Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6, 7802-7812 2024 Publisher: Learning Gate DOI: 10.55214/25768484.v8i6.3708 © 2024 by the authors; licensee Learning Gate © 2024 by the authors; licensee Learning Gate * Correspondence: a-foroughi@qom.ac.ir Solving fuzzy fractional programming problems by VNS algorithm using modified Kerre's inequality Dhurgam Kalel Ibrahim Alsaad1, Aliasghar Foroughi1*, Khatere Ghorbani-Moghadam2, Reza Ghanbari3 1Department of Mathematics, University of Qom, Qom, Iran; aalimath2018@gmail.com (D.K.I.A.) a-foroughi@qom.ac.ir (A.F.). 2Mosaheb Institute of Mathematics, Kharazmi University, Tehran, Iran; k.ghorbani@khu.ac.ir (K.G.M.). 3Faculty of Mathematical Sciences, Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran; rghanbari@um.ac.ir (R.Z.). Abstract: Here, we propose a new variable neighbourhood search (VNS) algorithm for solving fractional fuzzy number linear programming problems (FFNLPPs). We make use of modified Kerre’s inequality for comparison of LR fuzzy numbers. In our proposed algorithm, we introduced a new local search defined based on descent directions, which are found by solving four crisp mathematical programming problems. In several methods, a fuzzy fractional optimization problem is converted to a crisp problem. Still, in our proposed method, using modified Kerre’s inequality, the fuzzy optimization problem is solved directly, without changing it to a crisp program. To show the effectiveness of our method, we compare our proposed algorithm with other available methods. Keywords: Fractional linear optimization, Modified Kerre’s inequality, VNS algorithm. 1. Introduction Charnes and Cooper [4] first formulated a linear fractional programming problem as an optimization problem of a ratio of linear functions subject to linear constraints. Such linear fractional objectives (i.e., ratio objectives that have a linear numerator and denominator) are helpful in production planning, financial and corporate planning, health care, hospital planning, and so forth. Examples of fractional objectives in production planning include inventory/sales, output/employee, etc. For single objective, linear fractional programming, the Charnes and Cooper [4] transformation can transform the problem into a linear programming problem. In many real-world situations, some parameters of a linear program are given by experts. However, experts and decision-makers frequently are not aware of the precise values of the parameters. Since some optimization problems contain parameters with imprecise values (see [3]), fuzzy number linear programs (FNLPs) are handy tools for modelling and solving real-world problems . Arya et al. [1] presented a fuzzy based branch-bound approach is attempted for solving multi- objective linear fractional (MOLF) optimization problems. The original MOLF optimization problem is converted into an equivalent fuzzy multi-objective linear fractional (FMOLF) optimization problem. Then, branch and bound techniques were applied to the FMOLF optimization problem. The feasible space of FMOLF optimization problem was bounded by triangular simplex space. The weak duality theorem was used to generate the bound for each partition and feasibility conditions were applied to neglect one partition in each step. After finite number of steps, a fuzzy efficient (Pareto-optimal) solution was obtained for FMOLF optimization problem which was also efficient (Pareto-optimal) solution of the original MOLF optimization problem. Some theoretical validations were also established for the proposed approach on FMOLF optimization problem. Chinnadurai and Muthukumar [5] proposed using the (α,r) acceptable