TRIANGULAR INTUITIONISTIC FUZZY TRIPLE BONFERRONI HARMONIC MEAN OPERATORS AND APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING

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1 Iranian Journal of Fuzzy Systems Vol. 3 No pp TRIANGULAR INTUITIONISTIC FUZZY TRIPLE BONFERRONI HARMONIC MEAN OPERATORS AND APPLICATION TO MULTI-ATTRIBUTE GROUP DECISION MAKING S. P. WAN AND Y. J. ZHU Abstract. As an special intuitionistic fuzzy set defined on the real number set triangular intuitionistic fuzzy number TIFN is a fundamental tool for quantifying an ill-nown quantity. In order to model the decision maer s overall preference with mandatory requirements it is necessary to develop some Bonferroni harmonic mean operators for TIFNs which can be used to effectively intergrate the information of attribute values for multi-attribute group decision maing MAGDM with TIFNs. The purpose of this paper is to develop some Bonferroni harmonic operators of TIFNs and apply to the MAGDM problems with TIFNs. The weighted possibility means of TIFN are firstly defined. Hereby a new lexicographic approach is presented to ran TIFNs sufficiently considering the ris preference of decision maer. The sensitivity analysis on the ris preference parameter is made. Then three inds of triangular intuitionistic fuzzy Bonferroni harmonic aggregation operators are defined including a triangular intuitionistic fuzzy triple weighted Bonferroni harmonic mean operator TIFTWBHM operator a triangular intuitionistic fuzzy triple ordered weighted Bonferroni harmonic mean TIFTOWBHM operator and a triangular intuitionistic fuzzy triple hybrid Bonferroni harmonic mean TIFTHBHM operator. Some desirable properties for these operators are discussed in detail. By using the TIFTWBHM operator we can obtain the individual overall attribute values of alternatives which are further integrated into the collective ones by the TIFTHBHM operator. The raning order of alternatives is generated according to the collective overall attribute values of alternatives. A real investment selection case study verifies the validity and applicability of the proposed method.. Introduction In order to describe and depict uncertainty of real world Zedeh [37] initiated the concept of fuzzy sets FSs in 965 which can be used to represent the fuzziness nature. The fuzzy decision maing analysis appears. However the decision maing problems often involve many incomplete information and relate to many complex factors such as economy politics psychological behavior ideology and so on [8-20]. Therefore there often exist some hesitation degrees in the judgments of Received: September 205; Revised: April 206; Accepted: June 206 Key words and phrases: Multi-attribute group decision maing Triangular intuitionistic fuzzy number Possibility mean Bonferroni mean Harmonic mean.

2 8 S. P. Wan and Y. J. Zhu decision maer DM [2-24]. For example for a real investment selection problem because of incompleteness and uncertainty of information in the evaluation of the candidate s prospects the evaluation value can be expressed by triangular intuitionistic fuzzy number TIFN [ ] ; which means that the minimum value is 3.33 the maximum value is 3.75 and the most possible value is Meanwhile the maximum membership degree for the most possible value 3.53 is 70% the minimum non-membership degree for the most possible value 3.53 is 20% and the indeterminacy is 0%. That is to say the DM has a hesitation degree for the estimation on this judgement this hesitation influences the decision maing on the investment selection. As a generalization of fuzzy numbers TIFN is an special intuitionistic fuzzy set IFS [] defined on the real number set which seems to suitably describe an ill-nown quantity [0-22]. Shu et al. [6] defined a TIFN in a similar way to that of the fuzzy number and introduced an algorithm for intuitionistic fuzzy fault tree analysis. Li [] corrected some errors in the definition of the four arithmetic operations for the TIFNs in [6]. There exist several investigations on the raning of TIFNs and application to MADM and MAGDM. Based on the concept of a ratio of the value index to the ambiguity index Li [2] discussed the concept of the TIFN raning method and applied them to MADM problems in depth. Li et al. [3] defined the values and ambiguities for TIFN as well as the value-index and ambiguity-index developed a value and ambiguity based raning method and applied to MADM with TIFNs. Nan et al. [4] defined the raning order relations of TIFNs which were applied to matrix games with payoffs of TIFNs. Wan [25] introduced the possibility mean variance and variance coefficient of TIFNs and proposed a method based on possibility variance coefficient for MADM with TIFNs. Wan et al. [26] defined the crisp weighted possibility mean of TIFNs and the Hamming distance for TIFNs and extended the classic VlseKriterumsa Optimizaca I Kompromisno Resenje VIKOR method to solve MAGDM with TIFNs. Wang et al. [30] introduced new arithmetic operations and logic operators for TIFNs and applied them to fault analysis of a printed circuit board assembly system. Wan and Dong [27] proposed the possibility method for MAGDM with TIFNs and incomplete weight information. Wan et al. [9] developed some new generalized aggregation operators for TIFNs and applied to MAGDM. Dong and Wan [6] studied the MAGDM problems in which the attribute values are the TIFNs the attribute weights are completely unnown and the weights of DMs are given by linguistic variables. Dong and Wan [7] investigated the prioritized multi-criteria group decision maing with TIFNs. Wan et al. [29] developed two triangular intuitionistic fuzzy generalized aggregation operators: TAIFGOWA operator and TAIFGHWA operator and applied them to MAGDM. The aforementioned research about IFNs mainly deals with the operation laws [ 6 30] aggregation operators [ ] raning methods [6 2-4] decision maing methods [6 30]. Moreover the study on aggregation operators of TIFNs concentrates the weighted arithemetic average operator and logic operators. Most of existing aggregation operators for IFNs consider the input arguments as independent i.e. the inter-relationship of the individual arguments has not been

