EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6753
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Type-2 Neutrosophic Aczel-Alsina Hamy Mean
Aggregation Operators for Multiple-Attribute

Decision-Making Problems
Karahan Kara1,2, Galip Cihan Yalçın2, Vladimir Simic3,∗, Dragan Pamucar4,5,6,∗

1 Department of Business, Faculty of Economics and Administrative Sciences,
İzmir Democracy University, 35140 İzmir, Türkiye
2 Department of Business, Faculty of Economics and Administrative Sciences,
OSTIM Technical University, 06374 Ankara, Türkiye
3 Széchenyi István University, Egyetem tér 1, 9026 Győr, Hungary
4 Department of Applied Mathematical Science, College of Science and Technology,
Korea University, Sejong, 30019, Republic of Korea
5 Faculty of Engineering, Doğuş University, 34775 Umraniye, Istanbul, Türkiye
6 UNEC Applied Artificial Intelligence Research Center, Azerbaijan State University of
Economics (UNEC), Baku, Azerbaijan

Abstract. This study introduces novel aggregation operators for Type-2 Neutrosophic Number
Sets (T2NNs) by integrating the Hamy Mean with the Aczel-Alsina t-norm and t-conorm opera-
tions. The Hamy Mean, a mathematical averaging technique, is particularly effective in contexts
characterized by uncertainty and ambiguity, such as fuzzy set theory. Leveraging this, two aggre-
gation operators are proposed: the T2NN Aczel-Alsina Hamy Mean (T2NNAAHM) and the T2NN
Aczel-Alsina Weighted Hamy Mean (T2NNAAWHM). The study presents the formal definitions,
underlying operations, theoretical foundations, and proofs of these operators. Their applicabil-
ity is demonstrated through a Multiple-Attribute Decision-Making (MADM) case study, followed
by comparative analyses to evaluate their performance and robustness. The findings indicate
that T2NNAAHM and T2NNAAWHM are effective tools for decision-making problems involving
T2NNs. This research contributes to the field by providing rigorously defined aggregation operators
that address uncertainty and ambiguity, thereby enhancing the methodological toolkit available
for complex decision-making scenarios.
2020 Mathematics Subject Classifications: 03B52, 03E72
Key Words and Phrases: Type-2 neutrosophic number, Aczel-Alsina, Hamy mean, T2NNAAHM,
T2NNAAWHM

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6753

Email addresses: karahan.kara@idu.edu.tr (K. Kara),
galipcihan.yalcin@ostimteknik.edu.tr (G. C. Yalçın), simic.vladimir@sze.hu (V. Simic),
dpamucar@gmail.com (D. Pamucar)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



K. Kara et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6753 2 of 20

1. Introduction

The concept of neutrosophic sets, which comprehensively extends and enhances the
treatment of uncertainty beyond the scope of classical set theory, is introduced into the
literature by Smarandache [1]. Neutrosophic sets are defined with consideration for three
values: truth, indeterminacy, and falsity. These sets consist of three components: T-
membership, I-membership, and F-membership. T-membership represents the degree of an
element’s belonging to the set, while I-membership characterizes the uncertainty regarding
the element’s membership. F-membership quantifies the degree of non-membership of an
element. Neutrosophic sets go beyond the binary consideration of membership and non-
membership, considering the level of uncertainty, thus enabling calculations based on
complex probabilities.

Type-2 Neutrosophic Number Sets (T2NNs) are developed as an extension of neutro-
sophic sets [2] to handle greater levels of uncertainty. T2NNs enable information process-
ing by going beyond Type-1 Neutrosophic Sets (T1NS). In T2NNs, membership functions
exist at two levels: Primary Membership Functions (PMFs) and Secondary Membership
Functions (SMFs). PMFs represent the primary degree of truth, indeterminacy, and fal-
sity of an element, while SMFs provide a secondary layer of truth, indeterminacy, and
falsity degrees. This dual-layer membership function structure allows for the modeling of
complex problems where boundaries between membership, non-membership, and indeter-
minacy are not well-defined.

In the literature, T2NNs have been applied in various decision problems, utilizing dif-
ferent ranking methods, such as supplier selection [2], public transportation pricing system
selection [3], sustainable policies selection [4], smart technology selection [5], sustainable
route selection [6], segmentation of brain tumor tissue structures [7], energy blockchain
system selection [8], selection of second-hand chemical tankers [9], Ro-Ro vessel selec-
tion [10], offshore wind farm site selection [11], autonomous vehicle type selection [12],
green supplier selection [13], architecture selection for 5G-radio access networks [14], and
evaluation of container port sustainability [15]. Notably, the Type-2 neutrosophic num-
bers weighted averaging (T2NNWA) operator has been frequently employed in T2NNs
aggregation in research [3–7, 10, 12, 13, 16–23]

The Aczel–Alsina t-norm and t-conorm operations, together with the Hamy Mean,
are fundamental tools for aggregation in fuzzy set theory [24]. Within the framework
of neutrosophic sets, these operations are frequently utilized across various averaging
techniques [25–28] and are often combined with the Hamy Mean to enhance aggrega-
tion performance [29, 30]. The main objective of this study is to develop novel aggre-
gation operators for Type-2 Neutrosophic Numbers (T2NNs) based on the Aczel–Alsina
t-norm and t-conorm operations in conjunction with the Hamy Mean, specifically de-
signed for Multiple-Attribute Decision-Making (MADM) applications [31, 32]. To achieve
this goal, two new aggregation operators are proposed: the Type-2 Neutrosophic Numbers
Aczel–Alsina Hamy Mean operator (T2NNAAHM) and the Type-2 Neutrosophic Numbers
Aczel–Alsina Weighted Hamy Mean operator (T2NNAAWHM). The key contributions of
this study are as follows:



K. Kara et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6753 3 of 20

(i) Explaining the applications of Aczel–Alsina operations based on T2NNs in MADM
problems;

(ii) Developing Hamy Mean-based aggregation operators for T2NNs;

(iii) Introducing both weighted and unweighted aggregation operators for T2NNs using
Aczel–Alsina operations and Hamy Mean;

(iv) Defining and proving the newly developed aggregation operators;

(v) Conducting robustness tests for the obtained findings and aggregation operators;

(vi) Comparing the results obtained with T2NNWA operator results.

This paper consists of eight sections. The second section elaborates on Aczel-Alsina
operations based on T2NNs. The third section discusses Hamy Mean-based Aczel-Alsina
operations of T2NNs. The fourth section presents the MADM approach with T2NNAAHM
and T2NNAAWHM operators. The fifth section applies the aggregation operators with
illustrative examples. In the sixth section, comparative analyses are conducted. The sev-
enth section contains the implications of this research. The eight section is also conclusion
section.

2. Aczel-Alsina Operations based on Type-2 Neutrosophic Number Sets

In this section, we furnish detailed definitions encompassing the Aczel-Alsina t-norm
and t-conorm, along with a comprehensive exposition of the corresponding mathematical
operations.

Definition 1. The Aczel-Alsina t-norm is presented in Eq. (1), and the Aczel-Alsina
t-conorm is expressed in Eq. (2) [33]:

T ξ
AA(x, y) =



TD(x, y), if ξ = 0

min(x, y), if ξ = ∞

e
−
(
(− ln(x))ξ+(− ln(y))ξ

)1/ξ
, otherwise.

(1)

T ∗ξ
AA(x, y) =



SD(x, y), if ξ = 0

max(x, y), if ξ = ∞

e
−
(
(− ln(1−x))ξ+(− ln(1−y))ξ

)1/ξ
, otherwise.

(2)

For each parameter ξ ∈ [0,∞];T ξ
AA and T ∗ξ

AA are dual to each other.
We will delve into the fundamental operational laws of the Aczel-Alsina t-norm and Aczel-Alsina t-conorm for

T2NNs. The Aczel-Alsina product for the Aczel-Alsina t-norm is defined in Eq. (3). The Aczel-Alsina sum for the
Aczel-Alsina t-norm is defined in Eq. (4):

Ñ1 ⊗ Ñ2 =

⟨ T
(
[T (T (Ñ1)(z), T (T (Ñ2)(z)))], [T (I(Ñ1)(z), T (I(Ñ2)(z)))], [T (F (Ñ1)(z), T (F (Ñ2)(z)))]

)
,

T ∗([I(T (Ñ1)(z), I(T (Ñ2)(z)))], [I(I(Ñ1)(z), I(I(Ñ2)(z)))], [I(F (Ñ1)(z), I(F (Ñ2)(z)))]
)
,

T ∗([F (T (Ñ1)(z), F (T (Ñ2)(z)))], [F (I(Ñ1)(z), I(I(Ñ2)(z)))], [F (F (Ñ1)(z), F (F (Ñ2)(z)))]
)
⟩

(3)



K. Kara et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6753 4 of 20

Ñ1 ⊕ Ñ2 =

⟨ T ∗
(
[T (T (Ñ1)(z), T (T (Ñ2)(z)))], [T (I(Ñ1)(z), T (I(Ñ2)(z)))], [T (F (Ñ1)(z), T (F (Ñ2)(z)))]

)
,

T ∗
(
[I(T (Ñ1)(z), I(T (Ñ2)(z)))], [I(I(Ñ1)(z), I(I(Ñ2)(z)))], [I(F (Ñ1)(z), I(F (Ñ2)(z)))]

)
,

T
(
[F (T (Ñ1)(z), F (T (Ñ2)(z)))], [F (I(Ñ1)(z), I(I(Ñ2)(z)))], [F (F (Ñ1)(z), F (F (Ñ2)(z)))]

)
⟩

(4)
Definition 2. Suppose that Ñ, Ñ1, and Ñ2 are three T2NNs over the universe Z. The operations among these

three T2NNs based on Aczel-Alsina T and Aczel-Alsina T ∗ are described as follows (ω > 0):

(i)

Ñ1 ⊗ Ñ2 =
[ e−

(
(− lnT (T (Ñ1)(z)))

