Multiple attribute group decision making based on interval neutrosophic uncertain linguistic variables

To deal with decision-making problems with interval neutrosophic uncertain linguistic information, the paper proposes a multiple attribute group decision-making method under an interval neutrosophic uncertain linguistic environment. Firstly, the concept of an interval neutrosophic uncertain linguistic set and an interval neutrosophic uncertain linguistic variable (INULV) is presented by combining an uncertain linguistic variable with an interval neutrosophic set. Secondly, we introduce the operation rules of INULVs and the score function, accuracy function and certainty function of an INULV. Thirdly, we develop an interval neutrosophic uncertain linguistic weighted arithmetic averaging (INULWAA) operator and an interval neutrosophic uncertain linguistic weighted geometric averaging (INULWGA) operator and investigate their properties. Fourthly, a group decision-making method is established based on the INULWAA and INULWGA operators to solve multiple attribute group decision-making problems with interval neutrosophic uncertain linguistic information. Finally, an illustrative example is provided to demonstrate the application of the developed approach.


Introduction
Multiple attribute decision making is an important research topic in decision theory, and then it has been applied widely in many fields, such as engineering and economic management. Recently, new decision-making methods have been presented under various fuzzy environments [6][7][8]. In complex decision-making problems, however, there is a lot of qualitative information, where the evaluation results of decision makers may easily be expressed by linguistic variables or uncertain linguistic variables because of time pressure, lack of knowledge, and the decision maker's limited attention and information processing capabilities. Thus, Zadeh [19] originally proposed the concept of the linguistic variable and applied it to fuzzy reasoning. After that, Herrera et al. [2] put forward a model of consensus for group decision making under a linguistic assessment. Herrera and Herrera-Viedma [3] also proposed a linguistic decision analysis to solve decision-making problems with linguistic information. Then, Xu [15] presented a linguistic hybrid arithmetic averaging operator and applied it to multiple attribute group decision-making problems with linguistic information. Meanwhile, Xu [16] further proposed goal programming models for multiple attribute decision making under a linguistic environment. Furthermore, Zeng and Su [10] put forward linguistic induced generalized aggregation distance operators and applied them to multiple attribute decision making. Agarwal and Palpanas [1] further introduced a linguistic rough set (LRS) by integrating linguistic quantifiers in the rough set framework. On the other hand, Xu [14] proposed the uncertain linguistic ordered weighted averaging (ULOWA) operator and uncertain linguistic hybrid aggregation (ULHA) operator and applied them to multiple attribute group decision-making problems with uncertain linguistic information. Also, Xu [17] developed some induced uncertain linguistic ordered weighted averaging (IULOWA) operators and applied them to multiple attribute group decision making under an uncertain linguistic environment. Furthermore, by combining a linguistic variable with an intuitionistic fuzzy set (IFS), Wang and Li [12] introduced the intuitionistic linguistic set (ILS) that consists of a linguistic part and an intuitionistic part. Further, they [12] proposed an intuitionistic two-semantic, a Hamming distance between two intuitionistic two-semantics, and a ranking method for alternatives in accordance with the comprehensive membership degree to the ideal solution for each alternative. Wang and Li [13] also introduced the operation rules of intuitionistic linguistic variables (ILVs), the expected value, score and accuracy functions of an ILV and developed the intuitionistic linguistic weighted arithmetic averaging (ILWAA) operator and intuitionistic linguistic weighted geometric averaging (ILWGA) operator, and then they applied the ILWAA and ILWGA operators to multiple attribute decision-making problems with intuitionistic linguistic information. Furthermore, Liu and Jin [4] proposed the concept of an intuitionistic uncertain linguistic variable (IULV), which is an extension of the ILV concept, and developed an intuitionistic uncertain linguistic weighted geometric averaging (IULWGA) operator, an intuitionistic uncertain linguistic ordered weighted geometric (IULOWG) operator, and an intuitionistic uncertain linguistic hybrid geometric (IULHG) operator, which generalizes both the IULWGA operator and the IULOWG operator, and then they applied these operators to multiple attribute group decision-making problems with IULVs. Liu et al. [5] further developed some intuitionistic uncertain linguistic Heronian mean operators, including an intuitionistic uncertain linguistic arithmetic Heronian (IULAH) operator, an intuitionistic uncertain linguistic weighted arithmetic Heronian (IUL-WAH) operator, an intuitionistic uncertain linguistic geometric Heronian (IULGH) operator, and an intuitionistic uncertain linguistic weighted geometric Heronian (IULWGH) operator, and then applied them to group decision making.
