Two Approximation Models of Fuzzy Weight Vector from a Comparison Matrix

In this study, our uncertain judgment on multiple items is denoted as a fuzzy weight vector. Its membership function is estimated from more than one interval weight vector. The interval weight vector is obtained from a crisp/interval comparison matrix by Interval AnalyticHierarchy Process (AHP).We redefine it as a closure of the crispweight vectorswhich approximate the comparison matrix. The intuitively given comparison matrix is often imperfect so that there could be various approaches to approximate it. We propose two of them: upper and lower approximation models.The former is based on weight possibility and the weight vector with it includes the comparison matrix. The latter is based on comparison possibility and the comparison matrix with it includes the weight vector.


Introduction
AHP (Analytic Hierarchy Process) [1] is one of the wellknown multicriteria decision-making methods and applied to various decision situations.The decision problem in AHP is structured hierarchically with the criteria and alternatives.The final decision is a priority weight vector of the alternatives reflecting the importance of the criteria.The priority vector of the alternatives represents the decision-makers preference and is obtained by synthesizing two kinds of weight vectors.They are the importance weight vector of the criteria and the priority weight vector of the alternatives under each criterion.Each weight vector is obtained from a pairwise comparison matrix given by a decision-maker intuitively.The core technique in AHP is to derive a weight vector consisting of multiple items, such as alternatives and criteria, from a pairwise comparison matrix.We focus on this technique and derive a fuzzy weight vector from any given comparison matrix.
The often-used techniques are the eigenvector, geometric mean, and logarithm least square methods, all of which basically derive a crisp weight vector from a crisp comparison matrix.Instead of a crisp weight vector, an interval/fuzzy vector could be obtained from a crisp comparison matrix reflecting the inconsistency among the crisp comparisons.
On the other hand, the comparisons could be given as interval/fuzzy since a decision-maker's thinking on the items is not always accurate and precise.In order to represent our vague judgment, interval/fuzzy numbers may be more preferable and suitable than crisp numbers.Moreover, when linguistic patterns among the items are given, the comparison matrix is constructed based on transitivity so as to be robust for rank reverse [2].In case of an interval/fuzzy comparison matrix, a crisp or interval/fuzzy weight vector could be obtained.In this way, there are four cases: a crisp or interval/fuzzy weight vector from a crisp or interval/fuzzy comparison matrix.We review some previously proposed ideas to distinguish ours from them.
A crisp weight vector is useful for ranking the items linearly.It is obtained not only from a crisp comparison matrix but also from an interval comparison matrix.Using the upper or lower bound of each interval comparison, multiple crisp weight vectors are derived [3] and the crisp weight vector is determined by taking a Euclidean center of them in [4].The other method to derive a crisp weight vector from an interval comparison matrix is based on fuzzy programming approach where a deviation parameter of each comparison is introduced to measure the satisfaction [5].Moreover, some methods to derive a crisp weight vector from a fuzzy comparison matrix have been proposed [6][7][8] and the applications have been shown and compared in [9].These obtained crisp weight vectors are understood as a summary of the given comparison matrix, although a part of its information may be curtailed.Then, in order to reflect the uncertainty of each interval comparison, an interval comparison matrix is summarized in an interval weight vector.The interval weight vector is obtained from the viewpoint of the stability of the rank order of items in [10], where well-known eigenvector method is applied.The upper and lower bounds of each interval weight are obtained from the devised interval comparison matrix to be less inconsistent with logarithmic goal programming in [11].Furthermore, the crisp comparison matrix is processed into an interval comparison matrix based on transitivity and the interval weight vector is obtained from the upper and lower bounds of the interval comparisons [12].Each interval comparison depends on two crisp comparisons and does not reflect the other comparisons so that all the given comparisons are not considered equally.In case of a fuzzy comparison matrix, the fuzzy weight vector is derived so as to be close to it in various ways, and they are reviewed in [13].For instance, the degree of consistency of the obtained weight vector is minimized in [14], the deviations based on geometric mean are measured in [15], and the deviations by logarithm are considered in [16].As an application, Fuzzy AHP was used for a selection problem in [17,18].The fuzzy comparison has been processed from a crisp comparison by assuming its uncertainty in a certainty degree [19].The primal reason why a weight vector is an interval/fuzzy seems to be the fuzziness of the given comparisons so that these interval/fuzzy weight vectors reflect the uncertainty of each comparison.