ξ+(− lnT (T (Ñ2)(z)))
ξ
)1/ξ

, e−
(
(− lnT (I(Ñ1)(z)))

ξ+(− lnT (I(Ñ2)(z)))
ξ
)1/ξ

, e−
(
(− lnT (F (Ñ1)(z)))

ξ+(− lnT (F (Ñ2)(z)))
ξ
)1/ξ

;

1− e−
(
(− ln I(T (Ñ1)(1−z)))ξ+(− ln I(T (Ñ2)(1−z)))ξ

)1/ξ
, 1− e−

(
(− ln I(I(Ñ1)(1−z)))ξ+(− ln I(I(Ñ2)(1−z)))ξ

)1/ξ
, 1− e−

(
(− ln I(F (Ñ1)(1−z)))ξ+(− ln I(F (Ñ2)(1−z)))ξ

)1/ξ
;

1− e−
(
(− lnF (T (Ñ1)(1−z)))ξ+(− lnF (T (Ñ2)(1−z)))ξ

)1/ξ
, 1− e−

(
(− lnF (I(Ñ1)(1−z)))ξ+(− lnF (I(Ñ2)(1−z)))ξ

)1/ξ
, 1− e−

(
(− lnF (F (Ñ1)(1−z)))ξ+(− lnF (F (Ñ2)(1−z)))ξ

)1/ξ
]

(ii)

Ñ1 ⊕ Ñ2 =
[ 1− e−

(
(− lnT (T (Ñ1)(1−z)))ξ+(− lnT (T (Ñ2)(1−z)))ξ

)1/ξ
, 1− e−

(
(− lnT (I(Ñ1)(1−z)))ξ+(− lnT (I(Ñ2)(1−z)))ξ

)1/ξ
, 1− e−

(
(− lnT (F (Ñ1)(1−z)))ξ+(− lnT (F (Ñ2)(1−z)))ξ

)1/ξ
;

1− e−
(
(− ln I(T (Ñ1)(1−z)))ξ+(− ln I(T (Ñ2)(1−z)))ξ

)1/ξ
, 1− e−

(
(− ln I(I(Ñ1)(1−z)))ξ+(− ln I(I(Ñ2)(1−z)))ξ

)1/ξ
, 1− e−

(
(− ln I(F (Ñ1)(1−z)))ξ+(− ln I(F (Ñ2)(1−z)))ξ

)1/ξ
;

e−
(
(− lnF (T (Ñ1)(z)))

ξ+(− lnF (T (Ñ2)(z)))
ξ
)1/ξ

, e−
(
(− lnF (I(Ñ1)(z)))

ξ+(− lnF (I(Ñ2)(z)))
ξ
)1/ξ

, e−
(
(− lnF (F (Ñ1)(z)))

ξ+(− lnF (F (Ñ2)(z)))
ξ
)1/ξ

]

(iii)

ωÑ =
[ 1− e−

(
(ω(− lnT (T (Ñ))(1−z)))ξ

)1/ξ
, 1− e−

(
(ω(− lnT (I(Ñ))(1−z)))ξ

)1/ξ
, 1− e−

(
(ω(− lnT (F (Ñ))(1−z)))ξ

)1/ξ
;

e−
(
(ω(− ln I(T (Ñ))(z)))ξ

)1/ξ
, e−

(
(ω(− ln I(I(Ñ))(z)))ξ

)1/ξ
, e−

(
(ω(− ln I(F (Ñ))(z)))ξ

)1/ξ
;

e−
(
(ω(− lnF (T (Ñ))(z)))ξ

)1/ξ
, e−

(
(ω(− lnF (T (Ñ))(z)))ξ

)1/ξ
, e−

(
(ω(− lnF (F (Ñ))(z)))ξ

)1/ξ
]

(iv)

Ñω =
[ e−

(
(ω(− lnT (T (Ñ))(z)))ξ

)1/ξ
, e−

(
(ω(− lnT (I(Ñ))(z)))ξ

)1/ξ
, e−

(
(ω(− lnT (F (Ñ))(z)))ξ

)1/ξ
;

1− e−
(
(ω(− ln I(T (Ñ))(1−z)))ξ

)1/ξ
, 1− e−

(
(ω(− ln I(I(Ñ))(1−z)))ξ

)1/ξ
, 1− e−

(
(ω(− ln I(F (Ñ))(1−z)))ξ

)1/ξ
;

1− e−
(
(ω(− lnF (T (Ñ))(1−z)))ξ

)1/ξ
, 1− e−

(
(ω(− lnF (T (Ñ))(1−z)))ξ

)1/ξ
, 1− e−

(
(ω(− lnF (F (Ñ))(1−z)))ξ

)1/ξ
]

Example 1:

Ñ = ⟨(0.20, 0.20, 0.10), (0.65, 0.80, 0.85), (0.45, 0.80, 0.70)⟩, Ñ1 = ⟨(0.35, 0.35, 0.10), (0.50, 0.75, 0.80), (0.50, 0.75, 0.65)⟩, Ñ2 = ⟨(0.40, 0.30, 0.35), (0.50, 0.45, 0.60), (0.45, 0.40, 0.60)⟩

be three T2NNs over the universe Z. Then, the Aczel-Alsina operations using Definition 2 can be applied
for ξ = 2 and ω = 3.

(a)

Ñ1 ⊗ Ñ2 =
[ e−

(
(− ln 0.35)2+(− ln 0.40)2

)1/2
, e−

(
(− ln 0.35)2+(− ln 0.30)2

)1/2
, e−

(
(− ln 0.10)2+(− ln 0.35)2

)1/2
;

1− e−
(
(− ln(1−0.50))2+(− ln(1−0.50))2

)1/2
, 1− e−

(
(− ln(1−0.75))2+(− ln(1−0.45))2

)1/2
, 1− e−

(
(− ln(1−0.80))2+(− ln(1−0.60))2

)1/2
;

1− e−
(
(− ln(1−0.50))2+(− ln(1−0.45))2

)1/2
, 1− e−

(
(− ln(1−0.75))2+(− ln(1−0.40))2

)1/2
, 1− e−

(
(− ln(1−0.65))2+(− ln(1−0.60))2

)1/2
]

Result: 〈(0.25,0.20,0.08),(0.62,0.78,0.84),(0.60,0.77,0.75)〉.



K. Kara et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6753 5 of 20

(b)

Ñ1 ⊕ Ñ2 =
[ 1− e−

(
(− ln(1−0.35))2+(− ln(1−0.40))2

)1/2
, 1− e−

(
(− ln(1−0.35))2+(− ln(1−0.30))2

)1/2
, 1− e−

(
(− ln(1−0.10))2+(− ln(1−0.35))2

)1/2
;

1− e−
(
(− ln(1−0.50))2+(− ln(1−0.50))2

)1/2
, 1− e−

(
(− ln(1−0.75))2+(− ln(1−0.45))2

)1/2
, 1− e−

(
(− ln(1−0.80))2+(− ln(1−0.60))2

)1/2
;

e−
(
(− ln 0.50)2+(− ln 0.45)2

)1/2
, e−

(
(− ln 0.75)2+(− ln 0.40)2

)1/2
, e−

(
(− ln 0.65)2+(− ln 0.60)2

)1/2
]

Result: 〈(0.49,0.43,0.36),(0.62,0.78,0.84),(0.35,0.38,0.51)〉.
(c)

ωÑ =
[ 1− e−

(
(3(− ln(1−0.20)))2

)1/2
, 1− e−

(
(3(− ln(1−0.20)))2

)1/2
, 1− e−

(
(3(− ln(1−0.10)))2

)1/2
;

e−
(
(3(− ln 0.65))2

)1/2
, e−

(
(3(− ln 0.80))2

)1/2
, e−

(
(3(− ln 0.85))2

)1/2
;

e−
(
(3(− ln 0.45))2

)1/2
, e−

(
(3(− ln 0.80))2

)1/2
, e−

(
(3(− ln 0.70))2

)1/2
]

Result: 〈(0.32,0.32,0.17),(0.47,0.68,0.75),(0.25,0.68,0.54)〉.
(d)

Ñω =
[ e−

(
(3(− ln 0.20))2

)1/2
, e−

(
(3(− ln 0.20))2

)1/2
, e−

(
(3(− ln 0.10))2

)1/2
;

1− e−
(
(3(− ln(1−0.65)))2

)1/2
, 1− e−

(
(3(− ln(1−0.80)))2

)1/2
, 1− e−

(
(3(− ln(1−0.85)))2

)1/2
;

1− e−
(
(3(− ln(1−0.45)))2

)1/2
, 1− e−

(
(3(− ln(1−0.80)))2

)1/2
, 1− e−

(
(3(− ln(1−0.70)))2

)1/2
]

Result: 〈(0.06,0.06,0.02),(0.84,0.94,0.96),(0.64,0.94,0.88)〉.

Theorem 1: Suppose that Ñ1 and Ñ2 are two type-2 neutrosophic number sets over the universe Z, then:

(a)
Ñ1 ⊕ Ñ2 = Ñ2 ⊕ Ñ1

(b)
Ñ1 ⊗ Ñ2 = Ñ2 ⊗ Ñ1

(c)
ω(Ñ1 ⊕ Ñ2) = ωÑ1 ⊕ ωÑ2, for ω > 0

(d)
(Ñ1 ⊗ Ñ2)

ω = Ñω
1 ⊗ Ñω

2 , for ω > 0

(e)
ω1Ñ1 ⊕ ω2Ñ1 = (ω1 + ω2)Ñ1, for ω1, ω2 > 0

(f)
Ñω1

1 ⊗ Ñω2
1 = Ñω1+ω2

1 , for ω1, ω2 > 0

Proof. [Proof for Ñ1 ⊕ Ñ2 = Ñ2 ⊕ Ñ1]

Ñ1 = ⟨(0.35, 0.35, 0.10), (0.50, 0.75, 0.80), (0.50, 0.75, 0.65)⟩, Ñ2 = ⟨(0.40, 0.30, 0.35), (0.50, 0.45, 0.60), (0.45, 0.40, 0.60)⟩

be two T2NNs over the universe Z. Then, the Aczel-Alsina operations can be applied using Theorem 1 for
ξ = 2 and ω = 3.

