Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 182 https://internationalpubls.com Solving Intuitionistic Fuzzy Unconstrained Optimization Problems Using Interval Newton’s Method S. Shilpa Ivin Emimal1, R. Irene Hepzibah 2* 1&2* (Affiliated to Bharathidasan University), T.B.M.L.College (Post Graduate and Research Department of Mathematics), Porayar 609 307, Tamil Nadu, India E-mail: 1shilpaivins@gmail.com, 2* ireneraj74@gmail.com Article History: Received: 19-04-2024 Revised: 08-06-2024 Accepted: 21-06-2024 Abstract: This research work studies the optimization of intuitionistic fuzzy valued functions in unconstrained problems. The Interval Newton's Method using an Intuitionistic approach addresses both single and multivariable optimization problems. The study incorporates a mathematical comprehension of interval intuitionistic fuzzy valued problems, as well as real- world examples to demonstrate their effectiveness. Furthermore, MATLAB code is provided to demonstrate the implementation of the Interval Newton's Method. Keywords: Unconstrained Optimization Problems, Interval Newton's Meth0d(INM), intuitionistic fuzzy numbers. 1. Introduction This paper focuses on unconstrained optimization problems ἰnvolving Intuitἰonistic Fuzzy-valued functions. It delves into both single-variable and multi-variable scenarios, providing explanations and illustrations to demonstrate the application of the Interval Newt0n Method in tackling these optimization problems. Helmut Ratschek[4]explored interval methods in global optimization, providing solutions for diverse optimization scenarios, including unc0nstrained, constrained, and n0nsmooth optimization. It underscores the importance of bisection techniques in interval-based global optimization algorithms, where the problem domain is recursively divided and also acknowledges advancements in bisection strategies over the last decade. Furthermore, it emphasized the significance of interval arithmetic in handling global optimization problems, as it offers global information compared to local optimization meth0ds that rely on limited, local data. Recently, we have had to optimize provided systems and deal with linear and NLPP. The phrase optimization refers to reaching the best possible outcome. The interval analysis appr0ach emerged between 1950 and 1960. Interval analysis, intr0duced with the devel0pment of c0mputational mathematics, is a meth0d of describing by substituting a sἰngle fἰxed p0int with an ἰnterval. In general, we tackle n0nlinear pr0gramming pr0blems that require only crisp integers in the goal and a limited c0-efficient. When dealing with such natural situations, challenges exist because of uncertainty and inexactness ἰn the derived parameters. Since 0ccurrence of the uncertainty and inexactness, it is inadequate to apply the classic meth0d to handle such problems. Karl Nickel [6]explained the application of the Newt0n technique and interval arithmetic in solving n- dimensional vector problems. It delves into the characteristics of the Newt0n operator, conditions for solution existence, and limitations on error tolerances for convergent sequences. The concept is further illustrated through an example in convex programming, showcasing its real-world implications. Ved https://link.springer.com/chapter/10.1007/978-1-4615-2025-2_14#auth-Helmut-Ratschek Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 