MAT 540 Complete Course MAT540 Complete Course

MAT 540 Complete Course MAT540 Complete Course

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MAT 540 Week 1 Discussion

"Class Introductions" Please respond to the following:

Please introduce yourself, including your educational and career goals, as well as some

personal information about yourself. In your introduction, please draw from your own

experience (or use a search engine) to give an example of how probability is used in your

chosen profession. If you get your information from an online or other resource, be sure to

cite the source of the information.

MAT 540 Week 1 Homework

Chapter 1

1. The Retread Tire Company recaps tires. The fixed annual cost of the recapping operation is

$65,000. The variable cost of recapping a tire is $7.5. The company charges$25 to recap a tire.

a. For an annual volume of 15, 000 tire, determine the total cost, total revenue, and profit.

b. Determine the annual break-even volume for the Retread Tire Company operation.

2. Evergreen Fertilizer Company produces fertilizer. The company’s fixed monthly cost is

$25,000,

and its variable cost per pound of fertilizer is $0.20. Evergreen sells the fertilizer for $0.45 per

pound. Determine the monthly break-even volume for the company.

3. If Evergreen Fertilizer Company in problem 2 changes the price of its fertilizer from $0.45 per

pound to $0.55 per pound, what effect will the change have on the break-even volume?



4. If Evergreen Fertilizer Company increases its advertising expenditure by $10,000 per year,

what

effect will the increase have on the break-even volume computed in problem 2?

5. Annie McCoy, a student at Tech, plans to open a hot dog stand inside Tech’s football stadium

during home games. There are 6 home games scheduled for the upcoming season. She must pay

the

Tech athletic department a vendor’s fee of $3,000 for the season. Her stand and other equipment

will cost her $3,500 for the season. She estimates that each hot dog she sells will cost her $0.40.

she

has talked to friends at other universities who sell hot dogs at games. Based on their information

and the athletic department’s forecast that each game will sell out, she anticipates that she will sell

approximately 1,500 hot dogs during each game.

a. What price should she charge for a hot dog in order to break even?

b. What factors might occur during the season that would alter the volume sold and thus the

break-even price Annie might charge?

6. The college of business at Kerouac University is planning to begin an online MBA program.

The

initial start-up cost for computing equipment, facilities, course development and staff recruitment

and development is $400,000. The college plans to charge tuition of $20,000 per student per year.

However, the university administration will charge the college $10,000 per student for the first

100

students enrolled each year for administrative costs and its share of the tuition payments.

a. How many students does the college need to enroll in the first year to break-even?

b. If the college can enroll 80 students the first year, how much profit will it make?

c. The college believes it can increase tuition to $25,000, but doing so would reduce enrollment to

50. Should the college consider doing this?

Chapter 11

7. The following probabilities for grades in management science have been determined based on

past

records:

The grades are assigned on a 4.0 scale, where an A is a 4.0, a B a 3.0, and so on. Determine the

expected grade and variance for the course.

8. An investment firm is considering two alternative investments, A and B, under two possible

future

sets of economic conditions good and poor. There is a .60 probability of good economic

conditions

occurring and a .40 probability of poor economic conditions occurring. The expected gains and

losses under each economic type of conditions are shown in the following table:

Using the expected value of each investment alternative, determine which should be selected.

9. The weight of the bags of fertilizer is normally distributed, with a mean of 45 pounds and a

standard deviation of 5 pounds. What is the probability that a bag of fertilizer will weigh between

38 and 50 pounds?

10. The polo Development Firm is building a shopping center. It has informed renters that their

rental

spaces will be ready for occupancy in 18 months. If the expected time until the shopping center is

completed is estimated to be 15 months, with a standard deviation of 5 months, what is the

probability that the renters will not be able to occupy in 18 months?

11. The manager of the local National Video Store sells videocassette recorders at discount prices.

If

the store does not have a video recorder in stock when a customer wants to buy one, it will lose

the

sale because the customer will purchase a recorder from one of the many local competitors. The

problem is that the cost of renting warehouse space to keep enough recorders in inventory to meet

all demand is excessively high. The manager has determined that if 85% of customer demand for

recorders can be met, then the combined cost of lost sales and inventory will be minimized. The

manager has estimated that monthly demand for recorders is normally distributed, with a mean of

175 recorders and a standard deviation of 55. Determine the number of recorders the manager

should order each month to meet 85% of customer demand.


In your own words, explain how to obtain the “expected value of perfect information” for any

payoff table, which has probabilities associated with each state of nature. Then, provide an

example, drawing from any of the payoff tables in Problems 1-17 in the back of Chapter 12. If no

probabilities are given for the states of nature, then assume equal likelihood.


Chapter 12

1. A local real estate investor in Orlando is considering three alternative investments; a motel, a

restaurant, or a theater. Profits from the motel or restaurant will be affected by the availability of

gasoline and the number of tourists; profits from the theater will be relatively stable under any

conditions. The following payoff table shows the profit or loss that could result from each

investment:

Determine the best investment, using the following decision criteria.

a. Maximax

b. Maximin

c. Minimax regret

d. Hurwicz (α = 0.4)

e. Equal likelihood

2. A concessions manager at the Tech versus A&M football game must decide whether to have

the

vendors sell sun visors or umbrellas. There is a 35% chance of rain, a 25% chance of overcast

skies,

and a 40% chance of sunshine, according to the weather forecast in college junction, where the

game is to be held. The manager estimates that the following profits will result from each

decision,

given each set of weather conditions:

a. Compute the expected value for each decision and select the best one.

b. Develop the opportunity loss table and compute the expected opportunity loss for each

decision.

3. Place-Plus, a real estate development firm, is considering several alternative development

projects.

These include building and leasing an office park, purchasing a parcel of land and building an

office building to rent, buying and leasing a warehouse, building a strip mall, and selling

condominiums. The financial success of these projects depends on interest rate movement in the

next 5 years. The various development projects and their 5- year financial return (in $1,000,000s)

given that interest rates will decline, remain stable, or increase, are in the following payoff table.

Place-Plus real estate development firm has hired an economist to assign a probability to each

direction interest rates may take over the next 5 years. The economist has determined that there is

a

0.45 probability that interest rates will decline, a 0.35 probability that rates will remain stable, and

a

0.2 probability that rates will increase.

a. Using expected value, determine the best project.

b. Determine the expected value of perfect information.

4. The

director

of career advising at Orange Community College wants to use decision analysis to

provide information to help students decide which 2-year degree program they should pursue. The

director has set up the following payoff table for six of the most popular and successful degree

programs at OCC that shows the estimated 5-Year gross income ($) from each degree for four

future economic conditions:

Determine the best degree program in terms of projected income, using the following decision

criteria:

a. Maximax

b. Maximin

c. Equal likelihood

d. Hurwicz (α=0.4)

5. Construct a decision tree for the following decision situation and indicate the best decision.

Fenton and Farrah Friendly, husband-and-wife car dealers, are soon going to open a new

dealership.

They have three offers: from a foreign compact car company, from a U.S. producer of full-sized

cars, and from a truck company. The success of each type of dealership will depend on how much

gasoline is going to be available during the next few years. The profit from each type of

dealership,

given the availability of gas, is shown in the following payoff table:

Decision Tree diagram to complete:

MAT 540 Week 2 Quiz

Question 1

Probabilistic techniques assume that no uncertainty exists in model parameters.

Question 2

In general, an increase in price increases the break even point if all costs are held constant.

Question 3

Parameters are known, constant values that are usually coefficients of variables in equations.

Question 4

Fixed cost is the difference between total cost and total variable cost.

Question 5

P(A | B) is the probability of event A, if we already know that event B has occurred.

Question 6

A binomial probability distribution indicates the probability of r successes in n trials.

Question 7

If events A and B are independent, then P(A|B) = P(B|A).

Question 8

If fixed costs increase, but variable cost and price remain the same, the break even point

Question 9

If the price increases but fixed and variable costs do not change, the break even point

Question 10

A model is a functional relationship that includes:

Question 11

The indicator that results in total revenues being equal to total cost is called the

Question 12

The expected value of the standard normal distribution is equal to

Question 13

The area under the normal curve represents probability, and the total area under the curve sums to

Question 14

In a binomial distribution, for each of n trials, the event

Question 15

Administrators at a university are planning to offer a summer seminar. The costs of reserving a

room, hiring an instructor, and bringing in the equipment amount to $3000.

Suppose that it costs $25 per student for the administrators to provide the course materials. If we

know that 20 people will attend, what price should be charged per person to break even? Note:

please report the result as a whole number, rounding if necessary and omitting the decimal point.

