Deriving Weights of Criteria from Inconsistent Fuzzy Comparison Matrices by Using the Nearest Weighted Interval Approximation

Deriving the weights of criteria from the pairwise comparison matrix with fuzzy elements is investigated. In the proposed method we first convert each element of the fuzzy comparison matrix into the nearest weighted interval approximation one. Then by using the goal programming method we derive the weights of criteria. The presented method is able to find weights of fuzzy pairwise comparison matrices in any form. We compare the results of the presented method with some of the existing methods. The approach is illustrated by some numerical examples.


Introduction
Weight estimation technique in multiple criteria decision making MADM problem has been extensively applied in many areas such as selection, evaluation, planning and development, decision making, and forecasting 1 .The conventional MADM requires exact judgments.
In the process of multiple criteria decision making, a decision maker sometimes uses a fuzzy preference relation to express his/her uncertain preference information due to the complexity and uncertainty of real-life decision making problem and the time pressure, lack of knowledge, and the decision maker's limited expertise about problem domain.The priority weights derived from a fuzzy preference relation can also be used as the weights of criteria or used to rank the given alternatives.
Xu and Da 2 utilized the fuzzy preference relation to rank a collection of interval numbers.Fan et al. 3 studied the multiple attribute decision-making problem in which the decision maker provides his/her preference information over alternatives with fuzzy preference relation.They first established an optimization model to derive the attribute weights and then to select the most desirable alternative s .Xu

and Da 4 developed an
In this paper we first introduce the nearest weighting interval approximation of a fuzzy number see Saeidifar 23 , and then by using some weighting functions we convert each fuzzy element of the pairwise comparison matrix to its nearest weighting interval approximation.Then we apply the goal programming method to derive weights of criteria.Goal programming was originally proposed by Charnes and Cooper 24 and is an important technique for DMs to consider simultaneously several objectives in finding a set of acceptable solution.
The structure of the rest of this paper is as follows.Section 2 provides some required preliminaries.Section 3 of the paper gives a goal programming approach for deriving weights of criteria.Some examples are presented in Section 4. The paper ends with conclusion.

Preliminaries
In this section we review some basic definitions about fuzzy numbers, fuzzy pairwise comparison matrix, and goal programming method.The membership function of a triangular fuzzy number is expressed as formula 2.1 :

Fuzzy Numbers
2.1

Comparison between Two Fuzzy Numbers
In this subsection, in order to compare two fuzzy numbers, we use the concept of ranking function.A ranking function is a function g : F R → R, which maps each fuzzy number into the real line, where a natural order exists.Asady and Zendehnam 25 proposed a defuzzification using minimizer of the distance between two the fuzzy number.
If A a, b, c be a triangular fuzzy number, then they introduced distance minimization of a fuzzy number A that denoted by M A which was defined as follows: This ranking function has the following properties.

Property 1.
If A and B be two fuzzy numbers, then,

2.3
Property 2. If A and B be two fuzzy numbers, then,

Fuzzy Pairwise Comparison Matrix
Suppose the decision maker provides fuzzy judgments instead of precise judgments for a pairwise comparison.Without loss of generality we assume that we deal with pairwise comparison matrix with triangular fuzzy numbers being the elements of the matrix.We consider a pairwise comparison matrix where all its elements are triangular fuzzy numbers as follows:

Goal Programming
Consider the following problem: where f 1 , . . ., f k are objective functions and X is nonempty feasible region.Model 2.7 is called multiple objective programming.Goal programming is now an important area of multiple criteria optimization.The idea of goal programming is to establish a goal level of achievement for each criterion.Goal programming method requires the decision maker to set goals for each objective that he/she wishes to obtain.A preferred solution is then defined as the one which minimizes the deviations from the set goals.Then GP can be formulated as the following achievement function:

2.8
The DMs for their goals set some acceptable aspiration levels, b i i 1, . . ., k , for these goals, and try to achieve a set of goals as closely as possible.The purpose of GP is to minimize the deviations between the achievement of goals, f i x , and these acceptable aspiration levels, b i i 1, . . ., k .Also, d i and d − i are, respectively, over-and underachievement of the ith goal.

