Standard algorithms for finding the shortest path in a graph require that the cost of a path be additive in edge costs, and typically assume that costs are deterministic. We consider the problem of uncertain edge costs, with potential probabilistic dependencies among the costs. Although these dependencies violate the standard dynamicprogramming decomposition, we identify a weaker stochastic consistency condition that justifies a generalized dynamic-programming approach based on stochastic dominance. We present a revised pathplanning algorithm and prove that it produces optimal paths under time-dependent uncertain costs. We illustrate the algorithm by applying it to a model of stochastic bus networks, and present sample performance results comparing it to some alternatives. For the case where all or some of the uncertainty is resolved during path traversal, we extend the algorithm to produce optimal policies. This report is based on a paper presented at the Eleventh Conference on Unc...

We consider the problem of trave ling with least expec ted dela y in networ ks whose link delays change probabilistically acc ording to Markov cha ins. This is a typical routing problem in dynamic computer communication networ ks. We formulate sever al optimization problems, posed on infinite and finite horizons, and consider them with and without using memory in the decision making proc ess. We prove that all these problems ar e, in genera l, intrac table. Howe ver, for networks with nodal stochastic delays, a simple polynomial optimal solution is prese nted. This is typical of high-spee d networks, in which the dominant delays are incurre d by the nodes. For more gene ral networks, a tracta ble ε-optimal solution is pre sented.

The K shortest paths problem has been extensively studied for many years. Efficient methods have been devised, and many practical applications are known. Shortest hyperpath models have been proposed for several problems in different areas, for example in relation with routing in dynamic networks. However, the K shortest hyperpaths problem has not yet been investigated. In this paper we present procedures for finding the K shortest hyperpaths in a directed hypergraph. This is done by extending existing algorithms for K shortest loopless paths. Computational experiments on the proposed procedures are performed, and applications in transportation, planning and combinatorial optimization are discussed.

We present new complexity results and efficient algorithms for optimal route planning in the presence of uncertainty. We employ a decision theoretic framework for defining the optimal route: for a given source S and destination T in the graph, we seek an ST-path of lowest expected cost where the edge travel times are random variables and the cost is a nonlinear function of total travel time. Although this is a natural model for route-planning on real-world road networks, results are sparse due to the analytic difficulty of finding closed form expressions for the expected cost (Fan, Kalaba & Moore), as well as the computational/combinatorial difficulty of efficiently finding an optimal path which minimizes the expected cost. We identify a family of appropriate cost models and travel time distributions that are closed under convolution and physically valid. We obtain hardness results for routing problems with a given start time and cost functions with a global minimum, in a variety of deterministic and stochastic settings. In general the global cost is not separable into edge costs, precluding classic shortest-path approaches. However, using partial minimization techniques, we exhibit an efficient solution via dynamic programming with low polynomial complexity.

This paper examines the value of real-time traffic information to optimal vehicle routing in a nonstationary stochastic network. We present a systematic approach to aid in the implementation of transportation systems integrated with real time information technology. We develop decisionmaking procedures for determining the optimal driver attendance time, optimal departure times, and optimal routing policies under stochastically changing traffic flows based on a Markov decision process formulation. With a numerical study carried out on an urban road network in Southeast Michigan, we demonstrate significant advantages when using this information in terms of total costs savings and vehicle usage reduction while satisfying or improving service levels for just-in-time delivery.

Wireless networks combined with location technology create new problems and call for new decision aids. As a precursor to the development of these decision aids, a concept of communication distance is developed and applied to six situations. This concept allows travel time and bandwidth to be combined in a single measure so that many problems can be mapped onto a weighted graph and solved through shortest path algorithms. The paper looks at the problem of intercepting an out-of-communication team member and describes ways of using planning to reduce communication distance in anticipation of a break in connection. The concept is also applied to ad hoc radio networks. A way of performing route planning using a bandwidth map is developed and analyzed. The general implications of the work to transportation planning are discussed.

We consider stochastic networks' in which link travel times are dependent, discrete random variables. We present methods' for computing bounds' on path travel times using stochastic dominance relationships among link travel times, and discuss techniques for controlling tightness of the bounds'. We apply these methods' to shortest-path problems, show that the proposed algorithm can provide bounds' on the recommended path, and elaborate on extensions of the algorithm for demonstrating the anytime property.

We consider routing problems in dynamic networks where arc travel times are both random and time dependent. The problem of finding the best route to a fixed destination is formulated in terms of shortest hyperpaths on a suitable time-expanded directed hypergraph. The latter problem can be solved in linear time, with respect to the size of the hypergraph, for several definitions of hyperpath length. Different criteria for ranking routes can be modeled by suitable definitions of hyperpath length. We also show that the problem becomes intractable if a constraint on the route structure is imposed.

The Shortest Path with Recourse Problem involves finding the shortest expected-length paths in a directed network each of whose arcs have stochastic traversal lengths (or delays) that become known only upon arrival at the tail of that arc. The traveler starts at a given source node, and makes routing decisions at each node in such a way that the expected distance to a given sink node is minimized. We develop an extension of Dijkstra’s algorithm to solve the version of the problem where arclengths are nonnegative and reset after each arc traversal. All known no-reset versions of the problem are NP-hard. We make a partial extension to the case where negative arclengths are present.

Abstract. The objective is to minimize expected travel time from any origin to a specific destination in a congestible network with correlated link costs. Each link is assumed to be in one of two possible conditions. Conditional probability density functions for link travel times are assumed known for each condition. Conditions over the traversed links are taken into account for determining the optimal routing strategy for the remaining trip. This problem is treated as a multi-stage adaptive feedback control process. Each stage is described by the physical state (the location of the current decision point) and the information state (the service level of the previously traversed links). Proof of existence and uniqueness of the solution to the basic dynamic programming equations and a solution procedure are provided. Key Words. Shortest path, stochastic networks, dynamic programming, adaptive feedback control, correlated link costs.