
Practice_Questions_for_Exams.pdf Download Attachment
Department of Decision Sciences
San Francisco State University
FINAL EXAM QUESTIONS
The sample problems below are organized by topic. Where possible, answers are given.
Disclaimers:
• These...Department of Decision Sciences
San Francisco State University
FINAL EXAM QUESTIONS
The sample problems below are organized by topic. Where possible, answers are given.
Disclaimers:
• These questions are questions that have appeared on previous years’ (final) examinations.
They are for practice only. There is no guarantee that the questions on your finals will be
the same, or that different professors will give the same types of questions.
• Not all 412/786 sections cover the same material. The questions may contain material
that is unfamiliar to you, or not contain material you covered. If you have concerns about
the topics for your final, talk to your professor.
• Short answer and multiple choice questions are not included in this handout.
• Some questions may have been slightly modified to make them more general.
Decision Analysis
1. Sheree is deciding on how to invest her $1000 tax refund. She reads that Merck is partnering
with a small biotech firm to produce a cancer drug which may or may not receive FDA approval
in the next six months. If the FDA approves it, the biotech firm’s stock will double in value, but
if it rejects it, the firm goes bankrupt and its stock becomes worthless. Merck is a huge
pharmaceutical, so approval will result in a 20% appreciation in stock, and rejection in a 10%
decrease. She is also considering hedging her bets and investing half her money in Merck and
half in the Biotech firm. She could also put her money in a safe mutual fund, with a small (but
guaranteed) 3% increase in value in the next six months.
a) Fill in the decision table, labeling all decisions, outcomes and payoffs. Express payoffs
in terms of expected portfolio value at the end of six months. Hint: the table has the
correct dimensions.
Outcomes
___________
Decisions
_____________
___________
___________
___________
_______
b) Given the payoff table what is Sheree’s best decision using the Laplace, Maximax, and
Maximin decision criteria? Please show the decision, not just the payoff amount for full
credit. (Merck, Biotech, Mutual Funds)
c) Sheree thinks that the chance of the FDA approving the drug is 40%. Compute the
expected values for each decision, and circle her best decision, assuming she is rational
and risk neutral. (all Biotech 800, Merck 1020, 50/50 910, Mutual Funds 1030)
d) Construct the opportunity table below and show what investment Sheree would make
under the Minimax Regret Decision Criteria (no points just for guessing!) (50/50)
DS 412/BUS 786 Final Review Sheet
Decision Analysis
Page 1
Department of Decision Sciences
San Francisco State University
2. Yonni is considering expanding his Yuppie Yoga franchise by building a new studio in
Russian hill that will cost 6 million dollars to build and operate (over the relevant timeframe of
the model). Customer demand is uncertain, however. There is a 0.6 probability it will be high,
yielding him 10 million in revenues, but if it is low he only will earn 5 million in revenues from a
halffull club. If demand is low, he does have the option of trying to raise it through a marketing
campaign. If successful, this raises his club's revenue's to 8 million dollars. However, the
campaign will cost 2 million and only has a 50% chance of success and if unsuccessful, club
revenues remain at 5 million. Also, Yonni can choose not to build (and his profit would be 0).
a) Draw a decision tree to show Yonni's decision process, wherein he attempts to maximize
his profits. Show all final payoffs and intermediate expected values. (Reminder: profit
= revenues costs. All costs that are relevant to the problem are mentioned explicitly!)
b) Assuming Yonni is a riskneutral, rational decision maker, what is his course of action
over the timeframe of the model? (build studio, don’t advertise)
c) What is the expected value of this decision tree? ($2 million)
3. Bratt's Bed and Breakfast, in a small historic New England town, must decide how to
subdivide (remodel) the large old home that will become their inn. There are three alternatives:
Option A would modernize all baths and combine rooms, leaving the inn with four suites, each
suitable for two to four adults each. Option B would modernize only the second floor; the results
would be six suites, four for two to four adults, two for two adults only. Option C (the status quo
option) leaves all walls intact. In this case, there are eight rooms available, but only two are
suitable for four adults, and four rooms will not have private baths. Below are the details of profit
and demand patterns that will accompany each option.
Annual profit under various demand patterns
Capacity p
Average p
0.5 $25,000 0.5
(Modernize all) $90,000
nd
0.4 $70,000 0.6
(Modernize 2 ) $80,000
(Status Quo)
$60,000
0.3 $55,000
0.7
a) Construct a decision tree.
b) Which option has the highest expected value? (Option B, $74,000)
4. The Slumber Mist Company is considering marketing a new water bed. If the new bed is
successful, it will mean a $4 million profit (present value) over the life of the product. If
unsuccessful, a $2 million loss on investment will be incurred. Slumber Mist is considering
whether to hire a market research company, Market Competition Inc. (MCI), to perform an
analysis. The results of 40 previous studies by MCI are given below:
Forecast success
Forecast failure
Actual outcome
Success
Failure
18
3
4
15
MCI charges $100,000 for its survey. Draw a decision tree for this problem. Describe the
optimal strategy in words. (use MCI; if success predicted, market; if failure predicted, don’t)
DS 412/BUS 786 Final Review Sheet
Decision Analysis
Page 2
Department of Decision Sciences
San Francisco State University
5. Steve has a choice of one of three ways of going to school. The time it takes for him to travel
not only depends on which way he travels, but also on the weather that day. Steve is a man of fixed
habits who likes always to take the same way to work each day, regardless of the weather that day.
The following is a Decision Table giving the minutes to school under each weather condition.
Sunny
25
Route 280
Highway 101 30
35
19th Avenue
a)
b)
c)
d)
Windy
55
35
40
Rainy
60
45
40
If Steve is a Pessimist, which way should he choose? (19th Ave)
Give another name for the Pessimist's Criterion.
Which way should Steve choose if he is an Optimist? (280)
If Steve believes that any type of weather is equally likely, which way should he go, using
expected travel time as the criterion? (101)
6. The XYZ Book Publishing Company receives manuscripts (drafts of books) from authors which
it then considers for publication. A book that sells well should bring a profit of $80,000 whereas a
poorselling book will lose $100,000. (These figures don’t include reviewing fees.) Until recently,
XYZ had sent every manuscript to a reviewer, who charged $3000 to read the manuscript and either
“recommended” or “did not recommend” it for publication (a positive recommendation indicates that
the reviewer believes the book would sell well if it were published). Until recently, the XYZ
Company received so few manuscripts that it automatically published them all, irrespective of
whether they were “recommended” by the reviewer or not. The following table summarizes what
happened in the past:
Actual sales of the book
High sales Low sales
3
1
Reviewer “recommended”
6
10
Reviewer “did not recommend”
(For example, there were 6 manuscripts which sold well even though they were not recommended by
the reviewers.) XYZ has been losing money recently, and has started to ask itself questions like
these: Do we need a reviewer? If we have a reviewer, shouldn't our decision to publish be based on
what the reviewer says?
a) Draw a single decision tree that will determine a better operating policy, and answer the
above questions.
b) Summarize in words the best strategy as indicated by the tree, and state the expected profit
that would be obtained with this strategy. (use reviewer, if recommends, publish; if not,
don’t)
c) If the optimal strategy was adopted for the next fifty manuscripts, what would XYZ's total
profit be, approximately? ($650,000)
DS 412/BUS 786 Final Review Sheet
Decision Analysis
Page 3
Department of Decision Sciences
San Francisco State University
7. A greengrocer sells fresh mixed fruit which he buys from the farm for $1.50 per lb in 100 lb lots.
He tries to sell it the same day at a price of $2.00 per lb. (If he has any fruit left unsold at the end of
the day he sells it at $0.50 per lb. to a local market; he can sell any amount of fruit at this lower
price.) The daily demand for fresh fruit is as follows:
Demand (lbs) Probability
100
0.4
200
0.4
300
0.2
Note that this table shows the demand for fresh fruit. There is an unlimited demand for old fruit.
a) Give the decision table for this problem.
b) What should the greengrocer do to maximize his expected profit? Record your calculations
in the above table. (purchase 100 lbs)
c) What should the greengrocer do if he is a pessimist/uses maximin? (purchase 100 lbs)
8. Selfless Cell Company (SCC) is trying to determine the best course of action to take with
customers whose bills are overdue. The choices are disconnecting service, waiting, sending letters,
and calling the customers. Waiting does not have an associated cost. Sending letters and
disconnecting service each cost $1, while calling customers costs $3. The average size of an
outstanding bill is $50. In all cases, SCC is willing to wait at most 2 months before disconnecting
service. Disconnecting service is the simplest option; however, there is no possibility of collecting on
the debt in this case. SCC can choose to wait for a month to see if the customer pays the bill without
prompting. There is a 50% of this happening. If payment is still not forthcoming after this month,
SCC must decide what to do next. The customer may have service discontinued, may receive a
reminder letter, or may receive a call. Customers receiving the letter will pay with probability 40%.
75% pay if they are called pay. SCC may also decide to send the customer a nice reminder letter.
70% of customers receiving the nice letter will pay. Those that do not pay may receive a phone call,
a nasty letter, or be disconnected. 80% of those receiving the call pay, while 60% of the nastyletterrecipients do.
Finally, SCC may choose to call overdue customers. 90% of those receiving a call will immediately
pay. If they do not, SCC can try calling them again (with a success probability of 5%), or can
disconnect their service. Construct a decision tree to help SCC decide the best course of action.
What are the best decisions to be made at each step if SCC wishes to maximize profits?
(send a nice letter, then call)
9. Randy Hearst sells copies of a daily newspaper for $0.75 each at a street corner. Every
morning, Randy buys newspapers from the local distributor for $0.50. If he runs out of
newspapers during the day, he cannot order any more that day. On the other hand, if he has any
papers unsold at the end of the day, he can return them to the distributor for a refund of $0.15.
From past experience, Randy has estimated that the daily demand for newspapers will be either
18, 19, or 20, with probabilities of 0.3, 0.4, and 0.3.
a) Set up the decision table.
b) How many newspapers should Randy buy every morning to maximize expected profit?
(19)
DS 412/BUS 786 Final Review Sheet
Decision Analysis
Page 4
Department of Decision Sciences
San Francisco State University
10. The CEO of a biotech company must decide whether to spend $2 million to continue with a
particular research project or to stop the development. The success of the project (as measured by
obtaining a patent) is not assured. At this point the CEO judges only a 70% chance of getting the
patent. If the patent is awarded, the company can either license the patent for an estimated profit
