A bi-objective model for uncertain multi-modal shortest path problems
- Yan Zhang^{1},
- Pei Liu^{1},
- Lixing Yang^{1}Email author and
- Yuan Gao^{1}
https://doi.org/10.1186/s40467-015-0032-x
© Zhang et al.; licensee Springer. 2015
Received: 25 December 2014
Accepted: 13 March 2015
Published: 8 April 2015
Abstract
This paper employs uncertain programming to investigate the uncertain multi-modal shortest path problem, in which the arc weights (arc travel time, arc travel costs) associated with different transport modes are characterized by uncertain variables. By using the chance-constrained programming approach, we firstly formulate a bi-objective optimization model to minimize the total travel time and travel costs simultaneously with the given confidence levels. Moreover, using the basic concepts and properties in the uncertainty theory, we transform the proposed model into its deterministic crisp equivalent with an explicit proof. Finally, some numerical experiments are implemented to show the performance of the proposed approaches on a multi-modal transportation network with three specific modes.
Keywords
Introduction
The shortest path problem is widely applied to network optimization and has been studied by a lot of researchers. Classical shortest path problems focus on networks with deterministic arc weights (lengths), and Dijkstra [1], Bellman [2], and Dreyfus [3] have proposed some efficient algorithms which are still referenced widely now. However, due to the failure, maintenance, and other uncertain factors, arc weights are usually non-deterministic in a busy transportation network. In view of this fact, some researchers introduced probability theory into the shortest path problem and used probability distributions to describe the existing indeterminacy, such as Frank [4], Loui [5], Mirchandani [6], Yang et al. [7], and Yang and Zhou [8]. However, this method is typically imprecise when we are lack of a priori data information with respect to networks. With this concern, Dubois and Prade [9] first introduced a fuzzy shortest path problem. In the fuzzy set theory, the decision can be estimated by experts based on their experiences and professional judgments. Some other routing optimization with fuzzy information can be referred to Ji et al. [10], Hernandes et al. [11], and Yang et al. [12].
However, both probability theory and fuzzy set theory may lead to counterintuitive results [13]. In this case, uncertainty theory was founded to rationally deal with belief degrees, which enlightened a new approach to describe non-deterministic phenomena. Up to now, the uncertainty theory has been applied to many classical optimization problems, such as solid transportation problem [14], project scheduling problem [15], maximum flow problem [16], optimal assignment problem [17], uncertain graph [18], etc. As for the shortest path problem, Liu [19] introduced three concepts of uncertain path according to different decision criteria, including the expected shortest path, α-shortest path, and the most shortest path. He formulated three types of uncertain programming models and converted them into deterministic optimization ones. Gao [20] studied the uncertainty distribution of the shortest path length and proposed an effective method to find the α-shortest path and the most shortest path in an uncertain network. He pointed out that there existed an equivalence relation between the α-shortest path in an uncertain network and the shortest path in a corresponding deterministic network. Moreover, an effective algorithm was proposed to find the α-shortest path and the most shortest path. Zhou et al. [21] discussed the inverse shortest path problem on the graph with uncertain edge weights. This problem was formulated as an uncertain programming and was reformulated into a deterministic programming model.
Multi-modal transportation refers to a trip consisting of two or more means of transport to guide the passengers reach their destinations. The multi-modal shortest path problem is a significant generalization of the traditional shortest path problem, and it has been extensively investigated by a lot of researchers. Lozano and Storchi [22] found the shortest viable path in a multi-modal network using label correcting techniques. They defined the viable path as a path whose sequence of modes is feasible with respect to a set of constraints. Ma [23] presented an A ^{∗} label setting algorithm to solve a constrained shortest path problem in a multi-modal network. In this work, each link is characterized with a vector of resource consumption besides the travel time. Ambrosino and Sciomachen [24] proposed an approach for computing shortest routes in multi-modal networks with objectives of minimizing the overall time, cost, and users’ discomfort. Liu et al. [25] designed an improved exact algorithm for a multi-criteria multi-modal shortest path problem with both arriving time window and transfer delaying. Galvez-Fernandez et al. [26] introduced a transfer graph approach, which was believed to better abstract the distributed nature of real transport information sources, to calculate the best paths in multi-modal networks. Yamani et al. [27] presented a fuzzy shortest path algorithm in multi-modal transportation networks, which concerned about not only the path cost but also the path time which consisted of travel time and delays.