Question 16

A production run of toothpaste requires a fixed cost of $100,000. The variable cost per unit is

$3.00. If 50,000 units of toothpaste will be sold during the next month, what sale price must be

chosen in order to break even at the end of the month? Note: please report the result as a whole

number, rounding if necessary and omitting the decimal point.

Question 17

A production process requires a fixed cost of $50,000. The variable cost per unit is $25 and the

revenue per unit is projected to be $45. Find the break-even point.

Question 18

Wei is considering pursuing an MS in Information Systems degree. She has applied to two

different universities. The acceptance rate for applicants with similar qualifications is 20% for

University X and 45% for University Y. What is the probability that Wei will be accepted by at

least one of the two universities? {Express your answer as a percent. Round (if necessary) to the

nearest whole percent and omit the decimal. For instance, 20.1% would be written as 20}

Question 19

Employees of a local company are classified according to gender and job type. The following

table summarizes the number of people in each job category.

Question 20

An automotive center keeps tracks of customer complaints received each week. The probability

distribution for complaints can be represented as a table (shown below). The random variable xi

represents the number of complaints, and p(xi) is the probability of receiving xi complaints.


Discuss Simulation

Select one (1) of the following topics for your primary discussion posting:

Identify the part of setting up a simulation in Excel that you find to be the most

challenging, and explain why. Identify resources that can help you with that.

Explain how simulation is used in the real world. Provide a specific example from your

own line of work, or a line of work that you find particularly interesting.


Chapter 14

1. The Hoylake Rescue Squad receives an emergency call every 1, 2, 3, 4, 5, or 6 hours, according

to

the following probability distribution. The squad is on duty 24 hours per day, 7 days per week:

a. Simulate the emergency calls for 3 days (note that this will require a “running” , or

cumulative,hourly clock), using the random number table.

b. Compute the average time between calls and compare this value with the expected value

of the time between calls from the probability distribution. Why are the result different?

2. The time between arrivals of cars at the Petroco Services Station is defined by the following

probability distribution:

a. Simulate the arrival of cars at the service station for 20 arrivals and compute the average

time

between arrivals.

b. Simulate the arrival of cars at the service station for 1 hour, using a different stream of

random

numbers from those used in (a) and compute the average time between arrivals.

c. Compare the results obtained in (a) and (b).

3. The Dynaco Manufacturing Company produces a product in a process consisting of operations

of

five machines. The probability distribution of the number of machines that will break down in a

week follows

a. Simulate the machine breakdowns per week for 20 weeks.

b. Compute the average number of machines that will break down per week.

4. Simulate the following decision situation for 20 weeks, and recommend the best decision.

A concessions manager at the Tech versus A&M football game must decide whether to have the

vendors sell sun visors or umbrellas. There is a 30% chance of rain, a 15% chance of overcast

skies,

and a 55% chance of sunshine, according to the weather forecast in college junction, where the

game is to be held. The manager estimates that the following profits will result from each

decision,

given each set of weather conditions:

5. Every time a machine breaks down at the Dynaco Manufacturing Company (Problem 3), either

1, 2,

or 3 hours are required to fix it, according to the following probability distribution:

Simulate the repair time for 20 weeks and then compute the average weekly repair time.

MAT 540 Week 3 Quiz 2

Question 1

If two events are not mutually exclusive, then P(A or B) = P(A) + P(B)

Question 2

Probability trees are used only to compute conditional probabilities.

Question 3

Seventy two percent of all observations fall within 1 standard deviation of the mean if the data is

normally distributed.

Question 4

Both maximin and minimin criteria are optimistic.

Question 5

The minimin criterion is optimistic.

Question 6

The Hurwicz criterion is a compromise between the maximax and maximin criteria.

Question 7

The Hurwicz criterion is a compromise between the minimax and minimin criteria.

Question 8

The chi-square test is a statistical test to see if an observed data fit a _________.

Question 9

Assume that it takes a college student an average of 5 minutes to find a parking spot in the main

parking lot. Assume also that this time is normally distributed with a standard deviation of 2

minutes. Find the probability that a randomly selected college student will take between 2 and 6

minutes to find a parking spot in the main parking lot.

Question 10

A professor would like to utilize the normal distribution to assign grades such that 5% of students

receive A's. If the exam average is 62 with a standard deviation of 13, what grade should be the

cutoff for an A? (Round your answer.)

Question 11

A business owner is trying to decide whether to buy, rent, or lease office space and has

constructed the following payoff table based on whether business is brisk or slow.

If the probability of brisk business is .40 and for slow business is .60, the expected value of

perfect information is:

Question 12



The maximin strategy is:

Question 13

The maximin criterion results in the

Question 14

Determining the worst payoff for each alternative and choosing the alternative with the best worst

is called

Question 15

A life insurance company wants to update its actuarial tables. Assume that the probability

distribution of the lifetimes of the participants is approximately a normal distribution with a mean

of 71 years and a standard deviation of 3.5 years. What proportion of the plan participants are

expected to see their 75th birthday? Note: Write your answers with two places after the decimal,

rounding off as appropriate.

Question 16

A brand of television has a lifetime that is normally distributed with a mean of 7 years and a

standard deviation of 2.5 years. What is the probability that a randomly chosen TV will last more

than 8 years? Note: Write your answers with two places after the decimal, rounding off as

appropriate.

Question 17

A manager has developed a payoff table that indicates the profits associated with a set of

alternatives under 2 possible states of nature.

Alt S1 S2

1 10 2

2 -2 8

3 8 5

What is the highest expected value? Assume that the probability of S2 is equal to 0.4.

Question 18



If the probability of brisk business is .40, what is the numerical maximum expected value?

Question 19

A manager has developed a payoff table that indicates the profits associated with a set of

alternatives under 2 possible states of nature.

Alt S1 S2

1 10 2

2 -2 8

3 8 5

Compute the expected value of perfect information assuming that the probability of S2 is equal to

0.4.

Question 20

A group of friends are planning a recreational outing and have constructed the following payoff

table to help them decide which activity to engage in. Assume that the payoffs represent their

level of enjoyment for each activity under the various weather conditions.

Weather

Cold Warm Rainy

S1 S2 S3

Bike: A1 10 8 6

Hike: A2 14 15 2

Fish: A3 7 8 9

If the probabilities of cold weather (S1), warm weather (S2), and rainy weather (S3) are 0.2, 0.4,

and 0.4, respectively what is the EVPI for this situation?


Discuss Forecasting Methods


Identify any challenges you have in setting up a time-series analysis in Excel. Explain

what they are and why they are challenging. Identify resources that can help you with that.

Explain how forecasting is used in the real world. Provide a specific example from your

own line of work, or a line of work that you find particularly interesting.


Chapter 15

1. The manager of the Carpet City outlet needs to make an accurate forecast of the demand for

Soft

Shag carpet (its biggest seller). If the manager does not order enough carpet from the carpet mill,

customer will buy their carpet from one of Carpet City’s many competitors. The manager has

collected the following demand data for the past 8 months:

a. Compute a 3-month moving average forecast for months 4 through 9.

b. Compute a weighted 3-month moving average forecast for months 4 through 9. Assign

weights of 0.55, 0.35, and 0.10 to the months in sequence, starting with the most recent

month.

c. Compare the two forecasts by using MAD. Which forecast appears to be more accurate?

2. The manager of the Petroco Service Station wants to forecast the demand for unleaded gasoline

next month so that the proper number of gallons can be ordered from the distributor. The owner

has

accumulated the following data on demand for unleaded gasoline from sales during the past 10

months:

a. Compute an exponential smoothed forecast, using an α value of 0.4

b. Compute the MAD.

3. Emily Andrews has invested in a science and technology mutual fund. Now she is considering

liquidating and investing in another fund. She would like to forecast the price of the science and

technology fund for the next month before making a decision. She has collected the following data

on the average price of the fund during the past 20 months:

a. Using a 3-month average, forecast the fund price for month 21.

b. Using a 3-month weighted average with the most recent month weighted 0.5, the next

most

recent month weighted 0.30, and the third month weighted 0.20, forecast the fund price for

month 21.

c. Compute an exponentially smoothed forecast, using α=0.3, and forecast the fund price

for

month 21.

d. Compare the forecasts in (a), (b), and (c), using MAD, and indicate the most accurate.

4. Carpet City wants to develop a means to forecast its carpet sales. The store manager believes

that

the store’s sales are directly related to the number of new housing starts in town. The manager has

gathered data from county records on monthly house construction permits and from store records

on monthly sales. These data are as follows:

a. Develop a linear regression model for these data and forecast carpet sales if 30

construction

permits for new homes are filed.

b. Determine the strength of the causal relationship between monthly sales and new home

construction by using correlation.

