The capacitated dynamic lot-sizing problem with startup and reservation costs: A forward algorithm solution

Abstract

Abstract Often in production environments, there are several products available to be processed on a machine. The decisions that must be made are: 1) which product should be processed next, and 2) how much time should elapse before a changeover occurs and a different product is chosen for processing. This problem is known as the multiple product cycling problem. The single product variation of this problem is addressed in this paper. It is assumed that the product to be processed next has already been chosen (decision #1). The second decision concerning production run length must now be determined. In the single product problem, there is a fixed startup cost associated with turning the machine on in a period when the machine was off in the previous period and a fixed reservation cost incurred in each period in which the machine is on. Ending inventory in a period is assessed a holding cost. The relationship of the single product problem to the multiple period problem is as follows. The startup cost is identical to the changeover cost in the multiple product problem. By processing a particular product, other products must wait. This effect is captured by the reservation cost which is the opportunity cost one would incur by having the machine dedicated to processing a particular product. A forward dynamic programming algorithm is presented which uses heuristic rules to detect empirical decision horizons. The entire problem is partitioned into smaller Subproblems which are then easily solved. The subproblems are created by the occurrence of startup regeneration points. These are periods where there is no ending inventory and the following period contains a machine startup. Computational experience is reported which shows that reductions in computational effort (measured in CPU time and the number of nodes in the decision tree evaluated) can be as high as 95 to 99% when using the decision horizon techniques and bounding procedures described in this paper. These results are based on the solution of 28 test problems that were each 20 periods in length. Because a forward algorithm was used, not only were the solutions to the 20-period problems obtained, but the solutions to the one-period problems through 19-period problems were obtained as well. Even though the procedure described in this paper cannot guarantee that it will always find the optimal solution, it did find the optimal solution in every problem tested. By being able to solve the single product problem presented in this paper quickly, the ability to rapidly generate optimal or near optimal solutions to the multiple product problem (which uses the procedure described in this paper as a subroutine) appears closer to realization.