Order Relation between Intervals and Its Application to Shortest Path Problem

Abstract

We expand the definition of Ishibuchi and Tanaka's order relation between intervals by means of the left and right limits, center, and width of an interval, and propose the new definition by introducing two kind of parameters, that is, one is the degree between partial and total order relation, the other is the degree of a decision maker's preference for the expectation and the pessimistic value. A variety of setting of the parameters enable him to control the number of maximal/minimal intervals and to reflect his preference in the order relation between intervals. We try to apply this definition to the shortest path problem which is one of the simple and basic network problems and has a wide range of applications. Let each arc in the network be transportation time or cost instead of distance. However, the time and cost fluctuate depending on traffic conditions, payload and so on. Therefore, each arc has better be represented as interval composed a pair of optimistic and pessimistic values. In order to solve the interval version of the problem, we expand the Dijkstra's algorithm, and propose a new algorithm for obtaining some incomparable interval lengths along the routes. When too many routes exist for the decision maker to select the best one, he can reduce the number of routes by altering of the parameters. Finally, A large scale example based on the proposed algorithm is shown and the effectiveness is demonstrated.