of $25 million or invest an additional $12 million to create a production and marketing system to
sell the product directly. If the CEO chooses the latter, he/she faces uncertainty of demand and
associated profits from sales as follows:
Demand level Probability Profit from Sales
High
0.25
$55 million
Medium
0.55
$33 million
Low
0.2
$15 million
a) Draw the decision tree.
b) Using the EMV rule, what is the optimal strategy? What is the EMV for this strategy?
(invest, then license if successful)
11. A high school student is trying to decide what to do after high school. She could get a job, where
she expects to make an average of $25,000 per year over the next 10 years. She assumes she will
have no problems finding a job. She can attend an Ivy League school, which will cost $30,000/year
for 4 years if she gets a degree in the liberal arts or business. If she decides to major in engineering,
she concludes (perhaps incorrectly) that her studies will be much more stressful, and that she should
assign an extra $10,000/year in costs. There is a 50% chance that she will get into an Ivy League
school. If she does not, she can either get a job immediately, or go to a local state school. If she gets
an Ivy League degree, she anticipates an average salary of $60,000/year if she does not get an
engineering degree, or $80,000/year if she does. There is an 80% chance she will get such a job after
she graduates. If not, she can find the type of job she would have gotten had she not gone to college.
If she attends a state school, her costs will be $10,000/year for liberal arts/business and $15,000/year
for engineering. She assumes there is a 95% chance she will be accepted at a state school, and a 75%
chance she will be able to find a position after graduating. She anticipates average salaries of
$50,000/year and $65,000/year, depending on her major. Assume there are no time value of money
concerns. Draw the decision tree and make recommendations on which decisions are the most
appropriate. (go to a state school, major in engineering)
12. Once a week, Nan's Newsstand purchases copies of Money Magazine from the publisher at a cost
of $2.00/copy. Nan then sells the magazines for $4.00/copy. Copies not sold by the end of the week
are discarded. Based on past sales, Nan estimates demand for Money Magazine as follows.
Copies Demanded/Week
Probability
60
10%
80
50%
100
40%
Nan must decide how many copies of Money Magazine to buy each week. There are no shortage or
penalty costs.
a) Draw Nan's decision tree below. Show the relevant decisions, outcomes & profits. Label
clearly.
b) Evaluate the tree. What decision maximizes Nan's expected weekly profit? (purchase 80)
c) Calculate the expected value of perfect information (EVPI). ($20)
DS 412/BUS 786 Final Review Sheet
Decision Analysis
Page 5
Department of Decision Sciences
San Francisco State University
Forecasting
1. City Cycles has just started selling the new XYZ10 mountain bike, and below are the
monthly sales. Amit wants to forecast by exponential smoothing (setting February's forecast
equal to January's sales) with alpha = 0.1 Barbara wants to use a 3period moving average.
Sales Amit Barbara Amit error Barbara Error
400 January
380
400
February
410
March
375
April
May
a) Is there a strong linear trend in sales over time?
b) Fill in the table with what Amit and Barbara each forecast for May and the earlier
months, as relevant.
c) Assume that May's figure turns out to be 405. Append the table with error columns then
calculate MAD for both Amit's and Barbara's method. (16.11 and 19.17)
d) Based on these calculations, which method seems more accurate?
2. Weekly sales of copy paper at Cubicle Suppliers are in the table below.
Week
1234567
Sales(cases) 19 24 29 34 21 20 25
a) Forecast week 8 using a threeperiod moving average. (22)
b) Forecast week 8 using the naïve forecast. (25)
c) Forecast week 8 using a weighted moving average using the weights: 0.5, 0.35, and 0.15
(where the largest weight is for most recent observation). (22.65)
3. Wine sales in the US are shown in the table below (in millions of gallons), where Year 1
represents 1993 and Year 8 represents 2000.
Year
Sales
1
449
2
458
3
464
4
500
5
520
6
526
7
551
8
565
a) What method should be used to forecast sales for 2001? (Trend projection or linear
regression)
b) Forecast wine sales for 2001. (583.57)
c) Using the same forecasting method, what forecasting error would have been made in
1999? (2.74)
DS 412/BUS 786 Final Review Sheet
Forecasting
Page 6
Department of Decision Sciences
San Francisco State University
4. Demand for cement over the last six months is given below. Forecast the demand for the next
month (month 7) using a 3month moving average. (14)
1
2
3
4
5
6
Month
Demand 12.0 14.1 15.3 12.7 14.9 14.4
5. This is a spreadsheet table for Exponential Smoothing calculations, using a Level Model. Don’t
complete this table until you have first read this question completely and have answered parts
(a) through (e).
C
4
1
2
3
6
7
E
F
Demand
(tons)
Yt
12.0
14.1
15.3
Month
t
5
D
G
Level
Lt
a) Insert the two missing column headings, including the algebraic symbols that represent
them.
In answering parts (b) through (e), assume that the smoothing constant α is in cell B3. Write each
answer in the space below the question, not in the table.)
b) What should be typed into cell F5?
c) What should be typed into cell D6?
d) What formula should be typed into cell F6? (Note that this cell will be copied into the cells
below it in column F.)
e) What formula should be typed into cell G6? (Note that this cell will be copied into the cells
below it in column G.)
Now, complete spreadsheet rows 5, 6 and 7 (months 1, 2 and 3) in the above table as it would appear
on the screen, assuming that the smoothing constant is 0.2. That is, calculate the number that will
appear in each cell in the table. Do not write formulas into the cells in the table. Round each number
to 1 decimal place.
6. The manager of an industrial pump manufacturing facility must forecast future demand. She
has generated 2 series of forecasts already, and wishes to compare them to a 2period moving
average forecast.
Time
1
2
3
4
5
6
Demand
492
470
485
493
498
492
Forecast 1
480
490
497
493
Forecast 2
Moving Average
478
488
492
493
a) Compute the 2period moving average forecast.
b) Compute the ME and MAD for all three methods. (2, 4.25, 6.25; 2.5, 4.75, 8)
c) Which forecasting method would you recommend for these data? Why?
DS 412/BUS 786 Final Review Sheet
Forecasting
Page 7
Department of Decision Sciences
San Francisco State University
7. National Tiger Inc., sells rice cookers. Monthly sales for a 7month period were as follows:
Feb Mar Apr May Jun Jul Aug
Month
15
20
18 22 20
Sales (in thousands) 19 18
Forecast September Sales volume using each of the following techniques (Note: round off all
values to two decimal places):
a) Naïve approach (for a stable series) (20)
b) A fivemonth moving average (19)
c) A weighted moving average using 0.6 for the most recent month, 0.3 and 0.1 for the next
most recent months. (20.4)
d) Exponential smoothing with a smoothing constant equal to 0.2, assuming a March
forecast of 19 (thousands). (19.26)
8. A cosmetic manufacturer’s marketing department has developed a linear trend equation that
can be used to predict annual sales of its popular Hand&Foot cream.
Ft = 80,000+15,000t
Where Ft = Annual sales (bottles)
t = 0 corresponding to year 1990
a) Are annual sales increasing or decreasing? By how much?
b) Predict annual sales for the year 2006 using the equation. (320,000)
9. A regression was run between the Annual Revenues (Y) of Celandine Corp. in millions of
dollars and the year (X). The following equation was obtained: Y = 16 + 2.4X, where X = 1 for
1987. An Rsquared of 0.87 was obtained.
a) Interpret the values of the regression parameters a and b with specific reference to this
problem.
b) Is this a good regression model? Why?
c) Forecast revenues for 2001 using the above model. (52)
10. Quarterly data on the number of tourists visiting a ski resort exhibits a pattern of seasonality
with trend, as shown below:
Qr. 1 Qr. 2 Qr. 3 Qr. 4
Year 1 6888 3276 1722 4914
Year 2 7176 3772 1794 5658
Using the decomposition principle, compute the mean seasonal factors. Then forecast the number
of tourists in year 3. (1.60, 0.80, 0.40, 1.20; 7040, 3520, 1760, 5280)
DS 412/BUS 786 Final Review Sheet
Forecasting
Page 8
Department of Decision Sciences
San Francisco State University
Inventory Management
1. Andronico’s in the financial center is open 300 days a year. They sell Organic Hummus; and
sales are approximately normally distributed with an average of 100 tubs/day and a standard
deviation of 25 tubs/day. The wholesale cost is $1.80/tub, and they calculate their annual holding
costs by using 20% of the wholesale cost. Orders cost $50 to place, regardless of number of tubs
ordered and take 4 days to arrive
a) What is the economic order quantity (EOQ) for Organic Hummus? (2887)
b) What are total annual cyclestock holding costs for Organic Hummus? ($519.66)
c) What are the total annual fixed order costs for Organic Hummus? ($519.57)
d) Now Andronico’s has decided to save money and manufacture their own Hummus inhouse, with a production rate of 600 tubs/day, their setup cost would only be $7.50/run,
and their annual holding cost would be $0.20/tub, but all other parameters remain
unchanged. What is the Economic Production Quantity? (1643)
2. A gourmet coffee shop in downtown SF is open 200 days a year and sells an average of 75
pounds of Kona Coffee beans a day. (Demand can be assumed to be distributed normally with a
standard deviation of 15 pounds/day.) After ordering (fixed cost = $16 per order), beans are
always shipped from Hawaii within exactly 4 days. Perpound annual holding costs for the beans
are $3.
a) What is the economic order quantity (EOQ) for Kona coffee beans? (400)
b) What are the total annual holding costs of cycle stock for Kona coffee beans? ($600)
c) What are the total annual fixed ordering costs for Kona coffee beans? ($600)
d) Assume that management has specified that no more than a 1% risk during stock out is
acceptable. What should our reorder point (ROP) be? (369.79)
e) What is the safety stock need to attain a 1% risk of stockout during lead time? (69.79)
f) What is the annual holding cost of maintaining the level of safety stock needed to support
a 1% risk?
g) If management specified that a 2% risk of stockout during lead time would be
acceptable, would our safety stock holding costs decrease or increase?
3. Groundz Coffee Shop uses 4 pounds of a specialty tea weekly; each pound costs $16.
Carrying costs are $1 per pound per week because space is very scarce. It costs the firm $8 to
prepare an order. Assume Groundz is open 52 weeks per year, and closed on Mondays.
a) How many pounds should Groundz order at a time? (8)
b) What is the total annual cost of managing this item? ($416 or $3744)
c) The lead time for orders to arrive is 1.5 (work) weeks. If the standard deviation of
weekly demand is 1.2 pounds, and Groundz wishes to achieve at least a 98% customer
satisfaction level, what is the reorder point? (9.03)
d) Groundz has been contacted by a new supplier who is offering to sell the same tea for
$12 per pound. In return, Groundz can place orders only once a month. Inventory and
ordering costs would remain the same. Is it more costeffective for Groundz to switch
suppliers? Should Groundz switch? (new total cost $3042.67, switch)
DS 412/BUS 786 Final Review Sheet
Inventory Management
Page 9
Department of Decision Sciences
San Francisco State University
4. A firm that makes electronic circuits has been ordering a certain raw material 250 ounces at a
time. The firm estimates that carrying cost is 30% per year, and that ordering cost is about $20
per order. The current price of the ingredient is $200 per ounce. The assumptions of the basic
EOQ model are thought to apply. For what value of annual demand is their action optimal?
(93,750 ounces)
5. The following diagram shows how the quantity in stock of a certain item has changed over
