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Email: [email protected] Patrick’s Paradox: Patrick’s luck had changed overnight – but not his skill at mathematical reasoning. The day after graduating from college he used the $20 that his grandmother had given him as a graduation gift to buy a lottery ticket. He knew his chances of winning the lottery were extremely low and it probably was not a good way to spend this money. But he also remembered from the class he took in business analytics that bad decisions some- times result in good outcomes. So, he said to himself, “What the heck? Maybe this bad decision will be the one with a good outcome.” And with that thought, he bought his lottery ticket. The next day Patrick pulled the crumpled lottery ticket out of the back pocket of his blue jeans and tried to compare his numbers to the winning numbers printed in the paper. When his eyes finally came into focus on the numbers they also just about popped out of his head. He had a winning ticket! In the ensuing days he learned that his share of the jackpot would give him a lump sum payout of about $500,000 after taxes. He knew what he was going to do with part of the money, buy a new car, pay off his college loans, and send his grandmother on an all-expenses paid trip to Hawaii. But he also knew that he couldn’t continue to hope for good outcomes to arise from more bad decisions. So, he decided to take half of his winnings and invest it for his retirement. A few days later, Patrick was sitting around with two of his fraternity buddies, Josh and Peyton, trying to figure out how much money his new retirement fund might be worth in 30 years. They were all business majors in college and remembered from their finance class that if you invest p dollars for n years at an annual interest rate of i percent then in n years you would have p(1+i)^n dollars. So, they figure that if Patrick invested $250,000 for 30 years in an investment with a 10% annual return, then in 30 years he would have $4,362,351 (i.e.,

$250,000(1+ 0.10)^30). But after thinking about it a little more, they all agreed that it would be unlikely for Patrick to find an investment that would produce a return of exactly 10% each year for the next 30 years. If any of this money is invested in stocks, then some years the return might be higher than 10% and some years it would probably be lower. So, to help account for the potential variability in the investment returns Patrick and his friends came up with a plan; they would assume he could find an investment that would produce an annual return of 17.5% seventy percent of the time and a return (or actually a loss) of -7.5% thirty percent of the time. Such an investment should pro- duce an average annual return of 0.7(17.5%)+ 0.3(-7.5%)=10%. Josh felt certain that this meant Patrick could still expect his $250,000 investment to grow to $4,362,351 in 30 years (because $250,000(1+0.10)^30 = $4,362,351).

After sitting quietly and thinking about it for a while, Peyton said that he thought Josh was wrong. The way Peyton looked at it, Patrick should see a 17.5% return in 70% of the 30 years (or 0.7(30)=21 years) and a -7.5% return in 30% of the 30 years (or 0.3(30)=9 years). So, according to Peyton, that would mean Patrick should have $250,000 (1+0.175)^2(1-0.075)^9=$3,664,467 after 30 years. But that’s $697,884 less than what Josh says Patrick should have. After listening to Peyton’s argument, Josh said he thought Peyton was wrong because his calculation assumes that the “good” return of 17.5% would occur in each of the first 21 years and the “bad” return of -7.5% would occur in each of the last 9 years. But Peyton countered this argument by saying that the order of good and bad returns does not matter. The commutative law of arithmetic says that when you add or multiply numbers, the order doesn’t matter (i.e.,

X+Y=Y+X and X*Y=Y*X). So, Peyton says that because Patrick can expect 21 “good” returns and 9 “bad” returns and it doesn’t matter in what order they occur, then the expected outcome of the investment should be $3,664,467 after 30 years.

Patrick is now really confused. Both of his friends’ arguments seem to make perfect sense logically—but they lead to such different answers, and they can’t both be right. What really worries Patrick is that he is starting his new job as a business analyst in a couple of weeks. And if he can’t reason his way to the right answer in a relatively simple problem like this, what is he going to do when he encounters the more difficult problems awaiting him the business world? Now he really wishes he had paid more attention in his business analytics class. So, what do you think? Who is right, Joshua or Peyton? And more importantly, why? Joshua was right. This mainly based on the higher returns from the invested fund, with a future value of $4,362,351 in the 30 years as compared to Peyton’s returns.

