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C - 1© 2011 Pearson Education C C Transportation Modeling PowerPoint presentation to accompany Heizer and Render Operations Management, 10e, Global Edition Principles of Operations Management, 8e, Global Edition PowerPoint slides by Jeff Heyl

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C - 2 © 2011 Pearson Education Outline Transportation Modeling Developing an Initial Solution The Northwest-Corner Rule The Intuitive Lowest-Cost Method The Stepping-Stone Method Special Issues in Modeling Demand Not Equal to Supply Degeneracy

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C - 3 © 2011 Pearson Education Learning Objectives When you complete this module you should be able to: 1.Develop an initial solution to a transportation models with the northwest-corner and intuitive lowest-cost methods 2.Solve a problem with the stepping- stone method 3.Balance a transportation problem 4.Solve a problem with degeneracy

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C - 4 © 2011 Pearson Education Transportation Modeling An interactive procedure that finds the least costly means of moving products from a series of sources to a series of destinations Can be used to help resolve distribution and location decisions

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C - 5 © 2011 Pearson Education Transportation Modeling A special class of linear programming Need to know 1.The origin points and the capacity or supply per period at each 2.The destination points and the demand per period at each 3.The cost of shipping one unit from each origin to each destination

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C - 6 Transportation example The following example was used to demonstrate the formulation of the transportation model. Bath tubes are produced in plants in three different cities—Des Moines, Evansville, and Fort Lauder tube is shipped to the distributers in railroad,. Each plant is able to supply the following number of units, Des Moines 100 units, Evansville 300 units and Fort Lauder 300 units tubes will shipped to the distributers in three cities Albuquerque, Boston, and Cleveland,the distribution requires the following unites ; Albuquerque 300 units, Boston 200 units and Cleveland 200 units © 2011 Pearson Education

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C - 7 © 2011 Pearson Education shipping cost per unit shipping cost per unit To From AlbuquerqueBostonCleveland Des Moines$5$4$3 Evansville$8$4$3 Fort Lauder$9$7$5 Table C.1

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C - 8 © 2011 Pearson Education Transportation Problem Fort Lauder (300 units capacity) Albuquerque (300 units required) Des Moines (100 units capacity) Evansville (300 units capacity) Cleveland (200 units required) Boston (200 units required) Figure C.1

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C - 9 © 2011 Pearson Education Transportation Matrix From To AlbuquerqueBostonCleveland Des Moines Evansville Fort Lauder Factory capacity Warehouse requirement 300 200 100 700 $5 $4 $3 $9 $8 $7 Cost of shipping 1 unit from Fort Lauder factory to Boston warehouse Des Moines capacity constraint Cell representing a possible source-to- destination shipping assignment (Evansville to Cleveland) Total demand and total supply Cleveland warehouse demand Figure C.2

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C - 10 © 2011 Pearson Education Check the equality of the total supply & demand (if not, then we need for dummy solution ) how many ship of unit should be transported from each demand to each distribution center can be expressed as (Xij) i ……..for demand and j …….for distribution center

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C - 11 Step 2 Establish the initial feasible solution The Northwest corner rule Least Cost Method Vogel’s Approximation Model © 2011 Pearson Education

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C - 12 © 2011 Pearson Education Northwest-Corner Rule Start in the upper left-hand cell (or northwest corner) of the table and allocate units to shipping routes as follows: 1.Exhaust the supply (factory capacity) of each row before moving down to the next row 2.Exhaust the (warehouse) requirements of each column before moving to the next column 3.Check to ensure that all supplies and demands are met

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C - 13 The steps of the northwest corner method are summarized here: 1. Allocate as much as possible to the cell in the upper left-hand corner, subject to the supply and demand constraints. 2. Allocate as much as possible to the next adjacent feasible cell. 3. Repeat step 1,2 until all units all supply and demand have been satisfied © 2011 Pearson Education

