A novel approach based on relaxation and reduction to solve the capacitated lot sizing problem

Capacitated lot sizing problem (CLSP) finds its applications in manufacturing, production as-well-as process industries. Typical size of the real life problems is much larger compared to the sizes that can be handled by commercial solvers, as CLSP is an NP-hard problem. Researchers have attempted different variants of CLSP in last few decades and provided many classical, exact, heuristic and metaheuristic solution approaches. In this paper, we attempt to solve the multi item capacitated dynamic lot sizing problem with backorders and setup times (CLSPBS). We apply Lagrangian relaxation on the material balance constraint, to reduce the problem into several single constraint continuous knapsack problems with upper bound on the real variables. Bounded variable linear program determines the values of real variables. For the negative or positive value of coefficients in the Lagrangian function, we fix the setup variables at zero or one respectively. Thus, the reduced problem is solved in polynomial time, as the main effort required is only sorting of numbers. We use the set up variable that are at one, to develop yet another good upper bound. Branch-and-bound procedure benefits immensely by using these bounds to make a warm start.