3 Triangular Intuitionistic Fuzzy Triple Bonferroni Harmonic Mean Operators and... 9 captured by those operators. On the contrary the Boneferoni mean BM originally introduced by Bonferroni [3] has the capability to capture the correlations between the input arguments to be aggregated. The BM was recently generalized through replacing the simple averaging by other mean type operators as well as associating differing importance with the arguments [36]. Xu and Yager [34] first studied BM under intuitionistic fuzzy environments and applied the weighted intuitionistic fuzzy BM to multicriteria decision maing. Beliaov et al. [2] proposed the generalized Bonferroni mean GBM operator which models the average of the conjunctive expressions and the average of remaining. Both the BM and GBM operators ignore some aggregation information and the weight vector of the aggregated arguments. To overcome this drawbac Xia et al. [32] developed a generalized weighted Bonferroni mean GWBM operator as the weighted version of the GBM they also developed the generalized Bonferroni geometric mean GWBGM operator and extended to intuitionistic fuzzy environment. Par and Par [5] also defined a generalized fuzzy weighted Bonferroni harmonic mean GFWBHM operator a generalized fuzzy ordered weighted Bonferroni harmonic mean GFOWBHM operator generalized uncertain weighted Bonferroni harmonic mean GUWBHM operator and a fuzzy weighted generalized harmonic mean FWGHM operator. In mathematics the harmonic mean HM is a special case of the power mean. As it tends strongly toward the least elements of the list it may alleviate the influence of large outliers and increase the influence of small values. Recently harmonic mean has been extended to the case of triangular fuzzy numbers TFNs. Xu [35] developed the fuzzy harmonic mean FHM operators such as fuzzy weighted harmonic mean FWHM operator fuzzy ordered weighted harmonic mean FOWHM operator and fuzzy hybrid harmonic mean FHHM. Sun and Sun [7] further extended the BM operator to fuzzy environment. They introduced fuzzy Bonferroni harmonic mean FBHM operator and fuzzy ordered Bonferroni harmonic mean FOBHM operator and applied to MADM. TIFNs are special subset of IFSs. Compared with the definition of IFS TIFN maes the membership and non-membership no longer just relative to a fuzzy concept of excellent or good by adding the triangular fuzzy number. Hence TIFNs have better capability to model ill-nown quantities than IFSs. TIFNs can be used to describe the supply chain environment pollution outsouring service provider selection water resource management environment assesset military weapon evaluation etc. [27-29]. Sun and Sun [7] Par and Par [5] studied the BM operator of triangular fuzzy number. But triangular fuzzy number does not consider the hesitant degree. The BM operator of triangular fuzzy number can not be directly applied to TIFNs. However there is no investigation on the extension of the BM and HM operators for TIFNs. As far as we now only Dutta and Guha [8] researched the BM for trapezoidal intuitionistic fuzzy numbers TrIFNs. They pointed out that the arithmetic mean does not allow us to model any ind of mandatory requirements while the geometric mean employs the partial satisfaction of all the criteria even those criteria are not mandatory. The ey characteristic of the BM is the behavior along some of the edges of the unit hypercube. In particular the BM can model mandatory requirements. In order to model the DM s overall preference with

4 20 S. P. Wan and Y. J. Zhu the mandatory requirements it is necessary to develop some Bonferroni harmonic mean operators for TIFNs which can be used to effectively intergrate the information of attribute values for MAGDM with TIFNs. Nevertheless there exist some difficulties and challenges such as how to reasonably define the triple Bonferroni harmonic mean operators for TIFNs and how to discuss the desirable properties of these operators. Consequently the aim of this paper is to extend the Boneferoni harmonic means under triangular intuitionistic fuzzy environment. Some triangular intuitionistic fuzzy Bonferroni harmonic mean aggregation operators are developed including a triangular intuitionistic fuzzy triple weighted Bonferroni harmonic mean TIFTWBHM operator a triangular intuitionistic fuzzy triple ordered weighted Bonferroni harmonic mean TIFTOWBHM operator and a triangular intuitionistic fuzzy triple hybrid Bonferroni harmonic mean TIFTHBHM operator. Then a new decision method based on TIFTWBHM and TIFTHBHM operators is proposed for solving the MAGDM problems with TIFNs. The main wors and contributions of this paper are summarized as follows: i The weighted possibility means of TIFN are defined. Hereby a new lexicographic approach is presented to ran the TIFNs sufficiently considering the ris preference of DM. The sensitivity analyses on the ris preference parameter are made. ii The TIFTWBHM TIFTOWBHM and TIFTHBHM operators developed in this paper extend most of the existing BM and HM operators. For example the GFWBHM GUWBHM and FWGHM operators defined in [5] are a special case of the TIFTWBHM operator and the triangular fuzzy weighted Bonferroni harmonic mean TFWBHM operator defined in this paper respectively. The FBHM operator defined in [7] is a specal case of the triangular intuitionistic fuzzy weighted Bonferroni harmonic mean TIFWBHM operator defined in this paper. The FHM operator and the uncertain weighted harmonic mean UWHM operator defined in [35] are a specal case of the triangular intuitionistic fuzzy weight harmonic mean TIFWHM operator defined in this paper. The BM for TrIFNs developed by Dutta and Guha [8] only considered the BM and ignored HM whereas the TIFTWBHM TIFTOWBHM and TIFTHBHM operators developed in this paper tae the BM and HM into consideration simultaneously. iii Existing BM HM operators and their extensions are mainly focused on the weighted average and the ordered weighted average operators. There were few studies about the hybrid aggregation operators. The TIFTHBHM operator can reflect the important degrees of both the given arguments and the ordered positions of the arguments. It is usually applied to integrate the individual comprehensive attribute values of alternatives into the collective ones which can sufficiently reflect the importance degrees of different experts. The rest of this paper is structured as follows. In Section 2 we define the weighted possibility means of TIFNs. Hereby a new lexicographic approach to raning TIFNs is presented. In Section 3 the TIFTWBHM TIFTOWBHM and TIFTHBHM operators are defined and their desirable properties are discussed in detail. A new method for MAGDM with TIFNs is developed in Section 4. In

5 Triangular Intuitionistic Fuzzy Triple Bonferroni Harmonic Mean Operators and... 2 section 5 a numerical example is illustrated. Short conclusions are made in section Weighted Possibility Means and New Raning Approach for TIFNs In this section the weighted possibility means of TIFNs are defined. Thereby a new lexicographic approach is developed to ran the TIFNs. 2.. Definition and Operation Laws of TIFNs. Definition 2.. [3] A TIFN ã = a a ā; ωã uã is an special IFS on a real number set R whose membership function and non-membership function are defined as: and µãx = νãx = x a a a ω ã if a x < a ωã if x = a ā x ā a ω ã if a < x ā 0 if x < a or x > ā a x+x auã a a if a x < a uã if x = a x a+ā xuã ā a if a < x ā if x < a or x > ā respectively depicted as in Figure below. The values ωã and uã represent the maximum degree of membership and the minimum degree of non-membership respectively such that they satisfy the conditions: 0 ωã 0 uã and ωã + uã. Let πãx = µãx νãx which is called an intuitionistic fuzzy index of an element x in ã. If a 0 and one of the three values a a and ā is not equal to 0 then the TIFN ã = a a ā is called a positive TIFN denoted by ã > 0.The TIFNs discussed in this paper are all positive TIFNs. Definition 2.2. [2] Let ã i = a i a i ā i ; ωãi uãi i = 2 be two TIFNs and λ 0. Then the operation laws for TIFNs are defined as follows: ã + ã 2 = a + a 2 a + a 2 ā + ā 2 ; ωã ωã2 uã uã2 ; 2 λã = λs λa λā ; ωã uã ; 3 ã ã 2 = a a 2 a a 2 ā ā 2 ; ωã ωã2 uã uã2 ; 4 ã /ã 2 = a /ā 2 a /a 2 ā /a 2 ; ωã ωã2 uã uã2 ; 5 ã λ = a λ aλ ā λ ; ωã uã ; 6 ã = ā a a ; ω ã uã ; where the symbols and mean min and max operators respectively Weighted Possibility Means of TIFNs. Definition 2.3. [2] Let 0 α ωã uã β and 0 α + β. The α- cut set and β- cut set of a TIFN ã = a a ā; ωã uã are respectively defined as ã α = {x µãx α} and ã β = {x υãx β } which can be calculated as follows:

6 22 S. P. Wan and Y. J. Zhu Figure. α-cut Set and β-cut Set TIFN and ã α = [a l α a u a aα ā aα α] = [a + ā ] ωã ã β = [a l β a u β] = [ βa + β u ãa uã ωã βa + β u ãā ] 2 uã The α-cut set and β-cut set are also depicted in Figure. Motivated by [0] we give the definitions of the weighted possibility means of TIFNs as follows. Definition 2.4. [26] The f weighted lower and upper possibility means of membership function for a TIFN ã are respectively defined as: m µ ã = m µ ã = ωã 0 ωã 0 fpos[ã a l α]a l αdα = fpos[ã a u α]a u αdα = ωã 0 ωã 0 fαa l αdα 3 fαa u αdα 4 where Pos means possibility [0] and the weighting function f : [0 ωã] R is nonnegative monotone increasing and satisfies the conditions: ωã fαdα = ωã and 0 f0 = 0. Definition 2.5. [26] The g weighted lower and upper possibility means of membership function for a TIFN ã are respectively defined as: m ν ã = m ν ã = uã uã gpos[ã a l β]a l βdβ = gpos[ã a u β]a u βdβ = uã uã gβa l βdβ 5 gβa u βdβ 6 where the weighting function g : [uã ] R is non-negative monotone decreasing and satisfies the conditions: uã gβdβ = uã and g = 0.

7 Triangular Intuitionistic Fuzzy Triple Bonferroni Harmonic Mean Operators and Definition 2.6. For a TIFN ã = a a ā; ωã uã and the f weighted possibility mean of membership function and g weighted possibility mean of non-membership function are respectively defined as: m µ ã θ = θm µ ã + θ m µ ã 7 and m ν ã θ = θm ν ã + θ m ν ã 8 where θ [0 ] is the ris preference parameter of DM and can reflect different importance to the weighted lower and upper possibility means. Different DMs have various preferences for the lower and upper possibility means. θ 0.5 ] implies that DM prefers the weighted upper possibility mean namely DM is pessimistic; θ [0 0.5 shows that DM prefers the weighted lower possibility mean namely DM is optimistic; θ = 0.5 indicates that DM is indifference between the weighted lower and upper possibility means namely DM is preference neutral. IF θ = 0 then m µ ã 0 = m µ ã; if θ = then m µ ã = m µ ã and m ν ã = m ν ã ; if θ = 0.5 then m µ ã 0.5 = 2 [m µã + m µã] and m ν ã 0.5 = 2 [m νã + m ν ã]. Thus if the TrIFN ã = a a ā; ωã uã degenerates to TFN ã = a a ā i.e. ωã = and uã = 0 then m µ ã 0.5 = 2 [m µ ã + m µã] is just the f weighted possibility mean of fuzzy number defined in Definition 2 of [0] see pp Therefore the f weighted possibility mean of fuzzy number defined in [0] is just a special case of that defined in this paper. Obviously m µ ã θ synthetically reflects the information on every membership degree and m µ ã 0.5 may be regarded as a central value that represents from the membership function point of view. Liewise m ν ã θ synthetically reflects the information on every non-membership degree and m ν ã 0.5 may be regarded as a central value that represents from the non-membership function point of view. Example 2.7. If f and g are chosen as follows: fα = 2α/ωã α [0 ωã] 9 and gβ = 2 β/ uã β [uã ] 0 respectively then according to the equations 3-6 we have Further by the equations 7 and 8 we have m µ ã = 3 a + 2aω ã m µ ã = 3 ā + 2aω ã 2 m ν ã = 3 2a + a u ã 3 m ν ã = 3 2a + ā u ã. 4 m µ ã θ = 3 [ θa + 2a + θ2a + ā]ω ã 5 m ν ã θ = 3 [ θa + 2a + θ2a + ā] u ã. 6

8 24 S. P. Wan and Y. J. Zhu Remar 2.8. If a TrIFN ã = a a ā; ωã uã degenerates to a TFN ã = a a ā it follows from the equations 2 or 3 4 and 5 or 6 with θ = 0.5 that the weighted lower possibility mean weighted upper possibility mean and weighted possibility mean of the TFN ã = a a ā are obtained as follows: M ã = a+2a/3 M ã = 2a+ā/3 and Mã = a+4a+ā/6 respectively.these results of TFN are the same as those of TFN in Examples 2. of [4]. The weighted possibility means have some useful properties outlined in Theorem 2.9. Theorem 2.9. Let ã i = a i a i ā i ; ωãi uãi i = 2 be two TIFNs with ωã = ωã2 and uã = uã2. Then for γ > 0 any τ > 0 the following equalities are valid: m µ γã + τã 2 θ = γm µ ã θ + τm µ ã 2 θ 7 m ν γã + τã 2 θ = γm ν ã θ + τm ν ã 2 θ. 8 Proof. Since γ > 0 and τ > 0. then by Definition 2.3 we get the α -cut set of TrIFN γã + τ b as γã + τã 2 α = [γa l α + τa l 2α γa u α + τa u 2α]. By the equation 7 ωã = ωã2 and uã = uã2 we obtain m µ γã + τã 2 θ = θm µ γã + τã 2 + θ m µ γã + τã 2 = θ + θ 0 ωã ωã2 0 ωã ωã2 fαγa l α + τa l 2αdα fαγa u α + τa u 2αdα = γ[ θm µ ã + θ m µ ã ] + τ[ θm µ ã 2 + θ m µ ã 2 ] = γm µ ã θ + τm µ ã 2 θ. Thus the equation 7 holds. By the same way the equation 8 can be proven. Namely Theorem 2.9 is proven. Especially if γ = τ = then by Theorem 2.9 the following equalities are valid: m µ ã + ã 2 θ = m µ ã θ + m µ ã 2 θ m ν ã + ã 2 θ = m ν ã θ + m ν ã 2 θ. Remar 2.0. The weighting functions f and g can be chosen as several forms for example fα = n + α n /ωã n α [0 ωã] gβ = n + β n / uã n β [uã ] where the power n is any positive integer such as n = 2 etc. These power forms of weighting functions are motivated by [0] see Examples -3 in [0]. Hence by introducing different f and g we can give different case-dependent importance to α -cut set and β -cut set of TIFNs ã. In real-life application the functions f and g can be selected according to the real need of the decision problems and the preferences of DMs. Hence the weighted

