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Abstract Deriving accurate fuzzy priorities is very important in multi-criteria decision making with vague information. In this paper, appropriate formulas for obtaining fuzzy priorities from additive fuzzy pairwise comparison matrices are introduced. The formulas are based on the proper fuzzy extension of the formulas for obtaining priorities from additive pairwise comparison matrices proposed by Fedrizzi & Brunelli (2010, Soft Comput., 14, 639–645) satisfying Tanino's characterization. Moreover, a new normalization condition for priorities reachable (unlike other normalization conditions formerly proposed in the literature) also for inconsistent additive pairwise comparison matrices is proposed and extended properly to additive fuzzy pairwise comparison matrices. Furthermore, a new definition of a consistent additive fuzzy pairwise comparison matrix independent of the ordering of objects in the matrix is given, and the consistency requirement is also employed directly into the formulas for obtaining fuzzy priorities. Triangular fuzzy numbers are used for the fuzzy extension in the paper, and a brief discussion on how to easily modify the formulas and the definitions presented in the paper in order to apply on intervals, trapezoidal fuzzy numbers or any other type of fuzzy numbers is provided. The theory is illustrated on numerical examples throughout the paper. 1. Introduction Methods based on pairwise comparisons of objects form a significant part of multi-criteria decision-making methods. Beside well-known multiplicatively reciprocal pairwise comparison matrices, fuzzy preference relations are often used to compare pairwisely a set of n objects o1,…,on by expressing the intensity of preference of one compared object over another one. In this case, the intensities of preference are defined on the interval [0,1]. Priorities w1,…,wn of the objects expressing the relative importance of each object with respect to the other objects are then obtained from a fuzzy preference relation. The first bases of the theory of fuzzy preference relations were given by Orlovski (1978), Nurmi (1981), Tanino (1984) and Kacprzyk (1986). Tanino (1984) defined additive transitivity of fuzzy preference relations and derived a characterization describing the relation between the pairwise comparisons of objects in the fuzzy preference relation and the priorities of objects. This additive transitivity and related characterization have played a very important role in the subsequent research, and play a key role also in this paper. Afterwards, many other papers on fuzzy preference relations followed, see, e.g., Chiclana et al. (1998), Fan et al. (2002), Chiclana et al. (2003), Xu & Da (2003), Herrera-Viedma et al. (2004), Xu (2004a,b), Lee & Tseng (2006), Herrera-Viedma et al. (2007), Xu (2007b), Lee et al. (2008), Lee & Yeh (2008), Xu & Chen (2008b), Fedrizzi & Brunelli (2009, 2010) and Fedrizzi & Giove (2013). For example, Fan et al. (2002) proposed an optimization model to derive priorities w1,…,wn of objects. Xu & Da (2003) provided an approach for improving consistency of fuzzy preference relations and presented an iterative algorithm to compute priorities with acceptable consistency. Herrera-Viedma et al. (2004) developed a method for constructing consistent fuzzy preference relations from only n−1 known pairwise comparisons of objects. Xu (2004a) proposed an optimization method for obtaining priorities from incomplete fuzzy preference relations where the normalization condition ∑i=1nwi=1,wi∈[0,1],i=1,…n, was required. After, Xu & Chen (2008b) proposed a method for completing an incomplete fuzzy preference relation based again on the normalization condition ∑i=1nwi=1,wi∈[0,1],i=1,…n. Fedrizzi & Brunelli (2009) showed that this normalization condition is not compatible with the characterization given by Tanino (1984), and they proposed another normalization condition. After, Fedrizzi & Brunelli (2010) proposed a very simple method for obtaining priorities from fuzzy preference relations satisfying Tanino's characterization. Recently, many papers dealing with the extension of fuzzy preference relations to intervals have been published, see, e.g. Xu (2007a), Xu & Chen (2008a), Alonso et al. (2008), Genç et al. (2010), Wang & Li (2012), Wang et al. (2012), Liu et al. (2012a), Hu et al. (2014) and Xu et al. (2014). Xu (2007a) proposed a method for obtaining crisp priorities from interval fuzzy preference relations based on Tanino's characterization. Later, Xu & Chen (2008a) introduced a definition of consistency for interval fuzzy preference relations based on Tanino's characterization, and they employed this consistency requirement into the linear programming models for deriving interval priorities from interval fuzzy preference relations. However, both Xu (2007a) and Xu & Chen (2008a) employed in their models the normalization condition ∑i=1nwi=1,wi∈[0,1],i=1,…n, which is not compatible with Tanino's characterization. Hu et al. (2014) later proposed a modification of these models by replacing Tanino's characterization with another characterization, and Xu et al. (2014) generalized the models by adding a parameter into the characterization. Wang & Li (2012) proposed another definition of consistency based on Tanino's characterization, and they employed this consistency requirement together with the normalization condition ∑i=1nwi=1,wi∈[0,1],i=1,…n, into a linear programming model for obtaining interval priorities of objects. Wang et al. (2012) introduced a definition of consistency using a particular characterization based on logarithms, and they used this characterization also as the constraints in linear programming models for obtaining the interval priorities. However, the interpretation of such characterization based on logarithms was not clarified. Liu et al. (2012a) proposed another definition of consistency for interval fuzzy preference relations and a method for obtaining interval priorities from these relations. However, both the definition of consistency and the method for obtaining interval priorities are dependent on the ordering of objects in the matrix. Thus, with a change of the ordering of the objects (which does not change any information about the intensities of preference between the objects), an interval fuzzy preference relation which was consistent does not have to be consistent anymore, and the interval priorities of the objects differ from the interval priorities obtained before the change of the ordering of the objects. The extension of fuzzy preference relations to fuzzy sets has not been studied with such attention yet. Herrera et al. (2005) proposed a fuzzy extension to triangular fuzzy numbers with assigned linguistic terms. Wu & Chiclana (2014) extended fuzzy preference relations to intuitionistic fuzzy sets and proposed a procedure to estimate missing pairwise comparisons based on multiplicative consistency. Krejčí (2015) proposed methods for obtaining multiplicative fuzzy priorities from fuzzy preference relations of triangular fuzzy numbers and employed various types of consistency in order to obtain consistent multiplicative fuzzy priorities. In this paper, fuzzy preference relations and methods for obtaining priorities from these relations are extended properly to triangular fuzzy numbers. Tanino's characterization is considered as a key property of fuzzy preference relations, because this characterization (unlike other characterizations proposed in the literature) has a clear interpretation related to the priorities of objects obtained from a fuzzy preference relation. The formulas for obtaining priorities from fuzzy preference relations proposed by Fedrizzi & Brunelli (2010) and satisfying Tanino's characterization are fuzzified properly using the constrained fuzzy arithmetic. Moreover, the problem of normalizing priorities is dealt with in this paper. It is pointed out that the normalization condition proposed by Fedrizzi & Brunelli (2010) is reachable only for fuzzy preference relations consistent according to Tanino's additive transitivity. However, in real decision-making problems, full