Indeed, due to the complexity of the real-world situations, it is in general difficult to explicitly determine travel time or travel costs in a large-scale multi-modal network. What is more, it is also complicated when we consider the transfer time and costs from one mode to another. Thus, investigating the multi-modal shortest path problem in the indeterministic environment becomes a significant and challenging issue for the practical applications. We here note that finding the shortest path in a multi-modal network has been studied in various forms such as static [22], dynamic [26,28], stochastic [29,30], fuzzy [27], constrained [23], and multi-criteria [24,25,31,32]. However, to the best of our knowledge, few studies have been considered in the uncertain environment within the framework of uncertain programming.
This paper aims to investigate the multi-modal shortest path problem, in which both travel time and travel costs are regarded as uncertain variables. Specifically, the travel time includes travel time on the travel arcs and transfer time spent on the mode change. The travel costs consist of cost on each arc and the fixed cost. We intend to formulate a chance-constrained programming model with two objectives minimizing the travel time and travel cost simultaneously. The results of this research will provide a fundamental framework for investigating the shortest path problems with both multi-modes and uncertainty characteristics.
The remainder of this paper is organized as follows. Section ‘Preliminaries’ introduces some basic concepts and properties of uncertainty theory used throughout this paper. Section ‘Model formulations’ makes a description of the problem and formulates a chance-constrained programming model with two objectives. In Section ‘Crisp equivalent of the model’, we demonstrate how to convert the model into its crisp equivalent. In Section ‘Numerical experiments’, some experiments are given to illustrate the performance of the proposed model. Finally, some conclusions are made in Section ‘Conclusions’.
Preliminaries
As this research will investigate the problem of interest within the framework of uncertainty theory, we next first introduce some basic knowledge in this field for the completeness of this paper. Uncertainty theory was proposed by Liu [13], and it provided an axiomatic system to handle the imprecise information. Some foundational definitions and results in uncertainty theory are introduced below.
Definition 2.1.
Based on the uncertainty space, a formal definition of the uncertain variable will be introduced for better describing the uncertainties mathematically.
Definition 2.2.
[13] An uncertain variable is a measurable function ξ from an uncertainty space (Γ,L,M) to the set of real numbers such that {ξ∈B} is an event for any Borel set B.
Definition 2.3.
Theorem 2.1.
Definition 2.4.
[33] Let ξ be an uncertain variable with regular uncertainty distribution Φ(x). Then, the inverse function Φ ^{−1}(α) is called the inverse uncertainty distribution of ξ.
Definition 2.5.
There are some kinds of special uncertain variables namely the linear uncertain variable, the zigzag uncertain variable, the normal uncertain variable and the lognormal uncertain variable. As an example, the following will describe corresponding properties of the zigzag uncertain variable.
Definition 2.6.
denoted by Z(a,b,c), where a,b, and c are real numbers with a<b< c.
Theorem 2.2.
To characterize the structural features of an uncertain variable, the concepts of critical values, including optimistic value and pessimistic value, are formally defined as follows.
Definition 2.7.
