Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1031 https://internationalpubls.com Singularly Perturbed Initial Value Problem for First Order Delay Differential Equation Magdm Problem using a Magdm Problem Manjumari M 1 1Assistant Professor, Department of Mathematics, Dhanalakshmi Srinivasan College of Arts and Science for Women(Autonomous), Perambalur, India. manjumaths87@gmail.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: This paper presents a novel approach to Dynamic Intuitionistic Fuzzy Multiple Attribute Group Decision Making (DIF-MAGDM) problems, where decision attributes are represented by intuitionistic fuzzy numbers collected over multiple time periods. To determine the unknown decision-maker weights, a singularly perturbed initial value problem is formulated, incorporating a transition parameter Ο„ for computing the weighting vectors. The proposed method employs the Dynamic Intuitionistic Fuzzy Weighted Averaging (DIFWA) operator to transform all dynamic intuitionistic fuzzy decision matrices into a column matrix. To rank and select the most suitable alternative, a proximity coefficient function is applied. The effectiveness and practicality of the approach are demonstrated through a numerical case study on supplier selection. The results validate the robustness and applicability of the proposed decision-making framework in dynamic and uncertain environments. Keywords: Intuitionistic Fuzzy Sets (IFSs), MAGDM, Dynamic Intuitionistic Fuzzy Weighted Averaging (DIFWA) operator, Singular Perturbation Problem. Introduction Multi-Attribute Group Decision-Making (MAGDM) plays a crucial role in modern decision theory, providing a framework for evaluating multiple alternatives based on multiple criteria. When intuitionistic fuzzy sets (IFSs) are incorporated into MAGDM and extended across multiple time periods, the process is referred to as Dynamic Intuitionistic Fuzzy Multi-Attribute Group Decision- Making (DIF-MAGDM). Since Atanassov [1] introduced the concept of IFSs, significant advancements have been made in their theoretical development and practical applications. Intuitionistic fuzzy sets have been widely utilized in various real-world decision-making scenarios [2]. Atanassov et al. [3] explored the intuitionistic fuzzy interpretations of multi-criteria and multi-measurement decision-making methods, while Bustine and Burillo [4] extended the concept to vague sets. Several operations on IFSs have been further refined by Biswas and Roy [5]. In parallel, singular perturbation theory in ordinary differential equations has been a subject of extensive research. Doolan et al. [6] introduced this theory, and Kadalbajoo [7] expanded its application to singularly perturbed differential equations, incorporating both positive and negative Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1032 https://internationalpubls.com shifts. Liu et al. [8] proposed several improvements to MAGDM and multiple criteria decision- making models using IFSs. Furthermore, Miller et al. [9] developed fitted numerical techniques for singular perturbation problems in both one and two dimensions. The study of distances between intuitionistic fuzzy sets for group decision-making was introduced by Szmidt and