A Bi-Objective Minimum Cost-Time Network Flow Problem

The Minimum Cost-Time Network Flow (MCTNF) problem deals with shipping the available supply through the directed network to satisfy demand at minimal total cost and minimal total time. Shipping cost is dependent on the value of flow on the arcs; however shipping time is a fixed time of using an arc to send flow. In this paper, a new Bi-Objective Minimum Cost-Time Flow (BOMCTF) problem is formulated. The first and second objective functions consider the total shipping cost and the total shipping fixed time, respectively. We utilize the weighted sum scalarization technique to convert the proposed model to a well-known fixed charge minimum cost flow problem with single objective function. This problem is a parametric mixed integer programming which can be solved by the existence methods. A numerical example is taken to illustrate the proposed approach.