In this paper, we consider the following two-machine no-wait flow shop scheduling problem with two competing agents \(F2~|~M_1\rightarrow M_2,~ M_2,~ p_{ij}^{A} = p,~ no\text{- }wait~|~C_{\max }^A:~ C_{\max }^B~\le Q \): Given a set of n jobs \(\mathcal {J} = \{ J_1, J_2, \ldots , J_n\}\) and two competing agents A and B. Agent A is associated with a set of \(n_A\) jobs \(\mathcal {J}^A = \{J_1^A, J_2^A, \ldots , J_{n_A}^A\}\) to be processed on the machine \(M_1\) first and then on the machine \(M_2\) with no-wait constraint, and agent B is associated with a set of \(n_B\) jobs \(\mathcal {J}^B = \{J_1^B, J_2^B, \ldots , J_{n_B}^B\}\) to be processed on the machine \(M_2\) only, where the processing times for the jobs of agent A are all the same (i.e., \(p_{ij}^A = p\)), \(\mathcal {J} = \mathcal {J}^A \cup \mathcal {J}^B\) and \(n = n_A + n_B\). The objective is to build a schedule \(\pi \) of the n jobs that minimizing the makespan of agent A while maintaining the makespan of agent B not greater than a given value Q. We first show that the problem is polynomial time solvable in some special cases. For the non-solvable case, we present an \(O(n \log n)\)-time \((1 + \frac{1}{n_A +1})\)-approximation algorithm and show that this ratio of \((1 + \frac{1}{n_A +1})\) is asymptotically tight. Finally, \((1+\epsilon )\)-approximation algorithms are provided.

在本文中，我们考虑以下具有两个竞争代理的两机无等待流水车间调度问题\(F2~|~M_1\rightarrow M_2,~ M_2,~ p_{ij}^{A} = p,~ no \text{- }wait~|~C_{\max }^A:~ C_{\max }^B~\le Q \) : 给定一组n个作业\(\mathcal {J} = \{ J_1, J_2, \ldots , J_n\}\)和两个竞争代理A和B 。代理A与一组要处理的\(n_A\)个作业\(\mathcal {J}^A = \{J_1^A, J_2^A, \ldots , J_{n_A}^A\}\)相关联首先在机器\(M_1\)上，然后在具有无等待约束的机器\(M_2\)上，并且代理B与一组\(n_B\)作业相关联\(\mathcal {J}^B = \{J_1^B, J_2^B, \ldots , J_{n_B}^B\}\)仅在机器\(M_2\)上处理，其中代理A的作业的处理时间都相同（即\(p_{ij}^A = p\) ）、 \(\mathcal {J} = \mathcal {J}^A \cup \mathcal {J}^B\)和\(n = n_A + n_B\) 。目标是建立一个包含n 个作业的调度\(\pi \) ，以最小化代理A的完工时间，同时保持代理B的完工时间不大于给定值Q 。我们首先证明该问题在某些特殊情况下是多项式时间可解的。对于不可解的情况，我们提出了一个\(O(n \log n)\)时间\((1 + \frac{1}{n_A +1})\)近似算法，并证明了\((1 + \frac{1}{n_A +1})\)是渐近紧的。 最后，提供了\((1+\epsilon )\)近似算法。