MATLAB Code for Solving Transportation Problems

Solving transportation problems using MATLAB constitutes a broad and complex topic. By modifying matrices A (supply), D (demand), and B (cost), we can solve various transportation scenarios. The transportation model assumes m origins A1, A2, ..., Am, each with specific supply quantities, and n destinations B1, B2, ..., Bn, each with specific demand requirements. The cost to transport one unit from origin Ai to destination Bj is denoted as cij. The core challenge involves determining optimal shipment quantities xij to minimize total transportation costs, typically organized in a transportation tableau. The mathematical formulation minimizes the objective function min Σ(cij * xij) subject to constraints (S.T.): supply constraints Σxij = ai for each origin, demand constraints Σxij = bj for each destination, and non-negativity constraints xij ≥ 0. When total supply equals total demand (Σai = Σbj), it constitutes a balanced transportation problem; otherwise, it's unbalanced. Unbalanced problems can be converted to balanced form by introducing dummy origins or destinations with zero transportation costs. While computationally complex, understanding the underlying principles and mastering MATLAB implementation techniques enables effective solutions. This problem shares similarities with assignment problems, production planning models, and other logistics optimization challenges that can be implemented using MATLAB's optimization toolbox functions like linprog for linear programming solutions. Key implementation aspects include: constructing the constraint matrix using sparse matrix techniques, handling unbalanced problems through dummy variable insertion, and utilizing MATLAB's optimization algorithms to efficiently solve large-scale transportation problems. The code typically involves matrix manipulation for cost coefficients and constraint setup, followed by calling optimization solvers with proper problem formulation.