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C - 14 © 2011 Pearson Education Northwest-Corner Rule 1.Assign 100 tubs from Des Moines to Albuquerque (exhausting Des Moines’s supply) 2.Assign 200 tubs from Evansville to Albuquerque (exhausting Albuquerque’s demand) 3.Assign 100 tubs from Evansville to Boston (exhausting Evansville’s supply) 4.Assign 100 tubs from Fort Lauder to Boston (exhausting Boston’s demand) 5.Assign 200 tubs from Fort Lauderdale to Cleveland (exhausting Cleveland’s demand and Fort Lauder’s supply)

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C - 15 © 2011 Pearson Education To (A) Albuquerque (B) Boston (C) Cleveland (D) Des Moines (E) Evansville (F) Fort Lauder Warehouse requirement 300200 Factory capacity 300 100 700 $5 $4 $3 $9 $8 $7 From Northwest-Corner Rule 100 200 Figure C.3 Means that the firm is shipping 100 bathtubs from Fort Lauder to Boston

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C - 16 © 2011 Pearson Education Northwest-Corner Rule Computed Shipping Cost Table C.2 This is a feasible solution but not necessarily the lowest cost alternative (not the optimal solution) Route FromToTubs ShippedCost per UnitTotal Cost DA100$5$ 500 EA20081,600 EB1004400 FB1007700 FC2005$1,000 Total: $4,200

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C - 17 © 2011 Pearson Education second Method The Lowest-Cost Method second Method The Lowest-Cost Method 1.Evaluate the transportation cost and select the square with the lowest cost (in case a Tie make an arbitrary selection ) 2.Depending upon the supply & demand condition, allocate the maximum possible units to lowest cost square 3.Delete the satisfied allocated row or the column (or both) 4.Repeat steps 1 and 3 until all units have been allocated

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C - 18 © 2011 Pearson Education The Lowest-Cost Method To (A) Albuquerque (B) Boston (C) Cleveland (D) Des Moines (E) Evansville (F) Fort Lauder Warehouse requirement 300200 Factory capacity 300 100 700 $5 $4 $3 $9 $8 $7 From 100 First, $3 is the lowest cost cell so ship 100 units from Des Moines to Cleveland and cross off the first row as Des Moines is satisfied Figure C.4

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C - 19 © 2011 Pearson Education The Lowest-Cost Method The Lowest-Cost Method To (A) Albuquerque (B) Boston (C) Cleveland (D) Des Moines (E) Evansville (F) Fort Lauderdale Warehouse requirement 300200 Factory capacity 300 100 700 $5 $4 $3 $9 $8 $7 From 100 Second, $3 is again the lowest cost cell so ship 100 units from Evansville to Cleveland and cross off column C as Cleveland is satisfied Figure C.4

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C - 20 © 2011 Pearson Education The Lowest-Cost Method To (A) Albuquerque (B) Boston (C) Cleveland (D) Des Moines (E) Evansville (F) Fort Lauderdale Warehouse requirement 300200 Factory capacity 300 100 700 $5 $4 $3 $9 $8 $7 From 100 200 Third, $4 is the lowest cost cell so ship 200 units from Evansville to Boston and cross off column B and row E as Evansville and Boston are satisfied Figure C.4

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C - 21 © 2011 Pearson Education The Lowest-Cost Method To (A) Albuquerque (B) Boston (C) Cleveland (D) Des Moines (E) Evansville (F) Fort Lauderdale Warehouse requirement 300200 Factory capacity 300 100 700 $5 $4 $3 $9 $8 $7 From 100 200 300 Finally, ship 300 units from Albuquerque to Fort Lauder as this is the only remaining cell to complete the allocations Figure C.4

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C - 22 © 2011 Pearson Education The Lowest-Cost Method To (A) Albuquerque (B) Boston (C) Cleveland (D) Des Moines (E) Evansville (F) Fort Lauderdale Warehouse requirement 300200 Factory capacity 300 100 700 $5 $4 $3 $9 $8 $7 From 100 200 300 Total Cost= $3(100) + $3(100) + $4(200) + $9(300) = $4,100 Figure C.4

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C - 23 © 2011 Pearson Education The Lowest-Cost Method To (A) Albuquerque (B) Boston (C) Cleveland (D) Des Moines (E) Evansville (F) Fort Lauderdale Warehouse requirement 300200 Factory capacity 300 100 700 $5 $4 $3 $9 $8 $7 From 100 200 300 Total Cost= $3(100) + $3(100) + $4(200) + $9(300) = $4,100 Figure C.4 This is a feasible solution, and an improvement over the previous solution, but not necessarily the lowest cost alternative

