MAT540 Complete Course Week 1 to Week 11 Latest,MAT540 Homework Week 8 Page 1 of 4 MAT540 Week 8 Homework Chapter 4 1. Betty Malloy, owner of the Eagle Tavern in Pittsburgh, is preparing

MAT540 Complete Course Week 1 to week 11 Latest

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MAT540 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

MAT540 Week 1 Homework

Chapter 1

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

$65,000. Thevariable cost of recappinga tire is $7.5.The company charges$25 torecap a tire.

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

b. Determine the annual break-evenvolume for the Retread TireCompany operation.

2. Evergreen Fertilizer Company produces fertilizer. The company’sfixed monthly cost is $25,000, andits variable cost per pound of fertilizer is $0.20.Evergreen sells the fertilizer for $0.45 per pound. Determinethe monthly break-evenvolume for the company.3. If Evergreen Fertilizer Company in problem 2 changesthe price of its fertilizer from $0.45per poundto $0.55 per pound, what effect will thechange have on the break-even volume?4. If Evergreen Fertilizer Company increases its advertising expenditure by $10,000per year, whateffect will the increase haveon the break-even volume computedin problem 2?5. Annie McCoy,a student at Tech, plans to open a hot dog stand insideTech’s football stadiumduring home games. There are6 home games scheduledfor the upcoming season. She must pay the Techathleticdepartmenta vendor’s fee of $3,000for the season.Her stand and other



equipment will cost her $3,500for the season. She estimatesthat each hot dog shesells will cost her $0.40. she has talkedto friends at otheruniversities who sell hot dogs at games. Based on their information and the athletic department’s forecastthat each game will sell out, she anticipates that she will sell approximately 1,500 hotdogs during each game.a. What price shouldshe 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-evenprice Annie might charge?

6. The college of businessat Kerouac University is planningto begin an online MBA program.The initialstart-upcost for computingequipment, facilities, course development and staff recruitment and developmentis $400,000.The college plans to charge tuition of $20,000 per studentper year. However, the universityadministration will charge the college $10,000 per student for the first 100 studentsenrolled each year foradministrativecosts and its shareof the tuitionpayments.a. How many students does thecollege need to enroll in thefirst year to break-even?

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

c. The college believes it canincrease tuition to $25,000, but doing so would reduceenrollment to

50. Should the collegeconsider doing this? Chapter 11

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

Grade Probability

A 0.1

B 0.2

C 0.4

D 0.2

F 0.10

1.00

The grades are assigned on a 4.0 scale, where anA is a 4.0, a B a 3.0, andso on. Determine the expectedgrade andvariance for thecourse.

8. An investment firm is considering two alternative investments, A and B, under two possiblefuturesets of economic conditions good and poor. There is a .60 probability of good economicconditions occurringand a .40 probability of poor economicconditions occurring. The expected gains and lossesunder eacheconomic typeof conditionsareshown in the following table:

EconomicConditions

Investment

Good Poor

A $380,000 -$100,000

B $130,000 $85,000

Using the expected valueofeach investment alternative, determine which shouldbe selected.

9. The weight of the bags of fertilizer is normallydistributed, with a mean of 45 pounds and astandarddeviation of 5 pounds.What is the probability that a bag of fertilizer will weigh between 38 and 50pounds?

10. The polo Development Firm is buildinga shopping center. It has informedrenters that their rentalspaces will be ready for occupancyin 18 months. If the expected time until the shoppingcenter is completedis estimatedto be 15 months,with a standarddeviation of 5 months, what is the probability that therenterswill not be able tooccupy in 18 months?

11. The manager of the local National Video Store sells videocassette recorders at discount prices. Ifthe store does not have a video recorderin stock when a customer wants to buy one, it will lose the salebecausethe customer will purchase a recorderfrom one of the many local competitors. The problem is thatthe cost of rentingwarehouse space to keep enough recorders in inventory to meet all demand is excessivelyhigh. The managerhas determined that if 85% of customerdemand for recorders can be met, then thecombined cost of lost sales and inventorywill be minimized. The manager has estimatedthat monthlydemand for recordersis normally distributed, with a mean of 175 recordersa

nd a standarddeviation of 55.Determinethe number of recordersthe 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 threealternative investments; a motel, arestaurant, or a theater. Profits from the motel or restaurant will be affectedby the availability of gasolineandthe number of tourists;profits from the theater will be relatively stable under any conditions. The followingpayoff table shows the profit or loss that could result from each investment:

Weather Conditions

Investment

Shortage Stable Supply Surplus

Motel $-7,500 $12,000 $23,000

Restaurant 3000 7,000 6,500

Theater 5000 6,000 4,000

Determine the best investment, using the following decisioncriteria.

