Operations Research Answers

A motor company manufacture and sell cars and motorbikes. The cost of manufacturing x motorbikes and y cars is given by 2 2 C x y x xy y ( , ) 100 100 400 = + + . Each motorbike is sold for N$36 000-00 and each car is sold for N$180 000-00. Use Cramer’s rule to determine the number of motorbikes and the number of cars that should be manufactured and sold for a maximum profit  and determine the maximum profit max . (10 marks) 4.2 Use the Jacobian to test for functional dependence between the cost and the revenue functions in 4.1. (7 marks) 4.3 One of the stationary points of the function ( ) 4 4 2 2 f x y x y x xy y , 2 4 2 = + − + − is ( 2, 2 − ). Use the Hessian to test whether the given point is a maximum, minimum or a saddle point.

As part of a campaign to promote its Summer Annual Clearance Sale, Cassy clothing cc. decided to buy television advertising time on Etu television. Their television advertising budget was N$102 000-00. Morning time costs N$3000-00 per minute, afternoon time costs N$1000-00 per minute, and evening (prime) time costs N$12 000-00 per minute. Because of previous commitments, Etu television could not offer Cassy clothing more than 6 minutes of prime time and a total of 25 minutes of advertising time over the two weeks in which the commercials run in the morning would be seen by 200 000 people, those run in the afternoon would be seen by 100 000 people and those run in the evening would be seen by 600 000 people. How much morning, afternoon, and evening advertising time should Cassy clothing buy to maximize exposure of its products? 

1.      A jewelry store makes necklaces and bracelets from gold and platinum. The store has 18 ounces of gold and 20 ounces of platinum. Each necklace requires 3 ounces of gold and 2 ounces of platinum, whereas each bracelet requires 2 ounces of gold and 4 ounces of platinum. The demand for bracelets is no more than four. A necklace earns $300 in profit and a bracelet, $400. The store wants to determine the number of necklaces and bracelets to make in order to maximize profit. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis.

As part of a campaign to promote its Summer Annual Clearance Sale, Cassy clothing cc. decided to buy television advertising time on Etu television. Their television advertising budget was N$102 000-00. Morning time costs N$3000-00 per minute, afternoon time costs N$1000-00 per minute, and evening (prime) time costs N$12 000-00 per minute. Because of previous commitments, Etu television could not offer Cassy clothing more than 6 minutes of prime time and a total of 25 minutes of advertising time over the two weeks in which the commercials run in the morning would be seen by 200 000 people, those run in the afternoon would be seen by 100 000 people and those run in the evening would be seen by 600 000 people. How much morning, afternoon, and evening advertising time should Cassy clothing buy to maximize exposure of its products?

A margarine factory has two machines capable of pressing sunflower seeds to oil. Together the two machines need to produce at least 900 litres of oil per day. Machine A always produce at least twice as much oil as machine B. The other processes involved in the factory determine that the two machines should produce at most 1500 litres of oil per day. The cost of producing a litre of oil from machine A and B is in the ratio 3:2. Using the graphical method determine how much oil should each machine produce at minimum cost and determine the minimum cost if the cost of producing a litre of oil from machine B is N$1-00.

(Use the graph paper. Scale: 1 cm=250 units on both axes)

A shopkeeper has a uniform demand of an item at the rate of 600 items per year. He buys from the supplier at a cost of Rs.8 per item. And the cost of ordering Rs. 12 each time. If the stock holding costs are 20% per year of stock value, how frequently should he replenish his stocks and what is the Optimal Order Quantity.

 Solve the following linear programming problem using the simplex method.

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑃 = 2100𝑦1 + 2400𝑦2 + 10𝑦3 − 70𝑦4

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜

25𝑦1 + 15𝑦2 + 𝑦3 ≥ 250

20𝑦1 + 30𝑦2 − 𝑦3 − 𝑦4 ≥ 300

𝑦1 ≥ 0, 𝑦2 ≥ 0, 𝑦3 ≥ 0 , 𝑦4 ≥ 0 

Use the simplex method to obtain the optimal solution of the dual of following linear programming model

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑃 = 70𝑥1 + 50𝑥2

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜

40𝑥1 + 30𝑥2 ≤ 2400

−20𝑥1 − 10𝑥2 ≥ 1000

𝑥1 ≥ 0, 𝑥2 ≥ 0

1.     A firm manufactures two products; the net profit on product 1 is Rupees 3 per unit and Rupees 5 per unit on product 2. The manufacturing process is such that each product has to be processed in two departments D1 and D2. Each unit of product1 requires processing for 1 minute at D1 and 3 minutes at D2; each unit of product 2 requires processing for 2 minutes at D1 and 2 minutes at D2. Machine time available per day is 860 minutes at D1 and 1200 minutes at D2. How much of product 1 and 2 should be produced every day so that total profit is maximum. Make the mathematical model for the given problem.

2.3 Solve the following linear programming graphically [5]

Minimize 𝑧 = 3𝑥 + 9𝑦

Subject to the constraints: 𝑥 + 3𝑦 ≥ 6

𝑥 + 𝑦 ≤ 10

𝑥 ≤ 𝑦

𝑥 ≥ 0; 𝑦 ≥ 0