Problems


[Page 240 ( continued )]
1.

Green Valley Mills produces carpet at plants in St. Louis and Richmond. The carpet is then shipped to two outlets, located in Chicago and Atlanta. The cost per ton of shipping carpet from each of the two plants to the two warehouses is as follows :

 

To (cost)

From

Chicago

Atlanta

St. Louis

$40

$65

Richmond

70

30


The plant at St. Louis can supply 250 tons of carpet per week; the plant at Richmond can supply 400 tons per week. The Chicago outlet has a demand of 300 tons per week, and the outlet at Atlanta demands 350 tons per week. The company wants to know the number of tons of carpet to ship from each plant to each outlet in order to minimize the total shipping cost. Solve this transportation problem.

2.

A transportation problem involves the following costs, supply, and demand:

 

To (cost)

 

From

1

2

3

4

Supply

1

$500

$750

$300

$450

12

2

650

800

400

600

17

3

400

700

500

550

11

Demand

10

10

10

10

 

Solve this problem by using the computer.


[Page 241]
3.

Given a transportation problem with the following costs, supply, and demand, find the optimal solution by using the computer:

 

To (cost)

 

From

1

2

3

Supply

A

$6

$7

$4

100

B

5

3

6

180

C

8

5

7

200

Demand

135

175

170

 

4.

Consider the following transportation problem:

 

To (cost)

 

From

1

2

3

Supply

A

$ 6

$9

$ M

130

B

12

3

5

70

C

4

8

11

100

Demand

80

110

60

 

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

5.

Solve the following linear programming problem:

6.

Consider the following transportation problem:

 

To (cost)

 

From

1

2

3

Supply

A

$ 6

$9

$ 7

130

B

12

3

5

70

C

4

8

11

100

Demand

80

110

60

 

Solve it by using the computer.


[Page 242]
7.

Steel mills in three cities produce the following amounts of steel :

Location

Weekly Production (tons)

A. Bethlehem

150

B. Birmingham

210

C. Gary

320

 

680


These mills supply steel to four cities, where manufacturing plants have the following demand:

Location

Weekly Demand (tons)

1. Detroit

130

2. St. Louis

70

3. Chicago

180

4. Norfolk

240

 

620


Shipping costs per ton of steel are as follows:

 

To (cost)

 

From

1

2

3

4

A

$14

$ 9

$16

$18

B

11

8

7

16

C

16

12

10

22


Because of a truckers' strike, shipments are prohibited from Birmingham to Chicago. Formulate this problem as a linear programming model and solve it by using the computer.

8.

In Problem 7, what would be the effect on the optimal solution of a reduction in production capacity at the Gary mill from 320 tons to 290 tons per week?

9.

Coal is mined and processed at the following four mines in Kentucky, West Virginia, and Virginia:

Location

Capacity (tons)

A. Cabin Creek

90

B. Surry

50

C. Old Fort

80

D. McCoy

60

 

280


These mines supply the following amount of coal to utility power plants in three cities:

Plant

Demand (tons)

1. Richmond

120

2. Winston-Salem

100

3. Durham

110

 

330



[Page 243]

The railroad shipping costs (in thousands of dollars) per ton of coal are shown in the following table. Because of railroad construction, shipments are prohibited from Cabin Creek to Richmond:

 

To (cost, in $1,000s)

From

1

2

3

A

$ 7

$10

$ 5

B

12

9

4

C

7

3

11

D

9

5

7


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

10.

Oranges are grown, picked, and then stored in warehouses in Tampa, Miami, and Fresno. These warehouses supply oranges to markets in New York, Philadelphia, Chicago, and Boston. The following table shows the shipping costs per truckload (in hundreds of dollars), supply, and demand. Because of an agreement between distributors , shipments are prohibited from Miami to Chicago:

 

To (cost, in $100s)

 

From

New York

Philadelphia

Chicago

Boston

Supply

Tampa

$ 9

$14

$12

$17

200

Miami

11

10

6

10

200

Fresno

12

8

15

7

200

Demand

130

170

100

150

 

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

11.

A manufacturing firm produces diesel engines in four citiesPhoenix, Seattle, St. Louis, and Detroit. The company is able to produce the following numbers of engines per month:

Plant

Production

1. Phoenix

5

2. Seattle

25

3. St. Louis

20

4. Detroit

25


Three trucking firms purchase the following numbers of engines for their plants in three cities:

Firm

Demand

A. Greensboro

10

B. Charlotte

20

C. Louisville

15



[Page 244]

The transportation costs per engine (in hundreds of dollars) from sources to destinations are shown in the following table. However, the Charlotte firm will not accept engines made in Seattle, and the Louisville firm will not accept engines from Detroit; therefore, those routes are prohibited:

 

To (cost, in $100s)

From

A

B

C

1

$ 7

$ 8

$ 5

2

6

10

6

3

10

4

5

4

3

9

11


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

12.

The Interstate Truck Rental firm has accumulated extra trucks at three of its truck leasing outlets, as shown in the following table:

Leasing Outlet

Extra Trucks

1. Atlanta

70

2. St. Louis

115

3. Greensboro

60

Total

245


The firm also has four outlets with shortages of rental trucks, as follows:

Leasing Outlet

Truck Shortage

A. New Orleans

80

B. Cincinnati

50

C. Louisville

90

D. Pittsburgh

25

Total

245


The firm wants to transfer trucks from those outlets with extras to those with shortages at the minimum total cost. The following costs of transporting these trucks from city to city have been determined:

 

To (cost)

 

From

A

B

C

D

1

$70

$80

$45

$90

2

120

40

30

75

3

110

60

70

80


Solve this problem by using the computer.


[Page 245]
13.

