Type-1 OWA Unbalanced Fuzzy Linguistic Aggregation Methodology. Application to Eurobonds Credit Risk Evaluation

Type-1 OWA Unbalanced Fuzzy Linguistic Aggregation Methodology. Application to Eurobonds Credit Risk Evaluation Francisco Chiclana∗ Centre for Computational Intelligence, De Montfort University, Leicester, UK Email: [email protected] Francisco Mata, Luis G. P´erez Department of Computer Science, University of Ja´en, Spain Email: [email protected]; [email protected] Enrique Herrera-Viedma Department of Computer Science, University of Granada, Spain Email: [email protected] June 1, 2017 Abstract In decision making, a widely used methodology to manage unbalanced fuzzy linguistic information is the linguistic hierarchy (LH), which relies on a linguistic symbolic computational model based on ordinal 2-tuple linguistic representation. However, the ordinal 2-tuple linguistic approach does not exploit all advantages of Zadeh’s fuzzy linguistic approach to model uncertainty because the membership function shapes are ignored. Furthermore, the LH methodology is an indirect approach that relies on the uniform distribution of symmetric linguistic assessments. These drawbacks are overcome by applying a fuzzy methodology based on the implementation of the Type-1 Ordered Weighted Average (T1OWA) operator. The T1OWA operator is not a symbolic operator and it allows to directly aggregate membership functions, which in practice means that the T1OWA methodology is suitable for both balanced and unbalanced linguistic contexts and with heterogeneous membership functions. Furthermore, the final output of the T1OWA methodology is always fuzzy and defined in the same domain of the original unbalanced fuzzy ∗Corresponding author 1 linguistic labels, which facilitates its interpretation via a visual joint representation. A case study is presented where the T1OWA operator methodology is used to assess the creditworthiness of European bonds based on real credit risk ratings of individual Eurozone member states modelled as unbalanced fuzzy linguistic labels. Keywords: T1OWA operator, Linguistic hierarchy, Unbalanced fuzzy linguistic assessments, Credit quality. 1 Introduction In most decision making processes there exists uncertainty concerning the suitability of each one of alternatives to choose from. Mathematically, uncertainty has been tackled using precise numeric assessment values or using linguistic assessment values in both its representation and measurement. The second approach, though, happens when experts’ sensations and feelings pervades the decision making problem. The fuzzy linguistic methodology, introduced by Zadeh in his seminal paper,28 has proved to be useful in providing a mathematical structured framework to deal with decision making problems with vagueness and imprecise pervading the information available, i.e. when precise numeric assessments are not available but linguistic assessments are instead. For these type of decision making problems, traditionally categorised as unstructured, can indeed be applied an structured methodology based on the implementation Zadeh’s concept of linguistic variable and its semantics to describe the meaning of each one of the elements of the considered universe of discourse, which is done using fuzzy sets membership functions. An important aspect to be taken into consideration within a linguistic methodology is cardinality of the corresponding linguistic term set,1as the higher the cardinality is, the higher the uncertainty discrimination among the elements of the universe of discourse is achieved. It is a common practice in decision making problems with linguistic assessments to assume linguistic term sets with uniform distribution of symmetric linguistic assessment on the discourse domain. Clearly this approach may be appropriate to problems where the distinction of uncertainty is proportional and equal among the set of linguistic terms, but not where this may not be the case. A typical example of this latter case is given in14 for describing the UK educational grading system (see Fig. 1). Clearly, the right-hand side of the scale has more terms than the left-hand side and consequently a triangular fuzzy set representation of the semantic of each