Three Models Exercises

A. Decision Table Exercises

Given the decision tables for each of the following exercises, find the following: the maxi-max

solution, the maxi-min solution, the expected monetary value, the mini-max regret solution and

the expected value of perfect information (EVPI)

1.

Scenarios

Alternative Good OK Bad

High 100 30 -40

Medium 80 50 -10

Low 20 20 20

Probabilities 0.3 0.5 0.2

2.

Scenarios

Alternative Good OK Bad

High 200 100 -100

Medium 150 100 -40

Low 10 10 10

Probabilities 0.1 0.4 0.5

3. In 1 above, find the certainty equivalent to each of the alternatives if the utility function is

given by U(X) = ln(X+40).

4. In 2 above, find the certainty equivalent to each of the alternatives if the utility function is

given by U(X) =√ .

B. Newsboy Exercises.

Discrete Cases

1. Given the following demand schedule for newspapers, find the optimal order size if papers

cost the newsboy $.35 and he can sell them for $.75 each.

a. Assume any unsold papers cannot be resold.

b. How does your answer change if the newsboy can sell unsold papers back to the

newspaper for $.10 each?

2. Given the following demand schedule for newspapers, find the optimal order size if papers

cost the newsboy $.50 and he can sell them for $1.50 each.

a. Assume any unsold papers cannot be resold.

k P(X=k)

55 0.05

56 0.05

57 0.15

58 0.3

59 0.3

60 0.1

61 0.05

b. How does your answer change if the newsboy can sell unsold papers back to the

newspaper for $.40 each?

k P(X=k)

20 0.1

21 0.2

22 0.3

23 0.2

24 0.15

25 0.05

Continuous Cases

1. A sandwich shop prepares sandwiches for delivery to customers that cannot find the time to

go out to lunch. Sandwiches cost $1 to prepare and the shop sells them for $5 apiece.

a. Assuming any sandwiches that are not sold are given away to the employees, so the shop

earns nothing for them. If the shop estimates demand is normally-distributed with mean of

500 and standard deviation of 100 sandwiches, find the optimal number of sandwiches the

shop should prepare each morning.

b. Now, if the shop is able to write off each unsold sandwich for $.50, how many should be

prepared each morning?

c. Find the daily earnings in b. above on a day on which demand just equals the expected

value of 500 doughnuts.

2. Krispie Crème (KC) bakes doughnuts early each morning for sale during the day. Doughnuts

are sold for $.60 each and cost KC $.15 each to prepare. Daily demand is estimated to

average 6000 doughnuts with a standard deviation of 2000 doughnuts. Find the optimal

number of doughnuts to prepare each morning if:

a. KC cannot sell unsold doughnuts the next day.

b. Day-old doughnuts are sold for $.10 each.

How much does KC earn on doughnut sales in part a. above if demand for the day is:

c. 7000 doughnuts?

d. 8000 doughnuts?

C. Inventory Model Exercises

1. A convenience store orders quarts of milk from a nearby dairy. The dairy charges $50 per

delivery regardless of the number of quarts per load. The store estimates the annual holding

cost of a quart of milk to be $.30 and sells 300,000 quarts per year.

Find the economic order quantity and the annual inventory costs associated with quarts of

milk.

2. The ABC Clothing Company sells a popular shirt that costs them $15 and retails for $30. At

an interest rate of 8 percent, they estimate the cost of capital tied up in inventory to be $1.20

per shirt per year. Other inventory overhead expense per shirt is estimated to be $.80 per year,

or $2 total holding cost per year.

If demand for the item is constant throughout the year, annual demand is estimated to be

50,000 shirts, and ordering costs are $20 per order, find the optimal order size, and the

inventory costs for the shirt.

3. The XYZ Manufacturing Company makes a special metal bracket that costs $10 per unit.

XYZ makes 20,000 units per day on its casting machine, and setup costs are $600 per run.

The annual cost of storing each item is estimated at 80 cents in interest expense, .08*$10 =

$.80, plus 70 cents in warehouse expense per unit per year, or $1.50 total holding costs.

Demand is assumed to be a constant 8000 units per day during the production cycle.

Find the optimal run size and total inventory costs for producing one million units per year.

