A Decision-Making Approach Based on a Multi Q-Hesitant Fuzzy Soft Multi-Granulation Rough Model

In this paper, we propose a new hybrid model, multi Q-hesitant fuzzy soft multi-granulation rough set model, by combining a multi Q-hesitant fuzzy soft set and multi-granulation rough set. We demonstrate some useful properties of these multi Q-hesitant fuzzy soft multi-granulation rough sets. Furthermore, we define multi Q-hesitant fuzzy soft (MkQHFS) rough approximation operators in terms of MkQHFS relations and MkQHFS multi-granulation rough approximation operators in terms of MkQHFS relations. We study the main properties of lower and upper MkQHFS rough approximation operators and lower and upper MkQHFS multi-granulation rough approximation operators. Moreover, we develop a general framework for dealing with uncertainty in decision-making by using the multi Q-hesitant fuzzy soft multi-granulation rough sets. We analyze the photovoltaic systems fault detection to show the proposed decision methodology.


Introduction
The notion of rough set theory was introduced by Pawlak in 1982 [1].It is a mathematical approach concerning uncertainty that comes from noisy, inexact or incomplete information.In rough set theory, the equivalence relation plays a significant role in creating the upper and lower approximations of the set.Currently, rough set approximations [2] have been constructed into fuzzy sets [3], intuitionistic fuzzy sets [4], hesitant fuzzy sets [5] and covering sets [6].The soft set theory, originally initiated by Molodtsov [7], is a general tool for dealing with uncertainty.Different from some traditional tools for dealing with uncertainties, such as the theory of fuzzy sets [3], the theory of probability and the theory of rough sets [1], the advantage of soft set theory is that it is free from the inadequacy of the parametrization tools of those theories.According to Molodtsov [7], the soft set theory applied successfully to many fields such as functions' smoothness, game theory, theory of measurement and so on.Maji and Roy [8] introduced the soft set into the decision-making problems with the help of the rough theory.Necessary and possible hesitant fuzzy sets, and probabilistic soft sets and dual probabilistic soft sets in decision-making have discussed in [9,10].Moreover, many new rough set models have been established by combining the Pawlak rough set with other uncertainty theories such as soft set theory.Feng [11] provided a framework to combine fuzzy set, rough set, and soft set all together, which gave rise to several interesting new concepts such as rough soft set, soft rough set and soft rough fuzzy set [12].Zhang et al. [13] proposed the notion of soft rough intuitionistic fuzzy sets and intuitionistic fuzzy soft rough sets, which are generalized soft rough set models.Akram et al. [14] presented a new hybrid model, a hesitant N-soft set model for group decision-making.Several research works have been done to solve different real life decision-making problems (see [15][16][17][18][19]).All of these models have always been described by the expression of a one-dimensional membership function that can not be able to deal with the information that appears in a two-dimensional universal set.From this point of view, the idea of Q-fuzzy sets was came out.Afterwards, the concept of multi Q-fuzzy soft sets [20][21][22][23][24] was established to combine the key feature of soft sets and Q-fuzzy sets with multi membership values.The notion of multi Q-hesitant fuzzy soft sets is the generalization of multi Q-fuzzy soft sets.This extension can easily handle the difficulty more objectively than other developed Q-fuzzy set approaches.The combination of multi Q-hesitant fuzzy soft sets and rough sets will be an improved model of hesitant fuzzy rough approaches that concern both areas theoretical and practical applications.Qian et al. [25] proposed the model of multi-granulation rough sets.The main idea of this model is based on defined multiple equivalence relations in a given universe that eliminated the restrictions that may occur through the single equivalence relations in classical rough sets [1] perfectly.The notions of multi-granulation fuzzy rough sets and multi-granulation hesitant fuzzy rough sets are presented by Sun et al. [26] and Zhang et al. [27], respectively, to solve decision-making problems.For other notations and terminologies not mentioned in this paper, the readers are referred to [28][29][30][31][32][33].
In the field of electrical engineering, photovoltaic systems fault detection is one of the challenging tasks that electrical experts have faced in recent years dealing with a substantial amount of uncertain information.Different experts would give their different judgments towards the systems fault detection data.Hence, by combining multi Q-hesitant fuzzy soft sets with multi-granulation rough sets, we constructed the concept of a multi Q-hesitant fuzzy soft multi-granulation rough set model and its application in photovoltaic systems fault detection through developing a new data analysis model in fault detection procedures under the framework of Q-hesitant fuzzy soft information.In this paper, we propose a new hybrid model, multi Q-hesitant fuzzy soft multi-granulation rough set model, by combining a multi Q-hesitant fuzzy soft set and a multi-granulation rough set.We present some of its fundamental properties.We develop a general framework for dealing with uncertainty decision-making by using the multi Q-hesitant fuzzy soft multi-granulation rough sets.We use the photovoltaic systems fault detection to indicate the principle steps of the decision methodology.
The presentation of the article is organized as follows: In Section 2, we recalled some basic concepts of rough sets, soft sets and hesitant fuzzy soft sets.In Section 3, we have presented multi Q-hesitant fuzzy soft sets and discussed some properties.In Section 4, we have introduced a rough set model based on multi Q-hesitant fuzzy soft relation and have examined some properties of this model.In Section 5, we have generalized the notion of multi Q-hesitant fuzzy soft rough sets into multi Q-hesitant fuzzy soft multi-granulation rough set model.In Section 6, we have established a general approach to decision-making based on multi Q-hesitant fuzzy soft multi-granulation rough sets and illustrated the principal steps of the proposed decision method by a numerical example.Finally, in Section 7, we have concluded the paper with a summary and outlook for further research.

