IHJPAS. 53 (4)2022 241 This work is licensed under a Creative Commons Attribution 4.0 International License. Solving of the Quadratic Fractional Programming Problems by a Modified Symmetric Fuzzy Approach Abstract The aims of the paper are to present a modified symmetric fuzzy approach to find the best workable compromise solution for quadratic fractional programming problems (QFPP) with fuzzy crisp in both the objective functions and the constraints. We introduced a modified symmetric fuzzy by proposing a procedure, that starts first by converting the quadratic fractional programming problems that exist in the objective functions to crisp numbers and then converts the linear function that exists in the constraints to crisp numbers. After that, we applied the fuzzy approach to determine the optimal solution for our quadratic fractional programming problem which is supported theoretically and practically. The computer application for the algorithm was tested, and finally compared modified symmetric fuzzy approach with the modified simplex approach which is shown in the table 1. Finally, the procedures of numeric results in the paper indicate that modified symmetric fuzzy approach is reliable and saves valuable time. Keywords: fuzzy system, modified simplex approach, modified symmetric fuzzy, quadratic fractional programming problems. 1. Introduction Mathematical programming finds many applications in the field of management. Optimization of resources in any organization is mostly addressed with the use of mathematical programming. QFPP, which deals with situations where a ratio between two mathematical Doi: 10.30526/35.4.2858 Article history: Received 17 May 2022, Accepted 25 July2022, Published in October 2022. Ibn Al Haitham Journal for Pure and Applied Sciences Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Maher A. Nawkhass Department of Mathematics, College of Education Salahaddin University-Erbil, Erbil, Kurdistan Region, Iraq. maher.nawkhass@su.edu.krd Nejmaddin A. Sulaiman Department of Mathematics, College of Education Salahaddin University- Erbil, Erbil, Kurdistan Region, Iraq. nejmaddin.sulaiman@su.edu.krd https://creativecommons.org/licenses/by/4.0/ mailto:maher.nawkhass@su.edu.krd mailto:maher.nawkhass@su.edu.krd mailto:nejmaddin.sulaiman@su.edu.krd IHJPAS. 53 (4)2022 242 functions is either maximized or minimized, is an important class of mathematical programming problems that has attracted a lot of research and interest because it is useful in production planning, economic and financial planning, medical services, and healthcare planning. There are many managerial decision-making situations where the uncertainties in working situations are best explained by use of fuzzy set theory. The nation of fuzzy set theory is developed by [1], since then a considerable number of scholars have expressed their interest in the application of fuzzy set theory. [2] established the notion of decision making in a fuzzy environment and their