/** # Model Dovetail The company Dovetail produces two kinds of matches: long and short ones. The company makes a profit of 3 (× \$1,000) for every 100,000 boxes of long matches, and 2 (× \$1,000) for every 100,000 boxes of short matches. The company has one machine that can produce both long and short matches, with a total of at most 9 (× 100,000) boxes per year. For the production of matches the company needs wood and boxes: three cubic meters of wood are needed for 100,000 boxes of long matches, and one cubic meter of wood is needed for 100,000 boxes of short matches. The company has 18 cubic meters of wood available for the next year. Moreover, Dovetail has 7 (× 100,000) boxes for long matches, and 6 (× 100,000) for short matches available at its production site. The company wants to maximize its profit in the next year. It is assumed that Dovetail can sell any amount it produces. We introduce the *decision variables* $x_1$ and $x_2$: * $x_1 =$ the number of boxes (× 100,000) of long matches to be made the next year, and * $x_2 =$ the number of boxes (× 100,000) of short matches to be made the next year. The company makes a profit of 3 (× \$1,000) for every 100,000 boxes of long matches, which means that for$x_1$(× 100,000) boxes of long matches, the profit is$3x_1$(× \$1,000). Similarly, for $x_2$ (× 100,000) boxes of short matches the profit is $2x_2$ (× \$1,000). Since Dovetail aims at maximizing its profit, and it is assumed that Dovetail can sell its full production, the *objective* of Dovetail is: $\max 3x_1 + 2x_2.$ The function$3x_1 + 2x_2$is called the *objective function* of the problem. It is a function of the decision variables$x_1$and$x_2$. If we only consider the objective function, it is obvious that the production of matches should be taken as high as possible. However, the company also has to take into account a number of *constraints*. First, the machine capacity is 9 (× 100,000) boxes per year. This yields the constraint: \begin{equation} x_1 + x_2 \leq 9. \end{equation} Second, the limited amount of wood yields the constraint: \begin{equation} 3x_1 + x_2 \leq 18. \end{equation} Third, the numbers of available boxes for long and short matches is restricted, which means that$x_1$and$x_2$have to satisfy: \begin{equation} x_1 \leq 7, \mbox{ and } x_2 \leq 6. \end{equation} The above inequalities are called *technology constraints*. Finally, we assume that only nonnegative amounts can be produced, i.e., $x_1, x_2 \geq 0.$ The inequalities$x_1 \geq 0$and$x_2 \geq 0$are called *nonnegativity constraints*. Taking together the six expressions formulated above, we obtain Model Dovetail: $\max3x_1+2x_2\mbox{subject to}x_1+x_2\leq93x_1+x_2\leq18x_1\leq7x_2\leq6x_1, x_2 \geq 0.\$
*/ var x1 >= 0; var x2 >= 0; maximize z: 3*x1 + 2*x2; subject to c11: x1 + x2 <= 9; subject to c12: 3*x1 + x2 <= 18; subject to c13: x1 <= 7; subject to c14: x2 <= 6; subject to c15: x1 + x2 >= 5; end;