consistency is often unreachable; with an increasing number of compared objects, keeping the consistency is very difficult or even impossible due to the limited scale for expressing the intensities of preference of one object over another. In such situations, even if the decision maker was asked to reconsider his or her preferences, it would not have to lead to a consistent fuzzy preference relation. Therefore, a normalization condition reachable for both consistent and inconsistent fuzzy preference relations is needed. Such a normalization condition is proposed in this paper and extended to fuzzy preference relations with triangular fuzzy numbers. Furthermore, a proper fuzzy extension of the consistency based on Tanino's additive transitivity independent of the ordering of objects is introduced, and the requirement of consistency of pairwise comparisons is also employed directly into the formulas for obtaining the fuzzy priorities. Even though the fuzzy preference relations are usually extended only to intervals (as is evident from the literature review in the previous paragraphs), this paper focuses on the extension to triangular fuzzy numbers. That is because triangular fuzzy numbers, unlike intervals, allow us to work with degrees of possibility of the intensities of preference on compared objects. Thus, when a decision maker is asked to provide the intensity of preference of one compared object over another one, he or she can provide the most possible intensity of preference, the lowest possible and the highest possible one. By this we obtain the representing values of a triangular fuzzy number modelling the intensity of preference of one object over another one. By contrast, when using intervals, the decision maker provides only the lowest possible and the highest possible intensity of preference, and all the values in the range between these two values are equally possible. It is important to realize that triangular fuzzy numbers are not an extension of intervals. In order to extend intervals, trapezoidal fuzzy numbers, which extend also crisp numbers and triangular fuzzy numbers, would have to be used. However, using triangular fuzzy numbers for the fuzzy extension in this paper is not a drawback of the proposed methodology. All the definitions and the formulas given in this paper for triangular fuzzy numbers can be easily modified in order to be applied on intervals, trapezoidal fuzzy numbers or any other type of fuzzy numbers. When intervals are dealt with, only the formulas regarding the lower and upper boundary values of the triangular fuzzy numbers are worked with. When trapezoidal fuzzy numbers are used, the formulas regarding the middle values of the triangular fuzzy numbers are generalized based on the corresponding formulas regarding the lower and upper boundary values. And finally, when general fuzzy numbers are applied, the formulas for computing the α-cuts of these fuzzy numbers are derived from the corresponding formulas regarding the lower and upper boundary values of triangular fuzzy numbers. In the literature, the fuzzy preference relations are also often called (additively) reciprocal relations (see, e.g. De Baets et al., 2006, Fedrizzi & Brunelli, 2009, 2010, Fedrizzi & Giove, 2013, Krejčí, 2015). Because the extension of the fuzzy preference relations by triangular fuzzy numbers is approached in this paper, the second terminology is used for the clarity, and these reciprocal relations will be called additive fuzzy reciprocal relations where the word fuzzy stands for fuzzy numbers (in our case triangular fuzzy numbers). The pairwise comparison matrices corresponding to the additive fuzzy reciprocal relations will be then called additive fuzzy pairwise comparison matrices. This terminology is also chosen in order to have an analogy to the fuzzy pairwise comparison matrices in fuzzy extension of Analytic Hierarchy Process (AHP) that are multiplicatively reciprocal. For simplicity, PCM and FPCM will be used instead of the expressions pairwise comparison matrix and fuzzy pairwise comparison matrix hereafter. The rest of the paper is organized as follows. In Section 2, basic notions of triangular fuzzy numbers and operations with them are given. In Section 3, additive PCMs are defined, and some methods for obtaining priorities from these matrices are reviewed. Further, the normalization of the priorities is discussed, and a normalization condition suitable both for consistent and inconsistent fuzzy preference relations is introduced. In Section 4, additive FPCMs are defined and a proper fuzzy extension of the methods from the previous section is done for computing the fuzzy priorities. In Section 5, consistency of an additive FPCM based on Tanino's additive transitivity is defined, and the requirement of consistency is also employed directly into the formulas for obtaining fuzzy priorities from the additive FPCM. Finally, a conclusion is done in Section 6. 2. Preliminaries 2.1. Basic notions of triangular fuzzy numbers In this subsection, the definition of a triangular fuzzy number is given, and basic notions of the standard fuzzy arithmetic and the constrained fuzzy arithmetic are provided. Let U be a non empty set. A fuzzy set S˜ on the set U is characterized by its membership function S˜:U→[0,1]. The set CoreS˜:={u∈U;S˜(u)=1} denotes the core of S˜, and the set SuppS˜:={u∈U;S˜(u)>0} denotes the support of S˜. A triangular fuzzy number c˜ is a fuzzy set on ℝ whose membership function is uniquely determined by a triple of real numbers cL≤cM≤cU in the following way: c˜(x)={x−cLcM−cL,cL<x<cM,1,x=cM,cU−xcU−cM,cM<x<cU,0,otherwise. (2.1) For a triangular fuzzy number c˜ whose membership function is given by (2.1), the notation c˜=(cL,cM,cU) will be used hereafter; the real numbers cL and cU are called the lower and upper boundary values, and cM is called the middle value of the triangular fuzzy number c˜. A triangular fuzzy number c˜=(cL,cM,cU) is said to be positive if cL>0. The core of triangular fuzzy number c˜=(cL,cM,cU) is a singleton set Corec˜={cM}, and the support is an open interval Suppc˜=(cL,cU). Further, the α-cut of c˜=(cL,cM,cU) is an interval c˜α={x∈Suppc˜;c˜(x)≥α}. Particularly for triangular fuzzy numbers, c˜α=[cαL,cαU], where cαL=αcM+(1−α)cL and cαU=αcM+(1−α)cU are called the lower and upper boundary values of the α-cut c˜α, respectively. When there are no interactions between fuzzy numbers in a given set, the arithmetic operations performed on these fuzzy numbers are based on the standard fuzzy arithmetic. The sum of two triangular fuzzy numbers c˜=(cL,cM,cU) and d˜=(dL,dM,dU) is a triangular fuzzy number c˜+d˜=(cL+dL,cM+dM,cU+dU), and the reciprocal (in additive sense) of a triangular fuzzy number c˜=(cL,cM,cU) is a triangular fuzzy number 1−c˜=(1−cU,1−cM,1−cL). In the case of any interaction between fuzzy numbers, the constrained fuzzy arithmetic by Klir & Pan (1998) should be applied on the arithmetic operations. The importance of employing the constrained fuzzy arithmetic in the fuzzy extension of the methods based on pairwise comparisons, such as additive PCMs and AHP, was emphasized e.g. by Enea & Piazza (2004), Fedrizzi & Krejčí (2015), Krejčí (2015), Krejčí (2016) and Krejčí et al. (2016). In the case of the arithmetic operations performed on the triangular fuzzy numbers, the simplified constrained fuzzy arithmetic can be applied in order to obtain the results in the form of a triangular fuzzy number. Let f be a continuous function, f:ℝn→ℝ, let c˜i=(ciL,ciM,ciU),i=1,…,n, be triangular fuzzy numbers, and let D be a relation in ℝn describing interactions between the variables. Then, according to the simplified constrained fuzzy arithmetic, c˜=fF(c˜1,…,c˜n),c˜=(cL,cM,cU), is a triangular fuzzy number whose triplet of representing values is obtained as cL=min{f(x1,…,xn);(x1,…,xn)∈D∩[c1L,c1U]×…×[cnL,cnU]},cM=f(c1M,…,cnM),cU=max{f(x1,…,xn);(x1,…,xn)∈D∩[c1L,c1U]×…×[cnL,cnU]}. (2.2) The simplest example of interactions in (2.2) is the case when the operands in the arithmetic operation represent a particular state of the same linguistic variable. In such case, any value under one operand can be combined only with the same value of the other (equal) operand. Thus, for example, for a linguistic variable c˜=(1,2,3), the difference (dL,dM,dU):=c˜−c˜ should not be computed as (1,2,3)−(1,2,3)=(−2,0,2) (the standard fuzzy arithmetic) but as dL=min{x−y;x∈[1,3],y∈[1,3],x=y}=0,dM=2−2=0,dL=max{x−y;x∈[1,3],y∈[1,3],x=y}=0, i.e. c˜−c˜=0. The simplified constrained fuzzy arithmetic will be applied later in the paper in order to preserve the additive reciprocity and the consistency of pairwise comparisons in additive FPCMs during the process of computing fuzzy priorities. 