$12\mathrm{.}3-1.*$ The Research and Development Division of the Progres$-$

sive Company has been developing four possible new product lines.

Management must now make a decision as to which of these four

products actually will be produced and at what levels. Therefore,

an operations research study has been requested to find the most

profitable product mix.

A substantial cost is associated with beginning the produc$-$

tion of any product, as given in the first row of the following

table. Management's objective is to find the product mix that max$-$

imizes the total profit $($total net revenue minus start$-$up costs$).$

Let the continuous decision variables x$\_(1),$x$\_(2),$x$\_(3),$ and x$\_(4)$ be the

production levels of products $1,2,3,$ and $4,$ respectively. Manage$-$

ment has imposed the following policy constraints on these variables:

Use the following modifications to consider the constraints. $$Management has imposed the following policy constraints on these variables: $1)$ No more than two of the products can be produced. $2)$ Product $3$ can be produced only if product $1$ is produced. $3)$ Product $4$ can be produced only if either product $1$ or $2$ is produced. $4)$ Either $5$x$1+3$x$2+6$x$3+4$x$46,000$ Or $4$x$1+6$x$2+3$x$3+5$x$46,000$

$($a$)$ Introduce auxiliary binary variables to formulate a mixed BIP

model for this problem.

c $($b$)$ Use the computer to solve this model.