concept of fuzzy decision making is used in mathematical programming [3]. Many authors have discussed the use of fuzzy set theory in QFPP e.g., [4, 5 ,6 ,7]. The fuzzy system plays a very significant role in the theory of linear programming (LP). Thus, researchers have shown their interest in the concept of the script for a linear program under a fuzzy environment as well e.g., [8], their works formulates a quadratic fractional bi-level (QFBL) programming problem with probabilistic constraints in both the first (ruler) and second (supporter) levels, with two-parameter exponential random variables with recognized probabilistic, and fuzziness as a triangular and trapezoidal fuzzy number. Using a chance- constrained programming approach, the issue is first turned into an a like deterministic quadratic fractional fuzzy bi-level programming model in the suggested model. Second, each objective function of the bi-level QFP issue has its own nonlinear membership function in the proposed model. In [9], the fuzzy environment with parabolic concave membership functions is used to investigate a certain form of convex fractional programming issue and its dual. An ambition level technique is used to determine appropriate duality outcomes. It differs from other research in that it uses parabolic concave membership functions to reflect the decision-level maker's of pleasure. The Fully Fuzzy Quadratic Fractional Programming problem (FFQFPP) is solved using a solution process in which all variables and parameters are triangular fuzzy integers. In this case, FFQFPP was converted into a Multi-Objective Quadratic Fractional Programming issue (MOQFPP). Using a mathematical programming method, MOQFPP is then turned into an analogous multi-objective quadratic programming problem. A fuzzy goal programming paradigm for handling bilevel programming issues with decision-makers' objectives in quadratic fractional form. The membership functions for the defined fuzzy goals of the decision-makers' objectives at both levels are established first in the suggested approach. Then, in order to find the most satisfactory answer in the decision-making environment, a fuzzy goal programming model is constructed to minimize the group regret of the degree of satisfaction of both decision-makers. In [10], the linear constraints of quadratic fractional objective programming problem (QFOPP) have been established and developed. The Wolfe's technique and a modified simplex approach were used to solve the specific case for this problem. To extend this work, we study and introduce a modified symmetric fuzzy approach to solve QFPP. An algorithm and practical technique with theoretical support are suggested, also solving such a problem by a modified simplex approach, and test the validity will be compared against both outcomes. 2. Fundamentals of Fuzzy set, and Fuzzy Algebra Operation Some fundamentals and algebra operations of the fuzzy environment that are utilized in this research are listed [11, 12]: IHJPAS. 