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A branch-and-bound algorithm for two-machine flow-shop scheduling problems with batch delivery costs

Manufacturers often dispatch jobs in batches to reduce delivery costs. However, this technique can have a negative effect on other scheduling-related objective functions such as minimising maximum tardiness. This paper aims to minimise the maximum tardiness and the sum of delivery costs of a two-machine flow-shop problem in a batch delivery system. It is proven in the literature that this problem is NP-complete. We present a mixed integer linear programming (MILP) model to solve the problem. As this is an MILP model, the commercial solver (the GAMS software using CPLEX solver) is not guaranteed to find the optimal solution for large-size problems at a logical CPU run time. Therefore, a branch-and-bound (B&B) algorithm is proposed for obtaining the optimum solution. Additionally, an upper bound (UB) heuristic with a quick processing time is presented. These methods are evaluated using randomly generated instances and the results indicate the efficiency of the B&B and UB heuristic algorithms.

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