A Novel Weighted Averaging Operator of Linguistic Interval-Valued Intuitionistic Fuzzy Numbers for Cognitively Inspired Decision-Making

An aggregation operator of linguistic interval-valued intuitionistic fuzzy numbers (LIVIFNs) is an important tool for solving cognitively inspired decision-making problems with LIVIFNs. So far, many aggregation operators of LIVIFNs have been presented. Each of these operators works well in its specific context. But they are not always monotone because their operational rules are not always invariant and persistent. Dempster-Shafer evidence theory, a general framework for modelling epistemic uncertainty, was found to provide the capability for operational rules of fuzzy numbers to overcome these limitations. In this paper, a weighted averaging operator of LIVIFNs based on Dempster-Shafer evidence theory for cognitively inspired decision-making is proposed. Firstly, Dempster-Shafer evidence theory is introduced into linguistic interval-valued intuitionistic fuzzy environment and a definition of LIVIFNs under this theory is given. Based on this, four novel operational rules of LIVIFNs are developed and proved to be always invariant and persistent. Using the developed operational rules, a new weighted averaging operator of LIVIFNs is constructed and proved to be always monotone. Based on the constructed operator, a method for solving cognitively inspired decision-making problems with LIVIFNs is presented. The application of the presented method is illustrated via a numerical example. The effectiveness and advantage of the method are demonstrated via quantitative comparisons with several existing methods. For the numerical example, the best alternative determined by the presented method is exactly the same as that determined by other comparison methods. For some specific problems, only the presented method can generate intuitive ranking results. The demonstration results suggest that the presented method is effective in solving cognitively inspired decision-making problems with LIVIFNs. Furthermore, the method will not produce counterintuitive ranking results since its operational rules are always invariant and persistent and its aggregation operator is always monotone.


List of Acronyms
The i-th (i ∈ {1, 2, ..., m}) alternative in a cognitively inspired DM problem a The LIVIFN under DSET a i The LIVIFN i under DSET or summary value of a ij with respect to j (j ∈ {1, 2, ..., n}) a ij The summary value of a kij with respect to k (k ∈ {1, 2, ..., n � }) a k The LIVIFN k under DSET The LIVIFN under DSET of C j (j ∈ {1, 2, ..., n}) of A i (i ∈ {1, 2, ..., m}) evaluated by E k (k ∈ {1, 2, ..., n � }) a + The LIVIFN + under DSET a − The LIVIFN − under DSET BF f ( ) A belief function with respect to f BF L  The lower belief function in a linguistic lower belief interval BF U  The upper belief function in a linguistic upper belief interval BI f ( ) A belief interval with respect to f BI L  A linguistic lower belief interval BI U  A linguistic upper belief interval b i The LIVIFN i under DSET C j The j-th (j ∈ {1, 2, ..., n}) criterion in a cognitively inspired DM problem E k The k-th (k ∈ {1, 2, ..., n � }) expert in a cognitively inspired DM problem f( ) A basic probability assignment H i The i-th (i ∈ {1, 2, ..., n}) hypothesis H j The j-th (j ∈ {1, 2, ..., n}) hypothesis h The maximum index value of all linguistic terms in a finite linguistic term set LIVIFWA( ) The LIVIFWA operator ( ) The LIVIFWA operator under DSET M k The k-th (k ∈ {1, 2, ..., n � }) decision matrix N k The k-th (k ∈ {1, 2, ..., n � }) normalised decision matrix P( ) The probability PF f ( ) A plausibility function with respect to f PF L  The lower plausibility function in a linguistic lower belief interval PF U  The upper plausibility function in a linguistic upper belief interval p L  The lower bound of the linguistic membership degree of an LIVIFN in N k (k ∈ {1, 2, ..., n � }) p U  The upper bound of the linguistic membership degree of an LIVIFN in N k (k ∈ {1, 2, ..., n � }) q L  The lower bound of the linguistic nonmembership degree of an LIVIFN in N k (k ∈ {1, 2, ..., n � }) q U  The upper bound of the linguistic nonmembership degree of an LIVIFN in N k (k ∈ {1, 2, ..., n � }) SV h ( ) The score value of an LIVIFN with respect to h h ( ) The score value of an LIVIFN under DSET with respect to h 2 The power set of The weight of i or a i (i ∈ {1, 2, ..., n}) w j The weight of C j (j ∈ {1, 2, ..., n}) w ′

