triangular intuitionistic fuzzy ahp and its application to ...

Inter national Journal of Pure and Applied Mathematics Volume 113 No. 10 2017, 253 – 261 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue

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TRIANGULAR INTUITIONISTIC FUZZY AHP AND ITS APPLICATION TO SELECT BEST PRODUCT OF NOTEBOOK COMPUTER G. Nirmala1 , G. Uthra2 1 Research scholar, Bharathiar University Assistant Professor, Dept. of Mathematics Sri Sai ram Engg College, Chennai-600040, Tamilnadu [email protected] 2 Assistant Professor, Dept. of Mathematics Pachaiyappa’s college, Chennai-600030, Tamilnadu [email protected] Abstract This paper proposes a method for AHP based on Triangular Intuitionistic Fuzzy Number (TIFN) and applying that to select the best product of Notebook computer. Here comparison judgments are represented as TIFN.From this number location index number and fuzziness index function are evaluated and TIFN are represented using the above two. The priority values are evaluated using arithmetic mean method. The magnitude of the priority values are calculated by using a formula. Based on that, the alternatives are ranked. The above mentioned method is applied to select the best product of Notebook computer. AMS Subject Classification: 94D. Key Words and Phrases: Analytic Hierarchy process, Intuitionistic Fuzzy Number,Triangular Intuitionistic Fuzzy Number.

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Introduction

Analytic Hierarchy Process [1],one of the most effective tool for dealing with complex decision making problems. In AHP, there is no way to express decision maker’suncertainty and vagueness. To overcome this issue, AHP was handled in Fuzzy Environment wherethe elements of the comparison matrices are Fuzzy numbers. In Fuzzy set theory, the preferences are represented by membership function. There is no way to express the decision makers’ lack of knowledge or hesitancy. The lack of knowledge can be represented as Hesitancy degree in Intuitionistic Fuzzy sets theory [2]. The theory of the IF set is more useful to deal with vagueness and uncertainty than that of Fuzzy set. IFAHP was first developed and used in [3] for environmental decision making problem.in which triangular IF number are used for comparison matrices.In [4],a recently published work on IFAHP ,the comparison elements are represented as IF values which comprises of degree of membership and non-membership where the authors mainly focused on consistency of comparison matrix They have given an iterative algorithm to check the consistency and introduced automotive algorithm to repair the inconsistent matrices. In this paper, AHP is investigated using TIF number. This paper is arranged as follows.Section-2 reviews some basic concepts briefly. Section -3 describes the methodology for Triangular Intuitionistic Fuzzy AHP. Section-4 applies the proposed methodology to select the best product of NoteBook computer.

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Preliminaries

Definition 1. Intuitionistic Fuzzy Set[5] Let X be an ordinary finite nonempty set .An IFS in A is an expression given by A = {(x, µA (x), γA (x))/x ∈ X} where µA : X → [0, 1], µA : X → [0, 1] with the condition 0 ≤ µA (x) + γA (x) ≤ 1 for all x in X. The number µA (x) and γA (x) denote, respectively the degree of membership and the degree of non-membership of the element in the set A.

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πA (x) = 1 − µA (x) − γA (x); µA (x) : X → [0, 1] represents the degree of hesitation or intuitionistic index or non-determinacy of x in A. Definition 2. Triangular Intuitionistic Fuzzy Number I ˜ A = (a1 , a2 , a3 ; a′1 , a2 , a′3 ) is a Triangular Intuitionistic Fuzzy Number (TIFN) with parameters a′1 ≤ a1 ≤ a2 ≤ a3 ≤ a′3 and denoted by A˜I = (a1 , a2 , a3 ; a′1 , a2 , a′3 ) having the membership function and non-membership functions as follows:   1 for x < a′1   0 for x < a 1       a2 −x ′ x−a1      a2 −a1 for a1 ≤ x ≤ a2  a2 −a′1 for a1 ≤ x ≤ a2 µA˜I (x) = 1 γA˜I (x) = 0 for x = a2 for x = a1     x−a a −x ′ 2 3      a3 −a2 for a2 ≤ x ≤ a3  a′3 −a2 for a2 ≤ x ≤ a3   0  for x > a3 1 for x > a′3 Note: If m = a2 represents the modal value, σ1 = (a2 − a1 ) represents the left spread and β1 = (a3 − a2 ) represents the right spread of membership function and σ2 = (a2 − a′1 ) represents the left spread and β2 = (a′3 − a2 ) represents the right spread of nonmembership function.

