1. EMIS 8374 Network Flow Models updated 29 January 2008

2. The Minimum Cost Network Flow Problem (MCNFP) Extremely useful model in OR & EM Important Special Cases of the MCNFP –Transportation and Assignment Problems –Maximum Flow Problem –Minimum Cut Problem –Shortest Path Problem Network Structure –BFS’s for MCNFP LP’s have integer values !!! –Problems can be formulated graphically slide 1

3. Elements of the MCNFP Defined on a network G = (N,A) N is a set of n nodes: {1, 2, …, n} –Each node i has an associated value b(i) b(i) node i is a demand node with a demand for –b(i) units of some commodity b(i) = 0 => node i is a transshipment node b(i) > 0 => node i is a supply node with a supply of b(i) units slide 2

4. Elements of the MNCFP A is a set of arcs that carry flow –Decision variable x ij determines the units of flow on arc (i,j) –The arc (i,j) from node i to node j has cost c ij per unit of flow on arc (i,j) upper bound on flow of u ij (capacity) lower bound on flow of l ij (usually 0) slide 3

5. Example MCNFP N = {1, 2, 3, 4} b(1) =5, b(2) = -2, b(3) = 0, b(4) = -3 A ={(1,2), (1,3), (2,3), (2,4), (3,4)} c 12 = 3, c 13 = 2, c 23 =1, c 24 = 4, c 34 = 4 l 12 = 2, l 13 = 0, l 23 = 0, l 24 = 1, l 34 = 0 u 12 = 5, u 13 = 2, u 23 = 2, u 24 = 3, u 34 = 3 slide 4

6. Graphical Network Flow Formulation b(j)b(j) b(i)b(i) i j (c ij, l ij, u ij ) arc (i,j) slide 5

7. Example MCNFP 5 14 (1, 0, 2) 3 2 0 -3 -2 (2, 0,2) (4, 1,3) (4, 0, 3) (3, 2, 5) slide 6

8. Requirements for a Feasible Flow Flow on all arcs is within the allowable bounds: l ij  x ij  u ij for all arcs (i,j) Flow is balanced at all nodes: flow out of node i - flow into node i = b(i) MCNFP: find a minimum-cost feasible flow slide 7

9. LP Formulation of MCNFP slide 8

10. LP for Example MCNFP Min 3X 12 + 2X 13 + X 23 + 4X 24 + 4X 34 s.t. X 12 + X 13 = 5{Node 1} X 23 + X 24 - X 12 = -2{Node 2} X 34 - X 13 - X 23 = 0 {Node 3} - X 24 - X 34 = -3 {Node 4} 2  X 12  5, 0  X 13  2, 0  X 23  2,1  X 24  3, 0  X 34  3 slide 9

11. Example Feasible Solution 5 14 (1, 0,2) 3 2 0 -3 -2 (2, 0,2) (4, 1,3) (4, 0,3) (3, 2,5) 5 3 0 0 0 Cost = 15 + 12 = 27 Arc flows shown in blue. slide 10

12. Optimal Solution for Example MCNFP 5 14 (1, 0,2) 3 2 0 -3 -2 (2, 0,2) (4, 1,3) (4, 0,3) (3, 2,5) 3 1 0 2 2 Cost = 25 Arc flows shown in blue. slide 11

13. Balanced Transportation Problems

14. Graphical Network Flow Formulation b(j)b(j) b(i)b(i) i j (c ij, u ij ) arc (i,j) l ij = 0 for all arcs. slide 13

15. D Dummy Node -3 CW +4 +1 +2 Supply Nodes I S G Demand Nodes A F slide 14

16. CW +4 +1 +2 Supply Nodes I S G Demand Nodes A F (13, 1) (35, 1) (9, 1) (42, 1) Dummy Node -3 (0,4) (0,2) (0,1) D slide 15

17. Shortest Path Problems Defined on a network with two special nodes: s and t A path from s to t is an alternating sequence of nodes and arcs starting at s and ending at t: s,(s,v 1 ),v 1,(v 1,v 2 ),…,(v i,v j ),v j,(v j,t),t Find a minimum-cost path from s to t slide 16

18. Shortest Path Example 1 1 23 4 510 7 7 1 st 1,(1,2),2,(2,3),3Length = 15 1,(1,2),2,(2,4),4,(4,3)Length = 13 1,(1,4),4,(4,3),3Length = 14 slide 17

