Zales Jeweller uses rubies and sapphires to produce two types of rings. A type 1 ring requires two rubies, three sapphires and one hour of jewller's labour time. A type 2 ring requires three rubies, two sapphires and two hours of jeweller's labour time. Each type 1 ring sells for R1200 and each type 2 ring sells for R1500. All rings produced by Zales can be sold. At present, Zales has 100 rubies, 120 sapphires and 70 hours of jeweller's labour time. Extra rubies can be bought at R300 per ruby. Market demand requires at least 20 type 1 rings and 24 type 2 rings.

(a) Formulate the problem as a linear program (15 Marks)

(b) Use solver to solve your linear program (20 Marks)

(c) Write down the optimal solution (8 Marks)

(d) Suppose that instead of R300, each ruby costs R750, Will Zales still buy extra rubies?. If so, will the production plan and profit change? (8 Marks)

(e) Suppose that Zales was required to produce at least 23 type 2 rings. What would Zales' profit now be? (7 Marks)

(f) What is the most that Zales will be willing to pay for another hour of jeweller's labour time? How many hours may be obtained at this price? (7 Marks)

(g) What is the most Zales will be willing to pay for another sapphire? (5 Marks)

(h) Zales discovers another 20 rubies in his safe. How will the availability of these extra rubies change his production plan and profit? (5 Marks)

(i) The market demand changes and Zales must now produce at least 22 type 1 rings and at least 23 type 2 rings. How will this change his production plan and profit? (5 Marks)