Abstract
As the generalization of the classical fuzzy number, the concept of Z-number introduced by Zadeh indicates more ability to depict the human knowledge and judgments of both restraint and reliability as an order pair of fuzzy numbers. In indeterminacy and inconsistent environment, a neutrosophic set is described by the truth, falsity, and indeterminacy degrees, but they lack measures related to reliability. To describe the hybrid information of combining the truth, falsity and indeterminacy degrees with their corresponding reliability degrees, this paper first proposes the concept of a neutrosophic Z-number (NZN) set, which is a new framework of neutrosophic values combined with the neutrosophic measures of reliability, as the generalization of the Z-number and the neutrosophic set. Then, we define the operations of neutrosophic Z-numbers (NZNs) and a score function for ranking NZNs. Next, we present NZN weighted arithmetic averaging (NZNWAA) and NZN weighted geometric averaging (NZNWGA) operators to aggregate NZN information and investigate their properties. Regarding the NZNWAA and NZNWGA operators and the score function, a multicriteria decision making (MDM) approach is developed in the NZN environment. Finally, an illustrative example about the selection problem of business partners is given to demonstrate the applicability and effectiveness of the developed MDM approach in NZN setting.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Introduction
It is known that fuzzy sets proposed by Zadeh [1] play an essential role in the current scientific and technical applications [2,3,4,5,6,7]. In 2011, Zadeh [8] further introduced the concept of Z-numbers to describe the restraint and reliability of the evaluation by an order pair of fuzzy numbers in uncertain situations. Compared with the classical fuzzy number, it is a more generalized notion closely related to reliability. Hence, the Z-number implies more ability to describe the human knowledge and judgments by an order pair of fuzzy numbers corresponding to the restriction and reliability. Since then, it has obtained a lot of attentions. Some researchers presented theoretical studies of Z-numbers, like Z*-numbers [9], arithmetic operations of discrete and continuous Z-numbers [10, 11], modeling of Z-number [12], approximate reasoning of Z-numbers [13], functions based on a Z-number set [14], total utility of Z-numbers [15] and so on; while other researchers developed some applications of Z-numbers, such as Z-evaluations [16], sensor data fusion using Z-numbers [17], decision making approaches with Z-numbers [18,19,20,21,22,23,24], Z-numbers-based stable strategies analysis in evolutionary game [25], Z-numbers-based medicine selection of the patients with mild symptoms of the COVID-19 [26], Z-numbers-based environmental assessment under uncertainty [27] and so on.
In indeterminate and inconsistent environment, neutrosophic sets [28, 29] are described independently by the truth, falsity, and indeterminacy membership degrees, but the aforementioned Z-numbers cannot depict them. Then, neutrosophic sets have been applied in various areas, such as image processing [30], decision making [31,32,33,34], medical diagnosis [35,36,37], and mechanical fault diagnosis [38]. However, the truth, falsity, and indeterminacy membership degrees in the neutrosophic set lack the reliability measures related to them. If the Z-number notion is extended to the neutrosophic set, we can describe the hybrid information of combining the truth, falsity and indeterminacy degrees with their corresponding reliability degrees by three order pairs of fuzzy numbers. In multicriteria decision making (MDM) problems, the information expressions and decision making methods are vital research topics [39,40,41,42]. Motivated based on the ideas of combining the Z-number with the neutrosophic set and enhancing MDM reliability, the objects of this study are to present a more generalized neutrosophic notion closely related to reliability and to use it for MDM problems. To do so, this paper proposes the concept of a neutrosophic Z-number (NZN) set, which is a new framework of neutrosophic values combined with the neutrosophic measures of reliability, as the generalization of the Z-number and the neutrosophic set. Then, we define the operations of neutrosophic Z-numbers (NZNs) and a score function for ranking NZNs and propose NZN weighted arithmetic averaging (NZNWAA) and NZN weighted geometric averaging (NZNWGA) operators to aggregate NZN information. Regarding the NZNWAA and NZNWGA operators and the score function, a MDM approach is developed in the NZN environment. An illustrative example is used to demonstrate the applicability and effectiveness of the developed MDM approach in NZN setting. However, the proposed NZN notion and the developed MDM approach based on the NZNWAA and NZNWGA operators and the score function of NTN shows the novelty of this study.
For the first time study, the main contributions of the article are included as follows:
-
(a)
The proposed NZN set can solve the information expression problem of the truth, falsity and indeterminacy values combined with their related reliability measures by the three order pairs of fuzzy numbers in indeterminate and inconsistent situations.
-
(b)
The defined operations and NZNWAA and NZNWGA operators of NZNs are to realize the aggregation problems of the NZN information and then the score function of NZN is to rank NZNs, which provide the useful mathematical tools for MDM problems in NZN setting.
-
(c)
The developed MDM approach not only enhances the MDM reliability but also provides a new effective way for MDM problems in NZN setting.
