#### Abstract

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.

#### 1. 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 decision- making 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 [1–6]. 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 [10–20].

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 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 , , when we can evaluate the “growth” of a company, we can utilize a HILFE . Obviously, , , and indicate that the “growth” of a company may be “medium”, “high,” and “very high”, and the intuitionistic fuzzy numbers “,” “,” and “” explain the degree that the “growth” of a company belongs to and does not belong to , , and , 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.

#### 2. 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 be a finite linguistic term set with odd cardinality, where represents a possible value for a linguistic term and is the cardinality of . For example, when , 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].(1)The set is ordered: , if and only if ;(2)there is the negation operator: , such that ;(3)maximum operator is , if ;(4)minimum operator is , if .

##### 2.1. Intuitionistic Linguistic Set

*Definition 1 (see [27]). *Let be the continuous form of and let be in a given domain; an intuitionistic linguistic set in can be defined as
where , , , and . and represent the membership degree and the nonmembership degree of the element to the linguistic variable , respectively. Let , , ; then, is called a hesitancy degree of to the linguistic variable .

In (1), is an intuitionistic linguistic variable (ILV). For convenience, is used to represent an ILV.

*Definition 2 (see [27]). *Let and be two ILVs and ; then, the operational laws of ILVs are defined as follows:(1);(2);(3);(4).

*Definition 3 (see [40]). *Let be an ILV; the score function and the accuracy function of are defined, respectively, as follows:

Theorem 4 (see [40]). *Let and be two ILVs; then, they can be compared by the following laws:**if , then ;**if , then
*

##### 2.2. 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 , which can be represented by the following mathematical symbol:
where is a set of some values in , denoting the possible membership degrees of the element to the set . For convenience, Liu et al. [37] called a hesitant fuzzy element (HFE) and the set of all HFEs.

*Definition 6 (see [41]). *Let , , and be any three HFEs, and ; then, some operational laws of HFEs are defined as follows:(1);(2);(3);(4).

*Definition 7 (see [37]). *For two HFEs and , let and be the score function and variance function of , respectively, where is the number of elements in ; then,(1)if , then is superior to , denoted by ;(2)if , then(a)if , then is inferior to , denoted by ;(b)if , then is equivalent to , denoted by .

#### 3. Hesitant Intuitionistic Linguistic Fuzzy Set

Based on the intuitionistic linguistic set and the hesitant fuzzy set, we propose the definition of the hesitant intuitionistic linguistic fuzzy set, as well as the operational laws, score function, and accuracy function in what follows.

*Definition 8. *Let be a nonempty set of the universe and a continuous linguistic term set of ; then, a hesitant intuitionistic linguistic fuzzy set (HILFS) on can be expressed by the mathematical symbol as follows:
where is a set of ILVs; that is, , denoting the possible membership degrees of the element to the set . For convenience, one calls a hesitant intuitionistic linguistic fuzzy element (HILFE) and the set of all HILFEs.

*Definition 9. *Let , , and be any three HILFEs, and ; then, the operational laws of HILFEs are defined as follows:(1);(2);(3);(4).Obviously, the above operational results are still HIFLEs.

Theorem 10. *Let and be two HILFEs, and ; the calculation rules are shown as follows:*(1)*;*(2)*;*(3)*;*(4)*.*

*Definition 11. *Let be a HILFE; then, the score function of can be represented as follows:
where is the number of ILVs in and is the cardinality of linguistic term set .

*Definition 12. *Let be a HILFE; then, the accuracy function of can be represented as follows:
where is the number of ILVs in and is the cardinality of linguistic term set .

Theorem 13. *Let and be two HILFEs and let and be the score value and accuracy degree of , respectively; then, one has the following.**If , then is smaller than , denoted by .**If , then one has the following: if , then is equal to , denoted by ;if , then is smaller than , denoted by .*

#### 4. 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.

##### 4.1. Hesitant Intuitionistic Linguistic Fuzzy Weighted Aggregation Operators

*Definition 14. *Let () be a collection of HILFEs and let be the weight vector of them, with , , and ; then, the hesitant intuitionistic linguistic fuzzy weighted average (HILFWA) operator is a mapping HILFWA: , and
where is a hesitant intuitionistic linguistic fuzzy set.

Theorem 15. *Let be a collection of HILFEs, and let be the weight vector of them, with , , and ; then; their aggregated value by the HILFWA operator is still a HILFE, and
**In particular, if , then the HILFWA operator reduces to the hesitant intuitionistic linguistic fuzzy average (HILFA) operator:
*

Some desirable properties of the HILFWA operator are given as follows.

Theorem 16 (idempotency). *If all HILFEs are equal and for all , then
*

*Proof. *Since for all , then

Theorem 17 (boundedness). *Let be a collection of HILFEs; if , , , , , and , then
*

*Proof. *Since , , , , , and , we have
Then,
That is,
where is the numbers of ILVs in and is the cardinality of linguistic term set . Therefore, according to Theorem 13, we obtain
Similarly,
Therefore,

*Definition 18. *Let () be a collection of HILFEs and let be the weight vector of them, with , , and ; then, the hesitant intuitionistic linguistic fuzzy weighted geometric (HILFWG) operator is a mapping HILFWG: , and
where is a hesitant intuitionistic linguistic fuzzy set.