5. The manager of Gilley’s Ice Cream Parlor needs an accurate forecast of the demand for ice

cream.

The store orders ice cream from a distributor a week ahead; if the store orders too little, it loses

business, and if it orders too much, the extra must be thrown away. The manager belives that a

major determinant of ice cream sales is temperature (i.e.,the hotter the weather, the more ice

cream

people buy). Using an almanac, the manager has determined the average day time temperature for

14 weeks, selected at random, and from store records he has determined the ice cream

consumption

for the same 14 weeks. These data are summarized as follows:

a. Develop a linear regression model for these data and forecast the ice cream consumption if the

average weekly daytime temperature is expected to be 85 degrees.

b. Determine the strength of the linear relationship between temperature and ice cream

consumption by using correlation.

c. What is the coefficient of determination? Explain its meaning


"Reflection to date" Please respond to the following:

In a paragraph, reflect on what you've learned so far in this course. Identify the most

interesting, unexpected, or useful thing you've learned and explain why

MAT 540 Week 5 Midterm Exam

Question 1

Deterministic techniques assume that no uncertainty exists in model parameters.

Question 2

A continuous random variable may assume only integer values within a given interval.

Question 3

An inspector correctly identifies defective products 90% of the time. For the next 10 products, the

probability that he makes fewer than 2 incorrect inspections is 0.736.

Question 4

A decision tree is a diagram consisting of circles decision nodes, square probability nodes, and

branches.

Question 5

A table of random numbers must be normally distributed and efficiently generated.

Question 6

Simulation results will always equal analytical results if 30 trials of the simulation have been

conducted.

Question 7

Data cannot exhibit both trend and cyclical patterns.

Question 8

Qualitative methods are the least common type of forecasting method for the long-term strategic

planning process.

A company markets educational software products, and is ready to place three new products on

the market. Past experience has shown that for this particular software, the chance of "success" is

80%. Assume that the probability of success is independent for each product. What is the

probability that exactly 1 of the 3 products is successful?



minutes. What time is exceeded by approximately 75% of the college students when trying to find

a parking spot in the main parking lot?

Question 11

The __________ is the maximum amount a decision maker would pay for additional information.

Question 12

Random numbers generated by a __________ process instead of a __________ process are

pseudorandom numbers.

Question 13

Two hundred simulation runs were completed using the probability of a machine breakdown from

the table below. The average number of breakdowns from the simulation trials was 1.93 with a

standard deviation of 0.20.

No. of breakdowns per week

Probability

Cumulative probability

0

.10

.10

1

.25

.35

2

.36

.71

3

.22

.93

4

.07

1.00

What is the probability of 2 or fewer breakdowns?

Question 14

A seed value is a(n)

Question 15

Pseudorandom numbers exhibit __________ in order to be considered truly random.

Question 16

Given the following data on the number of pints of ice cream sold at a local ice cream store for a

6-period time frame:

If the forecast for period 5 is equal to 275, use exponential smoothing with α = .40 to compute a

forecast for period 7.

Question 17

rob

14, and 15)estion worth 2 points, 1 hour time limit (chapters 1,ue units EXCEPT:The U.S.

Department of Agriculture estimates that the yearly yield of limes per acre is distributed as

follows:

Yield, bushels per acre

Probability

350

.10

400

.18

450

.50

500

.22

The estimated average price per bushel is $16.80.

What is the expected yield of the crop?

Question 18

__________ is a linear regression model relating demand to time.

Question 19

Coefficient of determination is the percentage of the variation in the __________ variable that

results from the __________ variable.

Question 20

Which of the following possible values of alpha would cause exponential smoothing to respond

the most slowly to sudden changes in forecast errors?

Question 21

__________ is a measure of the strength of the relationship between independent and dependent

variables.

Question 22

__________ is absolute error as a percentage of demand.

Correct Answer:

MAPD

Question 23

Consider the following graph of sales.

Which of the following characteristics is exhibited by the data?

Question 24

In exponential smoothing, the closer alpha is to __________, the greater the reaction to the most

recent demand.

Question 25

An online sweepstakes has the following payoffs and probabilities. Each person is limited to one

entry.

The probability of winning at least $1,000.00 is ________.

Question 26

An automotive center keeps tracks of customer complaints received each week. The probability

distribution for complaints can be represented as a table or a graph, both shown below. The

random variable xi represents the number of complaints, and p(xi) is the probability of receiving

xi complaints.

xi

0

1

2

3

4

5

6

p(xi)

.10

.15

.18

.20

.20

.10

.07

What is the average number of complaints received per week? Round your answer to two places

after the decimal.

Question 27

A fair die is rolled 8 times. What is the probability that an even number (2,4, 6) will occur

between 2 and 4 times? Round your answer to four places after the decimal.

Question 28

The drying rate in an industrial process is dependent on many factors and varies according to the

following distribution.

Compute the mean drying time. Use two places after the decimal.

Question 29

An investor is considering 4 different opportunities, A, B, C, or D. The payoff for each

opportunity will depend on the economic conditions, represented in the payoff table below.

Economic Condition

Poor Average Good Excellent

Investment (S1) (S2) (S3) (S4)

A 50 75 20 30

B 80 15 40 50

C -100 300 -50 10

D 25 25 25 25

If the probabilities of each economic condition are 0.5, 0.1, 0.35, and 0.05 respectively, what is

the highest expected payoff?

Question 30

The local operations manager for the IRS must decide whether to hire 1, 2, or 3 temporary

workers. He estimates that net revenues will vary with how well taxpayers comply with the new

tax code. The following payoff table is given in thousands of dollars (e.g. 50 = $50,000).

If he thinks the chances of low, medium, and high compliance are 20%, 30%, and 50%

respectively, what is the expected value of perfect information? Note: Please express your

answer as a whole number in thousands of dollars (e.g. 50 = $50,000). Round to the nearest

whole number, if necessary.

Question 31

A normal distribution has a mean of 500 and a standard deviation of 50. A manager wants to

simulate one value from this distribution, and has drawn the number 1.4 randomly. What is the

simulated value?

Question 32

Robert wants to know if there is a relation between money spent on gambling and winnings.

What is the coefficient of determination? Note: please report your answer with 2 places after the

decimal point.

Question 33



Compute a 3-period moving average for period 6. Use two places after the decimal.

Question 34

The following sales data are available for 2003-2008.

Determine a 4-year weighted moving average forecast for 2009, where weights are W1 = 0.1, W2

= 0.2, W3 = 0.2 and W4 = 0.5.

Question 35

The following data summarizes the historical demand for a product.

Month

Actual Demand

March

20

April

25

May

40

June

35

July

30

August

45

Use exponential smoothing with α = .2 and the smoothed forecast for July is 32. Determine the

smoothed forecast for August.

Question 36

The following data summarizes the historical demand for a product

Month

Actual Demand

March

20

April

25

May

40

June

35

July

30

August

45

If the forecasted demand for June, July and August is 32, 38 and 42, respectively, what is MAPD?

Write your answer in decimal form and not in percentages. For example, 15% should be written as

0.15. Use three significant digits after the decimal.

Question 37

Daily highs in Sacramento for the past week (from least to most recent) were: 95, 102, 101, 96,

95, 90 and 92. Develop a forecast for today using a 2 day moving average.

Question 38



Compute a 3-period moving average for period 4. Use two places after the decimal.

Question 39

Given the following data, compute the MAD for the forecast.

Year Demand Forecast

2001

16

18

2002

20

19

2003

18

24

Question 40

Consider the following annual sales data for 2001-2008.

Year

Sales

2001

2

2002

4

2003

10

2004

8

2005

14

2006

18

2007

17

2008

20

Calculate the correlation coefficient . Use four significant digits after the decimal.


Discuss LP Models


The objective function always includes all of the decision variables, but that is not

necessarily true of the constraints. Explain the difference between the objective function

and the constraints. Then, explain why a constraint need not refer to all the variables.

Pick any constraint from any problem in the text, and explain how to plot the line that

corresponds to that constraint.


Chapter 2

1. A Cereal Company makes a cereal from several ingredients. Two of the ingredients, oats

and rice, provide vitamins A and B. The company wants to know how many ounces of oats

and rice it should include in each box of cereal to meet the minimum requirements of 45

milligrams of vitamin A and 13 milligrams of vitamin B while minimizing cost. An ounce

of oats contributes 10 milligrams of vitamin A and 2 milligram of vitamin B, whereas an

ounce of rice contributes 6 milligrams of A and 3 milligrams of B. An ounce of oats costs

$0.06, and an ounce of rice costs $0.03.

a. Formulate a linear programming model for this problem.

b. Setup the LP model for Excel Solver.

