# Handover Management in 5G Vehicular Networks
## 3 Preliminaries
### The Mamdani Icosagonal Fuzzy Inference System  
The Mamdani Icosagonal Fuzzy Inference System (MIFIS) takes into consideration two inputs (\(Input_{1}\) and \(Input_{2}\)) to calculate the value of the \(Output\) using fuzzy logic. The values of both \(Input_{1}\) and \(Input_{2}\) are normalized within the range [0, 1]. Additionally, the \(MF_{Input_{1}}\), \(MF_{Input_{2}}\), \(MF_{Output}\) Membership Functions (MFs) are created, indicating the linguistic terms and the corresponding IVIFNs for the fuzzy representation of \(Input_{1}\), \(Input_{2}\), and \(Output\), respectively. As a result, for each crisp value, two membership degrees are determined in the respective MF, one for the upper icosagon and one for the lower icosagon.  
Regarding the implementation of the MIFIS method, initially, the Membership Functions (MFs) for the \(Input_{1}\), \(Input_{2}\), and \(Output\) parameters are determined. Furthermore, a knowledge base that contains \(R\) fuzzy rules is created. In this base, each rule \(r\in R\) is a simple if-then statement with a condition and a conclusion. The rule's condition consists of \(MF_{Input_{1}}\) and \(MF_{Input_{2}}\) membership functions, while its conclusion indicates an \(MF_{Output}\) membership function. Subsequently, the \(Input_{1}\) and \(Input_{2}\) crisp values are converted to degrees of membership indicated as \(Input_{1}^{\prime}\) and \(Input_{2}^{\prime}\) by a lookup in the \(MF_{Input_{1}}\) and \(MF_{Input_{2}}\) membership functions, respectively. Thereafter, fuzzy operations are performed in order for the fuzzified inputs to be combined. In particular, the rule's condition is considered a \(Z^{\prime}\) strength of the \(r\) rule to be calculated. In addition, the fuzzy operators \"AND\" and \"OR\" can be used in multiple conditions to be combined. In this case, the \"algebraic product\" is applied for the \"AND\" operator, while the \"algebraic sum\" is applied for the \"OR\" operator. In this work, Formula (19) was used for the estimation of the \"algebraic product\", while Formula (20) was used for the estimations of the \"algebraic sum\".  
\[Z^{\prime}_{u,i,r}=Input^{\prime}_{1}\vdots Input^{\prime}_{2}=Input^{\prime} _{1}\cdot Input^{\prime}_{2} \tag{19}\]  
\[Z^{\prime}_{u,i,r}=(Input^{\prime}_{1}+Input^{\prime}_{2})-(Input^{\prime}_{1 }\cdot Input^{\prime}_{2}) \tag{20}\]Then, the consequence \(MF_{Output^{e}_{t}}\) of the rule conclusion is estimated by applying an implication method. Specifically, the implication method trims the \(MF_{Output}\), height by taking into consideration the \(Z^{\prime}_{r}\) strength, by applying Formula (21), which implements the Min method.  
\[MF_{Output^{e}_{t_{flight}}}=min\{MF_{Output_{t_{flight}}},Z^{\prime}_{r}\} \tag{21}\]  
Subsequently, the R rules' consequences are combined by applying an aggregation method in the \(Output^{A}\) fuzzy set to be calculated. In particular, the aggregation method uses Formula (22), which implements the Max method.  
\[Output^{A}=MF_{Output^{e}_{1}}\cup MF_{Output^{e}_{2}}\cup\ldots\cup MF_{Output ^{e}_{R}} \tag{22}\]  
Finally, defuzzification is performed. Specifically, the \(Output^{A}\) fuzzy set is transformed to the crisp value \(Output\). In this case, the Weighted Average method is applied using Formula (23). In this formula, the \(\mu_{r}\) represents the height of each rule and \(h_{r}\) represents the centroid of each rule, as they are obtained from \(Output^{A}\). Moreover, the parameters \(U\) and \(L\) indicate the upper and the lower icosagation of each rule, respectively.  
\[Output=\frac{\sum\limits_{r=1}^{R}(h_{r}^{\mu_{r}}\cdot h_{r}^{\mu_{r}}+\mu_{ r}^{L}\cdot h_{r}^{L})}{\sum\limits_{r=1}^{R}(h_{r}^{\mu_{r}}+h_{r}^{L})} \tag{23}\]