1. Linear Programming Network Flow Problems Transportation Assignment Transshipment Production and Inventory

2. Linear Programming Network Flow Problems - Transportation Building Brick Company (BBC) manufactures bricks. One of BBC’s main concerns is transportation costs which are a very significant percentage of total costs. BBC has orders for 80 tons of bricks at three suburban locations as follows: NNorthwood – 25 tons WWestwood – 45 tons EEastwood – 10 tons BBC has two plants, each of which can produce 50 tons per week. BBC would like to minimize transportation costs. How should end of week shipments be made to fill the above orders given the following delivery cost per ton? $/tonNorthwoodWestwoodEastwood Plant 1243040 Plan 2304042

3. Linear Programming Network Representation - BBC 1 Northwood 2 Westwood 3 Eastwood 1 Plant 1 2 Plant 2 50 25 45 10 Plants (Origin Nodes) DestinationsTransportation Cost per Unit Distribution Routes - arcsDemandSupply $24 $30 $40 $30 $40 $42

4. Linear Programming Define Variables - BBC Let: x ij = # of units shipped from Plant i to Destination j

5. Linear Programming General Form - BBC Min 24x 11 +30x 12 +40x 13 +30x 21 +40x 22 +42x 23 s.t. x 11 +x 12 +x 13 <= 50 x 21 +x 22 + x 23 <= 50 x 11 + x 21 = 25 x 12 + x 22 = 45 x 13 + x 23 = 10 x ij >= 0 for i = 1, 2 and j = 1, 2, 3 Plant 1 Supply Plant 2 Supply North Demand West Demand East Demand

6. Linear Programming Network Flow Problems Transportation Problem Variations TTotal supply not equal to total demand Total supply greater than or equal to total demand Total supply less than or equal to total demand MMaximization/ minimization Change from max to min or vice versa RRoute capacities or route minimums UUnacceptable routes

7. Linear Programming Network Flow Problems - Transportation Building Brick Company (BBC) manufactures bricks. One of BBC’s main concerns is transportation costs which are a very significant percentage of total costs. BBC has orders for 80 tons of bricks at three suburban locations as follows: NNorthwood – 25 tons WWestwood – 45 tons EEastwood – 10 tons BBC has two plants, each of which can produce 50 tons per week. BBC would like to minimize transportation costs. How should end of week shipments be made to fill the above orders given the following delivery cost per ton? Suppose demand at Eastwood grows to 50 tons. $/tonNorthwoodWestwoodEastwood Plant 1243040 Plan 2304042

8. Linear Programming Network Representation - BBC 1 Northwood 2 Westwood 3 Eastwood 1 Plant 1 2 Plant 2 50 25 45 10 Plants (Origin Nodes) DestinationsTransportation Cost per Unit Distribution Routes - arcsDemandSupply $24 $30 $40 $30 $40 $42 50

9. Linear Programming General Form - BBC Min 24x 11 +30x 12 +40x 13 +30x 21 +40x 22 +42x 23 s.t. x 11 +x 12 +x 13 <= 50 x 21 +x 22 + x 23 <= 50 x 11 + x 21 = 25 x 12 + x 22 = 45 x 13 + x 23 = 10 x ij >= 0 for i = 1, 2 and j = 1, 2, 3 Plant 1 Supply Plant 2 Supply North Demand West Demand East Demand Min 24x 11 +30x 12 +40x 13 +30x 21 +40x 22 +42x 23 s.t. x 11 +x 12 +x 13 = 50 x 21 +x 22 + x 23 = 50 x 11 + x 21 <= 25 x 12 + x 22 <= 45 x 13 + x 23 <= 50 x ij >= 0 for i = 1, 2 and j = 1, 2, 3

10. Linear Programming Network Flow Problems Transportation Problem Variations TTotal supply not equal to total demand Total supply greater than or equal to total demand Total supply less than or equal to total demand MMaximization/ minimization Change from max to min or vice versa RRoute capacities or route minimums UUnacceptable routes

11. Linear Programming Network Flow Problems - Transportation Building Brick Company (BBC) manufactures bricks. BBC has orders for 80 tons of bricks at three suburban locations as follows: NNorthwood – 25 tons WWestwood – 45 tons EEastwood – 10 tons BBC has two plants, each of which can produce 50 tons per week. BBC would like to maximize profit. How should end of week shipments be made to fill the above orders given the following profit per ton? $/tonNorthwoodWestwoodEastwood Plant 1243040 Plan 2304042

12. Linear Programming Network Representation - BBC 1 Northwood 2 Westwood 3 Eastwood 1 Plant 1 2 Plant 2 50 25 45 10 Plants (Origin Nodes) DestinationsTransportation Cost per Unit Distribution Routes - arcsDemandSupply $24 $30 $40 $30 $40 $42 Profit per Unit

13. Linear Programming General Form - BBC Min 24x 11 +30x 12 +40x 13 +30x 21 +40x 22 +42x 23 s.t. x 11 +x 12 +x 13 <= 50 x 21 +x 22 + x 23 <= 50 x 11 + x 21 = 25 x 12 + x 22 = 45 x 13 + x 23 = 10 x ij >= 0 for i = 1, 2 and j = 1, 2, 3 Plant 1 Supply Plant 2 Supply North Demand West Demand East Demand Max