The study is organized as the following structures. Section “Neutrosophic Z-number set” presents the notion of a NZN set, operations of NZNs, and a score function of NZN for comparing NZNs. Section “Two weighted aggregation operators of neutrosophic Z-numbers” proposes the NZNWAA and NZNWGA operators and presents their properties. A MDM approach based on the NZNWAA and NZNWGA operators and the score function is developed in section “MDM approach using the NZNWAA and NZNWGA operators and the score function”. In section “An illustrative example and relative comparative analysis”, an illustrative example and the relative comparative analysis are presented to demonstrate the applicability and effectiveness of the developed MDM approach in NZN setting. Lastly, conclusions and further study are presented in section “Conclusion”.
Neutrosophic Z-number set
In 2011, Zadeh [8] firstly introduced the concept of Z-number by an order pair of fuzzy numbers Z = (V, R) associated with a real-valued uncertain variable X, where the first component V is a fuzzy restriction on the values that X can take and the second component R is a measure of reliability for V.
Based on an extension of the Z-number concept [8] and the neutrosophic set, we can give the definition of a NZN set.
Definition 1
Set X as a universe set. Then a NZN set in a universe set X is defined as the following form:
where T(V, R)(x) = (TV(x), FR(x)), I(V, R)(x) = (IV(x), IR(x)), F(V, R)(x) = (FV(x), FR(x)): X → [0, 1]2 are the order pairs of truth, indeterminacy and falsity fuzzy values, then the first component V is neutrosophic values in a universe set X and the second component R is neutrosophic measures of reliability for V, along with the conditions \( 0 \le T_{V} (x) + I_{V} (x) + F_{V} (x) \le 3 \) and \( 0 \le T_{R} (x) + I_{R} (x) + F_{R} (x) \le 3 \).
For the convenient representation, the element \( \left\langle {x,T(V,R)(x),I(V,R)(x),F(V,R)(x)} \right\rangle \) in SZ is simply denoted as \( s_{Z} = \left\langle {T(V,R),I(V,R),F(V,R)} \right\rangle = \left\langle {(T_{V} ,T_{R} ),(I_{V} ,I_{R} ),(F_{V} ,F_{R} )} \right\rangle \), which is named NZN.
Definition 2
Let \( s_{Z1} = \left\langle {T_{1} (V,R),I_{1} (V,R),F_{1} (V,R)} \right\rangle = \left\langle {\left( {T_{V1} ,T_{R1} } \right),\left( {I_{V1} ,I_{R1} } \right),\left( {F_{V1} ,F_{R1} } \right)} \right\rangle \) and \( s_{Z2} = \left\langle {T_{2} (V,R),I_{2} (V,R),F_{2} (V,R)} \right\rangle = \left\langle {\left( {T_{V2} ,T_{R2} } \right),\left( {I_{V2} ,I_{R2} } \right),\left( {F_{V2} ,F_{R2} } \right)} \right\rangle \) be two NZNs and λ > 0. Then, we give the following relations:
-
1.
sZ1 ⊇ sZ2 ⇔ TV1 ≥ TV2, TR1 ≥ TR2, IV1 ≤ IV2, IR1 ≤ IR2, FV1 ≤ FV2, and FR1 ≤ FR2;
-
2.
sZ1 = sZ2 ⇔ sZ1 ⊇ sZ2 and sZ2 ⊇ sZ1;
-
3.
\( s_{Z1} \cup s_{Z2} = \Big\langle (T_{V1}^{{}} \vee T_{V2}^{{}} ,T_{R1}^{{}} \vee T_{R2}^{{}} ),(I_{V1}^{{}} \wedge I_{V2}^{{}} ,I_{R1}^{{}} \wedge I_{R2}^{{}} ),(F_{V1}^{{}} \wedge F_{V2}^{{}} ,F_{R1}^{{}} \wedge F_{R2}^{{}} ) \Big\rangle \);
-
4.
\( s_{Z1} \cap s_{Z2} = \Big\langle (T_{V1}^{{}} \wedge T_{V2}^{{}} ,T_{R1}^{{}} \wedge T_{R2}^{{}} ),(I_{V1}^{{}} \vee I_{V2}^{{}} ,I_{R1}^{{}} \vee I_{R2}^{{}} ),(F_{V1}^{{}} \vee F_{V2}^{{}} ,F_{R1}^{{}} \vee F_{R2}^{{}} ) \Big\rangle \);
-
5.
\( (s_{Z1} )^{C} = \left\langle {\left( {F_{V1} ,F_{R1} } \right),\left( {1 - I_{V1} ,1 - I_{R1} } \right),\left( {T_{V1} ,T_{R1} } \right)} \right\rangle \) (Complement of sZ1);
-
6.
