Q5. (Sensitivity Analysis: Adding a new constraint) (2.5 marks) Consider the following LP max z = 6x1 + x2 s.t. xl + x2 c 5 2x1 + x2 c 6 xi, X2 0 with the following final optimal Simplex tableau
basis xi x2 Si s2 rhs z 0 2 0 3 18 si xi 0 1 0.5 0.5 1 0 -0.5 0.5 2 3
where si and s2 are the slack variables in the first and second constraints, respectively.
(a) Please find the optimal solution if we add the new constraint 3x1 + x2 c 10 into the LP.
(b) Please find the optimal solution if we add the new constraint x1 — x2 6 into the LP.
(c) Please find the optimal solution if we add the new constraint 8x1 + x2 c 12 into the LP.
Q6. (LINGO) (2.5 marks) A boat production company must determine how many sailboats should be produced during each of the next four quarters. The demand during each of the next four quarters is as follows: first quarter, 40 sailboats; second quarter, 60 sailboats; third quarter, 75 sailboats; fourth quarter, 25 sailboats. The company must meet demands on time. At the beginning of the first quarter, it has an inventory of 10 sailboats. At the beginning of each quarter, it must decide how many sailboats should be produced during that quarter. For simplicity, we assume that sailboats manufactured during a quarter can be used to meet demand for that quarter. During each quarter, it can produce up to 40 sailboats with regular-time labor at a total cost of $400 per sailboat. By having employees work overtime during a quarter, the company can produce additional sailboats with overtime labor at a total cost of $450 per sailboat. At the end of each quarter (after production has occurred and the current quarter's demand has been satisfied), a carrying or holding cost of $20 per sailboat is incurred. An LP can be formulated to determine a production schedule minimising the sum of production and inventory costs during the next four quarters. Solve this LP problem using LINGO with the SET model. (Please show the LINGO code and the solution report yielded by LINGO.)