Semi-continuous network flow problems

Abstract

We consider semi-continuous network flow problems, that is, a class of network flow problems where some of the variables are restricted to be semi-continuous. We introduce the semi-continuous inflow set with variable upper bounds as a relaxation of general semi-continuous network flow problems. Two particular cases of this set are considered, for which we present complete descriptions of the convex hull in terms of linear inequalities and extended formulations. We consider a class of semi-continuous transportation problems where inflow systems arise as substructures, for which we investigate complexity questions. Finally, we study the computational efficacy of the developed polyhedral results in solving randomly generated instances of semi-continuous transportation problems.

Appendix

Given an integer $t\ge 1$, let $T:=\{1,\ldots ,t\}$. For each $r\in T$, consider $\pi ^r\in \mathbb R ^n$ and $\pi ^r_0,\pi ^r_1\in \mathbb R$. We are mainly interested in the case $\pi ^r_0< \pi ^r_1$, although this is not required in what follows. Given a closed convex set $C\subseteq \mathbb R ^n$, for each $Q\in \mathcal T :=2^T$, consider the set

$$\begin{aligned} C^Q:=\{x\in C:\ \pi ^rx\le \pi ^r_0\ \forall \,\, r\in Q,\ \pi ^rx\ge \pi ^r_1\ \forall \,\, r\notin Q\}. \end{aligned}$$

We call the set $\cup _{Q\in \mathcal T } C^Q$ a $t$-branch split disjunction as defined in [10]. Let

$$\begin{aligned} C^{\pi ,\pi _0,\pi _1}:=\textit{conv}\left( \cup _{Q\in \mathcal T } C^Q\right) . \end{aligned}$$

When $t=1$, the closedness of $C^{\pi ,\pi _0,\pi _1}$ was addressed in [3]. We extend this result for any $t\ge 1$.

Proposition 23

$C^{\pi ,\pi _0,\pi _1}$ is a closed convex set. Moreover, if $C$ is a polyhedron, so is $C^{\pi ,\pi _0,\pi _1}$.

Proof

Let $C_\infty$ be the recession cone of $C$, and for each $Q\in \mathcal T$, let $C^Q_\infty :=C^Q+C_\infty$. Also, let $\mathcal T ^*:=\{Q\in \mathcal T :\ C^Q\ne \emptyset \}$. If $\mathcal T ^*$ is empty, then the result holds. Thus, assume $\mathcal T ^*$ is nonempty.

Claim: $C^{\pi ,\pi _0,\pi _1}=\textit{conv}\left( \cup _{Q\in \mathcal T ^*} C^Q_\infty \right) .$

The forward inclusion is easy as $\cup _{Q\in \mathcal T } C^Q\subseteq \cup _{Q\in \mathcal T ^*} C^Q_\infty$.

For the reverse inclusion, consider $x\in \textit{conv}\left( \cup _{Q\in \mathcal T ^*} C^Q_\infty \right)$. We can write $x=\sum _{Q\in \mathcal T ^*}\lambda ^Q(x^Q+y^Q)$, where $x^Q\in C^Q$, $y^Q\in C_\infty$, and $\lambda ^Q\ge 0$ for each $Q\in \mathcal T ^*$, and $\sum _{Q\in \mathcal T ^*}\lambda ^Q=1$. If we show that for any $Q\in \mathcal T ^*$, $x^Q+y^Q$ belongs to $C^{\pi ,\pi _0,\pi _1}$, then the result follows. To that end, fix $Q\in \mathcal T ^*$ and let

$$\begin{aligned} R^-&{:=}&\{r\in T:\ \pi ^ry^Q<0\},\\ R^+&{:=}&\{r\in T:\ \pi ^ry^Q>0\},\\ R^=&{:=}&\{r\in T:\ \pi ^ry^Q=0\}.\\ \end{aligned}$$

Note that there exists finite $\lambda \ge 1$ such that $\pi ^r(x^Q+\lambda y^Q)\le \pi ^r_0$ for all $r\in R^-$ and $\pi ^r(x^Q+\lambda y^Q)\ge \pi ^r_1$ for all $r\in R^+$. Also, recall that $x^Q$ satisfies $\pi ^rx^Q\le \pi ^r_0$ for all $r\in Q$ and $\pi ^rx^Q\ge \pi ^r_1$ for all $r\notin Q$. Thus $x^Q+\lambda y^Q$ belongs to $C^{Q^{\prime }}$, where $Q^{\prime }:=R^-\cup (R^=\cap Q)$. Finally, note that $x^Q+y^Q\in \textit{conv}(\{x^Q,x^Q+\lambda y^Q\})$, which implies $x^Q+y^Q\in C^{\pi ,\pi _0,\pi _1}$ as desired. $\Diamond$

By the claim, $C^{\pi ,\pi _0,\pi _1}$ is the convex hull of the union of nonempty closed convex sets having the same recession cone. By Corollary 9.8.1 of [13], $C^{\pi ,\pi _0,\pi _1}$ is a closed convex set. Moreover, if $C$ is a polyhedron, then $C^{\pi ,\pi _0,\pi _1}$ is the convex hull of the union of nonempty polyhedra having the same recession cone, which is a polyhedron [1]. $\square$