entropy measure integrated fuzzy gowa operators

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Dr. D. Rajya Lakshmi, ... OWA operators, provide a parameterized class of mean type aggregation ... concept has proven its effectiveness and efficiency in finding well defined and practical solution. ... 12. 11 nm. (4). )w. ,......., w,w( w m. 2. 1 i = (5). In, this step, we use entropy based .... manual and economical efforts. To make ...

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ENTROPY MEASURE INTEGRATED FUZZY GOWA OPERATORS APPROACH FOR MULTI-ATTRIBUTE DECISION MAKING. K. Nikitha, C.S.E. Dept.,GITAM University, Medak Dist., Andhra Pradesh, India Email: [email protected]

T. Babu Rao, Dept. of Mech. Engg. GITAM University Medak Dist., Andhra Pradesh, India Email: [email protected]

Dr. D. Rajya Lakshmi, I.T. Dept.,GITAM University, Visakhapatnam, Andhra Pradesh, India Email: [email protected] Abstract: This paper presents a novel integrated approach to make an optimal decision for a multi attribute decision making problem. It completes within three phases as analysis of alternatives and their effective attributes in the first phase, formulation of the problem begins with the identification of individual influences among the attributes and determination of their weights accordingly in the second phase. In this consequence, the weights of the attributes are expressed in terms of Shannon’s entropy concept. It is a well known concept to determine the weights and here is used to deal with the fuzzy and vagueness with the data. Finally, with the help of GOWA operators, alternatives are sorted in preference order. In the process of rating the alternatives, IF sets being expressed by unifying entropy weights and GOWA operators. Hence, this methodology derives more objective and effective evaluation decisions and provides decision makers more information to make subtle decisions. The resulted decision has been validated by means of a multi-objective optimization technique Grey Relation Analysis (GRA). A case study is demonstrated to illustrate the decision makers, the practicality and effectiveness of this novel method. Keywords: multi-attribute decision making, Shannon's entropy weights, IF sets, GOWA operators, GRA. 1. Introduction In the present days, multi-attribute decision making (MADM) has been one of the fastest growing areas depending on the challenges in the business sector. A multi-attribute decision making can be generally described as a finite number of pre-specified alternatives with conflicting attributes [Figueira, J. et al. (2004)]. These alternatives may be classified based on qualitative and/or quantitative nature. In the real world, many alternatives exist for every problem, but, in consequence of selection, they may have vagueness with attribute data. Hence, the selection of an alternative based on fuzzy or uncertain data can be described as a complicated multi-attribute task. Bellman and Zadeh [Bellman, R.E. and Zadeh, L.E (1970)] were the first to relate fuzzy set theory on decision making problems. Chen and Hwang [Chen, S. J. and Hwang, C. L. (1992)] proposed and formulated a fuzzy decision matrix to a quantitative decision matrix by converting the linguistic terms to fuzzy numbers and then the fuzzy numbers to crisp scores. Quan Zhang [Zhang, Q. et al. (2007)] provided optimization model in order to assess the ranking values of the alternatives. Many researchers have proposed several methodologies by minimizing the vagueness with the data [Hwang, C. L and Yoon, K. (1981)] [Triantaphyllou, E and Lin, C. T. (1996)][ Triantaphyllou, E. (2000)][ Feng Kong and Hongyan Liu, (2005)]. Sharbafi et. al. [Sharbafi, M. A. et al. (2006)] suggested fuzzy decision making to improve the performance of genetic algorithm. Shannon's entropy [Shannon, C.E. and Weaver, W. (1947)] is a measure of disorder, or more precisely unpredictability. Zeleny [Zeleny, M. (1982)] highlighted the entropy concept for deciding the

