Shortest path problem with uncertain arc lengths

Uncertainty theory provides a new tool to deal with the shortest path problem with nondeterministic arc lengths. With help from the operational law of uncertainty theory, this paper gives the uncertainty distribution of the shortest path length. Also, it investigates solutions to the α-shortest path and the most shortest path in an uncertain network. It points out that there exists an equivalence relation between the α-shortest path in an uncertain network and the shortest path in a corresponding deterministic network, which leads to an effective algorithm to find the α-shortest path and the most shortest path. Roughly speaking, this algorithm can be broken down into two parts: constructing a deterministic network and then invoking the Dijkstra algorithm.     

Network flow problems with fuzzy arc lengths.

Network flow problems cover a wide range of engineering and management applications. Many streamlined solution methods have been devised for solving different types of the problems. This paper investigates the network flow problems in that the arc lengths of the network are fuzzy numbers. Based on the integer-solution property of the network flow problem, the Yager ranking indices can be calculated for the fuzzy arcs to change the fuzzy formulation of the problem to a crisp formulation. Consequently, the conventional streamlined solution methods can still be applied to find an optimal solution. This optimal solution is proved to be the same as that derived from an exhaustive comparison of all possible solutions. Two examples, one shortest path and one transshipment, discussed in some previous studies illustrate that the method proposed in this paper is able to find the optimal solution. To show that the proposed method is useful in solving real-world problems, the problem of multimedia transmission over the Internet is exemplified.     

Solving fuzzy shortest path problems by neural networks

In this paper, we introduce the neural networks for solving fuzzy shortest path problems. The penalization of the neural networks is realized after transforming into crisp shortest path model. The procedure and efficiency of this approach are shown with numerical simulations.     

用于求解井下最短逃生路径问题的离散萤火虫算法

针对煤矿井下避灾路线最短路径求解问题,提出了一种新的离散萤火虫算法。该算法通过采用转移概率方法初始化萤火虫个体,并提出一种新的有效编码和解码方式,重新定义萤火虫的空间距离、最大荧光亮度和相对荧光亮度等,使得萤火虫个体的状态可表示为一条从起点到目标点的有效路径。为增加解的多样性及防止计算结果陷入局部最优解,以一定概率对萤火虫代表的路径执行扰动操作,经过多次迭代计算后,可得到所要求解的最短路径。实验结果表明,该算法在种群规模较小、迭代次数较少的情况下可以收敛到最优解,具有较强的收敛性和灵活性,可用于求解任何实际的最短路径问题。     

Solving the fuzzy shortest path problem using multi-criteria decision method based on vague similarity measure

Many researchers have focused on the fuzzy shortest path problem in a network with non-deterministic information due to its importance to various applications. The goal of this paper is to select the shortest path in multi-constrained network using multi-criteria decision method based on vague similarity measure. In our approach, each arc length represents multiple metrics. The multi-constraints are equivalent to the concept of multi-criteria based on vague sets. We propose a similarity measure of vague sets in which the positive constraints and the negative constraints are defined. Furthermore, the procedures are developed to obtain the “best” and “worst” ideal paths. We evaluate similarity degrees between all candidate paths and two ideal paths with the proposed similarity measure. Through comparing the relative degrees of paths, it is shown that the path with the largest relative degree is the shortest path. Finally, we conduct two sets of numerical experiments—using Matlab to verify the feasibility and correctness of the proposed algorithm and developing a routing decision simulation system (RDSS) to demonstrate that the proposed approach is reasonable and effective.     

A biologically inspired solution for fuzzy shortest path problems

By considering the uncertainty that exists in the edge weights of the network, fuzzy shortest path problems, as one of the derivative problems of shortest path problems, emerge from various practical applications in different areas. A path finding model, inspired by an amoeboid organism, Physarum polycephalum, has been shown as an effective approach for deterministic shortest path problems. In this paper, a biologically inspired algorithm called Fuzzy Physarum Algorithm (FPA) is proposed for fuzzy shortest path problems. FPA is developed based on the path finding model, while utilizing fuzzy arithmetic and fuzzy distance to deal with fuzzy issues. As a result, FPA can represent and handle the fuzzy shortest path problem flexibly and effectively. Distinct from many existing methods, no order relation has been assumed in the proposed FPA. Several examples, including a tourist problem, are given to illustrate the effectiveness and flexibility of the proposed method and the results are compared with existing methods.     

