Search

In this work we address a game theoretic variant of the shortest path problem, in which two decision makers (players) move together along the edges of a graph from a given starting vertex to a given destination. The two players take turns in deciding in each vertex which edge to traverse next. The decider in each vertex also has to pay the cost of the chosen edge. We want to determine the path where each player minimizes its costs taking into account that also the other player acts in a selfish and rational way. Such a solution is a subgame perfect equilibrium and can be determined by backward induction in the game tree of the associated finite game in extensive form. We show that the decision problem associated with such a path is PSPACE-complete even for bipartite graphs both for the directed and the undirected version. The latter result is a surprising deviation from the complexity status of the closely related game Geography . On the other hand, we can give polynomial time algorithms for directed acyclic graphs and for cactus graphs even in the undirected case. The latter is based on a decomposition of the graph into components and their resolution by a number of fairly involved dynamic programming arrays. Finally, we give some arguments about closing the gap of the complexity status for graphs of bounded treewidth. [ABSTRACT FROM AUTHOR]

Abstract: A supply chain network design needs to consider the future probability of reconfiguration due to some problems of disaster or price changes. The objective of this article is to design a reconfigurable supply chain network by optimizing inventory allocation and transportation routing. A two-stage programming is composed according to Benders decomposition by allocating inventory in advance and anticipating the changes of transportation routings; thus the transportation routing is stochastic in nature. In addition, the fuzzy shortest path is developed to solve the problem complexity in terms of the multi-criteria of lead time and capacity with an efficient computational method. The results and analysis indicate that the proposed two-stage programming with fuzzy shortest path surpasses the performance of shortest path problem with time windows and capacity constraint (SPPTWCC) in terms of less computational time and CPU memory consumption. Finally, management decision-making is discussed among other concluding remarks. [Copyright &y& Elsevier]