- (1)The optimistic value of ξ is defined by:$$\begin{array}{@{}rcl@{}} \xi_{\sup}(\alpha)=\sup\{r|M\{\xi\geq r\}\geq \alpha\}, \alpha\in (0,1], \end{array} $$(5)
- (2)The pessimistic value of ξ is defined by:$$\begin{array}{@{}rcl@{}} \xi_{\inf}(\alpha)=\inf\{r|M\{\xi\leq r\}\geq \alpha\}, \alpha\in (0,1]. \end{array} $$(6)
Model formulations
Notations used in formulations
Notations | Definition |
---|---|
V | = set of nodes in the multi-modal network. |
E | = set of arcs in the multi-modal network. |
M | = set of the transport modes, M={p,b,m,t}. |
p | = the private mode. |
b | = the bus mode. |
m | = the metro mode. |
t | = the transfer mode. |
i,j | = index of nodes, i,j∈N. |
(i,j,k) | = connections between the adjacent nodes of mode k. |
ξ _{ ijk } | = travel time/transfer time on arc (i,j,k). |
c _{ ijk } | = travel cost on arc (i,j,k), k∈M∖{t}. |
d _{ k } | = fixed cost when entering mode k. |
x _{ ijk } | = 1, if arc (i,j,k) is on the path, 0 otherwise, (i,j,k)∈E. |
y _{ k } | = 1, if x _{ ijk }=1 for a certain (i,j,k)∈E, 0 otherwise, k∈M∖{t}. |
In this paper, we intend to formulate a bi-objective integer programming model to minimize both the travel time and travel costs. Different decision criteria have been used to evaluate the shortest path problem under uncertain environment. One of the widely used criteria is the critical value. Uncertain programming using the critical value is called the chance-constrained programming. Enlightened by this, we shall formulate a chance-constrained programming model, which is shown as follows:
In the above model, α and β are predetermined confidence levels. The objective function implies optimizing the total travel time and total travel cost. Constraints (12) and (13) mean that the total travel time and travel cost will be less than \(\overline {f_{1}}\) and \(\overline {f_{2}}\) with confidence levels α and β, respectively. Constraints (14), (15), and (16) ensure that a feasible path can be generated in the network. Constraint (17) implies the total transfer times will not exceed the given limit N. It is noted that the travel time and travel cost on arcs are imprecise due to the complexity of the actual network. Hence, the corresponding variables like ξ _{ ijk },c _{ ijk }, and d _{ k } are treated as uncertain variables.
Crisp equivalent of the model
It is easy to know that if the involved uncertain variables are complex, the model may be difficult to be dealt with. So it is necessary for us to transform the model into its crisp equivalent. In this section, we shall introduce how the model can be transformed into a deterministic form.
Theorem 4.1.
- (I)
(ξ+η)_{sup}(α)=ξ _{sup}(α)+η _{sup}(α),
- (II)
(ξ+η)_{inf}(α)=ξ _{inf}(α)+η _{inf}(α).
Theorem 4.2.
- (I)
(k ξ)_{sup}(α)=k·ξ _{sup}(α),
- (II)
(k ξ)_{inf}(α)=k·ξ _{inf}(α).
Theorem 4.3.
- (I)
(k _{1} ξ+k _{2} η)_{sup}(α)=k _{1}·ξ _{sup}(α)+k _{2}·η _{sup}(α),
- (II)
(k _{1} ξ+k _{2} η)_{inf}(α)=k _{1}·ξ _{inf}(α)+k _{2}·η _{inf}(α).
It is worth noting that the critical value of an uncertain variable is associated with the uncertainty distribution according to the Measure Inversion Theorem. Just for completeness, two theorems will be cited to illustrate this point.
Theorem 4.4.
[34] Suppose that ξ is an uncertain variable with continuous uncertainty distribution Φ(x) when 0<Φ(x)<1, and g(x,ξ)=h(x)−ξ, α is a confidence level in (0,1). Then, we have M{g(x,ξ)≤0}≥α if and only if h(x)≤f _{ ξ }(α), where f _{ ξ }(α)=Φ ^{−1}(1−α).
Theorem 4.5.
[34] Suppose that ξ is a continuous uncertain variable with increasing uncertainty distribution Φ(x) when 0<Φ(x)<1, g(x,ξ)=h(x)−ξ and α is a confidence level in (0,1). Then, M{g(x,ξ)≥0}≥α if and only if h(x)≥f _{ ξ }(α), where f _{ ξ }(α)=Φ ^{−1}(α).
