		\begin{algorithmic} 
			\State \textbf{Part 1 (Initialization)}:
			$\mathcal{L}_0:= 
			\{\mathcal{F}_y^z \cup \mathcal{M}^{\tau}_{zU}\}$;
			\ $\mathcal{B} = \{\eqref{VI5}\};$ \ $\mathcal{L'}_0:= \mathcal{L}_0 \cup \mathcal{B}$;
			\ $\mathcal{U}_0:= \{\eqref{VI3};\eqref{VI4} \}$;
			\ $\mathcal{A}_0:= C \setminus \mathcal{L}_0$; $\mathcal{V}^L_0 = \emptyset$;
			\ $\mathcal{V}^U_0 = \emptyset$.
			\State \textbf{Part 2 (Iterative Procedure): At node $k$} 
			\State \quad \textbf{Step 1: Solution of nodal relaxation problem:} 
		$	\mathbf{OA-MILP}_k: \; \min \eqref{obj_lin} \quad \text{s.to} \quad (x,y,\gamma,z,U,\mu,\tau,\omega) \in \mathcal{A}_k  \ . $
			 
			
			\State \quad \textbf{Step 2: Set Update:} \\
			
			\begin{itemize}
				\item \textbf{If the objective value corresponding to $X^*_k$ is not better than that of the incumbent}, the node is pruned.
				\item \textbf{If the objective value corresponding to $X^*_k$ is better than that of the incumbent}, then:
				\begin{itemize}
					
					
					
					\item \textbf{If  $X^*_k$ is fractional}, apply user callback:
					
					\begin{itemize}
						\item If $X_k^{*}$ violates any constraint in $\mathcal{U}_o$: 
						\begin{itemize}
							\item Move violated constraints to $\mathcal{V}_k^{U}$ and discard $X_k^{*}$.
							
							\item Update sets of user cuts and active constraints for each open node $o \in \mathcal {O}$
							\[
							\mathcal{U}_o \leftarrow \mathcal{U}_o  \setminus  \mathcal{V}^U_k \quad \text{and} \quad
							\mathcal{A}_o \leftarrow \mathcal{A}_o  \cup \mathcal{V}^U_k.
							\]   
						\end{itemize}
					\item If no valid inequality in $\mathcal{U}_o$ is violated by $X^*_k$, then branching constraints are entered to cut off $X^*_k$ and the next open node is processed. 
				\end{itemize}
			
		 \item 	\textbf{If $X^*_k$ is integer-valued},
    check for possible violation of lazy constraints: 
    \begin{itemize}
    \item If $X^*_k$ violates any constraint in $\mathcal{L}^{'}_o$:
    
    \begin{itemize}
    \item Move violated lazy constraints to $\mathcal{V}^L_k$ and discard $X^*_k$. 

    \item  Update sets of lazy and active constraints for each open node $o  \in \mathcal {O}$: 
\[
    \mathcal{L}^{'}_o \leftarrow \mathcal{L}^{'}_o \setminus \mathcal{V}^L_k 
    \qquad \text{and} \qquad 
    \mathcal{A}_o \leftarrow \mathcal{A}_o  \cup \mathcal{V}^L_k.
    \]
\end{itemize}
    
    \item If $X^*_k$ does not violate any constraint in $\mathcal{L}^{'}_o$, $X^*_k$ becomes the incumbent and node $k$ is pruned.
\end{itemize}
\end{itemize}
		
	\end{itemize}
	\State \textbf{Part 3 (Termination):} The algorithm stops when $\mathcal{O} = \emptyset$.
\end{algorithmic}