assessment can only be captured with non-uniform distribution of non-symmetric linguistic labels, which in literature has been named as unbalanced linguistic representation of information.2,14,18 A methodology proposed in literature and widely used to address decision making problems with unbalanced linguistic information is the linguistic hierarchy methodology (LH) introduced in6and later applied to improve the precision in processes of computing with words in multi-granular linguistic contexts in 2 FDCBA FDABC Figure 1: Semantic representation of the UK educational grading system in.9–11,17, 21 The aggregation of unbalanced linguistic information using the LH methodology was presented in.14 In summary, the LH methodology consists of building a representation structure with several levels, each one representing a different granularity set of uniform and symmetric linguistic terms that keeps the precedent level modal points in order to achieve a smooth transition between successive levels. Transformation functions are introduced to map linguistic labels of a level to linguistic labels at a different level without loss of information. Doing this, the unbalanced linguistic labels are mapped with its appropriate symmetric linguistic labels within the structure, and are transformed to a common domain with maximum granularity, which ultimately are aggregated using the 2-tuple linguistic computational model. Thus, the LH methodology deals with unbalanced linguistic information using an indirect approach via the already common and known uniform distribution of symmetric linguistic assessment on the universe of discourse. The LH methodology relies on a linguistic symbolic computational model based on ordinal scales and indexes, the 2-tuple linguistic representation, and therefore it does not exploit the advantages of Zadeh’s fuzzy linguistic approach. To avoid this issue, an alternative approach to process unbalanced linguistic information is possible by using the Type-1 Ordered Weighted Average (T1OWA) operator,31 which was developed applying Zadeh’s extension principle to Yager’s OWA operator.27 The T1OWA operator is not a symbolic operator; it allows to directly aggregate the whole linguistic terms because its computation involves the whole membership functions of the fuzzy sets used to appropriately represent the meaning of the linguistic terms, which in practice means that it can be suitable for both balanced and unbalanced linguistic contexts and with heterogeneous types of membership shapes (triangular, trapezoidal, gaussian, etc.). As a consequence, the output of the T1OWA operator is of the same type than the linguistic terms, i.e. a fuzzy set on the same universe of discourse. The T1OWA operator has been successfully applied to aggregate fuzzy linguistic information with fuzzy linguistic weights5, 32 and to address consensus reaching processes in multi-granular fuzzy linguistic contexts.3,20 Thus, the T1OWA operator is most appropriate to be implemented in decision making problems where uncertainty is linked to fuzzy set theory rather than probability theory,24–26 and in particular to contexts with unbalanced fuzzy linguistic information. This is 3 the focus of the present paper, which aims to present a T1OWA based methodology to deal with decision making problems with unbalanced fuzzy linguistic information by using as example the study of credit risk on a potential Eurobonds rating based on real credit risks of Eurozone member states as opposed to previous effort based on mock examples.32 Credit risk evaluation of corporations or the debt issuance of a state or government is usually carried out by rating agencies, with the three big ones being Standard & Poor’s (S&P), Moody’s and Fitch Group. Each agency utilises its own methodology and criteria to measure the creditworthiness of corporations evaluated and its own scale based on a combination of letters, numbers and/or positive and negative signs to assess the overall credit risk level of the corporation or state. Rating agencies rely on economic experts, mathematical models or a combination of both to arrive at their final credit risk assessment. Financial information is obtained from both public and private institutions as well as from