3. Additive pairwise comparison matrices In this section, additive PCMs are defined and the methods for obtaining the additive priorities of objects from such matrices are discussed. An additive PCM of n objects o1,…,on is a square matrix A=(aij)i,j=1n,aij∈[0,1], that is additively reciprocal, i.e. aij=1−aji,i,j=1,…,n. The elements aij,i,j=1,…,n, of the matrix express the intensity of preference of object oi over object oj: aij=1if oi is absolutely preferred to oj,aij∈(0.5,1)if oi is preferred to oj,aij=0.5if oi and oj are indifferent,aij∈(0,0.5)if oj is preferred to oi,aij=0if oj is absolutely preferred to oi. (3.1) In the literature, relations represented by these PCMs are called either (additively) reciprocal relations (De Baets et al., 2006; Fedrizzi & Brunelli, 2009, 2010; Fedrizzi & Giove, 2013) or fuzzy preference relations (Bezdek et al., 1978; Nurmi, 1981; Tanino, 1984; Kacprzyk, 1986; Gavalec et al., 2015). In this paper, the first name is preferred, and that is why the corresponding PCM A=(aij)i,j=1n is called an additive PCM. An additive PCM A=(aij)i,j=1n is called (additively) consistent if Tanino's additive-transitivity property (Tanino, 1984) aij=aik+akj−0.5, i,j,k=1,…,n, (3.2) is satisfied. Proposition 3.1 (Tanino, 1984) An additive PCM A=(aij)i,j=1n is additively consistent if and only if there exists a non-negative vector w=(w1,…,wn),|wi−wj|≤1, i,j=1,…,n, such that aij=0.5+0.5(wi−wj), i,j=1,…,n. (3.3) Proposition 3.1 says that, when an additive PCM A=(aij)i,j=1n of n objects is additively consistent, there exist priorities w1,…,wn of the objects using which we can determine precisely the original pairwise comparisons aij in the PCM A by applying Tanino's characterization (3.3). Moreover, Tanino's characterization (3.3) implies another very interesting relation between the priorities w1,…,wn and the pairwise comparisons aij,i,j=1,…,n, of an additively consistent PCM; the difference between each two priorities equals to the difference between the corresponding pairwise comparisons in the matrix, i.e. wi−wj=aij−aji,i,j=1,…,n. For example, for the intensity of preference aij=0.7, we know immediately that the difference between the priorities wi and wj is 0.4 ( wi−wj=aij−aji=0.7−(1−0.7)=0.4). Many other consistency conditions based on the notion of transitivity have been introduced for additive PCMs; for an overview, see, e.g., Herrera-Viedma et al. (2004), Chiclana et al. (2009) and Krejčí (2015). In this paper, however, the focus is put only on the traditional consistency condition based on Tanino's additive-transitivity property (3.2), because this transitivity property has a very clear interpretation related to the priorities obtained from an additive PCM. Fedrizzi & Brunelli (2010) proved that, for an additively consistent PCM A=(aij)i,j=1n, the only vector of priorities (up to an additive constant) satisfying (3.3) is w=(w1,…,wn) such that wi=2n∑j=1naij, i=1,…,n. (3.4) Proposition 3.2 Given an additive PCM A=(aij)i,j=1n, the priorities w1,…,wn obtained from A by formula (3.4) are such that ∑i=1nwi=n. (3.5) Proof. ∑i=1nwi=∑i=1n2n∑j=1naij=2n∑i=1n∑j=1naij=2n(∑i=1naii+∑i=1n∑j=1j≠inaij)=2n(n2+n(n−1)2)=n. □ Remark 3.1 Property (3.5) of the priorities given by (3.4) is independent of the additive consistency. That is, the priorities of objects can be obtained also from inconsistent additive PCMs by formulas (3.4), and their sum still equals n. However, such priorities do not satisfy (3.3) anymore. The expression 0.5+0.5(wi−wj) (3.6) gives only an approximate value of the actual pairwise comparison aij in the matrix. Nevertheless, it is a standard procedure to obtain the priorities of objects from an inconsistent PCM in this way. As was shown by Fedrizzi & Brunelli (2010), there exist infinitely many priority vectors satisfying Proposition 3.1. These priority vectors can be generated from (3.4) by adding an arbitrary constant. Notice that we cannot multiply the priorities as it is done in AHP where the ratios of the priorities estimate the original pairwise comparisons in the matrix. In our case, the original pairwise comparisons aij in the additive PCM A=(aij)i,j=1n are estimated by the differences between the priorities wi and wj by means of (3.6) and, thus, these differences have to remain unchanged. In order to reach uniqueness, a normalization condition is applied on the priority vectors. In AHP, the normalization condition ∑i=1nwi=1, wi∈[0,1], i=1,…,n, (3.7) is usually applied on the priorities obtained from multiplicatively reciprocal PCMs. It is worth to note that the condition (3.7) is reachable independently of the requirement of consistency of the multiplicatively reciprocal PCM, i.e. even the priorities obtained from an inconsistent multiplicatively reciprocal PCM can be normalized so that they satisfy (3.7). The normalization condition (3.7) has also been applied on the priorities obtained from additive PCMs (see, e.g. Xu, 2004a, 2007a; Xu & Chen, 2008a,b; Wang & Li, 2012; and the list of other papers provided by Fedrizzi & Brunelli, 2009). However, Fedrizzi & Brunelli (2009) showed that the normalization condition (3.7) is incompatible with Proposition 3.1. Furthermore, they proposed a new normalization condition in the form mini=1,…,nwi=0, wi∈[0,1], i=1,…,n. (3.8) However, the normalization condition (3.8) is reachable only for additively consistent PCMs. For inconsistent additive PCMs, in general, the normalized priorities satisfying the constraint mini=1,…,nwi=0 do not satisfy the constraint wi∈[0,1], i=1,…,n, as is illustrated on the following example. Example 3.1 Let us assume the additive PCM A=(0.50.910.10.50.700.30.5), (3.9) which is not additively consistent. The priorities of objects obtained by formula (3.4) are in the form w1=2415,w2=1315,w3=815. By applying the normalization constraint mini=1,…,nwi=0, we obtain normalized priorities in the form w1=1615,w2=515,w3=0. Clearly, w1>1 which violates the normalization constraint wi∈[0,1], i=1,…,n. Proposition 3.3 Given an additive PCM A=(aij)i,j=1n,n≥3, the property wi∈[0,1], i=1,…,n, is not reachable for the priorities (3.4) under any normalization condition. Proof. There exist infinitely many priority vectors obtainable from (3.4) by adding an arbitrary constant. In order to modify the priorities so that wi∈[0,1], i=1,…,n, a suitable constant c has to be added to the priorities (3.4). Furthermore, we know that the differences between the priorities do not change by adding a constant to them; (wi+c)−(wj+c)=wi−wj,i,j∈{1,…,n}. Clearly, the priorities (3.4) could be normalized so that wi+c∈[0,1], i=1,…,n, only if |wi−wj|≤1,i,j=1,…,n. However, it will be shown that |wi−wj|≤1,i,j=1,…,n, is not reachable in general. Let oi,i∈{1,…,n}, be such that it is absolutely preferred to all other objects, and let oj,j∈{1,…,n}, be such that all other objects are absolutely preferred to oj. Then, wi−wj=2n∑k=1naik−2n∑k=1najk=2n((0.5+n−1)−(0.5+0))=2n−2n>1, for n≥3. □ According to Proposition 3.3, the property wi∈[0,1], i=1,…,n, cannot be guaranteed for inconsistent additive FPCMs under any normalization condition. However, in many multi-criteria decision-making problems, it is difficult to reach additive consistency of additive PCMs especially because of the restricted scale [0,1] used for expressing the intensities of preference of one compared object over another. In general, the higher the dimension of an additive PCM is, the more difficult reaching the consistency is. Even when the decision maker is asked to reconsider his/her preferences, it does not have to lead to a consistent additive PCM. Therefore, in real-life applications, priorities