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:

maxz$=2x_1-2x_2+3x_3$

subject $to-x_1+x_2+x_{3}4$

$2x_1-x_2+x_{3}2$

$x_1+x_2+3x_{3}12$

$x_1,x_2,x_{3}0.$

$($Resource $1)$

$($Resource $2)$

$($Resource $3)$

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.