An Investment Planning

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# An Investment Planning - PowerPoint PPT Presentation

An Investment Planning. In a barber shop business Paxton Zhou Paul Kim. Basic parameters. Customers arrives according to a Poisson distribution Male and female has the same arrival rate of 3 customers per hour Processing time for each customer is uniformly distributed

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### An Investment Planning

In a barber shop business

Paxton Zhou

Paul Kim

Basic parameters
• Customers arrives according to a Poisson distribution
• Male and female has the same arrival rate of 3 customers per hour
• Processing time for each customer is uniformly distributed
• Male: [15, 75] min per customer
• Female: [21, 81] min per customer
• 5 barbers; 2 only handle male hair, 2 only handle female hair, the fifth barber (the master) can handle both male and female hair and is 2x faster than the other barbers
The no-queue scenario

Male

Baber 1

Male Barber 2

Master

Customers arrive Poisson (λ=3) for both male and female

Female Barber 3

Female Barber 4

• We assume that customers have no patience—if no barber is free they will just leave and go to another barber shop near by
• As a result, we assume immediate blocking when all “machines” are busy

List scheduling (L-S) method:

• FCFS
• When a customer arrives
• If male, assigned to male only barbers (B1, B2) first
• If no male barber available, assign to master barber (B5)
• If master barber is busy, the customer leaves for another barber shop
• If Female, assign to female only barbers first (B3, B4)
• If no female barber available, assign to master barber
• If master barber is busy, the customer leaves for another barber shop
• The barber shop opens for 8 hours. Any customer who arrives after 8 is not admitted. But barbers still need to work overtime to serve, for example, a customer who arrives at 7:50 and still need an hour to process
Run simulation 10,000 times and take average

Result of List-scheduling (L-S)

• The Average make span is (in hours): 8.666184804397565
• The Average # of Male customer served: 17.578
• The Average # Female Customer Served: 16.978
• The Average # of Total Customer Served: 34.556
Let’s think of how to improve the system to serve more customers, while minimizing the makespan
• The list-scheduling method assign customers to gender constrained barbers (barbers who only handle male or female)
• Can we improve the system by prioritizing the master barber, who is 2x faster and can handle both male and female?

Fastest Barber First (FBF) method:

• FCFS
• When a customer arrives
• If male, assigned to B5
• If B5 is busy, assign to B1 or B2, whoever is available
• If B1 and B2 are busy, the customer leaves for another barber shop
• If Female, assigned to B5
• If B5 is busy, assign to B3 or B4, whoever is available
• If B3 and B4 are busy, the customer leaves for another barber shop
• Same as above, the barber shop opens for 8 hours only
• There seems to be a very minor improvement
• We introduce another master barber B6 and see whether prioritizing fastest machines would improve the system

Compare Fastest Barber First with 2 master barbers(FBF-2) to List Scheduling with 2 master barbers (L-S-2)

• FCFS
• FBF-2: Assign new arrivals to B5 or B6 first, then to gender constraint barbers B1~4
• L-S-2: Assign new arrivals to B1~4, if these gender constraint barbers are busy, assign to B5 or B6
• Same as above, the barber shop opens for 8 hours only
Run L-S-2, FBF-2 10,000 times and take average, compare with previous methods
• There is no improvement in terms of # of customers served, actually, fewer (very little) customers are served
• There very limited reduction in makespan
• Hiring more barbers does not reduce makespan!
The queue scenario

Male

Baber 1

Male Barber 2

Master

Customers arrive Poisson (λ=3) for both male and female

Waitingseats

Female Barber 1

Female Barber 2

• We assume that customers are willing to wait for at most 30 minutes—if no barber is free then they will just leave and go to another barber shop near by
Introducing a Queue/ waiting area
• We compare the performance of L-S, FBF. L-S-2, FBF-2
• Having the queue does not make FBF any better compared to L-S
• FBF results in 2% less customers being served
Should we have a queue/ build a waiting area?
• Having a queue does not substantially increases makespan, but we can serve more customers  keep the queue!
• Do not hire one more master barber: only to serve 3~4 more customers is not worth it as an average barber serves more than 8 under the 5 barber system
• Having a queue improves the system by serving more customers, while not dramatically increasing the makespan!.

If the Poission arrival rate is very low, then whether or not do we have a queue does not make a big difference

• When a customer came in, one barber maybe idle and probably have been waiting for a long time
• But if the arrival rate is high (much higher than λ=3 per hour, i.e a holiday),:
• Having a queue will allow the barber shop to serve more customers (more arrivals will be “stored “ in the queue for later processing)
• The only drawback is that the makespan could be higher (customers in the queue may force barbers to work longer)
• We run a simulation under λ=10, and see if having a queue will benefit us. We also compare L-S to FBF

λ=10

• Having a queue allows us to serve 5 more customers per day by increasing 30mins makespan this is desirable as each customer has an average processing time of more than 30min
• L-S-2 is not desirable
• Hiring an extra master barber allows us to serve 12 more customers. This may be desirable, as originally, a master barber processes more than 12 customers we now serve more customers without suffering form diminishing return

A growing queue!

Under low arrival rate
• This is intuitive
• A queue is not required
• No need to hire an extra master barber not many customers out there to serve
• FBF is optimal (example: the shop is quite empty and most of the barbers idle, a customer arrives at t=7:50 you want to assign him/her to the available fastest barber to reduce makespan)
Our conclusion:
• Under normal arrival rate; we need to implement a queue; we do not hire an extra master barber; and either L-S or FBF is optimal
• Under high arrival rate, we need a queue; we may hire an extra barber; FBF-2 is optimal
• Under low arrival rate, a queue is not necessary; we do not need to hire an extra master barber; FBF is optimal