Fuzzy Generalized Prioritized Weighted ... - Wiley Online Library

Fuzzy Generalized Prioritized Weighted Average Operator and its Application to Multiple Attribute Decision Making Rajkumar Verma,∗ Bhu Dev Sharma† Department of Mathematics, Jaypee Institute of Information Technology (Deemed University), Noida-201307, Uttar Pradesh, India

The prioritized weighted average (PWA) operator was originally introduced by Yager. The prominent characteristic of the PWA operator is that it takes into account prioritization among attributes and decision makers. By combining the idea of generalized mean and PWA operator, we propose a new prioritized aggregation operator called fuzzy generalized prioritized weighted average (FGPWA) operator for aggregating triangular fuzzy numbers. The properties of the new aggregation operator are studied out and their special cases are examined. Furthermore, based on the FGPWA operator, an approach to deal with multiple attribute group decision making problems under triangular fuzzy environments is developed. Finally, a practical example is provided to illustrate the C 2013 Wiley Periodicals, Inc. multiple attribute group decision making process.

1.

INTRODUCTION

Information aggregation is an essential process of gathering relevant information from various sources. It extensively exists in many areas including engineering, artificial intelligence, medical diagnosis, decision making, pattern recognition, soft computing, and so on. In literature, many techniques have been developed to aggregate data information such as the maximum and minimum operators, the weighted arithmetic average operator, the weighted geometric mean operator, the weighted harmonic mean operator, and so on.1–3 Yager4, 5 introduced the ordered weighted averaging (OWA) operator for aggregating the data information. The fundamental aspect of the OWA operator is a reordering step in which the input arguments are rearranged in descending order. Since its introduction, the OWA operator has been applied in a wide range of problems.6–19 Chiclana et al.20 and Xu and Da21 introduced the ordered weighted geometric (OWG) operators, which are based on the OWA operator and the ∗ †

Author to whom all correspondence should be addressed; e-mail: rkver83@gmail.com. e-mail: bhudev.sharma@jiit.ac.in

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 29, 26–49 (2014) C 2013 Wiley Periodicals, Inc. View this article online at wileyonlinelibrary.com. • DOI 10.1002/int.21626

FUZZY PRIORITIZED WEIGHTED AVERAGE OPERATOR

27

geometric mean. A further interesting extension of the OWA operator is the generalized OWA (GOWA) operator22 that uses generalized means23 in the OWA operator. A GOWA operator in fact generalizes many situations, including OWA operator and its particular cases, the OWG operator, the ordered weighted harmonic averaging (OWHA) operator, and the generalized mean (GM). However, in some situations, the input arguments take the form of fuzzy numbers, rather than being crisp numbers, due to the increasing complexity of the socioeconomic environment, time pressure, not sufficient level of knowledge of the problem domain, and the ambiguity of human thinking. To aggregate fuzzy information, Xu13 and Wang and Fan24 proposed the fuzzy ordered weighted average (FOWA) operator. Xu25 introduced the fuzzy ordered weighted geometry (FOWG) operator. Xu14 proposed some fuzzy harmonic mean operators (FWHA), such as the fuzzy weighted harmonic mean operator, the fuzzy ordered weighted harmonic mean operator (FOWHM), and the fuzzy hybrid harmonic mean (FHHM) operator. Merigó and Casanovas11 proposed the fuzzy GOWA operator and found its application in strategic decision making. Furthermore, Wei26 developed the fuzzy induced ordered weighted harmonic mean (FIOWHM) operator and applied it to the group decision making. Triangular fuzzy numbers (TFNs) are very useful tools to deal with uncertainty and the concept of vagueness. In the last 10 years, many multiple attribute group decision making theories and methods have been proposed under triangular fuzzy environment with the assumption that the attributes and the decision makers are at the same priority level. However, in the real life multiple attribute group decision making problems, attributes and decision makers have different priority levels in general. To overcome this issue, motivated by the idea of prioritized weighted aggregation (PWA) operators,27, 28 Zhao et al.29 developed some fuzzy prioritized aggregation operators, such as the fuzzy prioritized weighted average (FPWA) operator, the fuzzy prioritized weighted geometric (FPWG) operator, and the fuzzy prioritized weighted harmonic (FPWH) operator, and formulated some approaches to solve multiple attribute group decision making problems under triangular fuzzy environment. In the present communication, motivated by the concept of generalized mean23 and prioritized weighted aggregation operator,27, 28 we propose a new aggregation operator called the fuzzy generalized prioritized weighted average (FGPWA) operator. The main advantage of the FGPWA operator is that it does not only take into account prioritization among the attributes and experts but also has a flexible parameter. It also includes a wide range of fuzzy prioritized aggregation operators, such as the FPWA operator, the FPWG operator, and the FPWH operator as particular cases. The paper is organized as follows: In Section 2, some basic concepts related to fuzzy set, fuzzy numbers, and prioritized weighted average (PWA) operator are briefly given. In Section 3, we introduce the new aggregation operator called the fuzzy generalized weighted average (FGPWA) operator and discuss its particular cases. Some properties of FGPWA operator are also studied here. In Section 4, a FGPWA operator based approach is developed for solving triangular fuzzy multiple attribute group decision making problems in which the attributes and decision makers are in different priority. Finally, in Section 5, a numerical example is International Journal of Intelligent Systems

DOI 10.1002/int

28

VERMA AND SHARMA

presented to illustrate the proposed approach to multiple attribute and to demonstrate its practicality and effectiveness. Our conclusions are presented in Section 6. 2.

PRELIMINARIES

In this section, we briefly review some basic concepts related to fuzzy set, fuzzy numbers and the PWA operator, which will be needed in the following analysis. DEFINITION 1 (Fuzzy Set30 ). A fuzzy set A defined in a universe of discourse X is mathematically represented as A = { x, μA (x)| x ∈ X} ,

(1)

where μA (x) : X → [0, 1] is the membership function of A and the number μA (x) describes the degree of membership of x ∈ X in the set A. DEFINITION 2 (Convex Fuzzy Set31 ). A fuzzy set A defined in a universe of discourse X is convex if and only if for all x1 , x2 ∈ X, μA (λx1 + (1 − λ) x2 ) ≥ min (μA (x1 ) , μA (x2 ))

where

λ ∈ [0, 1] .

(2)

DEFINITION 3 (Normal Fuzzy Set31 ). A fuzzy set A defined in a universe of discourse X is called a normal fuzzy set if ∃ xi ∈ X

such that μA (xi ) = 1.

(3)

DEFINITION 4 (Fuzzy Number31 ). A fuzzy number A is a fuzzy set defined in the set of real numbers, , whose membership function is both convex and normal DEFINITION5 (Triangular Fuzzy Number31, 32 ). A triangular number a˜ can be defined by a triplet a L , a M , a U , where 0 < a L ≤ a M ≤ a U , a L , and a U stand for the lower ˜ respectively, and a M for the modal value. and upper values of the support of a, And the membership function μa˜ (x) is defined as follows ⎧ 0, ⎪ ⎪ ⎪ ⎪ x − aL ⎪ ⎪ , ⎨ M L μa˜ (x) = a U − a ⎪ a −x ⎪ ⎪ , ⎪ ⎪ ⎪ aU − aM ⎩ 0,

x < aL, aL ≤ x ≤ aM , (4) a

M

≤x≤a , U

x > aU .

32, 33 DEFINITION ). Let Operations on Triangular Fuzzy Numbers L M6 (Arithmetical U L M U ˜ a˜ = a , a , a and b = b , b , b be two triangular fuzzy numbers, then some arithmetical operations defined on set of triangular fuzzy numbers are as follows:

International Journal of Intelligent Systems

DOI 10.1002/int


(i) (ii) (iii) (iv) (v) (vi)

a˜ ⊕ b˜ = a L , a M , a U ⊕ bL , bM , bU = a L + bL , a M + bM ,a U + bU , a˜ ⊗ b˜ = a L ,a M , a U ⊗ bL ,bM , bU = a L b L , a M bM , a U bU δ ⊗ a˜ = δ ⊗ a L , a M , a U = δa L , δa M , δa U , δ ≥ 0 δ ⊗ a˜ = δ ⊗ a L , a M , a U = δa U , δa M , δa L , δ < 0 δ δ δ δ a˜ δ = a L , a M , a U = a L , a M , a U , δ ≥ 0 δ δ δ δ a˜ δ = a L , a M , a U = a U , a M , a L , δ < 0.

29

34 D 7 (Expected Value of a Triangular Fuzzy Number ). Let a˜ = EFINITION L M U a , a , a be a triangular fuzzy number, then the expected value of a˜ is defined as follows:

˜ = E (a)

aL + aM + aU 3

(5)

.