必经点约束型最短路径问题的研究

为了解决网络路由中带有必经点约束的网络拓扑中最优路径的问题,提出了采用改进的蚁群算法和Dijkstra算法对最优路径进行规划。首先,针对含有必经点约束的问题,在信息素初始化时增加在必经点上的信息素。其次,为了能够更好地找到最优解,采用了狼群分配算法进行信息素更新,提高了收敛速度,并且采用了Dijkstra算法对蚂蚁找到的最优路径进行二次优化。最后,分别用不同节点和必经点规模的网络进行实验,并与基本的蚁群算法进行了比较,证明了改进蚁群算法的正确性和高效性。     

Extended dominance and a stochastic shortest path problem

In the context of stochastic networks, we study the problem of finding a path P   that combines in a reasonable way the mean m(P)$m(P)$ and variance v(P)$v(P)$ of its length. Specifically we study a separable objective function that combines these two path measures: namely, z(P)=f(m(P))+g(v(P))$z(P)=f(m(P))+g(v(P))$, where f$f$ is an increasing convex function and g is an increasing concave function. A new type of dominance (e-dominance), stronger than the standard form of dominance, is then introduced, and it is shown to satisfy a certain form of Bellman's optimality principle. This means that it is possible to modify existing label-setting and label-correcting methods by using e-dominance, and without sacrificing optimality. Computational experience with these enhanced labeling algorithms has been promising. Test results for a variety of sample problems show that the e-dominance criterion can often significantly reduce the number of nondominated path vectors, compared to the standard dominance criterion. We observe a consequent reduction in both computation time and storage requirements.     

Solving shortest path problem using particle swarm optimization

This paper presents the investigations on the application of particle swarm optimization (PSO) to solve shortest path (SP) routing problems. A modified priority-based encoding incorporating a heuristic operator for reducing the possibility of loop-formation in the path construction process is proposed for particle representation in PSO. Simulation experiments have been carried out on different network topologies for networks consisting of 15–70 nodes. It is noted that the proposed PSO-based approach can find the optimal path with good success rates and also can find closer sub-optimal paths with high certainty for all the tested networks. It is observed that the performance of the proposed algorithm surpasses those of recently reported genetic algorithm based approaches for this problem.     

Parallel algorithms for the single source shortest path problem

We present several parallel algorithms for the problem of finding shortest paths from a specified vertex (called the source) to all others in a weighted directed graph. The algorithms are for machines ranging from array processors, multipleinstruction multiple-data stream machines to a special network of processors. These algorithms have been designed by “parallelizing” two classic sequential algorithms — one due to Dijkstra (1959), the other due to Moore (1957). Our interest is not only in obtaining speeded-up parallel versions of the algorithms but also in exploring the design principles, the commonality of correctness proofs of the different versions, and the subjective complexity of explaining and understanding these versions.     

Fuzzy Dijkstra algorithm for shortest path problem under uncertain environment

A common algorithm to solve the shortest path problem (SPP) is the Dijkstra algorithm. In this paper, a generalized Dijkstra algorithm is proposed to handle SPP in an uncertain environment. Two key issues need to be addressed in SPP with fuzzy parameters. One is how to determine the addition of two edges. The other is how to compare the distance between two different paths with their edge lengths represented by fuzzy numbers. To solve these problems, the graded mean integration representation of fuzzy numbers is adopted to improve the classical Dijkstra algorithm. A numerical example of a transportation network is used to illustrate the efficiency of the proposed method.     

