Minimizing the expense transmission time from the source node to demand nodes

An undirected graph \(G=(V,A)\) by a set V of n nodes, a set A of m edges, and two sets \(S,\ D\subseteq V\) consists of source and demand nodes are given. This paper presents two new versions of location problems which are called the \(f(\sigma )\)-location and \(g(\sigma )\)-location problems. We define an \(f(\sigma )\)-location of the network N as a node \(s\in S\) with the property that the maximum expense transmission time from the node s to the destinations of D is as cheap as possible. The \(f(\sigma )\)-location problem divides the range \((0,\infty )\) into intervals \(\displaystyle \cup _{i}{(a_i,b_i)}\) and finds a source \(s_i\in S\), for each interval \((a_i,b_i)\), such that \(s_i\) is a \(f(\sigma )\)-location for each \(\sigma \in (a_i,b_i)\). Also, define a \(g(\sigma )\)-location as a node s of S with the property that the sum of expense transmission times from the node s to all destinations of D is as cheap as possible. The \(g(\sigma )\)-location problem divides the range \((0,\infty )\) into intervals \(\displaystyle \cup _{i}{(a_i,b_i)}\) and finds a source \(s_i\in S\), for each interval \((a_i,b_i)\), such that \(s_i\) is a \(g(\sigma )\)-location for each \(\sigma \in (a_i,b_i)\). This paper presents two strongly polynomial time algorithms to solve \(f(\sigma )\)-location and \(g(\sigma )\)-location problems.