INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
Online ISSN 1841-9844, ISSN-L 1841-9836, Volume: 17, Issue: 1, Month: February, Year: 2022
Article Number: 4646, https://doi.org/10.15837/ijccc.2022.1.4646

CCC Publications 

M-generalised q-neutrosophic extension of CoCoSo method

Z. Turskis, R. Bausys, F. Smarandache, G. Kazakeviciute-Januskeviciene and E. K. Zavadskas

Zenonas Turskis
Institute of Sustainable Construction
Vilnius Gediminas Technical University
LT–10223 Vilnius, Sauletekio al. 11, Lithuania
zenonas.turskis@vgtu.lt

Romualdas Bausys
Department of Graphical Systems
Vilnius Gediminas Technical University
LT–10223 Vilnius, Sauletekio al. 11, Lithuania
*Corresponding author: romualdas.bausys@vilniustech.lt

Florentin Smarandache
Department of Mathematics
University of New Mexico
705 Gurley Ave., Gallup, New Mexico 87301, USA
smarand@unm.edu

Giruta Kazakeviciute-Januskeviciene
Department of Graphical Systems
Vilnius Gediminas Technical University
LT–10223 Vilnius, Sauletekio al. 11, Lithuania
giruta.kazakeviciute-januskeviciene@vilniustech.lt

Edmundas Kazimieras Zavadskas
Institute of Sustainable Construction
Vilnius Gediminas Technical University
LT–10223 Vilnius, Sauletekio al. 11, Lithuania
edmundas.zavadskas@vilniustech.lt

Abstract
Nowadays fuzzy approaches gain popularity to model multi-criteria decision making (MCDM)

problems emerging in real-life applications. Modern modelling trends in this field include eval-
uation of the criteria information uncertainty and vagueness. Traditional neutrosophic sets are
considered as the effective tool to express uncertainty of the information. However, in some cases,
it cannot cover all recently proposed cases of the fuzzy sets. The m-generalized q-neutrosophic sets
(mGqNNs) can effectively deal with this situation. The novel MCDM methodology CoCoSomGqNN
is presented in this paper. An illustrative example presents the analysis of the effectiveness of dif-
ferent retrofit strategy selection decisions for the application in the civil engineering industry.

Keywords: Multi-criteria decision making, CoCoSo, Neutrosophic sets, retrofit strategy.



https://doi.org/10.15837/ijccc.2022.1.4646 2

1 Introduction
Multi-criteria decision making (MCDM) approaches introduce efficient tools to solve conflict situ-

ations for the managers. The MCDM process can be distinguished into the following steps:

a) Defining management system governing criteria which are associated with the compulsory decision-
making goals;

b) Establishing the discrete alternative systems to reach the prescribed necessary goals;

c) Performing multicriteria analysis by any MCDM method;

d) Acknowledging one alternative as the best by the ranking results. Therefore, the application
of the MCDM techniques facilitates the decision-makers to choose the best alternative, which
provides a suitable compromise between all possibly conflicting criteria.

