Question

2. David buys fruits and vegetables wholesale and retails them at Davids 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 1 cent to get rid of each pound of unsold bananas. Suppose that the weekly demand for bananas is uniformly distributed between 1000 and 5000 pounds. (a) How many pound of bananas should David order each week? (b) What is the optimal expected weekly profit? (c) Now assume that the demand for bananas is exponentially distributed with mean 3000. How many pound of bananas should be ordered? What is the optimal expected weekly profit? (d) Now suppose that the demand was normally distributed with mean 3000 pounds and variance 500 pounds2. How many pound of bananas should be ordered?

Answer 1

Here the week by week request is consistently dispersed somewhere in the range of 500 and 1500 pounds.

Price tag =20 pennies per pound

Moving cost = 50 pennies for every pound

Rescue cost for bananas = 1 pennies for each pound

(a) Here if X is the week by week interest for bananas in a given week.

at that point as it is consistently appropriated

Pr(X) = 1/(5000 - 1000) = 1/4000 ; 1000 < X < 5000

F(X) = (X - 1000)/4000 ; 1000 < X < 5000

thus, here we need to discover 0.58 quartile

thus,

(X - 1000)/4000 = 0.58

X = 4000 * 0.58 + 1000 = 3320 pounds

(b) Expected week by week benefit when we arrange 3320 pounds and as we probably am aware the normal interest would be 4000 pounds

Here expected benefit = 4000 * 0.50 - 3320 * 0.20 - (3320 - 4000) * 0.02 = $ 1349.6

(c) Here now the exponential interest with mean = 3000 pounds

so if X is the week after week request of some random week.

f(x) = (1/3000) e-x/3000 = 0.0003 e-x/3000

F(X) = 1 - e-x/3000

so for 0.58 quartile

0.58 = 1 - e-x/3000

e-x/3000 = 0.42

x=