Research Article  Open Access
Lagrangian Relaxation for the Multiple Constrained Robust Shortest Path Problem
Abstract
The study focuses on a multiple constrained reliable path problem in which travel time reliability and resource constraints are collectively considered. Nonlinear optimization model is developed to model constrained robust shortest path problem. The dual nature of the proposed problem is deduced based on the Lagrangian duality theory. An efficient algorithm based on Lagrangian dual relaxation is designed to solve constrained robust shortest path problem. An extension problem that considers multiple constraints is discussed. Numerical studies indicate that the proposed algorithm is efficient in terms of obtaining the closetooptimal solutions within reasonable computational times.
1. Introduction
1.1. Motivation
The constrained shortest path problem corresponds to the extension of the shortest path problem, and this is intended to obtain a shortest path that meets several constraints [1, 2]. The deficiency of classical constrained shortest path problems is that they assume the weights of each link as deterministic and not stochastic [3]. However, the weights of each link, such as link travel time, can be stochastic in reality [4–8]. Therefore, constrained shortest path problems should be extended from a deterministic network to a stochastic network.
In a stochastic network, the constrained shortest path problem defines the expected value of path travel time as the objective function of path that does not consider travel time reliability [9, 10]. However, several studies indicated that travelers care about saving travel time while improving travel time reliability in the stochastic network [11–15]. The unconstrained robust shortest path problem is extensively investigated [16–25].
However, to the best of the author’s knowledge, there is a lack of studies on the constrained robust shortest path problem. The author of this study made preliminary research results related to the constrained robust shortest path problem in the early stage, which only focused on the reliable shortest path problem with only one single constraint [26]. The study extends the problem to the general situation: multiple constrained robust shortest path problem. To fill the gap, the study focuses on the multiple constrained robust shortest path problem. A constrained robust shortest path problem aids in revealing the route choice behavior by considering reliability under a few resource constraints in a traffic network. However, the computational complexity of the problem significantly increases due to the nonlinear and nonadditive nature of the objective function and a few complex constraints.
1.2. Literature Review
The constrained shortest path problem is examined extensively. The constrained shortest path problem is an NPhard problem and cannot be directly solved by conventional labeling algorithms [27, 28]. Classical methods for dealing with the constrained shortest path problem include the path ranking method, dynamic programmingbased method, Lagrangian relaxationbased method, and heuristic method.
Path ranking method. Handler and Zang [3] first adopted a th shortest path algorithm to solve the constrained shortest path problem. The basic idea involves determining shortest paths until a first feasible path that satisfies constraints is obtained. The approach is easy and exact with respect to identifying the optimal path in a small network. The disadvantage of the method is that the computational cost and storage of algorithm increase with , and the defect causes the algorithm to be impractical in a large network.
Dynamic programmingbased method [29–35]. Modified label algorithms based on dynamic programming are used to solve the constrained shortest path problem with respect to a one side constraint or multiple side constraints. The advantage of the method is that conventional label algorithms are directly changed to solve the constrained shortest path problem, and the algorithm performs very well in solving reasonably sized networks. However, the algorithm performs very poorly in large networks due to the “curse of dimensionality” of dynamic programming.
Lagrangian relaxationbased method [3, 9, 36–40]. The main idea of the method involves dualizing the side constraints into the objection function. The original problem is translated into a Lagrangian dual problem based on Lagrangian duality theory. The corresponding optimal value of Lagrangian dual problem is the lower bound of the primal problem. The advantage of Lagrangian relaxationbased method is that it is effective in terms of a large sized network. The disadvantage of the method is that it obtains nearoptimal solutions that occasionally do not satisfy accuracy requirements and the defect is eliminated by combining this with unconstrained shortest path algorithms. Handler and Zang [3] first developed a Lagrangian relaxationbased method to solve the constrained shortest path problem and obtained nearoptimal solutions. Carlyle et al. [38] presented a Lagrangian relaxationbased method combined with enumerating nearshortest paths to solve the constrained shortest path problem. Muhandiramge and Boland [39] developed an iterative algorithm in which nodes and edges were eliminated to solve the Lagrangian dual problems. Wang et al. [9] examined a stochastic constrained shortest path problem and proposed a mixed algorithm based on Lagrangian relaxation to solve it although they only focused on obtaining the least expected travel time path and did not consider travel time reliability.
