UNIVERSITY OF TORONTO Rotman School of Management RSM 270
2021 Winter Term
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. Which of the following quantities is the most appropriate performance criterion for measuring the efficiency of a process?
A. Effectiveness.
B. Productivity.
C. Total Input consumed. D. Total output generated.
2. Consider the following chart. Company “a” aims to maintain its efficiency level and improve its effectiveness level. Which company among the existing companies should be its benchmark? Only one company meets this criterion.
A. Company e
B. Company b
C. Company c
D. Company d
3. Consider a facility with 4 employees, each working 8 hours per day. The facility produces 1,600 ice creams per day. What is the labour productivity of this facility?
A. 30 ice creams per worker hour
B. 200 ice creams per worker hour
C. 400 ice creams per worker hour
D. 50 ice creams per worker hour
A. Less than zero.
B. Between zero and one.
C. Equal to one.
D. Greater than one.
general, one expects the utilization of a non-bottleneck resource to be
A. The throughput rate is determined solely based on the capacity rate of the bottleneck subprocess.
B. The throughput rate is determined based on the capacity rate of the bottleneck subprocess and the
input rate of the process.
C. The bottleneck subprocess determines the capacity rate of the process.
D. Choices b and c.
6. Which of the following statements regarding unpredictable variability is False?
A. Unpredictable variability can lead to an increase in the inventory.
B. Unpredictable variability can be reduced by collecting more information.
C. Little’s law does not hold for a process whose inter-arrival times or service times have
unpredictable variability.
D. Both the inter-arrival times and the service times can have unpredictable variability.
7. Consider cars waiting at the drive-up queue of a fast-food restaurant. We plot the number of cars waiting in the queue over time. In this graph, the horizontal axis represents the time, t, and the vertical
a multistage process,
axis represents the number of cars. The area under the curve over a given time interval divided by the length of the time interval indicates
A. the average waiting time of the cars in the queue.
B. the average inventory of cars in the queue.
C. the maximum time a car must wait.
D. the inventory of cars waiting in the queue at time t.
8. Under which of the following conditions, the slope of the inventory build-up diagram is (always) equal to zero?
A. Inventory being zero.
B. Input rate being less than the capacity rate.
C. Input rate being equal to the capacity rate.
D. Input rate being greater than the capacity rate.
9. When the inter-arrival times are exponentially distributed, we have
B. Ca = 0.5.
10. Consider a coffee shop that is modeled as an M/M/1 queueing system. On average, 10 customers arrive at the coffee shop per hour. The average service rate of this coffee shop is 15 customers per hour. What is the average waiting time of a customer in the queue (in hour)?
B. 2/15 C. 3/10 D. 4/3
11. Given the same average customer arrival rate () and service rate (), which of the following queueing systems has the lowest average queue length? (Assuming 0 < < )
C. M/M/2 D. D/D/1
12. Which of the following is the downside of adding an inventory buffer in front of a process?
A. Decreased utilization of resources
B. Increased throughput rate
C. Increased average flow time
D. Decreased throughput rate
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). Project Better Place is building an electric car infrastructure in Denmark. Similar to current gas stations, battery switching stations will be available roadside. The driver pulls in and the depleted battery is quickly replaced with a fresh one without anyone having to leave the vehicle. The process takes less time than it does to fill a tank of liquid fuel.
Consider a switching station with four employees, three of them, Steve, Simon, and Stefan, responsible to switch batteries. The switching station is open from 7:00am to 9:00pm. On average, it takes 20 seconds to disconnect and remove the old battery, 30 seconds to install a new battery, and 10 seconds to bill the customer. When a customer arrives, (s)he first checks in with Steve’s younger sister Michelle. The check-in takes only 10 seconds. Next, one of the three “switchers” serves the customer and performs all three activities – removing the old battery, installing a new battery, and billing the customer – consecutively.
a. (3 points) What is the flow time of the customers in seconds?
Flow Time = 10+20+30+10=70 seconds
b. (4 points) What is the maximum number of customers that can be served per hour at the switching station?
Michelle’s capacity =1/10 c/sec = 6 c/minute = 360 c/hour 1 switcher capacity = 1/60 c/sec = 1 customer/minute
3 switcher’s capacity = 3*1 = 3 c/minute
The capacity of the whole process: 3c/minute = 180 c/hour.
Assume on average every 25 seconds a new customer arrives.
c. (3 points) How many customers depart every hour on average (throughput rate)?
min(capacity,throughput rate)= min (3600/25, 180)= 3600/25=144 customers/hour
d. (3 points) What is the utilization of each switcher?
