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EMIS 8373: Integer Programming

EMIS 8373: Integer Programming. “Easy” Integer Programming Problems: Network Flow Problems updated 11 February 2007. The Minimum Cost Network Flow Problem (MCNFP). Extremely useful model in OR & EM Important Special Cases of the MCNFP

reuben-simmons
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EMIS 8373: Integer Programming

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

  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

  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) < 0 => 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. Elements of the MNCFP • A is a set of arcs that carry flow • Decision variable xij determines the units of flow on arc (i,j) • The arc (i,j) from node i to node j has • cost cij per unit of flow on arc (i,j) • upper bound on flow of uij (capacity) • lower bound on flow of ij (usually 0)

  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)} c12 = 3, c13 = 2, c23 =1, c24 = 4, c34 = 4 12 = 2, 13 = 0, 23 = 0, 24 = 1, 34 = 0 u12 = 5, u13 = 2, u23 = 2, u24 = 3, u34 = 3

  6. Graphical Network Flow Formulation (cij, ij, uij) i j arc (i,j) b(j) b(i)

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

  8. Requirements for a Feasible Flow • Flow on all arcs is within the allowable bounds: ij xij uijfor 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. LP Formulation of MCNFP

  10. LP for Example MCNFP Min 3X12 + 2 X13 + X23 + 4 X24 + 4 X34 s.t. X12 + X13 = 5 {Node 1} X23 + X24 – X12 = -2 {Node 2} X34 – X13 - X23 = 0 {Node 3} – X24 - X34 = -3 {Node 4} 2  X12 5, 0  X13 2, 0  X23 2, 1  X24 3, 0  X34 3,

  11. Example Feasible Solution -2 (3, 2,5) (4, 1,3) 2 5 3 5 -3 1 4 (1, 0,2) 0 0 0 (4, 0,3) (2, 0,2) 3 Cost = 15 + 12 = 27 0

  12. Optimal Solution for Example -2 (3, 2,5) (4, 1,3) 2 3 1 5 -3 1 4 (1, 0,2) 0 2 2 (4, 0,3) (2, 0,2) 3 Cost = 25 0

  13. Transportation Problems

  14. Graphical Network Flow Formulation (cij, uij) i j arc (i,j) b(j) b(i) ij=0

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

  16. -1 A -1 C -1 W -1 Supply Nodes Demand Nodes +4 I F +1 +2 G S -3 Dummy Node

  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,n1),n1,(n1,n2),…,(ni,nj),nj,(nj,t),t • Find a minimum-cost path from s to t

  18. Shortest Path Example 5 10 1 2 3 s t 7 1 7 4 1,(1,2),2,(2,3),3 Length = 15 1,(1,2),2,(2,4),4,(4,3) Length = 13 1,(1,4),4,(4,3),3 Length = 14

  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

  20. Shortest Path as MCNFP 0 (5,1,0) (10,0,1) 1 2 3 1 -1 (1,0,1) (7,0,1) 4 (7,0,1) 0 1 0 1 2 3 1 1 0 4

  21. 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. Graphical Network Flow Formulation (cij) i j arc (i,j) b(j) b(i) ij= 0, uij=1

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

  24. LP Formulation

  25. 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. 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. Data for Fly-by-Night Example

  28. Network Representation 2 D C 4 5 s t SF NY 4 6 7 5 H A

  29. Graphical Network Flow Formulation (uij) i j arc (i,j) b(j) b(i) ij=0 cij=0

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

  31. Formulation as MCNFP (0,0,2) D C (0,0,4) (0,0,5) SF NY (0,0,4) (0,0,7) (0,0,6) (0,0,5) H A (-1,0,)

  32. MCNFP Solution (0,0,2) D C (0,0,4) (0,0,5) 2 2 4 SF NY (0,0,4) 2 (0,0,7) (0,0,6) 5 (0,0,5) H A 7 5 (-1,0,) 9

  33. LP Formulation

  34. 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. p1 d1 4000 -2400 p2 d2 4000 -2200 p3 d3 4000 -2700 p4 d4 4000 -2500 d0 I1 I0 I3 I4 I2 1000 -5700 -1500 Network Flow Formulation

  36. Arc Parameters • All arcs have ij = 0 and uij =  • Arcs (pi, d0) have cost 0. • Arcs (Ii, di+1) and (Ii,Ii+1) have cost 120.

  37. p1 d1 4000 -2400 p2 d2 4000 -2200 p3 d3 4000 -2700 p4 d4 4000 -2500 d0 I1 I0 I3 I4 I2 1000 -5700 -1500 Backorder Cost of $200/unit/month

  38. Parameters for Backorder Arcs • All arcs have ij = 0 and uij = 

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