1. David buys fruits and vegetables wholesale and retails them at David\'s Produ

Question

1. David buys fruits and vegetables wholesale and retails them at David's Produce on La Vista Road. One of the difficult decisions is the amount of bananas to buy. Let us make some simplifying assumptions, and assume that David purchases bananas once a week at 20 cents per pound and retails them at 50 cents per pound during the week. Bananas that are more than a week old are too ripe to sell and David will pay workers to take them away. It costs 2 cent to get rid of each pound of unsold bananas. Suppose that the weekly demand for bananas is uniformly distributed between 500 and 1500 pounds. (a) Find the 0.58 quantile of the weekly demand for bananas. (b) If you order the amount you found in part (a), what is the expected weekly profit? (c) Now assume that the demand for bananas is exponentially distributed with mean 1000 and 860 pounds are ordered. Find the 0.58 quantile of the weekly demand for bananas.

Explanation / Answer

Here the weekly demand is uniformly distributed between 500 and 1500 pounds.

Purchase price =20 cents per pound

Selling price = 50 cents per pound

Salvage price for bananas = 2 cents per pound

(a) Here if X is the weekly demand for bananas in a given week.

then as it is uniformly distributed

Pr(X) = 1/(1500 - 500) = 1/1000 ; 500 < X < 1500

F(X) = (X - 500)/1000 ; 500 < X < 1500

so, here we have to find 0.58 quartile

so,

(X - 500)/1000 = 0.58

X = 1000 * 0.58 + 500 = 1080 pounds

(b) Expected weekly profit when we order 1080 pounds and as we know the expected demand would be 1000 pounds

Here expected profit = 1000 * 0.50 - 1080 * 0.20 - (1080 - 1000) * 0.02 = $ 282.4

(c) Here now the exponential demand with mean = 1000 pounds

so if X is the weekly demand of any given week.

f(x) = (1/1000) e-x/1000 = 0.001 e-x/1000

F(X) = 1 - e-x/1000

so for 0.58 quartile

0.58 = 1 - e-x/1000

e-x/1000 = 0.42

x = 867.50

Related Questions

Navigate

• Browse (All)
• Browse 1
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.