Ñ1 ⊕ Ñ2 = Ñ2 ⊕ Ñ1



K. Kara et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6753 6 of 20

⇒
[ 1− e−

(
(− ln(1−0.35))2+(− ln(1−0.40))2

)1/2
, 1− e−

(
(− ln(1−0.35))2+(− ln(1−0.30))2

)1/2
, 1− e−

(
(− ln(1−0.10))2+(− ln(1−0.35))δ

)1/2
;

1− e−
(
(− ln(1−0.50))2+(− ln(1−0.50))2

)1/2
, 1− e−

(
(− ln(1−0.75))2+(− ln(1−0.45))2

)1/2
, 1− e−

(
(− ln(1−0.80))2+(− ln(1−0.60))2

)1/2
;

e−
(
(− ln 0.50)2+(− ln 0.45)2

)1/2
, e−

(
(− ln 0.75)2+(− ln 0.40)2

)1/2
, e−

(
(− ln 0.65)2+(− ln 0.60)2

)1/2
]

=
[ 1− e−

(
(− ln(1−0.40))2+(− ln(1−0.35))2

)1/2
, 1− e−

(
(− ln(1−0.30))2+(− ln(1−0.35))2

)1/2
, 1− e−

(
(− ln(1−0.35))2+(− ln(1−0.10))δ

)1/2
;

1− e−
(
(− ln(1−0.50))2+(− ln(1−0.50))2

)1/2
, 1− e−

(
(− ln(1−0.45))2+(− ln(1−0.75))2

)1/2
, 1− e−

(
(− ln(1−0.60))2+(− ln(1−0.80))2

)1/2
;

e−
(
(− ln 0.45)2+(− ln 0.50)2

)1/2
, e−

(
(− ln 0.40)2+(− ln 0.75)2

)1/2
, e−

(
(− ln 0.60)2+(− ln 0.65)2

)1/2
]

Result: 〈(0.49,0.43,0.36),(0.62,0.78,0.84),(0.35,0.38,0.51)〉=〈(0.49,0.43,0.36),(0.62,0.78,0.84),(0.35,0.38,0.51)〉.
Proof. [Proof for Ñ1 ⊗ Ñ2 = Ñ2 ⊗ Ñ1]

[ e−
(
(− ln 0.35)2+(− ln 0.40)2

)1/2
, e−

(
(− ln 0.35)2+(− ln 0.30)2

)1/2
, e−

(
(− ln 0.10)2+(− ln 0.35)2

)1/2
;

1− e−
(
(− ln(1−0.50))2+(− ln(1−0.50))2

)1/2
, 1− e−

(
(− ln(1−0.75))2+(− ln(1−0.45))2

)1/2
, 1− e−

(
(− ln(1−0.80))2+(− ln(1−0.60))2

)1/2
;

1− e−
(
(− ln(1−0.50))2+(− ln(1−0.45))2

)1/2
, 1− e−

(
(− ln(1−0.75))2+(− ln(1−0.40))2

)1/2
, 1− e−

(
(− ln(1−0.65))2+(− ln(1−0.60))2

)1/2
]

[ e−
(
(− ln 0.40)2+(− ln 0.35)2

)1/2
, e−

(
(− ln 0.30)2+(− ln 0.35)2

)1/2
, e−

(
(− ln 0.35)2+(− ln 0.10)2

)1/2
;

1− e−
(
(− ln(1−0.50))2+(− ln(1−0.50))2

)1/2
, 1− e−

(
(− ln(1−0.45))2+(− ln(1−0.75))2

)1/2
, 1− e−

(
(− ln(1−0.60))2+(− ln(1−0.80))2

)1/2
;

1− e−
(
(− ln(1−0.45))2+(− ln(1−0.50))2

)1/2
, 1− e−

(
(− ln(1−0.40))2+(− ln(1−0.75))2

)1/2
, 1− e−

(
(− ln(1−0.60))2+(− ln(1−0.65))2

)1/2
]

Result: 〈(0.25,0.20,0.08),(0.62,0.78,0.84),(0.60,0.77,0.75)〉=〈(0.25,0.20,0.08),(0.62,0.78,0.84),(0.60,0.77,0.75)〉.
Proof: The others proof is straightforward.

3. Hamy Mean-based Aczel-Alsina Operations of Type-2
Neutrosophic Number Sets

3.1. Type-2 neutrosophic numbers Aczel–Alsina Hamy Mean Oper-
ator
Within this sub-section, we introduce the T2NNAAHM operator, which is formulated in accordance with
the operational principles delineated in Definition 2. Definition 3. HM operator is defined as Eq. (5) [30, 34]:

HMτ (θ1, θ2, . . . , θn) =

∑
1≤i1<···<iτ≤n

( τ∏
i=1

θiµ
)1/τ

(n
τ

) (5)

In this context, {
Cτ

n =
n!

τ !(n− τ)!
, τ = 1, 2, . . . , n

}
states the binomial coefficient. The following conditions can be satisfied for the HM operator:

(a)
HMτ (θ1, θ2, . . . , θn) = θ if θ = θi, i = 1, 2, . . . , n

(b)
HMτ (θ1, θ2, . . . , θn) ≤ HMτ (δ1, δ2, . . . , δn) if θi ≤ δi, i = 1, 2, . . . , n

(c)

HMτ
AM(θ1, θ2, . . . , θn) =

1

n

n∑
i=1

θiµ for the arithmetic mean (AM),



K. Kara et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6753 7 of 20

(d)

HMτ
GM(θ1, θ2, . . . , θn) =

(
n∏

i=1

θiµ

)1/n

for the geometric mean (GM).

Theorem 2: Suppose that

Ñn =
⟨(

TT (Ñn)(z), TI(Ñn)(z), TF (Ñn)(z)
)
,
(
IT (Ñn)(z), II(Ñn)(z), IF (Ñn)(z)

)
,
(
FT (Ñn)(z), FI(Ñn)(z), FF (Ñn)(z)

)⟩
is a type-2 neutrosophic number set over the universe Z, then the Type-2 neutrosophic numbers Aczel–Alsina
Hamy Mean operator is defined as Eq. (6) and Eq. (7).

T2NNAAHM(θ1, θ2, . . . , θn) =

∑
1≤i1<···<iτ≤n

( τ∏
µ=1

θiµ

)1/τ
(n
τ

) (6)

⟨1− e
−
(

1
Cτ
n

(∑
1≤i1<···<iτ≤n

(
−ln(1−e

−( 1
τ

∑τ
µ=1(− lnT

T (ѵ)
(z))ξ)1/ξ

)
)ξ))1/ξ

, 1− e
−
(

1
Cτ
n

(∑
1≤i1<···<iτ≤n

(
−ln(1−e

−( 1
τ

∑τ
µ=1(− lnT

I(ѵ)
(z))ξ)1/ξ

)
)ξ))1/ξ

, 1− e
−
(

1
Cτ
n

(∑
1≤i1<···<iτ≤n

(
−ln(1−e

−( 1
τ

∑τ
µ=1(− lnT

F (ѵ)
(z))ξ)1/ξ

)
)ξ))1/ξ

e
−
(

1
Cτ
n

(∑
1≤i1<···<iτ≤n

(
−ln(1−e

−( 1
τ

∑τ
µ=1(− ln(1−I

T (ѵ)
(z)))ξ)1/ξ

)
)ξ))1/ξ

, e
−
(

1
Cτ
n

(∑
1≤i1<···<iτ≤n

(
−ln(1−e

−( 1
τ

∑τ
µ=1(− ln(1−I

I(ѵ)
(z)))ξ)1/ξ

)
)ξ))1/ξ

, e
−
(

1
Cτ
n

(∑
1≤i1<···<iτ≤n

(
−ln(1−e

−( 1
τ

∑τ
µ=1(− ln(1−I

F (ѵ)
(z)))ξ)1/ξ

)
)ξ))1/ξ

e
−
(

1
Cτ
n

(∑
1≤i1<···<iτ≤n

(
−ln(1−e

−( 1
τ

∑τ
µ=1(− ln(1−F

T (ѵ)
(z)))ξ)1/ξ

)
)ξ))1/ξ

, e
−
(

1
Cτ
n

(∑
1≤i1<···<iτ≤n

(
−ln(1−e

−( 1
τ

∑τ
µ=1(− ln(1−F

I(ѵ)
(z)))ξ)1/ξ

)
)ξ))1/ξ

, e
−
(

1
Cτ
n

(∑
1≤i1<···<iτ≤n

(
−ln(1−e

−( 1
τ

∑τ
µ=1(− ln(1−F

F (ѵ)
(z)))ξ)1/ξ

)
)ξ))1/ξ

⟩

(7)
Proof. For Theorem 2.

