$($please answer on apaper $)$

Wild West produces two types of cowboy hats. Type $1$ hat requires twice as much

labor time as a type $2.$ If all the available labor time is dedicated to type $2$ alone, the

company can produce a total $400$ type $2$ hats a day. The respective market limits for

the two types are $150$ and $200$ hats per day. The profit is $$8$ per type $1$ and $$5$ per type

$2$ hat. Formulate the model as a linear program model

a$)$ Plot the constraints and identify the feasible solution space

b$)$ What is the optimum values of the variables?

c$)$ Give $2$ values of the variables that are feasible $($not necessarily optimal$)$

d$)$ What is the optimum value of the objective function?

e$)$ Identify the constraints as binding, non$-$binding, and redundant, explain

f$)$ What is the maximum allowable increase in resource $1?$

g$)$ What is the maximum allowable decrease in resource $2?$

h$)$ Compute the unit worth of resource $1$

i$)$ Compute the shadow price of resource $2$

j$)$ Which resource should be given priority to increase? why?

k$)$ If you want to solve the above problem using the simplex method, what is the

number of iterations needed? Explain

Comment on your answer in part $($k$)$

$2-$

a$)$ Write the following constraint in a standard format

$x1-22-2$

b$)$ for the following problem

min$4x1+x2$

$st$

$41+326$

$x1+224$

Formulate the LP including the Artificial variables in the constraints and the objective

function