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Joy company currently maintains plant in Gazipur, Mymensing and Faridpur that supply major distribution centers in Dhaka, Rajshahi, Comilla and Chittagong. Because of an expanding demand, Joy has decided to open a forth plant and has narrowed the choice to one of two areas – Feni or Savar. The pertinent production and distribution costs, as well as the plant capacities and distribution demands, are shown in the accompanying table: Distribution

Plant Gazipur Mymensing Faridpur Feni (Proposed) Savar (Proposed)

Forecasted Demand

Dhaka

Rajshahi

Comilla

Chittagong

Capacity

Unit Production cost

Tk. 25 35 36 60 35

Tk. 55

Tk. 60

15000

Tk. 48

30 45 38 30

Tk. 40 50 26 65 41

40 66 27 50

6000 14000 11000 11000

50 52 53 49

10000

12000

15000

9000

Which of the new possible plant should be opened?

The fancy group owns factories in four towns (W, X, which distribute to three Fancy retail dress shops (in C). Factory availabilities, projected stores demands, shipping cost are summarized in the table that follows no cost data implies impossible distribution): To From W X Y Z Store Demand

Y and Z) A, B and and unit (cell with

A

B

C

Factory availability

Tk. 8 5 7 ---110

---6 9 3 34

Tk. 4 8 ---7 31

72 38 46 19

Use the Vogel Approximation method to establish an initial feasible solution and calculate its cost. Is your solution optimal? Apply MODI technique or Stepping Stone method to determine the optimality. 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

June

320

340

Determine a production schedule for PM that will minimize total cost. In Previous Problem, 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

June

380

500

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. 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.

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

Weekly Production (tons)

A. Bethlehem

150

B. Birmingham

210

C. Gary

320

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

Shipping costs per ton of steel are as follows:  

To (cost) 3

 

From

1

2

4

A

$14

$ 9

B

11

8

7

16

C

16

12

10

22

$16 $18

Because of a truckers' strike, shipments are prohibited from Birmingham to Chicago.

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:   European Plant

U.S. Distributor 7. Texas

8. Virginia

 

9. Ohio Supply (1,000s)

1. Germany

$2,600

2. Belgium

2,200

2,100

2,600

6.3

3. Italy

1,800

2,200

2,500

4.5

2.1

3.7

Demand (1,000s)

$1,900 $2,300

5.2

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

 

U.S. Distributor 7. Texas

8. Virginia

9. Ohio

4. Puerto Rico

$800

$700

$900

5. Mexico

600

800

1,100

6. Panama

900

700

1,200

Factory

1. Draw a network of the problem. Formulate a linear programming to determine the optimal shipments of laptops that will meet demand at the U.S. distributors at the minimum total cost.

A rental car company has an imbalance of cars at seven of its locations. The following network shows the locations of concern (the nodes) and the cost to move a car between locations. A positive number by a node indicates an excess supply at the node, and a negative number indicates an excess demand.

Develop a linear programming model for restoring the proper balance at the locations.

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

 

Formulate a linear programming to determine the optimal shipments between the European ports and the warehouses and the distribution centers to minimize total transportation costs.

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:  

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

C

7

2 2

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

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:  

U.S. Port

European Port

4. Norfolk

5. Jacksonville

6. New 7. Orleans 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

2,175

2,050

2,475

160

2,060

 

 

Inland Port

U.S. Port

8. Dallas

9. Kansas 10. Front Intermodal City Royal (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 (containers)

 

Capacity

  170

240

140

Capacity

 

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. Royal

390

1,200

450

310

60

105

50

120

Demand

Front 1,350 85

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.

