodes in the optimal path of DRSP in stage t. Because the road information is timely updated, which results in the other far nodes hardly on the optimal path in the following stage. From this point, the Dynamic Greedy Robust Shortest Path (DGRSP) model is proposed, which is a modification and improvement of the DRSP model. By paying more attention to the nearby nodes, DGRSP first constructs the optimal path from $s\left(T\right)$ to the nearby nodes and then expands the path to the other nodes. We call it the greedy strategy.

4.1. Problem Description

For the traveler in stage t, the optimization horizon is still $\left\{s\right(T),s(T+1),\cdots ,s(v),\cdots ,$$s\left(n_p\right)\}$, where $T=t+k$, $t=1,2,3\cdots ,n_p-k$. The objective of DGRSP for any node $s\left(v\right)$ in the optimal path is calculated by

$$\underset{s\left(v\right)\in V,{min}_{s(v-1)\in V}{\mathrm{RC}}_{s\left(T\right)s(v-1)}}{min}{\mathrm{RC}}_{s\left(T\right)s\left(v\right)}$$

(9)

Note that the DGRSP in stage t is also a DRSP if all the subpath $\left\{s\right(T),\cdots ,f\}$, of the DRSP $\left\{s\right(T),\cdots ,f,\cdots ,e\}$, is also a DRSP from $s\left(T\right)$ to f. Theorem 2 proves the relationship between DGRSP and DRSP.

Theorem 2.

The DGRSP is also a DRSP in stage t in the DI graph $G(V,A)$, if $\{s\left(T\right),f^{\prime }\}$ is still a DRSP for any node $f^{\prime }$ in DRSP $\{s\left(T\right),f^{\prime },f\}$.

Proof. 

First, path $\left\{s\right(T\left)\right\}$ is obviously the DRSP and DGRSP since it has only one node.

Second, suppose the path $\left\{s\right(T),\cdots ,s(v-1\left)\right\}$ is the DRSP and DGRSP, then the path $\left\{s\right(T),\cdots ,s(v-1),s(v\left)\right\}$ achieved by Equation (9) is a DGRSP. Obviously, it is also a DRSP. Otherwise, there are two conditions.

(1) Suppose there exists another node $s\left(v^{\prime }\right)$ to form a DRSP path $\{s\left(T\right),\cdots ,s(v-1),s\left(v^{\prime }\right),s\left(v\right)\}$. Then, the path $\{s\left(T\right),\cdots ,s(v-1),s\left(v^{\prime }\right),s\left(v\right)\}$ should have a smaller RC than the path $\left\{s\right(T),\cdots ,s(v-1),s(v\left)\right\}$. Contradictions arise since Equation (9) is violated.

(2) Suppose there exists another node $s(v^{\prime }-1)$ to form a DRSP path $\{s\left(T\right),\cdots ,s(v^{\prime }-1),s\left(v\right)\}$. Then, the path $\left\{s\right(T\left)\right\}$, $\left\{s\right(T),s(T+1\left)\right\}$, ⋯, $\{s\left(T\right),s(T+1),\cdots ,s(v^{\prime }-1)\}$ are all DRSPs with minimized RC and should be identified by Equation (9). The RC of $\{s\left(T\right),\cdots ,s(v^{\prime }-1),s\left(v\right)\}$ should not be smaller than that of $\left\{s\right(T),\cdots ,s(v-1),s(v\left)\right\}$. Contradictions arise.

In conclusion, the DGRSP is also a DRSP in stage t, if $\{s\left(T\right),f^{\prime }\}$ is still a DRSP for any node $f^{\prime }$ in DRSP $\{s\left(T\right),f^{\prime },f\}$.    □

4.2. Nested Dijkstra Algorithm for DGRSP

The Nested Dijkstra (ND) is a dynamic programming algorithm developed by Xu and Zhou [40]. It can effectively solve the DGRSP here.

