International Journal of Analysis and Applications Volume 17, Number 1 (2019), 76-104 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-17-2019-76 DIMINISHING CHOQUET HESITANT 2-TUPLE LINGUISTIC AGGREGATION OPERATOR FOR MULTIPLE ATTRIBUTES GROUP DECISION MAKING ISMAT BEG1∗, RAJA NOSHAD JAMIL2 AND TABASAM RASHID2 1Lahore School of Economics, Lahore-53200, Pakistan 2University of Management and Technology, Lahore-54770, Pakistan ∗Corresponding author: ibeg@lahoreschool.edu.pk Abstract. In this article, we develop a diminishing hesitant 2-tuple averaging operator (DH2TA) for hesi- tant 2-tuple linguistic arguments. DH2TA work in the way that it aggregate all hesitant 2-tuple linguistic elements and during the aggregation process it also controls the hesitation in translation of the resultant aggregated linguistic term. We develop a scalar product for hesitant 2-tuple linguistic elements and based on the scalar product a weighted diminishing hesitant 2-tuple averaging operator (DWH2TA) is introduced. Moreover, combining Choquet integral with hesitant 2-tuple linguistic information, the diminishing Chouqet hesitant 2-tuple average operator (DCH2TA) is defined. The proposed operators higher reflect the corre- lations among the elements. After investigating the properties of these operators, a multiple attribute decision making method based on DCH2TA operator is proposed. Finally, an example is given to illustrate the significance and usefulness of proposed method. 1. Introduction Different procedures wherein problems that manage indefinite and vague data mostly involves the vul- nerability of their definition structures. Utilizing numerical modelling to represent such indeterminate data would not be reliably adequate. In these conditions wherein the vulnerability would not be of probabilistic Received 2018-10-17; accepted 2018-11-26; published 2019-01-04. 2010 Mathematics Subject Classification. 91B06; 94D05; 90B50; 91B10; 46S40; 03E72. Key words and phrases. hesitant 2-tuple model; aggregation operator; choquet integral; multiple attribute group decision making; supply chain management. c©2019 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 76 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-76 Int. J. Anal. Appl. 17 (1) (2019) 77 nature and the capacities are unclear, it is hard to give numerical specific information. Usually the decision makers that take an interest in this kind of issues utilize linguistic descriptors to particular their evaluation and identified with the uncertain potential they have concerning the issues [38,40]. Therefore, the utilization of linguistic demonstration in problems managing non-probabilistic instability shows up justification and has made effective result in particular fields, for example: situation realization [35], decision models [6, 9, 49, 64], information retrieval [25, 30], risk evaluation [18], engineering analysis [39, 40], sensory evaluation [10, 36], performance appraisal [1, 2], data mining [27], social alternative [19] and waste management [51]. These accomplishments have not been possible without systems to complete the improvement of computing with words (CW) [34] that implies the utilization of linguistic knowledge. The accompanying algorithm indicates how these translations to be functional. Algorithm 1.1. Step 1. Input data in the form of linguistic terms or 2-tuple linguistic terms Step 2. Translation into equivalent numeric value Step 3. Manipulation Step 4. Retranslation into linguistic terms / 2-tuple linguistic terms accordingly Step 5. Output data These ideas for CW have an edge on probability theory [26, 33], the uncertainty models, in these problems are alternatively involving the imprecision and vagueness of the linguistic descriptors. For this reason other tools as fuzzy logic [71] and the fuzzy linguistic process [72] used specific computational models for CW, for instance: • The linguistic computational model created on membership functions such as [13, 37] these models based on the fuzzy linguistic approach and makes the computations instantly on the membership function of the linguistic terms by way of utilizing the extension principle [16, 31]. • Foundation of the linguistic symbolic computational models are on ordinal scales [65]. These models represent the understanding in keeping with the fuzzy linguistic technique and makes use of the ordered structure of the linguistic term set to achieve symbolic computations in such ordered lin- guistic scales. Equivalent tactics founded on this mode of computing has been discussed in [14, 62]. It notable that this mannequin has been frequently applied to decision making practices due to its easy adaptation and effortlessness for decision makers [65]. Linguistic models seek after the computational plan showed by means of Yager in [66, 68] can be seen in general algorithm 1.1, that features out the significance of the interpretation and translation approaches in CW and likewise Mendel and Wu [42] highlighted similar techniques in computing with perceptions . In the article author discussed that firstly, taking data linguistically and translates into a computing Int. J. Anal. Appl. 17 (1) (2019) 78 tool for manipulative structure. In the second stage, incorporates taking the outcomes from the control, computing device, arrange and change them into linguistic information as an approach to be reasonable by method for individuals that is without uncertainty, one of the essential desire of CW [42]. These linguistic computational units present a most important weak point, in view that they carried out the translation step as an approximation method to precise the outcome in the usual expression area (initial term set) scary a lack of accuracy [23]. To obstruct such inaccuracy in the translation step was once offered the 2-tuple linguistic computational model [22, 41]. It is a typical mannequin that broadens the utilization of records adjusting the fuzzy linguistic strategy representation with including a parameter with essential linguistic representation as an approach to show signs of improvement exactness of the linguistic calculations after the re-interpretation step, holding the CW plan stated in algorithm 1.1 and the work out the capacity of the result. In recent times, numerous aggregation operators have been produced for the 2-tuple linguistic model to assess diverse decision making issues [59]. Herrera and Mart́ınez [22] proposed the 2-tuple arithmetic weighted averaging operator, the 2-tuple