Caption an equation paragraph - latex

Question

I am trying to include different equation paragraphs in my LaTex document. Is it somehow possible to visually separate the paragraphs (e.g. by horizontal lines) and give them different captions?
I don't want the captions to emerge in the table of content, just underneath the equations.
\documentclass{article}
\usepackage[utf8]{inputenc}
\begin{document}
\section{Mathematical Formulation}
\begin{equation}
\operatorname{Max} F=\sum_{u,v \in V \mid u<v} \left(d_{uv} + d_{vu}\right) x_{uv}
\end{equation}
such that:
\begin{equation}
\sum_{l \in L_{u v}} y_{l} \geq x_{u v} \hspace{4em} u, v \in V \mid u<v
\end{equation}
\begin{equation}
\sum_{l \in \tilde L_{i j}} y_{l} \geq 1 \hspace{5em} \ \lbrack i,j \rbrack \in E
\end{equation}
%Next paragraph here
\begin{equation}
\operatorname{Min} F=\sum_{u \in V} \sum_{l \in L} \sum_{(i, j) \in \vec {E_{l}}} t_{l i j} x_{u l i j}
\end{equation}
\begin{equation}
\sum_{l \in L} \sum_{(i, v) \in \vec{E}_{l}} x_{u l i v}-\sum_{l \in L} \sum_{(v, j) \in \vec{E}_{l}} x_{u l v j}=d_{u v} \hspace{4em} u,v \in V \mid u \ne v
\end{equation}
\end{document}
The caption should look similar to the one for my GAMS code:

Answer 1

You could create a custom command like this:
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\newcounter{foo}
\newcommand{\mycaption}[1]{
{
\medskip
\centering
\setcounter{foo}{\theequation}
\addtocounter{foo}{1}
Equation \thefoo: #1
\vskip-0.5\baselineskip
\noindent\hrulefill
}
}
\begin{document}
\section{Mathematical Formulation}
\begin{equation}
\operatorname{Max} F=\sum_{u,v \in V \mid u<v} \left(d_{uv} + d_{vu}\right) x_{uv}
\end{equation}
such that:
\begin{equation}
\sum_{l \in L_{u v}} y_{l} \geq x_{u v} \hspace{4em} u, v \in V \mid u<v
\end{equation}
\begin{equation}
\sum_{l \in \tilde L_{i j}} y_{l} \geq 1 \hspace{5em} \ \lbrack i,j \rbrack \in E
\end{equation}
%Next paragraph here
\mycaption{some text}
\begin{equation}
\operatorname{Min} F=\sum_{u \in V} \sum_{l \in L} \sum_{(i, j) \in \vec {E_{l}}} t_{l i j} x_{u l i j}
\end{equation}
\begin{equation}
\sum_{l \in L} \sum_{(i, v) \in \vec{E}_{l}} x_{u l i v}-\sum_{l \in L} \sum_{(v, j) \in \vec{E}_{l}} x_{u l v j}=d_{u v} \hspace{4em} u,v \in V \mid u \ne v
\end{equation}
\end{document}
Or if you prefer a separate counter:
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\newcounter{foo}
\newcommand{\mycaption}[1]{
{
\medskip
\centering
% \setcounter{foo}{\theequation}
\addtocounter{foo}{1}
Model \thefoo: #1
\vskip-0.5\baselineskip
\noindent\hrulefill
}
}
\begin{document}
\section{Mathematical Formulation}
\begin{equation}
\operatorname{Max} F=\sum_{u,v \in V \mid u<v} \left(d_{uv} + d_{vu}\right) x_{uv}
\end{equation}
such that:
\begin{equation}
\sum_{l \in L_{u v}} y_{l} \geq x_{u v} \hspace{4em} u, v \in V \mid u<v
\end{equation}
\begin{equation}
\sum_{l \in \tilde L_{i j}} y_{l} \geq 1 \hspace{5em} \ \lbrack i,j \rbrack \in E
\end{equation}
%Next paragraph here
\mycaption{some text}
\begin{equation*}
\operatorname{Min} F=\sum_{u \in V} \sum_{l \in L} \sum_{(i, j) \in \vec {E_{l}}} t_{l i j} x_{u l i j}
\end{equation*}
\begin{equation}
\sum_{l \in L} \sum_{(i, v) \in \vec{E}_{l}} x_{u l i v}-\sum_{l \in L} \sum_{(v, j) \in \vec{E}_{l}} x_{u l v j}=d_{u v} \hspace{4em} u,v \in V \mid u \ne v
\end{equation}
\end{document}