Approximation algorithms for the three-machine proportionate mixed shop scheduling
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A mixed shop is a manufacturing infrastructure designed to process a mixture of a set of flow-shop 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 M3 | prpt | C_, 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 polynomial-time approximation scheme (FPTAS). On the negative side, while the F3 | prpt | C_ 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.
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