Higher order hesitant fuzzy Choquet integral operator and its application to
multiple criteria decision making

B. Farhadinia111Corresponding author. U. Aickelin† H.A. Khorshidi†

Dept. Math., Quchan University of Technology, Iran.
<EMAIL_ADDRESS>
† Dept. Computing and Information Systems, University of Melbourne, Australia.
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

Abstract. Generally, the criteria involved in a decision making problem are
interactive or inter-dependent, and therefore aggregating them by the use of
traditional operators which are based on additive measures is not logical.
This verifies that we have to implement fuzzy measures for modelling the
interaction phenomena among the criteria. On the other hand, based on the
recent extension of hesitant fuzzy set, called higher order hesitant fuzzy set
(HOHFS) which allows the membership of a given element to be defined in forms
of several possible generalized types of fuzzy set, we encourage to propose
the higher order hesitant fuzzy (HOHF) Choquet integral operator. This concept
not only considers the importance of the higher order hesitant fuzzy
arguments, but also it can reflect the correlations among those arguments.
Then, a detailed discussion on the aggregation properties of the HOHF Choquet
integral operator will be presented. To enhance the application of HOHF
Choquet integral operator in decision making, we first assess the appropriate
energy policy for the socio-economic development. Then, the efficiency of the
proposed HOHF Choquet integral operator-based technique over a number of
exiting techniques is further verified by employing another decision making
problem associated with the technique of TODIM (an acronym in Portuguese of
Interactive and Multicriteria Decision Making).

Keywords: Higher order hesitant fuzzy set (HOHFS), Choquet integral, Multiple
criteria decision making.

##  1 Introduction

The theory of extending and generalizing fuzzy sets is very prospective tool
in different research domains. One of interesting extensions is the concept of
hesitant fuzzy set (HFS) which was proposed by Torra and Narukawa [28]. HFS is
quite suit for the case where the decision maker has a set of possible values,
rather than a margin of error (like that in intuitionistic fuzzy sets [2]) or
some possibility distribution on the possible values (like that is considered
in defining type-2 fuzzy sets [19]). By the way, HFS [28] and its extension
which is called the generalized HFS (G-HFS) [22], have their own drawbacks
because of the reason that the membership degrees of an element to a given set
can be expressed only by crisp numbers or intuitionistic fuzzy sets. The
evidence to this fact is that in a real and practical decision making problem
where the information provided by a group of experts may not be described only
by hesitant fuzzy sets (HFSs) or just by one kind of HFS extensions. A more
specific situation occurs when a number of experts are intended to return
their evaluations in the form of hesitant multiplicative sets, and others are
interested to describe their evaluation by the help of hesitant fuzzy
linguistic term sets. Therefore, it is difficult for the decision maker to
provide exact hesitant multiplicative set or hesitant fuzzy linguistic term
set for the membership degrees. Such a difficulty is avoided by considering
the higher order HFS (HOHFS) [10] in order to describe the membership degrees.
The concept of HOHFS not only encompasses fuzzy sets, intuitionistic fuzzy
sets, type 2 fuzzy sets and HFSs, but also it extends the concept of
generalized hesitant fuzzy set (G-HFS) [22].

It does worth to say that the topic of developing information aggregations in
fuzzy multiple criteria decision making (MCDM) has been an interesting
research topic [8, 9, 11, 24, 30] so far. As it is known, most of the existing
fuzzy aggregation operators only consider those situations in which all the
elements are independent. Moreover, in many real-world situations, we observe
that the elements are usually interactive or interdependent, that is, they are
correlative.

The Choquet integral [7] with respect to fuzzy measures [25] is an aggregation
function that has been employed successfully in MCDM problems where it
determines the relative importance of decision criteria as well as their
interactions [17]. There exist a growing number of studies on the Choquet
integral, and it has been investigated by many researchers. Yager [41]
proposed the induced Choquet ordered averaging operator, and moreover, he
introduced the concept of Choquet aggregation based on the order induced
aggregation. By extending the concept of Choquet integral to that for fuzzy
context, Meyer and Roubens [18] applied this concept to multiple criteria
decision making problems. Yu et al. [35] defined a hesitant fuzzy aggregation
operator by taking the concept of Choquet integral into account, and they
implemented it in modelling a hesitant-based multiple criteria decision making
problem. Tan and Chen [27] and Tan [26] proposed respectively the
intuitionistic and the interval-valued intuitionistic fuzzy Choquet integral
operators. Bustince et al. [3] investigated a multiple criteria decision
making technique by considering the concept of interval-valued Choquet
integral. Wang et al. [31] investigated multiple criteria group decision
making problems in which the Choquet integral aggregation operators together
with interval two-tuple linguistic information play the main role. Some more
studies have been given by Abdullah et al. [1], Candeloro et al. [4], Pasi et
al. [20], etc.

By the way, inspired by the concept of Choquet integral, we will develop here
a higher order hesitant fuzzy Choquet integral operator for aggregating higher
order hesitant fuzzy (HOHF) information in MCDM, and investigate different
properties of this operator. As will be shown in Section 3, such a proposed
concept not only is well applicable to the case where the decision makers have
a hesitation among several possible memberships with uncertainties in a
general form (i.e., HOHFSs), but also it models the interaction phenomena
among the criteria (i.e., the Choquet integral property).

The present paper is organized as follows: The concept of higher order HFS
(HOHFS) as an extension of HFSs is reviewed in Section 2, and then the higher
order hesitant fuzzy (HOHF) Choquet integral operator is presented together
with a detailed discussion on the aggregation properties of the HOHF Choquet
integral operator. In Section 3, we apply the proposed HOHF Choquet integral
operator to MCDM problems involving the higher order hesitant fuzzy
information, and then a comparative analysis is then given to demonstrate the
effectiveness of the proposed operator. Finally, conclusion is drown in
Section 4.

##  2 The HOHF Choquet integral operator

Among the several extensions of the theory of HFSs, we deal in brief with the
higher order hesitant fuzzy set (HOHFS) which was first introduced by
Farhadinia [10].

######  Definition 2.1.

[10] Let $X$ be the universe of discourse. A generalized type of fuzzy set
(G-Type FS) on $X$ is defined as

$\displaystyle\widetilde{A}=\\{\langle x,\widetilde{A}(x)\rangle:x\in X\\},$
(1)

where

$\displaystyle\widetilde{A}:X\rightarrow\psi([0,1]).$

Here, $\psi([0,1])$ denotes a family of crisp or fuzzy sets that can be
defined with in the universal set $[0,1]$.
We mention here some special cases of G-Type FS which are encountered by Klir
and Yuan in [15]:

  * •

if $\psi([0,1])=[0,1]$, then the G-Type FS $\widetilde{A}$ reduces to an
ordinary fuzzy set;

  * •

if $\psi([0,1])=\varepsilon([0,1])$ denoting the set of all closed intervals,
then the G-Type FS $\widetilde{A}$ reduces to an interval-valued fuzzy set;

  * •

if $\psi([0,1])={\cal F}([0,1])$ denoting the set of all ordinary fuzzy sets,
then the G-Type FS $\widetilde{A}$ reduces to a type II fuzzy set;

  * •

if $\psi([0,1])=L$ denoting a partially ordered Lattice, then the G-Type FS
$\widetilde{A}$ reduces to a lattice fuzzy set.

######  Definition 2.2.

[28] Let $X$ be the universe of discourse. A hesitant fuzzy set (HFS) on $X$
is symbolized by

$H=\\{\langle x,h(x)\rangle:x\in X\\},$

where $h(x)$, referred to as the hesitant fuzzy element (HFE), is a set of
some values in $[0,1]$ denoting the possible membership degree of the element
$x\in X$ to the set $H$.

As can be seen from Definition 2.2, HFS expresses the membership degrees of an
element to a given set only by several real numbers between 0 and 1, while in
many real-world situations assigning exact values to the membership degrees
does not describe properly the imprecise or uncertain decision information.
Thus, it seems to be difficult for the decision makers to rely on HFSs for
expressing uncertainty of an element.
To overcome the difficulty associated with expressing uncertainty of an
element to a given set, the concept of HOHFS allows the membership degrees of
an element to be expressed by several possible G-Type FSs.

######  Definition 2.3.

[10] Let $X$ be the universe of discourse. A higher order hesitant fuzzy set
(HOHFS) on $X$ is defined in terms of a function when it is applied to $X$, it
then returns a set of G-Type FSs. A HOHFS is denoted by

$\displaystyle\widetilde{H}=\\{\langle x,\widetilde{h}(x)\rangle:x\in X\\},$
(2)

where $\widetilde{h}(x)$, referred to as the higher order hesitant fuzzy
element (HOHFE), is a set of some G-Type FSs denoting the possible membership
degree of the element $x\in X$ to the set $\widetilde{H}$. In this regards,
the HOHFS $\widetilde{H}$ is also represented as

$\displaystyle\widetilde{H}=\\{\langle
x,\\{\widetilde{h_{\widetilde{h}}}^{(1)}(x),...,\widetilde{h_{\widetilde{h}}}^{(|\widetilde{h}(x)|)}(x)\\}\rangle:x\in
X\\},$

where all
$\widetilde{h_{\widetilde{h}}}^{(1)}(x),...,\widetilde{h_{\widetilde{h}}}^{(|\widetilde{h}(x)|)}(x)$
are G-Type FSs on $X$, and $|\widetilde{h}(x)|$ is the number of G-type FSs in
$\widetilde{h}(x)$.

The noteworthy feature about Definition 2.3 is that it encompasses all the
above-mentioned generalizations of ordinary fuzzy sets which have been
discussed in detail by Farhadinia [10].