ND is constructed with a Dijkstra algorithm nested by another Dijkstra algorithm, where the Dijkstra algorithm refers to Dijkstra [1]. The inner Dijkstra algorithm computes the robust cost (RC) of a path, while the external Dijkstra algorithm determines the final solution for the DGRSP (see Figure 5).

Theorem 3, proven by Xu and Zhou [40], shows that the inner Dijkstra algorithm can calculate out the RC. Theorem 4 proves that ND returns the optimal solution of the DGRSP. Theorem 5 is the time complexity of the ND algorithm.

Theorem 3

(Xu and Zhou [40]). The robust cost of all paths can be determined by the inner Dijkstra algorithm with a time complexity of $O\left(n^2\right)$.

Theorem 4.

The ND algorithm returns the optimal solution of DGRSP.

Proof. 

First, for any nodes directly connected to $s\left(T\right)$, the ND algorithm returns the solution with the minimized RC.

Second, suppose, for any node $v-1$, the ND algorithm returns the solution with the minimized RC. Then, for node v, ND returns the solution with the minimized RC under the consideration of the minimized RC solution path from $s\left(T\right)$ to $v-1$.

Finally, the optimal solution of DGRSP from $s\left(T\right)$ to e is returned by the ND algorithm.    □

Theorem 5.

The ND algorithm returns the optimal solution of DGRSP with the time complexity of $O\left(n^4\right)$.

Proof. 

First, the inner Dijkstra algorithm calculates the RC for a path costs time of $O\left(n^2\right)$ (see in Theorem 2).

Second, the external Dijkstra algorithm needs to consider all the arcs in the graph. The arc number is at most $O\left(n^2\right)$, and the time-complexity is also $O\left(n^2\right)$.

Thus, the total time complexity of the ND algorithm is $O\left(n^4\right)$.    □

The pseudo-code for the ND algorithm is shown in detail as follows (Algorithm 1): it is applied in the numerical study in Section 6.

Algorithm 1: ND algorithm

$i:=s\left(T\right)$;      %Initialization while  node e unlabeled   do      %Terminal check     $n_i$:=a set of unlabeled nodes directly connected to i;     for iteration  $\forall j\in n_i$   do     %Update the RC of nodes in $n_i$         Compute ${\mathrm{RC}}_j$ by inner Dijkstra from $s\left(T\right)$ through i to j;         if  no previous reserved arc from $s\left(T\right)$ directly or indirectly to j;             then reserve arc $(i,j)$;         else then compare ${\mathrm{RC}}_j$ with the RC of previous path to j;             reserve the smaller one instead of the previous one;         end if     end     $n_u$:=a set of unlabeled nodes;     ${\mathrm{RC}}_{\min }:=+\infty$;     for iteration  $\forall i_x\in n_u$   do     %Find unlabeled nodes with min RC         if  ${\mathrm{RC}}_{i_x}<{\mathrm{RC}}_{\min }$   then   $i:=i_x$;         end if     end for     Label i and reserve the corresponding arc directly or indirectly to $s\left(T\right)$; end return the RC of all nodes and all the reserved arcs;      %Final result

5. DMSP Model in DI Graph

The Dynamic Mean Shortest Path (DMSP) model in stage t has the optimization horizon of $s\left(T\right),s(T+1),\cdots ,s\left(n_p\right)$, where $T=t+k$, $t=1,2,3\cdots ,n_p-k$. Suppose $c_{ij}\left(t\right)$ has the equal possibility in the interval of $[{\underline{c}}_{ij}\left(t\right),{\overline{c}}_{ij}\left(t\right)]$, then we let $c_{ij}^m\left(t\right)={\displaystyle \frac{{\underline{c}}_{ij}\left(t\right)+{\overline{c}}_{ij}\left(t\right)}{2}}$, that is, the uncertain arc $(i,j)$ has the mean cost of $c_{ij}^m\left(t\right)$ in stage t. The MIP formulations are shown in (10)–(12).

