Silva and Sons Ltd. (SSL)2 is the largest coconut processor in Sri Lanka. SSL buys coconuts at 300 rupees per thousand to produce two grades (fancy and granule) of desiccated (dehydrated)

coconut for candy manufacture, coconut shell flour used as a plastics filler, and charcoal.

Nuts are first sorted into those good enough for desiccated coconut (90%) versus those good only for their shells. Those dedicated to desiccated coconut production go to hatcheting/pairing to remove the meat and then through a drying process. Their shells pass on for use in flour and charcoal. The 10% of nuts not suitable for desiccated coconut go directly to flour and charcoal.

SSL has the capability to hatchet 300,000 nuts per month and dry 450 tons of desiccated coconut per month. Every 1000 nuts suitable for processing in this way yields 0.16 ton of desiccated coconut, 18% of which is fancy grade and the rest granulated. Shell flour is ground from coconut shells; 1000 shells yield 0.22 ton of flour. Charcoal also comes from shells; 1000 shells yield 0.50 ton of charcoal. SSL can sell fancy desiccated coconut at 3500 rupees per ton over hatcheting and drying cost, but the market is limited to 40 tons per month. A contract requires SSL delivery of at least 30 tons of granulated-quality desiccated coconut per month at 1350 rupees per ton over hatcheting and drying, but any larger amounts can be sold at that price. The market for shell flour is limited to 50 tons per month at 450 rupees each. Unlimited amounts of charcoal can be sold at 250 rupees per ton.

(a) Explain how this coconut production planning problem can be modeled as the LP max 3500s1 + 1350s2 + 450s3

+ 250s4 - 300p1 - 300p2 s.t. 0.10p1 - 0.90p2 = 0 0.82s1 - 0.18s2 = 0 p1 … 300 s1 + s2 … 450 s1 … 40 s2 Ú 30 s3 … 50 0.16p1 - s1 - s2 = 0 0.11p1 + 0.11p2 - 0.50s3 - 0.22s4 = 0 p1, p2, s1, s2, s3, s4 Ú 0

(b) State the dual of your primal linear program.

(c) Enter and solve the given primal LP with the class optimization software.

(d) Use your computer output to determine a corresponding optimal dual solution.

(e) Verify that your computer dual solution is feasible in the stated dual and that it has the same optimal solution value as the primal.

(f) On the basis of your computer output, determine how much SSL should be willing to pay to increase hatcheting capacity by 1 unit (1000 nuts per month).

(g) On the basis of your computer output, determine how much SSL should be willing to pay to increase drying capacity by 1 unit (1 ton per week).

(h) On the basis of your computer output, determine or bound as well as possible the profit impact of a decrease in hatcheting capacity (thousands of nuts per month) to 250. Do the same for a capacity of 200.

(i) On the basis of your computer output, determine or bound as well as possible the profit impact of an increase in hatcheting capacity (thousands of nuts per month) to 1000. Do the same for a capacity of 2000.

(j) The company now has excess drying capacity.

On the basis of your computer output, determine how low it could go before the optimal plan was affected.

(k) The optimum now makes no shell flour.

On the basis of your computer output, determine at what selling price per ton it would begin to be economical to make and sell flour.

(l) On the basis of your computer output, determine or bound as well as possible the profit impact of a decrease in the selling price of granulated desiccated coconut to 800 rupees per ton. Do the same for a decrease to 600 rupees.

(m) On the basis of your computer output, determine or bound as well as possible the profit impact of an increase to 400 rupees per ton in the price of charcoal. Do the same for an increase to 600 rupees.

(n) On the basis of your computer output, determine whether the primal optimal solution would change if we dropped the constraint on drying capacity.

(o) On the basis of your computer output, determine whether the primal optimal solution would change if we added a new limitation that the total number of nuts available per month cannot exceed 400,000. Do the same for a total not to exceed 200,000.