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Graphically solving operations research with linear programming

State all the assumptions and show all the work. Define your decision variables clearly. Briefly explain the constraints and objectives functions and define all unit of measure.

Consider the following LP model:

Max Z = x1 + x2
s.t.
x1 + 3x2 <=8
x1 + x2 <=4
x1 , x2 =>0

a) Solve the problem graphically (without any solver). What are the optimal values of x1 and x2 and the optimal objective function value?

b) Consider the following constraint: 2x1 + 3x2 <= b

i. For b = 12: Add this constraint to your plot. What happens to the feasible region? What is the new optimal solution and new optimal objective value?
ii. For b = 6: Add this constraint to your plot. What happens to the feasible region? What is the new optimal solution and new optimal objective value?

c) What happens to the optimal values of x1 and x2 and the objective function if the right hand side of constraint (1) is increased slightly? A qualitative answer is sufficient, but make sure to give reasons for your conclusion.

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