The problem of finding a minimal cost-reliability ratio path is considered. The optimal solution to this problem is shown to map into an extreme supported non-dominated objective point in the objective space of the biobjective shortest path problem. Different forms of reliability are presented. We assume that this reliability does not change over time. We employ a parametric network simplex alg...

The Minimum Cost Path Problem with Relays (MCPPR) consists of finding a minimum cost path from a source to a destination, along which relay nodes are located at a certain cost, subject to a weight constraint. This paper first models the MCPPR as a particular bicriteria path problem involving an aggregated function of the path and relay costs, as well as a weight function. A variant of this prob...

the problem of finding the minimum cost multi-commodity flow in an undirected and completenetwork is studied when the link costs are piecewise linear and convex. the arc-path model and overflowmodel are presented to formulate the problem. the results suggest that the new overflow model outperformsthe classical arc-path model for this problem. the classical revised simplex, frank and wolf and a ...

Proof: (by induction). We want to show that one SAP doesn’t introduce negative cycles in Gf . Initially there are no negative cost cycles. Feasible prices can be computed by using shortest path distances from s. After finding the shortest s-t path, it has reduced cost 0. Every arc on the path has reduced length 0. This demonstrates that the triangle inequality property is tight on shortest path...