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Chapter 10 Transportation and Assignment Models

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

10-1

2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Learning ObjectivesStudents will be able to Structure special LP problems using the transportation and assignment models. Use the N.W. corner, VAM, MODI, and stepping-stone method. Solve facility location and other application problems with transportation methods. Solve assignment problems with the Hungarian (matrix reduction) methodTo accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

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Chapter Outline10.1 Introduction 10.2 Setting Up a Transportation Problem 10.2 Developing an Initial Solution:Northwest Corner Rule 10.4 Stepping-Stone Method: Finding a Least-Cost Solution 10.5 MODI Method 10.6 Vogels Approximation Method 10.7 Unbalanced Transportation ProblemsTo accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

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Chapter Outline continued10.8 Degeneracy in Transportation Problems 10.9 More Than One Optimal Solution 10.10 Maximization Transportation Problems 10.11 Unacceptable or Prohibited Routes 10.12 Facility Location Analysis 10.13 Approach of the Assignment Model 10.14 Unbalanced Assignment Models 10.15 Maximization Assignment ProblemsTo accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

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Specialized Problems Transportation Problem Distribution of items from several sources to several destinations. Supply capacities and destination requirements known.

Assignment Problem One-to-one assignment of people to jobs, etc.

Specialized algorithms save time!


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Importance of Special Purpose Algorithms Fewer, less complicated, computations than with simplex Less computer memory required Produce integer solutions


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Transportation ProblemDes Moines (100 units) capacity Cleveland (200 units) required

Albuquerque (300 units) required

Evansville (300 units) capacity

Boston (200 units) required

Ft. Lauderdale (300 units) capacity


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Transportation CostsTo (Destinations)

From (Sources)

Albuquerque Cleveland Boston Des Moines $5 $4 $3 Evansville Fort Lauderdale $8 $9 $4 $7 $3 $5


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Unit Shipping Cost:1Unit, Factory to WarehouseAlbuquerque Cleveland Boston Factory (A) (C) (B) Capacity 5 4 3 Des Moines (D) Evansville (E) Fort Lauderdale (F) Warehouse Req. 8 4 3

9

7

5


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Total Demand and Total SupplyAlbuquerque Boston (A) (B) Des Moines (D) Evansville (E) Fort Lauderdale (F) Warehouse Req. 300 200 200 Cleveland Factory (C) Capacity 100

300

300

700


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Transportation Table For Executive Furniture Corp.Albuquerque Boston (A) (B) Des Moines (D) Evansville (E) Fort Lauderdale (F) Warehouse Req. 300 5 4 Cleveland Factory (C) Capacity 3 100

8

4

3

300

9

7

5

300

200

200

700


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Initial Solution Using the Northwest Corner Rule Start in the upper left-hand cell and allocate units to shipping routes as follows: Exhaust the supply (factory capacity) of each row before moving down to the next row. Exhaust the demand (warehouse) requirements of each column before moving to the next column to the right. Check that all supply and demand requirements are met.


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Initial Solution North West Corner RuleAlbuquerque Boston (A) (B) Des Moines (D) Evansville (E) Fort Lauderdale (F) Warehouse Req. 300 5 100 8 4 3 4 Cleveland Factory (C) Capacity 3 100

200

100

300

9 100

7

200

5

300

200

200

700


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The Stepping-Stone Method 1. Select any unused square to evaluate. 2. Begin at this square. Trace a closed path back to the original square via squares that are currently being used (only horizontal or vertical moves allowed). 3. Place + in unused square; alternate - and + on each corner square of the closed path. 4. Calculate improvement index: add together the unit cost figures found in each square containing a +; subtract the unit cost figure in each square containing a -. 5. Repeat steps 1 - 4 for each unused square.To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

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Stepping-Stone Method The Des Moines-toCleveland RouteAlbuquerque Boston (A) (B) Des Moines (D) Evansville (E) Fort Lauderdale (F) Warehouse Req. 300 100 5 4 Cleveland Factory (C) Capacity Start 3 100

-

+8 4 3 100 7 5

200

300

+9

+

100

200

300

200 200 700


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Stepping-Stone Method An Improved SolutionAlbuquerque Boston (A) (B) Des Moines (D) Evansville (E) Fort Lauderdale (F) Warehouse Req. 100 5 4 Cleveland Factory (C) Capacity 3 100

8 100 9 100 200

4

3

300

7

200

5

300

300

200

200

700


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Third and Final SolutionAlbuquerque Boston (A) (B) Des Moines (D) Evansville (E) Ft Lauderdale (F) Warehouse Req. 200 100 5 4 Cleveland Factory (C) Capacity 3 100