The objective (10) searches for a path p with minimized mean cost at stage t, where $\delta$ is a binary vector representing a path by constraint (11). If ${\delta }_{ij}=1$, arc $(i,j)$ is on path p; 0 otherwise.

Constraint (12) initializes the problem. ${\delta }_{ij}\ge 0$ implies that ${\delta }_{ij}\in \{0,1\}$; otherwise, constraint (11) is violated.

$$\left(DMSP\right)min\sum _{(i,j)\in A}{c}_{ij}^m\left(t\right){\delta }_{ij}$$

(10)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill s.t.\sum _{j:(i,j)\in A}{\delta }_{ij}-\sum _{j:(j,i)\in A}{\delta }_{ji}=\left\{\begin{array}{cc}-1,\hfill & \mathrm{if}i=e\hfill \\ 1,\hfill & \mathrm{if}i=s\left(T\right)\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.\forall i\ne j\end{array}\end{array}$$

(11)

$${\delta }_{ij}\ge 0\forall (i,j)\in A,i\ne j$$

(12)

Theorem 6.

The Dijkstra algorithm (or inner Dijkstra algorithm in ND algorithm) can be used to return the optimal solution of the DMSP model in stage t with the time-complexity of $O\left(n^2\right)$.

Proof. 

In stage t, the DMSP problem is equivalent to an SP problem in the SD graph. The inner Dijkstra algorithm finds the SP in a scenario and can be used to solve the DMSP problem.

The inner Dijkstra algorithm needs to consider all the arcs in the graph. The arc number is at most $O\left(n^2\right)$, and the time complexity is then $O\left(n^2\right)$. □

The Dijkstra algorithm is used to solve the DMSP model in the numerical study in Section 6.

6. Numerical Study

The following presents proof-of-concept implementations of three models, as well as the corresponding solution methods, in the DI graph. Suppose $C_{ij}=\{c_{ij}\left(t\right):\forall t=1,2,3,\cdots ,n\}$. Since more traffic information is achieved as time passes, we assume $[{\underline{c}}_{ij}\left(1\right),{\overline{c}}_{ij}\left(1\right)]\supseteq [{\underline{c}}_{ij}\left(2\right),{\overline{c}}_{ij}\left(2\right)]\supseteq \cdots \supseteq [{\underline{c}}_{ij}\left(n\right),{\overline{c}}_{ij}\left(n\right)]\supseteq C_{ij}$. That is to say, we first assume a much bigger uncertainty, and then shorten the uncertainty with time for the schedule. In the real-world application scenario, it is necessary to carefully shorten the uncertainty with time to avoid the violation of this assumption.

There are four main metrics, which are cost, regret, regret ratio and CPU time to evaluate the effectiveness of the models and solution methods (see also in

Table 4. Final solutions for five methods (in real roads).