experts with great knowledge and experience. Although rating agencies work with information that is quite precise, it is obvious that there also exit economic factors outside their control that generate uncertainty regarding their recommendations and evaluations. The uncertainty that arises when experts analyse all the available economic information may make more difficult the precise assessment of credit risk. Indeed, credit risk assessments tend to include intuitions and feelings of experts that emanate from the mentioned uncertainty. Thus, there are well founded grounds to support the use of fuzzy linguistic approaches in this context. The information can be associated to unbalanced linguistic labels whose meanings can be represented using fuzzy set membership functions. The structure of the rest of the paper is as follows: Section 2 reviews succinctly the basic concept a linguistic variable and its semantics as well as the 2-tuple LH methodology. Section 3 presents a new fuzzy alternative to manage unbalanced fuzzy linguistic information based on the T1OWA operator. A detailed description of its expression for aggregation fuzzy sets is given in Section 3.2, while Section 3.3 presents an example of aggregation of unbalanced linguistic labels using the T1OWA operator and applied to assess the creditworthiness of European bonds based on real credit risk ratings of individual Eurozone member states. The paper is closed with Section 4 where conclusions are drawn. 2 Unbalanced linguistic labels: the indirect ordinal 2-tuple LH approach In his seminal paper published in 1996,29 Zadeh explicitly stated that the rationale for computing with words (CWW) might be supported by a necessity when numbers are not able to be used to quantify the imprecision of the information available; or by a tolerance of imprecision that allows for words instead of numbers, which 4 might be costly to get. Later in30 an additional rationale was added when words are simply used to summarise numerical information. In CWW, the words are modelled into well-defined mathematical objects, which in turn are manipulated with sound mathematical computational tools. Indeed, words in CWW are considered labels of fuzzy sets with specified membership functions, which are computationally manipulated using fuzzy arithmetics, i.e. traditional mathematical arithmetics transformed via the extension principle. Linguistic variables are employed extensively in applications of fuzzy logic, and they are formally represented as a 5-tuple hL, T(L), U, S, Mi28 where: (i) Lis the name of the variable; (ii) T(L) is a finite term set of (primary) labels or words (a collection of linguistic values); (iii) Uis a universe of discourse or base variable; (iv) Sis the syntactic rule which generates the terms in T(L); and (v) Mis a semantic rule which associates with each linguistic value Xits meaning M(X) : U→[0,1]. Usually, T(L) is denoted as Lwhen there is no risk of confusion. A ‘compatibility function’28 or semantic rule associates with each element of the base variable its compatibility with each linguistic value. This interpretation of the meaning of a linguistic label coincides with that of a fuzzy set, and as mentioned above linguistic labels are formally represented as fuzzy subsets of their base variable. A very popular approach to represent and aggregate linguistic information is using a linguistic symbolic computational model based on indexes,4,7, 8 which is based on an ordinal interpretation of the linguistic label meaning. In,15 a more general symbolic approach was introduced: the 2–tuple linguistic model, which up to now has been used as the LH methodology computational model for unbalanced linguistic information. Sections 3.2 and 3.3 will present a fuzzy computational approach to unbalanced linguistic information based on the T1OWA operator. 