Question 2 (23 marks) A farmer has a 250-acre farm in the New Territories on whi

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Question 2 (23 marks) A farmer has a 250-acre farm in the New Territories on which he is to plant two kinds of flowers-mums and gladioli for three months These flowers are to be sold during the Lunar New Year Fairs. For each acre of mums and gladioli planted, the aggregate expenses are $1,000 and $2,000 respectively. The harvest of each acre of mums planted requires 200 square feet of storage and can earm a profit of $300, and the harvest of each acre of gladioli planted requires 80 square feet of storage and yields a profit of $450. The storage house beside the fam is 42,000 square feet in area. The farmer has $400,000 available as start- up capital to run the business. The model is formulated and input to an Excel spreadsheet to obtain the optimal solution using 'Solver'. The corresponding linear programming sensitivity report is given in the following table Adjustable cells Final value 100 150 Reduced Objective Allowable Allowable Name Mums coefficient increase decrease 150 150 cost 300 450 75 150 Constraints Final value Shadow Constraint Allowable Allowable R H. side increase Name Farm area Capital Storage space decrease 50 150 0.15 250 31.25 100000 83333.33333 E+30 32000 10000 a If the price of gladioli drops and the expected profit changes to 330 with each acre of gladioli planted, would the optimal solution obtained change and what is the effect on the total profit? (4 marks) b There is a resort beside the farm. If the resort would like to rent 20 acres of land from the farmer for three months at a price of $200 per acre, should the farmer accept the offer? Interpret the respective shadow price to explain why or why not. (4 marks) If the famer is able to increase his start-up capital to $490,000 by borrowing S90,000 from his friend and promising to pay her back $100,000 after the Lunar New Year, would it be worth doing so? Why or why not? What is the new total profit that the farmer can earn if he decided to borrow the money? c (7 marks) d Suppose that just before the farmer is to start planting, the price of fertilizer rises sharply and the profits with each acre of mums and gladioli planted are now $260 and $400 respectively. Without solving the linear programming model again, comment briefly on the consequences for the total profit brought by the changes. (8 marks)

Explanation / Answer

a.

If the price of gladioli drops and the expected profit changes to $330 from 450 with each acre of gladioli planted, the optimal solution obtained does not change as the allowable increase of the objective coefficient of Gladioli is 150. The total profit will be reduced by 150 * 120 = $18,000

b.

The shadow price of the farm area constraint is 150. It denotes that one acre change in the right hand side will change the profit by $150 and this change is valid within the allowable increase and decrease. If the farmer gives 20 acres for rent to the resort, the expected profit from farming will reduce by 20 * 150 = $3,000. The rent he will receive is 20 * 200 = $4000 for three months. At a net of he will earn $1,000 more by giving rent. So he can give 20 acres for rent to the resort.

c.

The allowable increase for the right hand side of the capital constraint is $100,000 and the shadow price is 0.15. It indicates that change in capital by $1 will change the profit by $0.15 and it is valid for increase upto $100,000. So adding $90,000 to the capital will give additional profit of 0.15 * 90,000 = $13,500. It is higher than the additional amount $10,000 he has to pay to his friend after three months. So it is worth to borrow $90,000 from his friend and promising to pay her back $100,000 afte the Lunar New year. The new total profit that the farmer can earn if he decided to borrow the money is (300*100) + (450 * 150) + 13,500 = $111,000.

d.

If the profits with each acre of mums and gladioli planted are now $260 and $400 respectively for per acre, the optimal solution will not change as the change in the coefficients are within the allowable decrease respectively. The total profit will get reduced as a consequence of reduction in the individual profits.

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