The PWA operator was originally introduced by Yager27, 28 as follows: DEFINITION 8 (PWA Operator27, 28 ). Let G = {G1 , G2 , . . . , Gn } be a collection of attributes and let there be a prioritization between the attributes expressed by the linear ordering G1 G2 G3 · · · Gn , indicating that attribute Gi has a higher priority than Gj , if i < j . Also let Gi (x) be the performance value of any alternative x under attribute Gi that satisfies Gi (x) ∈ [0, 1]. If PWA (G1 (x) , G2 (x) , . . . , Gn (x)) =

n

wi Gi (x).

(6)

i=1

(i = 2, 3, . . . , n), and T1 = 1, then where wi = nTi Ti , Ti = i−1 j =1 Gj (x) i=1 PWA(G1 (x), G2 (x), . . . , Gn (x)) is called the PWA operator. When using the above-defined PWA operator, it is assumed that the available information includes crisp numbers. However in real life decision making, some times the available information is vague or imprecise and it may not be possible to analyze it with crisp numbers. In these situations, it is contingent to use another approach such as fuzzy numbers to assess the uncertainty. In the next section, to aggregate the triangular fuzzy information, we propose the FGPWA operator and discuss its particular cases. Some desirable properties of FGPWA are also discussed there. 3.

FGPWA OPERATOR

We start with the following formal definition: 9 (FGPWA Operator). Given a set of triangular fuzzy numbers,a˜ i = DEFINITION (i = 1, 2, . . . , n), the FGPWA operator is defined as follows: aiL , aiM , aiU


DOI 10.1002/int

30

VERMA AND SHARMA

FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) =

i=1

=

1/λ

Ti ⊕ n n

i=1

T1

n i=1

Ti

Ti

a˜ iλ

T2 a˜ 1λ ⊕ n i=1

Tn a˜ 2λ ⊕ · · · ⊕ n

Ti

i=1

1/λ Ti

a˜ nλ (7)

E a˜ j (i = 2, 3, . . . , n), T1 = 1, E a˜ j is the expected value of where Ti = i−1 L M j =1 a˜ j = aj , aj , ajU , and λ ∈ (−∞, 0) ∪ (0, ∞) is a parameter. Next, based on the operational laws of TFNs, we can derive the following theorem: (i = 1, 2, . . . , n) be a set of triangular fuzzy THEOREM 1. Let a˜ i = aiL , aiM , aiU numbers, then the aggregated value by using the FGPWA operator is also a triangular fuzzy number, and F GP W A (a˜ 1 , a˜ 2 , . . . , a˜ n )

=

T1

n i=1

⎡⎛

n

Ti ⎛

T2 a˜ 1λ ⊕ n i=1

Ti

Tn a˜ 2λ ⊕ · · · ⊕ n i=1

1/λ Ti

a˜ nλ

(8)

⎤ 1/λ M λ ,⎥ ai ⎥ T ⎥ i i=1 ⎥ ⎥ ⎥ ⎦

⎞⎞1/λ n

L λ Ti ⎠ ⎠

n , ai

i ⎢⎝ ⎝ T ⎢ n ⎢ i=1 Ti =⎢ i=1 ⎢ n

1/λ ⎢ U λ Ti ⎣

n ai i=1 Ti i=1

i=1

E a˜ j (i = 2, 3, . . . , n), T1 = 1, E a˜ j is the expected value of where Ti = i−1 j =1 a˜ j = ajL , ajM , ajU , and λ ∈ (−∞, 0) ∪ (0, ∞) is a parameter. The FGPWA operator has the following properties: THEOREM 2 (Idempotency). Let a˜ i = aiL , aiM , aiU (i = 1, 2, . . . , n) be a set of (i = 2, 3, . . . , n), T1 = 1, E a˜ j be E a˜ triangular fuzzy numbers,Ti = i−1 L Mj =1 U j the expected value of a˜ j = aj , aj , aj , and λ ∈ (−∞, 0) ∪ (0, ∞) be a parameter. the triangular fuzzy numbers a˜ i (i = 1, 2, . . . , n) are equal, that is, a˜ i = a˜ = If all a L , a M , a U ∀ i, then ˜ FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) = a.


DOI 10.1002/int

(9)

31


Proof. By Definition 9, we have

T1 FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) = n

=

i=1

Ti

T1

n

T2

⊕ n

a˜ 1λ

Ti

i=1

T2 a˜ λ ⊕ n

i=1 Ti i=1 Ti

n 1/λ Ti λ ˜ ˜ = i=1 = a. a n i=1 Ti

a˜ 2λ

1/λ

Tn

⊕ · · · ⊕ n

i=1

Tn a˜ λ ⊕ · · · ⊕ n i=1

Ti Ti

a˜ nλ 1/λ a˜ λ (10)

This proves the theorem.

THEOREM 3 (Boundedness). Let a˜ i = (i = 1, 2, . . . , n) be a set of i−1 (i = 2, 3, . . . , n), T1 = 1, E a˜ j be triangular fuzzy numbers, Ti = j =1 E a˜ j L M U the expected value of a˜ j = aj , aj , aj , and λ ∈ (−∞, 0) ∪ (0, ∞) be a parameter. Also, let [aiL , aiM , aiU ]

a˜ − = min a˜ i = min aiL , min aiM , min aiU ,

(11)

a˜ + = max a˜ i = max aiL , max aiM , max aiU .

(12)

a˜ − ≤ FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) ≤ a˜ + .

(13)

i

i

i

i

and

i

i

i

i

Then

Since min aiL ≤ aiL ≤ max aiL , min aiM ≤ aiM ≤ max aiM , and min aiM ≤

Proof.

i

i

i

aiM ≤ max aiM ∀ i.

i

i

i

First, let λ ∈ (0, ∞), then

n

Ti

n

i=1

i=1 Ti

λ aiL

1/λ ≤

n

Ti

n

i=1

i=1 Ti

λ 1/λ

= max aiL .

max aiL i

i

(14) On the other hand,

n

i=1

Ti

n

i=1 Ti

L λ ai

1/λ ≥

n i=1

Ti

n

i=1 Ti

λ 1/λ

min aiL i

= min aiL . i

(15) International Journal of Intelligent Systems

DOI 10.1002/int

32

VERMA AND SHARMA

that is, min aiL i

≤

n

Ti

n

i=1

i=1 Ti

n

Ti

L λ ai

1/λ ≤ max aiL . i

(16)

Similarly, we have min aiM i

≤

min aiU ≤ i

n

i=1

i=1 Ti

n

Ti

i=1

n

i=1

Ti

M λ ai

U λ ai

1/λ ≤ max aiM

(17)

≤ max aiU .

(18)

i

1/λ i

Let FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) = a˜ = a L , a M , a U , then max aiL + max aiM + max aiU aL + aM + aU i i i ˜ = ≤ = E a˜ + E (a) 3 3

(19)

and min aiL + min aiM + min aiU aL + aM + aU i i i ˜ = ≥ = E a˜ − . E (a) 3 3 − + ˜ > E a˜ , we have ˜ < E a˜ and E (a) If E (a) a˜ − < FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) < a˜ + .

(20)

(21)

˜ = E a˜ + , that is, if If E (a) max aiL + max aiM + max aiU aL + aM + aU i i i = 3 3 then, we have a L = max aiL , i

a M = max aiM , i

a U = max aiU . i

(22)

So that FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) = a˜ + . International Journal of Intelligent Systems

DOI 10.1002/int

(23)


˜ = E a˜ − , that is, if If E (a)

33

min aiL + min aiM + min aiU aL + aM + aU i i i = 3 3 then, we have a L = min aiL , a M = min aiM , a U = min aiU . i

i

i

(24)

Thus, it follows that FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) = a˜ − .

(25)

From Equations (21), (23), and (25), we get a˜ − ≤ FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) ≤ a˜ + ,

when λ ∈ (0, ∞) .

(26)

a˜ − ≤ FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) ≤ a˜ + when λ ∈ (−∞, 0) .

(27)

Similarly, we can get

Finally from Equations (26) and (27), we get, a˜ − ≤ FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) ≤ a˜ + when λ ∈ (−∞, 0) ∪ (0, ∞) .

(28)

This proves the theorem.

and a˜ i = THEOREM 4 (Monotonicity). Let a˜ i = aiL , aiM , aiU L M U (i = 1, 2, . . . , n) be two sets of triangular fuzzy numai , ai , ai i−1 i−1 (i = 2, 3, . . . , n), T1 = T1 = 1, bers, Ti = j =1 E a˜ j , Ti = j =1 E a˜ j E a˜ j be the expected value of a˜ j = ajL , ajM , ajU , E a˜ j the expected value of a˜ j = ajL , ajM , ajU , and λ be a parameter such that λ ∈ (−∞, 0) ∪ (0, ∞), if a˜ i ≤ a˜ i for all i, then FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) ≤ FGPWA a˜ 1 , a˜ 2 , . . . , a˜ n .