Based on the aforementioned theorems, we can easily deduce the following conclusions:
Theorem 4.6.
With the aid of relevant theorems given above, the following theorem intends to show the equivalent model.
Theorem 4.7.
Proof.
Based on Equation 20, the pessimistic value of an uncertain variable is the inverse function with respect to α. Hence, we can easily obtain the equivalent model. The proof is thus completed.
Remark.
In the shortest path problem, the optimal objectives are related to the confidence levels α,β, i.e., if the confidence levels satisfy α _{1}<α _{2} and β _{1}<β _{2}, then the corresponding optimal objectives satisfy: \(\min ~\overline {f}_{1}<\min ~\overline {f}_{1}'\) and \(\min ~\overline {f}_{2}<\min ~\overline {f}_{2}'\), because the uncertainty distributions of uncertain variables are monotone nondecreasing functions. Here, \(\overline {f}_{1}\), \(\overline {f}_{1}'\), \(\overline {f}_{2}\), and \(\overline {f}_{2}'\) indicate the optimal objective values with α=α _{1}, α=α _{2}, β=β _{1}, and β=β _{2}.
Corollary 4.1.
Numerical experiments
Time and costs on travel arcs of the multi-modal network
Arcs | Arc time ( ξ _{ ijk } ) | Arc costs ( c _{ ijk } ) | Arcs | Arc time ( ξ _{ ijk } ) | Arc costs ( c _{ ijk } ) |
---|---|---|---|---|---|
(0,33,1) | Z(4, 5, 8) | Z(8, 9,12) | (42,44,2) | Z(1,2,5) | Z(1,2,4) |
(33,34,1) | Z(3, 4, 9) | Z(7, 9,14) | (43,44,2) | Z(2,5,6) | Z(2,4,5) |
(34,35,1) | Z(15,17,20) | Z(10,12,13) | (44,45,2) | Z(3,5,8) | Z(2,3,5) |
(35,36,1) | Z(12,14,18) | Z(9,10,13) | (45,46,2) | Z(2,3,5) | Z(2,3,6) |
(36,37,1) | Z(14,17,18) | Z(12,14,17) | (27,14,3) | Z(1,3,4) | Z(1,2,4) |
(37,38,1) | Z(9,12,14) | Z(11,14,16) | (27,28,3) | Z(1,2,4) | Z(1,2,4) |
(38,39,1) | Z(22,25,27) | Z(18,24,26) | (14,15,3) | Z(1,2,4) | Z(1,3,4) |
(39,46,1) | Z(11,15,17) | Z(11,14,16) | (15,16,3) | Z(2,3,5) | Z(1,3,4) |
(0, 1,2) | Z(4, 7, 9) | Z(1, 3, 4) | (16,17,3) | Z(1,2,4) | Z(1,2,5) |
(1, 2,2) | Z(3, 7, 8) | Z(2, 3, 6) | (17,18,3) | Z(2,3,5) | Z(1,2,4) |
(1, 3,2) | Z(4, 5, 7) | Z(2, 5, 6) | (28,22,3) | Z(3,4,6) | Z(3,4,7) |
(2, 4,2) | Z(5, 7, 8) | Z(2, 4, 5) | (22,32,3) | Z(4,5,7) | Z(2,3,6) |
(3, 5,2) | Z(4, 7, 8) | Z(3, 6, 7) | (32,19,3) | Z(5,7,8) | Z(4,6,7) |
(4, 6,2) | Z(2, 5, 7) | Z(1, 3, 4) | (28,29,3) | Z(1,2,4) | Z(2,4,7) |
(5, 6,2) | Z(6, 9,10) | Z(2, 5, 6) | (29,23,3) | Z(1,3,4) | Z(2,4,7) |
(5,13,2) | Z(3, 5, 6) | Z(2, 3, 5) | (23,30,3) | Z(1,2,4) | Z(2,5,7) |
(3,12,2) | Z(1, 3, 4) | Z(1, 3, 4) | (30,31,3) | Z(2,3,5) | Z(2,4,5) |