9 Triangular Intuitionistic Fuzzy Triple Bonferroni Harmonic Mean Operators and possibility mean of membership non-membership function not only reflects the information on every membership non-membership degree and represents a mean value of membership non-membership function but also exhibits great flexibility and facility for different ris preferences of DMs. In what follows the weighting functions f and g are respectively chosen as the equations 9 and 0 for computation convenience A New Lexicographic Raning Approach of TIFNs Based on Weighted Possibility Means. The possibility means of fuzzy numbers are similar to the mean of random variables. They can be used to quantitatively characterize the values of fuzzy numbers. Obviously the greater the possibility means the bigger the corresponding fuzzy number. Let m µ ã i θ and m ν ã i θ be the weighted possibility means of the membership and non-membership functions for TIFNs ã i = a i a i ā i ; ωãi uãi i = 2 respectively. Thereby a lexicographic approach to raning two TIFNs ã and ã 2 can be summarized as follows: If m µ ã θ < m µ ã 2 θ then ã is smaller than ã 2 denoted by ã < ã 2 ; 2 If m µ ã θ > m µ ã 2 θ then ã is bigger than ã 2 denoted by ã > ã 2 ; 3 If m µ ã θ = m µ ã 2 θ then a if m ν ã θ < m ν ã 2 θ then ã < ã 2 ; b if m ν ã θ > m ν ã 2 θ then ã > ã 2 ; c if m ν ã θ = m ν ã 2 θ then ã and ã 2 represent the same information denoted by ã = ã 2. Remar 2.. Nan et al. [4] utilized the membership function average index S µ ã = ωãa + 2a + ā/4 and the non-membership function average index S v ã = uãa + 2a + ā/4 of TIFN ã = a a ā; ωã uã to ran TIFNs. It is easily seen from the equations 5 and 6 that S µ ã and S v ã are respectively equal to m µ ã θ and m v ã θ if θ = 3ā 2a a/[4ā a]. Therefore the raning approach [4] may be regarded as a special case of the proposed approach in this paper. In other words this paper generalizes the raning approach of [4]. Example 2.2. Consider two TIFNs ã = 3 6 8; and ã 2 = 5 7 9; By the equations 5 and 6 we obtain ã > ã 2 if θ > ; ã < ã 2 if θ < However using the raning approach of [4] we can calculate the average indexes as follows: S µ ã = 3.67 S v ã = S µ ã 2 = S v ã 2 = Because S µ ã < S µ ã 2 the raning order obtained by [4] is ã < ã 2 which is the same as the raning order of the case θ < Namely the raning result obtained by [4] is just a special case of that obtained by this paper. Example 2.2 implies that when chosen different values of ris preference parameter θ we can get different raning orders. The new raning approach of TIFNs

10 26 S. P. Wan and Y. J. Zhu proposed in this paper sufficiently taes into the ris preference of DM consideration which can mae the raning result more reasonable. In contrast the raning approach [4] failed to consider the ris preference of DM. To analyze the effect of ris preference parameter θ on the raing of TIFNs we mae sensitivity analyses for m µ ã θ and m v ã θ with respect to θ respectively. Theorem 2.3. Let θ be a perturbation of the ris preference parameter θ with 0 θ + θ. If m µ ã θ m µ ã 2 θ then m µ ã θ + θ m µ ã 2 θ + θ if and only if max{m µ ã θ m µ ã 2 θ/η 2 η θ} θ θ η 2 > η θ θ θ η 2 = η θ θ min{m µ ã θ m µ ã 2 θ/η 2 η θ} η 2 < η where η i = 3 ā i a i ωãi i = 2. Proof. Let m µ ã θ + θ m µ ã 2 θ + θ then by the equation 5 we get 3 [ θ θa + 2a + θ + θ2a + ā ]ωã 3 [ θ θa 2 + 2a 2 + θ + θ2a 2 + ā 2 ]ωã2. Namely m µ ã θ m µ ã 2 θ η 2 η θ. Since 0 θ and 0 θ + θ we obtain θ θ θ. Then if η 2 > η we have θ m v ã θ m v ã 2 θ/η 2 η. Thus max{m v ã θ m v ã 2 θ/η 2 η θ} θ θ; if η 2 < η we have θ m v ã θ m v ã 2 θ/η 2 η. Thus θ θ min{m v ã θ m v ã 2 θ/η 2 η θ}; if η 2 = η we have θ θ θ. This completes the proof of Theorem 2.3. By the same way we have the following Theorem 2.4. Theorem 2.4. Let θ be a perturbation of the ris preference parameter θ with 0 θ + θ. If m v ã θ m v ã 2 θ then m v ã θ + θ m v ã 2 θ + θ if and only if max{m v ã θ m v ã 2 θ/ η 2 η θ} θ θ ξ 2 > ξ θ θ θ ξ 2 = ξ θ θ min{m v ã θ m v ã 2 θ/η 2 η θ} ξ 2 < ξ where ξ i = 3 ā i a i uãi i = 2.

11 Triangular Intuitionistic Fuzzy Triple Bonferroni Harmonic Mean Operators and Triangular Intuitionistic Fuzzy Triple Bonferroni Harmonic Mean Operators HM is the reciprocal of arithmetic mean of reciprocal which is a conservative average to be used to provide for aggregation lying between the max and min operators and is widely used as a tool to aggregate central tendency data [35]. In this section we develop three inds of triangular intuitionistic fuzzy triple Bonferroni harmonic mean operators. 3.. Related Bonferroni Mean Operators and Bonferroni Harmonic Mean Operators. The BM was originally introduced by Bonferroni [3] which was defined as follows: Definition 3.. [3] Let p q 0 and a i i = 2 n be a set of nonnegative real numbers. If BM pq a a 2... a n = nn =i j then BM pq is called the Bonferroni mean BM operator. Definition 3.2. [3 2] Let p q r 0 and a i i = 2 n be a set of nonnegative numbers. If TBM pqr a a 2 a n = a p i nn n 2 aq j ar 20 a p i aq j =i j p+q then TBM pqr is called the Triple Bonferroni mean TBM operator. It is noted that both BM and TBM operators do not consider the weight vector of the aggregated arguments. To overcome this drawbac Xia et al. [32] defined a generalized weighted BM operator. Definition 3.3. [32] Let p q r 0 and a i i = 2 n be a set of nonnegative numbers with the weight vector w = w w 2 w n T such that w i > 0 i = 2 n and n w i =. If i= GWBM pqr a a 2 a n = =i j w iw jw a p i aq j ar 9 2 then GWBM pqr operator. is called the generalized weighted Bonferroni mean GWBM Xia et al. [32] also extended the GWBM operator for real numbers to suit the case for the intuitionistic fuzzy sets. To aggregate the triangular fuzzy correlated information based on the BM and weighted HM operators Sun and Sun [7] developed the Bonferroni harmonic mean operator for TFNs which is called fuzzy weighted Bonferroni harmonic mean FWBHM operator. This operator considers the weight vector of the aggregated arguments.

