Question

An online store would like to decrease the inventory level of its warehouse by selling the old model of a particular mobile phone. They cannot cut down the price directly because the contract that they signed with the supplier does not allow it. Therefore, they plan to give coupons to the customers to buy the mobile phone with 20% discount of the original price. The original price is $8,000 (it means that the customers can buy the mobile phone with the price of $6,400 each by using the coupons) and assume that no one will buy it with the original price since it is too expensive. For this model of mobile, there are 2,000 units in the warehouse now and the store planned to give more than 2,000 coupons to the public since some of them may not be used. Assume that all the coupons will be taken by the public. The procurement cost of each mobile phone for the store is $5,000. The store has no duty to provide mobile phones to the coupon holders; i.e., if the phones are out of stock and a customer comes to the store with a coupon and orders a phone, the store is not required to fulfill the order.

The forecasted number of unused coupons is as follow (assume that it can be the values listed below only, i.e. cannot be 150):

Number of not used coupons Probability

0 5%

100 5%

200 5%

300 10%

400 15%

500 20%

600 15%

700 10%

800 5%

900 5%

1000 5%

a) If the store gives out 2,300 coupons, what is the probability that there are customers who would like to use the coupons, but the store cannot honor them?

b) If the store gives out 2,600 coupons, what is the probability that the store cannot sell all the mobile phones?

c) If the store cannot sell the mobile phones, they have to return them to the supplier with a refund of $3,000 each. If the customers would like to use the coupons, but the store cannot honor them, then the store will encounter a loss of goodwill which costs the store $800 on average for each mobile phone. What is the optimal number of coupons that the store should give out in order to maximize the revenue?

d) Referring to part c, if the store decides to give out 2,200 coupons, find the expected revenue.

Answer 1

a)

Giving out 2,300 coupons mean 300 coupons extra.

Not used	P(Q)	Additional coupons	Actual sales	Bumped
0	0.05	300	2000	300
100	0.05	300	2000	200
200	0.05	300	2000	100
300	0.10	300	2000	0
400	0.15	300	1900	0
500	0.20	300	1800	0
600	0.15	300	1700	0
700	0.10	300	1600	0
800	0.05	300	1500	0
900	0.05	300	1400	0
1000	0.05	300	1300	0

So, the probability that a customer comes with a coupon and finds no product = 0.05+0.05+0.05 = 0.15

b)

Giving out 2,600 coupons mean 600 coupons extra.

Not used	P(Q)	Additional coupons	Actual sales
0	0.05	600	2000
100	0.05	600	2000
200	0.05	600	2000
300	0.10	600	2000
400	0.15	600	2000
500	0.20	600	2000
600	0.15	600	2000
700	0.10	600	1900
800	0.05	600	1800
900	0.05	600	1700
1000	0.05	600	1600

So, the probability that not all phones are sold = 0.10+0.05+0.05+0.05 = 0.25

c)

Cost of underage, C_u = Cost of one less additional coupon than unused = Cost of one lost sales = 5000 - 3000 = 2000
Cost of overage, C_o = Cost of one more additional coupon than unused = 800

So, optimal in-stock probability = C_u / (C_o+C_u) = 2000/(2000+800) = 0.714

Q	P(Q)	F(Q)
0	5%	5%
100	5%	10%
200	5%	15%
300	10%	25%
400	15%	40% < 0.714
500	20%	60% < 0.714
600	15%	75% > 0.714
700	10%	85%
800	5%	90%
900	5%	95%
1000	5%	100%

So, the optimal number of coupons to give out = 2000+600 = 2,600

d)

If 2200 coupons are distributed, then additional coupons = 200.

Q	P(Q)	Additional coupons	Actual sales	Lost goodwill	Profit from sales	Cost of goodwill	Net profit
0	0.05	200	2000	200	$2,800,000	$160,000	$2,640,000
100	0.05	200	2000	100	$2,800,000	$80,000	$2,720,000
200	0.05	200	2000	0	$2,800,000	$0	$2,800,000
300	0.10	200	1900	0	$2,660,000	$0	$2,660,000
400	0.15	200	1800	0	$2,520,000	$0	$2,520,000
500	0.20	200	1700	0	$2,380,000	$0	$2,380,000
600	0.15	200	1600	0	$2,240,000	$0	$2,240,000
700	0.10	200	1500	0	$2,100,000	$0	$2,100,000
800	0.05	200	1400	0	$1,960,000	$0	$1,960,000
900	0.05	200	1300	0	$1,820,000	$0	$1,820,000
1000	0.05	200	1200	0	$1,680,000	$0	$1,680,000
					Expected profit		$2,347,000