(12,13,2) | Z(3, 5, 6) | Z(1, 3, 4) | (31,20,3) | Z(2,3,5) | Z(1,2,4) |
(13, 8,2) | Z(2, 5, 6) | Z(1, 4, 5) | (18,19,3) | Z(2,4,5) | Z(1,2,4) |
(6, 7,2) | Z(2, 3, 5) | Z(1, 2, 4) | (19,20,3) | Z(2,3,5) | Z(2,4,5) |
(8, 9,2) | Z(3, 5, 6) | Z(2, 4, 5) | (20,21,3) | Z(3,4,6) | Z(2,3,5) |
(7, 9,2) | Z(2, 3, 6) | Z(1, 3, 4) | (15,22,3) | Z(2,3,5) | Z(1,2,4) |
(8,10,2) | Z(1, 2, 4) | Z(1, 2, 5) | (22,23,3) | Z(1,2,5) | Z(2,3,5) |
(9,11,2) | Z(2, 3, 5) | Z(1, 2, 4) | (23,24,3) | Z(1,2,4) | Z(2,5,6) |
(10,11,2) | Z(1, 2, 4) | Z(1, 2, 4) | (24,25,3) | Z(3,4,6) | Z(2,5,7) |
(11,40,2) | Z(4, 7, 9) | Z(1, 3, 4) | (25,26,3) | Z(1,2,4) | Z(2,3,5) |
(40,41,2) | Z(1, 3, 5) | Z(1, 2, 4) | (26,21,3) | Z(1,2,5) | Z(1,2,4) |
(41,42,2) | Z(1, 2, 4) | Z(1, 2, 4) | (21,46,3) | Z(1,2,4) | Z(1,2,5) |
(41,43,2) | Z(2, 4, 5) | Z(1, 3, 4) |
Transfer time on transfer arcs of the multi-modal network
Arcs | Arc time ( ξ _{ ijk } ) | Arcs | Arc time ( ξ _{ ijk } ) |
---|---|---|---|
(33, 1,4) | Z(3, 4, 6) | (6,15,4) | Z(5, 6, 8) |
(34, 3,4) | Z(4, 5, 7) | (8,16,4) | Z(3, 4, 6) |
(34,27,4) | Z(3, 4, 6) | (16, 8,4) | Z(4, 6, 7) |
(27, 3,4) | Z(4, 6, 7) | (10,31,4) | Z(5, 8, 9) |
(3,27,4) | Z(2, 3, 5) | (31,10,4) | Z(6, 7, 9) |
(36,12,4) | Z(4, 5, 7) | (7,32,4) | Z(8,10,11) |
(35, 4,4) | Z(2, 3, 5) | (32, 7,4) | Z(7, 8,10) |
(28, 2,4) | Z(2, 3, 5) | (11,18,4) | Z(2, 3, 5) |
(2,28,4) | Z(3, 4, 6) | (18,11,4) | Z(2, 5, 6) |
(37,29,4) | Z(4, 6, 9) | (20,42,4) | Z(1, 2, 4) |
(5,23,4) | Z(8, 9,11) | (42,20,4) | Z(2, 5, 6) |
(23, 5,4) | Z(7, 9,10) | (25,44,4) | Z(2, 3, 6) |
(15, 6,4) | Z(4, 5, 7) | (44,25,4) | Z(1, 2, 4) |
With this method, the LINGO optimization software is employed to generate the shortest path from origin node 0 to destination node 46 in the multi-modal network. It is noted that α, β and δ are three parameters which will influence the optimal objective values. To capture the corresponding relations between the parameters and the objective functions, we shall implement two sets of experiments as follows:
Optimal solutions with different α and β ( δ=5 )
Parameters | Optimal objective values | Shortest path | |
---|---|---|---|
Single | α=0.6 | g _{1}(x)=37.2 | 0→33→34→27→28→29→23→24→25→26→21→46 |
Objective | β=0.6 | g _{2}(x,y)=35.8 | 0→1→3→12→13→8→10→31→20→21→46 |
Double | δ=5 | g _{1}(x)=42.0 | 0→1→2→28→22→23→30→31→20→21→46 |
Objectives | g _{2}(x,y)=38.6 | ||
Single | α=0.7 | g _{1}(x)=42.4 | 0→33→34→27→28→29→23→24→25→26→21→46 |
Objective | β=0.7 | g _{2}(x,y)=39.6 | 0→1→3→12→13→8→10→31→20→21→46 |