τ∏
µ=1

θµ =


e
−
(∑τ

µ=1(− ln(T
T (ѵ)

(z)))ξ
)1/ξ

e
−
(∑τ

µ=1(− ln(1−T
I(ѵ)

(z)))ξ
)1/ξ

e
−
(∑τ

µ=1(− ln(1−T
F (ѵ)

(z)))ξ
)1/ξ

1− e
−
(∑τ

µ=1(− ln(1−I
T (ѵ)

(z)))ξ
)1/ξ

1− e
−
(∑τ

µ=1(− ln(1−I
I(ѵ)

(z)))ξ
)1/ξ

1− e
−
(∑τ

µ=1(− ln(I
F (ѵ)

(z)))ξ
)1/ξ

1− e
−
(∑τ

µ=1(− ln(1−F
T (ѵ)

(z)))ξ
)1/ξ

1− e
−
(∑τ

µ=1(− ln(1−F
I(ѵ)

(z)))ξ
)1/ξ

1− e
−
(∑τ

µ=1(− ln(F
F (ѵ)

(z)))ξ
)1/ξ


then

(∏τ
µ=1 θµ

)1/τ
=


e
−
(

1
τ

∑τ
µ=1(− ln(T

T (ѵ)
(z)))ξ

)1/ξ

e
−
(

1
τ

∑τ
µ=1(− ln(T

I(ѵ)
(z)))ξ

)1/ξ

e
−
(

1
τ

∑τ
µ=1(− ln(T

F (ѵ)
(z)))ξ

)1/ξ

1− e
−
(

1
τ

∑τ
µ=1(− ln(1−I

T (ѵ)
(z)))ξ

)1/ξ

1− e
−
(

1
τ

∑τ
µ=1(− ln(1−I

I(ѵ)
(z)))ξ

)1/ξ

1− e
−
(

1
τ

∑τ
µ=1(− ln(1−I

F (ѵ)
(z)))ξ

)1/ξ

1− e
−
(

1
τ

∑τ
µ=1(− ln(1−F

T (ѵ)
(z)))ξ

)1/ξ

1− e
−
(

1
τ

∑τ
µ=1(− ln(1−F

I(ѵ)
(z)))ξ

)1/ξ

1− e
−
(

1
τ

∑τ
µ=1(− ln(1−F

F (ѵ)
(z)))ξ

)1/ξ


then


1− e

−

(∑
1≤i1<···<iτ≤n(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(T
T (ѵ)

(z)))ξ

)1/ξ))

)ξ
1/ξ

1− e

−

(∑
1≤i1<···<iτ≤n(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(T
I(ѵ)

(z)))ξ

)1/ξ))

)ξ
1/ξ

1− e

−

(∑
1≤i1<···<iτ≤n(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(T
F (ѵ)

(z)))ξ

)1/ξ))

)ξ
1/ξ

1− e

−

(∑
1≤i1<···<iτ≤n(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(1−I
T (ѵ)

(z)))ξ

)1/ξ))

)ξ
1/ξ

1− e

−

(∑
1≤i1<···<iτ≤n(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(1−I
I(ѵ)

(z)))ξ

)1/ξ))

)ξ
1/ξ

1− e

−

(∑
1≤i1<···<iτ≤n(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(1−I
F (ѵ)

(z)))ξ

)1/ξ))

)ξ
1/ξ

e

−

(∑
1≤i1<···<iτ≤n(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(1−F
T (ѵ)

(z)))ξ

)1/ξ))

)ξ
1/ξ

e

−

(∑
1≤i1<···<iτ≤n(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(1−F
I(ѵ)

(z)))ξ

)1/ξ))

)ξ
1/ξ

e

−

(∑
1≤i1<···<iτ≤n(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(1−F
F (ѵ)

(z)))ξ

)1/ξ))

)ξ
1/ξ



then


1− e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
−ln
(
1−e

−((1/τ)
∑τ

µ=1(− lnT
T (ѵ)

(z))ξ)1/ξ))ξ
)1/ξ

1− e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
−ln
(
1−e

−((1/τ)
∑τ

µ=1(− lnT
I(ѵ)

(z))ξ)1/ξ))ξ
)1/ξ

1− e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
−ln
(
1−e

−((1/τ)
∑τ

µ=1(− lnT
F (ѵ)

(z))ξ)1/ξ))ξ
)1/ξ

1− e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
−ln
(
1−e

−((1/τ)
∑τ

µ=1(− ln(1−I
T (ѵ)

(z)))ξ)1/ξ))ξ
)1/ξ

1− e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
−ln
(
1−e

−((1/τ)
∑τ

µ=1(− ln(1−I
I(ѵ)

(z)))ξ)1/ξ))ξ
)1/ξ

1− e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
−ln
(
1−e

−((1/τ)
∑τ

µ=1(− ln(1−I
F (ѵ)

(z)))ξ)1/ξ))ξ
)1/ξ

e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
−ln
(
1−e

−((1/τ)
∑τ

µ=1(− ln(1−F
T (ѵ)

(z)))ξ)1/ξ))ξ
)1/ξ

e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
−ln
(
1−e

−((1/τ)
∑τ

µ=1(− ln(1−F
I(ѵ)

(z)))ξ)1/ξ))ξ
)1/ξ

e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
−ln
(
1−e

−((1/τ)
∑τ

µ=1(− ln(1−F
F (ѵ)

(z)))ξ)1/ξ))ξ
)1/ξ



Consider the first condition;

T̃Ñ (z) =

⟨
1− e

−

(
1

Cτ
n

∑
1≤i1<···<iτ≤n

(
−ln

(
1−e

−
(
(1/τ)

∑τ
µ=1(− lnTT (ѵ)(z))ξ

)1/ξ))ξ

)1/ξ

,



K. Kara et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6753 8 of 20

1 - e
−

(
1

Cτ
n

∑
1≤i1<···<iτ≤n

(
−ln

(
1−e

−
(
(1/τ)

∑τ
µ=1(− lnTI (ѵ)(z))ξ

)1/ξ))ξ

)1/ξ

,

1− e

−

(
1

Cτ
n

∑
1≤i1<···<iτ≤n

(
−ln

(
1−e

−
(
(1/τ)

∑τ
µ=1(− lnTF (ѵ)(z))ξ

)1/ξ))ξ

)1/ξ⟩
, ϵ ∈ [0, 1].

=⇒ 0 ≤ T̃Ñ (z) ≤ 1.

ĨÑ (z) =

⟨
e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
− ln
(
1−e

−
(

1
τ

∑τ
µ=1(− ln(1−I

T (ѵ)
(z)))ξ

)1/ξ))ξ)1/ξ

,

e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
− ln
(
1−e

−
(

1
τ

∑τ
µ=1(− ln(1−I

I(ѵ)
(z)))ξ

)1/ξ))ξ)1/ξ

,

e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
− ln
(
1−e

−
(

1
τ

∑τ
µ=1(− ln(1−I

F (ѵ)
(z)))ξ

)1/ξ))ξ)1/ξ⟩
, ϵ ∈ [0, 1].

=⇒ 0 ≤ ĨÑ (z) ≤ 1.

F̃Ñ (z) =

⟨
e
−
(

1
Cτ
n

(∑
1≤i1<···<iτ≤n

(
− ln(1−e

−( 1
τ

∑τ
µ=1(− lnF

T (ѵ)
(z))ξ)1/ξ

)
)ξ))1/ξ

,

e
−
(

1
Cτ
n

(∑
1≤i1<···<iτ≤n

(
− ln(1−e

−( 1
τ

∑τ
µ=1(− lnF

I(ѵ)
(z))ξ)1/ξ

)
)ξ))1/ξ

,

e
−
(

1
Cτ
n

(∑
1≤i1<···<iτ≤n

(
− ln(1−e

−( 1
τ

∑τ
µ=1(− lnF

F (ѵ)
(z))ξ)1/ξ

)
)ξ))1/ξ⟩

, ϵ ∈ [0, 1]

=⇒ 0 ≤ F̃Ñ (z) ≤ 1.

Consider the second condition;

T̃Ñ (z)3 + ĨÑ (z)3 + F̃Ñ (z)3

=
⟨(
1− e

−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
− ln
(
1−e

−
(

1
τ

∑τ
µ=1(− lnT

T (ѵ)
(z))ξ

)1/ξ))ξ)1/ξ)3
,

(
1− e

−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
− ln
(
1−e

−
(

1
τ

∑τ
µ=1(− lnT

I(ѵ)
(z))ξ

)1/ξ))ξ)1/ξ)3
,

(
1− e

−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
− ln
(
1−e

−
(

1
τ

∑τ
µ=1(− lnT

F (ѵ)
(z))ξ

)1/ξ))ξ)1/ξ)3⟩
+
⟨
e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
− ln
(
1−e

−
(

1
τ

∑τ
µ=1(− ln(1−I

T (ѵ)
(z)))ξ

)1/ξ))ξ)1/ξ

,

e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
− ln
(
1−e

−
(

1
τ

∑τ
µ=1(− ln(1−I

I(ѵ)
(z)))ξ

)1/ξ))ξ)1/ξ

,

e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
− ln
(
1−e

−
(

1
τ

∑τ
µ=1(− ln(1−I

F (ѵ)
(z)))ξ

)1/ξ))ξ)1/ξ)3
+
⟨
e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
− ln
(
1−e

−
(

1
τ

∑τ
µ=1(− ln(1−F

T (ѵ)
(z)))ξ

)1/ξ))ξ)1/ξ

,

e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
− ln
(
1−e

−
(

1 ∑τ
µ=1(− ln(1−F

I(ѵ)
(z)))ξ

)1/ξ))ξ)1/ξ

,

e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
− ln
(
1−e

−
(

1
τ

∑τ
µ=1(− ln(1−F

F (ѵ)
(z)))ξ

)1/ξ))ξ)1/ξ)3



K. Kara et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6753 9 of 20

=⇒ 0 ≤ T̃Ñ (z)3 + ĨÑ (z)3 + F̃Ñ (z)3 ≤ 3.

Theorem 3: The Type-2 neutrosophic numbers Aczel–Alsina Hamy Mean operator meets the following
properties:
Monotonicity: Set Ñk ≤ Ñ∗

k . If T̃Ñ (z) ≤ T̃Ñ (z)∗, ĨÑ (z) ≥ ĨÑ (z)∗, and F̃Ñ (z) ≥ F̃Ñ (z)∗, then

T2NNAAHM(Ñ1, Ñ2, . . . , Ñs) ≤ T2NNAAHM(Ñ∗
1 , Ñ

∗
2 , . . . , Ñ

∗
k ).

Proof: For Theorem 3 (Monotonicity) Due to Ñk ≤ Ñ∗
k , there exists T̃Ñ (z) ≤ T̃Ñ (z)∗, ĨÑ (z) ≥ ĨÑ (z)∗,

and F̃Ñ (z) ≥ F̃Ñ (z)∗. Thus,

T2NNAAHM(M̃1, M̃2, . . . , M̃s) ≤ T2NNAAHM(M̃∗
1 , M̃

∗
2 , . . . , M̃

∗
k ).

