Empirical Versus Analytical Solutions to Full Fuzzy Linear Programming

Stanojević, Bogdana; Stanojević, Miroslav

doi:10.1007/978-3-030-53651-0_19

Cited by 2 publications

(7 citation statements)

References 22 publications

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“…Within our current study, Formula (10) is used in its particular form that replaces "∨" by the operator "max" and "∧" by "⊗", representing any operator of a generalized product. In this way, our approach differs from the one suggested in [6], and succeeds in providing improved solutions in the sense of thinner representations of the fuzzy set optimal values (see Propositions 1 and 2 and their proofs for detailed explanations).…”

Section: Problem Formulation and Solution Approachmentioning

confidence: 99%

“…The second variant of our approach is an improved Monte Carlo simulation algorithm. This variant enhanced the methodology presented in [6], since it reduces the universes from which the random selections are made. In short, during the simulation, the values of the parameters are randomly equal either to the left or right endpoint of their corresponding level sets.…”

Section: Second Variantmentioning

confidence: 99%

“…Then, using a lexicographic method, they derived the compromise objective values to the three crisp objectives that are the components of the previously obtained triangular fuzzy number. As proven in [6], the solution derived in this manner is not in accordance with the extension principle, since fuzzy arithmetic and fuzzy optimization were performed in two sequential steps.…”

Section: Introductionmentioning

confidence: 99%

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Empiric Solutions to Full Fuzzy Linear Programming Problems Using the Generalized “min” Operator

Stanojević,

Nǎdǎban

2023

Mathematics

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Solving optimization problems in a fuzzy environment is an area widely addressed in the recent literature. De-fuzzification of data, construction of crisp more or less equivalent problems, unification of multiple objectives, and solving a single crisp optimization problem are the general descriptions of many procedures that approach fuzzy optimization problems. Such procedures are misleading (since relevant information is lost through de-fuzzyfication and aggregation of more objectives into a single one), but they are still dominant in the literature due to their simplicity. In this paper, we address the full fuzzy linear programming problem, and provide solutions in full accordance with the extension principle. The main contribution of this paper is in modeling the conjunction of the fuzzy sets using the “product” operator instead of “min” within the definition of the solution concept. Our theoretical findings show that using a generalized “min” operator within the extension principle assures thinner shapes to the derived fuzzy solutions compared to those available in the literature. Thinner shapes are always desirable, since such solutions provide the decision maker with more significant information.

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Section: Problem Formulation and Solution Approachmentioning

confidence: 99%

Section: Second Variantmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Empiric Solutions to Full Fuzzy Linear Programming Problems Using the Generalized “min” Operator

Stanojević,

Nǎdǎban

2023

Mathematics

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“…was first given in [29] for the general linear case. The graph of the membership function given in (11) represents in fact the projection of the graph of the membership function (12) on the plane (Ox i , Oα), i = 1, n, where the axes Ox i and Oα contain the values of the variable x i and the membership degrees α, respectively.…”

Section: Fuzzy Transportation Problemsmentioning

confidence: 99%

“…Ezzati et al [41] addressed FF-LP problems by applying first fuzzy arithmetic to the TFN values of the decision variables and coefficients in order to derive the TFN value of the objective function; then, they constructed a three-objective crisp problem and solved it by a lexicographic method. Stanojević and Stanojević [29] proposed empirical solutions to FF-LP problems based on a Monte Carlo simulation that fully comply with the extension…”

Section: Fuzzy Linear Programming Problemsmentioning

confidence: 99%

Reinstatement of the Extension Principle in Approaching Mathematical Programming with Fuzzy Numbers

Stanojević

Nădăban

2021

Mathematics

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Optimization problems in the fuzzy environment are widely studied in the literature. We restrict our attention to mathematical programming problems with coefficients and/or decision variables expressed by fuzzy numbers. Since the review of the recent literature on mathematical programming in the fuzzy environment shows that the extension principle is widely present through the fuzzy arithmetic but much less involved in the foundations of the solution concepts, we believe that efforts to rehabilitate the idea of following the extension principle when deriving relevant fuzzy descriptions to optimal solutions are highly needed. This paper identifies the current position and role of the extension principle in solving mathematical programming problems that involve fuzzy numbers in their models, highlighting the indispensability of the extension principle in approaching this class of problems. After presenting the basic ideas in fuzzy optimization, underlying the advantages and disadvantages of different solution approaches, we review the main methodologies yielding solutions that elude the extension principle, and then compare them to those that follow it. We also suggest research directions focusing on using the extension principle in all stages of the optimization process.

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Empirical Versus Analytical Solutions to Full Fuzzy Linear Programming

Cited by 2 publications

References 22 publications

Empiric Solutions to Full Fuzzy Linear Programming Problems Using the Generalized “min” Operator

Empiric Solutions to Full Fuzzy Linear Programming Problems Using the Generalized “min” Operator

Reinstatement of the Extension Principle in Approaching Mathematical Programming with Fuzzy Numbers

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