The Nearest Interval Approximation
In this section, we introduce an interval operator of a fuzzy number, which is called the nearest weighted possibilistic interval approximation.First we introduce an f-weighted distance quantity on the fuzzy numbers, and then we obtain the interval approximations for a fuzzy number.
such that the functions f, f are nonnegative, monotone increasing and satisfies the following normalization condition: Definition 2.4 see 23 .Let A be a fuzzy number with A α a α , a α and f α f α , f α being a weighted function.Then the interval is the nearest weighted interval approximation to fuzzy number A.
Remark 2.5.The function f α can be understood as the weight of our interval approximation; the property of monotone increasing of function f α means that the higher the cut level is, the more important its weight is in determining the interval approximation of fuzzy numbers.
In applications, the function f α can be chosen according to the actual situation.
Corollary 2.6.Let A a, b, c be a triangular fuzzy number and let f α nα n−1 , nα n−1 be a weighting function.Then

2.10
Example 2.7.Let A 3, 4, 7 be a triangular fuzzy number and also let f 1 α 2α, 2α and f 2 α 4α 3 , 4α 3 be two weighting functions.Then the nearest weighted intervals to A is as follows see Figure 2 : .

2.11
Example 2.8.Let A 3, 7, 8, 13 be a trapezoidal fuzzy number and also let f 1 α 2α, 2α and f 2 α 4α 3 , 4α 3 be two weighting functions.Then the nearest weighted interval to A is as follows see Figure 3 :

Deriving the Weights of Criteria
In the conventional case, if a pairwise comparison matrix A be reciprocal and consistent then the weights of each criterion are simply calculated as w i a ij / n k 1 a kj , i 1, . . ., n.In the case of inconsistent matrix, we must obtain the importance weights w i , i 1, . . ., n such that a ij w i /w j or equivalently a ij w j − w i 0. Therefore in the case of uncertainty, for deriving the weights of criteria from inconsistent fuzzy comparison matrix we follow the following procedure.
Step 1.First by formula 2.10 we convert each fuzzy element a ij a L ij , a M ij , a U ij of the pairwise comparison matrix to the nearest weighted interval approximation a ij a L ij , a U ij .Hence the fuzzy pairwise comparison matrix A is converted to an interval pairwise comparison matrix A.
Step 2. Now we must calculate the weight vector where deviation variables p − ij , p ij and q − ij , q ij are nonnegative real numbers but cannot be positive at the same time, that is, p − ij p ij 0 and q − ij q ij 0. Now we apply the goal programming 8 Advances in Operations Research method.It is desirable that the deviation variables p ij and q ij are kept to be as small as possible, which leads to the following goal programming model:

3.2
By solving model 3.2 the optimal weight vector W w 1 , . . ., w n which shows the importance of each criterion will be obtained.We can use these weights in the process of solving a multiple criteria decision-making problem.Also, these weights show which criterion is more important than others.Proof.Consider W w 1 , . . ., w n which has the condition n i 1 w i 1, w i ≥ 0, i 1, . . ., n.Then we define Remark 3.2.For ranking of these criteria, we assign rank 1 to the criterion with the maximal value of w i , and so forth, in a decreasing order of w i .
Remark 3.3.The proposed method is able to derive the weights of criteria when the elements of the pairwise comparison matrix are fuzzy in any form see Example 3 in Section 4.3 .

Special Case: The Case of Matrix with Crisp Elements
In the case of matrix with crisp data, in order to derive the weights of criteria from the inconsistent pairwise comparison matrix, the goal programming model 3.2 can be converted to the following model: w j , p ij , q ij ≥ 0, i,j 1, . . ., n,

3.4
where p ij and q ij are deviation variables.By solving model 3.4 the optimal weight vector w j , j 1, . . ., n, which shows the importance of each criterion will be obtained.

