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1 ENGM 792

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2 Prototype Example K-Log Lumber Mill Warehouse

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3 Prototype Example K-Log Lumber Mill Warehouse 10 7 8

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4 Prototype Example K-Log Lumber Mill Warehouse 10 7 8 6 11 12

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5 Prototype Example K-Log Lumber Mill Warehouse 10 7 8 6 11 12 13 7 5

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6 Prototype RC DO OC SF AL SP 10 7 8 13 7 5 6 11 12

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7 Prototype RC DO OC SF AL SP 10 7 8 13 7 5 6 11 12 150 80 120 130 100 120 Supply Demand

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8 Prototype 1 6 5 4 2 3 10 7 8 13 7 5 6 11 12 150 80 120 130 100 120 Supply Demand

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9 Prototype Min Z = Transportation Costs s.t. Total amount shipped from plant i = Capacity at i Demand at each Warehouse is satisfied

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10 Prototype Min Z = 10X 14 + 7X 15 + 8X 16 + 13X 24 + 7X 25 + 5X 26 + 6X 34 + 11X 35 + 12X 36

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11 Prototype Min Z = 10X 14 + 7X 15 + 8X 16 + 13X 24 + 7X 25 + 5X 26 + 6X 34 + 11X 35 + 12X 36 s.t. X 14 + X 15 + X 16 = 130 X 24 + X 25 + X 26 = 100 X 34 + X 35 + X 36 = 120

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12 Prototype Min Z = 10X 14 + 7X 15 + 8X 16 + 13X 24 + 7X 25 + 5X 26 + 6X 34 + 11X 35 + 12X 36 s.t. X 14 + X 15 + X 16 = 130 X 24 + X 25 + X 26 = 100 X 34 + X 35 + X 36 = 120 X 14 + X 24 + X 34 = 150 X 15 + X 25 + X 35 = 80 X 16 + X 26 + X 36 = 120

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13 Prototype (re-index warehouse) Min Z = 10X 11 + 7X 12 + 8X 13 + 13X 21 + 7X 22 + 5X 23 + 6X 31 + 11X 32 + 12X 33 s.t. X 11 + X 12 + X 13 = 130 X 21 + X 22 + X 23 = 100 X 31 + X 32 + X 32 = 120 X 11 + X 21 + X 31 = 150 X 12 + X 22 + X 32 = 80 X 13 + X 23 + X 33 = 120

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14 General Formulation Transportation Problem MinZcX st Xsim Xdjn ij j n i m j n i i m j 11 1 1 12 12..,,,...,,,,..., Also, requires that supply matches demand.

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15 General Format Transportation Problem Also, requires that supply matches demand.

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16 Excel Solver Setup

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17 Excel Solver Setup

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18 Excel Solver Setup Note Excel Solver does not use a special transportation problem method. It just solves the problem with the usual LP software. For larger problems Excel Solver will be considerably slower than software designed to for transportation problems

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19 Transportation Tableau

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20 Transportation Tableau Total Demand = Total Supply

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21 Initial Feasible Solution Northwest Cornerrequires m+n-1 basic variables Vogel’s Approximation Russel’s Approximation (Not done for class)

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22 Initial Feasible Solution Northwest Corner

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23 Initial Feasible Solution Northwest Corner

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24 Initial Feasible Solution Total Cost = 10(130) + 13(20) + 7(80) + 11(0) + 12(120) = $3,560

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25 Clever Idea Suppose we can find a loop to move units around.

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26 Clever Idea Suppose we can find a loop to move units around.

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27 Clever Idea Suppose we can find a loop to move units around.

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28 Clever Idea Suppose we can find a loop to move units around.

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29 Clever Idea Suppose we can find a loop to move units around.

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30 Clever Idea For each unit I can move around the loop, I can save -5 + 12 - 11 + 7 = 3 per unit of flow

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31 Clever Idea I can move at most 80 units around this loop

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32 Clever Idea I can move at most 80 units around this loop

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33 Clever Idea Total Cost = 10(130) + 13(20) + 11(80) + 5(80) + 12(40) = $3,320 = $3,560 - 3(80)

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34 Finding the Best Loop Basic Cell c ij = u i + v j Nonbasic Celld ij = c ij - u i – v j Note: book doesn’t use d’s page 321

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35 Transportation Algorithm Arbitrarily select u 2 = 0

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36 Transportation Algorithm 13 = 0 + v 1 v 1 = 13 7 = 0 + v 2 v 2 = 7

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37 Transportation Algorithm 10 = u 1 + 13 u 1 = -3 11 = u 3 + 7 u 3 = 4