Therefore, the application of the MCDM techniques facilitates the decision-makers to embrace
the alternative of the highest quality, which provides a suitable compromise between all possibly
conflicting criteria. During the last decade, MCDM methods have actively been applied for the
solution of the various real-life problems. In [5], the authors point out the following the most popular
MCDM methods, such as SAW, WASPAS, VIKOR, TOPSIS, ELECTRE PROMETHEE methods.
Recently, in [28] was proposed a new MCDM technique, which composes a combined compromise
solution (CoCoSo). This method constructs the compromise solution applying various aggregation
approaches. This method was under intensive research during the last years. In [39], the authors
solved the problem of the ideal sustainable supplier choice by combining two methods: CoCoSo and
the BWM method. In [14] was proposed a hybrid MCDM approach which is constructed by combining
the CoCoSo and CRITIC approaches and applied to evaluate the 5G industries. In [25], the authors
combined SWARA and CoCoSo methods to solve the drug cold chain logistics supplier evaluation and
the location selection for a logistics center problems. This relatively new (CoCoSo) technique was
applied to study the sustainability aspects in the OPEC countries [3]. The extension of the CoCoSo
approach was proposed in [4]. The essence of this extension is the modelling environment in the form
of normalized weighted geometric Bonferroni mean functions. This approach was employed to solve
the supply chain problem. In [11], stochastic multi-criteria acceptability analysis was performed within
the multi-criteria decision-making framework, namely the CoCoSo approach, to handle the problem
of the renewable energy investments with stochastic information. This hybrid approach is based on
the DIBR method, which defines interrelationships between the ranked criteria, and the D’CoCoSo
approach, which is modelled under fuzzy Dombi environment. This technique was applied to solve
prioritizing aspects of the circular economy concepts for urban mobility [12]. The two-stage decision-
making technique, which was realized by the data envelopment analysis (DEA) and the RCoCoSo
method, which was modelled under rough full consistency (R-FUCOM) environment, was discussed
in [30]. The tourism attraction selection problem was utilised by applying the probabilistic linguistic
term set. For the solution, IDOCRIW and CoCoSo approaches are implemented [9]. Different versions
of the hybrid MCDM proposals based on the CoCoSo technique have been presented in [23], [10], [24],
[6].

Modern research in the area of multi-criteria decision-making tries to model the indeterminacy
and inconsistency of the initial decision making information, which prevails in most cases of real-life
applications. Therefore, different fuzzy set variations have been proposed starting width the pioneering
work in [33]. Existing challenges and future development trends of the fuzzy modelling importance in
the decision-making field have been discussed in [27]. During the last years have been proposed the
different extensions of the CoCoSo method applying the environments of the different fuzzy sets.

In [29], the authors modelled the best supplier problem in construction management applying grey
numbers and solving this problem with the DEMATEL, BWM, and CoCoSo methods. In [26] was
considered the problem of the third party logistics service providers applying a hesitant linguistic fuzzy
set environment to construct the extent of the CoCoSo method. In [13], the authors applied a q-rung
orthopair fuzzy set to develop a hybrid MCDM extension including CoCoSo and CRITIC methods
and tested this approach on the financial risk evaluation problem. In [1], the authors suggested a



https://doi.org/10.15837/ijccc.2022.1.4646 3

novel interval-valued intuitionistic set model for CoCoSo method extension and solved the problem of
the sustainable evolution in the fabrication region by means of green growth indicators. In [17], was
proposed a CoCoSo method extension applying a single-valued neutrosophic set. This technique was
applied to deal with the problem of the equipment selection for the waste management field including
sustainability aspects. Various fuzzy set environments have been implemented to model multi-criteria
decision-making problems [15], [16], [31], [32], [37], [21], [7], [7]. [8], [35], [38].

In 1999, Smarandache [19] proposed the notion of the neutrosophic sets. Neutrosophic sets (NS)
are based on the generalization of the fuzzy logic that takes into account the knowledge of neural
thought. Therefore, the greater amount of uncertainty can be analyzed. Each parameter of the
analyzed problem can be represented by the scope of the truth (T), the scope of the indeterminacy (I)
and the scope of the falsity (F) using the NS logic. In 2019, a notion of neutrosophic sets, which can be
considered as the generalization of the following fuzzy sets: intuitionistic fuzzy (IFS), q-rung orthopair
fuzzy and Pythagorean fuzzy sets was proposed by Smarandache [20]. Based on this concept, the new
extensions of the classical MCDM methods, like MULTIMOORA, WASPAS, PROMETHEE [34], [18],
[2], were developed by applying the environment of the m-generalized q-neutrosophic sets (mGqNSs).

The structure of the paper is established as follows. The second chapter presents the basic foun-
dations of the m-generalised q- neutrosophic sets. The third chapter provides the proposed extension
of the CoCoSo method under the environment m-generalised q- neutrosophic set. The problem for-
mulation together with the solution procedure of the case study are presented in the fourth chapter.
The paper ends with conclusions.