Kacprzyk [10, 11], while Xu and Yager [12–15] developed the Dynamic Intuitionistic Fuzzy Weighted Averaging (DIFWA) operator to enhance decision-making processes. The foundational concept of fuzzy sets, which provides a powerful approach to managing uncertainty, was first introduced by Zadeh [16]. Building upon these advancements, this paper presents an approach for solving DIF-MAGDM problems by incorporating intuitionistic fuzzy numbers across various time periods. The proposed method leverages the DIFWA operator and a proximity coefficient function to facilitate effective decision-making, as demonstrated through a numerical case study on supplier selection. PRELIMINARIES This section discusses key concepts of Dynamic Intuitionistic Fuzzy Multi-Attribute Group Decision- Making (DIF-MAGDM) and the different types of aggregation operators. These operators are essential for integrating intuitionistic fuzzy information from multiple decision-makers over various time periods, ensuring a systematic approach to decision-making. The section highlights various aggregation techniques, including the Dynamic Intuitionistic Fuzzy Weighted Averaging (DIFWA) operator, which plays a crucial role in managing uncertainty in group decision-making processes. Definition Let 𝜎(𝑑1), 𝜎(𝑑2), . . . . . . . . . , 𝜎(𝑑𝑝)be a collection of IFNs collected at p different periods π‘‘π‘˜(π‘˜ = 1,2, . . . . . . , 𝑝), π‘Žπ‘›π‘‘πœ†(𝑑) = (πœ†(𝑑1), πœ†(𝑑2), . . . . . , πœ†(𝑑𝑝)) 𝑇 be the weight vector of the periods π‘‘π‘˜(π‘˜ = 1,2, . . . . . . , 𝑝), then we call π·πΌπΉπ‘Šπ΄πœ†(𝑑) (𝛼(𝑑1), 𝛼(𝑑1), . . . . . , 𝛼(𝑑𝑝)) = πœ†(𝑑1)𝛼(𝑑1) βŠ•, πœ†(𝑑2)𝛼(𝑑2) βŠ•, . . . . . , πœ†(𝑑𝑝)𝛼(𝑑𝑝) A dynamic intuitionistic fuzzy weighted a averaging (DIFWA) operator. π·πΌπΉπ‘Šπ΄πœ†(𝑑) (𝛼(𝑑1), 𝛼(𝑑1),… . . , 𝛼(𝑑𝑝)) = (1 βˆ’ ∏(1 βˆ’ πœ‡πœŽ(π‘‘π‘˜)) 𝑝 π‘˜=1 πœ†(π‘‘π‘˜) , βˆπ›ΎπœŽ(π‘‘π‘˜) πœ†(π‘‘π‘˜) 𝑝 π‘˜=1 ,∏(1 βˆ’ πœ‡πœŽ(π‘‘π‘˜)) 𝑝 π‘˜=1 πœ†(π‘‘π‘˜) βˆ’ ∏ π›ΎπœŽ(π‘‘π‘˜) πœ†(π‘‘π‘˜) 𝑝 π‘˜=1 ) Where πœ†(π‘‘π‘˜) β‰₯ 0, π‘˜ = 1,2, . . . . . , 𝑝, βˆ‘ πœ†(π‘‘π‘˜) 𝑝 π‘˜=1 = 1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1033 https://internationalpubls.com A METHOD OF USING INTUITIONISTIC FUZZY INFORMATION IN GROUP DECISION MAKING: Step 1: To create a collective intuitionistic fuzzy decision matrix =(rij)mxn. , combine all of the individual intuitionistic fuzzy decision matrices using the DIFWA operator. Step 2: Define 𝒂+ = (π’‚πŸ +, π’‚πŸ +, . . . . . , π’‚π’Ž + )𝑻 and π’‚βˆ’ = (π’‚πŸ βˆ’, π’‚πŸ βˆ’, . . . . . , π’‚π’Ž βˆ’ )𝑻 as the intuitionistic fuzzy negative ideal solution (IFNIS) and the intuitionistic fuzzy ideal solution (IFIS), respectively, where π‘Žπ‘– + = (1,0,0)(𝑖 = 1,2, . . . , π‘š) are the m smallest and π‘Žπ‘– βˆ’ = (0,1,0)(𝑖 = 1,2, . . . , π‘š) m biggest IFNs. Additionally, for ease of illustration, we indicate the options π‘₯𝑖(𝑖 = 1,2, . . . , 𝑛)𝑏𝑦π‘₯𝑖 = (π‘Ÿπ‘–1, π‘Ÿπ‘–2, . . . , π‘Ÿπ‘–π‘š)𝑇, 𝑖 = 1,2, . . . . . , 𝑛. Step 3: Determine the proximity coefficient, where, �̃�𝑖 = (1,0)between the positive ideal value �̃�𝑖 and the collective overall preference values. Step 4: Sort each option based on the proximity coefficient 𝐢(π‘₯𝑖)(𝑖 = 1,2, . . . . , 𝑛),; the better option is indicated by a higher number π‘₯𝑖 . SINGULAR PERTURBATION PROBLEMS: To