Test Times		1	2	3	4	5	6	7	8	9	10	Average
optimal	cost	18,250	19,131;	17,936	18,792	19,407	18,595	18,504	17,707	17,749	17,812	18,388
	time(s)	<0.1	<0.1	<0.1	<0.1	<0.1	<0.1	<0.1	<0.1	<0.1	<0.1	<0.1
Static RSP	cost	22,361	20,743	19,995	20,797	20,442	21,094	21,345	19,757	19,006	19,868	20,541
	regret	4110	1612	2058	2005	1035	2499	2841	2050	1257	2056	2152
	ratio	22.5%	8.4%	11.5%	10.7%	5.3%	13.4%	15.4%	11.6%	7.1%	11.5%	11.7%
	time(s)	6.8	0.2	0.3	0.3	0.3	0.3	0.3	0.3	0.3	0.3	0.9
DRSP	cost	18,321	20,121	18,211	21,112	20,211	21,237	21,443	20,253	20,582	19,659	20,115
	regret	71	990	275	2320	804	2642	2939	2546	2833	1847	1727
	ratio	0.4%	5.2%	1.5%	12.4%	4.1%	14.2%	15.9%	14.4%	16.0%	10.4%	9.4%
	time(s)	0.5	0.5	0.5	0.5	0.6	0.6	0.6	0.5	0.8	0.6	0.6
DGRSP	cost	18,321	19,505	18,983	20,836	21,257	20,236	20,558	20,110	19,286	19,124	19,822
	regret	71	374	1047	2044	1851	1641	2055	2402	1537	1312	1433
	ratio	0.4%	2.0%	5.8%	10.9%	9.5%	8.8%	11.1%	13.6%	8.7%	7.4%	7.8%
	time(s)	2.5	2.5	3.2	2.8	2.6	2.8	3.2	2.9	3.2	3.1	2.9
DMSP	cost	18,321	19,505	18,306	20,836	20,762	20,236	20,558	20,110	19,286	18,983	19,690
	regret	71	374	370	2044	1355	1641	2055	2402	1537	1171	1302
	ratio	0.4%	2.0%	2.1%	10.9%	7.0%	8.8%	11.1%	13.6%	8.7%	6.6%	7.1%
	time(s)	<0.1	<0.1	<0.1	<0.1	<0.1	<0.1	<0.1	<0.1	<0.1	<0.1	<0.1

Bold characters explain less cost. ‘time’ denotes the average time to make each decision.

and

Table 5. Final solutions for five methods (in the generated network).

Number of Nodes		50	100	150	200	250	300	Average
optimal	cost	48	109	171	212	281	337	193
	time(s)	<0.1	<0.1	<0.1	<0.1	<0.1	<0.1	<0.1
Static RSP	cost	52	110	175	219	289	343	218
	regret	4	1	4	7	8	6	5
	ratio	8.3%	0.9%	2.3%	3.3%	2.9%	1.8%	2.6%
	time(s)	0.6	1.2	2.7	3.9	10.0 *	10.0 *	4.7
DRSP	cost	48	110	175	216	282	338	195
	regret	0	1	4	4	1	1	2
	ratio	0.0%	0.9%	2.3%	1.9%	0.3%	0.3%	1.0%
	time(s)	0.2	0.4	0.9	1.4	5.1 *	4.0 *	2.0
DGRSP	cost	48	111	172	213	283	339	194
	regret	0	2	1	1	2	2	1
	ratio	0.0%	1.8%	0.6%	0.5%	0.7%	0.6%	0.5%
	time(s)	0.1	0.2	0.6	1.2	2.3	3.7	1.3
DMSP	cost	48	111	172	213	283	339	194
	regret	0	2	1	1	2	2	1
	ratio	0.0%	1.8%	0.6%	0.5%	0.7%	0.6%	0.5%
	time(s)	<0.1	<0.1	<0.1	<0.1	<0.1	<0.1	<0.1

Bold characters explain less cost. ‘time’ is the average time to make decision. * mark the maximum decision time larger than 10.

). Cost and regret measure the absolute cost and regret of the final path. The regret ratio measures the relative regret compared to the optimal offline path cost. CPU time measures the time cost of the algorithm to return the result. The implementations are carried out by Matlab 7.11.0 and CPLEX Optimization Studio 12.5 in a personal computer with Intel(R) Core(TM) i3-2310M CPU @ 2.10 GHz 2.10 GHz, RAM 4.00 GB and Win 7 OS.

