Operations Research Answers

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

A pharmacy has determined that a healthy person should receive 70 units of proteins, 100 units of carbohydrates and 20 units of fat daily. If the store carries the six types of health food with their ingredients as shown in the table below, what blend of foods satisfies the requirements at minimum cost to the pharmacy? Make a mathematical model for the given problem

Foods Protein units Carbohydrates units Fat units Cost per unit

A 20 50 4 2

B 30 30 9 3

C 40 20 11 5

D 40 25 10 6

E 45 50 9 8

F 30 20 10 8

A farmer has 10 acres to plant in rice and corn. He has to plant at least 7 acres. However, he has only PhP1200 to spend and each acre of rice costs PhP200 to plant and each acre of corn costs PhP100 to plant. Moreover, the farmer has to get the planting done in 12 hours and it takes an hour to plant an acre of rice and 2 hours to plant an acre of corn. If the profit is PhP500 per acre of rice and PhP300 per acre of corn, how many acres of each should be planted to maximize profits?

A trust fund is planning to invest up to PhP6000 in two types of bonds: A and B. Bond A is safer than bond B and carries a dividend of 8 percent, and bond B carries a dividend of 10 percent. Suppose that the fund's rules state that no more than PhP4000 may be invested in bond B, while at least PhP1500 must be invested in bond A. How much should be invested in each type of bond to maximize the fund's return?

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A manufacturer produces two different models -x and y of the same product.Model x makes contribution of Rs 50 per unit and model y Rs 30 per unit towards total profit.Raw materials r1 and r2 are required for production.At least 18kg of r1 and 12kg of r2 must be used daily.Also at most 34hours of labour are to be utilised.A quantity of 2kg of r1 is needed for model x and 1kg of r1 for model y.For each of X and Y 1kg of r2 is required.It takes 3 hours to manufacture model x and 2hrs to manufacture y. How many units should be produced to maximise the profit?