Question

A supplier to a music store buys compact discs at $1 per unit and sells them to the music store at $5 per unit. The music store sells each disc to the end consumer at $10. At the retail price, the market demand distribution is give as follows

Demand	Prob.
1	0.15
2	0.20
3	0.35
4	0.30

1. With no contract in place, determine how much the music store should order to maximize it’s expected profit? (8 points)

1. (4 points) What is the total expected profit of the supply chain corresponding to the order quantity calculated from question 1? Show the calculations for the expected profit at the music store and supplier.

2. Now, the supplier to the music store agrees to buyback discs that have not sold at $2.50 per disc. What is the total expected profit of the supply chain if the music store orders 4 CDs? Show the calculations for the expected profit at the music store and supplier. (7 points)

3. The supplier agrees to sell each disc to the music store at $1 instead of $5. The music store agrees to share 35 % of the revenue from each disc sold. What is the total expected profit of the supply chain if the music store orders 2 CDs? Show the calculations for the expected profit at the music store and supplier. (6 points)

Answer 1

1.

For supplier -
Cost - $1
Sell Price - $5

For music store -
Cost - $5
Sell Price - $10

For the music store, the profit is given by $5 per unit of CD Sold.

The following table demonstrates the profit obtained

CDs Ordered	CDs Sold	Profit	Probability
1	1	5	1
2	1	0	0.15
2	2	10	0.2 + 0.35 + 0.3 = 0.85
3	1	-5	0.15
3	2	5	0.2
3	3	15	0.35 + 0.3 = 0.65
4	1	-15	0.15
4	2	0	0.2
4	3	10	0.35
4	4	20	0.3

To find the profit based on the number of CDs ordered, we calculate the expectation value of the profit, conditioned on the number of CDs ordered. Therefore, the required values are -

CDs Ordered	Expected Profit
1
2
3
4

Therefore, to maximize profit, the store must order 3 CDs.

In this case, the Music store has expected profit of $10, while the supplier has profit of $12. Therefore, the supply chain profit is $22.

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2. The supplier agrees to buyback discs at $2.5 per disc. In this case, we can draw the profit table for 4CDs ordered as follows.

CDs Ordered	CDs Sold	Profit (Music Store)	Profit Supplier	Supply Chain profit	Probability
4	1				0.15
4	2				0.2
4	3				0.35
4	4				0.3

Thus, the expected profit is

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3. The supplier now agrees to CDs to the store at $1 and to sharing 35% revenue from each disc sold. Therefore now, the profit per disc sold is $9.

Now, if the music store orders 2 CDs, we can draw the following table for calculating profit

CDs Ordered	CDs Sold	Profit	Profit (Store)	Profit (Supplier)	Probability
2	1				0.15
2	2				0.85

Therefore, we have

Expected profit at music store -
Expected profit at supplier -