Solving Decision Problems with Dependent Criteria by New Fuzzy Multicriteria Method in Excel

We propose a new MCDM method based on fuzzy pair-wise comparisons and a feedback between the criteria. The evaluation of the weights of criteria, the variants as well as the feedback between the criteria is based on the data given in pair-wise comparison matrices. Extended arithmetic operations with fuzzy numbers are used as well as ordering fuzzy relations to compare fuzzy outcomes. An illustrating numerical example is presented to clarify the methodology. A special SW – Microsoft Excel add-in named FVK was developed for applying the proposed method. Comparing to other software products, FVK is free, able to work with fuzzy data and utilizes capabilities of widespread spreadsheet Microsoft Excel


Introducation
In this paper we propose a new method for solving fuzzy MCDM problems.In some sense the method is parallel to, or, extension of the Analytic hierarchy process (AHP) with feedback between criteria.Instead of the classical eigenvector prioritization method employed in the AHP, here, a fuzzy approach is based on the logarithmic least squares method, i.e. on the geometric-mean aggregation.In (Lootsma, 1993(Lootsma, , 1996)), it was demonstrated that some notorious occurrences of rank reversal in AHP can be avoided by the geometric-mean aggregation.We also accept the reasons discussed in (Buckley, Feuring, & Hayashi, 2001), and other papers cited therein, for using geometric mean instead of Saaty's procedures.Our approach is however different to the method by (van Laarhoven & Pedrycz, 1983), and (Lootsma, 1996).
The interface between hierarchies, multiple objectives and fuzzy sets have been investigated by the author of AHP T.L. Saaty as early as in (Saaty, 2001).Later on, in (Saaty, 1991(Saaty, , 2001)), the author extends the AHP to a more general process with feedback called Analytic Network Process (ANP).In (Mikhailov & Singh, 2003) a new method based on ANP and fuzzy data is proposed, which is, however, essentially different to the approach used in this paper.Recently, (Büyüközkan, Ertay, Kahraman, & Ruyan, 2004), (Enea & Piazza, 2004) and (Mohanty, Agarwal, Choudhury, & Tiwari, 2005), proposed another versions of fuzzy ANP and applied their methods in practice.Here, we propose a relatively simple method based on the original approaches by (Buckley, 1985;Buckley et al., 2001) and (van Laarhoven & Pedrycz, 1983).

~ 2 ~
When applying classical MCDM approach on a real decision problem, e.g. when you want to select the best project from the given group of applications, or, to buy the best product for your personal use, say a car or digital camera, one usually meets among others two difficulties: • When evaluating pair-wise comparisons (e.g. on the 9-point AHP scale) natural uncertainty is not incorporated; • Decision criteria are not independent each other as they should be.
Here, we solve these difficulties by proposing a new method which incorporates uncertainty adopting pair-wise comparisons by triangular fuzzy numbers, and takes into account interdependences between decision criteria, see (Dubois & Prade, 2001).
First, in classical AHP the pairs of elements are evaluated on the scale { 1 / 9 , 1 / 8 , ..., 1 / 2 , 1, 2, ...,9}.Here, we extend this scale to the closed interval [ 1 / σ , σ], where σ is arbitrary positive number greater than 1.Moreover, our approach allows for multiple representations of uncertain human preferences: both crisp evaluations, interval, and also fuzzy judgments in the simple form of triangular fuzzy numbers.The proposed method offers a solution from incomplete sets of pair-wise comparisons.By using the geometric average method we derive the resulting weights -relative importance of the elements in the form of triangular fuzzy numbers.When calculating the weights we minimize the measure of uncertainty, i.e. the spreads of fuzzy weights, (Aguarón & Moreno-Jiménez, 2003).This feature, which is new comparing to previous methods, is very important as too large fuzziness of the weights could deteriorate the results.
Second, it was already mentioned above, the decision criteria are frequently dependent to each other.For instance, when buying a best product for our personal use, e.g. a car, the price of the car as a decision criterion is interdependent with some other criteria: the power of engine, maintenance cost, quality of design, prestige of the car etc. Evaluating the criteria individually without taking into account dependency between the criteria the result may become misleading.In AHP this problem has been solved by proposing a new methodology -ANP -a general approach for dealing with feedback among decision elements.Here we solve the problem analogically, however, not in the same generality as we do not have suitable tools for handling matrices with fuzzy elements, particularly inverse matrices, eigenvectors etc.We consider only dependences among decision criteria solving the inverse matrix problem by approximating it with the first few terms of Taylor expansion.This approach will help us to overcome the difficulty with interdependent criteria.
The paper is structured as follows: Section 2 presents the classical AHP/ANP theory in the suitable vector notation, in Section 3 pair-wise comparison matrices with triangular fuzzy elements are introduced, and in Section 4 we describe an algorithm for calculating fuzzy weights from fuzzy pair-wise comparison matrices.In Section 5 we deal with the problem of inconsistency, we introduce a new inconsistency index for fuzzy pair-wise comparison matrices and finally, in Section 6 we analyze an illustrating exampledecision making situation with 3 decision criteria and 3 variants.Also, we supply an illustrating example using a SW tool -Microsoft Excel add-in named FVK developed for applying the proposed method to demonstrate capabilities of the method.

