This study guide is meant to give you an idea of what to expect on the final exam. There will three or four questions that will cover the material on linear and integer programming. These questions may arbitrarily combine the model elements listed below. One of these questions will be spreadsheet/solver-based, and the rest will be algebra-based.
Linear Programming Model Elements

Resource allocation/production planning, diet

Process models (one process making multiple things, one operation feeding material into another, etc.)

Multi-period inventory/production

Transportation models

Blending constraints

Investment models

Project scheduling, including crashing (note also the trick of minimizing the maximum of a number of things, as we did with “declare done” in project scheduling)
(Mixed) Integer Programming Model Elements

Definition of integer and binary variables

Knapsack constraints

Logical constraints (one of n possibilities must or may be true, etc.)

Assignment models, grids of binary variables

Fixed charge models, logical upper and lower bounds

Set covering constraints
There will also be two questions on probability and simulation, based on YASAI spreadsheets.
These questions will draw on the following topics:
Probability, Simulation, and YASAI

Random variables (what are they?), and specifically

Poisson

Binomial (including setting the first argument to 1 to get a 0/1 result)

Uniform

Normal

From a table (GENTABLE)

How to use YASAI, including

PARAMETER (including testing all combinations of several parameters)

SIMOUTPUT

YASAI dialog box

Interpretation of output report, including percentiles
OM Final Exam Study Guide Page 1 of 37 Spring 2002


Using a 0/1 output to estimate the probability of an event

Static models (like NEWSBOY)

Dynamic models (like INVENTORY)

Using the central limit theorem to approximate a sum of lots of independent things as a single Normal random variable
We will prepare a separate review handout on the probability material. It may also be helpful to recall how the following Excel functions work:
IF INDEX SUMPRODUCT
INT MIN TRANSPOSE
MAX SUM SQRT
The exam will be three hours long. The ground rules for the test are similar to midterm, except that two (double-sided) sheets of notes are permitted. You may want to use these sheets to remember the syntax of various Excel and YASAI functions. The sheets must be handwritten in your own handwriting.
The only other materials allowed in the exam will be a calculator (which should not be absolutely necessary, and a dictionary, if English is not your native language.
The rest of this handout contains sample problems. Notes:

We used to divide questions for linear and integer programming into separate categories. We may now ask questions that combine topics from linear and integer programming (as in
Producing Biotech Chemicals below).

Some of the simulation problems were based on a different simulation package, @Risk, which we no longer use. It is quite similar to YASAI, although the output reports have a different format. We have tried to adapt the questions to YASAI as much as possible. You should ignore the “minimum” and “maximum” columns in the @Risk output reports
You may also wish review the midterm 1 and midterm 2 practice materials, and in particular the problems DigiNav , Simulating Semiconductor Fabrication, and Selling Dresses by Catalog .
OM Final Exam Study Guide Page 2 of 37 Spring 2002

Insurance Advertising
The Great Benefit insurance company sells automobile and house insurance. In order to increase the number of new policies written, the firm plans to advertise on TV and to send letters to new area residents. Earlier experience shows that each $1000 spent on TV ads results in 25 new automobile insurance policies and 6 new house insurance policies. Each letter to a new resident has an 0.3% chance of yielding a new house insurance policy and an 0.4% chance of generating a new automobile insurance policy. The letters cost $0.25 each. Great Benefit wants to generate at least 3000 expected new insurance contracts via the advertising campaign, under the condition that at least 30% of them are for house insurance.
Algebraically write a linear programming model to minimize the cost of Great Benefit’s promotion. Clearly define your variables. Feel free to skip algebraic and numeric simplifications. Use only continuous variables (do not worry about variables having fractional values).
Issuing Bonds
Based on current commitments from and to customers, the Enormous Construction Company is expecting the following cash flow from operations over the next four years:
Year 1 2 3 4
Cash Flow $(120) $20 $20 $110
All dollar amounts are in millions, and parentheses indicate a negative number (cash outflow).
The firm currently has $55 million on hand. The firm has a policy of planning to have at least
$25 million on hand at all times in order to cover contingencies and emergencies. Cash on hand earns 4.5% interest per year.
To cover its capital needs, the firm can issue bonds and/or take short-term loans from the money market. Money market loans can be taken in any of years 1 through 3, and must be repaid in full the following year, along with 10% interest. There are three kinds of bonds the firm is allowed to issue: A, B, and C. At most $60 million of each kind of bond can be issued. The cash flow profiles of the bonds are below. For example, each dollar of type C bonds issued gives the company one dollar in year 1, in return for three payments of $0.40 in years 2 through 4.
Year 1 Year 2 Year 3 Year 4
Bond A $ 1.000 $(0.085) $(0.085) $(1.085)
Bond B $ 1.000 $(0.080) $(1.080) $ -
Bond C $ 1.000 $(0.400) $(0.400) $(0.400)
The firm would like to find the combination of bonds and short term borrowing that gives it the maximum possible cash balance in year 4. For this purpose, they are using the spreadsheet shown below (the optimal solution is displayed):
OM Final Exam Study Guide Page 3 of 37 Spring 2002

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2
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A
Cash Flow from Operations
Minimum Cash Balance
Bond A
Bond B
Bond C
Short Term Interest Rates
On Money Market Loans
On Cash Balances
Value of Bonds Issued
Bond A
Bond B
Bond C
Money Market Loans
B
Year 1
$ (120)
$ 25
$ 1.000
$ 1.000
$ 1.000
10.0%
4.5%
$ -
Year 1
$ 13.36
C
Year 2
$ 20
$ 25
$ (0.085)
$ (0.080)
$ (0.400)
Year 2
$ -
D
Year 3
$ 20
$ 25
$ (0.085)
$ (1.080)
$ (0.400)
E
Year 4
$ 110
$ 25
$ (1.085)
$ -
$ (0.400)
Initial Cash Balance
$ 55
Maximum Value
that Can Be Issued
Bond A
Bond B
Bond C
$
$
$
60
60
60
Year 3 Year 4
$ -
Initial Cash
Cash Flow from Operations
Cash Flow from Bonds
Money Market Loan Repayment
Cash Balance
$ 55.00
$ (120.00)
$ 76.64
$ -
$ 25.00
$ (6.43)
$ (14.69)
$ 26.13
$ 20.00
$ (23.07)
$ -
$ 25.00
$ 26.12
$ 110.00
$ (65.10)
$ (2.14)
$ 68.88
The formulas in cells A20:C24 (except B22:C22) are:
17
18
19
20
21
22
23
24
A B
Year 1
Money Market Loans 13.3578431372545
0
C
Year 2
Initial Cash =D9
Cash Flow from Operations =B2
=(1+$B10)*B24
=C2
Cash Flow from Bonds
Money Market Loan Repayment 0 =-(1+$B9)*B18
Cash Balance =B18+SUM(B20:B23) =C18+SUM(C20:C23)
Cells D20:D24 and E20:E24 contain formulas obtained by copying C20:C24.
(a) What formula should you place in cell B22, to compute the total combined cash flow from year 1 bond sales and/or payments? Make sure your answer will yield correct results when copied to cells C22:E22 (for years 2, 3, and 4, respectively).
(b) Specify the information needed to operate Solver on this model. What is the target cell, and should it be maximized or minimized? What are the changing cells? What are all the constraints? Indicate whether or not you would use the “assume nonnegative”
Solver option.
OM Final Exam Study Guide Page 4 of 37 Spring 2002

Computer Assembly
Garden State Computers ( GSC
) makes PC’s. The demand for the next four months is
Month Demand
1
2
3
4
500
600
700
800
The PC's can be assembled in two factories, Factory 1 and Factory 2. Assembling a single PC in
Factory 1 requires 2 hours of labor and costs $400. A PC assembled in Factory 2 requires 3 hours of labor and costs $300. For each month, the number of available labor hours is 800 in Factory 1 and 600 in Factory 2. It costs $100 to hold a PC in inventory for a month. At the beginning of
Month 1, GSC has 200 PC’s on hand.
Algebraically formulate a linear program to minimize the cost of satisfying demand.
Clearly define all your variables.
Mixing Fertilizers
Grow-It, Inc. mixes sludge and nitrogen to produce two kinds of fertilizer, Fertilizer 1 and
Fertilizer 2. Sludge costs $15/ton and nitrogen costs $10/ton. Fertilizer 1 must contain at least
15% sludge and 60% nitrogen, and sells for $70/ton. Fertilizer 2 must contain at least 70% sludge and 10% nitrogen, and sells for $40/ton. 8,000 tons of sludge are currently available, along with 10,000 tons of nitrogen. Under the assumption that they can sell their entire production of both fertilizers, Grow-It would like to set up a production plan to maximize profits, using the following spreadsheet:
(a) What formula should they enter in cell E20, to compute the profit from the production plan?
(b) What formula should they enter in cell B23, to indicate the minimum amount of sludge that must be mixed into Fertilizer 1? Make sure your answer will also yield correct formulas when copied to cells C23, B24, and C24.
(c) Specify the information needed to operate Solver on this model. What is the target cell, and should it be maximized or minimized? What are the changing cells? What are the constraints? Should you use the “assume linear model” option?
OM Final Exam Study Guide Page 5 of 37 Spring 2002

