A stochastic production planning problem with nonlinear cost
Most production planning models are deterministic and often assume a linear relation between production volume and production cost. In this paper, we investigate a production planning problem in a steel production process considering the energy consumption cost which is a nonlinear function of the p...
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| Vydané v: | Computers & operations research Ročník 39; číslo 9; s. 1977 - 1987 |
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| Médium: | Journal Article |
| Jazyk: | English |
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01.09.2012
Elsevier Pergamon Press Inc |
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| ISSN: | 0305-0548, 1873-765X, 0305-0548 |
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| Abstract | Most production planning models are deterministic and often assume a linear relation between production volume and production cost. In this paper, we investigate a production planning problem in a steel production process considering the energy consumption cost which is a nonlinear function of the production quantity. Due to the uncertain environment, the production demands are stochastic. Taking a scenario-based approach to express the stochastic demands according to the knowledge of planners on the demand distributions, we formulate the stochastic production planning problem as a mixed integer nonlinear programming (MINLP) model.
Approximated with the piecewise linear functions, the MINLP model is transformed into a mixed integer linear programming model. The approximation error can be improved by adjusting the linearization ranges repeatedly. Based on the piecewise linearization, a stepwise Lagrangian relaxation (SLR) heuristic for the problem is proposed where variable splitting is introduced during Lagrangian relaxation (LR). After decomposition, one subproblem is solved by linear programming and the other is solved by an effective polynomial time algorithm. The SLR heuristic is tested on a large set of problem instances and the results show that the algorithm generates solutions very close to optimums in an acceptable time. The impact of demand uncertainty on the solution is studied by a computational discussion on scenario generation. |
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| AbstractList | Most production planning models are deterministic and often assume a linear relation between production volume and production cost. In this paper, we investigate a production planning problem in a steel production process considering the energy consumption cost which is a nonlinear function of the production quantity. Due to the uncertain environment, the production demands are stochastic. Taking a scenario-based approach to express the stochastic demands according to the knowledge of planners on the demand distributions, we formulate the stochastic production planning problem as a mixed integer nonlinear programming (MINLP) model. Approximated with the piecewise linear functions, the MINLP model is transformed into a mixed integer linear programming model. The approximation error can be improved by adjusting the linearization ranges repeatedly. Based on the piecewise linearization, a stepwise Lagrangian relaxation (SLR) heuristic for the problem is proposed where variable splitting is introduced during Lagrangian relaxation (LR). After decomposition, one subproblem is solved by linear programming and the other is solved by an effective polynomial time algorithm. The SLR heuristic is tested on a large set of problem instances and the results show that the algorithm generates solutions very close to optimums in an acceptable time. The impact of demand uncertainty on the solution is studied by a computational discussion on scenario generation. Most production planning models are deterministic and often assume a linear relation between production volume and production cost. In this paper, we investigate a production planning problem in a steel production process considering the energy consumption cost which is a nonlinear function of the production quantity. Due to the uncertain environment, the production demands are stochastic. Taking a scenario-based approach to express the stochastic demands according to the knowledge of planners on the demand distributions, we formulate the stochastic production planning problem as a mixed integer nonlinear programming (MINLP) model. Approximated with the piecewise linear functions, the MINLP model is transformed into a mixed integer linear programming model. The approximation error can be improved by adjusting the linearization ranges repeatedly. Based on the piecewise linearization, a stepwise Lagrangian relaxation (SLR) heuristic for the problem is proposed where variable splitting is introduced during Lagrangian relaxation (LR). After decomposition, one subproblem is solved by linear programming and the other is solved by an effective polynomial time algorithm. The SLR heuristic is tested