Theorem 3.4. In the case of crisp data, the pairwise comparison matrix A is consistent if and only if
Proof.Let us first prove that if d * 0, then matrix A is consistent.Since d * 0, we have p ij q ij 0. Therefore a ij w j − w i 0 and hence a ij w i /w j .This gives a ij a jk a ik , and we conclude that matrix A is consistent.
Conversely, suppose that matrix A is consistent.That is a ij a jk a ik , i, j, k 1, . . ., n.

3.5
Now, if we define then it is easy to check that W, p ij , q ij is feasible for model 3.4 .Since model 3.4 has minimization form, we conclude that d * 0.
Proof.By Theorem 3.1, proof is evident.

Illustrating Example
In this section we present an illustrating example showing that the proposed approach is a convenient tool not only for calculating the weights of criteria of a pairwise comparison matrices with fuzzy elements but also for calculating the weights of criteria of crisp pairwise comparison matrices.

4.2
By solving model 4.2 , we obtain the optimal vector W w 1 , w 2 , w 3 .We assign rank 1 to the criteria with the maximal value of w j , and so forth, in a decreasing order of w j .The result is shown in Table 1.The optimal objective of model 4.2 is d * 0.249, which shows that the pairwise comparison matrix A is inconsistent by Theorem 3.4.
In this example the rank order of these criteria is as The results of ranking these criteria are shown in last column of Table 1.

Example 2: Matrix with Fuzzy Elements in Triangular Form
Consider 3 × 3 reciprocal matrix A with triangular fuzzy elements: Now we convert the above fuzzy matrix to the equivalent interval approximation pairwise comparison matrix.We consider two cases.

4.5
We construct the goal programming model for the above interval approximation pairwise comparison matrix as model 4.6 : Min p 12 q 12 p 13 q 13 p 21 q 21 p 23 q 23 p 31 q 31 p 32 q 32 s.t.

4.6
By solving the goal programming model 4.6 , we obtain the weight vector W 0.64, 0.24, 0.12 .We can use these weights in the process of solving a multiple criteria decision-making problem.Also, these weights show that criterion 1 is important than others see Table 2 .

4.7
Similar to model 4.6 , by constructing the corresponding goal programming model and solving it, we obtain the weight vector as shown in Table 3.
It can be seen that in two above cases we derive the weights of criteria when the elements of their pairwise comparison matrix are in the form of triangular fuzzy numbers.

Example 3: Matrix with Fuzzy Elements in any Form
Consider 3 × 3 reciprocal matrix A with fuzzy elements in any form: where We see that there is a trapezoidal fuzzy number and there is a fuzzy number in general form.

4.10
Similar to model 4.6 , by constructing the corresponding goal programming model and solving it, we obtain the weight vector as Table 4.
We can use these weights in the process of solving a multiple criteria decision-making problem.Also, these weights show that criterion 1 is more important than others.Note 1.We claim that none of the existing methods can find the weights for such pairwise comparison matrices as Example 3 see Section 4.3 .

