Milan Hladík and Michal Černý. The shape of the optimal value of a fuzzy linear programming problem. In P. Melin et al., editor, Fuzzy Logic in Intelligent System Design, Advances in Intelligent Systems and Computing, pp. 281–286, Springer, Cham, 2018.
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We investigate the shape of the optimal value of a linear programming problem with fuzzy-number coefficients. We build on the classical and also very recent results from interval linear programming as well as from parametric programming. We show that under general assumptions the optimal value is a well-defined fuzzy number. Its shape is piecewise polynomial provided the shape of the input fuzzy coefficients are polynomial. We also show in particular that the optimal value shape is triangular as long as the following conditions are satisfied: the input fuzzy numbers are triangular and affect only the objective function or the right-hand side, and the problem is so called basis stable.
@inCollection{HlaCer2018a,
author = "Milan Hlad\'{\i}k and Michal {\v{C}}ern\'{y}",
title = "The shape of the optimal value of a fuzzy linear programming problem",
editor = "Melin et al., P.",
feditor = "Melin, Patricia and Castillo, Oscar and Kacprzyk, Janusz and Reformat, Marek and Melek, William",
booktitle = "Fuzzy Logic in Intelligent System Design",
fbooktitle = "Fuzzy Logic in Intelligent System Design: {Theory} and Applications",
publisher = "Springer",
address = "Cham",
series = "Advances in Intelligent Systems and Computing",
volume = "648",
pages = "281-286",
year = "2018",
doi = "10.1007/978-3-319-67137-6_31",
isbn = "978-3-319-67137-6",
issn = "2194-5365",
url = "https://link.springer.com/chapter/10.1007/978-3-319-67137-6_31",
bib2html_dl_html = "https://doi.org/10.1007/978-3-319-67137-6_31",
abstract = "We investigate the shape of the optimal value of a linear programming problem with fuzzy-number coefficients. We build on the classical and also very recent results from interval linear programming as well as from parametric programming. We show that under general assumptions the optimal value is a well-defined fuzzy number. Its shape is piecewise polynomial provided the shape of the input fuzzy coefficients are polynomial. We also show in particular that the optimal value shape is triangular as long as the following conditions are satisfied: the input fuzzy numbers are triangular and affect only the objective function or the right-hand side, and the problem is so called basis stable.",
keywords = "Interval analysis; Fuzzy number; Linear programming",
}
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