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Lexicographic multi-objective linear programming using grossone methodology: Theory and algorithm

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  • Cococcioni, Marco
  • Pappalardo, Massimo
  • Sergeyev, Yaroslav D.

Abstract

Numerous problems arising in engineering applications can have several objectives to be satisfied. An important class of problems of this kind is lexicographic multi-objective problems where the first objective is incomparably more important than the second one which, in its turn, is incomparably more important than the third one, etc. In this paper, Lexicographic Multi-Objective Linear Programming (LMOLP) problems are considered. To tackle them, traditional approaches either require solution of a series of linear programming problems or apply a scalarization of weighted multiple objectives into a single-objective function. The latter approach requires finding a set of weights that guarantees the equivalence of the original problem and the single-objective one and the search of correct weights can be very time consuming. In this work a new approach for solving LMOLP problems using a recently introduced computational methodology allowing one to work numerically with infinities and infinitesimals is proposed. It is shown that a smart application of infinitesimal weights allows one to construct a single-objective problem avoiding the necessity to determine finite weights. The equivalence between the original multi-objective problem and the new single-objective one is proved. A simplex-based algorithm working with finite and infinitesimal numbers is proposed, implemented, and discussed. Results of some numerical experiments are provided.

Suggested Citation

  • Cococcioni, Marco & Pappalardo, Massimo & Sergeyev, Yaroslav D., 2018. "Lexicographic multi-objective linear programming using grossone methodology: Theory and algorithm," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 298-311.
    • Handle: RePEc:eee:apmaco:v:318:y:2018:i:c:p:298-311
    DOI: 10.1016/j.amc.2017.05.058
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    References listed on IDEAS

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    1. Amodio, P. & Iavernaro, F. & Mazzia, F. & Mukhametzhanov, M.S. & Sergeyev, Ya.D., 2017. "A generalized Taylor method of order three for the solution of initial value problems in standard and infinity floating-point arithmetic," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 141(C), pages 24-39.
    2. Lolli, Gabriele, 2015. "Metamathematical investigations on the theory of Grossone," Applied Mathematics and Computation, Elsevier, vol. 255(C), pages 3-14.
    3. Pourkarimi, L. & Zarepisheh, M., 2007. "A dual-based algorithm for solving lexicographic multiple objective programs," European Journal of Operational Research, Elsevier, vol. 176(3), pages 1348-1356, February.
    4. Sergeyev, Yaroslav D., 2009. "Evaluating the exact infinitesimal values of area of Sierpinski’s carpet and volume of Menger’s sponge," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3042-3046.
    5. Sergeyev, Yaroslav D., 2007. "Blinking fractals and their quantitative analysis using infinite and infinitesimal numbers," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 50-75.
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    Citations

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    Cited by:

    1. Tohmé, Fernando & Caterina, Gianluca & Gangle, Rocco, 2020. "Computing Truth Values in the Topos of Infinite Peirce’s α-Existential Graphs," Applied Mathematics and Computation, Elsevier, vol. 385(C).
    2. Fiaschi, Lorenzo & Cococcioni, Marco, 2021. "Non-Archimedean game theory: A numerical approach," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    3. Renato Leone & Giovanni Fasano & Massimo Roma & Yaroslav D. Sergeyev, 2020. "Iterative Grossone-Based Computation of Negative Curvature Directions in Large-Scale Optimization," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 554-589, August.
    4. Lili Gong & Wu Cao & Kangli Liu & Jianfeng Zhao & Xiang Li, 2018. "Spatial and Temporal Optimization Strategy for Plug-In Electric Vehicle Charging to Mitigate Impacts on Distribution Network," Energies, MDPI, vol. 11(6), pages 1-16, May.
    5. Mustafa Sivri & Hale Gonce Kocken & Inci Albayrak & Sema Akin, 2019. "Generating a set of compromise solutions of a multi objective linear programming problem through game theory," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 29(2), pages 77-88.
    6. Lorenzo Fiaschi & Marco Cococcioni, 2022. "A Non-Archimedean Interior Point Method and Its Application to the Lexicographic Multi-Objective Quadratic Programming," Mathematics, MDPI, vol. 10(23), pages 1-34, November.
    7. Falcone, Alberto & Garro, Alfredo & Mukhametzhanov, Marat S. & Sergeyev, Yaroslav D., 2021. "A Simulink-based software solution using the Infinity Computer methodology for higher order differentiation," Applied Mathematics and Computation, Elsevier, vol. 409(C).

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