Having introduced HOHFEs of a HOHFS $\widetilde{H}=\\{\langle
x,\widetilde{{h}}(x)\rangle:x\in X\\}$, we now turn our attention to the
interpretation of HOHFS $\widetilde{H}$ as the union of all HOHFEs i.e.,

$\displaystyle\widetilde{H}=\bigcup_{\widetilde{{h}}(x)\in\widetilde{H}}\\{\widetilde{{h}}(x)\\}$
(3)

which is fundamental in the study of HOHFS aggregation operators within the
next parts of the paper.

Hereafter, to simplify the notation, we use only $\widetilde{{h}}$ instead of
$\widetilde{{h}}(x)$ in the following descriptions, that is, we assume that

$\displaystyle\widetilde{H}=\bigcup_{\widetilde{{h}}\in\widetilde{H}}\\{\widetilde{{h}}\\}.$

######  Definition 2.4.

Let
$\widetilde{H}_{1}=\bigcup_{\widetilde{{h}}_{1}\in\widetilde{H}_{1}}\\{\widetilde{{h}_{1}}\\}$
and
$\widetilde{H}_{2}=\bigcup_{\widetilde{{h}}_{2}\in\widetilde{H}_{2}}\\{\widetilde{{h}_{2}}\\}$
be two HOHFSs. We give the definition of operation $\odot$ on HOHFSs based on
its counterpart on HOHFEs as follows:

$\displaystyle\widetilde{H}_{1}\odot\widetilde{H}_{2}=\bigcup_{\widetilde{h_{1}}\in\widetilde{H_{1}},\widetilde{h_{2}}\in\widetilde{H_{2}}}\\{\widetilde{{h_{1}}}\odot\widetilde{{h_{2}}}\\}=\bigcup_{\widetilde{h_{1}}\in\widetilde{H_{1}},\widetilde{h_{2}}\in\widetilde{H_{2}}}\\{\bigcup_{\widetilde{h_{h1}}\in\widetilde{h_{1}},\widetilde{h_{h2}}\in\widetilde{h_{2}}}\\{\widetilde{{h_{h1}}}\odot\widetilde{{h_{h2}}}\\}~{}\\},$
(4)

where the last operation $\odot$ stands for the operation on G-type FSs.

Note that, if there is no confusion, all the operations performing over
HOHFSs, HOHFEs and G-type FSs are denoted by the same symbol. Furthermore,
that property holds for G-type FSs is denoted by $\widetilde{(p_{G})}$, and
that is defined on HOHFEs is symbolized by $\widetilde{(p)}$.

######  Theorem 2.5.

Let
$\widetilde{h}_{1}=\bigcup_{\widetilde{{h}}_{h1}\in\widetilde{h}_{1}}\\{\widetilde{{h}_{h1}}\\}$
and
$\widetilde{h}_{2}=\bigcup_{\widetilde{{h}}_{h2}\in\widetilde{h}_{2}}\\{\widetilde{{h}_{h2}}\\}$
be two HOHFEs. If the operation $\odot$ on G-type FSs has the property
$\widetilde{(p_{G})}$, then the operation $\odot$ on HOHFEs has the
counterpart property, denoted by $\widetilde{(p)}$.

Proof. The proof is obvious from Definition 2.4 Suppose that, for example, the
operation $\odot$ on G-type FSs is commutative, that is, for any G-type FSs
$\widetilde{h_{h1}}\in\widetilde{h_{1}}$ and
$\widetilde{h_{h2}}\in\widetilde{h_{2}}$, we have

$\displaystyle\widetilde{{h}_{h1}}\odot\widetilde{{h}_{h2}}=\widetilde{{h}_{h2}}\odot\widetilde{{h}_{h1}}.\quad\quad(\textrm{Property}~{}\widetilde{(p_{G})})$

Then, one can see from Definition 2.4 that

$\displaystyle\widetilde{h}_{1}\odot\widetilde{h}_{2}=\bigcup_{\widetilde{h_{h1}}\in\widetilde{h_{1}},\widetilde{h_{h2}}\in\widetilde{h_{2}}}\\{\widetilde{{h_{h1}}}\odot\widetilde{{h_{h2}}}\\}=\bigcup_{\widetilde{h_{h1}}\in\widetilde{h_{1}},\widetilde{h_{h2}}\in\widetilde{h_{2}}}\\{\widetilde{{h_{h2}}}\odot\widetilde{{h_{h1}}}\\}=\widetilde{h}_{2}\odot\widetilde{h}_{1}.\quad\quad(\textrm{Property}~{}\widetilde{(p)})$

This means that the operation $\odot$ on HOHFEs is accordingly commutative.
By similar reasoning, one can easily prove that the HOHFE operator inherits
all the properties of the G-type FS operator. $\Box$

As a corollary of Theorem 2.5 , it can be observed that HOHFS operators
inherit subsequently all operational properties of G-type FS operators.

######  Corollary 2.6.

Let
$\widetilde{H}_{1}=\bigcup_{\widetilde{{h}}_{1}\in\widetilde{H}_{1}}\\{\widetilde{{h}_{1}}\\}$
and
$\widetilde{H}_{2}=\bigcup_{\widetilde{{h}}_{2}\in\widetilde{H}_{2}}\\{\widetilde{{h}_{2}}\\}$
be two HOHFSs. If the operation $\odot$ on HOHFEs has the property
$\widetilde{(p)}$, then the operation $\odot$ on HOHFSs has the mentioned
property, denoted by $\widetilde{(P)}$.

Ranking fuzzy information play a key role in fuzzy decision-making procedure.
This fact is very encouraging for introducing a score function of HOHFSs. To
compare the HOHFSs, we define the following comparison laws:

######  Definition 2.7.

Let $X=\\{x_{1},x_{2}...,x_{n}\\}$. For HOHFS $\widetilde{H}=\\{\langle
x,\widetilde{{h}}(x)\rangle:x\in
X\\}=\bigcup_{\widetilde{h}\in\widetilde{H}}\\{\bigcup_{\widetilde{h_{h}}\in\widetilde{h}}\\{\widetilde{{h_{h}}}\\}\\}$,
the score function ${\cal S}(.)$ is defined by

$\displaystyle{\cal
S}(\widetilde{H})=\frac{1}{n}\sum_{i=1}^{n}S_{\widetilde{h}}(\widetilde{h}(x_{i}))=\frac{1}{n}\sum_{i=1}^{n}(~{}\frac{1}{|\widetilde{h}(x_{i})|}\sum_{\widetilde{h_{h}}\in\widetilde{h}(x_{i})}S_{\widetilde{{h_{h}}}}(\widetilde{{h_{h}}}(x_{i}))~{}),$
(5)

where $S_{\widetilde{h}}(.)$ is a score function of HOHFEs,
$|\widetilde{h}(x_{i})|$ is the number of G-type FSs in $\widetilde{h}(x_{i})$
and $S_{\widetilde{{h_{h}}}}(.)$ is a score function of G-type FSs. For two
HOHFSs $\widetilde{H_{1}}$ and $\widetilde{H_{2}}$, if ${\cal
S}(\widetilde{H}_{1})<{\cal S}(\widetilde{H}_{2})$ then
$\widetilde{H_{1}}\prec\widetilde{H_{2}}$; if ${\cal
S}(\widetilde{H}_{1})={\cal S}(\widetilde{H}_{2})$ then
$\widetilde{H_{1}}\approx\widetilde{H_{2}}$.

Similar comparison rules can be considered for HOHFEs and G-type FSs such that

  * •

for two HOHFEs $\widetilde{h_{1}}$ and $\widetilde{h_{2}}$ if
$S_{\widetilde{h}}(\widetilde{h}_{1})<S_{\widetilde{h}}(\widetilde{h}_{2})$
then $\widetilde{h_{1}}\prec\widetilde{h_{2}}$; if
$S_{\widetilde{h}}(\widetilde{h}_{1})=S_{\widetilde{h}}(\widetilde{h}_{2})$
then $\widetilde{h_{1}}\approx\widetilde{h_{2}}$;

  * •

for two G-type FSs $\widetilde{h_{h}}^{(1)}$ and $\widetilde{h_{h}}^{(2)}$ if
$S_{\widetilde{h_{h}}}(\widetilde{h}_{h}^{(1)})<S_{\widetilde{h_{h}}}(\widetilde{h}_{h}^{(2)})$
then $\widetilde{h_{h}}^{(1)}\prec\widetilde{h_{h}}^{(2)}$; if
$S_{\widetilde{h_{h}}}(\widetilde{h}_{h}^{(1)})=S_{\widetilde{h_{h}}}(\widetilde{h}_{h}^{(2)})$
then $\widetilde{h_{h}}^{(1)}\approx\widetilde{h_{h}}^{(2)}$.

In most of the real world applications, we usually face to the elements in the
universe of discourse which may have a different importance. This impulses us
to consider the weight of each element $x_{i}\in X$. Assume that the weight of
$x_{i}\in X$ is $\mu(\\{x_{i}\\})$, $(i=1,...,n)$ where $\mu$ is a fuzzy
measure.

######  Definition 2.8.

[25] A fuzzy measure on $X$ is a set function $\mu$ from the power set of $X$
into $[0,1]$, satisfying the following conditions:

$\displaystyle(1)~{}\mu(\emptyset)=0,~{}\mu(X)=1;$
$\displaystyle(2)~{}\textrm{For}~{}\textrm{any}~{}A,B\subseteq
X,~{}\textrm{if}~{}A\subseteq B,~{}\textrm{then}~{}\mu(A)\leq\mu(B);$
$\displaystyle(3)~{}\textrm{For}~{}\textrm{any}~{}A,B\subseteq X,~{}\mu(A\cup
B)=\mu(A)+\mu(B)-\rho\mu(A)\mu(B),~{}\textrm{where}~{}A\cap
B=\emptyset~{}\textrm{and}~{}\rho\in(-1,\infty).$

A fuzzy measure $\mu$ on $X$ is said to be

  * •

Additive if $\mu(A\cup B)=\mu(A)+\mu(B)$ for all $A,B\subseteq X$ where $A\cap
B=\emptyset$;

  * •

Subadditive if $\mu(A\cup B)\leq\mu(A)+\mu(B)$ for all $A,B\subseteq X$ where
$A\cap B=\emptyset$;

  * •

Superadditive if $\mu(A\cup B)\geq\mu(A)+\mu(B)$ for all $A,B\subseteq X$
where $A\cap B=\emptyset$.

