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Chapter 7: Transportation Models Skip Ship Routing & Scheduling (pp. 212-214) Service Selection Shortest Path Transportation Problem Vehicle Routing & Scheduling –One route: TSP –Multiple routes: VRP Consolidation

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Service Selection (Mode Selection) Most important factors: –Dependability (on-time delivery). –Cost. –Safety. –Tracking. Different modes have different costs and characteristics. Lowest transportation cost is not always best.

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Service Selection Tradeoff Transportation Cost vs. Inventory Cost. Shorter transit time: Higher transportation cost. Fewer days held Lower inventory cost. Usually, Shorter transit time Smaller vehicles. More frequent trips Higher transportation cost. Fewer units held Lower inventory cost.

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Service Selection for Competing Suppliers One buyer purchases 1000 cwt from each of two competing suppliers: A and B. Both use rail transport, but could use truck transport. Supplier profit = $20/cwt - transport cost. Transport CostDelivery Time Rail $2/cwt 6 days Truck $5/cwt 3 days Buyer offers to switch 100 cwt to supplier A from B for each day decrease in delivery time. For supplier A: SalesProfit Rail (current)1000 cwt 1000 cwt ($20/cwt - $2/cwt) = $18,000 Truck1300 cwt 1300 cwt ($20/cwt - $5/cwt) = $19,500

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Service Selection for Competing Suppliers What if supplier B also switches to truck? Buyer should give each equal business: SalesProfit Supplier A1000 cwt 1000 cwt ($20/cwt - $5/cwt) = $15,000 Supplier B1000 cwt 1000 cwt ($20/cwt - $5/cwt) = $15,000 So both suppliers are worse off than before! ($15,000 profit vs. $18,000 using rail)

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Shortest Path Model Network includes: –Nodes: cities, customers, demand points –Arcs or Links: Transportation links –Number for each link to represent travel cost, time or distance. A F D B E C 3 4 6 2 9 4 7 5 6

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Shortest Path Problem Given: –A network with a specified origin and destination. –The distance (or travel time or cost) for each link. Determine the shortest path from the origin to the destination. Solution:Labeling algorithm (one of many) –Nodes are labeled as "solved" or "unsolved". –Solved = shortest path from the origin to that node is known.

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Shortest Path Labeling Algorithm 1. The origin is a solved node. All others are unsolved. 2. For each solved node, find the one unsolved node that is nearest and calculate the minimum total distance (origin to solved node + solved node to nearest unsolved node). 3. Make the unsolved node with the smallest total distance a solved node. 4. Repeat steps 2 and 3 until the destination is a solved node. 5. Trace the shortest path.

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Shortest Path Example 1 A F D B E C 30 20 22 6 17 10 5 18 27 16 12 Find the shortest path from A to F.

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Shortest Path Example 1 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 20 A F D B E C 30 20 22 6 17 10 5 18 27 16 12 *

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Shortest Path Example 1 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 20 C 20 A-C A F D B E C 30 20 22 6 17 10 5 18 27 16 12 * *

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Shortest Path Example 1 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 20 C 20 A-C A C A F D B E C 30 20 22 6 17 10 5 18 27 16 12 * *

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Shortest Path Example 1 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 20 C 20 A-C A B 22 C A F D B E C 30 20 22 6 17 10 5 18 27 16 12 * *

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Shortest Path Example 1 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 20 C 20 A-C A B 22 C D 26 A F D B E C 30 20 22 6 17 10 5 18 27 16 12 * *

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Shortest Path Example 1 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A F D B E C 30 20 22 6 17 10 5 18 27 16 12 * * *

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Shortest Path Example 1 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A B C A F D B E C 30 20 22 6 17 10 5 18 27 16 12 * * *

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Shortest Path Example 1 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B C A F D B E C 30 20 22 6 17 10 5 18 27 16 12 * * *

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Shortest Path Example 1 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 C A F D B E C 30 20 22 6 17 10 5 18 27 16 12 * * *

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Shortest Path Example 1 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 C D 26 A F D B E C 30 20 22 6 17 10 5 18 27 16 12 * * *

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Shortest Path Example 1 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 C D 26 D 26 C-D A F D B E C 30 20 22 6 17 10 5 18 27 16 12 * * * *

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Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 D 26 C-D C D 26 B C D A F D B E C 30 20 22 6 17 10 5 18 27 16 12 * * * *