Throughput rate through each switcher/(effective capacity) = (2.4)/3/(1) = 0.8
e. (3 points) What is the utilization of Michelle?
Throughput rate/ capacity = 2.4/6 = 0.4 = 40%.
f. (4 points) What is the average utilization of all employees?
Average utilization = (2.4/6 +3*2.4/3)/4 = 0.7 = 70%
A regular customer of the switching station has suggested that the billing task be transferred to Michelle.
g. (4 points) What is the new capacity rate of the entire process? Please enter your answer in the units of customers per hour.
180 customers/hour.
h. (3 points) Who is the new bottleneck?
Michelle or (check in + billing).
2. Inventory Build-up and Little’s Law (22 points). The Ledger Milk Company owns a cow farm and employs one employee. Every morning, the employee milks the cows at a constant rate starting at 6am. It takes the employee (exactly) 3 hours to milk the cows. At the end of the 3-hour period, the employee has milked 1080 liters of milk. The milk is stored in a cooled tank (of sufficient size) until the milk truck arrives to pump out the milk. The milk truck (of sufficient size) arrives at 9:30am and pumps out 72 liters of milk per minute from the cooled tank. Once the tank is empty, the milk truck leaves.
(a) (6 points) Draw an inventory build-up diagram for the amount of milk stored in the cooled tank starting at 6am until the cooled tank empties out.
From 6:00am to 9:00am, the employee milks 1080/(3*60) = 6 liters per minute. This milk is stored in the cooled tank. For the next 30 minutes (9:00am to 9:30am), the inventory in the storage tank does not change. At 9:30am, the truck arrives and at the rate of 72 liters per minute the milk is pumped out. Thus, the cooled tank is emptied at 9:45am.
(b) (6 points) What is the average inventory of milk in liters in the cooled tank between 6am and the time the cooled tank empties out?
Average Inventory = ( 0.5*(9-6)*1080 + (9.5-9)*1080 + 0.5*(9.75-9.5)*1080 ) / 3.75 = 612 liters.
(c) (5 points) Calculate the average throughput rate of the cooled tank between 6am and the time it empties out. Please enter your answer in the units of liters per hour.
A total of 1080 liters flow through the process during a time span of 3.75 hours. Therefore, the average throughput rate = 1080 / 3.75 = 288 liters/hr.
(d) (5 points) How much time on average does the milk spend in the cooled tank? Please enter your answer in the units of hours in up to two decimal points.
By Little’s Law, we have:
Avg. waiting time = (avg. inventory) / (avg. throughput rate) = 2.125 hour
3. Queuing (27 points). A commerce student has invented a burrito vending machine that can create fresh burritos automatically. The student decides to open a new business in the food court of . 4 burrito machines will be operating at the same location, and a single line is designed for waiting students. Given the location, it is estimated that the average time between arriving customers is 3 minutes. The standard deviation of the inter-arrival times is 9 minutes. Each customer will only order one burrito. Each machine can make 10 burritos per hour. The service time is deterministic as the machine currently produces only one type of burrito.
(a) (4 points) What is the average customer arrival rate in customers/hour?
The average customer arrival rate is λ = 60/3 = 20 customers/hour.
(b) (3 points) What is the coefficient of variation (CV) of the inter-arrival times?
Ca = 9/3 = 3.
(c) (3 points) What is the coefficient of variation (CV) of the service times?
(d) (5 points) On average, how long do students have to wait in line (in minutes)? \
The service rate μ = 10 and τ = λ/cμ = 20/(4 · 10) = 0.5. Using the multi-server PK formula, we have Iq = 1.005. So, Tq = Iq/λ = 0.0503 hours = 3.0159 minutes.
Next, assume the machine has been upgraded and it can make three types of burritos now. In the meanwhile, the variability of the service times increases. The standard deviation of the service times of the upgraded machine is 6 minutes (the average service time has not changed).
(e) (4 points) What is the coefficient of variation (CV) of the service times for the upgraded machine?
Note that the average service time is 60/10 = 6 minutes. So, Cs = 6/6 = 1.
(f) (5 points) On average, how long do students have to wait in line after introducing the upgraded machine (in minutes, assuming the arrival process is the same as before)?
With the same PK formula, Iq = 1.117. So, Tq = Iq/λ = 0.05585 hours = 3.35 minutes.
(g) (3 points) The upgraded burrito machine is so popular that the average arrival rate increases to 50 customers/hour. Can the business maintain the demand in the long run? Why or why not?
The total service rate (capacity rate) of the system is 40 customers/hour. So, the new input rate is greater than the capacity rate, the system cannot sustain itself in the long run.
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