Cross Over, Break-Even Analysis and Decision Trees The City of High Point is buying new school buses for the local school system. High Point has found two models of school buses that it is interested in. Eagle Mover costs $80,000 to buy and uses diesel fuel, with an average fuel efficiency of 10 miles per gallon. Eagle Mover has an operating cost of $.28 per mile. Yellow Transport, a hybrid bus, costs $105,000 to buy and uses diesel fuel and battery power, getting an average of 22 miles per gallon. Yellow Transport has an operating cost of $.32 per mile. The distance traveled annually is determined to be 25,000 miles, with the expected life of either bus to be 10 years. The average diesel price is $3.50 per gallon. a) Based on life cycle cost, which bus is the better choice? Yellow Transport b) How many miles does the school district need to put on a bus for costs to be equal?12 miles c) What is the crossover point in years? 6 years

Blue Star is starting a new distribution service that delivers auto parts to the service departments of auto dealerships in the local area. Blue Star has found two light-duty trucks that would do the job well, so now it needs to pick one to perform this new service. The Ford TriVan costs $28,000 to buy and uses regular unleaded gasoline, with an average fuel efficiency of 24 miles per gallon. The TriVan has an operating cost of $.20 per mile. The Honda CityVan, a hybrid truck, costs $32,000 to buy and uses regular unleaded gasoline and battery power; it gets an average of 37 miles per gallon. The CityVan has an operating cost of $.22 per mile. The distance traveled annually is estimated to be 22,000 miles, with the life of either truck expected to be 8 years. The average gas price is $4.25 per gallon.

a) Based on life cycle cost, which model truck is the best choice? Honda CityVan b) How many miles does Blue Star need to put on a truck for the costs to be equal? 13 miles c) What is the crossover point in years? 5 years

The national coffee store Farbucks needs to decide in August how many holiday-edition insulated coffee mugs to order. Because the mugs are dated, those that are unsold by January 15 are considered a loss. These premium mugs sell for $23.95 and cost $6.75 each. Farbucks needs to spend $20,000 to start the production. Farbucks is uncertain of the demand. They believe that there is a 25% chance that they will sell 10,000 mugs, a 50% chance that they will sell 15,000, and a 25% chance that they will sell 20,000.

What is the minimum number of mugs that is needed to be sold so Farbucks makes a profit. 20,000 mugs

Build a decision tree for this process.

Decision Tree

START

Sell 10,000 (0.25)

Sell 15,000 (0.5) Sell 20,000 (0.25)

No sell (0.00)

END

What is the expected income for the company? 20,000*$23.95=$479,000.

Zan Azlett and Angela Zesiger have joined forces to start A&Z Lettuce Products, a processor of packaged shredded lettuce for institutional use. Zan has years of food processing experience, and Angela has extensive commercial food preparation experience. The process will consist of opening crates of lettuce and then sorting, washing, slicing, preserving, and finally packaging the prepared lettuce. Together, with help from vendors, they think they can adequately estimate demand, fixed costs, revenues, and variable cost per 5-pound bag of lettuce. They think a largely manual process will have monthly fixed costs of $37,500 and variable costs of $3.75 per bag. A more mechanized process will have fixed costs of $75,000 per month with variable costs of $3.00 per 5-pound bag. They expect to sell the shredded lettuce for $5.00 per 5-pound bag.

