# Problems

[Page 469 ( continued )]
1.

The Hickory Cabinet and Furniture Company makes chairs. The fixed cost per month of making chairs is \$7,500, and the variable cost per chair is \$40. Price is related to demand, according to the following linear equation:

v = 400 1.2 p

[Page 470]

Develop the nonlinear profit function for this company and determine the price that will maximize profit, the optimal volume, and the maximum profit per month.

2.

Graphically illustrate the profit curve developed in Problem 1. Indicate the optimal price and the maximum profit per month.

3.

The Rainwater Brewery produces beer. The annual fixed cost is \$150,000, and the variable cost per barrel is \$16. Price is related to demand, according to the following linear equation:

v = 75,000 1,153.8 p

Develop the nonlinear profit function for the brewery and determine the price that will maximize profit, the optimal volume, and the maximum profit per year.

4.

The Rolling Creek Textile Mill makes denim. The monthly fixed cost is \$8,000, and the variable cost per yard of denim is \$0.35. Price is related to demand, according to the following linear equation:

v = 17,000 5,666 p

Develop the nonlinear profit function for the textile mill and determine the optimal price, the optimal volume, and the maximum profit per month.

5.

The Grady Tire Company recaps tires. The weekly fixed cost is \$2,500, and the variable cost per tire is \$9. Price is related to demand, according to the following linear equation:

v = 200 4.75 p

Develop the nonlinear profit function for the tire company and determine the optimal price, the optimal volume, and the maximum profit per week.

6.

Andy Mendoza makes handcrafted dolls, which he sells at craft fairs. He is considering mass-producing the dolls to sell in stores. He estimates that the initial investment for plant and equipment will be \$25,000, while labor, materials, packaging, and shipping will be about \$10 per doll. He has determined that sales volume is related to price, according to the following linear equation:

v = 4,000 80 p

Develop the nonlinear profit function for Andy and determine the price that will maximize profit, the optimal volume, and the maximum profit per month.

7.

The Rainwater Brewery produces beer, which it sells to distributors in barrels. The brewery incurs a monthly fixed cost of \$12,000, and the variable cost per barrel is \$17. The brewery has developed the following profit function and demand constraint:

maximize Z = vp \$12,000 17 v

subject to

v = 1,800 15 p

Solve this nonlinear programming model for the optimal price ( p ).

8.

The Beaver Creek Pottery Company has developed the following nonlinear programming model to determine the optimal number of bowls ( x 1 ) and mugs ( x 2 ) to produce each day:

maximize Z = \$7 x 1 0.3 x 1 2 + 8 x 2 0.4 x 2 2

subject to

4 x 1 + 5 x 2 = 100 hr.

Determine the optimal solution to this nonlinear programming model.

[Page 471]
9.

The Evergreen Fertilizer Company produces two types of fertilizers, Fastgro and Super Two. The company has developed the following nonlinear programming model to determine the optimal number of bags of Fastgro ( x 1 ) and Super Two ( x 2 ) that it must produce each day to maximize profit, given a constraint for available potassium:

maximize Z = \$30 x 1 2 x 1 2 + 25 x 2 0.5 x 2 2

subject to

3 x 1 + 6 x 2 = 300 lb.

Determine the optimal solution to this nonlinear programming model.

10.

The Rolling Creek Textile Mill produces denim and brushed cotton cloth. The company has developed the following nonlinear programming model to determine the optimal number of yards of denim ( x 1 ) and brushed cotton ( x 2 ) to produce each day in order to maximize profit, subject to a labor constraint:

maximize Z = \$10 x 1 0.02 x 1 2 + 12 x 2 0.03 x 2 2

subject to

0.2 x 1 + 0.1 x 2 = 40 hr.

Determine the optimal solution to this nonlinear programming model.

11.

In the investment portfolio selection model in this chapter, what if Jessica Todd decided that she wanted to maximize her return while incurring a portfolio variance of no more than 0.020? Analyze this scenario by using the Excel spreadsheet in Exhibit 10.13 to determine what Jessica's portfolio and return would be.

12.

The Riverwood Paneling Company makes two kinds of wood paneling: Colonial and Western. The company has developed the following nonlinear programming model to determine the optimal number of sheets of Colonial paneling ( x 1 ) and Western paneling ( x 2 ) to produce to maximize profit, subject to a labor constraint:

maximize Z = \$25 x 1 0.8 x 1 2 + 30 x 2 1.2 x 2 2

subject to

x 1 + 2 x 2 = 40 hr.

Determine the optimal solution to this nonlinear programming model.

13.

Interpret the meaning of the Lagrange multiplier in Problem 12.

14.

Metro Telcom Systems develops, sells, and installs computer systems. The company has divided its customer base into five regions, and it has 15 representatives who sell and install the company's systems. The company wants to allocate salespeople to regions so that they maximize daily sales revenue. However, whereas the sales increase as the number of salespeople allocated to a region increases , they do so at a declining rate, according to the following nonlinear formula:

total sales = a ( b / x )

Following are the a and b parameters for daily sales in each region.

Region

1

2

3

4

5

a

\$15,000

\$24,000

\$8,100

\$12,000

\$21,000

b

9,000

15,000

5,300

7,600

12,500

[Page 472]

Because some of the regions are in urban areas and some are not, the representatives' daily expenses will differ between regions. The company has a daily expense budget of \$6,500, and the daily expenses (including travel costs) per representative for each region average \$355 for region 1, \$540 for region 2, \$290 for region 3, \$275 for region 4, and \$490 for region 5. Formulate and solve a nonlinear programming model for this problem to determine the number of representatives to allocate to each region to maximize daily sales.

