Polynomial-time algorithms to solve the single-item capacitated lot sizing problem with a 1-breakpoint all-units quantity discount
•We investigated a single item capacitated lot sizing problem with quantity discount.•We defined the concept of irregular orders in the optimal solution of the problem.•We developed two exact algorithms with polynomial time complexity and a heuristic. This study investigates the single-item capacita...
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| Published in: | Computers & operations research Vol. 134; p. 105373 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
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New York
Elsevier Ltd
01.10.2021
Pergamon Press Inc |
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| ISSN: | 0305-0548, 1873-765X, 0305-0548 |
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| Abstract | •We investigated a single item capacitated lot sizing problem with quantity discount.•We defined the concept of irregular orders in the optimal solution of the problem.•We developed two exact algorithms with polynomial time complexity and a heuristic.
This study investigates the single-item capacitated lot sizing problem considering a 1-breakpoint all-units quantity discount. Assuming the fixed ordering cost and the undiscounted unit purchase price to be non-increasing over the periods, the lot sizing problem under study is a special case of that with piecewise concave production cost function investigated by Koca et al. (2014). Koca’s DP algorithm solves our problem with a time complexity of O(T7), where T denotes the number of periods. It, however, fails in the case of large-sized instances in acceptable runtimes due to its higher complexity. Hence, the present study is a response to the need for more efficient algorithms to overcome this shortcoming. First, some properties of the optimal solution are proved. Second, these properties are exploited in an implicit enumeration exact algorithm. Third, certain speed-up techniques are employed to reduce the time complexity of the proposed algorithm from O(T5) to O(T4). Finally, a heuristic algorithm is presented for the problem. The proposed algorithms are compared with Koca’s DP algorithm and the commercial solver used to solve some test problems. Based on the runtimes observed, the exact algorithms proposed in this study are found to outperform both Koca’s DP one and the commercial solver. Moreover, the heuristic algorithm developed herein is found to be faster than both the exact algorithms in finding optimal solutions to most problem instances. |
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| AbstractList | •We investigated a single item capacitated lot sizing problem with quantity discount.•We defined the concept of irregular orders in the optimal solution of the problem.•We developed two exact algorithms with polynomial time complexity and a heuristic.
This study investigates the single-item capacitated lot sizing problem considering a 1-breakpoint all-units quantity discount. Assuming the fixed ordering cost and the undiscounted unit purchase price to be non-increasing over the periods, the lot sizing problem under study is a special case of that with piecewise concave production cost function investigated by Koca et al. (2014). Koca’s DP algorithm solves our problem with a time complexity of O(T7), where T denotes the number of periods. It, however, fails in the case of large-sized instances in acceptable runtimes due to its higher complexity. Hence, the present study is a response to the need for more efficient algorithms to overcome this shortcoming. First, some properties of the optimal solution are proved. Second, these properties are exploited in an implicit enumeration exact algorithm. Third, certain speed-up techniques are employed to reduce the time complexity of the proposed algorithm from O(T5) to O(T4). Finally, a heuristic algorithm is presented for the problem. The proposed algorithms are compared with Koca’s DP algorithm and the commercial solver used to solve some test problems. Based on the runtimes observed, the exact algorithms proposed in this study are found to outperform both Koca’s DP one and the commercial solver. Moreover, the heuristic algorithm developed herein is found to be faster than both the exact algorithms in finding optimal solutions to most problem instances. This study investigates the single-item capacitated lot sizing problem considering a 1-breakpoint all-units quantity discount. Assuming the fixed ordering cost and the undiscounted unit purchase price to be non-increasing over the periods, the lot sizing problem under study is a special case of that with piecewise concave production cost function investigated by Koca et al. (2014). Koca's DP algorithm solves our problem with a time complexity of O(T7), where T denotes the number of periods. It, however, fails in the case of large-sized instances in acceptable runtimes due to its higher complexity. Hence, the present study is a response to the need for more efficient algorithms to overcome this shortcoming. First, some properties of the optimal solution are proved. Second, these properties are exploited in an implicit enumeration exact algorithm. Third, certain speed-up techniques are employed to reduce the time complexity of the proposed algorithm from O(T5) to O(T4). Finally, a heuristic algorithm is presented for the problem. The proposed algorithms are compared with Koca's DP algorithm and the commercial solver used to solve some test problems. Based on the runtimes observed, the exact algorithms proposed in this study are found to outperform both Koca's DP one and the commercial solver. Moreover, the heuristic algorithm developed herein is found to be faster than both the exact algorithms in finding optimal solutions to most problem instances. |
| ArticleNumber | 105373 |
| Author | Malekian, Yaser Bijari, Mehdi Mirmohammadi, S. Hamid |
| Author_xml | – sequence: 1 givenname: Yaser surname: Malekian fullname: Malekian, Yaser email: Malekiyan@gmail.com – sequence: 2 givenname: S. Hamid surname: Mirmohammadi fullname: Mirmohammadi, S. Hamid email: h_mirmohammadi@iut.ac.ir – sequence: 3 givenname: Mehdi surname: Bijari fullname: Bijari, Mehdi email: bijari@iut.ac.ir |
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| Keywords | Polynomial exact algorithm All-units quantity discount Capacitated lot sizing problem Implicit enumeration algorithm |
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| Snippet | •We investigated a single item capacitated lot sizing problem with quantity discount.•We defined the concept of irregular orders in the optimal solution of the... This study investigates the single-item capacitated lot sizing problem considering a 1-breakpoint all-units quantity discount. Assuming the fixed ordering cost... |
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| SubjectTerms | Algorithms All-units quantity discount Capacitated lot sizing problem Complexity Cost function Enumeration Heuristic methods Implicit enumeration algorithm Lot sizing Operations research Polynomial exact algorithm Polynomials Production costs Run time (computers) Solvers |
| Title | Polynomial-time algorithms to solve the single-item capacitated lot sizing problem with a 1-breakpoint all-units quantity discount |
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