2.1 Ordinal linguistic representation using the 2-tuple linguistic model This linguistic model adds the concept of symbolic translation to the symbolic representation model based on indexes, which is used to represent the output of symbolic aggregation operators by means of a pair of values called linguistic 2–tuple: (si, αi), with sibeing one of the original linguistic terms (i.e. si∈S={s0, ..., sg}) and αi∈[−.5, .5) is the symbolic translation. The aim of this representation structure is to achieve that the symbolic aggregation output is identical to the one obtained using the symbolic representation model based on indexes while at the same time preventing loss of information by making use of information previously discarded by such symbolic representation model. Formally, let β∈[0, g] be the result of a symbolic aggregation of the indexes of a set of labels in a linguistic term set S={s0, ..., sg}, and i=round(β)∈ {0, . . . , g}. The value αi=β−i∈[−0.5,0.5) is 5 called a symbolic translation, and the pair of values (si, αi) is called the 2–tuple linguistic representation model. Thus, the following isomorphism can be established between the 2-tuple set associated with S, hSi=S×[−0.5,0.5), and the closed interval [0, g]: ∆(β) = (si, α),with      i=round(β), α=β−i, The inverse function is ∆−1(si, α) = i+α, and the corresponding symbolic computational model was presented in.16 2.2 The ordinal 2-tuple linguistic hierarchy A LH may be seen as a hierarchy structure of different levels of linguistic term sets with different granularity, which are denoted as l(t,n(t)) with trepresenting the LH level and n(t) the granularity of the linguistic term set at that level. Assumption are that the cardinality of all linguistic term sets is odd, and graphically are represented using symmetrical and uniform distributed triangular membership functions on the domain [0,1] as Fig. 2 shows. Both the process to build a LH and its computational model are explained below. Building linguistic hierarchies. The granularity of each linguistic term set is increased from one level (t) to the next (t+1) using the following expression17 and as illustrated in Fig. 2: l(t, n(t)) →l(t+ 1,2·n(t)−1). Figure 2: LH with four levels of granularity 3, 5, 9 and 17, respectively An issue associated to this approach is that the granularity of levels increases very rapidly, which has 6 been partially resolved applying the least common multiple approach to all granularities of the LH as it was proposed in.12,13 Computational model. The LH computational model is based on the following symbolic 2-tuple transformation,17 TF t t0:l(t, n(t)) −→ l(t0, n(t0)) TF t t0(sn(t) i, αn(t))=∆ ∆−1(sn(t) i, αn(t))·(n(t0)−1) n(t)−1!. The aim of such transformation function is that linguistic terms, independently of its shape and semantic, can be mapped to a unique expression domain, and consequently be amenable to be manipulated with the 2-tuples computational model. Obviously, this approach disregard the membership functions and so linguistic labels are modelled via their corresponding symbolic ordinal 2-tuple representations and not treated as fuzzy sets. Unbalanced linguistic information is managed within the LH methodology and 2-tuple computational model by first dividing the unbalanced linguistic term set into three term subsets, the one containing all labels below the central one (left lateral set), the one containing all the labels above the central one (right lateral set) and the one containing the central label (central set). In a second step, the granularities of the left lateral set and the right lateral set are compared against the (half) the granularity value for each LH level so that the closest symmetrical and uniform distributed LH labels is found to represent the unbalanced linguistic information. After this mapping has been completed, the symbolic aggregation based on the 2-tuple computational model is applied to process the information, with its output being finally retranslated into the original unbalanced linguistic term set. 3 Unbalanced fuzzy linguistic labels: the direct T1OWA approach In this section, a fuzzy approach to manage unbalanced fuzzy linguistic information will be presented based on the use of T1OWA operator. The advantages of using this route are: (i) it is fuzzy and not ordinal because the membership function characterising the fuzzy linguistic labels are fully used in the computation process; (ii) the shape of the membership function is not restricted to be triangular type but could be of any type; (iii) there is no need to translate and retranslate unbalanced information using an indirect balanced framework, i.e. it is a direct cardinal approach to dealing with unbalanced information compared to the indirect ordinal 2-tuple LH methodology; (iv) the final output will be a fuzzy set on the same domain than the original unbalanced fuzzy linguistic labels and it can be interpreted easily when compared with them 7 graphically. If necessary, a defuzzification process could be applied, for example by computing the centroid of the solution fuzzy set or using the 2-tuple representation model. Anyway it is proved that an equivalent set of values to the corresponding 2-tuple representation approach is obtained. 