(29)

Proof. It directly follows from Theorem 3. L M U (i = 1, 2, . . . , n) be a set of triangular THEOREM 5. Let a˜ i = ai , ai , ai ˜ (i E = 2, 3, . . . , n), T1 = 1, E a˜ j be the exfuzzy numbers, Ti = i−1 a j j =1 pected value of a˜ j = ajL , ajM , ajU , and λ ∈ (−∞, 0) ∪ (0, ∞) be a parameter. If International Journal of Intelligent Systems

DOI 10.1002/int

34

VERMA AND SHARMA

b˜ = bL , bM , bU is a triangular fuzzy number, then

˜ a˜ 2 ⊗ b, ˜ . . . , a˜ n ⊗ b˜ = FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) ⊗ b. ˜ FGPWA a˜ 1 ⊗ b,

(30)

Proof. According to the operational laws on triangular fuzzy number given in Definition 6, we have a˜ i ⊗ b˜ = aiL bL , aiM bM , aiU bU .

(31)

Then, ˜ a˜ 2 ⊗ b, ˜ . . . , a˜ n ⊗ b˜ FGPWA a˜ 1 ⊗ b, ⎡ =⎣

n

Ti

n

i=1

i=1

n

=⎣

Ti

n

i=1 Ti

i=1

⎡

Ti

n

Ti

n

i=1 Ti

i=1

n

Ti

n

i=1 Ti

i=1

L L λ ai b

1/λ n

Ti

n , i=1

i=1

aiU b

U λ

L λ ai

1/λ L

b ,

1/λ

ai

1/λ ,

⎤ 1/λ ⎦,

(32)

n

Ti

n

i=1 Ti

i=1

U λ

Ti

M M λ ai b

M λ ai

1/λ bM ,

⎤ bU ⎦ .

On the other hand, FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) ⊗ b˜ ⎡ n

Ti ⎣

n = i=1

i=1

n

i=1

Ti

Ti

n

i=1

Ti

L λ ai

1/λ n

Ti

n , i=1

U λ

ai

i=1

Ti

M λ

1/λ

ai

⎤ 1/λ ⎦ ⊗ bL , bM , bU


DOI 10.1002/int

,

(33)


⎡ n

Ti ⎣

n =

i=1

i=1

n

i=1

Ti

Ti

n

i=1 Ti

aiL

λ

U λ ai

1/λ bL ,

n

i=1

1/λ

Ti

n i=1

Ti

M λ ai

35

1/λ bM ,

⎤ bU ⎦ .

Thus ˜ a˜ 2 ⊗ b, ˜ . . . , a˜ n ⊗ b˜ = FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) ⊗ b. ˜ FGPWA a˜ 1 ⊗ b, This proves the theorem. L M U THEOREM 6. Let a˜ i = ai , ai , ai (i = 1, 2, . . . , n) be a set of triangular fuzzy (i = 2, 3, . . . , n), T1 = 1, E a˜ j be the expected E a˜ j numbers, Ti = i−1 j =1 value of a˜ j = ajL , ajM , ajU , and λ ∈ (−∞, 0) ∪ (0, ∞) be a parameter, if δ > 0 and b˜ = bL , bM , bU is a triangular fuzzy number, then

(i) FGPWA a˜ 1 ⊗ b˜ δ , a˜ 2 ⊗ b˜ δ , . . . , a˜ n ⊗ b˜ δ = (FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n )) ⊗ b˜ δ , (34) ˜ a˜ 2 ⊗δ b, ˜ . . . , a˜ n ⊗δ b˜ = (FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ))⊗ δ b˜ (ii) FGPWA a˜ 1 ⊗δ b, (35)

Proof. In the following, we only prove (i); result (ii) can be proved analogously. According to the operational laws on triangular fuzzy number (TFN) given in Definition 6, we have δ L δ M δ U δ , b b˜ = b , b and δ δ δ δ . a˜ i ⊗ b˜ = aiL bL , aiM bM , aiU bU


DOI 10.1002/int

(36)

36

VERMA AND SHARMA

Then, δ δ δ FGPWA a˜ 1 ⊗ b˜ , a˜ 2 ⊗ b˜ , . . . , a˜ n ⊗ b˜ ⎡ n

n

δ λ 1/λ δ λ 1/λ T T i i

n

n aiL bL aiM bM =⎣ , , T T i i i=1 i=1 i=1 i=1 ⎤ n

1/λ λ δ Ti ⎦

n aiU bU (37) T i i=1 i=1 ⎡ n

1/λ 1/λ n

L λ L δ M λ M δ T T i i

n

n , ai b , ai b =⎣ i=1 Ti i=1 Ti i=1 i=1 ⎤ n

1/λ U λ U δ Ti

n ai b ⎦. T i i=1 i=1 On the other hand, δ FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) ⊗ b˜ ⎡ 1/λ 1/λ n

n

L λ M λ T T i i

n

n ai ai =⎣ , , T T i i i=1 i=1 i=1 i=1 1/λ⎤ n

U λ Ti ⎦ ⊗ bL δ , bM δ , bU δ

n (38) ai i=1 Ti i=1 ⎡ n

1/λ 1/λ n

L λ L δ M λ M δ T T i i

n

n , =⎣ ai b , ai b i=1 Ti i=1 Ti i=1 i=1 ⎤ n

1/ λ U δ ⎥ U λ Ti

n b ⎦. ai T i i=1 i=1 Thus FGPWA a˜ 1 ⊗ b˜ δ , a˜ 2 ⊗ b˜ δ , . . . , a˜ n ⊗ b˜ δ = (FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n )) ⊗ b˜ δ .

This proves the theorem. International Journal of Intelligent Systems

DOI 10.1002/int

37 L M U (i = 1, 2, . . . , n) be a set of triangular fuzzy THEOREM 7. Let a˜ i = ai , ai , ai ˜ (i E a = 2, 3, . . . , n), T1 = 1, E a˜ j be the exnumbers, where Ti = i−1 j j =1 pected value of a˜ j = ajL , ajM , ajU , and λ ∈ (−∞, 0) ∪ (0, ∞) be a parameter, if δ > 0, then FUZZY PRIORITIZED WEIGHTED AVERAGE OPERATOR

FGPWA (δ a˜ 1 , δ a˜ 2 , . . . , δ a˜ n ) = δ (FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ))

(39)

Proof. According to the operational laws on triangular fuzzy numbers given in Definition 6, we have δ (a˜ i ) = δaiL , δaiM , δaiU .

(40)

Then, FGPWA (δ a˜ 1 , δ a˜ 2 , . . . , δ a˜ n ) ⎡ 1/λ 1/λ n

n

L λ M λ T T i i

n

n =⎣ δai δai , , i=1 Ti i=1 Ti i=1 i=1

n

Ti

n

i=1

i=1

Ti

⎡ n

Ti

n = ⎣δ i=1

i=1

δ

n

Ti

n i=1

i=1

U λ δai

Ti

Ti

⎤ 1/λ ⎦

L λ ai

U λ ai

(41)

1/λ ,δ

⎤ 1/λ ⎦.

n

i=1

Ti

n

i=1

Ti

M λ ai

1/λ ,

On the other hand, δ (FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n )) ⎡ 1/λ 1/λ n

n

L λ M λ T T i i

n

n = δ⎣ ai ai , , i=1 Ti i=1 Ti i=1 i=1

n

i=1

Ti

n

i=1

Ti

U λ ai

⎤ 1/λ ⎦,


(42) DOI 10.1002/int

38

⎡ n

Ti ⎣

n = δ i=1

i=1

δ

n

i=1

VERMA AND SHARMA

Ti

Ti

n

i=1

Ti

L λ

1/λ

ai

U λ ai

,δ

⎤ 1/λ ⎦.

n

Ti

n

i=1

i=1

Ti

M λ ai

1/λ ,

Thus, FGPWA (δ a˜ 1 , δ a˜ 2 , . . . , δ a˜ n ) = δ (FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n )) . This proves the theorem. L M U (i = 1, 2, . . . , n) be a set of triangular fuzzy THEOREM 8. Let a˜ i = ai , ai , ai i−1 (i = 2, 3, . . . , n), T1 = 1, E a˜ j be the exnumbers, where Ti = j =1 E a˜ j L M U pected value of a˜ j = aj , aj , aj , and λ ∈ (−∞, 0) ∪ (0, ∞) be a parameter. If δ > 0 and b˜ = bL , bM , bU is a triangular fuzzy number, then ˜ δ a˜ 2 ⊗ b, ˜ . . . , δ a˜ n ⊗ b˜ = δ (FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n )) ⊗ b. ˜ FGPWA δ a˜ 1 ⊗ b, (43) Proof. The proof simply follows from Theorem 5 and Theorem 7. L M U (i = 1, 2, . . . , n) be a set of triangular fuzzy THEOREM 9. Let a˜ i = ai , ai , ai ˜ (i E a = 2, 3, . . . , n), T1 = 1, E a˜ j be the exnumbers, where Ti = i−1 j j =1 pected value of a˜ j = ajL , ajM , ajU , and λ ∈ (−∞, 0) ∪ (0, ∞) be a parameter. L M U If δ, η > 0 and b˜ = b , b , b is a triangular fuzzy number, then

(i) FGPWA δ a˜ 1 ⊗ b˜ η , δ a˜ 2 ⊗ b˜ η , . . . , δ a˜ n ⊗ b˜ η = δ (FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n )) ⊗ b˜ η ,

(44)

˜ δ a˜ 2 ⊗ ηb, ˜ . . . , δ a˜ n ⊗ ηb˜ (ii) FGPWA δ a˜ 1 ⊗ ηb, ˜ = δ (FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n )) ⊗ ηb.

