Operations Management Math

INSTRUCTIONS:
1. Your name and student number must be clearly displayed on the assignment.
2. This assignment contains five questions with subparts. Marks for each question are given.
3. This assignment booklet contains 5 pages.
4. The assignment will count for 7.5 percent of your final grade. You are required to answer ALL questions. SHOW YOUR WORK FOR FULL CREDIT.
5. Where appropriate, please round your calculations to two significant digits (e.g. 0.21 or 0.0015)). Do not round until the end for best accuracy.
6. If you have to, make reasonable assumptions, but explain/justify any assumptions made in your answers.
Questions
1. A fast food restaurant has customers arrive at the checkout counter on average every 2.4 minutes. The service rate is 45 customers per hour. 20 marks
a) What is the utilization?
b) What is probability that there are zero customers at the counter (either in line or being served)?
c) What is the probability that there are two or more customers at the checkout counter?
d) What is the probability that there are 1 or 2 customers at the counter?
e) What is the average number of customers waiting in line?
f) If the store wanted the average time in the system to be 5 minutes, what would the service rate have to be?
g) What is the probability that checkout time will take between 1 and 2 minutes?
h) What is the probability that checkout time will take more than 2 minutes?
2. An operation manager of a chemical manufacturing plant must determine the lot size for a chemical that has a steady demand of 1,450 litres per day. The production rate is 6,850 litres per day, setup cost is
$200, annual holding cost is $0.21 per litre, and the plant operates 350 days per year. 20 marks
a. Determine the production order quantity (POQ), the number of orders and the time between orders. What is the total annual setup cost? What is the total holding cost per year? Using an appropriately labelled diagram, graph setup cost, holding cost, and total inventory cost, and show the optimal production order quantity and the minimum total inventory cost. What is the setup time in hours, based on a $25 per hour setup labour cost?
b. Determine the duration of the active production interval, t. Determine the maximum inventory. Determine demand during the production interval.
c. As part of its new JIT program, the company has signed a long-term contract with its customers and the company will take orders electronically for the chemical. The optimal lot size (production order quantity) will drop to 20,000 litres. Determine the new setup cost per order and the number of orders (setups). What is the total annual cost of managing the inventory with the new policy? What is the setup time in hours, based on a $25 per hour setup labour cost with the new policy?
3. You are asked to determine a location for a distribution centre serving three urban centres. They are: 20 marks
1. Guelph, population 1,200, at grid location (8.8, 2.8);
2. Cambridge, population 1,400, at grid location (3.5, 8.1); and
3. Kitchener, population 900, at grid location (7.8, 6.2).
a) Determine the centre of gravity of the three locations.
b) It is determined that planners would prefer to have the depot in either Cambridge or Kitchener (although it will serve all three centres). Using metropolitan distances, determine the load-distance scores for both Cambridge and Kitchener and make a recommendation as to which town should have the depot.
c) You are asked to provide input on a layout for a new office. The old layout is thought of as being inefficient. You are asked to redesign the layout and evaluate your redesign using load distance scores. The current layout is shown below. The overall shape cannot change; you can merely move the individual departments around. (8 marks)
Trips between the different departments are as follows: A→B:4 trips per day, A→C : 5, A→D : 7, A→E: 9 B→A : 5, B→C : 2, B→D: 4, B→E: 2
C→A : 4, C→B : 5, C→D : 1, C→E : 8
D→A : 8, D→B : 4, D→C : 3, D→E : 4
E→A : 8, E→B : 3, E→C : 5, E→D : 3
4). 20 marks
a) A technology company is making a decision about ordering a new tablet to sell during the holiday season. They are deciding whether or not to undertake market research to evaluate the potential market for the product. They have only one opportunity to order the product for this season and are committed to making either a small order or a large order.
If they choose not to undertake the research, they can make one of two choices: place a large order or place a small order. If they place an order the probability that demand is high is 50% and that demand is low is 50%.
If they undertake the market research, the chance that they will get favourable results is 45% and unfavourable results is 55%. If they get unfavourable results, the probability of high demand is 25% and of low demand is 75%. If they get favourable results, the probability of high demand is 80% and of low demand is 20%. The cost of doing the research is $10,000.
If they place a small order and demand is low the net return is $10,000. If they place a small order and demand is high the net return is $100,000. If they place a large order and demand is low the return is $- 50,000. If they place a large order and demand is high the net return is $200,000.
What are the right decisions for the firm to make relative to the market research and the size of the order? What are the associated expected values. If you were asked to minimize the maximum loss, what would your decision be?
b) Now consider only the “no market research” branch of your decision tree (there should be one). If someone could tell you definitively whether demand would be high or low (without doing market research), what would be the value of knowing (i.e. the expected value of perfect information).
5. a) You are in charge of ordering sandwiches for a lunch stand at Maple Leaf Square in Toronto. The sandwiches are made fresh and unsold ones are donated to a local shelter at the end of the day. The sandwiches sell for $8.50 and cost you $3.75 to order. Your demand is uniformly distributed with a minimum of 100 sandwiches and a maximum of 600. How many sandwiches should you order every day? 5 marks
b) The operations manager for Fine Foods Distribution has narrowed the search for a new facility to seven communities. Fixed costs (land, buildings, and equipment) and variable costs (labour, materials etc.) are shown below. 15 marks
Distribution Centre Costs
Community Fixed Costs ($millions) Variable Costs ($)
Kingston $2,400 16
Peterborough 3,000 15
London 1,000 18
Kitchener 3,200 16
Guelph 2,600 17
Barrie 800 20
Collingwood 2,000 17
Which of the communities can be eliminated from further consideration because they are dominated (both variable and fixed costs are higher) by another community?
i. Plot the total cost curves for all remaining communities on a single graph. Identify on the graph the approximate range over which each community provides the lowest cost.
ii. Using locational break-even analysis, calculate the break-even quantities to determine the range over which each community provides the lowest cost.
iii. If the company wants to produce 600,000 units, which location is preferred and what is the total cost?

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