# Fuzzy Neural Network with Ordered Fuzzy Numbers for Life Quality Technologies
## 3 Ordered Fuzzy Numbers  
When the precise amount of something could not be given, and is presented by such words as \"little\", \"more\", \"some\", we could use the term. The branch of artificial intelligence that deals with this problem is called fuzzy logic. It was described by [PERSON] in the paper \"Fuzzy sets\" in Information and Control journal in 1965 [38]. In this paper, he defines the term of a fuzzy set. This fuzzy set allows us to describe imprecise data. These data are presented by means of values from the interval from 0 to 1. A number that is assigned to the set defines its degree of membership in this set. The author used a three-valued logic defined by [PERSON] [39]. The use of this theory, the L-R representation of fuzzy numbers, was presented by [PERSON] and [PERSON] [40, 41, 42]. Another method for fuzzy set presentation was developed by [PERSON] [43] and his team. This method is called Ordered Fuzzy Numbers. The advantage of this presentation is that it allows us to link the change of trend to a fuzzy number. It gives more opportunities for arithmetic operation on fuzzy numbers, which was described in the next publications [44, 45, 46, 47]. The following definitions and equations provide basic information about Ordered Fuzzy Numbers, which will be used in the proposed fuzzy network. It is worth noting that this arithmetic is very simple, which means that it may be calculated in a fast and easy way. Definition 1: A is an ordered fuzzy number \(A\) when it is an ordered pair of functions  
\[A=(x_{up},x_{down}) \tag{1}\]_where \(x_{up},x_{down}:[0,1]\longrightarrow R\) are continuous functions._  
Figure 1 presents the respective parts of the functions called: \(up\) and \(down\). The OFN presentation of fuzzy numbers is presented in Figure 2.  
As it could be noticed, the continuity of the two parts (UP and DOWN) shows that they are limited in the following values: \(UP=(l_{A},1_{A}^{-})\) and \(DOWN=(1_{A}^{+},p_{A})\). When both functions within the OFN are continuous, then their inverse functions \(x_{UP}^{-1}\) and \(x_{down}^{-1}\) are defined by their limits UP and DOWN. This condition provides the following equations:  
\[l_{A}:x_{UP}(0),1_{A}^{-}:x_{UP}(1),1_{A}^{+}:x_{down}(1),p_{A}:x_{down}(0) \tag{2}\]  
If a constant function equal to 1 is added within the interval \([1_{A}^{-},1_{A}^{+}]\) are defined by their limits UP and DOWN. On this assumption, the following equations are valid:  
\[l_{A}:x_{UP}(0),1_{A}^{-}:x_{UP}(1),1_{A}^{+}:x_{down}(1),p_{A}:x_{down}(0) \tag{3}\]  
when a constant function (equal to 1) is added within the interval \([1_{A}^{-},1_{A}^{+}]\), then the result is functions UP and DOWN in one range (Figure 2, where:\(\mu_{down}=x_{down},\mu_{up}=x_{up}\)). This result could be treated as a carrier. Then, the function of membership in the fuzzy set is defined on the R set by the following equations:  
Figure 1: The functions up and down in Ordered Fuzzy Number representation. Figure 2: A fuzzy representation of Ordered Fuzzy Number. \[\mu_{A}(x)=0\text{ for }x\
otin[l_{A},p_{A}]\text{,}\] \[\mu_{A}(x)=x_{UP}^{-1}\text{ for }x\in UP, \tag{4}\] \[\mu_{A}(x)=x_{DWN}^{-1}\text{ for }x\in DOWN\text{,}\]  
The fuzzy set defined in the OFN way acquires an additional parameter order, whereas the following interval is the carrier:  
\[UP\cup[1_{A}^{-},1_{A}^{+}]\cup DOWN \tag{5}\]  
The following conditions defined the limit values for UP and DOWN functions in such way:  
\[ \begin{array}{l}\mu_{A}(l_{A})=0\\ \mu_{A}(l_{A}^{-})=1\\ \mu_{A}(l_{A}^{+})=1\\ \mu_{A}(p_{A})=0\end{array}  \tag{6}\]  
OFN can be described by four real following values:  
\[A=(l_{A},l_{A}^{-},l_{A}^{+},p_{A})\]  
Figure 3 presents an example of OFN, including their characteristic points in two orders: positive and negative. Functions \(f_{A},g_{A}\) correspond to parts \(up_{A},down_{A}\subseteq R^{2}\), which gives:  
\[up_{A}=\{(f_{A}(y),y):y\in[0,1]\]  
\[down_{A}=\{(g_{A}(y),y):y\in[0,1] \tag{7}\]  
Orientation corresponds to the order of graphs \(f_{A}\) and \(g_{A}\).