183 https://internationalpubls.com P. Madan [15]The Interval Newt0n Meth0d, especially the Hansen-Greenberg Appr0ach, is a technique for solving systems of n0nlinear equations by linearizing the system and making different changes t0 enhance efficiency. This appr0ach uses preconditioning, the Hansen-Sengupta step, and interval iterations to provide more accurate and efficient solutions for solving complicated mathematical problems. The paper's proposed procedural enhancements attempt to increase algorithm performance by adding improved approximations, a sequential overrelaxation methodology, and interval iterations as alternatives to existing methods. The discussed Interval Newt0n Meth0d aims to efficiently solve systems of nonlinear equations by integrating three subalg0rithms: a Gauss-Seidel-type step, a real Newton iteration, and linearized equation solving via elimination. This approach improves upon existing techniques like Krawczyk's method, offering better solution efficiency. The Newt0n method is a numerical method for solving equations that starts with an initial guess and then uses iterative steps to c0nverge t0wards the true solution. It is based on the idea of appr0ximating a function as a straight line tangent to the curve at a given point. To implement the interval Newt0n method, we will make use of interval arithmetic, where the result of an 0peration on intervals is an interval that bounds the possible values of the output. This allows to compute rigorous bounds on the solution of an equation. In this implementation, the `f` and `df` arguments are functions that represent the original function and its derivative, respectively. The `x0` argument is the initial guess for the root of the equation. The `epsilon` argument specifies the desired precision of the solution. The `interval` function is used to compute an interval from a single value. The method iteratively updates the value of `x` using the interval Newt0n iteration formula until the absolute value of `f(x)` is less than `epsilon`, indicating a suitable approximation has been found. The `df interval` is used to compute the slope of the tangent line at each iteration. The impact of employing a local minimizer to update the upper bound on the global minimum is also considered. This approach, named LISS_LP, is demonstrated through various global optimization problems, particularly focusing on those originating from chemical engineering. LISS_LP proves effective in solving problems with a vast number of local optima and those involving a relatively large number of variables. In 2017, Lei Wang described the Newt0n iteration-based interval uncertainty analysis approach for interdisciplinary systems, known as NI-IUAM, this technique divides complicated systems into distinct disciplines and uses Newton iteration equations to calculate the upperI and l0wer limits of connected state variables at each iterative step. NI-IUAM requires simply the boundaries of unknown parameters, removing the requirement for particular distribution formats and perhaps reducing the need for raw data. Additionally, NI-IUAM benefits from Newton iteration's super-linear convergence, which accelerates the convergence process. In 2021, Zhiping f0cused on studying the impact of uncertainties in n0nlinear structural system, a Newt0n iteration-based