A Novel Multiple Attribute Satisfaction Evaluation Approach with Hesitant Intuitionistic Linguistic Fuzzy Information

This paper investigates the multiple attribute decision making (MADM) problems in which the attribute values take the form of hesitant intuitionistic linguistic fuzzy element (HILFE). Firstly, motivated by the idea of intuitionistic linguistic variables (ILVs) and hesitant fuzzy elements (HFEs), the concept, operational laws, and comparison laws of HILFE are defined.Then, some aggregation operators are developed for aggregating the hesitant intuitionistic linguistic fuzzy information, such as hesitant intuitionistic linguistic fuzzy weighted aggregation operators, hesitant intuitionistic linguistic fuzzy ordered weighted aggregation operators, and generalized hesitant intuitionistic linguistic fuzzy weighted aggregation operators. Moreover, some desirable properties of these operators and the relationships between them are discussed. Based on the hesitant intuitionistic linguistic fuzzy weighted average (HILFWA) operator and the hesitant intuitionistic linguistic fuzzy weighted geometric (HILFWG) operator, an approach for evaluating satisfaction degree is proposed under hesitant intuitionistic linguistic fuzzy environment. Finally, a practical example of satisfaction evaluation for milk products is given to illustrate the application of the proposed method and to demonstrate its practicality and effectiveness.


Introduction
Multiattribute decision making (MADM), which addresses the problem of making an optimal choice that has the highest degree of satisfaction from a set of feasible alternatives that are characterized in terms of their attributes, both quantitative and qualitative, is a usual task in human activities.Due to the inherent vagueness of human preferences as well as the objects being fuzzy and uncertain or data about the decisionmaking problems domain, the attributes involved in the decision problems are not always expressed as crisp numbers, and some of them are more suitable to be denoted by fuzzy numbers [1][2][3][4][5][6].The fuzzy set theory originally proposed by Zadeh [7] is a very useful tool to describe uncertain information.However, in some real decision-making situations, the fuzzy set is imprecise resulting from characterizing the fuzziness just by a membership degree.On the basis of the fuzzy set theory, Atanassov [8,9] proposed the intuitionistic fuzzy set characterized by a membership function and a nonmembership function.Obviously, the intuitionistic fuzzy set can describe and characterize the fuzzy essence of the objective world more exquisitely, and it has received more and more attention since its appearance [10][11][12][13][14][15][16][17][18][19][20].
However, in the real world, decision makers usually cannot completely express their opinions by quantitative numbers, and some of them are more appropriately described by qualitative linguistic terms.Since linguistic variables [21] have been proposed, so far, a number of linguistic approaches have been defined such as 2-tuple linguistic [22], interval-valued 2-tuple linguistic [23], uncertain linguistic [24], trapezoid fuzzy linguistic [25], and trapezoid fuzzy 2-tuple linguistic [26] approaches.In order to express the uncertainty and ambiguity as accurate as possible, Wang and Li [27] proposed the concept of intuitionistic linguistic 2 Mathematical Problems in Engineering set based on linguistic variables and intuitionistic fuzzy set, which can overcome the defects for intuitionistic fuzzy set, which can only roughly represent criteria's membership and nonmembership to a particular concept, such as "good" and "bad, " and for linguistic variables which usually implies that membership degree is 1, and the nonmembership degree and hesitation degree of decision makers cannot be expressed.
In real decision-making process, we often encounter such situation that the decision makers are hesitant among a set of possible values which makes the outcome of decision making inconsistent.To solve this problem, the hesitant fuzzy set (HFS), an extension of fuzzy set [7], was proposed by Torra and Narukawa [28] and Torra [29], which permits the membership degree of an element to a given set to be represented by several possible numerical values.To accommodate more complex environment, several extensions of HFS have been presented, such as interval-valued hesitant fuzzy set (IVHFS) [30,31], hesitant triangular fuzzy set (HTFS) [32], hesitant multiplicative set (HMS) [33], hesitant fuzzy linguistic term set (HFLTS) [34], and hesitant fuzzy uncertain linguistic set (HFULS) [35].In particular, considering that the human judgments including preference information may be stated by a linguistic variable or an uncertain linguistic variable which permits the membership having a set of possible crisp values, Lin et al. [36] proposed the concepts of hesitant fuzzy linguistic set (HFLS) and hesitant fuzzy uncertain linguistic set (HFULS).Furthermore, Liu et al. [37] developed the hesitant intuitionistic fuzzy linguistic set (HIFLS) and the hesitant intuitionistic fuzzy uncertain linguistic set (HIFULS) which permit the possible membership degree and nonmembership degree of an element to a linguistic term and an uncertain linguistic term having sets of intuitionistic fuzzy numbers.
To the best of our knowledge, the existing methods under hesitant fuzzy environment are not suitable for dealing with MADM problems under hesitant intuitionistic linguistic fuzzy environment.In fact, when decision makers give their assessments on attributes which are in the form of intuitionistic linguistic variables (ILVs), they may also be hesitant among several possible ILVs.Therefore, inspired by the idea of the HFS, based on the ILVs, we propose a new fuzzy variable called hesitant intuitionistic linguistic fuzzy element (HILFE) which is composed of a set of ILVs.The main advantage of the HILFE is that it can describe the uncertain information by several linguistic variables in qualitative and intuitionistic fuzzy numbers adopted to demonstrate how much degree that an attribute value belongs and does not belong to a linguistic term in quantitative.For example, for a predefined linguistic set  = { 0 = extremely low,  1 = very low,  2 = low,  3 = medium,  4 = high,  5 = very high,  6 = extremely high}, when we can evaluate the "growth" of a company, we can utilize a HILFE {⟨ 3 , (0.6, 0.3)⟩, ⟨ 4 , (0.6, 0.2)⟩, ⟨ 5 , (0.5, 0.4)⟩}.Obviously,  3 ,  4 , and  5 indicate that the "growth" of a company may be "medium", "high, " and "very high", and the intuitionistic fuzzy numbers "(0.6, 0.3), " "(0.6, 0.2), " and "(0.5, 0.4)" explain the degree that the "growth" of a company belongs to and does not belong to  3 ,  4 , and  5 , respectively.
The remainder of this paper is organized as follows: some basic definitions of intuitionistic linguistic set and hesitant fuzzy set are briefly reviewed in Section 2. In Section 3, the concept, operational laws, score function, and accuracy function of the hesitant intuitionistic linguistic fuzzy element are defined.In Section 4, some hesitant intuitionistic linguistic fuzzy aggregation operators are proposed, and then some desirable properties of the proposed operators are investigated.In Section 5, we develop an approach to evaluate satisfaction degree with hesitant intuitionistic linguistic fuzzy information based on the proposed operators.In Section 6, a numerical example is given to illustrate the application of the proposed method.The paper is concluded in Section 7.