The Shotz Beer Company has breweries in two cities; the breweries can supply the following numbers of barrels of draft beer to the company's distributors each month:

Brewery

Monthly Supply (bbl)

A. Tampa

3,500

B. St. Louis

5,000

Total

8,500


The distributors, which are spread throughout six states, have the following total monthly demand:

Distributor

Monthly Demand (bbl)

1. Tennessee

1,600

2. Georgia

1,800

3. North Carolina

1,500

4. South Carolina

950

5. Kentucky

1,250

6. Virginia

1,400

Total

8,500


The company must pay the following shipping costs per barrel:

 

To (cost)

From

1

2

3

4

5

6

A

$0.50

$0.35

$0.60

$0.45

$0.80

$0.75

B

0.25

0.65

0.40

0.55

0.20

0.65


Solve this problem by using the computer.

14.

In Problem 13, the Shotz Beer Company management has negotiated a new shipping contract with a trucking firm between its Tampa brewery and its distributor in Kentucky. This contract reduces the shipping cost per barrel from $0.80 per barrel to $0.65 per barrel. How will this cost change affect the optimal solution?

15.

Computers Unlimited sells microcomputers to universities and colleges on the East Coast and ships them from three distribution warehouses. The firm is able to supply the following numbers of microcomputers to the universities by the beginning of the academic year:

Distribution Warehouse

Supply (microcomputers)

1. Richmond

420

2. Atlanta

610

3. Washington, DC

340

Total

1,370



[Page 246]

Four universities have ordered microcomputers that must be delivered and installed by the beginning of the academic year:

University

Demand (microcomputers)

A. Tech

520

B. A & M

250

C. State

400

D. Central

380

Total

1,550


The shipping and installation costs per microcomputer from each distributor to each university are as follows:

 

To (cost)

From

A

B

C

D

1

$22

$17

$30

$18

2

15

35

20

25

3

28

21

16

14


Solve this problem by using the computer.

16.

In Problem 15, Computers Unlimited wants to better meet demand at the four universities it supplies . It is considering two alternatives: (1) expand its warehouse at Richmond to a capacity of 600, at a cost equivalent to an additional $6 in handling and shipping per unit; or (2) purchase a new warehouse in Charlotte that can supply 300 units with shipping costs of $19 to Tech, $26 to A & M, $22 to State, and $16 to Central. Which alternative should management select, based solely on transportation costs (i.e., no capital costs)?

17.

Computers Unlimited in Problem 15 has determined that when it is unable to meet the demand for microcomputers at the universities it supplies, the universities tend to purchase microcomputers elsewhere in the future. Thus, the firm has estimated a shortage cost for each microcomputer demanded but not supplied that reflects the loss of future sales and goodwill. These costs for each university are as follows:

University

Cost/Microcomputer

A. Tech

$40

B. A & M

65

C. State

25

D. Central

50


Solve Problem 15 with these shortage costs included. Compute the total transportation cost and the total shortage cost.

18.

A severe winter ice storm has swept across North Carolina and Virginia, followed by over a foot of snow and frigid, single-digit temperatures . These weather conditions have resulted in numerous downed power lines and power outages, causing dangerous conditions for much of the population. Local utility companies have been overwhelmed and have requested assistance from unaffected utility companies across the Southeast. The following table shows the number of utility trucks with crews available from five different companies in Georgia, South Carolina, and Florida; the demand for crews in seven different areas that local companies cannot get to; and the weekly cost (in thousands of dollars) of a crew going to a specific area (based on the visiting company's normal charges, the distance the crew has to come, and living expenses in an area):


[Page 247]
 

Area (cost, in $1,000s)

 

Crews

NC-E

NC-SW

NC-P

NC-W

VA-SW

VA-C

VA-T

Crew Available

GA-1

15.2

14.3

13.9

13.5

14.7

16.5

18.7

12

GA-2

12.8

11.3

10.6

12.0

12.7

13.2

15.6

10

SC-1

12.4

10.8

9.4

11.3

13.1

12.8

14.5

14

FL-1

18.2

19.4

18.2

17.9

20.5

20.7

22.7

15

FL-2

19.3

20.2

19.5

20.2

21.2

21.3

23.5

12

Crews Needed

9

7

6

8

10

9

7

 

Determine the number of crews that should be sent from each utility to each affected area to minimize total costs.

19.

A large manufacturing company is closing three of its existing plants and intends to transfer some of its more skilled employees to three plants that will remain open . The number of employees available for transfer from each closing plant is as follows:

Closing Plant

Transferable Employees

1

60

2

105

3

70

Total

235


The following number of employees can be accommodated at the three plants remaining open:

Open Plants

Employees Demanded

A

45

B

90

C

35

Total

170


Each transferred employee will increase product output per day at each plant, as shown in the following table:

 

To (employees)

From

A

B

C

1

5

8

6

2

10

9

12

3

7

6

8


The company wants to transfer employees to ensure the maximum increase in product output. Solve this problem by using the computer.


[Page 248]
20.

The Sav-Us Rental Car Agency has six lots in Nashville, and it wants to have a certain number of cars available at each lot at the beginning of each day for local rental. The agency would like a model it could quickly solve at the end of each day that would tell it how to redistribute the cars among the six lots in the minimum total time. The times required to travel between the six lots are as follows:

 

To (min.)

From

1

2

3

4

5

6

1

12

17

18

10

20

2

14

10

19

16

15

3

14

10

12

8

9

4

8

16

14

12

15

5

11

21

16

18

10

6

24

12

9

17

15


The agency would like the following number of cars at each lot at the end of the day. Also shown is the number of available cars at each lot at the end of a particular day. Determine the optimal reallocation of rental cars by using any initial solution approach and any solution method:

 

Lot (cars)

Cars

1

2

3

4

5

6

Available

37

20

14

26

40

28

Desired

30

25

20

40

30

20


21.

