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

Abstract

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.