Consider the following problem

max

x$1,$x$2$

Z $=5$x$1+$c$2$x$2$

$$

$$

s$.$t$.$

$$

$3$x$1+$a$12$x$2>=60,$

$2$x$1+$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?