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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Vydáno v:Communications in nonlinear science & numerical simulation Ročník 84; s. 105177
Hlavní autoři: Cococcioni, Marco, Cudazzo, Alessandro, Pappalardo, Massimo, Sergeyev, Yaroslav D.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Amsterdam Elsevier B.V 01.05.2020
Elsevier Science Ltd
Témata:
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
Lexicographic optimization
Numerical infinitesimals
Multi-objective optimization
Mixed-integer linear programming
Grossone methodology
ISSN:1007-5704, 1878-7274
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Shrnutí:•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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ISSN:1007-5704
1878-7274
DOI:10.1016/j.cnsns.2020.105177