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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-1 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Chapter 10 Transportation and Assignment Models

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-2 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Learning Objectives Students 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) method

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-3 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Chapter Outline 10.1 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 Vogel’s Approximation Method 10.7 Unbalanced Transportation Problems

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-4 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Chapter Outline - continued 10.8 Degeneracy in Transportation Problems 10.9 More Than One Optimal Solution 10.10Maximization Transportation Problems 10.11 Unacceptable or Prohibited Routes 10.12Facility Location Analysis 10.13Approach of the Assignment Model 10.14Unbalanced Assignment Models 10.15Maximization Assignment Problems

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-5 © 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 One-to-one assignment of people to jobs, etc. Specialized algorithms save time!

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-6 © 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

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-7 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Transportation Problem Des Moines (100 units) capacity Cleveland (200 units) required Boston (200 units) required Evansville (300 units) capacity Ft. Lauderdale (300 units) capacity Albuquerque (300 units) required

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-8 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Transportation Costs From (Sources) To (Destinations) Albuquerque Boston Cleveland Des Moines Evansville Fort Lauderdale $5 $8 $9 $4 $7 $3 $5

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-9 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Unit Shipping Cost:1Unit, Factory to Warehouse Des Moines (D) Evansville (E) Fort Lauderdale (F) Warehouse Req. Albuquerque (A) Boston (B) Cleveland (C) Factory Capacity 543 57 48 9 3

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-10 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Total Demand and Total Supply Des Moines (D) Evansville (E) Fort Lauderdale (F) Warehouse Req. Albuquerque (A) Boston (B) Cleveland (C) Factory Capacity 300200 700 300 100

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-11 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Transportation Table For Executive Furniture Corp. Des Moines (D) Evansville (E) Fort Lauderdale (F) Warehouse Req. Albuquerque (A) Boston (B) Cleveland (C) Factory Capacity 300200 700 300 100 543 3 57 48 9

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-12 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 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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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-13 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Initial Solution North West Corner Rule Des Moines (D) Evansville (E) Fort Lauderdale (F) Warehouse Req. Albuquerque (A) Boston (B) Cleveland (C) Factory Capacity 300200 700 300 100 543 3 57 48 9 200 100 200

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-14 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 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.

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-15 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Stepping-Stone Method - The Des Moines-to- Cleveland Route Des Moines (D) Evansville (E) Fort Lauderdale (F) Warehouse Req. Albuquerque (A) Boston (B) Cleveland (C) Factory Capacity 300200 700 300 100 543 3 57 48 9 200 100 200 -+ - + + - Start

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-16 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Stepping-Stone Method An Improved Solution Des Moines (D) Evansville (E) Fort Lauderdale (F) Warehouse Req. Albuquerque (A) Boston (B) Cleveland (C) Factory Capacity 300200 700 300 100 543 3 57 48 9 200 100

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-17 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Third and Final Solution Des Moines (D) Evansville (E) Ft Lauderdale (F) Warehouse Req. Albuquerque (A) Boston (B) Cleveland (C) Factory Capacity 300200 700 300 100 543 3 57 48 9 200 100 200 100

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-18 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 MODI Method: 5 Steps 1. Compute the values for each row and column: set R i + K j = C ij for those squares currently used or occupied. 2. After writing all equations, set R 1 = 0. 3. Solve the system of equations for R i and K j values. 4. Compute the improvement index for each unused square by the formula improvement index: C ij - R i - K j 5. Select the largest negative index and proceed to solve the problem as you did using the stepping-stone method.

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-19 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Vogel’s Approximation 1. 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.

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-20 © 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

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-21 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Unbalanced Problem Demand Less than Supply Factory 1 Factory 2 Factory 3 Customer Requirements Customer 1 Customer 2 Dummy Factory Capacity 15080150380 80 130 170 85 0 0 09 1015 3

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-22 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Unbalanced Problem Supply Less than Demand Factory 1 Factory 2 Dummy Customer Requirements Customer 1 Customer 2 Customer 3 Factory Capacity 15080150380 80 130 170 8516 7 00 1015 0

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-23 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Degeneracy Factory 1 Factory 2 Factory 3 Customer Requirements Customer 1 Customer 2 Customer 3 Factory Capacity 100 300 80 120 100 543 3 57 48 9 80 20

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-24 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Degeneracy - Coming Up! Factory 1 Factory 2 Factory 3 Customer Requirements Customer 1 Customer 2 Customer 3 Factory Capacity 1508050280 80 130 70 85 7 9 16 10 15 3 70 80 50 30

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-25 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Stepping-Stone Method - The Des Moines-to- Cleveland Route Des Moines (D) Evansville (E) Fort Lauderdale (F) Warehouse Req. Albuquerque (A) Boston (B) Cleveland (C) Factory Capacity 300200 700 300 100 543 3 57 48 9 Start 200 100 200 -+ - + + -

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-26 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 The Assignment Problem Project Person123 Adams$11$14$6 Brown$8$10$11 Cooper$9$12$7

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-27 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 The Assignment Method 1. 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 assignment (step 4)

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-28 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 The Assignment Method - continued 3. 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 table PG 10.13b

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-29 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Hungarian Method Initial Table PersonProject 123 Adams 11146 Brown81011 Cooper9127

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-30 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Hungarian Method Row Reduction PersonProject 123 Adams Brown Cooper 580 023 250

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-31 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Hungarian Method Column Reduction PersonProject 123 Adams560 Brown003 Cooper230

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-32 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Hungarian Method PersonProject 123 Adams Brown Cooper 560 003 230 Testing Covering Line 2 Covering Line 1

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-33 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Hungarian Method PersonProject 123 Adams340 Brown005 Cooper010 Revised Opportunity Cost Table

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-34 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Hungarian Method PersonProject 123 Adams 340 Brown 005 Cooper 010 Testing Covering Line 1 Covering Line 2 Covering Line 3

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-35 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Hungarian Method Assignments PersonProject 123 Adams 6 Brown10 Cooper 9

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-36 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Maximization Assignment Problem Project 123Dummy Adams$11$14$6$0 Brown$8$10$11$0 Cooper$9$12$7$0 Davis$10$13$8$0

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To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 10-37 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Maximization Assignment Problem Project 123Dummy Adams$32$0$8$14 Brown$6$4$3$14 Cooper$5$2$77$14 Davis$4$1$6$14

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