Case Problem GLOBAL SHIPPING AT ERKEN APPAREL INTERNATIONAL Erken Apparel International manufactures clothing items around the world. It has currently contracted with a U.S. retail clothing wholesale distributor for men's goatskin and lambskin leather jackets for the next Christmas season. The distributor has distribution centers in Indiana, North Carolina, and Pennsylvania. The distributor supplies the leather jackets to a discount retail chain, a chain of mall boutique stores, and a department store chain. The jackets arrive at the distribution centers unfinished, and at the centers the distributor adds a unique lining and label specific to each of its customer. The distributor has contracted with Erken to deliver the following number of leather jackets to its distribution centers in late fall:

Distribution Center Goatskin Jackets Lambskin Jackets Indiana

1,000

780

North Carolina

1,400

950

Pennsylvania

1,600

1,150

Erken has tanning factories and clothing manufacturing plants to produce leather jackets in Spain, France, Italy, Venezuela, and Brazil. Its tanning facilities are in Mende in France, Foggia in Italy, Saragosa in Spain, Feira in Brazil, and El Tigre in Venezuela. Its manufacturing plants are in Limoges, Naples, and Madrid in Europe and in Sao Paulo and Caracas in South America. Following are the supplies of available leather from each tanning facility and the processing capacity at each plant (in pounds) for this particular order of leather jackets:

Tanning Factory Goatskin Supply (lb.) Lambskin Supply (lb.) Mende

4,000

4,400

Foggia

3,700

5,300

Saragosa

6,500

4,650

Feira

5,100

6,850

El Tigre

3,600

5,700

Plant

Production Capacity (lb.)

Madrid

7,800

Naples

5,700

Limoges

8,200

Sao Paulo 7,600 Caracas

6,800

In the production of jackets at the plants, 37.5% of the goatskin leather and 50% of the lambskin leather is waste (i.e., it is discarded during the production process and sold for other byproducts). After production, a goatskin jacket weighs approximately 3 pounds, and a lambskin jacket weighs approximately 2.5 pounds (neither with linings, which are added in the United States). Following are the costs per pound, in U.S. dollars, for tanning the uncut leather, shipping it, and producing the leather jackets at each plant:

 

Plant ($/lb.)

Tanning Factory Madrid Naples Limoges Sao Paulo Caracas Mende

$24

$22

$16

$21

$23

Foggia

31

17

22

19

22

Saragosa

18

25

28

23

25

Feira

16

18

El Tigre

14

15

Note that the cost of jacket production is the same for goatskin and lambskin. Also, leather can be tanned in France, Spain, and Italy and shipped directly to the South American plants for jacket production, but the opposite is not possible due to high tariff restrictions (i.e., tanned leather is not shipped to Europe for production). Once the leather jackets are produced at the plants in Europe and South America, Erken transports them to ports in Lisbon, Marseilles, and Caracas and then from these ports to U.S ports in New Orleans, Jacksonville, and Savannah. The available shipping capacity at each port and the transportation costs from the plants to the ports are as follows:

 

Port ($/lb.)

Plant

Lisbon Marseilles Caracas

Madrid

0.75

1.05

Naples

3.45

1.35

Limoges

2.25

0.60

Sao Paulo

1.15

Caracas

0.20

 

Port ($/lb.)

Capacity (lb.) 8,000 5,500

9,000

The shipping costs ($/lb.) from each port in Europe and South America to the U.S. ports and the available truck and rail capacity for transport at the U.S. ports are as follows:

 

U.S. Port ($/lb.)

Port

New Orleans Jacksonville Savannah

Lisbon

$2.35

$1.90

$1.80

Marseilles

3.10

2.40

2.00

Caracas

1.95

2.15

2.40

5,200

7,500

Capacity (lb.) 8,000

The transportation costs ($/lb.) from the U.S. ports to the three distribution centers are as follows:

 

Distribution Center ($/lb.)

U.S. Port

Indiana North Carolina Pennsylvania

New Orleans $0.65

$0.52

$0.87

Jacksonville

0.43

0.41

0.65

Savannah

0.38

0.34

0.50

Erken wants to determine the least costly shipments of material and jackets that will meet the demand at the U.S. distribution centers. Develop a transshipment model for Erken that will result in a minimum cost solution.