Theorem 19. *Let be a collection of HILFEs and let be the weight vector of them, with , , and ; then, their aggregated value by the HILFWG operator is still a HILFE, and
**In particular, if , then the HILFWG operator reduces to the hesitant intuitionistic linguistic fuzzy geometric (HILFG) operator:
**Similar to the HILFWA operator, the HILFWG operator also has the properties of idempotency and boundedness under some conditions, which can be proved similar to Theorems 16 and 17.*

Lemma 20 (see [42]). *Let , , , and ; then,
**
with equality if and only if .*

Theorem 21. *Let be a collection of HILFEs and let be the weight vector of them, with , , and , and let be the balancing coefficient which plays a role of balance; then, one has
*

*Proof. *According to Lemma 20, for any , , we have
That is,
where and are the numbers of ILVs in and , respectively, and is the cardinality of linguistic term set . Therefore, according to Theorem 13, we obtain

##### 4.2. Hesitant Intuitionistic Linguistic Fuzzy Ordered Weighted Aggregation Operators

*Definition 22. *Let () be a collection of HILFEs and let be the aggregation-associated vector such that , , and ; then, the hesitant intuitionistic linguistic fuzzy ordered weighted average (HILFOWA) operator is a mapping HILFOWA: , and
where is a hesitant intuitionistic linguistic fuzzy set. is the th largest element in ().

Theorem 23. *Let be a collection of HILFEs and let be the aggregation-associated vector such that , , and ; then, their aggregated value by the HILFOWA operator is still a HILFE, and
*

Similar to the HILFWA operator, the HILFOWA operator also has the properties of idempotency and boundedness under some conditions, which can be proved similar to Theorems 16 and 17. Furthermore, the HILFOWA operator also has the property of commutativity.

Theorem 24 (commutativity). *Let be a collection of HILFEs. If is any permutation of , then
*

*Proof. *Since is a permutation of , we have , for all . Then, based on Definition 22, we obtain

*Definition 25. *Let () be a collection of HILFEs and let be the aggregation-associated vector such that , , and ; then, the hesitant intuitionistic linguistic fuzzy ordered weighted geometric (HILFOWG) operator is a mapping HILFOWG: , and
where is a hesitant intuitionistic linguistic fuzzy set. is the th largest element in ().

Theorem 26. *Let be a collection of HILFEs and let be the aggregation-associated vector such that , , and ; then, their aggregated value by the HILFOWG operator is still a HILFE, and
*

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.

Theorem 27. *Let be a collection of HILFEs, let be the aggregation-associated vector such that , , and , and let be the balancing coefficient which plays a role of balance; then, one has
**
which can be proved similar to Theorem 21.*

##### 4.3. Hesitant Intuitionistic Linguistic Fuzzy Hybrid Aggregation Operators

*Definition 28. *Let () be a collection of HILFEs, let be the weight vector of them, with , , and , and let be the balancing coefficient which plays a role of balance; then, based on the aggregation-associated vector such that , , and , the hesitant intuitionistic linguistic fuzzy hybrid average (HILFHA) operator is a mapping HILFHA: , and
where is a hesitant intuitionistic linguistic fuzzy set. is the th largest element of hesitant intuitionistic linguistic fuzzy weighted arguments (, ).

Theorem 29. *Let be a collection of HILFEs, let be the weight vector of them, with , , and , and let be the aggregation-associated vector such that , , and ; then, their aggregated value by the HILFHA operator is still a HILFE, and
**In particular, if , then the HILFHA operator reduces to the HILFWA operator in (8); if , then the HILFHA operator reduces to the HILFOWA operator in (28).*

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.

*Definition 30. *Let () be a collection of HILFEs, let be the weight vector of them, with , , and , and let be the balancing coefficient which plays a role of balance; then, based on the aggregation-associated vector such that , , and , the hesitant intuitionistic linguistic fuzzy hybrid geometric (HILFHG) operator is a mapping HILFHG: , and
where is a hesitant intuitionistic linguistic fuzzy set. is the th largest element of hesitant intuitionistic linguistic fuzzy weighted arguments (, ).

Theorem 31. *Let be a collection of HILFEs, let be the weight vector of them, with , , and , and let be the aggregation-associated vector such that , , and ; then, their aggregated value by the HILFHG operator is still a HILFE, and
**In particular, if , then the HILFHG operator reduces to the HILFWG operator in (20). If , then the HILFHG operator reduces to the HILFOWG operator in (32).*

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.

