**1. 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 [15–19]). 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 [20–24] 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 [28–33].

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.