of objects have to be often elicited from inconsistent additive PCMs. This calls for a normalization condition applicable also on the priorities obtained from these inconsistent additive PCMs (remember that for multiplicatively reciprocal PCMs, there is such a normalization condition-(3.7)). The normalization condition (3.8) can be weakened as mini=1,…,nwi=0, (3.10) which is reachable for any additive PCMs (i.e. not only for additively consistent ones). By applying this normalization condition on the priorities obtained by formula (3.4), we can directly derive formulas for obtaining normalized priorities from an additive PCM as wi=2n∑j=1naij−mink∈{1,…,n}2n∑j=1nakj=2n(∑j=1naij−mink∈{1,…,n}∑j=1nakj), i=1,…,n. (3.11) Normalization condition (3.10) works well for additive PCMs. However, as will be shown in the following section, this normalization condition is not suitable for the fuzzy extension, i.e. for obtaining fuzzy priorities from additive FPCMs. The problem is that the condition (3.10) does not keep any information about the interactions between the priorities wi,i=1,…,n, which is indispensable for a proper fuzzy extension of the method. It only says that the smallest priority equals 0. For the fuzzy extension, a normalization condition such as (3.7) would be appropriate since it holds information about the interactions between all priorities. However, as was discussed earlier, the normalization condition (3.7) is not compatible with Proposition 3.1. We could weaken the requirements. We know that additive PCMs are not additively consistent in most cases and, thus, the expression (3.6) only approximates the original pairwise comparisons in the matrix. Moreover, according to Proposition 3.3, the constraint wi∈[0,1], i=1,…,n, is unreachable. Thus, we can just apply normalization condition ∑i=1nwi=1 (3.12) without any further constraints on the priorities. By applying this normalization condition on the priorities obtained by formulas (3.4), we derive formulas for obtaining normalized priorities from an additive PCM as wi=2n∑j=1naij−n−1n, i=1,…,n. (3.13) Proposition 3.4 Given an additive PCM A=(aij)i,j=1n, the priorities w1,…,wn obtained from A by formula (3.13) are such that ∑i=1nwi=1 (3.14) and −1<wi≤1, i=1,…,n. (3.15) Proof. ∑i=1nwi=∑i=1n(2n∑j=1naij−n−1n)=2n∑i=1n∑j=1naij−(n−1)=1, which proves (3.14). The value of priority wi,i∈{1,…,n}, obtained by formula (3.13) depends only on the pairwise comparisons in the i-th row of the matrix, i.e. on the intensities of preference of object oi over the other objects. To prove the inequality wi≤1, we just need to show that the priority of object oi will not exceed 1 even for the highest possible intensities of preference of object oi over all other objects. Let oi,i∈{1,…,n}, be absolutely preferred to all other objects. Then, wi=2n∑j=1naij−n−1n=1n(2∑j=1naij−n+1)=1n(2(n−1+0.5)−n+1)=1. Similarly, to prove the inequality −1<wi, we just need to show that the priority of object oi will be greater than −1 even for the lowest possible intensities of preference of object oi over all other objects. Let oi,i∈{1,…,n}, be absolutely preferred by all other objects. Then, wi=2n∑j=1naij−n−1n=1n(2(0+0.5)−n+1)=2−nn=−1+2n>−1. □ Remark 3.2 Also a more general characterization than Tanino's characterization (3.3) has appeared in the literature (see, e.g., Xu et al., 2009; Liu et al., 2012b; Xu et al. 2014): aij=0.5+β(wi−wj), β≥maxi=1,…,n{n2−∑j=1naij}>0 (3.16) together with priorities wi=1nβ∑j=1naij−12β+1n (3.17) satisfying this characterization and normalization condition ∑i=1nwi=1,wi∈[0,1]. More particularly, Xu et al. (2009) proposed to set β=n2, and Xu et al. (2011) and Hu et al. (2014) assumed β=n−12. It is true that by assuming the characterization (3.16) the obtained normalized priorities (3.17) are always non-negative. However, the priorities do not have an intuitive interpretation; aij−aji=0.5+β(wi−wj)−0.5−β(wj−wi)=2β(wi−wj), which means that the difference of priorities gives us 12β-th of the difference between the corresponding pairwise comparisons in the additive PCM, which is very difficult to interpret. Particularly, for β=n2 we obtain wi−wj=1n(aij−aji), and for β=n−12 we obtain wi−wj=1n−1(aij−aji). Notice that, for β=12, the characterization (3.16) equals to Tanino's characterization (3.3) and the corresponding priorities (3.17) equal to priorities (3.13) with a clear and intuitive interpretation wi−wj=aij−aji. Thus, in this paper, Tanino's characterization is preferred over the characterization (3.16), even though the non-negativity of the priorities wi,i=1,…,n, is not guaranteed. Actually, possible negativity of some priorities is not a problem at all because the scale on which the priorities are given is an interval scale; the differences between the priorities are meaningful. For example the normalized priorities w1=1415,w2=315,w3=−215 obtained from the additive PCM (3.9) by the formula (3.13) tell us that, e.g. a23−a32 is estimated as w2−w3=13 or that a23 is estimated as 0.5+0.5515=23. 4. Additive fuzzy pairwise comparison matrices and formulas for obtaining fuzzy priorities In this section, additive FPCMs are defined, and methods for obtaining fuzzy priorities of objects from such matrices are proposed. Definition 4.1 An additive FPCM of n objects is a square matrix A˜=(a˜ij)i,j=1n whose elements a˜ij=(aijL,aijM,aijU) are triangular fuzzy numbers defined on the interval [0,1]. Furthermore, the matrix is additively reciprocal, i.e. a˜ji=1−a˜ij=(1−aijU,1−aijM,1−aijL),i,j=1,…,n, and aii=0.5,i=1,…,n. Remark 4.1 It is necessary that the elements on the main diagonal of an additive FPCM A˜ are crisp numbers, namely aii=0.5,i=1,…,n. This necessity results from the fact that on the main diagonal of an additive FPCM an object is always compared with itself. Thus, because the compared objects are identical, they are indifferent and there is no vagueness in this comparison (see, e.g., Krejčí, 2015; Krejčí et al., 2016). Since additive FPCMs are formed by triangular fuzzy numbers, also the priorities of objects obtained from these matrices are expected to be triangular fuzzy numbers. Note that some authors proposed to derive crisp priorities from FPCMs which is not coherent with the acknowledgment of the vagueness of information modelled by fuzzy numbers in the FPCM. In order to obtain fuzzy priorities from an additive FPCM, fuzzy extension of the formulas from the previous section has to be done properly. Particularly, the additive reciprocity of pairwise comparisons needs to be preserved. At the same time, all the vagueness of the fuzzy pairwise comparisons in the original additive FPCM has to be captured by the resulting fuzzy priorities. Keeping in mind these requirements, formulas for computing the representing values of the fuzzy priorities of objects based on the constrained fuzzy arithmetic (Klir & Pan, 1998) need to be derived. In the following, notation v˜i will be used for non-normalized fuzzy priorities and w˜i for normalized fuzzy priorities in order to distinguish them easily. 