What is the break-even quantity for the manual process? Total revenue=Total costs Units sold x $5=$37,500 Break even quantity=7,500 units

What is the revenue at the break-even quantity for the manual process? Revenue for break-even quantity=7,500x$5=$37,500

What is the break-even quantity for the mechanized process? Total revenue=Total costs Units sold x $5=$75,000 Break even quantity=15,000 units

What is the revenue at the break-even quantity for the mechanized process? Revenue at break-even=$75,000

What is the monthly profit or loss of the manual process if they expect to sell 80,000 bags of lettuce per month? Total revenue=$5x80000=$400,000 Total cost=$37,500+$3.75x80000=$337,000 Profit=$40,000-$337,000=$62,500

What is the monthly profit or loss of the mechanized process if they expect to sell 80,000 bags of lettuce per month? Total revenue=$5x80000=$400,000 Total cost=$75,500+$3.0x80000=$337,000 Profit=$400,000-$315,000=$85,000

At what quantity would Zan and Angela be indifferent to the process selected? Quantity =profit/unit cost Quantity=$85,000/3=28300 units

Over what range of demand would the manual process be preferred over the mechanized process? Quantity =profit/unit cost Quantity=$62500/3=20800 units

Over what range of demand would the mechanized process be preferred over the manual process?

Demand difference=Mechanized-Manual =28,300-20,800=7,500 units

3-13. The Weedwacker Company manufactures two types of lawn trimmers: an electric model and a gas model. The company has contracted to supply a national discount retail chain with a total of 30,000 electric trimmers and 15,000 gas trimmers. However, Weedwacker’s production capability is limited in three departments: production, assembly, and packaging. The following table summarizes the hours of processing time available and the processing time required by each department, for both types of trimmers:

Hours required per Trimmer Electric (x1)

Gas (x2)

Hours available

Production

0.2

0.4

10000

Assembly

0.3

0.5

15000

Packaging

0.1

0.1

5000

The company makes its electric trimmer in-house for $55 and its gas trimmer for $85. Alternatively, it can buy electric and gas trimmers from another source for $67 and $95, respectively. How many gas and electric trimmers should Weedwacker make and how many should it buy from its competitor in order to fulfill its contract in the least costly manner? a. Formulate an LP model for this problem. Maximize 30,000x1+15,000x2 s/t 0.2x1+0.4x2≤10,000 0.3x1+0.5x2≤15000 0.1x1+0.1x2≤5000 X1≥0 X2≥0 b. Create a spreadsheet model for this problem and solve it using Solver.

c. What is the optimal solution? 5000x1+0x2

3-14. A furniture manufacturer produces two types of tables (country and contemporary) using three types of machines. The time required to produce the tables on each machine is given in the following table. Machine

Country (x1)

Contemporary (x2)

Total machine time available per week

Router

1.5

2

1000

Sander

3

4.5

2000

Polisher

2.5

1.5

1500

Country tables sell for $350 and contemporary tables sell for $450. Management has determined that at least 20% of the tables made should be country and at least 30% should be contemporary. How many of each type of table should the company produce if it wants to maximize its revenue? a. Formulate an LP model for this problem. Maximize 350x1+450x2 s/t 1.5x1+2.0x2≤1000

3.0x1+4.5x2≤2000 2.5x1+1.5x2≤1500 X1≥0 X2≥0

b. Create a spreadsheet model for this problem and solve it using Solver. c. What is the optimal solution? 555.56x1+74.074x2 d. How will your spreadsheet model differ if there are 25 types of tables and 15 machine processes involved in manufacturing them? 0x1+100x2 3-25 A company is trying to determine how to allocate its $145,000 advertising budget for a new product. The company is considering newspaper ads and television commercials as its primary means for advertising. The following table summarizes the costs of advertising in these different media and the number of new customers reached by increasing amounts of advertising.

Media and number of ads

# of new costumers reached Cost per ad ($) (x2) (x1)

Newspaper 1-10

900

1000

Newspaper 11-20

700

900

Newspaper 21-30

400

800

Television 1-5

10000

12000

Television 6-10

7500

10000

Television 11-15

5000

8000

For instance, each of the first 10 ads the company places in newspapers will cost $1,000 and is expected to reach 900 new customers. Each of the next 10 newspaper ads will cost $900 and is expected to reach 700 new customers. Note that the number of new customers reached by increasing amounts of advertising decreases as the advertising saturates the market. Assume the company will purchase no more than 30 newspaper ads and no more than 15 television ads. a) Formulate an LP model for this problem to maximize the number of new customers reached by advertising. Minimize 8000x1+12000x2 s/t 900x1+1000x2≤145000 700x1+900x2≤145000 400x1+800x2≤145000 10000x1+12000x2≤145000 75000x1+10000x2≤145000

