A Column Generation Algorithm for the Resource-Constrained Order Acceptance and Scheduling on Unrelated Parallel Machines

In this paper, we investigate the resource-constrained order acceptance and scheduling on unrelated parallel machines that arise in make-to-order systems. The objective of this problem is to simultaneously select a subset of orders to be processed and schedule the accepted orders on unrelated machines in such a way that the resources are not overutilized at any time. We first propose two formulations for the problem: mixed integer linear programming formulation and set partitioning. In view of the complexity of the problem, we then develop a column generation approach based on the set partitioning formulation. In the proposed column generation approach, a differential evolution algorithm is designed to solve subproblems efficiently. Extensive numerical experiments on different-sized instances are conducted, and the results demonstrate that the proposed column generation algorithm reports optimal or near-optimal solutions that are evidently better than the solutions obtained by solving the mixed integer linear programming formulation.

Fuzzy Order Acceptance and Scheduling on Identical Parallel Machines

In this study, we focus on the fuzzy order acceptance and scheduling problem in identical parallel machines (FOASIPM), which is a scheduling and optimization problem to decide whether the firm should accept or outsource the order. In general, symmetry is a fundamental property of optimization models used to represent binary relations such as the FOASIPM problem. Symmetry in optimization problems can be considered as an engineering tool to support decision-making. We develop a fuzzy mathematical model (FMM) and a Genetic Algorithm (GA) with two crossover operators. The FOASIPM is formulated as an FMM where the objective is to maximize the total net profit, which includes the revenue, the penalty of tardiness, and the outsourcing. The performance of the proposed methods is tested on the sets of data with orders that are defined by fuzzy durations. We use the signed distance method to handle the fuzzy parameters. While FMM reaches the optimal solution in a reasonable time for datasets with a small number of orders, it cannot find a solution for datasets with a large number of orders due to the NP-hard nature of the problem. Genetic algorithms provide fast solutions for datasets with a medium and large number of orders.