interval analysis meth0d (NI-IAM) to quantify these uncertainties by considering uncertain-but-b0unded parameters as interval variables. The meth0d simplifies the uncertain 0nlinear pr0blem into a series of uncertain linear issues through interval iteration schemes, TaylorSeriesExpansion, and the LagrangianMultiplierMethod. Numerical examples dem0nstrate the effectiveness and applicability of the prop0sed meth0d, and its compatibility with the perturbati0n-based pr0babilistic method is also studied. https://link.springer.com/article/10.1007/s11433-017-9068-5#auth-Lei-Wang-Aff1 https://link.springer.com/article/10.1007/s11433-017-9068-5#auth-Lei-Wang-Aff1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 184 https://internationalpubls.com Fuzzy Sets, initially introduced by Zadeh in 1965, have significantly impacted various fields due to their ability to represent the ambiguity in real-world scenarios. Triangular Fuzzy Numbers, a crucial aspect of fuzzy set theory, have been discussed in detail, along with their operations. Later, Krassimir Atanassov introduced Intuitionistic Fuzzy Sets in 1983, further expanding upon Zadeh's foundation. In addition to degrees of membership, intuitionistic fuzzy sets also incorporate degrees of non- membership. This means that for each element, there are not only measures of how much it belongs to the set, but also measures of how much it does not belong to the set. This dual perspective offers a more comprehensive representation of uncertainty and ambiguity in real-world situations. Intuitionistic fuzzy sets have found applicati0ns in vari0us fields, including decisi0n-making, pattern rec0gnition, image pr0cessing, medical diagnosis, and more. They provide a powerful framework for modeling and reasoning with uncertain and imprecise information, offering more nuanced solutions to problems where traditional crisp sets or classical fuzzy sets may fall short. Throughout the last few decades, various academics have created 0ptimization pr0blems with fuzzy valued 0bjective issues. Ganesan et al. (2017) presented a c0mparable solutἰ0n to fuzzy unc0nstraἰned optimἰzati0n problems using trἰangular fuzzy numbers (2020). Through0ut the last few decades, different researchers have created 0ptimization pr0blems with fuzzy valued objective problems. [Ben et al. [1]. Debski [2], Dennis and Shanable [3], ChengHand K0valyov [10], Timothy [11], Shi [21]. M Furthermore, unc0nstrained pr0blems are s0lvedHusingHdifferential calculus. Ganesan et al.,[12] propose a novel methodology for addressing single-variable problems without constraints using Newton's interval analysis method, which differs from the standard method of employing calculus.. Sophia porchelviHet al[8] extended to bivariate pr0blems using Newt0n’s meth0d. Irene hepzibah,Shilpa[5] extended into fuzzy single and bivariate unconstrained problems by using fuzzy Newt0n's interval analysis meth0d. This paper presents a novel approach to solving non-linear, unc0nstrained 0ptimization problems using the Intuitionistic Newt0n Meth0d and interval analysis. It intr0duces interval arithmetic operations for addressing intuitionistic fuzzy unconstrained optimization issues. Section 3 focuses on applying Newton's method and Interval Newt0n's method for unc0nstrained 0ptimization, while secti0n 4 provides illustrative examples. The study concludes with some remarks in section 5. 