A Quadratic Programming with Triangular Fuzzy Numbers

Abstract

Quadratic Programming (QP) is a mathematical modeling technique designed to optimize the usage of limited resources and has been widely applied to solve real world problems. In conventional quadratic programming model the parameters are known constants. However in many practical situations, it is not reasonable to require that the constraints or the objective function in quadratic programming problems be specified in precise, crisp terms. In such situations, it is desirable to use some type of Fuzzy Quadratic Programming (FQP) problem. In this paper a new approach is proposed to derive the fuzzy objective value of fuzzy quadratic programming problem, where the constraints coefficients and the right-hand sides are all triangular fuzzy numbers. The proposed method is solved using MATLAB TM toolbox and the numerical results are presented.

Penentuan Solusi Optimal Pemrograman Kuadratik Menggunakan Metode Beale (Studi Kasus: Produksi Padi Kalimantan Barat)

“…Pemrograman kuadratik atau Quadratic Programming (QP) merupakan pemrograman nonlinear, dengan fungsi tujuan berupa non-linear yaitu fungsi kuadrat, dan fungsi kendalanya itu berupa fungsi linear [12], [16]. Permasalahan pemrograman kuadratik adalah mencari nilai minimum atau maksimum dari fungsi kuadrat dengan batasan yang merupakan pertidaksamaan atau persamaan linear [17]. Berikut merupakan bentuk umum masalah pemrograman kuadratik menggunakan notasi matriks [12]:…”

Alternative solution optimization quadratic form with fuzzy number parameters

“…This section shares some of the required definitions of fuzzy set theory [6], [8], [9], [11]. Definition 2.2.1 [9] If R is a real line, then the fuzzy set A in R is defined as the set of ordered pairs 𝐴 = {(𝑥, 𝜇 𝐴 (𝑥))| 𝑥 ∈ 𝑅}, where 𝜇 𝐴 (𝑥) it is a fuzzy set membership function that maps each R element to a membership value between 0 and 1.…”

“…This section shares some of the required definitions of fuzzy set theory [6], [8], [9], [11]. Definition 2.2.1 [9] If R is a real line, then the fuzzy set A in R is defined as the set of ordered pairs 𝐴 = {(𝑥, 𝜇 𝐴 (𝑥))| 𝑥 ∈ 𝑅}, where 𝜇 𝐴 (𝑥) it is a fuzzy set membership function that maps each R element to a membership value between 0 and 1. Mathematically it can be written as follows: The quadratic programming problem (1.5), if some or all the parameters were fuzzy numbers, the problem can be fuzzy quadratic programming, so the general form of fuzzy quadratic programming as follows:…”

“…Therefore, the applicable fuzzy set theory is used to model. Fuzzy quadratic programming became interested in researchers as Abbasi Molai in 2012 [8], and then In 2017, Mirmohseni [9] carried out a program-related development. This research focuses on quadratic programs with fuzzy triangular numbers.…”

Unconstrained Quadratic Programming Problem with Uncertain Parameters

“…For allocating resources, the quadratic programming problem, which has an objective function in the quadratic form and subject to a set of constraints, is employed. Consequently, a quantitative decision could be made by referring to the optimal solution obtained from solving the quadratic programming problem, especially with the fuzzy parameters [8,9,10]. Also, the computational techniques for solving the quadratic programming problem under the probabilistic environment [11] and the related robust solution [12] are actively studied.…”