Comparison of aCCuraCy in ranking alternatives performing generaliZeD fuZZy average funCtions

The paper defines the notions of point, interval and triangular intuitionistic fuzzy numbers expressing the degree of membership and non-membership in the fuzzy set. The generalized fuzzy weighted average function is introduced according to operation rules on intuitionistic fuzzy numbers. In special cases, the generalized weighted average coincides with an arithmetic average or a geometric average. The generalized fuzzy weighted average function could be applied for solving problems in multiple criteria decision making. research on the stability of the generalized weighted averaging operator of ranking alternatives was performed applying the Monte Carlo method. The aim of the conducted research is to establish the types of intuitionistic fuzzy numbers and the exponent values of the generalized weighted averaging operator having the least error probabilities considering alternatives ranking. Computations were performed involving 3, 4 and 5 experts. In the case of 5 experts, initial decision matrices having high, middle and low separability alternatives were examined. Decision matrices created by the experts were modelled generating random intuitionistic fuzzy numbers according to uniform and normal distribution. The example of applying such methodology was shown to solve a real problem of ranking possible redevelopment alternatives for derelict rural buildings. keywords: intuitionistic fuzzy number, generalized weighted averaging operator, multiple criteria decision making, Monte Carlo method. reference to this paper should be made as follows: Kosareva, N.; Krylovas, a. 2013. Comparison of accuracy in ranking alternatives performing generalized fuzzy average functions, Technological and Economic Development of Economy 19(1): 162–187. Jel Classification: D81, C44, C63, o18. Technological and economic developmenT oF economY iSSn 2029-4913 print/iSSn 2029-4921 online Copyright © 2013 vilnius Gediminas Technical University (vGTU) Press http://www.tandfonline.com/TTeD 2013 Volume 19(1): 162–187 doi:10.3846/20294913.2012.763072 Corresponding address: N. Kosareva e-mail: Natalja.Kosareva@vgtu.lt introduction The current paper mainly deals with Multiple Criteria Decision Making (MCDM) in a fuzzy environment. MCDM is a branch of operations research (or) aimed at making the best decision according to several criteria. The quality of the reached decision relates to maximizing profit or utility and minimizing loss or cost. In MCDM, we have a finite set of alternatives or projects to select the best one according to their adequacy to a finite set of attributes. each attribute has its importance expressed by weight. In the decision matrix, project adequacy to attributes is expressed by crisp numbers. There are many methods for determining the order of alternatives in terms of a set of attributes or criteria. several well-known methods for dealing with multiple criteria decision making problems are, for example, Multiplicative exponential Weighting (MeW), simple additive Weighting (saW), Technique for ordering Preference by similarity to Ideal solution (ToPsIs), a method of multiple criteria Complex Proportional assessment of Projects (CoPras), additive ratio assessment method (aras), Multi-objective optimization by ratio analysis method (Moora), eleCTre, etc. MCDM methods were overviewed and classified according to available information and their application for solving economical decision problems (Hwang, yoon 1981; Figueira et al. 2005; Zavadskas, Turskis 2011). Under real conditions, vague or imprecise information creates difficulties in assigning a crisp value of a subjective judgment – an element of the decision matrix. such information is better determined using fuzzy numbers. sometimes the subjective judgment is defined as a linguistic variable, i.e. the variable the values of which are expressed in linguistic terms (Zimmermann 1985). Fuzzy numbers appropriately express linguistic variables. a fuzzy multiple criteria decision making (FMCDM) theory is an appropriate solution in such circumstances. Fuzzy numbers in the fuzzy multiple criteria decision making approach (in our case, intuitionistic fuzzy numbers) are the elements of the decision matrix (Deng 2009). other approach is to deal with attribute weights as intuitionistic fuzzy numbers (liu 2009). The ordered weighted averaging (oWa) operator was introduced by yager (1988). Zhao et al. (2010) developed some new generalized aggregation operators such as a generalized intuitionistic fuzzy ordered weighted averaging operator. Merigo and Wei (2011) investigated an uncertain