Sometimes, in MCDM, we need an aggregation function to determine the relative
importance of decision criteria considering their interactions. Such a need
could be satisfied by implementing the Choquet integral [7] which is defined
on the basis of fuzzy measures.

######  Definition 2.9.

[7] Let $f$ be a positive real-valued function on $X$, and $\mu$ be a fuzzy
measure on $X$. The Choquet integral of $f$ with respect to $\mu$ is defined
by

$\displaystyle
C_{\mu}(f)=\sum_{i=1}^{n}(\mu(A_{\sigma(i)})-\mu(A_{\sigma(i-1)})f_{\sigma(i)}$
(6)

where $(\sigma(1),\sigma(2),...,\sigma(n))$ is a permutation of $(1,2,...,n)$
such that $f_{\sigma(1)}\geq f_{\sigma(2)}\geq...\geq f_{\sigma(n)}$,
$A_{\sigma(k)}=\\{x_{\sigma(1)},x_{\sigma(2)},...,x_{\sigma(k)}\\}$ for $k\geq
1$, and $A_{\sigma(0)}=\emptyset$.

The Choquet integral $C_{\mu}(.)$ has some considerable aggregation
properties, such as idempotence, compensativeness and comonotonic additivity
[17]. Moreover, it generalizes the weighted arithmetic mean (WAM) and the
ordered weighted averaging (OWA) [40]. In view of the Choquet integral
formulation (6), we here introduce the HOHF Choquet integral:

######  Definition 2.10.

Let $\mu$ be a fuzzy measure on $X$, and $\widetilde{h}(x_{i}),~{}(i=1,...,n)$
be a collection of HOHFEs on $X$. The HOHF Choquet integral of
$\widetilde{h}(x_{i}),~{}(i=1,...,n)$ with respect to $\mu$ is defined by

$\displaystyle
HOHFEC_{\mu}(\widetilde{h}(x_{1}),...,\widetilde{h}(x_{n}))=\oplus_{i=1}^{n}[(\mu(A_{\sigma(i)})-\mu(A_{\sigma(i-1)})\widetilde{h}(x_{\sigma(i)})]\hskip
142.26378pt$ $\displaystyle\hskip
28.45274pt=\bigcup_{\widetilde{h_{h}}(x_{\sigma(1)})\in\widetilde{h}(x_{\sigma(1)}),...,\widetilde{h_{h}}(x_{\sigma(n)})\in\widetilde{h}(x_{\sigma(n)})}\\{\oplus_{i=1}^{n}[(\mu(A_{\sigma(i)})-\mu(A_{\sigma(i-1)})\widetilde{h_{h}}(x_{\sigma(i)})]\\},$
(7)

where $(\sigma(1),\sigma(2),...,\sigma(n))$ is a permutation of $(1,2,...,n)$
such that
$\widetilde{h}(x_{\sigma(1)})\succ\widetilde{h}(x_{\sigma(2)})\succ...\succ\widetilde{h}(x_{\sigma(n)})$,
$A_{\sigma(k)}=\\{x_{\sigma(1)},x_{\sigma(2)},...,x_{\sigma(k)}\\}$ for $k\geq
1$, and $A_{\sigma(0)}=\emptyset$.

######  Proposition 2.11.

If for any G-type FSs $\widetilde{h_{h}}^{(1)}$ and $\widetilde{h_{h}}^{(2)}$
of HOHFE $\widetilde{h}$, the addition
$\widetilde{h_{h}}^{(1)}\oplus\widetilde{h_{h}}^{(2)}$ and the scalar
multiplication $\lambda\widetilde{h_{h}}^{(1)}$ where $\lambda>0$ are also
G-type FSs, then the HOHF Choquet integral of
$\widetilde{h}(x_{i}),~{}(i=1,...,n)$ with respect to $\mu$ introduced in
Definition 2.10 is a HOHFE.

Proof. The proof is obvious by putting together the definition of the HOHF
Choquet integral of $\widetilde{h}(x_{i}),~{}(i=1,...,n)$ with respect to
$\mu$ introduced in Definition 2.10 and Theorem 2.5. $\Box$

######  Example 2.12.

Let $X=\\{x_{1},x_{2}\\}$, and $\widetilde{h}(x_{i}),~{}(i=1,2)$ be a
collection of HOHFEs on $X$ whose G-type FSs are in the form of IFSs given by

$\displaystyle\widetilde{h}(x_{1})=\\{\widetilde{h_{h}}^{(1)}(x_{1})=\langle
0.2,0.5\rangle,\widetilde{h_{h}}^{(2)}(x_{1})=\langle
0.3,0.4\rangle\\},\quad\quad\widetilde{h}(x_{2})=\\{\widetilde{h_{h}}^{(1)}(x_{2})=\langle
0.3,0.5\rangle\\},$

and a fuzzy measure $\mu:X\rightarrow[0,1]$ is given by

$\displaystyle\mu(\emptyset)=0,\quad\mu(\\{x_{1}\\})=0.2,\quad\mu(\\{x_{2}\\})=0.3,\quad\mu(\\{x_{1},x_{2}\\})=0.6.$

If we take the G-type FSs score function as the hesitancy degree, i.e.,

$\displaystyle
S_{\widetilde{h_{h}}}(\widetilde{h_{h}})=S_{\widetilde{h_{h}}}(\langle\mu_{\widetilde{{h_{h}}}},\nu_{\widetilde{{h_{h}}}}\rangle)=1-\mu_{\widetilde{{h_{h}}}}-\nu_{\widetilde{{h_{h}}}},$

then the HOHFE scores $S_{\widetilde{h}}(\widetilde{h}(x_{1}))=0.3$ and
$S_{\widetilde{h}}(\widetilde{h}(x_{2}))=0.1$ indicate that
$\widetilde{h}(x_{1})\succ\widetilde{h}(x_{2})$ and therefore
${\sigma(1)}=1,{\sigma(2)}=2$.
Now from (7) the HOHF Choquet integral of $\widetilde{h}(x_{i}),~{}(i=1,2)$
with respect to $\mu$ is calculated as follows:

$\displaystyle
HOHFEC_{\mu}(\widetilde{h}(x_{1}),\widetilde{h}(x_{2}))=\bigcup_{\widetilde{h_{h}}(x_{1})\in\widetilde{h}(x_{1}),\widetilde{h_{h}}(x_{2})\in\widetilde{h}(x_{2})}\\{\oplus_{i=1}^{2}[(\mu(A_{\sigma(i)})-\mu(A_{\sigma(i-1)})\widetilde{h_{h}}(x_{\sigma(i)})]\\}$
$\displaystyle=\\{[(\mu(A_{1})-\mu(A_{0})]\widetilde{h_{h}}^{(1)}(x_{1})\oplus[(\mu(A_{2})-\mu(A_{1})]\widetilde{h_{h}}^{(1)}(x_{2}),$
$\displaystyle\hskip
113.81102pt[(\mu(A_{1})-\mu(A_{0})]\widetilde{h_{h}}^{(2)}(x_{1})\oplus[(\mu(A_{2})-\mu(A_{1})]\widetilde{h_{h}}^{(1)}(x_{2})\\}$
$\displaystyle=\\{[(\mu(\\{x_{1}\\})-\mu(\emptyset)]\widetilde{h_{h}}^{(1)}(x_{1})\oplus[(\mu(\\{x_{2}\\})-\mu(\\{x_{1}\\})]\widetilde{h_{h}}^{(1)}(x_{2}),$
$\displaystyle\hskip
113.81102pt[(\mu(\\{x_{1}\\})-\mu(\emptyset)]\widetilde{h_{h}}^{(2)}(x_{1})\oplus[(\mu(\\{x_{2}\\})-\mu(\\{x_{1}\\})]\widetilde{h_{h}}^{(1)}(x_{2})\\}$
$\displaystyle=\\{[0.2]\langle 0.2,0.5\rangle\oplus[0.6-0.2]\langle
0.3,0.5\rangle,$ $\displaystyle\hskip 113.81102pt[0.2]\langle
0.3,0.4\rangle\oplus[0.6-0.2]\langle 0.3,0.5\rangle\\}.$

By the use of the following relations (see e.g. [44])

$\displaystyle\widetilde{{h_{h1}}}\oplus\widetilde{{h_{h2}}}=\langle\mu_{\widetilde{{h_{h1}}}},\nu_{\widetilde{{h_{h1}}}}\rangle\oplus\langle\mu_{\widetilde{{h_{h2}}}},\nu_{\widetilde{{h_{h2}}}}\rangle=\langle\mu_{\widetilde{{h_{h1}}}}+\mu_{\widetilde{{h_{h2}}}}-\mu_{\widetilde{{h_{h1}}}}.\mu_{\widetilde{{h_{h2}}}},~{}~{}\nu_{\widetilde{{h_{h1}}}}.\nu_{\widetilde{{h_{h2}}}}\rangle;$
$\displaystyle\widetilde{{h_{h1}}}^{\lambda}=\langle(\mu_{\widetilde{{h_{h1}}}})^{\lambda},1-(1-\nu_{\widetilde{{h_{h1}}}})^{\lambda}\rangle,\quad\lambda>0,$

in which $\mu_{\widetilde{{h_{h2}}}}$ and $\nu_{\widetilde{{h_{h2}}}}$ stand
respectively for the membership and the non-membership functions, we get

$\displaystyle\widetilde{h}_{{}_{HOHFEC}}:=HOHFEC_{\mu}(\widetilde{h}(x_{1}),\widetilde{h}(x_{2}))=\\{\langle
0.1708,0.6598\rangle,\langle 0.1927,0.6310\rangle\\}.$

######  Proposition 2.13.

(Idempotency). If for any G-type FS $\widetilde{h_{h}}$ of HOHFE
$\widetilde{h}$, it holds
$\lambda_{1}\widetilde{h_{h}}\oplus\lambda_{2}\widetilde{h_{h}}=(\lambda_{1}+\lambda_{2})\widetilde{h_{h}}$
where $\lambda_{1},\lambda_{2}>0$, and if all HOHFEs
$\widetilde{h}(x_{i}),~{}(i=1,...,n)$ are equal, that is, for all $i$,
$\widetilde{h}(x_{i})=\overline{\widetilde{h}}$, then the HOHF Choquet
integral of $\widetilde{h}(x_{i}),~{}(i=1,...,n)$ with respect to $\mu$
introduced in Definition 2.10 is equal to the HOHFE
$\overline{\widetilde{h}}$.