the whole of 2005. The times of deliveries are marked by “D”. The times when orders were
placed for more stock are indicated by “O”. Te last delivery occurred on 12/31/05. The company
uses the Q,R system in which a fixed quantity Q is ordered whenver the quantity in stock reaches
the reorder level R. However, the company is not necessarily using the best values for Q and R.
It costs $5 to place an order for more stock, and $6 to hold one unit in stock for a whole year.
O
12/31/04
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
D
O
O
O
DO
OD O
DO DO D O
D
O
D
12/31/05
D
What is the value of Q the company is currently using? (12)
What is the value of R that the company is currently using? (8)
What was the annual demand D in 2005? (127)
How many order cycles were there in 2005? (11)
What was the demand during the lead time in each of these order cycles? (I want a series
of numbers, not a definition.) (10, 11, 4, 9, 4, 8, 7, 7, 6, 6, 5)
What is the average demand during the lead time (to one decimal)? (7)
What value of Q should the company have used? (This item is not divisible, so you
should round to the nearest integer.) (15)
State the famous formula that you used to answer the previous question.
What Service Level did the company in achieve in 2005 (expressed as a percentage, to
one decimal place)? (0.95)
The company has decided that it would like to achieve a Service Level of 80% in the
future. Therefore, what should be its reorder level, R? Base your answer on the actual
demand during the lead time that you derived in question (c). (8.9 or 9)
What is the safety stock (SS) corresponding to the reorder level you derived in the
previous question (to one decimal place)? (1.9)
DS 412/BUS 786 Final Review Sheet
Inventory Management
Page 10
Department of Decision Sciences
San Francisco State University
6. The Italian eatery at the Student Union orders premade, frozen calzones from a gourmet food
distributor. They cost $2.50 apiece and can be sold to students for $4 apiece. Fixed Order costs
are $10 per order, and orders always take 4 days to arrive. The demand over a term (for which
the eatery is open 100 days) averages 80 calzones a day, with a standard deviation of 20 calzones
per day. Holding costs are 10% of the Eatery’s purchase price.
a) What is the economic order quantity for calzones, and how often do we expect to place
this order? (800, 10)
b) What is the reorder point (ROP) for calzones if the management has specified that the
chance of a stock out during a cycle is 15.87%? (360)
c) Assume that we placed an order 3 days ago, and our inventory of calzones is at 85. What
is the chance we run out of calzones before the next order comes in? (40.13%)
d) What do we expect our holding costs of cycle stock and our fixed order costs to be per
term if we use the EOQ? (100, 100)
e) Bonus: The Eatery has decided to make calzones themselves. These are made
periodically in large batches, and everything not used that day is frozen. The daily
production rate, p, is 160 calzones. Assume that the holding cost, H, is now $.40/calzone
per term and S, the setup cost, is $16 per run. All other parameters remain unchanged.
What is the EPQ, and how often do we start a production cycle? How many days do we
run production? (1131, 14.14, 7.07)
7. The buyer for a department store must decide on the quantity of a designer women’s handbag
to procure in Italy for the upcoming fall fashion season. The unit cost of the handbag to the store
is $30.00 and the handbag will sell for $150.00. Any handbags not sold by the end of the season
are purchased by a discount firm for $15.00. Based on historical data of similar handbags, the
buyer believes that the demand for the handbag can be modeled by a normal distribution with
mean = 150 and standard deviation = 25.
a) What is the shortage cost per unit (Cs)? ($120)
b) What is the excess cost per unit (Ce)? ($15)
c) What is the optimal number of handbags to purchase? (181)
8. A popular newsstand in a large metropolitan area is attempting to determine how many
copies of the Sunday paper it should purchase each week. Demand for the newspaper on Sundays
can be approximated by a Normal distribution with μ = 450 and σ = 100. The newspaper costs
the newsstand 50 cents/copy and sells for $2/copy. Any copies that go unsold can be taken to a
recycling center, which will pay 5 cents/copy.
a) How many copies of the Sunday paper should be ordered? (524)
b) The newsstand actually orders 550 copies every week. Since there is no question on the
cost of excess as outlined above, what is the implied cost of shortage, given the actual
order size? What might be a reason for this difference in shortage costs? ($2.39)
9. Patty paints pink houses for a living. Patty's demand for pink paint is 5 gallons per week. It
costs her $15 each time she drives to the paint store to buy some paint. Holding costs are
$0.75/gallon/week. The paint store charges $25/gallon if Patty buys less than 20 gallons of paint
but only $22/gallon if she buys 20 or more gallons at once. How much pink paint should Patty
purchase each time she goes to the paint store? Be sure to show your total cost calculations.
(20 gallons at a time, $6305/year vs. $7051.54/year)
DS 412/BUS 786 Final Review Sheet
Inventory Management
Page 11
Department of Decision Sciences
San Francisco State University
10. One popular item sold at Harry’s High End Hardware is packs of GE light bulbs. Annual
demand is 2,340 packs. Annual holding costs are $1/pack. Ordering costs are $28 per order. The
delivery leadtime for light bulbs is 1 week. Harry has observed that the demand for packs during
the lead time follows a Normal(μ = 45, σ = 12) distribution. While open 52 weeks per year,
Harry tries to provide a 95% service level. Find Harry’s optimal order quantity and reorder point
(including safety stock). Explain in a sentence how this inventory policy works. (Q* = 362,
R* = 65; Order 362 packs of light bulbs whenever inventory falls to 65 packs.)
11. Expected demand during lead time = 300 units, standard deviation of demand during
lead time = 30 units. Assuming that demand during lead time is normally distributed, determine
each of the following;
a) The ROP that will provide a risk of stockout of 1 percent during lead time. (369.9)
b) The safety stock needed to attain a 1 percent risk of stockout during lead time. (69.9)
c) Would a stockout risk of 2 percent require more or less safety stock than a 1 percent risk?
Explain.
12. Peet’s Coffees in Menlo Park, California, sells Melitta Number 101 coffee filters at a fairly
steady rate of about 60 boxes of filters monthly. The filters are ordered from a supplier in
Trenton, New Jersey. Peet’s pays $2.8 per box of filters and estimates that fixed costs of
employee time for placing and receiving orders amount to about $20 per order. Peet’s estimates
that the annual holding cost per box is 22 percent of the cost of filters.
a) What is the optimal order quality? (216)
b) How often should these orders be placed? (3.33/year)
c) What is the total annual cost (holding cost + ordering cost), assuming they adopt an
optimal policy? ($133.19)
13. Historical records on the use of spare parts for several large hydraulic presses are to serve as
an estimate of usage for spares of a newly installed press (as shown in the table). Shortage costs
involve downtime expenses and special ordering costs. These average $4,200 per unit short.
Spares cost $800 each, and unused parts have zero salvage value.
Number of Spares Used Relative Frequency
0
0.2
1
0.4
2
0.25
3
0.1
4
0.05
a) What is the service level? (0.84)
b) What is the optimal stocking level? (2)
c) Suppose that the optimal stocking level is three spare parts and the shortage cost remains
at $4,200 per unit, what is the implied range of excess cost per unit? ($221 to $741)
DS 412/BUS 786 Final Review Sheet
Inventory Management
Page 12
Department of Decision Sciences
San Francisco State University
14. A hospital uses trays at the constant rate of 200 per month; these are purchased from an
outside supplier for $8 each. The cost of placing an order is $48, while the annual inventory
holding cost is 18% of the value.
a) Find the optimal order quantity for trays. (400)
b) What is the total cost per year (order cost plus inventory holding cost)? ($576)
c) Suppose the hospital currently buys trays in batches of 200. What is the annual saving
achieved by adopting the optimal policy of part a? ($144)
15. A fan manufacturer uses a periodic review inventory system to purchase electric motors. The
review period is 2 weeks and the lead time is 1 week. Demand for electric motors is normally
distributed with a mean of 500 per week and a standard deviation of 80 per week. The target
inventory is 1762 motors.
a) What is the safety stock?
b) What service level is achieved?
16. SwellSound Corp. produces and sells CD players. The demand for CD players is
360,000/year. Each CD player requires a laser. These lasers are purchased from an outside
supplier. The cost of placing an order for lasers is $375, while the inventory holding cost is
$4.80/laser/year.
a) Find the optimal order quantity for lasers. (7500)
b) What is the total cost/year? ($36,000)
c) Suppose the ordering process is streamlined so that the cost of placing an order is reduced
by 36%. What annual cost saving is achieved? The firm always uses an optimal policy
that minimizes total cost. ($7,200)
DS 412/BUS 786 Final Review Sheet
Inventory Management
Page 13
Department of Decision Sciences
San Francisco State University
Linear Programming
1. A fertilizer manufacturer has to fulfill supply contracts to its two main customers (650 tons to
Cust_A and 800 tons to Cust_B). It can meet this demand by shipping from existing inventory from
any of its 3 warehouses. W1 has 400 tons of inventory on hand, W2 has 500 tons and W3 has 600 tons.
They would like to arrange the shipping for the lowest cost possible, where the per ton transit costs are
as follows:
W1
W2
W3
6.25
6.5
Cust_A 7.5
7
8
Cust_B 6.75
a) Explain what each of the six decision variables is: (writing V1 again doesn’t help)
V1 ________________________________________________________
V2 ________________________________________________________
V3 ________________________________________________________
V4 ________________________________________________________
V5 _________________________________________________________
V6 _________________________________________________________
b) Write out the objective function in terms of the variables (V1, V2, etc…) and the objective
coefficients
c) Are we MAXIMIZING or MINIMIZING the objective function?
d) Aside from nonnegativity of the variables, what are the 5 constraints? Write a short
description for each constraint, followed by writing out the formula (don’t forget to indicate
which type of equality/inequality)
After formulating and entering the linear program in Excel, the Solver gives you the following
sensitivity report:
Adjustable Cells
Final Reduced Objective Allowable Allowable
Cost
Coefficient Increase Decrease
Name Value
V1
0
1.5
7.5
1E+30
1.5
V2
100
0
6.25
0.25
0.75
V3
550
0
6.5
0.75
0.25
V4
400
0
6.75
0.5
1E+30
V5
400
0
7
0.75
0.5
V6
0
0.75
8
1E+30
0.75
Cell
$B$6
$C$6
$D$6
$E$6
$F$6
$G$6
Constraints
Cell
$H$7
$H$8
$H$9
$H$10
$H$11
e)
f)
g)
h)
i)
Final Shadow Constraint Allowable Allowable
Name Value
Price
R.H. Side
Increase Decrease
C1
650
6.5
650
50
550
C2
800
7.25
800
50
400
C3
400
0.5
400
400
50
C4
500
0.25
500
550
50
C5
550
0
600
1E+30
50
How many of the constraints are binding? (4)
How much slack/surplus is there with the nonbinding constraint(s)? (50 slack)
What is the range of optimality on Variable V3? (6.25 to 7.25)
If we could ship 10 tons less to Customer A (Constraint C1), how much money might we be
able to save? (65)
If we could chose to short either Cust_ A or Cust_B by 10 tons (C1 or C2), which would we
prefer to short? Why? (Explain using LP terminology to get credit) (Cust_B)
DS 412/BUS 786 Final Review Sheet
Linear Programming
Page 14
Department of Decision Sciences
San Francisco State University
2. Dr. Cholette hires undergraduate and grad assistants to help her grade quizzes. The going