probabilistic ordered weighted averaging (UPoWa) operator. Han and liu (2011) were solving unknown attribute weights and hybrid multiple attribute decision-making problems under risk. Zavadskas and Turskis (2011) presented a comprehensive overview of multiple criteria decision making methods in an uncertain environment along with their classification and applications. a methodology of fuzzy sets introduced by Zadeh (1965) has been extended and enriched and nowadays is being widely applied in many fields of scientific research such as knowledge management systems, project management, manufacturing and organizational strategy, evaluating investment direction and magnitude, etc. MCDM is one of the branches where the fuzzy set theory has been found a wide application area. atanassov (1986) formulated a concept of an intuitionistic fuzzy set (IFs) as the generalization of the notion of the fuzzy set. He stated that IFss had essentially higher describing possibilities than fuzzy sets. The IFs has gained wide recognition as a useful tool for modelling 163 Technological and Economic Development of Economy, 2013, 19(1): 162–187 uncertain phenomena. atanassov and Gargov (1989) extended the IFs to the interval valued intuitionistic fuzzy set (IvIFs) and defined operation rules for intuitionistic fuzzy numbers (IFNs). Xu (2007a) proposed some methods for aggregating interval valued intuitionistic fuzzy information that can be applied for reaching a solution to decision making problems. Zhang and liu (2010) used the triangular intuitionistic fuzzy number and weighted arithmetic and geometric averaging operators for decision making. Wei et al. (2012) investigated multiple attribute group decision making problems where both attribute weights and expert weights take the form of real numbers and attribute values take the form of interval intuitionistic trapezoidal fuzzy numbers. In fuzzy MCDM (FMCDM), the values of fuzzy information aggregation operators are the set of intuitionistic fuzzy numbers (one for each alternative). Thus, we obtain the problem of ranking fuzzy alternatives. Due to the fact that fuzzy numbers are not linearly ordered, ranking them is one of the fundamental problems of fuzzy decision making. This problem is still important in the case of intuitionistic fuzzy numbers. The widely used approach to compare fuzzy numbers is their defuzzification into crisp numbers. Then, ranking based on these crisp numbers is done. The purpose of our research is to compare various methods of aggregating fuzzy information in multiple criteria decision making. For this purpose, the Monte Carlo simulation method was applied. Zanakis et al. (1998) employed this technique for comparing various MCDM methods. The tasks we intended to accomplish included: – comparing alternative ranking results obtained with the help of the weighted arithmetic averaging operator and weighted geometric averaging operator in cases of various types of intuitionistic fuzzy numbers (point intuitionistic fuzzy numbers, interval valued intuitionistic fuzzy numbers and triangular intuitionistic fuzzy numbers); – comparing the stability of the generalized weighted averaging operator of ranking alternatives obtained from different types of intuitionistic fuzzy numbers, initial decision matrices and exponent values of the generalized weighted averaging operator. 1. intuitionistic fuzzy numbers The notion of an intuitionistic fuzzy set was introduced by (atanassov 1986). Definition 1.1. let X be a finite non empty set. an intuitionistic fuzzy set on X is an expression given by { } , ( ), ( ) | A A A x x x x X = < μ ν > ∈ where ( ) : [0;1], ( ) : [0;1] A A x X x X μ → ν → and 0 ( ) ( ) 1 A A x x ≤μ + ν ≤ for all x X ∈ . ( ) A x μ is the membership degree and ( ) A x ν is the non-membership degree of element x A ∈ . ( ) 1 ( ) ( ) A A A x x x π = −μ − ν is the degree of uncertainty (indeterminacy) associated with the membership of element x in A . In the special case of ( ) 1 ( ) ( ) 0, A A A x x x x X π = −μ + ν = ∀ ∈ we have fuzzy set A instead of the IFs. We will restrict our consideration to intuitionistic fuzzy numbers, henceforth our set X would be real line X R = . Burillo et al. (1994) defined an intuitionistic fuzzy number as follows. 164 N. Kosareva, A. Krylovas. Comparison of accuracy in ranking alternatives performing ...


introduction
The current paper mainly deals with Multiple Criteria Decision Making (MCDM) in a fuzzy environment. MCDM is a branch of operations research (or) aimed at making the best decision according to several criteria. The quality of the reached decision relates to maximizing profit or utility and minimizing loss or cost.