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C - 24 © 2011 Pearson Education The Lowest-Cost Method Computed Shipping Cost Table C.2 This is a feasible solution but not necessarily the lowest cost alternative (not the optimal solution) Route FromToTubs ShippedCost per UnitTotal Cost DC100$3$ 300 EC2003 600 EB2004800 FA30092700 Total: $4,100

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C - 25 Testing the initial feasible solution Test for optimality Optimal solution is achieved when there is no other alternative solution give lower cost. Two method for the optimal solution :- Stepping stone method Modified Distribution method (MODM) © 2011 Pearson Education

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C - 26 © 2011 Pearson Education Stepping-Stone Method 1.Proceed row by row and Select a water square (a square without any allocation) to evaluate 2.Beginning at this square, trace/forming a closed path back to the original square via squares that are currently being used 3.Beginning with a plus (+) sign at the unused corner (water square), place alternate minus and plus signs at each corner of the path just traced

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C - 27 © 2011 Pearson Education 4.Calculate an improvement index (the net cost change for the path )by first adding the unit-cost figures found in each square containing a plus sign and subtracting the unit costs in each square containing a minus sign 5.Repeat steps 1 though 4 until you have calculated an improvement index for all unused squares.

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C - 28 Evaluate the solution from optimality test by observing the sign of the net cost change I. The negative sign (-) indicates that a cost reduction can be made by making the change II.Zero result indicates that there will be no change in cost III.The positive sign (+) indicates an increase in cost if the change is made IV.If all the signs are positive, it means that the optimal solution has been reached V.If more than one squares have a negative signs then the water squared with the largest negative net cost change is selected for quicker solution, in case of tie; choose one of them randomly © 2011 Pearson Education

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C - 29 © 2011 Pearson Education 6. If an improvement is possible, choose the route (unused square) with the largest negative improvement index 7. On the closed path for that route, select the smallest number found in the squares containing minus signs, adding this number to all squares on the closed path with plus signs and subtract it from all squares with a minus sign.Repeat these process until you have evaluate all unused squares 8. Prepare the new transportation table and check for the optimality

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C - 30 © 2011 Pearson Education $5 $8$4 + - + - Stepping-Stone Method To (A) Albuquerque (B) Boston (C) Cleveland (D) Des Moines (E) Evansville (F) Fort Lauderdale Warehouse requirement 300200 Factory capacity 300 100 700 $5 $4 $3 $9 $8 $7 From 100 200 + - - + 1100 201 99 100200 Figure C.5 Des Moines- Boston index = $4 - $5 + $8 - $4 = +$3

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C - 31 © 2011 Pearson Education Stepping-Stone Method To (A) Albuquerque (B) Boston (C) Cleveland (D) Des Moines (E) Evansville (F) Fort Lauder Warehouse requirement 300200 Factory capacity 300 100 700 $5 $4 $3 $9 $8 $7 From 100 200 Figure C.6 Start +- + - + - Des Moines-Cleveland index = $3 - $5 + $8 - $4 + $7 - $5 = +$4

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C - 32 © 2011 Pearson Education Stepping-Stone Method To (A) Albuquerque (B) Boston (C) Cleveland (D) Des Moines (E) Evansville (F) Fort Lauderdale Warehouse requirement 300200 Factory capacity 300 100 700 $5 $4 $3 $9 $8 $7 From 100 200 Evansville-Cleveland index = $3 - $4 + $7 - $5 = +$1 (Closed path = EC - EB + FB - FC) Fort Lauder-Albuquerque index = $9 - $7 + $4 - $8 = -$2 (Closed path = FA - FB + EB - EA)

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C - 33 © 2011 Pearson Education Stepping-Stone Method To (A) Albuquerque (B) Boston (C) Cleveland (D) Des Moines (E) Evansville (F) Fort Lauder Warehouse requirement 300200 Factory capacity 300 100 700 $5 $4 $3 $9 $8 $7 From 100 200 Figure C.7 + +- - 1.Add 100 units on route FA 2.Subtract 100 from routes FB 3.Add 100 to route EB 4.Subtract 100 from route EA