Comparing with the Existing Methods
In this section, we provide four numerical examples to illustrate the potential applications of the proposed method.And also we use them for comparing the proposed method with some of the existing methods.These methods propose some methods to derive weights for fuzzy pairwise comparison matrices.Among the existing methods, we consider the following methods.
i Wang and Chin 16 proposed an eigenvector method EM to generate interval or fuzzy weight estimate from an interval or fuzzy comparison matrix.
ii Wang and Chin 18 proposed a sound yet simple priority method for fuzzy AHP which utilized a linear goal programming model to derive normalized fuzzy weights for fuzzy pairwise comparison matrices.
iii Taha and Rostam 19 proposed a decision support system for machine tool selection in flexible manufacturing cell using fuzzy analytic hierarchy process fuzzy AHP and artificial neural network.A program is developed in that model to iv Aya ǧ and Özdemir 20 proposed a fuzzy ANP-based approach to evaluate a set of conceptual design alternatives developed in an NPD environment in order to reach to the best one satisfying both the needs and expectations of customers, and the engineering specifications of company.
Consider the following fuzzy comparison matrix which is derived from Wang and Chin 16 : By constructing the corresponding goal programming model and solving it, we obtain the weight vector as shown in Table 5.We consider the case that we use the weighting function f α 3α 2 , 3α 2 .Consider the following fuzzy comparison matrix which is derived from Wang and Chin 18 : By constructing the corresponding goal programming model and solving it, we obtain the weight vector as shown in Table 6.
By constructing the corresponding goal programming model and solving it, we obtain the weight vector as shown in Table 7.
Consider the following fuzzy comparison matrix which is derived from Taha and Rostam 19 : By constructing the corresponding goal programming model and solving it, we obtain the weight vector as shown in Table 8.
In two previous examples we see that both of the Aya ǧ and Özdemir method and the Taha and Rostam method produce the exact nonfuzzy weights, and again we can see that the results of proposed method and their methods are very close.
Note 2. The above-mentioned methods are not able to derive weights of fuzzy pairwise comparison matrices as Example 3 see Section 4.3 .But the presented method is able to find weights of fuzzy pairwise comparison matrices in any form.

Conclusion
Finding the weights of criteria has been one of the most important issues in the field of decision making.In this paper, we have investigated the problem of deriving the weights of criteria from the pairwise comparison matrix with fuzzy elements.In the presented method we first convert the elements of the fuzzy comparison matrix into the nearest weighted interval approximation ones.Then by using the goal programming method we derive the weights of criteria.The presented method is able to find weights of fuzzy pairwise comparison matrices in any form.Also it is shown that the results of proposed method and the existing methods are very close.We saw that the existing methods are not able to derive weights of fuzzy pairwise comparison matrices in any form such as Example 3 see Section 4.3 , but the presented method is able to find weights of such fuzzy pairwise comparison matrices.The approach is illustrated by using some examples.

Fuzzy
numbers are one way to describe the vagueness and lack of precision of data.The theory of fuzzy numbers is based on the theory of fuzzy sets which Zadeh 12 introduced in 1965.Definition 2.1.A fuzzy number is a fuzzy set like A : R → I 0, 1 which satisfies i A is continuous, ii A x 0 outside some interval a, d , iii there are real numbers b, c such that a ≤ b ≤ c ≤ d, and 1 A x is increasing on a, b , 2 A x is decreasing on c, d , 3 A x 1, b ≤ x ≤ c.We denote the set of all fuzzy numbers by F R .Definition 2.2.A triangular fuzzy number is denoted as A a, b, c ; see Figure 1.

a b c 1 Figure 1 :
Figure 1: Triangular fuzzy number.

Table 1 :
The result of proposed method for Example 2 see Section 4.1 .

Table 2 :
The result of proposed method for Example 2 see Section 4.2 .

Table 3 :
The result of proposed method for Example 2 see Section 4.2 .

Table 4 :
The result of proposed method for Example 3 see Section 4.3 .

Table 5 :
The obtained weights of proposed method and Wang and Chin 16 method for A 1 .

Table 6 :
The obtained weights of proposed method and Wang and Chin 18 method for A 2 .In two previous examples we see that both of the Wang and Chin methods produce the fuzzy weights, and when we defuzzificate them by ranking function M • , we can see that the results of proposed method and their methods are very close.Now, consider the following fuzzy comparison matrix which is derived from Aya ǧ andÖzdemir 20 :

Table 7 :
The obtained weights of proposed method and Aya ǧ and Özdemir method for A 3 .

Table 8 :
The obtained weights of proposed method and Taha and Rostam method for A 4 .