Multicriteria Decision-Making Method and Application in the Setting of Trapezoidal Neutrosophic Z-Numbers

)e information expression and modeling of decision-making are critical problems in the fuzzy decision theory and method. However, existing trapezoidal neutrosophic numbers (TrNNs) and neutrosophic Z-numbers (NZNs) and their multicriteria decision-making (MDM) methods reveal their insufficiencies, such as without considering the reliability measures in TrNN and continuous Z-numbers in NZN. To overcome the insufficiencies, it is necessary that one needs to propose trapezoidal neutrosophic Z-numbers (TrNZNs), their aggregation operations, and an MDM method for solving MDM problems with TrNZN information. Hence, this study first proposes a TrNZN set, some basic operations of TrNZNs, and the score and accuracy functions of TrNZN and their ranking laws. )en, the TrNZN weighted arithmetic averaging (TrNZNWAA) and TrNZN weighted geometric averaging (TrNZNWGA) operators are presented based on the operations of TrNZNs. Next, an MDM approach using the proposed aggregation operators and score and accuracy functions is established to carry out MDM problems under the environment of TrNZNs. In the end, the established MDM approach is applied to an MDM example of software selection for revealing its rationality and efficiency in the setting of TrNZNs. )e main advantage of this study is that the established approach not only makes assessment information continuous and reliable but also strengthens the decision rationality and efficiency in the setting of TrNZNs.


Introduction
In fuzzy decision-making problems, various new fuzzy decision-making methods [1][2][3] have received many applications under neutrosophic, simplified neutrosophic hesitant fuzzy, and bipolar neutrosophic environments. en, triangular and trapezoidal fuzzy numbers are usually used for real decision-making problems because they can be depicted by the continuous fuzzy numbers of membership functions rather than exact/discrete fuzzy values. Hence, some researchers extended triangular fuzzy numbers to intuitionistic fuzzy sets (IFSs) and presented triangular intuitionistic fuzzy sets (TIFSs), where the values of the membership and nonmembership functions are triangular fuzzy numbers, and some triangular intuitionistic fuzzy aggregation operators for multicriteria decision-making (MDM) problems with triangular intuitionistic fuzzy information [4][5][6][7]. As the extension of TIFSs, Ye [8] introduced a trapezoidal intuitionistic fuzzy set (TrIFS), in which the values of its membership and nonmembership functions are trapezoidal fuzzy numbers rather than triangular fuzzy numbers, and some prioritized weighted aggregation operators of trapezoidal intuitionistic fuzzy numbers (TrIFNs) for MDM problems with TrIFNs. However, TIFSs and TrIFSs cannot depict inconsistence and indeterminacy information. Hence, Ye [9] generalized TrIFS and proposed a trapezoidal neutrosophic set (TrNS), in which the values of its truth, falsity, and indeterminacy membership functions are trapezoidal fuzzy numbers, to express incomplete, indeterminate, and inconsistent information, and then he presented some basic operations of trapezoidal neutrosophic numbers (TrNNs), score and accuracy functions of TrNNs, and TrNN weighted arithmetic averaging (TrNNWAA) and TrNN weighted geometric averaging (TrNNWGA) operators for MDM problems in the setting of TrNNs. en, some researchers utilized the integrated approach [10] and defuzzification method [11] for the evaluation and MDM problems with interval-valued TrNNs. Further, Giri et al. [12] applied TOPSIS method in MDM problems with interval-valued TrNNs. Also, Jana et al. [13] and Khatter [14] presented some basic operations of interval-valued TrNNs, score and accuracy functions of an interval-valued TrNN, and the interval-valued TrNNWAA and TrNNWGA operators for MDM problems in the setting of interval-valued TrNNs. e notion of a Z-number introduced by Zadeh [15] is described by a fuzzy number and its reliability measure to strengthen the reliability of the fuzzy information. After that, Z-numbers have been used for many areas [16][17][18][19][20][21][22]. Based on the truth, falsity, and indeterminacy Z-numbers, Du et al. [23] extended the Z-number concept and proposed neutrosophic Z-numbers (NZNs) to enhance the reliability of the neutrosophic information, and then they presented basic operations of NZNs, score and accuracy functions of NZN, and the NZN weighted geometric averaging (NZNWGA) and NZN weighted arithmetic averaging (NZNWAA) operators and further established their MDM method under the environment of NZNs. However, TrNN is described only by the trapezoidal fuzzy numbers of its truth, falsity, and indeterminacy membership functions without considering their reliability measures, while NZN is depicted only by exact/discrete truth, falsity, and indeterminacy Z-numbers rather than continuous Z-numbers. Hence, TrNN and NZN and their MDM methods reveal their insufficiencies in their information expressions and applications. To express both the continuous Z-numbers of truth, falsity, and indeterminacy membership functions and the reliability measures in MDM problems, it is necessary that this study needs to propose an MDM method based on trapezoidal neutrosophic Z-numbers (TrNZNs) to make up such insufficiencies of existing information expressions and MDM methods in the environments of TrNNs and NZNs. To do so, the main aims of this article are (1) to propose a TrNZN set and some basic operations of TrNZNs, (2) to introduce score and accuracy functions of TrNZN for ranking TrNZNs, (3) to put forward the TrNZNWAA and TrNZNWGA operators for aggregating TrNZNs, (4) to develop a MDM approach using the proposed aggregation operators and score and accuracy functions for solving MDM problems under the environment of TrNZNs, and (5) to apply the established MDM approach to an MDM example of software selection for revealing its efficiency in the setting of TrNZNs. e rest of the article is composed of the following sections. Section 2 introduces some basic notions of TrNNs as preliminaries of this study. Section 3 proposes a TrNZN set, basic operations of TrNZNs, the score and accuracy functions of TrNZN, and their ranking laws of TrNZNs. en, the TrNZNWAA and TrNZNWGA operators and their relative properties are presented in section 4. Section 5 develops an MDM approach using the TrNZNWAA and TrNZNWGA operators and score and accuracy functions of TrNZNs. In Section 6, the developed MDM approach is applied to an MDM example of software selection to indicate its efficiency in the setting of TrNZNs. In the end, conclusions and further study are contained in Section 7.