4.1. Non-normalized fuzzy priorities By applying properly the fuzzy extension of the formula (3.4), the formulas for computing the representing values of the non-normalized fuzzy priorities v˜i=(viL,viM,viU),i=1,…,n, from an additive FPCM A˜=(a˜ij)i,j=1n,a˜ij=(aijL,aijM,aijU), are obtained in the form viL=min{2n∑j=1naij;apq∈[apqL,apqU],apq=1−aqp,p,q=1,…,n}, (4.1) viM=2n∑j=1naijM, (4.2) viU=max{2n∑j=1naij;apq∈[apqL,apqU],apq=1−aqp,p,q=1,…,n}. (4.3) That is the middle values viM,i=1,…,n, are simply obtained as priorities from the additive PCM AM=(aijM)i,j=1n by applying the formula (3.4). In order to obtain the lower and upper boundary values of v˜i,i∈{1,…,n}, we need to search among all the additive PCMs constructed from the elements from the closures of the supports of the fuzzy numbers in the original additive FPCM A˜=(a˜ij)i,j=1n. For each such a matrix, we compute the priority vi by using the formula (3.4). The lower boundary value viL,i∈{1,…,n}, is then obtained as the minimum of these priorities, and the upper boundary value viU is obtained as the maximum of these priorities. Because the function optimized in the formulas (4.1) and (4.3) is increasing in all variables, the formulas can be further simplified so that no optimization is needed: viL=2n∑j=1naijL, viU=2n∑j=1naijU. (4.4) Remark 4.2 It is worth to note that the elimination of the optimization problems in the formulas (4.1) and (4.3) and their replacement by very simple formulas (4.4) was possible to do only because the constraints of the optimization problems have no effect on the optima; the reciprocity condition aij=1−aji has no influence since only pairwise comparisons from the i-th row of the additive FPCM are present in the optimized function. Usually, however, when the constrained fuzzy arithmetic is applied to derive fuzzy priorities from FPCMs, the formulas containing an optimization problem cannot be further simplified. As an example, the formulas for obtaining multiplicative fuzzy priorities from multiplicatively reciprocal FPCMs proposed by Enea & Piazza (2004) and Krejčí et al. (2016) and the formulas for obtaining multiplicative fuzzy priorities from additive FPCMs proposed by Krejčí (2015) are referred to. Similarly, later in this paper, optimization problems for obtaining normalized fuzzy priorities from additive FPCMs that cannot be further simplified will be given. Remark 4.3 As was already mentioned in the introduction, all the formulas and the definitions in this paper can be easily adapted in order to apply on intervals, trapezoidal fuzzy numbers, or any other type of fuzzy numbers. For example, for an additive interval PCM A¯=(a¯ij)i,j=1n,a¯ij=[aijL,aijU], the formulas for computing the interval priorities v¯i=[viL,viU] would be in the form (4.1), (4.3) or (4.4). Similarly, let us assume an additive fuzzy trapezoidal PCM A˜=(a˜ij)i,j=1n, where the trapezoidal fuzzy numbers a˜ij=[aijL,aijM,aijN,aijU] are defined as a˜ij(x)={x−aijLaijM−aijL,aijL<x<aijM,1,aijM≤x≤aijN,aijU−xaijU−aijN,aijN<x<aijU,0,otherwise. (4.5) The formulas (4.1) and (4.3) or (4.4) would be used for computing the representing values viL,viU of the fuzzy trapezoidal priorities v˜i=[viL,viM,viN,viU], and analogously, the formulas for computing the representing values viM,viN would be in the form viM=2n∑j=1naijM, viN=2n∑j=1naijN. (4.6) Analogously, all the formulas and the definitions in the rest of the paper can be modified. There are interactions between the fuzzy priorities v˜i,i=1,…,n, obtained by formulas (4.1)–(4.3). The property (3.5) valid for the priorities (3.4) obtained from an additive PCM is extended to the fuzzy priorities as ∀viα∈v˜iα ∃vjα∈v˜jα,j=1,…,n,j≠i: viα+∑j=1j≠invjα=n, (4.7) for all α∈[0,1] and i=1,…,n. This interaction property will be formulated properly and proved later. First, the following proposition is needed. Proposition 4.1 Let v˜i=(viL,viM,viU),i=1,…,n, be triangular fuzzy numbers. The interaction property (4.7) between the fuzzy numbers is valid if and only if ∑j=1nvjM=n, viU+∑j=1j≠invjL≤n, viL+∑j=1j≠invjU≥n, ∀i∈{1,…,n}. (4.8) Proof. First, let us show that (4.7) implies (4.8). For α=1, the α- cuts of v˜i,i=1,…,n, are singleton sets viM. Thus, for viM ∃vjM,j=1,…,n,j≠i:viM+∑j=1j≠invjM=n, which means ∑j=1nvjM=n. Furthermore, from (4.7), it follows that for viU,i∈{1,…,n}, ∃vj∈[viL,viU],j=1,…,n,j≠i:viU+∑j=1j≠invj=n. Because vjL≤vj, then clearly viU+∑j=1j≠invjL≤n. Analogously, for viL,i∈{1,…,n}, ∃vj∈[viL,viU],j=1,…,n,j≠i:viL+∑j=1j≠invj=n. Because vjU≥vj, then clearly viL+∑j=1j≠invjU≥n. Now, let us show that (4.8) implies (4.7). The property ∑j=1nvjM=n implies (4.7) for α=1. Furthermore, from viU+∑j=1j≠invjL≤n and viL+∑j=1j≠invjU≥n, it follows that ∀vi∈[viL,viU]vi+∑j=1j≠invjL≤n and vi+∑j=1j≠invjU≥n. Therefore, ∃vj∈[vjL,vjU]: vi+∑j=1j≠invj=n, which implies (4.7) for α=0. The proof of the validity of (4.7) for α∈(0,1) is analogous; we just need to show that the inequalities (4.8) hold also for the α-cuts [viαL,viαU], i.e. viαU+∑j=1j≠invjαL≤n, viαL+∑j=1j≠invjαU≥n, ∀i∈{1,…,n}. (4.9) Then it is enough to take the α-cuts [viαL,viαU] of v˜i,i=1,…,n, for [viL,viU],i=1,…,n, in the above part of the proof. Using the definition of α-cuts and formulas (4.4), we have viαU+∑j=1j≠invjαL=αviM+(1−α)viU+∑j=1j≠in[αvjM+(1−α)vjL]=αn+(1−α)2n∑k=1naikU +∑j=1j≠in(1−α)2n∑k=1najkL≤αn+(1−α)2n[n−1+n2+(n−1)(n−2)2]=n and viαL+∑j=1j≠invjαU=αviM+(1−α)viL+∑j=1j≠in[αvjM+(1−α)vjU]=αn+(1−α)2n∑k=1naikL +∑j=1j≠in(1−α)2n∑k=1najkU≥αn+(1−α)2n[n−1+n2+(n−1)(n−2)2]=n which proves the inequalities (4.9). □ Now, by utilizing Proposition 4.1, we can prove the following proposition. Proposition 4.2 Let v˜i=(viL,viM,viU),i=1,…,n, be triangular fuzzy numbers obtained from an additive FPCM by formulas (4.2) and (4.4). Then (4.7) holds for all α∈[0,1] and i=1,…,n. Proof. By utilizing Proposition 4.1, it is sufficient to show that the fuzzy priorities obtained by formulas (4.2) and (4.4) satisfy (4.8). ∑j=1nvjM=∑j=1n2n∑k=1najkM=2n(0.5n+n(n−1)2)=n,viU+∑j=1j≠invjL=2n∑k=1naikU+∑j=1j≠in2n∑k=1najkL=2n(∑k=1naikU+∑j=1j≠in∑k=1najkL)=2n(0.5n+(n−1)+∑j=1j≠in∑k=1k≠ik≠jnajkL) ≤2n(0.5n+(n−1)+(n−1)(n−2)2)=n,viL+∑j=1j≠invjU=2n∑k=1naikL+∑j=1j≠in2n∑k=1najkU=2n(∑k=1naikL+∑j=1j≠in∑k=1najkU) =2n(0.5n+(n−1)+∑j=1j≠in∑k=1k≠ik≠jnajkU) ≥2n(0.5n+(n−1)+(n−1)(n−2)2)=n. □ The interaction property (4.7) corresponds to the fact that, for any α∈[0,1],i∈{1,…,n}, any possible value viα∈v˜iα is obtained from a particular additive PCM by using formula (3.4). This means that the additive reciprocity is never violated. Since the matrix is additively reciprocal, the sum of viα and vjα,j=1,…,n,j≠i, obtained from this matrix by formula (3.4) is always n. Thus, for any possible value viα∈v˜iα,i∈{1,…,n}, there always exists a set of corresponding possible values vjα∈v˜jα,j≠i, such that ∑i=1nviα=n and the corresponding pairwise comparisons are additively reciprocal. The following example is given to illustrate better this interaction property. Example 4.1 Let us assume the additive FPCM A˜=(0.5(0.6,0.8,0.9)(0.8,0.9,1)(0.1,0.2,0.4)0.5(0.5,0.6,0.8)(0,0.1,0.2)(0.2,0.4,0.5)0.5). (4.10) The fuzzy priorities of objects obtained by formulas (4.2) and (4.4) are v˜1=(1915,2215,2415), v˜2=(1115,1315,1715), v˜3=(715,1015,1215). (4.11) Let us fix for example possible value v1=2415 of v˜1. In order not to violate the additive reciprocity of pairwise comparisons, the possible value v1 must have been obtained from the matrix (0.50.910.10.5...0...0.5) (4.12) since 23(0.5+0.9+1)=2415=v1. Notice that in order to obtain the priority vi,i∈{1,…,n}, by formula (3.4) we do not need to know all the pairwise comparisons in the additive PCM; the pairwise comparisons in the i-th row are sufficient. The possible values v2,v3 of the fuzzy priorities v˜2,v˜3 corresponding to the possible value v1 then can be obtained only from matrices (0.50.910.10.5x01−x0.5), x∈[0.5,0.8], in order to preserve the additive reciprocity of pairwise comparisons. The sum of the priorities obtained from such matrices is always equal to 3: ∑i=13vi=2415+23(0.1+0.5+x)+23(0+(1−x)+0.5)=3. 4.2. Normalized fuzzy priorities-fuzzy extension of the normalization condition ∑i=1nwi=1 By applying properly the fuzzy extension of the formula (3.13), the formulas for computing the representing values of the normalized fuzzy priorities w˜i=(wiL,wiM,wiU),i=1,…,n, from an additive FPCM A˜=(a˜ij)i,j=1n,a˜ij=(aijL,aijM,aijU), are obtained in the form wiL=min{2n∑j=1naij−n−1n;apq∈[apqL,apqU],apq=1−aqp,p,q=1,…,n}, (4.13) wiM=2n∑j=1naijM−n−1n, (4.14) wiU=max{2n∑j=1naij−n−1n;apq∈[apqL,apqU],apq=1−aqp,p,q=1,…,n}. (4.15) Analogously as in the case of formulas (4.1) and (4.3), also the formulas (4.13) and (4.15) can be simplified so that no optimization is needed: wiL=2n∑j=1naijL−n−1n, wiU=2n∑j=1naijU−n−1n. (4.16) The property (3.14) valid for the priorities (3.13) obtained from an additive PCM is extended to the fuzzy priorities (4.14) and (4.16) as ∀wiα∈w˜iα ∃wjα∈w˜jα,j=1,…,n,j≠i: wiα+∑j=1j≠inwjα=1, (4.17) for all α∈[0,1] and i=1,…,n. Similarly to Propositions 4.1 and 4.2, the following propositions are formulated. Proposition 4.3 Let w˜i=(wiL,wiM,wiU),i=1,…,n, be triangular fuzzy numbers. Then the interaction property (4.17) between the fuzzy numbers is valid if and only if ∑j=1nwjM=1, wiU+∑j=1j≠inwjL≤1, wiL+∑j=1j≠inwjU≥1, ∀i∈{1,…,n}. (4.18) Proof. The proof is analogous