UNIVERSITY OF TORONTO Rotman School of Management RSM 270
2021 Winter Term
Makeup Midterm Examination
Students found committing any dishonest practices will be immediately dismissed from the examination, and subject to such discipline as the Faculty or the University may impose. Examples of dishonest practices include (but are not limited to):
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• making use of books and other media except those authorized by the examiners, • speaking or communicating with other students,
• purposely displaying notes or memoranda to other students.
Please, sign below to acknowledge that you have read the above instructions and you have not cheated in this examination:
Part I: Multiple Choice (24 marks)
• This part contains 12 multiple-choice questions.
• Each correct response will earn 2 marks. Incorrect responses will be worth zero marks, without
additional deduction.
• There is only 1 correct answer for each question. If more than one answer appears correct, one will
be more generally correct or better than the others; If none appear correct, one will be the best or closest to being correct.
1. Productivity increases when:
A. Inputs increase while output remains the same.
B. Inputs decrease while output remains the same.
C. Input remains the same while output decreases.
D. Inputs increase while output decreases.
2. Consider a process consisting of a sequence of tasks in series. The flow time of a non-bottleneck resource is decreased. Then, we can most clearly conclude that
A. the capacity of the process increases.
B. the throughput of the process increases.
C. the (theoretical) flow time of the process decreases. D. none of the above.
3. Consider a process with a buffer. If the input rate of the process is less than its capacity rate in the short run, the output rate (in the short run) must be equal to the input rate.
A. The utilization of all subprocesses is the same.
B. The utilization of the bottleneck subprocess is always 100%.
C. The bottleneck subprocess has a utilization greater than 100%.
D. The bottleneck subprocess has the highest utilization.
5. Consider a process consisting of a sequence of subprocesses in series. Tripling the capacity rate of the bottleneck subprocess
A. Triples the utilization of this subprocess
B. Keeps the utilization of this subprocess at 100%.
C. Reduces the utilization of this subprocess to below 100%.
D. Reduces the utilization of this subprocess to 33.3%.
E. May have various implications on other subprocesses and I need more information for example
on the input rate to answer this question.
6. The capacity of a process with unlimited buffer space
A. can influence its flow time.
B. can be negative.
C. depends on the variability of the inter-arrival times.
D. always follows an exponential distribution.
a (long-run) stable multistage process
A. even if the inter-arrival times and service times have unpredictable variability.
B. only if the inter-arrival times and service times are exponentially distributed.
C. holds exactly when the inter-arrival times and service times are deterministic, and it holds
approximately when the inter-arrival times and service times have unpredictable variability.
8. The average flow time (the waiting time plus the service time) of an emergency operating room is 8 hours and there are on average 10 patients in the emergency operating room (either waiting or being served). On average, how many patients arrive to this emergency room every hour?
B. 0.8 C. 80 D. 8
9. The average inventory of the server Is in a single-server queueing system is equal to A. 𝜆/𝜇
B. 𝜇/𝜆 C. 𝜆−𝜇 D. 𝜇−𝜆 E. 𝜆×𝜇
know that Little’s Law holds
10. What is the utilization of a single-server queueing system with an average inter-arrival time of 60 seconds and average service time of 50 seconds?
E. There is no relationship between average inter-arrival time and cycle time
11. Increasing the coefficient of variation of the inter-arrival times in a single-server queueing system, holding all other parameters constant,
A. Increases the average flow time.
B. Decreases the average inventory.
C. Increases the capacity.
D. None of the above.
12. Which of the following statements is correct?
A. The average waiting time of the customers is higher in an M/M/1 queueing system compared to
an M/G/1 queueing system.
B. The average waiting time of the customers is lower in an M/M/1 queueing system compared to an
M/G/1 queueing system.
C. The average waiting time of the customers is higher in an M/M/1 queueing system compared to a
G/G/1 queueing system.
D. The average waiting time of the customers is lower in an M/M/1 queueing system compared to a
G/G/1 queueing system.
E. I require more information to answer this question.
Part II: Written Questions (76 marks)
• This part contains 3 questions, individually marked. Answer all questions in the boxes provided.
• Round your answers (intermediate and final) to 2 decimal places.
1. Process Analysis (27 points). Paul owns a preventive healthcare clinic, which provides annual health check-up to customers. When a customer arrives, (s)he is welcomed by the receptionist (employee A), who registers the customer. The registration takes 3 minutes. Then, the customer sequentially goes through the following four tests: lab-work (performed by employee B), ultrasound (performed by employee C), x-ray (performed by employee D), and physical examination (performed by employee E). The tests take 2 minutes, 15 minutes, 10 minutes, and 5 minutes, respectively. After all the tests are done, the customer receives a detailed report on his/her health condition.
a. (3 points) What is the flow time of the entire process for a customer in minutes?
Flow Time = 3+2+15+10+5=35 minutes.
b. (3 points) What is the maximum number of patients that can be served per hour?
Employee A (registration) capacity=1/3 c/minute=20 customer/hour Employee B (Lab work) capacity=1/2 *60=30 c/hour
Employee C (Ultrasound) capacity =1/15*60=4 customer/hour Employee D (X-ray) capacity= 1/10*60=6 customer/hour Employee E (Phys Exam)=1/5*60=12 customer/hour
The capacity of the whole process= min( 20,30, 4,6, 12)= 4 c/hour
c. (3 points) Which station is the bottleneck?