optimal value for a linear fractional programming problem with fuzzy coefficients and fuzzy decision variables, as well as developing a method for computing them. To obtain acceptable (α,r) optimal values, they take an α-cut on the objective function and r-cut on the 7803 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 7802-7812, 2024 DOI: 10.55214/25768484.v8i6.3708 © 2024 by the authors; licensee Learning Gate constraints. They then formulated an equivalent bi-objective linear fractional programming problem to calculate the upper and lower bounds of the fully fuzzy linear fractional programming (LFP) problem. Using the upper and lower bounds obtained, they constructed the membership functions of the optimal values numerically. Wang et al. [33] obtained the solution to bi-level linear fractional programming problem (BLFP) using an optimization algorithm based on the duality gap of the lower level problem. In their algorithm, the bi-level linear fractional programming problem was transformed into an equivalent single-level programming problem by forcing the dual gap of the lower level problem to zero. Then, by obtaining all vertices of a polyhedron, the single-level programming problem could be converted into a series of linear fractional programming problems. Pal et al. [17], presented a goal programming (GP) procedure for fuzzy multi-objective linear fractional programming (FMOLFP) problems. In the proposed approach, which is motivated by Mohamed (Fuzzy Sets and Systems 89 (1997) 215), GP model for achievement of the highest membership value of each of fuzzy goals defined for the fractional objectives is formulated. In the solution process, the method of variable change on the under- and over- deviational variables of the membership goals associated with the fuzzy goals of the model is introduced to solve the problem efficiently using linear goal programming (LGP) methodology. Yang et al. [34], presented to solve fuzzy multi-objective linear fractional programming (FMOLFP) problems through an approach based on superiority and inferiority measures method (SIMM). In the model for the proposed approach, each of fuzzy goals defined for the fractional objectives and some constraints have fuzzy numbers. To achieve the highest membership value, SIMM was adopted to deal with fuzzy number in constraints, then a linear goal programming methodology was introduced to solve the problem in which the fractional objectives was fuzzy goals. A case of agricultural planting structures optimization problem was solved to illustrate the application of the algorithm. Sakawa and Kato [24], by considering the experts’ vague or fuzzy understanding of the nature of the parameters in the problem formulation process, multi-objective linear fractional programming problems with block angular structure involving fuzzy numbers are formulated. Sakawa and Kato [24] proposed the following model: min 𝑧1(𝑥, �̃�1, �̃�1). . . min 𝑧𝑘(𝑥, �̃�𝑘 , �̃�𝑘)𝑠. 𝑡. (1 ) �̃�1𝑥1 + ⋯ + �̃�𝑝𝑥𝑝 ≤ �̃�0, �̃�1𝑥1 ≤ �̃�1, . . . �̃�𝑝𝑥𝑝 ≤ �̃�𝑝, 𝑥𝑗 ≥ 0, 𝑗 = 0, ⋯ , 𝑝. Where, 𝑥𝑗, 𝑗 = 1, ⋯ , 𝑝, are 𝑛𝑗 dimensional column vectors of decision variables, �̃�1𝑥1 + ⋯ + �̃�𝑝𝑥𝑝 ≤ �̃�0 are 𝑚0-dimensional coupling constraints, and �̃�𝑗, 𝑗 = 1, ⋯ , 𝑝 are 𝑚0 × 𝑛 fuzzy coefficient matrices. �̃�𝑗𝑥𝑗 ≤ �̃�𝑗 are 𝑚𝑗-dimensional block constraints with respect to 𝑥𝑗, �̃�𝑗, 𝑗 = 1, ⋯ , 𝑝 are 𝑚𝑗 × 𝑛𝑗 fuzzy coefficient matrices and �̃�𝑗, 𝑗 = 1, ⋯ , 𝑝, are 𝑚𝑗-dimensional column fuzzy vectors. Furthermore, 𝑧1(𝑥, �̃�1, �̃�1), ⋯ , 𝑧𝑘(𝑥, �̃�𝑘 , �̃�𝑘) are 𝑘 distinct fuzzy linear fractional objective functions defined by : 𝑧𝑖(𝑥, �̃�𝑖 , �̃�𝑖) = 𝑝𝑖(𝑥,𝑐̃𝑖) 𝑞𝑖(𝑥,�̃�𝑖) = 𝑐̃𝑖1𝑥1+⋯+𝑐̃𝑖𝑝𝑥𝑝+𝑐̃𝑖,𝑝+1 �̃�𝑖1𝑥1+⋯+�̃�𝑖𝑝𝑥𝑝+�̃�𝑖,𝑝+1 . Using the a-level sets of fuzzy numbers, the corresponding non-fuzzy α-multi-objective linear fractional programming problem is introduced. The fuzzy goals of the decision maker for the objective functions are quantified by eliciting the corresponding membership functions, including nonlinear ones. Through the introduction of extended pareto optimality concepts, if the decision maker specifies the degree α and the reference membership values, the corresponding extended pareto optimal solution can be obtained by solving the minimax problems for which the Dantzig-Wolfe decomposition method and Ritter’s partitioning procedure are applicable. Jiao et al. [14], presented a branch-and-bound algorithm for globally solving a wide class of generalized linear fractional programming problems (GLFP). This class included such problems as: minimizing a sum, or error for product of a finite number of ratios of linear functions, linear multiplicative programming, polynomial programming, etc. over nonconvex feasible region. First a problem (Q) was derived equivalent to problem (GLFP). In the algorithm, lower bounds were derived by solving a sequence of linear relaxation programming problems, which was 7804 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 7802-7812, 2024 DOI: 10.55214/25768484.v8i6.3708 © 2024 by the authors; licensee Learning Gate based on the construction of the linear lower bounding functions for the objective function and constraint functions of the problem (Q) over the feasible region. Shen et al. [26], proposed an interactive fuzzy programming method for seeking a satisfactory solution for multi-objective two-level linear fractional programming problems in which the decision makers in the upper and lower levels have several objectives, by first setting up the fuzzy goals for several objectives of each decision maker and seeking a satisfactory solution for the degree of satisfaction of two decision makers in a cooperative manner. In the proposed method, the decision maker at the upper level sets the minimal satisfactory levels for each fuzzy goal. The decision-maker at the lower level determines the aspiration levels. The minimal satisfactory levels were treated as a constraint and the solution closest to the aspiration levels of the decision maker at the lower level is computed. The ratio of the aggregative degrees of satisfaction of the decision makers in the upper and lower levels with the obtained solution is evaluated by using partial information on the preference of the decision makers. Chun-Feng and Pei-Ping [6], presented an efficient branch and bound method for general linear fractional problem (GFP). First, using a transformation technique, an equivalent problem (EP) of GFP was derived, then by exploiting structure of EP, a linear relaxation programming (LRP) of EP was obtained. To implement the algorithm, primary computation involve solving a sequence of linear programming problem, which could be solved efficiently. The proposed algorithm converged to the global maximum through the successive refinement of the solutions of a series of linear programming problems. D’Amato and Bernstein [8], used retrospective cost optimization to determine linear fractional transformations (LFTs). This method used an adaptive controller in feedback with a known system model. The goal was to identify the feedback portion of the LFT by adapting the controller with a retrospective cost. D’Amato and Bernstein [8] demonstrated this method on numerical examples of increasing complexity, ranging from linear examples with unknown feedback terms to nonlinear examples. Sheikhi and Ebadi [27] presented