We present several new results for a single machine time-dependent scheduling problem of minimizing the total completion time of a set of linearly deteriorating jobs with unit basic processing times. First, we show new properties of cyclic transformations of V-shaped sequences for this problem. Next, applying the results, we prove a new necessary condition of schedule optimality for the considered problem, which decreases the previous bound on the cardinality of the set containing all possible optimal schedules by a multiplicative factor which is at most proportional to the reciprocal of the square root of the number of jobs. Finally, we compare the strength of the new and the previous necessary conditions by estimation of the numbers of schedules satisfying the respective conditions.

中文翻译：

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开放时间相关调度问题的新结果

我们为单个机器时间相关调度问题提供了几个新结果，该问题最小化一组具有单位基本处理时间的线性恶化作业的总完成时间。首先，我们展示了这个问题的 V 形序列循环变换的新特性。接下来，应用这些结果，我们证明了所考虑问题的调度最优性的新必要条件，它通过乘法因子减少了包含所有可能最优调度的集合的基数的先前界限，该乘法因子至多与工作数量的平方根。最后，我们通过估计满足各自条件的调度数量来比较新的和以前的必要条件的强度。