Midterm-Practice-Exam-A-Solutions

1 COMM204 Section 921 Logistics and Operations Management Midterm Examination 09:00am-11:00am, Tuesday May 28, 2019 (Practice Exam A) Note: • Do not turn to the second page before the invigilator announces the start. • This is a closed book exam – no notes of any kind are permitted, except a one- sided cheat sheet (one 8.5-by-11-inch paper). • Simple (that is non-graphical) calculators are permitted • The examination takes 2 hours. • The exam has two parts (10 pages, including this page). Part I is multiple choice questions with total 30 points. Part II is short questions with total 70 points. The total points are 100 points. • Please answer all questions. • Sign the Acknowledgement of UBC Plagiarism Policy before start. • Good luck! Acknowledge of UBC Plagiarism Policy In taking this examination, I acknowledge and accept the UBC Plagiarism Policy. Student ID: _______________ NAME (printed) ____________________________________________ NAME (signed) ____________________________________________

2 Part I. Multiple choice questions. (30 points) Instructions: Answer all the 6 questions. Each has 5 points. Circle one (and only one) answer from the multiple choices. 1. Which of the following is not a typical type of process? A. Job Shop B. Batch flow C. Commodity Flow D. Project E. Continuous process Answer: C 2. Which of the following is not the assumption of the PK formula we learned in class: A. Customers may arrive, look and leave the system B. There is only one single server C. There is only one queue D. All units arrive independently of each other E. The queue may be unlimitedly long Answer: A 3. Which of the following(s) about PK formula is (are) true? I. It shows that higher utilization implies higher queue length II. We can relate it to OM triangle III. It shows that without variability the queue length will be zero A. I B. II C. I and II D. II and III E. I, II and III Answer: E 4. If the long-run average input rate is 120 units per hour and the short-run average input rate is 160 units per hour, which of the following statement(s) are (is) impossible? I. The long-run average capacity rate is 150 units per hour II. The long-run average capacity rate is 100 units per hour III. The short-run average capacity rate is 100 units per hour IV. The long-run average throughput rate is 100 units per hour

4 Part II. Short questions Question 1 [25 points] Daffy Dave’s Sub Shop makes custom sandwiches to order. They are analyzing the process at their shop. The general flow of the process takes the following four steps: As shown above, there are two workers but only one cashier in the shop who has to take the order as well as to bag the order. A customer can order either a beef or a cheese sandwich. A beef sandwich takes more time because adding beef takes an extra minute than adding the cheese. However, beef sandwiches make a profit of $1.2 per order, while cheese sandwiches only make a profit of $1 per order. Suppose Dave’s shop is open 8 hours a day. (a) Plot a linear flow chart to map the process in Dave’s shop. Your chart should contain a decision point as there are two different types of orders (beef and chees). [5 points] Activity Unit Load Resources Take the Order 1 min/order Cashier Add ingredients 3min/order (beef) or 2min/order (cheese) Worker 1 Add toppings 2 min/order Worker 2 Bag the order 1.5 min/order Cashier