to the proof of Proposition 4.1. □ Proposition 4.4 Let w˜i=(wiL,wiM,wiU),i=1,…,n, be triangular fuzzy numbers obtained from an additive FPCM by formulas (4.14) and (4.16). Then (4.17) holds for all α∈[0,1] and i=1,…,n. Proof. The proof is analogous to the proof of Proposition 4.2. □ Proposition 4.5 Let w˜i=(wiL,wiM,wiU),i=1,…,n, be triangular fuzzy numbers obtained by formulas (4.14) and (4.16). Then −1<w˜i≤1, i=1,…,n. (4.19) Proof. It is sufficient to prove inequalities −1<wiL and wiU≤1,i=1,…,n. The proof is analogous to the proof of Proposition 3.4. □ Remark 4.4 As was mentioned in Section 3, for an additive PCM, any vector derived from the priority vector (3.4) by adding an arbitrary constant is again a priority vector. For the case of additive FPCMs, the formulas (4.14) and (4.16) for obtaining normalized fuzzy priorities are in fact obtained from the formulas (4.2) and (4.4) by adding constant −n−1n, i.e. w˜i=v˜i−n−1n. That is, the shape of the triangular fuzzy numbers and the distances between them remain unchanged by applying the normalization condition (4.17); the whole set of triangular fuzzy numbers is just shifted back on the scale of real numbers by −n−1n. Example 4.2 Let us assume the additive FPCM (4.10). The fuzzy priorities obtained by formulas (4.2) and (4.4) are in form (4.11), and the normalized fuzzy priorities obtained by formulas (4.14) and (4.16) are in form w˜1=(915,1215,1415), w˜2=(115,315715), w˜3=(−315,0,215). (4.20) The fuzzy priorities are depicted in Fig. 1; the non-normalized fuzzy priorities (4.11) are depicted by grey colour and the normalized fuzzy priorities (4.20) are depicted by black colour. As is evident from the figure, the normalized fuzzy priorities have the same shape as the original non-normalized fuzzy priorities; they are just moved backwards by −23. Figure 1. View largeDownload slide Normalized fuzzy priorities. Figure 1. View largeDownload slide Normalized fuzzy priorities. 4.3. Normalized fuzzy priorities-fuzzy extension of the normalization condition mini∈{1,…,n}wi=0 The aim of this subsection is to demonstrate the inappropriateness of the fuzzy extension of the normalization condition (3.10) for obtaining additive fuzzy priorities from an additive FPCM. The fuzzy extension of the formula (3.11) is done, and it is shown that the fuzzy extension of the normalization condition mini∈{1,…,n}wi=0 distorts the information obtained in the original non-normalized fuzzy priorities. The formulas for obtaining the representing values of the normalized fuzzy priorities w˜i=(wiL,wiM,wiU), i=1,…,n, from an additive FPCM A˜=(a˜ij)i,j=1n by applying the fuzzy extension of the formula (3.11) are given as wiL=2nmin{∑j=1naij−mink=1,…,n{∑j=1nakj};apq∈[apqL,apqU],apq=1−aqp,p,q=1,…,n}, (4.21) wiM=2n[∑j=1naijM−mink=1,…,n{∑j=1nakjM}], (4.22) wiU=2nmax{∑j=1naij−mink=1,…,n{∑j=1nakj};apq∈[apqL,apqU],apq=1−aqp,p,q=1,…,n}. (4.23) Unlike the formulas (4.13) and (4.15), the formulas (4.21) and (4.23) cannot be further simplified. That is because in this case wiL and wiU do not depend only on the pairwise comparisons in the i-th row of the additive FPCM A˜ but also on pairwise comparisons in the other rows. This leads to the activation of the additive-reciprocity constraints. As a result, the optimized function is not increasing in its variables anymore. Moreover, unlike the normalized fuzzy priorities (4.14) and (4.16), the normalized fuzzy priorities (4.21)–(4.23) are not derived from the fuzzy priorities (4.2) and (4.4) by just adding an arbitrary constant. As was already mentioned in Section 3, the fuzzy extension of the normalization condition mini∈{1,…,n}wi=0 is not appropriate because it does not carry any information about the interactions between the fuzzy priorities. The requirement of the smallest possible priority of every set of possible priorities w1,…,wn being equal to 0 distorts the information obtained in the non-normalized fuzzy priorities (4.2) and (4.4), and it leads to a change in the shape of the representing triangular fuzzy numbers. The problem is described more in detail in the following illustrative example. Example 4.3 Let us again assume the additive FPCM (4.10). The non-normalized fuzzy priorities obtained by formulas (4.2) and (4.4) are in form (4.11), and the normalized fuzzy priorities obtained by formulas (4.21)–(4.23) are in form w˜1=(1015,1215,1715), w˜2=(115,315,1015), w˜3=(0,0,115). (4.24) In Fig. 2, the normalized fuzzy priorities (4.24) are graphically represented by black colour, and the original non-normalized fuzzy priorities (4.11) are represented by grey colour. As can be clearly seen from the figure, the triangular fuzzy numbers get distorted by the inappropriate normalization. This means that also the distances between the triangular fuzzy numbers change. However, this is unacceptable since the distances between the fuzzy priorities are required to remain unchanged. Let us briefly analyse the interpretation of the fuzzy extension of the normalization condition mini∈{1,…,n}wi=0 and show its infeasibility. For example, for α=0.5, the α-cuts of the fuzzy priorities (4.24) are in form w˜1α=[1115,2930], w˜2α=[215,1330], w˜3α=[0,130]. According to the normalization condition, for any possible value wiα∈w˜iα,i∈{1,2,3}, there should exist possible values wjα∈w˜jα,j≠i, such that mink∈{1,2,3}wk=0. Let us assume, e.g. w3α=130. Then, the corresponding possible values w1α,w2α can be chosen only from intervals [1115,2930] and [215,1330], respectively. However, it is impossible to satisfy the condition mink∈{1,2,3}wk=0. This means that the possible value w3α=130 of the fuzzy priority w˜3 with the degree of membership α=0.5 is unfeasible; it is not obtainable from the additive FPCM (4.10) under the normalization condition mink∈{1,2,3}wk=0. In a similar way, we find out that ∀α∈[0,1) all possible values w3α∈(w3αL,w3αU]=(0,w3αU] are unfeasible. Figure 2. View largeDownload slide Normalized fuzzy priorities. Figure 2. View largeDownload slide Normalized fuzzy priorities. 5. Employing consistency into the formulas for obtaining fuzzy priorities In the previous section, formulas for obtaining both non-normalized and normalized fuzzy priorities from an additive FPCM were proposed. However, the problematic issue of verifying consistency of the decision maker's preferences has not been discussed yet. In this section, a definition of consistency for additive FPCMs is given, and a consistency condition is also employed directly into the formulas for obtaining fuzzy priorities from an additive FPCM. By applying properly the fuzzy extension of the definition of additive consistency (3.2) for additive PCMs, the consistency of additive FPCMs is defined in the following way. Definition 5.1 Let A˜=(a˜ij)i,j=1n,a˜ij=(aijL,aijM,aijU), be an additive FPCM. A˜ is said to be additively consistent if ∀aijα∈a˜ijα ∃aikα∈a˜ikα,∃akjα∈a˜kjα: aijα=aikα+akjα−0.5, (5.1) for each α∈[0,1] and for each i,j,k∈{1,…,n}. Remark 5.1 Definition 5.1 says that an additive FPCM A˜ is called additively consistent if for any possible value aijα of fuzzy pairwise comparison a˜ij,i,j∈{1,…,n}, with a degree of membership α∈[0.1], there exist possible values aikα of a˜ik and akjα of a˜kj,k=1,…,n, with the same degree of membership α such that they preserve additive consistency according to (3.2). The definition of additive consistency is independent of the ordering of objects in the additive FPCM, e.g. by changing the order of objects compared in the additive FPCM, the conclusion about the consistency does not change. Checking the additive consistency by verifying the validity of (5.1) for each α∈[0,1] and for each i,j,k∈{1,…,n} would be a challenging task. The following proposition provides us with a useful tool for verifying the additive consistency of additive FPCMs. Proposition 5.1 An additive FPCM A˜=(a˜ij)i,j=1n,a˜ij=(aijL,aijM,aijU), is consistent if and only if aijM=aikM+akjM−0.5, aijL≥aikL+akjL−0.5, aijU≤aikU+akjU−0.5, ∀i,j,k∈{1,…,n}. (5.2) Proof. First, let us show that the property (5.1) for each α∈[0,1] and for each i,j,k∈{1,…,n} implies (5.2). For α=1 and i,j,k∈{1,…,n}, we get that for aijM ∃aikM,∃akjM:aijM=aikM+akjM−0.5, which is the