## 3.8 Exercises

Exercise 3.1 True or False: The Time Between option of the RECORD module will calculate and record the difference between a specified attribute’s value and the current simulation time.

Exercise 3.2 Provide the missing information for steps for the modeling questions.
• What is the $$\underline{\hspace{3cm}}$$?

• What are the elements of the $$\underline{\hspace{3cm}}$$?

• What information is known by the $$\underline{\hspace{3cm}}$$?

• What are the required performance $$\underline{\hspace{3cm}}$$?

• What are the $$\underline{\hspace{3cm}}$$ types?

• What information must be recorded or remembered for each $$\underline{\hspace{3cm}}$$ instance?

• How are entities ($$\underline{\hspace{3cm}}$$ instances) introduced into the system?

• What are the $$\underline{\hspace{3cm}}$$ that are used by the entity types?

• Which entity types use which $$\underline{\hspace{3cm}}$$ and how?
• What are the process flows? Sketch the process or make an $$\underline{\hspace{3cm}}$$

• Develop $$\underline{\hspace{3cm}}$$ for the situation

• Implement the model in Arena

Exercise 3.3 The method of $$\underline{\hspace{3cm}}$$ assumes that a random sample is formed when repeatedly running the simulation.

Exercise 3.4 Indicate whether or not the statistical quantity should be classified as tally-based or time-persistent.
Classification Description
The number of jobs waiting to be processed by a machine
The number of jobs completed during a week
The number in customers in queue:
The time that the resource spends serving a customer
The number of items in sitting on a shelf waiting to be sold
The waiting time in a queue
The number of patients processed during the first hour of the day
The time that it takes a bus to complete its entire route
The number of passengers departing the bus at Dickson street
The amount of miles that a truck travel empty during a week

Exercise 3.5 What module is used to compute statistical results across replications?

$$\underline{\hspace{3cm}}$$

Exercise 3.6 True or False A replication is the generation of one sample path which represents the evolution of the system from its initial conditions to its ending conditions.

Exercise 3.7 In a $$\underline{\hspace{3cm}}$$ horizon simulation, a well defined ending time or ending condition can be specified which clearly demarks the end of the simulation.

Exercise 3.8 True or False We use the Arena function, TAVG(), for writing out statistics on the average of observation-based variables.

Exercise 3.9 Which of the following are finite horizon situations? Select all that apply.
1. Bank: bank doors open at 9 am and close at 5 pm
2. Military battle: simulate until force strength reaches a critical value
3. A factory where we are interested in measuring the steady state throughput
4. A hospital emergency room which is open 24 hours a day, 7 days a week

Exercise 3.10 Compute the required sample size necessary to ensure a 95% confidence interval with a half-width of no larger than 30 minutes for the total time to produce a part. Given the half-width of the system time after 10 runs is 47.25, compute the sample size using the half-width ratio method. Round the answer to the next highest integer.

Exercise 3.11 Compute the required sample size necessary to ensure a 95% confidence interval with a half-width of no larger than 30 minutes for the total time to produce a part. Given the half-width of the system time after 10 runs is 47.25. Find the approximate number of replications needed in order to have a 99% confidence interval that is within plus or minus 2 minutes of the true mean system time using the half-width ratio method.

Exercise 3.12 Assume that the following results represent the summary statistics for a pilot run of 10 replications from a simulation for the system time in minutes. $$\overline{x} = 78.2658 \; \pm 9.39$$ with confidence 95%. Find the approximate number of additional replications in order to have a 99% confidence interval that is within plus or minus 2 minutes of the true mean system time.

Exercise 3.13 Suppose $$n=10$$ observations were collected on the time spent in a manufacturing system for a part. The analysis determined a 95% confidence interval for the mean system time of $$[18.595, 32.421]$$.
1. Find the approximate number of samples needed to have a 95% confidence interval that is within plus or minus 2 minutes of the true mean system time.
2. Find the approximate number of samples needed to have a 99% confidence interval that is within plus or minus 1 minute of the true mean system time.

Exercise 3.14 Suppose a pilot run of a simulation model estimated that the average waiting time for a customer during the day was 11.485 minutes based on an initial sample size of 15 replications with a 95% confidence interval half-width of 1.04. Using the half-width ratio sample size determination techniques, recommend a sample size to be 95% confident that you are within $$\pm$$ 0.10 of the true mean waiting time in the queue. The half-width ratio method requires a sample size of what amount?