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 thevendors sell sun visorsor umbrellas. There is a 35%chance of rain, a 25% chanceof overcast skies, and a40% chance of sunshine,according to the weatherforecast in collegejunction, 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

Decision

Rain 0.35

Weather Conditions

Overcast 0.25

Sunshine 0.40

Sun visors $-400 $-200 $1,500

Umbrellas 2,100 0 -800

a. Compute the expected valuefor eachdecision and select thebest 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 leasingan office park, purchasing a parcel of land and building an officebuildingto rent, buying and leasing a warehouse, building a strip mall, and sellingcondominiums. Thefinancialsuccess of these projects depends on interestrate movement in the next 5 years. The variousdevelopment projects and their 5- year financialreturn (in $1,000,000s) given that interestrates will decline,remain stable, or increase, are in the followingpayoff table. Place-Plus real estate development firm hashired an economistto assign a probability to each direction interest rates may take over the next 5 years.The economisthas determined that there is a

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

1.2 probability that rates will increase.

a. Using expected value, determinethe best project.

b. Determine the expectedvalue of perfect information.

Interest Rate

Project

Decline Stable Increase

Office park $0.4 $1.55 $3.5

Office building 2.5 1.8 2.75

Warehouse 1.7 1.45 1.5

Mall 0.8 2.3 3.5

Condominiums 3.2 1.5 0.5

4. The director of career advising at OrangeCommunity College wants to use decision analysis toprovide information to help studentsdecide which 2-year degree program they should pursue. The directorhas set up the followingpayoff table for six of the most popularand successful degree programsat OCCthat shows the estimated 5-Year gross income ($) from each degree for four future economic conditions:

Economic Conditions

Degree Program

Recession Average Good Robust

Graphic design 150,000 175,000 220,000 200,000

Nursing 160,000 180,000 205,000 215,000

Real estate 125,000 165,000 220,000 210,000

Medical technology 135,000 180,000 210,000 270,000

Culinary technology 110,000 145,000 235,000 205,000

Computer information 130,000 150,000 190,000 245,000

technology

Determine the best degree programin terms of projectedincome, using the followingdecision criteria:

a. Maximax

b. Maximin

c. Equal likelihood

d. Hurwicz (α=0.4)

5. Construct a decisiontree for the following decisionsituation and indicate the best decision.

Fenton and Farrah Friendly, husband-and-wifecar dealers,aresoon going to opena new dealership. They havethree offers:from a foreigncompact car company, from a U.S. producerof full-sized cars, and from a truckcompany.The success of each type of dealership will depend on how much gasolineis going to be availableduring the next few years.The profit from each type of dealership, giventhe availability of gas,is shown in thefollowingpayoff table:

Gasoline Availability

Dealership

Shortage 0.7

Surplus 0.3

Compact cars $25,000 $150,000

Full-sized cars -90,000 650,000

Trucks 125,000 170,000


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 emergencycall every 1, 2, 3, 4, 5, or 6 hours, accordingtothe following probability distribution. The squad is on duty 24 hours perday, 7 days per week:

Time Between Emergency Calls (hr.)

Probability

1 0.15

2 0.10

3 0.20

4 0.25

5 0.20

6 0.10

1.00

a. Simulate the emergency calls for 3 days (note that this will requirea “running” , or cumulative, hourlyclock),using the randomnumber table.

b. Compute the average time between calls and comparethis value with the expectedvalue of the timebetweencalls from the probability distribution. Why are theresult different?

2. The time between arrivals of cars at the Petroco Services Station is definedby the followingprobability distribution:

Time Between Emergency Calls (hr.)

Probability

1 0.35

2 0.25

3 0.20

4 0.20

1.00

Simulate the arrival of cars at the service station for 20 arrivalsand compute the average timebetweenarrivals.

a. Simulate the arrival of cars at the service station for 1 hour, using a differentstream of randomnumbersfrom those used in (a) and computethe average time betweenarrivals.

b. Compare the results obtainedin (a)and (b).

