Two-machine flow shop scheduling to minimize the sum of maximum earliness and tardiness

This paper presents optimal scheduling in a two-machine flow shop, in which the objective function is to minimize the sum of maximum earliness and tardiness (n/2/P/ETmax). Since this problem tries to minimize earliness and tardiness, the results can be useful for different production systems such as just in time (JIT). This objective function has already been considered for n jobs and m machines, but the proposed algorithms are not efficient to solve large scale problems. In this paper, neighborhood conditions are developed and the dominant set for any optimal solution is determined. The branch-and-bound (B&B) method is used to solve the problem and the proper upper and lower bounds are also introduced. A number of effective lemmas are introduced to develop an algorithm which is more efficient than those already known. To show the effectiveness of the proposed algorithm, 380 problems of different sizes are randomly generated and solved. More than 82% of the problems are shown to reach optimal solutions.