Question

Florence's Flowers sells bouquets of flowers near the entrance to a subway station. Florence must purchase fresh flowers each morning, at a cost of $5 per bouquet. She sells them for $20 over the course of the day. At the end of the day, she must dispose of the unsold bouquets by selling them to a discount florist for a price of $2.

a) What service level will maximize her expected profits?

Florence's best estimate of the distribution of demand is shown in the table below.

Demand          Probability

51                       0.1

52                       0.1

53                       0.1

54                       0.1

55                       0.1

56                       0.1

57                       0.1

58                       0.1

59                       0.1

60                       0.1

b) What is the average demand (expected value of demand)?

c) What order quantity will maximize her expected profits?

d) Florence has conducted some research to generate a better forecast of demand. She now believes that daily demand is normally distributed with a mean of 55 and a standard deviation of 2. Now, what order quantity will maximize her expected profits?

Answer 1

(a)

Cost of underage, Cu = 20 - 5 = 15
Cost of overage, Co = 5 - 2 = 3

So, Critical ratio = Cu / (Co + Cu) = 15 / (3+15) = 0.8333

So, the service level at which profit is maximized is 83.33%

(b)

Average (expected) demand = 0.1 * (51+52+53+54+55+56+57+58+59+60) = 55.5

(c)

Demand	Prob	CumProb
51	0.1	0.1
52	0.1	0.20
53	0.1	0.30
54	0.1	0.40
55	0.1	0.50
56	0.1	0.60
57	0.1	0.70
58	0.1	0.80 < 0.833
59	0.1	0.90 > 0.833
60	0.1	1.00
61	0.1	1.10

Note that the cumulative probability crosses the critical ratio at Demand = 59. So, the optimal order quantity is 59

(d)

The value of z corresponding to the optimum service level = normsinv(0.833) = 0.966

Optimal order quantity = Mean + z* SD = 55 + 0.966*2 = 56.9 i.e. 57 units.