(45)

Proof. These results follow directly from Theorems 5, 6, and 7. Special cases of FGPWA operator are shown as follows: (1) If λ = 1, then the FGPWA operator reduces to the FPWA operator:29 T1 FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) = n

i=1 Ti

T2 a˜ 1 ⊕ n

i=1 Ti


Tn a˜ 2 ⊕ · · · ⊕ n

DOI 10.1002/int

i=1

Ti

a˜ n .

(46)

39

FUZZY PRIORITIZED WEIGHTED AVERAGE OPERATOR (2) If λ → 0, then the FGPWA operator reduces to the FPWG operator:29 T

n 1 i=1 Ti

FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) = a˜ 1

T

n 2 i=1 Ti

nTn i=1 Ti

⊗ a˜ 2

⊗ · · · ⊗ a˜ n

(47)

.

(3) If λ = −1, then the FGPWA operator reduces to the FPWHA operator:29 FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) =

nT1

i=1 Ti

a˜ 1

⊕

1

nT2

i=1 Ti

⊕ ···

a˜ 2

nTn

.

(48)

i=1 Ti

a˜ n

(4) If λ = 2, then the FGPWA operator gives

FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) =

T1

n i=1

Ti

T2 a˜ 12 ⊕ n

Ti

i=1

Tn a˜ 22 ⊕ · · · ⊕ n i=1

Ti

a˜ n2

1/ 2 (49)

which we call the fuzzy prioritized weighted quadratic average operator. (5) If λ = 3, then the FGPWA operator gives

FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) =

T1

n i=1

Ti

T2 a˜ 13 ⊕ n

Ti

i=1

Tn a˜ 23 ⊕ · · · ⊕ n i=1

Ti

a˜ n3

1/ 3 (50)

which we call the fuzzy prioritized weighted cubic average operator. (6) If λ = 1 and the priority levels of the aggregated arguments reduced to the same levels, then the FGPWA operator reduces to the fuzzy weighted average operator n ⊕ wi a˜ i .

FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) =

(51)

i=1

(7) If λ → 0 and the priority levels of the aggregated arguments are reduced to the same levels, then the FGPWA operator reduces to the fuzzy weighted geometric operator

FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) =

wi

n

⊗ a˜ i

.

(52)

i=1

(8) If λ = −1 and the priority levels of the aggregated arguments are reduced to the same levels, then the FGPWA operator reduces to the fuzzy weighted harmonic average operator:14 ⎛ ⎜ FGPWA (a˜ 1 , a˜ 2 , . . . , a˜ n ) = ⎝

⎞ 1 n

⊕

i=1

wi a˜ i

⎟ ⎠.

(53)

In the next section, we suggest applications of the FGPWA operator, to solve multiattribute decision making problems with triangular fuzzy information and also give an illustrative example. International Journal of Intelligent Systems

DOI 10.1002/int

40

VERMA AND SHARMA

4.

APPLICATIONS OF FGPWA OPERATOR IN MULTIPLE ATTRIBUTE GROUP DECISION MAKING

Let us consider a multiple attribute group decision making problem involving a set of options X = {X1 , X2 , . . . , Xm } to be considered under a set of attributes G = {G1 , G2 , . . . , Gn } and let there be a prioritization between the attributes expressed by the linear ordering G1 G2 ·!· · Gn (indicating " that the attribute Gi has a higher priority than Gj , if i < j ), D = D1 , D2 , . . . , Dq be the set of decision makers, and also, let there be a prioritization between the decision makers expressed by the linear ordering D1 D2 · · · Dn , indicating that the decision Dη has a higher maker (k) M (k) U (k) (k) L (k) priority than Dς , if η < ς. Let A˜ = a˜ ij m×n = aij , aij , aij (k) (k) (k) be an be a triangular fuzzy decision matrix, and a˜ ij(k) = aijL , aijM , aijU attribute value provided by the decision maker Dk ∈ D, for the alternative Xi ∈ X with respect to the attribute Gj ∈ G. Using the FGPWA operator, we now formulate an algorithm to solve multiple attribute group decision making problems with triangular fuzzy information: Step 1. First we transform the triangular fuzzy decision matrices A˜ (k) = a˜ ij(k) = m×n (k) (k) (k) into normalized triangular fuzzy decision matrices R˜ (k) = , aijU aijL , aijM (k) (k) U (k) r˜ij(k) using the following formulas: = rijL , rijM , rij m×n

⎧ (k) (k) # n U (k) L = aijL , ⎪ i=1 aij ⎪ rij ⎪ ⎪ ⎨ # (k) (k) n M (k) = aijM , rM i=1 aij ⎪ ij ⎪ ⎪ # ⎪ ⎩ U (k) U (k) n L (k) = aij . rij i=1 aij

(54)

for the benefit attribute Gj , i = 1, 2, . . . , m; j = 1, 2, . . . , n and k = 1, 2, . . . , q. And ⎧ (k) # U (k) # n # (k) rijL aijL , = 1 aij ⎪ i=1 1 ⎪ ⎪ ⎪ ⎨ # # # (k) (k) (k) n aijM , = 1 aijM rM i=1 1 ⎪ ij ⎪ ⎪ # # # ⎪ U (k) (k) n ⎩ U (k) aij . = 1 aijL rij i=1 1

(55)

for cost attribute Gj , i = 1, 2, . . . , m; j = 1, 2, . . . , n and k = 1, 2, . . . , q. Step 2. Calculate the values of Tij(k) (k = 1, 2, . . . , q) as follows: Tij(k) =

k−1 $

γ E r˜ij (k = 2, 3, . . . , q) ,

(56)

γ =1

Tij(1) = 1. International Journal of Intelligent Systems

(57) DOI 10.1002/int

41

FUZZY PRIORITIZED WEIGHTED AVERAGE OPERATOR Step 3. Utilize the FGPWA operator: (q) r˜ij = rijL , rijM , rijU = FGPWA r˜ij(1) , r˜ij(2) , . . . , r˜ij =

Tij(1)

q

k=1

Tij(k)

r˜ij(1)

λ

(q) λ λ Tij(2) Tij (q) r˜ij(2) ⊕ · · · ⊕ q r˜ij ⊕ q (k) (k) T T k=1 ij k=1 ij

1/λ (58)

⎡⎛ ⎛ ⎛ λ ⎞⎞1/λ ⎛ λ ⎞⎞1/λ (k) (k) q q Tij(k) rijL Tij(k) rijM ⎢⎜ ⎜ ⎜ ⎟⎟ ⎜ ⎟⎟ ⎢ = ⎣⎝ ⎝ q ⎠⎠ , ⎝ ⎝ q ⎠⎠ , T (k) T (k) k=1

k=1

ij

k=1

k=1

ij

⎛ ⎛ λ ⎞⎞1/λ⎤ (k) q Tij(k) rijU ⎜ ⎜ ⎟⎟ ⎥ ⎝ ⎝ q ⎠⎠ ⎥ (k) ⎦, T k=1 ij k=1 ˜ (k) to aggregate all the individual triangular fuzzy decision matrix R = (k) (k = 1, 2, . . . , q) into the collective triangular fuzzy decision matrix r˜ij m×n ˜ R = r˜ij , i = 1, 2, . . . , m, j = 1, 2, . . . , n m×n

Step 4. Calculate the values Tij Tij =

j −1 $

(i = 1, 2, . . . , m, j = 1, 2, . . . , n), as follows:

E (˜r1ν )

(i = 1, 2, . . . , m, j = 2, 3, . . . , n) ,

(59)

ν=1

Ti1 = 1, i = 1, 2, . . . , m.

(60)

Step 5. Aggregate all triangular fuzzy values r˜ij (j = 1, 2, . . . , n) for each option Xi (i = 1, 2, . . . , m) by the FGPWA operator: r˜i = riL , riM , riU = GFPWA (˜ri1 , r˜i2 , . . . , r˜in ) 1/λ Ti1 Ti2 Tin (˜ri1 )λ ⊕ n (˜ri2 )λ ⊕ · · · ⊕ n (˜rin )λ = n j =1 Tij j =1 Tij j =1 Tij

(61)

⎡⎛ λ ⎞1/λ ⎛ n λ ⎞1/λ ⎛ n ⎞1/λ⎤ n Tij rijM Tij rijM Tij rijL ⎠ ,⎝ ⎠ ,⎝ ⎠ ⎦,

n

n

n = ⎣⎝ j =1 Tij j =1 Tij j =1 Tij j =1 j =1 j =1 i = 1, 2, . . . , m to derive the overall triangular fuzzy preference values r˜i (i = 1, 2, . . . , m) of the alternative Xi . Step 6. Calculate the expected values as follows: E (˜ri ) =

riL + riM + riU , 3

i = 1, 2, . . . , m.