On an exact method for the constrained shortest path problem

The constrained shortest path (CSP) is a well known NP-Hard problem. Besides from its straightforward application as a network problem, the CSP is also used as a building block under column-generation solution methods for crew scheduling and crew rostering problems. We propose an exact solution method for the CSP capable of handling large-scale networks in a reasonable amount of time. We compared our approach with three different state-of-the-art algorithms for the CSP and found optimal solutions on networks with up to 40,000 nodes and 800,000 arcs. We extended the algorithm to effectively solve the auxiliary problems of a multi-activity shift scheduling problem and a bus rapid transit route design problem tackled with column generation. We obtained significant speedups against alternative column generation schemes that solve the auxiliary problem with state-of-the-art commercial (linear) optimizers. We also present a first parallel version of our algorithm that shows promising results.     

An efficient fault-containing self-stabilizing algorithm for the shortest path problem

Shortest path finding has a variety of applications in transportation and communication. In this paper, we propose a fault-containing self-stabilizing algorithm for the shortest path problem in a distributed system. The improvement made by the proposed algorithm in stabilization times for single-fault situations can demonstrate the desirability of an efficient fault-containing self-stabilizing algorithm. For single-fault situations, the worst-case stabilization time of the proposed algorithm is O(Δ), where Δ is the maximum node degree in the system, and the contamination number of the proposed algorithm is 1.     

Stochastic shortest path with unlimited hops

We present new results for the Stochastic Shortest Path problem when an unlimited number of hops may be used. Nodes and links in the network may be congested or uncongested, and their states change over time. The goal is to adaptively choose the next node to visit such that the expected cost to the destination is minimized. Since the state of a node may change, having the option to revisit a node can improve the expected cost. Therefore, the option to use an unbounded number of hops may be required to achieve the minimum expected cost. We show that when revisits are prohibited, the optimal routing problem is np-hard. We also prove properties about networks for which continual improvement may occur.We study the related routing problem which asks whether it is possible to determine the optimal next node based on the current node and state, when an unlimited number of hops is allowed. We show that as the number of hops increases, this problem may not converge to a solution.     

基于云计算的混合并行遗传算法求解最短路径

为提高最短路径求解问题的效率,提出一种基于云计算的细粒度混合并行遗传算法求解最短路径的方法。方法采用云计算中H adoop的Map Reduce并行编程模型,提高编码效率,同时将细粒度并行遗传算法和禁忌搜索算法结合,提高了寻优算法的计算速度和局部寻优能力,进而提高最短路径的求解效率。仿真结果表明,该方法在计算速度和性能上优于经典遗传算法和并行遗传算法,是一种有效的最短路径求解方法。     

Order relation between intervals and its application to shortest path problem

It is true that intervals are frequently partially ordered and cannot be compared. Nevertheless, varous definitions for ranking intervals have been proposed. In this paper, we propose a new definition for order relation between intervals by introducing a parameter called “a degree between partial and total order”, and apply it to the shortest path problem with arcs represented as intervals. In order to solve this problem, we modify the Dijkstra's algorithm, and propose a new algorithm obtaining some incomparable interval solutions. Finally, a numerical example is shown.     

An ant colony optimization algorithm for the bi-objective shortest path problem

Multi-objective shortest path problem (MOSP) is an extension of a traditional single objective shortest path problem that seeks for the efficient paths satisfying several conflicting objectives between two nodes of a network. MOSP is one of the most important problems in network optimization with wide applications in telecommunication industries, transportation and project management. This research presents an algorithm based on multi-objective ant colony optimization (ACO) to solve the bi-objective shortest path problem. To analyze the efficiency of the algorithm and check for the quality of solutions, experimental analyses are conducted. Two sets of small and large sized problems that generated randomly are solved. Results on the set problems are compared with those of label correcting solutions that is the most known efficient algorithm for solving MOSP. To compare the Pareto optimal frontiers produced by the suggested ACO algorithm and the label correcting algorithm, some performance measures are employed that consider and compare the distance, uniformity distribution and extension of the Pareto frontiers. The results on the set of instance problems show that the suggested algorithm produces good quality non-dominated solutions and time saving in computation of large-scale bi-objective shortest path problems.