hourly rate for undergraduates is $8/hour and for graduates is $12/hour. She has 80 quizzes to
grade, and she knows that grad students work faster (20 quizzes per hour) than undergraduates
(who can grade only 10 quizzes/hour). She also must comply with the following departmental
labor guidelines:
• No more than 8 hours in total spent grading quizzes.
• Must use at least 2 hours of undergrad labor
• Must use at least 1 hour of grad labor
She would like to keep her labor cost as low as possible, but needs to get all the quizzes graded
and comply with all labor guidelines (even if it means employing students to just sit there and
surf the web after all the quizzes have been graded). Solve this as a graphical LP. Plot and label
the constraints on the graph and lightly shade in the feasible region. Points will be deducted for
sloppiness. (Try using the edge of your notebook or a paper to draw a straight line, rather than
scribble it freehand!). (2 hours undergrad, 3 hours grad, total cost $52)
3. Kings Department Store has 625 rubies, 800 diamonds, and 700 emeralds from which they
will make bracelets and necklaces that they have advertised in their Christmas brochure. Each of
the rubies is approximately the same size and shape as the diamonds and the emeralds. Kings
will net a profit of $250 on each bracelet, which is made with 2 rubies, 3 diamonds, and 4
emeralds, and $500 on each necklace, which includes 5 rubies, 7 diamonds, and 3 emeralds.
Formulate as an LP problem to maximize profit?
4. Wilson Creek Farm has 200 acres of land available for planting three crops: corn, soybeans, and
wheat. The production yield, water requirements, and labor requirements for a salable crop are given
below. The owner expects to have only 35,000 gallons of water available per week to use for the
crops, and during the growing season he will only have 8000 personhours of labor available. The
expected profit per bushel of each crop is $1.00 for corn, $1.60 for soybeans, and $3.00 for wheat.
The owner can use any mix of crops, and he does not necessarily have to use all of the 200 acres of
his farm. Formulate a linear program that will maximize the farmer’s profit.
Crop
Corn
Soybeans
Wheat
Yield in bushels
per acre
300
200
80
Water required (in gallons per
acre per week)
200
150
125
Labor required in personhours per acre
35
40
30
5. Solve graphically the following linear program. On your diagram, indicate the Feasible Region
and the Optimal Solution. Find the values of X, Y and the objective function in the optimal solution.
Maximize
3X + Y
subject to
2 X + Y ≤ 40
X + 3 Y ≤ 80
X ≥ 0, Y ≥ 0
(X = 20, Y = 0)
DS 412/BUS 786 Final Review Sheet
Linear Programming
Page 15
Department of Decision Sciences
San Francisco State University
6. A manufacturer produces a product at three plants and sells it through two marketservice
warehouses. She wishes to maximize her total profit. The following data have been provided below.
Formulate this problem as a linear program.
Plant
A
B
C
Variable production cost per
unit
$0.40
0.35
0.45
From plant A
From plant B
From plant C
Max. annual capacity
(units)
40,000
30,000
45,000
Shipping cost per unit to warehouse
1
2
$0.20
0.20
0.20
0.10
0.45
0.30
Warehouse
Selling price per unit
1
2
$0.60
0.80
Max. annual demand
(units)
40,000
60,000
7. Solve graphically the following linear program. On your diagram, indicate the Feasible
Region and the Optimal Solution. Find the values of X, Y and the objective function in the
optimal solution. (X = 0, Y = 20)
Max
X + 3 Y ≤ 75
s.t.
3 X + Y ≤ 40
X, Y ≥ 0
8. For a cafeteria that is open 24 hours a day, the minimum number of waitresses required is
given below. Waitresses work 8hour shifts starting at Midnight, 4am, 8am, Noon, 4pm. Note
that no waitresses start work at 8pm. The regular pay is $15/hour from 8am to 5pm. From 5pm
to midnight, they earn 50% overtime and from midnight to 8am they earn double pay. Formulate
a linear program to minimize total daily cost. Note that these pay periods do not coincide neatly
with the 4hour periods given in the table.
Time of Day
Minimum Number of Waitresses
Midnight to 4am
4
4am to 8am
8
8am to noon
10
Noon to 4pm
7
4pm to 8pm
12
8pm to Midnight
4
DS 412/BUS 786 Final Review Sheet
Linear Programming
Page 16
Department of Decision Sciences
San Francisco State University
9. The Appleville School District has two high schools, each of which has a capacity of 4000
students. Approximately onethird of the high school students in the district are members of
racial minorities. The distrct is divided into four distinct communities. The number of students
in each community expected to attend a public high school next year and the distances from the
center of each community to each high school are given below. Historically, the two high schools
have been racially unbalanced, with school A having a disproportionately high enrollment of
minority students and school B having a disproportionately high enrollment of majority students.
To satisfy a court agreement to achieve better racial balance between the schools, each high
school must have at least 24% and no more than 44% of its enrollment made up of minority
students. The school district would like to determine how many students of each type (majority
and minority) should be sent from each community to each high school to minimize total student
bus miles traveled. Formulate this problem as a linear program.
Community No. of majority No. of minority Miles to high Miles to high
students
students
school A
school B
1
1900
250
3.4
1.5
2
1700
400
2.4
2.2
3
800
650
1.1
2.9
4
550
1250
1.7
2.8
Total
4950
2550
13. An appliance manufacturer produces two models of microwave ovens: H and W. Both
models require fabrication and assembly work; each H uses 4 hours of fabrication and 2 hours of
assembly, and each W uses 2 hours of fabrication and 6 hours of assembly. There are 600
fabrication hours available this week and 480 hours of assembly. The profit per unit is $40 for H
and $30 for W.
a) What is the mathematical LP formulation of the problem?
b) What are the optimal quantities of H and W and the maximum profits (using graphical
approach)? Show all work. (H = 132, W = 36)
10. A company has 3 manufacturing plants (in Atlanta, Tulsa, and Springfield) that produce a
product that is then shipped to the distribution centers; A, B, C and D. Each distribution center
needs to meet the demand of 10 truckloads of product each week. The capacity of each plant and
the shipping costs per truckload between plants and distribution centers are given in the table.
The company needs to determine how much to ship from each plant to each distribution center to
minimize total shipping cost, while not exceeding capacity and while meeting demand.
A
Distribution center
B
C
D
$800
$1100
$600
$1300
$1400
$1200
Plant
Atlanta
Tulsa
Springfield
$400
$600
$800
$700
$1000
$900
Plant Capacity (in
truckloads of
product each week)
13
18
12
Formulate an LP model. (Note: You don’t have to solve the problem)
DS 412/BUS 786 Final Review Sheet
Linear Programming
Page 17
Department of Decision Sciences
San Francisco State University
11. Kevin builds two kinds of birdhouses, one for wrens and a second for bluebirds. Each wren
birdhouse takes 4 hours of labor and 2 units of lumber. Each bluebird house requires 4 hours of
labor and 12 units of lumber. The craftsman has available 60 hours of labor and 120 units of
lumber. Wren houses yield a profit of $6 each and bluebird houses yield a profit of $15 each.
The linear programming formulation is:
Let xW = number of birdhouses for wrens, and xB = number of birdhouses for bluebirds.
Maximize 6xW + 15xB
Subject to:
4xW + 4xB < 60
2xW + 12xB < 120
xW, xB > 0
The following report is the sensitivity report:
Microsoft Excel 10.0 Sensitivity Report
Worksheet: [birdhousea.xls]Solution
Report Created: 5/25/2004 1:13:54 PM
Adjustable Cells
Cell
Name
$B$2 Quantity to make Wren Birdhouse
$C$2 Quantity to make Bluebird Birdhouse
Final Reduced Objective Allowable Allowable
Value
Cost
Coefficient Increase Decrease
6
0
6
9
3.5
9
0
15
21
9
Constraints
Cell
Name
$D$6 Hours of labor Used
$D$7 Units of Lumber Used
Final Shadow Constraint Allowable Allowable
Value
Price
R.H. Side
Increase Decrease
60
1.05
60
180
20
120
0.9
120
60
90
a) What is the optimal solution? (6 wren houses, 9 bluebird houses)
b) What is the optimal value of the objective function? (171)
c) If Kevin has 20 more units of lumber, how much would the optimal value of the objective
function increase? (18)
d) If the profit per Wren birdhouse increased to $12, would the optimal solution change?
Why or why not? (no)
12. A firm operates two farms of comparable productivity. Each farm has a certain amount of
usable acreage and a supply of labor hours to plant and tend the crops. Farm 1 has 600 acres and
2000 labor hours available per month. The corresponding figures are 800 acres and 2500 labor
hours per month for Farm 2. The firm is considering two crops for planting: corn and peas. The
maximum acreage that can be devoted to each crop is as follows: 900 acres for corn and 700
acres for peas. Corn requires 3 labor hours per acre per month, while peas require 5 labor hours
per acre per month. The expected profits are $250 per acre for corn, and $350 per acre for peas.
The firm needs to determine how many acres to devote to each crop on each farm. The objective
is to maximize total profit. Formulate as a Linear Programming problem. Do not solve.
DS 412/BUS 786 Final Review Sheet
Linear Programming
Page 18
Department of Decision Sciences
San Francisco State University
13. The Distribution Unlimited Co. will be producing the same new product at two different
factories, and then the product must be shipped to two warehouses, where either factory can
supply either warehouse. The distribution network available for shipping this product is shown
below, where F1 and F2 are the two factories, W1 and W2 are the two warehouses, and DC is a
distribution center. The amounts to be shipped from F1 and F2 are shown to their right, and the
amounts to be received at W1 and W2 are shown to their right. Each arrow represents a feasible
shipping lane. Thus F1 can ship directly to W1 and has three possible routes (F1DCW2, F1F2DCW2, and F1W1W2) for shipping to W2. Factory F2 has just one route to W2 (F2DCW2)
and one to W1 (F2DCW2W1). The cost per unit shipped through each shipping lane is shown
next to the arrow. Also shown next to F1F2 and DCW2 are the maximum amounts that can be
shipped through these lanes. The other lanes have sufficient shipping capacity to handle
everything these factories can send. The Distribution Unlimited Co must decide how much to
ship through each shipping lane while trying to minimize costs.
50 units
available
$900/unit
F1
W1
30 units
needed
$400/unit
$200/unit
10 units max
$200/unit
DC
F2
$100/unit
80 units max
$300/unit
W2
60 units
needed
40 units
available
a)
b)
c)
d)
e)
f)
$300/unit
What is the goal of the problem, in words?
What are the decisions that need to be made, in words?
What are the constraints, in words?
What are the decision variables? Explicitly define them.
Formulate the (linear) objective function in terms of the variables defined in part d).
Formulate the (linear) constraints in terms of the variables defined in part d).
14. Solve the following LP problem graphically. Clearly identify the feasible region (FR). What
are the optimal solution and its objective function value?
Maximize
subject to
X
2X
–X
X
X
–
+
+
Y
4Y
2Y
Y
=
≥
≤
≤
≥
≥
Z
8
6
6
0
0
(X = 6, Y = 0, Z = 6)
DS 412/BUS 786 Final Review Sheet
Linear Programming
Page 19
Department of Decision Sciences
San Francisco State University
MRP
1. The following table lists the components used in assembling FGA. Also included for each
component are the following info: the onhand supply, lead time, and direct components. (Note:
In industry the “SA” prefix is sometimes used to denote SubAssembly, and “FG” is used to
denote Finished Goods.)
Item
FGA
OnHand
0
SAB
SAC
SAD
E
F
0
0
0
10
5
LT
Components
(weeks)
1
SAB, SAC(2), SAD(2)
1
SAD(2)
2
E, F(2)
2
E(3)
1
3