Definition 3. Location Index and Fuzziness Index[6] ˜I ˜I = (a, a The parametric representation ˜ ; a′ , a ˜′ ). of A is given ′ as A ˜ ˜ a(1)+˜ a(1) a (1)+˜ a′ (1) The number a0 = ˜ 2 or a0 = ˜ 2 are said to be a location index number of membership function and nonmembership function. a• (a0 − a), a• (a0 − a ˜) are called the ˜ left fuzziness and right fuzziness index functions for membership function respectively and a′• (a0 −a′ ), a•′ = (a0 −˜ a′ ) are called the left ˜ fuzziness and right fuzziness index functions for non-membership function respectively. Hence every triangular Intuitionistic fuzzy number A˜I = (a1 , a2 , a3 ; a′1 , a2 , a′3 ) can also be represented by A˜I = (a0 , a• , a• ; a0 , a′• , a•′ ) Definition 4. Arithmetic operations on triangular intuitionistic fuzzy numbers For an arbitrary triangular in˜I = tuitionistic fuzzy numbers A˜I = (a0 , a• , a• ; a0 , a′• , a•′ ) and B • ′ •′ (b0 , b• , b ; b0 , b• , b ) and ∗ = {+, −, ×, ÷} the arithmetic operations on the triangular intuitionistic fuzzy numbers are defined by A˜I ∗ ˜ I = (a0 ∗ b0 , a• ∨ b• , a• ∨ b• ; a0 ∗ b0 , a′• ∨ b′• , a•′ ∨ b•′ ) B

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Definition 5. Ranking of triangular intuitionistic fuzzy number For an arbitrary triangular intuitionistic fuzzy numbers A˜I = (a0 , a• , a• ; a0 , a′• , a•′ ) with parametric form is A˜I = (a, a ˜ ; a′ , a ˜′ ), the ˜ ˜ magnitude of the triangular intuitionistic fuzzy number Z 1 1 • •′ ′ I (a + a• + 8a0 + a + a• )f (r)dr M ag(A˜ ) = 2 0 where f (r) is a non-negative function on [0, 1] with R 1 and increasing 1 f (0) = 0, f (1) = 1 and 0 f (r)dr = 2 . This function f (r) can be considered as a weighting function. In real life applications, f (r) can be chosen by decision maker. Here we use f (r) = r .Hence • a + a•′ + 8a0 − a• − a′• I ˜ M ag(A ) = 4 The magnitude of a triangular Intuitionistic fuzzy number A˜I synthetically reflects the information on every membership degree and meaning of its magnitude is usual and natural. M ag(A˜I ) are used to rank fuzzy number. For any two triangular intuitionistic fuzzy ˜ I = (b0 , b• , b• ; b0 , b′• , b•′ ) numbers A˜I = (a0 , a• , a• ; a0 , a′• , a•′ ) and B ˜ I by comparing M ag(A˜I ) and in F (R).The ranking of A˜I and B I ˜ M ag(B ) onR follows: ˜ I iff M ag(A˜I ) M ag(B ˜I ) (i) A˜I B ˜ I iff M ag(A˜I ) M ag(B ˜I ) (ii) A˜I B ˜ I iff M ag(A˜I ) ≈ M ag(B ˜I ) (iii) A˜I ≈ B

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Procedure for TIFAHP

Procedure for Triangular Intuitionistic Fuzzy AHP Step 1: Construct a hierarchical structure for the problem to be solved Step 2: The Decision maker provides his/her preferences in the form of Triangular Intuitionistic Fuzzy number for comparison matrices with respect to goal and criteria. Step 3: Calculate the location index number and fuzziness index functions for TIF Number and Represent the TIFN using this both.

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StepP4: Determine the priority vectors using

wi =

n ˜I j=1 Aij Pn ˜I i=1 j=1 Aij

Pn

for all the comparison matrices.

