Branch and Bound Method to Minimize Three Objective Functions with Unequal Release Dates


In this paper, a single machine scheduling problem is considered to minimize Multiple Objectives Function (MOF). The aim in this study is to find the optimal solution for the sum of the number of tardy jobs (∑_(j=1)^n▒U_j ), maximum tardiness (T_max) and makespan (C_max) with unequal release dates. Ten special cases are derived and proved that yield optimal solutions. A Branch and Bound algorithm is proposed with two lower bounds (LB_1, LB_2) and two upper bounds (UB_1, UB_2) that introduced in this paper, in order to find the exact (optimal) solution for it with two dominance rules which helped in reducing the number of branches in the search tree. Results of extensive computational tests show the proposed (BAB) algorithm is effective in solving problem up to (40) jobs at a time less than or equal to (30) minutes. In general, this problem is strongly NP-hard, and to the best of our knowledge this problem was not studied before.