Transportation Assignment Transshipment Inventory

1. MT 235 1 Chapter 5 Transportation Assignment Transshipment Inventory

2. MT 235 2 Network Flow Problems Transportation Assignment Transshipment Production and Inventory

3. MT 235 3 Network Flow Problems Different authors hold different conventions. For example, I suggest the convention of denominating supplies as negative values (in this case, all constraints are greater-than-or-equal). Conversely, supplies can be denominated as positive values in which case their constraints become less-than-or-equals. Some indicate demands as equal constraints. While this is conceptually correct, the logical equivalent for a cost minimization problem is greater-than-or-equal. Do you see why? The bottom line is that there are often multiple ways to correctly model a business situation. It is up to you to understand the underlying logic so that valid interpretations of the results can be made.

4. MT 235 4 Transportation Problem Variations 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 Maximization/ minimization Change from max to min or vice versa Route capacities or route minimums Unacceptable routes

5. MT 235 5 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: Northwood – 25 tons Westwood – 45 tons 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?

6. MT 235 6 Network Representation - BBC

7. MT 235 7 Define Variables - BBC Let: xij = # of units shipped from Plant i to Destination j

8. MT 235 8 General Form - BBC Min 24x11 + 30x12 + 40x13 + 30x21 + 40x22 + 42x23 s.t. 1x11 + 1x12 + 1x13 + 0x21 + 0x22 + 0x23 <= 50 0x11 + 0x12 + 0x13 + 1x21 + 1x22 + 1x23 <= 50 1x11 + 0x12 + 0x13 + 1x21 + 0x22 + 0x23 >= 25 0x11 + 1x12 + 0x13 + 0x21 + 1x22 + 0x23 >= 45 0x11 + 0x12 + 1x13 + 0x21 + 0x22 + 1 x23 >=10 xij >= 0 for i = 1, 2 and j = 1, 2, 3

9. MT 235 9 Network Flow Problems Transportation Assignment Transshipment Production and Inventory

10. MT 235 10 Assignment Problem Variations 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 Maximization/ minimization Change from max to min or vice versa Unacceptable assignments

11. MT 235 11 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. How should the contractors be assigned to minimize total distance (and total cost)?

12. MT 235 12 Network Representation - ABC

13. MT 235 13 Define Variables - ABC Let: xij = 1 if contractor i is assigned to Project j and equals zero if not assigned

14. MT 235 14 General Form - ABC Min 50x11+36x12+16x13+28x21+30x22+18x23+35x31+32x32+20x33+25x41+25x42+14x43 s.t. 1x11 + 1x12 + 1x13 + 0x21 + 0x22 + 0x23 + 0x31 + 0x32 + 0x33 + 0 x41 + 0x42 + 0x43 <=1 0x11 + 0x12 + 0x13 + 1x21+ 1x22 + 1x23 + 0x31 + 0x32 + 0x33 + 0 x41 + 0x42 + 0x43 <=1 0x11 + 0x12 + 0x13 + 0x21 + 0x22 + 0x23 + 1x31 + 1x32 + 1x33 + 0 x41 + 0x42 + 0x43 <=1 0x11 + 0x12 + 0x13 + 0x21 + 0x22 + 0x23 + 0x31 + 0x32 + 0x33 + 1x41 + 1x42 + 1x43 <=1 1x11 + 0x12 + 0x13 + 1x21 + 0x22 + 0x23 + 1x31 + 0x32 + 0x33 + 1x41 + 0x42 + 0x43 >=1 0x11 + 1x12 + 0x13 + 0x21 + 1x22 + 0x23 + 0x31 + 1x32 + 0x33 + 0 x41 + 1x42 + 0x43 >=1 0x11 + 0x12 + 1x13 + 0x21 + 0x22 + 1x23 + 0x31 + 0x32 + 1x33 + 0 x41 + 0x42 + 1x43 >=1 xij >= 0 for i = 1, 2, 3, 4 and j = 1, 2, 3

