# Mathematical Optimization in Flexible Job Shop Scheduling: Modelling, Analysis, and Case Studies

[Doktorsavhandling]

The major theme of this thesis is the mathematical optimization modelling of the flexible job shop scheduling problem. This problem is present in the so-called multitask cell—a production cell at GKN Aerospace's facility located in Trollhättan, Sweden—which has served as a case study during the work with this thesis. The thesis consists of two parts, of which the first considers the major theme which runs through the first three appended papers. The second part of the thesis corresponds to the fourth appended paper and deals with the computational complexity of flow shop scheduling problems with deteriorating jobs. The basic flow shop scheduling problem is a special case of the flexible job shop scheduling problem.

In the first paper, two mathematical optimization models are formulated for the general flexible job shop scheduling problem. One of the models—a so-called time-indexed model—is incorporated into an iterative procedure, which is then shown to solve instances of the problem much faster than when the model is directly solved. The iterative procedure outperforms two other mathematical optimization models regarding the computation time required to find the best solution. We have studied two main types of objective functions: the makespan and a weighted average of completion times and tardiness of the jobs; we observed a large difference in the performance of the scheduling models between the objectives. This implies that any evaluation of scheduling algorithms must be done with respect to an objective that is relevant for the application for which they are intended. The makespan objective is the most widely used within research, however, it is seldom relevant in real applications. We recommend as a suitable objective to minimize the weighted average of the completion times and the total tardiness where the tardiness weight decreases with the job's due date.

In the second paper, we study the scheduling of the multitask cell and propose a means to schedule the cell including fixture availability and preventive maintenance activities. A suitable definition of the tardiness weights is proposed, and the quality of the resulting schedules, when applied in a dynamic environment, is discussed. A time-indexed model is formulated including three variants of modelling the preventive maintenance activities. Real instances collected from the multitask cell are solved by a version of the iterative procedure employing the time-indexed model. We are able to produce near-optimal schedules for the real data instances within 15 minutes, which is regarded as an acceptable time frame.

The third paper continues the work presented in the second paper: the time-indexed model is further developed and the possibility to schedule operations with unmanned processing during night shifts is explored. We are able to substantially increase the output of the cell with the inclusion of unmanned processing during night shifts. For a number of real scenarios for the coming shift collected from the multitask cell we compare the schedules produced by our iterative procedure—with and without the night shift possibilities included—with those constructed by the FIFO (first–in, first–out) and by a built-in (in the multitask cell) critical ratio (CR) priority dispatching rule. The objective values obtained after running our iterative procedure for 15 minutes were on average 0.71 times the corresponding values computed by the FIFO or CR rules.

The fourth appended paper establishes a complexity result for a flow shop scheduling problem in which the jobs' processing times are functions of the jobs' starting times.

**Nyckelord: **Flexible job shop scheduling, Mixed integer linear programming (MILP), Time-indexed formulation, Makespan, Tardiness, Fixture availability, Preventive maintenance, Night shift, Unmanned time window, Dynamic scheduling, Priority rules, Dispatching rules, Critical ratio

Denna post skapades 2013-08-20. Senast ändrad 2013-09-25.

CPL Pubid: 181850