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Chapter 10 Transportation and Assignment Models
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
101
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 steppingstone 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
102
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Chapter Outline10.1 Introduction 10.2 Setting Up a Transportation Problem 10.2 Developing an Initial Solution:Northwest Corner Rule 10.4 SteppingStone Method: Finding a LeastCost 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
103
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
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
104
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Specialized Problems Transportation Problem Distribution of items from several sources to several destinations. Supply capacities and destination requirements known.
Assignment Problem Onetoone assignment of people to jobs, etc.
Specialized algorithms save time!
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
105
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Importance of Special Purpose Algorithms Fewer, less complicated, computations than with simplex Less computer memory required Produce integer solutions
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
106
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
107
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Transportation CostsTo (Destinations)
From (Sources)
Albuquerque Cleveland Boston Des Moines $5 $4 $3 Evansville Fort Lauderdale $8 $9 $4 $7 $3 $5
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
108
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
109
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
1010
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
1011
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Initial Solution Using the Northwest Corner Rule Start in the upper lefthand 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.
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
1012
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
1013
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
The SteppingStone 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
1014
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
SteppingStone Method The Des MoinestoCleveland 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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
1015
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
SteppingStone 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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
1016
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
1017
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
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 steppingstone method. 1018To 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
1019
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Special Problems in Transportation Method Unbalanced Problem Demand Less than Supply Demand Greater than Supply
Degeneracy More Than One Optimal Solution
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
1020
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
1021
2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
1022
2003 by Prentice Hall, Inc. Upper Saddle River