DOI 10.1002/int

(62)

42

VERMA AND SHARMA Table I. Triangular fuzzy decision matrix A(1) .

X1 X2 X3 X4 X5

G1

G2

G3

G4

[0.80, 0.85, 0.90] [0.88, 0.90, 0.93] [0.95, 0.97, 0.98] [0.82, 0.85, 0.88] [0.78, 0.79, 0.81]

[0.72, 0.76, 0.80] [0.67, 0.77, 0.83] [0.90, 0.93, 0.95] [0.97, 0.98, 1.00] [0.78, 0.79, 0.81]

[0.91, 0.93, 0.96] [0.60, 0.67, 0.70] [0.77, 0.79, 0.82] [0.98, 0.99, 1.00] [0.83, 0.85, 0.88]

[0.62, 0.65, 0.68] [0.69, 0.72, 0.75] [0.93, 0.95, 0.96] [0.97, 0.99, 1.00] [0.80, 0.85, 0.90]

Table II. Triangular fuzzy decision matrix A(2) .

X1 X2 X3 X4 X5

G1

G2

G3

G4

[0.42, 0.47, 0.52] [0.50, 0.52, 0.55] [0.57, 0.59, 0.60] [0.44, 0.47, 0.50] [0.40, 0.41, 0.43]

[0.34, 0.38, 0.42] [0.29, 0.39, 0.45] [0.52, 0.55, 0.57] [0.59, 0.60, 0.62] [0.40, 0.41, 0.43]

[0.53, 0.55, 0.58] [0.22, 0.29, 0.32] [0.39, 0.41, 0.44] [0.60, 0.61, 0.62] [0.45, 0.47, 0.50]

[0.24, 0.27, 0.30] [0.31, 0.34, 0.37] [0.55, 0.57, 0.58] [0.59, 0.61, 0.62] [0.56, 0.59, 0.61]

Table III. Triangular fuzzy decision matrix A(3) .

X1 X2 X3 X4 X5

G1

G2

G3

G4

[0.65, 0.70, 0.75] [0.73, 0.75, 0.78] [0.80, 0.82, 0.83] [0.67, 0.70, 0.73] [0.63, 0.64, 0.66]

[0.57, 0.61, 0.65] [0.52, 0.62, 0.68] [0.75, 0.78, 0.80] [0.82, 0.83, 0.85] [0.63, 0.64, 0.68]

[0.76, 0.78, 0.81] [0.45, 0.52, 0.50] [0.62, 0.64, 0.67] [0.83, 0.84, 0.85] [0.68, 0.70, 0.76]

[0.47, 0.50, 0.53] [0.54, 0.57, 0.60] [0.78, 0.80, 0.81] [0.82, 0.84, 0.85] [0.79, 0.82, 0.80]

Table IV. Normalized triangular fuzzy decision matrix R (1) .

X1 X2 X3 X4 X5

G1

G2

G3

G4

[0.178, 0.195, 0.213] [0.196, 0.206, 0.220] [0.211, 0.223, 0.232] [0.182, 0.195, 0.208] [0.173, 0.181, 0.192]

[0.164, 0.180, 0.198] [0.153, 0.182, 0.205] [0.205, 0.220, 0.235] [0.221, 0.232, 0.250] [0.178, 0.187, 0.205]

[0.209, 0.220, 0.235] [0.138, 0.158, 0.171] [0.177, 0.187, 0.201] [0.225, 0.234, 0.245] [0.190, 0.201, 0.215]

[0.146, 0.152, 0.164] [0.158, 0.168, 0.181] [0.212, 0.222, 0.231] [0.222, 0.231, 0.241] [0.215, 0.227, 0.239]

Step 7. Rank all the alternatives Xi (i = 1, 2, . . . , m) and select the best one(s) in accordance with the expected values E (˜ri ) (i = 1, 2, . . . , m). To demonstrate the applicability of the proposed method to multiple attribute group decision making, we consider a university faculty recruitment group decision making problem below.

Example. The department of mathematics in a university wants to appoint outstanding mathematics teachers. The appointment is done by a committee of three decision makers, vice-chancellor D1 , dean of academics D2 , and human resource officer D3 . After preliminary screening, five teachers Xi (i = 1, 2, 3, 4, 5) remain for further evaluation. Panel of decision makers made strict evaluation for five teachers Xi (i = 1, 2, 3, 4, 5) according to the following four attributes: (1) G1 , the past experience, (2) G2 , the research capability, (3) G3 , subject knowledge, (4) International Journal of Intelligent Systems

DOI 10.1002/int

43

FUZZY PRIORITIZED WEIGHTED AVERAGE OPERATOR Table V. Normalized triangular fuzzy decision matrix R (2) .

X1 X2 X3 X4 X5

G1

G2

G3

G4

[0.162, 0.191, 0.223] [0.192, 0.211, 0.236] [0.219, 0.240, 0.258] [0.169, 0.191, 0.215] [0.154, 0.167, 0.185]

[0.137, 0.163, 0.196] [0.116, 0.167, 0.210] [0.209, 0.236, 0.266] [0.237, 0.258, 0.290] [0.161, 0.176, 0.201]

[0.215, 0.236, 0.265] [0.089, 0.124, 0.146] [0.159, 0.176, 0.201] [0.224, 0.262, 0.283] [0.183, 0.202, 0.228]

[0.097, 0.113, 0.133] [0.125, 0.143, 0.164] [0.222, 0.240, 0.258] [0.238, 0.256, 0.276] [0.226, 0.248, 0.271]

Table VI. Normalized triangular fuzzy decision matrix R (3) .

X1 X2 X3 X4 X5

G1

G2

G3

G4

[0.173, 0.194, 0.216] [0.194, 0.209, 0.224] [0.213, 0.227, 0.239] [0.179, 0.194, 0.210] [0.168, 0.177, 0.190]

[0.156, 0.175, 0.198] [0.142, 0.178, 0.207] [0.205, 0.224, 0.243] [0.224, 0.239, 0.258] [0.172, 0.184, 0.207]

[0.212, 0.224, 0.243] [0.125, 0.149, 0.150] [0.173, 0.184, 0.201] [0.231, 0.241, 0.255] [0.189, 0.201, 0.228]

[0.130, 0.142, 0.156] [0.150, 0.161, 0.176] [0.217, 0.227, 0.238] [0.228, 0.238, 0.250] [0.220, 0.232, 0.235]

Table VII. Collective triangular fuzzy decision matrix R ∗ (for λ = −1).

X1 X2 X3 X4 X5

G1

G2

G3

G4

[0.175, 0.194, 0.215] [0.195, 0.207, 0.222] [0.212, 0.226, 0.236] [0.180, 0.194, 0.209] [0.170, 0.179, 0.191]

[0.160, 0.177, 0.198] [0.146, 0.180, 0.206] [0.206, 0.223, 0.240] [0.224, 0.237, 0.257] [0.175, 0.185, 0.204]

[0.210, 0.223, 0.240] [0.128, 0.152, 0.167] [0.174, 0.185, 0.201] [0.225, 0.239, 0.252] [0.189, 0.201, 0.217]

[0.137, 0.145, 0.159] [0.152, 0.164, 0.178] [0.216, 0.225, 0.236] [0.224, 0.234, 0.246] [0.217, 0.231, 0.244]

Table VIII. Collective triangular fuzzy decision matrix R ∗ (for λ → 0).

X1 X2 X3 X4 X5

G1

G2

G3

G4

[0.175, 0.194, 0.215] [0.195, 0.207, 0.222] [0.213, 0.226, 0.237] [0.180, 0.194, 0.209] [0.170, 0.179, 0.191]

[0.160, 0.177, 0.198] [0.147, 0.180, 0.206] [0.206, 0.223, 0.240] [0.224, 0.237, 0.257] [0.175, 0.185, 0.204]

[0.210, 0.223, 0.240] [0.130, 0.153, 0.167] [0.174, 0.185, 0.201] [0.225, 0.239, 0.252] [0.189, 0.201, 0.217]

[0.138, 0.146, 0.159] [0.153, 0.164, 0.178] [0.217, 0.225, 0.236] [0.224, 0.234, 0.246] [0.217, 0.231, 0.244]

G4 , the teaching skill. During the evolution process, the university vice-chancellor (D1 ) has the absolute priority for decision making, and the dean of academics comes next. The prioritization relationship for the attributes is as follows:G1 G2 G3 G4 . The three decision makers evaluated the candidates Xi (i = 1, 2, 3, 4, 5) with respect to the attributes Gj (j = 1, 2, 3, 4) and provided their evaluation values in terms of triangular fuzzy numbers constructed the following three triangular and (q) (q) (q = 1, 2, 3) (see Tables I–III) fuzzy decision matrices A = aij 5×4

Step 1. Since all the attributes Gj (j = 1, 2, 3, 4) are of the benefit type, we utilize Equation (54) to calculate the normalized triangular fuzzy decision matrices R (q) (q = 1, 2, 3). The results are shown in Tables IV–VI. International Journal of Intelligent Systems

DOI 10.1002/int

44

VERMA AND SHARMA Table IX. Collective triangular fuzzy decision matrix R ∗ (for λ = 0.4).