\( s_{Z1} \oplus s_{Z2} = \Big\langle \left( {T_{V1} + T_{V2} - T_{V1} T_{V2} ,T_{R1} + T_{R2} - T_{R1} T_{R2} } \right),\left( {I_{V1} I_{V2} ,I_{R1} I_{R2} } \right),\left( {F_{V1} F_{V2} ,F_{R1} F_{R2} } \right) \Big\rangle \);
-
7.
\( {s_{Z1}} \otimes {s_{Z2}} = \left\langle \begin{gathered} \left( {{T_{V1}} {T_{V2}} ,{T_{R1}} {T_{R2}} } \right),\left( {{I_{V1}} + {I_{V2}} -{ I_{V1}} {I_{V2}}, {I_{R1}} + {I_{R2}} -{ I_{R1}} {I_{R2}} } \right), \hfill \\ \left( {{F_{V1}} + {F_{V2}} -{ F_{V1}} {F_{V2}} ,{F_{R1}} + {F_{R2}} - {F_{R1}} {F_{R2}} } \right) \hfill \\ \end{gathered} \right\rangle \);
-
8.
\( \lambda s_{Z1} = \Big\langle \left( {1 - (1 - T_{V1} )^{\lambda } ,1 - (1 - T_{R1} )^{\lambda } } \right),\left( {I_{V1}^{\lambda } ,I_{R1}^{\lambda } } \right),\left( {F_{V1}^{\lambda } ,F_{R1}^{\lambda } } \right) \Big\rangle \);
-
9.
\( s_{Z1}^{\lambda } = \Big\langle \left( {T_{V1}^{\lambda } ,T_{R1}^{\lambda } } \right),\left( {1 - (1 - I_{V1} )^{\lambda } ,1 - (1 - I_{R1} )^{\lambda } } \right),\left( {1 - (1 - F_{V1} )^{\lambda } ,1 - (1 - F_{R1} )^{\lambda } } \right) \Big\rangle \).
To compare NZNs \( s_{Zi} = \big\langle T_{i} (V,R),I_{i} (V,R),F_{i} (V,R) \big\rangle = \left\langle {\left( {T_{Vi} ,T_{Ri} } \right),\left( {I_{Vi} ,I_{Ri} } \right),\left( {F_{Vi} ,F_{Ri} } \right)} \right\rangle \) (i = 1, 2), we introduce a score function:
Thus, if Y(sZ1) ≥ Y(sZ2), there is the ranking sZ1 ≥ sZ2.
Example 1
Set two NZNs as sZ1 = < (0.7, 0.8), (0.1, 0.7), (0.3, 0.8) > and sZ2 = < (0.6, 0.9), (0.3, 0.8), (0.2, 0.7) > . Then, their ranking is given as follows:
By Eq. (1), we have Y(sZ1) = (2 + 0.7 × 0.8−0.1 × 0.7−0.3 × 0.8)/3 = 0.75 and Y(sZ2) = (2 + 0.6 × 0.9−0.3 × 0.8−0.2 × 0.7)/3 = 0.72. Since Y(sZ1) > Y(sZ2), their ranking is sZ1 > sZ2.
Two weighted aggregation operators of neutrosophic Z-numbers
Based on the operations (6)–(9) in Definition 2, we can propose the two weighted aggregation operators of NZNs in this section.
NZNWAA operator
Based on the operations (6) and (8) in Definition 2, we can present the NZNWAA operator of NZNs.
Definition 3
Let \( s_{Zi} = \left\langle {T_{i} (V,R),I_{i} (V,R),F_{i} (V,R)} \right\rangle = \left\langle {\left( {T_{Vi} ,T_{Ri} } \right),\left( {I_{Vi} ,I_{Ri} } \right),\left( {F_{Vi} ,F_{Ri} } \right)} \right\rangle \) (i = 1, 2, …, n) be a group of NZNs and NZNWAA: Ωn → Ω. Then, the NZNWAA operator is defined as
where λi (i = 1, 2, …, n) is the weight of sZi with 0 ≤ λi ≤ 1 and \( \sum\nolimits_{i = 1}^{n} {\lambda_{i} = 1} \).
Theorem 1
Let \( s_{Zi} = \left\langle {T_{i} (V,R),I_{i} (V,R),F_{i} (V,R)} \right\rangle = \left\langle {\left( {T_{Vi} ,T_{Ri} } \right),\left( {I_{Vi} ,I_{Ri} } \right),\left( {F_{Vi} ,F_{Ri} } \right)} \right\rangle \) (i = 1, 2, …, n) be a group of NZNs. Then, the collected value of the NZNWAA operator is a NZN, which is obtained by the following formula:
where λi is the weight of sZi (i = 1, 2, …, n) with 0 ≤ λi ≤ 1 and \( \sum\nolimits_{i = 1}^{n} {\lambda_{i} = 1} \).
Proof
Regarding mathematical induction, Eq. (3) is verified below.
-
1.
