Article Preview
Top1. Introduction
With the aim of increasing its profitability and ensuring its perenniality, in a modern evolutionary and competing world, a serious economic company must offer its customers a harmonious and effective service. To full fill its obligations, the company must install a system of telecommunication including technical transmission supports between its different centers (nodes). This system is confronted with the tough need for a compromise between cost and quality. In actual decision-making situations, a major concern is that most decision problems involve multiple objectives.
Within this framework, we were proposed to deliver a mathematical model and develop a data-processing solution respecting the problem constraints. For this purpose, our problem is divided into two parts. First of all, one must determine an optimal topology of the tele-exploitation network. The Mathematical model suggested for this part is based on graphs. Indeed, during the design phase of tele-exploitation system, one needs to inter-connect N centers (nodes) between them, using an arrangement of 
transmission supports, each one connects two nodes. Among all possible arrangements, the one using a minimum length of transmission supports is often the most desirable. One can model this problem using a non-directed graph
, where 
represents the set of nodes, and 
represents the set of possible inter-connections between each pair of nodes. Each edge 
is provided with a weight 
which represents the length of transmission support connecting 
to 
. The goal is then to find a “Minimum-Weight Spanning Tree (MWST)” of 
for which the total weight is minimum. This is easily obtained by a polynomial algorithm such as the Kruskal algorithm (Lust et al., 2013).
The main problem lies in the second part, for which we were asked to determine an optimal assignment of transmission supports to connect nodes between them in order to optimize the current operating telecommunication system.
To guarantee its optimal safety, the company for which this work was dedicated accepted to set up a system of tele-exploitation which is a set of transmission supports for information and order, as did many industrial companies. The installation of this system will make it possible to quickly reach essential information needed for the correct economic operation. To collect and exchange all this information, electronics components must be installed at each node, allowing transferring the information to the other nodes, via transmission supports such as, telephone lines, optical fiber, GSM, etc. Let notice, that the search space has 
feasible solutions, where 
designs the number of supports and 
the number of edges in the network. Figure1 represents an example of an assignment of five different supports on a very small input network instance, where the supports are represented by different colors. The number of solutions is 
.