3.1 Fuzzy linguistic representation model The representation of linguistic information using fuzzy numbers, i.e convex normal fuzzy subsets of the real line, is commonly refereed to as the cardinal representation in contrast to the ordinal representation covered above. In this framework, a linguistic label is characterised by a membership function on the unit interval [0,1] that maps each value in [0,1] to a degree of performance representing its compatibility with the linguistic assessment,28 examples of which are given in Fig. 1. It is not difficult to see that there exists a one-to-one mapping between the ordinal approach based on the 2-tuple representation of a term set of linguistic labels and the set of centroid elements of the fuzzy numbers used in the cardinal representation of the same term set of linguistic labels.22 Indeed, denoting the centroid of the linguistic term sh∈Sby v(sh), the semantic of the linguistic labels underlies a ranking relation that implies v(lh)>v(lk) when h > k. Without loss of generality it can be assumed that v(s0) = 0 and v(sg) = 1, otherwise the centroids are replaced by the values v(sh)−v(s0) v(sg)−v(s0). Denoting the symbolic 2–tuple representation of shby ah= ∆−1((sh,0)), the mapping δ(ah) = v(lh).(1) is the restriction of a continuous and strictly increasing function δ: [0, s]−→ [0,1] such that δ(0) = 0 and δ(s) = 1, i.e. a bijective function δexists and it can be used to derive the ordinal 2-tuple representation of a linguistic term set from the set of centroids of the fuzzy numbers used in a cardinal representation of the same linguistic term set. Obviously, it is not possible to derive a cardinal representation of a linguistic term set from an ordinal 2-tuple representation model. Furthermore, the type of membership function used in the cardinal representation model is not restricted to triangular type but could be trapezoidal or gaussian type, i.e. it could be of any type as long as it is convex and normal verifying that v(lh)>v(lk) when h>k. Thus, the cardinal fuzzy approach to linguistic information representation is general, flexible and appropriate to capture uncertainty, which is not the case with the ordinal approach. 3.2 The T1OWA operator In contrast to Yager’s OWA operator27 that is able to aggregate crisp numbers with crisp weights, the T1OWA operator was introduced in31 to directly aggregate fuzzy sets with uncertainty weights. Thus, given a set {A1,· · · , An}of type-1 fuzzy sets on Rthat are to be aggregated using the set of type-1 fuzzy weights sets 8 defined on the domain of discourse [0,1], {W1,· · · , W n}, the T1OWA operator output is a fuzzy set Y: Φ(A1,· · · , An) = Y with membership function µY(y) = sup n X k=1 ¯wiaσ(i)=y wi∈U, ai∈X µW1(w1)∧ · · · ∧ µWn(wn)∧µA1(a1)∧ · · · ∧ µAn(an)(2) where ¯wi=wi Pn i=1 wi ;σis a permutation function such that aσ(i)≥aσ(i+1),∀i= 1,· · · , n −1. Expression (2) has been proved to be too expensive from a computational point of view, what inevitably implied that its practical application in real world decision making problems was curtailed. This issue, however, was overcome with the development of a fast approach to T1OWA operations based on the horizontal representation of a fuzzy sets via their corresponding family of crisp α-level sets, in what it is known as the representation theorem of fuzzy sets.28 For each α∈[0,1], the α-level T1OWA operator to aggregates the α-level sets A1 α,· · · , An αwith α-level weight sets W1 α, . . . , Wn αis is given as ΦαA1 α,· · · , An α=       n P i=1 wiaσ(i) n P i=1 wi wi∈Wi α, ai∈Ai α,∀i       (3) where Wi α={w|µWi(w)≥α},Ai α={x|µAi(x)≥α}, and σis a permutation