# Search of the Shortest Path in a Communication Network with Fuzzy Cost Functions
## 3 Fuzzy Modeling of a Communication Network  
Fuzzy graphs can be classified into five main groups according to the fuzziness of the vertex set, the edge set and/or the edge weights. For our network model, we know the vertex and the edge sets as they correspond to the nodes and the connection lines. However, the weights of the edges (the costs in our network) are fuzzy numbers, as we cannot determine their exact value. According to [4], these features lead to a Type-V fuzzy graph.  
In this section, we describe the Type-V fuzzy graph associated to the network. We also provide the definition of a fuzzy number and its possible interpretation to ease a better understanding of our approach. At this point, it is necessary to clarify that the definition of a triangular fuzzy number is not equivalent to that of a triangular probability distribution. While the probability distribution responds to a binary logic (the excluded third principle is respected), the fuzzy number responds to a multivalued logic. We can say that the membership function that defines the fuzzy number does not provide the probability that the number takes a certain value, but rather the ratio in which the number is that value. Thus, processing uncertainty and the properties of both fuzzy and probabilistic approaches are different. [PERSON] distinguishes them as \"linguistics\" and \"stochastic\" uncertainty, respectively, [26; 27].  
Definition 1: (**Type-V fuzzy graph**) A Type-V fuzzy graph is based on known vertexes and edges. However, the costs of the edges has some uncertainty. It can be interpreted as a crisp graph in which the costs of the edges are fuzzy numbers.  
Our network is described as a Type-V fuzzy graph \(\tilde{G}\), which is defined as follows:  
* \(\tilde{G}=(V,E,\mathfrak{c})\) is a triplet where,
* \(V\) is the set of vertexes
* \(E\) is the set of edges
* \(\mathfrak{c}\) is the set of fuzzy costs. \(\tilde{C}_{ij}\in\mathfrak{c}\): fuzzy cost of edge \((i,j)\), \(i,j\in V\)  
Figure 1 depicts an example of a Type-V fuzzy graph.  
\(\bullet\)\(\tilde{G}\) is connected, that is, there is a path between each pair of different nodes in \(\tilde{G}\).  
\(\hat{C}_{e_{ij}}\in\mathcal{C}\), \(e_{ij}\in E\), is the fuzzy cost of edge \(e_{ij}\). The cost of path \(P\), denoted as \(\hat{C}_{P}\), is a function of the cost of its constituent edges, i.e.,: \[\hat{C}_{P}=f(\hat{C}_{e_{ij}}),\quad\forall e_{ij}\in P\]  
There are several ways to define a fuzzy number, but for the goal of this paper, we adopt the definition referred to as a \(f_{L}\)-\(f_{R}\) positive fuzzy number.  
Definition 2: (\(\mathbf{f_{L}-f_{R}}\) Positive fuzzy number) The membership function of a positive fuzzy number \(\hat{C}_{(u,p)}=(a,b,c,d,\omega)\), which is associated to the cost of an edge \(e=\langle u,v\rangle\) being \(u,v\in V\), is:  
\[\mu_{\hat{C}_{(u,\varphi)}}(x)=\left\{\begin{array}{ll}f_{L}(x)&a\leq x\leq b \\ \omega&b\leq x\leq c\\ f_{R}(x)&c\leq x\leq d\\ 0&\text{otherwise}\end{array}\right. \tag{1}\]  
where \(f_{L}:\;[a,b]\rightarrow[0,\omega]\) and \(f_{R}:\;[c,d]\rightarrow[0,\omega]\) are a strictly increasing and decreasing functions, respectively. They are also continuous functions with \(\omega\in[0,1]\).  
We assume \(f_{L}\) and \(f_{R}\) to be linear functions. Thus, according to the values of \(\omega\), the cost of each edge is either normalized or generalized, and according to \(a\), \(b\), \(c\) and \(d\), this cost is a triangular or trapezoidal fuzzy number (see Figure 2).  
* \(\hat{C}_{(u,v)}\) normalized triangular fuzzy number (\(\omega=1\), \(b=c\)).
* \(\hat{C}_{(u,v)}\) normalized trapezoidal fuzzy number (\(\omega=1\), \(a\
eq b\
eq c\
eq d\)).
* \(\hat{C}_{(u,v)}\) generalized triangular fuzzy number (\(\omega\
eq 1\), \(b=c\)).
* \(\hat{C}_{(u,v)}\) generalized trapezoidal fuzzy number (\(\omega\
eq 1\), \(a\
eq b\
eq c\
eq d\)).  
In what follows from this work, we focus on considering normalized triangular fuzzy numbers \(\bar{A}=(a,b,c)\).