4. A car manufacturer sells 25,000 of one of its models each year. The company can assemble

2000 cars per day and it sells 600 per day during the production cycle. The cost associated

with holding a car is estimated to be 10% of the $20,000 vehicle manufacturing cost or $2000.

Each production setup costs $7000.

Linear Programming Exercises

1. Find the optimal levels of each of the two activities in the following two problems.

a. Max profit = 80 X + 100 Y subject to:

X + Y ≤ 600

2X + 3Y ≤ 1500

b. Min cost = 200 X + 300 Y subject to

X + 4Y ≥ 50

3X + 2Y ≥ 90

X + Y = 35

2. Fine Furniture Mfg., Inc. makes tables and chairs. Each day it employs 60 hours of assembly

time and 40 hours of finishing time. Each table requires four hours of assembly time and two

hours of finishing time; each chair requires two hours of assembling and four hours of

finishing.

If the profit on each table is $80 and profit on each chair is $60, how many of each should be

produced per day and what is daily profit?

3. Carpet Manufacturers, Inc. makes carpets and rugs. Each month it employs 600 hours of

cutting time, 600 hours of sewing time and 660 hours of finishing time. Each carpet requires

three hours of cutting, two hours of sewing and three hours of finishing time; each rug

requires two hours of cutting time, three hours of sewing and three hours of finishing.

If the profit on each carpet is $250 and on each rug is $200, how many of each should be

produced per month and what is monthly profit?

4. Special Design Autos (SDA) manufactures two products, SUVs and minivans. These

automobiles are hand assembled, painted, and test driven before shipping. Currently SDA

has the following weekly constraints: 12,000 hours of assembly time, 600 hours of painting

time and 260 hours of testing time. Requirements for each type of vehicle are as follows:

Each SUV requires 500 hours of assembly, 20 hours of painting, and 10 hours of testing.

Each van requires 250 hours of assembly, 20 hours of painting, and 8 hours of testing.

If each SUV returns $3000 of profit and each minivan yields $2000 of profit, find the optimal

number of each vehicle to manufacture each week and the total weekly profit.

5. Fred's Coffee sells two blends of beans: Yusip Blend and Exotic Blend. Yusip Blend is one-

half Costa Rican beans and one-half Ethiopian beans. Exotic Blend is one-quarter Costa Rican

beans and three-quarters Ethiopian beans. Profit on the Yusip Blend is $3.50 per pound, while

profit on the Exotic Blend is $4.00 per pound. Each day Fred receives a shipment of 200

pounds of Costa Rican beans and 330 pounds of Ethiopian beans to use for the two blends.

How many pounds of each blend should be prepared each day to maximize profit? What is the

maximum profit?

6. The Mapple store sells Mapple computers and printers. The computers are shipped in 12-cubic-

foot boxes and printers in 8-cubic-foot boxes. The Mapple store estimates that at least 30

computers can be sold each month and that the number of computers sold will be at least 50%

more than the number of printers. The computers cost the store $1000 each and are sold for a

profit of $1000. The printers cost $300 each and are sold for a profit of $350. The store has a

storeroom that can hold 1000 cubic feet and can spend $70,000 each month on computers and

printers. How many computers and how many printers should be sold each month to maximize

profit? What is the maximum profit?

7. The Appliance Barn has 2400 cubic feet of storage space for refrigerators. Large refrigerators

come in 60-cubic-foot packing crates and small refrigerators come in 40-cubic-foot crates. Large

refrigerators can be sold for a $250 profit and the smaller ones can be sold for $150 profit. How

many of each type of refrigerator should be sold to maximize profit and what is the maximum

profit if:

a) At least 50 refrigerators must be sold each month.

b) At least 40 refrigerators must be sold each month.

c) There are no restrictions on what must be sold.

8. Shannon's Chocolates produces semisweet chocolate chips and milk chocolate chips at its plants

in Wichita, KS and Moore, OK. The Wichita plant produces 3000 pounds of semisweet chips

and 2000 pounds of milk chocolate chips each day at a cost of $1000, while the Moore plant

produces 1000 pounds of semisweet chips and 6000 pounds of milk chocolate chips each day at a

cost of $1500.