Proof. Suppose that for all $i,~{}(i=1,...,n)$, we have
$\widetilde{h}(x_{i})=\overline{\widetilde{h}}$. From definition of the HOHF
Choquet integral of $\widetilde{h}(x_{i}),~{}(i=1,...,n)$ with respect to
$\mu$ introduced in Definition 2.10, we have

$\displaystyle HOHFEC_{\mu}(\widetilde{h}(x_{1}),...,\widetilde{h}(x_{n}))$
$\displaystyle=$
$\displaystyle\oplus_{i=1}^{n}[(\mu(A_{\sigma(i)})-\mu(A_{\sigma(i-1)})\widetilde{h}(x_{\sigma(i)})]$
$\displaystyle=[(\mu(A_{\sigma(1)})-\mu(A_{\sigma(0)})]\overline{\widetilde{h}}\oplus...\oplus[(\mu(A_{\sigma(n)})-\mu(A_{\sigma(n-1)})]\overline{\widetilde{h}}.$

If for any G-type FS $\widetilde{h_{h}}$ of HOHFE $\widetilde{h}$, it holds
$\lambda_{1}\widetilde{h_{h}}\oplus\lambda_{2}\widetilde{h_{h}}=(\lambda_{1}+\lambda_{2})\widetilde{h_{h}}$
where $\lambda_{1},\lambda_{2}>0$, then by Theorem 2.5 and the fact that
$\mu(A_{\sigma(0)}=\emptyset)=0,\mu(A_{\sigma(n)}=X)=1$, one gets

$\displaystyle HOHFEC_{\mu}(\widetilde{h}(x_{1}),...,\widetilde{h}(x_{n}))$
$\displaystyle=([(\mu(A_{\sigma(1)})-\mu(A_{\sigma(0)})]+...+[(\mu(A_{\sigma(n)})-\mu(A_{\sigma(n-1)})])\overline{\widetilde{h}}=\overline{\widetilde{h}}.~{}\Box$

######  Lemma 2.14.

Suppose that the score functions $S_{\widetilde{h}}$ on HOHFEs and
$S_{\widetilde{{h_{h}}}}$ on G-type FSs are those introduced in Definition
2.7, that is,

$\displaystyle
S_{\widetilde{h}}(\widetilde{h})=\frac{1}{|\widetilde{h}|}\sum_{\widetilde{h_{h}}\in\widetilde{h}}S_{\widetilde{{h_{h}}}}(\widetilde{{h_{h}}}).$
(8)

If for each HOHFE
$\widetilde{h}=\bigcup_{\widetilde{h_{h}}\in\widetilde{h}}\\{\widetilde{h_{h}}\\}$,
$S_{\widetilde{{h_{h}}}}$ satisfies
$S_{\widetilde{{h_{h}}}}(\lambda\widetilde{{h_{h}}})=\lambda
S_{\widetilde{{h_{h}}}}(\widetilde{{h_{h}}})$, for any $\lambda>0$. Then,
$S_{\widetilde{h}}$ also satisfies the latter property, i.e.,

$\displaystyle S_{\widetilde{h}}(\lambda\widetilde{h})=\lambda
S_{\widetilde{h}}(\widetilde{h}),\quad\forall\lambda>0.$ (9)

Proof. From definition of $S_{\widetilde{h}}$ given by (8) and the mentioned
property of $S_{\widetilde{{h_{h}}}}$, one gets for any $\lambda>0$

$\displaystyle
S_{\widetilde{h}}(\lambda\widetilde{h})=\frac{1}{|\lambda\widetilde{h}|}\sum_{\widetilde{h_{h}}\in\widetilde{h}}S_{\widetilde{{h_{h}}}}(\lambda\widetilde{{h_{h}}})=\frac{1}{|\widetilde{h}|}\sum_{\widetilde{h_{h}}\in\widetilde{h}}S_{\widetilde{{h_{h}}}}(\lambda\widetilde{{h_{h}}})=\frac{1}{|\widetilde{h}|}\sum_{\widetilde{h_{h}}\in\widetilde{h}}\lambda
S_{\widetilde{{h_{h}}}}(\widetilde{{h_{h}}})=\lambda
S_{\widetilde{h}}(\widetilde{h}).~{}\Box$

Hereafter, the HOHFE score function $S_{\widetilde{h}}$ is said to be
$\lambda$-invariance if it satisfies the property (9).

######  Lemma 2.15.

Assume that the HOHFE score function $S_{\widetilde{h}}$ is
$\lambda$-invariance. For any two HOHFEs $\widetilde{h}_{1}$ and
$\widetilde{h}_{2}$, if $\widetilde{h}_{1}\succ\widetilde{h}_{2}$, then
$\lambda\widetilde{h}_{1}\succ\lambda\widetilde{h}_{2}$, for any $\lambda>0$.

Proof. As follows from Definition 2.7, we find that
$\widetilde{h}_{1}\succ\widetilde{h}_{2}$ whenever
$S_{\widetilde{h}}(\widetilde{h}_{1})>S_{\widetilde{h}}(\widetilde{h}_{2})$.
On the other hand, with the $\lambda$-invariance property of the HOHFE score
function $S_{\widetilde{h}}$ in mind, the relation
$S_{\widetilde{h}}(\widetilde{h}_{1})>S_{\widetilde{h}}(\widetilde{h}_{2})$
gives rise to
$S_{\widetilde{h}}(\lambda\widetilde{h}_{1})>S_{\widetilde{h}}(\lambda\widetilde{h}_{2})$
for any $\lambda>0$, which implies that
$\lambda\widetilde{h}_{1}\succ\lambda\widetilde{h}_{2}$. This completes the
proof. $\Box$

######  Proposition 2.16.

(Monotonicity). Let $\mu$ be a fuzzy measure on $X$, and
$\widetilde{h}_{1}(x_{i})$ and $\widetilde{h}_{2}(x_{i}),~{}(i=1,...,n)$ be
two collection of HOHFEs on $X$. Assume that the HOHFE score function
$S_{\widetilde{h}}$ is $\lambda$-invariance, and
$(\sigma(1),\sigma(2),...,\sigma(n))$ denotes a permutation of $(1,2,...,n)$
such that
$\widetilde{h}_{1}(x_{\sigma(1)})\succ\widetilde{h}_{1}(x_{\sigma(2)})\succ...\succ\widetilde{h}_{1}(x_{\sigma(n)})$
and
$\widetilde{h}_{2}(x_{\sigma(1)})\succ\widetilde{h}_{2}(x_{\sigma(2)})\succ...\succ\widetilde{h}_{2}(x_{\sigma(n)})$.
If $\widetilde{h}_{1}(x_{\sigma(i)})\succ\widetilde{h}_{2}(x_{\sigma(i)})$ for
$(i=1,...,n)$, then

$\displaystyle
HOHFEC_{\mu}(\widetilde{h}_{1}(x_{1}),...,\widetilde{h}_{1}(x_{n}))\succ
HOHFEC_{\mu}(\widetilde{h}_{2}(x_{1}),...,\widetilde{h}_{2}(x_{n})).$

Proof. The assertion follows by applying Lemma 2.15 to Definition 2.10. $\Box$

######  Proposition 2.17.

(Boundedness). Let $\mu$ be a fuzzy measure on $X$,
$\widetilde{h}(x_{i}),~{}(i=1,...,n)$ be a collection of HOHFEs on $X$, and
$(\sigma(1),\sigma(2),...,\sigma(n))$ be a permutation of $(1,2,...,n)$ such
that
$\widetilde{h}_{1}(x_{\sigma(1)})\succ\widetilde{h}_{1}(x_{\sigma(2)})\succ...\succ\widetilde{h}_{1}(x_{\sigma(n)})$.
Then, the $\lambda$-invariance property of the HOHFE score function
$S_{\widetilde{h}}$ gives rise to

$\displaystyle\widetilde{h}_{Max}\succ
HOHFEC_{\mu}(\widetilde{h}_{1}(x_{1}),...,\widetilde{h}_{1}(x_{n}))\succ\widetilde{h}_{Min},$

where
$\widetilde{h}_{Min}=\min\\{\widetilde{h}_{1}(x_{1}),...,\widetilde{h}_{1}(x_{n})\\}=\widetilde{h}_{1}(x_{\sigma(n)})$
and
$\widetilde{h}_{Max}=\max\\{\widetilde{h}_{1}(x_{1}),...,\widetilde{h}_{1}(x_{n})\\}=\widetilde{h}_{1}(x_{\sigma(1)})$.

