A Weighted Sum Method for Optimizing Multi-Objective Transportation Problems  

Rajesh Shrivastava¹, Surendra Singh Rajput²

¹ Professor, Department of Mathematics, Dr Shyama Prasad Mukherjee Science & Commerce College, Bhopal, India.

² Research Scholar, Department of Mathematics, Dr Shyama Prasad Mukherjee Science & Commerce College, Bhopal, India.   

Abstract

This study addresses the optimization of multi-objective transportation problems with the motivation to minimize transportation cost, time, and risk. A weighted sum method is employed to convert multiple objectives into a single objective function. Three cases were considered by assigning varying priority weights to each objective, reflecting the importance of cost, time, and risk in transportation planning. The transformed transportation matrices were solved using the Vogel Approximation Method, followed by the MODI Method to obtain the optimal solutions. The results show that different weight assignments lead to significant variations in the optimal transportation cost, ranging from Rs 1406.00 to Rs 1689.00. When cost was prioritised (w₁=0.5, w₂=0.3, w₃=0.2), the optimal cost was Rs 1406.00; when time was prioritised (w₁=0.2, w₂=0.6, w₃=0.2), the cost increased to Rs 1440.00; and when risk was prioritised (w₁=0.1, w₂=0.2, w₃=0.7), the cost rose to Rs 1689.00. These findings demonstrate that prioritising risk reduction substantially increases transportation cost, highlighting the trade-offs decision-makers must navigate when balancing these competing objectives. The weighted sum method provides a flexible framework for incorporating decision-maker preferences, though its limitations, including the inability to generate non-convex Pareto frontiers, are acknowledged.      

Keywords: Transportation problem, Multi-objective, Weight sum method, Optimal solution

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