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objective weights of attributes. These weights are concerned with formulation of the interrelationship between the criteria which we desire to model. The entropy weights which are based on mathematical computation gives the reliable decision unlike subjective weights which are vaguely measured with linguistic terms. The concept Intuitionistic Fuzzy Set, which is more expressive in terms of vagueness, is a generalization of fuzzy set. Atanassov [Atanassov, K. (1986)][ Atanassov, K. (1989)] introduced and developed Intuitionistic Fuzzy sets with triangular norm-based membership degrees. IF sets provide a pair of mappings ranging from 0 to 1 and the sum of membership and non membership ranges i.e. μ + ν ≤ 1 . The considerable improvement with the IF sets over fuzzy sets is its geometrical interpretation and interval based membership calculations. Such a generalization of fuzzy sets gives an additional possibility to represent imperfect knowledge in real world problems like human testimonies, opinions, etc. Therefore, IF sets concept is more suitable to solve multiattribute decision making problems and give appropriate ratings to the alternatives. R. Yager [Yager, R.R. (1988)][Yager, R.R.(2004)] introduced a new aggregation technique based on the ordered weighted averaging (OWA) operators. OWA operators, provide a parameterized class of mean type aggregation operators and is used to aggregate experts opinions, with the corresponding weight vector [Deng-Feng Li (2010)]. This is different from the classical weighted average technique in which coefficients are not associated directly with a particular attribute but rather to an ordered position. Additional to OWA operators, generalization of Ordered Weighted Averaging operators (GOWA) is considered by a vector of weights, as well as the power to which the arguments are raised. Finally, the best predictive model is selected by finding the one which has the minimum entropy. Grey relational analysis is used to evaluate the best alternative for decision making problem. Finally, an example is shown to demonstrate that the GOWA operators approach result is satisfactory and effective evaluation. 2. Proposed Methodology In this paper, a novel approach is presented to solve a decision making problem with fuzzy or uncertain data using GOWA operators approach integrated with Shannon entropy concept. The distinctive aspect of GOWA as compared to traditional approaches is, it does not make any presumption about the formulation and gives feasible solutions to handle difficult problems. Also the generated decision helps to directly have an interpretation of the attributes affecting the alternatives. More details of this methodology were discussed in section 3. In this work, the process of selecting a computer centre in an academia that best suits to the requirements was explicitly formulated as a MADM problem. This is an important phase in new information system sector as the improper selection might severely affect work productivity and flexibility. Now a days, depending upon the type of work production, a considerable number of alternative processes are available and choosing the best among the existing alternatives has really a quandary to the Educational Institution. Several factors such as investment costs, Contribution, Effort, and Outsourcing etc are the considerable factors during the selection. In general practice, most of the attributes are qualitative by nature and often the decision depends on expertise or on the past experiences. Therefore, it is very complex to elicit the complete, precise, and reliable knowledge from the experts. It can be noted that the classical MADM methods such as Fuzzy TOPSIS approach; Fuzzy AHP, Hierarchical Fuzzy TOPSIS etc., [Fasanghari, M. et al. (2008)][Saaty, R.W. (1987)] are not very efficient for handling decision making problems because they don’t find accurate result with minimal computational complexity due to the involvement of several assumptions made by the decision maker. Because of the above, GOWA operators approach is proposed in this paper for finding precise solution in MADM. Especially entropy concept has proven its effectiveness and efficiency in finding well defined and practical solution. The proposed methodology of integrating entropy weights and GOWA approach is depicted in Fig.1. 3. GOWA Operators Approach with entropy weights 3.1. Entropy weights Entropy in information theory is a measure of uncertainty formulated using probability theory. Entropy method calculates the objective weights of the attributes without any consideration to the preferences of decision maker. To determine weights by entropy measure, the decision matrix Rij with attributes and alternatives is considered. The amount of decision information of each attribute can be measured by the entropy value ei as: n

e i = − k  R ij lnR ij i =1

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(1)

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-1

Where k = ln[m] is constant. Entropy weight is a parameter that describes how much different alternatives approach one another with respect to a certain attribute [Sanghyun Park and Vijay S. P. (2006)]. The degree of divergence di of the average information contained by each criterion Ci (for i=1,2,.........m) can be calculated as: (2) d i = 1 - ei The higher the divergence di, the more important the attribute is for decision making problem under consideration [Szmidt, E. and Kacprzyk, J. (2002)]. The objective weight for each attributes Ci(for i=1,2,.........m) is given by: m

w i = di /  d k

(3)

k =1

satisfying, w i ∈ [0,1 ] , satisfying (for i=1,2,….,m) and  w i = 1 m

i =1

3.2. GOWA Operators Algorithm Approach In GOWA Operators approach, we perform the following actions: i. Construct a decision matrix by conversion from linguistic terms into crisp scores. Assume decision matrix or decision table with attributes Ci (for i=1,2,..........m), alternatives Aj (for j=1,2,..........n) and weights of attributes, wi (for i=1,2,..........m) as in Eq.(4) and Eq.(5). The decision matrix R= {Rij, i=1,2,...m; j=1,2,....n} represents the utility ratings of alternative Aj with respect to selection criteria Ci.  R11 R12 .. .. R1n     R 21 R 22 .. .. R 2n  (4) . . . .  R m× n =  .  . . . . .    .. .. R R R mn   m1 m2 (5) wi = (w1 , w 2 ,......., w m) In, this step, we use entropy based objective weights found based on Eq.(1), Eq.(2) and Eq.(3). ii. Construct IF sets mij , n ij for all the values of i and j in the decision matrix which defines the degree of membership and degree of non-membership respectively and 0≤ μij+ νij≤1. The degree of membership and degree of non-membership are chosen as follows:  R ij  μ ij =  α i  R imax 