a) Given this information, show the BillofMaterial associated with 1 unit of FGA
b) What is the total lead time (in weeks) associated with making an item of FGA, assuming
we had no starting onhand for any part? (6)
c) If we wanted to make one FGA, would we need to order any more of either E or F?
Why/ Why not? (yes, no)
2. Draw the part explosion diagram, with low level coding, for the following indented bill of
materials.
Number required

Lead time
Part number at next level up  Component
(weeks)
    A
1

A
1
B
2

B
2
F
2

C
1
E
3

D
2
C
4

E
1
E
1

F
1
D
2

F
2

F
3

(Continuation of previous question) The Master Production Schedule calls for 5 items of item A to
be delivered in week 8. There are no units on hand of any of the items. Complete the MRP tables
for the 6 parts A thru F.
DS 412/BUS 786 Final Review Sheet
MRP
Page 20
Department of Decision Sciences
San Francisco State University
3. A company produces product A, which consists of 2 units of component B, as well as 3 units
of component C. Each unit of C consists of one unit of D, as well as 3 units of B.
a) Draw the product structure tree.
b) Given the following information, construct the timephase plan for the production of A.
When you are finished, we should also know the inventory status of all components in
period 7.
A
QUANTITY
A
B
C
D
0
ONHAND
20
400
200
100
1
2
LOT SIZE
lotforlot
150
100
lotforlot
3
120
4
5
SCHEDULED RECEIPTS
300 in period 4
50 in period 2
6
7
200
LEAD TIME
1
2
1
1
4. Consider a bicycle whose product structure tree is shown below.
Bicycle


Wheel (2)


Seat (1)

Hub (1)

Chain (1)

Pedal (2)

Spoke (36)
Note the following important information:
• The Master Production Schedule requires 40 bicycles in week 4 and 55 in week 7.
• 50 wheels and 1760 spokes are already on hand.
• 16 wheels are scheduled to be received in week 2.
• All items have a 1week leadtime.
Using the LFL rule, develop the MRP tables for bicycle, wheel and spoke only.
5. A piano has 52 white keys and 36 black keys. The Hayama piano company expects to sell 70
pianos in the month of June. It currently has 8 pianos in stock. It also has 726 white keys and
418 black keys in stock. Safety stock is 450 for white keys and 300 for black keys. How many
white keys and how many black keys should Hamaya purchase? (2948 white, 2114 black)
6. A product A is assembled using 2 units of B and 3 units of C. Each unit of B requires 2 units
of D, while each unit of C requires 1 unit of D. There are currently 4 units of A, 12 units of B, 15
units of C, and 24 units of D in stock.
a) If there is a demand for 50 units of A in week 6, how many units of each item should be
produced? (46 A, 80 B, 123 C, 259 D)
b) If the lead times are 1 week for A, 2 weeks for B, 1 week for C, and 1 week for D, when
should production of D be commenced? (week 2)
DS 412/BUS 786 Final Review Sheet
MRP
Page 21
Department of Decision Sciences
San Francisco State University
Project Management
1. Below is a table of activities associated with a project, their durations, and what activities
each must precede.
activity
A (start)
B
C
E
F(end)
a)
b)
c)
d)
Duration
1 wk
1 wk
4 wks
2 wks
2 wks
Precedes
B,C
E
F
F