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Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 D 26 C-D C D 26 B E 32 C E 32 D E 31 A F D B E C 30 20 22 6 17 10 5 18 27 16 12 * * * *

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Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 D 26 C-D C D 26 B E 32 C E 32 E 31 D-E D E 31 A F D B E C 30 20 22 6 17 10 5 18 27 16 12 * * * * *

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Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 D 26 C-D C D 26 B E 32 C E 32 E 31 D-E D E 31 B D E A F D B E C 30 20 22 6 17 10 5 18 27 16 12 * * * * *

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Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 D 26 C-D C D 26 B E 32 C E 32 E 31 D-E D E 31 B F 49 D F 44 E F 47 A F D B E C 30 20 22 6 17 10 5 18 27 16 12 * * * * *

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Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 D 26 C-D C D 26 B E 32 C E 32 E 31 D-E D E 31 B F 49 D F 44 F 44 D-F E F 47 A F D B E C 30 20 22 6 17 10 5 18 27 16 12 * * * * * *

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Trace Shortest Path Backwards Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 D 26 C-D C D 26 B E 32 C E 32 E 31 D-E D E 31 B F 49 D F 44 F 44 D-F E F 47 A-C-D-F

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Check Answer A F D B E C 30 20 22 6 17 10 5 18 27 16 12 A-C-D-F Length = 20+6+18 = 44

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Shortest Path Example 2 Find the shortest path from A to K. A K H E 4 6 3 6 9 1 6 1 3 5 11 B C D G J F I 2 3 3 5 4 4

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A K H E 4 6 3 6 9 1 6 1 3 5 B C D G J F I 2 3 3 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 4 C 4 A-C 0

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A K H E 4 6 3 6 9 1 6 1 3 5 11 B C D G J F I 2 3 3 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 4 C 4 A-C A C 0 4

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A K H E 4 6 3 6 9 1 6 1 3 5 11 B C D G J F I 2 3 3 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 4 C 4 A-C A B 6 B 6 A-B C E 8 0 4 6

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A K H E 4 6 3 6 9 1 6 1 3 5 11 B C D G J F I 2 3 3 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 4 C 4 A-C A B 6 B 6 A-B C E 8 B C 0 4 6

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A K H E 4 6 3 6 9 1 6 1 3 5 11 B C D G J F I 2 3 3 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 4 C 4 A-C A B 6 B 6 A-B C E 8 B E 9 C E 8 E 8 C-E 0 4 6 8

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A K H E 4 6 3 6 9 1 6 1 3 5 11 B C D G J F I 2 3 3 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 4 C 4 A-C A B 6 B 6 A-B C E 8 B E 9 C E 8 E 8 C-E B C E 0 4 6 8

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A K H E 4 6 3 6 9 1 6 1 3 5 11 B C D G J F I 2 3 3 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 4 C 4 A-C A B 6 B 6 A-B C E 8 B E 9 C E 8 E 8 C-E B D 15 C F 10 E H 9 H 9 E-H 0 4 6 8 9

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A K H E 4 6 3 6 9 1 6 1 3 5 11 B C D G J F I 2 3 3 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath B C E H 0 4 6 8 9

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A K H E 4 6 3 6 9 1 6 1 3 5 11 B C D G J F I 2 3 3 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath B D 15 C F 10 F 10 C-F E I 14 H D 12 0 4 6 8 9 10

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A K H E 4 6 3 6 9 1 6 1 3 5 11 B C D G J F I 2 3 3 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath B D 15 C F 10 F 10 C-F E I 14 H D 12 B C E H F 0 4 6 8 9 10

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A K H E 4 6 3 6 9 1 6 1 3 5 11 B C D G J F I 2 3 3 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath B D 15 C F 10 F 10 C-F E I 14 H D 12 B D 15 C I 15 E I 14 H D 12 D 12 H-D F I 14 0 4 6 8 9 10 12

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A K H E 4 6 3 6 9 1 6 1 3 5 11 B C D G J F I 2 3 3 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath C E H F D 0 4 6 8 9 10 12

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A K H E 4 6 3 6 9 1 6 1 3 5 11 B C D G J F I 2 3 3 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath C I 15 E I 14 H K 14 K 14 H-K F I 14 D J 15 0 4 6 8 9 10 12 14 Shortest Length = 14