2 Foundations of the m-generalised q-neutrosophic set
First, the m-generalised q-neutrosophic set (mGqNS) prevailing definitions and the algebraic op-

erations are presented. The algebraic operation presentation is constructed by applying m-generalised
q-neutrosophic numbers (mGqNNs). These neutrosophic numbers form the basis of the proposed
CoCoSo method extension, namely CoCoSo-mGqNS.

2.1 Definitions

Definition 1. The modelled objects form the set X, in which every single object is x ∈ X. In the
present research, the objects are represented by a neutrosophic set. Given that X is a set of criteria
modelled under the m-generalised q-neutrosophic environment, x is a single criterion value.

Let q ≥ 1, m = 1 ‖ 3. The three membership functions define m-generalised q-neutrosophic set
Amq:

Tmq,Imq,Fmq : X → [0,r], here (0 ≤ r ≤ 1)

Given Tmq is the m-generalised function representing the scope of the truth, Imq is the m-generalised
indeterminacy membership function and Fmq is the m-generalised falsity membership function. In such
a way, the m-generalised q-neutrosophic set is prescribed by the following expression:

Amq = {〈Tmq(x),Imq(x),Fmq(x)〉 : x ∈ X}

These three membership functions also must comply with the following requirements:

0 ≤ Tmq(x),Imq(x),Fmq(x) ≤ 1, x ∈ X;

0 ≤ (Tmq(x))q + (Imq(x))q + (Fmq(x))q ≤
3
m
, x ∈ X;

The different m and q values represent different fuzzy sets.

Definition 2. The m-generalised q-neutrosophic number, mGqNN, is represented by the following
expression:

Nmq = 〈t, i,f〉



https://doi.org/10.15837/ijccc.2022.1.4646 4

Definition 3. The summation of the two mGqNNs, when Nmq1 = 〈t1, i1,f1〉 and Nmq2 = 〈t2, i2,f2〉
are the m-generalised q-neutrosophic numbers, we can designate as follows:

Nmq1 ⊕Nmq2 = 〈(1 − (1 − t
q
1)(1 − t

q
2)

1
q , i1i2,f1f2〉 (1)

The multiplication between the two mGqNNs can be performed as follows:

Nmq1 ⊗Nmq2 = 〈t1t2, (1 − (1 − i
q
1)(1 − i

q
2))

1
q , (1 − (1 −fq1 )(1 −f

q
2 ))

1
q 〉 (2)

The multiplication operation of the m-generalised q-neutrosophic number and a real number λ ≥ 0
is performed as follows:

λ ·Nmq1 = 〈(1 − (1 − t
q
1)
λ)

1
q , iλ1,f

λ
1 〉 (3)

When λ ≥ 0, the power function of mGqNN can be determined by:

Nλmq1 = 〈t
λ
1, (1 − (1 − i

q
1)
λ)

1
q , (1 − (1 −fq1 )

λ)
1
q 〉 (4)

The complementary m-generalised q-neutrosophic numbers is calculated as:

Ncmq = 〈f1, 1 − i1, t1〉 (5)

Definition 4. For the deneutrosophication step, the score value S(Nmq) can be calculated as follows:

S(Nmq) =
3 + 3tq − 2iq −fq

6
(6)

In this case, for Nmq1 = 〈t1, i1,f1〉 and Nmq2 = 〈t2, i2,f2〉 the comparison operations will be
completed by the following condition:

If S(Nmq) ≥ S(Nmq) , then Nmq ≥ Nmq
If S(Nmq) = S(Nmq) , then Nmq = Nmq

(7)

3 Combined compromise solution (CoCoSo-mGqNN) method
A new extension of the CoCoSo method under m-generalised q-neutrosophic set environment,

namelly CoCoSo-mGqNN, is designed. The main steps of this approach can be expressed as follows:

(1) Solution procedure of the CoCoSo method starts with the constructed initial decision-making
matrix which can be defined as follows:

xij =



x11 x12 · · · x1l
x21 x22 · · · x2l
...