Preliminaries of TrNSs
In this section, we introduce preliminaries of TrNSs, including TrNNs, operations of TrNNs, two TrNN weighted aggregation operators, and score and accuracy functions of TrNNs for ranking TrNNs.

Trapezoidal Neutrosophic Z-Number (TrNZN) Sets
To make trapezoidal neutrosophic information reliable, this section gives the following definitions of a TrNZN set, operations of TrNZNs, score and accuracy functions of TrNZN, and ranking laws of TrNZNs.
)> as two TrNZNs. en they are defined as the following basic operations: For ranking TrNZNs, the score and accuracy functions of TrNZN are defined according to the expected value of a trapezoidal fuzzy number and score and accuracy functions of TrNN [9].
en the score and accuracy functions of the TrNZN z 1 can be defined as follows: Based on equations (7) and (8), ranking laws between two TrNZNs are given by the following definition.
))> as two TrNZNs. en, the ranking laws between two TrNZNs are defined as follows:

Weighted Aggregation Operators of TrNZNs
Regarding information aggregation in MDM problems, one usually utilizes the weighted arithmetic and geometric averaging operators as the most basic information aggregation approaches. To aggregate TrNZNs, therefore, this section proposes the two following weighted aggregation operators of TrNZNs based on the basic operations of TrNZNs in Definition 2.