2. A Furniture Company produces chairs and tables from two resources- labor and wood. The

company has 125 hours of labor and 45 board-ft. of wood available each day. Demand for

chairs is limited to 5 per day. Each chair requires 7 hours of labor and 3.5 board-ft. of

wood, whereas a table requires 14 hours of labor and 7 board-ft. of wood. The profit

derived from each chair is $325 and from each table, $120. The company wants to

determine the number of chairs and tables to produce each day in order to maximize profit.

Formulate a linear programming model for this problem.


b. Setup the LP model for Excel Solver.

3. Kroeger supermarket sells its own brand of canned peas as well as several national brands.

The store makes a profit of $0.28 per can for its own peas and a profit of $0.19 for any of

the national brands. The store has 6 square feet of shelf space available for canned peas,

and each can of peas takes up 9 square inches of that space. Point-of-sale records show

that each week, the sales of its own brand is less than twice of the sales of the national

brands. The store wants to know how many cans of its own brand of peas of peas and how

many cans of the national brands to stock each week on the allocated shelf space in order

to maximize profit.


b. Setup the LP model for Excel Solver

4. Set up the LP model for Excel Solver:

Minimize Z=8X1 + 6X2

Subject to

4x1 + 2x2 ≥ 20

-6x1 + 4x2 ≤12

x1 + x2 ≥ 6

x1,x2 ≥ 0


Discuss sensitivity analysis


Identify any challenges you have in setting up a linear programming problem in Excel, and

solving it with Solver. Explain exactly what the challenges are and why they are

challenging. Identify resources that can help you with that.

Explain what the shadow price means in a maximization problem. Explain what this tells

us from a management perspective.


Chapter 3

4. Southern Sporting Good Company makes basketballs and footballs. Each product is

produced from two resources rubber and leather. Each basketball produced results in a

profit of $11 and each football earns $15 in profit. The resource requirements for each

product and the total resources available are as follows:

Product

Resource Requirements per Unit

Rubber (lb.) Leather (ft2)

Basketball 2.8 3.7

Football 1.5 5.2

Total resources available 600 900

a. Find the optimal solution.

b. What would be the effect on the optimal solution if the profit for the basketball changed

from $11 to $12?

c. What would be the effect on optimal solution if 400 additional pounds of rubber could

be obtained? What would be the effect if 600 additional square feet of leather could be

obtained?

2. A company produces two products, A and B, which have profits of $9 and $7,

respectively. Each unit of product must be processed on two assembly lines, where the

required production times are as follows:

Product

Resource Requirements per Unit

Line 1 Line 2

A 11 5

B 6 9

Total Hours 65 40

a.Formulate a linear programming model to determine the optimal product mix that will

maximize profit.

b. What are the sensitivity ranges for the objective function coefficients?

c. Determine the shadow prices for additional hours of production time on line 1 and line

2 and indicate whether the company would prefer additional line 1 or line 2 hours.

3. Formulate and solve the model for the following problem:

Irwin Textile Mills produces two types of cotton cloth denim and corduroy. Corduroy is a

heavier grade of cotton cloth and, as such, requires 8 pounds of raw cotton per yard,

whereas denim requires 6 pounds of raw cotton per yard. A yard of corduroy requires 4

hours of processing time; a yard od denim requires 3.0 hours. Although the demand for

denim is practically unlimited, the maximum demand for corduroy is 510 yards per

month. The manufacturer has 6,500 pounds of cotton and 3,000 hours of processing time

available each month. The manufacturer makes a profit of $2.5 per yards of denim and

$3.25 per yard of corduroy. The manufacturer wants to know how many yards of each

type of cloth to produce to maximize profit. Formulate the model and put it into standard

form. Solve it

a. How much extra cotton and processing time are left over at the optimal solution? Is the

demand for corduroy met?

b. If Irwin Mills can obtain additional cotton or processing time, but not both, which

should it select? How much? Explain your answer.

4. The Bradley family owns 410 acres of farmland in North Carolina on which they grow

corn and tobacco. Each acre of corn costs $105 to plant, cultivate, and harvest; each acre

of tobacco costs $210. The Bradleys’ have a budget of $52,500 for next year. The

government limits the number of acres of tobacco that can be planted to 100. The profit

from each acre of corn is $300; the profit from each acre of tobacco is $520. The

Bradleys’ want to know how many acres of each crop to plant in order to maximize their

profit.

a. Formulate the linear programming model for the problem and solve.

b. How many acres of farmland will not be cultivated at the optimal solution? Do the

Bradleys use the entire 100-acre tobacco allotment?

c. The Bradleys’ have an opportunity to lease some extra land from a neighbor. The

neighbor is offering the land to them for $110 per acre. Should the Bradleys’ lease the

land at that price? What is the maximum price the Bradleys’ should pay their neighbor

for the land, and how much land should they lease at that price?

d. The Bradleys’ are considering taking out a loan to increase their budget. For each dollar they

borrow, how much additional profit would they make? If they borrowed an additional

$1,000, would the number of acres of corn and tobacco they plant change?


Question 1

Surplus variables are only associated with minimization problems.

Question 2

A feasible solution violates at least one of the constraints.

Question 3

Graphical solutions to linear programming problems have an infinite number of possible

objective function lines.

Question 4

A linear programming model consists of only decision variables and constraints.

Question 5

If the objective function is parallel to a constraint, the constraint is infeasible.

Question 6

If the objective function is parallel to a constraint, the constraint is infeasible.

Question 7

In a linear programming problem, all model parameters are assumed to be known with certainty.

Question 8

In a linear programming problem, a valid objective function can be represented as

Question 9

Cully furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big

shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300

and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this

week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is

$300 and for each medium shelf is $150. What is the maximum profit?

Question 10

Cully furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big

shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300

and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this

week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is

$300 and for each medium shelf is $150. What is the storage space constraint?

Question 11

The production manager for the Coory soft drink company is considering the production of 2

kinds of soft drinks: regular (R) and diet (D). Two of her limited resources are production time (8

hours = 480 minutes per day) and syrup (1 of her ingredients) limited to 675 gallons per day. To

produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4

minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for

diet soft drink are $2.00 per case. What is the objective function?

Question 12

A graphical representation of a linear program is shown below. The shaded area represents the

feasible region, and the dashed line in the middle is the slope of the objective function.

If this is a maximization, which extreme point is the optimal solution?

Question 13

The following is a graph of a linear programming problem. The feasible solution space is shaded,

and the optimal solution is at the point labeled Z*.

Which of the following points are not feasible?

Question 14



The equation for constraint DH is:

Question 15

The production manager for the Coory soft drink company is considering the production of 2

kinds of soft drinks: regular and diet. Two of her limited resources are production time (8 hours

= 480 minutes per day) and syrup (1 of her ingredients) limited to 675 gallons per day. To



diet soft drink are $2.00 per case. For the production combination of 135 cases of regular and 0

cases of diet soft drink, which resources will not be completely used?

Question 16



This linear programming problem is a:

Question 17

In a linear programming problem, the binding constraints for the optimal solution are:

5x1 + 3x2 ≤ 30

2x1 + 5x2 ≤ 20

Which of these objective functions will lead to the same optimal solution?

Question 18

Consider the following minimization problem:

Min z = x1 + 2x2

s.t. x1 + x2 ≥ 300

2x1 + x2 ≥ 400

2x1 + 5x2 ≤ 750

x1, x2 ≥ 0

Find the optimal solution. What is the value of the objective function at the optimal solution?

Note: The answer will be an integer. Please give your answer as an integer without any decimal

point. For example, 25.0 (twenty five) would be written 25

Question 19

A graphical representation of a linear program is shown below. The shaded area represents the

feasible region, and the dashed line in the middle is the slope of the objective function.

What would be the new slope of the objective function if multiple optimal solutions occurred

along line segment AB? Write your answer in decimal notation.

Question 20

Consider the following linear programming problem:

Max Z = $15x + $20y

Subject to: 8x + 5y ≤ 40

0.4x + y ≥ 4

x, y ≥ 0

At the optimal solution, what is the amount of slack associated with the first constraint?


Practice setting up linear programming models for business applications

Select an even-numbered LP problem from the text, excluding 14, 20, 22, 36 (which are part of

your homework assignment). Formulate a linear programming model for the problem you select.