Shannon has an order from Food Box Supermarkets for at least 30,000 pounds of semisweet

chips and 60,000 pounds of milk chocolate chips. How should Shannon schedule its production

so that it can fill the order at minimum cost? What is the minimum cost?

[Examples 5-8 were found at:

http://www.algebra.com/algebra/homework/coordinate/word/THEO-2012-01-26.lesson

9. The objective is to find a minimum-cost diet that contains at least 300 calories, not more than 10

grams of protein, not less than 10 grams of carbohydrates, and not less than 8 grams of fat. In

addition, the diet should contain at least 0.5 units of fish and no more than 1 unit of milk.

Nutritional data was gathered for six foods: bread, milk, cheese, potato, fish and yogurt. Use

Excel Solver to create a dietary regimen that meets the objectives outlined above. Do you feel

you could live with such a diet? [NB: a unit of each food is 100g]

Food name cost ($) prot (gm) fat (gm) carb (gm) cal

Bread 0.5 10 3.4 49 270

Milk 0.1 3.4 1 5 42

Cheese 1 32 30 0.4 413

Potato 0.25 0.1 2.6 12 58

Fish 1.5 1.7 20 0 96

Yogurt 0.5 1.6 5.3 7 63

10. For your own interest, add foods to the list above, as well as constraints, and generate a diet of

your very own. Nutritional information can be found at the following website:

http://www.nutritionvalue.org/. [This exercise was adapted from

11. A farmer has 10 acres to plant in wheat and rye. He has to plant at least 7 acres. However, he has

only $1200 to spend and each acre of wheat costs $200 to plant and each acre of rye costs $100

to plant. Moreover, the farmer has to get the planting done in 12 hours and it takes an hour to

plant an acre of wheat and 2 hours to plant an acre of rye. If the profit is $500 per acre of wheat

and $300 per acre of rye how many acres of each crop should be planted to maximize profits?

12. A gold processor has two sources of gold ore, source A and source B. In order to keep his plant

running at least three tons of ore must be processed each day. Ore from source A costs $20 per

ton to process, and ore from source B costs $10 per ton to process. Costs must be kept to less

than $80 per day. Moreover, federal regulations require that the amount of ore from source B

cannot exceed twice the amount of ore from source A. If ore from source A yields 2 oz. of gold

per ton, and ore from source B yields 3 oz. of gold per ton, how many tons of ore from both

sources must be processed each day to maximize the amount of gold extracted subject to the

above constraints?

13. A publisher has orders for 600 copies of a certain text from San Francisco and 400 copies from

Sacramento. The company has 700 copies in a warehouse in Novato and 800 copies in a

warehouse in Lodi. It costs $5 to ship a text from Novato to San Francisco, but it costs $10 to

ship it to Sacramento. It costs $15 to ship a text from Lodi to San Francisco, but it costs $4 to

ship it from Lodi to Sacramento. How many copies should the company ship from each

warehouse to San Francisco and Sacramento to fill the order at the least cost? [This is actually a

transportation problem, but use solver to solve it as a LP problem. Then go back and solve it

using VAM and MODI, discussed in transportation models.]

[Exercises 11 – 13 were found at Steve Wilson’s (CSU Sonoma) website:

http://www.sonoma.edu/users/w/wilsonst/courses/math_131/lp/.]

14. LP formulation - another example from the Internet (with some revision):

[http://stackoverflow.com/questions/11975658/how-do-units-flow-through-matrix-operations]

Bob’s bakery sells bagels and muffins.

To bake a dozen bagels Bob needs 5 cups of flour, 2 eggs, and one cup of sugar.

To bake a dozen muffins Bob needs 4 cups of flour, 4 eggs and 1.2 cups of sugar.

Bob can sell bagels at $10/dozen and muffins at $15/dozen.

Bob has 400 cups of flour, 300 eggs and 96 cups of sugar.

How many bagels and muffins should Bob bake in order to maximize his revenue?

Transportation and Assignment Exercises

A. For each of the following transportation problems, find the minimum cost solution and

calculate the cost.

1.

