We consider the problem of minimizing the makespan on batch processing identical machines, subject to compatibility constraints, where two jobs are compatible if they can be processed simultaneously in a same batch. These constraints are modeled by an undirected graph $G$, in which compatible jobs are represented by adjacent vertices. We show that several subproblems are polynomial. We propose some exact polynomial algorithms to solve these subproblems. To solve the general case, we propose a mixed-integer linear programming (MILP) formulation alongside with heuristic approaches. Furthermore, computational experiments are carried out to measure the performance of the proposed methods.

翻译：我们考虑在兼容性约束下最小化批处理相同机器上的最大完工时间问题，其中两个作业若能在同一批次中同时处理则被视为兼容。这些约束通过无向图 $G$ 建模，图中相邻顶点表示兼容的作业。我们证明若干子问题是多项式可解的，并提出精确的多项式算法来求解这些子问题。针对一般情形，我们提出了混合整数线性规划（MILP）公式及启发式方法。此外，我们通过计算实验评估所提出方法的性能。