It is a human nature to make a decision and we often prefer to do it by ourselves.We do not prefer a system to do it.Since his/her final decision does not always equal the derived result which s/he refers to in making a decision, there is no need to summarize it as a rigid value, crisp weight vector.Instead, it is more important to represent a weight vector of the items in a decision-maker's mind as it is.For a decisionmaker, the decision aiding system to derive what s/he cannot easily notice but reflects his/her uncertain judgment would be helpful.The given comparison matrix does not represent the whole decision-maker's thinking, and its imperfectness is embodied in inconsistency among the intuitively given comparisons, as well as fuzziness of each comparison.In Interval AHP [20,21], the interval weight vector is obtained by reflecting inconsistency among the crisp comparisons.Because of its formulation, the upper and lower bounds of the interval weights are emphasized more than the values within an interval weight.This paper derives a fuzzy weight vector whose membership function is estimated by the interval weight vectors from an interval/crisp comparison matrix.The goal is neither to summarize an interval comparison matrix in a crisp weight vector nor to reflect the fuzziness of each interval comparison into an interval weight vector.We redefine the interval weight vector in Interval AHP based on the relations between the comparison matrix and the weight vector from possibilistic view.The interval weight vector reflects uncertainty in a decision-maker's thinking by inconsistency among the comparisons and fuzziness of the comparisons.
This paper is organized as follows.Sections 2 and 3 explain a comparison matrix and a fuzzy weight vector, respectively.In Section 4, two approaches to derive interval weight vector from an interval/crisp comparison matrix are proposed based on the idea that the given comparisons are not perfect.Then, in Section 5, two example comparison matrices are given to illustrate the proposed approaches to derive the fuzzy weight vectors.The last section concludes this paper.

Pairwise Comparison Matrix and Weight Vector
A decision-maker compares a pair of items and gives the comparison based on his/her intuitive judgment.A crisp comparison is used to denote a decision-maker's judgment as quantitative data.The interval comparison is more general than the crisp one since an interval becomes crisp when its upper and lower bounds are equal.We start with an interval pairwise comparison matrix of  items.A decision-maker can focus on the compared items without regarding the other items at the same time.It is easy for him/her to compare a pair of items, although s/he has to repeat comparing for all the pairs.The comparisons are inconsistent with each other since s/he compares an item several times intuitively.The given comparison matrix surely corresponds to a decision-maker's thinking on the items, and it is reasonable to derive the weight vector of the items from it.However, it is not the best to make the comparison matrix and the weight vector mathematically equal because of the imperfectness of the given comparisons.They do not represent the whole decision-maker's thinking, and it is unknown how good and how much reliable they are.Therefore, there could be various weight vectors to approximate a comparison matrix, and this study proposes two approaches to obtain interval weight vector  from interval/crisp comparison matrix .

Membership Function of Fuzzy Weight Vector
The membership function of a triangular/trapezoidal fuzzy number is estimated from two kinds of its -level sets.Let us assume them as interval whose membership values to fuzzy number X are  1 and  2 , respectively.The membership function  X is denoted as follows. where A decision-maker's thinking in this study is denoted as a fuzzy weight vector W = ( W1 , . . ., W ) , whose -level set is interval weight vector  = ( 1 , . . .,   )  , where   = [  ,   ], ∀.We can find the following fact about an interval weight vector [20].The sum of the lower bounds of the interval weights of all items,  = ∑    , is surely assigned to one of the items, while the left, 1 − , may be assigned to more than two items.Therefore,  represents a sure degree of  and can be considered as its membership value of interval weight vector  to fuzzy weight vector W. In other words,  is an -level set of W with  = .In order to estimate fuzzy weight vector by (2), we need at least two kinds of level sets.In the next section, two approaches to derive the representative interval weight vectors from an interval/crisp comparison matrix in Section 2 are proposed.

Interval Weight Vectors from Interval Pairwise Comparison Matrix
. .Definition of Interval Weight Vector.The interval weight vector  = ([  ,   ], ∀)  corresponding to the given interval comparison matrix  in ( 1) is defined as follows in this study [22].
where  is a small positive number and R denotes the relation between comparison matrix  = [  ] and weight vector  = (  , ∀)  .The interval weight vector by ( 3) is a closure of the normalized crisp weight vectors, and it is also normalized in interval sense:  ∈ N, where N denotes a set of the normalized interval weight vectors as follows [23,24].
The first two kinds of inequalities reduce the redundancy to make the sum of the crisp values in the intervals one.A crisp weight vector is an extreme case of interval weight vector with   =   =   , ∀, and the first two kinds of inequalities are replaced into crisp normalization ∑    = 1.