Chapter 4

1. Betty Malloy, owner of the Eagle Tavern in Pittsburgh, is preparing for Super Bowl Sunday,

and

she must determine how much beer to stock. Betty stocks three brands of beer- Yodel, Shotz, and

Rainwater. The cost per gallon (to the tavern owner) of each brand is as follows:

The tavern has a budget of $2,000 for beer for Super Bowl Sunday. Betty sells Yodel at a rate of

$3.00 per gallon, Shotz at $2.50 per gallon, and Rainwater at $1.75 per gallon. Based on past

football games, Betty has determined the maximum customer demand to be 400 gallons of

Yodel,

500 gallons of shotz, and 300 gallons of Rainwater. The tavern has the capacity to stock 1,000

gallons of beer; Betty wants to stock up completely. Betty wants to determine the number of

gallons

of each brand of beer to order so as to maximize profit.


b. Solve the model by using the computer.

2. As result of a recently passed bill, a congressman’s district has been allocated $3 million for

programs and projects. It is up to the congressman to decide how to distribute the money. The

congressman has decide to allocate the money to four ongoing programs because of their

importance to his district- a job training program, a parks project, a sanitation project, and a

mobile library. However, the congressman wants to distribute the money in a manner that will

please the most voters, or, in other words, gain him the most votes in the upcoming election. His

staff’s estimates of the number of votes gained per dollar spent for the various programs are as

follows.

In order also to satisfy several local influential citizens who financed his election, he is obligated

to

observe the following guidelines:

None of the programs can receive more than 30% of the total allocation

The amount allocated to parks cannot exceed the total allocated to both the sanitation

project and the mobile library.

The amount allocated to job training must at least equal the amount spent on the

sanitation

project.

Any money not spent in the district will be returned to the government; therefore, the

congressman

wants to spend it all. Thee congressman wants to know the amount to allocate to each program to

maximize his votes.



3. Anna Broderick is the dietician for the State University football team, and she is attempting to

determine a nutritious lunch menu for the team. She has set the following nutritional guidelines

for each lunch serving:

Between 1,300 and 2,100 calories

At least 4 mg of iron

At least 15 but no more than 55g of fat

At least 30g of protein

At least 60g of carbohydrates

No more than 35 mg of cholesterol

She selects the menu from seven basic food items, as follows, with the nutritional contributions

per pound and the cost as given:

The dietician wants to select a menu to meet the nutritional guidelines while minimizing the total

cost per serving.

a. Formulate a linear programming model for this problem and solve.

b. If a serving of each of the food items (other than milk) was limited to no more than a

half

pound, what effect would this have on the solution?

4. Dr. Maureen Becker, the head administrator at Jefferson County Regional Hospital, must

determine a schedule for nurses to make sure there are enough of them on duty throughout the

day. During the day, the demand for nurses varies. Maureen has broken the day in to twelve 2-

hour periods. The slowest time of the day encompasses the three periods from 12:00 A.M. to

6:00

A.M., which beginning at midnight; require a minimum of 30, 20, and 40 nurses, respectively.

The demand for nurses steadily increases during the next four daytime periods. Beginning with

the 6:00 A.M.- 8:00 A.M. period, a minimum of 50, 60, 80, and 80 nurses are required for these

four periods, respectively. After 2:00 P.M. the demand for nurses decreases during the afternoon

and evening hours. For the five 2-hour periods beginning at 2:00 P.M. and ending midnight, 70,

70, 60, 50, and 50 nurses are required, respectively. A nurse reports for duty at the beginning of

one of the 2-hour periods and works 8 consecutive hours (which is required in the nurses’

contract). Dr. Becker wants to determine a nursing schedule that will meet the hospital’s

minimum requirement throughout the day while using the minimum number of nurses.



5. The production manager of Videotechnics Company is attempting to determine the

upcoming 5-month production schedule for video recorders. Past production records

indicate that 2,000 recorders can be produced per month. An additional 600 recorders can

be produced monthly on an overtime basis. Unit cost is $10 for recorders produced

during regular working hours and $15 for those produced on an overtime basis.

Contracted sales per month are as follows:

Inventory carrying costs are $2 per recorder per month. The manager does not want any

inventory carried over past the fifth month. The manager wants to know the monthly

production that will minimize total production and inventory costs.



MAT 540 Week 8 Assignment 1

Assignment 1. Linear Programming Case Study

Your instructor will assign a linear programming project for this assignment according to the

following specifications.

It will be a problem with at least three (3) constraints and at least two (2) decision variables. The

problem will be bounded and feasible. It will also have a single optimum solution (in other

words, it won’t have alternate optimal solutions). The problem will also include a component

that involves sensitivity analysis and the use of the shadow price.

You will be turning in two (2) deliverables, a short writeup of the project and the spreadsheet

showing your work.

Writeup.

Your writeup should introduce your solution to the project by describing the problem. Correctly

identify what type of problem this is. For example, you should note if the problem is a

maximization or minimization problem, as well as identify the resources that constrain the

solution. Identify each variable and explain the criteria involved in setting up the model. This

should be encapsulated in one (1) or two (2) succinct paragraphs.

After the introductory paragraph, write out the L.P. model for the problem. Include the objective

function and all constraints, including any non-negativity constraints. Then, you should present

the optimal solution, based on your work in Excel. Explain what the results mean.

Finally, write a paragraph addressing the part of the problem pertaining to sensitivity analysis

and shadow price.

Excel.

As previously noted, please set up your problem in Excel and find the solution using Solver.

Clearly label the cells in your spreadsheet. You will turn in the entire spreadsheet, showing the

setup of the model, and the results.


Discuss characteristics of integer programming problems


Explain how the applications of Integer programming differ from those of linear

programming. Give specific instances in which you would use an integer programming

model rather than an LP model. Provide real-world examples.

Identify any challenges you have in setting up an integer programming problem in Excel,

and solving it with Solver. Explain exactly what the challenges are and why they are



Chapter 5

1. Rowntown Cab Company has 70 drivers that it must schedule in three 8-hour shifts. However,

the

demand for cabs in the metropolitan area varies dramatically according to time of the day. The

slowest period is between midnight and 4:00 A.M. the dispatcher receives few calls, and the calls

that are received have the smallest fares of the day. Very few people are going to the airport at

that

time of the night or taking other long distance trips. It is estimated that a driver will average $80

in

fares during that period. The largest fares result from the airport runs in the morning. Thus, the

drivers who sart their shift during the period from 4:00 A.M. to 8:00 A.M. average $500 in total

fares, and drivers who start at 8:00 A.M. average $420. Drivers who start at noon average $300,

and

drivers who start at 4:00 P.M. average $270. Drivers who start at the beginning of the 8:00 P.M.

to

midnight period earn an average of $210 in fares during their 8-hour shift.

To retain customers and acquire new ones, Rowntown must maintain a high customer service

level.

To do so, it has determined the minimum number of drivers it needs working during every 4-

hour

time segment- 10 from midnight to 4:00 A.M. 12 from 4:00 to 8:00 A.M. 20 from 8:00 A.M. to

noon, 25 from noon to 4:00 P.M., 32 from 4:00 to 8:00 P.M., and 18 from 8:00 P.M. to midnight.

a. Formulate and solve an integer programming model to help Rowntown Cab schedule

its

drivers.

b. If Rowntown has a maximum of only 15 drivers who will work the late shift from

midnight to 8:00 A.M., reformulate the model to reflect this complication and solve it

c. All the drivers like to work the day shift from 8:00 A.M. to 4:00 P.M., so the company

has decided to limit the number of drivers who work this 8-hour shift to 20. Reformulate

the model in (b) to reflect this restriction and solve it.

2. Juan Hernandez, a Cuban athlete who visits the United States and Europe frequently, is

allowed to

return with a limited number of consumer items not generally available in Cuba. The items,

which

are carried in a duffel bag, cannot exceed a weight of 5 pounds. Once Juan is in Cuba, he sells

the

items at highly inflated prices. The weight and profit (in U.S. dollars) of each item are as

follows:

Juan wants to determine the

combination of

items he should pack in his

duffel bag to maximize

his profit. This problem is an example of a type of integer programming problem known as a

“knapsack” problem. Formulate and solve the problem.

3. The Texas Consolidated Electronics Company is contemplating a research and development

program encompassing eight research projects. The company is constrained from embarking on

all

projects by the number of available management scientists (40) and the budget available for

R&D

projects ($300,000). Further, if project 2 is selected, project 5 must also be selected (but not vice

versa). Following are the resources requirement and the estimated profit for each project.

Formulate the integer programming model for this problem and solve it using the computer.

4. Corsouth Mortgage Associates is a large home mortgage firm in the southeast. It has a poll of

permanent and temporary computer operators who process mortgage accounts, including posting

payments and updating escrow accounts for insurance and taxes. A permanent operator can

process

220 accounts per day, and a temporary operator can process 140 accounts per day. On average,

the

firm must process and update at least 6,300 accounts daily. The company has 32 computer

workstations available. Permanent and temporary operators work 8 hours per day. A permanent

operator averages about 0.4 error per day, whereas a temporary operator averages 0.9 error per

day.

