# An Investment Planning In a barber shop business Paxton Zhou Paul Kim.

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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 Master Male Barber 2 Male Baber 1 Female Barber 4 Female Barber 3 Customers arrive Poisson (λ=3) for both male and female 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

Run FBF 10,000 times and take average, compare with L-S method 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 Master Male Barber 2 Male Baber 1 Female Barber 2 Female Barber 1 Customers arrive Poisson (λ=3) for both male and female 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 Waiting seats

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

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!. Should we have a queue/ build a waiting area?

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

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 λ=10 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