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Trace Shortest Path Backwards Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath B D 15 C F 10 F 10 C-F E I 14 H D 12 B D 15 C I 15 E I 14 H D 12 D 12 H-D F I 14 C I 15 E I 14 H K 14 K 14 H-K F I 14 D J 15

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Trace Shortest Path Backwards Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 4 C 4 A-C A B 6 B 6 A-B C E 8 B E 9 C E 8 E 8 C-E B D 15 C F 10 E H 9 H 9 E-H A-C-E-H-K

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Check Answer A-C-E-H-K Length = 4+4+1+5 = 14 A K H E 4 6 3 6 9 1 6 1 3 5 11 B C D G J F I 2 3 3 5 4 4

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Shortest Path Example 3 Find the shortest path from A to K. A K H E 4 6 3 16 7 5 6 1 3 12 10 B C D G J F I 12 16 8 5 4 4

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A K H E 4 6 3 7 5 6 1 3 12 10 B C D G J F I 12 16 8 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 4 C 4 A-C A B 6 B 6 A-B C E 8 B E 9 C E 8 E 8 C-E 0 4 6 8 First 3 steps are same as Example 2!

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A K H E 4 6 3 16 7 5 6 1 3 12 10 B C D G J F I 12 16 8 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 4 C 4 A-C A B 6 B 6 A-B C E 8 B E 9 C E 8 E 8 C-E B C E 0 4 6 8

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A K H E 4 6 3 16 7 5 6 1 3 12 10 B C D G J F I 12 16 8 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 4 C 4 A-C A B 6 B 6 A-B C E 8 B E 9 C E 8 E 8 C-E B D 13 C I 14 E H 13 0 4 6 8 Tie for minimum distance Select both!

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A K H E 4 6 3 16 7 5 6 1 3 12 10 B C D G J F I 12 16 8 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath A C 4 C 4 A-C A B 6 B 6 A-B C E 8 B E 9 C E 8 E 8 C-E B D 13 D 13 B-D C I 14 E H 13 H 13 E-H 0 4 6 8 13

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A K H E 4 6 3 16 7 5 6 1 3 12 10 B C D G J F I 12 16 8 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath C D E H 0 4 6 8 13

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A K H E 4 6 3 16 7 5 6 1 3 12 10 B C D G J F I 12 16 8 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath C I 14 D G 29 E I 14 H K 25 0 4 6 8 13 Tie for minimum distance Select both!

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A K H E 4 6 3 16 7 5 6 1 3 12 10 B C D G J F I 12 16 8 5 4 4 Nearest Total Minimum Solved Unsolved Distance Nearest DistancePath C I 14 I 14 C-I D G 29 E I 14 I 14 E-I H K 25 0 4 6 8 13 There are two equal shortest paths from the origin to I!

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Example 3 Answer A-C-E-I-F-J-K A-C-I-F-J-K Length = 4+4+6+4+5+1 = 24 Length = 4+10+4+5+1 = 24 A K H E 4 6 3 16 7 5 6 1 3 12 10 B C D G J F I 12 16 8 5 4 4

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Shortest Path Software in LogWare ROUTE module. For each node, enter: –Node number and name. –X and Y coordinates if desired. For each link (arc), enter: –“From node” number. –“To node” number. –Cost. –Save data. Click Solve to get shortest paths from node 1 to all other nodes.

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LogWare

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ROUTE Module in LogWare Otherwise, click “Open file” and open Rfl01.dat. If possible, click “Add row”; then enter data.

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ROUTE Module: Edit as desired Now, Delete and Add rows and edit data. Save before solving.

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ROUTE Module: Solution

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Transportation Problem Given: –m origins (sources for product flows). –n destinations (sinks for product flows). –Supply at each origin. –Demand at each destination. –Shipping cost per unit of product from each origin to each destination. Determine the minimum total cost shipping pattern to satisfy demand. –We will solve using TRANLP module of LogWare.

