A Two-Phase Normalized Deviation Algorithm for Multi-Objective Fractional Solid Transportation Problems with SDR-Based Decomposition

Abstract

A two-phase normalized deviation algorithm is proposed for solving the Multi-Objective Fractional Solid Transportation Problem (MOFSTP). The approach integrates a Source–Destination–Routing (SDR) based decomposition framework to separately optimize the numerator and denominator components of each fractional objective. The obtained solutions are normalized using bounded reference values to ensure comparability across objectives. A unified compromise model is then constructed through a weighted deviation function without introducing additional deviational variables. The proposed formulation preserves Pareto efficiency while reducing computational complexity associated with traditional goal programming and weighted sum approaches. Numerical experiments demonstrate that the method provides stable trade-off solutions and improved objective balance under varying preference structures. The results confirm the effectiveness of the proposed algorithm in handling multi-objective fractional transportation models arising in logistics and resource allocation systems.