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In this paper, a mixed integer linear programming (MILP) model is proposed to minimize the total tardiness cost and the extra energy cost. Decomposition approaches based on logic-based Benders decomposition (LBBD) are developed by reformulating the studied problem into a master problem and some independent sub-problems. The master problem is relaxed by only making assignment decisions. The sub-problems are to find optimal schedules in the job-to-machine assignments given by the master problem. Moreover, MILP and heuristic based on Tabu search are used to solve the sub-problems. To evaluate the performance of our methods, three groups of test instances were generated inspired by both real-world applications and benchmarks from the literature. The computational results demonstrate that the proposed decomposition approaches can compute competitive schedules for medium- and large-size problems in terms of solution quality. In particular, the LBBD with Tabu search performs the best among the suggested four methods.<\/jats:p>","DOI":"10.1007\/s40747-021-00601-9","type":"journal-article","created":{"date-parts":[[2021,12,13]],"date-time":"2021-12-13T16:02:46Z","timestamp":1639411366000},"page":"1339-1354","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Decomposition approaches for parallel machine scheduling of step-deteriorating jobs to minimize total tardiness and energy consumption"],"prefix":"10.1007","volume":"8","author":[{"given":"Xiao","family":"Wu","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5520-7701","authenticated-orcid":false,"given":"Peng","family":"Guo","sequence":"additional","affiliation":[]},{"given":"Yi","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Yakun","family":"Wang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,12,13]]},"reference":[{"key":"601_CR1","doi-asserted-by":"publisher","first-page":"113348","DOI":"10.1016\/j.eswa.2020.113348","volume":"157","author":"M Abedi","year":"2020","unstructured":"Abedi M, Chiong R, Noman N, Zhang R (2020) A multi-population, multi-objective memetic algorithm for energy-efficient job-shop scheduling with deteriorating machines. 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