3. The Dynaco Manufacturing Company produces a product in a processconsisting of operations of fivemachines. The probability distribution of the number of machines that will break down in a week follows:

Machine Breakdowns Per Week

Probability

0 0.10

1 0.20

2 0.15

3 0.30

4 0.15

5 0.10

1.00

a. Simulate the machine breakdownsper week for 20 weeks.

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

4. Simulate the following decisionsituation for 20 weeks,and recommendthe best decision.

A concessions manager at the Tech versus A&M football game must decide whether to have the vendorssell sun visorsor umbrellas. There is a 30%chance of rain, a 15% chanceof overcast skies, and a 55%chance of sunshine,according to the weatherforecast in collegejunction, where the game is to be held. Themanager estimates

that the following profits will result from each decision,given each set of weather conditions:

Decision Weather Conditions

Rain Overcast Sunshine

0.35 0.25 0.40

Sun visors $-400 $-200 $1,500

Umbrellas 2,100 0 -800

5. Every time a machinebreaks down at the Dynaco Manufacturing Company (Problem3), either 1, 2, or 3hours are requiredto fix it, according to thefollowing probability distribution:

Repair Time (hr.) Probability

1 0.20

2 0.50

3 0.30

1.00

Simulate the repair timefor 20 weeks and then computethe averageweekly repair time.

MAT 540 Week 4 Discussion

Discuss Forecasting MethodsSelect one (1) of the following topics for your primary discussion posting:

· 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.

MAT 540 Week 4 Homework

MAT540 Homework Week 4 Page 1 of 5 MAT540 Week 4 Homework 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: Month Demand for Soft Shag Carpet (1,000 yd.) 1 10 2 9 3 8 4 9 5 10 6 12 7 14 8 11 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: MAT540 Homework Week 4 Page 2 of 5 Month Gasoline Demanded (gal.) October 775 November 835 December 605 January 450 February 600 March 700 April 820 May 925 June July 1500 1200 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: Month Fund Price 1 $55 ¾ 2 54 ¼ 3 55 1/8 4 58 1/8 5 53 3/8 6 51 1/8 7 56 ¼ 8 59 5/8 9 62 ¼ 10 59 ¼ 11 62 3/8 12 57 1/1 MAT540 Homework Week 4 Page 3 of 5 13 58 1/8 14 62 ¾ 15 64 ¾ 16 66 1/8 17 68 ¾ 18 60.5 19 65.875 20 72.25 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: Monthly Carpet Sales (1,000 yd.) Monthly Construction Permits 9 17 14 25 10 8 12 7 15 14 9 7 24 45 21 19 20 28 MAT540 Homework Week 4 Page 4 of 5 29 28 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: Week Average Temperature (Degrees) Ice Cream Sold (gal.) 1 68 80 2 70 115 3 73 91 4 79 87 5 77 110 6 82 128 7 85 164 8 90 178 9 85 144 10 92 179 11 90 144 12 95 197 13 80 144 14 75 123 MAT540 Homework Week 4 Page 5 of 5 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.

MAT 540 Week 5 Discussion

"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 whyMAT 540 Week 6 Discussion

Discuss LP ModelsSelect one (1) of the following topics for your primary discussion posting:

· 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.


MAT540 Homework Week 6 Page 1 of 2 MAT540 Week 6 Homework 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. Solve the model by using graphical analysis.

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. a. Formulate a linear programming model for this problem. b. Solve the model by using graphical analysis. (Do not round the answers) c. How much labor and wood will be unused if the optimal numbers of chairs and tables are produced?

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 store never sales more than half as many cans of its own brand as it does 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. a. Formulate a linear programming model for this problem. b. Solve the model by using graphical analysis. MAT540 Homework Week 6 Page 2 of 2 4. Solve the following linear programming model graphically: Minimize Z=8X1 + 6X2 Subject to 4X1 + 2X2 20 -6X1 + 4X2 X1 + X2 X1 , X2


Chapter 3

1. 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?

