LOGISTICS OF CAPACITATED TRANSPORTATION PROBLEMS

Abstract

In the context of modern global logistics, enterprises face increasing challenges in managing transportation costs efficiently under strict infrastructure constraints. The classical transportation problem, while fundamental to operations research, assumes ideal conditions–unlimited route capacities and direct shipments–that rarely exist in practice. This discrepancy leads to theoretical models that yield «optimal» solutions which are unimplementable in real-world scenarios involving road repairs, limited warehouse throughput, or specific delivery priorities. This study addresses the critical need for adapting mathematical optimization models to these realistic constraints. The purpose of the research is to analyze the impact of additional constraints–specifically arc capacity limits and priority servicing of specific destinations– on the optimal distribution plan and total transportation costs. The study aims to quantify the “cost of constraints” and demonstrate how mathematical modeling can support decision-making in disrupted supply chains. The research methodology is based on mathematical modeling of transportation networks using linear programming. The study utilizes the LINDO (Linear, Interactive, and Discrete Optimizer) software environment to solve optimization tasks. The modeling process involves two scenarios: a basic unconstrained transportation problem and a capacitated variation with priority subsets. The approach integrates theoretical foundations established by L. Kantorovich and F.L. Hitchcock with modern algorithmic solutions for capacitated networks. Additionally, the study employs dual problem analysis to determine shadow prices, providing an economic interpretation of the limiting factors. The results of the numerical experiment, conducted on a balanced transportation network with four sources and five destinations, demonstrate that introducing capacity constraints on specific arcs and imposing priority delivery conditions significantly alters the optimal distribution plan. The system is forced to reroute flows through more expensive paths to satisfy the new boundary conditions. In the case study, these constraints led to an increase in total transportation costs from 195 to 240 monetary units, representing a 23 % rise in expenses. This quantitative finding validates the theoretical assumption that restrictions on the feasible region in linear programming generally worsen the objective function value. Sensitivity analysis further revealed that specific «bottleneck» routes have high shadow prices, indicating where infrastructure investment would yield the highest return. The practical value of this research lies in providing a proven methodology for simulating logistical bottlenecks. By calculating the cost difference between unconstrained and constrained models, logistics managers can make informed financial decisions–for example, determining whether investing in infrastructure to remove a bottleneck is more costeffective than paying higher transportation costs. The proposed model is adaptable for multi-product and multi-stage logistics networks and serves as a basis for decision support systems in regional logistics centers. The findings of this study also provide a foundation for future research into dynamic routing algorithms under uncertainty conditions and autonomous transport integration.