Data604 - Assignment# 7

Problem 6 page 179

In model 5.2, we assumed that the inspectors schedules did not overlap. We will now modify this model by having the shifts overlap as follows:

First Shift: 8:00 AM - 12:00 PM, 12:00 PM - 1:00 PM meal break, 1:00 PM - 5:00 PM

Second Shift: 4:00 PM - 8:00 PM, 8:00 PM - 9:00 PM meal break, 9:00 PM - 1:00 AM

Third Shift: 12:00 AM - 4:00 AM, 4:00 AM - 5:00 AM meal break, 5:00 AM - 9:00 AM

We will set up the schedule with overlapping shifts by having value of 2 in the hour that overlap.

Model 5.2

Schedule Setting

We will run this model with 25 repetitions, 25 days warm-up for 125 days and compare prior model (5.2) with this new one.

Results with Schedule overlap

Results without Scedule overlap

The steady-state average time for a boards in the system is slighly less for the new model and the utilization percentage is less as well. This would be as expected since the queue at inspection during meal time will be absorbed quicker.

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Problem # 7 page 179

We will now modify model 5.2 so that when board failed inspection more than 2 times it will be be routed to the bad lot.

We will track the number of times a board pass through rework, once boards has gone through reworks twice, if the board failed instpection again it will be discarded and sent to a reject sink.

We will start by creating a state variable (NumberRework) associated with the Server Rework. We will also create a tally statistic to keep track of this number.

As per the book example, we have the following ratio for good part=0.66, bad part=0.08, and rework=.26, we will change each weight as follows: good part = 0.66 x (NumberRework < 3), bad part = 0.08 x (NumberRework < 3), and rework = 0.26 x (NumberRework < 3). The condition for reject sink will be NumberRework == 3.

Model

Results of Experiment

From the results, we have no more than 3 rework (maximum = 3) and the rejet sink processed 2,018.7 parts.

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Problem 8 page 180

In the description of Model 5-3, we indicated that as part of our model verification, we had predicted the proportions of parts that would go to the fast, medium, and slow fine-pitch placement machines (38%, 33%, and 29%; respectively). Develop a queuing model to estimate these proportions.

The fast, medium, and slow fine-pitch machines have the processing time defined by triangular distributions of triangular(8,9,10)
triangular(10, 12, 14)
triangular(12, 14, 16), respectively.

In addition, the fast pitch machine is subject to random failure with uptime being exponentially distributed with with mean of 3 hours and repair times being exponentially distributed with a mean of 30 minutes. The medium and slow machine do not fail.

We will compute the average processing time for each machine. This will indicate how many part would each machine be able to process in a given time. As long as the machine is available, part have the same probability to get routed to each of the machine.

Since the machine have processing time given by triangular distribution (a, b, c), the mean would be given by the following:

\({ T }_{ s }=\frac { a\quad +\quad b\quad +\quad c }{ 3 }\)

All calculation will be done in R.

In addition the probability that fast machine failed is given by:

\({ p }_{ f }=\frac { 30 }{ 3\times (60) } =\frac { 1 }{ 6 }\)

All caculations will be done in R.

fast <- c(8, 9, 10)
medium <- c(10, 12, 14)
slow <- c(12, 14, 16)  
p_fail <- 1/6

time_fast <- sum(fast)/3
time_slow <- sum(slow)/3
time_medium <- sum(medium/3)

mu_fast <- (1/time_fast)*(1-p_fail)
mu_slow <- 1/time_slow
mu_medium <- 1/time_medium

mu <- c(mu_fast, mu_medium, mu_slow)

mu/sum(mu)

## [1] 0.3743316 0.3368984 0.2887701

Problem 9 page 180

Consider a pharmacy where customers come to have a prescription filled. Customers can either have their doctor fax their prescriptions ahead of time and come at a later time to pick up their prescriptions or they can walk in with the prescriptions and wait for them to be filled. Fax-in prescriptions are handled directly by the pharmacist who fills the prescriptions and leaves the filled prescriptions in a bin behind the counter.

Historical records show that approximately 69% of customers have faxed their prescriptions ahead of time and the times for a pharmacist to fill a prescription are triangularly distributed with parameters (2,5,8) minutes. If an arriving customer has faxed his/her prescription in already, a cashier retrieves the filled prescription from the bin and processes the customer’s payment. Historical records also estimate that the times required for the cashier to retrieve the prescriptions and process the payment are triangularly distributed with parameters (2,4,6) minutes. If an arriving customer has not faxed the prescription ahead of time, the cashier processes payment and sends the prescriptions to a pharmacist, who fills the prescription. The distributions of the cashier times and pharmacist times are the same as for the fax-in customers (triangular (2,4,6) and triangular (2,5,8), respectively). Fax-in and customer arrival rates vary during the day as do the pharmacy staffing levels. [The below table] gives the arrival and staffing data where C is the number of cashiers, P is the number of pharmacists, ??1 is the arrival rate for the fax-in prescriptions, and ??2 is the arrival rate for the fax-in customers.

Time-Period	C	P	??1	??2
8:00 am - 11:00 am	1	2	10	12
11:00 am - 3:00 pm	2	3	10	20
3:00 pm - 7:00 pm	2	2	10	15
7:00 pm - 10:00 pm	1	1	5	12

Develop a Simio model of this pharmacy. Performance metrics of interest include the average time fax-in prescriptions take to be filled, the average time customers spend in the system, and the scheduled utilizations of the cashiers and pharmacists. Assume that the Pharmacy opens ar 8:00 a.m. and closes at 10:00 p.m. and you can ignore faxes and customers that are still in the system at closing time (probably not the best customer service!). Use 500 replications for your analysis and generate the SMORE plots for the performance metrics of interest.

Model

We will use weight (.59) and conversely (.41) to direct the Fax-in vs walk-in entities. The arrival rate for each will be change based on time of day and will be specified using rate tables. The Capaciy of the pharmacist and cashiers as well as they work-hours will be specified on Workschedules.

WorkSchedules

Model

Schedule

Rate

Experiment

Pharmacy Utilization

Cashier Utilization