53 (4)2022 243 2.1. Fundamentals of Fuzzy set Fuzzy set: A fuzzy set Α̃ in Χ is a set of ordered pairs: Α̃ = {(𝓍, 𝜇Α̃(𝓍))|𝓍 ∈ Χ}, 𝜇Α̃(𝓍) is titled by the membership function of 𝓍 in Α̃. If Χ is a assemble of contraption defined generally by 𝓍. The support: 𝑆(Α̃) is the crisp set of all 𝓍 ∈ Χ if 𝜇Α̃(𝓍) > 0 is called by the support of fuzzy set Α̃. 𝛼-level set :The fragile set of elements that pertinence to the fuzzy set Α̃ at the lower to the grade 𝛼 is named by 𝛼-level set: Α𝛼 = {𝓍 ∈ Χ|𝜇Α̃(𝓍) ≥ 𝛼}, Α𝛼 ′ = {𝓍 ∈ Χ|𝜇Α̃(𝓍) > 𝛼} is defined by "strong 𝛼- level set" or "strong 𝛼-cut". 2.2. Fuzzy Algebra Operation Algebraic sum: The algebraic sum (likelihood sum) ∁̃= Α̃ + Β̃ is illustrated as ∁̃= {(𝓍, 𝜇Α̃+Β̃(𝓍))|𝓍 ∈ Χ}, where 𝜇Α̃+Β̃(𝓍) = 𝜇Α̃(𝓍) + 𝜇Β̃(𝓍) − 𝜇Α̃(𝓍). 𝜇Β̃(𝓍). Bounded sum: The bounded sum ∁̃= Α̃ ⨁ Β̃ is defined as ∁̃= {(𝓍, 𝜇Α̃ ⨁ Β̃(𝓍))|𝓍 ∈ Χ}, where 𝜇Α̃ ⨁ Β̃(𝓍) = min {1, 𝜇Α̃(𝓍) + 𝜇Β̃(𝓍)}. Bounded difference: The bounded difference ∁̃= Α̃ ⊖ Β̃ is defined as ∁̃= {(𝓍, 𝜇Α̃⊖Β̃(𝓍))|𝓍 ∈ Χ}, where 𝜇Α̃⊖Β̃(𝓍) = min {1, 𝜇Α̃(𝓍) + 𝜇Β̃(𝓍)}. 3. Mathematical Model Formula 3.1. Linear Programming models A special kind of decision model introduced by [13], is considered as Linear programming models, the restrictions define the decision space; the objective function defines the "target" (utility function), and the kind of choice is decision making under conditions. The standard linear programming model is as follows: 𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑓(𝑥) = 𝑐𝑇 𝑥 Such that: 𝐴𝑥 ≤ 𝑏 𝑥 ≥ 0 With 𝑐, 𝑥 ∈ 𝑅𝑛, 𝑏 ∈ 𝑅𝑛, 𝐴 ∈ 𝑅𝑚×𝑛 3.2. Linear Fractional Programming Formula The linear fractional programming problem (LFPP) introduced by [14] can be stated in the following manner: 𝑀𝑎𝑥. 𝑊 = 𝑐1 ′ 𝓍 + 𝛽 𝑐2 ′ 𝓍 + 𝛿 subject to: 𝐴𝓍 = 𝑏, 𝓍 ≥ 0 . (2) (1) IHJPAS. 53 (4)2022 244 Where (i) 𝑥, 𝑐1, and 𝑐2 are 𝑛 × 1 column vectors. (ii) 𝐴 is 𝑛 × 𝑚 matrix. (iii) 𝑏 is an 𝑚 × 1 vectors. (iv) The prime { ′ } on the vectors 𝑐1, and 𝑐2 indicate as a transpose of vectors and (v) 𝛽, 𝛿 are comparatively numeric. The procedure solution of such problems solved by [15]. 3.3. Quadratic Programming Problem (QPP) Quadratic Programming is a particular kind of mathematical optimization problem It is a problem of reducing or maximizing a quadratic function of numerous variables under linear constraints. The QPP can be written as follows:: Max. 𝑧 (or Min. z) = 𝑥𝑇 𝐺𝑥 + 𝑐𝑇 𝑥 + 𝛼 Subject to: 𝐴𝑥 ( ≤ ≥ = ) b 𝑥 ≥ 0 where 𝐴 = (𝑎𝑖𝑗 )𝑚×𝑛 matrix of coefficients, for 𝑖 = 1,2, … , 𝑚 and 𝑗 = 1,2, … , 𝑛, 𝑏 = (𝑏1, 𝑏2, … , 𝑏𝑚) 𝑇, 𝑥 = (𝑥1, 𝑥2, … , 𝑥𝑛 ) 𝑇,𝑐𝑇 = (𝑐1, 𝑐2, … , 𝑐𝑛), 𝛼 is scalar and 𝐺 = (𝑔𝑖𝑗 )𝑛×𝑛 is assumed negative definite of the problem in maximization, and positive definite of problem in minimization, The objective function is quadratic, whereas the restrictions are linear. [16]. 3.4. Quadratic Fractional Programming Problem (QFPP) This section discusses a specific situation of QFPP's mathematical form, which contains a quadratic function in the numerator and a denominator that can be represented as the product of two linear components, as the following: 𝑀𝑎𝑥. 