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Transportation Problem Example 3 origins (sources) and 4 destinations (sinks) Origin SupplyDestination Demand 1 300 cwt. 1 400 cwt. 2 900 cwt. 2 300 cwt. 3 800 cwt. 3 700 cwt. 4 600 cwt. Shipping cost ($/cwt): D1D2D3D4 O1 3342 O2 2437 O3 2515

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Transportation Problem Example 3 origins (sources) and 4 destinations (sinks) 300 O1 D4 600 D3 700 D2 300 D1 400 3 3 4 2 2 4 3 7 5 1 5 2 800 O3 900 O2

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Transportation Problem Example A feasible solution: flows are in blue. 300 O1 D4 600 D3 700 D2 300 D1 400 3 3 4 2 2 4 3 7 5 1 5 2 800 O3 900 O2 300 100 300 500 200 600 Cost = 300x3+100x2+300x4+500x3+200x5+600x5 = 7500

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Solving Transportation Problems Place data in Transportation Matrix. From/To T1 T2T3T4Supply F1 3 3 4 2 300 F2 2 4 3 7 900 F3 2 5 1 5 800 Demand 400 300700600 Enter data into TRANLP and solve. 1.Open a file. 2. Change Problem label and specify number of rows and columns. 3. Enter data (use Backspace to erase entries). 4. Save data. 5. Click Solve.

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TRANLP in LogWare If possible, enter “No. of rows” and “No. of columns”. If not, then click “Open file” and open TRAN01.dat.

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File TRAN01.dat Now, enter “No. of rows” and “No. of columns”. Then, edit data. Save before solving.

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TRAN01.dat Solution Solution. Click “Report” for more...

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TRANLP Output Problem label: Example OPTIMUM SUPPLY SCHEDULE ----------- Cell ------------ Unit Cell Units Source name Sink name cost cost allocated F1 T1 3.00.00 0 F1 T2 3.00.00 0 F1 T3 4.00.00 0 F1 T4 2.00 600.00 300 Totals 600.00 300 Source capacity = 300 Slack capacity = 0 F2 T1 2.00 800.00 400 F2 T2 4.00 1,200.00 300 F2 T3 3.00.00 0 F2 T4 7.00 1,400.00 200 Totals 3,400.00 900 Source capacity = 900 Slack capacity = 0 F3 T1 2.00.00 0 F3 T2 5.00.00 0 F3 T3 1.00 700.00 700 F3 T4 5.00 500.00 100 Totals 1,200.00 800 Source capacity = 800 Slack capacity = 0 Total allocated = 2,000 Slack required = 2,000 Total cost = 5,200.00

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TRANLP Output Problem label: Example OPTIMUM SUPPLY SCHEDULE ----------- Cell ------------ Unit Cell Units Source name Sink name cost cost allocated F1 T1 3.00.00 0 F1 T2 3.00.00 0 F1 T3 4.00.00 0 F1 T4 2.00 600.00 300 Totals 600.00 300 Source capacity = 300 Slack capacity = 0 F2 T1 2.00 800.00 400 F2 T2 4.00 1,200.00 300 F2 T3 3.00.00 0 F2 T4 7.00 1,400.00 200 Totals 3,400.00 900 Source capacity = 900 Slack capacity = 0 F3 T1 2.00.00 0 F3 T2 5.00.00 0 F3 T3 1.00 700.00 700 F3 T4 5.00 500.00 100 Totals 1,200.00 800 Source capacity = 800 Slack capacity = 0 Total allocated = 2,000 Slack required = 2,000 Total cost = 5,200.00 Optimal Cost Optimal flows

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Optimal Solution Optimal solution: flows are in blue. 300 O1 D4 600 D3 700 D2 300 D1 400 3 3 4 2 2 4 3 7 5 1 5 2 800 O3 900 O2 300 400 300 200 700 100 Cost = 300x2+400x2+300x4+200x7+700x1+100x5 = 5200

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TRANLP Output Problem label: Example OPTIMUM SUPPLY SCHEDULE ----------- Cell ------------ Unit Cell Units Source name Sink name cost cost allocated F1 T1 3.00.00 0 F1 T2 3.00.00 0 F1 T3 4.00.00 0 F1 T4 2.00 600.00 300 Totals 600.00 300 Source capacity = 300 Slack capacity = 0 F2 T1 2.00 800.00 400 F2 T2 4.00 1,200.00 300 F2 T3 3.00.00 0 F2 T4 7.00 1,400.00 200 Totals 3,400.00 900 Source capacity = 900 Slack capacity = 0 F3 T1 2.00.00 0 F3 T2 5.00.00 0 F3 T3 1.00 700.00 700 F3 T4 5.00 500.00 100 Totals 1,200.00 800 Source capacity = 800 Slack capacity = 0 Total allocated = 2,000 Slack required = 2,000 Total cost = 5,200.00 slack capacity=0 means all is sent from every source Total allocated = Slack required means each destination receives what it needs.