differentiate between regular and singular perturbation problems, first take a look at a family of boundary value problems π‘βˆˆ that, depending on a small parameter πœ€. can have a solution π‘¦πœ€(π‘₯)π‘œπ‘“π‘ƒπœ€constructed by a well-known "method of perturbation"; that is, as a power series πœ€ in which the problem's solution 𝑝0 (obtained by putting ∈= 0 in π‘βˆˆ ) is the first term 𝑦0. A power series expansion as βˆˆβ†’ 0 is said to be a regular perturbation issue when it converges evenly in x. is referred to as a single perturbation problem when π‘¦βˆˆ(π‘₯) it lacks a uniform limit in x as βˆˆβ†’ 0, as this regular perturbation procedure fails. A singly perturbed delay differential equation is an ordinary differential equation that includes at least one delay term and has its highest derivative multiplied by a small parameter. These equations frequently arise in various applications, such as control theory variation problems, depolarization in Stein's model, human pupil-light reflex modeling, and the first exit time problem in neuronal variability activation. This study focuses on boundary value problems for singularly perturbed linear second-order differential-difference equations, where a small parameter multiplies the highest-order derivative. The delay term u(xβˆ’1) is expressed using a Taylor series expansion at x=1. To address the numerical challenges posed by these equations, a numerical method based on a piecewise uniform Shishkin- type mesh is implemented within a classical finite difference scheme. A carefully constructed piecewise uniform Shishkin mesh is derived to ensure numerical stability and accuracy. It is proven that the proposed approach achieves first-order convergence in the Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1034 https://internationalpubls.com maximum norm, regardless of the small parameter. The Taylor series expansion at x is employed to effectively represent the delay term u(xβˆ’1), ensuring an accurate numerical solution. 𝑳𝒖(𝒙) = βˆ’πœΊπ’–β€² + 𝒂(𝒙)𝒖(𝒙) + 𝒃(𝒙)𝒖(𝒙 βˆ’ 𝟏) = 𝒇(𝒙)𝒐𝒏[𝟎,𝟐], with𝒖 = 𝝓𝒐𝒏[βˆ’πŸ,𝟎]𝒂𝒏𝒅𝒖(𝟐) = 𝒍. Where πœ™ is sufficiently smooth on [-1, 0]. For all π‘₯ ∈ [0,2] , it is assumed that a(x) and b(x) satisfy, 𝒂(𝒙) + 𝒃(𝒙) > 𝜢, and𝒃(𝒙) < 𝟎, π’‡π’π’“π’”π’π’Žπ’†π’“π’†π’‚π’π’π’–π’Žπ’ƒπ’†π’“πœΆ > 𝟎 Shishkin mesh: To construct an effective piecewise uniform mesh within the domain of definition, it is essential to account for the boundary layer behavior near x=0. The mesh should be refined in this region while remaining coarser elsewhere. If the total number of mesh points is N, they are distributed such that N/2 points are allocated to the inner region, where the solution exhibits rapid variation, and the remaining N/2 points are placed in the outer region, where the solution changes more gradually. The resulting mesh structure is outlined as follows. Where the parameter 𝜏 is defined as 𝜏 = π‘šπ‘–π‘› { 1 2 , πœ€ 𝛼 𝑙𝑛 𝑁} Discrete problem: The finite difference method is now applied to formulate the discrete two-point boundary value problem on an arbitrary grid. πΏπ‘π‘ˆ(π‘₯𝑗) = πœ€π·βˆ’π‘ˆ(π‘₯𝑗) + π‘Ž(π‘₯𝑗)π‘ˆ(π‘₯𝑗) + 𝑏(π‘₯𝑗)π‘ˆ(π‘₯𝑗 βˆ’ 1) = 𝑓(π‘₯𝑗)π‘“π‘œπ‘Ÿπ‘Žπ‘™π‘™(π‘₯𝑗), 𝑗 = 1,𝑁 πΏπ‘π‘ˆ = 𝑓, π‘ˆ(0) = 𝑒(0), π‘ˆ(1) = 𝑒(1). Where π‘ˆ(π‘₯𝑗 βˆ’ 