- In this research, a bi-criteria group scheduling problem is investigated in hybrid flow shop (HFS) environments, where the parallel machines in each stage are unrelated, meaning not identical. The objective of the problem is to minimize a linear combination of the total weighted completion times as a means of complying with the interests of the producer, and the total weighted tardiness as a means of complying with the interests of customers. The underlying assumptions of the problem include the group technology assumptions (GTA) that require all jobs within a group to be processed successively and on the same machine. The runtime of these jobs are dynamic and progressively decrease as the worker learns how to perform similar jobs. A sequence-dependent setup time is considered for switching between different groups on the same machine. Although all jobs have to move in unidirectional paths through the HFS, some may skip some of the stages. Furthermore, in order to capture more realistic features of the scheduling problems, the jobs are assumed to be released into the system at dynamic times, and the machines, as well, are assumed to be available at dynamic times. The problem is formulated as a mixed-integer linear programming (MILP) model. The MILP model for small sizes of the problem is solved to optimality using CPLEX. However, since the problem is strongly NP-hard, it is not possible to find its optimal solution within a reasonable time as the problem size increases to medium to large.
Several meta-heuristic algorithms based on tabu search (TS), simulated annealing (SA), and genetic algorithm (GA) are developed to find the optimal/near optimal solutions for this problem. Three alterations of algorithms are developed for TS and SA-based algorithms (referred to as local search algorithms) i.e. non-permutation, partial permutation and local searches with embedded progressive perturbations. Two alternatives are also considered for GA-based algorithms (referred to as population-based algorithms) i.e. simple GA and bi-level GA. The performances of these algorithms are compared to each other in order to identify which algorithm, if any, outperforms the others. Nevertheless, the performances of all algorithms are evaluated with respect to a tight lower bound (LB) obtained based on a branch-and-price (B&P) technique developed in this research.
The B&P technique uses Dantzig-Wolfe decomposition to divide the original problem into a master problem and several sub-problems. Although, the sub-problems are smaller than the original problem, they are still strongly NP-hard and cannot be optimally solved within a reasonable amount of time. However, an optimal dispatching rule is proposed that drastically reduces the number of variables and constraints in these sub-problems, and enables the B&P algorithm to find tight lower bounds even for large-size instances of the problem. A comparison between these lower bounds and the ones obtained from CPLEX reveals the impressive performance of the B&P algorithm, i.e. an average of 233% improvement for the largest size of the problems that have been tested. Evaluation of the proposed algorithms with respect to these tight lower bounds uncovers the outstanding performance of all the proposed algorithms, while identifying the bi-level GA as the best performing algorithm in dealing with the HFS scheduling problem. This algorithm reports a remarkable performance with an average deviation of only 2% from the optimal solution for small-size sample problems, and an average gap of 23% from the lower bound for the largest sizes of the tested problems. The largest problem tested in this research consists of a total of 1858 binary variables and 14654 constraints.