5000x1+8000x2≤145000 X1≥0 X2≥0 b) Implement your model in a spreadsheet and solve it. c) What is the optimal solution? 0x1+12.083x2 d) Suppose the number of new customers reached by 11–20 newspaper ads is 400 and the number of new customers reached by 21–30 newspaper ads is 700. Make these changes in your spreadsheet and reoptimize the problem. What is the new optimal solution? What (if anything) is wrong with this solution and why? 0x1+12.083x2 The optimal solution does not change.

$250,000(1+ 0.10)^30). But after thinking about it a little more, they all agreed that it would be unlikely for Patrick to find an investment that would produce a return of exactly 10% each year for the next 30 years. If any of this money is invested in stocks, then some years the return might be higher than 10% and some years it would probably be lower. So, to help account for the potential variability in the investment returns Patrick and his friends came up with a plan; they would assume he could find an investment that would produce an annual return of 17.5% seventy percent of the time and a return (or actually a loss) of -7.5% thirty percent of the time. Such an investment should pro- duce an average annual return of 0.7(17.5%)+ 0.3(-7.5%)=10%. Josh felt certain that this meant Patrick could still expect his $250,000 investment to grow to $4,362,351 in 30 years (because $250,000(1+0.10)^30 = $4,362,351).

After sitting quietly and thinking about it for a while, Peyton said that he thought Josh was wrong. The way Peyton looked at it, Patrick should see a 17.5% return in 70% of the 30 years (or 0.7(30)=21 years) and a -7.5% return in 30% of the 30 years (or 0.3(30)=9 years). So, according to Peyton, that would mean Patrick should have $250,000 (1+0.175)^2(1-0.075)^9=$3,664,467 after 30 years. But that’s $697,884 less than what Josh says Patrick should have. After listening to Peyton’s argument, Josh said he thought Peyton was wrong because his calculation assumes that the “good” return of 17.5% would occur in each of the first 21 years and the “bad” return of -7.5% would occur in each of the last 9 years. But Peyton countered this argument by saying that the order of good and bad returns does not matter. The commutative law of arithmetic says that when you add or multiply numbers, the order doesn’t matter (i.e.,

X+Y=Y+X and X*Y=Y*X). So, Peyton says that because Patrick can expect 21 “good” returns and 9 “bad” returns and it doesn’t matter in what order they occur, then the expected outcome of the investment should be $3,664,467 after 30 years.

Patrick is now really confused. Both of his friends’ arguments seem to make perfect sense logically—but they lead to such different answers, and they can’t both be right. What really worries Patrick is that he is starting his new job as a business analyst in a couple of weeks. And if he can’t reason his way to the right answer in a relatively simple problem like this, what is he going to do when he encounters the more difficult problems awaiting him the business world? Now he really wishes he had paid more attention in his business analytics class. So, what do you think? Who is right, Joshua or Peyton? And more importantly, why? Joshua was right. This mainly based on the higher returns from the invested fund, with a future value of $4,362,351 in the 30 years as compared to Peyton’s returns.

Cross Over, Break-Even Analysis and Decision Trees The City of High Point is buying new school buses for the local school system. High Point has found two models of school buses that it is interested in. Eagle Mover costs $80,000 to buy and uses diesel fuel, with an average fuel efficiency of 10 miles per gallon. Eagle Mover has an operating cost of $.28 per mile. Yellow Transport, a hybrid bus, costs $105,000 to buy and uses diesel fuel and battery power, getting an average of 22 miles per gallon. Yellow Transport has an operating cost of $.32 per mile. The distance traveled annually is determined to be 25,000 miles, with the expected life of either bus to be 10 years. The average diesel price is $3.50 per gallon. a) Based on life cycle cost, which bus is the better choice? Yellow Transport b) How many miles does the school district need to put on a bus for costs to be equal?12 miles c) What is the crossover point in years? 6 years