5 (b) Suppose there are sufficiently many orders for each type of sandwiches. What is the daily profit if Dave’s shop only sells beef sandwiches? What about only selling cheese sandwiches? Which type of sandwiches brings in a higher profit, beef or cheese? [10 points] For beef sandwiches, the unit load is (1+1.5=)2.5min, 3min, 2 min for cashier, worker 1, and worker 2, respectively. So worker 1 is the bottleneck resource and Capacity rate of beef sandwiches = capacity rate of worker 1 = # $%&’ ∗ 1 = 20 beef sandwiches/ℎ࠵? = 160 beef sandwiches/day So if selling beef sandwiches only, the profits is 1.2$*160/day=192 $/day. For cheese sandwiches, the unit load is 2.5min, 2min, and 2min for cashier, worker 1, and worker 2, respectively. So cashier is the bottleneck resource and Capacity rate of cheese sandwiches = capacity rate of cashier = # ;.=%&’ ∗ 1 = 24 cheese sandwiches/ℎ࠵? = 192 cheese sandwiches/day So if selling cheese sandwiches only, the profits is 1$*192/day=192/day. Thus, beef sandwiches and cheese sandwiches have equal daily profit. (c) Consider a product mix of one beef sandwich and two cheese sandwiches. What is the bottleneck of producing the product mix? What is the capacity rate? [10 points] For the product mix of one beef and two cheese sandwiches, the unit load is (2.5 + 2*2.5=)7.5min, (3 + 2*2=)7min, (2+2*2=)6min for cashier, worker 1, and worker 2, respectively. So cashier is the bottleneck resource and Capacity rate of the product mix = capacity rate of cashier = # @.=%&’ ∗ 1 = 8 product mix/ℎ࠵? = 64 product mix/day Or equivalently: (8 beef sandwiches + 16 cheese sandwiches)/hr = (64 beef sandwiches + 128 cheese sandwiches)/day

7 (b) On average, how many cars are waiting on the drive-way between 7am-12pm? [5 points] If your diagram is bar chart (discrete time), Average inventory = (0+0+6+5+2)/5=2.6 If your diagram is a line chart (continuous time), Average inventory = [ (1/2)*1*6 + (1/2)*(6+5) + (1/2)*(5+2) ] / 5 = 2.4 Question 3. (Queueing Analysis) [30 points] Newt Philly needs to decide where to get a haircut. He has narrowed the choice down to two local hair salons – Large Hair Salon (LHS) and Small Hair Cutters (SHC). A new customer walks into LHS every 15 minutes (with a standard deviation of 15 minutes). At SHC, a customer walks in every hour (with a standard deviation of 1 hour). LHS has a staff of 4 barbers (which follows a G/G/4 queuing model), while SHC has 1 barber (which follows a G/G/1 queuing model). A typical service time at either salon lasts 30 minutes (with a standard deviation of 30 minutes). Hint: For a G/G/c queueing model, ࠵? J = K L (NOP) #RK × T U L VT W L ; , ࠵? = Y Z[ , ࠵? is the number of the servers. (a) If Newt walks into LHS, how long on average must he wait in line before he can see a barber? (Only include the waiting time, not any service time) [10 points] 0 1 2 3 4 5 6 7 7 8 9 10 11 12 Inventories Inventories

8 The average inter-arrival time is 15 min, with a standard deviation of 15 min. Thus, the coefficient of variation (CV) of inter-arrival times at LHS is ࠵? ^ = _(^) ‘(^) = #= abc #= abc = 1 . The input rate at LHS is λ = # #=%&’ = 4 customers/hr. The average service time is 30 min, with a standard deviation of 30 min. Thus, the CV of service time at both salons is ࠵? e = _(e) ‘ e = $f abc $f abc = 1 . The capacity rate at both salons is µ = # $f%&’ = 2 customers/hr. The utilization at LHS: ρ = λ /cμ = 4/(4*2) = 0.5. By PK formula, the average no. of customers waiting: ࠵? J = K L NOP #RK × T U L VT W L ; = f.= L hOP #Rf.= ∗ #V# ; =0.1117/0.5 = 0.2234 By Little’s Law, the average waiting time: T j = I j /λ = 0.2234/4 = 0.05585 hr = 3.35 minutes (b) If Newt goes to SHC, how long on average must he wait in line before his haircut starts? [10 points] At SHC, the input rate is λ = 1 customer/hr. The average inter-arrival time is # l = 1 hr. Since the standard deviation is 1 hr, the CV of inter- arrival times at SHC is ࠵? ^ = _(^) ‘(^) = # mn # mn = 1 . Since μ = 2, c = 1 ρ = λ /cμ = 1/(1*2) = 0.5 I j = o L pOP #Ro × q r L Vq s L ; = f.= L POP #Rf.= ∗ #V# ; = 0.5 T j = I j /λ = 0.5/1 = 0.5 hr = 30 minutes