A number of scheduling and assignment problems are presented involving the execution of periodic operations in a multiprocessor environment. We consider the computational complexity of these problems and propose approximation algorithms for operations with identical periods as well as for operations with arbitrary integer periods.

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Periodic multiprocessor scheduling

This paper presents a stepwise partial enumeration algorithm for determining the economic lot schedule of a multiproduct single facility system on a repetitive basis. The algorithm is based on a simple “sufficient feasibility” test. The algorithm starts from a feasible schedule determined from the common cycle solution and continues stepwise partial enumerations to improve the schedule until no appreciable improvement occurs. Performance of the algorithm is demonstrated by solving six problems recently solved by Haessler. The algorithm gives better or equal solutions to the problems.

A Stepwise partial enumeration algorithm for the economic lot scheduling problem

In a supply chain, it is an important issue for logistic managers to offset the replenishment cycles of multiple products sharing a warehouse so as to minimize the maximum warehouse space requirement (MWSR). Most of the studies in the literature assume that warehouses replenish at the beginning of some basic planning period. In this paper, we relax this assumption by allowing the warehouse to replenish at any time. In order to solve this problem, we conduct theoretical analysis based on Fourier series and Fourier transforms and propose a procedure that is used to calculate MWSR efficiently for any given replenishment schedule. Then, we employ this procedure in a genetic algorithm (GA) to search for the optimal replenishment schedule. Using randomly generated instances, we show that the proposed GA significantly outperforms a previously published heuristic.

A genetic algorithm for determining optimal replenishment cycles to minimize maximum warehouse space requirements

We consider two economic order quantity models where multiple items use a common resource. In the tactical model, the objective is to establish and coordinate order quantities to minimize total inventory ordering and carrying costs without ever exceeding a capacity constraint on the resource. In the strategic model, the objective is to establish and coordinate order quantities to minimize the average cost which includes in addition to the ordering and inventory costs, a cost proportional to the peak usage of the resource. The common resource may be space in an automated warehouse or total capital invested in inventory. We show that a lower bound on the peak resource usage, known for certain subsets of policies, is valid for any feasible policy. We use this to derive lower bounds on the optimum average cost for both models and show that simple heuristics for either model have bounded worst-case performance ratios. More sophisticated heuristics require an effort to solve the embedded staggering problem of time-phasing the arrival of the orders to minimize the maximum use of the resource. We develop a heuristic for the staggering problem and use it to obtain efficient heuristics for the strategic and tactical models.

Single Resource Multi-Item Inventory Systems

Vehicle minimization for periodic deliveries

This paper studies the scheduling of multiple activities that require periodic processing on a single facility for an infinite horizon. It first formulates this scheduling problem as an integer linear programming (ILP) model, and then uses the Chinese Remainder Theorem to efficiently partition the original into smaller independent models. For a small problem involving about 10 activities, the solution can be obtained very simply by hand. An example is given to illustrate the approach.

Kyung Soo Park

Papers

Optimal Scheduling of Periodic Activities