Heuristic method [34, 40–45]. In order to improve the speed of calculation in large networks, various heuristic methods are proposed. The advantage of the method is that heuristic techniques embedded in an algorithm improve the computational efficiency of the algorithm specifically in large networks. However, in the heuristic method, it is difficult to obtain an exact solution of the constrained shortest path problem, and it is occasionally difficult to even obtain a closetooptimal solution. The heuristic technique is only used as an auxiliary means to improve the computational efficiency of an algorithm.
The reliable shortest path problem is examined by several researchers. It is formulated in various forms based on the objection function of path including the most reliable path problem (Frank, 1969; Nie and Wu, 2009a), reliable path problem (Chen and Ji, 2005; Chen et al., 2012; [19]), and meanstandard deviation shortest path problem [18, 20, 21, 23, 46, 47]. The first two problems are complementary to each other. The reliable path problem is translated into the meanstandard deviation shortest path problem when travel times are normally distributed. The third problem defines the objective function of path as the sum of mean and standard deviation of stochastic travel time, and they are only constructed based on the stochastic parameter that efficiently reduces the computation cost and memory usage. The study focuses on the third problem with resource constraints and extends studies where the third problem is discussed.
Sen et al. [46] considered the reliable path problem by defining the objection function as the sum of a mean and a variance of path. The solution to the problem is relatively simple due to the additive nature of the objective function although its disadvantage is that the mean and variance are not in an order of magnitude. Xing and Zhou [18] considered the meanstandard deviation path problem that directly reflects the reliability of the path although the solution to the problem is complicated due to the nonlinear and nonadditive nature of the objective function. A Lagrangian dual relaxation algorithm is constructed to solve the problem. Zeng et al. [47] focused on the reliable path problem in a stochastic network with spatial correlation. The Cholesky decomposition is used to resolve the correlation of link travel time, and a similar Lagrangian dual relaxation algorithm is developed to solve it. Khani and Boyles [20] discussed the relationship between the meanstandard deviation path problem and the meanvariance path problem and developed an iterative algorithm to solve the meanstandard deviation path problem. Wu [21] extended the meanstandard deviation path problem from a stochastic and static network to a stochastic and timedependent network and introduced stochastic dominance theory to solve the problem. Zhang et al. [23] reconstructed the meanstandard deviation path problem to a convex problem. They analyzed duality gaps in different Lagrangian relaxation approaches and selected a novel Lagrangian relaxation algorithm to solve it.
1.3. Contribution
The study first focuses on a constrained robust shortest path problem, and the specific contributions are as follows:
The study first proposes a multiple constrained robust shortest path problem to obtain the meanstandard deviation shortest path subject to a few resource constraints in stochastic network. A reliable objective function and a few resource constraints are simultaneously considered. The constrained robust shortest path problem is formulated as a mixed 01 convex nonlinear programming model.
The dual problem of constrained robust shortest path problem is derived based on Lagrangian duality theory, and the dual problem is divided into two independent subproblems that are easy to solve.
An efficient algorithm is constructed to solve a constrained robust shortest path problem. A gradient descent algorithm is developed to solve the dual problem by gradually reducing the gap between the upper and lower bounds. A th shortest path algorithm is used to obtain the feasible path.
An extension problem that considers multiple constraints is further discussed, and the corresponding dual problem is also derived.
A numerical test is developed on a small network, mediumscale transportation network, and largescale network. The numerical results are analyzed to verify the effectiveness and stability of the proposed algorithm.
The study is organized as follows. In Section 2, the constrained robust shortest path problem is discussed and formulated as a mixed 01 convex nonlinear programming model. In Section 3, the dual problem is derived. In Section 4, an efficient algorithm is constructed. In Section 5, an extension problem that considers multiple constraints is discussed. Numerical experiments are elucidated in Section 6. Finally, Section 7 details a few conclusions and discusses future research directions.
2. Problem Statement
We consider a directed graph that consists of a set of nodes () and a set of links (). The travel time on each link is a random variable , and its mean and standard deviation are and , respectively. Additionally, each link possesses a resource weight . A link travel time distribution is independent.