Bayville has built a new elementary school, increasing the town's total to four schoolsAddison, Beeks, Canfield, and Daley. Each has a capacity of 400 students. The school board wants to assign children to schools so that their travel time by bus is as short as possible. The school board has partitioned the town into five districts conforming to population densitynorth, south, east, west, and central. The average bus travel time from each district to each school is shown as follows:

 

Travel Time (min.)

 

District

Addison

Beeks

Canfield

Daley

Student Population

North

12

23

35

17

250

South

26

15

21

27

340

East

18

20

22

31

310

West

29

24

35

10

210

Central

15

10

23

16

290


Determine the number of children that should be assigned from each district to each school to minimize total student travel time.

22.

In Problem 21, the school board has determined that it does not want any of the schools to be overly crowded compared with the other schools. It would like to assign students from each district to each school so that enrollments are evenly balanced between the four schools. However, the school board is concerned that this might significantly increase travel time. Determine the number of students to be assigned from each district to each school such that school enrollments are evenly balanced. Does this new solution appear to significantly increase travel time per student?


[Page 249]
23.

The Easy Time Grocery chain operates in major metropolitan areas on the East Coast. The stores have a " no-frills " approach, with low overhead and high volume. They generally buy their stock in volume at low prices. However, in some cases they actually buy stock at stores in other areas and ship it in. They can do this because of high prices in the cities they operate in compared with costs in other locations. One example is baby food. Easy Time purchases baby food at stores in Albany, Binghamton, Claremont, Dover, and Edison and then trucks it to six stores in and around New York City. The stores in the outlying areas know what Easy Time is up to, so they limit the number of cases of baby food Easy Time can purchase. The following table shows the profit Easy Time makes per case of baby food, based on where the chain purchases it and at which store it is sold, plus the available baby food per week at purchase locations and the shelf space available at each Easy Time store per week:

 

Easy Time Store (profit/case)

 

Purchase Location

1

2

3

4

5

6

Supply

Albany

$ 9

$ 8

$11

$12

$7

$ 8

26

Binghamton

10

10

8

6

9

7

40

Claremont

8

6

6

5

7

4

20

Dover

4

6

9

5

8

10

40

Edison

12

10

8

9

6

7

45

Demand

25

15

30

18

27

35

 

Determine where Easy Time should purchase baby food and how the food should be distributed to maximize profit.

24.

Suppose that in Problem 23 Easy Time can purchase all the baby food it needs from a New York City distributor at a price that will result in a profit of $9 per case at stores 1, 3, and 4; $8 per case at stores 2 and 6; and $7 per case at store 5. Should Easy Time purchase all, none, or some of its baby food from the distributor rather than purchase it at other stores and truck it in?

25.

In Problem 23, if Easy Time could arrange to purchase more baby food from one of the outlying locations, which should it be, how many additional cases could be purchased, and how much would this increase profit?

26.

The Roadnet Transport Company has expanded its shipping capacity by purchasing 90 trailer trucks from a competitor that went bankrupt. The company subsequently located 30 of the purchased trucks at each of its shipping warehouses in Charlotte, Memphis, and Louisville. The company makes shipments from each of these warehouses to terminals in St. Louis, Atlanta, and New York. Each truck is capable of making one shipment per week. The terminal managers have indicated their capacity of extra shipments. The manager at St. Louis can accommodate 40 additional trucks per week, the manager at Atlanta can accommodate 60 additional trucks, and the manager at New York can accommodate 50 additional trucks. The company makes the following profit per truckload shipment from each warehouse to each terminal. The profits differ as a result of differences in products shipped, shipping costs, and transport rates:


[Page 250]
 

Terminal (profit)

Warehouse

St. Louis

Atlanta

New York

Charlotte

$1,800

$2,100

$1,600

Memphis

1,000

700

900

Louisville

1,400

800

2,200


Determine how many trucks to assign to each route (i.e., warehouse to terminal) in order to maximize profit.

27.

During the Gulf War in Iraq, large amounts of military mat riel and supplies had to be shipped daily from supply depots in the United States to bases in the Middle East. The critical factor in the movement of these supplies was speed. The following table shows the number of planeloads of supplies available each day from each of six supply depots and the number of daily loads demanded at each of five bases. (Each planeload is approximately equal in tonnage.) Also included are the transport hours per plane, including loading and fueling, actual flight time, and unloading and refueling:

 

Military Base

 

Supply Depot

A

B

C

D

E

Supply

1

36

40

32

43

29

7

2

28

27

29

40

38

10

3

34

35

41

29

31

8

4

41

42

35

27

36

8

5

25

28

40

34

38

9

6

31

30

43

38

40

6

Demand

9

6

12

8

10

 

Determine the optimal daily flight schedule that will minimize total transport time.

28.

PM Computer Services produces personal computers from component parts it buys on the open market. The company can produce a maximum of 300 personal computers per month. PM wants to determine its production schedule for the first 6 months of the new year. The cost to produce a personal computer in January will be $1,200. However, PM knows the cost of component parts will decline each month so that the overall cost to produce a PC will be 5% less each month. The cost of holding a computer in inventory is $15 per unit per month. Following is the demand for the company's computers each month:

Month

Demand

Month

Demand

January

180

April

210

February

260

May

400

March

340

June

320


Determine a production schedule for PM that will minimize total cost.


[Page 251]
29.

In Problem 28, suppose that the demand for personal computers increases each month, as follows:

Month

Demand

Month

Demand

January

410

April

620

February

320

May

430

March

500

June

380


In addition to the regular production capacity of 300 units per month, PM Computer Services can also produce an additional 200 computers per month by using overtime. Overtime production adds 20% to the cost of a personal computer.

Determine a production schedule for PM that will minimize total cost.

30.

