Two-phase time minimization transportation problem with the restricted flow

Motivated by the hierarchical structure within transportation systems, this paper explores a two-phase time minimization transportation problem with restricted flow ($2p-T{P}_F$). In this problem, the transportation of products occurs in two distinct phases due to the partition of source–destination links into two separate levels: Level-1 and Level-2 links, with a specified amount of commodity being transported in each phase. For the transportation of a specific quantity of goods during the first phase, only Level-1 links are utilized. Following this, during the second phase of transportation, only Level-2 links are utilized. Transportation is carried out concurrently via numerous source–destination links relevant to each phase. This paper proposes an iterative algorithm (Algorithm-$T{P}_F$) to find an optimal solution for a two-phase time minimization transportation problem that minimizes the sum of Phase-1 and Phase-2 transportation times. The proposed algorithm solves a solid time minimization transportation problem$\setminus$its restricted version at each iteration. Various theoretical results are proven to support the convergence of the algorithm. Numerical examples of various sizes are provided to support the theoretical results. Computational experiments conducted on randomly generated instances demonstrate the algorithm’s efficiency and convergence. The proposed algorithm offers an alternative method for solving two-phase time minimization transportation problems.