on a large set of problem instances and the results show that the algorithm generates solutions very close to optimums in an acceptable time. The impact of demand uncertainty on the solution is studied by a computational discussion on scenario generation. [PUBLICATION ABSTRACT] Most production planning models are deterministic and often assume a linear relation between production volume and production cost. In this paper, we investigate a production planning problem in a steel production process considering the energy consumption cost which is a nonlinear function of the production quantity. Due to the uncertain environment, the production demands are stochastic. Taking a scenario-based approach to express the stochastic demands according to the knowledge of planners on the demand distributions, we formulate the stochastic production planning problem as a mixed integer nonlinear programming (MINLP) model. Approximated with the piecewise linear functions, the MINLP model is transformed into a mixed integer linear programming model. The approximation error can be improved by adjusting the linearization ranges repeatedly. Based on the piecewise linearization, a stepwise Lagrangian relaxation (SLR) heuristic for the problem is proposed where variable splitting is introduced during Lagrangian relaxation (LR). After decomposition, one subproblem is solved by linear programming and the other is solved by an effective polynomial time algorithm. The SLR heuristic is tested on a large set of problem instances and the results show that the algorithm generates solutions very close to optimums in an acceptable time. The impact of demand uncertainty on the solution is studied by a computational discussion on scenario generation. |
| Author | Che, Ping Liu, Jiyin Tang, Lixin |
| Author_xml | – sequence: 1 givenname: Lixin surname: Tang fullname: Tang, Lixin email: qhjytlx@mail.neu.edu.cn organization: Liaoning Key Laboratory of Manufacturing System and Logistics, The Logistics Institute, Northeastern University, 110004 Shenyang, China – sequence: 2 givenname: Ping surname: Che fullname: Che, Ping email: cheping@mail.neu.edu.cn organization: Liaoning Key Laboratory of Manufacturing System and Logistics, The Logistics Institute, Northeastern University, 110004 Shenyang, China – sequence: 3 givenname: Jiyin surname: Liu fullname: Liu, Jiyin email: j.y.liu@lboro.ac.uk organization: Liaoning Key Laboratory of Manufacturing System and Logistics, The Logistics Institute, Northeastern University, 110004 Shenyang, China |
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| Cites_doi | 10.1016/S0377-2217(03)00014-6 10.1287/mnsc.42.5.738 10.1016/0377-2217(90)90300-Z 10.1016/j.ejor.2005.12.008 10.1016/S0305-0483(03)00059-8 10.1016/j.jmateco.2007.04.011 10.1016/S0305-0548(00)00076-9 10.1016/0377-2217(86)90328-0 10.1016/S0377-2217(00)00116-8 10.1287/opre.48.1.91.12450 10.1016/S0167-6377(97)00005-9 10.1016/j.ijpe.2008.08.034 10.1016/j.ijpe.2009.04.007 10.1016/S0925-5273(98)00060-7 10.1016/S0167-6377(98)00050-9 10.1016/S0098-1354(01)00676-7 10.1016/j.ijpe.2005.09.001 10.1016/j.ijpe.2005.02.015 10.1287/moor.16.1.119 10.1287/mnsc.5.1.89 10.1080/00207540500435116 10.1016/S0898-1221(02)00146-3 10.1016/S0377-2217(00)00240-X 10.1287/opre.35.3.329 |
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| Keywords | MINLP Stepwise Lagrangian relaxation Stochastic production planning Variable splitting Scenario-based approach Inventory Energy consumption Non linear programming Production function Polynomial method Linear model Optimization Lagrange multiplier Uncertain system Production planning Relaxation method Production process Script Quantity production Linear programming Mixed integer programming Stochastic programming Polynomial time Production management Supply demand balance Deterministic model Non linear model Heuristic method Production cost Approximation error Mixed problem Piecewise linearization Piecewise-linear techniques |
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| Snippet | Most production planning models are deterministic and often assume a linear relation between production volume and production cost. In this paper, we... |
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| SubjectTerms | Algorithms Applied sciences Demand Energy consumption Exact sciences and technology Heuristic Integer programming Inventory Inventory control, production control. Distribution Linear programming Marketing Mathematical analysis Mathematical models Mathematical programming Mathematics MINLP Nonlinearity Operational research and scientific management Operational research. Management science Probability and statistics Production planning Sampling theory, sample surveys Scenario-based approach Sciences and techniques of general use Statistics Steel production Stepwise Lagrangian relaxation Stochastic models Stochastic production planning Stochasticity Studies Variable splitting |
| Title | A stochastic production planning problem with nonlinear cost |
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