National Foods Company has five plants where it processes and packages fruits and vegetables. It has suppliers in six cities in California, Texas, Alabama, and Florida. The company has owned and operated its own trucking system in the past for transporting fruits and vegetables from its suppliers to its plants. However, it is now considering outsourcing all its shipping to outside trucking firms and getting rid of its own trucks. It currently spends $245,000 per month to operate its own trucking system. It has determined monthly shipping costs (in thousands of dollars per ton) of using outside shippers from each of its suppliers to each of its plants, as shown in the following table:

 

Processing Plants ($1,000s/ton)

 

Suppliers

Denver

St. Paul

Louisville

Akron

Topeka

Supply (tons)

Sacramento

$3.7

$4.6

$4.9

$5.5

$4.3

18

Bakersfield

3.4

5.1

4.4

5.9

5.2

15

San Antonio

3.3

4.1

3.7

2.9

2.6

10

Montgomery

1.9

4.2

2.7

5.4

3.9

12

Jacksonville

6.1

5.1

3.8

2.5

4.1

20

Ocala

6.6

4.8

3.5

3.6

4.5

15

Demand (tons)

20

15

15

15

20

90


Should National Foods continue to operate its own shipping network or sell its trucks and outsource its shipping to independent trucking firms?

31.

In Problem 30, National Foods would like to know what the effect would be on the optimal solution and the company's decision regarding its shipping if it negotiates with its suppliers in Sacramento, Jacksonville, and Ocala to increase their capacity to 25 tons per month. What would be the effect of negotiating instead with its suppliers at San Antonio and Montgomery to increase their capacity to 25 tons each?

32.

Orient Express is a global distribution company that transports its clients ' products to customers in Hong Kong, Singapore, and Taipei. All the products Orient Express ships are stored at three distribution centersone in Los Angeles, one in Savannah, and one in Galveston. For the coming month the company has 450 containers of computer components available at the Los Angeles center, 600 containers available at Savannah, and 350 containers available at Galveston. The company has orders for 600 containers from Hong Kong, 500 containers from Singapore, and 500 containers from Taipei. The shipping costs per container from each U.S. port to each of the overseas ports are shown in the following table:


[Page 252]
 

Overseas Port (cost/container)

U.S. Distribution Center

Hong Kong

Singapore

Taipei

Los Angeles

$300

$210

$340

Savannah

490

520

610

Galveston

360

320

500


Orient Express, as the overseas broker for its U.S. customers, is responsible for unfulfilled orders, and it incurs stiff penalty costs from overseas customers if it does not meet an order. The Hong Kong customers charge a penalty cost of $800 per container for unfulfilled demand, Singapore customers charge a penalty cost of $920 per container, and Taipei customers charge $1,100 per container. Formulate and solve a transportation model to determine the shipments from each U.S. distribution center to each overseas port that will minimize shipping costs. Indicate what portion of the total cost is a result of penalties.

33.

Binford Tools manufactures garden tools. It uses inventory, overtime, and subcontracting to absorb demand fluctuations. Expected demand, regular and overtime production capacity, and subcontracting capacity are provided in the following table for the next four quarters for its basic line of steel garden tools:

Quarter

Demand

Regular Capacity

Overtime Capacity

Subcontracting Capacity

1

9,000

9,000

1,000

3,000

2

12,000

10,000

1,500

3,000

3

16,000

12,000

2,000

3,000

4

19,000

12,000

2,000

3,000


The regular production cost per unit is $20, the overtime cost per unit is $25, the cost to subcontract a unit is $27, and the inventory carrying cost is $2 per unit. The company has 300 units in inventory at the beginning of the year.

Determine the optimal production schedule for the four quarters to minimize total costs.

34.

Al, Barbara, Carol, and Dave have joined together to purchase two season tickets to the Giants' home football games. Because there are eight home games, each person will get tickets to two games. Each person has ranked the games they prefer from 1 to 8, with 1 being most preferred and 8 least preferred, as follows:

 

Person

Game

Al

Barbara

Carol

Dave

1. Cowboys

1

2

1

4

2. Redskins

3

4

4

1

3. Cardinals

7

8

8

7

4. Eagles

2

7

5

3

5. Bengals

5

6

6

8

6. Packers

6

3

2

5

7. Saints

8

5

7

6

8. Jets

4

1

3

2



[Page 253]

Determine the two games each person should get tickets for that will result in the groups' greatest degree of satisfaction. Do you think the participants would think your allocation is fair?

35.

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 food stores and sold through catalogs. The shipping costs ($/1,000 lb.) from the European ports to the U.S. cities and the available supplies (1,000 lb.) at the European ports are provided in the following table:

 

U.S. City

 

European Port

4. Norfolk

5. New York

6. Savannah

Supply

1. Hamburg

$420

$390

$610

55

2. Marseilles

510

590

470

78

3. Liverpool

450

360

480

37


The transportation costs ($/1,000 lb.) from each U.S. city of the three distribution centers and the demands (1,000 lb.) at the distribution centers are as follows:

 

Distribution Center

Warehouse

7. Dallas

8. St. Louis

9. Chicago

4. Norfolk

$75

$63

$81

5. New York

125

110

95

6. Savannah

68

82

95

 

60

45

50


Determine the optimal shipments between the European ports and the warehouses and the distribution centers to minimize total transportation costs.

36.

