The quickest path problem with interval lead times

Abstract

In this paper, a version of the quickest path problem is considered in which arc capacities are fixed but lead times are expressed as intervals on real line. By considering order relations that represent the decision maker's preference between interval transmission times, we show that the problem can be solved by transforming it into a sequence of bicriteria shortest path problems in properly defined networks. Scope and purpose The quickest path problem has been proposed to cope with flow problems through networks whose arcs are characterized by both lead times and arc capacities. Basically, it consists in finding a path in a network to transmit a given amount of items from a source node to a sink with minimum transmission time, when the transmission time depends on both the traversal times of the arcs and the rates of flow along arcs. This model assumes that the coefficients of the problem are deterministic and fixed in value. Typically, arc capacities are specified in a precise way by the physical framework of the network. However, lead times can fluctuate depending on traffic conditions. In such a case, it might be useful and preferable to use interval numbers to represent the inexactness of the coefficients. This paper is focused on the solution procedure of the quickest path problem when lead times are expressed as intervals on real line. Taking into account order relations between intervals to compare paths, we show that the interval quickest path problem can be solved by successively determining efficient paths in networks which have fewer and fewer arcs.