Stochastic binary problems with simple penalties for capacity constraints violations
This paper studies stochastic programs with first-stage binary variables and capacity constraints, using simple penalties for capacities violations. In particular, we take a closer look at the knapsack problem with weights and capacity following independent random variables and prove that the proble...
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| Vydané v: | Mathematical programming Ročník 138; číslo 1-2; s. 199 - 221 |
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| Hlavní autori: | , , , |
| Médium: | Journal Article |
| Jazyk: | English |
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Berlin/Heidelberg
Springer-Verlag
01.04.2013
Springer Nature B.V Springer Verlag |
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| ISSN: | 0025-5610, 1436-4646, 1436-4646 |
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| Abstract | This paper studies stochastic programs with first-stage binary variables and capacity constraints, using simple penalties for capacities violations. In particular, we take a closer look at the knapsack problem with weights and capacity following independent random variables and prove that the problem is weakly
-hard in general. We provide pseudo-polynomial algorithms for three special cases of the problem: constant weights and capacity uniformly distributed, subset sum with Gaussian weights and strictly positively distributed random capacity, and subset sum with constant weights and arbitrary random capacity. We then turn to a branch-and-cut algorithm based on the outer approximation of the objective function. We provide computational results for the stochastic knapsack problem (i) with Gaussian weights and constant capacity and (ii) with constant weights and capacity uniformly distributed, on randomly generated instances inspired by computational results for the knapsack problem. |
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| AbstractList | This paper studies stochastic programs with first-stage binary variables and capacity constraints, using simple penalties for capacities violations. In particular, we take a closer look at the knapsack problem with weights and capacity following independent random variables and prove that the problem is weakly $${\mathcal{N}\mathcal{P}}$$-hard in general. We provide pseudo-polynomial algorithms for three special cases of the problem: constant weights and capacity uniformly distributed, subset sum with Gaussian weights and strictly positively distributed random capacity, and subset sum with constant weights and arbitrary random capacity. We then turn to a branch-and-cut algorithm based on the outer approximation of the objective function. We provide computational results for the stochastic knapsack problem (i) with Gaussian weights and constant capacity and (ii) with constant weights and capacity uniformly distributed, on randomly generated instances inspired by computational results for the knapsack problem. This paper studies stochastic programs with first-stage binary variables and capacity constraints, using simple penalties for capacities violations. In particular, we take a closer look at the knapsack problem with weights and capacity following independent random variables and prove that the problem is weakly NP-hard in general. We provide pseudo-polynomial algorithms for three special cases of the problem: constant weights and capacity uniformly distributed, subset sum with Gaussian weights and strictly positively distributed random capacity, and subset sum with constant weights and arbitrary random capacity. We then turn to a branch-and-cut algorithm based on the outer approximation of the objective function. We provide computational results for the stochastic knapsack problem (i) with Gaussian weights and constant capacity and (ii) with constant weights and capacity uniformly distributed, on randomly generated instances inspired by computational results for the knapsack problem. This paper studies stochastic programs with first-stage binary variables and capacity constraints, using simple penalties for capacities violations. In particular, we take a closer look at the knapsack problem with weights and capacity following independent random variables and prove that the problem is weakly -hard in general. We provide pseudo-polynomial algorithms for three special cases of the problem: constant weights and capacity uniformly distributed, subset sum with Gaussian weights and strictly positively distributed random capacity, and subset sum with constant weights and arbitrary random capacity. We then turn to a branch-and-cut algorithm based on the outer approximation of the objective function. We provide computational results for the stochastic knapsack problem (i) with Gaussian weights and constant capacity and (ii) with constant weights and capacity uniformly distributed, on randomly generated instances inspired by computational results for the knapsack problem. (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) This paper studies stochastic programs with first-stage binary variables and capacity constraints, using simple penalties for capacities violations. In particular, we take a closer look at the knapsack problem with weights and capacity following independent random variables and prove that the problem is weakly ...-hard in general. We provide pseudo-polynomial algorithms for three special cases of the problem: constant weights and capacity uniformly distributed, subset sum with Gaussian weights and strictly positively distributed random capacity, and subset sum with constant weights and arbitrary random capacity. We then turn to a branch-and-cut algorithm based on the outer approximation of the objective function. We provide computational results for the stochastic knapsack problem (i) with Gaussian weights and constant capacity and (ii) with constant weights and capacity uniformly distributed, on randomly generated instances inspired by computational results for the knapsack problem.[PUBLICATION ABSTRACT] This paper studies stochastic programs with first-stage binary variables and capacity constraints, using simple penalties for capacities violations. In particular, we take a closer look at the knapsack problem with weights and capacity following independent random variables and prove that the problem is weakly NP -hard in general. We provide pseudo-polynomial algorithms for three special cases of the problem: constant weights and capacity uniformly distributed, subset sum with Gaussian weights and strictly positively distributed random capacity, and subset sum with constant weights and arbitrary random capacity. We then turn to a branch-and-cut algorithm based on the outer approximation of the objective function. We provide computational results for the stochastic knapsack problem (i) with Gaussian weights and constant capacity and (ii) with constant weights and capacity uniformly distributed, on randomly generated instances inspired by computational results for the knapsack problem. |
| Author | Labbé, M. Fortz, B. Louveaux, F. Poss, M. |
| Author_xml | – sequence: 1 givenname: B. surname: Fortz fullname: Fortz, B. organization: Department of Computer Science, Faculté des Sciences, Université Libre de Bruxelles – sequence: 2 givenname: M. surname: Labbé fullname: Labbé, M. organization: Department of Computer Science, Faculté des Sciences, Université Libre de Bruxelles – sequence: 3 givenname: F. surname: Louveaux fullname: Louveaux, F. organization: Department of Business Administration, University of Namur (FUNDP) – sequence: 4 givenname: M. surname: Poss fullname: Poss, M. email: mjposs@gmail.com organization: CMUC, Department of Mathematics, University of Coimbra |
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| Keywords | Pseudo-polynomial algorithm Mixed-integer non-linear programming 90C11 90C15 Branch-and-cut algorithm Stochastic programming Knapsack problem 90C09 |
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| SubjectTerms | Algorithms Approximation Assignment problem Branch-and-cut algorithm Business & economic sciences Calculus of Variations and Optimal Control; Optimization Combinatorics Computation Computer Science Fines & penalties Full Length Paper Gaussian Knapsack problem Linear programming Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematics Mathematics and Statistics Mathematics of Computing Mixed-integer non-linear programming Méthodes quantitatives en économie & gestion Numerical Analysis Operations Research Optimization Pseudo-polynomial algorithm Quantitative methods in economics & management Random variables Sciences économiques & de gestion Stochastic programming Stochasticity Theoretical Travel Traveling salesman problem Violations |
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| Title | Stochastic binary problems with simple penalties for capacity constraints violations |
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