Multi-Criteria Decisions and AHP/ANP
In classical hierarchical analysis we usually consider a three-level hierarchical decision system: On the first level we consider a decision goal G, on the second level, we have n independent evaluation criteria: , where w(C i ) is a positive real number -a weight, usually interpreted as a relative importance of criterion C i subject to the goal G. On the third level, m variants (alternatives) of the decision outcomes V 1 , V 2 ,...,V m are considered such that again , where w(V r ,C i ) is a non-negative real number -an evaluation (weight) of V r subject to the criterion C i , i = 1,2,...,n.It is advantageous to formulate the above mentioned weights into the matrix form, see e.g.(Saaty, 2001).
Let W 1 be the n×1 matrix (weighing vector of the criteria), i.e.
, and W 3 be the m×n The columns of this matrix are evaluations of variants according to the criteria.Moreover, W 3 is a column-stochastic matrix, i.e. the sums of columns are equal to one.Then Z = W 3 W 1 is an m×1 matrix, i.e. the resulting priority vector of weights of the variants.The variants can be ranked according to these priorities.
In real decision systems there exist typical interdependences among criteria or variants.Decision systems with dependences have been extensively investigated by the Analytic Network Process (ANP), see (Saaty, 2001).Consider now the dependences among the criteria.The interdependences of the criteria are characterized by n×n matrix Here, the k-th column of W 2 is interpreted as the vector of weights characterizing influences of the individual criteria on the k-th criterion and w(C k ,C k ) = 0 as the influence of C k onto itself is zero.
Moreover, n-1 elements of this vector of weights form the vector where the elements of w (k) are calculated from the following (n-1)×(n-1) pair-wise comparison matrix In (Saaty, 2001), it has been shown that if the sum of every column of matrix is the resulting priority vector of weights of variants applicable for the decision making process, i.e. for ranking the variants.
Usually, the matrix W 2 is close to the matrix with zero elements and the dependences among criteria are weak, it can be approximated by the first few terms of Taylor's expansion ( ) Then by substituting the first four terms from ( 5) to (4) we get where I is the unit matrix.In the next section, formula (6) will be used for computing fuzzy evaluations of the variants.

Fuzzy Numbers and Fuzzy Matrices
A triangular fuzzy number a ~ can be equivalently expressed by a triple of real numbers, i.e.
, where a L is the Lower number, a M is the Middle number, and a U is the Upper number, a L ≤ a M ≤ a U .If a L = a M = a U , then a ~ is said to be the crisp number (non-fuzzy number).Evidently, the set of all crisp numbers is isomorphic to the set of real numbers.In order to distinguish fuzzy and non-fuzzy numbers we shall denote the fuzzy numbers, vectors and matrices by the tilde above the symbol.As usual, the membership function of a ~ is assumed to be piece-wise linear.
It is well known that the arithmetic operations +,-, * and / can be extended to fuzzy numbers by the Extension principle, see e.g.(Buckley et al., 2001) It should be noted, that the above formulae for multiplication and also for division are not obtained from the Extension principle.The above stated operations are only approximate ones, it means that the result of e.g.multiplication is a triangular fuzzy number with piece-wise linear membership function, whereas the membership function of the exact operation defined by the Extension principle is non-linear.
If all elements of an m×n matrix A are triangular fuzzy numbers we call A the matrix with triangular fuzzy elements and this matrix is composed of triples as follows: Particularly, let A ~ be an n×n matrix with triangular fuzzy elements.We say that A ~ is reciprocal, if the following condition is satisfied: where