The company wants to limit errors to 15 per day. A permanent operator is paid $120 per day

wheras

a temporary operator is paid $75 per day. Corsouth wants to determine the number of permanent

and temporary operators it needs to minimize cost. Formulate, and solve an integer programming

model for this problem and compare this solution to the non-integer solution.

5. Globex Investment Capital Corporation owns six companies that have the following estimated

returns (in millions of dollars) if sold in one of the next 3 years:

To generate operating funds, the company must sell at least $20 million worth of assets in year 1,

$25

million in year 2, and $35 million in year 3. Globex wants to develop a plan for selling these

companies

during the next 3 years to maximize return.

Formulate an integer programming model for this problem and solve it by using the computer.

MAT 540 Week 9 Quiz

Question 1

A constraint for a linear programming problem can never have a zero as its right-hand-side

value.

Question 2

Product mix problems cannot have "greater than or equal to" (≥) constraints.

Question 3

In a transportation problem, a demand constraint (the amount of product demanded at a given

destination) is a less-than-or equal-to constraint (≤).

Question 4

Fractional relationships between variables are permitted in the standard form of a linear program.

Question 5

A systematic approach to model formulation is to first construct the objective function before

determining the decision variables.

Question 6

The standard form for the computer solution of a linear programming problem requires all

variables to be to the right and all numerical values to be to the left of the inequality or equality

sign

Question 7

Compared to blending and product mix problems, transportation problems are unique because

Question 8

In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2,

an 3 which have selling prices of $15, $47.25, and $110, respectively. The investor has up to

$50,000 to invest. The stockbroker suggests limiting the investments so that no more than

$10,000 is invested in stock 2 or the total number of shares of stocks 2 and 3 does not exceed

350, whichever is more restrictive. How would this be formulated as a linear programming

constraint?

Question 9


an 3 which have selling prices of $15, $47.25, and $110, respectively. The investor stipulates that

stock 1 must not account for more than 35% of the number of shares purchased. Which

constraint is correct?

Question 10

When systematically formulating a linear program, the first step is

Question 11

The production manager for the Softy soft drink company is considering the production of 2

kinds of soft drinks: regular and diet. Two of her resources are constraint production time (8

hours = 480 minutes per day) and syrup (1 of her ingredient) limited to 675 gallons per day. To



diet soft drink are $2.00 per case. What is the optimal daily profit?

Question 12

The production manager for the Softy soft drink company is considering the production of 2

kinds of soft drinks: regular and diet. Two of her resources are production time (8 hours = 480

minutes per day) and syrup (1 of the ingredients) limited to 675 gallons per day. To produce a

regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3

gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are

$2.00 per case. What is the time constraint?

Question 13

Let xij = gallons of component i used in gasoline j. Assume that we have two components and

two types of gasoline. There are 8,000 gallons of component 1 available, and the demand

gasoline types 1 and 2 are 11,000 and 14,000 gallons respectively. Write the supply constraint

for component 1.

Question 14

A croissant shop produces 2 products: bear claws (B) and almond filled croissants (C). Each bear

claw requires 6 ounces of flour, 1 ounce of yeast, and 2 TS of almond paste. An almond filled

croissant requires 3 ounces of flour, 1 ounce of yeast, and 4 TS of almond paste. The company

has 6600 ounces of flour, 1400 ounces of yeast, and 4800 TS of almond paste available for

today's production run. Bear claw profits are 20 cents each, and almond filled croissant profits

are 30 cents each. What is the optimal daily profit?

Question 15



$50,000 to invest. The expected returns on investment of the three stocks are 6%, 8%, and 11%.

An appropriate objective function is

Question 16

Small motors for garden equipment is produced at 4 manufacturing facilities and needs to be

shipped to 3 plants that produce different garden items (lawn mowers, rototillers, leaf blowers).

The company wants to minimize the cost of transporting items between the facilities, taking into

account the demand at the 3 different plants, and the supply at each manufacturing site. The table

below shows the cost to ship one unit between each manufacturing facility and each plant, as

well as the demand at each plant and the supply at each manufacturing facility.

What is the demand constraint for plant B?

Question 17

The owner of Black Angus Ranch is trying to determine the correct mix of two types of beef

feed, A and B which cost 50 cents and 75 cents per pound, respectively. Five essential

ingredients are contained in the feed, shown in the table below. The table also shows the

minimum daily requirements of each ingredient.

Ingredient

Percent per pound in Feed A

Percent per pound in Feed B

Minimum daily requirement (pounds)

1

20

24

30

2

30

10

50

3

0

30

20

4

24

15

60

5

10

20

40

The constraint for ingredient 3 is:

Question 18

A systematic approach to model formulation is to first

Question 19

Quickbrush Paint Company makes a profit of $2 per gallon on its oil-base paint and $3 per

gallon on its water-base paint. Both paints contain two ingredients, A and B. The oil-base paint

contains 90 percent A and 10 percent B, whereas the water-base paint contains 30 percent A and

70 percent B. Quickbrush currently has 10,000 gallons of ingredient A and 5,000 gallons of

ingredient B in inventory and cannot obtain more at this time. The company wishes to use linear

programming to determine the appropriate mix of oil-base and water-base paint to produce to

maximize its total profit. How many gallons of oil based paint should the Quickbrush make?

Note: Please express your answer as a whole number, rounding the nearest whole number, if

appropriate.

Question 20

Kitty Kennels provides overnight lodging for a variety of pets. An attractive feature is the quality

of care the pets receive, including well balanced nutrition. The kennel's cat food is made by

mixing two types of cat food to obtain the "nutritionally balanced cat diet." The data for the two

cat foods are as follows:

Kitty Kennels wants to be sure that the cats receive at least 5 ounces of protein and at least 3

ounces of fat per day. What is the cost of this plan? Express your answer with two places to the

right of the decimal point. For instance, $9.32 (nine dollars and thirty-two cents) would be

written as 9.32


Discussion assignment and transshipment problems


Explain the assignment model and how it facilitates in solving transportation problems.

Determine the benefits to be gained from using this model.

Identify any challenges you have in setting up an transshipment model in Excel, and

solving it with Solver. Explain exactly what the challenges are and why they are



Chapter 6

1. Consider the following transportation problem:

Formulate this problem as a linear programming model and solve it by the using the computer.

2. Consider the following transportation problem:

Solve it by using the computer.

3. World foods, Inc. imports food products such as meats, cheeses, and pastries to the United

States

from warehouses at ports in Hamburg, Marseilles and Liverpool. Ships from these ports deliver

the

products to Norfolk, New York and Savannah, where they are stored in company warehouses

before being shipped to distribution centers in Dallas, St. Louis and Chicago. The products are

then

distributed to specialty foods stores and sold through catalogs. The shipping costs ($/1,000 lb.)

from

the European ports to the U.S. cities and the available supplies (1000 lb.) at the European ports

are

provided in the following table:

The transportation costs ($/1000 lb.) from each U.S. city of the three distribution centers and the

demands

(1000 lb.) at the

distribution centers are as follows:

Determine the optimal shipments between the European ports and the warehouses and the

distribution centers to minimize total transportation costs.

4. The Omega Pharmaceutical firm has five salespersons, whom the firm wants to assign to five

sales

regions. Given their various previous contacts, the sales persons are able to cover the regions in

different amounts of time. The amount of time (days) required by each salesperson to cover each

city is shown in the following table:

Which salesperson should be assigned to each region to minimize total time? Identify the optimal

assignments and compute total minimum time.


Question 1

If exactly 3 projects are to be selected from a set of 5 projects, this would be written as 3 separate

constraints in an integer program.

Answer

True

False

Question 2

A conditional constraint specifies the conditions under which variables are integers or real

variables.

Answer

True

False

Question 3

If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a conditional

constraint.

Answer

True

False

Question 4

If we are solving a 0-1 integer programming problem with three decision variables, the constraint

x1 + x2 + x3 ≤ 3 is a mutually exclusive constraint.

Answer

True

False

Question 5

The solution to the LP relaxation of a maximization integer linear program provides an upper

bound for the value of the objective function.

Answer

True

False

Question 6

If we are solving a 0-1 integer programming problem with three decision variables, the constraint

x1 + x2 ≤ 1 is a mutually exclusive constraint.

Answer

True

False

Question 7

In a __________ integer model, some solution values for decision variables are integers and

others can be non-integer.

Answer

total

0 - 1

mixed

all of the above

Question 8

If we are solving a 0-1 integer programming problem, the constraint x1 = x2 is a __________

constraint.

Answer

multiple choice

mutually exclusive

conditional

corequisite

Question 9

Assume that we are using 0-1 integer programming model to solve a capital budgeting problem

and xj = 1 if project j is selected and xj = 0, otherwise.