a novel method for solving fractional transportation problems (FTPs) with fuzzy numbers using a ranking function. The proposed method introduces a transformation technique that converts an FTP with fuzzy numbers into an FTP with crisp numbers by employing the robust ranking technique. Then, they formulated two transportation problems, one for maximization and another for depreciation, based on the given FTP. See other works in fuzzy in [2, 11, 12, 19, 20, 22, 21, 25]. A type of fractional fuzzy number linear programming problem (FFNLPP) can be described as follows: max �̃�𝑥+�̃� 𝑐̃𝑥+�̃� 𝑠. 𝑡. (2 ) 𝐴𝑥 ≤ ℎ, 𝑥 ≥ 0, where, �̃� ∈ 𝐹(ℝ𝑛), �̃� ∈ 𝐹(ℝ𝑛), �̃� ∈ 𝐹(ℝ), �̃� ∈ 𝐹(ℝ), 𝑥 ∈ ℝ𝑛, 𝐴 ∈ ℝ𝑚×𝑛 and ℎ ∈ ℝ𝑚, with 𝐹(ℝ) being the set of all fuzzy numbers. For solving (2), we propose a new fuzzy VNS algorithm with its local search intended to find a feasible solution by using increasing directions, leading to the value of the objective function being more significant than the current solution. In our proposed algorithm, we compare the fuzzy value of the objective function using the modified Kerre’s inequality. Using Kerre’s inequality, we can solve the problem (2) directly without changing it to the crisp problem [10]. The rest of our research is structured in the following manner. In Section 2, we provide some necessary definitions, properties of fuzzy ordering and fundamental aspects of modified Kerre’s method. We present our model in Section 3. In Section 4, we first talk about our VNS algorithm for solving our proposed model, then to make the algorithm more clear we implement our proposed algorithm on an example. We test our algorithm on several examples in Section 5, and conclude in Section 6. 2. Preliminaries Here, we provide an overview of the basic principles of fuzzy set theory. 2.1. Definitions and Notation Definition 1 [35] Let ℝ be a collection of objects denoted by 𝑥. Then �̃� is called a fuzzy set in ℝ, if �̃� is a set of ordered pairs , 7805 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 7802-7812, 2024 DOI: 10.55214/25768484.v8i6.3708 © 2024 by the authors; licensee Learning Gate �̃� = {(𝑥, 𝜇�̃�(𝑥)) | 𝑥 ∈ ℝ}, where 𝜇�̃�(𝑥) ∈ [0,1] is called the membership function or grade of membership of 𝑥 in �̃� . Definition 2 [29] A fuzzy number is a fuzzy quantity �̃� satisfying the following conditions : 1 . 𝜇�̃�(𝑥) = 1, for exactly one 𝑥 . 2 . The support {𝑥: 𝜇�̃�(𝑥) > 0} of �̃� is bounded . 3 . The 𝛼-cuts of �̃� are closed intervals . Definition 3 [35] A fuzzy number �̃� is of LR-type if there exist shape functions 𝐿 (for left), 𝑅 (for right), and scalars 𝛼 > 0 and 𝛽 > 0 such that 𝜇�̃�(𝑥) = { 𝐿 ( 𝑎−𝑥 𝛼 ) , 𝑥 ≤ 𝑎, 𝑅 ( 𝑥−𝑎 𝛽 ) , 𝑥 ≥ 𝑎. The mean value of �̃�, 𝑎, is a real number, and 𝛼 and 𝛽 are called the left and right spreads, respectively. Here, �̃� is denoted by (𝑎 − 𝛼/𝑎/𝑎 + 𝛽)𝐿𝑅 . Remark 1 Based on Definition 3, another representation of an LR fuzzy number �̃� is �̃� = (𝐴𝐿 , 𝐴𝑅), where 𝐴𝐿 is a shape function for the left arm and 𝐴𝑅 is a shape function for the right arm . Definition 4 [3] A triangular fuzzy number �̃� is defined by three real numbers 𝑎 < 𝑏 < 𝑐, where the base of the triangle is the interval [𝑎, 𝑐] and its vertex is at 𝑥 = 𝑏 . Remark 2 Another representation of a triangular fuzzy number �̃� is �̃� = (𝑀𝐿 , 𝑀𝑅), where 𝑀𝐿 and 𝑀𝑅 are the functions for the left arm and the right arm of triangular fuzzy number �̃� = (𝑎/𝑏/𝑐), respectively . Note 1 In the rest of our work, triangular fuzzy numbers will be represented as �̃� = (𝑎/𝑏/𝑐). We also denote a real number a by �̃� = (𝑎/𝑎/𝑎) . Theorem 1 [35] Let �̃� = (𝑚𝐿/𝑚/𝑚𝑅)𝐿𝑅, let �̃� = (𝑛𝐿/𝑛/𝑛𝑅)𝐿𝑅, and let 𝜆 ∈ ℝ+. Then , 1 . 𝜆�̃� = (𝜆𝑚𝐿/𝜆𝑚/𝜆𝑚𝑅)𝐿𝑅 . 