If n = 2, according to the operations (6) and (8) in Definition 2 we yield the following result:
-
2.
If n = m, Eq. (3) has the following form:
-
3.
If n = m + 1, according to the operations (6) and (8) in Definition 2 and Eqs. (4) and (5), there is the following result:
Based on the above results, Eq. (3) can keep for any n.
Thus, the verification is finished.□
Theorem 2
The NZNWAA operator of Eq. (3) implies the following properties:
-
1.
Idempotency: Set \( s_{Zi} = \left\langle {T_{i} (V,R),I_{i} (V,R),F_{i} (V,R)} \right\rangle = \left\langle {\left( {T_{Vi} ,T_{Ri} } \right),\left( {I_{Vi} ,I_{Ri} } \right),\left( {F_{Vi} ,F_{Ri} } \right)} \right\rangle \) (i = 1, 2,…, n) as a group of NZNs. If sZi= sZ (i = 1, 2,…, n), there is \( {\rm NZNWAA}\left( {s_{Z1} ,s_{Z2} , \cdots ,s_{Zn} } \right) = s_{Z} \).
-
2.
Boundedness: Set \( s_{Zi} = \left\langle {T_{i} (V,R),I_{i} (V,R),F_{i} (V,R)} \right\rangle = \left\langle {\left( {T_{Vi} ,T_{Ri} } \right),\left( {I_{Vi} ,I_{Ri} } \right),\left( {F_{Vi} ,F_{Ri} } \right)} \right\rangle \) (i = 1, 2, …, n) as a group of NZNs and let
Then, \( s_{Z\hbox{min} } \le {\rm NZNWAA}(s_{Z1} ,s_{Z2} , \cdots ,s_{Zn} ) \le s_{Z\hbox{max} } \) can keep.
-
3.
Monotonicity: Set \( s_{Zi} = \big\langle T_{i} (V,R),I_{i} (V,R),F_{i} (V,R) \big\rangle = \left\langle {\left( {T_{Vi} ,T_{Ri} } \right),\left( {I_{Vi} ,I_{Ri} } \right),\left( {F_{Vi} ,F_{Ri} } \right)} \right\rangle \) and \( s_{Zi}^{*} = \left\langle {T_{i}^{*} (V,R),I_{i}^{*} (V,R),F_{i}^{*} (V,R)} \right\rangle = \left\langle {\left( {T_{Vi}^{*} ,T_{Ri}^{*} } \right),\left( {I_{Vi}^{*} ,I_{Ri}^{*} } \right),\left( {F_{Vi}^{*} ,F_{Ri}^{*} } \right)} \right\rangle \) (i = 1, 2, …, n) as two groups of NZNs. When \( s_{Zi} \le s_{Zi}^{*} \), there is \({\rm NZNWAA}(s_{Z1} ,s_{Z2} , \cdots ,s_{Zn} ) \le NZNWAA(s_{Z1}^{*} ,s_{Z2}^{*} , \cdots ,s_{Zn}^{*} ) \).
Proof
-
1.
If sZi = sZ (i = 1, 2,…, n), the result of Eq. (3) is given by
-
2.
Since sZmin and sZmax are given by the minimum NZN and the maximum NZN, the inequality sZmin ≤ sZi ≤ sZmax exists. Thus, there is \( \sum\nolimits_{i = 1}^{n} {\lambda_{i} s_{Z\hbox{min} } } \le \sum\nolimits_{i = 1}^{n} {\lambda_{i} s_{Zi} } \le \sum\nolimits_{i = 1}^{n} {\lambda_{i} s_{Z\hbox{max} } } \). Based on the above property (1), \( s_{Z\hbox{min} } \le \sum\nolimits_{i = 1}^{n} {\lambda_{i} s_{Zi} \le } s_{Z\hbox{max} } \) can exist, i.e., there is \( s_{Z\hbox{min} } \le NZNWAA\left( {s_{Z1} ,s_{Z2} , \cdots ,s_{Zn} } \right) \le s_{Z\hbox{max} } \).
-
3.
Since \( s_{Zi} \le s_{Zi}^{*} \), there is \( \sum\nolimits_{i = 1}^{n} {\lambda_{i} s_{Zi} } \le \sum\nolimits_{i = 1}^{n} {\lambda_{i} s_{Zi}^{*} } \), i.e., \( {\rm NZNWAA}\left( {s_{Z1} ,s_{Z2} , \cdots ,s_{Zn} } \right) \le NZNWAA\left( {s_{Z1}^{*} ,s_{Z2}^{*} , \cdots ,s_{Zn}^{*} } \right) \).
Thus, the verification of all properties is completed.□
NZNWGA operator
Based on the operations (7) and (9) in Definition 2, we can present the NZNWGA operator of NZNs.