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1
2
3
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21
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18
19
A
Cost/Ton 15
Tons Available 8000
Fertilizer 1
Fertilizer 2
Fertilizer 1
Fertilizer 2
Fertilizer 1
Fertilizer 2
Total Used
Cost
Fertilizer 1
Fertilizer 2
70
40
0.15
0.7
B
Sludge
Sale Price
C
Nitrogen
Minimum Percentages
Sludge Nitrogen
0.6
0.1
D E
Amounts Mixed
Sludge Nitrogen
6560
1440
9840
160
=SUM(B16:B17) =SUM(C16:C17)
=B2*B18 =C2*C18
Total
Made Revenue
=SUM(B16:C16) =B6*D16
=SUM(B17:C17) =B7*D17
Profit
Minimum Amount Required
Sludge Nitrogen
Production and Shipment Optimization
10
10000
Your firm makes a product in three plants. The unit production costs and monthly capacities for these four plants are:
Production
Cost/Unit Capacity
Plant 1 $ 35.00 1200
Plant 2 $ 23.00 1400
Plant 3 $ 34.00 1500
The product is shipped to three regional distributors. The unit shipping costs to the regions are:
Unit Shipping Cost to
Region 1 Region 2 Region 3
Plant 1 $ 5.00 $ 8.00 $ 4.00
Plant 2 $ 8.50 $ 6.00 $ 5.00
Plant 3 $ 4.00 $ 7.65 $ 5.00
OM Final Exam Study Guide Page 6 of 37 Spring 2002

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7
You sell the product to the distributors for $50/unit in all regions. The amount shipped to each region should be less than or equal to the monthly demand for that region. This demand depends on the amount spent on local advertising, and is computed by the following formulas:
Region 1: 1000 units + ($ spent on region 1 advertising)/5
Region 2: 900 units + ($ spent on region 2 advertising)/4
Region 3: 800 units + ($ spent on region 3 advertising)/4 .
You also have a policy that not more than 50% of sales volume should come from any single region. For the coming month, a maximum of $196,000 can be spent on production, shipping, and advertising combined.
Subject to the above constraints, you would like to maximize profits for the coming month. A spreadsheet model is on the next page. The changing cells are D12:F12 (advertising expenditures for each region) and C18:E20 (shipments). Shaded cells contain formulas, although only the values are shown.
1
2
A B C
Production
Cost/Unit Capacity
D E F
Region 1
Unit Shipping Cost to
Region 2 Region 3
Plant 1
Plant 2
Plant 3
$
$
$
35.00
23.00
34.00
1200
1400
1500
Demand without Advertising
Advertising cost per Extra Sale
Shipments
From
Selling Price
Expense Budget
Advertising Expenditure
Plant 1
Plant 2
Plant 3
Total
Demand
Region 1
0
0
1150
1150
$ 5.00
$ 8.50
$ 4.00
1000
5
$ 50.00
$ 196,000
$ 750.00
1150
$ 8.00
$ 6.00
$ 7.65
900
4
$ -
900
To
Region 2
0
900
0
Region 3
1200
500
350
900 2050
Max for any one region
Production Cost
Advertising Cost
$ 125,200.00
$ 5,750.00
Shipping Cost $ 19,050.00
Total Cost =SUM(D24:D26)
Revenue
Profit
$ 4.00
$ 5.00
$ 5.00
800
4
Max per Region
50%
$ 5,000.00
2050
Total
=SUM(C18:E18)
=SUM(C19:E19)
=SUM(C20:E20)
=SUM(C21:E21)
2050
$ 205,000.00
$ 55,000.00
OM Final Exam Study Guide Page 7 of 37 Spring 2002

(a) What formula should be in cell D14, to compute region 1’s demand for the product? Make sure the formula will yield correct demands for regions 2 through 4 if copied to cells
E14:F14.
(b) What formula should be in cell C21, to compute the total units shipped to region 1? Make sure the formula will yield correct values for regions 2 through 4 if copied to cells D21:E21.
(c) What formula should be in cell F22, for the maximum number of units that can be shipped to any one region without violating the 50% sales volume rule?
(d) What formulas should be in cells D24, D25, and D26, to compute total production, advertising, and shipping costs, respectively?
(e) What formulas should be in cells F24 and F26, to compute total revenue and total profit, respectively?
(f) What target cell would you use in Solver? Would you maximize or minimize it? What constraints would you use? Would you use the Assume Nonnegative option? Would you use the Assume Linear Model option?
Goofy Park
The Dizzy Corporation is planning to develop a Goofy theme park. The sooner they can open the park, the sooner the cash will start rolling in. However, there are a number of tasks they must complete before they can open the park. The following table contains data on these tasks:
Task Symbol Task Description
Task Length
(weeks)
Tasks that Must
Precede This Task
A
B
C
D
E
F
G
H
Prepare Approved Site Plan
Put in Road
Put in Parking
Put in Rides
Put in Food Court
Hire Staff
Train Staff
Advertise Grand Opening
4
2
2
8
5
2
2
8 none
A
B
B
B
C
D, E, F
A
(a) Algebraically formulate (but do not solve) a linear programming problem that will find the shortest possible schedule for completing all of the tasks in the project.
Clearly define your decision variables.
(b) Suppose that in order to open for the lucrative Memorial Day weekend, the Dizzy
Corporation wants to complete the park in only 12 weeks. The following table lists the opportunities that Dizzy is considering to accomplish this speed-up:
Task Symbol Largest Possible Reduction (weeks) Cost per Week Saved
A
C
D
2
1
4
$ 10,000
$ 5,000
$ 12,000
OM Final Exam Study Guide Page 8 of 37 Spring 2002

Change your linear programming formulation so as to find the minimum cost way of meeting the 12-week completion requirement.
(c) Suppose now, that in addition to the previous speed up possibilities, Dizzy is offered an opportunity to buy a prefabricated food court instead of building its own. This prefabricated food court can be installed in one week (thus saving four weeks) and costs $30,000 extra. Unlike the other speed-up possibilities, this option is an “all or nothing” choice. Add this new possibility to the optimization problem you developed in part (c).
Memory Manufacturing and Purchasing
Bulk Memory Associates assembles commodity RAM memory chips into memory modules for use in personal computers and other electronic devices. There is currently a market for four kinds of memory modules, known as the Type 1 SIMM , Type 2 SIMM , DIMM , and X-DIMM .
Each module requires three kinds of resources to assemble: labor time, inserter machine time, and memory chips. The resource requirements and current market prices for the four kinds of modules are as follows:
Type 1
SIMM
Type 2
SIMM DIMM X-DIMM
Labor time (hours per module) 0.03
Inserter machine time (hours per module) 0.025
0.02
0.025
0.04
0.05
0.06
0.05
Memory chips per module 4 4 8 8
Each module sells for $ 75.00 $ 70.00 $ 155.00 $ 165.00
This week, up to 120 hours of labor time and 125 hours of machine time are available at a cost of
$20.50 and $15.00 per hour, respectively.
The firm has a policy of buying memory chips from three different suppliers. In any given week, no more than 40% of the chips purchased may come from any single supplier. The suppliers have the following number of chips available this week, at the indicated prices:
Supplier Name Sleeman Chang Malaya
Cost per chip $ 14.50 $ 12.60 $ 12.25
Chips available 9000 8500 8000
Write an algebraic linear programming model to maximize the firm’s profit for this week.
Clearly define all your decision variables. You may skip algebraic and arithmetic simplifications, if any arise.
Grading Language Tests
Duke Test Center administers French language tests to job candidates who are applying for work in international operations. Employees called raters grade the tests. The center employs five senior raters and nine junior raters.
OM Final Exam Study Guide Page 9 of 37 Spring 2002

Each French test consists of three parts: written , computer , and tape-recorded. The time, in minutes, required by each kind of rater to grade each type of test is as follows:
Test Type
Rater
Written Computer Recorded
Senior 20 5 15
Junior 30 6 18
Each rater can work up to 40 hours in the coming week. 800 candidates have just taken the test, and the results have to be announced after a week. Work can also be assigned to outside parttime raters, who charge $25 for each written test, $15 for each computer test, and $20 for each recorded test.
For quality control purposes, senior raters must grade at least 10% of all tests rated in-house in each category.
Formulate an algebraic linear programming model to minimize the amount paid to outside raters, while still meeting the deadline for grading all the tests. Clearly define all your decision variables. You may skip algebraic and arithmetic simplifications, if any arise.
Atlas Valve Co
Atlas Valve Company makes four kinds of industrial plumbing valves, called A, B, C, D, and E.
Atlas sells its products to a distributor who is willing to buy up to 100 of each kind of valve this month. Data on the production processes for the valves are as follows:
Unit
Product Profit
A
Production Production
Line
Setup
Hours
$ 741.00 16
Line
Unit
Time
2.00
Unit
Manual
Labor
4.00
Unit
Raw
Materials
15
B
C
D
E
$ 800.00
$ 620.00
$ 545.00
$ 1,025.00
10
15
8
20
2.40
1.50
1.25
3.00
3.00
3.00
2.30
3.50
20
30
12
40
For example, each type A valve makes the firm a profit of $741, while consuming 2.00 hours of time on the production line, 4.00 hours of manual labor, and 15 pounds of raw materials. In addition, 16 hours of production line time are required to set up the line to make type A valves; this time is incurred for the first type A valve made, but not for subsequent ones.
The firm has 4000 pounds of raw material, 320 hours of production line time, and 600 hours of manual labor available this month. The firm has a policy that at most four different products should be made in any given month. Furthermore, certain technical restrictions on the
OM Final Exam Study Guide Page 10 of 37 Spring 2002

production equipment dictate that if type B valves are made this month, then type D valves cannot be made this month.
Write an algebraic linear or integer programming model to maximize the firm’s profit for this month. Clearly define all your decision variables. You may skip algebraic and arithmetic simplifications, if any arise.
Investment Planning
Investment Partners, Inc. has $400,000 cash to invest. They have identified 5 attractive investment opportunities: assets 1, 2, and 3, and short-term corporate bonds A and B. Assets 1, 2, and 3 can be bought only in 1999, but may then be sold in any of the years 2000-2003. They are indivisible and must be traded in their entirety. Current prices of these assets and estimated sales prices for the next 4 years are given in the table below. The table also provides cash flows associated with each dollar invested in the two bond types (for example, each dollar invested in bond A in 1999 yields 7 cents of interest in 2000 and 2001, plus $1.07 in 2002). Bond investments may be made in arbitrary amounts. Uninvested cash earns interest at the rate of 6% per year.
150
160
170
180
200
100
105
110
120
125
130 -$1.00
135
150
160
170
$0.07
$0.07
$1.07
$0.00
$0.00
-$1.00
$0.08
$0.08
$1.08
0
40
100
120
150
The last column of the table shows liabilities that Investment Partners must pay in the years
2000-2003.
We assume that all transactions (purchases/sales of assets, purchases of bonds, coupon payments and payments of liabilities) take place on the last day of each year. Cash obtained from sales is immediately available for the payment of liabilities and for reinvestment on the same day, so purchases of type B bonds may be financed from sales made in the year 2000.
To determine the investment strategy that maximizes the cash balance at the end of the year
2003, Investment Partners is using the following spreadsheet. The `??’ entries hide formulae.
(a) What formula should be in cell B22, to compute the number of times asset 1 is sold?
Make sure your answer will yield correct results when copied to cells C22:D22.
OM Final Exam Study Guide Page 11 of 37 Spring 2002

