A new Lagrangian relaxation algorithm for hybrid flowshop scheduling to minimize total weighted completion time

Abstract

We investigate the problem of scheduling n jobs in s-stage hybrid flowshops with parallel identical machines at each stage. The objective is to find a schedule that minimizes the sum of weighted completion times of the jobs. This problem has been proven to be NP-hard. In this paper, an integer programming formulation is constructed for the problem. A new Lagrangian relaxation algorithm is presented in which precedence constraints are relaxed to the objective function by introducing Lagrangian multipliers, unlike the commonly used method of relaxing capacity constraints. In this way the relaxed problem can be decomposed into machine type subproblems, each of which corresponds to a specific stage. A dynamic programming algorithm is designed for solving parallel identical machine subproblems where jobs may have negative weights. The multipliers are then iteratively updated along a subgradient direction. The new algorithm is computationally compared with the commonly used Lagrangian relaxation algorithms which, after capacity constraints are relaxed, decompose the relaxed problem into job level subproblems and solve the subproblems by using the regular and speed-up dynamic programming algorithms, respectively. Numerical results show that the new Lagrangian relaxation method produces better schedules in much shorter computation time, especially for large-scale problems. Scope and purpose Traditional research on the resolution of hybrid flowshop scheduling has focused on branch and bound algorithms or heuristics to minimize makespan, maximum completion time or maximum lateness, etc. However, little research has been done on minimizing the sum of weighted completion time of all jobs. Further, in many practical scheduling environments, the schedulers expect to obtain a good schedule with acceptable quality in a short computation time. Although the commonly used Lagrangian relaxation method that decomposes the problem into job level subproblems and uses regular dynamic programming to solve the subproblems, has been used to solve similar scheduling problems, it is time-consuming to solve the problem in this paper, even with a small number of jobs. For this reason, this paper considers an alternative Lagrangian decomposition method for hybrid flowshop scheduling. In addition, we combine and generalize the existing approach for parallel identical machine subproblems in the new approach.