In MCDM, we have a finite set of alternatives or projects to select the best one according to their adequacy to a finite set of attributes. each attribute has its importance expressed by weight. In the decision matrix, project adequacy to attributes is expressed by crisp numbers. There are many methods for determining the order of alternatives in terms of a set of attributes or criteria. several well-known methods for dealing with multiple criteria decision making problems are, for example, Multiplicative exponential Weighting (MeW), simple additive Weighting (saW), Technique for ordering Preference by similarity to Ideal solution (ToPsIs), a method of multiple criteria Complex Proportional assessment of Projects (CoPras), additive ratio assessment method (aras), Multi-objective optimization by ratio analysis method (Moora), eleCTre, etc. MCDM methods were overviewed and classified according to available information and their application for solving economical decision problems (Hwang, yoon 1981;Figueira et al. 2005;Zavadskas, Turskis 2011).
Under real conditions, vague or imprecise information creates difficulties in assigning a crisp value of a subjective judgment -an element of the decision matrix. such information is better determined using fuzzy numbers. sometimes the subjective judgment is defined as a linguistic variable, i.e. the variable the values of which are expressed in linguistic terms (Zimmermann 1985). Fuzzy numbers appropriately express linguistic variables. a fuzzy multiple criteria decision making (FMCDM) theory is an appropriate solution in such circumstances. Fuzzy numbers in the fuzzy multiple criteria decision making approach (in our case, intuitionistic fuzzy numbers) are the elements of the decision matrix (Deng 2009). other approach is to deal with attribute weights as intuitionistic fuzzy numbers (liu 2009). The ordered weighted averaging (oWa) operator was introduced by yager (1988). Zhao et al. (2010) developed some new generalized aggregation operators such as a generalized intuitionistic fuzzy ordered weighted averaging operator. Merigo and Wei (2011) investigated an uncertain probabilistic ordered weighted averaging (UPoWa) operator. Han and liu (2011) were solving unknown attribute weights and hybrid multiple attribute decision-making problems under risk. Zavadskas and Turskis (2011) presented a comprehensive overview of multiple criteria decision making methods in an uncertain environment along with their classification and applications. a methodology of fuzzy sets introduced by Zadeh (1965) has been extended and enriched and nowadays is being widely applied in many fields of scientific research such as knowledge management systems, project management, manufacturing and organizational strategy, evaluating investment direction and magnitude, etc. MCDM is one of the branches where the fuzzy set theory has been found a wide application area. atanassov (1986) formulated a concept of an intuitionistic fuzzy set (IFs) as the generalization of the notion of the fuzzy set. He stated that IFss had essentially higher describing possibilities than fuzzy sets. The IFs has gained wide recognition as a useful tool for modelling uncertain phenomena. atanassov and Gargov (1989) extended the IFs to the interval valued intuitionistic fuzzy set (IvIFs) and defined operation rules for intuitionistic fuzzy numbers (IFNs). Xu (2007a) proposed some methods for aggregating interval valued intuitionistic fuzzy information that can be applied for reaching a solution to decision making problems. Zhang and liu (2010) used the triangular intuitionistic fuzzy number and weighted arithmetic and geometric averaging operators for decision making. Wei et al. (2012) investigated multiple attribute group decision making problems where both attribute weights and expert weights take the form of real numbers and attribute values take the form of interval intuitionistic trapezoidal fuzzy numbers.
In fuzzy MCDM (FMCDM), the values of fuzzy information aggregation operators are the set of intuitionistic fuzzy numbers (one for each alternative). Thus, we obtain the problem of ranking fuzzy alternatives. Due to the fact that fuzzy numbers are not linearly ordered, ranking them is one of the fundamental problems of fuzzy decision making. This problem is still important in the case of intuitionistic fuzzy numbers. The widely used approach to compare fuzzy numbers is their defuzzification into crisp numbers. Then, ranking based on these crisp numbers is done.
