$13\mathrm{.}8-7.$ The MFG Company produces a certain subassembly in each of two separate plants. These subassemblies are then brought to a third nearby plant where they are used in the production of a certain product. The peak season of demand for this product is approaching, so to maintain the production rate within a desired range, it is necessary to use temporarily some overtime in making the subassemblies. The cost per subassembly on regular time $($RT$)$ and on overtime $($OT$)$ is shown in the following table for both plants, along with the maximum number of subassemblies that can be produced on RT and on OT each day.

Let $\backslash ($ x$\_\{1\}\backslash )$ and $\backslash ($ x$\_\{2\}\backslash )$ denote the total number of subassemblies produced per day at plants $1$ and $2,$ respectively. The objective is to maximize $\backslash ($ Z$=$x$\_\{1\}+$x$\_\{2\}\backslash ),$ subject to the constraint that the total daily cost not exceed $\backslash (\backslash$$ $60,000\backslash ).$ Note that the mathematical programming formulation of this problem $($with $\backslash ($ x$\_\{1\}\backslash )$ and $\backslash ($ x$\_\{2\}\backslash )$ as decision variables$)$ has the same form as the main case of the separable programming model described in Sec. $13\mathrm{.}8,$ except that the separable functions appear in a constraint function rather than the objective function. However, the same approach can be used to reformulate the problem as a linear programming model where it is feasible to use OT even when the RT capacity at that plant is not fully used.

$($a$)$ Formulate this linear programming model.