This paper focuses on developing the optimal solution or a lower bound for N-job, M-machine Permutation Flowshop Scheduling (PFS) problem in a manufacturing system with the objective of minimizing the makespan using Lagrangian Relaxation (LR) technique. Even though LR technique is considered, in general, as a good method to obtain a lower bound, research in this direction with respect to our problem under study appears scarce. We address this gap by developing two MILP based Lagrangian Relaxation models, namely, Lagrangian Relaxation Method 1 (called Proposed Lagrangian Lower Bound Program (PLLBP)) and Alternate Lagrangian Relaxation Method 1 (called ALR) to find the optimal solution or a lower bound on the makespan. Basically, we develop these LR methods to overcome the possible limitation of the general LR procedure involving the sub-gradient approach. Benchmark PFS problem instances are used to evaluate the performance of these methods. It is observed that the PLLBP outperforms the ALR, and it provides better lower bounds than the lower bounds (in most instances) reported in the literature. Even though the PLLBP is superior in terms of solution quality, it has a limitation in that it cannot execute problem instances beyond 500 jobs due to the associated computational effort.

本文着重于为N -job，M开发最佳解或下界制造系统中的机器排列Flowshop调度（PFS）问题，其目的是使用拉格朗日松弛（LR）技术使制造期最小化。尽管通常考虑使用LR技术作为获得下界的一种好方法，但针对我们正在研究的问题，朝这个方向进行的研究似乎很少。我们通过开发两个基于MILP的拉格朗日弛豫模型（即拉格朗日弛豫方法1（称为拟议拉格朗日下界规划（PLLBP））和替代拉格朗日弛豫方法1（称为ALR））来找到此最佳解或下界完成时间。基本上，我们开发这些LR方法，以克服涉及次梯度方法的常规LR程序的可能限制。基准PFS问题实例用于评估这些方法的性能。可以观察到PLLBP优于ALR，并且它提供了比文献中报道的下限（在大多数情况下）更好的下限。即使PLLBP在解决方案质量方面是优越的，它也有一个局限性，即由于相关的计算工作，它不能执行超过500个工作的问题实例。