A multi-objective approach for PH-graphs with applications to stochastic shortest paths

• Published in: Mathematical methods of operations research (Heidelberg, Germany) Vol. 93; no. 1; pp. 153 - 178
• Main Authors: Buchholz, Peter, Dohndorf, Iryna
• Format: Journal Article
• Language: English
• Published: Berlin, Heidelberg Springer 01.02.2021; Springer Berlin Heidelberg; Springer Nature B.V
• Subjects: Algorithms; Budgets; Business and Management; Calculus of Variations and Optimal Control; Optimization; Computational geometry; Convexity; Graphs; Markov decision processes; Markov processes; Mathematics; Mathematics and Statistics; Multicriteria optimization; Operations research; Operations Research/Decision Theory; Optimization; Original Article; Pareto optimum; PH graphs; Phase type distributions; Policies; Random variables; Shortest-path problems; Stochastic shortest path problems; Multicriteria optimization; PH graphs; Markov decision processes; Phase type distributions; Stochastic shortest path problems
• ISSN: 1432-5217; 1432-2994; 1432-5217
• DOI: 10.1007/s00186-020-00729-3

Stochastic shortest path problems (SSPPs) have many applications in practice and are subject of ongoing research for many years. This paper considers a variant of SSPPs where times or costs to pass an edge in a graph are, possibly correlated, random variables. There are two general goals one can aim for, the minimization of the expected costs to reach the destination or the maximization of the probability to reach the destination within a given budget. Often one is interested in policies that build a compromise between different goals which results in multi-objective problems. In this paper, an algorithm to compute the convex hull of Pareto optimal policies that consider expected costs and probabilities of falling below given budgets is developed. The approach uses the recently published class of PH-graphs that allow one to map SSPPs, even with generally distributed and correlated costs associated to edges, on Markov decision processes (MDPs) and apply the available techniques for MDPs to compute optimal policies.