Draw an Activity on Nodes Diagram of the project, including activity durations.
Define the Critical path, by listing all critical activities in chronological order.
What is the project duration (in weeks)? (7)
What is the slack (in weeks) associated with any and all noncritical paths through the
project? (1)
2. A family has decided to remodel their kitchen. They will be doing so without the help of an
interior designer, though they will be having contractors do all the work. The project will involve
replacing the cabinets, the floors, and the countertop, as well as painting the walls. No new
appliances will be purchased. Before the work can be begun, the family must measure the
kitchen, decide on a layout, and find a budget. These activities take 0.5 days, 2 weeks, and 1
week, respectively. The family already knows which vendor they wish to purchase cabinets
from, but they must still find the general contractor, as well as the stores from which they will
purchase the floors and the countertop. Finding the contractor takes 1 week, while the other 2
activities take 2 weeks each. Once the stores have been selected, it takes 2 days and 3 days,
respectively, to pick the flooring and countertop. Choosing cabinets is slightly more involved and
will take a total of a week. Once the cabinets have been purchased, they will be manufactured
and delivered in 3 weeks. Packing up the existing kitchen takes 2 (very intense) days, while
demolition of the kitchen takes an additional 3 days. Installation of the new cabinets takes 1.5
weeks (including trim and detail work). Once the cabinets are installed, the countertops can be
installed, which will take 1 day. After this is complete, walls can be painted, which takes 2 days.
Finally, the flooring can be installed, taking another 2 days. Unpacking everything into the
kitchen takes 2 days.
a) Explicitly list the activities that must be completed, along with their expected durations,
so the kitchen project can be considered complete. For simplicity, assume that a week
consists of 5 days.
b) Construct a network diagram for the activities listed in part a).
c) Determine the critical path for the network using the ES/EF/LS/LF algorithm. Are there
several critical paths? What is the length of the “project”? (You do not have to list all
the times in a table, it is sufficient to have them in the graph if they are legible.)
DS 412/BUS 786 Final Review Sheet
Project Management
Page 22
Department of Decision Sciences
San Francisco State University
3. Consider a project having the following seven activities:
Activity
a)
b)
c)
d)
Immediate
Predecessor
Optimistic
Time (weeks)
Most likely
Time (weeks)
Pessimistic
Time (weeks)
A
none
2
3
4
B
A
4
4
8
C
A
3
5
7
D
B
5
5
5
E
B, C
3
6
7
F
D
4
5
9
G
E, F
3
3
7
What is the expected time for each activity?
Draw the network & find the expected project completion time. (21.84)
What is the critical path? (ABDFG)
What is the probability that the project will be completed in less than 24 weeks? (0.951)
4. Consider the following AON network. All activity times are in weeks.
5
3
D
A
13
Start
14
B
F
4
8
E
C
a) What is the critical path? (BF)
b) What is the project duration? (27)
c) Consider the following costs of shortening (crashing) the activities. Find the least costly
way to shorten the project by one week. What is the cost? ($1000)
Activity
A
B
C
D
E
F
Crash cost per week
$700
$1000
$1500
Cannot be shortened
Cannot be shortened
$3000
DS 412/BUS 786 Final Review Sheet
Project Management
Page 23
Department of Decision Sciences
San Francisco State University
5. A project consists of seven activities. The precedence relationships are given in the following
table.
Activity
Time (days)
Immediate Predecessors
A
5
B
18
A
C
13
A
D
10
B
E
4
D
F
11
C,D
G
9
E,F
a) Construct an AON diagram for this project.
b) Find the critical path and calculate the length of project. (ABDFG, 53)
6. Consider the following project network:
a) Fill in the missing values for the expected duration and variances.
Activity Optimistic Most likely Pessimistic Expected Variance
6
8
10
8
A
4
9
13
2.25
B
8
6
7
7
C
5
4
4.5
4.5
D
0.5
3
5.5
3
0.694444
E
b) What is the expected duration of the project? Do you have any concerns about the ability
to complete the project within this time period? (12.5)
c) What is the probability the project will be able to complete within 14 days? (0.98574)
d) Given the following crashing costs, how much will it cost to reduce the project’s duration
by 6 days, at minimum possible cost? (An activity can be crashed down to 1 day’s
duration. Crashing fractional days is possible; in this case, the crashing cost is prorated.)
($88.14)
Activity
Crash Cost/Day
$15
A
$8
B
$10
C
$2
D
$5
E
DS 412/BUS 786 Final Review Sheet
Project Management
Page 24
Department of Decision Sciences
San Francisco State University
7. Computer Nerds, Inc., is planning a project to develop a commercial software package for
PC’s. They have come up with the following list of activities, times and precedence
relationships.
Activity Time
Immediate
Symbol
Activity Description
Predecessors
(weeks)
––
P
Perform market survey
3
P
D
Design graphic icons
4
P
F
Develop flow chart
2
D, F
S
Design input/output screens
6
U
Code Module 1
5
F
F
V
Code Module 2
3
W
Code Module 3
U
7
X
Code Module 4
U, V
5
T
Merge modules & Test program
S, W, X
8
a)
b)
c)
d)
Draw the project as either a CPMnetwork or a PERTnetwork (your choice).
What are the critical activities? Use the 2pass method. (PFUWT)
How many weeks will the project be delayed if activity V is delayed by 5 weeks? (1)
Suppose the direct cost for crashing each activity is given in the table below. Also, the indirect
costs charged to the project total $10,000/week. Based on this information, which activities
should be crashed and for how many weeks? (W by 2 weeks)
Activity
P
13
Crash cost per week (in $1,000)
Max. # of weeks activity can be crashed 2
DF
9 17
41
U
VW
N/A 9 7.5
0
12
Crash Time
(days)
3
1
4
3
Normal Cost
($)
300
250
400
150
XT
8 11
31
8. A project is comprised of 4 tasks as shown below.
Task
Predecessor
A
B
C
D
B
A, C
Normal
Time (days)
4
3
7
5
Crash Cost
($)
500
325
550
250
Which tasks should be crashed to meet a project deadline of 13 days at minimum cost? Find the
total cost of crashing. (B, $150 or $75, depending on your interpretation)
DS 412/BUS 786 Final Review Sheet
Project Management
Page 25
Department of Decision Sciences
San Francisco State University
Quality Management
1. Odwalla’s OJ is packaged in 250 ml bottles and has a process standard deviation of
10 ml. In monitoring the fill process, 6 samples (of 25 bottles each) were collected and averaged.
a)
b)
c)
d)
Sample
Sample mean
s1
249
s2
252
s3
253
s4
248
s5
245
s6
253
What is the total number of bottles sampled? (150)
What is the standard deviation of the distribution of sample means? (2ml)
Compute the 3sigma control limits. Is the process in control, given these control limits?
Why/Why not? (244, 256, yes)
Compute the 2sigma control limits. Is the process in control, given these control limits?
Why/Why not? (246, 254, no)
2. An avantgarde clothing manufacturer runs a series of high profile, risqué ads on a billboard
on Hwy101 and regularly collects protest calls from people who are offended by them. They
have no idea how many people in total see the ad, but they have been collecting statistics on
complaints.
type
Description
#complaints
Offensive
10
R
racially/ethnically
4
M
Demeaning to men
14
W
Demeaning to women
I
6
Ad is Incomprehensible
2
O
other
a) Depict this data with a pareto chart. Also depict the cumulative complaint line.
b) What percent of the total complaints can be attributed to the most prevalent complaint?
(39%)
The ad agency also tracks the complaints by week received.
week
#complaints
1
4
2
5
3
4
4
11
5
3
6
9
c) What type of control chart would we use to monitor this process? And why? (cchart)
d) What are the 3sigma control limits for this process? Assume the historical complaint
rate is unknown. (0, 13.3)
e) Is the process mean in control, according to the control limits? Why/Why not?
f) Assume now that the historical complaint rate has been 4 calls a week. What would the
3sigma control limits for this process be now? Is the process in control? (0, 10, no)
DS 412/BUS 786 Final Review Sheet
Quality Management
Page 26
Department of Decision Sciences
San Francisco State University
3. A state department of tourism and recreation collects data on the number of cars with outofstate license plates in a state park. (The group's position is that more outofstate plates means the
state's advertising programs are working.) The sample size is fixed at n=100 each day. Data
from the previous 20 days indicate the following number of outofstate license plates:
Day Plates Day Plates Day Plates Day Plates
1
24
6
31
11
22
16
22
2
33
7
15
12
15
17
34
3
18
8
18
13
25
18
10
4
21
9
23
14
33
19
23
5
26
10
13
15
16
20
13
a) Calculate the overall proportion of “tourists” (cars with outofstate plates). (21.75%)
b) Calculate the LCL and UCL for these data. (9.37%, 34.13%)
c) Plot the control chart. Are all points within the control limits? (yes)
4. A part that connects two levels should have a distance between the two holes of 4". It has
been determined that Xbar and Rcharts should be set up to determine if the process is in
statistical control. The following ten samples of size four were collected. Calculate the control
limits, plot the control charts, and determine if the process is in control.
Mean
4.01
Sample 1
3.98
Sample 2
4.00
Sample 3
3.99
Sample 4
4.03
Sample 5
(3.97 to 4.03, 0 to 0.096, yes)
Range
0.04
0.06
0.02
0.05
0.06
Sample 6
Sample 7
Sample 8
Sample 9
Sample 10
Mean
3.97
4.02
3.99
3.98
4.01
Range
0.02
0.02
0.04
0.05
0.06
5. A mean chart is used to maintain the diameter of plastic tubing. The process is known to
have a standard deviation of 0.15cm. Five samples of three observations each are used to
construct the mean chart.
Sample
1
2
3
4
5
1
10.0
10.1
9.8
9.9
9.8
Observations (centimeters)
2
3