Preliminaries
To facilitate the following discussion, some basic definitions related to intuitionistic linguistic set and hesitant fuzzy set are briefly reviewed in this section.
Let  = { 0 ,  1 , . . .,   } be a finite linguistic term set with odd cardinality, where   represents a possible value for a linguistic term and  + 1 is the cardinality of .For example, when  = 6, a set of seven terms  can be given as follows.
In general, for any linguistic term set , it is required that   and   must satisfy the following properties [38,39].

Hesitant Fuzzy Set
Definition 5 (see [29]).Let  be a fixed set; then, a hesitant fuzzy set (HFS) on  is in terms of a function that when applied to  returns a subset of [0, 1], which can be represented by the following mathematical symbol: where h() = ⋃ r()∈ h() {r()} is a set of some values in [0, 1], denoting the possible membership degrees of the element  ∈  to the set .For convenience, Liu et al. [37] called h() a hesitant fuzzy element (HFE) and  the set of all HFEs.
Theorem 10.Let ℎ 1 and ℎ 2 be two HILFEs, and  ≥ 0; the calculation rules are shown as follows: (1) Definition 11.Let ℎ be a HILFE; then, the score function (ℎ) of ℎ can be represented as follows: where #ℎ is the number of ILVs in ℎ and +1 is the cardinality of linguistic term set .
Definition 12. Let ℎ be a HILFE; then, the accuracy function (ℎ) of ℎ can be represented as follows: where #ℎ is the number of ILVs in ℎ and +1 is the cardinality of linguistic term set .
Theorem 13.Let ℎ 1 and ℎ 2 be two HILFEs and let (ℎ  ) and (ℎ  ) be the score value and accuracy degree of ℎ  ( = 1, 2), respectively; then, one has the following. ( ( , then one has the following:

Hesitant Intuitionistic Linguistic Fuzzy Aggregation Operators
Motivated by the operational laws of HILFEs, in the following, some aggregation operators are developed for aggregating the hesitant intuitionistic linguistic fuzzy information.
Similar to the HILFOWA operator, the HILFOWG operator also has the properties of idempotency, boundedness, and commutativity under some conditions, which can be proved similar to Theorems 16, 17, and 24.
Similar to the HILFWA operator, the HILFHA operator also has the properties of idempotency and boundedness under some conditions, which can be proved similar to Theorems 16 and 17.
Similar to the HILFWG operator, the HILFHG operator also has the properties of idempotency and boundedness under some conditions, which can be proved similar to Theorems 16 and 17.
Similar to the HILFWA operator, the GHILFWA operator also has the properties of idempotency and boundedness under some conditions, which can be proved similar to Theorems 16 and 17.
Similar to the HILFWG operator, the GHILFWG operator also has the properties of idempotency and boundedness under some conditions, which can be proved similar to Theorems 16 and 17.
Similar to the HILFOWA operator, the GHILFOWA operator also has the properties of idempotency, boundedness, and commutativity under some conditions, which can be proved similar to Theorems 16, 17, and 24.
Similar to the HILFOWG operator, the GHILFOWG operator also has the properties of idempotency, boundedness, and commutativity under some conditions, which can be proved similar to Theorems 16, 17, and 24.

Numerical Example
In this section, a practical example of satisfaction evaluation for milk products is adapted to illustrate the application of the MADM method proposed in Section 5 and to demonstrate its feasibility and effectiveness in a realistic scenario.
To strengthen the competitiveness and enlarge the product lines, a milk and dairy company needs to know the consumer satisfaction of its products at first, so the market department organizes investigations in several supermarkets.There is a panel with four milk products: (1)  1 is the milk beverage; (2)  2 is the yoghourt; (3)  3 is the cheese; (4)  4 is the pasteurized milk.The milk and dairy company must make a decision according to the following four attributes: (1)  1 is the price; (2)  2 is the taste; (3)  3 is the packaging; (4)  4 is the storability, whose weight vector is given as  = (0.30, 0.35, 0.10, 0.25)  .The four possible alternatives { 1 ,  2 ,  3 ,  4 } are evaluated by using the linguistic term set  = { 0 = extremely poor,  1 = very poor,  2 = poor,  3 = fair,  4 = good,  5 = very good,  6 = extremely good} under the above four attributes.The hesitant intuitionistic linguistic fuzzy decision matrix  = (ℎ  ) 4×4 is constructed as shown in Table 1.
12 Mathematical Problems in Engineering In the following, we utilize the proposed MADM method to rank the milk products according to the customer satisfaction evaluation with hesitant intuitionistic linguistic fuzzy information.
For the same problem, if we utilize the decision making method based on the hesitant fuzzy linguistic term set proposed by Rodriguez et al. [34], then the hesitant fuzzy linguistic matrix is constructed as shown in Table 2.
Then, we can obtain the overall assessment value ℏ  of the milk products   ( = 1, 2, 3, 4) by the hesitant fuzzy linguistic weighted averaging (HFLWA) operator proposed by Zhang and Wu [35].Consider (57) Thus, we can obtain the score values of the milk products   ( = 1, 2, 3, 4) as shown in Figure 3.
Since (ℏ 4 ) > (ℏ 2 ) > (ℏ 1 ) > (ℏ 3 ), thus the ranking of all milk products is obtained as  4 ≻  2 ≻  1 ≻  3 .Therefore, the most desirable milk product is  4 (pasteurized milk) as well, which demonstrates the feasibility and validity of the method proposed in this paper.Moreover, we can know the membership degree and the nonmembership degree of the possible assessment value from the results obtained by the new methods proposed in this paper.

Conclusions
With respect to MADM problems in which the attribute values take the form of HILFE, this paper studies the MADM approach under hesitant intuitionistic linguistic fuzzy environment.Firstly, the concept, operational laws, and comparison laws of HILFE are proposed.Then, some aggregation operators are developed for aggregating the hesitant intuitionistic linguistic fuzzy information, such as hesitant intuitionistic linguistic fuzzy weighted aggregation operators, hesitant intuitionistic linguistic fuzzy ordered weighted aggregation operators, hesitant intuitionistic linguistic fuzzy hybrid aggregation operators, generalized hesitant intuitionistic linguistic fuzzy weighted aggregation operators, and generalized hesitant intuitionistic linguistic fuzzy ordered weighted aggregation operators.Based on the proposed HILFWA operator and HILFWG operator, an approach is proposed to solve MADM problems under hesitant intuitionistic linguistic fuzzy environment.Finally, a practical example is given to illustrate the application of the proposed method.The main advantage of our approach is that it can describe the uncertain information by several intuitionistic linguistic variablesin which linguistic variables demonstrate whether an attribute is good or bad in qualitative and intuitionistic fuzzy numbers are adopted to demonstrate how much degree that an attribute value belongs and does not belong to a linguistic variable in quantitative.In future research, we will focus on expanding the hesitant intuitionistic linguistic decision-making approach to other domains such as supplier selection, location choice, project selection, and green supply chain evaluation.