Introduction
Cognitively inspired decision-making (DM) is a cognitive process of selecting the best alternative from a certain number of alternatives based on summary values of one or multiple criteria of all alternatives, in which the values of criteria are evaluated by one or a group of domain experts [1][2][3].There are two critical steps in this process.The first is to quantify the evaluation results from domain experts, while the second is to generate a ranking of all alternatives via comprehensively considering all criteria.
For the quantification of evaluation results, the main challenge is to pinpoint cognition and judgments that are close to the human brain to improve DM quality.Cognitive computation, a computing system that imitates human thought processes, has been presented to achieve computing that functions like the human brain [4].In the era of big data, a variety of technological advancements is used in cognitive computation to handle the enormous volume of data with complicated structures [5,6].Since human thoughts are generally complex and vague, it is difficult to describe the data in form of crisp values.To effectively represent human cognitive process in DM, researchers proposed the use of fuzzy sets [7].So far, over thirty different types of fuzzy sets have been presented [8].Representative examples are fuzzy set [9], intuitionistic fuzzy set [10], interval-valued intuitionistic fuzzy set [11], linguistic intuitionistic fuzzy set [12], Pythagorean fuzzy set [13], generalised orthopair fuzzy set [14], and linguistic interval-valued intuitionistic fuzzy set (LIVIFS) [15,16].Using these fuzzy sets, many methods for solving cognitively inspired DM problems [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] have been proposed within academia.
LIVIFS, which was presented on the basis of intervalvalued intuitionistic fuzzy set, linguistic term set [35,36], and linguistic intuitionistic fuzzy set, is one of the most important types of fuzzy sets for quantifying evaluation results in cognitively inspired DM.An LIVIFS can be defined by an element and a membership degree and a non-membership degree of the element to the LIVIFS, where each degree is denoted by an interval of two linguistic terms.A pair composed of a membership degree and a non-membership degree is usually called a linguistic interval-valued intuitionistic fuzzy number (LIVIFN).Through such definition, an LIVIFS can effectively reflect the characteristics of human cognitive performance including acceptance, rejection, and hesitation.Compared with fuzzy set, intuitionistic fuzzy set, interval-valued intuitionistic fuzzy set, linguistic intuitionistic fuzzy set, Pythagorean fuzzy set, and generalised orthopair fuzzy set, LIVIFS provides stronger expressive capability and is more flexible for domain experts, since it allows them to give evaluation results using two intervals of linguistic terms (i.e.LIV-IFNs).Because of these features, application of LIVIFS to express the evaluation results in cognitively inspired DM [15,16,27,33,34,[37][38][39][40][41][42][43] has received extensive attention and is still gaining importance and popularity.
For the generation of a ranking, there are usually two approaches.The first is to use traditional DM methods, such as analytic hierarchy process, TOPSIS method, ELECTRE method, PROMETHEE method, MABAC method, and MOORA method.The second is to adopt aggregation operators (AOs), such as weighted averaging (WA) operator, Heronian mean operator, Bonferroni mean operator, Maclaurin symmetric mean operator, and Muirhead mean operator.In general, an AO has better traceability when solving cognitively inspired DM problems than a traditional DM method, since it can produce summary values of multiple criteria and a ranking of all alternatives, while a traditional DM method can only generate a ranking [44].To date, there have been many AOs of LIVIFNs for cognitively inspired DM.Representative examples include: a prioritised weighted averaging operator, a prioritised weighted geometric operator, a prioritised ordered weighted averaging operator, and a prioritised ordered weighted geometric operator presented by [27]; a WA operator, a weighted geometric operator, an ordered weighted averaging operator, an ordered weighted geometric operator, a hybrid average operator, and a hybrid geometric operator presented by [16]; a weighted Maclaurin symmetric mean operator presented by [37]; an Archimedean power weighted Muirhead mean operator presented by [33]; an Archimedean prioritised 'and' operator and an Archimedean prioritised 'or' operator presented by [34]; a neutrosophic Dombi hybrid weighted geometric operator presented by [42]; a partitioned weighted Hamy mean operator presented by [41]; a copula weighted Heronian mean operator presented by [43]; a Hamacher weighted averaging operator and a Hamacher weighted geometric operator presented by [40].
Most of the operators above use the operational rules (ORs) of LIVIFNs based on algebraic t-norm and t-conorm to perform their operations, while the remaining operators adopt the ORs based on other types of Archimedean t-norm and t-conorm.The ORs based on Archimedean t-norm and t-conorm make the operators general and flexible, but also make them produce counterintuitive ranking results for some cognitively inspired DM problems with LIVIFNs, because they are not always invariant and persistent and the AOs based on them are not always monotone.For example, assume a decision maker needs to select a proper additive manufacturing machine from two alternative machines M 1 and M 2 to build a part with certain material on the basis of the following conditions: the selection criteria include predicted strength ( C 1 ) and predicted hardness ( C 2 ) of the as-built part; the weight of C 1 ( w 1 ) and the weight of C 2 ( w 2 ) are respectively given as w 1 = 0.4 and w 2 = 0.6 ; the value of C 1 of M 1 ( 11 ), the value of C 2 of M 1 ( 12 ), the value of C 1 of M 2 ( 21 ), and the value of C 2 of M 2 ( 22 ) are all given by LIVIFNs under nine linguistic terms extremely small (0), very small (1), small (2), slightly small (3), medium (4), slightly large (5), large (6), very large (7), and extremely large (8): 11 = ( [2,4], [1,2]) ; 21 = ( [3,4], [1,3]) [1,3]) .According to these conditions, it is quite intuitive for the decision maker to choose M 2 to build the part, because 11 is less than 21 according to the rules for comparing two LIVIFNs used in the operators above.However, a counterintuitive result " M 1 is better than M 2 " will be obtained if any of the operators above (denoted as o) is used to solve this problem, because o( 11 , 12 ) is greater than o( 21 , 22 ) is obtained after using the operator and compari- son rules (The reason will be explained in detail in the second comparison in Sect.'Comparisons with Existing Methods').
Based on the analysis above, the motivations of this paper are threefold: 1.To overcome the limitation that the existing ORs of LIV-IFNs are not always invariant and persistent, Dempster-Shafer evidence theory (DSET) [45,46] is introduced to develop novel ORs of LIVIFNs.DSET, also known as theory of belief functions, is a general framework for modelling epistemic uncertainty.In this framework, there are four fundamental components: a basic probability assignment that describes the occurrence rate of criteria in basic events; a belief function that expresses the belief of a focal element; a plausibility function that expresses the uncertainty of a focal element; a belief interval that consists of a belief function and a plausibility function.As demonstrated in [47][48][49][50][51], fuzzy numbers can be converted into belief intervals without loss of information, and the ORs of fuzzy numbers under DSET are always invariant and persistent [52][53][54][55][56][57][58].Because of these characteristics, the developed novel ORs of LIV-IFNs are always invariant and persistent; 2. To address the issue that the existing AOs of LIVIFNs are not always monotone and could generate counterintuitive ranking results, the developed ORs are applied to construct a new WA operator of LIVIFNs.Benefiting from the advantages of the ORs, the constructed AO is always monotone and therefore will not produce counterintuitive ranking results.For example, an intuitive result " M 2 is better than M 1 " will be obtained if the constructed AO is used to solve the problem above; 3. To solve cognitively inspired DM problems with LIV-IFNs, a DM method based on the constructed AO is presented.This method has the advantages of the developed ORs and constructed AO.
The novelties of the paper lie in the following aspects:

Brief Introduction of LIVIFS
An LIVIFS needs to be defined on a continuous linguistic term set, which is defined below: Definition 1 [35,36] Let h be some fixed natural number.Then {0, 1, ..., h} is called a finite linguistic term set (which consists of h + 1 linguistic terms denoted by 0, 1, ..., h) and {i ∈ ℝ | 0 ≤ i ≤ h} is called a continuous linguistic term set (if i ∈ {0, 1, ..., h} , then i is called an original linguistic term; otherwise, i is called a virtual linguistic term).
For the sake of clarity and convenience of description, the following conventions will be adopted in the whole paper: Every h has the same meaning, that is, the maximum index value of all linguistic terms in a finite linguistic term set {0, 1, ..., h} ; All LIVIFSs and all LIV- IFNs are defined on a continuous linguistic term set A formal definition of LIVIFS is given below: Definition 2 [16] An LIVIFS A over a finite universal set , and U (x) + U (x) ≤ h for any x ∈ X .L (x), U (x) is the linguistic membership degree of x to A. L (x), U (x) is the linguistic non-membership degree of x to A.
In an LIVIFS A, for some x in a finite universal set X, L (x), U (x) , L (x), U (x) is called an LIVIFN.An LIVIFN is generally denoted as = L , U , L , U .To compare two LIVIFNs, the score and accuracy values of them are required.Two functions for respectively calculating these values are defined as follows: Definition 3 [16] Let be an arbitrary LIVIFN.The score value of with respect to h and the accuracy value of are respectively calculated by the following two equations: Based on the score and accuracy functions, rules for comparing two LIVIFNs are defined below: Definition 4 [16] Let 1 and 2 be two arbitrary LIVIFNs.The following notations are used to express five relationships between two LIVIFNs: 1 2 represents 1 is less than 2 ; 1 2 represents 1 is greater than 2 ; 1 2 represents 1 is equal to 2 ( is an ordering, i.e. it is reflexive, symmetric, and transitive); 1 2 represents 1 is less than or equal to 2 ; 1 2 represents 1 is greater than or equal to 2 .The comparison rules are: (1)