𝑍 = (𝑐1 𝑇𝑥+𝛼)(𝑐2 𝑇𝑥+𝛽) (𝑒1 𝑇𝑥+𝛾)(𝑒2 𝑇+𝛿) = (𝑤1)(𝑤2) (𝑤3)(𝑤4) Subject to: 𝐴𝑥 [ ≥ ≤ = ] 𝑏 𝑥 ≥ 0 𝐴 is 𝑚 × 𝑛 matrix, where 𝑥 is 𝑛 × 1 column vector𝑠 of decision variables, 𝑐1, 𝑐2, 𝑒 are 𝑛 × 1 column vectors of constants, 𝑏 is 𝑚 × 1 column vectors of constants. 𝛼, 𝛽, 𝛾 and 𝛿 are scalars, the prime (𝑇) over the vectors 𝑐1, 𝑐2, 𝑒1 and 𝑒2 denoted the transpose of the vectors. 𝑤1𝑤2 and 𝑤3𝑤4 are the values of numerator and denominator of objective function respectively 3.5. Symmetric Fuzzy linear programming We will suppose that the decision-maker can select an ambition level, z, for the value of the objective function he or she wishes to reach, and that each restriction is modelled as a fuzzy set in model (1). Our fuzzy album then transforms into: (3) (4) IHJPAS. 53 (4)2022 245 Find x such that 𝑐𝑇 𝑥 ≧ 𝑧 𝐴𝑥 ≦ 𝑏 𝑥 ≥ 0 Here ≦ signifies the fuzzified form of ≤ and has the linguistic interpretation "essentially smaller than or equal to.” ≧ signifies the fuzzified form of ≥ and has the linguistic interpretation "essentially greater than or equal to." The objective function of a model (1) might have to be written as a minimizing goal to consider z as an upper bound founded by [13]. 3.6. Zimmermann’s Approach: A Symmetric Model Principle for determining the fuzzy choice to solve the problem's fuzzy system of inequalities (FSI) (1). As a result, the following neat linear programming issue arises. 𝑀𝑎𝑥 𝛼 subject to, 𝜇0(𝑐 𝑇 𝓍) = (1 − 𝑧−𝑐𝑇 𝑝0 ) ≥ 𝛼 𝜇𝑖 (𝐴𝑖 𝓍) = (1 − 𝐴𝑖𝓍 −𝑏𝑖 𝑝𝑖 ) ≥ 𝛼 , 𝑖 = 1, … , 𝑚 𝛼 ∈ [0,1] 𝑥 ≥ 0 or 𝑀𝑎𝑥 𝛼 subject to 𝑐𝑗 𝑇 𝓍 ≥ 𝑧0 − (1 − 𝛼)𝑝0 𝐴𝑖 𝓍 ≥ 𝑏𝑖 + (1 − 𝛼)𝑝𝑖 . , 𝑖 = 1, … , 𝑚 𝑥 ≥ 0 , 𝜆 ∈ [0,1] 4. Modified Simplex Approach Development Dantzig invented the simplex approach in 1947. The Simplex technique is a systematic approach that involves moving from one basic variable solution (on the vertex) to another in a predefined order in order to elicit the value of the objective function. This leaping from vertex to vertex operation is repeated. If the goal function improves with each leap, no basis will ever be duplicated, and there will be no need to return to the vertex because it has already been covered. The algorithm must lead to the optimum vertex in a finite number of steps since the number of vertices is finite. (6) (5) IHJPAS. 53 (4)2022 246 For addressing linear programming problems, the Simplex method is an iterative (step-by-step) approach. It entails the following: Having a fundamental workable solution to constraint equations. ii) Determine whether it is the best option. iii) Using a set of criteria to improve the initial trial solution and continuing the procedure until an ideal solution is found. For more details. 4.1. Modified Simplex Approach to Solve Quadratic Fractional Programming Problem This section deals with the solution of the quadratic fractional programming problem developed by [17], which the solution procedure has some similarity. This technology can be used to high-speed computing with success. This strategy can be used if the problem's constraints are linear functions. i.e., the issue is of the following existence: 𝑀𝑎𝑥. 