1) = π‘ˆ(π‘₯𝑗) βˆ’ π·βˆ’π‘ˆ(π‘₯𝑗) π·βˆ’π‘ˆ(π‘₯𝑗) = π‘ˆ(π‘₯𝑗) βˆ’ π‘ˆ(π‘₯π‘—βˆ’1) β„Žπ‘— Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1035 https://internationalpubls.com MAGDM PROBLEM DETERMINATION EXPERTS WEIGHTS USING A SINGULARLY PERTURBED INITIAL VALUE PROBLEM: Problem proposed by decision maker. The unknown decision-maker weights are determined by solving a singularly perturbed initial value problem. βˆ’πœ€π‘’β€² + 1.5𝑒 βˆ’ 0.1𝑒(π‘₯ βˆ’ 1) = 1π‘“π‘œπ‘Ÿπ‘₯ ∈ (0,1) βˆͺ (1,2) 𝑒(0) = 1π‘“π‘œπ‘Ÿπ‘₯ ∈ [βˆ’1,0] The numerical solution presented above can be obtained using the classical finite difference scheme. The normalization of weighting vectors is shown in Tables 1 and 2, and the weighting vectors can be calculated for different values of Ο„. Table 1: Numerical solution of βˆ’πœΊπ’–β€² + 𝒂(𝒙)𝒖(𝒙) + 𝒃(𝒙)𝒖(𝒙 βˆ’ 𝟏) X 𝑒(π‘₯) Normalize 0.1406080 0.9994382 0.33333 0.1407457 0.9994377 0.33333 0.1408834 0.9994371 0.33333 Table 2: Numerical solution of βˆ’πœΊπ’–β€² + 𝒂(𝒙)𝒖(𝒙) + 𝒃(𝒙)𝒖(𝒙 βˆ’ 𝟏) X 𝑒(π‘₯) Normalize 0.1410212 0.9994366 0.20152 0.1117446 0.9955917 0.20075 0.2093871 0.9917987 0.19998 0.3070296 0.9880568 0.19923 0.4046720 0.9843654 0.19849 Hence, the weighting vectors given by the decision maker is calculated as πœ” = (0.33333,0.33333,0.33333)𝑇 , 𝑀 = (0.20152,0.20075,0.19998,0.19923,0.19849)𝑇 The two mesh algorithm is used to calculate the order of convergence and the maximum pointwise error. Table 1: Values of π‘«πœΊ 𝑡, 𝑫𝑡, 𝑷𝑡, π‘·βˆ—π’‚π’π’…π‘ͺ𝑷 π‘΅π’‡π’π’“πœΊπŸ = 𝜼 πŸ– π’‚π’π’…πœΆ = 𝟎. πŸ— 𝜼 128 256 512 1024 2048 πŸβˆ’πŸ‘ 0.625E-01 0.147E-02 0.862E-03 0.491E-03 0.275E-03 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1036 https://internationalpubls.com πŸβˆ’πŸ’ 0.391E-02 0.147E-02 0.862E-03 0.491E-03 0.275E-03 πŸβˆ’πŸ“ 0.244E-03 0.147E-02 0.862E-03 0.491E-03 0.275E-03 πŸβˆ’πŸ” 0.153E-04 0`147E-02 0.862E-03 0.491E-03 0.275E-03 πŸβˆ’πŸ• 0.954E-06 0.147E-02 0.862E-03 0.491E-03 0.275E-03 𝑫𝑡 0.147E-02 0.862E-03 0.491E-03 0.275E-03 𝑷𝑡 0.771E+00 0.813E+00 0.838E+00 π‘ͺ𝑷 𝑡 0.256E+00 0.256E+00 0.249E+00 0.237E+00 The order of convergence=0.7713155E+00 The error constant=0.2559836E+00 Figure 1: NUMERICAL SOLUTION OFβˆ’πœΊπ’–β€² + 𝒂(𝒙)𝒖(𝒙) + 𝒃(𝒙)𝒖(𝒙 βˆ’ 𝟏) = 𝒇(𝒙) NUMERICAL ILLUSTRATION: A company is looking to invest in a renewable energy source and has identified four potential alternatives: geothermal, solar, wind, and biomass. To evaluate these options, the company has established five key criteria: risk factor, industry growth rate, payback reliability, social benefits, and demand change. Given the increasing energy demand, growing environmental awareness, and strong government support for renewable energy projects in recent years, the company recognizes the need for a Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1037 https://internationalpubls.com dynamic assessment approach. As a result, renewable energy sources have been evaluated over the past six years in two-year intervals to account for changing conditions. To ensure a well-informed decision, a three-member decision-making group has been assembled to analyze the data and determine the most suitable investment option. 