12 28 S. P. Wan and Y. J. Zhu Definition 3.4. [7] Let a i i = 2 n be a collection of TFNs with the weight vector w = w w 2 w n T such that w i > 0 i = 2 n and n w i =. If FWBHM pq ã ã 2 ã n = = n = n = w i w j ã p i ãq j w i w j a p i aq j p+q p+q n = w i w j a p i aq j p+q n = i= w i w j ā p i āq j p+q where p q > 0 then FWBHM pq is called the triangular fuzzy weight Bonferroni harmonic mean TFWBHM operator Triangular Intuitionistic Fuzzy Triple Weighted Bonferroni Harmonic Mean Operator. Based on the GBM and TFWBHM operators we give the definition of the TIFTWBHM operator. Definition 3.5. Let ã i = a i a i ā i ; ωãi vãi i = 2 n be a collection of TIFNs with the weight vector w = w w 2 w n T such that w i > 0 i = 2 n and n w i =. If i= TIFTWBHM pqr ã ã 2 ã n = =i j w iw jw ã p i ãq j ãr where p q > 0 then TIFTWBHM pqr is called the TIFTWBHM operator. Remar 3.6. The term generalized used by Xia et al. [32] is different from that used by Beliaov et al. [2] we thin that the term triple originated by Bonferroni [3] is more suitable than the term generalized. Thus we use the term triple in this paper. Theorem 3.7. Let ã i = a i a i ā i ; ωãi vãi i = 2 n be a collection of TIFNs. Their aggregation result by TIFTWBHM operator is also a TIFN and = w i w j w =i j a p i aq j ar min i n {ωã i TIFTWBHM pqr ã ã 2 ã n w i w j w =i j a p i aq j ar w i w j w =i j ā p i āq j ār 22 } max i n {uã i }. 23 Proof. By Definition 2.2 we have w iw jw ã p = wiwjw i ãqjãr ā p i āqjār =i j w i w j w ã p i ãq j ãr = min i n {ωã i } max i n {uã } i wiwjw a p i aq j ar =i j w i w j w ā p i āq j ār wiwjw a p ; min i aq j ar t= {ωã t} max t= {uã t} =i j w i w j w a p i aq j ar =i j ; w i w j w a p ; i aq j ar

13 Triangular Intuitionistic Fuzzy Triple Bonferroni Harmonic Mean Operators and =i j =i j w i w j w ã p i ãq j ãr w i w j w a p i aq j ar = =i j w i w j w ā p i āq j ār ; min i n {ωã i } max i n {uã } i =i j w i w j w a p i aq j ar Thus =i j = w iw jw ã p i ãq j ãr =i j =i j w iw jw ā p i āq j ār w iw jw a p i aq j ar which completes the proof of Theorem 3.7. =i j w iw jw a p i aq j ar ; min {ω ã i } max {u ã i } i n i n The TIFTWBHM operator has some desirable properties such as idempotency boundedness commutativity monotonicity etc. Theorem 3.8. Idempotency: If all ã i j = 2 n are equal i.e. ã j = ã then TIFTWBHM pqr ã ã ã n = ã. then 2 Boundedness: Let ã + = max {a i } max {a i} max {ā i}; max {ω ã i } min {u ã i } i n i n i n i n i n ã = min {a i } min {a i} min {ā i}; min {ω ã i } max {u ã i } i n i n i n i n i n a TIFTWBHM pqr ã ã 2 ã n a Commutativity: Let a a 2 a n be any permutation of ã ã 2 ã n then TIFTWBHM pqr ã ã 2 ã n = TIFTWBHM pqr ã a 2 a n. Theorem 3.9. MonotonicityLet ã i = a i a i ā i ; ωãi uãi and b i = b i b i b i ; ω bi u bi i = 2 n be two collections of T IF Ns if 25 then a i b i a i b i ā i b i ωãi ω bi uãi u bi TIFTWBHM pqr ã ã 2 ã n TIFTWBHM pqr b b 2 b n

14 30 S. P. Wan and Y. J. Zhu Proof. By the equation 26 for all i j we have Let Thus a = b = a p i aq j ar bp i bq j br ap i aq j ar b p i bq j br ā p i āqjār b p b q i j b r w i w j w n a p w i w j w i aq j ar b p =i j i bq j br =i j =i j =i j w i w j w n a p i aq j ar = w i w j w b p i bq j br w i w j w n ā p w i w j w i āqjār bp b q =i j i j b r min {ω ã i } min i n i n {ω bi } max {u ã i } max } i n i n {u bi max i n {u ã i } max i n {u bi }. TIFTWBHM pqr ã ã 2 ã n = ã = a a ā; ωã uã TIFTWBHM pqr b b 2 b n = b = b b b; ω b u b w i w j w n a p w i w j w a = i aq j ar a p =i j i aq j ar w i w j w ā = ā p i āqjār =i j =i j =i j ωã = min i n {ω ã i } uã = max i n {u ã i } w i w j w b p b = i bq j br b = n =i j n =i j w i w j w bp b q i j b r ω b = min i n {ω bi } u b = max i n {u bi }. w i w j w b p i bq j br 3 [ θa + 2a + θ2a + ā]ω ã 3 [ θb + 2b + θ2b + b]ω b 3 [ θa + 2a + θ2a + ā] u ã 3 [ θb + 2b + θ2b + b] u b. In terms of the equations 5 and 6 we get m µ TIFTWBHM pqr ã ã 2 ã n θ m µ TIFTWBHM pqr b b 2 b n θ m v TIFTWBHM pqr ã ã 2 ã n θ m v TIFTWBHM pqr b b 2 b n θ. According to the lexicographic raning approach of TIFNs in subsection 3.2 the equation 27 holds.

15 Triangular Intuitionistic Fuzzy Triple Bonferroni Harmonic Mean Operators and... 3 Theorem 3.0. Let ã i i = 2 n be a collection of TIFNs > 0 then TIFTWBHM pqr ã ã 2 ã n = TIFTWBHM pqr ã ã 2 ã n. Theorem 3.. Let b be a TIFN and ã i i = 2 n be a collection of TIFNs then TIFTWBHM pqr ã b ã2 b ãn b = TIFTWBHM pqr ã ã 2 ã n b. Proof. By Definition 3.5 we have = TIFTWBHM pqr ã b ã2 b ãn b = w iw jw a p i =i j bp a q j bq a r br ; w iw jw w iw jw a p i =i j bp a q j bq a r br ā p i =i j b p ā q j b q ā r b r b w i w j w =i j a p i aq j ar min i n {ωã i } ω b max i n {uã i } u b = T IF T W BHM pqr ã ã 2 ã n b. min i n {ω ã i } ω b max i n {u ã i } u b b w i w j w =i j a p i aq j ar b w i w j w =i j ā p i āq j ār Hence the proof of Theorem 3. is completed Triangular Intuitionistic Fuzzy Triple Ordered Weighted Bonferroni Harmonic Mean. Chen et al. [5] developed an ordered weighted HM OWHM operator for real numbers. Definition 3.2. [5] An OWHM operator of dimension n is a mapping OWHM : R n R and OWHM w a a 2 a n = n w i a σi where w = w w 2 w n T is the weight vector associated with OWHM satisfying that w i > 0 i = 2 n and n i= i= w i = σ σ2 σn is a permutation of 2 n such that a σi a σi for all i = 2 3 n. Xu [35] defined a fuzzy ordered Bonferroni harmonic mean operator for TFNs. To suit the case of TIFNs we define the TIFTOWBHM operator as follows: Definition 3.3. Let ã i = a i a i ā i ; ωãi uãi i = 2 n The triple ordered weighted Bonferroni harmonic mean operator of dimension n is defined as TIFTOWBHM pqr ã ã 2 ã n = w iw jw ã p =i j σiãqσjãr σ ;