Proof. The application of Proposition 2.16 gives rise to the assertion. $\Box$

######  Proposition 2.18.

If for any G-type FS $\widetilde{h_{h}}$ of HOHFE $\widetilde{h}$, it holds
$(\widetilde{h_{h}}^{(1)}\oplus\widetilde{h_{h}}^{(2)})\oplus\widetilde{h_{h}}^{(3)}=\widetilde{h_{h}}^{(1)}\oplus(\widetilde{h_{h}}^{(2)}\oplus\widetilde{h_{h}}^{(3)})$
and
$\lambda_{1}\widetilde{h_{h}}\oplus\lambda_{2}\widetilde{h_{h}}=(\lambda_{1}+\lambda_{2})\widetilde{h_{h}}$
where $\lambda_{1},\lambda_{2}>0$, then the HOHF Choquet integral of
$\widetilde{h}(x_{i}),~{}(i=1,...,n)$ with respect to $\mu$ introduced in
Definition 2.10 satisfies

$\displaystyle
HOHFEC_{\mu}(\kappa\widetilde{h}_{1}(x_{1})\oplus\widetilde{\overline{h}},...,\kappa\widetilde{h}_{1}(x_{n})\oplus\widetilde{\overline{h}})=\kappa
HOHFEC_{\mu}(\widetilde{h}_{1}(x_{1}),...,\widetilde{h}_{1}(x_{n}))\oplus\widetilde{\overline{h}},\quad\kappa>0.$

Proof. The proof is much like that of Proposition 2.13 and therefore is
omitted. $\Box$

##  3 MCDM with the HOHF Choquet integral operator

In this section, we present a new method for MCDM, in which the evaluations of
the alternatives are given by HOHFEs and the interaction among the criteria
are allowed.
The MCDM procedure with HOHF Choquet integral operator can be described as
follows: suppose that $Y=\\{y_{1},y_{2},...,y_{m}\\}$ is an alternative set of
$m$ alternatives. In a MCDM problem, the decision maker is interested in
choosing the best one(s) from $Y$ according to the criteria set
$X=\\{x_{1},x_{2},...,x_{n}\\}$. In the HOHFE MCDM, the evaluation of each
alternative on each criterion is a HOHFE. By using the score function
$S_{\widetilde{h}}$ introduced in Definition 2.7, all HOHFEs associated with
each alternative are re-ordered such that
$\widetilde{h}(x_{\sigma(1)})\succ\widetilde{h}(x_{\sigma(2)})\succ...\succ\widetilde{h}(x_{\sigma(n)})$
where $(\sigma(1),\sigma(2),...,\sigma(n))$ is a permutation of $(1,2,...,n)$.
Taking into account the correlations of the HOHFEs, the evaluations of an
alternative can be aggregated to its overall evaluation by the HOHF Choquet
integral operator. According to a given total order relation on aggregated
HOHFEs, the decision maker can rank those overall evaluations and get the best
alternative(s).

###  3.1 A practical example

In what follows, we are going to demonstrate the practicality of implementing
the proposed concept of HOHFE in a higher order hesitant fuzzy multi-attribute
decision making problem.
Energy plays an important role in socio-economic development. Thus selecting
an appropriate policy for energy is critical for economic development and
environment. Assume that there exist five alternatives (energy projects)
${y}_{i},(i=1,2,3,4,5)$ which are invested in accordance with four criteria:
$x_{1}$: technological; $x_{2}$: environmental; $x_{3}$: socio-political;
$x_{4}$: economic. A number of decision makers are invited to evaluate the
performance of the above five alternatives. For an alternative under a
criterion, the decision makers give their evaluations anonymously in the form
of HOHFSs. In this regard, the results evaluated by the decision makers are
contained in a higher order hesitant fuzzy decision matrix which is shown in
Table 1.

Table 1. Higher order hesitant fuzzy decision matrix

| $x_{1}$ | $x_{2}$ | $x_{3}$ | $x_{4}$
---|---|---|---|---
${y}_{1}$ | $\\{(0.3,0.4,0.5)\\}$ | $\\{(0.4,0.5,0.6),\\{0.7,0.8,0.9\\}\\}$ | $\\{(0.5,0.7,0.7),(0.7,0.8,0.9)\\}$ | $\\{(0.2,0.3,0.4),\\{0.3,0.4,0.5\\}\\}$
${y}_{2}$ | $\\{(0.1,0.2,0.3),(0.2,0.3,0.4)\\}$ | $\\{(0.5,0.6,0.7),\\{0.3,0.4,0.5\\}\\}$ | $\\{(0.2,0.4,0.6),(0.7,0.8,0.9)\\}$ | $\\{(0.1,0.4,0.7),(0.6,0.7,0.8)\\}$
${y}_{3}$ | $\\{(0.1,0.2,0.3),\\{0.3,0.4,0.5\\}\\}$ | $\\{0.7,0.8,0.9\\}$ | $\\{(0.2,0.3,0.4),(0.5,0.6,0.7)\\}$ | $\\{(0.4,0.5,0.6)\\}$
${y}_{4}$ | $\\{0.2,0.3,0.4\\}$ | $\\{(0.3,0.4,0.5),(0.2,0.4,0.6)\\}$ | $\\{(0.5,0.6,0.7),\\{0.3,0.4,0.5\\}\\}$ | $\\{(0.1,0.2,0.3),(0.3,0.4,0.5)\\}$
${y}_{5}$ | $\\{(0.2,0.4,0.6),(0.7,08,0.9)\\}$ | $\\{0.7,0.8,0.9\\}$ | $\\{(0.3,0.4,0.5)\\}$ | $\\{(0.5,0.7,0.7),(0.7,0.8,0.9)\\}$

In Table 1, the notation $A=(a_{1},a_{2},a_{3})$ is used to describe a fuzzy
event by the help of a triangular fuzzy number in which the values $a_{1}$,
$a_{2}$ and $a_{3}$ denote respectively the smallest possible value, the most
promising value and the largest possible value, and moreover,
$\\{\widetilde{h_{\widetilde{h}}}^{(1)},\widetilde{h_{\widetilde{h}}}^{(2)},\widetilde{h_{\widetilde{h}}}^{(3)}\\}$
indicates a HFE. In this setting and as a consequence, we observe that the
second and third elements of the last row of Table 1 are conceptually
different. The second array of Table 1 indicates a HFE, while, the third array
is used to indicate a triangular fuzzy number.

Suppose that the fuzzy measures of criteria $X=\\{x_{1},x_{2},x_{3},x_{4}\\}$
are given as follows:

$\displaystyle\mu(\emptyset)=0;$
$\displaystyle\mu(\\{x_{1}\\})=0.2,~{}\mu(\\{x_{2}\\})=0.4,~{}\mu(\\{x_{3}\\})=0.3,~{}\mu(\\{x_{4}\\})=0.1;$
$\displaystyle\mu(\\{x_{1},x_{2}\\})=0.4,~{}\mu(\\{x_{1},x_{3}\\})=0.2,~{}\mu(\\{x_{1},x_{4}\\})=0.1,~{}\mu(\\{x_{2},x_{3}\\})=0.3,$
$\displaystyle\mu(\\{x_{2},x_{4}\\})=0.3,~{}\mu(\\{x_{3},x_{4}\\})=0.1;$
$\displaystyle\mu(\\{x_{1},x_{2},x_{3}\\})=0.3,~{}\mu(\\{x_{1},x_{2},x_{4}\\})=0.1,~{}\mu(\\{x_{1},x_{3},x_{4}\\})=0.2,~{}\mu(\\{x_{2},x_{3},x_{4}\\})=0.2;$
$\displaystyle\mu({X})=1.$

For re-ordering HOHFEs given in Table 1, we recall the HOHFE score function
$S_{\widetilde{h}}$ introduced in Definition 2.7, where

$\displaystyle
S_{\widetilde{h}}(\widetilde{h})=\frac{1}{|\widetilde{h}|}\sum_{\widetilde{h_{h}}\in\widetilde{h}}S_{\widetilde{{h_{h}}}}(\widetilde{{h_{h}}}).$
(10)

Here, the G-type FS score function $S_{\widetilde{{h_{h}}}}$ is taken as

  * •

(For triangular fuzzy number):

$\displaystyle
S_{\widetilde{{h_{h}}}}({(a_{1},a_{2},a_{3})})=\frac{1}{3}(a_{1}+a_{2}+a_{3}),$
(11)

which was proposed by Lee and Li in [16];

  * •

(For hesitant fuzzy element):

$\displaystyle
S_{\widetilde{{h_{h}}}}(\\{\widetilde{h_{\widetilde{h}}}^{(1)},...,\widetilde{h_{\widetilde{h}}}^{(|\widetilde{h}|)}\\})=\frac{1}{|\widetilde{h}|}(\\{\widetilde{h_{\widetilde{h}}}^{(1)}+...+\widetilde{h_{\widetilde{h}}}^{(|\widetilde{h}|)}\\}),$
(12)

Although, the corresponding score functions of HFEs and triangular fuzzy
numbers may differ from each other, but in this example, both of them have a
same rule. However, by using the HOHFE score function $S_{\widetilde{h}}$
given by (10), we re-arrange the HOHFEs corresponding to each criterion in
descending order as follows:

$\displaystyle
y_{1}:\quad\widetilde{h}(x_{\sigma(1)}=x_{3})\succ\widetilde{h}(x_{\sigma(2)}=x_{2})\succ\widetilde{h}(x_{\sigma(3)}=x_{1})\succ\widetilde{h}(x_{\sigma(4)}=x_{4});$
$\displaystyle
y_{2}:\quad\widetilde{h}(x_{\sigma(1)}=x_{3})\succ\widetilde{h}(x_{\sigma(2)}=x_{4})\succ\widetilde{h}(x_{\sigma(3)}=x_{2})\succ\widetilde{h}(x_{\sigma(4)}=x_{1});$
$\displaystyle
y_{3}:\quad\widetilde{h}(x_{\sigma(1)}=x_{2})\succ\widetilde{h}(x_{\sigma(2)}=x_{4})\succ\widetilde{h}(x_{\sigma(3)}=x_{3})\succ\widetilde{h}(x_{\sigma(4)}=x_{1});$
$\displaystyle
y_{4}:\quad\widetilde{h}(x_{\sigma(1)}=x_{3})\succ\widetilde{h}(x_{\sigma(2)}=x_{2})\succ\widetilde{h}(x_{\sigma(3)}=x_{1})\succ\widetilde{h}(x_{\sigma(4)}=x_{4});$
$\displaystyle
y_{5}:\quad\widetilde{h}(x_{\sigma(1)}=x_{2})\succ\widetilde{h}(x_{\sigma(2)}=x_{4})\succ\widetilde{h}(x_{\sigma(3)}=x_{1})\succ\widetilde{h}(x_{\sigma(4)}=x_{3}).$

If we employ the HOHF Choquet integral operator, we then are able to aggregate
all HOHFEs in the i-$th$ row of the HOHFE decision matrix into an overall
value. For instance,

$\displaystyle\widetilde{h}_{y_{1}}=HOHFEC_{\mu}(\widetilde{h}(x_{1}),...,\widetilde{h}(x_{4}))=$
$\displaystyle[(\mu(A_{\sigma(1)})-\mu(A_{\sigma(0)}))\widetilde{h}(x_{\sigma(1)})]\oplus[(\mu(A_{\sigma(2)})-\mu(A_{\sigma(1)}))\widetilde{h}(x_{\sigma(2)})]$
$\displaystyle\oplus[(\mu(A_{\sigma(3)})-\mu(A_{\sigma(2)}))\widetilde{h}(x_{\sigma(3)})]\oplus[(\mu(A_{\sigma(4)})-\mu(A_{\sigma(3)}))\widetilde{h}(x_{\sigma(4)})]$
$\displaystyle=[(\mu(\\{x_{\sigma(1)}\\})-\mu(\emptyset))\widetilde{h}(x_{\sigma(1)})]\oplus[(\mu(\\{x_{\sigma(1)},x_{\sigma(2)}\\})-\mu(\\{x_{\sigma(1)}\\}))\widetilde{h}(x_{\sigma(2)})]$
$\displaystyle\oplus[(\mu(\\{x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)}\\})-\mu(\\{x_{\sigma(1)},x_{\sigma(2)}\\}))\widetilde{h}(x_{\sigma(3)})]\oplus[(\mu(X)-\mu(\\{x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)}\\}))\widetilde{h}(x_{\sigma(4)})]$
$\displaystyle=[(\mu(\\{x_{3}\\})-\mu(\emptyset))\widetilde{h}(x_{3})]\oplus[(\mu(\\{x_{3},x_{2}\\})-\mu(\\{x_{3}\\}))\widetilde{h}(x_{2})]$
$\displaystyle\oplus[(\mu(\\{x_{3},x_{2},x_{1}\\})-\mu(\\{x_{3},x_{2}\\}))\widetilde{h}(x_{1})]\oplus[(\mu(X)-\mu(\\{x_{3},x_{2},x_{1}\\}))\widetilde{h}(x_{4})]$
$\displaystyle=[(0.3-0)\\{(0.5,0.7,0.7),(0.7,0.8,0.9)\\}]\oplus[(0.3-0.3)\\{(0.4,0.5,0.6),\\{0.7,0.8,0.9\\}\\}]$
$\displaystyle\oplus[(0.3-0.3)\\{(0.3,0.4,0.5)\\}]\oplus[(1-0.3)\\{(0.2,0.3,0.4),\\{0.3,0.4,0.5\\}\\}].$

Let us now apply the following arithmetic operations

  * •

(For triangular fuzzy number [14]):

$\displaystyle\lambda A=\lambda(a_{1},a_{2},a_{3})=(\lambda a_{1},\lambda
a_{2},\lambda a_{3}),$ $\displaystyle A\oplus
B=(a_{1},a_{2},a_{3})\oplus(b_{1},b_{2},b_{3})=(a_{1}+b_{1}-a_{1}b_{1},a_{2}+b_{2}-a_{2}b_{2},a_{3}+b_{3}-a_{3}b_{3});$

  * •

(For hesitant fuzzy element [11]):

$\displaystyle\lambda\\{\widetilde{h_{\widetilde{h}}}^{(1)},...,\widetilde{h_{\widetilde{h}}}^{(|\widetilde{h}|)}\\}=\\{\lambda\widetilde{h_{\widetilde{h}}}^{(1)},...,\lambda\widetilde{h_{\widetilde{h}}}^{(|\widetilde{h}|)}\\},\quad\lambda\geq
0,$
$\displaystyle\\{\widetilde{h_{\widetilde{h_{1}}}}^{(1)},...,\widetilde{h_{\widetilde{h_{1}}}}^{(|\widetilde{h}|)}\\}\oplus\\{\widetilde{h_{\widetilde{h_{2}}}}^{(1)},...,\widetilde{h_{\widetilde{h_{2}}}}^{(|\widetilde{h}|)}\\}$
$\displaystyle\quad\quad=\\{\widetilde{h_{\widetilde{h_{1}}}}^{(1)}+\widetilde{h_{\widetilde{h_{2}}}}^{(1)}-\widetilde{h_{\widetilde{h_{1}}}}^{(1)}\widetilde{h_{\widetilde{h_{2}}}}^{(1)},...,\widetilde{h_{\widetilde{h_{1}}}}^{(|\widetilde{h}|)}+\widetilde{h_{\widetilde{h_{2}}}}^{(|\widetilde{h}|)}-\widetilde{h_{\widetilde{h_{1}}}}^{(|\widetilde{h}|)}\widetilde{h_{\widetilde{h_{2}}}}^{(|\widetilde{h}|)}\\},$

to the latter findings. Then, one achieves that

$\displaystyle\widetilde{h}_{y_{1}}$ $\displaystyle=$
$\displaystyle[\\{(0.15,0.21,0.21),(0.21,0.24,0.27)\\}]\oplus[\\{(0.14,0.21,0.28),\\{0.21,0.28,0.35\\}\\}]$
$\displaystyle=$
$\displaystyle\\{(0.27,0.37,0.43),(0.32,0.40,0.47),\\{0.21,0.28,0.35\\}\\}.$

The HOHFE score value of $\widetilde{h}_{y_{1}}$ is then equal to

$\displaystyle
S_{\widetilde{h}}(\widetilde{h}_{y_{1}})=\frac{1}{|\widetilde{h}_{y_{1}}|}\sum_{\widetilde{h_{{h}_{y_{1}}}}\in\widetilde{h}_{y_{1}}}S_{\widetilde{{h_{h}}}}(\widetilde{{h_{{h}_{y_{1}}}}})=0.3456.$

Similarly, the other HOHFE score values are obtained as:

$\displaystyle S_{\widetilde{h}}(\widetilde{h}_{y_{2}})=0.4915,\quad
S_{\widetilde{h}}(\widetilde{h}_{y_{3}})=0.4364,\quad
S_{\widetilde{h}}(\widetilde{h}_{y_{4}})=0.2739,\quad
S_{\widetilde{h}}(\widetilde{h}_{y_{5}})=0.5601.$

Therefore,

$\displaystyle
S_{\widetilde{h}}(\widetilde{h}_{y_{5}})>S_{\widetilde{h}}(\widetilde{h}_{y_{2}})>S_{\widetilde{h}}(\widetilde{h}_{y_{3}})>S_{\widetilde{h}}(\widetilde{h}_{y_{1}})>S_{\widetilde{h}}(\widetilde{h}_{y_{4}}),$

which are led to

$\displaystyle y_{5}\succ y_{2}\succ y_{3}\succ y_{1}\succ y_{4},$

and hence the most appropriate energy project is $y_{5}$.

###  3.2 Comparisons and further discussions

Here, we adopt a multiple criteria group decision making problem from [21]
which is originally based on extending TODIM method by encountering the
Choquet integral within a multiset hesitant fuzzy environment.
Peng et al. [21] considered an investment company for investing in a project.
They supposed five alternatives ${y}_{1}$: car company; ${y}_{2}$: food
company; ${y}_{3}$: computer company; ${y}_{4}$: arms company; and ${y}_{5}$:
TV company which need to be invested. In this setting, the desired decision
has to be made according to the four criteria including $x_{1}$: the
environment impact which is referred to as the impact on the companys
environment $x_{2}$: the risk factor including product risk and development
environment risk; $x_{3}$: the growth prospects including increased
profitability and returns; and $x_{4}$: social-political impact that is
referred to as the governments and local residents support for company.
Peng et al. [21] used multiset hesitant fuzzy sets to evaluate alternatives by
two decision makers. In this contribution to have a complete picture about the
performance of the proposed approach and that of Peng et al. [21], we re-state
Peng et al’s [21] decision matrix in the following form of higher order
hesitant fuzzy decision matrix:

Table 2. Higher order hesitant fuzzy decision matrix

| $x_{1}$ | $x_{2}$ | $x_{3}$ | $x_{4}$
---|---|---|---|---
${y}_{1}$ | $\\{(0.4,0.5,0.7)\\}$ | $\\{(0.5,0.5,0.8)\\}$ | $\\{(0.6,0.6,0.9)\\}$ | $\\{0.5,0.6\\}$
${y}_{2}$ | $\\{(0.6,0.7,0.8)\\}$ | $\\{0.5,0.6\\}$ | $\\{(0.6,0.7,0.7)\\}$ | $\\{0.4,0.5\\}$
${y}_{3}$ | $\\{0.6,0.8\\}$ | $\\{(0.2,0.3,0.5)\\}$ | $\\{0.6,0.6\\}$ | $\\{0.5,0.7\\}$
${y}_{4}$ | $\\{(0.5,0.5,0.7)\\}$ | $\\{0.4,0.5\\}$ | $\\{0.8,0.9\\}$ | $\\{(0.3,0.4,0.5)\\}$
${y}_{5}$ | $\\{0.6,0.7\\}$ | $\\{0.5,0.7\\}$ | $\\{0.7,0.8\\}$ | $\\{(0.3,0.3,0.4)\\}$

some of its arrays are in form of triangular fuzzy numbers and some of them
are HFEs. Needless to say that preserving both forms of arrays appears to
maintain the implication of HOHFEs.

