A bi-objective branch-and-bound and an $∊$-constraint algorithms are proposed for the unit-time job shop scheduling problem. These two objectives are the minimization of the makespan and the total completion time. We model this problem as a bi-objective mixed graph coloring using both the chromatic number and the sum of path-endpoints coloring (i.e. sum of the colors assigned to the endpoints of maximal paths) to determine the optimal set of the non-dominated solutions. A new lower bound is also constructed for the sum of path-endpoints coloring which is used alongside an existing lower bound from the literature on two bounding procedures.

Computational experiments on benchmark data sets show that the proposed lower bound improves considerably upon the known lower bound from the literature. Besides, our algorithms are found to find an optimal set of non-dominated solutions for most of the tested benchmarks within a reasonable amount of CPU time. In addition, two interesting performance metrics are introduced to compare and assess the effectiveness and the efficiency of our algorithms.

双目标分支定界法和 $∊$针对单位时间作业车间调度问题，提出了一种约束算法。这两个目标是最小化制造时间和总完成时间。我们将此问题建模为使用色数和路径端点着色之和（即，分配给最大路径端点的颜色之和）两者的双目标混合图着色，以确定非受控解的最佳集合。还为路径端点着色的总和构造了一个新的下界，将其与文献中有关两个边界过程的现有下界一起使用。

在基准数据集上进行的计算实验表明，与文献中已知的下限相比，提出的下限有了很大的提高。此外，我们的算法还可以在合理的CPU时间内为大多数经过测试的基准找到最佳的非支配解决方案集。另外，引入了两个有趣的性能指标来比较和评估我们算法的有效性和效率。