Distributionally robust stochastic shortest path problem (original) (raw)

Stochastic Shortest Path Problem with Uncertain Delays

Proceedings of the 1st International Conference on Operations Research and Enterprise Systems, 2012

This paper considers a stochastic version of the shortest path problem, the Stochastic Shortest Path Problem with Delay Excess Penalty on directed, acyclic graphs. In this model, the arc costs are deterministic, while each arc has a random delay, assumed normally distributed. A penalty occurs when the given delay constraint is not satisfied. The objective is to minimize the sum of the path cost and the expected path delay penalty. In order to solve the model, a Stochastic Projected Gradient method within a branch-and-bound framework is proposed and numerical examples are given to illustrate its effectiveness. We also show that, within given assumptions, the Stochastic Shortest Path Problem with Delay Excess Penalty can be reduced to the classic shortest path problem.

Stochastic Shortest Path Problem with Delay Excess Penalty

Electronic Notes in Discrete Mathematics, 2010

We study and solve a particular stochastic version of the Restricted Shortest Path Problem, the Stochastic Shortest Path Problem with Delay Excess Penalty. While arc costs are kept deterministic, arc delays are assumed to be normally distributed and a penalty per time unit occurs whenever the given delay constraint is not satisfied. The objective is to minimize the sum of path cost and total delay penalty.

An analysis of stochastic shortest path problems

Mathematics of Operations Research, 1991

We consider a stochastic version of the classical shortest path problem whereby for each node of a graph, we must choose a probability distribution over the set of successor nodes so as to reach a certain destination node with minimum expected cost. The costs of transition between successive nodes can be positive as well as negative. We prove natural generalizations of the standard results for the deterministic shortest path problem, and we extend the corresponding theory for undiscounted finite state Markovian decision problems by removing the usual restriction that costs are either all nonnegative or all nonpositive.

A note on the stochastic shortest-route problem

Naval Research Logistics Quarterly

This paper develops an algorithm for a "shortest route" network problem in which it is desired to find the path which yields the shortest expected distance through the network. It is assumed that if a particular arc is chosen, then there is a finite probability that an adjacent arc will be traversed instead. Backward induction is used and appropriate recursion formulae are developed. A numerical example is provided.

Robust shortest path problems with uncertain costs

2008

Abstract Data coming from real-world applications are very often affected by uncertainty. On the other hand, it is difficult to translate uncertainty in terms of combinatorial optimization. In this paper we study a combinatorial optimization model to deal with uncertainty in arc costs in shortest path problems. We consider a model where feasible arc cost scenarios are described via a convex polytope. We present a computational complexity result and we discuss exact algorithms for two different robust optimization criteria.

Shortest paths in stochastic networks

Proceedings. 2004 12th IEEE International Conference on Networks (ICON 2004) (IEEE Cat. No.04EX955), 2004

This paper discusses the sensitivity of network flows to uncertain link state information for various routing protocols. We show that the choice of probability distribution for the link metrics for a given network can have markedly different effects on the probabilities of path selection. Exact results are obtained for these probabilities but their computation is NP-hard. We provide simulation results for three networks to illustrate the sensitivity of shortest paths to different link metric distributions. We provide results for mean path costs and the k-shortest path algorithm as a comparison.

The shortest path problem in the stochastic networks with unstable topology

SpringerPlus, 2016

Background The deterministic shortest path problem has been studied extensively and applied in many fields of optimization; there are polynomial time algorithms to solve the deterministic shortest path problem (Dijkstra 1959; Bellman 1958; Orlin et al. 2010). However, paths in networks should be reliable to transmit flow from a source node to a destination node especially in delay sensitive networks. The best connection helps to avoid traffic congestion in networks. So, the arrival probability is used to evaluate the reliability of paths and it has been considered as an optimality index of the stochastic shortest path length (Bertsekas and Tsitsiklis 1991; Fan et al. 2005; Kulkarni 1986; Shirdel and Abdolhosseinzadeh 2016). The stochastic shortest path problem (SSP) is defined as the best path with stochastic optimality condition. Liu (2010) assumed the arc lengths to be uncertain variables. Pattanamekar et al. (2003) considered the individual travel time variance and the mean travel time forecasting error. Also, Hutson and Shier (2009) and Rasteiro and Anjo (2004) supposed two criteria: mean and variance of path length. Fan et al. (2005) assumed known conditional probabilities for link travel times that each link

On the robust shortest path problem

Computers & Operations Research, 1998

The shortest path (SP) problem in a network with nonnegative arc lengths can be solved easily by Dijkstra's labeling algorithm in polynomial time. In the case of significant uncertainty of the arc lengths, a robustness approach is more appropriate. In this paper, we study the SP problem under arc length uncertainties. A scenario approach is adopted to characterize uncertainties. Two robustness criteria are specified: the absolute robust criterion and the robust deviation criterion. We show that under both criteria the robust SP problem is NP-complete even for the much more restricted layered networks of width 2, and with only 2 scenarios. A pseudo-polynomial algorithm is devised to solve the robust SP problem in general networks under bounded number of scenarios. Also presented is a more effecient algorithm for layered networks. However, in the case of unlimited number of scenarios, we show that the robust SP problem is strongly NP-hard. A simple heuristic for finding a good robust shortest path is provided, and the worst case performance is analyzed.

Optimal paths in graphs with stochastic or multidimensional weights

Abst ract This paper formulates a stochastic and a multidimensional oplimal path problem, each as an extension of the shortest path problem. Conditions when existing shortest path methods apply are noted. In each problem instance, a utility function defines preference among candidate paths.

The stochastic quickest path problem via minimal paths

2010

The quickest path problem, a version of the shortest path problem, is to find a single quickest path that sends a given amount of data from the source to the sink with minimum transmission time. More specifically, the capacity of each arc in a network is assumed to be deterministic. However, in many real-life networks, such as computer systems, telecommunication systems, etc., the capacity of each arc is stochastic due to failure, maintenance, etc. Such a network is named a stochastic-flow network.