Employee C or ultrasound.
In the following parts, assume that on average patients arrive every 20 minutes.
d. (3 points) What is the throughput rate of the clinic? Please state your answer in the units of customers per hour.
Throughput rate = min(1/20*60,4)=3 customers/hour
e. (3 points) What is the utilization of the lab-work employee (employee B)?
Throughput rate/ capacity of lab-work employee = 3/30 = 0.1 = 10%
f. (3 points) What is the average utilization of all employees?
A Utilization= 3/20
B utilization=3/30
C utilization= 3/4
D utilization = 3/6
E utilization = 3/12
Average utilization = (.15+.1+.75+.5+.25)/5=1.75/5=.35
g. A talented RSM270 student realizes that ultrasound is the bottleneck of the entire process. She suggests Paul to add another ultrasound machine (employee C can handle both machines). What would be the
1. (3 points) New bottleneck?
X-ray (employee D)
2. (3 points) The new capacity rate of the entire process? Please state your answer in the units of customers per hour.
6 customers/hour.
3. (3 points) New flow time of the entire process for a customer in minutes?
The flow time is still 35 minutes.
2. Inventory build-up and Little’s Law (22 points). offers fresh avocado rolls to students every weekday between 11am and 2pm. Suppose that opens at 11:00AM. Customers arrive at the rate of 100 customers per hour between 11:00am and 11:30am. Customers arrive at the rate of 200 customers per hour between 11:30am and 1:00pm. Customers arrive the rate of 120 customers per hour between 1:00pm and 2:00pm. Suppose that the arriving customers form a line in front of the restaurant and wait until they are either served or turned away. The restaurant can serve 180 customers per hour. The restaurant closes at 2:00pm regardless of the number of customers waiting in line and the unsatisfied customers are turned away.
(a) (6 points) Draw an inventory build-up diagram for the number of customers waiting in the line between 11am and 2pm.
The build-up rate between 11am and 11:30am is 100-180 customers per hour. Therefore, the inventory (number of customers in the line) remains zero between 11am and 11:30am. The build-up rate between 11:30am and 1pm is 200-180=20 customers per hour. Therefore, inventory increases linearly from zero at 11:30am to 30 at 1pm. The build-up rate between 1pm and 2pm is 120-180=-60. Therefore, inventory decreases linearly from 30 customers at 1pm to zero customers at 1:30pm. Then, inventory remains zero between 1:30pm and 2pm.
(b) (6 points) What is the average number of customers waiting in the line between 11am and 2pm?
Average Inventory = ( 0.5*(13-11.5)*30 + 0.5*(13.5-13)*30 ) / 3 = 10 customers.
(c) (5 points) Calculate the average throughput rate of the restaurant between 11am and 2pm. Please enter your answer in the units of customers per hour in up to two decimal points.
A total of 100*0.5+200*1.5+120 = 470 are served in a time span of 3 hours. Therefore, the average throughput rate = 470 / 3 = 156.67 customers/hr
(d) (5 points) How much time on average do the customers spend waiting in line? Please enter your answer in minutes in up to two decimal points.
By Little’s Law, we have:
Avg. waiting time = (avg. inventory) / (avg. throughput rate) = 3.83 minutes
3. Queuing (27 points). Consider a bike rental shop that maintains a fleet of (homogeneous) bikes in Toronto Islands. We assume each new customer will only rent one bike and must return the bike to the shop after the ride.
The shop owns (and rents out) 30 bikes. The bike rental shop charges a flat price of $10 per ride. The customers ride the bikes for exactly 1 hour (constant). Based on the historical data, the shop has indicated that the customer arrival process has an average inter-arrival time of 2.5 minutes with a standard deviation of 5 minutes. Currently, the shop has one single line for new customers.
a. (4 points) What is the average customer arrival rate in customers/hour?
The average customer arrival rate is λ = 60/2.5 = 24 customers/hour.
b. (4 points) What is the average utilization of each bike?
The service rate is μ = 1 customer/hour for each bike. The utilization equals to τ = λ/(cμ) = 0.8.
c. (3 points) What is the coefficient of variation (CV) of the inter-arrival times?
Ca = 5/2.5 = 2.
d. (4 points) Can we model the bike rental shop as an M/D/30 queue?
No. While the service time is deterministic, the inter-arrival time does not follow an exponential distribution because Ca is not equal to 1. So, we cannot model the system as an M/D/30 queue.
e. (4 points) What is the average inventory of the queue (queue length)?
Using the PK formula for multi-server queues, we have Iq = 1.7256.
f. (4 points) How long does each customer spend in the queue waiting for a bike (in minutes)?
The average waiting time is Tq = Iq/λ = 1.7256/24 = 0.0719 hours = 4.3139 minutes.
g. (4 points) Next, assume the ride durations follow an exponential distribution with an average ride duration of 1 hour. What is the average inventory of the queue (queue length) in this case?
Now, the new Cs =1. So, Iq = 2.1569.
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