In real decision-making problems, there is often incomplete, indeterminate and inconsistent information. Thus, the neutrosophic set proposed by Smarandache [9] can be better to express this kind of information. Therefore, Ye [18] proposed the concepts of interval neutrosophic linguistic set (INLS) and interval neutrosophic linguistic variable (INLV) by combining a linguistic variable with an interval neutrosophic set (INS). Since an INLV consists of a linguistic part and an interval neutrosophic part, we can also consider that the linguistic part of the INLV is the linguistic variable represented by decision maker's judgment to an evaluated object and the interval neutrosophic part of the INLV is the subjective evaluation value on the reliability of the given linguistic variable, which is expressed by a truth-membership degree interval and an indeterminacy-membership degree interval and a falsitymembership degree interval. However, an INLS is an extension of an ILS by replacing the intuitionistic part of the ILS with interval neutrosophic part, while the linguistic part in the INLS is still the linguistic variable rather then the uncertain linguistic variable that easily expresses the qualitative information. Hence, the INLS cannot represent and deal with interval neutrosophic uncertain linguistic information. To overcome the shortcoming, the INLS should be extended by expressing the linguistic part of an INLS with an uncertain linguistic variable to propose the concepts of an interval neutrosophic uncertain linguistic set (INULS) and an interval neutrosophic uncertain linguistic variable (INULV), which are composed of an uncertain linguistic part and an interval neutrosophic part. Therefore, INULSs can easily express and better handle the uncertain information and inconsistent information than ILSs, IULSs and INLSs. As the further extension of author's previous work [18], the purposes of this paper are: (1)   Then, the discrete linguistic term set S = {s 0 , s 1 , s 2 , s 3 , s 4 , s 5 , s 6 } can be extended to a continuous linguistic set S ¼ fs a ja 2 Rg, which also satisfy the aforementioned characteristics, to minimize the linguistic information loss in the operation process. For any two linguistic variables s i and s j for s i ; s j 2 S, the operation rules are defined as follows [15,16]: Definition 1 [14,17] Supposes ¼ ½s a ; s b , where s a ; s b 2 S with a B b are the lower limit and the upper limit ofs, respectively. Thens is called an uncertain linguistic variable. Lets 1 ¼ ½s a1 ; s b1 ands 2 ¼ ½s a2 ; s b2 be any two uncertain linguistic variables, then their operation rules are defined as follows [14,17]: 1.s 1 Ès 2 ¼ ½s a1 ; s b1 È ½s a2 ; s b2 ¼ ½s a1þa2 ; s b1þb2 ; 2.s 1 s 2 ¼ ½s a1 ; s b1 ½s a2 ; s b2 ¼ ½s a1Âa2 ; s b1Âb2 ; 3.s 1 =s 2 ¼ ½s a1 ; s b1 =½s a2 ; s b2 ¼ ½s a1=b2 ; s b1=a2 if a2 = 0 and b2 = 0; 4. ks 1 ¼ k½s a1 ; s b1 ¼ ½s ka1 ; s kb1 for k C 0. 5.s 1 ð Þ k ¼ ½s ða1Þ k ; s ðb1Þ k for k C 0.

Some concepts of INSs
Smarandache [9] firstly proposed a neutrosophic set, which generalizes fuzzy sets, IFSs and interval valued intuitionistic fuzzy sets (IVIFSs), and gave the following definition of a neutrosophic set from philosophical point of view.
Definition 2 [9] Let X be a space of points (objects), with a generic element in X denoted by x. A neutrosophic set A in X is characterized by a truth-membership function T A (x), an indeterminacy-membership function I A (x) and a falsity-membership function F A (x). The functions T A (x), I A (x) and F A (x) are real standard or nonstandard subsets of For easy applications in real science and engineering areas, Wang et al. [11] introduced the concept of an INS, which is a subclass of a neutrosophic set, and gave the following definition of an INS.
Definition 3 [11] Let X be a space of points (objects) with generic elements in X denoted by x. An INS A in X is characterized by a truth-membership function T A (x), an indeterminacy-membership function I A (x), and a fal- Obviously, the sum of T A (x), I A (x) and F A (x) satisfies the Then, some relations of INSs are introduced as follows [11]:

Interval neutrosophic linguistic set
Based on combining INSs and linguistic variables, Ye [18] firstly proposed INLSs and gave the following definition: Definition 4 [18] Let X be a finite universal set, then an INLS in X is defined as and F A (x) express, respectively, the truth-membership degree interval, indeterminacy-membership degree interval, and falsity-membership degree interval of the element x in X to the linguistic variable s h(x) .
Þi is called an INLV and A can also be viewed as the collection of INLVs. Thus, the INLS can also be expressed by ð ÞÞi be any two INLVs for a 1 , a 2 [ S and any real number k C 0, then their operation rules are defined as follows [18]: Þi be any two INLVs for a 1 , a 2 [ S and any real numbers k, k 1 , k 2 C 0, then they satisfy the following properties [18]: Then, Ye [18] defined the score function, accuracy function and certainty function of an INLV, which are important indexes for ranking alternatives in decisionmaking problems.
Then, the score function, accuracy function and certainty function for the INLV a are defined, respectively, as follows: Definition 7 [18]. Let a 1 and a 2 be any two INLVs for a 1 , a 2 [ S. Then, the ranking method can be defined as follows: Let a j (j = 1, 2, …, n) be a collection of INLVs and W = (w 1 , w 2 , …, w n ) T be the weight vector of a j (j = 1, 2, …, n) with w j e[0,1] and P n j¼1 w j ¼ 1. Then Ye [18] proposed the INLWAA and INLWGA operators, respectively, as follows: INLWAA a 1 ;a 2 ;ÁÁÁ;a n ð Þ ¼ sP n j¼1 wjhðajÞ ; 1 À Y n j¼1 ð1 À inf Tða j ÞÞ wj ; INLWGAða 1 ; a 2; Á Á Á ; a n Þ 3 Interval neutrosophic uncertain linguistic set In real decision making, there is a lot of qualitative information, which is easily expressed by linguistic variables or uncertain linguistic variables by decision makers. An ILS consists of the intuitionistic part and the linguistic part. Then, an IULS composes of the intuitionistic part and the uncertain linguistic part. Hence, the IULS only extends the linguistic part of the ILS and can express the truth-membership degree and falsity-membership degree belonging to an uncertain linguistic variable, but cannot express the truth-membership degree, indeterminacy-membership degree, and falsity-membership degree belonging to an uncertain linguistic variable. Furthermore, an INLS consists of the interval neutrosophic part and the linguistic part and can handle the information expressed by the truthmembership degree, indeterminacy-membership degree and falsity-membership degree belonging to the linguistic variable, but cannot represent and deal with the information expressed by the truth-membership degree, indeterminacy-membership degree and falsity-membership degree belonging to the uncertain linguistic variable.   sup Tðã 1 ÞÞ k ; ½inf I k ðã 1 Þ; sup I k ðã 1 Þ; ½inf F k ðã 1 Þ; sup F k ðã 1 ÞÞi; 4.ã k 1 ¼ h½s h k ðã 1 Þ; s q k ðã 1 Þ ; ð½inf T k ðã 1 Þ; sup T k ðã 1 Þ; ½1 À ð1 À inf Iðã 1 ÞÞ k ; 1 À ð1 À sup Iðã 1 ÞÞ k ; ½1 À ð1À inf F ðã 1 ÞÞ k ; 1 À ð1 À sup Fðã 1 ÞÞ k Þi: Obviously, the above operation results are still INULVs.
Then, the score function, accuracy function, and certainty function for the INULVã are defined, respectively, as follows: HðãÞ CðãÞ Based on Definitions 7 and 11, a ranking method between INULVs can be given as follows.

