Bilevel time minimizing transportation problem

For a given time minimizing transportation problem comprising mm sources and nn destinations, the set of mm sources is to be optimally partitioned into two mutually disjoint subsets L1L1 and L2L2 where, L1L1 contains m1m1 sources called Level-I sources and L2L2 contains the remaining (m−m1)(m−m1) sources termed as Level-II sources. First, the Level-I decision maker sends the shipment from Level-I sources to partially meet the demand of destinations. Later, the Level-II decision maker sends the material from the Level-II sources to meet the left over demand of the destinations. A finite number of cost minimizing transportation problems are solved to judiciously generate a few of Cm1m partitions of the set of mm sources. The aim of this study is to find an optimal partition of the set of mm sources such that the sum of times of transportation in the Level-I and Level-II shipments is the minimum. The proposed polynomial bound algorithm to find the global minimizer has been successfully coded in C++ and run on a variety of randomly generated test problems differing in input data.