An efficient algorithm for two-stage capacitated time minimization transportation problem with restricted flow

This paper discusses a two-stage capacitated time minimization transportation problem with the restricted flow (TSCTMTP-F) in which the transportation takes place in two stages and only a specified amount of commodity is transported in both stages. The total amount F_1 is transported during Stage-I and F_2 during Stage-II, and the objective is to minimize the sum of the transportation times for Stage-I and Stage-II. In 2017, Kaur et al. (2017) studied this problem and developed a polynomially bounded iterative algorithm (Algorithm-A) to solve TSCTMTP-F. However, their proposed algorithm has some flaws and may not always yield an optimal solution to the problem TSCTMTP-F. An improved iterative algorithm (Algorithm-C) is proposed in this paper that guarantees an optimal solution to the problem. Various theoretical results prove the convergence and efficiency of Algorithm-C to obtain an optimal solution to the problem TSCTMTP-F. Numerical problems are included in the support of theory along with a counter-example for which Algorithm-A fails to obtain its optimal solution. Computational experiments on a variety of test problems have been carried out to validate the convergence and efficiency of Algorithm-C.