14. Linear Programming Network Flow Problems Transportation Problem Variations TTotal supply not equal to total demand Total supply greater than or equal to total demand Total supply less than or equal to total demand MMaximization/ minimization Change from max to min or vice versa RRoute capacities or route minimums UUnacceptable routes

15. Linear Programming Network Flow Problems - Transportation Building Brick Company (BBC) manufactures bricks. One of BBC’s main concerns is transportation costs which are a very significant percentage of total costs. BBC has orders for 80 tons of bricks at three suburban locations as follows: NNorthwood – 25 tons WWestwood – 45 tons EEastwood – 10 tons BBC has two plants, each of which can produce 50 tons per week. BBC would like to minimize transportation costs. How should end of week shipments be made to fill the above orders given the following delivery cost per ton? BBC has just been instructed to deliver at most 5 tons of bricks to Eastwood from Plant 2. $/tonNorthwoodWestwoodEastwood Plant 1243040 Plan 2304042

16. Linear Programming Network Representation - BBC 1 Northwood 2 Westwood 3 Eastwood 1 Plant 1 2 Plant 2 50 25 45 10 Plants (Origin Nodes) DestinationsTransportation Cost per Unit Distribution Routes - arcsDemandSupply $24 $30 $40 $30 $40 $42 At most 5 tons Delivered from Plant 2

17. Linear Programming General Form - BBC Min 24x 11 +30x 12 +40x 13 +30x 21 +40x 22 +42x 23 s.t. x 11 +x 12 +x 13 <= 50 x 21 +x 22 + x 23 <= 50 x 11 + x 21 = 30 x 12 + x 22 = 45 x 13 + x 23 = 10 x 23 <= 5 x ij >= 0 for i = 1, 2 and j = 1, 2, 3 Plant 1 Supply Plant 2 Supply North Demand West Demand East Demand Route Max

18. Linear Programming Network Flow Problems Transportation Problem Variations TTotal supply not equal to total demand Total supply greater than or equal to total demand Total supply less than or equal to total demand MMaximization/ minimization Change from max to min or vice versa RRoute capacities or route minimums UUnacceptable routes

19. Linear Programming Network Flow Problems - Transportation Building Brick Company (BBC) manufactures bricks. One of BBC’s main concerns is transportation costs which are a very significant percentage of total costs. BBC has orders for 80 tons of bricks at three suburban locations as follows: NNorthwood – 25 tons WWestwood – 45 tons EEastwood – 10 tons BBC has two plants, each of which can produce 50 tons per week. BBC would like to minimize transportation costs. How should end of week shipments be made to fill the above orders given the following delivery cost per ton? BBC has just learned the route from Plant 2 to Eastwood is no longer acceptable. $/tonNorthwoodWestwoodEastwood Plant 1243040 Plan 2304042

20. Linear Programming Network Representation - BBC 1 Northwood 2 Westwood 3 Eastwood 1 Plant 1 2 Plant 2 50 25 45 10 Plants (Origin Nodes) DestinationsTransportation Cost per Unit Distribution Routes - arcsDemandSupply $24 $30 $40 $30 $40 $42 Route no longer acceptable

21. Linear Programming General Form - BBC Min 24x 11 +30x 12 +40x 13 +30x 21 +40x 22 +42x 23 s.t. x 11 +x 12 +x 13 <= 50 x 21 +x 22 + x 23 <= 50 x 11 + x 21 = 30 x 12 + x 22 = 45 x 13 + x 23 = 10 x ij >= 0 for i = 1, 2 and j = 1, 2, 3 Plant 1 Supply Plant 2 Supply North Demand West Demand East Demand x 13 = 10 24x 11 +30x 12 +40x 13 +30x 21 +40x 22 x 21 +x 22 <= 50

22. Linear Programming Network Flow Problems Transportation Assignment Transshipment Production and Inventory

23. Linear Programming Network Flow Problems - Assignment ABC Inc. General Contractor pays their subcontractors a fixed fee plus mileage for work performed. On a given day the contractor is faced with three electrical jobs associated with various projects. Given below are the distances between the subcontractors and the projects. HHow should the contractors be assigned to minimize total distance (and total cost)? Project SubcontractorsABC Westside503616 Federated283018 Goliath353220 Universal25 14

24. Linear Programming Network Representation - ABC 1A1A 2B2B 3C3C 1 West 2 Fed 1 1 1 1 1 Contractors (Origin Nodes) Electrical Jobs (Destination Nodes) Transportation Distance Possible Assignments - arcs DemandSupply 50 36 16 28 30 18 3 Goliath 4 Univ. 1 1 35 32 20 25 14

25. Linear Programming Define Variables - ABC Let: x ij = 1 if contractors i is assigned to Project j and equals zero if not assigned