...
...

...
xk1 xk,2 · · · xkl


 ; i = 1, 2, . . . ,k; j = 1, 2, . . . , l. (8)

(2) The compromise normalization approach is applied to normalize the decision-making matrix
elements [36]:

rij =
xij −min

i
xij

√
k(max

i
xij −min

i
xij)

for maximised criteria, (9)

rij =
max
i
xij −xij

√
k(max

i
xij −min

i
xij)

for minimised criteria. (10)

The standard compromise normalization equation is slightly modified to make it be applicable
for neutrosophic algebra.



https://doi.org/10.15837/ijccc.2022.1.4646 5

(3) Next, the neutrosophication of the decision matrix should be accomplished. This means that
crisp values rij of the decision matrix elements are replaced by the values (Nmq)ij of m-generalised
truth membership, m-generalised indeterminacy membership and m-generalised falsity member-
ship functions. This action is performed applying the standard crisp-to-neutrosophic mapping.
The novel m-generalised neutrosophic decision matrix is created at this step.

(4) At this step, the weighted comparability series and the power weighted comparability series for
each alternative are constructed. These series are denoted by Ri and Pi, respectively:

Ri =
l∑

j=1
(wj(Nmq)ij) (11)

the Ri series reflect weighted sum model:

Pi =
l∑

j=1
((Nmq)ij)

wj (12)

the Pi series correspond to the WASPAS multiplicative component.

(5) Three score assessment strategies are the basis for the calculations of the relative weights. These
strategies are expressed by the following equations (13)-(15).

kia =
S(Pi) + S(Ri)∑k

i=1(S(Pi) + S(Ri))
, (13)

kib =
S(Ri)

min
i
S(Ri)

+
S(Pi)

min
i
S(Pi)

, (14)

kic =
λS(Ri) + (1 −λ)S(Pi)

λmax
i
S(Ri) + (1 −λ)max

i
S(Pi)

; 0 ≤ λ ≤ 1. (15)

By equation (13), arithmetic mean values of the weighted sum model (WSM) and weighted
product model (WPM) are calculated. By equation (14), relative scores of WSM and WPM
are determined. These relatives scores are determined with respect to minimum values of S(Pi)
and S(Ri). By equation (15), the balanced compromise scores of WSM and WPM models are
calculated. In our case, value for λ = 0.5 is chosen.

(6) The final ranking function is constructed as follows:

ki = (kiakibkic)
1
3 +

1
3

(kia + kib + kic). (16)

4 Case study
A secondary school is the object of the research [22]. The school’s frame is a three-story reinforced

concrete frame (constructed in the eighth decade of the twentieth century). The 2478.60 m2 building
has 240 mm thick aerated concrete slab exterior walls and the 200 mm thick reinforced concrete
block plinth. Under part of the building was a local diesel boiler house. The building has plastered
and painted external and external surfaces. The local municipality decided to renovate the structure
according to the prepared project. They provide the following thermal insulation: 200 mm thick
mineral wool boards for walls and 200 mm thick polystyrene boards for the plinth. The project
envisages the roof insulated with 250 mm thick extruded polystyrene panels and an upper 20 mm
thick mineral wool panels. Besides, old windows and doors need replacement as planned.

The most significant part of construction projects are multifaceted, and decision-makers must
take into account tangible and intangible characteristics of plans, opinions, negotiations and dispute
resolution and projects.



https://doi.org/10.15837/ijccc.2022.1.4646 6

Decisions in the construction industry need compromise solutions [28], [29], [35]. Project managers
should select among feasible discrete alternatives in a specific construction site and specific objects
taking into account international and local perspectives. Designers presented nine possible project
implementation scenarios. The building renovation process could involve several phases and be selected
from other options with different employees (20, 40 or 60 for retrofitting at different speeds), see Table
1.