Existing ORs of LIVIFNs
Operations related to LIVIFNs can be performed using certain ORs of LIVIFNs.There are currently several sets of ORs of LIVIFNs.The most used one is based on algebraic t-norm and t-conorm, which is defined as follows: Definition 5 [16] Let , 1 , and 2 be three arbitrary LIV-IFNs and be an arbitrary positive number.The following notations are used to express two operations between 1 and 2 and two operations between and :  1 ⊕  2 represents 1 plus 2 ;  1 ⊗  2 represents 1 times 2 ; represents times ; represents the power of .These operations can be performed using the following rules: It is worth noting that will be equal to ([0, 0], [h, h]) and will be equal to ([h, h], [0, 0]) if = 0 .The four operations in the ORs above satisfy the following algebraic laws [16]: (3) where 1 and 2 are two arbitrary positive numbers.Further, there is yet no evidence that the four operations satisfy other algebraic laws, such as associativity of ⊕ : , and distributive law of , where 1 , 2 , 3 , and are four arbitrary LIVIFNs.

Existing WA Operator of LIVIFNs
An AO is a function for grouping together two or more values to achieve a summary value.The most common AO for solving cognitively inspired DM problems is the WA operator.A WA operator of LIVIFNs based on the ORs of LIVIFNs in Definition 5 is defined below: Definition 6 [16] Let i (i ∈ {1, 2, ..., n}) be n arbitrary LIV- IFNs, and w i be the weight of i such that 0 ≤ w i ≤ 1 and Σ n i=1 w i = 1 .The aggregation function is called the linguistic interval-valued intuitionistic fuzzy weighted averaging (LIVIFWA) operator.

Limitations of Existing ORs of LIVIFNs
The ORs of LIVIFNs in Eqs. ( 3) and ( 5) are found to generate counterintuitive ranking results for some cognitively inspired DM problems with LIVIFNs due to the following limitations: 1.The operation in the OR of LIVIFNs in Eq. ( 3) is not always invariant with respect to the score function in Eq. ( 1), the accuracy function in Eq. ( 2), and the comparison rules in Definition 4: For three arbitrary LIVIFNs 1 , 2 , and 3 , 1 2 cannot always imply The operation in the OR of LIVIFNs in Eq. ( 5) is not always persistent with respect to the score function in Eq. ( 1), the accuracy function in Eq. ( 2), and the comparison rules in Definition 4: For two arbitrary LIVIFNs 1 and 2 and an arbitrary positive number , 1 2 cannot always imply 1 2 .

Limitation of Existing WA Operator of LIVIFNs
The LIVIFWA operator in Eq. ( 13) is found to produce counterintuitive ranking results for some cognitively inspired DM problems with LIVIFNs because of the following limitation: The LIVIFWA operator in Eq. ( 13) is not always monotone with respect to the score function in Eq. ( 1), the accuracy function in Eq. ( 2), and the comparison rules in Definition 4: For three arbitrary LIVIFNs 1 , 2 , and 3 and certain weights w 1 and w 2 , 1 2 cannot always imply LIVIFWA( 1 , 3 ) LIVIFWA( 2 , 3 ).
A numerical example for illustrating the limitation above is given as follows:

LIVIFS Based on DSET
Five fundamental concepts in DSET are frame of discernment, basic probability assignment, belief function, plausibility function, and belief interval, which are respectively defined as follows: Definition 7 [45,46]  P(H i ∩ H j ) = 0 for any i, j ∈ {1, 2, ..., n} and i ≠ j ) and the probability of at least one hypothesis in being true is one (i.e.P( , then is called a frame of discernment.
Definition 8 [45,46] Let be a frame of discernment and 2 be the power set of .A basic probability assignment over is a mapping Definition 9 [45,46] Let be a frame of discernment, f be a basic probability assignment over and f () > 0 , and H be an element of the power set of (i.e. Definition 10 [45,46] Let be a frame of discernment, f be a basic probability assignment over , and H be an element of the power set of (i.e.H ∈ 2 ).A plausibility function of H with respect to Definition 11 [45,46] Let be a frame of discernment, f be a basic probability assignment over , H be an element of the power set of (i.e.H ∈ 2 ), BF f (H) be a belief function of H with respect to f, and PF f (H) be a plausibility function of H with respect to f.A belief interval over H with respect to f (denoted as BI f (H) ) is an interval whose lower bound is BF f (H) and upper bound is Based on the definitions above and the definition of interval-valued intuitionistic fuzzy set under DSET [51], the definition of LIVIFS in Definition 2 can be rewritten below: is called the linguistic lower belief interval of x to , and To compare two LIVIFNs under DSET, the score and accuracy values of them are needed.Two functions for respectively calculating these values are defined as follows: Definition 13 Let be an arbitrary LIVIFN and a be under DSET.The score and accuracy values of a with respect to h can be respectively calculated by the following two equations: (14) Based on the score and accuracy functions, rules for comparing two LIVIFNs under DSET are defined below: Definition 14 Let 1 and 2 be two arbitrary LIVIFNs and a 1 and a 2 be respectively 1 and 2 under DSET.The following notations are used to express five relationships between two LIVIFNs under DSET: a 1 a 2 represents a 1 is less than a 2 ; a 1 a 2 represents a 1 is greater than a 2 ; a 1 a 2 represents a 1 is equal to a 2 ; a 1 a 2 represents a 1 is less than or equal to a 2 ; a 1 a 2 represents a 1 is greater than or equal to a 2 .The comparison rules are:

ORs of LIVIFNs Based on DSET
To perform the operations related to LIVIFNs under DSET, a set of novel ORs of LIVIFNs under DSET is developed as follows: Definition 15 Let be an arbitrary LIVIFN, a be under DSET, i (i ∈ {1, 2, ..., n}) be n arbitrary LIVIFNs, a i be i under DSET, and be an arbitrary positive number.The following notations are used to express two operations between a 1 and a 2 and two operations between and a: a 1 ⊞ a 2 repre- sents a 1 plus a 2 ; a 1 ⊠ a 2 represents a 1 times a 2 ; a represents times a; a represents the power of a.These operations can be performed using the following rules: (15) ⊞ n i=1 a i = It is easy to prove that the four operations in the ORs above satisfy the following algebraic laws: where 1 and 2 are two arbitrary positive numbers.
The developed OR of LIVIFNs under DSET in Eq. ( 16) is free of the limitation of the OR of LIVIFNs in Eq. ( 3), as stated in the following theorem: Theorem 1 The operation in the OR of LIVIFNs under DSET in Eq. ( 16) is always invariant with respect to the score function in Eq. (14), the accuracy function in Eq. (15), and the comparison rules in Definition 14: For three arbitrary LIV-IFNs under DSET a 1 , a 2 , and a 3 , a 1 a 2 can always imply (a 1 ⊞ a 3 ) (a 2 ⊞ a 3 ).
According to the score function in Eq. ( 14) and the accuracy function in Eq. ( 15), we have Using the OR of LIVIFNs under DSET in Eq. ( 16), we obtain According to the score function in Eq. ( 14) and the accuracy function in Eq. ( 15), we have There are two possible situations where a 1 a 2 on the basis of the comparison rules in Definition 14: 1.
h (a 1 ) < h (a 2 ) : According to the expressions of h (a 1 ) and h (a 2 ) , we obtain  L Based on this, we further obtain from the expressions of h (a 1 ⊞ a 3 ) and h (a 2 ⊞ a 3 ) that h (a 1 ⊞ a 3 ) < h (a 2 ⊞ a 3 ) .Therefore, we can obtain from the comparison rules in Definition 14 that (a 1 ⊞ a 3 ) (a 2 ⊞ a 3 ); 2.
h (a 1 ) = h (a 2 ) and h (a 1 ) > h (a 2 ) : According to the expressions of h (a 1 ) , h (a 2 ) , h (a 1 ) , and Based on this, we further obtain from the expressions of h (a 1 ⊞ a 3 ) , h (a 2 ⊞ a 3 ) , h (a 1 ⊞ a 3 ) , and ) .Therefore, we can obtain from the comparison rules in Definition 14 that (a 1 ⊞ a 3 ) 2 ⊞ a 3 ).
On the basis of the two situations above, we can conclude that a 1 2 can always imply (a 1 ⊞ a 3 ) 2 ⊞ a 3 ) .◻ The developed OR of LIVIFNs under DSET in Eq. ( 18) is free of the limitation of the OR of LIVIFNs in Eq. ( 5), as stated in the following theorem: Theorem 2 The operation in the OR of LIVIFNs under DSET in Eq. ( 18) is always persistent with respect to the score function in Eq. ( 14), the accuracy function in Eq. (15), and the comparison rules in Definition 14: For two arbitrary LIVIFNs under DSET a 1 and a 2 and an arbitrary positive number , a 1 2 can always imply a 1 2 .
Proof Let According to the score function in Eq. ( 14) and the accuracy function in Eq. ( 15), we have Using the OR of LIVIFNs under DSET in Eq. ( 18), we obtain According to the score function in Eq. ( 14) and the accuracy function in Eq. ( 15), we have There are two possible situations where a 1 a 2 on the basis of the comparison rules in Definition 14: 1.
h (a 1 ) < h (a 2 ) : According to the expressions of h (a 1 ) and h (a 2 ) , we obtain  L Based on this, we further obtain from the expressions of h ( a 1 ) and h ( a 2 ) that h (a 1 ) < h (a 2 ) .Therefore, we can obtain from the comparison rules in Definition 14 that a 1 a 2 ; 2.
h (a 1 ) = h (a 2 ) and h (a 1 ) > h (a 2 ) : According to the expressions of h (a 1 ) , h (a 2 ) , h (a 1 ) , and h (a 2 ) , we obtain L 1 Based on this, we further obtain from the expressions of h ( a 1 ) , h ( a 2 ) , h ( a 1 ) , and h ( a 2 ) that h ( a 1 ) = h ( a 2 ) and h (a 1 ) > h (a 2 ) .Therefore, we can obtain from the comparison rules in Definition 14 that a 1 a 2 .
On the basis of the two situations above, we can conclude that a 1 a 2 can always imply a 1 a 2 .◻