Definition 4
Let \( s_{Zi} = \left\langle {T_{i} (V,R),I_{i} (V,R),F_{i} (V,R)} \right\rangle = \left\langle {\left( {T_{Vi} ,T_{Ri} } \right),\left( {I_{Vi} ,I_{Ri} } \right),\left( {F_{Vi} ,F_{Ri} } \right)} \right\rangle \) (i = 1, 2, …, n) be a group of NZNs and NZNWGA: Ωn → Ω. Then, the NZNWGA operator is defined as
where λi (i = 1, 2, …, n) is the weight of sZi with 0 ≤ λi ≤ 1 and \( \sum\nolimits_{i = 1}^{n} {\lambda_{i} = 1} \).
Theorem 3
Let \( s_{Zi} = \left\langle {T_{i} (V,R),I_{i} (V,R),F_{i} (V,R)} \right\rangle = \left\langle {\left( {T_{Vi} ,T_{Ri} } \right),\left( {I_{Vi} ,I_{Ri} } \right),\left( {F_{Vi} ,F_{Ri} } \right)} \right\rangle \) (i = 1, 2, …, n) be a group of NZNs. Then, the collected value of the NZNWGA operator is a NZN, which is obtained by the following formula:
where λi is the weight of sZi (i = 1, 2, …, n) with 0 ≤ λi ≤ 1 and \( \sum\nolimits_{i = 1}^{n} {\lambda_{i} = 1} \).
By the similar verification process of Theorem 1, we can also verify that the NZNWGA operator of Eq. (7) is true, which is not repeated here.
Theorem 4
The NZNWGA operator of Eq. (7) also implies the following properties:
-
1.
Idempotency: Set \( s_{Zi} = \left\langle {T_{i} (V,R),I_{i} (V,R),F_{i} (V,R)} \right\rangle = \left\langle {\left( {T_{Vi} ,T_{Ri} } \right),\left( {I_{Vi} ,I_{Ri} } \right),\left( {F_{Vi} ,F_{Ri} } \right)} \right\rangle \) (i = 1, 2,…, n) as a group of NZNs. If sZi= sZ (i = 1, 2,…, n), there is \( {\rm NZNWGA}\left( {s_{Z1} ,s_{Z2} , \cdots ,s_{Zn} } \right) = s_{Z} \).
-
2.
Boundedness: Set \( s_{Zi} = \big\langle T_{i} (V,R),I_{i} (V,R),F_{i} (V,R) \big\rangle = \left\langle {\left( {T_{Vi} ,T_{Ri} } \right),\left( {I_{Vi} ,I_{Ri} } \right),\left( {F_{Vi} ,F_{Ri} } \right)} \right\rangle \) (i = 1, 2, …, n) as a group of NZNs and let
Then, \( s_{Z\hbox{min} } \le NZNWGA(s_{Z1} ,s_{Z2} , \ldots ,s_{Zn} ) \le s_{Z\hbox{max} } \) can keep.
-
3.
Monotonicity: Set \( s_{Zi} = \big\langle T_{i} (V,R),I_{i} (V,R),F_{i} (V,R) \big\rangle = \left\langle {\left( {T_{Vi} ,T_{Ri} } \right),\left( {I_{Vi} ,I_{Ri} } \right),\left( {F_{Vi} ,F_{Ri} } \right)} \right\rangle \) and \( s_{Zi}^{*} = \left\langle {T_{i}^{*} (V,R),I_{i}^{*} (V,R),F_{i}^{*} (V,R)} \right\rangle = \left\langle {\left( {T_{Vi}^{*} ,T_{Ri}^{*} } \right),\left( {I_{Vi}^{*} ,I_{Ri}^{*} } \right),\left( {F_{Vi}^{*} ,F_{Ri}^{*} } \right)} \right\rangle \) (i = 1, 2, …, n) as two groups of NZNs. When \( s_{Zi} \le s_{Zi}^{*} \), there is \( {\rm NZNWGA}(s_{Z1} ,s_{Z2} , \ldots ,s_{Zn} ) \le NZNWGA\left( {s_{Z1}^{*} ,s_{Z2}^{*} , \ldots ,s_{Zn}^{*} } \right) \).
Obviously, the above properties corresponding to the NZNWGA operator can be also verified by the similar verification process of Theorem 2, which is not repeated here.
MDM approach using the NZNWAA and NZNWGA operators and the score function
Regarding the proposed NZNWAA and NZNWGA operators and the score function, this section develops a MDM approach to solve MDM problems with the evaluation information of both neutrosophic values and neutrosophic measures of reliability related to the neutrosophic values in NZN setting.