2. Preliminariesl In this section, some important definiti0ns are provided Definition 1: [13]: A fuzzy set à can be expressed as the pair 𝓍, 𝜇�̃�(𝓍), where x is an element from a classical set A and 𝜇�̃�(𝓍) is the membershℹp functi0n associated with x. This functi0n takes values within the interval [0, 1] and indicates x's degree of membershℹp in the fuzzy set A.The fuzzy set à can be symbolized by the notation {𝓍, 𝜇�̃�(𝓍): 𝓍 ∈ 𝜇�̃�(𝓍) ∈ [0,1]}. This notation emphasizes the link between the classical set A and the membershℹp functi0n 𝜇�̃�(𝓍). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 185 https://internationalpubls.com Definition 2: L-U Parametrℹc Representation of Intuiti0nℹstic Trℹangular Fuzzy Number A Intuiti0nistℹc triangular fuzzy number �̃�𝐼 𝐿𝑈 = (𝑚, 𝛼, 𝛽); (𝑚′, 𝛼′, 𝛽′) where 𝑚 = 𝑏, 𝛼 = 𝑏 − 𝑎 ≥ 0, 𝛽 = 𝑐 − 𝑏 ≥ 0.𝑚′ = 𝑏′, 𝛼′ = 𝑏′ − 𝑎′ ≥ 0, 𝛽′ = 𝑐′ − 𝑏′ ≥ 0. 𝜇 �̃� 𝐿𝑅 (x) = { 0 ; −∞ < 𝑥 ≤ 𝑚 ∝ 1 − 𝑚−𝑥 𝛼 ;𝑚 − 𝑑 ≤ 𝑥 < 𝑚 1 − 𝑥−𝑚 𝛽 ; 𝑚 ≤ 𝑥 < 𝑥 + 𝛽 0 ; 𝑚 + 𝛽 ≤ 𝑥 < ∞ 𝛾 �̃� 𝐿𝑈 (x) = { 1 ; −∞ < 𝑥 ≤ 𝑚 ∝ 𝑚−𝑥 𝛼 ;𝑚 − 𝑑 ≤ 𝑥 < 𝑚 𝑥−𝑚 𝛽 ;𝑚 ≤ 𝑥 < 𝑥 + 𝛽 1 ; 𝑚 + 𝛽 ≤ 𝑥 < ∞ 𝛾 �̃� 𝐿𝑈 (x) 1 O m 𝛼 𝛽 DEFINITION 3: RANKING FUNCTION An efficient method for computing Intuℹti0nistic Fuzzy numbers inv0lves the use of a ranking function R, which maps Intuiti0nistℹc Fuzzy numbers (f) to the real line (R) ,ℜ:f(R)→R. This function enables the establishment of a natural order within the set F(R) of Intuitionistic Fuzzy numbers defined on the real number set. With this ranking function, comparisons and ordering of Intuiti0nistic Fuzzy numbers can be effectively performed. R(A) = m+[(𝛽 − 𝛼)]/4+( 4𝑚′ + (𝛽′ − 𝛼′))/4) 2.2 Arithmetic operation for Triangular intuiti0nistic fuzzy Numbers : Let �̃�𝐼 = {(𝑎1, 𝑎2, 𝑎3); (𝑎1 ,, 𝑎2 ′, 𝑎3 ,)} and �̃�𝐼 = {(𝑏1, 𝑏2, 𝑏3); (𝑏1 ,, 𝑏2 , 𝑏3 ,)} are two TIFN , Addition: �̃�𝐼 + �̃�𝐼 = {𝑎1 + 𝑏1, 𝑎2 + 𝑏2, 𝑎3 + 𝑏3); ( 𝑎1 , + 𝑏1 ,, 𝑎2 ′ + 𝑏2 ′, 𝑎3 , + 𝑏3 ,) is also a TIFN Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 186 https://internationalpubls.com Subtraction �̃�𝐼 − �̃�𝐼 = {𝑎1 − 𝑏3, 𝑎2 − 𝑏2, 𝑎3 − 𝑏1); ( 𝑎1 , − 𝑏3 ,, 𝑎2 ′ − 𝑏2 ′, 𝑎3 , − 𝑏1 ,) is also a TIFN Multiplication �̃�𝐼 . �̃�𝐼 = (𝑎1. 𝑅(�̃� 𝐼), 𝑎2. 𝑅(�̃� 𝐼), 𝑎3. 𝑅(�̃� 𝐼)); (𝑎1 ,. 𝑅(�̃�𝐼), 𝑎2 ′. 𝑅(�̃�𝐼), 𝑎3 ,. 𝑅(�̃�𝐼)) is also a TIFN, where 𝑅(�̃�𝐼) = (𝑎1 + [(𝑎2 − 𝑎1)]/4) + ( 4𝑎1′ + (𝑎2′ − 𝑎3′))/4) Division �̃�𝐼/�̃�𝐼 = (𝑎1/𝑅(�̃� 𝐼), 𝑎2/𝑅(�̃� 𝐼), 𝑎3/𝑅(�̃� 𝐼)); (𝑎1 ,/𝑅(�̃�𝐼), 𝑎2 /𝑅(�̃�𝐼), 𝑎3 ,/𝑅(�̃�𝐼)) is also a TIFN, where 𝑅(�̃�𝐼) = (𝑎1 + [(𝑎2 − 𝑎1)]/4) + ( 4𝑎1′ + (𝑎2′ − 𝑎3′))/4) 2.3 INTERVAL VALUED INTUITI0NISTIC FUZZY NUMBER A Interval valued Intuiti0nℹstic fuzzy number �̃�𝐼 is pair of [𝑢, �̅� ] of 𝑢(𝑟), �̅�(𝑟); 0 ≤ 𝑟 ≤ 1 which satisfies the following conditi0ns: Stefen et al enhancing the efficiency of a n0nlℹnear-system-s0lver using a Comp0nentwℹse Newt0n Meth0d, intr0duce impr0vements to the meth0d, such as utilizing ℹndex- lists, combining with an Interval Newt0n Gauss-Seℹdel Step, and verifying solution unℹqueness and it pr0vides alg0rithmic descriptions, pr0perties 0f the meth0d, examples, and results to dem0nstrate the effectiveness of the pr0p0sed appr0ach in solving n0nlℹnear equations Let �̃� = [𝑣(𝑟), �̅�(𝑟)] x > 0; 𝑥 = [𝑥𝑣(𝑟), 𝑥�̅�(𝑟)] and x < 0; 𝑥 = [ 𝑥�̅�(𝑟), x𝑣(𝑟)] 3. Newt0n’s Meth0d For Unc0nstraℹned Intuiti0nℹstic Fuzzy Optℹmizati0n Now we consider an unc0nstrained ℹntuiti0nistic fuzzy 0ptimℹzation