Depending on relation R, various interval weight vectors can be derived from a comparison matrix.Since the comparisons do not represent the decision-maker's thinking perfectly, they are approximated by the weight vector in various ways.This study considers the imperfectness of the comparison matrix with weight or comparison possibility and correspondingly assumes approximation relation R. The upper approximation model in Section 4.2 introduces weight possibility for a weight vector to include the comparison matrix, while the lower approximation model in Section 4.3 introduces comparison possibility for the comparison matrix to include a weight vector.
. .Upper Approximation Model.The upper approximation model is based on the idea that the comparison matrix is imperfect because of its scarceness.The given comparison matrix is a part of a decision-maker's thinking so that the weight vector needs to cover it.Interval AHP [21] where the variables are interval weights [  ,   ], ∀.It is noted that (5) can derive the interval weight vector from a crisp comparison matrix,  = [  ], where   =   =   ∀, .The third constraint is for the interval normalization in (4).The first two kinds of constraints denote the inclusion relation: where a fraction of intervals is defined with the maximum range.The interval weight vector includes the comparison matrix so that ( 5) is called upper approximation model.By minimizing the widths of the interval weights, the weight vector approximates the comparison matrix as precisely as possible.It synthesizes the fact that the less uncertain interval weight is preferable since the sum of the widths of the interval weights represents how uncertain the interval weight is.
The LP problem ( 5) is feasible for any comparison matrix, although the uniqueness of the optimal weights [  ,   ], ∀, has not been mentioned in detail [21].In order to take all possible optimal weights into consideration, this study redefines the interval weight vector as (3) whose  R  is as follows.
The minimum weight possibility  for comparison matrix  is obtained by the following problem.
where the variable are  and weight vector (  , ∀)  .The optimal objective function value,  * , is unique to , although there may be various optimal weight vectors.In order to include the comparison matrix, crisp weight vector needs to be widened by  * at least.In other words, no crisp weight vector could be found in the comparison matrix if weight possibility was less than  * .
Then, the membership function of fuzzy weight vector W is estimated by (2) with two -level sets: interval weight vectors ( * ) and ( * ).The fuzzy weight vector is estimated more precisely by setting more variety of  between  * and  * .The weight possibility, , is introduced in order not to rely too much on the given imperfect comparisons.
. .Lower Approximation Model.The upper approximation model widens a crisp weight vector into the pseudo interval weight vector by weight possibility.This is one of the approaches to handle an imperfect comparison matrix, which do not represent the whole decision-maker's thinking.The other approach is based on comparison possibility and the given comparison matrix is narrowed down or widened into the processed comparison matrix under the condition that it includes a weight vector.Therefore, it is called lower approximation model.In Interval AHP [21], the lower approximation model has been formulated as a counterpart of the upper one (5) as follows.
where the variables are interval weights [  ,   ], ∀.Contrary to (5), the inclusion relation is The widths of the interval weights are maximized to approximate the comparison matrix as closely as possible.
However, this lower approximation model (10) may miss a possible normalized weight vector.A didactic example of missing weight vector is as follows.Let us denote the optimal solutions of (10) as [  ,   ],  = 1, 2, 3 ∈ N. It can be as follows: if The former weights ( 1 ,  2 ,  3 ), whose sum exceeds 1, are impossible, although the second constraint requires  1 ≤  12  2 .This is also shown with a didactic numerical example: where  is the given comparison matrix and  is obtained by (10).Crisp weight vector  = (0.35, 0.3, 0.35)  is not included in , although crisp comparisons by ,  12 = 7/6,  13 = 1,  23 = 6/7, are included in the interval comparisons of .
The conventional lower approximation model ( 10) has such a drawback as missing a normalized weight vector included in the comparison matrix.In order not to miss any, the interval weight vector for the lower approximation model,  * (1) = ([  (1),   (1)], ∀)  , is redefined by (3) as follows.
where  R  indicates the fact that the comparison matrix includes a weight vector.This problem is not always feasible.We may not find a normalized weight vector  = (  , ∀)  within comparison matrix , when the comparisons are very inconsistent with each other or crisp as   =   =   , ∀, ,  > ,   ̸ =     , ∃, , .When ( 12) is feasible, the membership value of interval weight vector (1) to the fuzzy weight vector is  W(  (1)) =  W(  (1)) = (1) = ∑    (1), ∀.