Weighted Arithmetic Averaging Operator of TrNZNs
. .,n) as a series of TrNZNs. en, the TrNZNWAA operator is defined as where λ j (j � 1, 2,. . .,n) is the weight of the jth TrNZN z j (j � 1, 2,. . .,n) for λ j ∈ [0, 1] and n j�1 λ j � 1. Based on the basic operations of TrNZNs in Definition 2 and equation (9), we have the following theorem. 1, 2,. . .,n) as a series of TrNZNs. en, the aggregated value of equation (9) is also TrNZN, which is yielded by the following equation: where Proof. e proof of equation (10) can be given by mathematical induction.
(1) Set n � 2. en there is the following result: Journal of Mathematics 5 (2) Set n � k. en, equation (10) can hold in the following equation: (3) Set n � k + 1. By equations (11) and (12), we can obtain Regarding the above results, equation (10) can hold for any n. us, the proof is completed.

Journal of Mathematics
I V1 j � 1 n w j , I V2 j � 1 n w j , I V3 j � 1 n w j , I V4 j � 1 n w j ⎛ ⎝ ⎞ ⎠ , ⎛ ⎝ I R1 j � 1 n w j , I R2 j � 1 n w j , I R3 j � 1 n w j , I R4 j � 1 n w j ⎛ ⎝ ⎞ ⎠⎞ ⎠ , . us, the proof of these properties is completed.  1, 2, . . ., n) as a series of TrNZNs. en, the TrNZNWGA operator is defined as where λ j (j � 1, 2,. . .,n) is the weight of the jth TrNZN z j for λ j ∈ [0, 1] and n j�1 λ j � 1. Regarding the basic operations of TrNZNs in Definition 2 and equation (16), we can give the theorem below. en, the aggregated value of the TrNZNWGA operator is also TrNZN, which is obtained by where λ j (j � 1, 2, . . ., n) is the weight of the jth TrNZN z j for λ j ∈ [0, 1] and n j�1 λ j � 1. Based on the similar proof process of eorem 1, we can verify eorem 3, which is omitted.

MDM Approach Using the TrNZNWAA and TrNZNWGA Operators and Score and Accuracy Functions
is section establishes an MDM approach by using the TrNZNWAA and TrNZNWGA operators and score and accuracy functions to handle MDM problems with TrNZN information.
Regarding an MDM problem with TrNZN information, a set of alternatives Q � {Q 1 , Q 2 , . . ., Q m } are commonly presented and satisfactorily assessed by a set of criteria S � {s 1 , s 2 , . . ., s n }. Each alternative over criteria is assessed by decision makers and then their given assessment values are expressed in the form of TrNZNs z ij � <((T Vij1 , T Vij2 , T Vij3 , T Vij4 ), (T Rij1 , T Rij2 , T Rij3 , T Rij4 )), ((I Vij1 , I Vij2 , I Vij3 , I Vij4 ), (I Rij1 , indicate the falsity degrees and reliability measures of the alternative Q i over the criteria s j , along with 0 ≤ T Vij4 + I Vij4 + F Vij4 ≤ 3 and 0 ≤ T Rij4 + I Rij4 + F Rij4 ≤ 3 for j � 1, 2, . . ., n and i � 1, 2, . . ., m. en, all the specified TrNZNs are constructed as their decision matrix Z � (z ij ) m×n . us, the TrNZNWAA and TrNZNWGA operators and the score and accuracy functions can be applied to MDM problems with TrNZN information, and then their MDM approach can be indicated by the following procedures: Step 1: the aggregated TrNZN z i for Q i (i � 1, 2, . . ., m) is obtained by applying the TrNZNWAA or TrNZNWGA operator:

Journal of Mathematics
Step 2: by equation (7), we calculate the score values of S(z i ). If necessary, we calculate the accuracy values of H(z i ) (i � 1, 2, . . ., m) by equation (8).
Step 3: all the alternatives Q i (i � 1, 2, . . ., m) are ranked corresponding to the score values (the accuracy values) and the best one(s) is chosen in the set of alternatives.