2 . −�̃� = (−𝑚𝑅/−𝑚/−𝑚𝐿)𝐿𝑅 . 3 . �̃� ⊕ �̃� = (𝑚𝐿 + 𝑛𝐿/𝑚 + 𝑛/𝑚𝑅 + 𝑛𝑅)𝐿𝑅 . 2.1.1. Modified Kerre’s Inequality Kerre’s inequality is recognized as a highly efficient method for assessing fuzzy number comparisons [32]. Ghanbari et al. [10] introduced a modified version of Kerre’s inequality and established straightforward formulas for comparing fuzzy triangular numbers, as stated in the following theorem. Theorem 2 [10] Consider �̃� = (𝑎/𝑏/𝑐) and �̃� = (𝑎′/𝑏′/𝑐′) as two triangular fuzzy numbers with 𝑏 ≤ 𝑏′. Then the following assertions hold: 1 . If 𝑐 ≤ 𝑎′, then 𝑟(�̃�, �̃�) = 𝑐′−𝑎′ 2 + 𝑐−𝑎 2 . (3 ) 2 . If 𝑏 = 𝑏′, then 𝑟(�̃�, �̃�) = 𝑐′+𝑎′ 2 − 𝑐+𝑎 2 . (4 ) 7806 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 7802-7812, 2024 DOI: 10.55214/25768484.v8i6.3708 © 2024 by the authors; licensee Learning Gate 3 . If 𝑏 < 𝑏′ and 𝑎′ < 𝑐 then 𝑟(�̃�, �̃�) = 𝑐′−𝑎′ 2 + 𝑐−𝑎 2 − �̅�(𝑐 − 𝑎′), (5 ) where �̅� = 𝑀𝑅(�̅�) = 𝑁𝐿(�̅�) in which �̅� is the length of the intersection point of 𝑀𝑅 and 𝑁𝐿 and defined as follows: �̅� = 𝑏′𝑐−𝑏𝑎′ (𝑏′−𝑎′)+(𝑐−𝑏) . So, �̅� in (5) is defined as follows: �̅� = (𝑐−𝑎′) (𝑏′−𝑎′)+(𝑐−𝑏) . (6 ) Note 2 If 𝑟(�̃�, �̃�) ≥ 0 then �̃� ≤ �̃�, otherwise �̃� ≥ �̃� [10 .] Note 3 Throughout our work, <, >, and = on fuzzy numbers are defined based on our modified Kerre’s inequality, and so we show them by <𝐾, >𝐾, and =𝐾, respectively [10]. 3. The Proposed Model We consider the following fractional programming problem: max �̃�𝑇𝑥+�̃� 𝑐𝑇𝑥+𝛽 (𝐹𝐹𝑁𝐿𝑃𝑃) 𝑠. 𝑡. (7 ) 𝐴𝑥 ≤ 𝑏, 𝑥 ≥ 0, where, �̃� ∈ 𝐹(ℝ𝑛), �̃� ∈ 𝐹(ℝ), 𝑐 ∈ ℝ𝑛, 𝛽 ∈ ℝ, �̃� and �̃� are 𝐿𝑅 fuzzy numbers and 𝑐𝑇𝑥 + 𝛽 > 0 for any feasible solution 𝑥. Let 𝑧 = 1 𝑐𝑇𝑥+𝛽 and 𝑦 = 𝑧𝑇𝑥. By getting the ideas from [4], the problem (7) can be converted to the following problem: max �̃�𝑇𝑦 + �̃�𝑧, 𝑠. 𝑡. (8 ) 𝐴𝑦 − 𝑏𝑧 ≤ 0, 𝑐𝑇𝑦 + 𝛽𝑧 = 1, 𝑦, 𝑧 ≥ 0, We can rewrite the problem (8) to the vector form as follows: max [�̃�, �̃�] [ 𝑦 𝑧 ] , 𝑠. 𝑡., (9 ) [𝐴, −𝑏𝑇] [ 𝑦 𝑧 ] ≤ 0, [𝑐𝑇 , 𝛽] [ 𝑦 𝑧 ] = 1[𝑦, 𝑧] ≥ 0, If we suppose 𝑋 = [ 𝑦 𝑧 ], �̃� = [�̃�, �̃�], 𝐵 = [𝐴, −𝑏𝑇] and 𝐻 = [𝑐𝑇 , 𝛽] then we have , max 𝑓(𝑋) = �̃�𝑋, 𝑠. 𝑡. (10 ) 𝐵𝑋 ≤ 0, 𝐻𝑋 = 1𝑋 ≥ 0. By solving the problem (10), we can find the so lution of the problem (7). Example 1 Consider the following programming problem: max (−43/−3/0)𝑥1+(−76/−48/45)𝑥2+(−22/15/41) 14𝑥1+55𝑥2+9 , 𝑠. 𝑡., (11 ) 100𝑥1 + 59𝑥2 ≤ 359,24𝑥1 + 41𝑥2 ≤ 229, 𝑥1, 𝑥2 ≥ 0. Let, 𝑧 = 1 14𝑥1+55𝑥2+9 and 𝑦 = 𝑧𝑥. So the problem (11) convert to the following problem: max (−43/−3/0)𝑦1 + (−76/−48/45)𝑦2 + (−22/15/41)𝑧, 𝑠. 𝑡., (12 ) 100𝑦1 + 59𝑦2 − 359𝑧 ≤ 0,24𝑦1 + 41𝑦2 − 229𝑧 ≤ 0,14𝑦1 + 55𝑦2 + 9𝑧 = 1, 𝑦1, 𝑦2, 𝑧 ≥ 0. 7807 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 7802-7812, 2024 DOI: 10.55214/25768484.v8i6.3708 © 2024 by the authors; licensee Learning Gate Finally, we can rewrite the problem (12) to the following problem : max ((−43, −3,0), (−76, −48,45), (−22,15,41)) ( 𝑦1 𝑦2 𝑧 ) , 𝑠. 𝑡., (13 ) [14,55,9] ( 𝑦1 𝑦2 𝑧 ) = 1 ( 100 59 −395 24 41 −229) ( 𝑦1 𝑦2 𝑧 ) ≤ 0, 𝑦1, 𝑦2, 𝑧 ≥ 0. In the next section we propose VNS algorithm for solving the problem (10). 4. Vns Algorithm for Solving Fuzzy Fractional Programming Problem with Fuzzy Objective Parameter The VNS algorithm was initially introduced by Mladenovic and Hansen [18]. It is a local search approach that systematically explores the search space by introducing deliberate modifications to neighbourhood structures. Considering the demonstrated effectiveness of VNS and its extensions in addressing combinatorial and continuous optimization problems [18], we propose the utilization of VNS. In each iteration, a specific neighborhood structure, denoted as k, is chosen following the sequence outlined in [10]. A random neighbor s^' is generated in this