function such that aσ(i)≥ aσ(i+1),∀i= 1,· · · , n −1. According to the Representation Theorem of type-1 fuzzy sets, the following typefuzzy set on Rcan be constructed: G=∪ 0<α≤1αΦαA1 α,· · · , An α(4) with membership function µG(x) = ∨ α:x∈Φα(A1 α,··· ,An α)α α(5) Fuzzy sets Yand G, which apparently seem to be different, were proved in32 to have the same membership functions and consequently are equal. This fundamental result is known as the Representation Theorem of Type-1 OWA Operators. Furthermore, this α-level approach was proved to be much faster than (2),32 which 9 4 Conclusions A fuzzy approach to directly fuse unbalanced linguistic information based on the T1OWA operator has been presented. In comparison to the existing approach to unbalanced linguistic information, the ordinal 2-tuple LH methodology, it is worth noting the following: (i) it allows for a soft interpretation of the linguistic information, implements and makes use of the whole membership functions characterising the linguistic label as fuzzy sets; (ii) the shape of the membership function is not restricted to be triangular type; (iii) there is no need to translate and retranslate unbalanced information as the 2-tuple LH methodology does; (iv) the final output is always fuzzy and defined in the same domain than the original unbalanced fuzzy linguistic labels, which facilitates its interpretation via their visual joint representation; (v) defuzzification could be applied if necessary, and indeed this process will always derive in an equivalent result to the 2-tuple LH methodology. The application of the T1OWA unbalanced fuzzy linguistic methodology has been illustrated in the evaluation of the creditworthiness and credit risk quality of a potential issuance of bonds at European Community level, that were the focus of many discussion within the EU during the hardest and most difficult years of the present economic crisis and that were known as Eurobonds. In the future, the T1OWA approach here presented will be compared with alternative linguistic tools that could be useful to manage unbalanced linguistic information, an example of which might derive from the work presented in.19 Acknowledgement This research work has been supported by the research projects grants (TIN2013-40658-P and TIN201675850-R) from the FEDER funds, and the University of Granada ‘Strengthening through Short-Visits’ (Ref. GENIL-SSV 2015) programme. References [1] P.P. Bonissone and K.S. Decker. Selecting uncertainty calculi and granularity: An experiment in tradingoff precision and complexity. Mach Intell Pattern Recognit, 4:217–247, 1986. [2] F.J. Cabrerizo, S. Alonso, and E. Herrera-Viedma. A consensus model for group decision making problems with unbalanced fuzzy linguistic information. International Journal of Information Technology and Decision Making, 8(1):109–131, 2009. 16 [3] F. J. Cabrerizo, F. Chiclana, R. Al-Hmouz, A. Morfeq, A. S. Balamash, and E. Herrera-Viedma. Fuzzy decision making and consensus: challenges. Journal of Intelligent & Fuzzy Systems, 29(3): 1109–1118, 2015. [4] A. Cid-Lopez, M. J. Hornos, R. A. Carrasco, E. Herrera-Viedma, and F. Chiclana. Linguistic multicriteria decision-making model with output variable expressive richness. Expert Systems with Applications, 83, 350–362, 2017. [5] F. Chiclana and S-M. Zhou. Type-reduction of general type-2 fuzzy sets: The type-1 owa approach. International Journal of Intelligent Systems, 28:505–522, 2013. [6] O. Cord´on, F. Herrera, and I. Zwir. Linguistic modeling by hierarchical systems of linguistic rules. IEEE Transactions on Fuzzy Systems, 10(1):2–20, 2001. [7] M. Delgado, J.L. Verdegay, and M.A Vila. Linguistic decision making models. International Journal of Intelligent Systems, 7:479–492, 1992. [8] M. Delgado, J.L. Verdegay, and M.A Vila. On aggregation operations of linguistic labels. International Journal of Intelligent Systems, 8:351–370, 1993. [9] Y. Dong, C.-C. Li, and F. Herrera. Connecting the numerical scale model to the unbalanced linguistic term sets. In FUZZ-IEEE’14, pages 455–462, 2014. [10] Y. Dong, C.