6.1. Real Roads Test

The real road data are selected from the examination questions in the National Postgraduate Mathematic Contest in Modeling in 2009 D in China. Figure 6 shows the real road network with 307 nodes (crossroads) and 458 arcs (roads). All of them are marked in Cartesian coordinates with the units in meters. A car runs from node s (14418, 8046) to node e (216, 414). The arc cost is restricted in the interval number of $[{\underline{c}}_{ij},{\overline{c}}_{ij}]=[{\displaystyle \frac{9-3{\xi }_{ij}}{10}}{d}_{ij},{\displaystyle \frac{11+3{\xi }_{ij}}{10}}{d}_{ij}]$. $d_{ij}$ is the distance between i and j, since the main arc cost, such as travel time cost and fuel cost, is related to the distance $d_{ij}$. ${\xi }_{ij}$ is a random variable with uniform distribution $f\left({\xi }_{ij}\right)=\left\{\begin{array}{cc}1,\hfill & {\xi }_{ij}\in [0,1]\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$, which models the uncertainty hitting leading the width change of the interval, and ${\xi }_{ij}={\xi }_{ji}$, $\forall (i,j)$. We suppose that the exact cost $c_{s\left(t\right)j}(\in [{\underline{c}}_{s\left(t\right)j},{\overline{c}}_{s\left(t\right)j}]$, $\forall j$) of arc $\left(s\right(t),j)$ is known if the traveler is located in node $s\left(t\right)$ in stage t because the road will be crossed immediately with the uncertainty being negligible (similar to Figure 3). The frozen nodes are ignored to strengthen the influence of the updated information (see Figure 2b).

Table 4 presents a comparison of five methods. The “optimal” denotes the offline optimal path, which is calculated by the Dijkstra algorithm in stage n when all the final arc costs are fixed. The “Static RSP” is calculated out in the DRSP model by CPLEX in stage 1 from s to e. The “DRSP” is calculated out in the DRSP model by CPLEX in each stage with updating road information. The “DGRSP” is achieved in the DGRSP model by the ND algorithm by updating road information. The “DMSP” is achieved in the DMSP model by the Dijkstra algorithm by updating road information.

It is easy to know that the DGRSP model and DMSP model give the best two solutions compared to the other methods. They generally have the least average-regret-ratio of DGSP 7.8% and DMSP 7.1%, where $\mathrm{regret}\mathrm{ratio}={\displaystyle \frac{\mathrm{regret}}{\mathrm{offline}\mathrm{optimal}\mathrm{cost}}}$. And CPLEX for Static RSP gives the worst solution with the maximum regret ratio because this model does not consider the updated information of the road cost. The computation time of four methods is all below 4 s, which is acceptable for decision.

Figure 6 illustrates a group of final solution paths by five methods in Table 4, while Figure 7 is the comparison of the regret ratio by four methods. The fluctuation happens since the arc cost $[{\underline{c}}_{ij},{\overline{c}}_{ij}]$ is randomly generated. Generally, the DGRSP model and DMSP model give the best solutions. Although DRSP makes use of the updated information, it is generally not better than DGRSP and DMSP in most test cases. Besides, we can see that all methods present their advantages in some cases, in which they are better than the other methods. So, an integration of the four methods in the application should also be a valuable consideration.

6.2. Generated Network Test

The following is the test of the generated network with n nodes. For long-distance transportation, the chosen route is often roughly in one direction. This network models the chosen nodes in the long and narrow range and roughly in one direction. The coordinate of the ith node is $(i+{(-1)}^i·{\displaystyle \frac{2n-i}{10}}·{\epsilon }_{i1},i+{(-1)}^{i+1}·{\displaystyle \frac{2n-i}{10}}·{\epsilon }_{i2})$, where ${\epsilon }_{ij}$ is a uniformly distributed random variable and models the uncertainty in this generation. We have $f\left({\epsilon }_{ik}\right)=\left\{\begin{array}{cc}1,\hfill & {\epsilon }_{ik}\in [0,1]\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$, $\forall i=1,2,\cdots ,n$, and $\forall k=1,2$. Here, each node connects at most four neighbor nodes since the reality road network has nodes connecting rarely more than four neighbor nodes. For node $i,j=1,2,\cdots ,n$, the cost of arc $(i,j)$ is in the interval of $[{\underline{c}}_{ij},{\overline{c}}_{ij}]$ with