We introduce comparison possibility  and process the given comparisons, [  ,   ], ∀, ,  >  into [  /,   ], where   / ≤   , ∀, ,  > .By reducing comparison possibility  ≤ 1 when ( 12) is feasible, the comparisons are narrowed down, and they are widened by increasing comparison possibility  ≥ 1.There are the other ways to narrow or to widen the range of the interval comparison such as increasing and decreasing the upper and lower bounds, respectively, by  > 0, [  + ,   − ], like in [4,5,19] and by 0 <  < 1, [(1 + )  , (1 − )  ].A decisionmaker compares a pair of items and gives the importance ratio of one over the other, such as "very" and "extremely."Then, for calculation, it is replaced by a numerical number in such a list as {1/9, 1/7, 1/5, 1/3, 1, 3, 5, 7, 9} or that with the even numbers between them in addition.The interval comparison matrix often consists of these numbers.When we compare two numbers, such as 3 and 5, 3 is multiplied by 5/3 into 5 and 5 is multiplied by 3/5.
where  = 1 corresponds to (12) where the variables are the crisp weights   , ∀, and comparison possibility .The given comparison matrix could be more focused in case of  * ≤ 1.Such a processed comparison would be given if we had asked a decision-maker for more precise comparison.On the other hand, the given comparison matrix could be widened in case of  * ≥ 1 to represent the whole decision-maker's thinking, such a processed comparison would be given if s/he had been stuck with something and overlooked the others.The processed comparison matrix with  * barely includes a normalized crisp weight vector.Nonlinear programming problem ( 14) is transformed into the following LP problem by logarithm.
where   = ln ,    = ln   ,    = ln   , ∀, ,  >  and    = ln   , ∀, and the variables are   and    , ∀.The last constraint of (15) means  1 = 1, which is different from the corresponding constraint of ( 14), ∑    = 1.The role of this constraint is to fix the weights whose ratios correspond to the given comparisons, as well as to normalize the weights.Hence, it can be replaced into making one of the weights 1 in (15).The optimal objective function value of (15),   * , is transformed into that of (14) as  * =    * .There is no need to transform the optimal weights,    , ∀, since the interval weight vector is obtained by (3).
The interval weights with comparison possibility  * are obtained by (3) as ( * ) = ([  ( * ),   ( * )], ∀)  , where All the crisp weight vectors included in the processed comparison matrix with comparison possibility  * are included in interval weight vector ( * ).It is noted that the interval ratio of two interval weights by ( 16) is not always included in the given interval comparison.Assume two normalized crisp weight vectors as  1 = ( 1 , ∀)  and  2 = ( 2 , ∀)  , whose interval comparison matrix includes  1 / 1 ⊆   and  2 / 2 ⊆   .The closure of two crisp weight vectors is interval weight vector  = ([  ,   ], ∀)  , where   = min( 1 ,  2 ) and   = max( 1 ,  2 ).Since interval weight [  ,   ] includes the values between  1 and  2 , interval comparison matrix does not always include interval ratio The membership value of ( * ) to fuzzy weight vector W is  W(  ( * )) =  W(  ( * )) = ( * ) = ∑    ( * ) ≤ 1, ∀, where ( * ) is not close enough to 1, differently from ( * ) in the upper approximation model.Therefore, the comparison possibility, , is reconsidered for each comparison in more detail as follows. min where   , ∀, ,  >  and   , ∀, are the variables and  * is the optimal value of ( 14).This is a nonlinear programming problem and solved by logarithm in the same way as (14).
Then, the membership function of the fuzzy weight vector is estimated by (2) with two -level sets, interval weight vectors (  * ) or ( * ) and (1) if  * ,   * ≤ 1.In case of  * > 1, (12) is infeasible so that the interval weight vector with the other comparison possibility,  ≥  * , needs to be obtained.The fuzzy weight vector is estimated more precisely by setting several other .The comparison possibility is introduced to estimate a decision-maker's thinking precisely from the given imperfect comparison matrix.In other words, each comparison is supplemented by narrowing down or widening.

Numerical Example
We consider two types of comparison matrices of three items,  1 and  2 , whose elements are interval and crisp, respectively, as follows.
The fuzzy weight vectors are obtained from these comparison matrices, which represent a part of the decision-maker's thinkings.Their imperfectness is taken into consideration by weight possibility or comparison possibility in the upper or lower approximation model, respectively.