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Transportation Problem In last problem total supply = total demand. –Each origin sends all it has. –Each destination receives all it demands. Other possibilities: –Total Supply > Total Demand Some origins will keep some of the supply. –Total Supply < Total Demand Some destinations will not receive all they demand.

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TRANLP Output #2 Problem label: Example OPTIMUM SUPPLY SCHEDULE ----------- Cell ------------ Unit Cell Units Source name Sink name cost cost allocated F1 T1 3.00.00 0 F1 T2 3.00.00 0 F1 T3 4.00.00 0 F1 T4 2.00 600.00 300 Totals 600.00 300 Source capacity = 300 Slack capacity = 0 F2 T1 2.00 800.00 400 F2 T2 4.00 1,200.00 300 F2 T3 3.00.00 0 F2 T4 7.00.00 0 Totals 2,000.00 700 Source capacity = 900 Slack capacity = 200 F3 T1 2.00.00 0 F3 T2 5.00.00 0 F3 T3 1.00 700.00 700 F3 T4 5.00 500.00 100 Totals 1,200.00 800 Source capacity = 800 Slack capacity = 0 Total allocated = 1,800 Slack required = 1,800 Total cost = 3,800.00 What is happening here?

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TRANLP Output #2 Problem label: Example OPTIMUM SUPPLY SCHEDULE ----------- Cell ------------ Unit Cell Units Source name Sink name cost cost allocated F1 T1 3.00.00 0 F1 T2 3.00.00 0 F1 T3 4.00.00 0 F1 T4 2.00 600.00 300 Totals 600.00 300 Source capacity = 300 Slack capacity = 0 F2 T1 2.00 800.00 400 F2 T2 4.00 1,200.00 300 F2 T3 3.00.00 0 F2 T4 7.00.00 0 Totals 2,000.00 700 Source capacity = 900 Slack capacity = 200 F3 T1 2.00.00 0 F3 T2 5.00.00 0 F3 T3 1.00 700.00 700 F3 T4 5.00 500.00 100 Totals 1,200.00 800 Source capacity = 800 Slack capacity = 0 Total allocated = 1,800 Slack required = 1,800 Total cost = 3,800.00 Total allocated = Slack required means each destination receives what it needs. Supply > Demand slack capacity=200 means 200 is left at origin 2

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TRANLP Output #3 Problem label: Example OPTIMUM SUPPLY SCHEDULE ----------- Cell ------------ Unit Cell Units Source name Sink name cost cost allocated F1 T1 3.00.00 0 F1 T2 3.00.00 0 F1 T3 4.00.00 0 F1 T4 2.00 600.00 300 Totals 600.00 300 Source capacity = 300 Slack capacity = 0 F2 T1 2.00 800.00 400 F2 T2 4.00 1,200.00 300 F2 T3 3.00.00 0 F2 T4 7.00 1,400.00 200 Totals 3,400.00 900 Source capacity = 900 Slack capacity = 0 F3 T1 2.00.00 0 F3 T2 5.00.00 0 F3 T3 1.00 700.00 700 F3 T4 5.00 500.00 100 Totals 1,200.00 800 Source capacity = 800 Slack capacity = 0 Total allocated = 2,000 Slack required = 2,200 Total cost = 5,200.00 What is happening here?

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TRANLP Output #3 Problem label: Example OPTIMUM SUPPLY SCHEDULE ----------- Cell ------------ Unit Cell Units Source name Sink name cost cost allocated F1 T1 3.00.00 0 F1 T2 3.00.00 0 F1 T3 4.00.00 0 F1 T4 2.00 600.00 300 Totals 600.00 300 Source capacity = 300 Slack capacity = 0 F2 T1 2.00 800.00 400 F2 T2 4.00 1,200.00 300 F2 T3 3.00.00 0 F2 T4 7.00 1,400.00 200 Totals 3,400.00 900 Source capacity = 900 Slack capacity = 0 F3 T1 2.00.00 0 F3 T2 5.00.00 0 F3 T3 1.00 700.00 700 F3 T4 5.00 500.00 100 Totals 1,200.00 800 Source capacity = 800 Slack capacity = 0 Total allocated = 2,000 Slack required = 2,200 Total cost = 5,200.00 Total allocated < Slack required means some destination(s) did not receive what they need. Can not tell which one(s) without input data. Supply < Demand

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