16 32 S. P. Wan and Y. J. Zhu = w i w j w w a p i w j w =i j σi aq σj ar a p σ =i j σi aq σj ar σ ; w i w j w ā p =i j σiāqσjār σ min i n {ωã i } max i n {uã } i 28 where w = w w 2 w n T is the weight vector associated with satisfying that w i > 0 i = 2 n and n = p q r 0 σ σ2 σn is a w i i= permutation of 2 n such that ã σi ã σi for all i = 2 3 n. Especially if r = 0 then the TIFTOWBHM operator is reduced to TIFTOWBHM pq0 ã ã 2 ã n = w i w j = a p p+q σi aq σj w i w j = ā p σiāq p+q σj w i w j = a p p+q σi aq σj min i n {ωã i } max i n {uã } 29 i which is called the triangular intuitionistic fuzzy ordered weighted Bonferroni harmonic mean TIFOWBHM operator. TIFTOWBHM operator is reduced to = TIFTOWBHM p00 ã ã 2 ã n = n i= w i a p σi p n i= w i a p σi p n i= w i ā p σi ; If r = q = 0 and p = then the w i w j w p ã p = σiã0σjã0 σ ; min p i n {ωã i } max i n {uã i } 30 which is called the triangular intuitionistic fuzzy ordered weighted harmonic TIF- OWH operator. If TIFNs ã i = a i a i ā i ; ωãi uãi are reduced to TFNs a i = a i a i ā i i = 2 n then: i The TIFTOWBHM i.e. the equation 28 becomes TIFTOWBHM pqr a a 2 a n = w i w j w =i j a p σi aq σj ar σ w i w j w =i j a p σi aq σj ar σ. w i w j w =i j ā p σiāq σjār σ Here we call it the TFTOWBHM operator which is just the GFOWBHM operator defined in [5]. ii The TIFOWBHM operator i.e. the equation 29 becomes TIFOWBHM pq0 a a 2 a n = n = w i w j p q a i a j p+q

17 Triangular Intuitionistic Fuzzy Triple Bonferroni Harmonic Mean Operators and = n = w i w j a p σi aq σj p+q n = w i w j a p σi aq σj p+q n = w i w j ā p σiāq σj which is just the FOBHM operator defined in [7]. iii The TIFOWH operator i.e. the equation 30 becomes TIFOWH a a 2 a n = n i= w i a p σi p n i= w i a p σi p n i= p+q w i ā p σi which is just the FOWHM operator defined in [35]. iv If the TFNs a i = a i a i ā i i = 2 n are reduced to the interval numbers a i = [a i ā i ] then the TIFTOWBHM operator i.e. the equation 28 is reduced to the GUOWBHM operator defined in [5] as follows: GUOWBHM pqr a a 2 a n = [ = w iw jw a p i aq j ar = p w iw jw ā p i āq j ār ]. Furthermore the TIFOWH operator i.e. the equation 30 is reduced to the UOWHM operator defined in [35]: UOWHM p a a 2 a n = = w i a i =. w i ā i v If a i = ā i = a i for all i = 2 n i.e. the TFNs a i = a i a i ā i are reduced to real numbers then the TIFTOWBHM operator i.e. the equation 30 is reduced to the triple ordered weighted Bonferroni harmonic mean TOWBHM operator: TOWBHM pqr a a 2 a n = w iw jw a p σi = aq σj ar σ whic is just the GOWBHM defined in [5]. The weight vector w = w w 2 w n T associated with the TIFTOWBHM operator can be determined according to actual needs. Moreover there are also many methods to obtain the associated weight vector see [9 33] for details. Similar to the TIFTWBHM operator the TIFTOWBHM operator has some desirable properties such as idempotency boundedness commutativity monotonicity omitted. Theorem 3.4. Let ã i i = 2 n be a collection of TIFNs > 0 then TIFTOWBHM pqr ã ã 2 ã n = TIFTOWBHM pqr ã ã 2 ã n. Theorem 3.5. Let b be a TIFNs and ã i i = 2 n be a collection of TIFNs then TIFTOWBHM pqr ã b ã2 b ãn b = TIFTOWBHM pqr ã ã 2 ã n b. 3

18 34 S. P. Wan and Y. J. Zhu Proof. By Definition 3.3 we have = TIFTOWBHM pqr ã b ã2 b ãn b = w iw jw a p i =i j bp a q j bq a r br ; w iw jw w iw jw a p i =i j bp a q j bq a r br ā p i =i j b p ā q j b q ā r b r min {ω ã i } i n ω b max {u ã i i n }u b b b w i w j w w a p i w j w =i j i aq j ar a p =i j i aq j ar min {ω ã i } i n ω b max {u ã i } u b. i n which completes the proof of Theorem 3.5. b w i w j w ā p =i j i āq j ār The TIFTOWBHM operator has some special cases as follows: If W = 0 0 T then TIFTOWBHM pqr W ã ã 2 ã n = max {ã i}; i n 2 If W = T then TIFTOWBHM pqr W ã ã 2 ã n = min {ã i}; i n 3 If W = n n n T then = n TIFTOWBHM pqr W ã ã 2 ã n = n ã p i =i j ãq j ãr n n a p =i j i aq j ar a p =i j i aq j ar ā p =i j i āq j ār min i n {ω ã i } max i n {u ã i } Triangular Intuitionistic Fuzzy Triple Hybrid Bonferroni Harmonic Mean Operator. By combining the advantages of the weighted harmonic mean operator and the ordered weighted harmonic mean operator Xu [35] developed the fuzzy hybrid harmonic mean FHHM operator for the triangular fuzzy numbers Sun and Sun [7] proposed the notation of the fuzzy hybrid Bonferroni harmonic mean FHBHM operator for TFNs as follows: Definition 3.6. [7] Let â i = a i a i ā i i = 2 n be a collection of TFNs. A FHBHM operator of dimension n is defined as FHBHM pq â â 2 â n = n w iw j ȧ = σi p ȧ σj p+q w iw j q ȧ σi p ȧ σj p+q w iw j q a σi p a σj p+q q = = ; ;