The travel time along path is also a random variable as follows:whereThe path is denoted as a vector . The mean and standard deviation of path travel time are as follows:
The objective function of path travel time is given as follows:where denotes the reliability coefficient. The resource consumption along path is also a variable as follows:
The resource consumption along path must be bound by constraint conditions (e.g., fuel consumption or power consumption). Subsequently, the following expression is obtained:The parameter denotes the upper limit value allowed for resource consumption along path.
Given the definition, the constrained robust shortest path problem is described as the following mathematical model: s.t.:
The constrained robust shortest path problem is more complex than the constrained shortest path problem due to the addition of the standard deviation in the objective function (8) that completely corresponds to a nonadditive function.
3. The Dual Problem
3.1. Model Reformulation
It is difficult to solve the constrained robust shortest path problem due to the nonlinear objective function (8) and constraint (9). The objective function (8) is a nonlinear and nonadditive function due to the standard deviation part. Our goal involves translating the nonadditive objective function into an additive objective function. We introduce artificial variable to eliminate the nonadditivity of the standard deviation in the objective function.
The objective function (8) is equivalent to the following functions:
Lemma 1 (see [18]). , where denotes the variance of the least expected travel time path in network with respect to the link weight .
Subsequently, the problem ((8)(10)) is reformulated as the following constrained shortest path problem. s.t.:
3.2. Lagrangian Dual Problem
Based on the Lagrangian duality theory, two hard constraints (14) and (15) are dualized into the objective function with two Lagrangian multipliers. Therefore, with respect to any Lagrangian multiplier vector and , where ,
The Lagrangian function is defined as follows:where
The first subproblem is expressed as follows:It corresponds to the classical shortest path problem with a new weight . Dijkstra’s algorithm (or Dijkstra’s Shortest Path First algorithm, SPF algorithm) is an algorithm for finding the classical shortest paths between nodes in a graph [28].
The second subproblem is expressed as follows: This is a nonlinear concave optimization problem, and it is easily solved through minimizing the concave function value at two interval endpoints: 0 and [32]; thus the optimal solution is .
Based on the Lagrangian duality theory [48], the following expression is obtained:
The Lagrangian dual problem of the constrained robust shortest path problem ((8)(10)) is as follows:
4. Solution Algorithm
An efficient algorithm is designed to solve constrained robust shortest path problem based on the algorithm suggested by Fisher [48]. A gradient descent iterative algorithm is proposed to solve the dual problem. Specifically, in each iteration, the first subproblem (23) and second subproblem (24) are solved, and the sum of two corresponding optimal values is set as the lower bound of the optimal value of the original problem ((8)(10)). Furthermore, the solution of the first subproblem (23) provides a potentially feasible solution for the original problem ((8)(10)), and the corresponding objective value of the original problem ((8)(10)) is set as the upper bound of the optimal value of the original problem ((8)(10)). However, the solution of the first subproblem (23) potentially does not satisfy the constraints and is not a feasible solution to the original problem ((8)(10)). Therefore, the shortest path algorithm is introduced to obtain a feasible solution of the original problem ((8)(10)). The shortest path algorithm is a generalization of the shortest path algorithm. The algorithm finds not only the shortest path, but also − 1 other paths in nondecreasing order of cost. is the number of shortest paths to find. Since 1957 there have been many papers published on the shortest path routing algorithm problem. Handler and Zang [3] first adopted the shortest path algorithm to solve the constrained shortest path problem. The basic idea involves determining shortest paths until a first feasible path that satisfies constraints is obtained. Since the shortest path algorithm itself can solve the constrained path problem, the proposed algorithm adopts this algorithm to accelerate and guarantee the convergence of the proposed algorithm. In general, the upper bound and lower bound of the proposed algorithm change during the iterative process. In extreme cases, the upper bound of the proposed algorithm remains unchanged during the iterative process if the upper bound of the original problem ((8)(10)) (that is, the optimal solution) is obtained by the shortest path algorithm, but the lower bound approximates the upper bound, which is the optimal value.
In the iterative process, the gap between the upper and lower bounds gradually decreases. Additionally, when the gap corresponds to the minimum gap or zero, the corresponding upper bound is a nearoptimal solution or an accurate solution of the original problem ((8)(10)). The detailed procedure of the algorithm is described in Algorithm 1.