-C. Li, Y. Xu, and X. Gu. Consensus-based group decision making under multi-granular unbalanced 2-tuple linguistic preference relations. Group Decision and Negotiation, 24:217–242, 2015. [11] Y. Dong, Y. Xu, and S. Yu. Computing the numerical scale of the linguistic term set for the 2-tuple fuzzy linguistic representation model. Fuzzy Systems, IEEE Transactions on, 17(6):1366–1378, 2009. [12] M. Espinilla, J. Liu, and L. Mart´ınez. An extended hierarchical linguistic model for decision-making problems. Computational Intelligence, 27(3):489–512, 2011. [13] M. Espinilla, R.M. Rodr´ıguez, L. Mart´ınez, F. Mata, and J. Liu. Optimizing the method for building an extended linguistic hierarchy. In Proceedings of ISDA 2009 - 9th International Conference on Intelligent Systems Design and Applications, pages 815–820, 2009. [14] F. Herrera, E. Herrera-Viedma, and L. Mart´ınez. A fuzzy linguistic methodology to deal with unbalanced linguistic term sets. IEEE Transactions on Fuzzy Systems, 16(2):354–370, 2008. [15] F. Herrera and L. Mart´ınez. A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Transactions on Fuzzy Systems, 8(6):746–752, 2000. 17 [16] F. Herrera and L. Mart´ınez. The 2-tuple linguistic computational model. advantages of its linguistic description, accuracy and consistency. International Journal of Uncertainty, Fuzziness and KnowledgeBased Systems, 9(Suppl.):33–49, 2001. [17] F. Herrera and L. Mart´ınez. A model based on linguistic 2-tuples for dealing with multigranularity hierarchical linguistic contexts in multiexpert decision-making. IEEE Transactions on Systems, Man and Cybernetics. Part B: Cybernetics, 31(2):227–234, 2001. [18] E. Herrera-Viedma and A.G. L´opez-Herrera. A model of information retrieval system with unbalanced fuzzy linguistic information. International Journal of Intelligent Systems, 22(11):1197–2014, 2007. [19] S. Massanet, J.V. Riera, J. Torrens, and E. Herrera-Viedma. A new linguistic computational model based on discrete fuzzy numbers for computing with words. Information Sciences, 258:277–290, 2014. [20] F. Mata, L.G. P´erez, S.-M. Zhou, and F. Chiclana. Type-1 owa methodology to consensus reaching processes in multi-granular linguistic contexts. Knowledge-Based Systems, 58:11–22, 2014. [21] J.A. Morente-Molinera, I.J. P´erez, M.R. Ure˜na, and E. Herrera-Viedma. On multi-granular fuzzy linguistic modelling in group decision making problems: a systematic review and future trends. KnowledgeBased Systems, pages 49–60, 2015. [22] P. P´erez-Asurmendi and F. Chiclana. Linguistic majorities with difference in support. Applied Soft Computing, 18:196 – 208, 2014. [23] Standard and Poor’s Corporation. Guide to Credit Rating Essentials. 2009. [24] J. Wu and F. Chiclana. Visual information feedback mechanism and attitudinal prioritisation method for group decision making with triangular fuzzy complementary preference relations. Information Sciences, 279, 716–734, 2014. [25] J. Wu and F. Chiclana. A risk attitudinal ranking method for interval-valued intuitionistic fuzzy numbers based on novel score and accuracy expected functions. Applied Soft Computing, 22, 772–786, 2014. [26] J. Wu, F. Chiclana, and E. Herrera-Viedma. Trust Based Consensus Model for Social Network in an Incomplete Linguistic Information Context. Applied Soft Computing, 35, 827–839, 2015. [27] R. R. Yager. On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Transactions on Systems, Man and Cybernetics, 18:183–190, 1988. [28] L. A. Zadeh. The concept of a linguistic variable and its applications to approximate reasoning. Part I. Information Sciences, 8:199–249, 1975. 18 [29] L. A. Zadeh. Fuzzy logic = computing with words. IEEE Transactions on Fuzzy Systems, 4(2):103–111, 1996. [30] L. A. Zadeh. Computing with Words. Principal Concepts and Ideas, volume 277 of Studies in Fuzziness and Soft Computing. Springer, Heidelberg, Germany, 2012. [31] S. Zhou, F. Chiclana, R. I. John, and J. M. Garibaldi. Type-1 OWA operators for aggregating uncertain information with uncertain weights induced by type-2 linguistic quantifiers. Fuzzy Sets and Systems, 159(24):3281–3296, 2008. [32] S. Zhou, F. Chiclana, R. I. John, and J. M. Garibaldi. Alpha-level aggregation: A practical approach to type-1 OWA operation for aggregating uncertain information with applications to breast cancer treatments. IEEE Transactions on Knowledge and Data Engineering, 23(10):1455–1468, 2011. 19