19 Triangular Intuitionistic Fuzzy Triple Bonferroni Harmonic Mean Operators and where ˆa σi = ȧ σi ȧ σi ȧ σi is the i th largest of the weighted TFNs â i â i = nw i â i i = 2 n w = w w 2 w n T is the weight vector of â â 2 â n satisfying w i > 0 and n w i = and n is the balancing coefficient. i= It should be pointed out that Definition 3.6 is not right since it did not consider the associated weight vector with the FHBHM operator. Generally the hybrid aggregation operator contains two inds of weight vector: the associated weight vector or position weight vector and the weight vector of the fused arguments. However there is only the weighted vector of the fused arguments in Definition 3.6. Namely the associated weight vector is confused as the weighted vector of the fused arguments. Analogously the FHHM operator defined by Xu [35] has the same error. Definition 3.7. Let ã i = a i a i ā i ; ωãi uãi i = 2 n be a collection of TIFNs. The triangular intuitionistic fuzzy triple hybrid Bonferroni harmonic mean operator of dimension n is defined as TIFTHBHM pqr wv ã ã 2 ã n = =i j =i j =i j w iw jw ã p =i j σi ã q σj ã r σ w i w j w ȧ σi p ȧ σj q ȧ σ r w i w j w ȧ σi p ȧ σj q ȧ σ r w i w j w ȧ σi p ȧ σj q ȧ σ r ; min i n {ω ã σi } max i n {u ã σi } 32 where w = w w 2 w n T is the weight vector associated with TIFTHBHM satisfying that w i > 0 i = 2 n and n w i = ã σi = ȧ σi ȧ σi ȧ σi ; i= ω ã σi u ã σi is the i th largest of the weighted TIFNs a i a i = nv i ã i i = 2 n and v = v v 2 v n T is the weight vector of ã ã 2 ã n satisfying that v i > 0 i = 2 n and n v i = and n is the balancing coefficient. i= Especially if v i = /n i = 2 n then the TIFTHBHM operator is reduced to the TIFTOWBHM operator. That is to say the TIFTOWBHM operator is a special case of the TIFTHBHM operator. If the TIFNs reduce to TFNs and r = 0 then the TIFTHBHM operator is reduced to the corrected FHBHM operator defined by Sun and Sun [7] i.e. the FHBHM operator is also a special case of the TIFTHBHM operator. The above three inds of triangular intuitionistic fuzzy triple hybrid Bonferroni harmonic mean operators can tae the given arguments and their relationships into =

20 36 S. P. Wan and Y. J. Zhu consideration. The TIFTWBHM operator emphasizes the importance of each argument the TIFTOWBHM operator stresses the importance of the ordered position of each argument while the TIFTHBHM operator reflects the important degrees of both the given arguments and the ordered position of the arguments. Furthermore the prominent characteristic of these operators is that they can tae the correlationships between the TIFN arguments into account. In MAGDM with TIFNs since the attribute values of alternatives are represented as TIFNs the TIFTWBHM operator can be directly used to integrate the attribute values of alternatives into the individual comprehensive attribute values of alternatives. Meanwhile there are multiple experts participating decision maing together different experts should be allocated various weights. Thus the TIFTHBHM operator can be used to integrate the individual comprehensive attribute values of alternatives into the collective ones by sufficiently considering the expert weights. 4. A New Method for MAGDM with TIFNs In this section employed the TIFTWBHM and TIFTHBHM operators a new decision maing method is developed to solve the MAGDM problems with TIFNs. 4.. Presentation of MAGDM Problems with TIFNs. Let A = {A A 2 A m } be an alternative set and C = {c c 2 c n } be the set of attributes. The attribute weight vector is w = w w 2 w n T satisfying that 0 w j j = 2 n and n j= w j =. There are s DMs or experts participating in decision maing denote the set of DMs by E = {e e 2 e s }. The weight vector of the DMs is v = v v 2 v s T satisfying that 0 v = 2 s and s v =. The rating of an alternative A i on an attribute = c j given by the DM e is a TIFN ã = a a ā ; ω ã u ã where ω ã and uã denote respectively the maximum membership degree and the minimum nonmembership degree of alternative A i on attribute c j given by the DM e satisfying 0 ωã 0 uã and 0 ωã + uã. Hence a MAGDM problem can be concisely expressed in matrix format as à = ã m n = 2 s which are referred to as TIFN decision matrices. Since the attributes are generally incommensurate the decision matrix needs to be normalized so as to transform the various attribute values into comparable values. The matrix à = ã m n is normalized into R = r m n where r = r r r ; ω r u r ω r = ωã u r = uã and r = a ā + j a ā + j ā ā + ; ω r u r for benefit attributes 33 j r = a j ā a j a a j ; ω r u r for cost attributes 34

21 Triangular Intuitionistic Fuzzy Triple Bonferroni Harmonic Mean Operators and where ā + j = max{ā i = 2.m; = 2 s} and a j = min{ i = 2.m; = 2 s} j = 2 n Decision Method Based on TIFTWBHM and TIFTHBHM Operators. In sum an algorithm and process for solving the MAGDM problems with TIFNs may be summarized as follows. Step : Normalize the decision matrix à into R by the equations 33 and 33; Step 2: Combined the TIFTWBHM operator with the attribute weight vector w = w w 2 w n T the individual overall attribute value of alternative A i given by DMe can be obtained as follows: = r i = r i r i r i ; ω r i u r = TIFTWBHM r i r i2 r in i tju= tju= w tw j w u r it p r q r iu r tju= w tw j w u rit p r q riu r w tw j w u r it p r q r iu i = 2 m; = 2 s. r Step 3: Utilize the TIFTHBHM operator to aggregate the individual overall attribute values r i = 2 s and get the collective overall attribute value r i of alternative A i as follows: s r i = TIFTHBHM pqr ωv r i r2 i rs i = ω tω j ω tj= ȧ σi i p ȧ σj i q ȧ σ i r 35 s ω tω j ω s ω tω j ω tj= ṙ σi i p ṙ σj i q ṙ σ i r tj= r σi i p r σj i q r σ i r ; min σ } max σ } 36 s {ω r i s {u r i where ω = ω ω 2 ω s T satisfying that ω i > 0 i = 2 s and ω r σi i u r σi i is the weight vector correlated with TIFTHBHM s i= ω i =. r σi i = ṙ σi i ṙ σi i r σi i ; is the ith largest of the weighted TIFNs r i = sv i r i = 2 s and v = v v 2 v s T is the weight vector of the DMs. Step 4: Use the equations 5 and 6 to calculate the weighted possibility means m µ r i θ and m v r i θ of r i = r i r i r i ; ω ri u ri. Then the raning order of alternatives can be obtained according to the lexicographic raning approach in Subsection A Real Investment Selection Case Study and Comparative Analysis In this section a real investment selection case study is illustrated to demonstrate the applicability and implementation process of the MAGDM method proposed in this paper. The comparison analyses of computational results are also conducted to show the superiority of the proposed method.