𝑧 (𝑜𝑟 𝑀𝑖𝑛. 𝑧) = (𝑐1 𝑇𝑥 + 𝛼)(𝑐2 𝑇 𝑥 + 𝛽) (𝑒1 𝑇𝑥 + 𝛾)(𝑒2 𝑇 + 𝛿) Subject to: 𝐴𝑥 ( ≤ ≥ = ) 𝑏 𝑥 ≥ 0 𝐴 is 𝑚 × 𝑛 matrix; 𝑥, 𝑐1, 𝑐1, 𝑒1 and 𝑒2 are 𝑛 × 1 column vectors; 𝑏 is 𝑚 × 1 column vectors; 𝛼, 𝛽, 𝛿 are 𝛾 scalars and prime (𝑇) denoted the transpose of the vector. Where 𝑧1 = (𝑐1 𝑇 𝑥 + 𝛼)(𝑐2 𝑇 𝑥 + 𝛽) 𝑧2 = (𝑒1 𝑇 𝑥 + 𝛾)(𝑒2 𝑇 + 𝛿) And supposed 𝑧1 1 = (𝑐1 𝑇𝑥 + 𝛼) 𝑧1 2 = (𝑐2 𝑇𝑥 + 𝛽) 𝑧2 1 = (𝑒1 𝑇 𝑥 + 𝛾) 𝑧2 2 = (𝑒2 𝑇 + 𝛿) To apply simplex process, first ∆𝑗1 and ∆𝑗2need to be found from the coefficients of the first linear vector and second linear vector of objective function respectively, by using the following formula: ∆𝑖𝑗 = 𝑐𝑖𝑗 − 𝑐𝐵𝑖 𝑥𝑖𝑗 , 𝑖 = 1, 2. 𝑗 = 1, 2, … , 𝑚 + 𝑛 𝑧1 = 𝑧1 (1)𝑧1 (2) IHJPAS. 53 (4)2022 247 𝑧2 = 𝑧2 (1)𝑧2 (2) 𝑧 = 𝑧1/𝑧2 the values of objective functions in this approach the formula is defined to find ∆𝑗 from ∆𝜉1𝑗 , ∆𝜉2𝑗 , 𝑧1, 𝑧2 following: ∆𝑗 = 𝑧1∆𝜉1𝑗 + 𝑧2∆𝜉2𝑗. 5. Modify Symmetric Fuzzy Approach (MSFA) to Solve Quadratic Fractional Programming Problems Our modified approach depends on the adopted "fuzzy" version of formula (4) is: find 𝑥 such that (𝑐1 𝑇𝑥+𝛼)(𝑐2 𝑇𝑥+𝛽) (𝑒1 𝑇𝑥+𝛾)(𝑒2 𝑇+𝛿) ≳ (𝑤1)(𝑤2) (𝑤3)(𝑤4) 𝐴𝑥 ≲ 𝑏 𝑥 ≧ 0 here 𝑐1, 𝑐2, 𝑒1and 𝑒2 are the vector of coefficients of numerator and denominator respectively of the ratio of the goal function, 𝑏 denoted as a vector of constraints, and 𝐴 is the factor of matrix. The sign " ≲" indicate the fuzzified type of " ≦ " and read out "basically smaller than or commensurate to". Note that (7) is wholly symmetric with observance to goal function and constraints, To solve the above problem, First choose an appropriate membership function for each of the fuzzy inequality of (7). In particular, let 𝑓0 denote the membership function for objective function and 𝑓𝑖 , 𝑖 = 1, … , 𝑚 denote the membership function for constraint, let 𝑝0 and 𝑝𝑖 , 𝑖 = 1, … , 𝑚 be the permissible tolerances for objective function and constraint and let 𝑓0 and 𝑓𝑖 , 𝑖 = 1, … , 𝑚 be a continuous and nondecreasing linear membership which is explained below: We have two cases that include only for 𝑓0: First case if (𝑤3)(𝑤4) > (𝑤1)(𝑤2) 𝑓0(𝑐 𝑇 𝓍) = { 1 1−(((𝑤1)(𝑤2)−(𝑤3)(𝑤4)) −((𝑐1 𝑇𝑥+𝛼)(𝑐2 𝑇𝑥+𝛽)−(𝑒1 𝑇𝑥+𝛾)(𝑒2 𝑇+𝛿))) 𝑝0 0 𝑓𝑜𝑟 ((𝑐1 𝑇 𝑥 + 𝛼)(𝑐2 𝑇 𝑥 + 𝛽) − (𝑒1 𝑇 𝑥 + 𝛾)(𝑒2 𝑇 + 𝛿)) > ((𝑤1)(𝑤2) − (𝑤3)(𝑤4 )) 𝑓𝑜𝑟 ((𝑤1)(𝑤2) − (𝑤3)(𝑤4)) − 𝑝0 ≤ ((𝑐1 𝑇 𝑥 + 𝛼)(𝑐2 𝑇 𝑥 + 𝛽) − (𝑒1 𝑇 𝑥 + 𝛾)(𝑒2 𝑇 + 𝛿)) ≤ ((𝑤1 )(𝑤2) − (𝑤3)(𝑤4)) 𝑓𝑜𝑟 ((𝑐1 𝑇 𝑥 + 𝛼)(𝑐2 𝑇 𝑥 + 𝛽) − (𝑒1 𝑇 𝑥 + 𝛾)(𝑒2 𝑇 + 𝛿)) < ((𝑤1)(𝑤2) − (𝑤3)(𝑤4)) − 𝑝0 With 𝑓𝑖 (𝐴𝑖 𝓍) = { 1 1−(𝐴𝑖𝓍 −𝑏𝑖) 𝑝𝑖 0 𝑓𝑜𝑟 𝐴𝑖 𝓍 < 𝑏𝑖 𝑓𝑜𝑟 𝑏𝑖 ≤ 𝐴𝑖 𝓍 ≤ 𝑏𝑖 𝑓𝑜𝑟 𝐴𝑖 𝓍 > 𝑏𝑖 + 𝑝𝑖 +𝑝𝑖 , i = 1, … , m Now, in order to solve the problem we dentify the fuzzy decision as following approach: The crisp linear programming denoted by (𝐿𝑃)(𝑓,𝜆) is: In case (𝑤3)(𝑤4) > (𝑤1)(𝑤2) we have: 𝑀𝑎𝑥 𝜆 subject to, (7) (9) (8) IHJPAS. 