Changes in the right$-$hand side. JOBCO produces two products on two machines. A unit of Product

$1$ requires $2$ hours on machine $1$ and $1$ hour on machine $2.$ For Product $2,$ a unit requires $1$ hour on

machine $1$ and $3$ hours on machine $2.$ The revenues per unit of products $1$ and $2$ are $$30$ and $$20,$

respectively. The total daily processing time available for each machine is $8$ hours.

Letting $x_1$ and $x_2$ represent the daily number of units of products $1$ and $2,$ respectively, the LP model

is given as

maxz$=30x_1+20x_2$

$s.t.$

$2x_1+x_{2}8$

$x_1+3x_{2}8$

$x_1,x_{2}0$

Changes in the objective coefficients. We use the same example with the previous section.

$($a$)$ There is a range for the coefficients of the objective function that will keep the optimum

solution unchanged. Find this range for the previous example.

$($b$)$ Suppose that the unit revenues for products $1$ and $2$ are changed to $$35$ and $$25,$ respectively.

Will the current optimum remain the same?

$($c$)$ Suppose that the unit revenue of product $2$ is fixed at its current value of $c_2=$$$20.$ What is the

associated range for $c_1,$ the unit revenue for product $1$ that will keep the optimum unchanged?