(6)

 R ij  ν ij = β i  R imax 

(7)

α i ∈ [ 0,1]and β i ∈ [ 0,1] , satisfying the conditions 0≤ α i + β i ≤ 1 . iii. The IF decision matrix representing MADM problem with IF sets can be expressed concisely as:

R = ( μ ij , ν ij ) mxn

 μ11 , ν11  μ , ν  21 21 = . .   μ m1 , ν m1  iv.

μ12 , ν12

..... μ1n , ν1n 

μ 22 , ν 22 . .

..... μ 2n , ν 2n . .

μ m2 , ν m2

..... μ mn , ν mn

Determine the score function Δ(rij ) ∈ [−1,1]

     

(8)

(9)

where Δ(rij ) ∈ [−1,1]

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The score function Δ ( rij ) determines the "net" membership degree. And the scores are organized in increasing or decreasing order. m g(a 1 , a 2 ,....., a n ) = (  w i d iλ )1/λ

(10)

i =1

where d i = μ i , ν i

is the ith largest one of all IF sets as (k=1,2,.........,m) using the ranking methods [Yager,

R.R. (1988)] of IF sets from Eq.(11), w is the weight vector which is correlative with g and λ ∈ (0,+ α) is a parameter which is always positive, since the negative power of di has no meaning. Then g is called GOWA operators with IF sets. From Eq.(10), v. m m w w g(a 1 , a 2 ,....., a n ) = [1 − ∏ (1 − μ iλ ) i ]1/λ ,1 − {1 − ∏ [1 − (1 − ν i ) i ]}1/λ (11) i =1

i =1

The following conclusions are derived. m When λ → 0, g(a , a ,....., a ) = ∏ d w i 1

2

n

i =1

i

The GOWA operator ‘g’, reduces to the OWG operator using IF sets. When m m w w λ → 1, [1 − ∏ (1 − μ1i ) i ]1,1 − {1 − ∏ [1 − (1 − ν i )1 ] i }1 i =1

i =1

When, λ → + α, , if w ∈ 0 for all the values of i, then g (a1 , a 2 ,......, a n ) = d n . The GOWA operator ‘g’,

i

reduces to the max operator using IF sets and dn is the largest one of all IF sets i=1,2,3....,m. For all the values of j (j=1,2,3....,n), determine the scores Δr j and the accuracies Δσ j , which are the vi. difference and sum of μ j and ν j respectively. vii.

Rank the order of all alternatives based on the scores and accuracies. i. If ΔA 1 > ΔA 2 , then A1 is greater than A2 . ii. If ΔA 1 = ΔA 2 , then a. If σ(A 1 ) > σ(A 2 ) , then A1 is greater than A2 . b. If σ(A1 ) < σ(A 2 ) , then A1 is lesser than A2 . c. If σ(A1 ) = σ(A 2 ) , then A1 is equal to A2 .

4. An Illustrative Example

The proposed entropy based fuzzy GOWA operator’s method has been applied to solve a general problem in an educational institute. It is very expensive to transfer the current computer centre to latest centre, in the view of manual and economical efforts. To make an optimum decision, four experts have been concerned in decision making to improve work productivity. The data di (for i=1,2,3,4) entered by the decision makers for the analysis are given in Table 1. Table 1 Alternatives represented by attributes in terms of linguistic terms D2 D3 D4 Attributes D1

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C1

High

C2

Very High

C3

Medium High

Very High Medium Low Very High

C4

Medium Low

Low

High

Very High

Very High

High

Medium Medium High

High

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Data input

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Identification of Attributes and Alternatives

Conversion of linguistic terms to crisp values if any

Calculations

Calculation of relative importance using entropy method

Determination of IF sets

Final Evaluation

Determination of scores and arranging them in ascending or descending order

Determination of GOWA operators with IF sets and various parameter values

Ranking of alternatives based on scores and accuracies

Fig.1. Block diagram of GOWA operators Approach with entropy weights

It describes the four decisions represented with four alternatives. The fuzzy comparison matrix for the attributes using triangular fuzzy numbers is given in Table 2. Table 2 Alternatives represented by attributes in terms of crisp values Attributes D D D D 1

