Adding equation numbers to aligned environment

Question

I need to typeset a text with several equations that: (#1) have to include the "maximize" and "subjet to" expressones in line with the corresponding equations; (#2) have to include three aligned right braces ; and (#3) the equations must be individually numbered.

After trying several routes (using optidef, rcases, plain arrays...) I found a clue in Align multiple equations within right braces

I've solved problem #2, and I've dealt with problem #1 by brute force, but I have no idea of how to solve requirement #3 (equation numbers).

\documentclass{memoir}
\usepackage{amsmath}
\usepackage{lipsum}
% ================================================
\begin{document}
%
\lipsum[1][1-5]
%
\begin{equation}
   \begin{aligned}
      \text{Maximize:}\qquad\qquad          W &= W ( U_1 , U_2 ) \\[\jot]
      \text{Subject to:}\qquad\qquad      U_1 &= U_1 \left( x_1 , y_1 \right)\\
                                          U_2 &= U_2 \left( x_2 , y_2 \right)\\[2\jot]
                                            x &= x \left( K_x , L_x \right)\\
                                            y &= y \left( K_y , L_y \right)\\[2\jot]
                                          K_x &+ K_y = \overline{K}    \\
                                          L_x &+ L_y = \overline{L}
   \end{aligned}
   \begin{aligned}
      &\left.\vphantom{\begin{aligned}
         \text{Maximize:}\qquad\qquad          W &= W ( U_1 , U_2 )\\[\jot]
                           \end{aligned}}\right.\\
      &\left.\vphantom{\begin{aligned}
         \text{Subject to:}\qquad\qquad      U_1 &= U_1 \left( x_1 , y_1 \right)\\
                                             U_2 &= U_2 \left( x_2 , y_2 \right)\\[2\jot]
                           \end{aligned}}\right\rbrace\quad\text{Preferences}\\
      &\left.\vphantom{\begin{aligned}
                                               x &= x \left( K_x , L_x \right)\\
                                               y &= y \left( K_y , L_y \right)\\[2\jot]
                           \end{aligned}}\right\rbrace\quad\text{Tecnology}\\
      &\left.\vphantom{\begin{aligned}
                                             K_x &+ K_y = \overline{K}    \\
                                             L_x &+ L_y = \overline{L}
                           \end{aligned}}\right\rbrace\quad\text{Resources}\\
   \end{aligned}
\end{equation}
%
\lipsum[1][1-5]
%
\end{document}

Which gives this output:

Any hint will be welcome.

Answer 1

You could overlap the 2 row entries using \raisebox. (This is how multirow works, more or less.)

If you want to move Maximize all the way to the left, replace align with flalign and add an extra & to the right.

\documentclass{memoir}
\usepackage{amsmath}
\usepackage{lipsum}
\newsavebox{\tempbox}
% ================================================
\begin{document}
%
\lipsum[1][1-5]
%
\savebox{\tempbox}{$\displaystyle \left. \begin{aligned} \mathstrut\ \mathstrut\end{aligned}\right\rbrace$}%
%
\begin{align}
      &\text{Maximize:}&        W &= W ( U_1 , U_2 )\[\jot]
      &\text{Subject to:}&      U_1 &= U_1 \left( x_1 , y_1 \right)  && \raisebox{-0.5\baselineskip}[0pt][0pt]{\usebox\tempbox Preferences}\
      &&                        U_2 &= U_2 \left( x_2 , y_2 \right)\[2\jot]
      &&                        x &= x \left( K_x , L_x \right) && \raisebox{-0.5\baselineskip}[0pt][0pt]{\usebox\tempbox Technology}\
      &&                        y &= y \left( K_y , L_y \right)\[2\jot]
      &&                        K_x &+ K_y = \overline{K} && \raisebox{-0.5\baselineskip}[0pt][0pt]{\usebox\tempbox Resources}\
      &&                        L_x &+ L_y = \overline{L}
\end{align}
%
\lipsum[1][1-5]
%
\end{document}

Answer 2

Speaking for myself, I find that a layout with lots of curly braces runs the serious risk of chopping up the visual "flow" of the elements in the multi-line equations, making the material more difficult rather than easier to grasp.

I would therefore like to propose two entirely different layouts for your consideration. As a side-effect of rearranging the material, numbering the equations is trivially easy.

The horizontal lines in the following screenshot are there just to delimit the vertical extent of the alignat and equation/alignat groups. Still speaking for myself, I find the second layout simpler and therefore more attractive.

\documentclass{memoir}
\usepackage{amsmath}
\begin{document}
\hrule
\begin{alignat}{3}
\text{Maximize}&\qquad&               &&          W &= W(U_1,U_2) \\[2\jot]
\text{Subject to}&&\text{Preferences}&\qquad&   U_1 &= U_1( x_1,y_1)\\
                  &&&&                          U_2 &= U_2( x_2,y_2)\\[\jot]
                  &&\text{Technology} &&          x &= x (K_x,L_x )\\
                  &&&&                            y &= y (K_y,L_y )\\[\jot]
                  &&\text{Resources} &&         K_x &+ K_y = \bar{K} \\
                  &&&&                          L_x &+ L_y = \bar{L}
\end{alignat}
\hrule
\begin{equation}
\text{Maximize} \quad W = W(U_1,U_2) 
\end{equation}
subject to the following conditions:
\begin{alignat}{2}
\text{Preferences}&\qquad&   U_1 &= U_1( x_1,y_1)\
                  &&         U_2 &= U_2( x_2,y_2)\[\jot]
\text{Technology} &&           x &= x (K_x,L_x )\
                  &&           y &= y (K_y,L_y )\[\jot]
\text{Resources}  &&         K_x &+ K_y = \bar{K} \
                  &&         L_x &+ L_y = \bar{L}
\end{alignat}
\hrule
\end{document}