Exercise 3.15 Assume that the following table represents the summary statistics for a pilot run of 10 replications from a simulation.
Simulation Statistics NPV P(NPV $$<$$ 0)
Sample Average 83.54 0.4
Standard Deviation 39.6 0.15
Count 10 10
1. Find the approximate number of additional replications to execute in order to have a 99% confidence interval that is within plus or minus 20 dollars of the true mean net present value using the normal approximation method.

2. Find the number of replications necessary to be 99% confident that you have an interval within plus or minus 2% of the true probability of negative present value.

Exercise 3.16 Consider a manufacturing system comprising two different machines and two operators. Each operator is assigned to run a single machine. Parts arrive with an exponentially distributed inter-arrival time with a mean of 3 minutes. The arriving parts are one of two types. Sixty percent of the arriving parts are Type 1 and are processed on Machine 1. These parts require the assigned operator for a one-minute setup operation. The remaining 40 percent of the parts are Type 2 parts and are processed on Machine 2. These parts require the assigned operator for a 1.5-minute setup operation. The service times (excluding the setup time) are lognormally distributed with a mean of 4.5 minutes and a standard deviation of 1 minute for Type 1 parts and a mean of 7.5 minutes and a standard deviation of 1.5 minutes for Type 2 parts. The operator of the machine is required to both setup and operate the machine.

Run your model for 20000 minutes, with 10 replications. Report the utilization of the machines and operators. In addition, report the total time spent in the system for each type of part.

Exercise 3.17 Incoming phone calls arrive according to a Poisson process with a rate of 1 call per hour. Each call is either for the accounting department or for the customer service department. There is a 30% chance that a call is for the accounting department and a 70% chance the call is for the customer service department. The accounting department has one accountant available to answer the call, which typically lasts uniformly between 30 minutes to 90 minutes. The customer service department has three operators that handle incoming calls. Each operator has their own queue. An incoming call designated for customer service is routed to operator 1, operator 2, or operator 3 with a 25%, 45%, and 30% chance, respectively. Operator 1 typically takes uniformly between 30 and 90 minutes to answer a call. The call-answering time of Operator 2 is distributed according to a triangular distribution with a minimum of 30 minutes, a mode of 60 minutes, and a maximum of 90 minutes. Operator 3 typically takes exponential 60 minutes to answer a call. Run your simulation for 8 hours and estimate the average queue length for calls at the accountant, operator 1, operator 3, and operator 3. Develop an simulation for this situation. Simulate 30 days of operation, where each day is 10 hours long. Report the utilization of the accountant and the operators as well as the average time that calls wait within the respective queues.

Exercise 3.18 Parts arrive to a small manufacturing cell at a rate of one part every 30 $$\pm$$ 20 seconds. Forty percent of the parts go to drill press, where one worker drills the hole in 60 $$\pm$$ 30 seconds. The rest of the parts go to the hole punch where one worker punches the hole in 45 $$\pm$$ 30 seconds. All parts must go to a deburring station, which takes 25 $$\pm$$ 10 seconds with one operator. For all parts, they then go through a heat treatment operation for 20 $$\pm$$ 10 minutes. For the purposes of this problem, there is always room within the heat treatment area to hold as many parts as necessary. The notation X $$\pm$$ Y indicates that the random variable is uniformly distributed over the range (X-Y, X+Y). Build a model that can estimate the following performance measures: 1) average number of parts in the manufacturing cell, and 2) average time spent in the manufacturing cell. Run your model for 20 replications of 100 parts.

Exercise 3.19 Referring to Exercise 3.16. Determine the number of replications to ensure that you are 95% confident that the true expected system time for part type 2 has a half-width of 0.5 minutes or less.

Exercise 3.20 Referring to Exercise 3.17. Determine the number of replications to ensure that you are 95% confident that the true waiting time for the accountant is within $$\pm$$ 2 minutes.

Exercise 3.21 Referring to Exercise 3.18. Determine the number of replications to ensure that you are 95% confident that the average number of parts within the system is within $$\pm$$ 1 units.

Exercise 3.22 Referring to Exercise 2.20. Determine the sample size necessary to estimate the mean shipment time for the truck only combination to within 0.5 hours with 95% confidence

Exercise 3.23 Referring to Exercise 2.22. Develop an model to estimate the average profit with 95% confidence to within plus or minus \$0.5.