# A Fuzzy Method for Joint Resource Allocation and Stable Pairing in D2D Communications
## 1 Introduction and Related Works
### Fuzzy Mathematics  
[PERSON] introduced fuzzy sets in his significant paper [20] in 1965. His idea was to introduce a generalization of the concept crisp sets. The idea was to use numbers in \([0,1]\) as the membership degrees of elements instead of \(0\) and \(1\). **Definition 1** ([21]). : _Suppose that \(X\) is a collection of objects. Then, \(A=\{(x,\mu(x)):\,x\in X\}\) is called a fuzzy subset of \(X\), where \(\mu:X\rightarrow[0,1]\) is a function called membership function which denotes the membership degrees of elements in \(X\)._  
In 1966, an application of fuzzy sets to find a pattern classification was introduced by [PERSON], [PERSON], and [PERSON] [22]. Four years later, a decision-making problem was solved by [PERSON] and [PERSON] [23]. In the 1970s and 1980s, decision making, control, and pattern classification received the most attention by researchers [24; 25; 26]. The fuzzy mathematics applications were expanded in the following years. Another important application of fuzzy sets appeared to use fuzzy controllers [27]. Fuzzy genetics algorithms and neuro-fuzzy systems were studied in the 1990s [28]. Pattern recognition, machine learning, and wireless sensor networks are other fields of study that benefit from fuzzy mathematics [28; 29; 30]. Moreover, the applications of fuzzy mathematics in D2D communications have mostly been in the field of resource allocation and power control. A continuous fuzzy power control scheme is studied in [31] in such a way that the cellular user can communicate in low power to achieve better D2D users connection quality. A resource allocation algorithm is modeled through fuzzy mathematics and game theory in [32] to introduce an uncertain relationship between the resource and user. In [33], a fuzzy clustering algorithm is proposed to divide D2D users into several groups to reuse the cellular user resources and repress the co-channel interference. In [34], a resource allocation based on fuzzy clustering is proposed to solve the spectrum scarcity problem. A fuzzy logic-based vehicle handover algorithm is studied in [35] to select a proper access network. In this study, we consider each transmitter as a fuzzy node in a D2D mobile network in such a way that a set of transmitters together with a membership function is regarded as a fuzzy set. The membership function assigns to each transmitter a number in \([0,1]\) based on the transmitter's data rate and battery level. Then, a decision-making function chooses the best transmitter with the highest fuzzy degree. We use triangular norms and co-norms to build our fuzzy membership function. **Definition 2** (Ref. [36]). : _Let \(T:[0,1]\times[0,1]\rightarrow[0,1]\) be a commutative, associative, and non-decreasing function which \(T(x,1)=x\), for all \(x\in[0,1]\). Then, \(T\) is called a triangular norm (\(t\)-norm)._  
For example, basic \(t\)-norms that will be called Lukasiewicz, Minimum, and Product \(t\)-norms are defined as follows, respectively. 1. \(T_{L}(x,y)=\max(x+y-1,0)\),
2. \(T_{M}(x,y)=\min(x,y)\),
3. \(T_{P}(x,y)=x.y\). [PERSON] and [PERSON] introduced \(t\)-conorms as dual operations of \(t\)-norms in 1961 [37]. **Definition 3** ([36]). : _Let \(S:[0,1]\times[0,1]\rightarrow[0,1]\) be an associative, commutative, and monotone function that \(S(x,0)=x\) for all \(x\in[0,1]\). Then, \(S\) is called triangular conorm (\(t\)-conorm)._  
Corresponding to basic \(t\)-norms, there are some \(t\)-conorms written as follows. 1. \(S_{L}(x,y)=\min(x+y,1)\), (Lukasiewicz t-conorm);
2. \(S_{M}(x,y)=\max(x,y)\), (Maximum \(t\)-conorm);
3. \(S_{P}(x,y)=x+y-x.y\), (Probabilistic sum). Another \(t\)-conorm, named the Einstein sum, is used in this paper to build our membership function [36]. The Einstein sum \(\tau:[0,1]\times[0,1]\rightarrow[0,1]\) is defined as follows for all \(x,y\in[0,1]\). \[\tau(x,y)=\frac{x+y}{1+xy}. \tag{1}\]  
In order to find a proper membership function, the Einstein sum is chosen after examining several \(t\)-norms and \(t\)-conorms.