Blue Star is starting a new distribution service that delivers auto parts to the service departments of auto dealerships in the local area. Blue Star has found two light-duty trucks that would do the job well, so now it needs to pick one to perform this new service. The Ford TriVan costs $28,000 to buy and uses regular unleaded gasoline, with an average fuel efficiency of 24 miles per gallon. The TriVan has an operating cost of $.20 per mile. The Honda CityVan, a hybrid truck, costs $32,000 to buy and uses regular unleaded gasoline and battery power; it gets an average of 37 miles per gallon. The CityVan has an operating cost of $.22 per mile. The distance traveled annually is estimated to be 22,000 miles, with the life of either truck expected to be 8 years. The average gas price is $4.25 per gallon.

a) Based on life cycle cost, which model truck is the best choice? Honda CityVan b) How many miles does Blue Star need to put on a truck for the costs to be equal? 13 miles c) What is the crossover point in years? 5 years

The national coffee store Farbucks needs to decide in August how many holiday-edition insulated coffee mugs to order. Because the mugs are dated, those that are unsold by January 15 are considered a loss. These premium mugs sell for $23.95 and cost $6.75 each. Farbucks needs to spend $20,000 to start the production. Farbucks is uncertain of the demand. They believe that there is a 25% chance that they will sell 10,000 mugs, a 50% chance that they will sell 15,000, and a 25% chance that they will sell 20,000.

What is the minimum number of mugs that is needed to be sold so Farbucks makes a profit. 20,000 mugs

Build a decision tree for this process.

Decision Tree

START

Sell 10,000 (0.25)

Sell 15,000 (0.5) Sell 20,000 (0.25)

No sell (0.00)

END

What is the expected income for the company? 20,000*$23.95=$479,000.

Zan Azlett and Angela Zesiger have joined forces to start A&Z Lettuce Products, a processor of packaged shredded lettuce for institutional use. Zan has years of food processing experience, and Angela has extensive commercial food preparation experience. The process will consist of opening crates of lettuce and then sorting, washing, slicing, preserving, and finally packaging the prepared lettuce. Together, with help from vendors, they think they can adequately estimate demand, fixed costs, revenues, and variable cost per 5-pound bag of lettuce. They think a largely manual process will have monthly fixed costs of $37,500 and variable costs of $3.75 per bag. A more mechanized process will have fixed costs of $75,000 per month with variable costs of $3.00 per 5-pound bag. They expect to sell the shredded lettuce for $5.00 per 5-pound bag.

What is the break-even quantity for the manual process? Total revenue=Total costs Units sold x $5=$37,500 Break even quantity=7,500 units

What is the revenue at the break-even quantity for the manual process? Revenue for break-even quantity=7,500x$5=$37,500

What is the break-even quantity for the mechanized process? Total revenue=Total costs Units sold x $5=$75,000 Break even quantity=15,000 units

What is the revenue at the break-even quantity for the mechanized process? Revenue at break-even=$75,000

What is the monthly profit or loss of the manual process if they expect to sell 80,000 bags of lettuce per month? Total revenue=$5x80000=$400,000 Total cost=$37,500+$3.75x80000=$337,000 Profit=$40,000-$337,000=$62,500

What is the monthly profit or loss of the mechanized process if they expect to sell 80,000 bags of lettuce per month? Total revenue=$5x80000=$400,000 Total cost=$75,500+$3.0x80000=$337,000 Profit=$400,000-$315,000=$85,000

At what quantity would Zan and Angela be indifferent to the process selected? Quantity =profit/unit cost Quantity=$85,000/3=28300 units

Over what range of demand would the manual process be preferred over the mechanized process? Quantity =profit/unit cost Quantity=$62500/3=20800 units

Over what range of demand would the mechanized process be preferred over the manual process?