first equality of (5.2). Further, for α=0 and for i,j,k∈{1,…,n}, we get that for aijL ∃aik∈[aikL,aikU],akj∈[akjL,akjU]:aijL=aik+akj−0.5. Because aik≥aikL,akj≥akjL, we get aijL≥aikL+akjL−0.5, which is the second inequality of (5.2). Similarly, for α=0 and for i,j,k∈{1,…,n}, we get that for aijU ∃aik∈[aikL,aikU],akj∈[akjL,akjU]:aijU=aik+akj−0.5. Because aik≤aikU,akj≤akjU, we get aijU≤aikU+akjU−0.5, which is the third inequality of (5.2). Now, let us show that the property (5.2) implies (5.1) for each α∈[0,1] and for each i,j,k∈{1,…,n}. From the first equation in (5.2) we obtain (5.1) for α=1 and for each i,j,k∈{1,…,n}. From the second and the third inequality in (5.2) we get ∀aij∈[aijL,aijU] inequalities aikL+akjL−0.5≤aij≤aikU+akjU−0.5, and thus ∃aik∈[aikL,aikU],∃akj∈[akjL,akjU]:aij=aik+akj−0.5, which proves property (5.1) for α=0 and for each i,j,k∈{1,…,n}. To prove the property (5.1) for α∈(0,1) and for each i,j,k∈{1,…,n}, we just need to show that the inequalities (5.2) are valid also for the α-cuts, α∈(0,1), i.e. aijαL≥aikαL+akjαL−0.5, aijαU≤aikαU+akjαU−0.5, ∀i,j,k∈{1,…,n}. After, we can just take the α-cuts [aijαL,aijαU],i,j=1,…,n, for [aijL,aijU] in the above part of the proof. From the definition of α- cuts, we have aijαL=αaijM+(1−α)aijL,aijαU=αaijM+(1−α)aijU, i,j=1,…,n. Thus ∀i,j,k∈{1,…,n}, aikαL+akjαL−0.5=αaikM+(1−α)aikL+αakjM+(1−α)akjL−0.5=α(aikM+akjM−0.5)+(1−α)(aikL+akjL−0.5)≤αaijM+(1−α)aijL=aijαL and aikαU+akjαU−0.5=αaikM+(1−α)aikU+αakjM+(1−α)akjU−0.5=α(aikM+akjM−0.5)+(1−α)(aikU+akjU−0.5)≥αaijM+(1−α)aijU=aijαU. □ Remark 5.2 It can be easily verified that the additive FPCM (4.10), from which both the non-normalized and normalized fuzzy priorities were computed in the examples in the previous section, is consistent according to (5.1); it is sufficient to check the validity of (5.2). In decision-making problems, often the consistency of PCMs is checked first, and only if the matrix is (sufficiently) consistent, the priorities of objects are elicited from the matrix. In our case it means that the additive FPCM has to be consistent according to Definition 5.1. Then, the fuzzy priorities of objects are elicited from the additive FPCM by formulas (4.2) and (4.4) or (4.14) and (4.16) only if the additive FPCM is consistent. Otherwise, the decision maker is asked to reconsider his or her preferences. However, the consistency condition (5.1) can be quite difficult to reach; with an increasing dimension n of an additive FPCM, keeping the consistency becomes very difficult (or even impossible) for a decision maker. And even after being asked to reconsider the preferences, the decision maker might have serious problems to reach the consistency by satisfying (5.1). The problem of violating the consistency is caused also by the fact that the scale [0,1] used for expressing the intensities of preference is restricted. This is not only a problem of additive FPCMs but also of additive PCMs. For example, if oi is absolutely preferred to oj and oj is absolutely preferred to ok, i.e. aij=1,ajk=1, which might very well happen in a real decision-making problem, we should have aik=aij+ajk−0.5=1.5 which is out of the scale [0,1]. Thus, in some cases a weaker form of the consistency (5.1) might be needed. An additive FPCM contains infinitely many additive PCMs (obtained by combining elements from the closures of the supports of the triangular fuzzy numbers from the original additive FPCM preserving the additive reciprocity) representing different preference scenarios. Each of these additive PCMs can be subjected to the consistency check using (3.2). Thus, the consistency could be checked for each such an additive PCM separately, and only the consistent pairwise comparisons could be considered further in the process of obtaining the fuzzy priorities. This consistency check can be done directly during the computation of the fuzzy priorities. Since in the formulas (4.1), (4.3) and (4.13), (4.15) every additive PCM obtained by combining elements from the closures of the supports of the triangular fuzzy numbers from the original additive FPCM preserving the additive reciprocity is examined, the consistency check can be also applied on the matrices in this phase of the computation. However, at least a minimum consistency requirement has to be satisfied by the additive FPCM A˜=(a˜ij)i,j=1n,a˜ij=(aijL,aijM,aijU), before applying this procedure to obtain the fuzzy priorities; the additive PCM of middle values AM=(aijM)i,j=1n is required to be consistent according to (3.2). Thus, it is ensured that the middle values viM,wiM,i=1,…,n, of the fuzzy priorities (i.e. the most possible priorities of objects) computed by (4.2) and (4.14), respectively, are obtained from a consistent additive PCM and that there exists at least one consistent additive PCM obtainable by combining particular elements from the closures of the supports of the triangular fuzzy numbers from the original additive FPCM. The above-mentioned procedure models the following decision-making situation. The decision maker is consistent in his or her decisions; the middle values of the triangular fuzzy numbers expressing the intensities of decision maker's preference satisfy the consistency requirement. At the same time, the imprecision of information in real decision-making problems and the vagueness of the intensities of preference are acknowledged by considering triangular fuzzy numbers instead of crisp numbers. By this the decision maker admits that his or her preference on a pair of compared objects can vary slightly from the most possible intensity of preference (the middle value of the corresponding triangular fuzzy number) over the range from the lowest to the highest possible intensity of preference (the lower and upper boundary values of the triangular fuzzy number). However, at the same time, the decision maker states that his or her preferences are always consistent with respect to the consistency condition (3.2). For an additive FPCM A˜=(a˜ij)i,j=1n,a˜ij=(aijL,aijM,aijU), consistent according to (5.1), the formulas for obtaining the non-normalized fuzzy priorities cv˜i=(cviL,cviM,cviU),i=1,…,n, with employed consistency condition (3.2) are given in form cviL=min{2n∑j=1naij;apq∈[apqL,apqU],apq=1−aqp,apq=apr+arq−0.5,p,q,r=1,…,n}, (5.3) cviM=2n∑j=1naijM, (5.4) cviU=max{2n∑j=1naij;apq∈[apqL,apqU],apq=1−aqp,apq=apr+arq−0.5,p,q,r=1,…,n}. (5.5) Similarly, the formulas for obtaining the normalized fuzzy priorities cw˜i=(cwiL,cwiM,cwiU),i=1,…,n, with employed consistency condition (3.2) are given in form cwiL=min{2n∑j=1naij−n−1n;apq∈[apqL,apqU],apq=1−aqp,apq=apr+arq−0.5,p,q,r=1,…,n}, (5.6) cwiM=2n∑j=1naijM−n−1n, (5.7) cwiU=max{2n∑j=1naij−n−1n;apq∈[apqL,apqU],apq=1−aqp,apq=apr+arq−0.5,p,q,r=1,…,n}. (5.8) The fuzzy priorities cv˜i=(cviL,cviM,cviU),i=1,…,n, obtained by formulas (5.3)–(5.5) will be called non-normalized consistent fuzzy priorities, and analogously, the fuzzy priorities cw˜i=(cwiL,cwiM,cwiU),i=1,…,n, obtained by formulas (5.6)–(5.8) will be called normalized consistent fuzzy priorities. Remark 5.3 Notice that the optimization problems (5.3) and (5.5) are derived from the optimization problems (4.1) and (4.3), respectively, by just adding additive-consistency constraints aij=aik+akj−0.5,i,j,k=1,…,n. However, by adding these constraints, the function optimized in (5.3) and (5.5) is not increasing in its variables anymore. Thus, the formulas (5.3) and (5.5), unlike the formulas (4.1) and (4.3), cannot be further simplified by removing the optimization. The same is valid for the optimization problems (5.6) and (5.8). Proposition 5.2 Let A˜=(a˜ij)i,j=1n,a˜ij=(aijL,aijM,aijU), be an additive FPCM consistent according to (5.1). Then, for the non-normalized fuzzy priorities v˜i=(viL,viM,viU),i=1,…,n, obtained by the formulas (4.1)–4.3) and the non-normalized consistent fuzzy priorities cv˜i=(cviL,cviM,cviU),i=1,…,n, obtained by the formulas (5.3)–5.5), the following holds: cviM=viM, cviL≥viL, cviU≤viU, ∀i∈{1,…,n}. (5.9) Proof. Clearly, the expression viL=2n∑j=1naijL is always smaller or equal to the expression (5.3), similarly, the expression viU=2n∑j=1naijU is always greater or equal to the expression (5.5). □ Proposition 5.3 Let A˜=(a˜ij)i,j=1n,a˜ij=(aijL,aijM,aijU), be an additive FPCM consistent according to (5.1). Then, for the normalized fuzzy priorities w˜i=(wiL,wiM,wiU),i=1,…,n, obtained by the formulas (4.13)–4.15) and the normalized consistent fuzzy priorities cw˜i=(cwiL,cwiM,cwiU),i=1,…,n, obtained by the formulas (5.6)–5.8) the following holds: cwiM=wiM, cwiL≥wiL, cwiU≤wiU, ∀i∈{1,…,n,}. (5.10) Proof. The proof is analogous to the proof of Proposition 5.2. □ Remark 5.4 The properties (5.9) and (5.10) are quite natural since the linear optimization problems (5.3), (5.5), (5.6) and (5.8) are derived from the linear optimization problems (4.1), (4.3), (4.13) and (4.15), respectively, by adding extra constraints (particularly the linear constraints aij=aik+akj−0.5,i,j,k=1,…,n). In the decision-making terminology, this means that by adding an additional information about the consistency of pairwise comparisons we obtain fuzzy priorities that are more precise (less vague). Remark 5.5 The normalized consistent fuzzy priorities (5.6)–5.8) can be obtained directly from the non-normalized consistent fuzzy priorities (5.3)–5.5) by just adding the constant −n−1n, i.e. cw˜i=cv˜i−n−1n. The following example illustrates the computation of the non-normalized and normalized consistent fuzzy priorities from an additive FPCM. Example 5.1 Let us assume the additive FPCM B˜=(0.5(0.5,0.8,0.9)(0.8,0.9,1)(0.9,1,1)(0.1,0.2,0.5)0.5(0.5,0.6,0.7)(0.6,0.7,0.9)(0,0.1,0.2)(0.3,0.4,0.5)0.5(0.6,0.6,0.9)(0,0,0.1)(0.1,0.3,0.4)(0.1,0.4,0.4)0.5). (5.11) It can be easily seen that the additive PCM BM=(bijM)i,j=1n is consistent according to (3.2). Thus, B˜ satisfies the minimum consistency requirement which is needed in order to use the formulas for obtaining consistent fuzzy priorities. In the first column of Table 1, the non-normalized fuzzy priorities obtained from B˜ by the formulas (4.1)–4.3) are given, and in the second column, the non-normalized consistent fuzzy priorities obtained by the formulas (5.3)–5.5) are given. As can be seen from the table, [cviL,cviU]⊆[viL,viU] for all i=1,…,4. The normalized fuzzy priorities w˜i,i=1,…,n, and the normalized consistent fuzzy priorities cw˜i,i=1,…,n, can be obtained from the non-normalized fuzzy priorities v˜i,i=1,…,n, and from the non-normalized consistent fuzzy priorities cv˜i,i=1,…,n, respectively, just by adding the constant −n−1n=−34. Table 1 Non-normalized and non-normalized consistent fuzzy priorities obtained from B˜ Fuzzy priorities obtained by formulas (4.1)–(4.3) (5.3)–(5.5) v˜1=(2720,3220,3420) cv˜1=(2820,3220,3320) v˜2=(1720,2020,2620) cv˜2=(1720,2020,2520) v˜3=(1320,1620,2120) cv˜3=(1520,1620,1920) v˜4=(720,1220,1520) cv˜4=(920,1220,1420) Fuzzy priorities obtained by formulas (4.1)–(4.3) (5.3)–(5.5) v˜1=(2720,3220,3420) cv˜1=(2820,3220,3320) v˜2=(1720,2020,2620) cv˜2=(1720,2020,2520) v˜3=(1320,1620,2120) cv˜3=(1520,1620,1920) v˜4=(720,1220,1520) cv˜4=(920,1220,1420) Table 1 Non-normalized and non-normalized consistent fuzzy priorities obtained from B˜ Fuzzy priorities obtained by formulas (4.1)–(4.3) (5.3)–(5.5) v˜1=(2720,3220,3420) cv˜1=(2820,3220,3320) v˜2=(1720,2020,2620) cv˜2=(1720,2020,2520) v˜3=(1320,1620,2120) cv˜3=(1520,1620,1920) v˜4=(720,1220,1520) cv˜4=(920,1220,1420) Fuzzy priorities obtained by formulas (4.1)–(4.3) (5.3)–(5.5) v˜1=(2720,3220,3420) cv˜1=(2820,3220,3320) v˜2=(1720,2020,2620) cv˜2=(1720,2020,2520) v˜3=(1320,1620,2120) cv˜3=(1520,1620,1920) v˜4=(720,1220,1520) cv˜4=(920,1220,1420) 6. Conclusion In this paper, the fuzzy extension of additive PCMs and methods for obtaining priorities from them based on the constrained fuzzy arithmetic was dealt with. First, the methodology related to obtaining priorities from additive PCMs satisfying Tanino's characterization was reviewed. The method proposed by Fedrizzi & Brunelli (2010) for obtaining non-normalized priorities was approved, and normalization of the priorities was dealt with. After, a proper fuzzy extension of the methodology to additive FPCMs with triangular fuzzy numbers based on the constrained fuzzy arithmetic was proposed. Furthermore, a brief discussion on how to easily modify all the definitions and formulas proposed in the paper in order to be applied on intervals, trapezoidal fuzzy numbers or any other type of fuzzy numbers was provided. It is known that Tanino's characterization is not compatible with the usual normalization condition ∑i=1nwi=1,wi∈[0,1],i=1,…n, for the priorities. However, unlike other characterizations proposed in the literature, whose interpretation is not clear, it provides us with a very clear interpretation of the resulting priorities; the difference of two priorities equals the difference of the corresponding pairwise comparisons of objects, i.e. aij−aji=wi−wj,i,j=1,…,n. Therefore, it is convenient to have a normalization condition compatible with Tanino's characterization. Fedrizzi & Brunelli (2009) proposed a normalization condition in the form min{w1,…,wn}=0,wi∈[0,1],i=1,…n. However, as was shown in this paper, this normalization condition is reachable only for additive PCMs consistent according to Tanino's additive–transitivity property. In many real decision-making problems, consistency is often very difficult to reach. In such situations, there is a need to obtain normalized priorities even from inconsistent additive PCMs. In this paper, the normalization condition ∑i=1nwi=1 reachable both for consistent and inconsistent additive PCMs was proposed. By applying this normalization condition, some of the normalized priorities can even be negative. However, possible negativity of some priorities does not cause any problems since the scale on which the priorities are given is an interval scale; the differences between the priorities matter. The main contribution of the paper is a proper fuzzy extension of the additive PCMs to triangular fuzzy numbers. Linear programming models based on the constrained fuzzy arithmetic were proposed for computing the lower and upper boundary values of the resulting non-normalized and normalized fuzzy priorities, and it was shown that these linear programming models can be reduced to very simple formulas (no optimization is needed). Moreover, the interactions between priorities ( ∑i=1nvi=n for non-normalized priorities and ∑i=1nwi=1 for normalized priorities) were transferred by the proper fuzzy extension to the fuzzy priorities in a more general form; all the values from the closures of the supports of the fuzzy priorities are mutually interdependent. Furthermore, the normalization of the fuzzy priorities proposed in this paper preserves the distances between the fuzzy priorities, which is one of the key properties of the priorities obtained from additive PCMs. Further, additive consistency of additive FPCMs independent of the ordering of objects was defined based on the fuzzy extension of Tanino's additive–transitivity property for additive PCMs, and a useful tool for verifying additive consistency of additive FPCMs was proposed. A weaker form of additive consistency was also introduced, and the additive-consistency condition was employed directly into the formulas for obtaining the consistent non-normalized and normalized fuzzy priorities. The weaker requirement of additive consistency in this form was introduced in order to deal with situations in which the decision maker provides imprecise/vague pairwise comparisons (in form of triangular fuzzy numbers) but at the same time the decision maker states that his or her preferences are always consistent. By employing the consistency condition in the form of constraints of the linear programming problems, the resulting consistent fuzzy priorities are narrower than the fuzzy priorities obtained from the additive FPCM by the original formulas. This is in conformity with the fact that by requiring all the pairwise comparisons to be consistent at all times, we add additional information which should lead to more precise (less vague) results. 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IMA Journal of Management Mathematics – Oxford University Press
Published: May 25, 2016
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