Problem

$1($

Sensitivity Analysis

$)($

$-$

$$pts

$.$

$)$

Consider the operations of a company that produces three products using three resources. In order to

maximize the profit, the following linear program was formulated:

max z

$=$

$2$

x

$12$

x

$2$

$+$

$3$

x

$3$

subject to

$$

$$x

$1$

$+$

$$x

$2$

$+$

$$x

$3$

$<$

$=$

$4($

Resource

$1)$

$2$

x

$1$

$$x

$2$

$+$

$$x

$3$

$<$

$=$

$2($

Resource

$2)$

x

$1$

$+$

$$x

$2$

$+$

$3$

x

$3$

$<$

$=$

$12($

Resource

$3)$

x

$1$

$,$

$$x

$2$

$,$

$$x

$3$

$>$

$=$

$0$

$.$

Here, x

$1$

$,$

$$x

$2$

$$and x

$3$

$$denote the number of units of each product type being produced, respectively. Each

constraint captures the utilization of the three resources considered.

i

$.$

$$Solve the above linear program using the Simplex algorithm.

ii

$.$

$$Obtain the primal solution, dual solution, and reduced cost at optimality.

iii. Using the matrix form, conduct the sensitivity analysis with respect to all the objective function

coefficients.

iv

$.$

$$Using the matrix form, conduct the sensitivity analysis with respect to all the right

$-$

hand side

parameters.

v

$.$

$$Interpret all the observations and results in the context of this resource allocation problem