8 200 9

4 100 7

3

300

100

5

300

300

200

200

700


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MODI Method: 5 Steps1. Compute the values for each row and column: set Ri + Kj = Cij for those squares currently used or occupied. 2. After writing all equations, set R1 = 0. 3. Solve the system of equations for Ri and Kj values. 4. Compute the improvement index for each unused square by the formula improvement index: Cij - Ri - Kj 5. Select the largest negative index and proceed to solve the problem as you did using the stepping-stone method. 10-18To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Vogels Approximation1. For each row/column of table, find difference between two lowest costs. (Opportunity cost) 2. Find greatest opportunity cost. 3. Assign as many units as possible to lowest cost square in row/column with greatest opportunity cost. 4. Eliminate row or column which has been completely satisfied. 4. Begin again, omitting eliminated rows/columns.To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

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Special Problems in Transportation Method Unbalanced Problem Demand Less than Supply Demand Greater than Supply

Degeneracy More Than One Optimal Solution


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Unbalanced Problem Demand Less than SupplyCustomer Dummy Customer Factory 1 2 Capacity 8 5 0 Factory 1 170 Factory 2 15 10 0

130

Factory 3

3

9

0

80

Customer Requirements

150

80

150

380


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Unbalanced Problem Supply Less than DemandCustomer Customer Customer Factory 2 1 3 Capacity Factory 1 8 5 16 170

Factory 2

15

10

7

130

Dummy

0

0

0

80

Customer Requirements 150

80

150

380


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DegeneracyCustomer Customer Customer Factory 2 1 3 Capacity Factory 1 100 5 4 3 100

Factory 2

8 100 9

4 20 7

3

120

Factory 3

80

5

80

Customer Requirements 100

100

100

300


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Degeneracy - Coming Up!Customer Customer Customer Factory 2 1 3 Capacity Factory 1 70 8 5 16 70

Factory 2

15 50 3 30 80

10

7

130

Factory 3

9

50

10

80

Customer Requirements

150

80

50

280


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Stepping-Stone Method The Des Moines-toCleveland RouteAlbuquerque Boston (A) (B) Des Moines (D) Evansville (E) Fort Lauderdale (F) Warehouse Req. 300 100 5 Cleveland Factory (C) Capacity 4 Start 3 100

-

+8 4 3 100 7 5

200

300

+9

+

100

200

300

200 200 700


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The Assignment Problem

Person Adams Brown Cooper

1 $11 $8 $9

Project 2 $14 $10 $12

3 $6 $11 $7


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The Assignment Method1. subtract the smallest number in each row from every number in that row subtract the smallest number in each column from every number in that column 2. draw the minimum number of vertical and horizontal straight lines necessary to cover zeros in the table if the number of lines equals the number of rows or columns, then one can make an optimal 10-27 assignment (step 4)To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

The Assignment Method - continued3. if the number of lines does not equal the number of rows or columns subtract the smallest number not covered by a line from every other uncovered number add the same number to any number lying at the intersection of any two lines return to step 2 4. make optimal assignments at locations of zeros within the tableTo accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

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PG 10.13b


Hungarian MethodInitial Table

Person 1 Adams Brown Cooper 11 8 9

Project 2 14 10 12 3 6 11 7


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Hungarian MethodRow Reduction

Person 1 Adams Brown Cooper 5 0 2

Project 2 8 2 5 3 0 3 0


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Hungarian MethodColumn Reduction

Person 1 Adams Brown CooperTo accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

Project 2 6 0 310-31

3 0 3 0 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

5 0 2

Hungarian MethodTesting

Person 1 Adams Brown Cooper

Project 2

Covering Line 2

3

5 0 210-32

6 0 3

0 3 0 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Covering Line 1


Hungarian MethodRevised Opportunity Cost Table Person 1 Adams Brown Cooper 3 0 0 Project 2 4 0 1 3 0 5 0


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Hungarian MethodTesting PersonCovering Covering Line 1 Project Line 3

1 Adams Brown Cooper 3 0 0

2 4 0 110-34

3 0 5 0 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Covering Line 2


Hungarian MethodAssignments

Person 1 Adams Brown CooperTo accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna

Project 2 3 6 10 910-35 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Maximization Assignment Problem

Adams Brown Cooper Davis

1 $11 $8 $9 $10

Project 2 3 Dummy $14 $6 $0 $10 $11 $0 $12 $7 $0 $13 $8 $0


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Maximization Assignment Problem

Adams Brown Cooper Davis

1 $32 $6 $5 $4

Project 2 3 Dummy $0 $8 $14 $4 $3 $14 $2 $77 $14 $1 $6 $14


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