neighborhood. Afterwards, a descent method is applied to s^'. If the best solution found by descent method, s^", is better than the best known solution, it is updated and the neighborhood structure is set to the first one. Otherwise, the search continues in the following neighborhood structure. When we explore the last neighborhood structure kmax, the search return to the first neighborhood. This process continues until a stop condition is reached. Some successful examples of application of VNS can also be found in [7, 30, 31]. Based on the definition of the objective function in the (𝐹𝐹𝑁𝐿𝑃𝑃), the value of the objective function for each feasible solution is a fuzzy number. Thus, to compare the solutions in the proposed algorithm (e.g., Step 1-2-3), we use modified Kerre’s inequality in Theorem 2. Using our modified Kerre’s inequality, the fuzzy optimization problem is solved directly without changing it to a crisp problem. Algorithm 1 FVNS for solving 𝑭𝑭𝑵𝑳𝑷𝑷 Inputs: Neighboring structures 𝑵𝒕 (𝒕 = 𝟏, 𝟐, ⋯ , 𝒕𝐦𝐚𝐱), an initial feasible solution 𝒙𝟎 and 𝒎𝒂𝒙𝒊𝒕𝒆𝒓 (maximum number of iterations). [output: ] Return 𝒙∗. [1. ] While 𝒌 < 𝒎𝒂𝒙𝒊𝒕𝒆𝒓 do : 1 . 𝒕 = 𝟏 . 2 . If 𝒕 < 𝒕𝐦𝐚𝐱 then : (a) Select one point in the neighborhood of 𝒙𝟎 and name it to be 𝒙𝟏 (𝒙𝟏 ∈ 𝑵𝒕(𝒙𝟎)) . (b) Apply local search on 𝒙𝟏 and name the new point as 𝒙𝟐 . (c) if �̃�(𝒙𝟐) ≤𝑲 �̃�(𝒙𝟎) then let 𝒕 = 𝒕 + 𝟏 and go to 1-2, else go to 1-3 . 3 . 𝒙𝟎 = 𝒙𝟐 . 4 . 𝒌 = 𝒌 + 𝟏 . Details of the steps involved in algorithm 1 are described below . 1 . Neighboring structure [10]: We define neighboring structure as follows : ||𝑥′ − 𝑥0|| < 𝑡𝜖, (14 ) 7808 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 7802-7812, 2024 DOI: 10.55214/25768484.v8i6.3708 © 2024 by the authors; licensee Learning Gate for 𝑡 = 1, ⋯ , 𝑡max, where 𝑥′ is in the 𝑡-th neighborhood of 𝑥0 and 𝜖 is an arbitrary parameter . 2 . Selection of a point in the neighborhood of 𝑥0: Let 𝑥′ be a point in the neighborhood of 𝑥0. Then, 𝑥′ is to satisfy in the following conditions : (a) (𝑡 − 1)𝜖 < ||𝑥′ − 𝑥0|| ≤ 𝑡𝜖 . (b) 𝐵𝑥′ ≤ 0 . (c) 𝐻𝑥′ = 1 . (d) 𝑥′ ≥ 0 . see details in [10 .] 3 . Local search[10]: Let 𝑥𝑘 represent a feasible solution during iteration 𝑘, we want to find a feasible solution with larger value of the objective function with respect to 𝑥𝑘. In other words, we must find 𝑠𝑘 such that �̃�𝑇(𝑥𝑘 + 𝑠𝑘) ≥𝑘 �̃�𝑇𝑥𝑘, where, { �̃�𝑇(𝑥𝑘 + 𝑠𝑘) = (𝐷𝐿𝑇 (𝑥𝑘 + 𝑠𝑘)/𝐷𝑇(𝑥𝑘 + 𝑠𝑘)/𝐷𝑅𝑇 (𝑥𝑘 + 𝑠𝑘)), �̃�𝑇𝑥𝑘 = (𝐷𝐿𝑇 𝑥𝑘/𝐷𝑇𝑥𝑘/𝐷𝑅𝑇 𝑥𝑘). Since the given results for the modified Kerre’s method depend on the mean value of the fuzzy numbers, to find 𝑠𝑘 we consider the following three cases: Case 1: 𝐷𝑇𝑥𝑘 < 𝐷𝑇(𝑥𝑘 + 𝑠𝑘) , Case 2: 𝐷𝑇(𝑥𝑘 + 𝑠𝑘) < 𝐷𝑇𝑥𝑘 , Case 3: 𝐷𝑇(𝑥𝑘 + 𝑠𝑘) = 𝐷𝑇𝑥𝑘 . (a) Let 𝐷𝑇𝑥𝑘 < 𝐷𝑇(𝑥𝑘 + 𝑠𝑘), based on Note 2, we must have 𝑟(�̃�𝑇𝑥𝑘 , �̃�𝑇(𝑥𝑘 + 𝑠𝑘)) ≥ 0, and thus we propose problem (𝑃1) as follows: max 𝑟(�̃�𝑇(𝑥𝑘), �̃�𝑇(𝑥𝑘 + 𝑠𝑘)), 𝑠. 𝑡. (15 ) (𝑃1) 𝐷𝑇𝑠𝑘 ≥ 0, 𝐵𝑠𝑘 ≤ −𝐵𝑥𝑘 , 𝐻𝑠𝑘 = 1 − 𝐻𝑥𝑘 , −𝑠𝑘 ≤ 𝑥𝑘 . Note that replacing the constraint 𝐷𝑇𝑠𝑘 > 0 with 𝐷𝑇𝑠𝑘 ≥ 0 has no impact on the solution of (𝑃1) . i. Let (�̃�𝑇𝑥𝑘)𝑅 and (�̃�𝑇(𝑥𝑘 + 𝑠𝑘))𝐿 have an intersection point. In this case, we have 𝐷𝑇𝑥𝑘 ≤ 𝐷𝑇(𝑥𝑘 + 𝑠𝑘) and 𝐷𝐿𝑇 (𝑥𝑘 + 𝑠𝑘) ≤ 𝐷𝑅𝑇 𝑥𝑘, and (15) turns into (𝑃1 ′) as follows: max (𝐷𝑅 − 𝐷𝐿)𝑇𝑥𝑘 + (𝐷𝑅−𝐷𝐿)𝑇 2 𝑠𝑘 − ((𝐷𝑅−𝐷𝐿)𝑇𝑥𝑘−𝐷𝐿𝑇 𝑠𝑘)2 (𝐷𝑅−𝐷𝐿)𝑇𝑥𝑘+(𝐷−𝐷𝐿)𝑇𝑠𝑘 , 𝑠. 𝑡., (16 ) (𝑃1 ′) − 𝐷𝑇𝑠𝑘 ≤ 0, 𝐷𝐿𝑇 𝑠𝑘 ≤ (𝐷𝑅 − 𝐷𝐿)𝑇𝑥𝑘, 𝐵𝑠𝑘 ≤ −𝐵𝑥𝑘 , 𝐻𝑠𝑘 = 1 − 𝐻𝑥𝑘 , −𝑠𝑘 ≤ 𝑥𝑘 . Note that (𝑃1 ′) is a quadratic fractional programming problem. By using a method that is proposed in [9] (Theorem and Section 3) we solve problem (𝑃1 ′) . ii. 𝐷𝑅𝑇 𝑥𝑘 ≤ 𝐷𝐿𝑇 (𝑥𝑘 + 𝑠𝑘), that means two numbers do not have any intersection point (see (3)). By using (3) we have the following program: max (𝐷𝑅 − 𝐷𝐿)𝑇𝑥𝑘 + (𝐷𝑅−𝐷𝐿)𝑇 2 𝑠𝑘 , 𝑠. 