An FPTAS for Quickest Multi-commodity Contraflow Problem with Asymmetric Transit Times

One of the challenges in operations research is to route numerous commodities from particular supply locations to the corresponding demand points across the lanes of a network infrastructure while maintaining capacity restrictions. The quickest multi-commodity flow problem would be one of those that reduces the time it takes to complete the process. Reorienting lanes toward demand sites can increase outbound lane capacity. The quickest multi-commodity contraflow problem is NP-hard computationally. We use a ∆-condensed time-expanded graph to propose an FPTAS for this problem by including the lane reversal technique. We look into asymmetric transit times on anti-parallel arcs to address the unequal road conditions and flow dependency.