WA Operator of LIVIFNs Based on DSET
Based on the developed ORs of LIVIFNs under DSET, a WA operator of LIVIFNs under DSET is constructed below: Definition 16 Let i (i ∈ {1, 2, ..., n}) be n arbitrary LIV- IFNs, a i be i under DSET, and w i be the weight such 0 ≤ w i ≤ 1 and Σ n i=1 w i = 1 .The aggregation function is called the LIVIFWA operator under DSET.
The LIVIFWA operator under DSET above has the property of monotonicity, as stated in the following theorem: Theorem 3 Let i (i ∈ {1, 2, ..., n}) be n arbitrary LIVIFNs and b i be i under DSET.
Proof According to the LIVIFWA operator under DSET in Eq. ( 26), we have Based on this, we can obtain from the score function in Eq. ( 14) and the accuracy function in Eq. ( 15) that h ( (a 1 , a 2 , ..., . According to the comparison rules in Definition 14, we have The constructed LIVIFWA operator under DSET does not have the properties of idempotency and boundedness.However, a small modification of its expression (multiplying by n, i.e. n (a 1 , a 2 , ..., a n ) ) can generate an LIVIFWA operator under DSET having idempotency and boundedness, as respectively stated in the following two theorems: Theorem 4 Let be an arbitrary LIVIFN and a be under DSET.
Proof According to the LIVIFWA operator under DSET in Eq. ( 26), we have a for all i ∈ {1, 2, ..., n} , we can obtain Proof On the basis of the proof of Theorem 3, it is easy to prove that n has the property of monotonicity.The constructed LIVIFWA operator under DSET in Eq. ( 26) is free of the limitation of the LIVIFWA operator in Eq. ( 13), as stated in the following theorem: The LIVIFWA operator under DSET in Eq. ( 26) is always monotone with respect to the score function in Eq. ( 14), the accuracy function in Eq. (15), and the comparison rules in Definition 14: For three arbitrary LIVIFNs under DSET a 1 , a 2 , and a 3 and certain weights w 1 and w 2 , a 1 2 can always imply (a 1 , a 3 ) 2 , a 3 ).
According to the score function in Eq. ( 14) and the accuracy function in Eq. ( 15), we have Using the LIVIFWA operator under DSET in Eq. ( 26), we obtain According to the score function in Eq. ( 14) and the accuracy function in Eq. ( 15), we have There are two possible situations where a 1 a 2 on the basis of the comparison rules in Definition 14: 1.
h (a 1 ) < h (a 2 ) : According to the expressions of h (a 1 ) and h (a 2 ) , we obtain  L Based on this, we further obtain from the expressions of h ( (a 1 , a 3 )) and . Therefore, we can obtain from the comparison rules in Definition 14 that , and h (a 2 ) , we obtain L 1 Based on this, we further obtain from the expressions of h ( (a 1 , a 3 )) , h ( (a 2 , a 3 )) , h ( (a 1 , a 3 )) , and h ( (a 2 , a 3 )) that . Therefore, we can obtain from the comparison rules in Definition 14 that (a 1 , a 3 ) (a 2 , a 3 ).
On the basis of the two situations above, we can conclude that a 1 a 2 can always imply (a 1 , a 3 ) (a 2 , a 3 ).◻