Demand difference=Mechanized-Manual =28,300-20,800=7,500 units

3-13. The Weedwacker Company manufactures two types of lawn trimmers: an electric model and a gas model. The company has contracted to supply a national discount retail chain with a total of 30,000 electric trimmers and 15,000 gas trimmers. However, Weedwacker’s production capability is limited in three departments: production, assembly, and packaging. The following table summarizes the hours of processing time available and the processing time required by each department, for both types of trimmers:

Hours required per Trimmer Electric (x1)

Gas (x2)

Hours available

Production

0.2

0.4

10000

Assembly

0.3

0.5

15000

Packaging

0.1

0.1

5000

The company makes its electric trimmer in-house for $55 and its gas trimmer for $85. Alternatively, it can buy electric and gas trimmers from another source for $67 and $95, respectively. How many gas and electric trimmers should Weedwacker make and how many should it buy from its competitor in order to fulfill its contract in the least costly manner? a. Formulate an LP model for this problem. Maximize 30,000x1+15,000x2 s/t 0.2x1+0.4x2≤10,000 0.3x1+0.5x2≤15000 0.1x1+0.1x2≤5000 X1≥0 X2≥0 b. Create a spreadsheet model for this problem and solve it using Solver.

c. What is the optimal solution? 5000x1+0x2

3-14. A furniture manufacturer produces two types of tables (country and contemporary) using three types of machines. The time required to produce the tables on each machine is given in the following table. Machine

Country (x1)

Contemporary (x2)

Total machine time available per week

Router

1.5

2

1000

Sander

3

4.5

2000

Polisher

2.5

1.5

1500

Country tables sell for $350 and contemporary tables sell for $450. Management has determined that at least 20% of the tables made should be country and at least 30% should be contemporary. How many of each type of table should the company produce if it wants to maximize its revenue? a. Formulate an LP model for this problem. Maximize 350x1+450x2 s/t 1.5x1+2.0x2≤1000

3.0x1+4.5x2≤2000 2.5x1+1.5x2≤1500 X1≥0 X2≥0

b. Create a spreadsheet model for this problem and solve it using Solver. c. What is the optimal solution? 555.56x1+74.074x2 d. How will your spreadsheet model differ if there are 25 types of tables and 15 machine processes involved in manufacturing them? 0x1+100x2 3-25 A company is trying to determine how to allocate its $145,000 advertising budget for a new product. The company is considering newspaper ads and television commercials as its primary means for advertising. The following table summarizes the costs of advertising in these different media and the number of new customers reached by increasing amounts of advertising.

Media and number of ads

# of new costumers reached Cost per ad ($) (x2) (x1)

Newspaper 1-10

900

1000

Newspaper 11-20

700

900

Newspaper 21-30

400

800

Television 1-5

10000

12000

Television 6-10

7500

10000

Television 11-15

5000

8000

For instance, each of the first 10 ads the company places in newspapers will cost $1,000 and is expected to reach 900 new customers. Each of the next 10 newspaper ads will cost $900 and is expected to reach 700 new customers. Note that the number of new customers reached by increasing amounts of advertising decreases as the advertising saturates the market. Assume the company will purchase no more than 30 newspaper ads and no more than 15 television ads. a) Formulate an LP model for this problem to maximize the number of new customers reached by advertising. Minimize 8000x1+12000x2 s/t 900x1+1000x2≤145000 700x1+900x2≤145000 400x1+800x2≤145000 10000x1+12000x2≤145000 75000x1+10000x2≤145000

5000x1+8000x2≤145000 X1≥0 X2≥0 b) Implement your model in a spreadsheet and solve it. c) What is the optimal solution? 0x1+12.083x2 d) Suppose the number of new customers reached by 11–20 newspaper ads is 400 and the number of new customers reached by 21–30 newspaper ads is 700. Make these changes in your spreadsheet and reoptimize the problem. What is the new optimal solution? What (if anything) is wrong with this solution and why? 0x1+12.083x2 The optimal solution does not change.