The project implementation contractor offered the following project implementation choices:

1. One-phase retrofitting process: The general contractor shall carry out all modernisation
works in one phase:

• Replacement of windows, exterior and interior doors; b) Insulation of basement and external
walls and roof;

• Installation of the roof;
• Renovation of heating, ventilation, air conditioning and electrical systems;
• Installation of a new lighting system.

The modification process may affect the efficiency of the project due to the multi-tasking
of the retrofitting methods.

2. Two-phase retrofitting process: The upgrading process takes place in two phases:

• The first phase includes replacing windows and exterior doors, repairing the roof, and
installing thermal insulation of exterior walls and the basement;

• The second phase includes installing thermal insulation of ground-level floors, updating
heating, ventilation, air conditioning and electric lighting systems.
There is a nine months break between the first and the second phases. The employees work
during the summer seasons (summer holidays).
There is a risk (depending on the number of employees) that contractor not all works will
finish in time, so construction work can adversely affect physical performance.

3. Three-phase retrofitting :

• The first phase includes replacing windows and exterior doors;
• The second phase includes repairing the roof and installing thermal insulation of exterior

walls and the basement;
• The third phase includes installing thermal insulation of ground-level floors, updating heat-

ing, ventilation, air conditioning and electric lighting systems.
The contractor will make a one-year break between the first and the second phases. The
employees work during the summer seasons (summer holidays).
There is a risk (depending on the number of employees) that contractor not all works will
finish in time, so construction work can adversely affect physical performance.

Many of the contradictory features reflect the solution to a complex problem. The authorities
usually determine the winners based on a single indicator, the lowest price at the local level. This
attitude of civil servants incorrectly describes the real economic benefits of renovating buildings and
other features of modernised buildings. After three steps of the Delphi method, the decision-makers
selected five key characteristics that assessed the renovation’s economic benefits and the impact of the
construction process on workers and visitors.

Costs with VAT, [€]: The construction works costs interlinks with the amount of work required to
complete the project, the current price level in the country (salary, prices of materials and equipment),
the number of potentially competitive contractors and their competitive advantages, construction time
and project implementation time. Longer project implementation time and an increasing number of



https://doi.org/10.15837/ijccc.2022.1.4646 7

Table 1: Initial decision-making matrix

Alternatives Price withVAT, €

Number of
working days

(8 working
hours per day)

Renovation’s
payback time,

[years]

Energy savings
through ten

years, [MWh]

People’s
satisfaction,

[scores]

A1 1309.50 800 21.86 3656.0 6.66
A2 1309.50 510 21.86 3656.0 11.38
A3 1309.50 320 21.86 3656.0 12.89
A4 1379.0 920 21.86 3524.4 7.46
A5 1379.0 640 22.60 3524.4 12.18
A6 1415.0 440 22.60 3524.4 13.69
A7 1415.0 980 23.18 3188.7 8.27
A8 1415.0 700 23.18 3188.7 12.98
A9 1415.0 500 23.18 3188.7 14.49

Optimality min, ↓ min, ↓ min, ↓ max, ↑ max, ↑

stages of construction processes increase construction costs. Longer-term constructions are expen-
sive because the project requires household and storage facilities to bring and take out bio-toilets,
temporary construction site’s fencing, lighting, and need to implement protection measures.

Duration of project implementation [working days]: depends on the number of outputs (person-
hours) and the number of workers on the construction site. This study examines the project implemen-
tation options (twenty, forty or sixty employees employed on the construction site). The contractors
comply with all the necessary construction technology and safety requirements. They cannot perform
the work faster than the established technological requirements in terms of time. For instance, it
cannot paint the facade until three weeks have elapsed since the end of the plastering. The mineral
plaster is undergoing chemical processes at that time.

Renewal payback period, [years]: The customers calculate this time according to the investment
required for the construction work and the energy savings. They consider the efficiency of the work
performed (such as replacing exterior doors and windows). The fuel’s calorific value, the boiler’s
efficiency, the level of fuel prices and laboratory tests of the research object are the basis for calculating
the economic payback time. Laboratory research shows actual energy savings by implementing specific
design modernisation solutions.

