Question

Einstein faces the decision of how many GiggleBloxx units to order for the coming holiday season. Members of the management team suggested order quantities of 25,000, 36,000, 48,000, or 65,000 units. The wide range of order quantities suggested indicates considerable disagreement concerning the market potential. The product management team asks you for analysis of the stock-out probabilities for various order quantities, an estimate of the profit potential, and help with making an order quantity recommendation. Einstein expects to sell GiggleBloxx for $14 based on a cost of $5 per unit. If inventory remains after the holiday season, Einstein will sell all surplus inventory for $3 per unit. After reviewing the sales history of similar products, Einstein's senior sales forecaster predicted an expected demand of 50,000 units with a .95 probability that demand would be between 30,000 units and 70,000 units.

1. Use the sales forecaster's prediction to describe a normal probability distribution that can be used to approximate the demand distribution. Sketch the distribution and show its mean and standard deviation.

2. Compute the probability of a stock-out for the order quantities suggested by members of the management team.

3. compute the projected profit for the order quantities suggested by the management team under three scenarios: worst case in which sales = 30,000 units, most likely case in which sales = 50,000 units, and best case in which sales = 70,000 units.

4. One of Einstein's managers felt that the profit potential was so great that the order quantity should have a 75% chance of meeting demand and only a 25% chance of any stock-outs. What quantity would be order under this policy, and what is the projected profit under the three sales scenarios?

5. Provide a recommendation for an order quantity and. note the associated profit projections.

Answer 1

Solution

Back-up Theory

Empirical rule, also known as 68 – 95 – 99.7 percent rule: applicable to Normal Distribution

P{(µ - σ) ≤ X ≤ (µ + σ)} = 0.68; ..........………………….............................................................…..……………….(1a)

P{(µ - 2σ) ≤ X ≤ (µ + 2σ)} = 0.95; ...............................……….......................................................................…….(1b)

P{(µ - 3σ) ≤ X ≤ (µ + 3σ)} = 0.997 ......................................………………….....................................................….(1c)

i.e., Mean ± 1 Standard Deviation holds 68% of the observations; ……………………....................................….(1d)

Mean ± 2 Standard Deviations holds 95% of the observations …….....................................................………….(1e)

and Mean ± 3 Standard Deviations holds 99.7% of the observations. …....................................................…….(1f).

Now, to work out the solution,

Let

X = demand

Part (1)

Vide (1b) or (1e), given, ‘expected demand of 50,000 units with a .95 probability that demand would be between 30,000 units and 70,000 units.’ =>

Mean demand µ = 50000 Answer 1

Noting that (70000 – 50000) = (50000 – 30000) = 20000,

Standard deviation of demand σ = 20000/2 = 10000 Answer 2

Part (2)

25000 = µ - 2.5σ

36000 = µ - 1.4σ

48000 = µ - 0.2σ

65000 = µ + 1.5σ

Referring to standard Normal Probability Tables, the corresponding probabilities of stock out are:

1 – 0.0062 = 0.9938 Answer 3

1 – 0.0808 = 0.9192 Answer 4

1 – 0.4207 = 0.5793 Answer 5

0.0668 Answer 6

[Stock-out occurs when demand is more than the order quantity]

Part (3)

Profit per unit sold = selling price per unit – cost per unit = (14 - 5) = 9.

Projected profit for sales of 30,000 units = 30000 x 9 = $270000 Answer 6

Projected profit for sales of 50,000 units = 50000 x 9 = $450000 Answer 7,

Projected profit for sales of 70,000 units = 70000 x 9 = $630000 Answer 8

Part (4)

Let Q be the order quantity that should have a 75% chance of meeting demand and only a 25% chance of any stock-outs. i.e., P(Demand ≤ Q) = 0.75 and P(Demand > Q) = 0.25

Then, referring to standard Normal Probability Tables, Q should be µ + 0.6745σ = 50000 + 6745 = 56745.

Thus, the order quantity should be 56745 Answer 9

Profit = 56745 x 9 = $510705 if demand is more than or equal to order quantity since in this scenario full order quantity would be sold and as we have already seen, profit per unit sold is $9. [Since stock-out cost is not given, cost of loss sales is taken as zero.]

Profit = (9 x Demand) – (56745 – Demand) x loss due to stock clearance

= (9 x Demand) – 2(56745 – Demand) if demand is less than order quantity since in this scenario only demand quantity would be sold and the left over has to be sold at a discounted price of $3 thus making a loss of $2.

So,

Projected profit for sales of 30,000 units = (30000 x 9) – (2 x 26745) = $216510 Answer 10

Projected profit for sales of 50,000 units = (50000 x 9) – (2 x 6745) = $436510 Answer 11,

Projected profit for sales of 70,000 units = 56745 x 9 = $510705 Answer 12

DONE