. .Fuzzy Weight Vector by Upper Approximation Model.The minimum weight possibility by ( 7) is  1 * = 0.07 and  2 * = 0.029, respectively.The maximum weight possibility is  1 * =  2 * = 1/3 because  = 3.The corresponding interval weight vectors by (8) and (9), where relation  R  in (3) indicates that a pseudo weight vector includes the given comparison matrix as in (6), are obtained as follows.
For instance,  1 ( 1 * ) is a closure of 6 kinds of normalized crisp weight vectors.They are obtained by minimizing and maximizing the crisp weight of each item so as to include the comparison matrix with weight possibility  1 * as in (8) and are all equal in this example.The membership value of  1 ( 1 * ) to fuzzy weight vector W1 is 1 so that it is -level set with  = 1.In the same way, the interval weight vectors in (19) are considered as -level sets of the fuzzy weight vectors and their membership values are shown below them.In addition to  ⋅ * and  ⋅ * , we set several other weight possibilities  between them and obtain the corresponding interval weights by (8) replacing  * into .Then, the membership functions of fuzzy weight vectors W1 and W2 by the upper approximation models are obtained by (2).They are illustrated as the dotted lines in Figures 2 and 3, respectively.It is noted that very small lower bounds make approximation relation  R  by (6) meaningless.
. .Fuzzy Weight Vector by Lower Approximation Model.In the lower approximation model, relation  R  in (3) indicates that a processed comparison matrix includes a crisp weight vector as in (13).The minimum comparison possibility by (14) As for interval comparison matrix  1 , the processed comparison matrix with  1  * , ∀, ,  >  is more focused than that with  1 * so that the sure degree of the former weight vector is more than that of the latter one.As for crisp comparison matrix  2 , the other interval weight vectors by setting  are needed to estimate the membership function of the fuzzy weight vector.For both comparison matrices, we set several other comparison possibilities  ≥  * in approximation relation  R  by (13), and from each processed comparison matrix the interval weight vector is obtained by (16) replacing  * into .Then, the membership functions fuzzy weight vectors W1 and W2 by the lower approximation model are obtained by (2) and illustrated as the bold lines in Figures 2  and 3, respectively.
. .Comparing Upper and Lower Approximations.The dotted and bold lines are similar in both figures since they are two kinds of the approximations of the same comparison matrix,  1 or  2 , respectively.The imperfectness of the given comparison matrix can be well handled by both weight possibility and comparison possibility.We find that the bold lines are wider than the dotted lines.It indicates that the lower approximation model with comparison possibility tends to derive a vague fuzzy weight vector.We can tell neither whether the comparison matrix is given by chance or by well-consideration nor whether a decision-maker is an expert or not.Therefore, unknown uncertainty in the comparison matrix is considered by weight or comparison possibility, and the correspondingly fuzzy weight vector is obtained.The decrease of weight or comparison possibility increases the membership value of the weight vector.When the membership value is more than 0.8, we can find the linear order of three items of both comparison matrices by both models.The interval weight vectors at high membership value could be the results by removing inconsistency of the given comparison matrix.On the contrary, the weight vectors at the lower membership value could be the results by considering the imperfectness more positively.

Conclusion
The approaches to obtain a fuzzy weight vector from an interval/crisp comparison matrix have been proposed.We estimated the membership function of a fuzzy vector from more than one interval weight vector.The membership value of the interval weight vector is the sum of the lower bounds of the interval weights, which represents its sure degree.We redefined the interval weight vector as a closure of the crisp weight vectors.The crisp weight vector could approximate the comparison matrix in various ways since the comparison matrix does not represent the whole decision-maker's thinking.We proposed upper and lower approximation models and introduced weight possibility and comparison possibility, respectively.In the upper approximation model, a crisp weight vector with weight possibility should include the comparison matrix.In the lower approximation model, the comparison matrix with comparison possibility should include a crisp weight vector.As the weight or comparison possibility increases, the incompleteness of the comparison matrix is taken into consideration more positively.We have applied the proposed approaches to the crisp and interval comparison matrices.They could be applied to a fuzzy comparison matrix by focusing on its -level set since it is an interval/crisp comparison matrix.However, it remains an open question how to accumulate the fuzzy weight vectors at various -level sets.

Figure 1 :
Figure 1: Membership function from two -level sets.
This study is based on this inverse relation, and comparison [  ,   ] is narrowed down or widened into [  /,   ] with  < 1 or  > 1, respectively.With comparison possibility  the relation  R  for the lower approximation model is rewritten as follows.