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22
9
10
11
12
13
1
2
3
4
5
6
7
8
(b) What formula should be in cell E17, to compute the cash flow resulting from the purchases of assets in the year 1999?
(c) What formula should be in cell E18, to compute the cash flow from the sales of assets in the year 2000? Make sure your answer will yield correct results when copied to cells E19:E21.
(d) What formula should be in cell F17, to compute the cash flow from bond investments in in the year 2000? Make sure your answer will yield correct results when copied to cells F18:F21.
(e) What formula should be in cell G17, to compute the cash balance at the end of
1999?
(f) What formula should be in cell G18, to compute the cash balance at the end of the year 2000? Make sure your answer will yield correct results when copied to cells
G19:G21.
(g) Specify the information needed to operate Solver on this model. What is the target cell, and should it be maximized or minimized? Which cells are the changing cells?
What are all the constraints?
A G
Year
1999
2000
2001
2002
2003
B C D E F
Price (in k$) of
Cash Flows per $1
Invested
Asset 1 Asset 2 Asset 3 Bond A Bond B
150 100 130 -$1.00
$0.00
160
170
105
110
135
150
$0.07
$0.07
-$1.00
$0.08
180
200
120
125
160 $1.07
170 $0.00
$0.08
$1.08
Liability (k$)
0
40
100
120
150
Interest Rate 6% Initial Cash (k$)
Invest (k$) In
Bond A Bond B
27 221
400
Year
1999
2000
2001
2002
2003
Times Sold
Buy/Sell Assets
Asset 1 Asset 2 Asset 3
1 0 1
1
0
0
0
0
0
0
0
0
1
0
0
??
??
??
Cash Flows (in k$) Cash Balance
Assets Bonds (k$)
??
??
??
??
??
??
??
??
??
??
??
??
??
??
??
OM Final Exam Study Guide Page 12 of 37 Spring 2002

Production/Inventory Planning with Setup Costs
Your firm expects the following demand for one of its product over the next six months:
Month Demand
4
5
6
1
2
3
500
150
250
740
650
250
We assume these demand levels are known with certainty. At the beginning of month one, there are 100 units of the product in inventory.
During any month that you wish to run your production line, you must pay a “setup” cost of
$5,000. Once the production line is set up, you must produce between 200 and 600 units
(inclusive) at a cost of $35 per unit. In months when you don’t run the production line, there is no production cost, and no units can be produced.
If there is unsold inventory left over at the end of the month, it must be stored in the warehouse.
Opening the warehouse for a month incurs a fixed cost of $2,000 for security, lights, and insurance, regardless of how much inventory is kept. In addition, there is a further cost of $5.00 per month per unit stored, assessed on the amount of inventory in the warehouse at the end of each month. The warehouse can hold up to 300 units. The warehouse is not allowed to be open at the end of months in which there is no production.
You are using the spreadsheet model below to try to find the cheapest way to meet the demand for the product. All formulas in the model have been obscured by “???”.
(a) What formula should be in cell E13, to compute the inventory left at the end of month one? Make sure your answer will compute correct ending inventories for months 2 through 6 if copied to cells E14:E18.
(b)
What formula should be in cell F13, giving a “logical lower bound” on month one production? Make sure your answer will compute correct bounds for months 2 through 6 if copied to cells F14:F18.
(c) What formula should be in cell G13, giving a “logical upper bound” on month one production? Make sure your answer will compute correct bounds for months 2 through 6 if copied to cells G14:G18.
(d)
What formula should be in cell H13, giving a “logical upper bound” on month one ending inventory? Make sure your answer will compute correct bounds for months
2 through 6 if copied to cells H14:H18.
(e) What formula should be in cell I13, to compute the total cost incurred in month one? Make sure your answer will compute correct costs for months 2 through 6 if copied to cells I14:I18.
OM Final Exam Study Guide Page 13 of 37 Spring 2002

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3
4
1
2
5
6
A
1
2
3
B
Month Demand
500
150
250
C D E F
Production Inventory
Setup Cost 5,000.00
Unit Cost
Minimum, if Setup
Maximum
$ 35.00
200
600
$
0
5.00
300
G H
4
5
6
740
650
250
Logical Logical Logical
Produce Open Ending
Lower
Bound on
Upper Upper
Bound on Bound on
Month Any?
Production Warehouse?
Inventory Production Production Inventory
0 100
1
1
1
0
1
1
550
0
440
600
600
250
1
1
1
0
0
0
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
???
Total
I
Cost
(f) What formula should be in cell I19, to compute the total cost over six months?
(g) How would you set up Solver to find the cheapest production plan? Specifically, what is the target cell, and would you maximize or minimize it? What are the changing cells? What are all the constraints? How would you set the Assume
Linear Model option? How would you set the Assume Nonnegative option?
Housing Developments
???
???
???
???
???
???
???
The West Tudor Township plans to establish three new housing developments. They have received offers from seven developers. The details of the offers are shown in the table below.
Offer
Houses Townhouses
Number Size (sq.ft.) Price (k$) Number Size (sq.ft.) Price (k$)
40 2600 370 1
4
5
2
3
6
20
25
35
2000
2300
3000
180
200
420
40
30
65
1200
1400
1600
140
150
160
75 1000 130
7 50 1800 190
Each project is a “package” that can be built only in its entirety; for example, it is not possible to build the houses from project 2 without also building its townhouses, and vice versa . The prices listed are per unit of housing. The total “value” of a project is the sum of the sales prices of all the housing units it contains.
Township tax revenues are proportional to housing value. Therefore, the township wants to maximize the total value of the housing built. However, it must obey the following conditions:
OM Final Exam Study Guide Page 14 of 37 Spring 2002


No more than 150 new housing units may be built.

There should be at least one project containing some units of 1500 sq. feet or less.

There should be at least one project containing units between 1500 and 2500 sq. feet.

There should be exactly one project containing units larger than 2500 sq. feet.
Algebraically write an integer programming model to maximize the value of West Tudor’s new developments. Clearly define your variables. Feel free to skip algebraic and numeric simplifications.
Siting Restaurants
Fast Food Conglomerate Incorporated (FFCI) operates four chains of restaurants: Burger Man ,
Burger Deluxe , Burrito Barn , and Eat-za Pizza . The firm has acquired 8 sites around
Indianapolis, an area where it previously did not have a presence. A capital budget of up to $3.7 million has been allocated to build restaurants on the sites. FFCI is trying to decide how many of each kind of restaurant to build. At most four restaurants can be built for any one chain. To maintain the premium image of the Burger Deluxe chain, the firm will not build more Burger
Deluxe restaurants than Burger Man restaurants.
Burger Man restaurants cost $400,000 each to build. If FFCI builds one Burger Man , its annual profits should be $110,000. If they build two, total annual profits are expected to be $200,000 from the two restaurants combined. If they build three, total annual profits should be $270,000, and if they build four, total annual profits should be $330,000. The table in cells C5:E9 of the spreadsheet below gives similar data for the other three restaurant chains.
FFCI would like to build the combination of restaurants that yields the highest annual profit. To
12
13
14
15
16
17
18
19
20
21
9
10
11
6
7
8
1
4
5
2
3 this end, they are using the spreadsheet below. The decision variables are binary and are in cells
B12:E15. For example, cell D14 is meant to be one if FFCI builds exactly three Burrito Barns , and otherwise zero. The shaded cells contain formulas, although only values are displayed; the remaining cells contain either numbers or formulas as indicated.
A
Number of Sites Available
Construction Budget
B
8
$ 3,700
C
(All $ in 1,000's)
D E F
Unit Construction Cost
Annual Profit From 1 Site
Annual Profit From 2 Sites
Annual Profit From 3 Sites
Annual Profit From 4 Sites
Number of Sites Used
1
2
3
4
Number of Restaurants Built
Construction Cost
Total Annual Profit
Burger Man
$ 400
$ 110
$ 200
$ 270
$ 330
Burger Deluxe Burrito Barn
$ 500 $ 350
$ 140
$ 270
$ 375
$ 450
$
$
$
$
100
200
295
350
Eat-za Pizza
$ 600
$ 300
$ 500
$ 550
$ 600
Burger Man
0
1
0
Burger Deluxe
0
1
0
Burrito Barn
0
0
1
Eat-za Pizza
1
0
0
0 0 0 0
=SUM(B12:B15) =SUM(C12:C15) =SUM(D12:D15) =SUM(E12:E15)
2
$ 800
$ 1,065
2
$ 1,000
3
$ 1,050
1
$ 600
Total
=SUM(B18:E18)
=SUM(B19:E19)
OM Final Exam Study Guide Page 15 of 37 Spring 2002