15.

Blue Ridge Power Company has a maximum capacity of 2.5 million kwh of electric power available on a daily basis. The demand (in millions of kwh) for power from its customers for high-demand (peak) hours and low-demand (off-peak) hours is determined by the following formulas:

high demand: 5.8 0.06 p h + 0.005 p l

low demand: 3.0 0.11 p l + 0.008 p h

The variable p l equals the price per kwh during low-demand hours, and p h is the price per kwh during high-demand (peak) hours.

Formulate and solve a nonlinear programming model to determine the price structure (per kwh) that will maximize revenue.

16.

The Metro Police Department has partitioned the city into four quadrants. The department has 20 police patrol cars available for each shift per day to assign to the quadrants. The department wants to assign patrol cars to the quadrants during a shift so that the crime rate is minimized while providing average response times to calls of less than 10 minutes. As patrol cars are assigned to a quadrant, the crime rate and response time decrease, but at a decreasing rate according to the following formula:

y = a + ( b / x )

The a and b parameters for daily crime rate ( expressed as crimes per 1,000 population) and response time (in minutes) are provided in the following table:

Crime Rate

Response Time (min.)

a

b

a

b

1

0.24

0.15

4

11

2

0.37

0.21

8

8

3

0.21

0.12

6

10

4

0.48

0.30

3

9

Formulate and solve a nonlinear programming model to determine the number of police patrol cars to assign to each quadrant that will result in the minimum overall crime rate.

17.

The Burger Doodle restaurant chain purchases ingredients from four different food suppliers. The company wants to construct a new central distribution center to process and package the ingredients it uses in its menu items before shipping them to their various restaurants . The suppliers transport the food items in 40- foot tractor-trailer trucks . The coordinates of the four suppliers and the annual number of truckloads that will be transported to the distribution center are as follows :

Coordinates

Supplier

x

y

A

200

200

65

B

100

500

120

C

250

600

90

D

500

300

75

[Page 473]

Determine the set of coordinates for the new distribution center that will minimize the total miles traveled from the suppliers.

18.

Home-Base, a home improvement and building supply chain, is going to build a new warehouse facility to serve its stores in six North Carolina citiesCharlotte, Winston-Salem, Greensboro, Durham, Raleigh, and Wilmington. The coordinates of these cities (in miles), using Columbia, South Carolina, as the graphical origin (0, 0) and the annual truckload trips that supply each store are as follows:

Coordinates

Store

x

y

Charlotte

15

85

160

Winston-Salem

42

145

90

Greensboro

88

145

105

Durham

125

140

35

Raleigh

135

125

60

Wilmington

180

18

75

Determine the set of coordinates for the new warehouse that will minimize the total miles traveled to the stores and identify on a map the closest town to these coordinates.

19.

An investment adviser is helping a couple plan a retirement portfolio. The adviser has recommended three stocksAllied Electronics, Bank United, and Consolidated Computers. Following are the annual return and variance for each stock and the covariance between stocks:

Stock

Annual Return

Covariance

Allied Electronics

.14

.10

Bank United

.10

.04

Consolidated Computers

.12

.08

Stock Combination ( i, j )

Covariance

A,B

.4

A,C

.7

B,C

.3

The couple wants a total portfolio return of at least .11. Determine the proportion of each stock to include in the portfolio to minimize the overall risk.

20.

Mark Decker has identified four stocks for his portfolio, and he wants to determine the percentage of his total available funds he should invest in each stock. The alternative stocks include an Internet company, a computer software company, a computer manufacturer, and an entertainment conglomerate. He wants a total annual return of .12. From historical data, he has determined the average annual return and variance for each of the funds, as follows:

Stock

Annual Return

Variance

1. Internet

.18

.112

2. Software

.12

.061

3. Computer

.10

.045

4. Entertainment

.15

.088

[Page 474]

He has also estimated the covariances between stocks, as follows:

Stock Combination ( i, j )

Covariances

1,2

.9

1,3

.7

1,4

.3

2,3

.8

2,4

.4

3,4

.2

Determine the percentage of Mark's total funds that he should invest in each stock to minimize his overall risk.

21.

The Accentuate Consulting Firm has eight projects for which it has contracted with clients to develop computer systems and software. The firm has 35 team members it can assign to the projects. The firm has developed the following profit functions for each project, where x i equals the number of team members assigned to a project:

Project

Profit

1

\$250,000 x 1 0.50

2

370,000 x 2 0.30

3

140,000 x 3 0.70

4

500,000 x 4 0.25

5

230,000 x 5 0.33

6

170,000 x 6 0.60

7

280,000 x 7 0.45

8

315,000 x 8 0.40

These functions take into account the project completion time, cost, and probability of successful completion based on the number of team members assigned. The functions reflect the fact that profit increases but at a decreasing rate as team members are assigned. Thus, as additional team members are assigned to a project, the marginal effect decreases. Determine the optimal number of team members to assign to each project in order to maximize profit.

Introduction to Management Science (10th Edition)
ISBN: 0136064361
EAN: 2147483647
Year: 2006
Pages: 358

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