The constraint (x1 + x2 + x3 + x4 ≤ 2) means that __________ out of the 4 projects must be

selected.

Answer

exactly 2

at least 2

at most 2

none of the above

Question 10

The Wiethoff Company has a contract to produce 10000 garden hoses for a customer. Wiethoff

has 4 different machines that can produce this kind of hose. Because these machines are from

different manufacturers and use differing technologies, their specifications are not the same.

Write a constraint to ensure that if machine 4 is used, machine 1 will not be used.

Answer

Y1 + Y4 ≤ 0

Y1 + Y4 = 0

Y1 + Y4 ≤ 1

Y1 + Y4 ≥ 0

Question 11

The Wiethoff Company has a contract to produce 10000 garden hoses for a customer. Wiethoff

has 4 different machines that can produce this kind of hose. Because these machines are from

different manufacturers and use differing technologies, their specifications are not the same.

Write the constraint that indicates they can purchase no more than 3 machines.

Answer

Y1 + Y2 + Y3+ Y4 ≤ 3

Y1 + Y2 + Y3+ Y4 = 3

Y1 + Y2 + Y3+ Y4 ≥3

none of the above

Question 12

If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a __________

constraint.

Answer

multiple choice

mutually exclusive

conditional

corequisite

Question 13

You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each

site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are:

Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7.

Restriction 2. Evaluating sites S2 or

S4 will prevent you from assessing site S5.

Restriction 3. Of all the sites, at least 3 should be assessed.

Assuming that Si is a binary variable, write the constraint(s) for the second restriction

Answer

S2 +S5 ≤ 1

S4 +S5 ≤ 1

S2 +S5 + S4 +S5 ≤ 2

S2 +S5 ≤ 1, S4 +S5 ≤ 1

Question 14

Binary variables are

Answer

0 or 1 only

any integer value

any continuous value

any negative integer value

Question 15

In a capital budgeting problem, if either project 1 or project 2 is selected, then project 5 cannot

be selected. Which of the alternatives listed below correctly models this situation?

Answer

x1 + x2 + x5 ≤ 1

x1 + x2 + x5 ≥1

x1 + x5 ≤ 1, x2 + x5 ≤ 1

x1 - x5 ≤ 1, x2 - x5 ≤ 1

Question 16

The solution to the linear programming relaxation of a minimization problem will always be

__________ the value of the integer programming minimization problem.

Answer

greater than or equal to

less than or equal to

equal to

different than

Question 17

If we are solving a 0-1 integer programming problem, the constraint x1 + x2 ≤ 1 is a __________

constraint.

Answer

multiple choice

mutually exclusive

conditional

corequisite

Question 18

You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each

site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are:

Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7.

Restriction 2. Evaluating sites S2 or

S4 will prevent you from assessing site S5.

Restriction 3. Of all the sites, at least 3 should be assessed.

Assuming that Si is a binary variable, the constraint for the first restriction is

Answer

S1 + S3 + S7 ≥ 1

S1 + S3 + S7 ≤1

S1 + S3 + S7 = 2

S1 + S3 + S7 ≤ 2

Question 19

Consider the following integer linear programming problem

Max Z = 3x1 + 2x2

Subject to: 3x1 + 5x2 ≤ 30

5x1 + 2x2 ≤ 28

x1 ≤ 8

x1 ,x2 ≥ 0 and integer

Find the optimal solution. What is the value of the objective function at the optimal solution.


point. For example, 25.0 (twenty-five) would be written 25

Question 20

Max Z = 3x1 + 5x2

Subject to: 7x1 + 12x2 ≤ 136

3x1 + 5x2 ≤ 36

x1, x2 ≥ 0 and integer

Find the optimal solution. What is the value of the objective function at the optimal solution.


point. For example, 25.0 (twenty-five) would be written 25


"Reflection to date" Please respond to the following:

• In a paragraph, reflect on what you've learned in this course. Identify the most interesting,

unexpected, or useful thing you’ve learned, and explain how it can be applied to your work or

daily life in some manner.

MAT 540 Week 11 Final Exam

Question 1

In a transshipment problem, items may be transported from destination to destination and from

source to source.

Answer

True

False

Question 2

Excel can be used to simulate systems that can be represented by both discrete and continuous

random variables.

Answer

True

False

Question 3

In an unbalanced transportation model, supply does not equal demand and one set of constraints

uses ≤ signs.

Answer

True

False

Question 4

Fractional relationships between variables are not permitted in the standard form of a linear

program.

Answer

True

False

Question 5

A cycle is an up and down movement in demand that repeats itself in less than 1 year.

Answer

True

False

Question 6

In a total integer model, all decision variables have integer solution values.

Answer

True

False

Question 7



The conservative (maximin) strategy is:

Answer

Buy

Rent

Lease

Brisk.

Question 8

Using the minimax regret criterion to make a decision, you

Answer

Construct a table of regrets. Look at the maximum regret for each decision. Select the

decision with the smallest maximum regret.

Look at the worst payoff for each possible decision and select the decision with the largest

worst payoff

Construct a table of regrets. Look at the minimum regret for each decision. Select the

decision with the smallest minimum regret.

Run in circles, scream and shout

Question 9

Using the maximin criterion to make a decision, you

Answer

Construct a table of regrets. Look at the maximum regret for each decision. Select the

decision with the smallest maximum regret.

Look at the worst payoff for each possible decision and select the decision with the largest

worst payoff

Look at the best payoff for each possible decision and select the decision with the largest

best payoff

Consult an astrological table to forecast the state of nature

Question 10

The probability of observing x

successes in a fixed number of trials is a problem related to

Answer

the normal distribution

the binomial distribution

conditional probability

the Poisson distribution

Question 11

Events that cannot occur at the same time in any trial of an experiment are:

Answer

exhaustive

dependent

independent

mutually exclusive

Question 12

Steinmetz furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each

big shelf costs $100 and requires 100 cubic feet of storage space, and each medium shelf costs

$50 and requires 80 cubic feet of storage space. The company has $25000 to invest in shelves

this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf

is $85 and for each medium shelf is $75. What is the constraint on money to invest?

Answer

Max Z = 85B + 75M

100B + 50M ≤ 25000

100B + 50M ≥ 25000

100B + 80M = 18000

Question 13

Steinmetz furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each

big shelf costs $100 and requires 100 cubic feet of storage space, and each medium shelf costs

$50 and requires 80 cubic feet of storage space. The company has $25000 to invest in shelves

this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf

is $85 and for each medium shelf is $75. What is the storage space constraint?

Answer

Max Z = 75B + 85M

100B + 50M ≥ 25000

100B + 80M ≤ 18000

100B + 80M = 18000

Question 14

Given the following linear programming problem that minimizes cost.

Min Z = 2x + 8y

Subject to 8x + 4y ≥ 64

2x + 4y ≥ 32

y ≥ 2

What is the sensitivity range for the third constraint, y ≥ 2?

Answer

0 to 4

2 to 5.33

0 to 5.33

4 to 6.33

Question 15

The production manager for Beer etc. produces 2 kinds of beer: light (L) and dark (D). Two

resources used to produce beer are malt and wheat. He can obtain at most 4800 oz of malt per

week and at most 3200 oz of wheat per week respectively. Each bottle of light beer requires 12

oz of malt and 4 oz of wheat, while a bottle of dark beer uses 8 oz of malt and 8 oz of wheat.

Profits for light beer are $2 per bottle, and profits for dark beer are $1 per bottle. What is the

optimal weekly profit?

Answer

$1000

$900

$800

$700

Question 16



$50,000 to invest.

An appropriate part of the model would be

Answer

15X1 + 47.25X2 +110 X3 ≤ 50,000

MAX Z =15X1 + 47.25X2 + 110X3

X1 + X2 +X3 ≤ 50,000

MAX Z = 50(15)X1 + 50 (47.25)X2 + 50 (110)X3

Question 17

Let xij = gallons of component i used in gasoline j. Assume that we have two components and

two types of gasoline. There are 8,000 gallons of component 1 available, and the demands for

gasoline types 1 and 2 are 11,000 and 14,000 gallons respectively. Write the supply constraint

for component 1.

Answer

x11 + x21 ≤ 8000

x12 + x21 ≥ 8000

x11 + x12 ≤ 8000

x11 + x12 ≥ 8000

Question 18

If we are solving a 0-1 integer programming problem, the constraint x1 = x2 is a __________

constraint.

Answer

multiple choice

mutually exclusive

conditional

corequisite

Question 19

The Kirschner Company has a contract to produce garden hoses for a customer. Kirschner has 5

different machines that can produce this kind of hose. Write the constraint that indicates they

have to use at least three of the five machines in their production.

