It is known that fuzzy sets proposed by Zadeh [
1] play an essential role in the current scientific and technical applications [
2‐
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‐
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‐
34], medical diagnosis [
35‐
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‐
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:
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.
$$ S_{Z} = \left\{ {\left\langle {x,T(V,R)(x),I(V,R)(x),F(V,R)(x)} \right\rangle |x \in X} \right\}, $$
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.
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:
$$ Y(s_{Zi} ) = \frac{{2 + T_{Vi} T_{Ri} - I_{Vi} I_{Ri} - F_{Vi} F_{Ri} }}{3}\;\;{\text{for}}\;\;Y(s_{Zi} ) \in [0,1] $$
(1)
Thus, if
Y(
sZ1) ≥
Y(
sZ2), there is the ranking
sZ1 ≥
sZ2.
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.