The purpose of our research is to compare various methods of aggregating fuzzy information in multiple criteria decision making. For this purpose, the Monte Carlo simulation method was applied. Zanakis et al. (1998) employed this technique for comparing various MCDM methods. The tasks we intended to accomplish included: -comparing alternative ranking results obtained with the help of the weighted arithmetic averaging operator and weighted geometric averaging operator in cases of various types of intuitionistic fuzzy numbers (point intuitionistic fuzzy numbers, interval valued intuitionistic fuzzy numbers and triangular intuitionistic fuzzy numbers); -comparing the stability of the generalized weighted averaging operator of ranking alternatives obtained from different types of intuitionistic fuzzy numbers, initial decision matrices and exponent values of the generalized weighted averaging operator.

intuitionistic fuzzy numbers
The notion of an intuitionistic fuzzy set was introduced by (atanassov 1986). Definition 1.1. let X be a finite non empty set. an intuitionistic fuzzy set on X is an expression given by In the special case of ( ) 1 ( ) ( ) 0, of the IFs. We will restrict our consideration to intuitionistic fuzzy numbers, henceforth our set X would be real line X R = . Burillo et al. (1994) defined an intuitionistic fuzzy number as follows.   Fig. 1 shows two intuitionistic fuzzy numbers, the degree of the uncertainty of the first number (on the left figure) is it is a special case of the fuzzy number. The figure on the right depicts an intuitionistic fuzzy number with 0 be two intuitionistic fuzzy numbers. The operation rules of IFss are defined as follows (atanassov, Gargov 1989):  Gargov (1989) introduced the notion of an interval valued intuitionistic fuzzy number. Bustince and Burillo (1995) presented a theorem that allows constructing interval valued intuitionistic fuzzy sets from intuitionistic fuzzy sets and to recover intuitionistic fuzzy sets used for constructing the interval valued intuitionistic fuzzy set from different operators. Definition 1.3. let X be a finite non empty set. The interval valued intuitionistic fuzzy set on X is an expression given by intuitionistic fuzzy numbers operation rules on which are defined as follows:   Xu (2007a) defined aggregation operators of IFNs as follows. Definition 1.4. suppose ( 1,2,..., ) is a set of triangular intuitionistic fuzzy numbers and : n f ω Ω → Ω , Ω is a set of all triangular intuitionistic fuzzy numbers. If 1 2 1 ( , ,..., )  ∑ is the weight vector of ( 1,2,..., ) and is called the weighted geometric averaging operator of triangular intuitionistic fuzzy numbers. Two formulas for aggregation operations in a set of triangular intuitionistic fuzzy numbers have been established (Zhang, liu 2010). The proof of these formulas follows immediately from formulas (1)-(2) and operation rules on triangular intuitionistic fuzzy numbers. suppose ([ , , ],[ , , ]), is a set of triangular intuitionistic fuzzy numbers. Then, the result of the arithmetic averaging operator is a triangular intuitionistic fuzzy number and 1 2 1 ( , ,..., ) is a set of triangular intuitionistic fuzzy numbers. Then, the result of the geometric averaging operator is a triangular intuitionistic fuzzy number and 1 2 1

ranking intuitionistic fuzzy numbers
a number of researchers have analyzed the problem of comparing fuzzy numbers. Intuitionistic fuzzy weighted averaging operators are used for aggregating individual opinions of decision makers to have a combined opinion. as a result, intuitionistic fuzzy numbers (one for each alternative) are obtained. one of the most frequently used comparison methods is the defuzzification of fuzzy numbers, i.e. transforming them to crisp numbers can be easily compared. Chen and Tan (1994) provided a score function to defuzzificate intuitionistic fuzzy numbers.
be an intuitionistic fuzzy number. The score function S of intuitionistic fuzzy number A  is represented as follows: The larger is the score of A S  , the greater is intuitionistic fuzzy number A  . Hong and Choi (2000) proposed improved comparison technique based on the score function and accuracy function.
be an intuitionistic fuzzy number. The accuracy function H of intuitionistic fuzzy number A  is represented as follows: , . If the score function values of two IFNs coincide, then, the larger is the accuracy of A H  , the greater is intuitionistic fuzzy number A  . Xu (2007b) generalized these definitions for interval valued intuitionistic fuzzy numbers.
be an interval valued intuitionistic fuzzy number. The score function A S  of A  is represented as follows: be an interval valued intuitionistic fuzzy number. The accuracy function A H  of A  is represented as follows: . according to Zhang and liu (2010), the score function and accuracy function of the triangular intuitionistic fuzzy number are defined in a similar way.
is a triangular intuitionistic fuzzy number. Then, is a triangular intuitionistic fuzzy number. Then, Based on score function A S  and accuracy function A H  Xu (2007b) proposed an order relation between two intuitionistic fuzzy values defined as follows.