Suppose that in a MDM problem a set of m alternatives Q = {Q1, Q2, …, Qm} is presented and assessed by a set of n criteria X = {x1, x2, …, xn}. Then, the impotence of each criterion xi (i = 1, 2, …, n) is considered by the weight λi, which is constructed as the weight vector λ = (λ1, λ2, …, λn). Decision makers are requested to give the suitability assessment of each criterion xi (i = 1, 2, …, n) for each alternative Qj (j = 1, 2, …, m) by both the truth, falsity, indeterminacy fuzzy values and the measures of corresponding reliabilities, which are constructed as a NZN \( s_{Zji} = \left\langle {T_{ji} (V,R),I_{ji} (V,R),F_{ji} (V,R)} \right\rangle = \left\langle {\left( {T_{Vji} ,T_{Rji} } \right),\left( {I_{Vji} ,I_{Rji} } \right),\left( {F_{Vji} ,F_{Rji} } \right)} \right\rangle \), where \( T_{Vji} ,I_{Vji} ,F_{Vji} \in [0,1] \) and \( T_{Rji} ,I_{Rji} ,F_{Rji} \in [0,1] \). Thus, the decision matrix of NZNs can be represented as SZ = (sZji)m×n. In this MDM problem, the decision process is described below:
or
- Step 2:
-
The score values of Y(sZj) (j = 1, 2, …, m) are calculated using Eq. (1)
- Step 3:
-
The alternatives are ranked based on the score values and the best one is chosen among ones
- Step 4:
-
End
An illustrative example and relative comparative analysis
An illustrative example
This part provides an illustrative example about the selection problem of business partners to demonstrate the applicability and effectiveness of the developed MDM approach with NZN information.
Suppose a manufacturing company needs to choose a suitable supplier in potential business partners. The expert panel provides a set of four suppliers/alternatives Q = {Q1, Q2, Q3, Q4} from potential business partners, which must satisfy the assessment requirements of the three criteria: (1) x1 is the cost of product; (2) x2 is the quality of product; (3) x3 is the quality of service. The weigh vector of the three criteria is specified as λ= (0.33, 0.35, 0.32) to indicate the importance of the three criteria. Then, the experts/decision makers are invited to assess the four suppliers/alternatives over the three criteria by the NZNs that are composed of their truth, falsity, indeterminacy fuzzy values and the measures of corresponding reliabilities. Thus, all NZNs can be constructed as the following NZN decision matrix:
On the one hand, the developed MDM approach using the NZNWAA operator can be used for this MDM problem and depicted by the following decision process:
- Step 1:
-
Using Eq. (8), the overall collected NZNs sZj (j = 1, 2, 3, 4) are given as follows:
- Step 2:
-
By Eq. (1), the score values of Y(sZj) for the alternative Qj (j = 1, 2, 3, 4) are yielded below:
- Step 3:
-
According to the score values Y(sZ2) > Y(sZ4) > Y(sZ1) > Y(sZ3), the four alternatives are ranked as Q2 > Q4 > Q1 > Q3. Hence, the best supplier is Q2
On the other hand, the developed MDM approach using the NZNWGA operator can be also used for this MDM problem and depicted by the following decision process:
- Step 1’:
-
By Eq. (9), the overall collected NZNs sZj (j = 1, 2, 3, 4) are obtained as follows:
- Step 2’:
-
By Eq. (1), the score values of Y(sZj) for the alternative Qj (j = 1, 2, 3, 4) are given as follows:
- Step 3’:
-
According to the score values Y(sZ2) > Y(sZ4) > Y(sZ1) > Y(sZ3), the four alternatives are ranked as Q2 > Q4 > Q1 > Q3. Thus the best supplier is Q2
Based on the developed MDM approach using the NZNWAA and NZNWGA operators and the score function, we can see that the above two kinds of ranking orders regarding the four alternatives and the best one are identical. Hence, the developed MDM approach is effective.
Relative comparison
For convenient comparison with existing relative method [33], assume we do not consider the assessment measures of corresponding reliabilities in the decision matrix SZ as a special case of the above example. Then, the NZN decision matrix is reduced to the following single-valued neutrosophic decision matrix:
Then, Eqs. (1), (8) and (9) are also reduced to existing score function, single-valued neutrosophic weighted arithmetic averaging (SVNWAA) and the single-valued neutrosophic weighted geometric averaging (SVNWGA) operators [33]:
Based on existing MDM method [33], decision results are given using Eqs. (10)–(12). Thus, all decision results obtained by existing MDM method [33] and the proposed MDM approach are shown in Table 1.
By comparing the developed MDM approach with existing neutrosophic MDM method [33] in Table 1, we see that there is the ranking difference between them. The developed MDM approach based on the NZNWAA or NZNWGA operator indicates that the ranking is Q2 > Q4 > Q1 > Q3 and the best alternative is Q2, while the MDM method based on the NZNWAA or SVNWGA operator [33] indicates that the ranking is Q4 > Q2 > Q1 > Q3 or Q4 > Q1 > Q2 > Q3 and the best alternative is Q4.