2

C1

0.8636

1.000

C2

1.000

C3

0.6667

C4

0.333

3

4

0.8636

1.000

0.333

1.000

0.8636

1.000

0.5

0.8636

0.249536

0.6667

1.000

The attributes considered here are i) Expenditure on Costs of hardware/software (C1), ii) Influence on the performance of the organization (C2), iii) Effort to transform from current system (C3), iv) Outsourcing software developer reliability (C4). The overall priority weights are calculated using Eq. (1)-(3) and are listed in Table 3. Table 3 Entropy based weights

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ei

di

wi

0.1827

0.8172

0.3687

0.3555

0.6445

0.2907

0.5363

0.4636

0.2091

0.7089

0.2910

0.1312

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Assume a and b be positive values [Venkata Rao, R. (2007)], such that a +b £ 1 . Using Eq.(6) and Eq.(7), for different values of a and b , the relative degrees of membership mij and the relative degrees of non-

{

n ij } .

membership n ij are calculated for Dj (j=1,2,3….,n) and are represented as IF sets mij , Table 4 IF sets Alternatives D1 D2 D3 D4

μ1 ,ν1

μ 2 ,ν 2

μ 3 ,ν 3

μ 4 ,ν 4

0.69,0.09 0.9,0.05 0.57,0.07 0.25,0.06

0.8,0.1 0.3,0.02 0.85,0.1 0.19,0.05

0.69,0.08 0.9,0.05 0.43,0.05 0.5,0.13

0.8,0.1 0.78,0.04 0.73,0.086 0.75,0.2

According to Eq.(9), the score functions are obtained as and are arranged descending order.

Δ(r11 ) = 0.6, Δ(r21 ) = 0.85, Δ(r31 ) = 0.5, Δ(r41 ) = 0.18 Δ(r 21 ) > Δ(r11 ) > Δ(r 31 ) > Δ(r 41 ) similarly, Δ(r 32 ) > Δ(r12 ) > Δ(r 22 ) > Δ(r 42 ) Δ(r 23 ) > Δ(r13 ) > Δ(r 33 ) > Δ(r 43 ) Δ(r 24 ) > Δ(r14 ) > Δ(r 34 ) > Δ(r 44 ) Table 5 Overall Assessments D1

D2

D3

D4

λ

r1

߂r1

σ(r1)

r2

߂r2

σ(r2)

r3

߂r3

σ(r3)

r4

߂r4

σ(r4)

0

(0.63,0.06)

0.56

0.69

(0.55,0.07)

0.47

0.63

(0.67,0.072)

0.6

0.74

(0.77,0.09)

0.68

0.86

1

(0.75,0.06)

0.69

0.82

(0.72,0.06)

0.65

0.79

(0.76,0.06)

0.69

0.83

(0.77,0.07)

0.7

0.85

2

(0.77,0.87)

0.68

0.85

(0.74,0.06)

0.68

0.81

(0.77,0.06)

0.7

0.84

(0.77,0.77)

0.7

0.85

α

(0.9,0.1)

0.75

0.95

(0.85,0.04)

0.75

0.95

(0.77,0.04)

0.7

0.8

(0.8,0.1)

0.7

0.9

(

)(

)(

)(

)

, 2 2 5 1 . 0 4 2 . 0 1 × 3 7 1 . 0 6 6 5 . 0 1 × 8 2 8 2 . 0 9 6 . 0 1 × 7 1 9 3 . 0 9 . 0 1 )(

)(

)(

)(

))

2 2 5 1 .

)(

0 6 6 6 0 . 0 +

3 7 1 .

0 6 6 6 0 . 0 +

)(

8 2 8 2 .

7 1 9 3 .

0 6 3 6 8 0 . 0 +

(((

0 5 0 . 0

= r1

1

for λ=1,

(

)

2 2 5 1 . 0 × 6 6 6 0 . 0 + 3 7 1 . 0 × 6 6 6 0 . 0 + 8 2 8 2 . 0 × 6 3 6 8 0 . 0 + 7 1 9 3 . 0 × 5 0 . 0

= r1

(

, 2 2 5 1 . 0 4 2 . 0 × 3 7 1 . 0 6 6 5 . 0 × 8 2 8 2 . 0 9 6 . 0 × 7 1 9 3 . 0 9 . 0

Table 5 represents the overall assessment of all decisions for different values of λ determined using Eq.(10). Finally, the score functions of all the values for j = 1, 2, 3...., n are determined and arranged in descending order. Here, for λ=0,