For example, if T̃Ñ (z) ≤ T̃Ñ (z)∗, T̃(T
Ñ

)(z) ≤ T̃(T
Ñ

)(z)
∗, T̃(I

Ñ
)(z) ≥ T̃(I

Ñ
)(z)

∗, and T̃(F
Ñ

)(z) ≥ T̃(F
Ñ

)(z)
∗,

then:

[
1− exp

(
−
(

1

Cτ
n

( ∑
1≤i1<···<iτ≤n

(
− ln

(
1− exp(−(

1

τ

τ∑
µ=1

(− ln(T(T
Ñ

)(z)))
ξ)1/ξ)

))ξ))1/ξ)]

≤
[
1− exp

(
−
(

1

Cτ
n

( ∑
1≤i1<···<iτ≤n

(
− ln

(
1− exp(−(

1

τ

τ∑
µ=1

(− ln(T ∗
(T

Ñ
)(z)))

ξ)1/ξ)
))ξ))1/ξ)]

.

[
1− exp

(
−
(

1

Cτ
n

( ∑
1≤i1<···<iτ≤n

(
− ln

(
1− exp(−(

1

τ

τ∑
µ=1

(− ln(T(I
Ñ

)(z)))
ξ)1/ξ)

))ξ))1/ξ)]

≥
[
1− exp

(
−
(

1

Cτ
n

( ∑
1≤i1<···<iτ≤n

(
− ln

(
1− exp(−(

1

τ

τ∑
µ=1

(− ln(T ∗
(I

Ñ
)(z)))

ξ)1/ξ)
))ξ))1/ξ)]

.

[
1− exp

(
−
(

1

Cτ
n

( ∑
1≤i1<···<iτ≤n

(
− ln

(
1− exp(−(

1

τ

τ∑
µ=1

(− ln(T(F
Ñ

)(z)))
ξ)1/ξ)

))ξ))1/ξ)]

≥
[
1− exp

(
−
(

1

Cτ
n

( ∑
1≤i1<···<iτ≤n

(
− ln

(
1− exp(−(

1

τ

τ∑
µ=1

(− ln(T ∗
(F

Ñ
)(z)))

ξ)1/ξ)
))ξ))1/ξ)]

.

Idempotency: Set

ѵ =
⟨
(T(T

ѵ
)(y), T(I

ѵ
)(y), T(F

ѵ
)(y)), (I(T

ѵ
)(y), I(I

ѵ
)(y), I(F

ѵ
)(y)),

(F(T
ѵ

)(y), F(I
ѵ

)(y), F(F
ѵ

)(y))
⟩
= Ñ, (µ = 1, 2, . . . , s).

For example, if Ñ1
(z) = T̃Ñ2

(z), T(T
Ñ1

)(z) = T(T
Ñ2

)(z), T(I
Ñ1

)(z) = T(I
Ñ2

)(z),

T(F
Ñ1

)(z) = T(F
Ñ2

)(z), then:[
1− exp

(
−
( 1

Cτ
n

( ∑
1≤i1<···<iτ≤n

(
− ln

(
1− exp

(
−

1

τ

( τ∑
µ=1

− ln(T(T
Ñ

)(z))
ξ
)1/ξ)))ξ))1/ξ)]

=

[
1− exp

(
−
( 1

Cτ
n

( ∑
1≤i1<···<iτ≤n

(
− ln

(
1− exp

(
−

1

τ

( τ∑
µ=1

− ln(
(
T(T

Ñ
)(z)

)∗
)ξ
)1/ξ)))ξ))1/ξ)]



K. Kara et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6753 10 of 20[
1− exp

(
−
( 1

Cτ
n

( ∑
1≤i1<···<iτ≤n

(
− ln

(
1− exp

(
−

1

τ

( τ∑
µ=1

− ln(T(I
Ñ

)(z))
ξ
)1/ξ)))ξ))1/ξ)]

=

[
1− exp

(
−
( 1

Cτ
n

( ∑
1≤i1<···<iτ≤n

(
− ln

(
1− exp

(
−

1

τ

( τ∑
µ=1

− ln((T(I
Ñ

)(z))
∗)ξ
)1/ξ)))ξ))1/ξ)]

[
1− exp

(
−
( 1

Cτ
n

( ∑
1≤i1<···<iτ≤n

(
− ln

(
1− exp

(
−

1

τ

( τ∑
µ=1

− ln(T(F
Ñ

)(z))
ξ
)1/ξ)))ξ))1/ξ)]

=

[
1− exp

(
−
( 1

Cτ
n

( ∑
1≤i1<···<iτ≤n

(
− ln

(
1− exp

(
−

1

τ

( τ∑
µ=1

− ln((T(F
Ñ

)(z))
∗)ξ
)1/ξ)))ξ))1/ξ)]

Boundedness: If the maximum and minimum T2NNs are given below:

Ñmax =
⟨(

max
µ

TT (ѵ)(y), max
µ

TI(ѵ)(y), max
µ

TF (ѵ)(y)
)
,(

minµ IT (ѵ)(y), minµ II(ѵ)(y), minµ IF (ѵ)(y)
)
,(

minµ FT (ѵ)(y), minµ FI(ѵ)(y), minµ FF (ѵ)(y)
)⟩

.

Ñmin =
⟨(

min
µ

TT (ѵ)(y), min
µ

TI(ѵ)(y), min
µ

TF (ѵ)(y)
)
,(

maxµ IT (ѵ)(y), maxµ II(ѵ)(y), maxµ IF (ѵ)(y)
)
,(

maxµ FT (ѵ)(y), maxµ FI(ѵ)(y), maxµ FF (ѵ)(y)
)⟩

.

Ñmin ≤ T2NNAAHM(Ñ1, Ñ2, . . . , Ñs) ≤ Ñmax.

Proof: For Theorem 3 (Boundedness)

By using Monotonicity and Idempotency conditions, we get:
T2NNAAHM min(Ñmin1 , Ñmin2 , . . . , Ñmins ) ≤ T2NNAAHM(Ñ1, Ñ2, . . . , Ñs),

T2NNAAHM Ñmax(Ñmax1 , Ñmax2 , . . . , Ñmaxs ) ≥ T2NNAAHM(Ñ1, Ñ2, . . . , Ñs).

3.2. Type-2 neutrosophic numbers Aczel–Alsina weighted Hamy Mean
Operator

Definition 4. Let Ñn =
⟨
(TT (Ñn)(z), TI(Ñn)(z), TF (Ñn)(z)),

(IT (Ñn)(z), II(Ñn)(z), IF (Ñn)(z)), (FT (Ñn)(z), FI(Ñn)(z), FF (Ñn)(z))
⟩

be a T2NN. Then, the T2NNAAWHM
operator is defined as in Eq.(8):

T2NNAAWHM(θ1, θ2, . . . , θn) =



∑
1≤i1<···<iτ≤n

(
1−

τ∑
µ=1

ωkµ

)( τ∏
i=1

θiµ

)1/τ
Cτ

n

, 1 ≤ τ ≤ n,

∏τ
i=1 θ

(1−ωkµ )/(n−1)

iµ
, τ = n

(8)

Theorem 4. Let

Ñn =
⟨
(TT (Ñn)(z), TI(Ñn)(z), TF (Ñn)(z)), (IT (Ñn)(z), II(Ñn)(z), IF (Ñn)(z)),

(FT (Ñn)(z), FI(Ñn)(z), FF (Ñn)(z))
⟩
be T2NNs. Then the Type-2 neutrosophic numbers Aczel–Alsina Weighted

Hamy Mean operator is defined as in Eqs. (9) and (10) (1 ≤ τ < n).

T2NNAAWHM(θ1, θ2, . . . , θn) =



K. Kara et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6753 11 of 20

⟨(
1− e

−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(T
T (ѵ)

(z)))ξ)1/ξ

))
)ξ)1/ξ)

,

1− e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(T
I(ѵ)

(z)))ξ)1/ξ

))
)ξ)1/ξ)

,

1− e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(T
F (ѵ)

(z)))ξ)1/ξ

))
)ξ)1/ξ))

,

(
e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(1−I
T (ѵ)

(z)))ξ)1/ξ

))
)ξ)1/ξ)

,

e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(1−I
I(ѵ)

(z)))ξ)1/ξ

))
)ξ)1/ξ)

,

e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(1−I
F (ѵ)

(z)))ξ)1/ξ

))
)ξ)1/ξ))

,

(
e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(1−F
T (ѵ)

(z)))ξ)1/ξ

))
)ξ)1/ξ)

,

e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(1−F
I(ѵ)

(z)))ξ)1/ξ

))
)ξ)1/ξ)

,

e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(1−F
F (ѵ)

(z)))ξ)1/ξ

))
)ξ)1/ξ))⟩

.