pr0blem 𝑥∈𝑋 𝑚𝑖𝑛�̃�(𝑥) where �̃�: 𝑋 → 𝐹(ℜ) is a fuzzy valued functi0n defined on 𝑋 ⊆ ℜ𝑛. The Newt0n meth0d is an iterative appr0ach for determining the r00t of a function f(x). Following an ℹnitial estimate to the root x_0, better guesses are provided. The Newt0n approach for single variable 0ptimization[12] is used to find the root. the Newt0n meth0d can be adapted to solve unc0nstraℹned Intuiti0nℹstic Fuzzy Optimizati0n pr0blems, taking into account the unique characteristics of Intuiti0nistic Fuzzy numbers and their associated membership functions. 𝑥𝑘+1 = 𝑥𝑘 − [ 𝑓′(𝑥𝑘) 𝑓′′(𝑥𝑘) ] 𝑢𝑛𝑡𝑖𝑙 |𝑥𝑘+1 − 𝑥𝑘| <∈ Newton method for tw0 variable 0ptimization[5] for 0btaining the root, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 187 https://internationalpubls.com Newton's approach for determining a function's critical points requires the existence of its first and second derivatives. Newton's approach to finding critical p0ints of a functi0n involves analyzing the behavi0r of its first and sec0nd derivatives. Critical points are l0cations where the function's rate of change, represented by the first derivative (also known as the derivative or slope), either changes its sign or becomes zero. To determine these points, we need the first derivative to identify where changes in the function's rate of change occur. Additionally, Newton's method requires the second derivative (the derivative of the first derivative) to assess the concavity of the function at these critical points. If the second derivative is positive, the function is concave up, and if it's negative, the function is concave down. This information helps in classifying critical points as local maxℹma (concave up and negative first derivative), local minℹma (concave down and positive first derivative), or points of inflection (concave up and positive or concave down and negative first derivative) If the second-order derivatives at iteration 𝑥𝑖+1are zero, the technique fails. The Newton method's initial derivatives are employed only to find the functions' roots. The sec0nd derivatives of the Newt0n method determine the maximum and least of a given functi0n. 3.1 INTERVAL NEWTON METHOD The Interval Newt0n Meth0d shares simℹlarities with the Bℹsection Method, as it also begins with a bracketing ℹnterval. In each iterati0n, it generates smaller and smaller intervals, all b0unded by the intersecti0n with previ0us ℹterati0ns. The f0rmula is as foll0ws 𝑥𝑘+1 = [𝑚(𝑥𝑘) − 𝑓′(𝑚(𝑥𝑘)) 𝑓′′(𝑥𝑘) ] ∩ (𝑥𝑘) (3.1.1) where x’s are the ℹnterval m(x) is the mℹdp0int of interval x, and f is unc0nstrained 0ptℹmization function. For an fuzzy unc0nstraℹned optimizati0n problem then (3.1.1) takes the f0rm �̃�𝑘+1 𝐼 = [𝑚(�̃�𝑘) 𝐼 − 𝑓′(𝑚(�̃�𝑘) 𝐼 𝑓′′(�̃�𝑘) 𝐼 ] ∩ (�̃�𝑘) 𝐼 (3.1.2) �̃�𝑘+1 𝐼 = [(𝑚(�̃�𝑇1) ,𝑚(�̃�𝑇2) , … ,𝑚(�̃�𝑇𝑛)) ; (𝑚(�̃�𝐹1) ,𝑚(�̃�𝐹2) , … ,𝑚(�̃�𝐹𝑛)) ] −[ 𝑓′[(𝑚(�̃�𝑇1) ,𝑚(�̃�𝑇2) ,…,𝑚(�̃�𝑇𝑛)) ;(𝑚(�̃�𝐹1) ,𝑚(�̃�𝐹2) ,…,𝑚(�̃�𝐹𝑛)) ] 𝑓′′([((�̃�𝑇1) ,(�̃�𝑇2) ,…,(�̃�𝑇𝑛)) ;((�̃�𝐹1) ,(�̃�𝐹2) ,…,(�̃�𝐹𝑛)) ]) ] ∩ ([((�̃�𝑇1) , (�̃�𝑇2) , … , (�̃�𝑇𝑛)) ; ((�̃�𝐹1) , (�̃�𝐹2) , … , (�̃�𝐹𝑛)) ]) where n den0tes the ℹntuiti0nistic fuzzy number. If n=3, it is denoted as ℹntuiti0nistic trℹangular fuzzy number, if n=4, it is denoted as ℹntuiti0nistic trapez0idal fuzzy number and so on. �̃�𝑘+1 𝐼 = [(( �̃�𝑇1𝐿 + �̃�𝑇1𝑈 2 ) , ( �̃�𝑇2𝐿 + �̃�𝑇2𝑈 2 ) , … , ( �̃�𝑇𝑛𝐿 + �̃�𝑇𝑛𝑈 2 )) ; (( �̃�𝐹1𝐿 + �̃�𝐹1𝑈 2 ) , ( �̃�𝐹2𝐿 + �̃�𝐹2𝑈 2 ) ,… , ( �̃�𝐹𝑛𝐿 + �̃�𝐹𝑛𝑈 2 )) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 188 https://internationalpubls.com − [ 𝑓′ [ (( �̃�𝑇1𝐿 + �̃�𝑇1𝑈 2 ) , ( �̃�𝑇2𝐿 + �̃�𝑇2𝑈 2 ) ,… , ( �̃�𝑇𝑛𝐿 + �̃�𝑇𝑛𝑈 2 )) ; (( �̃�𝐹1𝐿 + �̃�𝐹1𝑈 2 ) , ( �̃�𝐹2𝐿 + �̃�𝐹2𝑈 2 ) ,… , ( �̃�𝐹𝑛𝐿 + �̃�𝐹𝑛𝑈 2 )) ] 𝑓′′ ([ ((�̃�𝑇1𝐿 , �̃�𝑇1𝑈) , (�̃�𝑇2𝐿 , �̃�𝑇2𝑈) , … , (�̃�𝑇𝑛𝐿, �̃�𝑇𝑛𝑈)) ; ((�̃�𝐹1𝐿 , �̃�𝐹1𝑈) , (�̃�𝐹2𝐿 , �̃�𝐹2𝑈) , … , (�̃�𝐹𝑛𝐿, �̃�𝐹𝑛𝑈)) ]) ] ∩ ([ ((�̃�𝑇1) , (�̃�𝑇2) , … , (�̃�𝑇𝑛)) ; ((�̃�𝐹1) , (�̃�𝐹2) , … , (�̃�𝐹𝑛)) ]) For Bivariate, the f0rmula 𝑋𝑘+1 = 𝑁𝑘((𝑋𝑘, 𝑌𝑘)) ∩ (𝑋𝐾) where 𝑁𝑘((𝑋𝑘, 𝑌𝑘)) = [𝑚(𝑋𝑘) − 𝑓′(𝑚(𝑋𝑘),𝑚(𝑌𝑘)) 𝑓′′((𝑋𝑘,𝑌𝑘) ] 𝑋𝑘+1 = [𝑚(𝑋𝑘) − 𝑓′(𝑚(𝑋𝑘),𝑚(𝑌𝑘)) 𝑓′′(𝑋𝑘,𝑌𝑘) ] ∩ (𝑋𝑘) , 𝑓′′(𝑋𝑘, 𝑌𝑘) ≠ 0 (3.1.3) and 𝑌𝑖+1 = 𝑁𝑘((𝑋𝑘, 𝑌𝑘)) ∩ (𝑌𝐾) where 𝑁𝑘((𝑋𝑘, 𝑌𝑘)) = [𝑚(𝑌𝑘) − 𝑓′(𝑚(𝑋𝑘),𝑚(𝑌𝑘)) 𝑓′′((𝑋𝑘,𝑌𝑘) ] 𝑌𝑘+1 = [𝑚(, 𝑌𝑘) − 𝑓′(𝑚(𝑋𝑘),𝑚(𝑌𝑘)) 𝑓′′((𝑋𝑘,𝑌𝑘) ] ∩ (𝑌𝑘) , 𝑓′′(𝑋𝑘, 𝑌𝑘) ≠ 0 (3.1.4) where x’s are the ℹnterval m(x) is the mℹdp0int of interval x, and f is unc0nstraℹned 0ptimization function. �̃�𝑘+1 𝐼 = ( [ (𝑚(�̃�𝑇1) ,𝑚(�̃�𝑇2) , … ,𝑚(�̃�𝑇𝑛)) ; (𝑚(�̃�𝐹1) , 𝑚(�̃�𝐹2) , … ,𝑚(�̃�𝐹𝑛)) ] − [ 𝑓′ [ (𝑚(�̃�𝑇1, �̃�𝑇1) ,𝑚(�̃�𝑇2, �̃�𝑇2) , … ,𝑚(�̃�𝑇𝑛, �̃�𝑇𝑛)) ; (𝑚(�̃�𝐹1, �̃�𝑇1) ,𝑚(�̃�𝐹2, �̃�𝑇1) , … ,𝑚(�̃�𝐹𝑛,�̃�𝑇1)) ] 𝑓′′ ([ ((�̃�𝑇1, �̃�𝑇1) , (�̃�𝑇2, �̃�𝑇2) , … , (�̃�𝑇𝑛, �̃�𝑇𝑛)) ; ((�̃�𝐹1, �̃�𝐹1) , (�̃�𝐹2, �̃�𝐹2) , … , (�̃�𝐹𝑛,�̃�𝐹𝑛)) ]) ] ) ∩ [ ((�̃�𝑇1𝐿 , �̃�𝑇1𝑈) , (�̃�𝑇2𝐿 , �̃�𝑇2𝑈) , … , (�̃�𝑇𝑛𝐿, �̃�𝑇𝑛𝑈)) ; ((�̃�𝐹1𝐿, �̃�𝐹1𝑈) , (�̃�𝐹2𝐿 , �̃�𝐹2𝑈) , … , (�̃�𝐹𝑛𝐿, �̃�𝐹𝑛𝑈)) ] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 189 https://internationalpubls.com �̃�𝑘+1 𝐼 = [ [ ( ( �̃�𝑇1𝐿 + �̃�𝑇1𝑈 2 ) , ( �̃�𝑇2𝐿 + �̃�𝑇2𝑈 2 ) , … , ( �̃�𝑇𝑛𝐿 + �̃�𝑇𝑛𝑈 2 ) ) ; ( ( �̃�𝐹1𝐿 + �̃�𝐹1𝑈 2 ) , ( �̃�𝐹2𝐿 + �̃�𝐹2𝑈 2 ) , … , ( �̃�𝐹𝑛𝐿 + �̃�𝐹𝑛𝑈 2 ) ) ] − 𝑓′ ′ [ ( (( �̃�𝑇1𝐿+�̃�𝑇1𝑈 2 ),( �̃�𝑇1𝐿+�̃�𝑇1𝑈 2 )) , (( �̃�𝑇2𝐿+�̃�𝑇2𝑈 2 ),( �̃�𝑇2𝐿+�̃�𝑇2𝑈 2 )),… (( �̃�𝑇𝑛𝐿+�̃�𝑇𝑛𝑈 2 ),( �̃�𝑇𝑛𝐿+�̃�𝑇𝑛𝑈 2 )); ) , ( (( �̃�𝐹1𝐿+�̃�𝐹1𝑈 2 ),( �̃�𝐹1𝐿+�̃�𝐹1𝑈 2 )), (( �̃�𝐹𝑛𝐿+�̃�𝐹𝑛𝑈 2 ),( �̃�𝐹𝑛𝐿+�̃�𝐹𝑛𝑈 2 ))…. (( �̃�𝐹𝑛𝐿+�̃�𝐹𝑛𝑈 2 ),( �̃�𝐹𝑛𝐿+�̃�𝐹𝑛𝑈 2 )) ) ] 𝑓′′ ( ( [ ((�̃�𝑇1, �̃�𝑇1) , (�̃�𝑇2, �̃�𝑇2) , … , (�̃�𝑇𝑛, �̃�𝑇𝑛)) ; ((�̃�𝐹1, �̃�𝐹1), (�̃�𝐹2, �̃�𝐹2) , … , (�̃�𝐹𝑛,�̃�𝐹𝑛)) ] ) ) ] ∩ ([ ((�̃�𝑇1𝐿 , �̃�𝑇1𝑈) , (�̃�𝑇2𝐿 , �̃�𝑇2𝑈) , … , (�̃�𝑇𝑛𝐿, �̃�𝑇𝑛𝑈)) ; ((�̃�𝐹1𝐿, �̃�𝐹1𝑈) , (�̃�𝐹2𝐿 , �̃�𝐹2𝑈) , … , (�̃�𝐹𝑛𝐿, �̃�𝐹𝑛𝑈)) ]) �̃�𝑘+1 𝐼 = ( [ (𝑚(�̃�𝑇1) , 𝑚(�̃�𝑇2) , … ,𝑚(�̃�𝑇𝑛)) ; (𝑚(�̃�𝐹1) ,𝑚(�̃�𝐹2) , … ,𝑚(�̃�𝐹𝑛)) ] − [ 𝑓′ [ (𝑚(�̃�𝑇1, �̃�𝑇1) ,𝑚(�̃�𝑇2, �̃�𝑇2) , … ,𝑚(�̃�𝑇𝑛, �̃�𝑇𝑛)) ; (𝑚(�̃�𝐹1, �̃�𝑇1) ,𝑚(�̃�𝐹2, �̃�𝑇1) , … ,𝑚(�̃�𝐹𝑛,�̃�𝑇1)) ] 𝑓′′ ([ ((�̃�𝑇1, �̃�𝑇1) , (�̃�𝑇2, �̃�𝑇2) , … , (�̃�𝑇𝑛, �̃�𝑇𝑛)) ; ((�̃�𝐹1, �̃�𝐹1) , (�̃�𝐹2, �̃�𝐹2) , … , (�̃�𝐹𝑛,�̃�𝐹𝑛)) ]) ] ) ∩ [ ((�̃�𝑇1𝐿 , �̃�𝑇1𝑈) , (�̃�𝑇2𝐿 , �̃�𝑇2𝑈) , … , (�̃�𝑇𝑛𝐿, �̃�𝑇𝑛𝑈)) ; ((�̃�𝐹1𝐿 , �̃�𝐹1𝑈) , (�̃�𝐹2𝐿 , �̃�𝐹2𝑈) , … , (�̃�𝐹𝑛𝐿, �̃�𝐹𝑛𝑈)) ] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 190 https://internationalpubls.com �̃�𝑘+1 𝐼 = [ [ ( ( �̃�𝑇1𝐿 + �̃�𝑇1𝑈 2 ) , ( �̃�𝑇2𝐿 + �̃�𝑇2𝑈 2 ) , … , ( �̃�𝑇𝑛𝐿 + �̃�𝑇𝑛𝑈 2 ) ) ; ( ( �̃�𝐹1𝐿 + �̃�𝐹1𝑈 2 ) , ( �̃�𝐹2𝐿 + �̃�𝐹2𝑈 2 ) , … , ( �̃�𝐹𝑛𝐿 + �̃�𝐹𝑛𝑈 2 ) ) ] − 𝑓 ′ [ ( (( �̃�𝑇1𝐿+�̃�𝑇1𝑈 2 ),( �̃�𝑇1𝐿+�̃�𝑇1𝑈 2 )) , (( �̃�𝑇2𝐿+�̃�𝑇2𝑈 2 ),( �̃�𝑇2𝐿+�̃�𝑇2𝑈 2 )),… (( �̃�𝑇𝑛𝐿+�̃�𝑇𝑛𝑈 2 ),( �̃�𝑇𝑛𝐿+�̃�𝑇𝑛𝑈 2 )); ) , ( (( �̃�𝐹1𝐿+�̃�𝐹1𝑈 2 ),( �̃�𝐹1𝐿+�̃�𝐹1𝑈 2 )), (( �̃�𝐹𝑛𝐿+�̃�𝐹𝑛𝑈 2 ),( �̃�𝐹𝑛𝐿+�̃�𝐹𝑛𝑈 2 ))…. (( �̃�𝐹𝑛𝐿+�̃�𝐹𝑛𝑈 2 ),( �̃�𝐹𝑛𝐿+�̃�𝐹𝑛𝑈 2 )) ) ] 𝑓′′ ( ( [ ((�̃�𝑇1, �̃�𝑇1) , (�̃�𝑇2, �̃�𝑇2) , … , (�̃�𝑇𝑛, �̃�𝑇𝑛)) ; ((�̃�𝐹1, �̃�𝐹1), (�̃�𝐹2, �̃�𝐹2) , … , (�̃�𝐹𝑛,�̃�𝐹𝑛)) ] ) ) ] ∩ ([ ((�̃�𝑇1𝐿 , �̃�𝑇1𝑈) , (�̃�𝑇2𝐿 , �̃�𝑇2𝑈) , … , (�̃�𝑇𝑛𝐿, �̃�𝑇𝑛𝑈)) ; ((�̃�𝐹1𝐿 , �̃�𝐹1𝑈) , (�̃�𝐹2𝐿 , �̃�𝐹2𝑈) , … , (�̃�𝐹𝑛𝐿, �̃�𝐹𝑛𝑈)) ]) 𝑓′′ ( ( [ ((�̃�𝑇1, �̃�𝑇1) , (�̃�𝑇2, �̃�𝑇2) , … , (�̃�𝑇𝑛, �̃�𝑇𝑛)) ; ((�̃�𝐹1, �̃�𝐹1), (�̃�𝐹2, �̃�𝐹2) , … , (�̃�𝐹𝑛,�̃�𝐹𝑛)) ] ) ) ≠ 0𝐼 For ℹntuiti0nistic fuzzy unc0nstraℹned 0ptimization pr0blem then (3.1.3) and (3.1.4) takes the form �̃�𝑘+1 𝐼 = 𝑁𝑘 𝐼 ((�̃�𝑘, �̃�𝑘)) 𝐼 ∩ (�̃�𝐾) 𝐼 (3.1.5) and �̃�𝑘+1 𝐼 = 𝑁𝑘 𝐼 ((�̃�𝑘, �̃�𝑘)) 𝐼 ∩ (�̃�𝐾) 𝐼 (3.1.6) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 191 https://internationalpubls.com 3.2 ALGORITHMS 3.2.1 Algorithmi For Singlei Variablei Fuzzy Intuitionℹstici Intervali Newton’si Methodi 3.2.2 Algorithmi For Bivariatei Intuitionistici Fuzzyi Intervali Newton’si Methodi: 4. Numerical Illustrations Example 4.1: Let us c0nsider the f0llowing Unc0nstrained fuzzy 0ptimization Pr0blem (𝑓(�̃�) 𝐼 ) = (�̃�4)𝐼 − (48�̃�)𝐼 + (12)𝐼 𝑓(�̃�) = ((−1,1,3); (−1.1,1,2.9))𝑋4 − ((47,48,49); (46.9,48,49.1))𝑋 + ((10,12,14); (10.5,12,13.5) where 𝑋 ∈ ℜ. By empl0ying 0ur pr0posed arithmetic operati0ns, we first transf0rm all Intuℹti0nistic Triangular Fuzzy numbers into their l0catℹon ℹndex and fuzzℹness ℹndex functℹ0ns. then the parametric form for unc0nstraℹned 0ptimℹzation pr0blem is written as 𝑓(�̃�) = ((−1,1,3); (−1.1,1,2.9))𝑋3 − ((47,48,49); (46.9,48,49.1))𝑋 + ((10,12,14); (10.5,12,13.5) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 192 https://internationalpubls.com 𝑓′(�̃�) = 4((−1,1,3); (−1.1,1,2.9))𝑋3 − ((47,48,49); (46.9,48,49.1)) 𝑓′(�̃�)=((−1,1,3); (−1.1,1,2.9))𝑋3 − ((11.75,12,12.25); (11.725,12,12.2750) 𝑓′′(�̃�) =3((−1,1,3); (−1.1,1,2.9))𝑋2 The Interval analysis of Newt0n meth0d for unc0nstrained fuzzy 0ptimization is (�̃�𝑖+1) 𝐼 = 𝑁(�̃�) 𝐼 ∩ (�̃�𝑖) 𝐼 Here 𝑁(�̃�) 𝐼 = [𝑚(�̃� ) 𝐼 − 𝑓′(𝑚(�̃� ) 𝐼 𝑓′′(�̃� )𝐼 ] Let us take the ℹnitial ℹnterval �̃�0 = [((1,2,3); (1.1,2,2.9)), ((3,4,5); (3.1,4,4.9))] in Intuℹti0nistic triangular fuzzy number. 𝑚(�̃�0 ) 𝐼 = [(2,3,4); (2.1,3,2.9)] 𝑁(�̃�) 𝐼 = 𝑚(�̃�0 ) 𝐼 − [ (((−1,1,3); (−1.1,1,2.9)) ∗ [(2,3,4); (2.1,3,3.9)]3 −((11.75,12,12.25); (11.725,12,12.2750) 3[((1,2,3); (1.1,2,2.9)), ((3,4,5); (3.1,4,4.9))] 2 ] = [(2,3,4); (2.1,3,2.9)] − [ (((−1,1,3); (−1.1,1,2.9)) ∗ [(2,3,4); (2.1,3,3.9)]3 −((11.75,12,12.25); (11.725,12,12.2750) 3[((2,4,6); (2.2,4,5.8), (12,16,20); (12.4,16,19.6)] ] =[ ((0.5208,1.25,1.9791); (0.59791,1.25,1.9020)), (0.13021,0.31250.49479); (0.14947,0.3125,0.4755)) ] (�̃�1) 𝐼 = [((1,2,3); (1.1,2,2.9)), ((3,4,5); (3.1,4,4.9))] ∩ [ ((0.5208,1.25,1.9791); (0.59791,1.25,1.9020)), (0.13021,0.31250.49479); (0.14947,0.3125,0.4755)) ] (�̃�1)=[ ((1,2,3); (1.1,2,2.9)), ((3,4,5); (3.1,4,4.9)) ] ∩ [ (1.375,1.75,2.125); (1.4,1.75,2.1), (1.4687,2.6875,3.9062); (1.55,2.6875,3.825) ] (�̃�1) = [ (1,2,3); (1.1,2,2.9), (1.4687,2.6875,3.9062); (1.55,2.6875,3.825) ] (�̃�2) = [ (1.6693,2.2708,2.8724); (1.7184,2.2708,2.8232), (1.47524,2.0333,3.1315); (1.5429,2.3033,3.0638) ] Thus, f(x) is minimum at 𝑥 = 2.28675𝑖 , instead the elementary calculus technique produces minima at 𝑥 = 2.28675𝑖 . Example 2: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 193 https://internationalpubls.com Consider x and y as distinct diagnostic markers or criteria for medical diagnosis. These signals could be anything from specific lab readings to clinical complaints or imaging results. The equation then reflects the interaction