# Fuzzy Neural Network with Ordered Fuzzy Numbers for Life Quality Technologies
## 3 Ordered Fuzzy Numbers
Definition 2: A membership function of an OFN \(A\) is the function \(\mu_{A}:R\longrightarrow[0,1]\) defined for \(x\in R\) in the following way:  
if  
\[ \begin{array}{l}x\
otin supp_{A}\Rightarrow\mu_{A}(x)=0\\ x\in(1_{A}^{-},1_{A}^{+})\Rightarrow\mu_{A}(x)=1\\ x\in supp_{A}\bigwedge x\
otin(1_{A}^{-},1_{A}^{+})\Rightarrow\mu_{A}(x)=max( f_{A}^{-1}(x),g_{A}^{-1}(x))\end{array}  \tag{8}\]  
Such a definition of the membership function can be applied in the OFN notation for setting control rules in the same way as in classic fuzzy numbers. Figure 3: Fuzzy number that is ordered (**a**) positively or (**b**) negatively. **Definition 3**. : _The replacement of the part LIP (function \(f_{A}\)) with the part DOWN (function \(g_{A}\)) is a reversal of the OFN. This could be defined as follows:_  
\[B=A\mid^{-}\Leftrightarrow g_{B}=f_{A}\wedge f_{B}=g_{A} \tag{9}\]  
_where:_  
* _A is OFN (described by the pair of functions (_\(f_{A},g_{A}\)_),_
* _B is a result of the operation of reversal of the OFN_ \(A\)_,_
* _sign_ \"\(|^{-}\)_\" is a symbol that describes a reversed orientation of OFN,_
* _The result obtained in this operation is called a reversed OFN or a reversed orientation number._  
A set of key points of OFN is very important for number interpretations. This set is defined as follows:  
\[[f(0),f(1),g(1),g(0)]\]  
The basic arithmetic operations for A = (f, g) and B = (e, h) could be defined as another pair of affine functions, according to the following formulas:  
* addition \(A+B=(f+e,g+h)=C\) \[C\longrightarrow[f(0)+e(0),f(1)+e(1),g(1)+h(1),g(0)+h(0)]\] (10)
* scalar multiplication \(C=\lambda A=(\lambda f,\lambda g)\), \[C\longrightarrow[\lambda f(0),\lambda f(1),\lambda g(1),\lambda g(0)]\] (11)
* subtraction \(A-B=(f-e,g-h)=C\), \[C\longrightarrow[f(0)-e(0),f(1)-e(1),g(1)-h(1),g(0)-h(0)]\] (12)
* multiplication \(A\times B=(f\times e,g\times h)=C\), \[C\longrightarrow[f(0)\times e(0),f(1)\times e(1),g(1)\times h(1),g(0)\times h (0)]\] (13)  
Ordered Fuzzy Numbers in practice  
Ordered Fuzzy Numbers are applicable in many everyday life aspects, such as:  
* Smart home systems,
* Data transmission control,
* Servers management in cloud systems,
* Detection of attacks on IT systems. The application of the OFN solution accelerates the control of the heating system. The tests performed in the climatic chamber have confirmed that the results of data analysis with fuzzy logic with OFN algebra, flowing through the controller speeds up reaction so that the temperature can be kept closer to the set one [48]. In the data transmission that uses multiplexing of data transmission paths in the technology Multi Path Transmission Control, the application of an algorithm to predict the error rate in a given path, using OFNs allowed for a faster response and the choice of the different transmission path. With such solution, it is possible to decrease the number of data retransmissions in the network. Following up, there is the possibility to save electric power-transmitting. The same data with a lower retransmissions number saves the power required to send the data. This algorithm can be applied in today's networks, including even 5G networks [49]. Similar possibilities of saving energy necessary in everyday life can be achieved by using algorithms for predicting the demand for virtual machines in cloud systems. The proposed cloud algorithm to forecast servers use with the OFN's application was characterized by the possibility to shut down machines faster at a time when the need forcomputing power to handle incoming connections is decreased [50]. The faster shutdown of servers also means lower power consumption. One of the present Internet problems is the Denial of Service attack type, including Distributed Denial of Service [51]. Such an attack can lock each system available in the network, and lead to an immediate decline in the quality of life. The lock of bank systems can exclude the payments by bank cards (credit, debt) and paralyze trade. These days, such attacks are elements of hybrid wars.