According to Peng et al’s [21] assumption, the fuzzy measures of criteria
$X=\\{x_{1},x_{2},x_{3},x_{4}\\}$ are as follows:

$\displaystyle\mu(\emptyset)=0;$
$\displaystyle\mu(\\{x_{1}\\})=0.40,~{}\mu(\\{x_{2}\\})=0.25,~{}\mu(\\{x_{3}\\})=0.37,~{}\mu(\\{x_{4}\\})=0.20;$
$\displaystyle\mu(\\{x_{1},x_{2}\\})=0.60,~{}\mu(\\{x_{1},x_{3}\\})=0.70,~{}\mu(\\{x_{1},x_{4}\\})=0.56,~{}\mu(\\{x_{2},x_{3}\\})=0.68,$
$\displaystyle\quad\quad\quad\mu(\\{x_{2},x_{4}\\})=0.43,~{}\mu(\\{x_{3},x_{4}\\})=0.54;$
$\displaystyle\mu(\\{x_{1},x_{2},x_{3}\\})=0.88,~{}\mu(\\{x_{1},x_{2},x_{4}\\})=0.75,~{}\mu(\\{x_{1},x_{3},x_{4}\\})=0.84,~{}\mu(\\{x_{2},x_{3},x_{4}\\})=0.73;$
$\displaystyle\mu(X)=1.$

We re-arrange the HOHFEs corresponding to each criterion in descending order
by the use of HOHFE score function $S_{\widetilde{h}}$ given by (10) as
follows:

$\displaystyle
y_{1}:\quad\widetilde{h}(x_{\sigma(1)}=x_{3})\succ\widetilde{h}(x_{\sigma(2)}=x_{2})\succ\widetilde{h}(x_{\sigma(3)}=x_{1})\succ\widetilde{h}(x_{\sigma(4)}=x_{4});$
$\displaystyle
y_{2}:\quad\widetilde{h}(x_{\sigma(1)}=x_{1})\succ\widetilde{h}(x_{\sigma(2)}=x_{3})\succ\widetilde{h}(x_{\sigma(3)}=x_{2})\succ\widetilde{h}(x_{\sigma(4)}=x_{4});$
$\displaystyle
y_{3}:\quad\widetilde{h}(x_{\sigma(1)}=x_{1})\succ\widetilde{h}(x_{\sigma(2)}=x_{3})\succ\widetilde{h}(x_{\sigma(3)}=x_{4})\succ\widetilde{h}(x_{\sigma(4)}=x_{2});$
$\displaystyle
y_{4}:\quad\widetilde{h}(x_{\sigma(1)}=x_{3})\succ\widetilde{h}(x_{\sigma(2)}=x_{1})\succ\widetilde{h}(x_{\sigma(3)}=x_{2})\succ\widetilde{h}(x_{\sigma(4)}=x_{4});$
$\displaystyle
y_{5}:\quad\widetilde{h}(x_{\sigma(1)}=x_{3})\succ\widetilde{h}(x_{\sigma(2)}=x_{1})\succ\widetilde{h}(x_{\sigma(3)}=x_{2})\succ\widetilde{h}(x_{\sigma(4)}=x_{4}).$

Now, by the help of HOHF Choquet integral operator, we aggregate all HOHFEs in
the i-$th$ row of the HOHFE decision matrix into an overall value, for
example,

$\displaystyle\widetilde{h}_{y_{1}}=HOHFEC_{\mu}(\widetilde{h}(x_{1}),...,\widetilde{h}(x_{4}))=$
$\displaystyle[(\mu(A_{\sigma(1)})-\mu(A_{\sigma(0)}))\widetilde{h}(x_{\sigma(1)})]\oplus[(\mu(A_{\sigma(2)})-\mu(A_{\sigma(1)}))\widetilde{h}(x_{\sigma(2)})]$
$\displaystyle\oplus[(\mu(A_{\sigma(3)})-\mu(A_{\sigma(2)}))\widetilde{h}(x_{\sigma(3)})]\oplus[(\mu(A_{\sigma(4)})-\mu(A_{\sigma(3)}))\widetilde{h}(x_{\sigma(4)})]$
$\displaystyle=[(\mu(\\{x_{\sigma(1)}\\})-\mu(\emptyset))\widetilde{h}(x_{\sigma(1)})]\oplus[(\mu(\\{x_{\sigma(1)},x_{\sigma(2)}\\})-\mu(\\{x_{\sigma(1)}\\}))\widetilde{h}(x_{\sigma(2)})]$
$\displaystyle\oplus[(\mu(\\{x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)}\\})-\mu(\\{x_{\sigma(1)},x_{\sigma(2)}\\}))\widetilde{h}(x_{\sigma(3)})]\oplus[(\mu(X)-\mu(\\{x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)}\\}))\widetilde{h}(x_{\sigma(4)})]$
$\displaystyle=[(\mu(\\{x_{3}\\})-\mu(\emptyset))\widetilde{h}(x_{3})]\oplus[(\mu(\\{x_{3},x_{2}\\})-\mu(\\{x_{3}\\}))\widetilde{h}(x_{2})]$
$\displaystyle\oplus[(\mu(\\{x_{3},x_{2},x_{1}\\})-\mu(\\{x_{3},x_{2}\\}))\widetilde{h}(x_{1})]\oplus[(\mu(X)-\mu(\\{x_{3},x_{2},x_{1}\\}))\widetilde{h}(x_{4})]$
$\displaystyle=\\{(0.0765,0.0795,0.1205),\\{0.0100,0.0120\\}\\}.$

By applying the same procedure, we achieve

$\displaystyle\widetilde{h}_{y_{2}}=\\{(0.0550,0.0642,0.0672),\\{0.0533,0.0645\\}\\},$
$\displaystyle\widetilde{h}_{y_{3}}=\\{(0.0110,0.0165,0.0275),\\{0.0650,0.0750\\}\\},$
$\displaystyle\widetilde{h}_{y_{4}}=\\{(0.0210,0.0230,0.0310),\\{0.0713,0.0830\\}\\},$
$\displaystyle\widetilde{h}_{y_{5}}=\\{(0.0060,0.0060,0.0080),\\{0.1330,0.1633\\}\\}.$

Moreover, the HOHFE score function (11) results in

$\displaystyle S_{\widetilde{h}}(\widetilde{h}_{y_{1}})=0.2985,\quad
S_{\widetilde{h}}(\widetilde{h}_{y_{2}})=0.3041,\quad
S_{\widetilde{h}}(\widetilde{h}_{y_{3}})=0.1950,\quad
S_{\widetilde{h}}(\widetilde{h}_{y_{4}})=0.2293,\quad
S_{\widetilde{h}}(\widetilde{h}_{y_{5}})=0.3163.$

Therefore,

$\displaystyle
S_{\widetilde{h}}(\widetilde{h}_{y_{5}})>S_{\widetilde{h}}(\widetilde{h}_{y_{2}})>S_{\widetilde{h}}(\widetilde{h}_{y_{1}})>S_{\widetilde{h}}(\widetilde{h}_{y_{4}})>S_{\widetilde{h}}(\widetilde{h}_{y_{3}}).$

These findings indicate that

$\displaystyle y_{5}\succ y_{2}\succ y_{1}\succ y_{4}\succ y_{3},$

and hence the most appropriate energy project is $y_{5}$.

In what follows, we are going to verify the validity of proposed technique
compared to the other existing techniques including Xu’s [37], Wei’s [36],
Zhang et al.’s [46], Chen et al.’s [5], Xu’s [39], Farhadinia’s [10], Zhang
and Wei’s [47], Zhang and Xu’s [48], Wang et al.’s [33] and Peng et al.’s [21]
from a perspective of performance. Indeed, from the performance stand point of
view, we intend to discuss the resulted rankings of techniques macroscopy, and
not microscopy, that is, we do not intend to investigate the pros and cons of
each technique.
Before going more in depth with future analysis, let us first re-state here
from Farhadinia [12] the algorithm being implemented to sort decision making
techniques based on their outcomes.

######  Algorithm 3.1.

[12] (Sorting of decision making techniques based on their outcomes)

Step 1.

Take into account each outcome of decision making techniques (i.e., the
resulted ranking order of alternatives) as an individual preference $R_{k}$
$(k=1,...,m)$ together with the associated dominance-vector $\pi_{R_{k}}$.

Step 2.

Add up all the individual preference matrices component-wisely to get the
collective preference matrix $\overline{R}=[\overline{r}_{ij}]_{n\times n}$.

Step 3.

Extract the collective preference from the matrix
$\overline{R}=[\overline{r}_{ij}]_{n\times n}$ with respect to the collective
majority decision rule.