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38 Transportation Algorithm 12 = 4 + v 3 v 3 = 8

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39 Transportation Algorithm d 12 = 7 -(-3) - 7 = +3 3

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40 Transportation Algorithm d 13 = 8 -(-3) - 8 = +3 33

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41 Transportation Algorithm d 23 = 5 -0 - 8 = -3 33 3

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42 Transportation Algorithm d 31 = 6 -4 - 13 = -11 33 3 11

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43 Transportation Algorithm Note: -3 is the same thing we got earlier by finding a loop. 33 3 11

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44 Transportation Algorithm Let non-basic cell with largest -d ij enter basis. 33 3 11

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45 Transportation Algorithm Find a feasible loop.

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46 Transportation Algorithm Move the maximim unit flow around the loop.

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47 Transportation Algorithm Move the maximim unit flow around the loop. Total Cost = 10(130) + 13(20) + 7(80) + 12(120) = $3,560

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48 Transportation Algorithm Note that ui and vj must now be recomputed from new basis. Arbitrarily select v 1 = 0

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49 Class Problem Find u 1, u 2, u 3, v 2, v 3 d ij for non-basic cells

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50 Class Problem Find u 1, u 2, u 3, v 2, v 3 d ij for non-basic cells

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51 Class Problem Find u 1, u 2, u 3, v 2, v 3 d ij for non-basic cells

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52 Class Problem Find u 1, u 2, u 3, v 2, v 3 and d ij for non-basic cells 8 14

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53 Class Problem Find most -d ij. Find feasible loop for transfer. 14

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54 Class Problem Find most -d ij. Find feasible loop for transfer.

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55 Class Problem Total Cost = 10(130) + 7(80) + 5(20) + 6(20) + 12(120) = $3,280 = 3,560 - 20(14)

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56 Class Problem Arbitrarily select u 2 = 0. Find other multiplier values.

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57 Class Problem Arbitrarily select u 2 = 0. Find other multiplier values.

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58 Class Problem Arbitrarily select u 2 = 0. Find other multiplier values.

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59 Class Problem Arbitrarily select u 2 = 0. Find other multiplier values.

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60 Class Problem Arbitrarily select u 2 = 0. Find other multiplier values.

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61 Class Problem 118 3 Find all d ij values. Select largest –d ij to leave basis.

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62 Class Problem Find largest -d ij. Find feasible loop for transfer.

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63 Class Problem Total Cost = 10(50) + 7(80) + 5(100) + 6(100) + 12(20) = $2,400 = 3,280 - 11(80)

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64 Class Problem Arbitrarily select u 1 = 0. Find other multiplier values.

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65 Class Problem Arbitrarily select u 1 = 0. Find other multiplier values.

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66 Class Problem Arbitrarily select u 1 = 0. Find other multiplier values.

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67 Class Problem Arbitrarily select u 1 = 0. Find other multiplier values.

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68 Class Problem Arbitrarily select u 1 = 0. Find other multiplier values.

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69 Class Problem 8 Find all d ij values. Select largest –d ij to leave basis.

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70 Class Problem Find largest -d ij. Find feasible loop. 8

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71 Class Problem Find largest -d ij. Find feasible loop.

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72 Class Problem Total Cost = 10(30) + 7(80) + 8(20) + 5(100) + 6(120) = $2,240 = 2,400 - 8(20)

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73 Class Problem Arbitrarily select u 1 = 0.

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74 Class Problem Arbitrarily select u 1 = 0. Find other multipliers.

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75 Class Problem Arbitrarily select u 1 = 0. Find other multipliers.

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76 Class Problem optimal All d ij > 0 Solution is optimal. 63 88

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77 Class Problem Z = 10(30) + 7(80) + 8(20) + 5(100) + 6(120) = 2,240 63 88

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78 Initialization (Vogel’s)

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79 Initialization (Vogel’s) Table 8.17 H&L

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80 Initialization (Vogel’s) Table 8.17 H&L

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81 Initialization (Vogel’s) Table 8.17 H&L

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82 Initialization (Vogel’s) Table 8.17 H&L

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83 Initialization (Vogel’s) Table 8.17 H&L

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84 Initialization (Vogel’s) Table 8.17 H&L

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85 Dummy Warehouse Suppose total supply exceeds total demand.

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86 Dummy Warehouse Add dummy warehouse with 0 cost.

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87 Dummy Supplier Suppose total demand exceeds total supply.

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88 Dummy Supplier

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89 Final slide Transportation Problem Northwest corner Method Transportation Tableau Method Vogler’s approximation (Initialization)

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