Destination

Origin Albany Burlington Clifton Dexter Total

Elmira $7 $5 $2 $4 4000

Framingham $6 $3 $3 $3 6000

Griffon $4 $7 $6 $5 4000

Total 4000 4000 3000 3000 14000

2.

Destination

Origin Boston Chicago St. Louis Lexington Total

Cleveland $5 $4 $3 $4 4000

Bedford $6 $3 $3 $3 6000

York $4 $7 $6 $5 2500

Total 4000 4000 2500 2000 12500

3.

Destination

Origin Buenos Aires Chicago Toronto Rome Total

Guadalajara $3 $2 $7 $6 5000

Shanghai $7 $5 $2 $3 6000

Sao Paulo $2 $5 $4 $5 2500

Total 6000 4000 2500 1000 13500

4.

Destination

Origin Boston Chicago St. Louis Charleston Total

Cleveland $5 $3 $4 $8 5000

Savannah $7 $8 $12 $2 6000

New York $3 $6 $10 $8 2500

Total 4000 4000 2500 2000

5.

Destination

Origin Boston Chicago St. Louis Lexington Total

Cleveland $5 $4 $6 $3.50 6000

Bedford $6 $3 $3 $3 6000

York $4 $7 $9 $5 2500

Total 4000 4000 2500 2000 12500

6. (Executive Furniture Co.)

Destination

Origin Albuquerque Boston Cleveland Total

Des Moines $5 $4 $3 100

Evansville $8 $4 $3 300

Ft. Lauderdale $9 $7 $5 300

Total 300 200 200 700

7. Arden County, MD School Bussing Problem (from Render and Stair)

Schools

Zone

School

B

School

C

School

E

A 5 8 6 700

B 0 4 12 400

C 4 0 7 100

D 7 2 5 800

E 12 7 0 500

Total 900 900 900

Assignment Problems. For each of the following assignment problems, find the minimum cost or

time solution and calculate the cost or time.

1. Minimize total cost.

Task

Person 1 2 3

Cooper $9 $12 $7

Brown $8 $10 $11

Adams $11 $14 $6

2. Minimize total task time.

Machines

M1 M2 M3

J1 14 12 16

J2 11 17 21

J3 20 8 7

3. Minimize travel time for ACC officials.

Officials RALEIGH ATLANTA DURHAM CLEMSON

A 210 90 180 160

B 100 70 130 200

C 175 105 140 170

D 80 65 105 120

4. Minimize total task time.

Machine A Machine B Machine C Machine D

Task 1 5 8 6 9

Task 2 4 6 8 4

Task 3 10 10 9 8

Task 4 10 12 11 9

Network Models

A. Minimal Spanning Tree (MST) Examples – For each of the diagrams below,

find the minimal spanning tree that connects all of the nodes into the network. Be sure to

show the start node and end node in the correct order and the total length of the span. You

may start from any node.

1.

2.

3. LAN network. Connect all the office computers.

4. Airline hub and spokes. Connect all the airports.

5.

12 15

5

4 10 3 16 9

8 8

9 7

6

10 4 8 6

22

13

2

1

4

85

9

36

10

7

B. Shortest Path Exercises – For each of the following networks identify the

shortest path between the indicated nodes. You will need to show your work by

indicating for every node the shortest path between the starting node and that node. Do

this as was done in the lecture with a box of the form:

C

22

where C indicates the immediate predecessor node and 22 indicates the

cumulative distance from the start node to the current node.

1. Start at 1 and go to 7

2. Start at 1 and go to 7

3. Using the map below, find the shortest route from SFO to MIA and from LAX to BOS.

4. Start at A and go to H

20 20

105 5 15

1025 20

1530 5 10

25

A

B

D

C

E

F

G

H

A

B

D

C

E

F

G

H

SFO

LAX DFW

ORD

JFK

BOS

MIA

1846

1258

802

2704

1090 1235

1464 337

2342

1121

740

867

187

C. Maximal Flow Models – For each of the following flow diagrams, find the

maximum flow from the terminus on the left to that on the right. Be sure to indicate each

path and the maximum flow as well as the revised capacities after each path is used.

Read and watch the lectures to see haw I want this done.