of various diagnostic signs in evaluating a patient's status. In a medical context, x and y may indicate various clinical aspects or biomarkers associated with a specific disease or condition. For example, x might indicate blood pressure data, and y could represent blood glucose levels in a diabetic patient. The equation 𝑥3 − 3𝑥𝑦 + 𝑦3 could represent a diagnostic situation in which the values of certain indicators interact in a given manner, resulting in a particular diagnosis or clinical consequence. The equation may be analyzed to determine how changes in one diagnostic marker affect the interpretation of another, as well as how these interactions influence the diagnostic process. Let us c0nsider the following Unc0nstrained fuzzy 0ptimization Pr0blem with intuiti0nistic triangular(0.5,1,1.5); (0.4,1,1.6)𝑥3 − (2,3,4); (2.1,3,3.9)𝑥𝑦 + (0.5,1,1.5); (0.4,1,1.6)𝑦3, Initial interval intuiti0nistic value for 𝑋0 = [(0.5,1,1.5); (0.4,1,1.6), (1,2,3); (0.9,2,3.1)] 𝑎𝑛𝑑 𝑌0 = [(0.5,1,1.5); (0.4,1,1.6), (1,2,3); (0.9,2,3.1)] m(𝑋𝑘) = [(0.75,1.5,2.25); (0.65,1.5,2.35)] m(𝑌𝑘) = [(0.75,1.5,2.25); (0.65,1.5,2.35)] 𝑓′(𝑋𝑘) = ((0.375,0.75,1.125); (0.325,0.75,1.175)) and 𝑓′(𝑌𝑘) = ((0.375,0.75,1.125); (0.325,0.75,1.175)) 𝑓′′(𝑋𝑘) = [((1,2,3); (0.8,2,3.2)), ((2,4,6); (1.8,4,6.2))] and 𝑓′′(𝑌𝑘) = [((1,2,3); (0.8,2,3.2)), ((2,4,6); (1.8,4,6.2))] 𝑋1 = [ [(0.75,1.5,2.25); (0.65,1.5,2.35)] − ((0.375,0.75,1.125); (0.325,0.75,1.175)) [((1,2,3); (0.8,2,3.2)), ((2,4,6); (1.8,4,6.2))] ] ∩ [(0.5,1,1.5); (0.4,1,1.6), (1,2,3); (0.9,2,3.1)] 𝑋1 = [ ((0.5625,1.125,1.6875); (0.4875,1.125,1.7625)), ((0.6562,1.3125,1.9687); (0.5688,1.3125,2.0563) ] ∩ [(0.5,1,1.5); (0.4,1,1.6), (1,2,3); (0.9,2,3.1)] 𝑋1 = [ ((0.5625,1.125,1.6875); (0.4875,1.125,1.7625)), ((0.6562,1.3125,1.9687); (0.5688,1.3125,2.0563) ] And 𝑌1 = [ ((0.5625,1.125,1.6875); (0.4875,1.125,1.7625)), ((0.6562,1.3125,1.9687); (0.5688,1.3125,2.0563) ] 𝑋2 = [ ((0.5625,1.125,1.6875); (0.4875,1.125,1.7625)), ((0.5586,1.1172,1.6758); (0.4841,1.1172,1.7503)) ] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 194 https://internationalpubls.com 𝑌2 = [ ((0.5625,1.125,1.6875); (0.4875,1.125,1.7625)), ((0.5586,1.1172,1.6758); (0.4841,1.1172,1.7503)) ] In a hypothetical scenario where x denotes blood pressure and y represents blood glucose levels, a negative coefficient like -3 implies that an increase in blood pressure is associated with a decrease in blood glucose levels. This relationship could potentially indicate the presence of a specific medical condition or risk factor that influences both blood pressure and glucose levels in this manner. However, to confirm such a connection, further research and analysis involving larger datasets and expert input would be necessary. 5. Conclusion The proposed paper introduces an innovative approach to tackle interval-based Intuiti0nistic Fuzzy optimization problems. By employing triangular interval-valued fuzzy numbers and the interval Newt0n's meth0d, it aἰm to provide a reliable solution for these complex issues. The effectiveness of this meth0d is demonstrated through the presentation of relevant numerical examples. Acknowledgement The authors express their gratitude to the unanim0us referees f0r theἰr valuable suggestions, which significantly contributed to enhancing the quality 0f their research paper. 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