Energy savings over ten years, [MWh], determine the expected economic benefits and return on
investment. Calculating the value of this indicator over a more extended period is very difficult or
even impossible due to the significant change in inflation, the change in energy prices and the price
of energy resources, the changing rate requirements that govern the environmental impact and many
other reasons. The changing climate is also contributing to its contribution. Determining energy
savings by calculating energy consumption [MWh] rather than monetary value [€] partially reduces
this uncertainty.

People Satisfaction, [points]: Survey of members of interest groups (building staff and visitors)
is the basis for calculating this characteristic. Twenty-five stakeholders gave nine points for the best
available option and one for the worst. Decision-makers multiplied the final scores by 100 per cent at
the end. Determining weights of attributes is one of the most important and critically acclaimed issues
in multi-attribute decision-making problems. The decision-makers chose and applied a systematic
procedure for the study: the SWARA (Step-wise Weight Assessment Ratio Analysis) method.

The formulated initial decision-making matrix is presented in Table 1, and the normalized values
of the weights are w=(0.37, 0.12, 0.36, 0.1, 0.05). At the following step the proposed extension
CoCoSo-mGqNN was applied to perform the final ranking of the alternatives. The normalization
of the decision-making matrix was performed applying equations (9-10). The elements of the initial
decision-making matrix after the normalization step are presented in Table 2. The results of the
neutrosophication step are presented in Table 3. The intermediate results of the proposed extension
CoCoSo-mGqNN together with final ranking results are presented in Table 4.



https://doi.org/10.15837/ijccc.2022.1.4646 8

Table 2: Initial decision-making matrix
Alternatives X1 X2 X3 X4 X5

A1 0.3333 0.0909 0.3333 0.3333 0
A2 0.3333 0.2374 0.3333 0.3333 0.2009
A3 0.3333 0.3333 0.3333 0.3333 0.2652
A4 0.1137 0.0303 0.3333 0.2395 0.0341
A5 0.1137 0.1717 0.1465 0.2395 0.2350
A6 0 0.2727 0.1465 0.2395 0.2993
A7 0 0 0 0 0.0685
A8 0 0.1414 0 0 0.2691
A9 0 0.2424 0 0 0.3333

Optimality min, ↓ min, ↓ min, ↓ max, ↑ max, ↑

Table 3: Neurosophic decision-making matrix
Alternatives X1 X2 X3 X4 X5

A1

(0.3333,
0.7167,
0.6667)

(0.0909,
0.9091,
0.9091)

(0.3333,
0.7167,
0.6667)

(0.3333,
0.7167,
0.6667)

(0.0, 1.0, 1.0)

A2

(0.3333,
0.7167,
0.6667)

(0.2374,
0.8126,
0.7626)

(0.3333,
0.7167,
0.6667)

(0.3333,
0.7167,
0.6667)

(0.2009,
0.8491,
0.7991)

A3

(0.3333,
0.7167,
0.6667)

(0.3333,
0.7167,
0.6667)

(0.3333,
0.7167,
0.6667)

(0.3333,
0.7167,
0.6667)

(0.2652,
0.7848,
0.7348)

A4

(0.1137,
0.8931,
0.8863)

(0.0303,
0.9697,
0.9697)

0.3333,
0.7167,
0.6667)

(0.2395,
0.8105,
0.7605)

(0.0341,
0.9659,
0.9659)

A5

(0.1137,
0.8931,
0.8863)

(0.1717,
0.8641,
0.8283)

(0.1465,
0.8768,
0.8535)

(0.2395,
0.8105,
0.7605)

(0.2350,
0.8150,
0.7650)

A6 (0.0, 1.0, 1.0)
(0.2727,
0.7773,
0.7273)

0.1465,
0.8768,
0.8535)

(0.2395,
0.8105,
0.7605)

(0.2993,
0.7507,
0.7007)