MDM Example of Software Selection.
is section indicates an MDM example of software selection adapted from [9] to reveal the usability and efficiency of the established MDM approach under the environment of TrNZNs.
In an MDM example, an investment company needs to select a suitable software system from potential software systems, where five candidate software systems are provided preliminarily and denoted as a set of five alternatives Q � {Q 1 , Q 2 , Q 3 , Q 4 , Q 5 }. en, these alternatives must satisfy the requirements of the four criteria: s 1 (the contribution to organization performance), s 2 (the effort to transform from current system), s 3 (the costs of hardware/software investment), and s 4 (the outsourcing software developer reliability). Regarding the importance of the four criteria, the weight values of the four criteria are specified as the weight vector λ � (0.25, 0.25, 0.3, 0.2). us, decision makers/experts assess the satisfiability of the five alternatives over the four criteria by TrNZNs  (2) and (3)) and the score function of TrNNs (equation (4)) are introduced from [9], which are shown in Table 2.
Based on the decision results in Tables 1 and 2, we can see that the ranking orders based on the established MDM approach and the existing MDM approach [9] reveal their difference, but the best alternative Q 4 (the best software system) is identical. en, the reason for their ranking difference is that decision information in the existing MDM approach [9] only contains TrNNs without considering the reliability measures of TrNNs in this MDM example, while decision information in the established MDM approach contains both TrNNs and their reliability measures. Hence, different decision information can result in different ranking results. It is obvious that the reliability measures in this example can affect the ranking order of alternatives, which shows the efficiency and rationality of the established MDM approach under the environment of TrNZNs.
However, the different decision information and decision methods can have an impact on the ranking of alternatives in the MDM problem, which reveals their importance in MDM applications.
us, existing MDM methods [11][12][13][14]23] only contain the TrNN or NZN information without considering the reliability measures in TrNNs or continuous Z-numbers in NZNs; they may lose some useful decision information so as to result in decision distortion/unreasonable decision results, which reveal some insufficiencies, while the new established approach can contain much more information than existing MDM methods and overcome the insufficiencies. Furthermore, existing methods [11][12][13][14]23] also cannot deal with such MDM problems with TrNZNs.
Based on the above comparative analysis, the new established approach in setting of TrNZNs not only makes assessment information of TrNNs more reliable but also strengthens the effectiveness and continuity of decision information by comparison with existing MDM methods with TrNN and NZN information [9,[11][12][13][14]23], which reveals the highlighting advantages of the new established approach in the information representation and MDM applications. erefore, the new established approach not only extends existing methods but also demonstrates its superiority over them.

Conclusion
To make TrNN reliable, this paper presented a TrNZN set based on the truth, falsity, and indeterminacy trapezoidal Z-numbers as the generalization of the Z-number concept and then defined basic operations of TrNZNs, score and accuracy functions of TrNZNs, and ranking laws of TrNZNs. Next, the TrNZNWAA and TrNZNWGA operators were proposed to aggregate the TrNZN information. Furthermore, an MDM approach based on the two aggregation operators and score and accuracy functions was established in the setting of TrNZNs, in which the assessment values of alternatives over the criteria take the form of TrNZNs containing TrNNs and their reliability measures. Finally, an MDM example of software selection was provided to reveal the suitability and efficiency of the established MDM approach in the setting of TrNZNs.
e main advantage of this study is that the established method not only makes assessment information of TrNNs more reliable but also strengthens the decision rationality and efficiency in solving MDM problems with TrNZN information. However, the established method only uses the basic aggregation algorithms of TrNZNWAA and TrNZNWGA for MDM problems without considering the interactions of some evaluation criteria with each other, which implies the limitation of the proposed method in MDM applications. For capturing these relationships, the future study is to develop other aggregation algorithms and to use them for some other MDM problems including slope design schemes, energy and environmental managements, and medicine options.

Data Availability
ere are no underlying data supporting the results of your study.

Conflicts of Interest
e authors declare no conflicts of interest.