Algorithm
The proposed method of finding the best variant (or ranking all the variants) can be characterized in the following three steps: 1. Calculate the triangular fuzzy weights from the fuzzy pair-wise comparison matrices.
2. Calculate the aggregating triangular fuzzy evaluations of the variants.
~ 6 ~ Below we explain in details the individual steps.
Step 1: Calculate the triangular fuzzy weights from the fuzzy pair-wise comparison matrices.
We assume that both the importance of the criteria and values of the criteria and also the feedback between the criteria are given by the fuzzy weights calculated from the corresponding pair-wise comparison matrices with triangular fuzzy elements.
Let A ~ be an n×n reciprocal pair-wise comparison matrix with fuzzy elements (8) a w w a w w a w w here, we apply a modification of this well known logarithmic least-squares method for calculating Particularly, we solve the following optimization problem: Setting the derivatives of the function in (9) to zero (a necessary condition of optimality), it can be easily shown, that the optimal solution of problem ( 9), (10) shall satisfy the following relations: From ( 10) and ( 11) we obtain: Second, we want to find weights ..,n, with the minimal spread, i.e. the minimal measure of fuzziness: Therefore, by ( 12) and ( 13) we choose the resulting weights as follows: 1.The above stated method can be applied both for calculating the triangular fuzzy weights of the criteria and for eliciting relative triangular fuzzy values of the criteria for the individual variants.Moreover, it can be used also for calculating feedback impacts of some criteria on the other criteria.
2. The advantage of our approach is that it usually gives smaller spreads (i.e.fuzziness) of the calculated fuzzy weights.This fact follows from the following inequalities: for all i, j.

3.
Eventually, the values of the criteria can be given directly as triangular fuzzy numbers.Then, by the normalization they are transformed again to fuzzy weights.Particularly, assume that evaluations of variants according to some criterion are expressed by triangular fuzzy numbers, moreover, assume that the criterion is maximizing (i.e."bigger is better").Let ..,r, be a set of triangular fuzzy numbers, e.g.fuzzy evaluations of variants according to some criterion.We assume also . Then we "normalize" the values to obtain triangular fuzzy weights as follows: and Step 2: Calculate the aggregating triangular fuzzy evaluations of the variants.
Having calculated triangular fuzzy weights from all reciprocal matrices with triangular fuzzy elements as it was described above, we calculate the aggregated triangular fuzzy evaluation of the individual variants.For this purpose we use formula ( 6) applied to reciprocal matrices with triangular fuzzy elements.We calculate either are fuzzy evaluations of variants according to the criteria.Similarly, 2 W ~is calculated according to (1) -( 3), however, with triangular fuzzy elements.For addition and multiplication of triangular fuzzy numbers we use the fuzzy operations defined earlier.
Step 3: Select the "best" variant, or, order the variants.
In Step 2 we calculated m variants described as triangular fuzzy numbers, i.e. by the above formula we obtain m triangular fuzzy numbers ) ; ; ( ),..., ;  ; ( . Finally, we have to solve the problem of ranking fuzzy variants.As the set of triangular fuzzy numbers is not linearly ordered we have to use some ranking methods.There exist a number of sophisticated methods for ranking fuzzy numbers, for a comprehensive review of ranking methods see e.g.(Chen & Hwang, 1992).Here we present three relatively simple and well applicable ranking methods.
The first method for ranking a set of triangular fuzzy numbers is the well known center of gravity method (COG).This method is based on computing the x-th coordinates g i x of the center of gravity of every triangle given by the corresponding membership functions of i z ~, i = 1,2,...,n.Evidently, it By (19) the variants can be ordered from the best (with the highest value of (19)) to the worst (with the lowest value of ( 19)).The other two ranking methods are based on the α-cut of the triangular fuzzy number, where α∈[0,1] is an aspiration level given by DM.As the α-cut of a triangular fuzzy , we can rank the variants either by the left end points of the corresponding intervals, or by their right end points.In the former case we obtain in some sense a pessimistic ranking method (called L-dominant) whereas in the letter case we get an optimistic ranking method of alternatives (R-dominant), see (Ramik, 2007;Ramik & Perzina, 2010).Notice that all three above mentioned ranking methods coincide with the usual ordering of real numbers if the variants are crisp (i.e.non-fuzzy).

