1 Ardavan Asef-Vaziri Jan-2011Operations Management: Waiting Lines 2 Bank of San Pedro has only 1...

1Ardavan Asef-Vaziri Jan-2011Operations Management: Waiting Lines 2

Bank of San Pedro has only 1 teller. On average, 1 customer comes every 6 minutes, and it takes the teller an average of 3 minutes to serve a customer. To improve customer satisfaction, the bank is going to implement a unique policy called, “We Pay While You Wait.” Once implemented, the bank will pay each customer $3 per minute while a customer waits in line. (So the clock starts when a customer joins the line, and stops when the customer begins to talk to the teller.) Bank of San Pedro hired you as a consultant and you are responsible for estimating how much the “We Pay While You Wait” program will cost. Your preliminary study indicates there are, on average, 0.5 customers waiting in line. Assume linear cost. If a customer waits for ten seconds in line, Bank of San Pedro will pay $0.5. Assume that arrival follows Poisson and service time follows exponential distribution.

Problem 12


a) Compute the capacity of the tellera) 10 customers/hourb) 3.33 customers/hourc) 20 customers/hourd) 30 customers/houre) Cannot be determined

It takes the teller an average of 3 minutes to serve a customer

60/3 = 20 customers per hour

Problem 12


b) Calculate the proportion of the time the teller is busy. a)100%b)80%c)62%d)50%e)40%

R=10, Rp= 20

R/Rp= 10/20 = 0.5

= 50%

Problem 12


c) How long, on average, does a customer wait in line? a)6 minutesb)4.8 minutesc)3 minutesd)2.6 minutese)2 minutes

Indeed Ii was even given in the problem.

Ti= Ii/R

Ti= 0.5/10 = .05

0.05(60) = 3 minutes

5.02

11

)5.01(

5.0 2

Ii

Problem 12


d) Calculate the expected “hourly” cost of the “We Pay While You Wait” program. a)$9b)$36c)$60d)$90e)$140

Problem 12

Ii =0.5. Therefore, a half of a customer is always there. For each hour one customer gets 60(3) = $180. Thus 0.5 customer gets $90.Perhaps you do not believe me.Each customer waits, on average, 3 minutes.

He or she receives, on average, 3(3) =$9. There are 10 customers arriving per hour. The overall cost of this program is 9*10=$90.


e) Suppose each additional clerk costs X dollars per hour (including all other clerk related costs such as benefits, space and equipment hourly costs). Compute the maximum value of X if it is at our benefit to hire one additional clerk?

Problem 12

045.0)25.01(

25.0 )12(2

Ii

Ii reduced from 0.5 customers to 0.045 customers = 0.455

If we have two clerks, Rp increases from 20 to 40, and utilization drops from 0.5 to 10/40 = 0.25


Problem 12

The number of customers waiting in the line reduced by 0.455.

It means each hour, there are 0.455 less customers waiting in line.

The cost of each hour waiting per 1 customer is $180

The waiting cost of 0.455 customers is 180(0.455) = 81.96

If the additional clerk costs less than $81.96 per hour it is at our benefit to hire her.

e) Suppose each additional clerk costs $30 per hour. How many new clerk should we hire, one or two?

Obviously, it is at hour benefit to hire one clerk.

If we hire two clerks (to have 3 clerks), Rp increases to 60, and utilization drops 10/60 = 0.167


Problem 12

Ii is reduced from 0.045 customers to 0.008, a 0.037 customer reduction.

By adding the third clerk, there are 0.037 less customers waiting in line (each hour and always)

180(0.037) = about $6-$7

It is not at our benefit to hire the second clerk, pay $30 per hour capacity cost to reduce waiting cost by $6-$7 per hour.

And we will not pay more that $1-$2 to the forth clerk.

008.0)167.01(

167.0 )13(2

Ii


We did not need to do this much computations for the third clerk. With two clerks the total number of customers waiting in line was:

Problem 12

Ii was equal to 0.045 for c =2.

Even if we reduce the number of customers in the waiting line to 0, we have reduced the line by 0.045 customers.

0.045(180) = 8.1

It is not worth the cost $30 to benefit $8


American Vending Inc. (AVI) supplies vending food to a large university. Because students often kick the machines out of anger and frustration, management has a constant repair problem. The machines break down on an average of 3/hr, and the breakdowns are distributed in a Poisson manner. Downtime costs the company $25/hr/machine, and each maintenance worker gets $4 per hr. One worker can service machines at an average rate of 5/hr, distributed exponentially; 2 workers working together can service 7/hr, distributed exponentially; and a team of 3 workers can do 8/hr, distributed exponentially. What is the optimal maintenance crew size for servicing the machines?

Problem 13


U0.6000.4290.375

Problem 13

c=????c = 1U=?? U

UIi

c

1

)1(2

Members in a team Demand Capacity1 3 52 3 73 3 8

Ii0.900.320.23

Capacity Cost48

12

Down Time Cost22.58.05.6

Down time cost = 25Ii

Capacity Cost= 4(# of team members)

Total Cost

Total Cost26.516.017.6

U

U

1

2

2


Have I made any mistakes?Downtime costs the company $25 /hr/machine.When the machine is down?Until it is up.In the waiting line it is down. In the processor until the end of the process it is down. There fore, besides Ii, I also need Ip

Problem 13

Members in a team Demand Capacity U Ii

Waiting Cost

1 3 5 0.600 0.90 22.502 3 7 0.429 0.32 8.043 3 8 0.375 0.23 5.63

In Process Cost15.0010.719.38

Labor Cost Total Cost

4 41.508 26.7512 27.00


Lets check by using Ti and Tp instead of Ii and Ip

Problem 13

Members in a team Demand Capacity U Ii Ti

1 3 5 0.600 0.90 0.302 3 7 0.429 0.32 0.113 3 8 0.375 0.23 0.08

Waiting Cost

/Machine7.502.681.88

Waiting Cost

/Hour22.508.045.63

Processing time (hr) / Machine

0.200.140.13

In Process

Cost /Machine

5.003.573.13

In Process

Cost /hour15.0010.719.38

Labor Cost

Total Cost

4.00 41.508.00 26.7512.00 27.00

1 Ardavan Asef-Vaziri Jan-2011Operations Management: Waiting Lines 2 Bank of San Pedro has only 1...

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