X1 X2 X3 X4 X5

G1

G2

G3

G4

[0.175, 0.194, 0.215] [0.195, 0.207, 0.223] [0.213, 0.226, 0.237] [0.180, 0.194, 0.209] [0.170, 0.179, 0.191]

[0.160, 0.177, 0.198] [0.147, 0.180, 0.206] [0.206, 0.223, 0.241] [0.224, 0.237, 0.257] [0.175, 0.185, 0.204]

[0.210, 0.223, 0.240] [0.131, 0.153, 0.167] [0.174, 0.185, 0.201] [0.225, 0.239, 0.252] [0.189, 0.201, 0.218]

[0.139, 0.146, 0.160] [0.153, 0.164, 0.178] [0.212, 0.225, 0.236] [0.225, 0.236, 0.248] [0.217, 0.231, 0.243]

Table X. Collective triangular fuzzy decision matrix R ∗ (for λ = 0.8).

X1 X2 X3 X4 X5

G1

G2

G3

G4

[0.175, 0.194, 0.215] [0.195, 0.207, 0.223] [0.213, 0.226, 0.237] [0.180, 0.194, 0.209] [0.170, 0.179, 0.191]

[0.160, 0.177, 0.198] [0.147, 0.180, 0.206] [0.206, 0.223, 0.241] [0.224, 0.237, 0.258] [0.175, 0.185, 0.204]

[0.210, 0.223, 0.241] [0.131, 0.153, 0.167] [0.174, 0.185, 0.201] [0.225, 0.239, 0.252] [0.189, 0.201, 0.218]

[0.139, 0.147, 0.160] [0.153, 0.164, 0.176] [0.214, 0.225, 0.236] [0.225, 0.236, 0.248] [0.217, 0.231, 0.244]

Table XI. Collective triangular fuzzy decision matrix R ∗ (for λ = 1).

X1 X2 X3 X4 X5

G1

G2

G3

G4

[0.175, 0.194, 0.215] [0.195, 0.207, 0.223] [0.213, 0.226, 0.237] [0.180, 0.194, 0.209] [0.170, 0.179, 0.191]

[0.160, 0.177, 0.198] [0.147, 0.180, 0.206] [0.206, 0.223, 0.241] [0.224, 0.237, 0.258] [0.175, 0.185, 0.204]

[0.210, 0.223, 0.241] [0.131, 0.153, 0.167] [0.174, 0.185, 0.201] [0.225, 0.239, 0.252] [0.189, 0.201, 0.218]

[0.139, 0.147, 0.160] [0.153, 0.164, 0.179] [0.217, 0.225, 0.236] [0.226, 0.237, 0.249] [0.217, 0.231, 0.245]

Table XII. Collective triangular fuzzy decision matrix R ∗ (for λ = 2).

X1 X2 X3 X4 X5

G1

G2

G3

G4

[0.175, 0.194, 0.215] [0.195, 0.207, 0.223] [0.213, 0.226, 0.237] [0.180, 0.194, 0.209] [0.170, 0.179, 0.191]

[0.160, 0.177, 0.198] [0.148, 0.180, 0.206] [0.206, 0.223, 0.241] [0.224, 0.237, 0.258] [0.175, 0.185, 0.204]

[0.210, 0.223, 0.241] [0.132, 0.154, 0.168] [0.174, 0.185, 0.201] [0.225, 0.240, 0.253] [0.189, 0.201, 0.218]

[0.140, 0.147, 0.160] [0.154, 0.165, 0.179] [0.214, 0.225, 0.236] [0.225, 0.236, 0.248] [0.217, 0.231, 0.245]

Step 2. Utilize Equations (56) and (57) to calculate the Tij(1) , Tij(2) , and Tij(3) , given by

Tij(1)

Tij(3)

⎡ 1 1 ⎢1 1 ⎢ = ⎢1 1 ⎣1 1 1 1 ⎡ 0.037 ⎢0.044 ⎢ = ⎢0.053 ⎣0.037 0.030

1 1 1 1 1

⎤ 1 1⎥ ⎥ 1⎥ , 1⎦ 1

0.030 0.030 0.052 0.061 0.034

⎡

Tij(2)

0.052 0.019 0.038 0.060 0.041

0.195 ⎢0.207 ⎢ = ⎢0.222 ⎣0.195 0.182 ⎤ 0.018 0.024⎥ ⎥ 0.053⎥ . 0.059⎦ 0.056


0.181 0.180 0.220 0.234 0.190

0.221 0.156 0.188 0.235 0.202

DOI 10.1002/int

⎤ 0.154 0.169⎥ ⎥ 0.222⎥ , 0.231⎦ 0.227


45

∗

Table XIII. Collective triangular fuzzy decision matrix R (for λ = 7).

X1 X2 X3 X4 X5

G1

G2

G3

G4

[0.176, 0.194, 0.215] [0.195, 0.207, 0.223] [0.213, 0.227, 0.238] [0.180, 0.194, 0.209] [0.171, 0.179, 0.191]

[0.161, 0.178, 0.198] [0.150, 0.180, 0.206] [0.206, 0.224, 0.243] [0.225, 0.238, 0.261] [0.176, 0.185, 0.205]

[0.210, 0.224, 0.242] [0.135, 0.155, 0.166] [0.175, 0.186, 0.201] [0.225, 0.241, 0.255] [0.189, 0.201, 0.218]

[0.143, 0.149, 0.161] [0.155, 0.166, 0.179] [0.213, 0.226, 0.238] [0.226, 0.237, 0.251] [0.217, 0.232, 0.247]

Table XIV. Collective triangular fuzzy decision matrix R ∗ (for λ = 10).

X1 X2 X3 X4 X5

G1

G2

G3

G4

[0.176, 0.194, 0.215] [0.195, 0.207, 0.224] [0.213, 0.227, 0.239] [0.180, 0.194, 0.209] [0.171, 0.179, 0.191]

[0.161, 0.178, 0.198] [0.151, 0.180, 0.206] [0.206, 0.224, 0.244] [0.225, 0.239, 0.263] [0.176, 0.186, 0.205]

[0.210, 0.224, 0.243] [0.136, 0.156, 0.169] [0.175, 0.186, 0.201] [0.225, 0.242, 0.257] [0.189, 0.201, 0.218]

[0.144, 0.150, 0.162] [0.156, 0.166, 0.180] [0.215, 0.227, 0.240] [0.227, 0.239, 0.254] [0.218, 0.233, 0.250]

Table XV. Collective triangular fuzzy decision matrix R ∗ (for λ = 15).

X1 X2 X3 X4 X5

G1

G2

G3

G4

[0.176, 0.194, 0.215] [0.195, 0.207, 0.224] [0.213, 0.228, 0.240] [0.181, 0.194, 0.209] [0.171, 0.180, 0.191]

[0.162, 0.178, 0.198] [0.151, 0.181, 0.206] [0.206, 0.224, 0.246] [0.225, 0.241, 0.266] [0.176, 0.186, 0.205]

[0.210, 0.224, 0.245] [0.136, 0.156, 0.169] [0.175, 0.186, 0.201] [0.225, 0.244, 0.260] [0.189, 0.201, 0.219]

[0.145, 0.151, 0.162] [0.156, 0.166, 0.180] [0.215, 0.227, 0.240] [0.227, 0.239, 0.254] [0.218, 0.233, 0.250]

Table XVI. Collective triangular fuzzy decision matrix R ∗ (for λ = 25).