A sports apparel company has received an order for a college basketball team's national championship T-shirt. The company can purchase the T-shirts from textile factories in Mexico, Puerto Rico, and Haiti. The shirts are shipped from the factories to companies in the United States that silk-screen the shirts before they are shipped to distribution centers. Following are the production and transportation costs ($/shirt) from the T-shirt factories to the silk-screen companies to the distribution centers, plus the supply of T-shirts at the factories and demand for the shirts at the distribution centers:

 

Silk-screen Company

 

T-shirt Factory

4. Miami

5. Atlanta

6. Houston

Supply (1,000s)

1. Mexico

$4

$6

$3

18

2. Puerto Rico

3

5

5

15

3. Haiti

2

4

4

23



[Page 254]
 

Distribution Center

Silk-screen Company

7. New York

8. St. Louis

9. Los Angeles

4. Miami

$5

$7

$9

5. Atlanta

7

6

10

6. Houston

8

6

8

Demand (1,000s)

20

12

20


Determine the optimal shipments to minimize total production and transportation costs for the apparel company.

37.

Walsh's Fruit Company contracts with growers in Ohio, Pennsylvania, and New York to purchase grapes. The grapes are processed into juice at the farms and stored in refrigerated vats. Then the juice is shipped to two plants, where it is processed into bottled grape juice and frozen concentrate. The juice and concentrate are then transported to three food warehouses/distribution centers. The transportation costs per ton from the farms to the plants and from the plants to the distributors, and the supply at the farms and demand at the distribution centers are summarized in the following tables:

 

Plant

 

Farm

4. Indiana

5. Georgia

Supply (1,000 tons)

1. Ohio

$16

21

72

2. Pennsylvania

18

16

105

3. New York

22

25

83


 

Distribution Center

Plant

6. Virginia

7. Kentucky

8. Louisiana

4. Indiana

$23

$15

$29

5. Georgia

20

17

24

Demand (1,000 tons)

90

80

120


  1. Determine the optimal shipments from farms to plants to distribution centers to minimize total transportation costs.

  2. What would be the effect on the solution if the capacity at each plant were 140,000 tons?

38.

A national catalog and Internet retailer has three warehouses and three major distribution centers located around the country. Normally, items are shipped directly from the warehouses to the distribution centers; however, each of the distribution centers can also be used as an intermediate transshipment point. The transportation costs ($/unit) between warehouses and distribution centers, the supply at the warehouses (100 units), and the demand at the distribution centers (100 units) for a specific week are shown in the following table:


[Page 255]
 

Distribution Center

 

Warehouse

A

B

C

Supply

1

$12

$11

$7

70

2

8

6

14

80

3

9

10

12

50

Demand

60

100

40

 

The transportation costs ($/unit) between the distribution centers are

 

Distribution Center

Distribution Center

A

B

C

A

$

$8

$3

B

1

2

C

7

2


Determine the optimal shipments between warehouses and distribution centers to minimize total transportation costs.

39.

Horizon Computers manufactures laptops in Germany, Belgium, and Italy. Because of high tariffs between international trade groups, it is sometimes cheaper to ship partially completed laptops to factories in Puerto Rico, Mexico, and Panama and have them completed before final shipment to U.S. distributors in Texas, Virginia, and Ohio. The cost ($/unit) of the completed laptops plus tariffs and shipment costs from the European plants directly to the United States and supply and demand are as follows:

 

U.S. Distributor

 

European Plant

7. Texas

8. Virginia

9. Ohio

Supply (1,000s)

1. Germany

$2,600

$1,900

$2,300

5.2

2. Belgium

2,200

2,100

2,600

6.3

3. Italy

1,800

2,200

2,500

4.5

Demand (1,000s)

2.1

3.7

7.8

 

Alternatively, the unit costs of shipping partially completed laptops to plants for finishing before sending them to the United States are as follows:

 

Factory

European Plant

4. Puerto Rico

5. Mexico

6. Panama

1. Germany

$1,400

$1,200

$1,100

2. Belgium

1,600

1,100

900

3. Italy

1,500

1,400

1,200



[Page 256]
 

U.S. Distributor

Factory

7. Texas

8. Virginia

9. Ohio

4. Puerto Rico

$800

$700

$900

5. Mexico

600

800

1,100

6. Panama

900

700

1,200


Determine the optimal shipments of laptops that will meet demand at the U.S. distributors at the minimum total cost.

40.

The Midlands Field Produce Company contracts with potato farmers in Colorado, Minnesota, North Dakota, and Wisconsin for monthly potato shipments. Midlands picks up the potatoes at the farms and ships mostly by truck (and sometimes by rail) to its sorting and distribution centers in Ohio, Missouri, and Iowa. At these centers the potatoes are cleaned, rejects are discarded, and the potatoes are sorted according to size and quality. They are then shipped to combination plants and distribution centers in Virginia, Pennsylvania, Georgia, and Texas, where the company produces a variety of potato products and distributes bags of potatoes to stores. Exceptions are the Ohio distribution center, which will only accept potatoes from farms in Minnesota, North Dakota, and Wisconsin, and the Texas plant, which won't accept shipments from Ohio because of disagreements over delivery schedules and quality issues. Following are summaries of the shipping costs from the farms to the distribution centers and the processing and shipping costs from the distribution centers to the plants, as well as the available monthly supply at each farm, the processing capacity at the distribution centers, and the final demand at the plants (in bushels):

 

Distribution Center ($/bushel)

Farm

5. Ohio

6. Missouri

7. Iowa

Supply (bushels)

1. Colorado

$

$1.09

$1.26

1,600

2. Minnesota

0.89

1.32

1.17

1,100

3. North Dakota

0.78

1.22

1.36

1,400

4. Wisconsin

1.19

1.25

1.42

1,900

Processing Capacity (bushels)

1,800

2,200

1,600

 

 

Plant ($/bushel)

Distribution Center

8. Virginia

9. Pennsylvania

10. Georgia

11. Texas

5. Ohio

$4.56

$3.98

$4.94

$

6. Missouri

3.43

5.74

4.65

5.01

7. Iowa

5.39

6.35

5.70

4.87

Demand (bushels)

1,200

900

1,100

1,500


Formulate and solve a linear programming model to determine the optimal monthly shipments from the farms to the distribution centers and from the distribution centers to the plants to minimize total shipping and processing costs.