22 38 S. P. Wan and Y. J. Zhu 5.. A Real Investment Selection Case Study. The proposed method is applied to a real investment selection example. An investment company Shenzhen Capital Group SCGC for short was born in a hotbed of innovation the city of Shenzhen in China. The birth of SCGC itself is an innovation of the city government of Shenzhen. The city wants to create a shoulder to support high-tech industries and to promote business development. The scale of established government-baced funds is RMB 6.84 billion and the new funds being raised is RMB.77 billion. SCGC desires to invest a sum of money in the best option. After preliminary screening five possible candidates i.e. alternatives remain for further evaluation. Candidate A is a car company Zhejiang geely automobile co. ltd A 2 is a food company Oreo company A 3 is a computer company Lenovo Group ltd A 4 is an arms company Xi an days and defense technology Limited by Share Ltd and A 5 is a TV company Changhong company. The decision maing committee consists of three DMs: president e vice-president e 2 and department manager e 3. They assess the five candidates on the basis of four attributes including the ris analysis c the growth analysis c 2 the social-political impact analysis c 3 the environmental impact analysis c 4 the first attribute is cost attribute and the other three attributes are benefit attributes. The weight vector of the attributes is W = T the weight vector of the DMs is v = T. The ratings of the candidates with respect to attributes can be represented by TIFNs as in Tables -3. For example TIFN ; in Table indicates that the mar of the candidate with respect to the attribute is about 3.53 with the maximum satisfaction degree 0.7 while the minimum dissatisfaction degree 0.2. In other words the hesitation degree is 0.. This assessment value ; can be obtained by the following pre-processing phase: After negotiation and discussion the group of DMs all agree that the assessments on attribute c should use triangular fuzzy numbers in 0-point scale grades from 0 up to 0 are used in endpoints of mode lower and upper limits with 0 being worst and 0 being best to score. 2 Each DM respectively gives the lower mar a the most possible mar b and the upper mar c for alternative A on attribute c. 3 Meanwhile for the most possible mar each DM provides the maximum membership degree wã and the minimum non-membership degree uã according to his/her nowledge and experience or by statistical methods. Thus TrIFN ã = a b c; ωã uã is produced. The other TrIFNs in Tables -3 can be similarly explained and obtained. c c 2 c 3 c 4 A ; ; ; ;0.80. A ; ; ; ; A ; ; ; ;0.80. A ; ; ; ;0.70. A ; ; ; ; Table. TIFN Decision Matrix Given by DM e Step : According to the equations 33 and 34 the normalized TIFN decision matrices are obtained and listed in Tables 4-6.

23 Triangular Intuitionistic Fuzzy Triple Bonferroni Harmonic Mean Operators and c c 2 c 3 c 4 A ; ; ; ;0.80. A ; ; ; ; A ; ; ; ;0.80. A ; ; ; ;0.70. A ; ; ; ; Table 2. TIFN Decision Matrix Given by DM e 2 c c 2 c 3 c 4 A ; ; ; ;0.80. A ; ; ; ; A ; ; ; ;0.80. A ; ; ; ;0.70. A ; ; ; ; Table 3. TIFN Decision Matrix Given by DM e 3 c c 2 c 3 c 4 A ; ; ; ;0.80. A ; ; ; ; A ; ; ; ;0.80. A ; ; ; ;0.70. A ; ; ; ; Table 4. Normalized TIFN Decision Matrix Given by DM e c c 2 c 3 c 4 A ; ; ; ;0.80. A ; ; ; ; A ; ; ; ;0.80. A ; ; ; ;0.70. A ; ; ; ; Table 5. Normalized TIFN Decision Matrix Given by DM e 2 Step 2: Use TIFTWBHM operator to integrate the elements in the i-th row of the normalized matrix R to obtain the individual overall TIFNs of the alternative r i i = ; = 2 3. For example let p = q = r = 2 then r j can be calculated by the equation 35 as follows: r = ; r 2 = ; ; r 3 = ; r 4 = ; ; r 5 = ; r 2 = ; ; r 2 2 = ; r 2 3 = ; ; r 2 4 = ; r 2 5 = ; ; r 3 = ; r 3 2 = ; ; r 3 3 = ; r 3 4 = ; ; r 3 5 = ; Step 3: Combined the TIFTHBHM operator with the DMs weight vector v = T the collective overall attribute values of alternatives A i i = 2 5 can be obtained. For example using the method of Xu [33] to obtain the associated weight vector ω = T and taing p = q = r = 2 by the equation 36 we have r = r r r ; ω r u r = ; ; r 2 = r 2 r 2 r 2; ω r2 u r2 = ; ; r 3 = r 3 r 3 r 3; ω r3 u r3 = ; ; r 4 = r 4 r 4 r 4; ω r4 u r4 = ; ; r 5 = r 5 r 5 r 5; ω r5 u r5 = ;

24 40 S. P. Wan and Y. J. Zhu c c 2 c 3 c 4 A ; ; ; ;0.80. A ; ; ; ; A ; ; ; ;0.80. A ; ; ; ;0.70. A ; ; ; ; Table 6. Normalized TIFN Decision Matrix Given by DM e 3 Step 4: Ran A i i = according to the lexicographic raning approach. For example if θ = 0.5 we get the weighted possibility means of the collective overall attribute values of the alternatives by the equations 5 and 6 as follows: m µ r θ = m µ r 2 θ = m µ r 3 θ = m µ r 4 θ = m µ r 5 θ = 0.373m v r θ = m v r 2 θ = m v r 3 θ = m v r 4 θ = m v r 5 θ = Thus the raning of candidates is generated as A 4 A 5 A 3 A 2 A.The best candidate is A 4. For any other ris preference parameter values θ in the same way we can obtain the raning orders of candidates listed in Table 7. θ Raning orders Best candidates 0 A 3 A 5 A 4 A 2 A A 3 0. A 3 A 5 A 4 A 2 A A A 3 A 5 A 4 A 2 A A A 3 A 5 A 4 A 2 A A A 4 A 5 A 3 A 2 A A A 4 A 5 A 3 A 2 A A A 3 A 4 A 5 A 2 A A A 4 A 5 A 3 A 2 A A A 4 A 5 A 3 A 2 A A A 4 A 5 A 3 A 2 A A 3.0 A 4 A 5 A 2 A 3 A A 4 Table 7. Raning Orders For Different θ Values with p = q = r = 2 It can be seen from Table 7 that for different ris preference parameter values the raning orders of candidates are also not completely the same. For instance if θ [0 0.3] then the raning isa 3 A 5 A 4 A 2 A the best is A 3 ; if θ [ ] then the raning is A 4 A 5 A 3 A 2 A the best is A 4 ; if θ [0.7.0 then the raning is A 4 A 5 A 2 A 3 A the best is A 4. In the same way we can obtain the collective overall attribute values of candidates for any other preference parameter values pqrθ. The computation results and raning are listed in Tables 8 and 9. It can be seen from Tables 8 and 9 that for different parameter values pq r the raning orders of candidates are also not completely the same. For instance if θ [0 0.4 and p = q = r = then the raning is A 3 A 5 A 4 A 2 A the best is A 3 ; if θ [0 0.6 and p = q = r = 3 then the raning is A 3 A 5 A 2 A 4 A the best candidate is A 3. The above analysis suggests that the ris preference of DM for the weighted lower and upper possibility means indeed plays an important role in the decision maing. Since TIFN is a special ind of intuitionistic fuzzy set involving DM s ris preference to ran the TIFNs is very reasonable and necessary. When the ris preference parameter values θ are different the corresponding decision results