(9)

Proof: For Theorem 4 for 1 ≤ τ < n

τ∏
µ=1

θµ =



e
−
(∑τ

µ=1

(
− ln(TT

ѵ
(z))

)ξ
)1/ξ

, 1− e
−
(∑τ

µ=1

(
− ln(1−TI

ѵ
(z))

)ξ
)1/ξ

, 1− e
−
(∑τ

µ=1

(
− ln(1−TF

ѵ
(z))

)ξ
)1/ξ

1− e
−
(∑τ

µ=1

(
− ln(1−IT

ѵ
(z))

)ξ
)1/ξ

, 1− e
−
(∑τ

µ=1

(
− ln(1−II

ѵ
(z))

)ξ
)1/ξ

, e
−
(∑τ

µ=1

(
− ln(IF

ѵ
(z))

)ξ
)1/ξ

1− e
−
(∑τ

µ=1

(
− ln(1−FT

ѵ
(z))

)ξ
)1/ξ

, 1− e
−
(∑τ

µ=1

(
− ln(1−FI

ѵ
(z))

)ξ
)1/ξ

, e
−
(∑τ

µ=1

(
− ln(FF

ѵ
(z))

)ξ
)1/ξ


.

then  τ∏
µ=1

θµ

1/τ

=

(
e
−
(

1
τ

∑τ
µ=1(− ln(TT

ѵ
(z)))ξ

)1/ξ

, 1− e
−
(

1
τ

∑τ
µ=1(− ln(TI

ѵ
(z)))ξ

)1/ξ

, 1− e
−
(

1
τ

∑τ
µ=1(− ln(TF

ѵ
(z)))ξ

)1/ξ

,

1− e
−
(

1
τ

∑τ
µ=1(− ln(1−IT

ѵ
(z)))ξ

)1/ξ

, 1− e
−
(

1
τ

∑τ
µ=1(− ln(1−II

ѵ
(z)))ξ

)1/ξ

, e
−
(

1
τ

∑τ
µ=1(− ln(1−IF

ѵ
(z)))ξ

)1/ξ

,

1−e
−
(

1
τ

∑τ
µ=1(− ln(1−FT

ѵ
(z)))ξ

)1/ξ

, 1−e
−
(

1
τ

∑τ
µ=1(− ln(1−FI

ѵ
(z)))ξ

)1/ξ

, e
−
(

1
τ

∑τ
µ=1(− ln(1−FF

ѵ
(z)))ξ

)1/ξ)
then

∑
1≤i1<···<iτ≤n

(
1−

τ∑
µ=1

ωkµ

)( τ∏
i=1

θµ

)1/τ

=



K. Kara et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6753 12 of 20

⟨
(
1− e

−
(∑

1≤i1<···<iτ≤n(1−
∑τ

µ=1 ωkµ )(− ln(1−e
−( 1

τ

∑τ
µ=1(− ln(T

T (ѵ)
(z)))ξ)1/ξ

)))ξ
)1/ξ)

,

1− e
−
(∑

1≤i1<···<iτ≤n(1−
∑τ

µ=1 ωkµ )(− ln(1−e
−( 1

τ

∑τ
µ=1(− ln(T

I(ѵ)
(z)))ξ)1/ξ

)))ξ
)1/ξ

,

1− e
−
(∑

1≤i1<···<iτ≤n(1−
∑τ

µ=1 ωkµ )(− ln(1−e
−( 1

τ

∑τ
µ=1(− ln(T

F (ѵ)
(z)))ξ)1/ξ

)))ξ
)1/ξ

⟩

then ∑
1≤i1<···<iτ≤n

(∏τ
i=1 θiµ

)1/τ
Cτ

n

=

⟨1− e
−
(∑

1≤i1<···<iτ≤n

(
(1−

∑τ
µ=1 ωkµ

)(− ln(1−e
−((1/τ)

∑τ
µ=1(− lnTT (Nµ)(z))

ξ)1/ξ

))

)ξ

Cτ
n

)1/ξ

,

1− e
−
(∑

1≤i1<···<iτ≤n

(
(1−

∑τ
µ=1 ωkµ

)(− ln(1−e
−((1/τ)

∑τ
µ=1(− lnTI(Nµ)(z))

ξ)1/ξ

))

)ξ

Cτ
n

)1/ξ

,

1− e
−
(∑

1≤i1<···<iτ≤n

(
(1−

∑τ
µ=1 ωkµ

)(− ln(1−e
−((1/τ)

∑τ
µ=1(− lnTF (Nµ)(z))

ξ)1/ξ

))

)ξ

Cτ
n

)1/ξ

⟩

Proof: For Theorem 4, for τ = n

θ

1−ωkµ
n−1

iµ
=

⟨


e
−
(

1−ωkµ
n−1

(− ln(T
T (ѵ)

(z)))ξ
)1/ξ

, e
−
(

1−ωkµ
n−1

(− ln(T
I(ѵ)

(z)))ξ
)1/ξ

, e
−
(

1−ωkµ
n−1

(− ln(T
F (ѵ)

(z)))ξ
)1/ξ

1− e
−
(

1−ωkµ
n−1

(− ln(1−I
T (ѵ)

(z)))ξ
)1/ξ

, 1− e
−
(

1−ωkµ
n−1

(− ln(1−I
I(ѵ)

(z)))ξ
)1/ξ

, 1− e
−
(

1−ωkµ
n−1

(− ln(1−I
F (ѵ)

(z)))ξ
)1/ξ

1− e
−
(

1−ωkµ
n−1

(− ln(1−F
T (ѵ)

(z)))ξ
)1/ξ

, 1− e
−
(

1−ωkµ
n−1

(− ln(1−F
I(ѵ)

(z)))ξ
)1/ξ

, 1− e
−
(

1−ωkµ
n−1

(− ln(1−F
F (ѵ)

(z)))ξ
)1/ξ


⟩

then
τ∏

i=1

θ

1−ωkµ
n−1

iµ
=

⟨ (
e
−(
∑τ

µ=1

1−ωkµ
n−1

(− lnT
T (ѵ)

(z))ξ)1/ξ

, e
−(
∑τ

µ=1

1−ωkµ
n−1

(− lnT
I(ѵ)

(z))ξ)1/ξ

, e
−(
∑τ

µ=1

1−ωkµ
n−1

(− lnT
F (ѵ)

(z))ξ)1/ξ)
(
1− e

−(
∑τ

µ=1

1−ωkµ
n−1

(− ln(1−I
T (ѵ)

(z)))ξ)1/ξ

, 1− e
−(
∑τ

µ=1

1−ωkµ
n−1

(− ln(1−I
I(ѵ)

(z)))ξ)1/ξ

, 1− e
−(
∑τ

µ=1

1−ωkµ
n−1

(− ln(1−I
F (ѵ)

(z)))ξ)1/ξ)
(
1− e

−(
∑τ

µ=1

1−ωkµ
n−1

(− ln(1−F
T (ѵ)

(z)))ξ)1/ξ

, 1− e
−(
∑τ

µ=1

1−ωkµ
n−1

(− ln(1−F
I(ѵ)

(z)))ξ)1/ξ

, 1− e
−(
∑τ

µ=1

1−ωkµ
n−1

(− ln(1−F
F (ѵ)

(z)))ξ)1/ξ)
⟩

T2NNAAWHM(θ1, θ2, . . . , θn) =

⟨


e
−
(∑τ

µ=1

1−ωkµ
n−1

(
− ln(T

T (ѵ)
(z))

)ξ
)1/ξ

e
−
(∑τ

µ=1

1−ωkµ
n−1

(
− ln(T

I(ѵ)
(z))

)ξ
)1/ξ

e
−
(∑τ

µ=1

1−ωkµ
n−1

(
− ln(T

F (ѵ)
(z))

)ξ
)1/ξ

1− e
−
(∑τ

µ=1

1−ωkµ
n−1

(
− ln(1−I

T (ѵ)
(z))

)ξ
)1/ξ

1− e
−
(∑τ

µ=1

1−ωkµ
n−1

(
− ln(1−I

I(ѵ)
(z))

)ξ
)1/ξ

1− e
−
(∑τ

µ=1

1−ωkµ
n−1

(
− ln(1−I

F (ѵ)
(z))

)ξ
)1/ξ

1− e
−
(∑τ

µ=1

1−ωkµ
n−1

(
− ln(1−F

T (ѵ)
(z))

)ξ
)1/ξ

1− e
−
(∑τ

µ=1

1−ωkµ
n−1

(
− ln(1−F

I(ѵ)
(z))

)ξ
)1/ξ

1− e
−
(∑τ

µ=1

1−ωkµ
n−1

(
− ln(1−F

F (ѵ)
(z))

)ξ
)1/ξ


⟩

.

(10)



K. Kara et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6753 13 of 20

First condition: Ñ (z) : Z → D[0, 1], ĨÑ (z) : Z → D[0, 1], F̃Ñ (z) : Z → D[0, 1]

Second condition: 0 ≤ T̃Ñ (z)3 + ĨÑ (z)3 + F̃Ñ (z)3 ≤ 3

Consider the first condition:

T̃Ñ (z) =

⟨1− e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(TT (ѵ)(z)))ξ)1/ξ
))

)ξ
)1/ξ

1− e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(TI (ѵ)(z)))ξ)1/ξ
))

)ξ
)1/ξ

1− e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−((1/τ)
∑τ

µ=1(− ln(TF (ѵ)(z)))ξ)1/ξ
))

)ξ
)1/ξ

⟩
, ϵ ∈ [0, 1]

=⇒ 0 ≤ T̃Ñ (z) ≤ 1.

ĨÑ (z) =

⟨ e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
(1−

∑τ
µ=1 ωkµ )

(
−ln(1−e

−( 1
τ

∑τ
µ=1(− ln(1−I

T (ѵ)
(z)))ξ)1/ξ

)
))ξ)1/ξ

e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
(1−

∑τ
µ=1 ωkµ )

(
−ln(1−e

−( 1
τ

∑τ
µ=1(− ln(1−I

I(ѵ)
(z)))ξ)1/ξ

)
))ξ)1/ξ

e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n

(
(1−

∑τ
µ=1 ωkµ )

(
−ln(1−e

−( 1
τ

∑τ
µ=1(− ln(1−I

F (ѵ)
(z)))ξ)1/ξ

)
))ξ)1/ξ

⟩
, ϵ[0, 1]

=⇒ 0 ≤ ĨÑ (z) ≤ 1

F̃Ñ (z) =

⟨e

−

 1
Cτ
n

∑
1≤i1<···<iτ≤n

(
1−
∑τ

µ=1 ωkµ

)(
− ln

(
1−e

−( 1
τ

∑τ
µ=1(− ln(1−F

T (ѵ)
(z)))ξ)1/ξ

))ξ
1/ξ

,

e

−

 1
Cτ
n

∑
1≤i1<···<iτ≤n

(
1−
∑τ

µ=1 ωkµ

)(
− ln

(
1−e

−( 1
τ

∑τ
µ=1(− ln(1−F

I(ѵ)
(z)))ξ)1/ξ

))ξ
1/ξ

,

e

−

 1
Cτ
n

∑
1≤i1<···<iτ≤n

(
1−
∑τ

µ=1 ωkµ

)(
− ln

(
1−e

−( 1
τ

∑τ
µ=1(− ln(1−F

F (ѵ)
(z)))ξ)1/ξ

))ξ
1/ξ

⟩
, ϵ[0, 1]

