A bi-level problem are not found into the Pareto optimal solution of a bi-objective problem.Problem FormulationA bi-level programming problem is a mathematical programming problem which is composed of an upper-level and a lower-level problem. In this paper, the upper-level problem aims to minimize the connection cost, while the lower-level problem seeks to minimize the average message delay time. When solving the bi-level problem both the decision maker of the upperlevel (hereafter referred to as the leader) and the decision maker of the lower-level bmjopen-2015-010112 (hereafter referred to as the follower) interact to get the best solution. For a formal definition of the Bilevel Local Area Flavopiridol cancer network Design Problem (BLANDP) and based on the model proposed in [14], let N = 1,2,. . .,n be the set of users (e.g. routers) in the telecommunication network, and let G = (V, E) be an undirected graph, where V = v1, v2,. . ., vm is the set of vertices (clusters) and E = (vp, vq): p < q is the set of edges (bridges) that connect the clusters. For each cluster, the maximum traffic capacity Cp that can flow through it is known. Also, for each bridge the average response time bpq to route a package between the respective clusters is known. We assume that the traffic characteristics between users are known and summarized in the users traffic matrix S, where an element sij 2 S represents the traffic from user i 2 N to user j 2 N. Two cost elements are considered in the problem: connection cost between clusters wpq: (p, q) 2 E and connection cost between users and clusters ip: i 2 N, p 2 V.PLOS ONE | DOI:10.1371/journal.pone.0128067 June 23,5 /GA for the BLANDPConsider the following decision variables: ( 1; if user i is asigned to cluster p yip ?0; otherwise ( xpq ?1; 0; if cluster p is conected to cluster q otherwiseThe decision variable xpq is defined as x 2 X, where X is a set of spanning trees. From these decision variables other important elements related to the traffic in the network will be defined. To define these terms the introduction of the following concept is needed. A path between 0 2 V and r 2 V, i.e., path(0, r), is a sequence of vertex without repetition (vi-1, vi) 2 E for all i = 1,. . .,r. A vertex vk is called intermediate vertex in path(0, r), if v0,. . ., vk,. . ., vr. Similarly, we define the concept of intermediate edge as the set of all edges (p, q) in path(0, r). With the above definitions and concepts we precise the following terms, let: be the total offered traffic in jir.2014.0227 the network, which can be computed as G?N N XX i? j?sij or by G ?XXp2V q2VtpqT be the traffic matrix between clusters, which can be computed as T = YT SY, where Y is the clustering matrix, which assigns users to clusters. An element tpq of this matrix represents the traffic Chloroquine (diphosphate) molecular weight forwarded from cluster p 2 V to cluster q 2 V. L(x)k be the total traffic at cluster k 2 V, this can be computed as X X X tpk ?tkq ?t ??L ?p2V q2Vnfkg fp;q2Vjk2path ;q pq F(x)(p.q) be the total traffic which flows on the bridge (p, q) 2 E’ E, computed as X :q?F ??t fk;r2Vj ;q?path ;r kr??The leader’s optimization problem consists in determining the best allocation of users to clusters such that the connection costs are minimized. On the other hand, the follower’s optimization problem is to determine the subset of edges E’ E that while forming a spanning tree T = (V, E’) in G, minimize the average message delay time in the network. It should be noted that L(x)k and F(x)(p.q) strongly depend o.A bi-level problem are not found into the Pareto optimal solution of a bi-objective problem.Problem FormulationA bi-level programming problem is a mathematical programming problem which is composed of an upper-level and a lower-level problem. In this paper, the upper-level problem aims to minimize the connection cost, while the lower-level problem seeks to minimize the average message delay time. When solving the bi-level problem both the decision maker of the upperlevel (hereafter referred to as the leader) and the decision maker of the lower-level bmjopen-2015-010112 (hereafter referred to as the follower) interact to get the best solution. For a formal definition of the Bilevel Local Area Network Design Problem (BLANDP) and based on the model proposed in [14], let N = 1,2,. . .,n be the set of users (e.g. routers) in the telecommunication network, and let G = (V, E) be an undirected graph, where V = v1, v2,. . ., vm is the set of vertices (clusters) and E = (vp, vq): p < q is the set of edges (bridges) that connect the clusters. For each cluster, the maximum traffic capacity Cp that can flow through it is known. Also, for each bridge the average response time bpq to route a package between the respective clusters is known. We assume that the traffic characteristics between users are known and summarized in the users traffic matrix S, where an element sij 2 S represents the traffic from user i 2 N to user j 2 N. Two cost elements are considered in the problem: connection cost between clusters wpq: (p, q) 2 E and connection cost between users and clusters ip: i 2 N, p 2 V.PLOS ONE | DOI:10.1371/journal.pone.0128067 June 23,5 /GA for the BLANDPConsider the following decision variables: ( 1; if user i is asigned to cluster p yip ?0; otherwise ( xpq ?1; 0; if cluster p is conected to cluster q otherwiseThe decision variable xpq is defined as x 2 X, where X is a set of spanning trees. From these decision variables other important elements related to the traffic in the network will be defined. To define these terms the introduction of the following concept is needed. A path between 0 2 V and r 2 V, i.e., path(0, r), is a sequence of vertex without repetition (vi-1, vi) 2 E for all i = 1,. . .,r. A vertex vk is called intermediate vertex in path(0, r), if v0,. . ., vk,. . ., vr. Similarly, we define the concept of intermediate edge as the set of all edges (p, q) in path(0, r). With the above definitions and concepts we precise the following terms, let: be the total offered traffic in jir.2014.0227 the network, which can be computed as G?N N XX i? j?sij or by G ?XXp2V q2VtpqT be the traffic matrix between clusters, which can be computed as T = YT SY, where Y is the clustering matrix, which assigns users to clusters. An element tpq of this matrix represents the traffic forwarded from cluster p 2 V to cluster q 2 V. L(x)k be the total traffic at cluster k 2 V, this can be computed as X X X tpk ?tkq ?t ??L ?p2V q2Vnfkg fp;q2Vjk2path ;q pq F(x)(p.q) be the total traffic which flows on the bridge (p, q) 2 E’ E, computed as X :q?F ??t fk;r2Vj ;q?path ;r kr??The leader’s optimization problem consists in determining the best allocation of users to clusters such that the connection costs are minimized. On the other hand, the follower’s optimization problem is to determine the subset of edges E’ E that while forming a spanning tree T = (V, E’) in G, minimize the average message delay time in the network. It should be noted that L(x)k and F(x)(p.q) strongly depend o.