Promlem $1(20$ Points$).$ Ziba, Inc. manufactures Titanium and Carbon frame racquets. In the coming

production period, Ziba needs to decide how many of each type of racquets should be produced in order to

maximize profit. Each racquet goes through two production processes. Each Titanium frame requires $21$

minutes of process I time and $15$ minutes of process II time. Each Carbon frame requires $15$ minutes of

process I time and $20$ minutes of process II time. In the upcoming production period, at least $27$ hours are

available in process I and no more than $40$ hours are available in process II$.$ In the upcoming period the

demand for the Titanium frames is estimated to be at least $60$ units. Due to a decline in the demand for

Carbon frames, the company does not want to produce more than $45$ Carbon frame racquets. Furthermore,

management has decided that the production of the number of Carbon frames should be at least $20\%$ of total

production. Each Titanium racquet costs $$88$ to produce and sells for $$200.$ Each Carbon frame racquet

costs $$70$ to produce and sells for $$170.$

a$.$ Formulate an LP problem to determine the highest profit. Write your problem formulation.

Let your decision variables be:

$x1=$ Number of Titanium frame racquets to produce

$x2=$ Number of Carbon frame racquets to produce

b$.$ Graph the constraints and identify the region of feasible solutions.

Using the graphical procedure, solve the problem.

How many of each type of racquets should be produced and, what will be the maximum profit?

Are there any slack or surpluses at optimum? What are they, and what are their values?

c$.$ If the production costs of Titanium frame racquets were increased to $$125$ per unit, what would be the

optimum solution? Is there more than one optimum solution to the new problem? If yes, give at least two

optimum solutions and their respective slacks and or surpluses.