1. Initialization  
Set initial iteration , initial Lagrangian multipliers , initial  
lower bound , initial upper bound , maximum relative gap ,  
maximum value of iteration .  
2. Update the lower bound  
Solve obtain an optimal solution and optimal value .  
Solve obtain an optimal solution and optimal value .  
Set the lower bound  
3. Update the upper bound  
If satisfies all the side constraints (9) then set the upper bound:  
Else if, solve the problem ((8)(10)) by shortest path algorithm to obtain a feasible  
Solution , and set the upper bound:  
End  
4. Update Lagrangian multipliers  
Compute subgradient direction  
Updated Lagrangian multipliers  
where , ,  
the value of is suggested by Fisher [48].  
5. Convergence test  
If relative gap or stop; Otherwise , go to  
step 2. 
5. Extension to the Multiple Constraints
In this section, the reliable shortest path problem with a single constraint is extended to the problem with multiple constraints. In a stochastic network, each link possesses multiple resource weights, . Given the definition, the constrained robust shortest path problem is described as follows. s.t.:
The parameter denotes the upper value of the resource weights. The constrained robust shortest path problem ((27)(29)) is reformulated as the following constrained shortest path problem: s.t.:where , .
For any Lagrangian multiplier vector and , the Lagrangian function is defined as follows:where
The Lagrangian dual problem of the constrained robust shortest path problem ((30)(33)) is as follows:
A solution algorithm suggested in Section 4 is modified to solve the multiple constrained robust shortest path problem. Step 4 in the algorithm updating Lagrangian multipliers is modified as follows, and the other steps are similar to the solution algorithm suggested in Section 4.
Compute the subgradient direction
Update Lagrangian multiplierswhere , , and the values of are suggested by Fisher [48].
6. Numerical Experiments
This section verifies the validity and feasibility of the proposed model and algorithm. All algorithms were coded via Matlab and tested on a Windows10 (64) workstation with two 2.00 GHz Xeon CPUs and 4G RAM. Three networks are considered to represent different experimental environments and include a small network, Sioux Falls network [49], and largescale network [6].
6.1. A Small Network
A small network is constructed with 5 nodes and 6 links as shown in Figure 2. Link travel time is a random variable with its mean and standard deviation . Additionally, each link possesses two resource weights and . The parameters of each link are shown in Figure 1.
The initial parameters are set as the OD pair: , , , and . The cases of different constraints are computed. Table 1 shows the updated process of Lagrangian multiplier, bound, and gap based on iterations with two upper values of the resource weights: and . Figure 3 shows the updated process of the bound and relative gaps with respect to the iteration. Table 2 shows the updated process of the bound and gap based on iterations with and . Table 3 shows the updated process of the bound and gap based on iterations with and .



As shown in Table 1, the exact optimal value is 40, obtained at the iteration. The two resource weights of optimal path are 9 and 12, and this satisfies two constraints. As shown in Figure 2, the upper bounds are gradually close to the lower bounds, and an exact optimal value is finally determined when the relation gap is reduced to 0 in the iteration. As shown in Table 2, an exact optimal value is 47.73, and this is obtained at the iteration. The two resource weights of optimal path are 7 and 9, and this satisfies two constraints. As shown in Table 3, an exact optimal value is 38.87, and this is obtained at the iteration. The two resource weights of the optimal path are 11 and 12, and this satisfies two constraints. There are differences in the optimal value and optimal path with respect to the constraint conditions.
6.2. Sioux Falls Network
Sioux Falls network is a mediumscale transportation network that includes 24 zones, 24 nodes, and 76 links as shown in Figure 3 [49].
6.2.1. Experimental Design
The experimental design is set as follows. In the Sioux Falls network, link travel times are random variables, its mean is , and its standard deviation is . In order to simulate its stochastic properties, the expected values and variances are randomly obtained from certain intervals as shown in Table 4. Each link possesses a resource weight that is also randomly obtained from a certain interval.

6.2.2. Computational Results of Different OD Pairs
Table 5 shows the computational results of different OD pairs in which the initial parameters are set as , , , . The 10 OD pairs are randomly selected from all 24 nodes. As shown in Table 5, all relative gaps are less than the given value , and this implies that the algorithms are terminated within the maximum step . Two relative gaps are zero, and this implies that the accurate solution is finally obtained. All path lengths are less than the constraint value and satisfy the path length constraint, and this verifies the feasibility and accuracy of the proposed algorithm.