26. Linear Programming General Form - ABC Min 50x 11 +36x 12 +16x 13 +28x 21 +30x 22 +18x 23 +35x 31 +32x 32 +20x 33 +25x 41 +25x 42 +14x 43 s.t. x 11 +x 12 +x 13 <=1 x 21 +x 22 +x 23 <=1 x 31 +x 32 +x 33 <=1 x 41 +x 42 +x 43 <=1 x 11 +x 21 +x 31 +x 41 =1 x 12 +x 22 +x 32 +x 42 =1 x 13 +x 23 +x 33 +x 43 =1 x ij >= 0 for i = 1, 2, 3, 4 and j = 1, 2, 3

27. Linear Programming Network Flow Problems Assignment Problem Variations TTotal number of agents (supply) not equal to total number of tasks (demand) Total supply greater than or equal to total demand Total supply less than or equal to total demand MMaximization/ minimization Change from max to min or vice versa UUnacceptable assignments

28. Linear Programming Network Flow Problems Transportation Assignment Transshipment Production and Inventory

29. Linear Programming Network Flow Problems - Transshipment Thomas Industries and Washburn Corporation supply three firms (Zrox, Hewes, Rockwright) with customized shelving for its offices. Thomas and Washburn both order shelving from the same two manufacturers, Arnold Manufacturers and Supershelf, Inc. Currently weekly demands by the users are: 550 for Zrox, 660 for Hewes, 440 for Rockwright. Both Arnold and Supershelf can supply at most 75 units to its customers. Because of long standing contracts based on past orders, unit shipping costs from the manufacturers to the suppliers are: ThomasWashburn Arnold58 Supershelf74 The costs (per unit) to ship the shelving from the suppliers to the final destinations are: ZroxHewesRockwright Thomas158 Washburn344 Formulate a linear programming model which will minimize total shipping costs for all parties.

30. Linear Programming Network Representation - Transshipment 5 Zrox 6 Hewes 7 Rockwright 3 Thomas 4 Washburn 75 50 60 40 Warehouses (Transshipment Nodes) Retail Outlets (Destinations Nodes) Transportation Cost per Unit Distribution Routes - arcsDemandSupply $1 $5 $8 $3 $4 Transportation Cost per Unit 1 Arnold 2 Super S. $5 $8 $7 $4 Plants (Origin Nodes) Flow In 150 Flow Out 150 Resembles Transportation Problem

31. Linear Programming Define Variables - Transshipment Let: x ij = # of units shipped from node i to node j

32. Linear Programming General Form - Transshipment Min 5x 13 +8x 14 +7x 23 +4x 24 +1x 35 +5x 36 +8x 37 +3x 45 +4x 46 +4x 47 s.t. x 13 +x 14 <= 75 x 23 +x 24 <= 75 x 35 +x 36 +x 37 = x 13 +x 23 x 45 +x 46 +x 47 = x 14 +x 24 +x 35 +x 45 = 50 +x 36 +x 46 = 60 +x 37 +x 47 = 40 x ij >= 0 for all i and j Flow In 150 Flow Out 150

33. Linear Programming Network Flow Problems Transshipment Problem Variations TTotal supply not equal to total demand Total supply greater than or equal to total demand Total supply less than or equal to total demand MMaximization/ minimization Change from max to min or vice versa RRoute capacities or route minimums UUnacceptable routes

34. Linear Programming Network Flow Problems Transportation Assignment Transshipment Production and Inventory

35. Linear Programming Network Flow Problems – Production & Inventory A producer of building bricks has firm orders for the next four weeks. Because of the changing cost of fuel oil which is used to fire the brick kilns, the cost of producing bricks varies week to week and the maximum capacity varies each week due to varying demand for other products. They can carry inventory from week to week at the cost of $0.03 per brick (for handling and storage). There are no finished bricks on hand in Week 1 and no finished inventory is required in Week 4. The goal is to meet demand at minimum total cost. AAssume delivery requirements are for the end of the week, and assume carrying cost is for the end-of-the-week inventory. (Units in thousands)Week 1Week 2Week 3Week 4 Delivery Requirements58365270 Production Capacity60626466 Unit Production Cost ($/unit)$28$27$26$29

36. Linear Programming Network Representation – Production and Inventory 1 Week 1 62 Production NodesDemand NodesProduction Costs Production - arcs Demand Production Capacity 2 Week 2 3 Week 3 4 Week 4 64 66 60 5 Week 1 6 Week 2 7 Week 3 8 Week 4 36 52 70 58 $28 $27 $26 $29 $0.03 Inventory Costs

37. Linear Programming Define Variables - Inventory Let: x ij = # of units flowing from node i to node j

38. Linear Programming General Form - Production and Inventory Min 28x 15 +27x 26 +26x 37 +29x 48 +.03x 56 +.03x 67 +.03x 78 s.t. x 15 <= 60 x 26 <= 62 x 37 <= 64 x 48 <= 66 x 15 = 58+x 56 x 26 +x 56 = 36+x 67 x 37 +x 67 = 52+x 78 x 48 +x 78 = 70 x ij >= 0 for all i and j