This study focuses on a particular instance of the capacitated multi-index transportation problem, specifically the four-index variant. It involves transporting various products from multiple sources to multiple destinations using different means of transport chosen depending on reserved charges, while satisfying the constraints associated with each dimension of the model. Although this problem was previously studied by Zitouni and Keraghel (2003), the available methods still suffer from slow convergence and difficulty in handling degeneracy. This work proposes a new procedure that ensures a stable and feasible starting solution. The approach involves introducing dummy allocations with very small values to some null allocations, in order to handle cases of degeneracy. It is based on an extended version of the Vogel's approximation method, where the traditional selection criterion is modified to account for the four dimensions of the problem. This enhancement improves the initial distribution of quantities, reduces ineffective allocations and accelerates convergence. The proposed method maintains full compliance with all problem constraints including supply, demand, capacity and the structural logic of the four-index model. The methodology was evaluated through a comparative numerical study, applying the proposed method to a set of problem instances of varying sizes, and comparing it with Zitouni's approach in terms of convergence speed and number of iterations. The results show that the proposed method achieves a clear advantage in terms of faster convergence required to reach the solution. This highlights the relevance of the approach for complex logistical models and supply chain planning, where efficient solutions must be obtained within limited time. Moreover, the proposed approach lays the foundation for the development of more advanced multi-index optimization algorithms.
linear programming, transportation problem, capacitated transportation problem, multi-index problem
90C05, 90B06, 03B52