A7 (0.0, 1.0, 1.0) (0.0, 1.0, 1.0) (0.0, 1.0, 1.0) (0.0, 1.0, 1.0)
(0.0685,
0.9315,
0.9315)

A8 (0.0, 1.0, 1.0)
(0.1414,
0.8793,
0.8586)

(0.0, 1.0, 1.0) (0.0, 1.0, 1.0)
(0.2691,
0.7809,
0.7309)

A9 (0.0, 1.0, 1.0)
(0.2424,
0.8076,

0.07576)
(0.0, 1.0, 1.0) (0.0, 1.0, 1.0)

(0.3333,
0.7167,
0.6667)

Optimality min, ↓ min, ↓ min, ↓ max, ↑ max, ↑



https://doi.org/10.15837/ijccc.2022.1.4646 9

Table 4: Intermediate results of the CoCoSo-mGqNN and the final ranking of the alternatives
Alternatives R P ka kb kc k Rank

A1

(0.3139,
0.7498,
0.7061)

(0.9728,
0.0548,
0.0425)

0.1241 60.8264 0.9529 22.5646 3

A2

(0.3201,
0.7337,
0.6837)

(0.9960,
0.0163,
0.0106)

0.1288 63.7319 0.9895 23.6271 2

A3

(0.3306
0.7199

0.6699)

(0.9973,
0.0123,
0.0080)

0.1302 66.0184 1.0000 24.4313 1

A4

(0.2488,
0.8285,
0.7998)

(0.9786,
0.0485,
0.0404)

0.1168 45.1218 0.8715 17.0566 4

A5

(0.1628,
0.8712,
0.8478)

(0.9914,
0.0342,
0.0255)

0.1135 35.2094 0.8715 13.5805 5

A6

(0.1848,
0.8932,
0.8690)

(0.9937,
0.0364,
0.0270)

0.1115 30.6951 0.8561 11.9850 6

A7

(0.0253,
0.9965,
0.9965)

(0.8746,
0.4295,
0.4295)

0.0778 2.0000 0.5973 1.3446 9

A8

(0.1098,
0.9726,
0.9666)

(0.9689,
0.1596,
0.1405)

0.0969 9.3949 0.7439 4.2899 8

A9

(0.1533,
0.9586,
0.9478)

(0.9793,
0.1249,
0.1048)

0.1006 13.7132 0.7727 5.8837 7

5 Conclusion
Modern modelling trends in this field include evaluation of the uncertainty and vagueness of the

initial information. Traditional neutrosophic sets are considered as the effective tool to express un-
certainty of the information. However, in some cases, it cannot cover all recently proposed cases of
the fuzzy sets. The m-generalized q-neutrosophic sets were recently proposed to deal with this sit-
uation. The m-generalized q-neutrosophic sets can be considered as the generalisation of fuzzy set,
Pythagorean fuzzy set, intuitionistic fuzzy set, q-rung orthopair fuzzy set, single-valued neutrosophic
set, single-valued n-hyperspherical neutrosophic set and single-valued spherical neutrosophic set. In
this paper, the CoCoSo method extension under the environment of the m-generalized q-neutrosophic
numbers (mGqNN) is proposed. This novel extension has been tested for the selection of the best
retrofit strategy. The numerical example also showed that the CoCoSo-mGqNN extension provides
a robust approach that can be applied to deal with different fuzzy sets within the same MCDM
framework.

Author contributions

The authors contributed equally to this work.

Conflict of interest

The authors declare no conflict of interest.

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Cite this paper as:
Turskis Z.; Bausys R.; Smarandache F.; Kazakeviciute-Januskeviciene G.; Zavadskas E. K. (2022).
M-generalised q-neutrosophic extension of CoCoSo method, International Journal of Computers Com-
munications & Control, 17(1), 4646, 2022.

https://doi.org/10.15837/ijccc.2022.1.4646


	Introduction
	Foundations of the m-generalised q-neutrosophic set
	Definitions

	Combined compromise solution (CoCoSo-mGqNN) method
	Case study
	Conclusion