15. MT 235 15 Network Flow Problems Transportation Assignment Transshipment Production and Inventory

16. MT 235 16 Transshipment Problem Variations 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 Maximization/ minimization Change from max to min or vice versa Route capacities or route minimums Unacceptable routes

17. MT 235 17 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: 50 for Zrox, 60 for Hewes, 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:

18. MT 235 18 Network Representation - Transshipment

19. MT 235 19 Define Variables - Transshipment Let: xij = # of units shipped from node i to node j

20. MT 235 20 General Form – TransshipmentShowing supplies as negative values Min 5x13+8x14+7x23+4x24+1x35+5x36+8x37+3x45+4x46+4x47 s.t. 1x13 + 1x14 + 0x23 + 0x24 + 0x35 + 0x36 + 0x37 + 0x45 + 0x46 + 0x47 >= -75 0x13 + 0x14 + 1x23 + 1x24 + 0x35 + 0x36 + 0x37 + 0x45 + 0x46 + 0x47 >= -75 -1x13 + 0x14 - 1x23 + 0x24 + 1x35 + 1x36 + 1x37 + 0x45 + 0x46 + 0x47 = 0 0x13 - 1x14 + 0x23 - 1x24 + 0x35 + 0x36 + 0x37 + 1x45+ 1x46 + 1x47 = 0 0x13 + 0x14 + 0x23 + 0x24 + 1x35 + 0x36 + 0x37 + 1x45 + 0x46 + 0x47 = 50 0x13 + 0x14 + 0x23 + 0x24 + 0x35 + 1x36 + 0x37 + 0x45 + 1x46 + 0x47 = 60 0x13 + 0x14 + 0x23 + 0x24 + 0x35 + 0x36 + 1x37 + 0x45 + 0x46 + 1x47 = 40 xij >= 0 for all i and j

21. MT 235 21 General Form – TransshipmentShowing supplies as positive values Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 8x37 + 3x45 + 4x46 + 4x47 s.t. 1x13 + 1x14 + 0x23 + 0x24 + 0x35 + 0x36 + 0x37 + 0x45 + 0x46 + 0x47 <= 75 0x13 + 0x14 + 1x23 + 1x24 + 0x35 + 0x36 + 0x37 + 0x45 + 0x46 + 0x47 <= 75 -1x13 + 0x14 - 1x23 + 0x24 + 1x35 + 1x36 + 1x37 + 0x45 + 0x46 + 0x47 = 0 0x13 - 1x14 + 0x23 - 1x24 + 0x35 + 0x36 + 0x37 + 1x45+ 1x46 + 1x47 = 0 0x13 + 0x14 + 0x23 + 0x24 + 1x35 + 0x36 + 0x37 + 1x45 + 0x46 + 0x47 = 50 0x13 + 0x14 + 0x23 + 0x24 + 0x35 + 1x36 + 0x37 + 0x45 + 1x46 + 0x47 = 60 0x13 + 0x14 + 0x23 + 0x24 + 0x35 + 0x36 + 1x37 + 0x45 + 0x46 + 1x47 = 40 xij >= 0 for all i and j

22. MT 235 22 Network Flow Problems Transportation Assignment Transshipment Production and Inventory

23. MT 235 23 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. Assume delivery requirements are for the end of the week, and assume carrying cost is for the end-of-the-week inventory.

24. MT 235 24 Network Representation – Production and Inventory

25. MT 235 25 Define Variables - Inventory Let: xij = # of units flowing from node i to node j

26. MT 235 26 General Form - Production and Inventory Min 28x15+27x26+26x37+29x48+.03x56+.03x67+.03x78 s.t. x15 <= 60 x26 <= 62 x37 <= 64 x48 <= 66 x15 = 58+x56 x26 +x56 = 36+x67 x37 +x67 = 52+x78 x48 +x78 = 70 xij >= 0 for all i and j