St way to assign users to clusters, while the routing problem determines the segments where the clusters need to be interconnected through bridges leading us to a spanning tree. Typically the design of communication networks requires the existence of a spanning tree, in which each node must be able to communicate with every other node. However, [4] remarks that spanning trees solutions do not provide a reliable design but a minimum cost qhw.v5i4.5120 design. The LAN topology design has been a very active Peretinoin solubility research area in the last two decades. Many authors have proposed both exact methods (e.g. [5], [6] and [7]) and heuristics ([8], [9], [10], [4] and [11]) for the design of LANs and its variants, where genetic algorithms have had a strong preference over other meta-heuristics. One of the first descriptions of the operation and structure of LANs is found in [12]. Particularly relevant to our research are [13] and [14]. The former research proposes a model of nonlinear integer programming which minimizes the average delay in the network as performance criteria and applies a genetic AZD-8835 molecular weight algorithm to solve it. Starting with the approach presented in [13], in [14] a bi-level model is proposed and a Genetic algorithm based on Nash-equilibrium to solve the problem is designed. In this paper, a genetic algorithm that considers the Stackelberg equilibrium is proposed as a solution method for the bi-level topological design of a Local Area Network. The StackelbergGenetic procedure assumes that the follower rational reacts to a leader’s decision; this is, an acceptable spanning tree is selected by the follower due to the difficulty of finding its optimal response in an efficient way. This paper has two objectives: the primary objective is to investigate the performance of the Stackelberg-Genetic algorithm for solving the bi-level problem, and the secondary objective is to show the difference in solving bi-level problems with either Stackelberg or Nash approaches. The remainder of this section describes the previous contributions consisting in metaheuristics developed for this and related bi-level problems. Section 2 presents the bi-level mathematical model. Section 3 is devoted to describe the solution method proposed for obtaining high-quality bi-level solutions. Section 4 shows the computational experiments carried out on previously reported instances and in new generated ones. Also, in jir.2014.0227 this section a Nash-Genetic algorithm similar to the one described in [14] is used for solving the benchmark instances and the obtained results are discussed. The paper is finished with the associated conclusions remarking the importance of solving the bi-level problems with the appropriate methodology.Related literatureIn the last 25 years the field of bi-level optimization has received considerable attention reflected in a wide variety of applications, where metaheuristic algorithms have been considered as a good alternative for finding high quality solutions to considerable size bi-level problems. There are papers in the fields of environmental studies (see [15]), humanitarian logistics (see [16] and [17]), network design (see [18] and [19]), transportation (see [20] and [21]), tollPLOS ONE | DOI:10.1371/journal.pone.0128067 June 23,2 /GA for the BLANDPsetting (see [22] and [23]), location (see [24] and [25]), production planning (see [26]), among many others. For a description of more applications and solution methods we refer the reader to [27] and [28]. Table 1 summarizes so.St way to assign users to clusters, while the routing problem determines the segments where the clusters need to be interconnected through bridges leading us to a spanning tree. Typically the design of communication networks requires the existence of a spanning tree, in which each node must be able to communicate with every other node. However, [4] remarks that spanning trees solutions do not provide a reliable design but a minimum cost qhw.v5i4.5120 design. The LAN topology design has been a very active research area in the last two decades. Many authors have proposed both exact methods (e.g. [5], [6] and [7]) and heuristics ([8], [9], [10], [4] and [11]) for the design of LANs and its variants, where genetic algorithms have had a strong preference over other meta-heuristics. One of the first descriptions of the operation and structure of LANs is found in [12]. Particularly relevant to our research are [13] and [14]. The former research proposes a model of nonlinear integer programming which minimizes the average delay in the network as performance criteria and applies a genetic algorithm to solve it. Starting with the approach presented in [13], in [14] a bi-level model is proposed and a Genetic algorithm based on Nash-equilibrium to solve the problem is designed. In this paper, a genetic algorithm that considers the Stackelberg equilibrium is proposed as a solution method for the bi-level topological design of a Local Area Network. The StackelbergGenetic procedure assumes that the follower rational reacts to a leader’s decision; this is, an acceptable spanning tree is selected by the follower due to the difficulty of finding its optimal response in an efficient way. This paper has two objectives: the primary objective is to investigate the performance of the Stackelberg-Genetic algorithm for solving the bi-level problem, and the secondary objective is to show the difference in solving bi-level problems with either Stackelberg or Nash approaches. The remainder of this section describes the previous contributions consisting in metaheuristics developed for this and related bi-level problems. Section 2 presents the bi-level mathematical model. Section 3 is devoted to describe the solution method proposed for obtaining high-quality bi-level solutions. Section 4 shows the computational experiments carried out on previously reported instances and in new generated ones. Also, in jir.2014.0227 this section a Nash-Genetic algorithm similar to the one described in [14] is used for solving the benchmark instances and the obtained results are discussed. The paper is finished with the associated conclusions remarking the importance of solving the bi-level problems with the appropriate methodology.Related literatureIn the last 25 years the field of bi-level optimization has received considerable attention reflected in a wide variety of applications, where metaheuristic algorithms have been considered as a good alternative for finding high quality solutions to considerable size bi-level problems. There are papers in the fields of environmental studies (see [15]), humanitarian logistics (see [16] and [17]), network design (see [18] and [19]), transportation (see [20] and [21]), tollPLOS ONE | DOI:10.1371/journal.pone.0128067 June 23,2 /GA for the BLANDPsetting (see [22] and [23]), location (see [24] and [25]), production planning (see [26]), among many others. For a description of more applications and solution methods we refer the reader to [27] and [28]. Table 1 summarizes so.
