Solving the Lexicographic Multi-Objective Mixed-Integer Linear Programming Problem using branch-and-bound and grossone methodology
•Lexicographic Multi-Objective Mixed Integer Linear Programming problem (LMILP) is studied.•The Grossone methodology using numerical infinities and infinitesimals is applied to LMILP.•A Grossone-based Branch-and-Bound in combination with Grossone-based simplex is proposed.•Theoretical conditions for...
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| Published in: | Communications in nonlinear science & numerical simulation Vol. 84; p. 105177 |
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| Format: | Journal Article |
| Language: | English |
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Elsevier B.V
01.05.2020
Elsevier Science Ltd |
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| ISSN: | 1007-5704, 1878-7274 |
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| Abstract | •Lexicographic Multi-Objective Mixed Integer Linear Programming problem (LMILP) is studied.•The Grossone methodology using numerical infinities and infinitesimals is applied to LMILP.•A Grossone-based Branch-and-Bound in combination with Grossone-based simplex is proposed.•Theoretical conditions for the correctness of the proposed methodology are established.•Experiments show that the new methods are able to solve LMILPs with up to 200 objectives.
In the previous work (see [1]) the authors have shown how to solve a Lexicographic Multi-Objective Linear Programming (LMOLP) problem using the Grossone methodology described in [2]. That algorithm, called GrossSimplex, was a generalization of the well-known simplex algorithm, able to deal numerically with infinitesimal/infinite quantities.
The aim of this work is to provide an algorithm able to solve a similar problem, with the addition of the constraint that some of the decision variables have to be integer. We have called this problem LMOMILP (Lexicographic Multi-Objective Mixed-Integer Linear Programming).
This new problem is solved by introducing the GrossBB algorithm, which is a generalization of the Branch-and-Bound (BB) algorithm. The new method is able to deal with lower-bound and upper-bound estimates which involve infinite and infinitesimal numbers (namely, Grossone-based numbers). After providing theoretical conditions for its correctness, it is shown how the new method can be coupled with the GrossSimplex algorithm described in [1], to solve the original LMOMILP problem. To illustrate how the proposed algorithm finds the optimal solution, a series of LMOMILP benchmarks having a known solution is introduced. In particular, it is shown that the GrossBB combined with the GrossSimplex is able solve the proposed LMOMILP test problems with up to 200 objectives. |
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| AbstractList | In the previous work (see [1]) the authors have shown how to solve a Lexicographic Multi-Objective Linear Programming (LMOLP) problem using the Grossone methodology described in [2]. That algorithm, called GrossSimplex, was a generalization of the well-known simplex algorithm, able to deal numerically with infinitesimal/infinite quantities. The aim of this work is to provide an algorithm able to solve a similar problem, with the addition of the constraint that some of the decision variables have to be integer. We have called this problem LMOMILP (Lexicographic Multi-Objective Mixed-Integer Linear Programming). This new problem is solved by introducing the GrossBB algorithm, which is a generalization of the Branch-and-Bound (BB) algorithm. The new method is able to deal with lower-bound and upper-bound estimates which involve infinite and infinitesimal numbers (namely, Grossone-based numbers). After providing theoretical conditions for its correctness, it is shown how the new method can be coupled with the GrossSimplex algorithm described in [1], to solve the original LMOMILP problem. To illustrate how the proposed algorithm finds the optimal solution, a series of LMOMILP benchmarks having a known solution is introduced. In particular, it is shown that the GrossBB combined with the GrossSimplex is able solve the proposed LMOMILP test problems with up to 200 objectives. •Lexicographic Multi-Objective Mixed Integer Linear Programming problem (LMILP) is studied.•The Grossone methodology using numerical infinities and infinitesimals is applied to LMILP.•A Grossone-based Branch-and-Bound in combination with Grossone-based simplex is proposed.•Theoretical conditions for the correctness of the proposed methodology are established.•Experiments show that the new methods are able to solve LMILPs with up to 200 objectives. In the previous work (see [1]) the authors have shown how to solve a Lexicographic Multi-Objective Linear Programming (LMOLP) problem using the Grossone methodology described in [2]. That algorithm, called GrossSimplex, was a generalization of the well-known simplex algorithm, able to deal numerically with infinitesimal/infinite quantities. The aim of this work is to provide an algorithm able to solve a similar problem, with the addition of the constraint that some of the decision variables have to be integer. We have called this problem LMOMILP (Lexicographic Multi-Objective Mixed-Integer Linear Programming). This new problem is solved by introducing the GrossBB algorithm, which is a generalization of the Branch-and-Bound (BB) algorithm. The new method is able to deal with lower-bound and upper-bound estimates which involve infinite and infinitesimal numbers (namely, Grossone-based numbers). After providing theoretical conditions for its correctness, it is shown how the new method can be coupled with the GrossSimplex algorithm described in [1], to solve the original LMOMILP problem. To illustrate how the proposed algorithm finds the optimal solution, a series of LMOMILP benchmarks having a known solution is introduced. In particular, it is shown that the GrossBB combined with the GrossSimplex is able solve the proposed LMOMILP test problems with up to 200 objectives. |
| ArticleNumber | 105177 |
| Author | Pappalardo, Massimo Sergeyev, Yaroslav D. Cococcioni, Marco Cudazzo, Alessandro |
| Author_xml | – sequence: 1 givenname: Marco orcidid: 0000-0002-7020-1524 surname: Cococcioni fullname: Cococcioni, Marco email: marco.cococcioni@unipi.it organization: University of Pisa, Pisa, Italy – sequence: 2 givenname: Alessandro surname: Cudazzo fullname: Cudazzo, Alessandro email: alessandro@cudazzo.com organization: University of Pisa, Pisa, Italy – sequence: 3 givenname: Massimo surname: Pappalardo fullname: Pappalardo, Massimo email: massimo.pappalardo@unipi.it organization: University of Pisa, Pisa, Italy – sequence: 4 givenname: Yaroslav D. orcidid: 0000-0002-1429-069X surname: Sergeyev fullname: Sergeyev, Yaroslav D. email: yaro@dimes.unical.it organization: University of Calabria, Rende, Italy |
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| Snippet | •Lexicographic Multi-Objective Mixed Integer Linear Programming problem (LMILP) is studied.•The Grossone methodology using numerical infinities and... In the previous work (see [1]) the authors have shown how to solve a Lexicographic Multi-Objective Linear Programming (LMOLP) problem using the Grossone... |
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| SubjectTerms | Algorithms Grossone methodology Integer programming Lexicographic optimization Linear programming Lower bounds Mathematical problems Mixed integer Mixed-integer linear programming Multi-objective optimization Numerical infinitesimals Simplex method Upper bounds |
| Title | Solving the Lexicographic Multi-Objective Mixed-Integer Linear Programming Problem using branch-and-bound and grossone methodology |
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