Question

Pete owns a small business renting out bicycles in Vancouver. He rents each bike for $5 a day. After a year, his rental bikes are worn out and they become basically worthless. So each year Pete gives his old bikes to a local charity and replaces his entire fleet of bikes. Pete buys his bikes from a local manufacturer at the discounted price of $500 each.
Pete has estimated the following probability distribution of the number of bikes he can expect to rent on any given day:

Number of bikes rented per day	10	11	12	13	14
Probability	0.10	0.25	0.40	0.20	0.05

After Pete rents out a bike, he has to clean and tune it before he can rent it out again. The average cost of this maintenance is $2 per bike. This includes the cost of materials and the opportunity cost of Pete’s time.
Pete’s store is open five days a week year round, except during a four-week period each January when he goes skiing.
How many bikes should Pete buy each year to maximize his expected profit?

Answer 1

Total number of weeks in a year are 52 out of which he goes for skiing for 4 weeks and the store is open for 5 days a week.

So, Number of days it is open in a year is (52-4)*5= 240 days

If PETE buys 10 bikes-

Total cost =10*500=5000$

Total income from rent= 240*10*5=12000$

Total maintainance cost= 240*10*2=4800$

Profit= 12000-5000-4800=2200$

If PETE buys 11 bikes-

Total cost =500*11=5500$

Total income from rent=240*5*11=13200$

Total maintainance cost= 240*11*2=5280$

Profit= 13200-5500-5280=2420

If PETE buys 12 bikes-

Total cost = 500*12=6000$

Total income from rent=240*12*5=14400$

Total maintainance cost=240*12*2=5760$

Profit=14400$-6000$-5760$=2460

If PETE buys 13 bikes-

Total cost =500*13=6500

Total income from rent=240*13*5=15600$

Total maintainance cost=240*13*2=6240

Profit=15600-6500-6240=2860

If PETE buys 14 bikes-

Total cost =500*14=7000

Total income from rent=240*14*5=16800

Total maintainance cost=240*14*2=6720

Profit=16800-6720-7000=3080

EXPECTED PROFIT FOR EACH CASE-

10 BIKES=2200*0.1=220

11 BIKES=2420*0.25=605

12 BIKES=2460*0.4=984

13 BIKES=2860*0.2=572

14 BIKES=3080*0.05=154

So, pete should buy 12 bikes to maximize expected profit.