41.

KanTech Corporation is a global distributor of electrical parts and components. Its customers are electronics companies in the United States, including computer manufacturers and audio/visual product manufacturers. The company contracts to purchase components and parts from manufacturers in Russia, Eastern and Western Europe, and the Mediterranean, and it has them delivered to warehouses in three European ports, Gdansk, Hamburg, and Lisbon. The various components and parts are loaded into containers based on demand from U.S. customers. Each port has a limited fixed number of containers available each month. The containers are then shipped overseas by container ships to the ports of Norfolk, Jacksonville, New Orleans, and Galveston. From these seaports, the containers are typically coupled with trucks and hauled to inland ports in Front Royal (Virginia), Kansas City, and Dallas. There are a fixed number of freight haulers available at each port each month. These inland ports are sometimes called "freight villages," or intermodal junctions, where the containers are collected and transferred from one transport mode to another (i.e., from truck to rail or vice versa). From the inland ports, the containers are transported to KanTech's distribution centers in Tucson, Pittsburgh, Denver, Nashville, and Cleveland. Following are the handling and shipping costs ($/container) between each of the embarkation and destination points along this overseas supply chain and the available containers at each port:


[Page 257]
 

U.S. Port

 

European Port

4. Norfolk

5. Jacksonville

6. New Orleans

7. Galveston

Available Containers

1. Gdansk

$1,725

$1,800

$2,345

$2,700

125

2. Hamburg

1,825

1,750

1,945

2,320

210

3. Lisbon

2,060

2,175

2,050

2,475

160


 

Inland Port

 

U.S. Port

8. Dallas

9. Kansas City

10. Front Royal

Intermodal Capacity (containers)

4. Norfolk

$825

$545

$ 320

85

5. Jacksonville

750

675

450

110

6. New Orleans

325

605

690

100

7. Galveston

270

510

1,050

130

Intermodal Capacity (containers)

170

240

140

 

 

Distibution Center

Inland Port

11. Tucson

12. Denver

13. Pittsburgh

14. Nashville

15. Cleveland

8. Dallas

$450

$830

$565

$420

$960

9. Kansas City

880

520

450

380

660

10. Front Royal

1,350

390

1,200

450

310

Demand

85

60

105

50

120


Formulate and solve a linear programming model to determine the optimal shipments from each point of embarkation to each destination along this supply chain that will result in the minimum total shipping cost.

42.

In Problem 41, KanTech Corporation is just as concerned that its U.S. distributors receive shipments in the minimum amount of time as they are about minimizing their shipping costs. Suppose that each U.S. distributor receives one major container shipment each month. Following are summaries of the shipping times (in days) between each of the embarkation and destination points along KanTech's overseas supply chain. These times not only encompass travel time but also processing, loading, and unloading times at each port:


[Page 258]
 

U.S. Port

European Port

4. Norfolk

5. Jacksonville

6. New Orleans

7. Galveston

1. Gdansk

22

24

27

30

2. Hamburg

17

20

23

26

3. Lisbon

25

21

24

26


 

Inland Port

U.S. Port

8. Dallas

9. Kansas City

10. Front Royal

4. Norfolk

10

8

6

5. Jacksonville

12

9

8

6. New Orleans

8

7

10

7. Galveston

12

6

8


 

Distribution Center

Inland Port

11. Tucson

12. Denver

13. Pittsburgh

14. Nashville

15. Cleveland

8. Dallas

5

6

5

7

8

9. Kansas City

6

4

4

5

7

10. Front Royal

10

5

7

4

6


  1. Formulate and solve a linear programming model to determine the optimal shipping route for each distribution center along this supply chain that will result in the minimum total shipping time. Determine the shipping route and time for each U.S. distributor.

  2. Suppose the European ports can accommodate only three shipments each. How will this affect the solution in part (a)?

43.

Solve the following linear programming problem:

minimize Z

=

18 x 11 + 30 x 12 + 20 x 13 + 18 x 14 + 25 x 21 + 27 x 22 + 22 x 23

   

+ 16 x 24 + 30 x 31 + 26 x 32 + 19 x 33 + 32 x 34 + 40 x 41 + 36 x 42

   

+ 27 x 43 + 29 x 44 + 30 x 51 + 26 x 52 + 18 x 53 + 24 x 54

subject to

   
   



[Page 259]
44.

A plant has four operators to be assigned to four machines. The time (minutes) required by each worker to produce a product on each machine is shown in the following table:

 

Machine (min.)

Operator

A

B

C

D

1

10

12

9

11

2

5

10

7

8

3

12

14

13

11

4

8

15

11

9


Determine the optimal assignment and compute total minimum time.

45.

A shop has four machinists to be assigned to four machines. The hourly cost of having each machine operated by each machinist is as follows:

 

Machine (cost/hr.)

Machinist

A

B

C

D

1

$12

$11

$8

$14

2

10

9

10

8

3

14

8

7

11

4

6

8

10

9


However, because he does not have enough experience, machinist 3 cannot operate machine B.

  1. Determine the optimal assignment and compute total minimum cost.

  2. Formulate this problem as a general linear programming model.

46.

The Omega pharmaceutical firm has five salespersons, whom the firm wants to assign to five sales regions. Given their various previous contacts, the salespersons 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:

 

Region (days)

Salesperson

A

B

C

D

E

1

17

10

15

16

20

2

12

9

16

9

14

3

11

16

14

15

12

4

14

10

10

18

17

5

13

12

9

15

11


Which salesperson should be assigned to each region to minimize total time? Identify the optimal assignments and compute total minimum time.


[Page 260]
47.