𝑅1 = ( (0.6,0.3,0.1) (0.7,0.2,0.1) (0.3,0.6,0.1) (0.6,0.2,0.2) (0.4,0.5,0.1) (0.9,0.1,0.0) (0.1,0.8,0.1) (0.4,0.5,0.1) (0.7,0.1,0.2) (0.4,0.5,0.1) (0.5,0.4,0.1) (0.8,0.2,0.0) (0.5,0.4,0.1) (0.5,0.3,0.2) (0.5,0.2,0.3) (0.4,0.6,0.0) (0.2,0.6,0.1) (0.6,0.3,0.1) (0.6,0.2,0.1) (0.8,0.1,0.1) ) 𝑅2 = ( (0.9,0.1,0.0) (0.7,0.2,0.1) (0.8,0.2,0.0) (0.7,0.2,0.1) (0.2,0.6,0.2) (0.8,0.1,0.1) (0.1,0.8,0.1) (0.3,0.5,0.2) (0.7,0.3,0.0) (0.3,0.4,0.3) (0.5,0.4,0.1) (0.9,0.1,0.0) (0.6,0.3,0.1) (0.3,0.5,0.2) (0.5,0.3,0.2) (0.5,0.4,0.1) (0.3,0.7,0.0) (0.9,0.1,0.0) (0.7,0.1,0.2) (0.4,0.6,0.0) ) 𝑅3 = ( (0.5,0.5,0.0) (0.6,0.2,0.2) (0.7,0.3,0.0) (0.6,0.3,0.1) (0.6,0.3,0.1) (0.8,0.1,0.1) (0.8,0.2,0.0) (0.4,0.5,0.1) (0.3,0.4,0.3) (0.5,0.4,0.1) (0.5,0.4,0.1) (0.7,0.2,0.1) (0.5,0.4,0.1) (0.5,0.2,0.3) (0.6,0.2,0.2) (0.4,0.6,0.0) (0.3,0.7,0.0) (0.6,0.3,0.1) (0.6,0.2,0.2) (0.5,0.4,0.1) ) Step: 1Use the DIFWA operator to aggregate all individual intuitionistic fuzzy decision matrices into a collective intuitionistic fuzzy decision matrix R=(rij)mxn. 𝑅 = [ [(0.72855,0.24662,0.02482)], [(0.66980,0.20000,0.13019)], [(0.65239,0.33019,0.01741)], [(0.63657,0.22895,0.13448)], [(0.26813,0.66494,0.06692)] [(0.42310,0.44814,0.12876)], [(0.84125,0.10000,0.05874)], [(0.45486,0.50397,0.04116)], [(0.36836,0.46416,0.20075)], [(0.74801,0.20801,0.04397)] [(0.60209,0.22895,0.16896)], [(0.40560,0.43089,0.16350)], [(0.49999,0.40000,0.10000)], [(0.81828,0.15874,0.02297)], [(0.63657,0.15874,0.02297)] [(0.53584,0.36343,0.10073)], [(0.44065,0.31073,0.24862)], [(0.53584,0.22895,0.23522)], [(0.43537,0.52415,0.04047)], [(0.60851,0.28845,0.10304)] ] Using step2, we obtain π‘Ÿ1 = [ (0.72855,0.24662,0.02482), (0.66980,0.20000,0.13019), (0.65239,0.33019,0.01741), (0.63657,0.22895,0.13448), (0.26813,0.66494,0.06692) ] 𝑇 π‘Ÿ2 = [ (0.42310,0.44814,0.12876), (0.84125,0.10000,0.05874), (0.45486,0.50397,0.04116), (0.36836,0.46416,0.20075), (0.74801,0.20801,0.04397) ] 𝑇 π‘Ÿ3 = [ (0.60209,0.22895,0.16896), (0.40560,0.43089,0.16350), (0.49999,0.40000,0.10000), (0.81828,0.15874,0.02297), (0.63657,0.15874,0.02297) ] 𝑇 π‘Ÿ4 = [ (0.53584,0.36343,0.10073), (0.44065,0.31073,0.24862), (0.53584,0.22895,0.23522), (0.43537,0.52415,0.04047), (0.60851,0.28845,0.10304) ] 𝑇 STEP: 3 calculate the closeness coefficient of each alternatives Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1038 https://internationalpubls.com 𝑐(π‘₯𝑖) = βˆ‘ πœ”π‘—(1 βˆ’ 𝛾𝑖𝑗) 𝑛 𝑗=1 βˆ‘ πœ”π‘—(1 + πœ‹π‘–π‘—) 𝑛 𝑗=1 𝐢(οΏ½ΜƒοΏ½1, οΏ½ΜƒοΏ½ +) = 0.62019 𝐢(οΏ½ΜƒοΏ½2, οΏ½ΜƒοΏ½ +) = 0.59839 𝐢(οΏ½ΜƒοΏ½3, οΏ½ΜƒοΏ½ +) = 0.66079 𝐢(οΏ½ΜƒοΏ½4, οΏ½ΜƒοΏ½ +) = 0.57333 Rank all the alternatives π‘₯𝑖(𝑖 = 1,2,3,4) according to the closeness coefficient of 𝑐(π‘₯𝑖)(𝑖 = 1,2,3,4). π‘₯3 > π‘₯1 > π‘₯2 > π‘₯4. Conclusion: In this study, the numerical solution of the singularly perturbed initial value problem for the decision- making process is determined using the unknown weights of the decision-makers. To select the optimal alternative, the dynamic intuitionistic multi-attribute decision-making approach is applied, computing the proximity coefficient for each option. This method ensures a systematic and dynamic evaluation of alternatives, leading to a well-informed investment decision. Reference: [1] K.Atanassov, Intuitionistic Fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96, http://dx.doi.org/10.1016/S0165-0114(86)80034-3. [2] K.Atanassov, Intuitionistic