Step 5: Combine all the weights from the lowest level of the hierarchy to the highest level of the hierarchy, and rank the overall weightsusing magnitudeand choose the best alternative.

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Illustrative Example

Application of IFNAHP to select the best product of Notebook computer Notebook computers plays a vital role in human life in this era of technology because of its portability, ability and mobility. Therefore the selection of effective notebook computers to suit the need of buyers is essential [7].Nowdays many information sources have been available to select Notebook computers which suits the buyer’s requirements. They have mostly presented the features, prices, and pros and cons of each product and model of notebook computers. Mostof them haven’t decided that which one is the best or the worst to be bought or not bought, but they have just giveninformation and let the buyers compare and decide by themselves .In practice environment, the buyers have to face with a variety of notebook computers’ information types that are difficult to determine the decision alternatives. This problem is a multicriteria decision making (MCDM) problem. There have been many approaches proposed for MCDM problem. One of the techniques is Analytic Hierarchy process. For the problem of selecting Best Notebook computer, the first criteria choosing in this problem is platform which represents the Operating system, specs which represents CPU capacity, RAM capacity, hard disc capacity etc. Third criteria we are choosing are Battery life. The next criteria is Weight and size of the computer which is very important for portability .Next and much more important criteria is prize. Last criteria is based on monitor which involves Screen size, display and monitor resolution. Alternatives we are choosing are HP, Lenovo, Dell and ASUS. Data collection Six criteria and four alternatives are identified for our problem.

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The hierarchical structure of the problem is given in Figure [1].Totally 100 models, 25 models from each product are taken from sources like magazines, websites and brochures for comparison. Decision makers are asked to compare the models. Decision makers set of three person who have experience of more than five years in the field of computers. The collective judgments are taken as a matrix form. Totally 7 matrices are forms, one for criteria comparison with respect to goal, remaining 6 for comparison of Alternatives with respect to criteria. All comparison matrices whose elements TIF Numbers are given in Annexure-I .Weight evaluation of criteria are given in Annexure-II.Overall weight evaluation is given in Annexure-III.As the result of our evaluation Lenovo is the best, HP is second best followed by Asus, Dell.

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Conclusion

In this paper,a study was made on AHP in the Intuitionistic Fuzzy environment,where the elements of the comparison matrices are represented by TIFN. Since TIFN is powerful in expressing vagueness and uncertainty, this method has more advantages than AHP, FAHP and IFAHP. Since Triangular Intuitionistic fuzzy number is converted into its parametric form which is written in terms of r, this method is more flexible to the decision maker as the decision maker chooses the value for ranywhere between [0,1] . For any value of r, the ranking will be the same. This paperappliesthe proposed method to select the Best Notebook computer. The result is Lenovo is the best followed by HP, Asus, Dell.

References [1] T.L. Saaty, A scaling method for priorities in a hierarchical structure, J. Math. Psychol., 15 (1977), 224-281. [2] K.T. Atanassov, Intuitionistic Fuzzy sets, Fuzzy sets and system, 20, No. 1 (1986), 87-96. [3] Rehansadiq, Environmental decision making under uncertainty using Intuitionistic Fuzzy Analytic Hierarchy process, stoch Environmental Res Risk Assess, 23 (2009), 75-91.

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[4] Xu, H. Liao, Intuitionistic Fuzzy Analytic Hierarchy process, IEEE transactions on Fuzzy systems, 22, No. 4. [5] G.S. Mahabatra, T.K. Roy, Relaibility evaluation Using Triangular Intuitionistic Fuzzy Numbers Arithmetic operations, International Journal of Computer, Electrical, Automation, Control and Information Engineering, 3, No. 2 (2009). [6] K. Prabakaranand K. Ganesan, A new approach on solvingintuitionistic fuzzy linear programming problem, ARPN Journal of Engineering and Engineering and Applied Sciences, 10 (2015). [7] PhanarutSrichetta and WannasiriThurachon, Applying Fuzzy Analytic Hierarchy Process to Evaluate and Select Product of Notebook, Computers International Journal of Modeling and Optimization, 2, No. 2 (2012).

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