𝑡., (17 ) (𝑃2) − 𝐷𝑇𝑠𝑘 ≤ 0, −𝐷𝐿𝑇 𝑠𝑘 ≤ (𝐷𝐿 − 𝐷𝑅)𝑇𝑥𝑘, 𝐵𝑠𝑘 ≤ −𝐵𝑥𝑘 , 𝐻𝑠𝑘 = 1 − 𝐻𝑥𝑘 , −𝑠𝑘 ≤ 𝑥𝑘 . After removing the constant terms, we have: max (𝐷𝑅−𝐷𝐿)𝑇 2 𝑠𝑘 , 𝑠. 𝑡., (18 ) (𝑃2 ′) − 𝐷𝑇𝑠𝑘 ≤ 0, −𝐷𝐿𝑇 𝑠𝑘 ≤ (𝐷𝐿 − 𝐷𝑅)𝑇𝑥𝑘, 𝐵𝑠𝑘 ≤ −𝐵𝑥𝑘 , 𝐻𝑠𝑘 = 1 − 𝐻𝑥𝑘 , −𝑠𝑘 ≤ 𝑥𝑘 . 7809 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 7802-7812, 2024 DOI: 10.55214/25768484.v8i6.3708 © 2024 by the authors; licensee Learning Gate (b) Let 𝐷𝑇𝑥𝑘 > 𝐷𝑇(𝑥𝑘 + 𝑠𝑘). In this case if 𝑟(�̃�𝑇(𝑥𝑘 + 𝑠𝑘), �̃�𝑇𝑥𝑘) ≤ 0 then �̃�𝑇(𝑥𝑘 + 𝑠𝑘) >𝑘 �̃�𝑇𝑥𝑘. So, we propose the following problem: min 𝑟(�̃�𝑇(𝑥𝑘 + 𝑠𝑘), �̃�𝑇𝑥𝑘), 𝑠. 𝑡., (19 ) (𝑃3) 𝐷𝑇𝑠𝑘 ≤ 0, 𝐵𝑠𝑘 ≤ −𝐵𝑥𝑘 , 𝐻𝑠𝑘 = 1 − 𝐻𝑥𝑘 , −𝑠𝑘 ≤ 𝑥𝑘 . Note that replacing the constraint 𝐷𝑇𝑠𝑘 < 0 with 𝐷𝑇𝑠𝑘 ≤ 0 has no impact on the solution of (𝑃3) . i. Let (�̃�𝑇(𝑥𝑘 + 𝑠𝑘))𝑅 and (�̃�𝑇𝑥𝑘)𝐿 have an intersection point. In this case, we have 𝐷𝑇𝑥𝑘 ≥ 𝐷𝑇(𝑥𝑘 + 𝑠𝑘) and 𝐷𝐿𝑇 𝑥𝑘 ≤ 𝐷𝑅𝑇 (𝑥𝑘 + 𝑠𝑘), the problem (19) turns into (𝑃3 ′) as follows: min (𝐷𝑅 − 𝐷𝐿)𝑇𝑥𝑘 + (𝐷𝑅−𝐷𝐿)𝑇 2 𝑠𝑘 − ((𝐷𝑅−𝐷𝐿)𝑇𝑥𝑘+𝐷𝑅𝑇 𝑠𝑘)2 (𝐷𝑅−𝐷𝐿)𝑇𝑥𝑘+(𝐷𝑅−𝐷)𝑇𝑠𝑘 , 𝑠. 𝑡., (20 ) (𝑃3 ′) 𝐷𝑇𝑠𝑘 ≤ 0, −𝐷𝑅𝑇 𝑠𝑘 ≤ (𝐷𝑅 − 𝐷𝐿)𝑇𝑥𝑘𝐵𝑠𝑘 ≤ −𝐵𝑥𝑘 , 𝐻𝑠𝑘 = 1 − 𝐻𝑥𝑘 , −𝑠𝑘 ≤ 𝑥𝑘 . By using a method that is proposed in [9] (Theorem and Section 3) we solve problem (𝑃3 ′) . ii. 𝐷𝑅𝑇 (𝑥𝑘 + 𝑠𝑘) < 𝐷𝐿𝑇 𝑥𝑘, that means two numbers do not have any intersection point (see (3)). Since 𝐷𝑅𝑇 (𝑥𝑘 + 𝑠𝑘) < 𝐷𝐿𝑇 𝑥𝑘 and �̃�𝑇(𝑥𝑘 + 𝑠𝑘) is located completely on the left side of �̃�𝑇𝑥𝑘 and according to (5) 𝑟(�̃�𝑇(𝑥𝑘 + 𝑠𝑘), �̃�𝑇(𝑥𝑘)) > 0, so there is no increasing direction. (c) Let 𝐷𝑇𝑥𝑘 = 𝐷𝑇(𝑥𝑘 + 𝑠𝑘). So, according to (4), this case is divided into two cases : i. Suppose 𝑎 = 𝐷𝐿𝑇 𝑥𝑘, 𝑐 = 𝐷𝑅𝑇 𝑥𝑘, 𝑎′ = 𝐷𝐿𝑇 (𝑥𝑘 + 𝑠𝑘) and 𝑐′ = 𝐷𝑅𝑇 (𝑥𝑘 + 𝑠𝑘) in (4 :) max (𝐷𝑅+𝐷𝐿)𝑇 2 𝑠𝑘 , 𝑠. 𝑡., (21 ) (𝑃4) 𝐷𝑇𝑠𝑘 = 0, 𝐵𝑠𝑘 ≤ −𝐵𝑥𝑘 , 𝐻𝑠𝑘 = 1 − 𝐻𝑥𝑘 , −𝑠𝑘 ≤ 𝑥𝑘 . ii. Suppose 𝑎 = 𝐷𝐿𝑇 (𝑥𝑘 + 𝑠𝑘), 𝑐 = 𝐷𝐿𝑇 (𝑥𝑘 + 𝑠𝑘), 𝑎′ = 𝐷𝐿𝑇 𝑥𝑘, 𝑐′ = 𝐷𝑅𝑇 𝑥𝑘, in (4 :) min − (𝐷𝑅+𝐷𝐿)𝑇 2 𝑠𝑘 , 𝑠. 𝑡., (22 ) (𝑃5) 𝐷𝑇𝑠𝑘 = 0, 𝐵𝑠𝑘 ≤ −𝐵𝑥𝑘 , 𝐻𝑠𝑘 = 1 − 𝐻𝑥𝑘 , −𝑠𝑘 ≤ 𝑥𝑘 . 4. Stopping condition[10]: We stop the proposed FVNS algorithm when reach maxiter successive iterati ons without improvement. Note 4 The (21) and (22) problems have the same solution, so you can find the increasing direction by solving either one. Finally, to find s_k, we need to solve for P_1^', P_2, P_3^', and P_4. It is clear that P_1^', P_2, P_3^' and P_4 can be solved in parallel. After solving these problems, choose the best s_k. That is, the one given by P_1^', P_2, P_3^' and P_4 that causes the largest increase in the objective function. Example 2 Suppose the fuzzy fractional linear programming problem as follows : max 𝑓(𝑥) = (−97,−73,−31)𝑥1+(−37,−27,9)𝑥2+(−99,−27,42)𝑥3+(−77,−53,14) 64𝑥1+3𝑥2+12𝑥3+82 , 𝑠. 𝑡. (23 ) 86𝑥1 + 11𝑥2 + 86𝑥3 ≤ 280,73𝑥1 + 90𝑥2 + 17𝑥3 ≤ 343, 𝑥1, 𝑥2, 𝑥3 ≥ 0. With x_0 as the starting point (which can be generated randomly in the interval [1,5]), using Algorithm 4, after one iteration we obtain: 𝑥0 = [2,2,1], 𝑓(𝑥0) = [−3.6680, −2.3028,0.0987], 𝑥∗ = [0,0.3575,3.2101], 𝑓(𝑥∗) = [−3.3557, −1.2281,1.2504], 𝑟(𝑓(𝑥0), 𝑓(𝑥∗)) = 1.5517. (24 ) According to Note 2, we can conclude that 𝑓(𝑥∗) >𝑘 𝑓(𝑥0) . 7810 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 7802-7812, 2024 DOI: 10.55214/25768484.v8i6.3708 © 2024 by the authors; licensee Learning Gate 5. Numerical Results In this section, we demonstrate the effectiveness of the algorithm 4 we have developed. To evaluate its performance, we generated numerous test problems for the FFLPP. These test problems were created using a random generation process in the MATLAB 7.0 programming environment, executed on a notebook equipped with an Intel(R) Core(TM) i5-3210M CPU running at 2.5 GHz, with 4.00 GB of RAM. To introduce fuzzy random coefficients into the objective function, we initiated the process by generating three numbers within the range of [- 100,100]. Subsequently, these numbers were sorted in ascending order and employed as the components of the fuzzy number. Table 1. Objective function values obtained by various methods on the examples . 