(a) What formula should you place in cell B21, to compute the total combined annual profit from all restaurants?
(b) What formula should be in cell B18, the total number of Burger Man restaurants built?
Assume this cell will be copied to C18 through D18 (for the other chains).
(c) What formula should be in cell B19, the total construction cost for Burger Man restaurants? Assume this formula will be copied to C18 through D18 to compute the construction costs for the other chains.
(d) Specify the information needed to operate Solver on this model. What is the target cell, and should it be maximized or minimized? What are all the constraints?
Assigning People to Projects
A contractor has four construction projects underway. The following table shows the estimated time to complete a project when a specified number of foremen are assigned to it.
Project
Number of Foremen Assigned
1 2 3
A
B
C
D
5
7
9
11
3
5
8
8
2
3
7
8
The contractor has only six foremen and wishes to minimize the sum of the project completion times. At least one foreman must be assigned to each project.
Algebraically formulate this problem as an integer program. Define your variables.
Locating Restaurants
The Di Livio fancy restaurant chain is considering opening several new restaurants in River
County. There are six potential locations: towns A, B, C, D, E, and F. The populations (in thousands) of these towns and the distances between the towns (in miles), respectively, are given in the two tables below:
Town A B C D E F
Inhabitants 11 20 10 19 13 12
Town A B C D E F
A 0 7 4 10 11 14
B 7 0 6 3 8 10
C 4 6 0 5 7 12
D 10 3 5 0 4 7
E 11 8 7 4 0 3
F 14 10 12 7 3 0
OM Final Exam Study Guide Page 16 of 37 Spring 2002

The premises in towns B and E must be bought from a single owner as a bundle, and therefore Di
Livio can either locate restaurants in both B and E, or in neither of these towns.
(a) Suppose Di Livio’s goal is to locate its restaurants so that, for every inhabitant of towns A-F, the distance to the nearest Di Livio restaurant is at most 5 miles.
Formulate an algebraic integer programming model to minimize the number of restaurants required. Clearly define all your variables. You may skip algebraic and arithmetic simplifications, if any arise.
(b) Instead, suppose that Di Livio can locate only two new restaurants in the area.
Algebraically formulate an integer programming model to maximize the number of inhabitants of these towns who live within 5 miles of a Di Livio restaurant. Clearly define all your variables. You may skip algebraic and arithmetic simplifications, if any arise. [Note: this part of the question was intended to be a more challenging problem for stronger students, and was not assigned a large number of points.]
Planning Construction of a Mall
Garden State Malls (GSM), Inc. is planning to build a mall in Piscataway. The main activities of the project, along with their duration and prerequisites (if any) are tabulated below:
Activity Description Duration (days) Prerequisites
D
E
F
G
A
B
C
Licensing and surveying
Equipment acquisition
Foundation and frame
Inside construction
Outside walls and roofing
Installation and electricity
Landscaping and parking lots
25
10
30
22
17
10
14
None
None
A,B
C
C
D,E
E
H Inspection 5 F,G
(a) Algebraically formulate a linear programming problem to determine the minimum possible duration of the project as presented above. Clearly define all your decision variables. You may skip algebraic and arithmetic simplifications, if any arise.
Now suppose that some of the above activities can be shortened, as described in the following table:
Activity
B
Maximum
Reduction (days)
5
Fixed cost
($/reduction)
2000
Variable cost
($/day)
400
C
D
E
F
G
12
6
8
4
6
3000
800
1200
600
1000
250
300
200
150
350
OM Final Exam Study Guide Page 17 of 37 Spring 2002

For example, it is possible to shorten activity E by 5 days, since 5

8 (the maximum reduction), and doing so will cost
$2200

$1200 (fixed cost) + 5

$200 (variable cost)
GSM has budgeted $25,000 for speeding up completion of the project as much as possible.
(b) Algebraically formulate the problem of minimizing the project duration subject to the above budget. Use linear constraints wherever possible. Clearly define all your decision variables. You may skip algebraic and arithmetic simplifications, if any arise.
Refining Germanium
Your company produces high-performance semiconductor chips. The production process for these chips requires the rare earth metal germanium. Germanium can be purchased in “raw” form for $44 per kilo, and you must refine it before use. Your basic refinement process costs $23 per kilo of raw germanium processed, from which it produces 0.45 kilos of regular refined germanium, 0.30 kilos of premium refined germanium, and 0.25 kilos of defective germanium.
Some or all of the regular refined germanium produced by the basic process may be subjected to a second process, called remelting , which costs $18 per kilo. Remelting a kilo of regular refined germanium produces 0.9 kilos of premium refined germanium and 0.1 kilos of defective germanium.
The chip production line requires a total of 50 kilos of refined germanium this month, of which at least 35 kilos must be premium grade. Defective germanium cannot be used for chip production, and must be disposed of at a cost of $12 per kilo.
Write an algebraic linear programming model to minimize the cost of meeting this month’s refined germanium requirements. Clearly define all your decision variables. You may skip algebraic and arithmetic simplifications, if any arise.
Writing Modular Software
Specialized Software, Inc. (SSI) is considering marketing up to seven different software packages, denoted here simply as 1 through 7. Each package is constructed from a set of smaller pieces of software called modules . There are six different possible modules, which may be combined into the seven packages in the following way:
Software Package 1 2 3 4 5 6 7
Required Modules {A,B} {A,E} {B,C,D} {C,D} {C,D,E} {D,F} {E,F}
If SSI decides to market two or more packages that require the same module, the common module only needs to be written once. Thus, for example, if the firm markets packages 3, 4 and
5, the common modules C and D (as well as the activities not in common) only need to be written once. The tables below display the cost of writing each module and the total estimated
OM Final Exam Study Guide Page 18 of 37 Spring 2002

revenue from marketing each package. For example, if the firm decides to market just packages 1 and 2, its profit is the revenue from packages 1 and 2 minus the cost of writing the required modules A, B and E; that is, (in tens of thousands of dollars) (7

10)

(4

5

7)

1. You may ignore all costs besides those of writing the modules.
Module A B C D E F
Cost to Write ($10,000’s) 4 5 17 10 7 5
Software Package
Revenue ($10,000’s)
1
7
2
10
3
9
4
3
5
8
6
6
7
8
Algebraically formulate an integer programming model SSI can use to determine how to maximize its profits. Clearly define all your decision variables. You may skip algebraic and arithmetic simplifications, if any arise.
Pension Fund Planning
The pension fund manager of DRP, Inc . has up to $10,000,000 to invest. She has identified four reliable investment companies with a long record of successful operation: Olech, Spitzer ,
Martens and Aubin.
These funds hold mixes of assets in three categories: domestic large capitalization stocks, domestic small capitalization stocks, and foreign stocks, in the proportions given in the following table.
Olech Spitzer Martens Aubin
U.S. Large Stocks 60% 20% 40% 50%
U.S. Small Stocks 20% 60% 10% 30%
Foreign Stocks 20% 20% 50% 20%
Each of these companies is ready to invest DRP’s capital, as long as the amount invested is at least $2 million.
The percentage of the total invested capital in each of the three asset categories should be between the minimum and maximum values given below:
Minimum Maximum
U.S. Large Stocks 40% 45%
U.S. Small Stocks
Foreign Stocks
15%
25%
30%
35%
Subject to these constraints, the manager would like to maximize the total return on her investment, assuming the following annual rates of return:
Olech Spitzer Martens Aubin
15% 15% 14% 16%
To determine the best investment plan, the manager is using the spreadsheet model displayed on the next page. The shaded “??” cells contain formulas, whose values are not shown.
OM Final Exam Study Guide Page 19 of 37 Spring 2002

15
16
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23
24
25
26
27
28
7
8
9
10
11
12
13
14
3
4
5
6
1
2
(c) What formula should you place in cell G18, to compute the total amount invested?
(d) What formula should you place in cell B19, to compute the logical lower bound on the amount invested in Olech? Make sure it yields correct results for the other funds when copied to cells C19:E19.
(e) What formula should you place in cell B20, to compute the logical upper bound on the amount invested in Olech? Make sure it yields correct results when copied to cells C20:E20.
(f) What formula should you place in cell B24, to compute the total amount invested in large-cap stocks? Make sure your answer yields correct results for the other asset categories when copied to cells B25:B26.
A B C
U.S. Large Stocks
U.S. Small Stocks
Foreign Stocks
Olech
60%
20%
20%
D
Portfolio Composition
Spitzer Martens
20%
60%
20%
40%
10%
50%
E
Aubin
50%
30%
20%
F G
Capital (k$)
10,000
Minimum
Investment (k$)
2,000
Annual Returns
U.S. Large Stocks
U.S. Small Stocks
Foreign Stocks
Invest? (1--yes, 0--no)
Amount Invested (k$)
Minmum Amount Possible
Maximum Amount Possible
U.S. Large Stocks
U.S. Small Stocks
Foreign Stocks
Return (k$)
Olech
15%
Required Composition
Minimum Maximum
40%
15%
25%
45%
30%
35%
Olech
??
??
Amount
Invested (k$)
??
??
??
??
0
0
Spitzer
15%
Spitzer
1
2,000
??
??
Martens
14%
Martens
1
3,000
??
??
Required Bounds
Minimum Maximum
??
??
??
??
??
??
Aubin
16%
Aubin
1
5,000
??
??
Total Amount
Invested (k$)
??
(g) What formula should you place in cell C24, to compute the minimum amount that may be invested in large-cap stocks? Make sure your answer yields correct results when copied to the other cells in the range C24:D26, in order to compute minimum and maximum amounts for all asset categories.
(h) What formula should you place in cell B28, to compute the total return for the portfolio?
OM Final Exam Study Guide Page 20 of 37 Spring 2002