DM Method Based on the New WA Operator
A cognitively inspired DM problem with LIVIFNs is generally described by m alternatives A i (i ∈ {1, 2, ..., m}) , n cri- teria C j (j ∈ {1, 2, ..., n}) , a vector of weights of criteria (w 1 , w 2 , ..., w n ) such that 0 ≤ w j ≤ 1 is the weight of C j and Σ n j=1 w j = 1 , n ′ experts (an expert refers to a person with spe- cial knowledge, experience, or skills in the domain to which the DM problem belongs who provides evaluation values of criteria) E k (k ∈ {1, 2, ..., n � }) , a vector of weights of experts each kij is an LIVIFN which value of C j of A i evaluated by E k .The aim of solving such a problem is to determine the best alternative from A i on the basis of M k , (w � 1 , w � 2 , ..., w � n � ) , and (w 1 , w 2 , ..., w n ) .Using a DM method based on the constructed WA operator of LIVIFNs under DSET, the problem can be solved via the following steps: 1. Normalise the decision matrices M k .There are two types of criteria in multi-criterion decision-making, which are benefit and cost criteria.A benefit criterion is a criterion that has positive effect on the decisionmaking result (the larger its value, the more favourable the decision-making result), while a cost criterion is a criterion that affects the decision-making result adversely (the smaller its value, the more favourable the decision-making result).For example, total area and price belong to a benefit criterion and a cost criterion in selection of a house to buy, respectively, since the larger the total area and the lower the price, the more favourable the decision-making result.A DM problem may contain only benefit criteria, both benefit and cost criteria, or only cost criteria.When it contains cost criteria, specific rules are generally applied to normalise the values of cost criteria to obtain normalised decision matrices.For the studied DM problem with LIVIFNs, the normalisation rules and normalised decision matrices are expressed as follows: 2. Convert the LIVIFNs in the normalised decision matrices N k into LIVIFNs under DSET to obtain the follow- ing matrices: 3. Calculate the summary values of a kij using the LIVIFWA operator under DSET in Eq. ( 26) with the weight vector Calculate the summary values of a ij using the LIVIFWA operator under DSET in Eq. ( 26) with the weight vector (w 1 , w 2 , ..., w n ) : (30) a i = (a i1 , a i2 , ..., a in ) 5. Calculate the score values of a i using the score function in Eq. ( 14) and the accuracy values of a i using the accu- racy function in Eq. ( 15). 6. Rank a i according to the comparison rules in Defini- tion 14 and determine the best alternative based on the ranking results.
According to the steps above, the general flow of the proposed DM method is depicted in Fig. 1.
In this example, a decision maker needs to select a proper additive manufacturing machine from four alternative machines M 1 , M 2 , M 3 , and M 4 to build a part with certain material.The decision maker invited three domain experts E 1 , E 2 , and E 3 to evaluate the four alternative machines.The evaluation criteria include the predicted surface roughness ( C 1 ), predicted strength ( C 2 ), predicted elongation ( C 3 ), and predicted hardness ( C 4 ) of the as-built part.The weights of the three experts are respectively 0.4, 0.3, and 0.3.The weights of the four criteria are respectively 0.1, 0.3, 0.3, and 0.3.The three experts were asked to use LIVIFNs to express their evaluation results.There are nine available linguistic terms, which are extremely small (0), very small (1), small (2), slightly small (3), medium (4), slightly large (5), large (6), very large (7), and extremely large (8).The evaluation results are listed in Table 1.

Comparisons with Existing Methods
To verify the effectiveness of the proposed DM method, a comparison of the ranking results of the method and the DM methods with LIVIFNs presented by [16,27,37], and [33] is carried out.In this comparison, the problem in Sect.5.1 is taken as a benchmark.The linguistic intervalvalued intuitionistic fuzzy prioritised weighted averaging (LIVIFPWA) operator and the linguistic interval-valued intuitionistic fuzzy prioritised weighted geometric (LIVIF-PWG) operator are respectively used in the method of [27].
The LIVIFWA operator and the linguistic interval-valued intuitionistic fuzzy weighted geometric (LIVIFWG) operator are respectively used in the method of [16].The linguistic interval-valued intuitionistic fuzzy weighted Maclaurin symmetric mean (LIVIFWMSM) operator is used in the method of [37].The linguistic interval-valued intuitionistic fuzzy power weighted Muirhead mean (LIVIFPWMM) operator is used in the method of [33].To facilitate the comparison, the methods of [16,27,37], and [33] use the score function in Eq. ( 1), the accuracy function in Eq. ( 2), and the comparison rules in Definition 4 uniformly to generate the ranking results.The details and results of the comparison are listed in Table 5 and depicted in Fig. 2. It can be seen from Table 5 and Fig. 2 that the best additive manufacturing machine determined by the proposed method is exactly the same as that determined by all other methods.This demonstrates the effectiveness of the proposed method in solving practical cognitively inspired DM problems with LIVIFNs.The advantage of the proposed method is that the developed ORs of LIVIFNs under DSET are always invariant and persistent and the presented LIVIFWA operator under DSET is always monotone.This advantage has been proved in Theorem 1, Theorem 2, and Theorem 6.To show the advantage more intuitively and how it affects the DM results, another comparison of the ranking results of the DM methods in Table 5 is carried out.In this comparison, Example 3 in Sect.3.2 is taken as a benchmark.The AOs used in each method are the same as that in first comparison.Further, the methods of [16,27,37], and [33] also use the score function in Eq. ( 1), the accuracy function in Eq. ( 2), and the comparison rules in Definition 4 uniformly to generate the ranking results.The results of the comparison are listed in Table 6 and depicted in Fig. 3.As can be seen from Table 6 and Fig. 3, the proposed method can generate an intuitive ranking for Example 3, while other comparison methods  6 Details and results of the second comparison o stands for the used operator; a 1 , a 2 , and a 3 respectively stand for 1 , 2 , and 3 under DSET; For the method of [37], k = 2 in the aggregation by the LIVIFWMSM operator; For the method of [33], Q = (1, 2) in the aggregation by the LIVIFPWMM operator; The LIVIFPWA and LIV-IFPWG operators respectively reduce to the LIVIFWA and LIVIFWG operators, since the weights given in Example produce a counterintuitive ranking.This is because the proposed method uses the ORs of LIVIFNs under DSET that are always invariant and persistent, while all of other comparison methods use the ORs of LIVIFNs in Definition 5 that do not have these properties.