Two weighted aggregation operators for INULVs
Based on the operation rules in Definition 10 and the extension of Eqs. (4) and (5), we can propose the weighted arithmetic aggregation operator and weighted geometric aggregation operator for INULVs to aggregate interval neutrosophic uncertain linguistic information.
Proof The proof of Eq. (10) can be done by means of mathematical induction.
The decision steps are described as follows: Step 1: Obtain the integrated matrix D = (d ij ) m 9 n by the following aggregation formula: Step 2: Calculate the individual overall value of the INULVd i for A i (i = 1, 2, …, m) by the following aggregation formula: Step 3: Calculate the score function E(d i ) (i = 1, 2, …, m) (accuracy function H(d i ) and certainty function C(d i )) by applying Eq. (6) (Eqs. (7) and (8)).
Step 4: Rank the alternatives according to the values of E(d i ) (H(d i ) and C(d i )) (i = 1, 2, …, m) by the ranking method in Definition 12, and then select the best one(s).

Illustrative example
An illustrative example about investment alternatives adapted from [18] is used to demonstrate the applications of the proposed decision-making method under an interval neutrosophic uncertain linguistic environment. There is an investment company, which wants to invest a sum of money in the best option. To invest the money, there is a panel with four possible alternatives: (1) A 1 is a car company; (2) A 2 is a food company; (3) A 3 is a computer company; (4) A 4 is an arms company. The investment company must take a decision according to the three attributes: (1) C 1 is the risk; (2) C 2 is the growth; Assume that three experts or decision makers are required in the evaluation process and their weight vector is given as V = (0.37, 0.33, 0.3) T . Then, the evaluation information of an alternative A i (i = 1, 2, 3, 4) with respect to an attribute C j (j = 1, 2, 3) can be given by the three experts. For example, the INULV value of an alternative A 1 with respect to an attribute C 1 is given as \ s Step , the ranking order of four alternatives is A 2 1 A 4 1 A 3 1 A 1 . Thus, we can see that the alternative A 2 is still the best choice among all the alternatives.
From the above decision results, we can see that the two kinds of ranking orders and the best alternative are identical, which are in agreement with the results of the method in [18].
Compared with the author's previous method in [18], although the decision results are in accordance with the ones in [18], the method proposed in this paper differs from the method in [18] for the multiple attribute decisionmaking problem not only due to the fact that the method proposed in this paper uses the INULV information and the weighted arithmetic aggregation operator and the weighted geometric aggregation operator for INULVs in the group decision-making problem, but also due to the consideration of the uncertain linguistic variable represented by decision makers' judgment to an evaluated object and the subjective evaluation value on the reliability of the given uncertain linguistic variable, which includes the indeterminacy information besides truth and falsity information belonging to the uncertain linguistic variable in the INULV. However, the method in [18] is a special case of the proposed method in this paper. Therefore, the group decision-making method proposed in this paper is more general and more feasible than the decision-making method in [18] since the former is a generalization of the later. The advantage is that the former easily reflects the ambiguous nature of a group of decision makers' judgment to an evaluated object because the INULV can provide the uncertain linguistic variable which easily expresses the qualitative information and the reliability of the given uncertain linguistic variable by a group of decision makers (experts) in the group decision-making problem, while the decision-making method in [18] can only provide the exact linguistic value, which difficultly expresses the qualitative information in some situations, and the reliability of the given linguistic value by unique decision maker (expert). Therefore, the group decision-making method in this paper is superior to the decision-making method in [18].

Conclusion
To deal with group decision making problems with INULVs, this paper proposed a group decision-making method based on the INULWAA and INULWGA operators to handle group decision-making problems with interval neutrosophic uncertain linguistic information. First, INULSs and the operation rules of INULVs were proposed as the generalization of the concepts of INLSs and INLVs. Then, the score function, accuracy function and certainty function of an INULV were defined to rank INULVs. Furthermore, we proposed the INULWAA and INULWGA operators and investigated their properties, and then applied them to group decision-making problems with interval neutrosophic uncertain linguistic information. Finally, an illustrative example was provided to demonstrate the application of the proposed method. The developed group decision-making method is the extension of existing decision-making method [18] and more suitable for expressing indeterminate and inconsistent information and handling group decision-making problems with interval neutrosophic uncertain linguistic information. The group decision-making method in this paper is superior to the one in [18]. In the future, we shall continue working on the extension and application of the developed operators to other domains.