(i) Specify the information needed to operate Solver on this model. What is the target cell, and should it be maximized or minimized? Which cells are the changing cells?
What are all the constraints? Would you use the Assume Nonnegative and Assume
Linear Model options?
Setting Up Telemarketing Centers
ABC Corporation is deciding where to open telemarketing centers, and has identified seven candidate locations. ABC's telemarketing concentrates on eight area codes: 314, 316, 417, 501,
502, 606, 816, and 918. The following table shows the annual cost of operating a center at each possible location, and the area codes each location can serve:
Center Location Area Codes Location Can Serve Cost
Dallas, TX
Atlanta, GA
Louisville, KY
Denver, CO
316, 417, 501, 918
314, 502, 606, 816
314, 316, 417, 918
501, 502, 606
$ 500,000
$ 800,000
$ 400,000
$ 900,000
Little Rock, AR 314, 417, 502, 816
Memphis, TN 316, 417, 501, 606
St. Louis, MO 314, 502, 606, 816
$ 300,000
$ 450,000
$ 550,000
ABC would like to open between 4 and 5 centers, and for each area code, there must be at least one open center that can serve it. Where should they be located to minimize the total annual cost under the following constraints?

Centers cannot be opened in both Dallas, TX and Louisville, KY.

You can open a center in St. Louis, MO only if you open centers in both Dallas, TX and
Atlanta, GA.
Algebraically formulate an optimization model for this problem. Clearly define all your decision variables. You may skip algebraic and arithmetic simplifications, if any arise.
Producing Biotech Chemicals
Your firm makes a specialty reagent (chemical) used by biotechnology companies. You keep no inventory and supply all the reagent you make to a distributor. You make the reagent in three forms (also called “products”): regular , refined , and ultrapure . You make the refined form by processing the regular form, and you make the ultrapure form by processing the refined form.
Making a liter of the regular form of the reagent requires 3 hours of labor, 1 hour of reactor time, and 0.5 hours of distiller time, at a cost $400.
Making a liter of refined form requires 2 hours of labor, 0.5 hours or reactor time, and 1 hour of distiller time, at a cost of $200. It also consumes 1.1 liters of the regular form of the reagent.
OM Final Exam Study Guide Page 21 of 37 Spring 2002

Making a liter of the ultrapure reagent requires 1.5 hours of labor, 0.1 hours of reactor time, and
1.6 hours of distiller time, and incurs a cost of $300. In the process, you also consume 1.2 liters of the refined form of the reagent.
This week, 140 hours of labor, 40 hours of reactor time, and 40 hours of distiller time are available. Also, your distributor will purchase the products from you as follows:
Product Price per Liter
Maximum Amount Distributor will Buy (Liters)
Regular
Refined
$ 500
$ 1,000
100
50
Ultrapure $ 1,500 30
The distributor also has a policy that if they buy any of a given product from you, they must buy at least 10 liters. For example, you can either sell them either zero liters of regular reagent, or between 10 and 100 liters.
Finally, you have a policy that at most 80% of your revenue may come from any single product.
Algebraically formulate an optimization model to maximize your profits for the week.
Clearly define all your decision variables. You may skip algebraic and arithmetic simplifications, if any arise.
Bouquets
Each morning, Premium Flower Service receives a shipment of bouquets from its supplier at a unit cost of $10 per bouquet. It inspects the shipment carefully; each bouquet has a 4% chance
(independent of all other bouquets in the shipment) of failing to meet Premium’s rigorous quality standards. Bouquets failing inspection are returned to the supplier immediately for a credit of $9 each. Premium holds the remaining bouquets in inventory to meet the day’s demand, which is uncertain. Experience has shown that the number of orders per day is well modeled by a Poisson random variable with a mean value of 53. Each order filled produces revenue of $29 and incurs a delivery cost of $8. Any bouquets left over at the end of the day are “salvaged” by selling them to a local hotel chain for $6.50 each.
The supplier provides bouquets only in lots of 12 (any number of bouquets can be returned for credit, though). Premium is trying to decide whether ordering 36, 48, 60, 72 or 84 bouquets would give it the highest average profit. To this end, they have constructed the following spreadsheet simulation model:
OM Final Exam Study Guide Page 22 of 37 Spring 2002

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14
15
16
9
10
11
12
17
18
19
20
7
8
5
6
3
4
1
2
21
A
Probability of Failing Inspection
Average Demand
Unit Cost of Bouquets
Unit Immediate Return Credit
Unit Selling Price
Unit Delivery Cost
Unit Salvage Value
Order Size
Number Failing Inspection
Bouquets Available for Delivery
Actual Demand
Bouquets Delivered
Bouquets Left Over for Salvage
0.04
53
10
9
29
8
6.5
=B9-B10
=B11-B13
B
Cost of Bouquets Ordered
Delivery Costs
=B9*B3
=B13*B6
Revenue from Delivered Bouquets =B5*B13
Return Credits =B10*B4
Revenue from Salvaged Bouquets =B14*B7
Profit =SUM(B18:B20)-SUM(B16:B17)
C
36
48
60
72
84
D
Possible
Order
Sizes
(a) What formula should they enter in cell B10, the number of bouquets failing inspection?
(b) What formula should they enter for cell B12, the actual demand on a given day?
(c) What formula should they enter for cell B13, the number of orders filled?
(d) Which formulas in the cells above should be enclosed in calls to the function
SIMOUTPUT before running the simulation?
(e) Premium would like @Risk to automatically try the five possible order quantities 36 through 84. What formula should they enter in cell B9?
Mixing Funds
The pension fund manager of IBF, Incorporated has identified three reliable mutual funds with long records of successful operation: the Balanced Fund , the World Fund and the Income Fund .
These funds invest in three asset categories, U.S. stocks, foreign stocks and U.S. bonds, in the proportions given in the following table.
Funds
Asset Categories Balanced World Income
U.S. Stocks 70% 40% 20%
Foreign Stocks 10% 50% 10%
U.S. Bonds 20% 10% 70%
All three asset categories have random annual returns. Stock returns are well described by normal random variables with the parameters shown below.
U.S. Stocks
Foreign Stocks
Expected Value Standard Deviation
0.14 0.11
0.11 0.12
For bonds, the rate of return is equally likely to be any value between 0.04 and 0.08.
OM Final Exam Study Guide Page 23 of 37 Spring 2002

The manager wants to invest her pension capital in these three mutual funds. She is considering various fund compositions (portfolios) in the package offered to the IBF participants. For example, one possibility is to invest 20% of the capital in the Balanced Fund, 10% of the capital in the World Fund, and 70% in the Income Fund.
She is using the simulation spreadsheet to help find the composition that best suits the objectives of the pension fund participants. The output report should contain data on cells D19 and D20.
9
10
11
3
4
5
6
7
8
1
2
12
13
14
15
16
17
18
A B C D E F G
Funds
Asset Categories Balanced World Income
U.S. Stocks
Foreign Stocks
U.S. Bonds
70%
10%
20%
40%
50%
10%
20%
10%
70%
U.S. Stocks
Foreign Stocks
U.S. Bonds
H I
Annual Return Data
Expected
Value
Standard
Deviation
0.14
0.11
0.11
0.12
Minimum Maximum
0.04
0.08
U.S. Stocks
Foreign Stocks
U.S. Bonds
Package Composition
Balanced World Income
20% 10% 70%
Asset content
32.0%
14.0%
54.0%
Return
0.14
0.11
0.06
Possible Package Compositions
Balanced World Income
20% 10% 70%
30%
35%
40%
10%
15%
20%
60%
50%
40%
19 Total Return 0.093
20 Lower than 0.04?
0
(j) What formula should be in cell C11, to set the fraction of the capital invested in the
Balanced Fund? Make sure your answer will yield correct results when copied to cells D11:E11.
(k) What formula should be in cell C15, to compute the fraction of the capital invested in US stocks? Make sure your answer will yield correct results when copied to cells
C16:C17.
(l) What formulas should be in cells E15:E17, to simulate the annual returns of the three asset categories?
(m) What formula should be in cell E19, to simulate the annual return of the package?
(n) What formula should be in cell E20 in order to calculate the value “1” when the return is lower than 0.04 (the minimum return on bonds), and to calculate the value
“0” otherwise?
(o) Relevant portions of the @Risk detail statistics report are displayed below. Which fund composition has the highest expected return? What is the probability that the return of this package will be below 0.04? Briefly explain your reasoning.
(p) Of the packages that have at most a 10% risk of having a return lower than 0.04, which one has the highest expected return? Briefly explain your reasoning.
OM Final Exam Study Guide Page 24 of 37 Spring 2002

