# Search of the Shortest Path in a Communication Network with Fuzzy Cost Functions
## 5 Application to a Communications Network. Fuzzy Functions and Strategies
### Strategies for the Search of Path with Minimum Cost  
Here, we describe the set of well-known strategies that we will use to compare the competitiveness of the fuzzy model based methodology versus the traditional crisp based mode. All these strategies are designed to search the best communication path by the application of the Dijkstra algorithm. Therefore, we compare both methodologies based on the relative results for each strategy and not on their absolute terms. _Strategy 1._ Application of Dijkstra algorithm using the _Normalized Instantaneous Used Bandwidth_ as the cost function (Equation (6)). This strategy aims to find the path with the minimum sum of the _Normalized Instantaneous Used Bandwidths_ of their links. The instants at which this magnitude is measured will be given by \(t=t_{0}+nT\), where \(t_{0}\) is the instant of the first measurement, and the measured value will be considered constant during \(T\) (time interval between measurements). The total cost of a path is given by the algebraic sum of their links costs. Thus, this is a strategy of additive costs. _Strategy 2._ Application of Dijkstra algorithm using the Normalized Mean Used Bandwidth as the cost function (Equation (7)). With this strategy, we intend to find the path that minimizes the sum of the Normalized Mean Used Bandwidths of their links. In Strategies 1 and 2, we consider the magnitudes as crisp numbers. _Strategy 3._ Application of fuzzy Dijkstra algorithm using the _Fuzzy Normalized Used Bandwidth_. The cost function at the \(n\)-th time interval is defined in Equation (9),  
\[\overline{[bw]}_{ij}=\left([bw]_{ij}^{\min},\overline{[bw]}_{ij},[bw]_{ij}^{ \max}\right), \tag{9}\]  
where \([bw]_{ij}^{\min}\) and \([bw]_{ij}^{\max}\) are the minimum and maximum value of the Normalized Instantaneous Used Bandwidths measured in the \((n-1)\)-th time interval, respectively, and \(\overline{[bw]}_{ij}\) is the mean value of the variable during the \((n-1)\)-th time interval. In this instance, the fuzzy Dijkstra algorithm described in Section 4 was applied considering additive fuzzy costs. This strategy is directly comparable to Strategy 1 and 2. _Strategy 4._ (Shortest-Widest, SW) [10]. This strategy searches for the path with the maximum _Mean Residual Bandwidth_ from (Equation (8)). If there are several paths with the same maximum value, the strategy chooses the one with the minimum number of edges (hops). While the previous metrics are additive (the cost of the path is the sum of the costs of their links), the _Residual Bandwidth_ is concave (the cost of the path is the cost of the link with minimum cost). Note that this metric is not properly a cost, but quite the opposite: it is a metric that measures the goodness of a link. Therefore, the goodness of a path will be associated with that of its worst link (the one with minimal residual bandwidth). Figure 3 helps understanding how this strategy works. The residual bandwidth appears on each link. According to the SW, paths \(P_{A}\) and \(P_{C}\) are firstly chosen (paths with the maximum residual bandwidth in their worst link), and between those two, path \(P_{C}\) is selected, since it has the minimum number of hops. _Strategy 5._ (Fuzzy Shortest-Widest, FSW). It is the same than Strategy 4, but defining the _Fuzzy Mean Residual Bandwidth_ measurement of each link \((i,j)\) as a triangular fuzzy number \((r_{ij}^{\min},\bar{r}_{ij},r_{ij}^{\max})\) where \(r_{ij}^{\min}\), \(\bar{r}_{ij}\) and \(r_{ij}^{\max}\) are computed according to Equation (10),  
\[r_{ij}^{\min}=c_{ij}-[BW]_{ij}^{\max},\quad\bar{r}_{ij}=c_{ij}-\overline{[BW] }_{ij},\quad r_{ij}^{\max}=c_{ij}-[BW]_{ij}^{\min}, \tag{10}\]  
The path selection is performed following the fuzzy Dijkstra algorithm. This strategy is directly comparable with Strategy 4. _Strategy 6_: (Widest-Shortest, WS) [11]. This strategy consists of the inverse process of the SW. First, it searches for the path with the minimum number of hops. In case of  
Figure 3: SW applied to a 9-node network with explicit Mean Residual Bandwidth in every link.