Fuzzy Sets, Theory and Applications, Physica-Verlag, Heidelberg (1999), https://doi.org/10.1007/978-3-7908-1870-3_1 [3] K.Atanassov, G.Pasi, and R.R Yager, Intuitionistic fuzzy interpretations of multi-criteria multi- person and multi-measurement tool decision making, International Journal of Systems Science 36 (2005), 859–868, DOI:10.1080/00207720500382365 [4] H.Bustine, and P.Burillo, Vague sets are intuitionistic fuzzy sets, Fuzzy Sets and Systems 79(1996) 403–405, https://doi.org/10.1016/j.ijar.2007.08.008 [5] S.KDe, R.Biswas, and A.R Roy, Some operations on intuitionistic fuzzy sets, Fuzzy Sets and Systems 114 (2000), 477–484, DOI:10.1016/S0165-0114(98)00191-2 [6] E.PDoolan, J.J.H Miller and W.H.ASchilders,Uniform numerical methods for problems with initial and boundary layers. Boole Prees,Dublin,Ireland,(1980), DOI: 10.4236/ajcm.2018.82012 [7] M.KKadalbajoo, and Y.NReddy, Initial values technique for a class of nonlinear singular perturbation problems. Journal of optimization theory and applications 53(3),(1987), 395-406, DOI: 10.12691/ajna-2-2-3 [8] H.WLiu, and G.JWang, Multi-criteria decision-making methods based on intuitionistic fuzzy sets, European Journal of Operational Research 179 (2007), 220– 233, DOI:10.1016/j.ejor.2006.04.009 http://dx.doi.org/10.1080/00207720500382365 https://doi.org/10.1016/j.ijar.2007.08.008 http://dx.doi.org/10.1016/S0165-0114(98)00191-2 https://doi.org/10.4236/ajcm.2018.82012 http://dx.doi.org/10.1016/j.ejor.2006.04.009 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1039 https://internationalpubls.com [9] J.J.HMiller, E.ORiordan, and G.IShishkin, Fitted numerical methods for singular perturbation problems.Error estimates in the maximum norm for linear problems in one and two dimensions. World scientific publishing co, pvt.ltd.,Singapore,(1996). https://doi.org/10.1142/8410V .:; [10] E.Szmidt, and J.Kacprzyk, Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems 114 (2000), 505–518, https://doi.org/10.1016/S0165-0114(98)00244-9 [11] E.Szmidt, and J.Kacprzyk, Using intuitionistic fuzzy sets in group decision making, Control and Cybernetics 31(2002), 1037–1053, DOI: 10.1002/mp.17313ISBN: 2473-4209 [12]Z.SXu, An overview of methods for determining OWA weights, International Journal of Intelligent Systems 20 (2005), 843–865, DOI:10.1002/int.20097 [13]Z.SXu, Intuitionistic preference relations and their application in group decision making, Information Sciences 177 (2007), 2363–2379, DOI:10.1016/j.ins.2006.12.019 [14]Z.SXu, and R.RYager, Dynamic intuitionistic fuzzy multi-attribute decision making, International Journal of Approximate Reasoning,48(2008),246–262, doi:10.1016/j.ijar.2007.08.008 [15] R.RYager, OWA aggregation over a continuous interval argument with applications to decision making, IEEE Transactions on Systems, Man, and Cybernetics – Part 34 (2004), 1952–1963, DOI: 10.1109/tsmcb.2004.831154 [16]L.AZadeh, Fuzzy Sets, Information and Control 8 (1965), 338–353, DOI: https://doi.org/10.2307/2272014 https://doi.org/10.1142/8410V https://doi.org/10.1016/S0165-0114(98)00244-9 http://dx.doi.org/10.1002/int.20097 http://dx.doi.org/10.1016/j.ins.2006.12.019 https://doi.org/10.1109/tsmcb.2004.831154 https://doi.org/10.2307/2272014