Discussion
From the decision results of Table 1, it is obvious that the MDM methods with different decision information can affect the ranking orders. The reason resulting in the different ranking is that the developed MDM approach uses the hybrid assessment information of both neutrosophic values and neutrosophic measures of corresponding reliabilities, while existing neutrosophic MDM method [33] only uses the assessment information of single-valued neutrosophic values without considering the related reliability measures. Clearly, the introduced reliability measures not only can enhance the information expression and credibility of the evaluation results but also can impact on the ranking order of alternatives, which show the effectiveness and rationality of the developed MDM approach. Since NZNs indicate more ability to depict the human knowledge and judgments by neutrosophic values and the reliability measures related to the neutrosophic values, NZNs richen the measure information of reliability related to the neutrosophic values in indeterminate and inconsistent setting. Hence, the information expression of NZN is superior to that of the single neutrosophic value or the single Z-number in MDM problems. Then, the developed MDM approach in this paper indicates more generalized form to extend existing neutrosophic MDM theory and methods because the existing neutrosophic MDM method is only the special case of this study and cannot carry out the MDM problem with NZN information. Therefore, the developed MDM approach can overcome the flaw of existing neutrosophic MDM theory and methods and strengthen the MDM reliability and effectiveness, which show the highlighting advantages under the environment of NZNs.
Conclusion
In the original study, the proposed NZN set can solve the hybrid information expression problem of both neutrosophic values and corresponding reliability measures and overcome the flaw of missing reliability measures in existing neutrosophic set. Then, the proposed basic operations, score function, and NZNWAA and NZNWGA operators of NZNs provided effective and reasonable mathematical tools for the information aggregation of NZNs and MDM modeling in the setting of NZNs. Further, the developed MDM approach using the NZNWAA and NZNWGA operators and the score function can solve MDM problems with NZN information as the extension of existing neutrosophic MDM methods. The decision results of the illustrative example about the selection problem of business partners indicated the applicability and effectiveness of the developed MDM approach in NZN setting.
However, the developed MDM approach extends existing neutrosophic MDM theory and methods and provides a new way for solving MDM problems with NZNs. Since as the first time study, the ranking method based on the score function and the NZNWAA and NZNWGA operators are the most basic algorithms for MDM problems with NZNs, we should continue to propose the new aggregation operators (e.g., Bonferroni mean, Heronian mean, Dombi aggregation operators) and ranking methods of NZNs for improving neutrosophic MDM methods and to extend them to the applications of group decision making, medical diagnosis, pattern recognition, optimization programming etc.
References
Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353
De Luca A, Termini S (1972) A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory. Inf Control 20(4):301–312
Pal NR, Bezdek JC (1994) Measuring fuzzy uncertainty. IEEE Trans Fuzzy Syst 2(2):107–118
Garibaldi JM, Ozen T (2007) Uncertain fuzzy reasoning: a case study in modelling expert decision making. IEEE Trans Fuzzy Syst 15(1):16–30
Pourabdollah A, Wagner C, Aladi JH, Garibaldi JM (2016) Improved uncertainty capture for nonsingleton fuzzy systems. IEEE Trans Fuzzy Syst 24(6):1513–1524
Xiao F (2017) A novel evidence theory and fuzzy preference approach-based multi-sensor data fusion technique for fault diagnosis. Sensors 17:11. https://doi.org/10.3390/s17112504
Zheng H, Deng Y (2018) Evaluation method based on fuzzy relations between Dempster-Shafer belief structure. Int J Intell Syst 33(7):1343–1363
Zadeh LA (2011) A note on z-numbers. Inf Sci 181(14):2923–2932
Banerjee R, Pal SK (2015) Z*-numbers: augmented z-numbers for machine-subjectivity representation. Inf Sci 323:143–178
Aliev RA, Alizadeh AV, Huseynov OH (2015) The arithmetic of discrete z-numbers. Inf Sci 290:134–155
Aliev R, Huseynov O, Zeinalova L (2016) The arithmetic of continuous z-numbers. Inf Sci 373:441–460
Patel P, Khorasani ES, Rahimi S (2016) Modeling and implementation of z-number. Soft Comput 20(4):1341–1364
Aliev RA, Pedrycz W, Huseynov OH, Eyupoglu SZ (2017) Approximate reasoning on a basis of z-number-valued if-then rules. IEEE Trans Fuzzy Syst 25(6):1589–1600
Aliev RA, Pedrycz W, Huseynov OH (2017) Functions defined on a set of z-numbers. Inf Sci 423:353–375
