# Search of the Shortest Path in a Communication Network with Fuzzy Cost Functions
## 1 Introduction  
The management of a communication network is mainly oriented to maximize the ratio between sent and received information for any pair of source-destination nodes while reducing transmission delays. This objective is especially relevant in high occupancy conditions. Necessary resources to achieve this objective are algorithms and routing protocols, which seek to provide the optimal path between any pair of source-destination nodes. To evaluate the goodness level of a path, it is crucial to know the state of the network by assigning a cost to every link. In this way, we compute the cost of a particular path using variables such as Used Bandwidth, Residual Bandwidth (non-used bandwidth), Packet Delay (time spent in the delivery of an information packet), Packet Delivery Ratio (ratio between delivered and sent packets), Packet Loss Ratio (one minus the Package Delivery Ratio), etc. From this knowledge, we can establish the routing tables, which contain the path communicating nodes to transmit the information at each moment. Because of the dynamic behavior of a network, and consequently, the high variability in time of its performing conditions, the usual way to work is to compute the state variablesover a given time interval using measurements obtained in the immediately previous time interval. So, typical routing protocols update the state (cost) of links in two possible ways [1]:  
1. By time intervals. Each cost value (and therefore, the routing tables) is updated with a constant periodicity. This update is either to the instantaneous state value at the beginning of the new interval or to the mean value of costs over the previous time interval. 2. By thresholds. Some update thresholds are fixed so that the update of a state variable is carried out when the difference between its current real value and the one determined more recently exceeds the corresponding threshold. It is evident that, although these cost values are well defined real numbers (crisp numbers in our lingo), these procedures include a degree of uncertainty about the current state due to the fact that the variables change during a time interval. Consequently, we believe that considering this uncertainty in the decision-making process for the management should allow us to find better solutions. In this sense, some elements of Fuzzy Logic, in particular, the application of arithmetic operations and properties of fuzzy numbers, can be helpful. As a first approach, we interpret a triangular fuzzy number as a triplet of real numbers. The numbers at the ends delimit the interval of values that can be assigned to the fuzzy number (variable values with a probability value greater than 0 in the membership function) and the central number is the value of the variable with the highest value in the membership function. This triangular fuzzy number is a particular case of the more general trapezoidal fuzzy number described in Section 3 and in [2; 3]). An essential aspect of our work is to represent the cost measurements of the network by using fuzzy numbers in order to reasonably incorporate the aforementioned uncertainty. For this objective, option (B) has an apparent advantage over option (A) as when fixing the thresholds, we are also setting the upper and lower limits of the corresponding triangular fuzzy number. However, the central value of this fuzzy number, which would theoretically be considered the most feasible one, does not have a physical justification: it would respond to an instantaneous value from which we have no information about its feasibility. Moreover, if the central value is given, we compute the upper and lower values by adding and subtracting the threshold. The crisp number corresponding to this fuzzy cost function will always be constant and equal to the central value. Therefore, the meaning of fuzzification is lost. On the other hand, option (A) does not directly provide us with the extreme values of the normalized triangular fuzzy number.