X1 X2 X3 X4 X5

G1

G2

G3

G4

[0.177, 0.195, 0.216] [0.195, 0.207, 0.225] [0.213, 0.229, 0.244] [0.181, 0.195, 0.210] [0.172, 0.180, 0.191]

[0.163, 0.179, 0.198] [0.152, 0.181, 0.206] [0.205, 0.226, 0.250] [0.227, 0.244, 0.272] [0.177, 0.186, 0.205]

[0.211, 0.226, 0.249] [0.137, 0.157, 0.170] [0.176, 0.186, 0.201] [0.225, 0.247, 0.266] [0.189, 0.201, 0.219]

[0.145, 0.151, 0.163] [0.157, 0.167, 0.180] [0.215, 0.229, 0.243] [0.228, 0.242, 0.259] [0.218, 0.235, 0.255]

Table XVII. Collective triangular fuzzy decision matrix R ∗ (for λ = 50). 0 X1 X2 X3 X4 X5

G1

G2

G3

G4

[0.177, 0.195, 0.217] [0.195, 0.208, 0.228] [0.214, 0.232, 0.249] [0.181, 0.195, 0.210] [0.172, 0.180, 0.191]

[0.163, 0.179, 0.198] [0.152, 0.181, 0.206] [0.206, 0.228, 0.257] [0.230, 0.250, 0.280] [0.177, 0.186, 0.205]

[0.211, 0.229, 0.256] [0.138, 0.158, 0.170] [0.176, 0.186, 0.201] [0.225, 0.247, 0.273] [0.189, 0.201, 0.221]

[0.146, 0.152, 0.164] [0.157, 0.167, 0.180] [0.216, 0.232, 0.249] [0.231, 0.248, 0.267] [0.220, 0.240, 0.262]

Step 3. Use the FGPWA operator Equation (58) to aggregate all the individual decision matrices R (q) (q = 1, 2, 3) into the collective decision matrix R ∗ = rij 5×4 = aijL , aijM , aijU 5×4 ; taking different values of λ: λ = −1, λ → 0, λ = 0.4, λ = 0.8, λ = 1, λ = 2, λ = 7, λ = 10, λ = 15, λ = 25, and λ = 50, we get the following tables: Tables VII–XVII. International Journal of Intelligent Systems

DOI 10.1002/int

46

VERMA AND SHARMA

Table XVIII. Values of Tij (i = 1, 2, . . . , m, j = 1, 2, . . . , n) taking different values of λ. For λ = −1 ⎡ 1 0.1947 ⎢1 0.2080 ⎢ Tij = ⎢ ⎢1 0.2247 ⎣1 0.1943 1 0.1800 For λ = 0.4 ⎡ 1 0.1947 ⎢1 0.2083 ⎢ Tij = ⎢ ⎢1 0.2253 ⎣1 0.1943 1 0.1800 For λ = 1 ⎡ 1 0.1947 ⎢1 0.2083 ⎢ Tij = ⎢ ⎢1 0.2253 ⎣1 0.1943 1 0.1800 For λ = 7 ⎡ 1 ⎢1 ⎢ Tij = ⎢ ⎢1 ⎣1 1

0.1950 0.2083 0.2260 0.1943 0.1803

For λ = 15 ⎡ 1 0.1950 ⎢1 0.2087 ⎢ Tij = ⎢ ⎢1 0.2270 ⎣1 0.1947 1 0.1807 For λ = 50 ⎡ 1 0.1963 ⎢1 0.2103 ⎢ Tij = ⎢ ⎢1 0.2317 ⎣1 0.1953 1 0.1810

⎤ 0.0078 0.0055⎥ ⎥ 0.0094⎥ ⎥ 0.0111⎦

0.0347 0.0369 0.0501 0.0465 0.0338

0.0068

0.0347 0.0370 0.0503 0.0465 0.0338

⎤ 0.0078 0.0055⎥ ⎥ 0.0094⎥ ⎥ 0.0111⎦ 0.0069

0.0347 0.0370 0.0503 0.0466 0.0338

⎤ 0.0078 0.0055⎥ ⎥ 0.0094⎥ ⎥ 0.0111⎦ 0.0069

0.0349 0.0372 0.0507 0.0469 0.0340 0.0350 0.0374 0.0512 0.0475 0.0341 0.0353 0.0378 0.0534 0.0495 0.0343

⎤ 0.0079 0.0057⎥ ⎥ 0.0095⎥ ⎥ 0.0113⎦ 0.0069 ⎤ 0.0079 0.0058⎥ ⎥ 0.0096⎥ ⎥ 0.0115⎦ 0.0069

For λ → 0 ⎡ 1 0.1947 ⎢1 0.2083 ⎢ Tij = ⎢ ⎢1 0.2253 ⎣1 0.1943 1 0.1800 For λ = 0.8 ⎡ 1 0.1947 ⎢1 0.2083 ⎢ Tij = ⎢ ⎢1 0.2253 ⎣1 0.1943 1 0.1800 For λ = 2 ⎡ 1 0.1947 ⎢1 0.2083 ⎢ Tij = ⎢ ⎢1 0.2253 ⎣1 0.1943 1 0.1800

⎤ 0.0078 0.0055⎥ ⎥ 0.0094⎥ ⎥ 0.0111⎦

0.0347 0.0370 0.0502 0.0465 0.0338

0.0068

0.0347 0.0370 0.0503 0.0466 0.0338

⎤ 0.0078 0.0056⎥ ⎥ 0.0094⎥ ⎥ 0.0111⎦ 0.0069

0.0347 0.0371 0.0503 0.0466 0.0338

⎤ 0.0078 0.0056⎥ ⎥ 0.0094⎥ ⎥ 0.0111⎦ 0.0069

For λ = 10 ⎡ 1 0.1950 ⎢1 0.2083 ⎢ Tij = ⎢ ⎢1 0.2260 ⎣1 0.1943 1 0.1803

0.0349 0.0372 0.0507 0.0469 0.0340

For λ = 25 ⎡ 1 0.1960 ⎢1 0.2090 ⎢ Tij = ⎢ ⎢1 0.2287 ⎣1 0.1953 1 0.1810

0.0353 0.0376 0.0519 0.0484 0.0343

⎤ 0.0079 0.0057⎥ ⎥ 0.0095⎥ ⎥ 0.0113⎦ 0.0069 ⎤ 0.0081 0.0058⎥ ⎥ 0.0097⎥ ⎥ 0.0119⎦ 0.0070

⎤ 0.0082 0.0059⎥ ⎥ 0.0100⎥ ⎥ 0.0123⎦ 0.0070

Step 4. Use expressions in Equations (59) and (60) to calculate the Tij (i = 1, 2, . . . , m, j = 1, 2, . . . , n). It gives Table XVIII. Step 5. Aggregate all triangular preference values r˜ij (j = 1, 2, . . . , n) by using the FGPWA operator (Equation 61), to derive the overall triangular fuzzy preference values r˜i (i = 1, 2, . . . , m) of the teachers Xi (Table XIX): Step 6. Calculate the expected values E (˜ri ) of the teachers Xi (Table XX): Step 7. Rank all the teachers Xi (i = 1, 2, 3, 4, 5) in accordance with the expected values E (˜ri ) (i = 1, 2, 3, 4, 5) of the overall triangular fuzzy values r˜i (i = 1, 2, . . . , m) (Table XXI).

We thus find the change in order of the rankings and this brings in the role of parameter λ. International Journal of Intelligent Systems

DOI 10.1002/int


47

Table XIX. Preference values r˜i (i = 1, 2, . . . , m) of the teachers taking different values of λ. For λ = −1, r˜1 = [0.1730, 0.1914, 0.2123], r˜2 = [0.1818, 0.1997, 0.2169], r˜3 = [0.2092, 0.2235, 0.2351], r˜4 = [0.1874, 0.2014, 0.2170], r˜5 = [0.1714, 0.1806, 0.1937].

For λ → 0, r˜1 = [0.1732, 0.1916, 0.2125], r˜2 = [0.1836, 0.2002, 0.2172], r˜3 = [0.2101, 0.2237, 0.2360], r˜4 = [0.1881, 0.2020, 0.2176], r˜5 = [0.1715, 0.1807, 0.1938].

For λ = 0.4, r˜1 = [0.1733, 0.1917, 0.2126], r˜2 = [0.1842, 0.2004, 0.2181], r˜3 = [0.2101, 0.2238, 0.2362], r˜4 = [0.1884, 0.2023, 0.2179], r˜5 = [0.1715, 0.1807, 0.1939]. For λ = 1, r˜1 = [0.1734, 0.1918, 0.2127], r˜2 = [0.1849, 0.2007, 0.2183], r˜3 = [0.2103, 0.2239, 0.2363], r˜4 = [0.1889, 0.2027, 0.2186], r˜5 = [0.1715, 0.1808, 0.1940]. For λ = 7, r˜1 = [0.1758, 0.1932, 0.2138]. r˜2 = [0.1898, 0.2028, 0.2197], r˜3 = [0.2109, 0.2255, 0.2380], r˜4 = [0.1952, 0.2085, 0.2254], r˜5 = [0.1730, 0.1815, 0.1952]. For λ = 15, r˜1 = [0.1787, 0.1952, 0.2158], r˜2 = [0.1922, 0.2043, 0.2215], r˜3 = [0.2114, 0.2267, 0.2407], r˜4 = [0.2042, 0.2191, 0.2396], r˜5 = [0.1748, 0.1847, 0.1985]. For λ = 50, r˜1 = [0.1965, 0.2133, 0.2384], r˜2 = [0.1941, 0.2071, 0.2270], r˜3 = [0.2131, 0.2313, 0.2515], r˜4 = [0.2223, 0.2417, 0.2702], r˜5 = [0.1984, 0.2165, 0.2369].

