Please completely solve this problem and explain steps.

Question $3($inspired by H&L $7\mathrm{.}6-4).$

Consider the following problem

$ma{x}_{x_1,x_2}Z=5x_1+c_2{x}_2$

$s.t.\{\begin{array}{c}3x_1+a_{12}{x}_{2}60,\end{array}$

$2x_1+a_{22}{x}_{2}60,$

$x_{1}0,x_{2}0,$

where $x_1$ represents the level of activity $1$ and $x_2$ represents the level of activity $2.$ The values

of $c_2,a_{12},$ and $a_{22}$ have not been determined yet. Only activity $1$ needs to be undertaken soon

whereas activity $2$ will be initiated somewhat later. There are different scenarios that could

unfold between now and the time activity $2$ is undertaken that would lead to different values

for $c_2,a_{12},a_{22}.$ Therefore, the goal is to use all of this information to choose a value for $x_1$

now and to simultaneously determine a plan for choosing a value of $x_2$ later after seeing which

scenario has occurred.

Three scenarios are considered plausible possibilities. They are listed below, along with the

values of $c_2,a_{12},$ and $a_{22}$ that would result from each one:

Scenario $1$: $c_2=4,a_{12}=2,$ and $a_{22}=3.$

Scenario $1$: $c_2=6,a_{12}=3,$ and $a_{22}=4.$

Scenario $1$: $c_2=3,a_{12}=2,$ and $a_{22}=1.$

These three scenarios are considered equally likely.

$($a$)$ Use stochastic programming with recourse to formulate the appropriate model for this

problem.

$($b$)$ Solve the model by using AMPL $($do not forget to submit your AMPL code$).$ What is

the optimal plan?