Preliminaries
In this section, we recall some basic notions and definitions which will be used in this paper.

Definition 1 ([1]
).Let U be a non-empty finite universe and R be an equivalence relation on U. We use U/R to denote the family of all equivalence classes of R (or classifications of U), and [x] R to denote an equivalence class of R containing the element x ∈ U.The pair (U, R) is called an approximation space.For any X ⊆ U, we can define the lower and upper approximations of X as follows: The pair (R(X), R(X)) is referred to as the rough set of X.The rough set (R(X), R(X)) gives rise to a description of X under the present knowledge, i.e., the classification of U.
Furthermore, the positive region, negative region, and boundary region of X about the approximation space (U, R) are defined as follows, respectively: where ∼ X stands for complementation of the set X. Definition 2 ([7]).Let E be the set of parameters with the connection to the objects in U. A pair (F, E) is called a soft set over U, where F is a mapping given by F : E −→ P(U), P(U) is a set of all subsets of U.
This definition shows that a soft set over U is a parameterized family of subsets of the universe U.For e ∈ E, F(e) is regarded as the set of e-approximate elements of the soft set (F, E).Definition 3 ([5]).Given a non-empty subset A of X, a hesitant fuzzy set H X = {(x, h X (x) : x ∈ X)} on X satisfying the following condition: ∈ A is called a hesitant fuzzy set related to A (briefly, A-hesitant fuzzy set) on X and is represented by

Definition 4 ([34]
).Let H(U) be the set of all hesitant fuzzy sets in U. A pair ( F, Ã) is called a hesitant fuzzy soft set over U, where F is a mapping given by A hesitant fuzzy soft set is a mapping from parameters to H(U).It is a parameterized family of hesitant fuzzy subsets of U.For e ∈ A, F(e) may be considered as the set of e-approximate elements of the hesitant fuzzy soft set ( F, A).

Multi Q-Hesitant Fuzzy Soft Sets
We first introduce the notion of Q-hesitant fuzzy soft sets as a generalization of Q-fuzzy soft sets.Definition 5. Let U be a universal set and Q be non-empty set.A Q-hesitant fuzzy set A Q is a set given by fuzzy set, and the set of all Q-hesitant fuzzy sets over U × Q will be denoted by QHF(U × Q).Definition 6.Let U be a non-empty finite universe and Q be a non-empty set.For any A Let U be a universal set and Q be non-empty set, I be a unit interval [0, 1] and k be a positive integer.A multi Q-hesitant fuzzy set HQ in U × Q is a set defining by Let U be a universal set and be non-empty set, E be the set of parameters and M k QHF(U × Q) be the set of all multi Q-hesitant fuzzy sets on U × Q with the dimension k.Let A ⊆ E the pair (H Q , A) is called a multi Q-hesitant fuzzy soft set (M k QHFSS) over U, where (H Q , A) is given by the form The set of all multi Q-hesitant fuzzy soft sets over U × Q will be denoted by M k QHFSS(U × Q).

