Sensitivity Analysis and Programmatic Verification of Transportation Problems: A Comparison Between Manual Method and Software Solutions Using R and python: An Applied Study on Abnaa Al-Mashat Cement Brick Factory

Abstract

The transportation problem is one of the most important applications of linear programming in operations research, aiming to determine the optimal distribution plan that achieves the minimum possible cost while adhering to supply and demand constraints, making it an effective tool for supporting logistical decisions in industrial enterprises. This study aims to validate the manual solutions of the transportation problem and compare them with programmatic solutions using R and Python, in addition to conducting sensitivity analysis to assess the stability of the optimal solution and extract shadow prices and reduced costs, thereby supporting decision-making under potential changes in model data.

The study relied on a transportation model applied at the Abnaa Al-Mashat factory for cement brick manufacturing, utilizing supply, demand, and transportation cost data derived from a previous study (Alfaqihe-Alshrefe, 2025). The model was solved manually using the Northwest Corner, Least Cost, and Vogel's Approximation methods, in addition to the Two-Phase Simplex algorithm, and then verified programmatically using the lpSolve library in R and the PuLP library in Python.

The results showed a complete match between the manual and programmatic solutions, with the minimum total transportation cost reaching 155,650 monetary units. Sensitivity analysis also evaluated shadow prices and reduced costs and tested the stability of the optimal solution, showing that the optimal solution possesses a good degree of stability, and the shadow prices provided quantitative indicators to assess the impact of changes in resources and demand on the total cost.