Approximation algorithms for the three-machine proportionate mixed shop scheduling

Abstract

A mixed shop is a manufacturing infrastructure designed to process a mixture of a set of flowshop jobs and a set of open-shop jobs. Mixed shops are in general much more complex to schedule than flow-shops and open-shops, and have been studied since the 1980's. We consider the three machine proportionate mixed shop problem denoted as M 3 | prpt | C max , in which each job has equal processing times on all three machines. Koulamas and Kyparisis [European Journal of Operational Research, 243:70-74,2015] showed that the problem is solvable in polynomial time in some very special cases; for the non-solvable case, they proposed a 5/3-approximation algorithm. In this paper, we present an improved 4/3-approximation algorithm and show that this ratio of 4/3 is asymptotically tight; when the largest job is a flow-shop job, we present a fully polynomialtime approximation scheme (FPTAS). On the negative side, while the F 3 | prpt | C max problem is polynomial-time solvable, we show an interesting hardness result that adding one open-shop job to the job set makes the problem NP-hard if this open-shop job is larger than any flow-shop job. We are able to design an FPTAS for this special case too. Figure 5.1: A feasible schedule π for the constructed instance of the M 3 | prpt | C max problem, when the set S can be partitioned into two equal parts S 1 and S 2 . The partition of the flow-shop jobs {J 1 , J 2 } is correspondingly constructed. In the schedule, the jobs of J 1 and the jobs of J 2 are processed in the LPT order, respectively.

Improved Scheduling for the Three-Machine Proportionate Open Shop and Mixed Shop Minimum Makespan Problems

“…Lemma 5 is of course right for the special case in which there is only one open shop job J 1 and q 1 < p 1 < Q, and thus we extend the results of Lemma 6.2 and Lemma 6.3 in [21]. Besides, our proofs are more simple and straightforward.…”

“…Besides, if there is only one open shop job, i.e. n = 1 and p 1 ≥ Q, Theorem 4 in this paper degenerates into Lemma 6.1 in [21].…”

“…In the following two sections, we investigate the M 3|prpt|C max problem and focus on conditions under which the problem is solvable in this section. In the next section, we focus on the non-solvable case of the M 3|prpt|C max problem and present a (7/6 + )approximation algorithm for any given > 0, noting that the algorithms proposed in [1] and [21] supply the worstcase ratio bounds of 5/3 and 4/3, respectively. Following [1], we have that the makespan of an optimal algorithm satisfies C * max ≥ max{3p 1 , P + Q, 2q 1 + Q} for the M 3|prpt|C max problem.…”

“…Firstly, according to Lemma 3.1 and Lemma 3.2 in [21], we have the following corollary without a proof. Corollary 2: (1) If p 1 ≤ q 1 , then the M 3|prpt|C max problem is solvable in polynomial time when T n 2 ≤ q 1 or when there exists some p i satisfying T i 2 ≤ q 1 and p 1 + T n i+1 ≤ q 1 , and figure 8.…”

“…Recently, Liu et al presented a fully polynomial time approximation scheme (FPTAS) for the non-solvable cases of the M 3|prpt|C max problem when q 1 ≥ p 1 , or when p 1 > q 1 and n = 1 (i.e., there is only one open shop job which is larger than any flow shop job) [21]. Liu et al also designed an improved 4/3-approximation algorithm for the other nonsolvable cases of the M 3|prpt|C max problem and proved that the performance ratio of 4/3 is asymptotically tight [21].…”