Illustrating Case Study
In this section we analyze an illustrating example -decision making situation with 3 decision criteria ~ 10 ~ and 3 variants.Let us assume that the goal of our decision problem is to choose the "best" new shop for our company from 3 pre-selected ones according to 3 criteria: location, suitability for our business, and building design.Location (Criterion 1) concerns surrounding area around the shop.Suitability for our business (Criterion 2) refers to internal structure of the building with respect to our business needs.Building design (Criterion 3) is related to look and feel of the building.The criteria are evaluated by pair-wise comparisons with triangular fuzzy values, moreover, interdependency among them is considered.
First, we apply our method, i.e. the algorithm described in Section 4, for solving the decision problem.The evaluation of the weights of criteria, the variants according to criteria as well as the feedback between the criteria is based on the data from fuzzy pair-wise comparison matrices.Here, based on the same arguments as in the classical Saaty´s method, we use the scale -interval [ 1 , 9 9 ] for evaluating preferences between alternatives.We apply, however, triangular fuzzy numbers.
Second, for comparison our fuzzy approach with non-fuzzy one, we use also non-fuzzy evaluations in the pair-wise comparisons without feedback.In this case we apply the same approach, i.e. the method based on the geometric-mean aggregation, a particular case of the previous more general situation of crisp evaluations by using the middle numbers a M , i.e. a M = a L = a U .We demonstrate that the results are different comparing to the previous fuzzy approach.
Third, we solve the same problem by applying classical AHP by Expert Choice, i.e. we use non-fuzzy evaluations in the pair-wise comparisons without the feedback.Again, we compare the results with the previous cases.
A special SW (Microsoft Excel add-in) named FVK was developed for applying the proposed method.Comparing to other software products, FVK is able to work with fuzzy data with capabilities of MS Excel.Here, as a demonstration, we use outputs obtained by this SW.The DM should fill in the grey regions in the corresponding tables; the white regions are calculated automatically.
Step 1: Evaluate the pair-wise comparison matrices and calculate the corresponding weights by the geometric-mean aggregation.
The data for relative importance of the criteria are given in the following pair-wise comparison matrix: By ( 15 The data for evaluations of fuzzy feedbacks between the criteria are given in 3 pair-wise comparison matrices B k , see formula (3), however, with fuzzy triangular elements.The interpretation of e.g.B 1 is as follows: Criterion 1 is "approximately 2 times more influenced" by Criterion 2 than it is influenced by Criterion 3. The term "approximately 2 times more influenced" is evaluated by the triangular fuzzy number (1; 2; 3).
In the following step we have to rank the evaluations of the above fuzzy variants resulting in the "best" variant by using a proper way of ranking the triangular fuzzy numbers depicted in Figure 1.
By (19) the variants are ranked from the best (with the biggest value of center of gravity) to the worst (with the lowest value of COG).We can see that by COG method the best variant is Var 2. We can see that Var 1 is again evaluated as the best one, moreover, numerically the AHP evaluations of the variants are well comparable to the evaluations calculated by our method.

Conclusion
Considering feedback dependences between the criteria the total rank of the variants may change as we have demonstrated in the above example.Fuzzy (soft) evaluation of pair-wise comparisons may be considered more comfortable and appropriate for DMs.Here, triangular fuzzy numbers have been adopted.A triangular fuzzy number is a natural extension of the crisp number by allowing the left and right spread of uncertainty of the value in question.This flexibility may be advantageous for the DM.An extension to more flexible trapezoidal fuzzy numbers is, of course, possible.By a trapezoidal fuzzy number the DM will define an interval where his/her evaluations are not distinquishable.However, by incorporating trapezoidal fuzzy numbers into our method, the simplicity of the algorithm would be lost.
In comparison with other AHP software tools, e.g.Expert Choice, our software FVK is limited to 3 hierarchical levels (goal-criteria-variants) which is most frequent in practice.It cannot be easily extended to more than 3 levels, e.g by adding the level of sub-criteria.In such case, the problem can be reformulated to 3-level problem by combining criteria and sub-criteria.Moreover, the FVK enables to model dependences among criteria which is realistic (and also frequent).A presence of fuzziness in evaluations may change the final rank, when comparing to crisp evaluations of variants in the classical AHP, too.In contrast to the classical AHP which is based on the eigenvector aggregation, our method is based on the geometric-mean aggregation allowing for fuzzy pair-wise comparisons and feedback dependences between the criteria.The evaluation of the weights of criteria, the variants as well as the feedback between the criteria is based on the data given in pair-wise comparison matrices.Extended arithmetic operations with fuzzy numbers are used as well as ordering ~ 15 ~ fuzzy relations to rank fuzzy variants.
An illustrating numerical case study is presented to clarify the methodology.A special SW -Microsoft Excel add-in named FVK was developed by the authors of this paper both for solving tentative and real decision makimg problems.Comparing to other software products, FVK is able to work with fuzzy data as well as with capabilities of widespread spreadsheet Microsoft Excel.

) k = 1 , 2 ,First.
...,n, where coefficients C L , C M , C U are suitable positive constants satisfying the following requirements.Hence, from normalization condition and (11) we obtain: ) Particularly, if A ~ is a crisp (i.e.non-fuzzy) matrix, i.e.for all k, and, the solution -weights are crisp, too.
is a feedback among the criteria.www.todayscience.org/jbm.phpJournal of Business & Management Vol. 3, Issue 4of fuzzy weights of the individual criteria and the columns of matrix ) -(17) we calculate the corresponding triangular fuzzy weights, i.e. the relative fuzzy importance of the individual criteria: ~ 11 ~ The character of all three criteria is qualitative, they are evaluated by pair-wise comparisons with triangular fuzzy values given in the following 3 pair-wise comparison matrices: The corresponding fuzzy matrix 3 W ~ of fuzzy weights -evaluations of variants according to the individual criteria is calculated by (15) -(17) as follows:

Figure 1 .
Figure 1.Total evaluation of fuzzy variants