19. MCNFP Formulation of Shortest Path Problems Source node s has a supply of 1 Sink node t has a demand of 1 All other nodes are transshipment nodes Each arc has capacity 1 Tracing the unit of flow from s to t gives a path from s to t slide 18

20. Shortest Path as MCNFP: Graphical Formulation 2 1 1 3 4 (5,0,1)(10,0,1) 0 0 (7,0,1) (1,0,1) slide 19

21. Shortest Path as MCNFP: Graphical Solution 2 1 1 3 4 (5,0,1)(10,0,1) 0 0 (7,0,1) Arc flows shown in blue. 0 0 1 1 1 (1,0,1) slide 20

22. Shortest Path Example 2 In a rural area of Texas, there are six farms connected my small roads. The distances in miles between the farms are given in the following table. What is the minimum distance to get from Farm 1 to Farm 6? slide 21

23. Graphical Network Flow Formulation b(j)b(j) b(i)b(i) i j l ij = 0, u ij =1 arc (i,j) (c ij ) slide 22

24. Formulation as Shortest Path s t 1 24 3 9 10 5 6 6 8 4 5 5 4 2 3 1 0 0 0 0 slide 23

25. LP Formulation of Shortest Path Problem slide 24

26. “Greedy” Solution s t 1 24 3 9 10 5 6 6 8 4 5 5 4 2 3 1 0 0 0 0 x 13 = x 23 = x 35 = x 35 = 1, x ij = 0 for all other arcs. Objective function value = 19. slide 25

27. Shortest Path: Optimal Solution s t 1 24 3 9 10 5 6 6 8 4 5 5 4 2 3 1 0 0 0 0 x 13 = x 35 = x 56 = 1, x ij = 0 for all other arcs. Objective function value = 17. slide 26

28. Maximum Flow Problems Defined on a network –Source node s –Sink node t –All other nodes are transshipment Nodes –Arcs have capacities, but no costs Maximize the total flow from s to t slide 27

29. Example: Rerouting Airline Passengers Due to a mechanical problem, Fly-By-Night Airlines had to cancel flight 162 which is its only non-stop flight from San Francisco to New York. Formulate a maximum flow problem to reroute as many passengers as possible from San Francisco to New York. slide 28

30. Data for Fly-by-Night Example slide 29

31. Graphical Network Flow Formulation b(j)b(j) b(i)b(i) i j l ij = 0 arc (i,j) (u ij ) slide 30

32. Network Representation s t SF DC H 2 6 A 5 NY 5 4 4 7 slide 31

33. LP Formulation slide 32

34. Max Flow: Optimal Solution slide 33

35. MCNF Formulation of Maximum Flow Problems 1.Let arc cost = 0 for all arcs 2.Add an arc from t to s –Give this arc a cost of –1 and infinite capacity 3.All nodes are transshipment nodes 4.Circulation Problem slide 34

36. Max Flow Formulation as MCNFP SF DC H (0,0,2) (0,0,6) A (0,0,5) NY (0,0,5) (0,0,4) (0,0,7) (-1,0,  ) slide 35

37. LP Formulation of MCNFP Representation slide 36

38. MCNFP Solution SF DC H (0,0,2) (0,0,6) A (0,0,5) NY (0,0,5) (0,0,4) (0,0,7) 9 4 2 2 5 7 5 2 (-1,0,  ) slide 37

39. LP Formulation of MCNFP Representation: Optimal Solution slide 38

40. NSC Example Max production per month = 4,000 tons Inventory holding cost = $120/ton/month Initial inventory = 1,000 tons Final inventory = 1,500 tons slide 39

41. Network Flow Formulation D1D1 D2D2 D3D3 D4D4 P1P1 P2P2 P3P3 P4P4 4000 -2400 -2200 -2700 -2500 I4I4 -1500 I0I0 1000 D0D0 -5700 I1I1 I2I2 I3I3 slide 40

42. Network Flow Formulation: Arc Costs D1D1 D2D2 D3D3 D4D4 P1P1 P2P2 P3P3 P4P4 I4I4 I0I0 D0D0 I1I1 I2I2 I3I3 slide 41 7400 7500 7600 7800 7520 7620 7720 7920 120 All other arc costs are 0

43. Network Flow Solution D1D1 D2D2 D3D3 D4D4 P1P1 P2P2 P3P3 P4P4 4000 -2400 -2200 -2700 -2500 I4I4 -1500 I0I0 100 d0 -5700 I1I1 I2I2 I3I3 900 1800 2500 1500 1700 2300 900 2200 1300 2700 4000 slide 42 l ij = 0 and u ij =  for all arcs 2700