This paper addresses the scheduling problem on two identical parallel machines with a single server in charge of loading and unloading operations of jobs. Each job has to be loaded by the server before being processed on one of the two machines and unloaded by the same server after its processing. No delay is allowed between loading and processing, and between processing and unloading. The objective function involves the minimization of the makespan. This problem referred to as P2, S1|sj , tj |Cmax generalizes the classical parallel machine scheduling problem with a single server which performs only the loading (i.e., setup) operation of each job. For this NP-hard problem, no solution algorithm was proposed in the literature. Therefore, we present two mixedinteger linear programming (MILP) formulations, one with completion-time variables along with two valid inequalities and one with time-indexed variables. In addition, we propose some polynomial-time solvable cases and a tight theoretical lower bound. In addition, we show that the minimization of the makespan is equivalent to the minimization of the total idle times on the machines. To solve large-sized instances of the problem, an efficient General Variable Neighborhood Search (GVNS) metaheuristic with two mechanisms for finding an initial solution is designed. The GVNS is evaluated by comparing its performance with the results provided by the MILPs and another metaheuristic. The results show that the average percentage deviation from the theoretical lower-bound of GVNS is within 0.642%. Some managerial insights are presented and our results are compared with the related literature.

翻译：本文研究两台同速并行机在单台服务器负责作业装卸操作条件下的调度问题。每个作业需在任一台机器加工前由服务器装载，并在加工完成后由同一服务器卸载。装载与加工之间、加工与卸载之间均不允许存在时间延迟。目标函数为最小化最大完工时间。该问题标记为P2,S1|sj,tj|Cmax，是对经典并行机调度问题（仅由单台服务器执行每个作业的装载（即准备）操作）的推广。针对这一NP-hard问题，现有文献尚未提出求解算法。为此，我们提出两种混合整数线性规划（MILP）模型：一种采用完工时间变量并附加两个有效不等式，另一种采用时间索引变量。此外，我们提出若干多项式时间可解情形及一个紧的理论下界，并证明最小化最大完工时间等价于最小化机器总空闲时间。为求解大规模问题实例，设计了一种高效的通用变邻域搜索（GVNS）元启发式算法，包含两种初始解生成机制。通过将GVNS与MILP模型及另一种元启发式算法的求解结果进行对比评估，结果表明GVNS相对于理论下界的平均百分比偏差不超过0.642%。最后给出管理启示，并将研究结果与相关文献进行比较。