Multi item multi period capacitated lot sizing problem with backorders and setup considerations: strong and weak formulations

We investigate different formulations of the multi item, multi period capacitated lot sizing problem with inclusions of backorders, setup times and setup costs into it. The problem is closer to the realistic situations and is abbreviated as CLSP_BS in this work. Apart from the classical formulation, we give two variants of the transportation formulation of CLSP_BS. Objective values of these three formulations are exactly equivalent to each other, but they rank different in terms of computational times. When we compare the bounds obtained by LP relaxation of the classical and the two transportation formulations, it is observed that classical and one of the two transportation formulations are exactly equivalent; however the other transportation formulation generates a comparatively better bound. Based on this information on strength of bounds, we earmark the formulations of CLSP_BS as strong and weak. This knowledge about strong and weak formulations can prove to be fruitful while solving real life large sized problems. Limited computational experiences are shown here which establish the stated claims.