ordered weighted averaging operator and the extended 2-tuple weighted averaging operator. Xu et al. [63] developed the extended geometric mean operator, the extended arithmetic averaging operator, the extended ordered weighted averaging operator and the extended ordered weighted geometric operator . Jiang and Fan [28], proposed the 2-tuple ordered weighted averaging operator and the 2-tuple ordered weighted geometric operator. The extended 2-tuple ordered weighted averaging operator was proposed in [73]. The extended 2-tuple weighted geometric operator and the extended 2-tuple ordered weighted geometric operator have been developed in [60]. Herrera et al. [24] proposed an unbalanced linguistic computational model that helpful for calculating the 2-tuple fuzzy linguistic computational model to achieve processes of evaluating words for unbalanced term sets in an accurate mode without loss of information. Furthermore, Dong et al. [15] proposed a consistency improving model which preserves the utmost original knowledge and preferences in the process of improving consistency and it also guarantees that the elements in the optimal adjusted unbalanced linguistic preference relation are all simple unbalanced linguistic terms. Aggregation operators examined over, the attributes are thought to be autonomous of each other, which are differentiated by an independent axiom [43, 44, 57]. But in the real decision making practice, the charac- teristics of the problem are often dependent or correlated to each other. Choquet integral [11] is one of the valuable instrument to build up the model an issue, which utilize the properties as between reliance or con- nection to each other. Choquet integral has examined and connected all the basic properties of the decision making problems [32, 50, 67, 69]. Yager [67] proposed the induced Choquet ordered averaging operator to aggregate a group of real arguments while in [69], Yager combined the intuitionistic fuzzy sets with Choquet integral. The intuitionistic fuzzy Choquet integral operator obtained in [9]. Tan and Chen [53] developed Int. J. Anal. Appl. 17 (1) (2019) 79 the induced Choquet ordered averaging operator. Xu [64] proposed the intuitionistic fuzzy correlated aver- aging operator, the intuitionistic fuzzy correlated geometric operator, the interval-valued intuitionistic fuzzy correlated averaging operator and the interval-valued intuitionistic fuzzy correlated geometric operator to aggregate the intuitionistic fuzzy information and the interval-valued intuitionistic fuzzy information. Beg and Rashid [4] used Choquet integral, for selection of bike when the criteria include interactions among each others. Yang and Zhiping [70] proposed, 2-tuple correlated averaging operator, the 2-tuple correlated geometric operator and the generalized 2-tuple correlated averaging operator combined with Choquet inte- gral. Joshi et al. [29] developed novel hesitant probabilistic fuzzy linguistic ordered weighted averaging and hesitant probabilistic fuzzy linguistic ordered weighted geometric aggregation operators for ill structured and complex decision making problems. Torra [55] discussed that hesitant fuzzy set can deal with the conditions where the assessment of a selection under each and every criterion is represented by several feasible values, not by a margin of error, or some probability distribution on the possible values. For instance, decision maker gives the membership value of x into A, and so they wish to assign 0.23, 0.26 and 0.31, which is a hesitant fuzzy element {0.23, 0.26, 0.31} rather than the interval between 0.23 and 0.31. Use these qualities of the hesitant fuzzy set, Beg and Rashid [5] proposed hesitant 2-tuple linguistic information to take care of marginal error. As we observed in known literature for hesitant sets, the aggregation operators produced more hesitation during aggregation process with respect to given hesitant elements [74]. Due to this we develop new operational laws for hesitant 2-tuple. These operational laws reduce the hesitation during aggregate process. We used these operational laws to develop a diminishing hesitant 2-tuple averaging operator (DH2TA) for hesitant 2-tuple linguistic arguments. DH2TA worked in two ways, firstly it aggregated all hesitant 2-tuple linguistic elements and secondly it also reduces the hesitation in resultant 2-tuple linguistic term. In this article, to check applicability of our method we apply it to supply chain management area. In today globalization era, a suitable supplier selection is a core issue of supply chain management that effect the overall performance as without efficient suppliers it is impossible to produce low-cost and high quality products [3, 56]. Especially for organizations that spend a high level of their business income on parts and material supplies and whose material costs is a large part of aggregate costs. An organized and transparent approach regarding the choice of supplier is essential for these organizations. Supplier selection is a procedure by which suppliers are assessed, evaluated, and then selected to become a part of the company’s supply chain [8]. To overcome the supply chain risk, reduced the production cost, optimize inventory levels and the end profitability are major targets of supply chain management [12, 21]. There exist some well known method for supplier section for instant, matrix approach [20], vendor performance matrix approach [52], vendor profile analysis [54], analytic hierarchy process (AHP) [47, 48] and multiple objective programming (MOP) [17]. Int. J. Anal. Appl. 17 (1) (2019) 80 In this paper, we use the notion of hesitant 2-tuple linguistic information which was proposed by Beg and Rashid [5] to develop a diminishing hesitant 2-tuple averaging operator (DH2TA) for hesitant 2-tuple linguistic arguments. The rest of the paper is structured as follows: some basic concepts are presented to understand our proposal in section 2. In section 3 we propose some definition which is ranking the hesitant 2- tuple linguistic information In Section 4, we define diminishing hesitant 2-tuple averaging (DH2TA) operator, and discussed some properties of DH2TA. In Section 5, we merge Choquet integral with the operator DH2TA and developed a new operator diminishing Choquet hesitant 2-tuple average operator (DCH2TA) and also discussed different properties of DCH2TA. The multiple attribute decision making method based on DCH2TA is proposed in Section 6. In Section 7, a numerical example is given to illustrate the developed approach and to demonstrate its feasibility and practicality. Concluding remarks are given in last section. 