Multiobjective fuzzy stochastic linear programming problems with inexact probability distribution

Abstract

This study deals with multiobjective fuzzy stochastic linear programming problems with uncertainty probability distribution which are defined as fuzzy assertions by ambiguous experts. The problem formulation has been presented and the two solutions strategies are; the fuzzy transformation via ranking function and the stochastic transformation when α− cut technique and linguistic hedges are used in the uncertainty probability distribution. The development of Sen's method is employed to find a compromise solution, supported by illustrative numerical example.

Solving Stochastic Transportation Electricity Problem With Fuzzy Information on Probability Distribution Using Matlab Program

“…2.1 Basic Definitions 2.1.1 Definition of Fuzzy Set: Let 𝑋 be a universal set, 𝐴 ̃⊆ 𝑋. 𝐴 ̃is called a fuzzy/non-exact set that contains ordered pairs, 𝐴 ̃= {(𝑥, 𝜇 𝐴 ̃(𝑥)), ∀𝑥 ∈ 𝑋} where 𝜇 𝐴 ̃(𝑥) is membership function of 𝑥 ∈ 𝐴 ̃(i.e., a characteristic/indicator function for 𝐴 ̃that shows to what degree 𝑥 ∈ 𝐴 ̃), if the height of fuzzy set is one, then fuzzy set is normal, where the height of a fuzzy set is the largest membership value attained by any point in the set (Ameen, 2015;Dharani, K;Selvi, D, 2018;Mahdavi-Amiri, N;Nasseri, SH, 2006;Mahdavi-Amiri;NezamNasseri;Seyed Hadi, 2007; Sakawa, Fundamentals of fuzzy set theory, 1993; Sakawa, Interactive multiobjective linear programming with fuzzy parameters, 1993). Formulation (4.2-3) is one of the fuzzy set kinds.…”

“…The alpha−level set of fuzzy set 𝐴 ̃is a set 𝐴 ̃𝛼 = {𝑥 ∈ ℝ, 𝜇 𝐴 ̃(𝑥) ≥ 𝛼, 0 < 𝛼 ≤ 1}. The lower and upper bounds of alpha−level set of fuzzy set 𝐴 ̃are finite numbers represented by inf(𝑥 ∈ 𝐴 ̃𝛼) , sup(𝑥 ∈ 𝐴 ̃𝛼) respectively (Ameen, 2015;Dharani, K;Selvi, D, 2018).When Formulation (4.2-1) convert to Formulation (4.2-2) needs alphalevel set to finding efficient points via applying conditions of fuzzy set, and also used in analyzing cases for second condition of alpha-cut technique Formulation (4.1-3).…”

“…Fuzzy number is a convex fuzzy set 𝐴 ̃on ℝ if and only if its membership function is piecewise continuous, and there exist have three intervals [𝑎, 𝑏], [𝑏, 𝑐] and [𝑐, 𝑑] such that 𝐴 ̃is increasing on [𝑎, 𝑏], equal to 1 on [𝑏, 𝑐], decreasing on [𝑐, 𝑑], and equal to 0 elsewhere, ∀𝑎, 𝑏, 𝑐, 𝑑 ∈ ℝ (Ameen, 2015;Dharani, K;Selvi, D, 2018;Mahdavi-Amiri, N;Nasseri, SH, 2006;Mahdavi-Amiri;NezamNasseri;Seyed Hadi, 2007; Sakawa, Fundamentals of fuzzy set theory, 1993; Sakawa, Interactive multiobjective linear programming with fuzzy parameters, 1993). In and Formulation (4.2-4) shows seven difference convex fuzzy numbers.…”

“…An Stochastic Transportation Linear Programming Problems (STLPP) is a DTLPP when parameters are random and represented by probability distributions Masri, Hatem, 2005;Masri, Hatem, 2010;Masri, Hatem, 2009;Ameen, 2015;Guo, Haiying;Wang, Xiaosheng;Zhou, Shaoling, 2015;Hassan, Nasruddin, 2018). The probability distribution space (𝛺, 2 𝛺 , 𝑃) of an STLPP in many cases is unknown, undetermined, and un-specified since it has fuzzy information, unknown distribution, then should be determined/specified as first step in solution procedures.…”

“…The probability distribution space (𝛺, 2 𝛺 , 𝑃) of an STLPP in many cases is unknown, undetermined, and un-specified since it has fuzzy information, unknown distribution, then should be determined/specified as first step in solution procedures. Further, an STLPP's under fuzzy information on probability and described by fuzzy linear inequalities polyhedral set are called Stochastic Transportation Linear Programming Problems with Fuzzy Uncertainty Unknown Information on Probability Distribution Space STLPPFI (A. Edward Samuel;M. Venkatachalapathy, 2011;Masri, Hatem, 2005;Masri, Hatem, 2009;Masri, Hatem, 2010;Ameen, 2015;Appati, Justice Kwame;Gogovi, Gideon Kwadwo;Fosu, Gabriel Obed, 2015;Dharani, K;Selvi, D, 2018;Guo, Haiying;Wang, Xiaosheng;Zhou, Shaoling, 2015;Mahdavi-Amiri, N;Nasseri, SH, 2006;Mahdavi-Amiri;NezamNasseri;Seyed Hadi, 2007) and (Sengamalaselvi, 2017; Interactive multiobjective linear programming with fuzzy parameters, 1993; Sakawa, Fundamentals of fuzzy set theory, 1993).…”