In order to resolve the fuzzy linear programming problem, the fuzzy exponential barrier approach is the major strategy employed in this article. To overcome the problems with fuzzy linear programming, this method uses an algorithm. In this concept, a fuzzy inequality constraint is produced since the objective functions are convex.numerical examples are provided

Nagoor gani, A

Yogarani, R

English

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Ratio Mathematica Volume 46, 2023Solving fuzzy linear programmingproblems by using the fuzzy exponentialbarrier methodA.Nagoor gani*R.Yogarani†AbstractIn order to resolve the fuzzy linear programming problem, the fuzzyexponential barrier approach is the major strategy employed in this ar-ticle. To overcome the problems with fuzzy linear programming, thismethod uses an algorithm. In this concept, a fuzzy inequality con-straint is produced since the objective functions are convex.numericalexamples are provided.Keywords:fuzzy exponential barrier function, fuzzy exponential bar-rier convergence,fuzzy optimality solution.2020 AMS subject classifications: 54E20, 54H25. 1*Research Department of Mathematics, Jamal Mohamed College(Autonomous),Affiliated toBharathidasan University,Tiruchirappalli-620020,Tamil Nadu,India. ganijimc@yahoo.co.in.†Research Department of Mathematics, Jamal Mohamed College(Autonomous), Affiliated toBharathidasan University,Tiruchirappalli-620020, Tamil Nadu, India.yogaranimaths@gmail.com.1Received on September 15, 2022. Accepted on December 15, 2022.Published on March 20,2023. DOI: 10.23755/rm.v46i0.1088. ISSN: 1592-7415. eISSN: 2282-8214. ©A.Nagoor gani etal. This paper is published under the CC-BY licence agreement.310A.Nagoor gani and R.Yogarani1 IntroductionFuzzy set theory has aided mathematical modelling and control theory. Un-solved decision-making problems can be applied in real-world situations using avariety of techniques. The fuzzy set theory developed by A. et al. [2012] has pre-sented in 1965. Fuzzy linear programming was first established in 1997 by Tanakaand Asai [1984], and fuzzy numbers were published in Dubois [1983]. A functionwas proposed by Hsieh [1999] to cope with fuzzy arithmetical operations. Using atechnique that divides, adds, and multiplies fuzzy integers in addition to subtract-ing and adding them. The fuzzy linear programming problem, as formulated byMahadevi et al. [2009], A. and Yogarani [2021] and Fiacco and V. [1990] Zimmer-mann [1978] likewise has a duality. The process of defuzzifying involves convert-ing fuzzy values into clear crispvalues. Since a few years ago, these techniqueshave been thoroughly researched and applied in fuzzy systems Mahadevi et al.[2006]. A representative value from a given set, according to some characters,was the main objective of these processes. The defuzzification method establishesa link between all fuzzy sets and all real numbers. Given the high cost ofan in-feasible solution, Moengin and Parwadi [2011] et al. Fiacco and Anthony [1976]developed the exponential barrier function technique. The fuzzy exponential bar-rier approach is a different way to solve fuzzy linear programming problems. Thefuzzy exponential barrier technique includes a fuzzy objective function that spec-ifies a severe penalty for violating the fuzzy requirements in order to approximatefuzzy constraints. a fuzzy barrier function in fuzziness optimization problems thathas a value that grows exponentially with the separation from the viable zone. Thefuzzy exponential barrier parameter, a positive decreasing parameter used in thisprocess, establishes how close to the original fuzzy unconstrained problem is. Insection 2 of this study, some fundamental ideas of fuzzy set theory and the alge-braic operation of