=⇒ 0 ≤ F̃Ñ (z) ≤ 1

Consider the second condition:
T̃Ñ (z)3 + ĨÑ (z)3 + F̃Ñ (z)3 =



K. Kara et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6753 14 of 20

T̃Ñ (z)3 + ĨÑ (z)3 + F̃Ñ (z)3 =

⟨(1− e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−( 1
τ

∑τ
µ=1(− ln(T

T (ѵ)
(z)))ξ)1/ξ

))ξ

)1/ξ)
(
1− e

−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−( 1
τ

∑τ
µ=1(− ln(T

I(ѵ)
(z)))ξ)1/ξ

))ξ

)1/ξ)
(
1− e

−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−( 1
τ

∑τ
µ=1(− ln(T

F (ѵ)
(z)))ξ)1/ξ

))ξ

)1/ξ)

+

(
e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−( 1
τ

∑τ
µ=1(− ln(1−I

T (ѵ)
(z)))ξ)1/ξ

))ξ

)1/ξ)
(
e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−( 1
τ

∑τ
µ=1(− ln(1−I

I(ѵ)
(z)))ξ)1/ξ

))ξ

)1/ξ)
(
e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−( 1
τ

∑τ
µ=1(− ln(1−I

F (ѵ)
(z)))ξ)1/ξ

))ξ

)1/ξ)

+

(
e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−( 1
τ

∑τ
µ=1(− ln(1−F

T (ѵ)
(z)))ξ)1/ξ

))ξ

)1/ξ)
(
e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−( 1
τ

∑τ
µ=1(− ln(1−F

I(ѵ)
(z)))ξ)1/ξ

))ξ

)1/ξ)
(
e
−
(

1
Cτ
n

∑
1≤i1<···<iτ≤n(1−

∑τ
µ=1 ωkµ )(− ln(1−e

−( 1
τ

∑τ
µ=1(− ln(1−F

F (ѵ)
(z)))ξ)1/ξ

))ξ

)1/ξ)

⟩
, ϵ ∈ [0, 1]

=⇒ 0 ≤ T̃Ñ (z)3 + ĨÑ (z)3 + F̃Ñ (z)3 ≤ 3.

Theorem 5: The T2NNAAWHM operator meets the following properties:
Monotonicity: Let Ñk ≤ Ñ∗

k . If T̃Ñ (z) ≤ T̃ ∗
Ñ
(z), ĨÑ (z) ≥ Ĩ∗

Ñ
(z), and F̃Ñ (z) ≥ F̃ ∗

Ñ
(z), then

T2NNAAWHM(Ñ1, Ñ2, . . . , Ñs) ≤ T2NNAAWHM(Ñ∗
1 , Ñ

∗
2 , . . . , Ñ

∗
s ).

Proof: It is similar to the proof of the Monotonicity property of the T2NNAAHM operator.
Idempotency: Set

Ñk =
⟨
(TT (ѵ)(y), TI(ѵ)(y), TF (ѵ)(y)), (IT (ѵ)(y), II(ѵ)(y), IF (ѵ)(y)), (FT (ѵ)(y), FI(ѵ)(y), FF (ѵ)(y))

⟩
, (µ = 1, . . . , s).

If M̃µ = M̃ , then

T2NNAAWHM(1, Ñ2, . . . , Ñs) = Ñ.
Proof: It is similar to the proof for the idempotency property of the T2NNAAHM operator.
Boundedness: If the maximum and minimum T2NNs are given below:

(i) max =
⟨(

maxµ
(
TT

ѵ
(y)
)
,maxµ

(
TI

ѵ
(y)
)
,maxµ

(
TF

ѵ
(y)
))
,(

minµ
(
IT

ѵ
(y)
)
,minµ

(
II

ѵ
(y)
)
,minµ

(
IF

ѵ
(y)
))
,(

minµ
(
FT

ѵ
(y)
)
,minµ

(
FI

ѵ
(y)
)
,minµ

(
FF

ѵ
(y)
))⟩

(i) min =
⟨(

minµ
(
TT

ѵ
(y)
)
,minµ

(
TI

ѵ
(y)
)
,minµ

(
TF

ѵ
(y)
))
,(

maxµ
(
IT

ѵ
(y)
)
,maxµ

(
II

ѵ
(y)
)
,maxµ

(
IF

ѵ
(y)
))
,(

maxµ
(
FT

ѵ
(y)
)
,maxµ

(
FI

ѵ
(y)
)
,maxµ

(
FF

ѵ
(y)
))⟩

(i) Then, min ≤ T2NNAAWHM(Ñ1, Ñ2, . . . , Ñs) ≤ Ñmax can hold.

Proof : It is similar to the proof for the boundedness property of the T2NNAAHM operator.



K. Kara et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6753 15 of 20

4. T2NN Aczel-Alsina-based Hamy Mean MADM Model
In this section, the formulated T2NNAAHM and T2NNAAWHM operators are employed to introduce the

T2NN Aczel-Alsina-based Hamy Mean (T2NNAAHM) MADM model. A MADM problem consists of a set of r
alternatives Gi = {G1, G2, . . . , Gr} and a set of s attributes Kp = {k1, k2, . . . , ks}. The alternatives and criteria
are represented by

R = (Ñip)s×r for T2NNs.

Here, each entry Ñip is expressed as

Ñip = ⟨(TT (Ñip)
(y), TI(Ñip)

(y), TF (Ñip)
(y)), (IT (Ñip)

(y), II(Ñip)
(y), IF (Ñip)

(y)), (FT (Ñip)
(y), FI(Ñip)

(y), FF (Ñip)
(y))⟩.

Then, the weighting of attributes Kk (k = 1, 2, . . . , s) is specified by a weight vector

ω = (ω1, ω2, . . . , ωs), ωk ∈ [0, 1],

s∑
k=1

ωk = 1.

5. Results
Within this section, we incorporate a comprehensive study aimed at validating the newly developed aggregation

operators, namely T2NNAAHM and T2NNAAWHM, tailored for T2NNs. The study is adapted from the research
conducted by Abdel-Basset et al. [2], a deliberate choice to ensure the congruence and comparability of the resultant
outcomes.

This study focuses on the intricate task of selecting the best bank within the realm of banking transactions. We
consider a set of four distinct banking alternatives, denoted as Gi (i = 1, 2, . . . , 4), and five discernible attributes,
represented as Kp (p = 1, 2, . . . , 5). The five attributes include: reputation and elegance (G1), customer service
(G2), location of the bank and its branch network (G3), fee structures (G4), and promotional offers (G5).

According to Abdel-Basset et al. [2], the criterion weights are established as:

ωk = (0.20, 0.25, 0.30, 0.15, 0.10).

The decision matrix containing the relevant data is presented in Table 1.

Table 1: Decision matrix for four banks and five attributes with T2NNs
K1 K2 K3 K4 K5

G1 ⟨
(0.75, 0.80, 0.85)
(0.20, 0.15, 0.30)
(0.10, 0.15, 0.20)

⟩ ⟨
(0.65, 0.70, 0.75)
(0.40, 0.45, 0.50)
(0.35, 0.40, 0.35)

⟩ ⟨
(0.85, 0.90, 0.95)
(0.30, 0.35, 0.40)
(0.25, 0.40, 0.35)

⟩ ⟨
(0.50, 0.40, 0.55)
(0.10, 0.15, 0.30)
(0.10, 0.20, 0.20)

⟩ ⟨
(0.30, 0.45, 0.25)
(0.20, 0.10, 0.30)
(0.10, 0.25, 0.20)

⟩

G2 ⟨
(0.60, 0.50, 0.65)
(0.30, 0.25, 0.30)
(0.20, 0.30, 0.25)

⟩ ⟨
(0.65, 0.70, 0.75)
(0.10, 0.15, 0.20)
(0.05, 0.10, 0.15)

⟩ ⟨
(0.45, 0.35, 0.50)
(0.15, 0.10, 0.10)
(0.20, 0.30, 0.25)

⟩ ⟨
(0.45, 0.50, 0.60)
(0.30, 0.20, 0.30)
(0.25, 0.30, 0.25)

⟩ ⟨
(0.50, 0.45, 0.35)
(0.30, 0.25, 0.30)
(0.20, 0.30, 0.25)

⟩

G3 ⟨
(0.45, 0.50, 0.80)
(0.15, 0.30, 0.55)
(0.55, 0.20, 0.25)

⟩ ⟨
(0.40, 0.45, 0.50)
(0.15, 0.20, 0.25)
(0.10, 0.15, 0.20)

⟩ ⟨
(0.40, 0.45, 0.60)
(0.05, 0.20, 0.25)
(0.40, 0.20, 0.25)

⟩ ⟨
(0.45, 0.80, 0.90)
(0.40, 0.70, 0.55)
(0.55, 0.20, 0.40)

⟩ ⟨
(0.80, 0.50, 0.80)
(0.45, 0.30, 0.55)
(0.55, 0.20, 0.25)

⟩

G4 ⟨
(0.85, 0.70, 0.95)
(0.60, 0.50, 0.65)
(0.45, 0.15, 0.35)

⟩ ⟨
(0.60, 0.65, 0.70)
(0.35, 0.40, 0.45)
(0.30, 0.40, 0.45)

⟩ ⟨
(0.95, 0.70, 0.80)
(0.15, 0.10, 0.30)
(0.30, 0.35, 0.30)

⟩ ⟨
(0.90, 0.70, 0.95)
(0.60, 0.40, 0.65)
(0.45, 0.15, 0.35)

⟩ ⟨
(0.65, 0.70, 0.80)
(0.40, 0.35, 0.25)
(0.15, 0.15, 0.20)

⟩

The resolution of the task involving the identification of the optimal bank through the application of the
T2NNAAHM and T2NNAAWHM operators within the context of the MADM framework has been systematically
executed.

