# Search of the Shortest Path in a Communication Network with Fuzzy Cost Functions
## 5 Application to a Communications Network. Fuzzy Functions and Strategies
### Description of Cost Functions and Strategies  
In this section, we describe the different cost functions used in our experiments. These functions are defined both in their _classic form_ (as crisp numbers) and in their fuzzy version as triangular fuzzy numbers. Let us remember from the Introduction that our first objective is not to compare the efficiency of the different cost functions, but rather the competitiveness of the fuzzy model based methodology versus that the traditional mode-based one using crisp magnitudes, for each cost function.  
Additionally, as one of the main contributions of this paper and as our second objective, we propose a new fuzzy cost function that has no crisp equivalent, since it is an empirical adaptation of the generalized fuzzy number model described above. This new function will be explained in detail when presenting Strategy 8.  
The simulated classic cost functions are the following:  
* _Normalized Instantaneous Used Bandwidth_[30]: It is the value of the bandwidth occupied in a link at the instant \(t\), when weighted according to Equation (6). \[bw_{ij}(t)=\frac{BW_{ij}(t)}{C_{ij}}\frac{C_{\max}}{C_{ij}},\] (6) where, \[bw_{ij}(t)\] is the _Normalized Instantaneous Used Bandwidth_ on link \[(i,j)\] at the instant \[t\]\(BW_{ij}(t)\) is the _Instantaneous Used Bandwidth_ on link \((i,j)\) at the instant \(t\). \(C_{ij}\) is the capacity of link \((i,j)\). \(C_{\max}\) is the capacity of the link with the highest bandwidth in the network. This value will be updated in the routing tables with a predefined periodicity according with the measurement protocol. Therefore, it will be considered \"fixed\" during the update period.
* _Normalized Mean Used Bandwidth_, [5]: It is the mean value of the _Used Bandwidth_ in a particular link, over a considered time interval. This value is normalized by the capacity of the link and is weighted according to the ratio between the network's larger capacity and the current link capacity. It is defined in Equation (7). \[\overline{[bw]}_{ij}=\frac{\overline{[BW]}_{ij}}{C_{ij}}\frac{C_{\max}}{C_{ij}},\] (7) where, \(\overline{[bw]}_{ij}\) is the _Normalized Mean Used Bandwidth_ in link \((i,j)\) at a measurement time interval. \(\overline{[BW]}_{ij}\) is the _Mean Used Bandwidth_ in link \((i,j)\) at the measured period of time. \(C_{ij}\) and \(C_{\max}\) as defined above.
* _Mean Residual Bandwidth associated with a link (or a path)_: In a link \((i,j)\), the _Mean Residual Bandwidth_ (\(\overline{r}_{ij}\)) is defined by the difference between the link capacity and \(\overline{[BW]}_{ij}\) at the considered time interval, as shown in Equation (8). \[\overline{r}_{ij}=C_{ij}-\overline{[BW]}_{ij},\] (8) The Mean Residual Bandwidth of a path (\(\overline{r}_{P}\)) is given by the minimum among the Mean Residual Bandwidths of the links that are part pf the path, that is, the Residual Bandwidth of the \"worst\" link. Equations (6)-(8) will be transformed into triangular fuzzy numbers. In particular, in our experimental study, for each link \((i,j)\) at the \(n\)-th time interval, we will use triangular fuzzy numbers \((a,b,c)\), where \(a\) and \(c\) are computed from the minimum and maximum values of the variable measured at the \((n-1)\)-th time interval, and \(b\) is computed from the mean value of the variable over the \((n-1)\)-th time interval. Once the cost functions have been described, we define the different strategies for searching the shortest path by using the Dijkstra algorithm in both its crisp and fuzzy versions.