triangular fuzzy numbers are provided. A fuzzy exponentialbarrier technique is used to build an algorithm in section 3 to tackle the problemof fuzzy linear programming problem. Section 4 includes an example.2 Preliminaries2.1 : Fuzzy SetIf M̃ = {(x, µ̃M(x)) : x ∈ M,µM̃(x) ∈ [0, 1]} defines a fuzzy set M̃ The pair(x, µM̃(x))that makes up the membership function has two elements: x, which isa member of the classical set M̃ ,and the µM̃(x) belongs to the interval [0,1].2.2 : Triangular fuzzy numberThe triangular fuzzy number is the fuzzy set M̃ = (M1,M2,M3),M1 ≤ M2 ≤311Solving fuzzy linear programming problems by using the fuzzy exponentialbarrier methodM3 ∈ R, if the membership function of M̃ is defined byµM̃(y) =y−M1M2−M1 , M1 ≤ y ≤M2m3−ym3−M2 , M2 ≤ y ≤M30, otherwise3 : Fuzzy exponential barrier methodPrimal-dual of the fuzzy linear programming problemsmax Z̃ = f̃ t(ỹ)subject to Mj ỹ ≥ Ñk, j = 1, 2.Where M1 ∈ Rm×n, f̃ , ỹ ∈ Rn, Ñk ∈ Rmf̃ t = (f1, f2, f3), Ñ = (n1, n2, n3) (1)min Z̃ = Ñ(ỹ)MTj (x̃) ≤ f̃ ,M2 =MT1 ∈ Rm×n, ÑTk , f̃ tt= f̃ ,We may consider the rank of the matrix M1 without losing generality. The tri-angular fuzzy number is used to specify the fuzzy components in the precedingproblem. Consider that every problem has at least one feasible solution (1)The fuzzy exponential barrier function is considered byẼ(ỹ, γ) : Rn → Rfor every scalar δ > 0 as follows.Ẽ(ỹ, γ) = f̃ t(ỹ)− γm∑i=1e(Bi(ỹ))β(2)Consider Ẽ : Rn → (−∞,∞) is a function given byBi(ỹ) =Mj ỹ − Ñk ≤ 0,Mj ỹ − Ñk = 0forallj,Mj ỹ − Ñk > 0,Mj ỹ − Ñk 6= 0forallj.3.1 : Fuzzy exponential barrier lemmaThis lemma is based on the local and global behaviour of the unconstraintsmaxi-mizer of the fuzzy exponential barrier function. Statement: An exponential barrierparameter γk is a fuzzy sequence that increased in size. The fuzzy linear program-ming problems and fuzzy exponential barrier function then for each k conditionare below.(i).Ẽ(ỹk+1, γk+1) ≥ Ẽ(ỹk, γk)(ii).Ẽ(ỹk ≤ Ẽ(ỹk+1)(iii).f̃ t(ỹk) ≥ f̃ t(ỹk+1)(iv).f̃ t(ỹ∗) ≤ Ẽ(ỹk, γk) ≤ f̃ t(ỹk)312A.Nagoor gani and R.YogaraniProof :(i).Ẽ(ỹk+1, γk+1) =f̃ t(ỹk+1)− γk+1m∑i=1e(β(Mj ỹk+1−Ñk)) ≥ f̃ t(ỹk)− γkm∑i=1e(β(Mj ỹk−Ñk))= Ẽ(x̃k, γk)(ii).f̃ t(ỹk)− γkm∑i=1e(β(Mj ỹk+1−Ñk)) ≤ f̃ t(ỹk+1)γk+1m∑i=1e(β(Mj ỹk+1−Ñk)) (3)f̃ t(ỹk+1)− γk+1m∑i=1e(β(Mj ỹk+1−Ñk)) ≤ f̃ t(ỹk)− γkm∑i=1e(β(Mj ỹk−Ñk)) (4)The inequalities adding (3) and (4), we getf̃ t(ỹk)− γkm∑i=1e(β(Mj ỹk−Ñk))+f̃ t(ỹk+1)−m∑i=1e(βk+1(Mj ỹk−Ñk)) ≤ f̃ t(ỹk+1)−m∑i=1e(βk+1(Mj ỹk+1−Ñk))+ f̃ t(ỹk)− γkm∑i=1e(β(Mj ỹk−Ñk))...(f̃ t(ỹk)+ f̃ t(ỹk+1))− (f̃ t(ỹk+1)+ (f̃ t(ỹk)) + γkm∑i=1e(βk(Mj ỹk−Ñk))+ γk+1m∑i=1e(βk+1(Mj ỹk−Ñk)))≤ γkm∑i=1e(βk(Mj ỹk−Ñk))+ γk+1m∑i=1e(βk+1(Mj ỹk−Ñk))).However (f̃ t(ỹk) + f̃ t(ỹk+1))− (f̃ t(x̃k+1) + f̃ t(x̃k)) = 0.Then (γk+γk+1)γk∑mi=1 e(βk(Mj ỹk−Ñk)) ≥ (γk+γk+1)∑mi=1 e(βk+1(Mj ỹk−Ñk))313Solving fuzzy linear programming problems by using the fuzzy exponentialbarrier methodSince δk ≤ δk+1, Ẽ(ỹk+1) ≥ Ẽ(ỹk).(iii). From the proof of (i), it can be obtained thatf̃ t(ỹk)− γk∑mi=1 e(β(Mj ỹk−Ñk)) ≥ f̃ t(ỹk+1)− γk+1∑mi=1 e(β(Mj ỹk+1−Ñk)).Ẽ(ỹk) ≤ Ẽ(ỹk+1). Then f̃ t(ỹk+1) ≥ f̃ t(ỹk).(iv). From the proof of (ii)∑mi=1 e(β(Mj ỹk−Ñk)) ≤∑mi=1 e(β(Mj ỹk−Ñk)), f̃ t(ỹk) ≥f̃ t(ỹk+1).f̃ t(ỹ∗) ≥ f̃ t(ỹk)− γkm∑i=1e(β(Mj ỹk−Ñk)) = Ẽ(ỹk, γk) ≥ f̃ t(ỹk).3.2 : A Fuzzy exponential barrier convergence theoremSuppose that the fuzzy linear programming problem and the fuzzy exponentialbarrier functions are both continuous functions. In a fuzzy linear programmingproblem, an increasing series of positive fuzzy exponential barrier parameters k isnecessary such that {γk} → ∞, k → ∞ to ỹ of z̃there is an optimal solution. Ifthe limit pointxis within the boundaries of the range of ỹof z̃, then it exists.Proof :Let ỹ be a limit point of {ỹk}Ẽ(ỹk, γ̃k) = f̃ t(ỹk)− γkm∑i=1e(β(Mj ỹk−Ñk)) ≤ f̃ t(ỹ∗)limk→∞f̃ t(ỹk) = f̃ t(ỹ), limk→∞e(β(Mj ỹk−Ñk)) ≤ 0From the lemma 3.2 (iv) we get,limk→∞Ẽ(ỹk, γ̃k) = f̃ t(y∗)ỹ is feasible.3.2 fuzzy exponential barrier function algorithm1. Acknowledge the fuzzy objective function and constraints of the problem andrepharse the problem standard forms to reflect them. write max z̃ = f̃ t(ỹ). Sub-ject to Mj ỹk − Ñk ≤ 0.2.The fuzzy exponential barrier function defined by314A.Nagoor gani and R.YogaraniẼ(ỹ, γ̃) = f̃ t(ỹ)− γm∑i=1e(β(Mj ỹk−Ñk)).3.3. Maxinequality constraints with the fuzzy exponential barrier functionsmax Ẽ(ỹ, γ̃) = f̃ t(ỹ)− γm∑i=1e(β(Mj ỹk−Ñk)).4.First-order needed conditions for optimality γ →∞, which are used to findthe best solution to the specified fuzzy linear programming problems.5.Find Ẽ(ỹk, γ̃k) = maxx≥0 Ẽ(ỹ, γ̃), then minimize ỹk and γ = 1, k =1, 2, ......, k = I then stop.Alternatively, cancontinue to step 5.The fuzzy exponential barrier approach employs the same procedure as the earlieralgorithm for the dual fuzzy linear programming problems.4. Numerical example The primal fuzzy linear programming problemmin z̃ = (3.75, 4, 4.25)ỹ1 + (2.75, 3, 3.25)ỹ22ỹ1 + 3ỹ2 ≥ (5.75, 6, 6.25),4ỹ1 + ỹ2 ≥ (3.75, 4, 4.25).Solution : Figure 1: Finding an infeasible solution for primal fuzzy linear pro-gramming problem Using the fuzzy exponential barrier function algorithm we get,The fuzzy exponential barrier method is given byẼ(ỹ, γ) = ((3.75, 4, 4.25)ỹ1 + (2.75, 3, 3.25)ỹ2 − γm∑i=1e(Bi(y))βAccording to this strategy, you can convert fuzzy linear programming problemsinto a standard form of unconstrained problems.ỹ1 = (0.45, 0.8, 1.15)−1.2104γ, ỹ2 = (0.0727, 0.6, 1.1273)−1.3951γx̃1 =0.0060γ+ (0.35, 0.60, 0.85), x̃2 = (0.9518, 1.60, 2.25) +0.0180γ,315Solving fuzzy linear programming problems by using the fuzzy exponentialbarrier methodFor different values of γj, j → ∞ the optimal values of ỹ1(γ), ỹ2(γ) are calcu-lated as listed in Tables (1) and (2) we get,Table1: Primal fuzzy linear programming problem solution(i)No δk ỹ1 ỹ21 10 (0.3290,0.6790,1.0290) (-0.0668,0.4605,0.9878)2 102 (0.4379,0.7879,1.1379) (0.0587,0.5860,1.1133)3 103 (0.4488,0.7988,1.1488) (0.0713,0.5986,1.1259)4 104 (0.4499,0.7999,1.1499) (0.0726,0.5999,1.1272)5 105 (0.45,0.8,1.15) (0.0727,0.6,1.1273)Table 2: Dual fuzzy linear programming problem solution(i)No γj ỹ1 ỹ21 10 (0.3506,0.6006,0.8506) (0.9518,1.6018,2.2518)2 102 (0.3501,0.6001,0.8501) (0.9502,1.6002,2.2502)3 103 (0.3500,0.600,0.8500) (0.9500,1.600,2.2500)4 104 (0.3500,0.600,0.8500) (0.9500,1.600,2.2500)5 105 (0.3500,0.600,0.8500) (0.9500,1.600,2.2500)The optimal value of the given problem (1) can be obtained byỹ1 = (0.45, 0.8, 1.15), ỹ2 = (0.0727, 0.6, 1.1273),min z̃ = (2.9908, 7.2, 11.4092).The corresponding given problem of the optimal values for x̃1, x̃2 are to be ob-tained byx̃1 = (0.35, 0.6, 0.85), x̃2 = (0.95, 1.6, 2.25),max z̃ = (4.25, 7.2, 10.15).3 ConclusionsThis article outlines a method for solving the primal-dual fuzzy linear pro-gramming problem more effectively by combining the fuzzy exponential barrierfunction and the fuzzy exponential barrier parameter.The table for the primal-dualproblem demonstrates how quickly the best solution can be reached using thefuzzy exponential barrier primal-dual algorithm we developed when the FuzzyExponential barrier parameter is used.316A.Nagoor gani and R.YogaraniFigure 1: Finding an infeasible solution for primal fuzzy linear programmingproblemReferencesN. 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Solving fuzzy linear programming problems by using the fuzzy exponential barrier method

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Solving fuzzy linear programming problems by using the fuzzy exponential barrier method

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