Makespan minimization in a job shop with a batch processing machine

A job shop with one batch processing and several discrete machines is analyzed. Given a set of jobs, their process routes, processing requirements, and size, the objective is to schedule the jobs such that the makespan is minimized. The batch processing machine can process a batch of jobs as long as the machine capacity is not violated. The batch processing time is equal to the longest processing job in the batch. The problem under study can be represented as Jm:batch:Cmax. If no batches were formed, the scheduling problem under study reduces to the classical job shop scheduling problem (i.e. Jm:: Cmax), which is known to be NP-hard. This research extends the scheduling literature by combining Jm::Cmax with batch processing. The primary contributions are the mathematical formulation, a new network representation and several solution approaches. The problem under study is observed widely in metal working and other industries, but received limited or no attention due to its complexity. A novel network representation of the problem using disjunctive and conjunctive arcs, and a mathematical formulation are proposed to minimize the makespan. Besides that, several algorithms, like batch forming heuristics, dispatching rules, Modified Shifting Bottleneck, Tabu Search (TS) and Simulated Annealing (SA), were developed and implemented. An experimental study was conducted to evaluate the proposed heuristics, and the results were compared to those from a commercial solver (i.e., CPLEX). TS and SA, with the combination of MWKR-FF as the initial solution, gave the best solutions among all the heuristics proposed. Their results were close to CPLEX; and for some larger instances, with total operations greater than 225, they were competitive in terms of solution quality and runtime. For some larger problem instances, CPLEX was unable to report a feasible solution even after running for several hours. Between SA and the experimental study indicated that SA produced a better average Cmax for all instances. The solution approaches proposed will benefit practitioners to schedule a job shop (with both discrete and batch processing machines) more efficiently. The proposed solution approaches are easier to implement and requires short run times to solve large problem instances.

The order selection and lot sizing problem in the make-to-order environment

This research is motivated by the need for considering lot sizing while accepting customer orders in a make-to-order (MTO) environment, in which each customer order must be delivered by its due date. Job shop is the typical operation model used in an MTO operation, where the production planner must make three concurrent decisions; they are order selection, lot size, and job schedule. These decisions are usually treated separately in the literature and are mostly led to heuristic solutions. The first phase of the study is focused on a formal definition of the problem. Mathematical programming techniques are applied to modeling this problem in terms of its objective, decision variables, and constraints. A commercial solver, CPLEX is applied to solve the resulting mixed-integer linear programming model with small instances to validate the mathematical formulation. The computational result shows it is not practical for solving problems of industrial size, using a commercial solver. The second phase of this study is focused on development of an effective solution approach to this problem of large scale. The proposed solution approach is an iterative process involving three sequential decision steps of order selection, lot sizing, and lot scheduling. A range of simple sequencing rules are identified for each of the three subproblems. Using computer simulation as the tool, an experiment is designed to evaluate their performance against a set of system parameters. For order selection, the proposed weighted most profit rule performs the best. The shifting bottleneck and the earliest operation finish time both are the best scheduling rules. For lot sizing, the proposed minimum cost increase heuristic, based on the Dixon-Silver method performs the best, when the demand-to-capacity ratio at the bottleneck machine is high. The proposed minimum cost heuristic, based on the Wagner-Whitin algorithm is the best lot-sizing heuristic for shops of a low demand-to-capacity ratio. The proposed heuristic is applied to an industrial case to further evaluate its performance. The result shows it can improve an average of total profit by 16.62%. This research contributes to the production planning research community with a complete mathematical definition of the problem and an effective solution approach to solving the problem of industry scale.