Proposed an Algorithm for Solving a Special Case of Multi-Index Multistage Transportation Problem

The transportation problem is a special case from applications of linear programming and plays a vital role in the logistics supply chain. Its primary objective is to minimize the total cost of transportation for a network that spans multiple sources and destinations while ensuring that the sources are available, and the destinations are fulfilled. This paper presents the mathematical formulations for four types of multi-index multistage transportation problems (MMTP1, MMTP2, MMTP3, and MMTP4), which cover the various conditions that apply to operating trucks fleets in real transportation networks. The algorithm introduced here is designed to solve MMTP1, a multi-index multistage transportation problem without any restrictions on transportation in the intermediate stages. The algorithm transforms the problem into a multi-index single-stage problem using dynamic programming to find the minimum transportation cost for each means of transport between the first stage's sources and the last stage's destinations. The LINGO software is then used to validate the optimal solution. An illustrative example is provided to demonstrate the effectiveness of the proposed algorithm.