6.2.3. Computational Results of Different Constraints
Tables 6–8 show the computational results of different constraints with the same OD pair as follows:, , and , respectively. The notation NO in Tables 6 and 7 indicates that the algorithm is not successful in obtaining the optimal value when the constrained value is less than the shortest path weight, a path that satisfies the constraint is absent, and the constrained robust shortest path problem does not include a feasible solution. In Tables 6–8, implies that the constrained robust shortest path problem is reduced to the reliable shortest path problem without constraints. The optimal path varies with respect to the constraint condition that is denoted by constraint value. For example, in Table 6, the optimal paths are different when the constraint values are =50 and 60. Similarly, in Table 7, the optimal paths are also different when the constraint values are =30 and 40. Finally, in Table 8, the optimal paths are also different when the constraint values are =40, 50, and 55. Specifically, differences exist in the optimal paths solved by the reliable shortest path problem with and without constraint. For example, in Table 8, the optimal paths are different when the constraint values are =55 and . Conversely, with increases in the constraint value and path length, the upper bound and lower bound of the optimal path exhibit a decreased trend. For example, in Table 8, the constraint value increases from 40 to 55 and the corresponding path length increases from 36 to 54 although the upper bound decreases from 72.44 to 57.12 and the lower bound decreases from 70.39 to 54.39. The relative gap is not significantly influenced by the constraint condition.



6.2.4. Computational Results of Different Reliability Coefficients
Table 9 shows the computational results of different reliability coefficients in which the initial parameters are set as the OD pair as follows: , , , and . With respect to the first case in Table 9, implies that the constrained robust shortest path problem is reduced to the constrained shortest path problem and differences exist between the optimal paths solved by the constrained robust shortest path problem and constrained shortest path problem. The optimal path varies with respect to the reliability coefficient that denotes travelers’ attitudes to path reliability. For example, in Table 9, the optimal paths are different when the reliability coefficient , 0.5, and 0.8.

6.3. LargeScale Network
6.3.1. Experimental Design
The grid network is a scalable network that is typically used to test traffic network equilibrium problem [6]. A grid network of level contains nodes and links as shown in Figure 4: . The experimental design is set as follows. In the grid network, link travel times are random variables, its mean is , and its standard deviation is . In order to simulate its stochastic properties, all means and standard variances are randomly selected from the interval . Each link possesses a resource weight that is also randomly selected from the interval .
The length constraint of path is randomly selected from the interval . The initial parameters are set as , , . The computational times of the algorithm are analyzed under different network scales. All CPU times were measured in seconds with the utility routine CPU time.
6.3.2. Computational Time of the Proposed Algorithm
Table 10 shows the average computational time of the proposed algorithm for different network scales. The network scale is represented by the level, number of nodes, and links of the grid network. It should be noted that the relative gap is the average of 10 computational results with the same OD pairs. The number of iterations and CPU time is also the average of 10 computational results for the same OD pairs.

Evidently, as shown in Table 10, the relative gaps are all less than the initial parameters . The CPU time significantly increases with respect to the network scale. For example, the CPU time increases from 4 S to 805 S when the level of grid network increases from 5 to 90. The number of iterations does not monotonically increase with respect to the network scale although it exhibits an increasing tendency with respect to the network scale. For example, when the level of grid network increases from 20 to 30, the number of iterations decreases from 23 to 21. The relative gaps also are not significantly influenced by the network scale. For example, when the level of grid network increases from 50 to 90, the relative gaps decrease from 4.28 to 3.19.
Figure 5 shows the computation times relative to the number of nodes of grid network. Figure 6 shows the number of iterations relative to the number of nodes of grid network. As shown in Figure 5, the computational times almost monotonically increase with respect to the number of nodes and links of network, and this indicates that the network scale significantly influences the computational efficiency of the algorithm. This is consistent with the expectations. Increased computational efforts are required for shortest path or kshortest path searches with increases in the network scale.
However, as shown in Figure 6, the number of iterations does not monotonically increase with respect to the number of nodes, and this implies that the number of nodes is an important factor that influences the number of iterations, and the number of iterations is also significantly influenced by other initial parameters such as the initial upper bound and lower bound.