53 (4)2022 248 𝑓0(𝑐 𝑇 𝓍) = ( 1−(((𝑤1)(𝑤2)−(𝑤3)(𝑤4)) −((𝑐1 𝑇𝑥+𝛼)(𝑐2 𝑇𝑥+𝛽)−(𝑒1 𝑇𝑥+𝛾)(𝑒2 𝑇+𝛿))) 𝑝0 ) ≥ 𝜆 𝑓𝑖 (𝐴𝑖 𝓍) = ( 1−(𝐴𝑖𝓍 −𝑏𝑖) 𝑝𝑖 ) ≥ 𝜆 , 𝑖 = 1, … , 𝑚 𝜆 ∈ [0,1] 𝑥 ≥ 0 Or 𝑀𝑎𝑥 𝜆 subject to, ((𝑤1)(𝑤2) − (𝑤3)(𝑤4)) ≤ 1 − 𝜆𝑝0 − ((𝑐1 𝑇 𝑥 + 𝛼)(𝑐2 𝑇 𝑥 + 𝛽) − (𝑒1 𝑇𝑥 + 𝛾)(𝑒2 𝑇 + 𝛿)) ∑ 𝑎𝑖𝑘 𝑥𝑘 𝑛 𝑖=1 ≤ 𝑏𝑘 + 1 − 𝜆(𝑝𝑘 ) , 𝑘 = 1, … , 𝑚 𝜆, 𝑥𝑘 ≥ 0 , 𝑘 = 1, … , 𝑚. Second case if (𝑤3)(𝑤4) < (𝑤1)(𝑤2) 𝑓0(𝑐 𝑇 𝓍) = { 1 1−(((𝑤3)(𝑤4)−(𝑤1)(𝑤2)) −((𝑐1 𝑇𝑥+𝛼)(𝑐2 𝑇𝑥+𝛽)−(𝑒1 𝑇𝑥+𝛾)(𝑒2 𝑇+𝛿))) 𝑝0 0 𝑓𝑜𝑟 ((𝑐1 𝑇 𝑥 + 𝛼)(𝑐2 𝑇 𝑥 + 𝛽) − (𝑒1 𝑇 𝑥 + 𝛾)(𝑒2 𝑇 + 𝛿)) > ((𝑤3)(𝑤4) − (𝑤1 )(𝑤2)) 𝑓𝑜𝑟 ((𝑤3)(𝑤4) − (𝑤1 )(𝑤2)) −𝑝0 ≤ ((𝑐1 𝑇 𝑥 + 𝛼)(𝑐2 𝑇 𝑥 + 𝛽) − (𝑒1 𝑇 𝑥 + 𝛾)(𝑒2 𝑇 + 𝛿)) ≤ ((𝑤3)(𝑤4) − (𝑤1)(𝑤2)) 𝑓𝑜𝑟 ((𝑐1 𝑇 𝑥 + 𝛼)(𝑐2 𝑇 𝑥 + 𝛽) − (𝑒1 𝑇 𝑥 + 𝛾)(𝑒2 𝑇 + 𝛿)) < ((𝑤3)(𝑤4) − (𝑤1)(𝑤2)) − 𝑝0 With 𝑓𝑖 (𝐴𝑖 𝓍) = { 1 1−(𝐴𝑖𝓍 −𝑏𝑖) 𝑝𝑖 0 𝑓𝑜𝑟 𝐴𝑖 𝓍 < 𝑏𝑖 𝑓𝑜𝑟 𝑏𝑖 ≤ 𝐴𝑖 𝓍 ≤ 𝑏𝑖 𝑓𝑜𝑟 𝐴𝑖 𝓍 > 𝑏𝑖 + 𝑝𝑖 +𝑝𝑖 , i = 1, … , m Now, in order to solve the problem we identify the fuzzy decision as following approach: The crisp linear programming denoted by (𝐿𝑃)(𝑓,𝜆) is: In case (𝑤3)(𝑤4) < (𝑤1)(𝑤2) we have: 𝑀𝑎𝑥 𝜆 subject to, 𝑓0(𝑐 𝑇 𝓍) = ( 1−(((𝑤3)(𝑤4)−(𝑤1)(𝑤2))−((𝑐1 𝑇𝓍+𝛽)−(𝑐2 𝑇𝑥+𝛿))) 𝑝0 ) ≥ 𝜆 𝑓𝑖 (𝐴𝑖 𝓍) = ( 1−(𝐴𝑖𝓍 −𝑏𝑖) 𝑝𝑖 ) ≥ 𝜆 , 𝑖 = 1, … , 𝑚 𝜆 ∈ [0,1] , 𝑥 ≥ 0 Or (10) (11) (12) IHJPAS. 53 (4)2022 249 𝑀𝑎𝑥 𝜆 subject to, ((𝑤3)(𝑤4) − (𝑤1)(𝑤2)) ≤ 1 − 𝜆𝑝0 − ((𝑐1 𝑇𝓍 + 𝛽) − (𝑐2 𝑇 𝑥 + 𝛿)) ∑ 𝑎𝑖𝑘 𝑥𝑘 𝑛 𝑖=1 ≤ 𝑏𝑘 + 1 − 𝜆(𝑝𝑘 ) , 𝑘 = 1, … , 𝑚 𝜆, 𝑥𝑘 ≥ 0 , 𝑘 = 1, … , 𝑚. 𝑤1𝑤2 and 𝑤3𝑤4 are the values of numerator and denominator of objective function respectively, and 𝑐𝑗 𝑇 𝓍 are 𝑛 × 1 column vectors for 𝑗 = 1,2 By using Simplex algorithm, we can solve problem (10 or 13), the optimum result to (10 or 13) is the optimal solution as well to (7) and as well to (4). Theorem 5.1: In problem (7), let the membership functions 𝑓𝑖 : ℝ → [0,1], (𝑖 = 0, … , 𝑚), be continuous and nondecreasing. Then the fuzzy solution of (7) is assumed by the parametric solution of the parametric fractional linear programming problem. 𝐏𝐫𝐨𝐨𝐟. We have to solve the following problem to obtain the solution of (7). (𝐿𝑃)(𝑓,𝜆) 𝑀𝑎𝑥 𝜆 subject to 𝑓0(𝑐 𝑇 𝓍) ≥ 𝜆 𝑓𝑖 (𝐴𝑖 𝓍) ≥ 𝜆 , 𝑖 = 1, … , 𝑚 𝑥 ≥ 0 , 𝜆 ∈ [0,1] Where 𝑓0(𝑐 𝑇 𝓍) = inf (𝑓0𝑗 (𝑐𝑗 𝑇 𝓍) ) and 𝑓i(𝐴𝑖 𝓍) = inf (𝑓ij(𝐴𝑖𝑗 𝓍) ) , 𝑖 = 1, … , 𝑚. But 𝑓ij is nondecreasing and continuous function, given that 𝑓𝑖𝑗 −1 exist, and 𝑓0𝑗 (𝑐𝑗 𝑇 𝓍) ≥ 𝜆 ⇒ 𝑐𝑗 𝑇 𝓍 ≥ 𝑓0𝑗 −1(𝜆) with 𝑓ij(𝐴𝑖𝑗 𝓍) ≥ 𝜆 ⇒ 𝐴𝑖𝑗 𝓍 ≥ 𝑓𝑖𝑗 −1(𝜆). Therefore, the problem (𝐿𝑃)(𝑓,𝜆) can be written as 𝑀𝑎𝑥 𝜆 subject to 𝑐𝑗 𝑇 𝓍 ≥ 𝑓0𝑗 −1(𝜆) 𝐴𝑖 𝓍 ≥ 𝑓𝑖𝑗 −1(𝜆). , 𝑖 = 1, … , 𝑚 𝑥 ≥ 0 , 𝜆 ∈ [0,1] Which is equivalent to (6) 𝑀𝑎𝑥 𝛼 subject to 𝑐𝑗 𝑇 𝓍 ≥ 𝑧0 − (1 − 𝛼)𝑝0 𝐴𝑖 𝓍 ≥ 𝑏𝑖 + (1 − 𝛼)𝑝𝑖 . , 𝑖 = 1, … , 𝑚 𝑥 ≥ 0 , 𝛼 ∈ [0,1] Thus, the fuzzy linear programming (10 or 13) can be solved by the (crisp) linear programming 𝑀𝑎𝑥 𝛼 subject to (13) IHJPAS. 