Algorithms for a two-machine no-wait flow shop scheduling problem with two competing agents

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.