9.9
10.0
10.1
10.3
10.0
10.1
9.9
9.8
9.9
9.9
a) Compute x . (9.97)
b) Compute threesigma limits for an x chart. (9.53, 10.23)
c) Is the process incontrol? Why or why not? (yes)
d) Suppose that the process mean shifts to 10.2. (Assume that the standard deviation does
not change.) What is probability of a Type II error? (That is, using the control limits
calculated in part (b) what is the probability that the mean chart will indicate an incontrol point in a sample of three observations.) (78.4%)
DS 412/BUS 786 Final Review Sheet
Quality Management
Page 27
Department of Decision Sciences
San Francisco State University
6. The specification for a plastic handle calls for a length of 7.0 inches ± 0.2 inches. The
standard deviation of the process is estimated to be 0.05 inches. The process is known to operate
at a mean thickness of 7.0 inches.
a) What are the upper and lower specification limits for this product? (6.8, 7.2)
b) What is the Cp for this process? (1.33)
c) Is this process capable of producing the desired part? Explain.
7. Ten samples of size four were taken from a process, and their weights measured. The sample
averages and sample ranges are in the following table. Construct and plot an Xbar and Rchart
using these data. Is the process in control? (no)
Sample
Mean
Range
1
20.01
0.45
2
19.98
0.67
3
20.25
0.3
4
19.9
0.3
5
20.35
0.36
6
19.23
0.49
7
20.01
0.53
8
19.98
0.4
9
20.56
0.95
10
19.97
0.79
8. Western Airlines merged with a smaller airline recently. Every day since the merger, a
random sample of 30 Western flights has been analyzed. A flight that takes off more than 10
minutes after its scheduled departure time is considered to be late; otherwise, it is not late.
Western’s performance over the past 10 days is shown in the table below.
Day
# Late Flights
1
4
2
2
3
5
4
9
5
2
6
3
7
3
8
1
9
2
10
8
What are the 95.5% control limits for p, the true percentage of late flights? No chart is required.
Has Western Airlines’ ontime performance been incontrol over the last 2 weeks? Explain
briefly. (0.7%, 25.3%; no, it’s been outofcontrol as 2 of the 10 samples are above the
UCL.)
9. Whole Grains LLC uses statistical process control to ensure that their healthconscious, lowfat, multigrain sandwich loaves have the proper weight. Over the past few days, they have taken
five random samples of four loaves and have found the following.
Sample #
Net Weight (oz.)
Loaf #1
Loaf #2
Loaf #3
Loaf #4
6.3
6.0
5.9
5.9
1
6.0
6.1
6.3
5.8
2
6.3
4.8
5.6
5.3
3
6.2
6.0
6.2
5.9
4
6.5
6.4
6.5
6.8
5
a) Calculate control limits for the range chart. (0, 1.41)
b) Calculate control limits for the mean chart. (5.61, 6.51)
c) Is the process in control? Explain. (no – sample 5’s average of 6.55 is above the UCL)
DS 412/BUS 786 Final Review Sheet
Quality Management
Page 28
Department of Decision Sciences
San Francisco State University
10. A carpet manufacturer wishes to set up a cchart for its manufacturing process. It inspects 10
carpets at random, and notes the number of defects in each carpet, as follows.
Carpet # 1 2 3 4 5 6 7 8 9 10
# of defects 2 3 0 2 1 2 1 3 4 2
a) Determine twosigma control limits for the mean number of defects per carpet. (0, 4.83)
b) Three carpets are subsequently inspected, and the # of defects are 1, 6, 2. Is the process
in control? Explain why. (yes)
11. DataRite Inc. is a large data processing center. It wishes to set up control limits for the
proportion of records which have data entry errors. Accordingly, the section supervisor chooses
100 records each day over a 10day period. The number of records (out of 100) which have
errors on each of the 10 days is shown below:
1 2 3 4 5 6 7 8 9 10
Carpet #
# of records 5 7 8 4 9 6 7 3 5 6
Set up twosigma control limits for the proportion of records with errors. (0.0126, 0.1074)
DS 412/BUS 786 Final Review Sheet
Quality Management
Page 29
Department of Decision Sciences
San Francisco State University
Aggregate Planning
1. It is the end of November and the South Central nuclear generator plant currently employs 350
fully trained workers and has these manpower needs over the next five months:
Dec
Jan
Feb
Mar
Apr
Month
Manpower needed (in hours) 40,000 45,000 35,000 50,000 45,000
According to state law a reactor employee can actually work no more than 130 hours per month. It is
plant policy that if more trained employees are available than are needed in any month, each worker
is still fully paid, even though he or she is not required to work the 130 hours.
The training of each new employee requires one month of oneonone classroom instruction before
he/she is permitted to work alone in the reactor. Therefore, the company must hire trainees one
month before they are actually needed. Each trainee teams up with a trained worker and requires 90
hours of that employee's time, meaning that 90 hours less of that technician's time are available that
month for actual reactor work.
Personnel department records indicate that 5% of the trained technicians at the start of any month
resign by the end of that month. An employee works the full month in which he resigns, is paid for
that month, then leaves the company without any additional compensation.
A trained worker earns $2000 per month (regardless of the number of hours actually worked, as
noted above). Trainees are paid $900 during their one month of instruction.
Formulate this staffing problem as a linear program.
2. Tavisbond Manufacturing Company makes highgrade pipe for the oil and chemical
industries. Tavisbond must plan its production for the next four months: March to September.
The forecast demands (in thousands of feet) for its pipe are below. Tavisbond can make 75,000
feet of pipe per month using regulartime production at a cost of $1.25 per foot. Tavisbond can
make up to 15,000 more feet of pipe each month using overtime production at a cost of $1.50 per
foot. Any pipe made in one month and sold in a subsequent month incurs an inventory holding
cost of $0.15 per foot per month. Tavisbond expects to end April with 5,000 feet of pipe in
inventory and would like to end August with 10,000 feet of pipe in inventory. Formulate this
program as a linear program to minimize total cost.
May June July August
70
80
90
100
DS 412/BUS 786 Final Review Sheet
Aggregate Planning Supplement
Page 30
Department of Decision Sciences
San Francisco State University
Capacity Planning
1. Ted plans to sell two different types of soy shakes at an upcoming concert at Stern Grove. He
expects that he can sell 50 strawberry shakes and 90 chocolate shakes during the entire 2 hours of
the concert. He can chose to rent one of two different types of machines, a ShakeMaker, or the
faster but more expensive SuperShaker. The time each type of shake takes for each machine is
shown below.
ShakeMaker
Berry
minutes
needed
3 / shake
SuperShaker
Berry
Chocolate
Minutes
needed
2/shake
3/shake
Chocolate
4 / shake
a) If Ted rents ShakeMakers, how many would he need to satisfy all expected demand?
(Hint Use the tables above to help with these questions you’ll find it helpful to fill in the
blank spaces as appropriate) (4.25 or 5)
b) If Ted rents SuperShakers, how many would Ted need to satisfy all expected demand?
(3.08 or 4)
c) If ShakeMakers costs $45/each to rent and SuperShakers costs $55/each to rent, which
type should Ted choose to rent in order to minimize costs? (SuperShaker)
2. Andre is investigating setting up a crepe stand on campus. He could rent space in the student
union (which costs $300/month in rent and overhead). His materials and labor costs are $1 per
crepe, and the sale price is $4 per crepe.
a) What is the break even quantity for this option? (I.e., How many crepes per month would
Andre have to sell before making a profit) (100)
b) Andre could use a portable crepe maker from a friend and set up a booth outside the
union. He’d have no rent or other general overhead costs (i.e. no fixed costs), but his
friend would demand $1.50 per crepe sold. What is the break even quantity for this
option? (0)
c) Assume that an informal survey shows that Andre could expect 350 crepes to be sold per
month. Which capacity option should he elect (Student_Union_Stand or
Portable_Crepe_Maker) and what would his total monthly profit be on the better option?
(SU, $750)
d) By how much (and in what direction) would the demand have to be different before he
would consider switching to the other capacity option? (150 less)
3. The local convenience store makes personal pan pizzas. Currently, their oven has a fixed cost
of $3000 per month and a variable cost of $0.50 per pizza. Pizzas currently sell for $5.00 each.
The owner is considering a brick oven. It has a fixed cost of $6000 per month and a variable cost
of $0.50 per pizza. Since brick oven cooked pizzas are the latest trend, the owner will be able to
charge $6.00 per pizza.
a) If the owner expects to sell 9000 pizzas per month, should she get the new oven? Show
your work. (yes)
b) For what range of volume of pizzas per month is the brick oven preferred over the current
oven? (BEP = 3000)
DS 412/BUS 786 Final Review Sheet
Capacity Planning Supplement
Page 31
Department of Decision Sciences
San Francisco State University
4. The National Bargainer Magazine is produced each week. The total number of pages depends
on the amount of advertising, which varies from week to week. Two printing companies are capable
of printing the magazine: ABC Co. charges $0.005 (half of one cent) for each page it prints, XYZ
Co. charges 2.5 cents per page. However, XYZ is located in the city where the magazine is
distributed and it charges $300 to ship the printed copies to the distributors each week, whereas ABC
is located 50 miles away and therefore has to charge $1000/week for shipping. (These shipping
charges are not dependent on the quantity shipped.)
a) Draw the breakeven diagram. Do all the right things: Choose sensible scales, label the axes,
label the lines.
b) What is the breakeven point in the above example? (I want a number, not a definition).
(35,000)