53 (4)2022 250 𝑐𝑗 𝑇 𝓍 ≥ 𝑧0 − (1 − 𝛼)𝑝0 𝐴𝑖 𝓍 ≥ 𝑏𝑖 + (1 − 𝛼)𝑝𝑖 . , 𝑖 = 1, … , 𝑚 𝑥 ≥ 0 , 𝛼 ∈ [0,1] Which is the same as problem (6) with 𝜆 = 𝛼 . Remark 5.1 If the problem (7) has fuzzy as well as crisp constraints, then in the alike (crisp) linear fractional programming problem, the original crisp constraints will not have any modification as for them the tolerances are nil. 6. Algorithm to Solve QFPP by Using MSFA To find the result of optimal solution for the QFPP of formulation (4), a procedure is given as below: Step One: Convert the main problem to the fuzzy version as formula 7 Step Two: Find the value of 𝑝0 and 𝑝𝑖 , 𝑖 = 1, … , 𝑚 which are the permissible tolerances for objective function and constraint by using formula (8, 11) and (9, 12), respectively. Step Three: Construct the crisp linear programming (𝐿𝑃)(𝑓,𝜆) by using formula (10, 13) Step Four: Optimize step 3 by using simplex algorithm. 7. An Illustrative Numerical Example and Results: In this section, we solve an example through MSFA and compared with MODIFIED SIMPLEX APPROACH. Example: Maximize Z = (4𝑥1+6𝑥2−2)(2𝑥1+3𝑥2+1) (6𝑥1+9𝑥2+3) 2 Subject to: 𝑥1 + 3𝑥2 ≤ 5 2𝑥1 + 𝑥2 ≤ 2 𝑥1, 𝑥2 ≥ 0 Step one: The fuzzy version of problem is: find 𝑥 such that: (4𝑥1+6𝑥2−2)(2𝑥1+3𝑥2+1) (6𝑥1+9𝑥2+3)(6𝑥1+9𝑥2+3) ≳ (𝑤1)(𝑤2) (𝑤3)(𝑤4) 𝑥1 + 3𝑥2 ≲ 5 2𝑥1 + 𝑥2 ≲ 5 𝑥1, 𝑥2 ≧ 0 Step Two: Find the value of 𝑝0 and 𝑝𝑖 , 𝑖 = 1, … , 𝑚 which are the permissible tolerances for objective function and constraint. Firstly, obtain the value of (𝑤1), (𝑤2), (𝑤3) and (𝑤4) by simplex algorithm as follows: 𝑀𝑎𝑥. 𝑤1 = (4𝑥1 + 6𝑥2 − 2) Subject to: 𝑥1 + 3𝑥2 ≤ 5 2𝑥1 + 𝑥2 ≤ 2 𝑥1, 𝑥2 ≥ 0 Optimal solution: 𝑥1 = 0.2, 𝑥2 = 1.6 𝑀𝑎𝑥. 𝑤1 = 8.4 𝑀𝑎𝑥. 𝑤1 = (2𝑥1 + 3𝑥2 + 1) Subject to: 𝑥1 + 3𝑥2 ≤ 5 2𝑥1 + 𝑥2 ≤ 2 𝑥1, 𝑥2 ≥ 0 Optimal solution: 𝑥1 = 0.2, 𝑥2 = 1.6 𝑀𝑎𝑥. 𝑤1 = 6,2 IHJPAS. 53 (4)2022 251 𝑀𝑎𝑥. 𝑤1 = (6𝑥1 + 9𝑥2 + 3) Subject to: 𝑥1 + 3𝑥2 ≤ 5 2𝑥1 + 𝑥2 ≤ 2 𝑥1, 𝑥2 ≥ 0 Optimal solution: 𝑥1 = 0.2, 𝑥2 = 1.6 𝑀𝑎𝑥. 𝑤1 = 18.6 𝑀𝑎𝑥. 𝑤1 = (6𝑥1 + 9𝑥2 + 3) Subject to: 𝑥1 + 3𝑥2 ≤ 5 2𝑥1 + 𝑥2 ≤ 2 𝑥1, 𝑥2 ≥ 0 Optimal solution: 𝑥1 = 0.2, 𝑥2 = 1.6 𝑀𝑎𝑥. 𝑤1 = 18.6 Now, find 𝑝0 by using case 1 at decision variable 𝑥1 = 0.2, 𝑥2 = 1.6 since (𝑤3)(𝑤4) > (𝑤1)(𝑤2), so can get it as follow: ( 1 − ((18.6)(18.6)) − (8.6)(6.2)) − ((4𝑥1 + 6𝑥2 − 2)(2𝑥1 + 3𝑥2 + 1) − (6𝑥1 + 9𝑥2 + 3)(6𝑥1 + 9𝑥2 + 3)) 𝑝0 ) ≥ 𝜆 → 𝑝0 ≥ 2 Now, find 𝑝1 by using formula (9) at decision variable 𝑥1 = 0.2, 𝑥2 = 1.6 as follow: ( 1 − ( (𝑥1 + 3𝑥2) − 5) 𝑝1 ) ≥ 𝜆 → 𝑝1 ≥ 2 For 𝑝2 by using formula (9) at decision variable 𝑥1 = 0.2, 𝑥2 = 1.6 as follow: ( 1 − ( 2𝑥1 + 𝑥2) − 5) 𝑝2 ) ≥ 𝜆 → 𝑝2 ≥ 2 Step Three: Construct the crisp linear programming (𝐿𝑃)(𝑓,𝜆) by using formula (10,13) 𝑀𝑎𝑥 𝜆 subject to, 2𝜆 − 6𝑥1 + 3𝑥2 ≤ 30.6 2𝜆 + 𝑥1 + 3𝑥2 ≤ 6 2𝜆 + 2𝑥1 + 𝑥2 ≤ 3 𝜆, 𝑥1, 𝑥2 ≥ 0 Step four: Optimize crisp linear programming (𝐿𝑃)(𝑓,𝜆) by using simplex algorithm, found that the decision variable is 𝒙𝟏 = 𝟎. 𝟐, 𝒙𝟐 = 𝟏. 𝟔 and 𝑴𝒂𝒙. 𝒁 = 𝟎. 𝟏𝟓 In additional, will solve the such numerical example by computational procedure of quadratic fractional algorithm introduced by [10], section 4.1 to compare between two methods (MSFA) and (Modified Simplex Approach). Now, Maximize Z = (4𝑥1+6𝑥2−2)(2𝑥1+3𝑥2+1) (6𝑥1+9𝑥2+3) 2 Subject to: 𝑥1 + 3𝑥2 ≤ 5 2𝑥1 + 𝑥2 ≤ 2 𝑥1, 𝑥2 ≥ 0 introduce the slack variables S1 ≥ 0 and S2 ≥ 0 the problem in the standard form becomes: 𝑀𝑎𝑥. 