Answer

Y1 + Y2 + Y3 + Y4

+ Y5 ≤ 3

Y1 + Y2 + Y3 + Y4

+ Y5 = 3

Y1 + Y2 + Y3 + Y4

+ Y5 ≥ 3

none of the above

Question 20

The following table represents the cost to ship from Distribution Center 1, 2, or 3 to Customer A,

B, or C.

The constraint that represents the quantity demanded by Customer B is:

Answer

6X1B + 2X2B + 8X3B ≤ 350

6X1B + 2X2B + 8X3B = 350

X1B + X2B + X3B ≤ 350

X1B + X2B + X3B = 350

Question 21

The following table represents the cost to ship from Distribution Center 1, 2, or 3 to Customer A,

B, or C.

The constraint that represents the quantity supplied by DC 1 is:

Answer

4X1A + 6X1B + 8X1C ≤ 500

4X1A + 6X1B + 8X1C = 500

X1A + X1B + X1C ≤ 500

X1A + X1B + X1C ≥500

Question 22



minutes. What percentage of the students will take between 2 and 6 minutes to find a parking

spot in the main parking lot?

Answer

11.13%

47.72%

43.32%

62.47%

Question 23

The metropolitan airport commission is considering the establishment of limitations on noise

pollution around a local airport. At the present time, the noise level per jet takeoff in one

neighborhood near the airport is approximately normally distributed with a mean of 100 decibels

and a standard deviation of 3 decibels. What is the probability that a randomly selected jet will

generate a noise level of more than 105 decibels? Note: please provide your answer to 2 places

past the decimal point, rounding as appropriate.

Answer

0.03

0.05

0.07

0.09

Question 24

In the Monte Carlo process, values for a random variable are generated by __________ a

probability distribution.

Answer

sampling from

running

integrating

implementing

Question 25

Consider the following graph of sales.

Which of the following characteristics is exhibited by the data?

Answer

Trend only

Trend plus seasonal

Trend plus irregular

Seasonal

Question 26

A bakery is considering hiring another clerk to better serve customers. To help with this

decision, records were kept to determine how many customers arrived in 10-minute intervals.

Based on 100 ten-minute intervals, the following probability distribution and random number

assignments developed.

Number of

ArrivalsProbability

Random

numbers

6 .1 .01 - .10

7 .3 .11 - .40

8 .3 .41 - .70

9 .2 .71 - .90

10 .1 .91 - .00

Suppose the next three random numbers were .18, .89 and .67. How many customers would

have arrived during this 30-minute period?

Answer

23

24

22

25

Question 27

Given an actual demand of 59, a previous forecast of 64, and an alpha of .3, what would the

forecast for the next period be using simple exponential smoothing?

Answer

36.9

57.5

60.5

62.5

Question 28

Nixon’s Bed and Breakfast has a fixed cost of $5000 per month and the revenue they receive

from each booked room is $200. The variable cost per room is $75. How many rooms do they

have to sell each month to break even? (Note: The answer is a whole number. Give the answer

as a whole number, omitting the decimal point. For instance, use 12 for twelve rooms).

Answer

Question 29

Students are organizing a "Battle of the Bands" contest. They know that at least 100 people will

attend. The rental fee for the hall is $200 and the winning band will receive $500. In order to

guarantee that they break even, how much should they charge for each ticket? (Note: Write your

answer with two significant places after the decimal and do not include the dollar “$” sign. For

instance, for five dollars, write your answer as 5.00).

Answer

Question 30

Joseph is considering pursuing an MS in Information Systems degree. He has applied to two

different universities. The acceptance rate for applicants with similar qualifications is 30% for

University X and 60% for University Y. What is the probability that Jim will not be accepted at

either university? (Note: write your answer as a probability, with two decimal places. If

necessary, round to two decimal places. For instance, a probability of 0.252 should be written

as 0.25).

Answer

Question 31

Tracksaws, Inc. makes tractors and lawn mowers. The firm makes a profit of $30 on each tractor

and $30 on each lawn mower, and they sell all they can produce. The time requirements in the

machine shop, fabrication, and tractor assembly are given in the table.

Formulation:

Let x = number of tractors produced per period

y = number of lawn mowers produced per period

MAX 30x + 30y

subject to 2 x + y ≤ 60

2 x + 3y ≤ 120

x ≤ 45

x, y ≥ 0

The graphical solution is shown below.

What is the shadow price for fabrication? Write your answers with two significant places after

the decimal and do not include the dollar “$” sign.

Answer

Question 32

Consider the following linear program, which maximizes profit for two products, regular (R),

and super (S):

MAX

50R + 75S

s.t.

1.2R + 1.6 S ≤ 600 assembly (hours)

0.8R + 0.5 S ≤ 300 paint (hours)

.16R + 0.4 S ≤ 100 inspection (hours)

Sensitivity Report:

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$B$7 Regular = 291.67 0.00 50 70 20

$C$7 Super = 133.33 0.00 75 50 43.75

Final Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease

$E$3 Assembly (hr/unit) 563.33 0.00 600 1E+30 36.67

$E$4 Paint (hr/unit) 300.00 33.33 300 39.29 175

$E$5 Inspect (hr/unit) 100.00 145.83 100 12.94 40

If downtime reduced the available capacity for painting by 40 hours (from 300 to 260 hours),

profits would be reduced by __________. Write your answers with two significant places after

the decimal and do not include the dollar “$” sign.

Answer

Question 33

Kalamazoo Kennels provides overnight lodging for a variety of pets. An attractive feature is the

quality of care the pets receive, including well balanced nutrition. The kennel’s cat food is made

by mixing two types of cat food to obtain the “nutritionally balanced cat diet.” The data for the

two cat foods are as follows:

Cat Food

Cost/

oz

protien

(%) fat (%)

Pet's Choice 0.35 40 15

Feline Chow 0.32 20 30

Kalamazoo Kennels wants to be sure that the cats receive at least 5 ounces of protein and at least

3 ounces of fat per day. What is the optimal cost of this plan? Note: Please write your answers

with two significant places after the decimal and do not include the dollar “$” sign. For

instance, $9.45 (nine dollars and fortyfive cents) should be written as 9.45

Answer

Question 34

Find the optimal Z value for the following problem. Do not include the dollar “$” sign with your

answer.

MAX Z = 5x1 + 8x2

s.t. x1 + x2 ≤ 6

5x1 + 9x2 ≤ 45

x1, x2 ≥ 0 and integer

Answer

Question 35

Suppose that x is normally distributed with a mean of 10 and a standard deviation of 3. Find P(x

≤ 6). Note: Round your answer, if necessary, to two places after the decimal. Please express

your answer with two places after the decimal.

Answer

Question 36

Ms. Hegel is considering four different opportunities, A, B, C, or D. The payoff for each

opportunity will depend on the economic conditions, represented in the payoff table below.

Investment

Economic Conditions

Poor

(S1)

Averag

e

(S2)

Good

(S3)

Excellent

(S4)

A 80 15 18 47

B 50 75 35 35

C -90 225 -50 12

D 36 25 25 27

Suppose all states of the world are equally likely (each state has a probability of 0.25). What is

the expected value of perfect information? Note: Report your answer as an integer, rounding to

the nearest integer, if applicable

Answer

Question 37



tax code. The probabilities of low, medium, and high compliance are 0.20, 0.30, and 0.50

respectively. What is the expected value of perfect information? Do not include the dollar “$”

sign with your answer. The following payoff table is given in thousands of dollars (e.g. 50 =

$50,000). Note: Please express your answer as a whole number in thousands of dollars (e.g. 50

= $50,000). Round to the nearest whole number, if necessary.

Answer

Question 38



tax code. The probabilities of low, medium, and high compliance are 0.20, 0.30, and 0.50

respectively. What are the expected net revenues for the number of workers he will decide to

hire? The following payoff table is given in thousands of dollars (e.g. 50 = $50,000). Note:

Please express your answer as a whole number in thousands of dollars (e.g. 50 = $50,000).

Round to the nearest whole number, if necessary.

Answer

Question 39

Recent past demand for product ABC is given in the following table.

MonthActual

Demand

May 33

June 32

July 39

August 37

The forecasted demand for May, June, July and August were 25, 30, 33, and 38 respectively.

Determine the value of MAD. Note: Please express the result as a number with 2 decimal

places. If necessary, round your result accordingly. For instance, 9.146, should be expressed as

9.15

Answer

Question 40

Consider the following decision tree. The objective is to choose the best decision among the two

available decisions A and B. Find the expected value of the best decision. Do not include the

dollar “$” sign with your answer.

Answer

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