1. Start at 1 and go to 6

2. Start at A and go to F

3. Start at A and go to H

4. Start at 1 and go to 10

10 4 10

8 12

10 6 10

12

20 8 8 6 10 5 7

10 9 10

10 4 8

15 8 2

10

12 10

A

B

D

C

E

F

G

H

12 15

5

4 10 3 16 9

8 8

9 7

6

10 4 8 6

22

13

2

1

4

85

9

36

10

7

Waiting Line (Queuing) Models Exercises

For each of the exercises in this section find the following parameters:

P0 = the probability that nobody is in the system, either being served or waiting in

line

Pw = the probability that a new arrival will have to wait for service,

Lq = the expected number of persons waiting in line to be served

Wq = the expected length of time one will wait in line to be served

Ws = the expected total time in the “system”; that is, in the line or being served,

and

Ls = the expected number of people in the system either in line or being served.

State the assumptions of the model. Tables and formulas can be found at the

following sites.

Waiting Line Tables

Po http://faculty.citadel.edu/silver/Po.pdf

Pw http://faculty.citadel.edu/silver/pw.pdf

Lq http://faculty.citadel.edu/silver/lq.pdf Waiting Line Formulas are found at http://faculty.citadel.edu/silver/waiting_line_formulas.htm

1. Three people arrive each minute at the airport and wait to use the check-in kiosk. On average

a traveler requires one minute to check in. Currently four kiosks are in service.

2. Four people arrive at a toll plaza each minute; drivers require, on average, one minute to pay

the toll. Five toll booths are in operation. What will happen if a sixth booth is opened?

3. On average five cars per hour come off the first stage of an assembly line; cars are moved

directly to a second stage of the line. The first station of the second stage can process an average

of two cars per minute and there are three different lines available for the second stage.

4. 80 people arrive at the bank each hour and 30 can be served per hour; that is, service times

average two minutes. There are three tellers currently available. How will the average waiting

time change if a fourth teller arrives for work? (Hint: Use the value of λ/µ on your tables closest

to the one you calculated.)

Project Management, PERT/CPM Exercises

For each of the following exercises construct a PERT chart, calculate the earliest start and

earliest finish for each activity, identify the critical path, and calculate the latest start and finish

for each activity.

1. From the Internet, a very simple project.

2. Again from the Internet, a slightly more difficult project.

3. Another Internet example.

4. Here’s an Internet exercise with a, m, b times; you may wish to try this one out and plug it into

POM/QM for windows.

5. Another gem from the Internet to chew on.

6. One of my own

Activity Immediate Expected

Letter Precdecessor Time

A --- 2

B --- 3

C A 2

D B 4

E C 4

F C 3

G D, E, F 5

7. Build a house. A massive undertaking; not for the faint of heart!

ACTIVITY

DESCRIPTION

DURATION (DAYS)

PREDECESSORS

(a, m, b) (DAYS)

A Site Work 7 - 4, 7, 10

B Foundation 30 A 28, 30, 35

C Framing 22 B 15, 22, 30

E Plumbing Rough-in 10 C 7, 10, 12

F Roofing 4 C 2, 4, 7

G Install Exterior Windows and Doors 2 F 1, 2, 3

H Electric Rough-in 8 G 5, 8, 10

I H.V.A.C. 8 G 5,8,9

J Siding 5 G 2,5,7

K Insulation 2 I, E, H 1,2,3

L Drywall 10 K 7,10,14

M Paint Exterior (Caulk/Prime/Paint) 3 J 2, 3, 5

N Paint Interior Drywall/Stain Trim 5 L 3, 5, 7

O Install Interior Trim 7 N 4, 7, 9

P Install Tile, Carpet, Hardwoods 10 O 8, 10, 12

Q Install Fixtures/Connect Appliances 6 P 3, 6, 8

R Install Kitchen Cabinets/countertops 4 P 3, 4, 5

S Exterior Landscaping 7 M 4, 7,1 0

T Electrical Final Trim 4 R 2, 4, 6

U Hardware (cabinets, doors) 6 R 3, 6, 8

V Walk through/final touchups 4 S, Q, U, T 2, 4, 6

W Cleaning 2 V 1, 2, 3

X Final Walk-through 1 W 5, 1, 2

Y Move in 4 x 3, 4, 5