$($Bonus $2$ points$).$ Suppose that the costs are $$900$ if selecting project $1$ to do$,$ $$800$ if selecting

project $2$ to do$,$ and $$700$ if selecting project $3$ to do$,$ respectively. Suppose that we have $$2,000$

available.

Write a constraint to represent: "Total cost of doing selected projects cannot be more than

$$2,{000}^{\text{'}\text{'}},$ in terms of $x_1,x_2,$ and $x_3$ as defined above.

Question $10,11,$ and $12$ are about a fixed charge problem, in which a fixed charge of

$$780$ is associated with product $1.$ The related integral variable $x_1$ and binary variable $Y_1$ are

defined as follows:

$x_1=$ number of units of product $1$ to be produced, an integer.

$Y_1=1$ if $x_{1}1,$ and $Y_1=0$ if $x_1=0$

What does the expression $"780Y_1"$ mean?

a$.$ The expense of the fixed charge is $$780.$

b$.$ The $$780$ fixed charge is spent if $Y_1=1$; and $$780$ is not spent if $Y_1=0.$

c$.$ It is the total cost of producing product $1,$ including both fixed and variable costs.

Write a constraint to represent: $"$If $x_1=0$ then $Y_1=0."$

Write a constraint to represent: $"$If $x_{1}1$ then $Y_1=1."$