An improved test set approach to nonlinear integer problems with applications to engineering design
Many problems in engineering design involve the use of nonlinearities and some integer variables. Methods based on test sets have been proposed to solve some particular problems with integer variables, but they have not been frequently applied because of computation costs. The walk-back procedure ba...
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| Published in: | Computational optimization and applications Vol. 62; no. 2; pp. 565 - 588 |
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| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
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Springer US
01.11.2015
Springer Nature B.V |
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| ISSN: | 0926-6003, 1573-2894 |
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| Abstract | Many problems in engineering design involve the use of nonlinearities and some integer variables. Methods based on test sets have been proposed to solve some particular problems with integer variables, but they have not been frequently applied because of computation costs. The walk-back procedure based on a test set gives an exact method to obtain an optimal point of an integer programming problem with linear and nonlinear constraints, but the calculation of this test set and the identification of an optimal solution using the test set directions are usually computationally intensive. In problems for which obtaining the test set is reasonably fast, we show how the effectiveness can still be substantially improved. This methodology is presented in its full generality and illustrated on two specific problems: (1) minimizing cost in the problem of scheduling jobs on parallel machines given restrictions on demands and capacity, and (2) minimizing cost in the series parallel redundancy allocation problem, given a target reliability. Our computational results are promising and suggest the applicability of this approach to deal with other problems with similar characteristics or to combine it with mainstream solvers to certify optimality. |
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| AbstractList | Many problems in engineering design involve the use of nonlinearities and some integer variables. Methods based on test sets have been proposed to solve some particular problems with integer variables, but they have not been frequently applied because of computation costs. The walk-back procedure based on a test set gives an exact method to obtain an optimal point of an integer programming problem with linear and nonlinear constraints, but the calculation of this test set and the identification of an optimal solution using the test set directions are usually computationally intensive. In problems for which obtaining the test set is reasonably fast, we show how the effectiveness can still be substantially improved. This methodology is presented in its full generality and illustrated on two specific problems: (1) minimizing cost in the problem of scheduling jobs on parallel machines given restrictions on demands and capacity, and (2) minimizing cost in the series parallel redundancy allocation problem, given a target reliability. Our computational results are promising and suggest the applicability of this approach to deal with other problems with similar characteristics or to combine it with mainstream solvers to certify optimality. |
| Author | Ucha, J. M. Gago-Vargas, J. Hartillo, I. Puerto, J. |
| Author_xml | – sequence: 1 givenname: J. surname: Gago-Vargas fullname: Gago-Vargas, J. organization: Dpto. de Álgebra, Univ. de Sevilla – sequence: 2 givenname: I. surname: Hartillo fullname: Hartillo, I. email: hartillo@us.es organization: Dpto. de Matemática Aplicada I, E.T.S. de Ingeniería Informática, Univ. de Sevilla – sequence: 3 givenname: J. surname: Puerto fullname: Puerto, J. organization: Dpto. de Estadística e Investigación Operativa, Facultad de Matemáticas, Univ. de Sevilla – sequence: 4 givenname: J. M. surname: Ucha fullname: Ucha, J. M. organization: Dpto. de Álgebra, Univ. de Sevilla |
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| Cites_doi | 10.1007/s12532-008-0001-1 10.1287/ijoc.8.2.173 10.1109/99.714603 10.1016/j.amc.2006.02.001 10.1007/s10107-002-0337-7 10.1007/s10107-005-0581-8 10.1016/S0096-3003(99)00217-9 10.1007/s10898-009-9477-0 10.1109/24.510811 10.1016/j.ejor.2010.11.007 10.1007/s10107-002-0322-1 10.1016/j.eswa.2010.09.018 10.1016/0166-218X(95)00026-N 10.1016/0167-6377(92)90008-Q 10.1016/j.cor.2013.05.019 10.1016/j.ress.2007.06.004 |
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| Keywords | Non-linear integer programming Gröbner basis Test set Chance constrained programming |
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| SubjectTerms | Algorithms Computation Convex and Discrete Geometry Costs Design engineering Engineering Engineering design Integer programming Integers Linear programming Management Science Mathematical analysis Mathematical models Mathematics Mathematics and Statistics Nonlinearity Operations Research Operations Research/Decision Theory Optimization Optimization techniques Statistics Studies Test sets |
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| Title | An improved test set approach to nonlinear integer problems with applications to engineering design |
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