multiple criteria decision making by aggregated fuzzy functions
The fuzzy multiple criteria decision making approach implies that denotes the satisfaction degree of project i s to attribute j r and denotes the non-satisfaction degree of project i s to attribute j r . The multiple criteria decision making method based on triangular intuitionistic fuzzy numbers, according to Zhang and liu (2010), could be accomplished following the below introduced steps.  (7) and (8), the values of the score function and, in case it is necessary, the values of accuracy function for f i I and g i I are calculated. It is a defuzzification procedure converting each aggregated triangular fuzzy number into a crisp value for ranking and further analysis. 4. Comparing projects. on the grounds of Proposition 2.1., the best project from m project set is selected. We can similarly operate with (point) intuitionistic fuzzy numbers and interval valued intuitionistic fuzzy numbers.

aggregated fuzzy functions based on the generalized averaging operator
We want to extend the notions of aggregation operations on a set of triangular intuitionistic fuzzy numbers.
where 1 1 1 ( , ,..., ) , is the weight vector of ( 1,2,..., ) , function p f ω is called the generalized weighted averaging operator with the exponent p of triangular intuitionistic fuzzy numbers. remark 1. If 1 p = , then, the generalized weighted averaging operator coincides with the arithmetic averaging operator described by (3). remark 2. The limit of the generalized weighted averaging operator, when 0 p → , coincides with the geometric averaging operator described by (4): is a set of triangular intuitionistic fuzzy numbers. Then, according to formula (9) and operation rules for triangular intuitionistic fuzzy numbers, the result of the generalized weighted averaging operator with exponent p is a triangular intuitionistic fuzzy number. a formula for calculating the generalized weighted averaging operator in the set of triangular IFNs proposed by Zhao et al. (2010) is as follows:

monte Carlo research on the stability of the generalized weighted averaging operator considering ranking alternatives
Initially design our research. suppose we have 5 experts and 5 alternative projects that must be ranked according to 5 criteria. There are 3 versions of initial decision matrices the elements ij D of which reflect the satisfaction degree of project , 1,2,...5 i s i = to attribute , 1,2,...5 j r j = and are represented by crisp numbers. Initial decision matrices reflect an objective (true) judgment of projects. The weights of criteria are considered as equal, i.e. their importance is equal to . Initial decision matrices are given in Table 1. The first matrix shows high, the second -medium and the third -low separability of alternatives: Nevertheless, we can easily check that for all 3 decision matrices and equal weights of criteria, the ranking results of alternatives, according to all our surveyed methods, would be the same: 1 2 3 4 5 s s s s s     . The elements of the Monte Carlo experiment involving 5 expert decision matrixes are intuitionistic fuzzy numbers randomly generated from the values of the initial matrixes given in Table 1 by uniform or normal distribution with different variance values.
In the case of fuzzy numbers, the standard procedures of generating a random number could be applied. as FNs have 2 components -the degree of membership a µ and the degree of non-membership a ν related by equation 1 a a µ + ν = -it is enough to generate only one component, for example, a µ . In the case of IFNs, there are two degrees of freedom -the degree of membership and the degree of uncertainty. Therefore, for IFNs, the procedure of generation must be applied twice. The method of an inverse cumulative distribution function, as described by Gentle (2003), has been used. supposedly, the judgment of each expert is a random number that does not differ significantly from an objective judgment. at the stage of planning our experiment, the idea of evaluating the degrees of membership and non-membership independently have been used (Dubois et al. 2005). The fuzzy number of expert evaluation ( , ) a a µ ν is generated by normal distribution from crisp number a in the following way. For the fixed σ value, random number 1 X from normal distribution with average 1 EX a = and standard deviation 1 DX = σ is generated. a µ is the realization of random variable 1 X . Then, the degree of uncertainty a π is formed, which is the realization of random variable 2 X having normal distribution with 2 2 0.1, 0.05 EX DX = = . Next, 1 a a a ν = − µ − π is calculated. Generation by uniform distribution was performed in a similar way. 1 X as a uniform random variable in the interval [ , ] a a − σ + σ and 2 X as a uniform random variable in the interval [0.05, 0.15] have been generated.