Double | δ=5 | g _{1}(x)=47.2 | 0→1→3→5→23→24→25→26→21→46 |
Objectives | g _{2}(x,y)=42.8 | ||
Single | α=0.8 | g _{1}(x)=46.4 | 0→1→2→28→29→23→24→25→26→21→46 |
Objective | β=0.8 | g _{2}(x,y)=43.4 | 0→1→3→12→13→8→10→31→20→21→46 |
Double | δ=5 | g _{1}(x)=50.8 | 0→1→3→5→23→24→25→26→21→46 |
Objectives | g _{2}(x,y)=46.2 | ||
Single | α=0.9 | g _{1}(x)=50.2 | 0→1→2→28→29→23→24→25→26→21→46 |
Objective | β=0.9 | g _{2}(x,y)=47.2 | 0→1→3→12→13→8→10→31→20→21→46 |
Double | δ=5 | g _{1}(x)=54.4 | 0→1→3→5→23→24→25→26→21→46 |
Objectives | g _{2}(x,y)=49.6 | ||
Single | α=1.0 | g _{1}(x)=54.0 | 0→1→2→28→29→23→24→25→26→21→46 |
Objective | β=1.0 | g _{2}(x,y)=51.0 | 0→1→3→12→13→8→10→31→20→21→46 |
Double | δ=5 | g _{1}(x)=59.0 | 0→1→3→5→23→30→31→20→21→46 |
Objectives | g _{2}(x,y)=52.0 |
Optimal solutions with different δ ( α=0.9 , β=0.9 )
δ | Objective values |
---|---|
1 | g _{1}(x)=50.2 |
g _{2}(x,y)=55.2 | |
2 | g _{1}(x)=51.4 |
g _{2}(x,y)=54.2 | |
3 | g _{1}(x)=51.4 |
g _{2}(x,y)=54.2 | |
4 | g _{1}(x)=54.0 |
g _{2}(x,y)=52.4 | |
5 | g _{1}(x)=54.4 |
g _{2}(x,y)=49.6 | |
6 | g _{1}(x)=55.6 |
g _{2}(x,y)=48.6 | |
7 | g _{1}(x)=56.8 |
g _{2}(x,y)=47.2 | |
8 | g _{1}(x)=56.8 |
g _{2}(x,y)=47.2 | |
9 | g _{1}(x)=56.8 |
g _{2}(x,y)=47.2 | |
10 | g _{1}(x)=56.8 |
g _{2}(x,y)=47.2 | |
11 | g _{1}(x)=56.8 |
g _{2}(x,y)=47.2 | |
12 | g _{1}(x)=56.8 |
g _{2}(x,y)=47.2 |
Conclusions
Uncertainty theory is a branch of mathematics to study the characteristics of nondeterministic phenomenon. In this paper, we applied uncertainty theory to investigating the multi-modal shortest path problem in which the arc time and arc costs are represented by uncertain variables. Considering two objectives, which are to minimize the total travel time and travel costs, we formulated a chance-constrained programming model for the problem of interest. For handling convenience, the model was then transformed into its deterministic crisp equivalent. Additionally, the bi-objective model was simplified into a single objective model by converting one objective function into a new constraint with the given threshold. Finally, the results of the numerical experiments deriving from the LINGO software demonstrated the performance of the proposed approaches. Actually, multi-modal shortest path problems have been widely solved by the label-setting algorithm [35], label-correcting algorithm [25], and genetic algorithm [36,37], which can also be considered in our further studies.
Authors’ Affiliations
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