)

)

For some special values of parameter, r j is determined and shown in Table 5. Corresponding scores and accuracies of r j (j= 1,2,3....,n) were also shown. Algorithm to rank the alternatives based on the scores and accuracies. Input scores:߂r1 , ߂r2, ߂r3, ߂r4 and accuracies: σ(r1), σ(r2), σ(r3), σ(r4) i. repeat steps iii and iv for n=1 to 4 ii. if ߂rn > ߂rn-1, then rn is greater than rn-1 iii. if ߂rn = ߂rn-1, then iv. a. if σ(rn)> σ(rn-1), rn is greater than rn-1 b. if σ(rn)< σ(rn-1), rn is lesser than rn-1 c. if σ(rn)= σ(rn-1), rn is equal to rn-1

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When a greater r value is same for more than one decision, then s value among these r values is considered and greater value of σ is taken as a good decision. From the Table 5, it is simple to say that the best selection is the decision D4. The order of decisions is D4>D3>D1>D2. 5. Grey Relational Analysis

Grey relation analysis was originally developed by Prof. Deng [Deng J. L. (1989)], and is used to solve problems based on uncertain information. It is well known technique for solving the multi- attribute optimization problems [Deng, J.L. (1988)][Deng, J.L. (2002)][Wen-de, Y.I. and Gui-wu, W.E.I. (2007)]. The results of GRA method are based on original data and the calculations are simple. The following steps are considered while applying grey relational analysis: Grey coefficient for the given data yields: γ(y 0 (j), y k (j)) =

Δmin + ξΔmax Δ oi (j) + ξΔmax

(12)

Where, a. i=1,2,....m; j=1,2,...n, n is the number of alternatives available for the given data and m is the number of attributes. y0(j) is the reference sequence (y0(j)=1, j=1,2,3....n); yi(j) is the specific comparison sequence. b. c.

Δ

d.

Δ min = min min y ( j ) − y ( j ) is the smallest value of yi(j). i ∀i ∈ k ∀j o

e.

Δ max = max max y ( j ) − y ( j ) is the largest value of yi(j). i ∀i ∈ k ∀j o

oi

( j ) = y ( j ) − y ( j ) is the absolute value of the difference between y0(j) and yi(j). o i

ξ is the distinguishing coefficient which is defined in the range 0 ≤ ξ ≤ 1 . Calculating the grey relational grade γ i , by averaging the grey relational coefficient yields: f.

γi =

1 j

n

γ

ik

(13)

k =1

From Table 2, the absolute value is calculated and is shown in Table 6. From this, ߂min is 0 and ߂max is 0.1364. Table 6 The absolute value of the difference between y0(j) and yi(j) ∆C1 ∆C2 ∆C3 ∆C4 0.1364 0 0.3 0.667 D1 0 0.667 0 0.75 D2 0.1364 0 0.5 0.3333 D3 0 0.1364 0.1 0 D4 The grey relational coefficient is calculated for all the attributes. (Ci where i=1,2,...m) as given in Eq.12. Also the grey relational grade is calculated as per Eq. 13. Table 7 Grey relational Coefficients and the grey relational grade GRC GRC GRC GRC GRC/Decisions C1 C2 C3 C4 Grade 0.333 1.000 0.428 0.359 0.530 D1 1.000 0.333 1.000 0.666 0.666 D2 0.333 1.000 0.333 0.529 0.549 D3 1.000 0.709 0.646 1.000 0.839 D4 The higher grade decision is the better decision in grey relational analysis and hence the D4 is the best choice.

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6. Conclusion

The proposed work suggested an optimum decision with minimum fuzziness for MADM problem based on the integration of an efficient Shannon's entropy concept for measuring the weights of alternatives with well known fuzzy GOWA operators for arranging the alternatives in priority order. Since, the selection of an appropriate alternative has become a complex issue in the presence of vagueness with the attributes. Today, for every problem large numbers of alternatives are available with many distinguished attributes. However, these attributes are represented quantitatively and/or qualitatively. It gives fuzziness to the decision maker during the selection. The entropy concept derives the weights for attributes accurately, with minimum computational complexity. Hence, this methodology helps to derive more objective and provides decision makers additional information to make subtle decisions. An example was demonstrated and the results were compared for correctness with GRA method are found as the results of the proposed method are well in agreement. Selection of an appropriate decision can be done effectively with the proposed method in the presence of fuzzy multiattribute decision making problems. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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