1. EMIS 8373: Integer Programming “Easy” Integer Programming Problems: Network Flow Problems updated 11 February 2007

2. slide 1 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

3. slide 2 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

4. slide 3 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 ij (usually 0)

5. slide 4 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 12 = 2, 13 = 0, 23 = 0, 24 = 1, 34 = 0 u 12 = 5, u 13 = 2, u 23 = 2, u 24 = 3, u 34 = 3

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

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

8. slide 7 Requirements for a Feasible Flow Flow on all arcs is within the allowable bounds: 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

9. slide 8 LP Formulation of MCNFP

10. slide 9 LP for Example MCNFP Min 3X 12 + 2 X 13 + X 23 + 4 X 24 + 4 X 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,

11. slide 10 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

12. slide 11 Optimal Solution for Example 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

13. Transportation Problems

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

15. slide 14 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

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

17. slide 16 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,n 1 ),n 1,(n 1,n 2 ),…,(n i,n j ),n j,(n j,t),t Find a minimum-cost path from s to t

18. slide 17 Shortest Path Example 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

19. slide 18 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

20. slide 19 Shortest Path as MCNFP 1 1 23 4 (5,1,0) (10,0,1) 1 1 23 4 10 1 0 0 0 (7,0,1) (1,0,1)

21. slide 20 Shortest Path Example In a rural area of Texas, there are six farms connected by 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?

22. slide 21 Graphical Network Flow Formulation b(j) b(i) i j ij = 0, u ij =1 arc (i,j) (c ij )

23. slide 22 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

24. slide 23 LP Formulation

25. slide 24 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

26. slide 25 Example: Rerouting Airline Passengers Due to a mechanical problem, Fly-By-Night Airlines had to cancel flight 162 - 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.

27. slide 26 Data for Fly-by-Night Example

28. slide 27 Network Representation s t SF DC H 2 6 A 5 NY 5 4 4 7

29. slide 28 Graphical Network Flow Formulation b(j) b(i) i j (u ij ) arc (i,j) ij =0 c ij =0

30. slide 29 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

31. slide 30 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,  )

32. slide 31 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,  )

33. slide 32 LP Formulation

34. slide 33 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

35. slide 34 Network Flow Formulation d1 d2 d3 d4 p1 p2 p3 p4 4000 -2400 -2200 -2700 -2500 I4 -1500 I0 1000 d0 -5700 I1I2I3

36. slide 35 Arc Parameters All arcs have ij = 0 and u ij =  Arcs (p i, d 0 ) have cost 0. Arcs (I i, d i+1 ) and (I i,I i+1 ) have cost 120.

37. slide 36 Backorder Cost of $200/unit/month d1 d2 d3 d4 p1 p2 p3 p4 4000 -2400 -2200 -2700 -2500 I4 -1500 I0 1000 d0 -5700 I1I2I3

38. slide 37 Parameters for Backorder Arcs All arcs have ij = 0 and u ij = 