Step 4.

Characterize the corresponding dominance-vector hesitant fuzzy sets to the
individual and the collective preferences. Then, determine the importance
weight of decision making techniques in accordance with the distance value of
their individual preference from the collective preference using a
corresponding distance measure.

Now, for determining the importance weights of decision making techniques
according to their distance values of their individual rankings from the
collective one, we implement here Algorithm 3.1 above. Indeed, Algorithm 3.1
is allocated to assess the best outcome(s) without regarding of how the
preference ordering is obtained or from what numerical values one concludes
the preference ordering.
Now, by considering the total ranking order $R$ on
$Y=\\{y_{1},y_{2},y_{3},y_{4},y_{5}\\}$, we can correspond to each of them the
dominance-vectors $\pi_{R_{k}}$ (for $k=X1,W1,...,Pro$) which are given in
Table 3.

Table 3. The dominance-vectors corresponding to the different techniques.

Techniques | Ranking orders | Dominance-vector
---|---|---
$(e_{X1})$: Xu’s [37] technique |  $R_{X1}$: $y_{4}\succ y_{5}\succ y_{2}\succ y_{1}\succ y_{3}$ | $\pi_{R_{X1}}=(2,3,1,4,5)$
$(e_{W1})$: Wei’s [36] technique |  $R_{W1}$: $y_{2}\succ y_{5}\succ y_{4}\succ y_{1}\succ y_{3}$ | $\pi_{R_{W1}}=(2,5,1,3,4)$
$(e_{Z1})$: Zhang et al.’s [46] technique |  $R_{Z1}$: $y_{5}\succ y_{2}\succ y_{1}\succ y_{4}\succ y_{3}$ | $\pi_{R_{Z1}}=(3,4,1,2,5)$
$(e_{C})$: Chen et al.’s [5] technique |  $R_{C}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{4}\succ y_{3}$ | $\pi_{R_{C}}=(4,3,1,2,5)$
$(e_{X2})$: Xu’s [39] technique |  $R_{X2}$: $y_{5}\succ y_{2}\succ y_{4}\succ y_{1}\succ y_{3}$ | $\pi_{R_{X2}}=(2,4,1,3,5)$
$(e_{F})$: Farhadinia’s [10] technique |  $R_{F}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{4}\succ y_{3}$ | $\pi_{R_{F}}=(4,3,1,2,5)$
$(e_{Z2})$: Zhang and Wei’s [47] technique |  $R_{Z2}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{4}\succ y_{3}$ | $\pi_{R_{Z2}}=(4,3,1,2,5)$
$(e_{Z3})$: Zhang and Xu’s [48] technique |  $R_{Z3}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{3}\succ y_{4}$ | $\pi_{R_{Z3}}=(4,3,2,1,5)$
$(e_{W2})$: Wang et al.’s [33] technique |  $R_{W2}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{3}\succ y_{4}$ | $\pi_{R_{W2}}=(4,3,2,1,5)$
$(e_{P})$: Peng et al.’s [21] technique |  $R_{P}$: $y_{5}\succ y_{2}\succ y_{1}\succ y_{3}\succ y_{4}$ | $\pi_{R_{P}}=(3,4,2,1,5)$
$(e_{Pro})$: The proposed technique |  $R_{Pro}$: $y_{5}\succ y_{2}\succ y_{1}\succ y_{4}\succ y_{3}$ | $\pi_{R_{Pro}}=(3,4,1,2,5)$

By considering all the individual preference matrices $R_{k}$
$(k=X1,W1,...,Pro)$, for instance, $R_{X1}$: $y_{4}\succ y_{5}\succ y_{2}\succ
y_{1}\succ y_{3}$ which is given by

$\displaystyle R_{X1}=\left(\begin{array}[]{ccccc}0&-1&1&-1&-1\\\
1&0&1&-1&-1\\\ -1&-1&0&-1&-1\\\ 1&1&1&0&1\\\ 1&1&1&-1&0\\\
\end{array}\right),$

the collective preference $\overline{R}$ is then achieved through the
aggregation process in which all the individual preference matrices are added
up component-wisely. The result is as follows:

$\displaystyle\overline{R}=\left(\begin{array}[]{ccccc}0&-1&11&5&-11\\\
1&0&11&9&-9\\\ -11&-11&0&-5&-11\\\ -5&-9&5&0&-9\\\ 11&9&11&9&0\\\
\end{array}\right),$

which returns $y_{5}\succ y_{2}\succ y_{1}\succ y_{4}\succ y_{3}$ as the
collective preference.
Now, with the distance

$\displaystyle{\mathcal{D}}({\pi_{R_{k}}},{\pi_{\overline{R}}})=\max\\{\max_{\pi_{R_{k}}}\min_{\pi_{\overline{R}}}|\pi_{R_{k}},\pi_{\overline{R}}|~{},~{}\min_{\pi_{R_{k}}}\max_{\pi_{\overline{R}}}|\pi_{R_{k}},\pi_{\overline{R}}|\\},\quad(k=X1,W1,...,Pro)$
(15)

at hand, we may obtain the distance of individual preferences from the
collective preference as those given in Table 4.

Table 4. The distance of individual preferences from the collective
preference.

Techniques | Ranking orders | Distance values
---|---|---
$(e_{X1})$: Xu’s [37] technique |  $R_{X1}$: $y_{4}\succ y_{5}\succ y_{2}\succ y_{1}\succ y_{3}$ | ${{\cal{D}}}(\pi_{R_{X1}},{\pi}_{\overline{R}})=4$
$(e_{W1})$: Wei’s [36] technique |  $R_{W1}$: $y_{2}\succ y_{5}\succ y_{4}\succ y_{1}\succ y_{3}$ | ${{\cal{D}}}(\pi_{R_{W1}},{\pi}_{\overline{R}})=4$
$(e_{Z1})$: Zhang et al.’s [46] technique |  $R_{Z1}$: $y_{5}\succ y_{2}\succ y_{1}\succ y_{4}\succ y_{3}$ | ${{\cal{D}}}(\pi_{R_{Z1}},{\pi}_{\overline{R}})=0$
$(e_{C})$: Chen et al.’s [5] technique |  $R_{C}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{4}\succ y_{3}$ | ${{\cal{D}}}(\pi_{R_{C}},{\pi}_{\overline{R}})=2$
$(e_{X2})$: Xu’s [39] technique |  $R_{X2}$: $y_{5}\succ y_{2}\succ y_{4}\succ y_{1}\succ y_{3}$ | ${{\cal{D}}}(\pi_{R_{X2}},{\pi}_{\overline{R}})=2$
$(e_{F})$: Farhadinia’s [10] technique |  $R_{F}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{4}\succ y_{3}$ | ${{\cal{D}}}(\pi_{R_{F}},{\pi}_{\overline{R}})=2$
$(e_{Z2})$: Zhang and Wei’s [47] technique |  $R_{Z2}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{4}\succ y_{3}$ | ${{\cal{D}}}(\pi_{R_{Z2}},{\pi}_{\overline{R}})=2$
$(e_{Z3})$: Zhang and Xu’s [48] technique |  $R_{Z3}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{3}\succ y_{4}$ | ${{\cal{D}}}(\pi_{R_{Z3}},{\pi}_{\overline{R}})=4$
$(e_{W2})$: Wang et al.’s [33] technique |  $R_{W2}$: $y_{5}\succ y_{1}\succ y_{2}\succ y_{3}\succ y_{4}$ | ${{\cal{D}}}(\pi_{R_{W2}},{\pi}_{\overline{R}})=4$
$(e_{P})$: Peng et al.’s [21] technique |  $R_{P}$: $y_{5}\succ y_{2}\succ y_{1}\succ y_{3}\succ y_{4}$ | ${{\cal{D}}}(\pi_{R_{P}},{\pi}_{\overline{R}})=2$
$(e_{Pro})$: The proposed technique |  $R_{Pro}$: $y_{5}\succ y_{2}\succ y_{1}\succ y_{4}\succ y_{3}$ | ${{\cal{D}}}(\pi_{R_{Pro}},{\pi}_{\overline{R}})=0$

The larger distance ${{\cal{D}}}$ indicates the more dis-similar between each
individual preference and the collective preference. This implies that the
larger value of distance shows lower performance of decision-making technique.
In view of Table 4, we are able to sort the considered techniques which are
labelled by $k=X1,W1,...,Pro$ as the following:

$\displaystyle\\{(e_{Pro}),(e_{Z1})\\}>\\{(e_{C}),(e_{X2}),(e_{F}),(e_{Z2}),(e_{P})\\}>\\{(e_{X1}),(e_{W1}),(e_{Z3}),(e_{W2})\\}.$

The above results verify that among the above decision-making techniques, the
proposed technique can be chosen as the one of the most suitable techniques.

However, these findings indicate that although the procedure of obtaining the
most appropriate alternative using different techniques may be different, but
their outcomes can be used for constructing an admissible assessment which was
called here as the collective preference.

##  4 Conclusion

The current contribution proposed the higher order hesitant fuzzy (HOHF)
Choquet integral operator that not only considers the importance of the
elements, but also it can reflect the correlations among the elements. Then,
we employed the HOHF Choquet integral operator to aggregate HOHFEs in a higher
order hesitant fuzzy MCDM problem. Eventually, a comparative analysis of
different techniques with the HOHF Choquet integral-based technique presented,
and it indicates that the latter technique is enough efficient. In future
work, we plan to study the other aggregation operators in the higher order
hesitant fuzzy setting for handling MCDM with higher order hesitant fuzzy
information. In addition, since the application potentials of HOHFS are
diverse, it can be investigated in other domains such as clustering, pattern
recognition, image processing, etc.

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