The Bunker Manufacturing firm has five employees and six machines and wants to assign the employees to the machines to minimize cost. A cost table showing the cost incurred by each employee on each machine follows:

 

Machine

Employee

A

B

C

D

E

F

1

$12

$7

$20

$14

$8

$10

2

10

14

13

20

9

11

3

5

3

6

9

7

10

4

9

11

7

16

9

10

5

10

6

14

8

10

12


Because of union rules regarding departmental transfers, employee 3 cannot be assigned to machine E, and employee 4 cannot be assigned to machine B. Solve this problem, indicate the optimal assignment, and compute total minimum cost.

48.

Given the following cost table for an assignment problem, determine the optimal assignment and compute total minimum cost:

 

Machine

Operator

A

B

C

D

1

$10

$2

$8

$6

2

9

5

11

9

3

12

7

14

14

4

3

1

4

2


49.

An electronics firm produces electronic components, which it supplies to various electrical manufacturers. Quality control records indicate that different employees produce different numbers of defective items. The average number of defects produced by each employee for each of six components is given in the following table:

 

Component

Employee

A

B

C

D

E

F

1

30

24

16

26

30

22

2

22

28

14

30

20

13

3

18

16

25

14

12

22

4

14

22

18

23

21

30

5

25

18

14

16

16

28

6

32

14

10

14

18

20


Determine the optimal assignment that will minimize the total average number of defects produced by the firm per month.


[Page 261]
50.

A dispatcher for Citywide Taxi Company has six taxicabs at different locations and five customers who have called for service. The mileage from each taxi's present location to each customer is shown in the following table:

 

Customer

Cab

1

2

3

4

5

A

7

2

4

10

7

B

5

1

5

6

6

C

8

7

6

5

5

D

2

5

2

4

5

E

3

3

5

8

4

F

6

2

4

3

4


Determine the optimal assignment(s) that will minimize the total mileage traveled.

51.

The Southeastern Conference has nine basketball officials who must be assigned to three conference games, three to each game. The conference office wants to assign the officials so that the total distance they travel will be minimized. The distance (in miles) each official would travel to each game is given in the following table:

 

Game

Official

Athens

Columbia

Nashville

1

165

90

130

2

75

210

320

3

180

170

140

4

220

80

60

5

410

140

80

6

150

170

190

7

170

110

150

8

105

125

160

9

240

200

155


Determine the optimal assignment(s) to minimize the total distance traveled by the officials.

52.

In Problem 51, officials 2 and 8 have had a recent confrontation with one of the coaches in the game in Athens. They were forced to eject the coach after several technical fouls. The conference office has decided that it would not be a good idea to have these two officials work the Athens game so soon after this confrontation, so they have decided that officials 2 and 8 will not be assigned to the Athens game. How will this affect the optimal solution to this problem?

53.

State University has planned six special catered events for the Saturday of its homecoming football game. The events include an alumni brunch, a parents' brunch, a booster club luncheon, a postgame party for season ticket holders, a lettermen's dinner, and a fund-raising dinner for major contributors. The university wants to use local catering firms as well as the university catering service to cater these events, and it has asked the caterers to bid on each event. The bids (in thousands of dollars) based on menu guidelines for the events prepared by the university are shown in the following table:


[Page 262]
 

Event

Caterer

Alumni Brunch

Parents' Brunch

Booster Club Lunch

Postgame Party

Lettermen's Dinner

Contributors' Dinner

Al's

$12.6

$10.3

$14.0

$19.5

$25.0

$30.0

Bon Apet ­t

14.5

13.0

16.5

17.0

22.5

32.0

Custom

13.0

14.0

17.6

21.5

23.0

35.0

Divine

11.5

12.6

13.0

18.7

26.2

33.5

Epicurean

10.8

11.9

12.9

17.5

21.9

28.5

Fouch ss

13.5

13.5

15.5

22.3

24.5

36.0

University

12.5

14.3

16.0

22.0

26.7

34.0


The Bon Apet ­t, Custom, and University caterers can handle two events, whereas each of the other four caterers can handle only one. The university is confident that all the caterers will do a high-quality job, so it wants to select the caterers for the events that will result in the lowest total cost.

Determine the optimal selection of caterers to minimize total cost.

54.

A university department head has five instructors to be assigned to four different courses. All the instructors have taught the courses in the past and have been evaluated by the students. The rating for each instructor for each course is given in the following table (a perfect score is 100):

 

Course

Instructor

A

B

C

D

1

80

75

90

85

2

95

90

90

97

3

85

95

88

91

4

93

91

80

84

5

91

92

93

88


The department head wants to know the optimal assignment of instructors to courses to maximize the overall average evaluation. The instructor who is not assigned to teach a course will be assigned to grade exams.

55.

The coach of the women's swim team at State University is preparing for the conference swim meet and must choose the four swimmers she will assign to the 800-meter medley relay team. The medley relay consists of four strokesbackstroke, breaststroke, butterfly , and freestyle. The coach has computed the average times (in minutes) each of her top six swimmers has achieved in each of the four strokes for 200 meters in previous swim meets during the season, as follows:


[Page 263]
 

Stroke (min.)

Swimmer

Backstroke

Breaststroke

Butterfly

Freestyle

Annie

2.56

3.07

2.90

2.26

Beth

2.63

3.01

3.12

2.35

Carla

2.71

2.95

2.96

2.29

Debbie

2.60

2.87

3.08

2.41

Erin

2.68

2.97

3.16

2.25

Fay

2.75

3.10

2.93

2.38


Determine for the coach the medley relay team and its total expected relay time.

56.