MAT540 Week 8Homework C hap t e r 4

1. Betty Malloy, owner of the Eagle Tavern in Pittsburgh, is preparing for Super Bowl Sunday, and shemust 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 eachbrand is as follows:

Brand Cost/Gallon

Yodel $1.50

Shotz 0.90

Rainwater 0.50

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 footballgames, 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 capacitytostock 1,000 gallons of beer;Betty wantsto stock up completely.Betty wants todetermine the number of gallons of eachbrandofbeer to orderso as to maximize profit.

a. Formulate a linear programming model for this problem. b. Solvethemodel by using the computer.

2. As result of a recently passed bill, a congressman’s district has been allocated $3 million for programs andprojects. It is up to the congressman to decide howto distribute the money. The congressman has decide toallocatethemoney to four ongoing programs because of their importance to his district- a job trainingprogram, a parks project, a sanitation project, and a mobile library. However, the congressman wants to distribute the moneyin a manner that will please the most voters, or, in other words, gain himthemost votes inthe upcoming election. His staff’s estimates of the number of votes gained per dollar spent for the various programs are asfollows.

Program Votes/Dollar

Job training 0.03

Parks 0.08

Sanitation 0.05

Mobile library 0.03

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

· None of theprograms canreceive more than 30% of thetotal allocation

· The amount allocated to parks cannot exceed the total allocated to both the sanitation project andthemobile 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 congressmanwants tospend it all. Thee congressman wants to know the amount to allocate to each programto maximize his votes.

a. Formulate a linear programmingmodel for this problem. b. Solvethemodel by using the computer.

3. Anna Broderick isthe dieticianfor the State Universityfootball team, and sheis attempting to determine anutritious lunch menu for the team. She has set the following nutritional guidelinesforeach lunchserving:

· Between 1,300 and 2,100calories

· At least 4 mg of iron

· At least 15 butno more than 55g of fat

· At least 30g of protein

· At least 60g of carbohydrates

· Nomorethan 35 mg of cholesterol

She selects the menu from seven basic food items, as follows, with the nutritional contributions per pound and the costas given:

Calories

(per lb.)

Iron

(mg/lb.)

Protein

(g/lb.)

Carbo-

hydrates

(g/lb.)

Fat

(g/lb.)

Cholesterol

(mg/lb)

Cost

($/lb.)

Chicken 500 4.2 17 0 30 180 0.85

Fish 480 3.1 85 0 5 90 3.35

Ground beef 840 0.25 82 0 75 350 2.45

Dried beans 590 3.2 10 30 3 0 0.85

Lettuce 40 0.4 6 0 0 0 0.70

Potatoes 450 2.25 10 70 0 0 0.45

Milk (2%) 220 0.2 16 22 10 20 0.82

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

a. Formulate alinear 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, whateffect would this have on the solution?

4. Dr. Maureen Becker, the head administrator at Jefferson County Regional Hospital, must determine aschedule for nurses to make sure there are enough of them on duty throughout the day.During the day, thedemand for nurses varies. Maureen has broken the day in to twelve 2- hour periods. The slowest time of theday encompasses the three periodsfrom12:00 A.M. to 6:00A.M., which beginning at midnight; require a minimum of 30, 20, and 40 nurses, respectively. The demandfor nurses steadily increases during the next four daytime periods. Beginning withthe6:00 A.M.- 8:00 A.M. period, a minimum of 50, 60, 80, and 80 nurses are required for these four periods, respectively. After 2:00P.M. the demand for nurses decreases during the afternoon and evening hours. For the five 2-hour periodsbeginning 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 beginningof one of the 2-hour periods and works 8 consecutive hours(whichisrequired in the nurses’ contract). Dr. Becker wants to determine a nursing schedule that will meet the hospital’sminimum requirement throughout the daywhile using theminimum number of nurses.

a. Formulate alinear programming model for this problem.

b. Solve themodel by using the computer.

5. The production manager of Videotechnics Company is attempting todetermine the upcoming 5-month production schedule for video recorders. Past production records indicate that 2,000 recorderscan be produced per month. An additional 600 recorders can be produced monthly on an overtime basis. Unit cost is $10 forrecorders producedduring regular working hours and $15 for those produced on an overtime basis. Contracted sales per month are as follows:

Month Contracted Sales (units)

1 1200

2 2100

3 2400

4 3000

5 4000

Inventory carrying costs are $2 per recorder per month. The manager doesnot want any inventory carried over past the fifth month. The manager wantsto know the monthly production that will minimize total production and inventory costs.

a. Formulate a linear programming modelforthis problem.

b. Solve the model by using thecomputer.