The realization of the generated decision matrix of high separability alternatives in the case of normal distribution when 0.2 σ = is presented in Table 2.  such decision matrixes were generated for all 5 experts. Next, the combined decision matrix is created from separate experts matrixes. The elements of the combined matrix are triangular intuitionistic fuzzy numbers ([ , , ], [ , , ]) The probability of ranking errors has been calculated as a proportion of wrong ranking results amongst all 20000 experiments. The higher is error probability, the less stable is the corresponding generalized weighted averaging operator. It seems to be clear that the more standard deviation σ is the higher is the probability of ranking error. The results of the experiment for the initial matrixes of high, medium and low separability in the case of generation by uniform distribution are presented in Table 3. each row (different σ values) contains the least error probability marked in bold font. The most stable generalized average operators for high and medium separability initial matrixes have been interval fuzzy with exponents 0.01 p = and 0.25 p = , whereas for a low separability initial matrix, triangular and interval generalized average operators with 0.01 p = have been accepted the most steady.
The dependence of error probability on σ taking into consideration various values of exponent p in the case of uniform distribution for intuitionistic interval fuzzy numbers of a high separability initial matrix is depicted in Figure 2.   The dependence of error probability on p taking into consideration various values of σ in the case of uniform distribution for the intuitionistic interval fuzzy numbers of the initial separability matrix is presented in Figure 3.
The results of the experiment on high, medium and low separability initial matrixes in the case of generating normal distribution are presented in Table 4. The most stable generalized averaging operators of the high separability initial matrix include the point generalized averaging operator with exponents 0.01 p = and 0.25 p = . For medium and low separability initial matrices, the point generalized averaging operator having 0.01 p = appeared to be the most stable. The dependence of error probability on σ taking into consideration various values of p in the case of normal distribution for the point intuitionistic fuzzy numbers of the low separability initial matrix is shown in Figure 4.
Next, the probability of ranking errors depending on the number of experts will be analyzed. Comparative analysis involving 3, 4 and 5 experts and the medium separability initial matrix has been performed. Different exponent values of generalized weighted averaging operators and various types of intuitionistic fuzzy numbers (triangular, interval and point) have been examined. The chosen numbers of experts are small because, as a rule, hiring   experts is rather expensive. Table 5 indicates the probability of ranking errors considering generalized averaging operators with uniform distribution in the cases of 3, 4 and 5 experts. For each row, the least probability of error is marked in bold font. The expectable result has been obtained -the bigger is the number of experts, the lower is the probability of ranking errors. Trends are very similar in the cases of 3, 4 and 5 experts -the most stable generalized averaging operators almost for all σ values have been those with a fuzzy interval and exponent 0.01 p = .   Table 6 shows the probability of ranking errors for generalized averaging operators with normal distribution in the cases of 3, 4 and 5 experts and the medium separability initial matrix. likewise in the case of uniform distribution, a higher number of experts results in the lower probability of ranking errors. In all cases, the most stable values of generalized averaging operators have been those including a fuzzy point with exponent p = 0.01. Figure 5 shows two graphs of the dependence of error probability on σ regarding various values of p in the cases of uniform and normal distribution and the medium separability initial matrix. Table 6. The probability of ranking errors for generalized averaging operators with normal distribution and the medium separability initial matrix in the cases of 3, 4 and 5 experts The results of the conducted research have disclosed that the least probability of ranking errors has not been noticed neither as regards arithmetic ( 1) p = nor geometric averaging operators ( 0) p = in the case of 5 alternatives and 5 criteria for 3, 4 and 5 experts when analyzing normally and uniformly distributed IFNs. Thus, the advantage of generalized averaging operators is a minor