Biggio's Department Store has six employees available to assign to four departments in the storehome furnishings, china, appliances, and jewelry . Most of the six employees have worked in each of the four departments on several occasions in the past and have demonstrated that they perform better in some departments than in others. The average daily sales for each of the six employees in each of the four departments are shown in the following table:

 

Department Sales

Employee

Home Furnishings

China

Appliances

Jewelry

1

$340

$160

$610

$290

2

560

370

520

450

3

270

350

420

4

360

220

630

150

5

450

190

570

310

6

280

320

490

360


Employee 3 has not worked in the china department before, so the manager does not want to assign this employee to china.

Determine which employee to assign to each department and indicate the total expected daily sales.

57.

The Vanguard Publishing Company hires eight college students as salespeople to sell encyclopedias during the summer. The company desires to allocate them to three sales territories. Territory 1 requires three salespeople, and territories 2 and 3 require two salespeople each. It is estimated that each salesperson will be able to generate the amounts of dollar sales per day in each of the three territories as given in the following table:

 

Territory

Salesperson

1

2

3

A

$110

$150

$130

B

90

120

80

C

205

160

175

D

125

100

115

E

140

105

150

F

100

140

120

G

180

210

160

H

110

120

90


Help the company allocate the salespeople to the three territories so that sales will be maximized.


[Page 264]
58.

Carolina Airlines, a small commuter airline in North Carolina, has six flight attendants that it wants to assign to six monthly flight schedules in a way that will minimize the number of nights they will be away from their homes . The numbers of nights each attendant must be away from home with each schedule are given in the following table:

 

Schedule

Attendant

A

B

C

D

E

F

1

7

4

6

10

5

8

2

4

5

5

12

7

6

3

9

9

11

7

10

8

4

11

6

8

5

9

10

5

5

8

6

10

7

6

6

10

12

11

9

9

10


Identify the optimal assignments that will minimize the total number of nights the attendants will be away from home.

59.

The football coaching staff at Tech focuses its recruiting on several key states, including Georgia, Florida, Virginia, Pennsylvania, New York, and New Jersey. The staff includes seven assistant coaches, two of whom are responsible for Florida, a high school talent-rich state, whereas one coach is assigned to each of the other five states. The staff has been together for a long time and at one time or another, all the coaches have recruited in all the states. The head coach has accumulated some data on the past success rate (i.e., percentage of targeted recruits signed) for each coach in each state, as shown in the following table:

 

State

Coach

GA

FL

VA

PA

NY

NJ

Allen

62

56

65

71

55

63

Bush

65

70

63

81

75

72

Crumb

46

53

62

55

64

50

Doyle

58

66

70

67

71

49

Evans

77

73

69

80

80

74

Fouch

68

73

72

80

78

57

Goins

72

60

74

72

62

61


Determine the optimal assignment of coaches to recruiting regions that will maximize the overall success rate and indicate the average percentage success rate for the staff with this assignment.

60.

Kathleen Taylor is a freshman at Roanoke College, and she wants to develop her schedule for the spring semester. Courses are offered with class periods either on Monday and Wednesday or Tuesday and Thursday for 1 hour and 15 minutes duration, with 15 minutes between class periods. For example, a course designated as 8M meets on Monday and Wednesday from 8:00 A.M. to 9:15 A.M. ; the next class on Monday and Wednesday (9M) meets from 9:30 to 10:45; the next class (11M) is from 11:00 A.M. to 12:15 P.M. ; and so on. Kathleen wants to take the following six freshman courses, with the available sections shown in order of her preference, based on the professor who's teaching the course and the time:


[Page 265]

Course

Sections Available

Math

11T, 12T, 9T, 11M, 12M, 9M, 8T, 8M

History

11T, 11M, 14T, 14M, 8T, 8M

English

9T, 11T, 14T, 11M, 12T, 14M, 12M, 9M

Biology

14T, 11M, 12M, 14M, 9M, 8T, 8M

Spanish

9T, 11M, 12M, 8T

Psychology

14T, 11T, 12T, 9T, 14M, 8M


For example, there are eight sections of math offered, and Kathleen's first preference is the 11T section, her second choice is the 12T section, and so forth.

  1. Determine a class schedule for Kathleen that most closely meets her preferences.

  2. Determine a class schedule for Kathleen if she wants to leave 11:00 A.M. to noon open for lunch every day.

  3. Suppose Kathleen wants all her classes on two days, either Monday and Wednesday or Tuesday and Thursday. Determine schedules for each and indicate which most closely matches her preferences.

61.

CareMed, an HMO health care provider, operates a 24-hour outpatient clinic in Draperton, near the Tech campus. The facility has a medical staff with doctors and nurses who see regular local patients according to a daily appointment schedule. However, the clinic sees a number of Tech students who visit the clinic each day and evening without appointments because their families are part of the CareMed network. The clinic has 12 nurses who work according to three 8-hour shifts. Five nurses are needed from 8:00 A.M. to 4:00 P.M. , four nurses work from 4:00 P.M. to midnight, and 3 nurses work overnight from midnight to 8:00 A.M. The clinic administrator wants to assign nurses to a shift according to their preferences and seniority (i.e., when the number of nurses who prefer a shift exceed the shift demand, the nurses are assigned according to seniority ). While the majority of nurses prefer the day shift, some prefer other shifts because of the work and school schedules of their spouses and families. Following are the nurses' shift preferences (where 1 is most preferred) and their years working at the clinic:

 

Shift

Nurse

8 A.M. to 4 P.M.

4 P.M. to Midnight

Midnight to 8 A.M.

Years' Experience

Adams

1

2

3

2

Baxter

1

3

2

5

Collins

1

2

3

7

Davis

3

1

2

1

Evans

1

3

2

3

Forrest

1

2

3

4

Gomez

2

1

3

1

Huang

3

2

1

1

Inchio

1

3

2

2

Jones

2

1

3

3

King

1

3

2

5

Lopez

2

3

1

2


Formulate and solve a linear programming model to assign the nurses to shifts according to their preferences and seniority.




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

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