2. Hesitant Fuzzy Sets Some important preliminary concepts are given in this section to understand our proposed aggregation operators. Hesitant fuzzy set was defined by Torra [55] to match the vagueness of real life, when some one is hesitant about membership value. Definition 2.1. [55] For a reference set X. The hesitant fuzzy set on X is defined by function that will give a subset of [0, 1] when applied to X. To be easily understood, Xia et al. [61] expressed the HFS by a mathematical symbol: E = {< x,hE(x) > |x ∈ X}, where hE(x) is set of values form [0, 1], known as the possible membership degrees of x to set E. Also h = hE(x) is called hesitant fuzzy element (HFE). To find order between two HFEs, Xia et al. [61] defined score function as follow: Definition 2.2. [61] Let e be a HFE and h ∈ e then score function ”S” of e is S (e) = 1 n(e) n(e)∑ i=1 hi where n(e) be total number of elements in e. Let e1 and e2 be two HFEs then, if S(e1) < S(e2) then e1 ≺ e2 and if S(e1) = S(e2) then e1 ≈ e2 Int. J. Anal. Appl. 17 (1) (2019) 81 Let e, e1 and e2 be elements of a hesitant fuzzy set A then following basic operations are introduced by Xia et al. [61]: (1) eα = ∪h∈e{hα} , α > 0. (2) αe = ∪h∈e{1 − (1 −h) α} ,α > 0. (3) h1 ⊕h2 = ∪h1∈e1,h2∈e2 {h1 + h2 −h1h2} . (4) h1 ⊗h2 = ∪h1∈e1,h2∈e2 {h1h2} . Next we study concise review of 2-tuple linguistic information and some important basic concepts which are necessary to develop the aggregation operator for hesitant 2-tuple linguistic information. Assume that L = {li | i = 2n+ 1,∀n ∈ N} where N be the set of natural number and li be representation of a possible value for linguistic variable. The set L have the following properties by [22]: P 1. The set L must be ordered: li ≥ lj if i ≥ j, P 2. The maximum of any two linguistic terms is max(li, lj) = li if li ≥ lj, P 3. The minimum of any two linguistic terms is min(li, lj) = li if li ≤ lj. The cardinality of the set L must be low enough that is not to impose unnecessary precision for users and it should be rich enough to allow discrimination of the performance of the individual criteria in the limited number of ranking. Psychologist [45] recommend the use of 7 ± 2 labels. Due to this point of view, a linguistic term set, L with seven labels can be defined as follows: L = {l0 = extremely unattractive (EU), l1 = fairly unattractive (FU), l2 = unattractive (U), l3 = normal (N), l4 = attractive (A), l5 = fairly attractive (FA), l6 = extremely attractive (EA)}. In the literature different models have been recommended for processing of linguistic information. In this paper, we have implemented 2-tuple linguistic representation model, which is based on symbolic translation [22]. Symbolic translation is defined as follow: Definition 2.3. [22] Let L = {l0, l1, ..., lg} be the set of linguistic terms, δi ∈ [0, g] for any i ∈{0, 1, ...,g}, j = round(δi) and ςj = δi − j =⇒ ςj ∈ [−0.5, 0.5), then ςj is called the value of the symbolic translation. Where round(δi) is the usual round operation on label index of set L. Definition 2.4. [22] Let L = {l0, l1, l2, ..., lg} be the set of linguistic terms set and δi be the number representing the aggregation result of symbolic operation. The function 4 used to obtain the 2-tuple linguistic information equivalent to δi is defined as: 4 : [0, g] −→ L× [−0.5, 0.5), 4(δi) = (lj, ςj) with   lj j = round(δi)ςj = δi − j ςj ∈ [−0.5, 0.5) Int. J. Anal. Appl. 17 (1) (2019) 82 Figure 1. Structure of different 2-tuple linguistic elements Inverse function of 4 is always exist and denoted by 4−1. 4−1 : L× [−0.5, 0.5) −→ [0, g], 4−1 (lj, ςj) = ςj + j = δi Example 2.1. Suppose we have different 2 − tuple elements x1 = (Nothing, 0.19), x2 = (Low, 0.43), x3 = (Medium,−0.22) and x4 = (V ery high,−0.19) then the structure of these elements is described in Figure 1. Definition 2.5. [22] Let (li, ςi) and (lj, ςj) be two 2-tuple linguistic elements, then order between them is according to an ordinary lexicographic order: (1) If i < j then (li, ςi) < (lj, ςj) , (2) If i = j then • if ςi < ςj then (li, ςi) < (lj, ςj) • if ςi = ςj then (li, ςi) = (lj, ςj) Definition 2.6. [58] A fuzzy measure α on the set X is a set function α : P(X) → [0, 1] satisfying the following conditions: (1) α(∅) = 0, α(X) = 1; (2) If B ⊆ C ⇒ α(B) ≤ α(C), ∀ B,C ⊆ X; (3) α(B ∪C) = α(B) + α(C) + λα(B)α(C) ∀ B,C ⊆ X and B ∩C = ∅, where λ ∈ (−1, +∞). Int. J. Anal. Appl. 17 (1) (2019) 83 By parameter λ the interaction between criteria can be represented. n⋃ i=1 xi = X for a finite set X. The λ− fuzzy measure α satisfied the following equation α (X) = α ( n⋃ i=1 xi ) =   1 λ { n∏ i=1 (1 + λα(xi)) − 1 } if λ 6= 0 n∑ i=1 α(xi) λ = 0 (2.1) where xi ∩xj = ∅ for all i,j = 1, 2, ...,n and i 6= j. The number α(xi) for a subset with a single element {xi} is called a fuzzy density. α (A) =   1 λ { n∏ i=1 (1 + λα(xi)) − 1 } if λ 6= 0 n∑ i=1 α(xi) λ = 0 (2.2) Based on above equation, the value of λ can be find from the following equation, if α(X) = 1 then, 1 = 1 λ { n∏ i=1 (1 + λα(xi)) − 1 } (2.3) In the above definition, if λ = 0, then the third condition reduces to the axiom of the additive measure i.e. α(B ∪C) = α(B) + α(C) ∀B,C ⊆ X and B ∩C = ∅ If the elements of B are independent, then α(B) = ∑ xi∈B α(xi) ∀B ⊆ X. 3. Hesitant 2-tuple Linguistic Information Hesitant 2-tuple linguistic information model is introduced by Beg and Rashid [5] to manage the conditions in which information described is in linguistic term and decision maker has some hesitation to decide its possible linguistic translations. Let X be a universe of discourse and L = {l0, l1, l2, ..., lg} be the linguistic term set then a hesitant 2-tuple linguistic term set in X is an expression E = {(x,h(x)) : x ∈ X} , where h(x) = (li, ςi,j) be the hesitant linguistic information by mean of 2-tuple, ςi,j is finite subset of [−0.5, 0.5) which represent the possible translations of li and j be the cardinality of ςi,j and i ∈{0, 1, 2, ...,g} Definition 3.1. Let h(x) = (li, ςi,j) be a H2TLE then score function S of h(x) is S (h(x)) = 1 j ∑ γ∈ςi,j γ where j is the cardinality of ςi,j To find order between two H2TLE we use the score function defined in definition 3.1 Definition 3.2. Let h1(x) = (li, ςi,j) and h2(x) = (lk, ςk,p) be two H2TLEs, then order between them is according to an ordinary lexicographic order: Int. J. Anal. Appl. 17 (1) (2019) 84 (1) If i < k then