20% Perc =
25% Perc =
30% Perc =
35% Perc =
40% Perc =
45% Perc =
50% Perc =
55% Perc =
60% Perc =
65% Perc =
70% Perc =
75% Perc =
80% Perc =
85% Perc =
90% Perc =
95% Perc =
Cell Name
E19 (Sim#1) Total Return / Return
E19 (Sim#2) Total Return / Return
E19 (Sim#3) Total Return / Return
E19 (Sim#4) Total Return / Return
E20 (Sim#1) Lower than 0.04? / Return
Minimum
-5.45E-02
-7.30E-02
-8.47E-02
-9.64E-02
0
Mean
9.26E-02
9.66E-02
0.1003559
0.1041521
0.0854606
Maximum
0.1910723
0.2114594
0.2251528
0.2431004
1
E20 (Sim#2) Lower than 0.04? / Return
E20 (Sim#3) Lower than 0.04? / Return
0
0
9.88E-02
0.1043285
E20 (Sim#4)
Name
Description
Cell
Minimum =
Maximum =
Lower than 0.04? / Return 0 0.1120977
Total Return / Return Total Return / Return Total Return / Return
1
Total Return / Return
Output (Sim#1)
E19
-5.45E-02
0.1910723
Output (Sim#2)
E19
-7.30E-02
0.2114594
Output (Sim#3)
E19
-8.47E-02
0.2251528
Output (Sim#4)
E19
-9.64E-02
0.2431004
Mean =
Std Deviation =
9.26E-02
3.90E-02
9.66E-02
4.38E-02
0.1003559
4.81E-02
0.1041521
5.24E-02
1
1
Variance =
Skewness =
Kurtosis =
Errors Calculated =
Mode =
5% Perc =
10% Perc =
15% Perc =
1.52E-03
-6.74E-02
2.938637
0
9.73E-02
2.76E-02
4.27E-02
5.09E-02
1.92E-03
-6.72E-02
2.956512
0
0.120571
2.44E-02
4.06E-02
5.00E-02
2.31E-03
-6.52E-02
2.960571
0
8.56E-02
2.15E-02
3.86E-02
0.0488772
2.74E-03
-0.0631323
2.963889
0
0.1006457
1.77E-02
3.69E-02
4.82E-02
0.05904
6.57E-02
7.16E-02
7.73E-02
8.35E-02
8.81E-02
9.31E-02
9.76E-02
0.1030465
0.1079706
0.1125525
0.1182759
0.1243901
0.132916
0.1430084
0.1577649
0.0596187
6.60E-02
7.31E-02
7.98E-02
8.58E-02
9.15E-02
9.74E-02
0.1031572
0.1076845
0.113685
0.1195623
0.1258519
0.1321911
0.1422836
0.1530751
0.1694962
6.00E-02
6.70E-02
7.45E-02
8.23E-02
8.86E-02
9.54E-02
0.1015068
0.1074711
0.1122065
0.1190225
0.125087
0.1323717
0.1394468
0.1500885
0.1618073
0.1804472
6.00E-02
6.78E-02
7.58E-02
8.44E-02
9.08E-02
9.91E-02
0.1059535
0.1110939
0.1172146
0.1244385
0.1308157
0.1383917
0.1473786
0.1580315
0.1713361
0.1911264
Gambling
Hans Idlemann plans on going to Atlantic City to play the quarter slot machines. He will start with $10.00 in quarters, and plans to play 100 times or until he has no quarters left, whichever happens first. Each time you play a quarter slot machine, you have a 90% chance of losing a quarter. You have a 9% chance of getting your quarter back, plus $1.00 in quarters. You also have a 1% chance of getting your quarter back, plus $10.00 in quarters.
The exhibits below and on the following pages show a simulation model of Hans’ gambling spree, along with the summary statistics and detail statistics reports from a 2000-iteration simulation. Note that in the particular scenario shown on the spreadsheet, Hans runs out of money on the 96 th play, and cannot continue playing afterwards. Rows 14 through 112 are all copies of row 13, although rows 23 through 91 are hidden. The output cells are B6, B8, and B9.
(Note: with the conversion from @Risk to YASAI, the formulas in cells B6, B8, and B9 should all be enclosed in calls to SIMOUTPUT; you should ignore the “minimum” and “maximum” columns).
(a) What formula should be in cell B13, to simulate the payoff from the first play?
Make sure your answer will yield correct results for all subsequent plays if copied to cells B14:B112.
(b) What formula should be in cell C13, to compute the amount of money Hans has left after the first play? Make sure your answer will yield correct results for all subsequent plays if copied to cells C14:C112.
OM Final Exam Study Guide Page 25 of 37 Spring 2002

3
4
1
2
5
6
7
8
9
10
11
101
102
103
104
105
106
107
108
109
110
111
112
95
96
97
98
99
100
20
21
22
92
93
94
16
17
18
19
12
13
14
15
A
Payoff
Probability
Starting $
Ending $ =C112
B
$ (0.25)
90%
$10.00
=IF(B6=0,1,0)
=IF(B6>B4,1,0)
Play
89
90
91
92
93
94
95
96
97
98
99
100
83
84
85
86
87
88
8
9
10
80
81
82
4
5
6
7
2
3
0
1
C
$ 1.00
9%
Payoff Money Left
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ 1.00
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ (0.25)
$ -
$ -
$ -
$ -
$1.75
$1.50
$1.25
$1.00
$0.75
$0.50
$0.25
$0.00
$0.00
$0.00
$0.00
$0.00
=B4
$9.75
$9.50
$9.25
$9.00
$8.75
$8.50
$8.25
$8.00
$7.75
$7.50
$2.75
$2.50
$2.25
$2.00
$1.75
$1.50
$2.50
$2.25
$2.00
D
$ 10.00
1%
OM Final Exam Study Guide Page 26 of 37 Spring 2002

(c) Estimate the expected (average) amount of money Hans has left after his gambling spree.
(d) Estimate the probability that Hans will have no money left at the end of his spree.
(e) Estimate the probability that Hans will leave the Casino with more money than he started with.
Iterations= 2000
Simulations= 1
Summary Statistics
Cell
B6
B8
B9
Name
Ending $
Minimum Mean
0 7.187375
Maximum
59
0
0
0.4465
0.305
1
1
10% Perc =
15% Perc =
20% Perc =
25% Perc =
30% Perc =
35% Perc =
40% Perc =
45% Perc =
50% Perc =
55% Perc =
60% Perc =
65% Perc =
70% Perc =
75% Perc =
80% Perc =
85% Perc =
90% Perc =
95% Perc =
@RISK Simulation of gambling.xls
Name
Description
Cell
Minimum =
Maximum =
Mean =
Std Deviation =
Variance =
Skewness =
Kurtosis =
Errors Calculated =
Mode =
5% Perc =
Run on 4/29/99, 2:01:15 PM
1
1
1
1
1
1
1
0
0
1
0
0
0
0
0
Output
B8
0
1
0.4465
0.4971295
0.2471378
0.2152357
1.046326
0
0
0
0
0
0
7.75
10.25
12.75
14.25
16.75
20.5
27
0
0
0
0
0.25
2.75
5.25
6.5
Ending $
Output
B6
0
59
7.187375
9.294309
86.38417
1.451631
4.996541
0
0
0
0
0
0
Simulations= 1 Iterations= 2000
0
1
1
1
1
1
1
0
0
0
0
0
0
0
0
Output
B9
0
1
0.305
0.4604074
0.211975
0.8470758
1.717537
0
0
0
0
0
0
OM Final Exam Study Guide Page 27 of 37 Spring 2002

Investing in Risky Stocks
You have set aside a pool of money to invest in risky stocks. Currently, you have $10,000 in this pool. Each month you can invest any fraction of this money. With probability 0.3, you “win” and the amount you invested is quadrupled (for example, if you invest $100, it becomes $400, increasing the size of the pool by $300). However, there is also a probability of 0.7 that you lose your entire investment.
You are considering the following investment strategies:

Each month, invest 5% of the pool.

Each month, invest 10% of the pool.

Each month, invest 20% of the pool.
You are using the simulation spreadsheet below to try to determine which investment policy would be the best for a 12-month period. You are using three simulations, each with 3000 iterations, to evaluate the three possible investment policies. You are curious as to whether, at the end of the 12 months, the pool will contain at least $12,000, an amount called the target .
Cell E11 contains =B5. Lines 13 through 23 are all copied from line 12.
(a) Cell B7 is meant to contain the investment percentage (5% for the first simulation,
10% for the second, 20% for the third). What formula should it contain?
Parts (b)-(e) request formulas for row 12, which concerns the first month. In each case, make sure your answers will yield correct answers for all subsequent months when row 12 is copied to rows 13 through 23.
A
Prob of Winning
Win Code (1=Yes, 0=No)
B
0.3
1
C
0.7
0
D
Target
E
$ 12,000
5
6
7
3
4
1
2
8
9
10
18
19
20
21
22
23
11
12
13
14
15
16
17
Multiple Won
Beginning Pool Size
Investment percentage
Month
7
8
9
10
11
12
5
6
3
4
0
1
2
3
$10,000
$10,000
$11,500
$10,925
$10,379
$11,936
$11,339
$13,040
$12,388
$11,768
$11,180
$10,621
$12,214
5%
Starting
Pool Size
Possible investment percentages
5% 10% 20%
Amount
Invested
$500
$575
$546
$519
$597
$567
$652
$619
$588
$559
$531
$611
Hit Target?
Win?
0
0
0
1
0
1
1
0
1
1
0
0
Ending
Pool Size
$10,000
$11,500
$10,925
$10,379
$11,936
$11,339
$13,040
$12,388
$11,768
$11,180
$10,621
$12,214
$14,046
1
OM Final Exam Study Guide Page 28 of 37 Spring 2002