7. Concluding Remarks
The constrained robust shortest path problem is first discussed to obtain the meanstandard deviation shortest path subject to a few resource constraints in a stochastic network in which a reliable objective function and a few resource constraints are simultaneously considered. The constrained shortest path problem is extended from a deterministic network to a stochastic network. It obtains a priori reliable shortest path subject to resource constraints in the stochastic network. The model is a truer reflection of the actual traffic routing selection than a single model. However, the computational complexity of the model significantly exceeds that of a single model due to the nonlinear and nonadditive nature of the objective function and a few complex constraints.
The constrained robust shortest path problem is formulated as a mixed 01 convex integer programming model. The objective function is defined as a linear combination of the mean and standard deviation to directly reflect the reliability of travel time. The resource constraint is defined in terms of inequality constraints. The proposed problem is complex due to the nonlinear and nonadditive nature of the objective function, and the inequality constraints that are extremely difficult to calculate in reality.
The dual problem of constrained robust shortest path problem is derived based on Lagrangian duality theory. The dual problem is divided into two independent subproblems. The first subproblem corresponds to the classical shortest path problem that is easy to solve. The second subproblem is a nonlinear concave optimization problem, and it is easily solved through minimizing the concave function value at two interval endpoints.
An efficient algorithm is constructed to solve a constrained robust shortest path problem. A gradient descent algorithm is developed to solve the dual problem by gradually reducing the gap between the upper and lower bounds. The first subproblem and second subproblem are solved in iterations. The sum of their corresponding objective values is set as the lower bound. The kth shortest path algorithm is used to obtain the feasible path, and the corresponding objective values are set as the lower bound. When the gap corresponds to the minimum gap or zero, the corresponding upper bound is a nearoptimal solution or an accurate solution.
An extension of constrained robust shortest path problem that considers multiple constraints is further discussed, and the corresponding dual problem is also derived. An efficient algorithm is introduced to solve this via modifying the proposed algorithm.
A numerical test is developed on a small network, mediumscale transportation network, and largescale network. The numerical results are analyzed. Numerical results indicate the following: the proposed algorithm solves the constrained robust shortest path problem and obtains the nearoptimal solution or accurate solution in different scale of networks. This indicates the effectiveness of the proposed algorithm. the computational times of proposed algorithm almost increase linearly with respect to the number of nodes. This indicates the stability of the proposed algorithm. there are differences between the reliable shortest paths obtained under resource unconstraint and constraint, differences also exist between reliable shortest paths obtained under different resource constraints, and the resource constraint significantly influences the choice of reliable shortest path.
The constrained robust shortest path problem can be very useful for route guidance systems and logistics optimization problems with complex constraints and reliablebased objectives in stochastic networks. In particular, due to the limitation of battery capacity, the electric vehicle navigation problem is a typical constraint robust shortest path problem. The algorithm can be applied to real road network as the core of invehicle route guidance systems in the future.
On the basis of the algorithm proposed in this paper, some extensions can be envisaged. First, in this study link travel times are assumed to follow a spatial independent distribution. Previous studies have found that travel times could be better approximated by correlated link travel times [6]. In future study correlations between link travel times can be incorporated either through statistical model or through traffic network simulation to capture a traffic network in a more realistic way. Second, in this study link travel times are assumed to be time independent; dynamic characteristics among traffic network are not incorporated in the model. Actually link travel times are not only random but also dynamic in realistic traffic network [50]. The constrained robust shortest path problem should be extended from static network to dynamic network. Third, the constrained robust shortest path problem was formulated based on the meanstandard deviation shortest path concept. However, other models of reliable shortest path have been developed in different application contexts, such as αreliable path [17]. The extension of the constrained robust shortest path problem to other reliable shortest path models will be another topic for further studies.
Data Availability
The data of traffic network used to support the findings of this study have been deposited at the website http://www.bgu.ac.il/~bargera/tntp.
Conflicts of Interest
The author declares no conflicts of interest.
Acknowledgments
This study is sponsored in part by National Science Foundation of China under grant No. 51508280 and by Startup Foundation of Nanjing Forestry University under grant No. GXL2015031, to which the author is very grateful.
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