h1(x) ≺ h2(x), (2) If i = k and • S(h1(x)) < S(h2(x)) then h1(x) ≺ h2(x) • S(h1(x)) = S(h2(x)) then h1(x) = h2(x) Definition 3.3. Let ( lip, ςip,jp ) be k H2TLEs. If k ∈ N,ip ∈ {0, 1, 2, ...,g} and jp be cardinality of ςip,jp, then minkp=1 ( lip, ςip,jp ) is defined as follow: (1) If all ( lip, ςip,jp ) are different due to different ip then, let min k p=1(ip) = i ∈ {0, 1, 2, ...,g}. If ςi,j be represent the translations of li and j be the cardinality of ςi,j, then by definition 3.2 k min p=1 ( lip, ςip,jp ) = (li, ςi,j) (2) If ip = i but ςip,jp are different for each ( lip, ςip,jp ) . If ςi,j is represent the translations of lip = li with S ( lip, ςip,jp ) = minkp=1 ( 1 jp ∑ γ∈ςip,jp γ ) then k min p=1 ( lip, ςip,jp ) = (li, ςi,j) (3) If all ( lip, ςip,jp ) are equal, such that ip = i and ςip,jp = ςi,j, then by definition 3.2 k min p=1 ( lip, ςip,jp ) = (li, ςi,j) (4) If all ( lip, ςip,jp ) H2TLEs are equal by definition 3.2 but still there is some possibility exist that ςip,jp are different, but score are same then min k p=1 ( lip, ςip,jp ) is one with maximum hesitation. Definition 3.4. Let ( lip, ςip,jp ) be k H2TLEs. If k ∈ N, ip ∈ {0, 1, 2, ...,g} and jp be cardinality of ςip,jp, then maxkp=1 ( lip, ςip,jp ) is defined as follow: (1) If all ( lip, ςip,jp ) are different due to different ip then, let max k p=1(ip) = i ∈ {0, 1, 2, ...,g}. If ςi,j be represent the translations of li and j be the cardinality of ςi,j, then by definition 3.2 k max p=1 ( lip, ςip,jp ) = (li, ςi,j) (2) If ip = i but ςip,jp are different for each ( lip, ςip,jp ) . If ςi,j is represent the translations of lip = li with S ( lip, ςip,jp ) = maxkp=1 ( 1 jp ∑ γ∈ςip,jp γ ) then k max p=1 ( lip, ςip,jp ) = (li, ςi,j) (3) If all ( lip, ςip,jp ) are equal, such that ip = i and ςip,jp = ςi,j, then by definition 3.2 k max p=1 ( lip, ςip,jp ) = (li, ςi,j) Int. J. Anal. Appl. 17 (1) (2019) 85 (4) If all ( lip, ςip,jp ) H2TLEs are equal by definition 3.2 but still there is some possibility exist that ςip,jp are different, but score are same then max k p=1 ( lip, ςip,jp ) is one with minimum hesitation. Definition 3.5. If (lk1, ςk1,j1 ) is a hesitant 2-tuple linguistic element, g be the upper limit of the linguistic term set and λ ≥ 0 is any scalar. Then the scalar product for H2TLE is defined as follows: λ (lk1, ςk1,j1 ) = (λlk1,λςk1,j1 ) = (li, ςi,j) (li, ςi,j) is calculated as follow Let β1 = { µ|µ = ((g + 1) ( 1 − ( 1 − g + γk1,j1 g + 1 )λ) ∀γk1,j1 ∈ ςk1,j1 } , then β2 = ⋃ {θ1|θ1 = round(µ), ∀µ ∈ β1}, i = ( ∑ θ1∈β2 θ1 + k1 ) |β2| + 1 , |β2| be cardinality of β2ςip,jp = {θ2|θ2 = different(µ−θ1)}, jp be cardinality of each ςip,jp ςi,j = {x|x ∈ ς1 ∩ ς2 } for all rq ∈ η = k⋃ p=1 ςip,jp, while ς1 = |η|⋃ q=1   min  rq, max   min ( k⋃ p=1 { max ( ςip,jp )}) , max ( k⋃ p=1 min {( ςip,jp )})       and ς2 = |η|⋃ q=1   max  rq, min   min ( k⋃ p=1 { max ( ςip,jp )}) , max ( k⋃ p=1 { min ( ςip,jp )})       , where round (∗) is usual round operation. 4. Diminishing Hesitant 2-tuple Averaging Operator Beg and Rashid [5] discussed a model which is characterized by a linguistic term and its possible symbolic translations. This model is more suitable for dealing with fuzziness and uncertainty than the 2-tuple linguistic arguments. In this section, we defined an operator for the hesitant 2-tuple linguistic elements to handle the situation, where experts face some hesitation to present its possible linguistic translations. Definition 4.1. If h1 = (li1, ςi1,j1 ) ,h2 = (li2, ςi2,j2 ) , ...,hk = (lik, ςik,jk) are k, 2-tuple hesitant linguistic terms where jp is the cardinality of ςip,jp, then diminishing hesitant 2-tuple averaging operator (DH2TA) is Int. J. Anal. Appl. 17 (1) (2019) 86 defined as DH2TA((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = (li, ςi,j) where i = round ( i1 + i2 + ... + ik k ) and ςi,j = {x|x ∈ ς1 ∩ ς2 } for all rq ∈ η = k⋃ p=1 ςip,jpwe have ς1 = |η|⋃ q=1   min  rq, max   min ( k⋃ p=1 { max ( ςip,jp )}) , max ( k⋃ p=1 { min ( ςip,jp )})       and ς2 = |η|⋃ q=1   max  rq, min   min ( k⋃ p=1 { max ( ςip,jp )}) , max ( k⋃ p=1 { min ( ςip,jp )})       where round (∗) be the round function and |η| be the cardinality of η. Example 4.1. Let h1 = (l2,{−0.3,−0.25,−0.1, 0.0, 0.2}), h2 = (l3,{−0.2,−0.1, 0.1 , 0.2, 0.25}) and h3 = (l3,{0.1, 0.23, 0.3}) be 2-tuple hesitant linguistic terms, then, DH2TA(h1,h2,h3) = (l3,{0.1, 0.2}). Theorem 4.1. Let (li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk) be k 2-tuple hesitant linguistic terms, where j1,j2, ...,jk be the cardinality of ςi1,j1, ςi2,j2, ..., ςik,jk respectively, i1,i2, ..., ik ∈{0, 1, 2, ...,m} If all k 2-tuple hesitant lin- guistic terms are equal i.e. li1 = li2 = ... = lik = li and also ςi1,j1 = ςi2,j2 = ... = ςik,jk = ςi,j then, DH2TA ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = (li, ςi,j) . Proof. As li1 = li2 = ... = lik = li and ςi1,j1 = ςi2,j2 = ... = ςik,jk = ςi,j therefore,( i1 + i2 + ... + ik k ) = ( i + i + ... + i k ) = i. and ςi,j = k⋃ p=1 ςip,jp. (4.1) Let ε1 = min ( k⋃ p=1 { max ( ςip,jp )}) , ε2 = max ( k⋃ p=1 { min ( ςip,jp )}) , r1 = max (ε1,ε2) and r2 = min (ε1,ε2) (4.2) Int. J. Anal. Appl. 17 (1) (2019) 87 from equation 4.1 and 4.2, ς1 = k⋃ p=1 {min (rp,r1)} = ςi,j and ς2 = k⋃ p=1 {max (rp,r2)} = ςi,j =⇒ ςi,j = ς1 ∩ ς2 where rp ∈ ςi,j Hence DH2TA ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = (li, ςi,j) . � Theorem 4.2. Let (li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk) be k, 2-tuple hesitant linguistic terms where jp be the cardinality of ςip,jp for ip = 0, 1, 2, ...,g, jp = 1, 2, ...,n and p = 1, 2, ...,k, then min((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) ≤ DH2TA ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) ≤ max ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) . Proof. As each 2-tuple hesitant linguistic term ( lip, ςip,jp ) consist of two parts such that lip be the hesitant linguistic information by mean of 2-tuple and ςip,jp is a finite subset of [−0.5, 0.5) which represent the possible translations of lip, where jp be the cardinality of ςip,jp and ip ∈{0, 1, 2, ...,g},then Case 1. If all ( lip, ςip,jp ) are different due to different ip then, by definition 4.1 DH2TA((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = (li, ςi,j) i = round ( i1 + i2 + ... + ik k ) clearly k min p=1 (ip) ≤ i ≤ k max p=1 (ip) therefore by definitions 3.2, 3.3,3.4 and 4.1 we have min((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) (4.3) ≤ DH2TA ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) ≤ max ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) Case 2. If ip = i for all ( lip, ςip,jp ) but ςip,jp are different for each p = 1, 2, 3...,k, then, round ( i1 + i2 + ... + ik k ) = round ( i + i + ... + i k ) = i (4.4) Let DH2TA((li1, ςi1,j1 ), (li2, ςi2,j2 ), ..., (lik, ςik,jk)) = (li, ςi,j), min((li1, ςi1,j1 ), (li2, ςi2,j2 ), ..., (lik, ςik,jk)) = (li, ς min i,j ), and max((li1, ςi1,j1 ), (li2, ςi2,j2 ), ..., (lik, ςik,jk)) = (li, ς max i,j ). Consider S (li, ςi,j) , S ( li, ς min i,j ) and S ( li, ς max i,j ) are scores of (li, ςi,j) , ( li, ς min i,j ) and ( li, ς max i,j ) re- spectively. Then, by definition of 3.2 and 4.1, Int. J. Anal. Appl. 17 (1) (2019) 88 Case 2.1. Let max(ς min i,j ) ≤ min(ς max i,j ) =⇒ max(ς min i,j ) ≤ r ≤ min(ς max i,j )∀r ∈ ςi,j =⇒ S ( li, ς min i,j ) ≤ S (li, ςi,j) ≤ S ( li, ς max i,j ) . Therefore,( li, ς min i,j ) ≤ DH2TA((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) ≤ ( li, ς max i,j ) (4.5) Case 2.2. Let max(ς min i,j ) ≥ min(ς max i,j ) =⇒ min(ς max i,j ) ≤ r ≤ max(ς min i,j )∀r ∈ ςi,j =⇒ S ( li, ς min i,j ) ≤ S (li, ςi,j) ≤ S ( li, ς max i,j ) . Therefore,( li, ς min i,j ) ≤ DH2TA((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) ≤ ( li, ς max i,j ) (4.6) Case 2.3. Let min(ς min i,j ) ≤ max(ς max i,j ) ≤ max(ς min i,j ) =⇒ ςi,j = ς max i,j =⇒ S ( li, ς max i,j ) = S (li, ςi,j). Therefore, DH2TA((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = ( li, ς max i,j ) (4.7) Equations 4.3,4.4,4.5,4.6 and 4.7 provide the required result. � Theorem 4.3. If ( lip, ςip,jp ) ≤ ( l ′ i ′ p , ς ′ i ′ p,j ′ p ) for ip, i ′ p ∈{0, 1, 2, ...,g} , jp,j ′ p ∈{1, 2, ...,n} and p = 1, 2, ...,k, then DH2TA ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) ≤ DH2TA (( l ′ i1 , ς ′ i1,j1 ) , ( l ′ i2 , ς ′ i2,j2 ) , ..., ( l ′ ik , ς ′ ik,jk )) Proof. Given that ( lip, ςip,jp ) ≤ ( l ′ ip , ς ′ ip,jp ) for all p = 1, 2, ...,k, Case 1. If, ∀p = 1, 2, ...,k, ip < i ′ p =⇒ (∑k p=1 ip k ) < (∑k p=1 i ′ p k ) =⇒ DH2TA ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) (4.8) < DH2TA (( l ′ i1 , ς ′ i1,j1 ) , ( l ′ i2 , ς ′ i2,j2 ) , ..., ( l ′ ik , ς ′ ik,jk )) Case 2. If ip = i ′ p which implies that (∑k p=1 ip k ) = (∑k p=1 i ′ p k ) = i. Let DH2TA((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = (li, ςi,j) , DH2TA((l ′ i1 , ς ′ i1,j1 ), (l ′ i2 , ς ′ i2,j2 ), ..., (l ′ ik , ς ′ ik,jk )) = (l ′ i, ς ′ i,j), min((li1, ςi1,j1 ), (li2, ςi2,j2 ), ..., (lik, ςik,jk)) = (li, ς min i,j ), min((l ′ i1 , ς ′ i1,j1 ), (l ′ i2 , ς ′ i2,j2 ), ..., (l ′ ik , ς ′ ik,jk )) = (l ′ i, ς ′min i,j ), max((li1, ςi1,j1 ), (li2, ςi2,j2 ), ..., (lik, ςik,jk)) = (li, ς max i,j ) and max((l ′ i1 , ς ′ i1,j1 ), (l ′ i2 , ς ′ i2,j2 ), ..., (l ′ ik , ς ′ ik,jk )) = (l ′ i, ς ′ max i,j ) Int. J. Anal. Appl. 17 (1) (2019) 89 therefore by theorem 4.2, ( li, ς min i,j ) ≤ DH2TA((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) ≤ ( li, ς max i,j ) , (4.9) and ( l ′ i, ς ′min i,j ) ≤ DH2TA (( l ′ i1 , ς ′ i1,j1 ) , ( l ′ i2 , ς ′ i2,j2 ) , ..., ( l ′ ik , ς ′ ik,jk )) ≤ ( l ′ i, ς ′ max i,j ) . (4.10) As, ( li, ς min i,j ) ≤ ( l ′ i, ς ′ min i,j ) , ( li, ς max i,j ) ≤ ( l ′ i, ς ′ max i,j ) and( lip, ςip,jp ) ≤ ( l ′ ip , ς ′ ip,jp ) , therefore S ( li, ς min i,j ) ≤ S ( l ′ i, ς ′ min i,j ) , S ( li, ς max i,j ) ≤ S ( l ′ i, ς ′ max i,j ) and S ( lip, ςip,jp ) ≤ S ( l ′ ip , ς ′ ip,jp ) for each p. Therefore by Theorem 4.2 DH2TA((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) ≤ DH2TA((l ′ i1 , ς ′ i1,j1 ), (l ′ i2 , ς ′ i2,j2 ), ..., (l ′ ik , ς ′ ik,jk )), which is required result. � Theorem 4.4. Let ( li′p , ςi′p,jp ) be a permutation of p hesitant 2-tuples linguistic elements of ( lip, ςip,jp ) ,where ip, i ′ p ∈{0, 1, 2, ...,g} , jp,j ′ p = {1, 2, ...,n} and p = 1, 2, ...,k then, DH2TA ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = DH2TA (( l ′ i1 , ς ′ i1,j1 ) , ( l ′ i2 , ς ′ i2,j2 ) , ..., ( l ′ ik , ς ′ ik,jk )) Proof. Let us consider (σ(1),σ(2), ...,σ(k)) be permutation of (1, 2, ...,k) such that( lip, ςip,jp ) σ(1) ≤ ( lip, ςip,jp ) σ(2) ≤ ... ≤ ( lip, ςip,jp ) σ(k) then, ( lip, ςip,jp ) σ(p) = ( li′p , ςi′p,jp ) σ(p) ∀p = 1, 2, ...,k therefore, (ip)σ(p) = ( i ′ p ) σ(p) =⇒ (∑k p=1 ip k ) = (∑k p=1 i ′ p k ) ∀p = 1, 2, ...,k (4.11) also, S ( (lip, ςip,jp)σ(p) ) = S (( li′p , ςi′p,jp ) σ(p) ) ∀p = 1, 2, ...,k (4.12) from equation 4.11 and 4.12 DH2TA ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = DH2TA (( l ′ i1 , ς ′ i1,j1 ) , ( l ′ i2 , ς ′ i2,j2 ) , ..., ( l ′ ik , ς ′ ik,jk )) . � Int. J. Anal. Appl. 17 (1) (2019) 90 5. Hesitant 2-tuple Linguistic Information Aggregation Operators Based on the Choquet Integral In this section, we develop diminishing Choquet hesitant 2-tuple average operator (DCH2TA) by selecting Choquet integral to find the weights for DWH2TA. We also discussed different properties of DCH2TA. Definition 5.1. Let h1 = (li1, ςi1,j1 ) , h2 = (li2, ςi2,j2 ) , ...,hk = (lik, ςik,jk) be k, 2-tuple hesitant linguistic terms where jp be the cardinality of ςip,jp for any finite natural number p. X be the set of attributes and α be the fuzzy measure on X, then diminishing Choquet hesitant 2-tuple average operator (DCH2TA) is defined as follow: DCH2TAα (h1,h2, ...,hk) (5.1) = DH2TA ({( α ( Hσ(p) ) −α ( Hσ(p−1) )) (li, ςi,j)σ(p) |p = 1, 2, ...,k }) , here (σ(1),σ(2), ...,σ(k)) be the permutation of (1, 2, ...,k) such that (li, ςi,j)σ(1) ≥ (li, ςi,j)σ(2) ≥ ... ≥ (li, ςi,j)σ(k), Xσ(p) is the attribute corresponding to (li, ςi,j)σ(p) and Hσ(p) = { xσ(l)|l ≤ p } for p ≥ 1, Hσ(0) = ∅. Theorem 5.1. Let (li1, ςi1,j1 ) = (li2, ςi2,j2 ) = ... = (lik, ςik,jk) be all equal k, hesitant 2-tuples linguistic elements such that i1 = i2 = ... = ik and ςi1,j1 = ςi2,j2 = ... = ςik,jk. If ip ∈ {0, 1, 2, ...,g} , jp ∈ {1, 2, ...,n} and p = 1, 2, ...,k, X is the set of attributes and α be the fuzzy measure on X, then diminishing Choquet hesitant 2-tuple average operator (DCH2TA) is always DCH2TAα ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = (li, ςi,j) , where (li, ςi,j) = wp ( lip, ςip,jp ) p , wp = α ( Hσ(p) ) −α ( Hσ(p−1) ) = 1 k and (σ(1),σ(2), ...,σ(k)) be the permutation of (1, 2, ...,k). Proof. Given that, ( lip, ςip,jp ) σ(p) = ( lip, ςip,jp ) σ(p−1) ∀p = 1, 2, 3, ...,k, therefore( lip ) σ(p) = ( lip ) σ(p−1) also ( ςip,jp ) σ(p) = ( ςip,jp ) σ(p−1) ∀p = 1, 2, 3, ...,k by definition 3.5 scalar product for wp−1 and wp are A1 =   (µ) σ(p) |(µ) σ(p) = (g + 1) ( 1 − ( 1 − g+(γip,jp)σ(p) g+1 )wp) |∀ ( γip,jp ) σ(p) ∈ ( ςip,jp ) σ(p)   and B1 =   (µ) σ(p−1) |(µ) σ(p−1) = (g + 1) ( 1 − ( 1 − g+(γip,jp)σ(p−1) g+1 )wp−1) |∀ ( γip,jp ) σ(p−1) ∈ ( ςip,jp ) σ(p−1)   Int. J. Anal. Appl. 17 (1) (2019) 91 this implies that, A1 = B1 as wp−1 = wp and ( lip, ςip,jp ) σ(p) = ( lip, ςip,jp ) σ(p−1) ∀p = 1, 2, 3, ...,k. (5.2) Let, A2 = ⋃{ θ1 σ(p) |θ1 σ(p) = round((µ) σ(p) ), ∀(µ) σ(p) ∈ A1 } and B2 = ⋃{ θ1 σ(p−1) |θ1 σ(p−1) = round((µ) σ(p−1) ), ∀(µ) σ(p−1) ∈ B1 } . By equation 5.2 A2 = B2. (5.3) As (ip)σ(p) = (ip)σ(p−1) therefore by equation 5.3,  ∑ θ1 σ(p) ∈A2 θ1 σ(p) + (ip)σ(p)   (|A2| + 1) =   ∑ θ1 σ(p−1) ∈B2 θ1 σ(p−1) + (ip)σ(p−1)   (|B2| + 1) = i and ( ςip,jp ) σ(p) = ( ςip,jp ) σ(p−1) for all p, =⇒ k⋃ p=1 ( ςip,jp ) σ(p) = k⋃ p=1 ( ςip,jp ) σ(p−1) = η1(say) (5.4) where, ( ςip,jp ) σ(p) = { θ2 σ(p) |θ2 σ(p) = different((µ) σ(p) −θ1 σ(p) ) } and ( ςip,jp ) σ(p−1) = { θ2 σ(p−1) |θ2 σ(p−1) = different((µ) σ(p−1) −θ1 σ(p−1) ) } . As ( ςip,jp ) σ(p) = ( ςip,jp ) σ(p−1) for all p, =⇒ max ( ςip,jp ) σ(p) = max ( ςip,jp ) σ(p−1) for all p therefore min ( k⋃ p=1 { max ( ςip,jp ) σ(p) }) = min ( k⋃ p=1 { max ( ςip,jp ) σ(p−1) }) and max ( k⋃ p=1 { min ( ςip,jp ) σ(p) }) = max ( k⋃ p=1 { min ( ςip,jp ) σ(p−1) }) . Int. J. Anal. Appl. 17 (1) (2019) 92 By equation 5.4 let rq ∈ η1 then, (ς1) σ(p) = |η1|⋃ q=1   min  rq, max   min ( k⋃ p=1 { max ( ςip,jp ) σ(p) }) , max ( k⋃ p=1 { min ( ςip,jp ) σ(p) })       (ς1) σ(p−1) = |η1|⋃ q=1   min  rq, max   min ( k⋃ p=1 { max ( ςip,jp ) σ(p−1) }) , max ( k⋃ p=1 { min ( ςip,jp ) σ(p−1) })       and (ς2) σ(p) = |η1|⋃ q=1   max  rq, min   min ( k⋃ p=1 { max ( ςip,jp ) σ(p) }) , max ( k⋃ p=1 { min ( ςip,jp ) σ(p) })       (ς2) σ(p−1) = |η1|⋃ q=1   max  rq, min   min ( k⋃ p=1 { max ( ςip,jp ) σ(p−1) }) , max ( k⋃ p=1 { min ( ςip,jp ) σ(p−1) })       by equations 5.2,5.3 and 5.4 we have, (ς1) σ(p) = (ς1) σ(p−1) and (ς2) σ(p) = (ς2) σ(p−1) therefore, (ς1) σ(p) ∩ (ς2) σ(p) = (ς1) σ(p−1) ∩ (ς2) σ(p−1) = ςi,j. (5.5) Given that wp = α ( Hσ(p) ) −α ( Hσ(p−1) ) = 1 k for all p, therefore by definition 3.5 and equation 5.5 we have, wp ( lip, ςip,jp ) p = wp−1 ( lip, ςip,jp ) p−1 = (li, ςi,j) (say) Therefore by theorem 4.1 we have required result, DCH2TAα ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = (li, ςi,j) � Theorem 5.2. Let ( lip, ςip,jp ) be k, hesitant 2-tuples linguistic elements, if ip ∈ {0, 1, 2, ...,g} , jp ∈ {1, 2, ...,n} and p = 1, 2, ...,k. X be the set of attributes and α be the fuzzy measure on X, then for any g use as upper limit of the linguistic term set, then diminishing Choquet hesitant 2-tuple average operator (DCH2TA) must satisfied, ( lip, min ( ςip,jp )) σ(k) ≤ DCH2TAα (h1,h2, ...,hk) ≤ ( lip, max(ςip,jp) ) σ(1) where (σ(1),σ(2), ...,σ(k)) be the permutation of (1, 2, ...,k) such that (lip, ςip,jp)σ(k) ≤ (lip, ςip,jp)σ(k−1) ≤ ... ≤ (lip, ςip,jp)σ(1). Int. J. Anal. Appl. 17 (1) (2019) 93 Proof. As, (lip, ςip,jp)σ(k) ≤ (lip, ςip,jp)σ(k−1) ≤ ... ≤ (lip, ςip,jp)σ(1) also ( lip, min ( ςip,jp )) σ(k) ≤ ( lip, ςip,jp ) σ(k) and ( lip, ςip,jp ) σ(1) ≤ ( lip, max ( ςip,jp )) σ(1) . Therefore ( lip, min ( ςip,jp )) σ(k) ≤ (lip, ςip,jp)σ(k) ≤ (lip, ςip,jp)σ(k−1) ≤ ... ≤ (lip, ςip,jp)σ(1) ≤ ( lip, max ( ςip,jp )) σ(1) . Because, 0 ≤ α ( Hσ(p) ) −α ( Hσ(p−1) ) ≤ 1 ∀p = 1, 2, 3, ...,k Therefore by definition 3.5 we have, Let τp = α ( Hσ(p) ) −α ( Hσ(p−1) )( lip, ςip,jp ) ∀p = 1, 2, ...,k τp =   µ2|µ2 = (g + 1) ( 1 − ( 1 − g+(γip,jp) g+1 )α(Hσ(p))−α(Hσ(p−1))) ∀ ( γip,jp ) ∈ ( ςip,jp ) where p = 1, 2, ...k   and βp = ⋃ {θ2|θ2 = round(µ2),∀µ2 ∈ τp} =⇒ λp = ( ∑ θ2∈βp θ2 + ip ) |βp| + 1 , |βp| be cardinality of βp. Clearly, (ip)σ(k) ≤ round (∑ λp k ) = i ′ ≤ (ip)σ(1) . (5.6) Let li′ be the linguistic term of DCH2TAα ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) . By definition 5.1 we ob- serve that translation of li′ , ( say ςi′,j′ ) may truncate the extreme values of ( ςip,jp ) σ(k) and ( ςip,jp ) σ(1) i.e. it must satisfied the following condition, min ( (ςip,jp)σ(k) ) ≤ γ ′ ip,jp ≤ max ( (ςip,jp)σ(1) ) ∀γ ′ ip,jp ∈ ςi′,j′ (5.7) therefore, from equation 5.6 and 5.7 we have, ( lip, min ( ςip,jp )) σ(k) ≤ DCH2TAα ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) ≤ ( lip, max(ςip,jp) ) σ(1) � Int. J. Anal. Appl. 17 (1) (2019) 94 Theorem 5.3. Let ( lip, ςip,jp ) ≤ ( l ′ i ′ p , ς ′ i ′ p,j ′ p ) for all p = 1, 2, ...,k if for ip, i ′ p ∈ {0, 1, 2, ...,g} , jp,j ′ p ∈ {1, 2, ...,n} . Let X be the set of attributes and α be the fuzzy measure on X, then, DCH2TAα((li1, ςi1,j1 ), (li2, ςi2,j2 ), ..., (lik, ςik,jk)) ≤ DCH2TAα((l ′ i ′ 1 , ς ′ i ′ 1,j ′ 1 ), (l ′ i2 , ς i ′ 2,j ′ 2 ), ..., (l ′ i ′ k , ς i ′ k ,j ′ k )) Proof. If lip = l ′ i ′ p then order of ( lip, ςip,jp ) and ( l ′ i ′ p , ς ′ i ′ p,j ′ p ) depend on possible translations of lip and l ′ i ′ p . As,( lip, ςip,jp ) σ(p) ≤ ( l ′ i ′ p , ς ′ i ′ p,j ′ p ) σ(p) ∀p = 1, 2, 3, ...,k. Therefore, ∑ γip,jp∈ ( ςip,jp ) σ(p) γip,jp n ≤ ∑ γ ′ i ′ p ,j ′ p ∈ ( ς ′ i ′ p ,j ′ p ) σ(p) γ ′ i ′ p,j ′ p n ∀p = 1, 2, 3, ...,k (5.8) We also know that 0 ≤ ( α ( Hσ(p) ) −α ( Hσ(p−1) )) ≤ 1 ∀p = 1, 2, 3, ...,k also 0 ≤ ( α ( H ′ σ(p) ) −α ( H ′ σ(p−1) )) ≤ 1 ∀p = 1, 2, 3, ...,k Implies that β1 = α ( Hσ(p) ) −α ( Hσ(p−1) )( lip, ςip,jp ) σ(p) =   µ|µ = (g + 1) ( 1 − ( 1 − g+(γip,jp)σ(p) g+1 )α(Hσ(p))−α(Hσ(p−1))) ∀ ( γip,jp ) σ(p) ∈ ( ςip,jp ) σ(p)   Let β2 = ⋃ {θ1|θ1 = round(µ),∀µ ∈ β1} =⇒ (i)σ(p) = ( ∑ θ1∈β2 θ1 + (ip)σ(p) ) |β2| + 1 , |β2| be cardinality of β2 (5.9) β ′ 1 = α ( Hσ(p) ) −α ( Hσ(p−1) )( lip, ςip,jp ) =   µ ′ |µ ′ = (g + 1)  1 −  1 − g+ ( γ ′ i ′ p,j ′ p ) σ(p) g+1   α ( H ′ σ(p) ) −α ( H ′ σ(p−1) )  ∀ ( γ ′ i ′ p,j ′ p ) σ(p) ∈ ( ς ′ i ′ p,j ′ p ) σ(p)   Int. J. Anal. Appl. 17 (1) (2019) 95 Let β ′ 2 = ⋃{ θ ′ 1|θ ′ 1 = round(µ ′ ),∀µ ′ ∈ β ′ 1 } =⇒ ( i ′ ) σ(p) = ( ∑ θ ′ 1∈β ′ 2 θ ′ 1 + ( i ′ p ) σ(p) ) |β′2| + 1 , |β ′ 2| be cardinality of β ′ 2 (5.10) From equations 5.8, 5.9 and 5.10 we have, DCH2TAα (h1,h2, ...,hk) ≤ DCH2TAα ( h ′ 1,h ′ 2, ...,h ′ k ) If ip < i ′ p then obviously from equation 5.9 and 5.10 we have (i)σ(p) ≤ ( i ′ ) σ(p) =⇒ DCH2TAα (h1,h2, ...,hk) ≤ DCH2TAα ( h ′ 1,h ′ 2, ...,h ′ k ) � Theorem 5.4. Let h ′ p = ( l ′ i ′ p , ς ′ i ′ p,jp ) p be a permutation of p hesitant 2-tuples linguistic elements of hp =( lip, ςip,jp ) p . Where ip = 0, 1, 2, ...,g, jp = 1, 2, ...,n and p = 1, 2, ...,k. X be the set of attributes and α be the fuzzy measure on X, then, DCH2TAα (h1,h2, ...,hk) = DCH2TAα ( h ′ 1,h ′ 2, ...,h ′ k ) Proof. Let us consider (σ(1),σ(2), ...,σ(k)) be permutation of (1, 2, ...,k) such that( lip, ςip,jp ) σ(1) ≤ ( lip, ςip,jp ) σ(2) ≤ ... ≤ ( lip, ςip,jp ) σ(k) then, ( lip, ςip,jp ) σ(p) = ( l ′ i ′ p , ς ′ i ′ p,j ′ p ) σ(p) such that ip = i ′ p and ςip,jp = ς ′ i ′ p,j ′ p ∀p = 1, 2, 3, ...,k (5.11) We also know that 0 ≤ ( α ( Hσ(p) ) −α ( Hσ(p−1) )) ≤ 1 ∀p = 1, 2, 3, ...,k Implies that β1 = α ( Hσ(p) ) −α ( Hσ(p−1) )( lip, ςip,jp ) σ(k) =   µ|µ = (g + 1) ( 1 − ( 1 − g+(γip,jp)σ(p) g+1 )α(Hσ(p))−α(Hσ(p−1))) ∀ ( γip,jp ) σ(p) ∈ ( ςip,jp ) σ(p)   . Let β2 = ⋃ {θ1|θ1 = round(µ),∀µ ∈ β1} =⇒ (i)σ(p) = ( ∑ θ1∈β2 θ1 + (ip)σ(k) ) |β2| + 1 , |β2| be cardinality of β2 (5.12) Int. J. Anal. Appl. 17 (1) (2019) 96 β ′ 1 = α ( Hσ(p) ) −α ( Hσ(p−1) )( lip, ςip,jp ) =   µ ′ |µ ′ = (g + 1)  1 −  1 − g+ ( γ ′ i ′ p,j ′ p ) σ(p) g+1   α ( H ′ σ(p) ) −α ( H ′ σ(p−1) )  ∀ ( γ ′ i ′ p,j ′ p ) σ(p) ∈ ( ς ′ i ′ p,j ′ p ) σ(p)   . Let β ′ 2 = ⋃{ θ ′ 1|θ ′ 1 = round(µ ′ ),∀µ ′ ∈ β ′ 1 } =⇒ ( i ′ ) σ(p) = ( ∑ θ ′ 1∈β ′ 2 θ ′ 1 + ( i ′ p ) σ(p) ) |β′2| + 1 , |β ′ 2| be cardinality of β ′ 2. (5.13) From equations 5.11, 5.12 and 5.13 β1 = β ′ 1 and β2 = β ′ 2∀p = 1, 2, 3, ...,k =⇒ DCH2TAα (h1,h2, ...,hk) = CH2TNα ( h ′ 1,h ′ 2, ...,h ′ k ) which is required proof. � 6. An Application of DCH2TA Operators to Multiple Attribute Decision Making In this section DCH2TA operator is applied to multiple attribute decision making problems based on hesitant 2-tuple linguistic information. Firstly, we developed a decision making method for utilization of DCH2TA operator. Let D = {D1,D2, ...,Dr} be the set of ”r” decision makers, X = {x1,x2, ...,xm} be the set of alternatives and Y = {y1,y2, ...,yn} be the set of attributes. Step 1. The decision makers developed the decision matrices Mp = [( l p ijk, ς p )] m×n , where ( l p ijk, ς p ) be the hesitant evaluation of the alternatives xi determined by the decision makers Dp based on attributes yj, where i = 1, 2, ...,m, j = 1, 2, ...,n, and p = 1, 2, ..,r, where ς p ⊂ [−0.5, 0.5) and k ∈{0, 1, 2, ...g}. Step 2. Find the matrix Magg = [ DH2TA ( l p ijk, ς p )] m×n , where, DH2TA ( l p ijk, ς p ) is an aggregate value of ( l p ijk, ς p ) (i = 1, 2, ...,m,j = 1, 2, ...,n) for all decision maker’s evaluation as follow: DH2TA ( (l1ijk1, ς 1), (l2ijk2, ς 2), ..., (lnijkn, ς n) ) = (lk, ςk) where k = round ( k1 + k2 + ... + kn n ) and ςk = {x|x ∈ ς1 ∩ ς2 } for all rq ∈ η = n⋃ j=1 ςjwe have Int. J. Anal. Appl. 17 (1) (2019) 97 ς1 = |η|⋃ q=1   min  rq, max   min ( n⋃ j=1 {max (ςp)} ) , max ( n⋃ j=1 min{(ςp)} )       and ς2 = |η|⋃ q=1   max  rq, min   min ( n⋃ j=1 {max (ςp)} ) , max ( j⋃ j=1 {min (ςp)} )       where round (∗) be the round function and |η| be the cardinality of η. Step 3. Confirm the fuzzy measures of attributes sets of B. We use the DCH2TA operator define in definition 5.1 to aggregate the values to find overall values (l, ς)i (i = 1, 2, ...,m) of alternatives Ai. (l, ς)i = DCH2TAα ((li1, ςi1), (li2, ςi2), ..., (lin, ςin)) = DH2TA ( wi1(liσ(1), ςiσ(1)),wi2(liσ(2), ςiσ(2)), ...,win(liσ(n), ςiσ(n)) ) where (σ(1),σ(2), ...,σ(n)) be the permutation of (1, 2, ...,n) such that (liσ(1), ςiσ(1)) ≥ (liσ(2), ςiσ(2)) ≥ ... ≥ (liσ(n), ςiσ(n)) and wij = α(Hiσ(j)) −α(Hiσ(j−1)) is the set of attributes corresponding to (liσ(1), ςiσ(1)),(liσ(2), ςiσ(2)),..., (liσ(n), ςiσ(n)). Step 4. Rank these aggregate values (l, ς)i (i = 1, 2, ...,m) in descending order according to the rule in definition 3.2 and select (l, ς)i with the largest value. 7. Illustrative example In order to demonstrate the significance of our newly proposed method, we consider an example where Mr. Robert, a food chain owner, wants to hire a supplier for raw food material for his chain. To save hedge risks, a three member committee (decision makers), D = {D1,D2,D3} has been created to select the most suitable supplier. Decision makers short listed five potential suppliers after initial analysis for supplier’s capabilities. Let S = {s1, s2, s3, s4, s5} be the set of short listed suppliers. During the supplier selection process, decision maker decide to consider the following set of attributes for judgments Y = {Y1(price), Y2(quality), Y3(delivery time), Y4(financial status of the company)}. In numerous practical group decision making problems in supply chain management, the contractor se- lection or determination of an accomplice for an endeavor in the field of production network administration, military framework effectiveness assessment, etc. decision makers normally need to give their preferences over alternatives. As preference information given by decision makers is normally imprecise. It might be due Int. J. Anal. Appl. 17 (1) (2019) 98 to hesitations, uncertainty or vagueness about preferences as a decision should be made under time pressure and lack of information or knowledge processing capacities especially when financial condition turns out to be more complex. The best choice for decision maker is to handle data in hesitant 2-tuple elements due to effectiveness of them in these particular situations. Consider that decision makers evaluate the alternatives with respect to the attributes in 2-tuple linguistic arguments to form decision matrices Mp where p = {1, 2, 3}. Step 1. Develop decision matrices Mp = [( l p ijk, ς p )] 5×4 , ςp ⊂ [−0.5, 0.5) M1 =   (M,{−0.3, 0.0, 0.2}) (G,{0.45, 0.32, 0.2}) (P,{0.2, 0.3}) (P,{−0.3, 0.1}) (P,{0.0, 0.2, 0.1}) (M,{−0.48,−0.2, 0.0}) (M,{−0.45, 0.1}) (G,{−0.2, 0.1, 0.2}) (G,{−0.3, 0.1, 0.2}) (M,{−0.0, 0.2}) (V G,{−0.2.0.0, 0.4}) (P,{−0.3, 0.1, 0.2}) (V G,{−0.1, 0.0, 0.2}) (P,{0.0, 0.2, 0.4}) (P,{−0.5,−0.3}) (M,{−0.45,−0.25}) (EG,{−0.4,−0.3, 0.1}) (P,{−0.1, 0.2, 0.3}) (V P,{−0.45,−0.2}) (G,{−0.4,−0.1, 0.0})   M2 =   (P,{−0.3,−0.1}) (V G,{−0.1, 0.0, 0.1}) (V P,{−0.2, 0.3}) (M,{0.1, 0.2, 0.4}) (V P,{0.4}) (P,{0.2, 0.3}) (G,{0.3, 0.4}) (V G,{−0.1,−0.45,−0.2}) (M,{0.0, 0.3}) (P,{−0.1, 0.2}) (G,{0.1, 0.3}) (V P,{−0.3,−0.2, 0.0}) (EG,{0.2, 0.4}) (M,{−0.4, 0.3}) (P,{0.2, 0.4}) (G,{0.1, 0.3, 0.4}) (G,{−0.2, 0.1}) (M,{−0.2, 0.15}) (P,{−0.1, 0.2}) (V G,{−0.1, 0.3})   M3 =   (G,{−0.5, 0.1, 0.2}) (V G,{0.2, 0.3}) (M,{0.1, 0.2}) (V P,{0.0, 0.1, 0.2}) (M,{−0.4,−0.1}) (P,{0.0, 0.2, 0.4}) (V G,{−0.3,−0.2}) (M,{−0.2,−0.1, 0.0}) (P,{−0.2, 0.0, 0.1}) (V G,{−0.05, 0.2}) (G,{0.0, 0.1, 0.25}) (V P,{−0.3,−0.2, 0.0}) (G,{−0.3,−0.1, 0.0}) (G,{0.0, 0.25, 0.45}) (P,{0.1, 0.2, 0.3}) (M,{−0.1, 0.2, 0.3}) (M,{−0.1, 0.1, 0.3}) (P,{−0.2,−0.1, 0.0}) (M,{0.1, 0.4, 0.45}) (EG,{−0.05, 0.25})   Step 2 Use the DH2TA operator to aggregate value of ( l p ijk, ς p ) (i = 1, 2, 3, 4, 5, j = 1, 2, 3, 4, p = 1, 2, 3 and k ∈{0, 1, 2, ..., 6}) for all decision maker’s evaluation as follow: Magg =  (M,{−0.3,−0.1}) (V G,{−0.1, 0.1, 0.2}) (P,{0.2}) (P,{0.1}) (P,{−0.1, 0.0, 0.1, 0.2, 0.4}) (P,{0.0, 0.2}) (G,{−0.2, 0.1, 0.3}) (G,{−0.2,−0.1}) (M,{0.1}) (M,{−0.05, 0.2}) (G,{0.1, 0.25}) (V P,{−0.3,−0.2, 0.0}) (V G,{0.0, 0.2}) (M,{0.0, 0.2, 0.25, 0.3}) (P,{−0.3, 0.1, 0.2}) (M,{−0.25,−0.1, 0.1}) (G,{−0.1, 0.1}) (P,{−0.1, 0.0) (P,{−0.2,−0.1, 0.1}) (V G,{−0.05, 0.0})   Int. J. Anal. Appl. 17 (1) (2019) 99 Step 3. To find the fuzzy measures for attributes of Y = {Y1(Price), Y2(Quality), Y3(Delivery time), Y4(Financial status of company)} and parameter λ. Let α(Y1) = 0.3, α(Y2) = 0.25 ,α(Y3) = 0.15 and α(Y4) = 0.29. Then by equation 2.3, λ = 0.00277 and by equation 2.2, α(Y1,Y2) = 0.5502, α(Y1,Y3) = 0.4501, α(Y1,Y4) = 0.5902, α(Y2,Y3) = 0.4001, α(Y2,Y4) = 0.5402, α(Y3,Y4) = 0.4401, α(Y1,Y2,Y3) = 0.7004, α(Y1,Y2,Y4) = 0.8406, α(Y1,Y3,Y4) = 0.7405, α(Y2,Y3,Y4) = 0.6904, α(Y1,Y2,Y3,Y4) = 1. To find DCH2TA aggregate value for the following elements, firstly we use wij = α(Hiσ(j))−α(Hiσ(j−1)) weight for each element. (l1σ(1), ς1σ(1)) = (V G,{−0.1, 0.1, 0.2}) (l1σ(2), ς1σ(2)) = (M,{−0.3,−0.1}) (l1σ(3), ς1σ(3)) = (P,{0.2}) (l1σ(4), ς1σ(4)) = (P,{0.1}) As H1σ(1) = {Y2}, H1σ(2) = {Y1,Y2} and H1σ(3) = {Y1,Y2,Y3}, H1σ(4) = {Y1,Y2,Y3,Y4} we can get w11 = 0.25, w12 = 0.3002 and w13 = 0.1502, w14 = 0.2996. (l, ς)1 = DCH2TAα ( (l1σ(1), ς1σ(1)), (l1σ(2), ς1σ(2)), (l1σ(3), ς1σ(3)), (l1σ(4), ς1σ(4)) ) = (M,{−0.0700,−0.0537,−0.0163, 0.2138}) S((l, ς)1) = 0.0184 Similarly, find the values of (l, ς)2, (l, ς)3, (l, ς)4 and (l, ς)5 are (l, ς)2 = DCH2TAα ( (l1σ(1), ς1σ(1)), (l1σ(2), ς1σ(2)), (l1σ(3), ς1σ(3)), (l1σ(4), ς1σ(4)) ) (l, ς)2 = (M,{−0.1967,−0.146,−0.0921}) S ((l, ς)2) = −0.1449 (l, ς)3 = DCH2TAα ( (l1σ(1), ς1σ(1)), (l1σ(2), ς1σ(2)), (l1σ(3), ς1σ(3)), (l1σ(4), ς1σ(4)) ) (l, ς)3 = (M,{−0.0674,−0.0072, 0.0924, 0.2177}) S ((l, ς)3) = 0.0589 (l, ς)4 = DCH2TAα ( (l1σ(1), ς1σ(1)), (l1σ(2), ς1σ(2)), (l1σ(3), ς1σ(3)), (l1σ(4), ς1σ(4)) ) (l, ς)4 = (M,{0.0462, 0.0654, 0.0955}) and S ((l, ς)4) = 0.0690 Int. J. Anal. Appl. 17 (1) (2019) 100 and (l, ς)5 = DCH2TAα ( (l1σ(1), ς1σ(1)), (l1σ(2), ς1σ(2)), (l1σ(3), ς1σ(3)), (l1σ(4), ς1σ(4)) ) (l, ς)5 = (M,{−0.3002,−0.2846,−0.2118,−0.0477,−0.0380,−0.0163}) S ((l, ς)5) = −0.1498 As by definition 3.2 (l, ς)4 > (l, ς)3 > (l, ς)1 > (l, ς)2 > (l, ς)5, hence s4 � s3 � s1 � s2 � s5. Therefore, the most suitable supplier’s option is s4, second, third and four position suppliers are s3,s1 and s2 respectively, while the worst suppliers option is s5 . 8. Discussion and conclusion Herrera and Mart́ınez [22], discussed a symbolic model and name it 2-tuple linguistic representation model. The 2-tuple linguistic model use words toward processing without loss of any information. In their proposed 2-tuple model, the linguistic term sets were consistent and symmetrically distributed. In view of the Herrera and Mart́ınez [22], the following models have been considered afterward: • Wang and Hao model [59], • Herrera et al. model [24], • Numerical scale model [15]. In each of these models, linguistic term sets examined consistently and symmetrically scattered. Moreover, the symbolic proportions over linguistic terms are precise qualities, and just a single linguistic term set is considered for translation of these qualities. But these models did not address where hesitation occurs between the translation of arguments. Beg and Rashid [5] introduced the concept of hesitant 2-tuple linguistic model to merge Herrera and Mart́ınez’s [22], 2-tuple linguistic model with Torra’s [55], hesitant fuzzy set. Hesitant 2-tuple linguistic model is very helpful for the situation where decision maker may hesitant to pick a possible value of translation for a linguistic term as it will not cause any loss of information in the process. Beg and Rashid [5] used hesitant 2-tuple linguistic model for the situation where the attributes in the decision making problem are evaluated by hesitant 2-tuple linguistic arguments and they used TOPSIS technique to illustrate hesitant 2-tuple linguistic model’s efficiency and feasibility in real-world decision making applications. As TOPSIS technique use maximum and minimum distance or similarity from all terms provide the best option accordingly. Some time the resultant value did not reflect the true picture and fail to find the best result over the argument. Particularly, where we have an interrelation between the arguments. Choquet integral [11] is the best choice where interrelationship is required. In today globalization era, choice of a suitable supplier for the business in the sense of supply chain management has become a key strategic consideration. But due to natural human hesitation, incomplete supplier information and performances and market uncertainty, a supplier selection process has become more Int. J. Anal. Appl. 17 (1) (2019) 101 complicated. Due to this, it is difficult for decision makers to express their conclusion on the suppliers with exact and crisp values and the evaluations are often expressed in linguistic terms. In such circumstances fuzzy set theory is a very appropriate tool to deal with this kind of problems. In this paper, we have observed a situation that the attributes within the selection for decision making problems are interactive or interdependent and analyze the values in the form of 2 tuple hesitant linguistic arguments. By utilizing the Choquet integral, we have developed DH2TA and DCH2TA aggregation operators. The properties of both operators are studied, such as commutativity, boundedness and monotonicity. We proved that DH2TA operator is an idempotent operator. We also utilized DCH2TA operator to the more than one attribute group decision making problems for hesitant 2-tuple linguistic understanding and proposed a method for group decision making problems. An illustrative example has been given to demonstrate the proposed decision making approach. We observe that DCH2TA is suitable for conditions where decision making problems are interdependent. The operator DCH2TA has the properties to reduce hesitation in aggregated value of hesitant 2-tuple linguistic elements. In real decision making problem, there involve the interrelationships between the arguments. Often Bonfeeroni mean operators (BM) [7] and Muirhead mean operators (MM) [46] used as the tools where interrelationships between arguments exist. As we observed that diminishing operational laws have the ability to reduce hesitation in resultant argument. 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