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C - 34 © 2011 Pearson Education The first modified transportation table by Stepping-Stone Method To (A) Albuquerque (B) Boston (C) Cleveland (D) Des Moines (E) Evansville (F) Fort Lauderdale Warehouse requirement 300200 Factory capacity 300 100 700 $5 $4 $3 $9 $8 $7 From 100 200 100 200 Figure C.8 Total Cost= $5(100) + $8(100) + $4(200) + $9(100) + $5(200) = $4,000 is this the optimal solution

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C - 35 Since the condition for the acceptability is met (m+n -1 = the used cells ) Repeat steps 1 though 4 until you have calculated an improvement index for all unused squares. Evaluate the unused cells as follow ; © 2011 Pearson Education

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C - 36 DB …… DB, DA,EA,EB= +4 -5+8 – 4 = +3 DC……..DC,DA,FA,FC= +3-5+9-5 = +2 EC ……..EC,FC,FA,EA = +3 -5 +9-8 = -1 FB…….FB,FA,EA,EB= +7-9+8-4 = +2 Since there is a negative sign appear, then the pervious solution is not the optimal solution and there is chance to modify the solution © 2011 Pearson Education

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C - 37 © 2011 Pearson Education The second modified transportation table by Stepping-Stone Method 100 To (A) Albuquerque (B) Boston (C) Cleveland (D) Des Moines (E) Evansville (F) Fort Lauderdale Warehouse requirement 300200 Factory capacity 300 100 700 $5 $4 $3 $9 $8 $7 From 200 100 0 Figure C.8 Total Cost= $5(100) + $4(200) + $3(100) + $9(200) + $5(100) = $39,000 is this the optimal solution ?

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C - 38 Re-evaluate the unused cells AD; DB,EB,EC,FC,FA,AD= +4-4+3-5+9-5 = +2 DC; DC,FC,FA,DA= +3 -5+9-5 = +2 EA; EA,EC,FC,FA =+8 -3+5-9 = +1 FB; FB,EB,EC,FC= +7 – 4 +3 – 5 = +1 Since there is no negative sign appear, then the pervious solution is the optimal solution and there is no chance to modify the solution © 2011 Pearson Education

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C - 39 The initial cost = $4,200 The first modified cost = $4,000 The second modified cost= $39,000 Optimal solution © 2011 Pearson Education

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C - 40 © 2011 Pearson Education Special Issues in Modeling Demand not equal to supply Called an unbalanced problem Common situation in the real world Resolved by introducing dummy sources or dummy destinations as necessary with cost coefficients of zero

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C - 41 © 2011 Pearson Education Special Issues in Modeling Figure C.9 New Des Moines capacity To (A) Albuquerque (B) Boston (C) Cleveland (D) Des Moines (E) Evansville (F) Fort Lauderdale Warehouse requirement 300200 Factory capacity 300 250 850 $5 $4 $3 $9 $8 $7 From 50 200 250 50 150 Dummy 150 0 0 0 Total Cost= 250($5) + 50($8) + 200($4) + 50($3) + 150($5) + 150(0) = $3,350

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C - 42 © 2011 Pearson Education Special Issues in Modeling Degeneracy We must make the test for acceptability to use the stepping-stone methodology, that mean the feasible solution must met the condition of the number of occupied squares in any solution must be equal to the number of rows in the table plus the number of columns minus 1 M (number of rows) + N (number of columns ) = allocated cells If a solution does not satisfy this rule it is called degenerate

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C - 43 © 2011 Pearson Education To Customer 1 Customer 2 Customer 3 Warehouse 1 Warehouse 2 Warehouse 3 Customer demand 100 Warehouse supply 120 80 100 300 $8 $7 $2 $9 $6 $9 $7 $10 From Special Issues in Modeling 0 100 80 20 Figure C.10 Initial solution is degenerate Place a zero quantity in an unused square and proceed computing improvement indices

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C - 44 © 2011 Pearson Education All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America.

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Transportation and Assignment Models

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