error in ranking probability. The following numerical example will show that in a marginal case, when at least for one alternative and one criteria (0;0;0) a µ = , the weighted geometric averaging operator will assign the lowest rank to this alternative despite of the values of other elements of the decision matrix. on the other hand, when at least for one alternative and one criteria (1;1;1) a µ = , the weighted arithmetic averaging operator will assign the highest rank to this alternative despite of the values of other elements of the decision matrix. Thus, in such marginal cases, the weighted geometric averaging operator and the weighted arithmetic averaging operator become insensitive and inappropriate. x x x = : , If we have m alternatives , 1,2,..., , the decision matrix is as follows 2 : Thus, numbers j m and j M can be chosen: Further, an example (antuchevičienė et al. 2010, 2011) where a decision on possible redevelopment alternatives to derelict rural buildings must be chosen from 3 alternatives will be analyzed: the reconstruction of rural buildings and adapting them to production (or commercial) activities (alternative 1 A ), improving and using them for farming (alternative x -average soil fertility in the area (points); 2 x -the quality of life of the local population (points); 3 x -population activity index (%); 4 x -GDP proportion with respect to the average GDP of the country (%); 5 x -material investment in the area (lt per resident); 6 x -foreign investment in the area (lt × 10 3 per resident); 7 x -building redevelopment costs (lt × 10 6 ); 8 x -an increase in the income of the local population (lt × 10 6 per year); 9 x -an increase in sales in the area (%); 10 x -an increase in employment in the area (%); 11 x -state income from business and property taxes (lt × 10 6 per year); 12 x -business outlook; 13 x -difficulties in changing the original purpose of a site; 14 x -the degree of contamination; 15 x -the attractiveness of the countryside (i.e. image, landscape, etc). among the criteria, considered 2 7 13 14 , , , x x x x are associated with cost/loss and therefore their lower value is better while the remaining criteria are associated with benefit and their higher value is better. Hence, * {2,7,13,14}, J = {1,3,4,5,6,8,9,10,11,12,15} J = . Three decision matrixes are given for three regions according to the concept of the spatial development of the country: the areas of active development, the areas of regressing development and 'buffer' areas. These matrixes are presented in Table 7.
. 1 x has a direct relationship with the alternatives (a greater value is better). Thus, following the application of formula (12)  Notice that if p = 1.0, 1 i I is the weighted arithmetic averaging operator, and if p = 0.0, 0 i I is the weighted geometric averaging operator. Then, for each p, the values of the score function are calculated using formulas (5)-(7) and ranking intuitionistic fuzzy numbers is accomplished. The results are given in Table 9. additionally, the ranking results obtained after applying 4 FMCDM methods -CoPras, ToPsIs, including vector and linear normalization and vIKor (antuchevičienė et al. 2011) are presented in Table 9. The ranking results of active development areas differ depending on the applied method. CoPras, ToPsIs with linear normalization and vIKor rank the alternatives in the following order: Difficulties in making some strict inferences in the case of ranking results of 'buffer' areas appear, because the results differ not only in different p values taking into account the method of generalized aggregated fuzzy average functions, but also considering various FMCDM methods. Nevertheless, only generalized aggregated fuzzy average functions having p = 0 (geometric mean) ranking results coincide with ranking results of CoPras and ToPsIs using vector normalization: 1 2 3 A A A   . Notice, that for p = 0 (geometric mean), the scores of alternative 3 A for active development areas, the areas of regressing development and 'buffer' areas are negative while the scores of other alternatives are positive. Thus, alternative 3 A will always be the worst. such situation of a geometric mean will be in the case when for at least one of indicators (0,0,0) j µ = for the given alternative. Therefore, we have 9 (0,0,0) µ = and 10 (0,0,0) µ = for x 9 and x 10 (see Table 7). suppose that for one alternative and at least one variable  Table 8 shows such variables ( 4 5 6 , , x x x ) for each of 3 alternatives, i.e. the values of the generalized aggregated fuzzy average function for 0 p ≠ are simple triangular fuzzy numbers (not intuitionistic fuzzy numbers) for all 3 alternatives.
Continued Table 9