𝐝𝐢𝐦 Obj(init) Obj (A) 1] 𝐫(𝐀𝐥𝟏), 10 × 20 [−12.5611/−6.5882/0.5839] [−4.7204/−3.8596/−2.6984] 4.081016 15 × 45 [−13.3151/−4.8021/3.0413] [−5.7192/−5.0795/−0.4860] -1.7628 20 × 50 [−83.3284/−29.3203/19.0412] [−24.2866/−10.7058/1.1153] 33.5785 25 × 75 [−37.0272/−18.1645/5.1532] [−14.7937/−5.1766/−2.6203] 15.0961 60 × 100 [−117.8211/−42.6375/32.4857] [−40.4461/−36.0957/−25.3402] 15.7779 25 × 200 [−76.6690/−72.7284/−29.6852] [−187.8614/−74.2193/41.3462] 21.5528 40 × 300 [−56.8188/−21.7506/13.0071] [−23.6253/−19.7209/−0.8734] 11.5796 300 × 600 [−222.6613/−88.4877/48.5212] [−86.3218/−81.8225/−77.2927] 11.6141 In Table 1, the dim column shows the dimensions of the test problems, the column entitled Obj (init) shows the value of the objective function corresponding to the initial solution of Algorithm 1, the column entitled Obj (Al4) shows the values of the objective functions obtained by Algorithm 1 and the column entitled r(Al1,) shows the comparison of the value of objective function of initial solution corresponding to Algorithm 1 with the ones obtained by Algorithm4 using (5). According to the obtained results shown in column labeled as r(Al1, ), it is observed (based on Note (2)) that the Algorithm 4 can improve the initial solution. All problems were also solved by the methods due to [27]. Sheikhi and Ebadi in [27] proposed method introduces a transformation technique that converts an FTP with fuzzy numbers into an FFNLPP with crisp numbers by employing the robust ranking technique. Details o the results are shown in Table 2. Table 2. Objective function values obtained by Sheikhi and Ebadi [27] on the examples . dim Obj(Al[27]) r(Obj(Al]), Obj(Al[27])) 10 × 20 [−10.7321/−4.1001/1.2150] 3.2697 15 × 45 [−14.8912/−6.1221/1.7841] 2.6305 20 × 50 [−80.1432/−35.5698/16.1201] 5.7479 25 × 75 [−30.8531/−20.1345/−10.5200] 6.5833 60 × 100 [−120.2510/−50.3217/25.1432] 12.1784 25 × 200 [−110.2002/−80.1214/−10.4879] 13.8171 40 × 300 [−60.1901/−30.3114/20.5417] 5.6262 300 × 600 [−150.1122/−90.3333/−20.5417] 0.0858 In Table 2, the dim column shows the dimension of problems, the column entitled Obj(Al [27]) shows the value of the objective function corresponding to the solution obtained by Algorithm proposed in [27], and the column entitled r(Al1, Al([27])) shows the comparison of the value of objective function corresponding to Algorithm 4 with the ones obtained by the methods using in [27]. According to the 7811 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 7802-7812, 2024 DOI: 10.55214/25768484.v8i6.3708 © 2024 by the authors; licensee Learning Gate obtained results shown in column labeled as r(Al1, Al([27])), it is observed that the objective function values obtained by Algorithm 1 are more significant than the ones due to other methods on all the test problems. The obtained results showed that the objective function values obtained by the proposed algorithm were significant than the ones obtained by the algorithms due to Sheikhi and Ebadi in [27] 6. Conclusions and Future Work In this paper, we considered a fractional fuzzy linear programming problem, with the assumption that the parameters of the objective function are fuzzy numbers, and we presented a new model with the name FFNLPP. We proposed an FVNS algorithm based on modified Kerre’s inequality for solving FFNLPP. 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