This paper considers unrelated parallel machine scheduling involving machine usage costs, in addition to classic job completion time-related costs. The usage cost of each machine is made up of a fixed usage cost and a variable usage cost proportional to the total processing time of the jobs assigned to it. These features model many practical situations where machine usage costs include, for example, rental fees when the machines are not owned but rented. To tackle this problem, four mathematical models based on the Shortest Weighted Processing Time (SWPT) rule are introduced. Additionally, the problem is formulated into a set-partitioning model, for which a branch-and-price algorithm is proposed with an appropriate branching strategy. This facilitates the development of an efficient pseudo-polynomial dynamic programming algorithm and a polynomial-time heuristic to solve the pricing problem. Extensive numerical experiments demonstrate the superior performance of the proposed branch-and-price algorithm over the four SWPT-based mathematical formulations and an existing branch-and-price algorithm designed for a special case. Notably, it can optimally solve instances involving up to 225 jobs and 15 machines within one hour. Moreover, statistical analyses reveal that the proposed polynomial-time heuristic significantly reduces the computation time, and the mathematical model based on the contribution of every job to the total weighted completion time exhibits the best overall performance.

中文翻译：

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一种具有机器使用成本的不相关并行机调度的分支价格算法

除了经典的作业完成时间相关成本之外，本文还考虑了涉及机器使用成本的不相关并行机器调度。每台机器的使用成本由固定使用成本和与分配给它的作业的总处理时间成比例的可变使用成本组成。这些特征模拟了许多实际情况，其中机器使用成本包括例如机器不是拥有而是租用时的租赁费用。为了解决这个问题，引入了四种基于最短加权处理时间（SWPT）规则的数学模型。此外，该问题被表述为集合划分模型，并针对该模型提出了具有适当分支策略的分支与价格算法。这有助于开发高效的伪多项式动态规划算法和多项式时间启发式算法来解决定价问题。大量的数值实验证明了所提出的分支与价格算法比四种基于 SWPT 的数学公式和针对特殊情况设计的现有分支与价格算法具有优越的性能。值得注意的是，它可以在一小时内优化解决涉及多达 225 个作业和 15 台机器的实例。此外，统计分析表明，所提出的多项式时间启发式显着减少了计算时间，并且基于每个作业对总加权完成时间的贡献的数学模型表现出最佳的整体性能。