$$[{\underline{c}}_{ij},{\overline{c}}_{ij}]=\left\{\begin{array}{cc}[1+2{\xi }_{ij},2+2{\xi }_{ij}],\hfill & \mathrm{if}i-j\in \{-2,-1,1,2\}\hfill \\ [\infty ,+\infty ],\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(13)

where ${\xi }_{ij}$ is a uniform distributed random variable modeling the width of the interval, that is, $f\left({\xi }_{ij}\right)=\left\{\begin{array}{cc}1,\hfill & {\xi }_{ij}\in [0,1]\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$, and ${\xi }_{ij}={\xi }_{ji}$, $\forall (i,j)$. Figure 8 shows an example of this network. The traveler (visitor) starts at the first node, and ends at the $(n-10)$th node. In the DI graph, the traveler is supposed to know the exact cost $c_{sj}(\in [{\underline{c}}_{sj},{\overline{c}}_{sj}]$, $\forall j$) of the arc $(s,j)$ if it is located in node s (similar to the real road in Section 6.1).

Table 5 illustrates the result of the five methods. The “optimal” denotes the offline optimal path calculated out by the Dijkstra algorithm in stage n. The “Static RSP” is calculated out in the DRSP model by CPLEX in stage 1 from s to e. The “DRSP” is calculated out in the DRSP model by CPLEX in each stage with updating road information. The “DGRSP” is achieved in the DGRSP model by the ND algorithm with updated road information. The “DMSP” is achieved in the DMSP model by the Dijkstra algorithm with updated road information.

The result is similar to that in the real road test. The DGRSP model and DMSP model give the best two solutions compared to the other methods. They generally have the lowest average-regret-ratio 0.5%. CPLEX for Static RSP gives the worst solution with the maximum regret ratio because this model does not consider the updated information on the road cost. The computation time of DGRSP and DMSP are all below 4 s, which is acceptable for the reality decision. However, the worst-case computation time of Static RSP and DRSP for nodes 250 and 300 reaches the maximum time of 10 s, which is not so acceptable for the commercial software in real-time reality application.

Figure 8 illustrates a group of final results of the five methods in Table 5. The fluctuation happens since the arc cost $[{\underline{c}}_{ij},{\overline{c}}_{ij}]$ is randomly generated. Figure 9 is the comparison of the four methods. The static RSP method has the highest regret ratio since it has not used the updated information, while DGRSP and DMSP methods have the lowest regret ratio. Although DRSP makes use of the updated information, it is generally not better than DGRSP and DMSP in most test cases. Besides, we can see that all the methods present their advantages in some cases in which they are better than the other methods. So, an integration of the four methods in the application should also be a valuable consideration.

7. Conclusions and Future Work

In this paper, three main methods are proposed for the RTSP problem where the traveler finds the SP with minimized regret. In addition, the static RSP is also used as a comparison. The numerical result validates that the DGRSP and DMSP models, generally, have the best solution with less regret than the other two models. The Static RSP has the worst solution since it has not used the updated information. Although DRSP makes use of the updated information, it is generally not better than DGRSP and DMSP in most test cases. There are two reasons. First, DGRSP and DMSP models in stage t pay more attentions on the arcs near to the visitor-position for optimization, which makes full use of the updated information. Second, the DRSP problem is NP-hard, which leads to non-optimality for the large number nodes cases. The computation times of DGRSP and DMSP are all below 4 s, which is acceptable for real-time transportation route planning in reality. Unfortunately, the computation time of Static RSP and DRSP for more than 250 nodes may exceed 10 s, which is not so attractive for real-time applications.

The models also have a few limitations. For example, the DRSP model may not work well for large-scale problems and fast solution methods are appreciated. That is to say, developing more efficient models and algorithms for RTSP is a future research direction. In addition, the integration of the above methods in an application is also another research direction.