(b) What formula should be in the cell B12, the pool size at the beginning of the first month?
(c) What formula should be in the cell C12, the amount invested in the first month?
(d) What formula should be in the cell D12, which should contain a 1 if you win in the first month, and otherwise a 0?
(e) What formula should be in the cell E12, the pool size at the end of the first month?
(f) Cell E7 is meant to contain a 1 if you have at least $12,000 in the pool at the end of
12 months, and otherwise 0. What formula should be in this cell?
Cell Name
E23 (Sim#1) Pool Size
E23 (Sim#2) Pool Size
E23 (Sim#3) Pool Size
E7 (Sim#1) Hit Target?
E7 (Sim#2) Hit Target?
E7 (Sim#3) Hit Target?
Minimum Mean Maximum
5,404 11,314 36,511
2,824 12,798 111,665
687 16,454 703,687
0 0.276 1
0 0.516
0 0.276
1
1
(g) The relevant portion of the @Risk output is displayed above. Which investment policy should you use to maximize your expected capital at the end? With this investment policy, what is the expected final pool size, and what is the probability of meeting the target, that is, having at least $12,000 in the pool at the end?
(h) Which investment policy should you use to maximize the probability of meeting the target? With this investment policy, what is the expected final capital, and what is the probability of meeting the target?
Shuttle Buses
Backenforth Shuttle Bus Service, Inc.
has entered into an agreement to take passengers from the local airport to the convention center. They must provide a shuttle bus every 20 minutes from
7:20 AM to 10:00 PM. During each 20-minute period between 7:00 AM and 10:00 PM, the number of passengers arriving at the airport bus stop is a Poisson random variable with average value 15.3. The passengers form a line at the bus stop and attempt to board the shuttle buses in a first-come, first-served manner.
Each passenger who boards the bus pays a fare of $5. Every time there is not enough room on the bus for everybody who is waiting, each passenger left behind has a 50% chance, independent all other passengers, of giving up and taking a taxi to the convention center instead. The rest continue to wait in line to see if they can board the next bus.
There are four different shuttle buses the company is considering using for the job. The smallest has 16 seats and costs $500 per day to operate. The larger buses have 20, 25, or 30 seats, and cost $600, $700, and $750 per day to operate, respectively.
OM Final Exam Study Guide Page 29 of 37 Spring 2002

The company is using the simulation spreadsheet below to try to determine which size bus would be best. They are using four simulations, each with 1000 iterations, to evaluate the four possible bus options. All the cells below line 8 displaying numbers, except G16, contain formulas. Lines
18 through 61 are all copies of line 17, although lines 32 through 48 are hidden. The simulation outputs are in cells F9:F11.
(a) Cell B10 is meant to contain the capacity of each kind of bus (16 for the first simulation, 20 for the second, etc.) What formula should it contain? Make sure your answer will yield a correct formula for daily operating cost when copied to cell
B11.
Parts (b)-(g) request formulas for row 17. In each case, make sure your answers will yield correct answers for all subsequent time periods if row 17 is copied to rows 18 through 61.
(b) What formula should be in cell B17, the number of passengers arriving at the airport bus stop between 7:00 AM and 7:20 AM?
(c) What formula should be in cell C17, the total number of passengers wishing to board the 7:20 AM bus?
(d) What formula should be in D17, the number of passengers boarding the 7:20 AM bus?
(e) What formula should be in cell E17, the number of passengers left behind at the bus stop at 7:20 AM?
(f) What formula should be in cell F17, the number of passengers left behind at the bus stop at 7:20 AM who then give up and took a taxi instead?
(g) What formula should be in cell G17, the number of passengers left behind at the bus stop at 7:20 AM who decide to wait for the next bus?
(h) What formulas should be in cells F9:F11? F9 should hold the average number of people left waiting at the bus stop, F10 should contain the total revenue for the day, and F11 should hold the day's operating profit.
(i) The summary statistics report from the @Risk run is reproduced below. Which bus should the firm use, and why? With this size bus, what is the average number of people left behind after each bus leaves the bus stop?
Cell Name Minimum Mean Maximum
F9 (Sim#1) Average Left at Bus Stop 0.4666667 1.738422
3.866667
F9 (Sim#2) Average Left at Bus Stop
F9 (Sim#3) Average Left at Bus Stop
0 0.2458 0.9111111
0 8.53E-03 0.1777778
F9 (Sim#4) Average Left at Bus Stop
F10 (Sim#1) Revenue
F10 (Sim#2) Revenue
0 0.0002
6.67E-02
2895 3248.925
3480
3035 3423.74
3760
F10 (Sim#3) Revenue
F10 (Sim#4) Revenue
F11 (Sim#1) Profit
F11 (Sim#2) Profit
F11 (Sim#3) Profit
F11 (Sim#4) Profit
3000 3448.67
3040 3451.09
2395 2748.925
2435 2823.74
2300 2748.67
2290 2701.09
3910
3825
2980
3160
3210
3075
OM Final Exam Study Guide Page 30 of 37 Spring 2002

Time
11:00 AM
11:20 AM
11:40 AM
12:00 PM
6:00 PM
6:20 PM
6:40 PM
7:00 PM
7:20 PM
7:40 PM
8:00 PM
8:20 PM
8:40 PM
9:00 PM
9:20 PM
9:40 PM
10:00 PM
7:00 AM
7:20 AM
7:40 AM
8:00 AM
8:20 AM
8:40 AM
9:00 AM
9:20 AM
9:40 AM
10:00 AM
10:20 AM
10:40 AM
10
11
12
13
14
15
53
54
55
56
57
58
28
29
30
31
49
50
51
52
59
60
61
20
21
22
23
24
25
26
27
16
17
18
19
6
7
4
5
1
2
3
8
9
A
Capacity
Daily Cost $
B
16
500.00
Chosen Option
Capacity
Daily Cost
16
$ 500.00
Arrived at
Bus Stop
14
9
18
18
20
17
10
18
14
10
11
19
12
14
13
17
13
21
18
14
16
17
22
18
11
21
13
22
C
Bus Options
20
$ 600.00
Total
Wanting to
Board Bus
15
16
17
23
21
12
19
14
14
13
19
14
21
20
21
17
22
14
9
18
19
23
22
11
18
15
10
11
$
D
Average Passengers arrived per time slot
Fare
Chance of Taking Taxi if Left at Bus Stop
25
700.00
Average Left at Bus Stop
Boarded
Bus
$
E
15.3
30
750.00
7
5
0
0
0
1
5
1
6
0
3
0
5
4
3
0
0
0
2
0
0
0
3
7
6
0
0
2
15
16
16
16
16
12
16
14
14
13
16
14
16
16
16
16
16
14
9
16
16
16
16
11
16
15
10
11
50%
Left at
Bus Stop
F
1.96
Revenue $ 3,205.00
Profit $ 2,705.00
G
Gave Up Waited and Took for Next a Taxi Bus
3
1
0
0
0
1
4
0
3
0
1
0
2
1
2
0
0
0
1
0
0
0
3
5
1
0
1
0
0
4
4
0
0
0
0
1
1
3
0
2
0
3
3
1
0
0
0
1
0
0
0
0
2
5
0
0
1
OM Final Exam Study Guide Page 31 of 37 Spring 2002

Temporary Water Supply for a Hotel
A temporary local water main problem will make the city water supply to the Ramilton hotel unfit to drink tomorrow. Ramilton’s management is considering renting a portable filtration unit to process the water to make it drinkable. There are three possible filtration units:
Option 1 Option 2 Option 3
Capacity (Gallons) 900 1000 1100
Cost $ 700 $ 900 $ 1,200
They are also considering not renting a filtration unit. If they don’t rent a unit, or if the unit’s capacity turns out to be insufficient to meet the hotel guests’ demand for water, the hotel will have to supply bottled water to every occupied room. Providing bottled water will cost the hotel a flat fee of $400, plus $5 per occupied room.
The hotel has 240 rooms. At present, they have 180 room reservations they consider “firm”, with a negligible chance of cancellation. They have another 35 reservations they consider “doubtful”, each with an independent 65% chance of resulting in an occupied room. They also expect to get some “last minute” requests for rooms, which they estimate to be Poisson with a mean of 30.
Occupied rooms consume random amounts of water independently from one another. Each occupied room consumes an average of 4 gallons of water, with a standard deviation of 3.1.
Management is using the spreadsheet simulation model below to help with the decision. All water amounts are in gallons. Management is interested in the expected total cost of each option, and the probability of not having enough filtered water to meet demand.
(a) What formulas should be in cells B13:B14, to respectively represent the filtering capacity and cost in each scenario, including “don’t rent”?
(b) What formulas should be in cells B19:B21, representing respectively:
 The number of “doubtful” reservations that actually result in occupied rooms

The number of last-minute requests for rooms

The number of occupied rooms
(c) What formulas should be in cells B24:B26, representing respectively:

The average of the total water demand for the occupied rooms, given the number of occupied rooms computed in cell B21

The standard deviation of the total water demand for the occupied rooms, given the number of occupied rooms computed in cell B21

The actual total water demand of the occupied rooms
(d) Cell B28 should contain 1 if the filtration unit is insufficient to meet water demand
(or no unit was rented), and should contain 0 if the filtration capacity is sufficient.
What formula should be in this cell?
(e) What formulas should be in cells B29:B30, respectively representing the cost (if any) of supplying each room with bottled water, and the total cost for the day?
(f) A simulation output report from YASAI appears below. Which filtration unit gives the lowest expected cost? If Ramilton rents this unit, what is the probability of having to supply bottled water?
OM Final Exam Study Guide Page 32 of 37 Spring 2002