c) Complete the blanks in the following sentence:
a) If the number of printed pages is less than ___________ the magazine should be printed by
_________; otherwise it should be printed by the other printing company. (35,000, XYZ)
d) A third printing company is thinking of competing for the business. It knows that it has to
charge $750 for shipping. What is the most it can charge per page to have any chance of
taking some business away from ABC or XYZ? (1.2 cents)
5. Kirian and Sarah plan to buy and run a Bed and Breakfast establishment in Northern
California. They are looking at 2 different inns to purchase. Inn A would run $2000/month in
fixed costs (mortgage, utilities, etc.) and has 6 rooms suitable for guests. Inn B is larger, with 8
rooms available for guests, but fixed costs are higher at $3000/month. They will charge $110 per
night per room let. The variable costs for either property would be only $24 per night per room
let (these costs are associated with cleaning and making breakfasts), regardless of which inn is
purchased.
a) What is the break even quantity (number of rooms let in a month) necessary to prevent
Inn A from losing money? (23.3 or 24)
b) What is the break even quantity (number of rooms let in a month) necessary to prevent
Inn B from losing money? (34.9 or 35)
While local city folk flock to the area on weekends, nobody comes up during the week. With
little revenue expected during the week, Kirian and Sarah plan to keep their Inn open only 10
nights each month. Furthermore, they expect to average exactly 50% occupancy (for example, an
8 room inn would let 4 rooms) on those open nights.
a) What would the monthly profit or loss be on Inn A? ($580)
b) What would the monthly profit or loss be on Inn B? ($440)
c) Which inn should Kirian and Sarah purchase?
d) Why? (valid business reason, please)
6. A small firm intends to increase the capacity of a bottleneck operation by adding a new
machine. Two alternatives, A and B, have been identified, and the associated costs and revenues
have been estimated. Annual fixed costs would be $40,000 for A and $30,000 for B; variable
costs per unit would be $10 for A and $11 for B; and revenue per unit would be $15.
a) Using alternative A, how many units must be produced and sold to realize a profit of
$5,000. (9000)
b) Calculate each alternative’s breakeven point in units. (8000, 7500)
c) Determine the range of output volume for which alternative A yields the higher profit.
(10000 and higher)
DS 412/BUS 786 Final Review Sheet
Capacity Planning Supplement
Page 32
Department of Decision Sciences
San Francisco State University
7. The UMC Inc. currently buys toasters from Russia for US$110 each. It is now trying to decide
whether to manufacture the toaster in its factory in Sweden or its factory in Tajikistan. The
development and production costs (in U.S. dollars) of making the toaster in these two countries are:
Onetime
development
cost (US$)
Sweden
Tajikistan
50,000
100,000
Production
cost per
module
(US$)
35
10
The company therefore has three alternative ways of obtaining the toaster, including purchasing it
from Russia (in which case nothing more needs to be spent on development).
a) Draw a breakeven diagram for this situation. Do all the right things: label the axes, label
the lines, scale each axis, use sensible scales, draw straight lines, etc. (Hint: Your diagram
should have THREE lines, one for each way of obtaining the toaster.)
b) Identify the breakeven point(s). (667, 2000, 10000)
8. A company producing custommade teddy bears is considering several options for expanding
their existing capacity. There are 3 possibilities. The first is a lowend machine, which would
take 10 minutes/bear for the (machined) manufacturing process. In addition, an average of 30
minutes of detail work would have to be done by hand (per bear). The second option is a highend machine, which would take 8 minutes/bear, and reduce the amount of hand detail work to 25
minutes/bear. The final option is to subcontract out the bears. The subcontractor is willing to
provide up to 400 bears per year for a flat fee of $2,000. Additional bears would cost $8 each.
There is no difference in bear quality between the 3 options. It costs $10,000 to buy the lowend
machine. Yearly maintenance is $1,000. The purchase price for the highend machine is
$15,000, while maintenance is $2,200. Management estimates the cost for running a machine at
$6/hour. Labor costs are $15/hour. Assume the factory runs 350 days/year for 8 hours/day.
a) If you expect a yearly demand of 12,000 bears, which option is the cheapest over a 3year
time horizon? (highend machine)
b) If the service times on both machines are Exponentially distributed, and the job arrivals
have a Poisson distribution with a rate as specified in part a), what is the expected time
between the job’s “arrival” at the factory to the time it is complete and can leave?
(Assume that there are more than enough workers to cover the hand detail work.) (uses
queueing theory – 63.3 minutes and 43.18 minutes)
c) At what arrival rates (demand levels) would the different options make sense, given a 3year time horizon? (buy always better than LE; 8842)
DS 412/BUS 786 Final Review Sheet
Capacity Planning Supplement
Page 33
Department of Decision Sciences
San Francisco State University
JustInTime
Best Industries Inc. manufactures automobile transmissions. It uses the kanban system of
inventory control. Each standardized container can hold 3 transmissions, and there are 10
containers in use. The demand rate for transmissions is 10/hour, and the lead time to produce a
batch of transmissions is 2 hr. 30 mins. What is the safety stock (efficiency) factor? (0.2)
DS 412/BUS 786 Final Review Sheet
JIT Supplement
Page 34
Department of Decision Sciences
San Francisco State University
Queueing Theory
1. A crew of mechanics at the Highway Department garage repair vehicles that break down at
an average of λ = 7 vehicles per day (approximately Poisson in nature). The mechanic crew can
service an average of μ = 11 vehicles per day with a repair time distribution that approximates an
exponential distribution.
a) What is the utilization rate for this service system? (0.6364)
b) What is the average time before the facility can return a breakdown to service? (That is,
how long does it take between the time the breakdown occurs and the time that the
vehicle can return to service?) (0.25 days)
c) How much of that time is spent waiting for service? (0.159 days)
d) How many vehicles are likely to be in the system at any one time? (1.75)
2. Many of a bank’s customers use its automatic teller machine to transact business after normal
banking hours. During the early evening hours in the summer months, customers arrive at a
certain location at the rate of one every other minute. This can be modeled using a Poisson
distribution. Each customer spends an average of 90 seconds completing his or her transactions.
Transaction time is exponentially distributed. Determine:
a) The average time customers spend at the machine, including waiting in line and
completing transactions. (6 minutes)
b) The probability that a customer will not have to wait upon arriving at the automatic teller
machine. (0.25)
c) The average number of customers waiting to use the machine. (2.25)
DS 412/BUS 786 Final Review Sheet
Queueing Theory Supplement
Page 35
Department of Decision Sciences
San Francisco State University
Simulation
1. The Seven Hills power generator breaks down very frequently. The time required to repair the
generator is as follows:
Repair time
required (hours)
1
2
3
Probability
.3
.5
.2
When he repairs the generator, the repairman may, if he wishes, take an extra 30 minutes to lubricate
the generator. This has an effect on the time the generator will then run until it fails again, as shown
in the following table. (For example, the chance that the generator will run continuously for exactly
2 hours is 35% if it was not lubricated but 10% if it was lubricated.)
Running time
Probability
between failures (hours)
If not lubricated If lubricated
0.5
.06
.05
1.0
.07
.05
1.5
.17
.05
2.0
.35
.10
2.5
.20
.25
3.0
.15
.30
3.5
.00
.20
a) Simulate the operation of the generator until the 4th breakdown has been repaired, assuming
that the repairman does not lubricate the generator. At the start of the simulation, assume
that the generator has just started running after a nonlubricated repair.
b) Simulate the operation of the generator until the 4th breakdown has been repaired, assuming
that the repairman does lubricate the generator. At the start of the simulation, assume that
the generator has just started running after a lubricated repair. (Remember to add 30 minutes
to each repair time in this case.)
On the basis of your simulation, what percentage of the time was the generator running in (a) and
(b)? Which of the two methods therefore seems more effective?
Remember to specify the random number ranges, and the random numbers used in your simulation.
DS 412/BUS 786 Final Review Sheet
Simulation Supplement
Page 36
Department of Decision Sciences
San Francisco State University
2. The number of inches of snow dropped by a winter storm in the Sierras has the following
distribution:
Number of inches of snow deposited Probability
1
0.05
2
0.10
3
0.20
4
0.30
5
0.25
6
0.10
In between storms, there is sunshine, causing the snow to melt according to the following
distribution:
Number of inches of snow that melt Probability
between successive storms
1
0.63
2
0.20
3
0.10
4
0.07
Assume the first storm of the winter has just finished and has deposited 4 inches of snow. Simulate
until the sixth storm has finished. What is the height of the snow pack at that point in time? Use the
following random numbers. Read them from left to right.
52 37 82 69 98 96 33 50 88 90 50 27 45 81 66 74 30 59 67 60 60 80 53 69 37 06 63 57 02 94 52 69
33 32 30
DS 412/BUS 786 Final Review Sheet
Simulation Supplement
Page 37
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