Remark 1. Clearly, ((H
Definition 14.The union of two multi Q-hesitant fuzzy soft sets of dimension k over U, (H Q , A) and (F Q , B) is the multi Q-hesitant fuzzy soft set (G Q , C), where C = A ∪ B, and for all e ∈ C, G Q (e) = H Q (e) ∪ F Q (e).
We write Definition 15.The intersection of of two multi Q-hesitant fuzzy soft sets of dimension k over U, (H Q , A) and (F Q , B) with A ∩ B = φ is the multi Q-hesitant fuzzy soft set (G Q , C), where C = A ∩ B, and for all e ∈ C, In this case, we write (H Definition 17.Let U be nonempty universe, Q be a nonempty set and E be the , are two multi Q-hesitant fuzzy soft sets, respectively, defined as follows: Example 2. Suppose that U = {u 1 , u 2 , u 3 } is the set of cars that Mr X wants to buy and Q = {q 1 , q 2 } represents the companies of the different cars.They form the universe (U,Q) and let E = {e 1 = size, e 2 = price, e 3 = colour} be the set of parameters.Consider a multi Q-hesitant fuzzy soft relation R Q : U × Q −→ E × Q with dimension k = 2 is given by Table 1.
The pair (R Theorem 2. Let (U, E, Q, R Q ) be multi Q-hesitant fuzzy soft approximation space.The lower and upper Q-hesitant fuzzy soft rough approximations operators R Q (A Q ) and R Q (A Q ), respectively, for any A Q , B Q ∈ M k QHF(E) satisfy the following properties: Proof. 1.By Definition 17, we have Similarly, it can be proved.

Multi Q-Hesitant Fuzzy Soft Multi-Granulation Rough Set
Definition 18.Let U be a universal set and Q be non-empty set, and E be the set of parameters and R Q j ,(j=1,2,. . . ,m) be multi Q m -hesitant fuzzy soft relations over be called multi Q-hesitant fuzzy soft multi-granulation approximation space, for any A Q ∈ M k QHF(E), the optimistic lower and upper approximation of A Q with respect to (U, E, Q, R Q j ) are defined as follows: , the optimistic lower and upper approximation satisfy the following properties: Proof. 1.By Definition 18, we have, Similarly, we can obtain that , the following properties are true:

2.
It can be proved similarly to 1.

Definition 19.
Let U be a universal set and Q be a non-empty set, and E be the set of parameters and R Q j ,(j=1,2,...,m) are multi Q m -hesitant fuzzy soft relations over (U × Q) × (E × Q), the triple(U, E, Q, R Q j ) is called multi Q-hesitant fuzzy soft multi-granulation approximation space, for any A Q ∈ M k QHF(E), and the pessimistic lower and upper approximation of A Q with respect to (U, E, Q, R Q j ) are defined as follows: (uq, eq) ∧ h i A Q (eq)} : uq ∈ U × Q}.
The pair (∑ m j=1 R Q j p (A Q ), ∑ m j=1 R Q j p (A Q )) is called an pessimistic multi Q-hesitant fuzzy soft multi-granulation rough set of A Q with respect to (U, E, Q, R Q j ).
Theorem 6.Let (U, E, Q, R Q j ) be multi Q-hesitant fuzzy soft multi-granulation approximation space and R Q j ∈ M k QHFSR((U × Q) × (E × Q),(i=1,2,...,m) be multi Q m hesitant fuzzy soft relations over (U × Q) × (E × Q), for any A Q , B Q ∈ M k QHF(E), the pessimistic lower and upper approximation satisfy the following properties:

Conclusions
A multi Q-hesitant fuzzy soft multi-granulation rough set is a new hybrid model, which is a combination of powerful topics: multi Q-hesitant fuzzy soft sets and multi-granulation rough sets.We have defined M k QHFS rough approximation operators in terms of M k QHFS relations and M k QHFS multi-granulation rough approximation operators in terms of M k QHFS relations.We have investigated the properties of lower and upper M k QHFS rough approximation operators and lower and upper M k QHFS multi-granulation rough approximation operators.Finally, we have developed a general framework for dealing with uncertainty decision-making by using the multi Q-hesitant fuzzy soft multi-granulation rough sets.We have used the photovoltaic systems fault detection to indicate the principle steps of the decision methodology.In the future, we will mainly focus on investigating uncertain measures and knowledge reductions of the M k QHFS rough sets.