16
17
18
19
20
21
22
23
10
11
12
13
14
15
1
2
3
4
5
6
7
8
9
24
25
26
27
28
29
30
A
Rooms
Firmly Booked
"Doubtful" Reservations
Average Last Minute Demand
Total Number in Hotel
Probability of Doubtfuls Arriving
Occupied Room Water Use
Mean
Standard Deviation
Filtration Unit
Capacity
Cost
Bottled Water Costs
Fixed
Per Room
Doubtfuls Arriving
Last Minute Demand
Occupied Rooms
Shortfall?
Shortfall Cost
Total Cost
$
B
$ 700.00
400.00
$
180
35
30
240
65%
5.00
4
3.1
28
44
240
C D E F
Option 1 Option 2 Option 3 Don't Rent
900 900
$ 700
Total Water Demand
Mean
Standard Deviation
960 (not shown)
48.025 (not shown)
Actual 1028.337
1000
$ 900
1100
$ 1,200
0
$ -
Scenario
1
2
3
4
1
$ 1,600.00
$ 2,300.00
Parameter
Filtration Unit Cost Filtration Unit Capacity
$700.00
900
$900.00
$1,200.00
$0.00
1000
1100
0
Output Name
Shortfall?
Shortfall?
Shortfall?
Shortfall?
Total Cost
Total Cost
Total Cost
Total Cost
Scenario
3
4
1
2
3
4
1
2
Observations
1000
1000
Mean
0.714
0.086
1000
1000
0.000
1.000
1000 1818.455
1000 1035.765
1000 1200.000
1000 1561.380
OM Final Exam Study Guide Page 33 of 37 Spring 2002

Managing Inventory
You manage a discount outlet that sells a product for $500. On “heavy” weeks, which randomly happen 40% of the time, demand for the product is Poisson with a mean of 100. On “light” weeks, which happen the rest of the time, demand is Poisson with a mean of 50.
Your regular stockroom has a capacity of 375 units. You can hold more than 375 units in inventory, but during any week in which that happens, you must pay $100 to rent additional storage space (this amount is a “flat fee” which does not depend on how much you exceed 375 units).
Each time you order inventory, you pay a $1,500 handling charge plus $300 per unit ordered.
Two weeks elapse between placing an order and receiving it.
Your inventory cost is $10 per unit per week, assessed on the average inventory level for the week, which you define to be the average of the week’s beginning and ending inventory levels.
Your ordering policy is characterized by two numbers R and Q , as follows. At the beginning of each week, let I be the amount of inventory in stock, and T be the amount of inventory ordered last week and currently in transit (if you didn’t order inventory last week, then
T = 0). In any case, if I + T

R , then you order an additional Q units. These units arrive at the beginning of the week after next. Your stockroom currently contains 225 units and there are no units in transit.
You are using the simulation spreadsheet model on the next page to evaluate all nine possible combinations of R = 200, 225, or 250 and Q = 200, 250, or 300. You base the analysis on a 26week period (half a year). Any inventory on hand, in transit, or just ordered at the end of the 26week period is assigned a “salvage” value of $300 per unit.
A week in which there is insufficient inventory to meet demand is called a “stockout”. In this case, the potential sales are lost; there is no “backordering”. A week in which the initial stock of items (including deliveries) is more than 375 units is called an “overflow”. You are interested in computing the expected total profit (with the “salvage” adjustment), the expected number of stockouts and overflows, and the probability of having any stockouts and overflows during the
26-week period.
(a) Cells D14:D15 should contain the values of R and Q, respectively, for each of the nine scenarios you want to test with YASAI. What formulas should be in these cells?
Parts (b)-(j) concern the formulas in row 20 of the spreadsheet, corresponding to week 1.
Make sure your answer will yield correct results for the remaining weeks when copied to rows 21 through 45. If you feel you have to add more columns to the spreadsheet, you may do so; simply specify what formulas they contain.
(b) What formula should be in cell B20, the amount of stock available at the start of the week (including deliveries)?
OM Final Exam Study Guide Page 34 of 37 Spring 2002

46
47
48
49
50
40
41
42
43
44
45
36
37
38
39
32
33
34
35
23
24
25
26
19
20
21
22
27
28
29
30
31
5
6
7
8
9
10
11
12
3
4
1
2
13
14
15
16
17
18
A B C
Heavy Week
Light Week
Order Cost
Unit cost
Selling price
Salvage value
Holding cost per unit
Stockroom capacity
Extra stockroom rental
Order Threshold
Order Quantity
D
Average
E
Demand Chance
$
100
50
1,500
$ 300
$ 500
$ 300
$ 10.00
375
$ 100.00
250
300
40%
60%
200
200
F
Trial Values
225
250
G
250
300
Week In Stock In Transit Ordered Demand Sold Left
222
160
409
318
264
156
195
432
388
284
223
180
359
308
225
106
314
220
165
414
361
320
217
95
300
247
21
22
23
24
25
26
17
18
19
20
13
14
15
16
6
7
4
5
2
3
0
1
8
9
10
11
12
0
0
300
0
0
300
0
0
0
0
300
0
0
300
0
0
0
0
300
0
0
0
300
0
0
0
0
0
300
0
0
300
0
0
0
0
300
0
0
300
0
0
0
0
300
0
0
0
300
0
0
0
0
300
119
92
94
55
51
53
41
103
122
96
53
52
63
44
104
61
43
121
51
86
62
51
91
54
108
47
62
51
91
54
108
47
63
44
104
61
43
121
51
86
119
92
94
55
51
53
41
103
122
95
53
52
160
109
318
264
156
109
132
388
284
223
180
59
308
222
225
106
14
220
165
114
361
320
217
95
0
247
195
Revenue $ 958,000
Cost $ 700,140
"Salvage" $ 122,700
Adjusted Profit $ 380,560
$
$
$
$
$
$
$
$
$
H
Cost
93,155
$
600
2,670
93,425
$
$
$
$
1,395
3,975
3,405
2,685
93,060
$
475
2,735
93,710
$
$
$
$
1,635
4,200
3,460
2,535
93,515
$
$
$
1,195
3,335
2,650
93,410
$
$
$
$
1,345
3,735
2,910
2,100
92,825
Number of
Were there any
I
Stockout
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
J
Overflow
1
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
Stockouts Overflows
1
1
4
1
OM Final Exam Study Guide Page 35 of 37 Spring 2002

(c) What formula should be in cell C20, the amount of stock ordered at the beginning of last week, and currently in transit?
(d) What formula should be in cell D20, the number of units ordered this week?
(e) What formula should be in call E20, to simulate customer demand for this week?
(f) What formula should be in cell F20, the number of units sold this week?
(g) What formula should be in cell G20, the number of units left in inventory at the end of the week?
(h) What formula should be in cell H20, the total cost incurred this week?
(i)
What formula should be in cell I20, to compute a “1” if there was a stockout this week, and a “0” otherwise?
(j) What formula should be in cell J20, to compute a “1” if there was an overflow this week, and a “0” otherwise?
(k) What formulas should be in cells D47:D50, which respectively represent:

The total revenue collected from customers

The total cost incurred

The salvage value of units left in stock, in transit, or just ordered at the end of week 26

The total profit for the 26-week period
(l) What formulas should be in cells I48:J49? Cell I48 is the number of stockouts during the period, and I49 should be 1 if there were any stockouts, and otherwise 0.
J48:J49 are similar, but refer to overflows instead of stockouts.
(m) The following page shows a simulation output report from YASAI. Which combination of R and Q gives the highest expected profit? With this choice, what is the probability of a stockout occurring during the 26-week period, and what are the chances of an overflow during the 26-week period? Of the policies whose expected number of stockouts is less than 1, which has the highest expected profit?
OM Final Exam Study Guide Page 36 of 37 Spring 2002

Scenario
1
2
3
4
5
6
7
8
9
Output Name
Number of Overflows
Number of Overflows
Number of Overflows
Number of Overflows
Number of Overflows
Number of Overflows
Number of Overflows
Number of Overflows
Number of Overflows
Were there any Overflows
Were there any Overflows
Were there any Overflows
Were there any Overflows
Were there any Overflows
Were there any Overflows
Were there any Overflows
Were there any Overflows
Were there any Overflows
Number of Stockouts
Number of Stockouts
Number of Stockouts
Number of Stockouts
Number of Stockouts
Number of Stockouts
Number of Stockouts
Number of Stockouts
Number of Stockouts
Were there any Stockouts
Were there any Stockouts
Were there any Stockouts
Were there any Stockouts
Were there any Stockouts
Were there any Stockouts
Were there any Stockouts
Were there any Stockouts
Were there any Stockouts
Adjusted Profit
Adjusted Profit
Adjusted Profit
Adjusted Profit
Adjusted Profit
Adjusted Profit
Adjusted Profit
Adjusted Profit
Adjusted Profit
Order Threshold Order Quantity
200 200
225
250
200
200
200
225
250
250
250
250
200
225
250
300
300
300
Scenario
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
Observations
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
4
5
6
7
2
3
8
9
1
3
4
5
1
2
6
7
7
8
9
5
6
1
2
3
4
8
9
3
4
5
6
1
2
5
6
3
4
7
8
9
9
1
2
7
8
0.899
0.571
0.283
$ 367,852
$ 370,713
$ 369,286
$ 366,733
$ 368,611
$ 366,177
$ 363,995
$ 364,271
$ 361,627
1.006
0.409
1.982
0.822
0.339
0.973
0.741
0.409
0.922
0.639
0.353
Mean
0.000
0.000
0.001
0.000
0.260
0.990
0.681
2.120
3.468
0.000
0.000
0.001
0.000
0.246
0.650
0.527
0.923
0.987
2.391
1.189
0.520
2.278
0.301
0.495
0.451
$ 21,911
$ 25,286
$ 26,755
$ 22,381
$ 25,874
$ 27,601
$ 22,533
$ 26,223
$ 27,587
0.267
0.113
1.129
0.981
0.716
1.342
0.984
0.600
1.255
0.859
0.589
0.162
0.438
0.492
0.268
0.481
0.478
Standard
Deviation
0.000
0.000
0.032
0.000
0.470
0.926
0.756
1.184
1.460
0.000
0.000
0.032
0.000
0.431
0.477
0.500
OM Final Exam Study Guide Page 37 of 37 Spring 2002