MAT540 Homework Week 8 Page 1 of 4 MAT540 Week 8 Homework 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: Brand

Cost/Gallon Yodel $1.50 Shotz 0.90 Rainwater 0.50 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. a. Formulate a linear programming model for this problem. 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. MAT540 Homework Week 8 Page 2 of 4 Program Votes/Dollar Job training 0.03 Parks 0.08 Sanitation 0.05 Mobile library 0.03 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. a. Formulate a linear programming model for this problem. b. Solve the model by using the computer. 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: MAT540 Homework Week 8 Page 3 of 4 Calories (per lb.) Iron (mg/lb.) Protein (g/lb.) Carbohydrates (g/lb.) Fat (g/lb.) Cholesterol (mg/lb) Cost ($/lb.) Chicken 500 4.2 17 0 30 180 0.85 Fish 480 3.1 85 0 5 90 3.35 Ground beef 840 0.25 82 0 75 350 2.45 Dried beans 590 3.2 10 30 3 0 0.85 Lettuce 40 0.4 6 0 0 0 0.70 Potatoes 450 2.25 10 70 0 0 0.45 Milk (2%) 220 0.2 16 22 10 20 0.82 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. a. Formulate a linear programming model for this problem. MAT540 Homework Week 8 Page 4 of 4 b. Solve the model by using the computer. 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: Month Contracted Sales (units) 1 1200 2 2100 3 4 5 2400 3000 4000 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. a. Formulate a linear programming model for this problem. b. Solve the model by using the computer.


MAT540 Homework Week 9 Page 1 of 3 MAT540 Week 9 Homework 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: MAT540 Homework Week 9 Page 2 of 3 Item Weight (lb.) Profit Denim jeans 2 $90 CD players 3 150 Compact discs 1 30 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. Project Expense ($1,000s) Management Scientists required Estimated Profit (1,000,000s) 1 50 6 0.30 2 105 8 0.85 3 56 9 0.20 4 45 3 0.15 5 90 7 0.50 6 80 5 0.45 7 78 8 0.55 8 60 5 0.40 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 MAT540 Homework Week 9 Page 3 of 3 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: Company Year Sold (estimated returns, $1,000,000s) 1 2 3 1 $14 $18 $23 2 9 11 15 3 18 23 27 4 16 21 25 5 12 16 22 6 21 23 28 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.


MAT540 Homework Week 10 Page 1 of 2 MAT540 Week 10 Homework Chapter 6 1. Consider the following transportation problem: From To (Cost) Supply 1 2 3 A 6 5 5 150 B 11 8 9 85 C 4 10 7 125 Demand 70 100 80 Formulate this problem as a linear programming model and solve it by the using the computer. 2. Consider the following transportation problem: From To (Cost) Supply 1 2 3 A 8 14 8 120 B 6 17 7 80 C 9 24 10 150 Demand 110 140 100 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 ($/1,000 lb.) from the European ports to the U.S. cities and the available supplies (1000 lb.) at the European ports are provided in the following table: MAT540 Homework Week 10 Page 2 of 2 From To (Cost) Supply 4. Norfolk 5. New York 6. Savannah 1. Hamburg 320 280 555 75 2. Marseilles 410 470 365 85 3. Liverpool 550 355 525 40 The transportation costs ($/1000 lb.) from each U.S. city of the three distribution centers and the demands (1000 lb.) at the distribution centers are as follows: Warehouse Distribution Center 7. Dallas 8. St. Louis 9. Chicago 4. Norfolk 80 78 85 5. New York 100 120 95 6. Savannah 65 75 90 Demand 85 70 65 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: Salesperson Region (days) A B C D E 1 20 10 12 10 22 2 14 10 18 11 15 3 12 13 19 11 14 4 16 12 14 22 16 5 12 15 19 26 23 Which salesperson should be assigned to each region to minimize total time? Identify the optimal assignments and compute total minimum time.

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MAT540 Complete Course Week 1 to Week 11 Latest,MAT540 Homework Week 8 Page 1 of 4 MAT540 Week 8 Homework Chapter 4 1. Betty Malloy, owner of the Eagle Tavern in Pittsburgh, is preparing