𝑊 = (4𝑥1+6𝑥2−2)(2𝑥1+3𝑥2+1) (6𝑥1+9𝑥2+3)(6𝑥1+9𝑥2+3) = 𝑧 (1)𝑧(2) 𝑧 (3)𝑧(4) subject to: IHJPAS. 53 (4)2022 252 𝑥1 + 3𝑥2 + S1 = 5 2𝑥1 + 𝑥2+S2 = 2 𝑥1, 𝑥2, S1, S2 ≥ 0 for starting table, we will find ∆1 = −72, ∆2 = −108, ∆3 = 0, ∆4 = 0. We choose min ∆𝑗 (∆2 in this case). Thus, z can be increased by taking X2 in the basis. The method to determine the leaving variables and also the new value of Xij, XB, ∆1, ∆2, ∆3 and ∆4 corresponding to improved solution will be the same as for ordinary simplex method. Thus, S1 will be the departing variable. Starting table c1B 4 6 0 0 c2B 2 3 0 0 C1B 6 9 0 0 C2B 6 9 0 0 B C1B C2B c1B c2B XB X1 X2 S1 S2 Min ( XB X1 ) S1 0 0 0 0 5 1 3 1 0 1.6 S2 0 0 0 0 2 2 1 0 1 2 𝑧1 (1) = c1BXB = −2 Δk1 1𝑗 -4 -6 0 0 𝑧1 (2) = c2BXB = 1 Δk1 2𝑗 -2 -3 0 0 𝑧2 (1) = 𝐶1BXB = 3 Δk2 1𝑗 -6 -9 0 0 𝑧2 (2) = 𝐶2BXB = 3 Δk2 2𝑗 -6 -9 0 0 𝑧1 = 𝑧1 (1)𝑧1 (2) = −2 ∆𝜉1𝑗 -8 -12 0 0 𝑧2 = 𝑧2 (1)𝑧2 (2) = 9 ∆𝜉2𝑗 0 0 0 0 𝑧 = 𝑧1 𝑧2 = −0.222 ∆𝑗 -72 -108 0 0 First iteration table Introducing X2 and dropping S1, we get the following table: c1B 4 6 0 0 c2B 2 3 0 0 C1B 6 9 0 0 C2B 6 9 0 0 B C1B C2B c1B c2B XB X1 X2 S1 S2 Min ( XB X1 ) X2 9 9 6 3 5/3 1/3 1 1/3 0 5 S2 0 0 0 0 0.33 1.66 0 -0.33 1 1.19 𝑧1 (1) = c1BXB = 8 Δk1 1𝑗 -2 0 2 0 𝑧1 (2) = c2BXB = 6 Δk1 2𝑗 -1 0 1 0 𝑧2 (1) = 𝐶1BXB = 18 Δk2 1𝑗 -3 0 3 0 𝑧2 (2) = 𝐶2BXB = 18 Δk2 2𝑗 -3 0 3 0 𝑧1 = 𝑧1 (1)𝑧1 (2) = 48 ∆𝜉1𝑗 -4 0 4 0 𝑧2 = 𝑧2 (1)𝑧2 (2) = 324 ∆𝜉2𝑗 0 0 0 0 ∆𝑗 -1296 0 1296 0 Second iteration table Introducing X1 and dropping S2, we get the following table: c1B 4 6 0 0 IHJPAS. 53 (4)2022 253 c2B 2 3 0 0 C1B 6 9 0 0 C2B 6 9 0 0 B C1B C2B c1B c2B XB X1 X2 S1 S2 Min ( XB X1 ) X2 9 9 6 3 1.6 0 1 0.39 -0.2 X1 6 6 4 2 0.2 1 0 -1.19 0.602 𝑧1 (1) = c1BXB = 8.4 Δk1 1𝑗 0 0 1.34 2 𝑧1 (2) = c2BXB = 6.2 Δk1 2𝑗 0 0 0.67 0.6 𝑧2 (1) = 𝐶1BXB =18.6 Δk2 1𝑗 0 0 2.01 1.8 𝑧2 (2) = 𝐶2BXB =18.6 Δk2 2𝑗 0 0 2.01 1.8 𝑧1 = 𝑧1 (1)𝑧1 (2) = 52.08 ∆𝜉1𝑗 0 0 2.68 2.4 𝑧2 = 𝑧2 (1)𝑧2 (2) =345.95 ∆𝜉2𝑗 0 0 0 0 𝑧 = 𝑧1 𝑧2 =0.15 ∆𝑗 0 0 926342 83020 Since all ∆𝑗 ≥ 0 we have reached the optimal solution: 𝐗𝟏 = 𝟎. 𝟐, 𝐗𝟐 = 𝟏. 𝟔, 𝐒𝟏 = 𝟎, 𝐒𝟐 = 𝟎, 𝑴𝒂𝒙. 𝑾 = 𝟎. 𝟏𝟓 . The fuzzy and unfuzzy results are shown in the following: Table 1- The result of numerical example solved in two methods Unfuzzy [Modified Simplex Approach] and Fuzzy [MSFA] Unfuzzy (Modified Simplex Approach) Fuzzy (MSFA) 𝑥1 = 0.2 𝑥2 = 1.6 𝑊 = 0.15 𝑥1 = 0.2 𝑥2 = 1.6 𝑊 = 0.15 Constraints: 1 5 1 6 2 2 2 3 We conclude from the solution we have same profit under regard all constraints. 8. Conclusion This paper used modify symmetric fuzzy approach. After converting to crisp linear programming, the maximum value of QFPP was discovered and compared to the modified simplex technique. The values of the objective function are used to compare these techniques, the study found that 𝑀𝑎𝑥. 𝑧 resulted of those two methods are same but modify symmetric fuzzy solve the problem directly without iteration while modified simplex approach take two iterations. 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