Step 1: An expert determines the matrix as seen in Table 1.
Step 2: When considering ξ = 1 and τ = 2, we computed the aggregated T2NNs and subsequently determined

the weighted aggregated T2NNs. Table 2 displays the results of the aggregated T2NN values, which were obtained
using the T2NNAAHM operator.

Table 3 provides an overview of the weighted aggregated T2NN values, which were calculated using the
T2NNAAWHM operator.



K. Kara et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6753 16 of 20

Table 2: Aggregated T2NN values for T2NNAAHM operator.
Alternatives T̃Ñ (y) ĨÑ (y) F̃Ñ (y)

G1 (0.7921, 0.8306, 0.8535) (0.3509, 0.3404, 0.5203) (0.0530, 0.1177, 0.1047)
G2 (0.7146, 0.6816, 0.7532) (0.3392, 0.2884, 0.3555) (0.0557, 0.1047, 0.0859)
G3 (0.6789, 0.7227, 0.8850) (0.3268, 0.4750, 0.5976) (0.2484, 0.0626, 0.1118)
G4 (0.9335, 0.8580, 0.9575) (0.5804, 0.4935, 0.6315) (0.1564, 0.0891, 0.1570)

Table 3: Weighted aggregated T2NN values for T2NNAAWHM operator.
Alternative T̃N (y) ĨN (y) F̃N (y)

G1 (0.5929, 0.6387, 0.6605) (0.2221, 0.2108, 0.3515) (0.1646, 0.2705, 0.2525)
G2 (0.5276, 0.4974, 0.5619) (0.2268, 0.1894, 0.2388) (0.1794, 0.2606, 0.2306)
G3 (0.5037, 0.5422, 0.7373) (0.2248, 0.3289, 0.4330) (0.4431, 0.1903, 0.2712)
G4 (0.7992, 0.6906, 0.8531) (0.4173, 0.3437, 0.4512) (0.3270, 0.2260, 0.3256)

Step 3: The computation of score functions and accuracy functions was carried out. The outcomes of these
computations, derived from the application of both the T2NNAAHM and T2NNAAWHM operators, were system-
atically organized and presented in Table 4.

Step 4: The ranking of the alternatives is established based on the score functions’ values.
Utilizing the T2NNAAHM operator, the ranking of the alternatives is as follows: G1 > G2 > G4 > G3.
Conversely, employing the T2NNAAWHM operator, the ranking is consistent, with the order G1 > G2 > G4 >

G3.

Table 4: The score and accuracy values of alternatives.
Alternatives T2NNAAHM Operator T2NNAAWHM Operator

Score function Accuracy function Score function Accuracy function
G1 0.7801 0.7284 0.7147 0.3931
G2 0.7673 0.6199 0.6923 0.2882
G3 0.7207 0.6309 0.6595 0.3076
G4 0.7430 0.7788 0.6977 0.4822

6. Comparative Analysis

6.1. Comparison of T2NNWA with the Novel Aggregation Operators
During this investigation, we introduced novel aggregation operators based on Hamy Mean, namely T2NNAAHM

and T2NNAAWHM, grounded in Aczel–Alsina t-norm and t-conorm operations. Following the validation of these
proposed aggregation operators, we have demonstrated their practical application through the comprehensive study
extracted from Abdel-Basset et al. [2].

Additionally, within the scope of this research, we compared these research findings with the T2NNWA operator
findings. Upon a comprehensive comparison of the findings derived from this research with those of Abdel-Basset
et al. [2], a remarkable alignment is evident. Specifically, for ξ = 1 and τ = 2, the aggregated T2NN values,
score functions, and accuracy function values derived from both the T2NNAAHM and T2NNWA operators exhibit
congruence.

A noticeable implication arises when considering the results obtained from the application of T2NNAAWHM
and T2NNWA. For ξ = 1 and τ = 2, the T2NNAAHM operator and the T2NNAAWHM operator assign the first
alternative as the best one. For a comprehensive comparative analysis, Table 5 is presented, which shows an overview
of the juxtaposition between T2NNWA, T2NNAAHM, and T2NNAAWHM for T2NNs.

6.2. Comparison of the Novel Aggregation Operators based on Parameter
Variations

The ξ parameter in the proposed T2NNAAHM and T2NNAAWHM operators yields varying outcomes under
different variations. These variations are examined in terms of the robustness of the alternative rankings obtained



K. Kara et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6753 17 of 20

Table 5: Comparison of aggregation operators for T2NNs.

Operator G1 G2 G3 G4 Alt. rank
T2NNWA 0.7461 0.7223 0.6906 0.7201 G1 > G2 > G4 > G3

T2NNAAHM 0.7801 0.7673 0.7207 0.7430 G1 > G2 > G4 > G3

T2NNAAWHM 0.7147 0.6923 0.6595 0.6977 G1 > G4 > G2 > G3

by comparing the generated score functions. Score function values and alternative rankings for various ξ parameter
variations in the case of T2NNAAHM are presented in Table 6.

Table 6: Score functions under various ξ for T2NNAAHM.

ξ G1 G2 G3 G4 Alt. rank
1 0.7801 0.7673 0.7208 0.7430 G1 > G2 > G4 > G3

2 0.7522 0.7307 0.6900 0.7214 G1 > G2 > G4 > G3

3 0.7434 0.7138 0.6743 0.7091 G1 > G2 > G4 > G3

10 0.7435 0.6843 0.6441 0.6925 G1 > G4 > G2 > G3

15 0.7472 0.6806 0.6406 0.6913 G1 > G4 > G2 > G3

25 0.7513 0.6781 0.6387 0.6909 G1 > G4 > G2 > G3

50 0.7548 0.6765 0.6380 0.6912 G1 > G4 > G2 > G3

75 0.7560 0.6760 0.6378 0.6914 G1 > G4 > G2 > G3

100 0.7566 0.6758 0.6377 0.6914 G1 > G4 > G2 > G3

150 0.7572 0.6755 0.6377 0.6915 G1 > G4 > G2 > G3

The score function values and alternative rankings for various ξ parameter variations in the case of T2NNAAWHM
are displayed in Table 7.

Table 7: Score functions under various ξ for T2NNAAWHM.

ξ G1 G2 G3 G4 Alt. rank
1 0.7148 0.6924 0.6596 0.6977 G1 > G4 > G2 > G3

2 0.7136 0.6875 0.6541 0.6926 G1 > G4 > G2 > G3

3 0.7177 0.6836 0.6491 0.6895 G1 > G4 > G2 > G3

10 0.7381 0.6746 0.6361 0.6876 G1 > G4 > G2 > G3

15 0.7440 0.6740 0.6353 0.6882 G1 > G4 > G2 > G3

25 0.7495 0.6741 0.6356 0.6892 G1 > G4 > G2 > G3

50 0.7540 0.6745 0.6365 0.6904 G1 > G4 > G2 > G3

75 0.7554 0.6747 0.6368 0.6908 G1 > G4 > G2 > G3

100 0.7562 0.6748 0.6370 0.6910 G1 > G4 > G2 > G3

150 0.7569 0.6748 0.6372 0.6912 G1 > G4 > G2 > G3

Upon close examination of the alternative rankings, it is evident that the alternative ranking remains unchanged
within the T2NNAAWHM operator. Conversely, within the T2NNAAHM operator, a shift between the second and
third ranks occurs when ξ = 10. Nevertheless, it is noteworthy that the ranking of the top-performing alternatives
remains consistent across all ξ values for all operators.

7. Implications
The findings of this research have several important implications for theory and practice:



K. Kara et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6753 18 of 20

(i) Introduction of novel operators: Two aggregation operators, T2NNAAHM and T2NNAAWHM, were devel-
oped for use with T2NNs, expanding the currently limited set of available operators.

(ii) Flexibility in aggregation: The T2NNAAHM operator provides an unweighted aggregation mechanism, while
the T2NNAAWHM operator enables weighted aggregation, thus broadening applicability across different
decision-making contexts.

(iii) Theoretical advancement: Both operators are grounded in Hamy Mean averaging, derived through Aczél–
Alsina t-norm and t-conorm operations, with assumptions and proofs rigorously established.

(iv) Comparative reliability: Results obtained from the proposed operators consistently aligned with those pro-
duced by the existing T2NNWA operator, thereby confirming the robustness and stability of the proposed
approach. utility: The proposed operators offer decision-makers alternative aggregation outcomes, allowing
for multiple perspectives in decision-making rather than a single deterministic result.

8. Conclusions
This study has advanced the field of MADM by introducing two novel aggregation operators for T2NNs: the

T2NNAAHM and T2NNAAWHM. Unlike the existing literature, which only includes the T2NNWA operator, the
proposed operators broaden the methodological toolkit by incorporating Hamy Mean-based aggregation founded on
Aczel–Alsina t-norms and t-conorms. Their mathematical development has been comprehensively detailed, ensuring
transparency of assumptions and proofs.

Empirical analyses demonstrated that both operators consistently produced outcomes comparable to those ob-
tained with the T2NNWA operator, including under parameter � variations. This stability underscores the robustness
of the Aczel–Alsina-based Hamy Mean MADM model. The ability to perform both weighted and unweighted ag-
gregation enhances the adaptability of the proposed methods in diverse decision-making contexts, making them
particularly valuable when handling linguistic expressions and uncertain information.

Looking forward, future research could extend this line of work by exploring additional aggregation operators
for T2NNs and systematically comparing their performance with both the newly developed operators and existing
benchmarks. Such comparative investigations will not only deepen the understanding of operator robustness but
also provide practitioners with more reliable tools for complex decision-making scenarios. By facilitating alternative
yet consistent decision outcomes, the novel operators proposed in this study hold strong potential for practical
applications as well as for enriching theoretical research in neutrosophic decision-making.

Acknowledgements
The authors thank the readers of European Journal of Pure and Applied Mathematics, for making our journal

successful.

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