Discussion Problem

We begin an adventure into some more complex modeling using Simio. Discuss some of the intermediate modeling techniques you intend to use in your final project, and discuss how you will use them.

According to the synopsis of Curb Your Enthusiasm, Season 6, Episode 3, The Ida Funkhouser Roadside Memorial:

Waiting online at the perfume store, Larry grows anxious and tries to second guess which line will move faster, switching back and forth with another man. But the woman in front of him hold things up when she tries different samples, and the man who’d been behind him, moves to the front of the other line and snags the last bottle of Cheryl’s favorite perfume. Enraged, Larry causes a scene in the store.

This video was referenced in an Operations Management course for its comedic value in relation to queuing theory. Although, this sitution is too simple to model for the final project, it does highlight some interesting points to consider when modeling a system which I hope to address in my final project:

Jockeying: When the customer enters one line and then switches to a different one in an effort to reduce the waiting time.
Reneging: The customer enters the line but decides to leave before being served.
Balking: The customer decides not to enter the waiting line.
Faffing: The time delay when a person gather his things after paying at the checkout.

Other intermediate modeling techniques that I intend to use are Tally Statistics, SMORE Plots, Input Analysis, and Comparing Multiple Alternative Scenarios.

Assignment Problems

Kelton Chapter 5 # 6, 7, 8, 9.

Problem 5.7.6

In Model 5-2, we assumed that the inspection workers did not overlap between shifts (I.e., there were never two inspectors working at the same time). Modify the model so that the inspectors do overlap for the first hour of one shift and last hour of the other shift. Compare the results of the two models. Does having the inspectors overlap help?

No Overlap	Overlap

Added shift of 12:00 PM to 1:00 PM with a value (capacity) of 2 to the DayPattern1 under the Data tab. The change did not appear to make a statistically significant difference in TIS, WIP, or NumTimes.

Problem 5.7.7

In the description of Model 5-2, we noted that there is no limit on the number of times that a board may be found in need of rework. Modify Model 5-2 so that if a board fails the inspection more than two times, it is rejected as a bad part. Count the number of boards that are rejected because of 3 or more failures of inspection.

Model	Boards Rejected
	Maxmium Times Processed

Modified the respective link weights of 0.66, 0.08, and 0.26 for good, bad, and rework inspection outcomes to:

\((\textrm{ModelEntity.TimesProcessed} < 3) * 0.66\)
\((\textrm{ModelEntity.TimesProcessed} < 3) * 0.08\)
\((\textrm{ModelEntity.TimesProcessed} < 3) * 0.26\)

Then added a Reject sink connected the inspection server with the link weight expression \((\textrm{ModelEntity.TimesProcessed} == 3)\). The results show no boards were processed more than thrice, and that the new Reject server rejected 9,840.0000 boards.

Problem 5.7.8

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.

\[T_s = \frac{a+c+b}{3} = \left\{ \frac { 8+9+10 }{ 3 } ,\frac { 10+12+14 }{ 3 } ,\frac { 12+14+16 }{ 3 } \right\} =\left\{ 9,12,14 \right\}, \quad p_\textrm{fail} = \left\{ \frac { 30 }{ 3 \left( 60 \right) } ,0,0 \right\} =\left\{ \frac { 1 }{ 6 } ,0,0 \right\}\]
\[\mu = \frac{N}{T_s}\left(1-p_\textrm{fail}\right) = \left\{ \frac { 1 }{ 9 } \left( 1-\frac { 1 }{ 6 } \right) ,\frac { 1 }{ 12 } \left( 1-0 \right) ,\frac { 1 }{ 14 } \left( 1-0 \right) \right\} =\left\{ \frac { 1 }{ 9 } \left( \frac { 5 }{ 6 } \right) ,\frac { 1 }{ 12 } \left( 1 \right) ,\frac { 1 }{ 14 } \left( 1 \right) \right\} =\left\{ \frac { 5 }{ 52 } ,\frac { 1 }{ 12 } ,\frac { 1 }{ 14 } \right\} \]
\[\frac { \mu }{ \sum { \mu } } =\left\{ \frac { \frac { 5 }{ 52 } }{ \frac { 5 }{ 52 } +\frac { 1 }{ 12 } +\frac { 1 }{ 14 } } ,\frac { \frac { 1 }{ 12 } }{ \frac { 5 }{ 52 } +\frac { 1 }{ 12 } +\frac { 1 }{ 14 } } ,\frac { \frac { 1 }{ 14 } }{ \frac { 5 }{ 52 } +\frac { 1 }{ 12 } +\frac { 1 }{ 14 } } , \right\} =\left\{ \frac { 5 }{ 52 } \left( \frac { 546 }{ 137 } \right),\frac { 1 }{ 12 } \left( \frac { 546 }{ 137 } \right),\frac { 1 }{ 14 } \left( \frac { 546 }{ 137 } \right) \right\} =\left\{ \frac { 105 }{ 274 } ,\frac { 91 }{ 274 } ,\frac { 39 }{ 137 } \right\}\]

N <- 1
a <- c(8, 10, 12)
c <- c(9, 12, 14)
b <- c(10, 14, 16)
time_s <- (a + c + b) / 3
p_fail <- c(30 / (3 * 60), 0, 0)
mu <- (N / time_s) * (1 - p_fail)
mu / sum(mu)

## [1] 0.3743316 0.3368984 0.2887701

Problem 5.7.9

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, \(\lambda_1\) is the arrival rate for the fax-in prescriptions, and \(\lambda_2\) is the arrival rate for the fax-in customers.

Time Period	\(C\)	\(P\)	\(\lambda_1\)	\(\lambda_2\)
8:00 a.m. - 11:00 a.m.	1	2	10	12
11:00 a.m. - 3:00 p.m.	2	3	10	20
3:00 p.m. - 7:00 p.m.	2	2	10	15
7:00 p.m. - 10:00 p.m.	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	Work Schedules	Rate Tables
Results

Two entities (Fax, Person), two sources (Fax-in, Walk-in), and two servers (Cashier Pharmacist). There are a series of three 0.68 Selection Weight connectors from the Fax-in source to the Pharmacist, then the Cashier, and finally the sink. There are a series of three (1-0.68) Selection Weight connectors from the Walk-in source to the Cashier, then the Pharmacist, and finally the sink. Capacity Type was set as WorkSchedule for \(C\) and \(P\) by entering the schedules under Data > Work Schedules. Arrival Mode was set as Time Varying Arrival Rate for \(\lambda_1\) and \(\lambda_2\) by entering the schedules under Data > Rate Tables. The performance metrics of interest produced by the model were as follows:

Pharmacist.InputBuffer.Contents.AverageTimeWaiting = 7.24253 minutes
ModelEntity1.Population.TimeInSystem.Average = 46.2844 minutes
Cashier.Capacity.Utilized.Average = 0.91735
Pharmacist.Capacity.Utilized.Average = 0.801275

RxFillTIme	TimeInSystem	UtilizationC	UtilizationP

Intermediate Modelling with Simio

Jose Zuniga