Determining Logistics Route Capacity Using Maximal Solutions of Two-Sided Max-Min Linear System with Python Implementation

Abstract

Max-min algebra, equipped with maximum and minimum operations, provides a natural framework for modeling systems with bottleneck constraints, such as logistics and transportation networks. An algorithm for solving two-sided linear systems of the form A ⊗ x = B ⊗ x in max-min algebra has been proposed, for which the maximal solution can be constructed explicitly for the 1 × n case under three conditions—unconstrained, with a given upper bound, and with a given lower bound—and generalized to the m × n case, terminating after at most m iterations and producing a unique maximal solution with complexity O(m²n). However, no implementation has been made available, which limits its use for larger, practically sized problems: the algorithm may require up to m computational cycles, so the computational burden grows with the matrix size—particularly the number of rows—and manual calculation becomes more prone to error. This paper addresses this gap by providing a Python implementation of the algorithm, along with a numerical application to a logistics distribution network. The Python implementation makes the algorithm accessible to practitioners, and the logistics example demonstrates how the model can be used to determine optimal road capacities. This work provides a practical computational tool for solving capacity planning problems in logistics and transportation.