Kang B, Deng Y, Sadiq R (2018) Total utility of z-number. Appl. Intell. 48(3):703–729
Yager RR (2012) On z-valuations using zadeh’s z-numbers. Int J Intell Syst 27(3):259–278
Jiang W, Xie C, Zhuang M, Shou Y, Tang Y (2016) Sensor data fusion with z-numbers and its application in fault diagnosis. Sensors 16(9):1–22
Kang B, Hu Y, Deng Y, Zhou D (2016) A new methodology of multicriteria decision-making in supplier selection based on z-numbers. Math Probl Eng 2016:1–17
Wang JQ, Cao YX, Zhang HY (2017) Multi-criteria decision-making method based on distance measure and choquet integral for linguistic z-numbers. Cogn Comput 9(6):82–842
Aliev RA, Huseynov OH, Serdaroglu R (2016) Ranking of Z-Numbers and its application in decision making. Int J Inf Technol Decis Making 15(06):1503–1519
Jabbarova AI (2017) Application of Z-number concept to supplier selection problem. Procedia Comput Sci 120:473–477
Aboutorab H, Saberi M, Asadabadi MR, Hussain O, Chang E (2018) ZBWM: the Z-number extension of Best Worst Method and its application for supplier development. Expert Syst Appl 107:115–125
Ding XF, Zhu LX, Lu MS, Wang Q, Feng YQ (2020) A novel linguistic Z-number QUALIFLEX method and its application to large group emergency decision making. Sci Program 2020:1631869. https://doi.org/10.1155/2020/1631869
Ye T, Bingyi K (2020) A modified method of generating Z-number based on OWA weights and maximum entropy. Soft Comput. https://doi.org/10.1007/s00500-020-04914-8
Kang B, Chhipi-Shrestha G, Deng Y, Hewage K, Sadiq R (2018) Stable strategies analysis based on the utility of z-number in the evolutionary games. Appl Math Comput 324:202–217
Ren Z, Liao H, Liu Y (2020) Generalized Z-numbers with hesitant fuzzy linguistic information and its application to medicine selection for the patients with mild symptoms of the COVID-19. Comput Ind Eng 145:106517. https://doi.org/10.1016/j.cie.2020.106517
Kang B, Zhang P, Gao Z, Chhipi-Shrestha G, Hewage K, Sadiq R (2020) Environmental assessment under uncertainty using Dempster-Shafer theory and Z-numbers. J Ambient Intel Hum Comput 11(5):2041–2060
Smarandache F (1998) Neutrosophy: neutrosophic probability, set, and logic. American Research Press, Rehoboth
Ye J (2014) A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J Intell Fuzzy Syst 26:2459–2466
Guo Y, Sengur A, Ye J (2014) A novel image thresholding algorithm based on neutrosophic similarity score. Measurement 58:175–186
Zhang HY, Wang JQ, Chen XH (2014) Interval neutrosophic sets and their application in multicriteria decision making problems. Sci World J 2014:15
Liu PD, Wang YM (2014) Multiple attribute decision making method based on single-valued neutrosophic normalized weighted Bonferroni mean. Neural Comput Appl 25(7–8):2001–2010
Peng JJ, Wang JQ, Wang J, Zhang HY, Chen XH (2016) Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems. Int J Syst Sci 47(10):2342–2358
Zhou LP, Dong JY, Wan SP (2019) Two new approaches for multi-attribute group decision-making with interval-valued neutrosophic Frank aggregation operators and incomplete weights. IEEE Access 7:102727–102750
Ye J (2015) Improved cosine similarity measures of simplified neutrosophic sets for medical diagnoses. Artif Intell Med 63(3):171–179
Alia M, Son LH, Thanhc ND, Minh NV (2018) A neutrosophic recommender system for medical diagnosis based on algebraic neutrosophic measures. Appl Soft Comput 71:1054–1071
Nguyen GN, Son LH, Ashour AS et al (2019) A survey of the state-of-the-arts on neutrosophic sets in biomedical diagnoses. Int J Mach Learn Cyber 10:1–13
Ye J (2017) Single valued neutrosophic similarity measures based on cotangent function and their application in the fault diagnosis of steam turbine. Soft Comput 21(3):817–825
Kumar RR, Kumar C (2018) A multi criteria decision making method for cloud service selection and ranking. Int J Ambient Comput Intell 9(3):1–14
Jayakumar L, Dumka A, Janakiraman S (2020) Energy aware dynamic mode decision for cellular D2D communications by using integrated multi-criteria decision making model. Int J Ambient Comput Intell 11(3):131–151
Du SG, Ye J, Yong R, Zhang FW (2020) Simplified neutrosophic indeterminate decision making method with decision makers’ indeterminate ranges. J Civ Eng Manag 26(6):590–598
Ye J, Du SG, Yong R, Zhang FW (2020) Weighted aggregation operators of fuzzy credibility cubic numbers and their decision making strategy for slope design schemes. Curr Chin Comput Sci. https://doi.org/10.2174/2665997201999200717165743
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that we have no conflict of interest regarding the publication of this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Du, S., Ye, J., Yong, R. et al. Some aggregation operators of neutrosophic Z-numbers and their multicriteria decision making method. Complex Intell. Syst. 7, 429–438 (2021). https://doi.org/10.1007/s40747-020-00204-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40747-020-00204-w