Case Study HW

Instructions: Read the following case study and answer the following questions. Your answers must be accurate and 100% original.

Case Study: National Capital Diary

Nelson Family Farms of Ottawa has a herd of 150 milk cows. At any given time 120 cows are actively producing milk. The herd consists entirely of Holsteins, the breed that comprises about 95% of milk-producing cows in Canada. Average daily milk production is 27 liters per cow. Two enterprising young graduates of Telfer school, Elizabeth and Jeff, who are looking for a business opportunity, establish National Capital Diary, a local diary that, initially, will process raw milk and resell it in 1-liter containers. They have signed an exclusive contract with Nelsons family farm to purchase all raw milk at a price of $72 per hectoliter. (1 hectoliter (hL) =100 liters) Raw milk is 3.5% cream and can be processed (by pasteurization, etc.) to produce whole milk (3.5% cream), reduced fat milk (2% cream), and skim milk (a negligible trace of cream). Each hectoliter of whole milk will contain 3.5 liters of cream and 96.5 liters of skim milk. Similarly, a hectoliter of 2% milk will contain 2 liters of cream and 98 liters of skim milk. National Capital Diary has surveyed local area merchants ad found that it can sell whole milk in 1-liter containers at $1.10 per liter, 2% milk at $1.05 per liter, and skim milk at $0.95 per liter. After separating the milk from the cream for the production of skim milk and 2% milk, the diary will have residual unused cream that it can sell at $2.00 per liter. Elizabeth and Jeff have already decided that all the available raw milk purchased from the Nelson family farm will be processed every day. They now have to decide how many 1-liter containers of each type of milk, including cream, to package in order to maximize daily profit. A survey of local demand for milk reveals that the diary should produce at least 400 liters each of whole milk and 2% milk, and at least 200 liters of skim milk daily. However the amount of 2% milk should not exceed 500 liters daily. Production and demand constraints also require that the amount of 2% milk produced should not exceed twice the quantity of whole milk. Also, the amount of skim milk produced should be equivalent to at least two-thirds of the amount of 2% milk.

Questions:

a. Formulate an LP model to determine the optimal production quantities for the various products. (Hint: define the following decision variables :

1. XW = the number of litres of whole milk to produce

2. XR = the number of litres of reduced fat (%2) milk to produce

3. XS = the number of litres of skim milk to produce

4. XC = the number of litres of cream to produce)

b. Specify the optimal solution. How many 1-liters containers of each type of milk should National Capital Diary produce daily? How many 1-liter containers of cream should be produced? what is the diaries maximum daily revenue ? What is the daily profit after subtracting the cost of purchasing the raw milk from the Nelsons farm?

c. If one of the milk-producing cows at Nelsons farm cannot produce milk for several days, how will this affect National Capital Diary’s profit for each day that the cow is not producing milk?

d. If the minimum 2% milk requirement is increased by 20%, determine the profit.

e. The selling price for skim milk goes up by $0.10. How will this affect the optimal solution? What is the new value of profit?

f. The minimum requirement for skim milk is 200 liters. How will a change in this lower limit affect profit? Specify the change in profit per unit in the right hand side of the constraint, and also the range within which this change in profit is valid. Explain your conclusion in your own words.

g. What percentage of total daily revenue can be attributed to either 2% or skim milk?

h. A neighboring farm that specializes in the production of butter uses all its cream to manufacture butter but has a surplus of skim milk. It offers to sell 400 liters of skim milk to National Capital Diary at $0.90 a liter. Should National Capital Diary accept this offer? How would revenue be affected?