Theorem 32. *Let be a collection of HILFEs, let be the weight vector of them, with , , and , let be the aggregation-associated vector such that , , and , and let be the balancing coefficient which plays a role of balance; then, one has
**
which can be proved similar to Theorem 21.*

##### 4.4. Generalized Hesitant Intuitionistic Linguistic Fuzzy Weighted Aggregation Operators

*Definition 33. *Let () be a collection of HILFEs and let be the weight vector of them, with , , and ; is a parameter such that ; then, the generalized hesitant intuitionistic linguistic fuzzy weighted average (GHILFWA) operator is a mapping GHILFWA: , and
where is a hesitant intuitionistic linguistic fuzzy set.

Theorem 34. *Let be a collection of HILFEs and let be the weight vector of them, with , , and ; then, their aggregated value by the GHILFWA operator is still a HILFE, and**In particular, if , then the GHILFWA operator reduces to the HILFWA operator in (8).*

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.

*Definition 35. *Let () be a collection of HILFEs and let be the weight vector of them, with , , and ; is a parameter such that ; then, the generalized hesitant intuitionistic linguistic fuzzy weighted geometric (GHILFWG) operator is a mapping GHILFWG: , and
where is a hesitant intuitionistic linguistic fuzzy set.

Theorem 36. *Let be a collection of HILFEs and let be the weight vector of them, with , , and ; then, their aggregated value by the GHILFWG operator is still a HILFE, and
**In particular, if , then the GHILFWG operator reduces to the HILFWG operator in (20).*

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.

Theorem 37. *Let be a collection of HILFEs, let be the weight vector of them, with , , and , and let be the balancing coefficient which plays a role of balance; then, one has
**
which can be proved similar to Theorem 21.*

##### 4.5. Generalized Hesitant Intuitionistic Linguistic Fuzzy Ordered Weighted Aggregation Operators

*Definition 38. *Let () be a collection of HILFEs and let be the aggregation-associated vector such that , , and ; is a parameter such that ; then, the generalized hesitant intuitionistic linguistic fuzzy ordered weighted average (GHILFOWA) operator is a mapping GHILFOWA: , and
where is a hesitant intuitionistic linguistic fuzzy set. is the th largest element in ().

Theorem 39. *Let be a collection of HILFEs and let be the aggregation-associated vector such that , , and ; then, their aggregated value by the GHILFOWA operator is still a HILFE, and**In particular, if , then the GHILFOWA operator reduces to the HILFOWA operator in (28).*

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.

*Definition 40. *Let () be a collection of HILFEs and let be the aggregation-associated vector such that , , and ; is a parameter such that ; then, the generalized hesitant intuitionistic linguistic fuzzy ordered weighted geometric (GHILFOWG) operator is a mapping GHILFOWG: , and
where is a hesitant intuitionistic linguistic fuzzy set. is the th largest element in ().

Theorem 41. *Let be a collection of HILFEs and let be the aggregation-associated vector such that , , and ; then, their aggregated value by the GHILFOWG operator is still a HILFE, and**In particular, if , then the GHILFOWG operator reduces to the HILFOWG operator in (32).*

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.

Theorem 42. *Let be a collection of HILFEs, let be the aggregation-associated vector such that , , and , and let be the balancing coefficient which plays a role of balance; then, one has
**
which can be proved similar to Theorem 21.*

#### 5. An Approach for Satisfaction Evaluation with Hesitant Intuitionistic Linguistic Fuzzy Information

For a MADM problem, let be a finite set of alternatives and let be the set of attributes. Suppose that all values assigned to alternatives with respect to attributes are expressed by a hesitant intuitionistic linguistic fuzzy decision matrix denoted by , where elements are HILFEs provided for the rating of the alternative () with respect to the attribute (), with . If the information about attribute weights is completely known as , with , , and , then to determine the most desirable alternative(s), the HILFWA operator or the HILFWG operator is utilized to propose an approach to MADM under hesitant intuitionistic linguistic fuzzy environment, which involves the following steps.

*Step **1.* Aggregate the hesitant intuitionistic linguistic fuzzy assessment values of the alternative () on all attributes () into the overall assessment value of the alternative () based on the HILFWA operator or the HILFWG operator in (50) and (51), respectively. Consider
or
where is in the form of HILFEs and it can be denoted by .

*Step **2.* Calculate the score values of overall assessment values () by
where is the number of ILVs in and is the cardinality of linguistic term set . If there is no difference between two score values and , then we need to calculate the accuracy degrees and of the overall assessment values and ( and ), respectively, according to
*Step **3.* Rank all feasible alternatives () according to Theorem 4 and select the most desirable alternative(s).

*Step **4.* End.

#### 6. 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) is the milk beverage; (2) is the yoghourt; (3) is the cheese; (4) is the pasteurized milk. The milk and dairy company must make a decision according to the following four attributes: (1) is the price; (2) is the taste; (3) is the packaging; (4) is the storability, whose weight vector is given as . The four possible alternatives are evaluated by using the linguistic term set , , , , , , under the above four attributes. The hesitant intuitionistic linguistic fuzzy decision matrix is constructed as shown in Table 1.

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.

*Step **1.* Aggregate the hesitant intuitionistic linguistic fuzzy assessment values of the milk products () on all attributes () into the overall assessment value of the milk products () based on the HILFWA operator in (50). Take the milk product for example; we have