For λ = 0.8, r˜1 = [0.1734, 0.1918, 0.2127], r˜2 = [0.1847, 0.2006, 0.2182], r˜3 = [0.2102, 0.2238, 0.2363], r˜4 = [0.1887, 0.2026, 0.2184], r˜5 = [0.1715, 0.1808, 0.1939]. For λ = 2, r˜1 = [0.1736, 0.1920, 0.2129], r˜2 = [0.1862, 0.2012, 0.2186], r˜3 = [0.2104, 0.2240, 0.2364], r˜4 = [0.1897, 0.2035, 0.2194], r˜5 = [0.1716, 0.1809, 0.1941]. For λ = 10, r˜1 = [0.1767, 0.1939, 0.2144], r˜2 = [0.1910, 0.2035, 0.2210], r˜3 = [0.2112, 0.2257, 0.2392], r˜4 = [0.1987, 0.2123, 0.2308], r˜5 = [0.1735, 0.1823, 0.1962]. For λ = 25, r˜1 = [0.1853, 0.2003, 0.2212], r˜2 = [0.1933, 0.2052, 0.2232], r˜3 = [0.2116, 0.2281, 0.2450], r˜4 = [0.2128, 0.2296, 0.2541], r˜5 = [0.1810, 0.1935, 0.2098].

Table XX. Expected values E (˜ri ) (i = 1, 2, 3, 4, 5) of the teachers taking different values of λ. λ = −1 λ→0 λ = 0.4 λ = 0.8 λ=1 λ=2 λ=7 λ = 10 λ = 15 λ = 25 λ = 50

E (˜r1 ) = 0.1922, E (˜r1 ) = 0.1924, E (˜r1 ) = 0.1925, E (˜r1 ) = 0.1926, E (˜r1 ) = 0.1926, E (˜r1 ) = 0.1928, E (˜r1 ) = 0.1943, E (˜r1 ) = 0.1950, E (˜r1 ) = 0.1966, E (˜r1 ) = 0.2023, E (˜r1 ) = 0.2161,




E (˜r4 ) = 0.2019, E (˜r4 ) = 0.2026, E (˜r4 ) = 0.2029, E (˜r4 ) = 0.2032, E (˜r4 ) = 0.2034, E (˜r4 ) = 0.2042, E (˜r4 ) = 0.2097, E (˜r4 ) = 0.2139, E (˜r4 ) = 0.2210, E (˜r4 ) = 0.2322, E (˜r4 ) = 0.2447, DOI 10.1002/int

E (˜r5 ) = 0.1819. E (˜r5 ) = 0.1820. E (˜r5 ) = 0.1820. E (˜r5 ) = 0.1821 E (˜r5 ) = 0.1821. E (˜r5 ) = 0.1822. E (˜r5 ) = 0.1832. E (˜r5 ) = 0.1840. E (˜r5 ) = 0.1860. E (˜r5 ) = 0.1948. E (˜r5 ) = 0.2173.

48

VERMA AND SHARMA Table XXI. Ranking of the teachers by taking different values of λ.

Values of λ

Ranking

λ = −1 λ→0 λ = 0.4 λ = 0.8 λ=1 λ=2 λ=7 λ = 10 λ = 15 λ = 25 λ = 50

X3 X3 X3 X3 X3 X3 X3 X3 X3 X4 X4

5.

X4 X4 X4 X4 X4 X4 X4 X4 X4 X3 X3

X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X5

X1 X1 X1 X1 X1 X1 X1 X1 X1 X1 X1

X5 X5 X5 X5 X5 X5 X5 X5 X5 X5 X2

CONCLUSIONS

In this paper, we explored multiple attribute group decision-making problems in which the attribute and decision makers are at different priority levels, and the decision information provided by decision makers takes the form of triangular fuzzy numbers. A new prioritized aggregation operator called FGPWA operator for aggregating triangular fuzzy numbers has been introduced involving a parameter. This aggregation operator includes many existing aggregation operators such as FPWA operator, FPWG operator, (FPWHA) average operator, and FHA operator as special cases. A fuzzy multiple attribute decision making approach based on FGPWA operator is proposed to solve multiple attribute decision making problems under triangular fuzzy environment. This more general study should trigger fineness in visualizing the levels at which decisions undergo change. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

Beliakov G, Pradera A, Calvo T. Aggregation functions: a guide for practitioners. Berlin, Germany: Springer–Verlag; 2007. Calvo T, Mayor G, Mesiar R. Aggregation operators: new trends and applications. Berlin, Germany: Physica–Verlag; 2002. Xu ZS, Da QL. An overview of operators for aggregating information. Int J Intell Syst 2003;18(9):953–969. Yager RR. On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans Syst Man Cybernet 1988;18(1):183–190. Yager RR. Families of OWA operators. Fuzzy Sets Syst 1993;59(2):125–148. Fodor J, Marichal JL, Roubens M. Characterization of the ordered weighted averaging operators. IEEE Trans Fuzzy Syst 1995;3(2):236–240. Merigó JM, Gil-Lafuente AM. Unification point in methods of selection of financial products. Fuzzy Econ Rev 2007;12(1):35–50. Merigó JM, Casanovas M. Induced aggregation operators in decision making with the Dempster-Shafer belief structure. Int J Intell Syst 2008;24(8):934–954. Merigó JM, Casanovas M. Using fuzzy numbers in heavy aggregation operators. World Acad Sci Eng Technol 2008;24:532–537. International Journal of Intelligent Systems

DOI 10.1002/int

FUZZY PRIORITIZED WEIGHTED AVERAGE OPERATOR 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

49

Merigó JM, Gil-Lafuente AM. The Induced generalized OWA operator. Inf Sci 2009;179(6):729–741. Merigó JM, Casanovas M. The fuzzy generalized OWA operator and its application in strategic decision making. Cybernet Syst 2010;41(5):359–370. Merigó JM, Gil-Lafuente AM. New decision making methods and their application in the selection of financial products. Inf Sci 2010;180(11):2085–2094. Xu ZS. A priority method for triangular fuzzy number complementary judgment matrix. Syst Eng Theory Pract 2003;23(10):86–89. Xu ZS. Fuzzy harmonic mean operator. Int J Intell Syst 2009;24(2):152–172. Yager RR. On weighted median aggregation. Int J Uncertain Fuzz 1994;2(1):101–113. Yager RR. Quantifier guided aggregation using OWA operators. Int J Intell Syst 1996;11(1):49–73. Yager RR, Kacprzyk J. The ordered weighted averaging operator: theory and applications. Boston, MA: Kluwer; 1997. Yager RR. Generalized OWA aggregation operators. Fuzzy Optim Decis Making 2004;3(1):93–107. Yager RR. Centered OWA operators. Soft Comput 2007;11(7):631–639. Chiclana F, Herrera F, Herrera-Viedma. The ordered weighted geometric operator: properties and applications. In: Proc Eighth Int Conf Information Processing and Management of Uncertainty in Knowledge Based Systems, Madrid, Spain; 2000:985–991. Xu ZS, Da QL. The ordered weighted geometric averaging operators. Int J Intell Syst 2002;17(7):709–716. Yager RR. Generalized OWA aggregation operators. Fuzzy Optim Decis Making 2004;3(1):93–107. Dyckhoff H, Pedrycz W. Generalized means as model of compensative connectives. Fuzzy Sets Syst 1984;14(2):143–154. Wang XR, Fan ZP. Fuzzy ordered weighted averaging (FOWA) operator and its applications. Fuzzy Syst Math 2003;17(4):67–72. Xu ZS. A fuzzy ordered weighted geometric operator and its applications in fuzzy AHP. Syst Eng Electron 2002;31(4):855–858. Wei G. FIOWHM operator and its application to multiple attribute group decision making. Exp Syst Appl 2011;38(4):2984–2989. Yager RR. Prioritized aggregation operators. Int J Approx Reason 2008;48(1):263–274. Yager RR. Prioritized OWA aggregation. Fuzzy Optim Decis Making 2009;8(3):245–262. Zhao X, Lin R, Wei G. Fuzzy prioritized operators and their application to multiple attribute group decision making. Appl Math Modell 2013;37(7):4759–4770. Zadeh LA. Fuzzy sets. Inf Control 1965;8(3):338–353. Dubois D, Prade H, Fuzzy sets and systems: theory and applications New York: Academic Press; 1980. Van Laaehoven PJM, Pedrycz W. A fuzzy extension of Saaty’s priority theory. Fuzzy Sets Syst 1983;11(1–3):199–227. Chen SH. Operations on fuzzy numbers with function principle. Tamkang J Manage Sci 1985; 6(1): 13–26. Zètènyi T. Fuzzy sets in psychology Amsterdam, The Netherlands: North-Holland; 1988.


DOI 10.1002/int