Conclusion
In this paper, a WA operator of LIVIFNs under DSET is presented to solve cognitively inspired DM problems with LIVIFNs.Firstly, an interpretation of LIVIFS under DSET is given.Based on this interpretation, four novel ORs of LIVIFNs are then developed.The characteristics of these ORs are highlighted and proved.After that, a new WA operator of LIVIFNs, i.e. the LIVIFWA operator under DSET, is constructed using the developed ORs.The properties of this operator is explored and its advantage is highlighted and proved.Finally, a method for solving cognitively inspired DM problems with LIVIFNs based on the constructed operator is proposed.The paper also introduces a numerical example to illustrate the application of the proposed method and documents quantitative comparisons with several existing methods to demonstrate the effectiveness and advantage of the method.
The main contributions of the paper are threefold: Future work will aim especially at improving the proposed method to be capable to solve the cognitively inspired DM problems with LIVIFNs where the weights of criteria are expressed by LIVIFNs.In some cognitively inspired DM problems with LIVIFNs, decision makers may use LIVIFNs to describe the degrees of importance of the considered criteria.The proposed method is applicable for the problems where the degrees of importance of criteria are expressed as decimals.It cannot be applied to the problems where the weights of criteria are in the form of LIVIFNs.To address this limitation, two rules for the power and division operations between two LIVIFNs under DSET would be developed and a new AO would be constructed using these rules.Further, it would be interesting to combine the developed novel ORs with the power average operator, Bonferroni mean operator, Maclaurin symmetric mean operator, and Muirhead mean operator under linguistic interval-valued intuitionistic fuzzy environment to construct some more powerful AOs of LIVIFNs.Last but not least, applications of the constructed AOs to solve more cognitively inspired DM problems would also be studied.
Acknowledgements The authors are very grateful to the editor and the four anonymous reviewers for their insightful comments for the improvement of the paper.The authors would also like to acknowledge the financial support by the National Natural Science Foundation of China (No. 52105511), the EPSRC UKRI Innovation Fellowship (Ref.EP/S001328/1) and the EPSRC Future Advanced Metrology Hub (Ref.EP/P006930/1).

i 1 An arbitrary positive number 2 An arbitrary positive number L
The i-th (i ∈ {1, 2, ..., n}) LIVIFN Θ A frame of discernment An arbitrary positive number The lower bound of the linguistic membership degree of an LIVIFN U The upper bound of the linguistic membership degree of an LIVIFN L The lower bound of the linguistic nonmembership degree of an LIVIFN U The upper bound of the linguistic nonmembership degree of an LIVIFN L The lower bound of the linguistic hesitancy degree of an LIVIFN U The upper bound of the linguistic hesitancy degree of an LIVIFN A An LIVIFS The LIVIFS A under DSET AV( ) The accuracy value of an LIVIFN h ( ) The accuracy value of an LIVIFN under DSET with respect to h A i

Fig. 1
Fig. 1 General flow of the proposed DM method

Fig. 2
Fig. 2 Graphical presentation of the results of the first comparison

1 .
Four ORs of LIVIFNs based on DSET are developed.Compared to the existing most used ORs of LIVIFNs, the developed ones are always invariant and persistent with respect to the score function, accuracy function, and comparison rules of LIVIFNs under DSET; 2. A WA operator of LIVIFNs based on DSET is constructed.Compared to the existing WA operator of LIVIFNs, the constructed one is always monotone with respect to the score function, accuracy function, and comparison rules of LIVIFNs under DSET; 3. A new method to solve cognitively inspired DM problems with LIVIFNs is proposed.This method has the advantages of the developed ORs and constructed AO.

Table 2
Elements

Table 4
Elements of the aggregated decision matrix