Two-Machine Flow Shop with a Fixed Non-Availability Interval on the Second Machine

2026-09-12

AI Summary

研究了第二台机器有固定不可用区间的两机流水车间问题，目标是最小化总完成时间。提出了多项式时间近似算法和伪多项式时间精确动态规划，并证明了该问题除非P=NP否则不存在FPTAS。

Abstract

This paper investigates a two-machine permutation flow shop in which the second machine is unavailable during one fixed interval $[s,t]$. We consider the non-resumable setting: an operation interrupted by the interval must restart from the beginning after the machine becomes available. The objective is to minimize the makespan. We establish three results. First, we give a polynomial-time $10/7$-approximation algorithm. Second, we develop a pseudopolynomial-time exact dynamic program. Third, we prove that the problem does not admit a fully polynomial-time approximation scheme (FPTAS) unless $\mathrm{P}=\mathrm{NP}$, even when the non-availability interval has unit length. Together, these results characterize a distinctive complexity profile: exact optimization is possible in pseudopolynomial time, whereas the usual route from such an algorithm to an FPTAS is impossible unless $\mathrm{P}=\mathrm{NP}$. They also reveal an approximability separation from the corresponding non-resumable problem with the interval on the first machine.

Problem

• two-machine flow shop
• non-availability interval
• makespan
• non-resumable

Innovation

• permutation flow shop
• non-availability interval
• approximation algorithm
• dynamic programming
• FPTAS