\begin{algorithmic}
\State \textbf{Part 1 (Initialization)}:
$\mathcal{L}_0:=
\{\mathcal{F}_y^z \cup \mathcal{M}^{\tau}_{zU}\}$;
\ $\mathcal{B} = \{\eqref{VI5}\};$ \ $\mathcal{L'}_0:= \mathcal{L}_0 \cup \mathcal{B}$;
\ $\mathcal{U}_0:= \{\eqref{VI3};\eqref{VI4} \}$;
\ $\mathcal{A}_0:= C \setminus \mathcal{L}_0$; $\mathcal{V}^L_0 = \emptyset$;
\ $\mathcal{V}^U_0 = \emptyset$.
\State \textbf{Part 2 (Iterative Procedure): At node $k$}
\State \quad \textbf{Step 1: Solution of nodal relaxation problem:}
$ \mathbf{OA-MILP}_k: \; \min \eqref{obj_lin} \quad \text{s.to} \quad (x,y,\gamma,z,U,\mu,\tau,\omega) \in \mathcal{A}_k \ . $
\State \quad \textbf{Step 2: Set Update:} \\
\begin{itemize}
\item \textbf{If the objective value corresponding to $X^*_k$ is not better than that of the incumbent}, the node is pruned.
\item \textbf{If the objective value corresponding to $X^*_k$ is better than that of the incumbent}, then:
\begin{itemize}
\item \textbf{If $X^*_k$ is fractional}, apply user callback:
\begin{itemize}
\item If $X_k^{*}$ violates any constraint in $\mathcal{U}_o$:
\begin{itemize}
\item Move violated constraints to $\mathcal{V}_k^{U}$ and discard $X_k^{*}$.
\item Update sets of user cuts and active constraints for each open node $o \in \mathcal {O}$
\[
\mathcal{U}_o \leftarrow \mathcal{U}_o \setminus \mathcal{V}^U_k \quad \text{and} \quad
\mathcal{A}_o \leftarrow \mathcal{A}_o \cup \mathcal{V}^U_k.
\]
\end{itemize}
\item If no valid inequality in $\mathcal{U}_o$ is violated by $X^*_k$, then branching constraints are entered to cut off $X^*_k$ and the next open node is processed.
\end{itemize}
\item  \textbf{If $X^*_k$ is integer-valued},
check for possible violation of lazy constraints:
\begin{itemize}
\item If $X^*_k$ violates any constraint in $\mathcal{L}^{'}_o$:
\begin{itemize}
\item Move violated lazy constraints to $\mathcal{V}^L_k$ and discard $X^*_k$.
\item Update sets of lazy and active constraints for each open node $o \in \mathcal {O}$:
\[
\mathcal{L}^{'}_o \leftarrow \mathcal{L}^{'}_o \setminus \mathcal{V}^L_k
\qquad \text{and} \qquad
\mathcal{A}_o \leftarrow \mathcal{A}_o \cup \mathcal{V}^L_k.
\]
\end{itemize}
\item If $X^*_k$ does not violate any constraint in $\mathcal{L}^{'}_o$, $X^*_k$ becomes the incumbent and node $k$ is pruned.
\end{itemize}
\end{itemize}
\end{itemize}
\State \textbf{Part 3 (Termination):} The algorithm stops when $\mathcal{O} = \emptyset$.
\end{algorithmic}
{
    "title": "drone-delivery network for opioid overdose -- nonlinear integer   queueing-optimization models and methods",
    "id": "2206.14340",
    "abstract": "we propose a new stochastic emergency network design model that uses a fleet of drones to quickly deliver naxolone in response to opioid overdoses. the network is represented as a collection of m/g/k queuing systems in which the capacity k of each system is a decision variable and the service time is modelled as a decision-dependent random variable. the model is an optimization-based queuing problem which locates fixed (drone bases) and mobile (drones) servers and determines the drone dispatching decisions, and takes the form of a nonlinear integer problem, which is intractable in its original form. we develop an efficient reformulation and algorithmic framework. our approach reformulates the multiple nonlinearities (fractional, polynomial, exponential, factorial terms) to give a mixed-integer linear programming (milp) formulation. we demonstrate its generalizablity and show that the problem of minimizing the average response time of a network of m/g/k queuing systems with unknown capacity k is always milp-representable. we design two algorithms and demonstrate that the outer approximation branch-and-cut method is the most efficient and scales well. the analysis based on real-life overdose data reveals that drones can in virginia beach: 1) decrease the response time by 78%, 2) increase the survival chance by 432%, 3) save up to 34 additional lives per year, and 4) provide annually up to 287 additional quality-adjusted life years.",
    "categories": "math.oc",
    "doi": "",
    "created": "2022-06-28",
    "updated": "2024-01-25",
    "authors": [
        "miguel lejeune",
        "wenbo ma"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2206.14340"
}
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