1. ENGM 732 Network Flow Programming Network Flow Models

2. Transportation Models (Flow, Cost) [External Flow] [2] [4] [3] [-3] (3,3) (1,1) (0,4) (2,2) (3,3) (0,3) (0,4) 1 2 3 4 5 6

3. Transportation Models (Flow, Cost) [External Flow] [2] [4] [3] [-3] (3,3) (1,1) (0,4) (2,2) (3,3) (0,3) (0,4) 1 2 3 4 5 6 Properties 1.All arcs have infinite capacity. 2.All nodes have nonzero fixed external flows. 3.The sum of the external flows over all nodes is zero.

4. Assignment Models (Flow, Cost) [External Flow] [1] [-1] (0,4) (1,1) (1,3) (1,2) (0,4) (0,2) 1 2 3 4 5 6 (0,8)

5. Assignment Models (Flow, Cost) [External Flow] [1] [-1] (0,4) (1,1) (1,3) (1,2) (0,4) (0,2) 1 2 3 4 5 6 (0,8) Assignment 1.All demands and supplies are unity. 2.Find the one-to-one pairing of the two sets that minimizes the sum of the pairing costs.

6. Shortest Path (Flow, Cost) [External Flow] [1] [-1] (0,3) (0,5) (0,4) (1,1) 2 1 4 5 3 (0,6) (1,2) (0,5) (1,4)

7. Shortest Path (Flow, Cost) [External Flow] [1] [-1] (0,3) (0,5) (0,4) (1,1) 2 1 4 5 3 (0,6) (1,2) (0,5) (1,4) Shortest Path 1.One node is the source. 2.One node is the sink. 3.Optimal path is the sequence of arcs such that the sum of the arc costs on the path are minimized.

8. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6) 1 2 3 4 5 6 (6,8) (3,3) (4,4) (4,10)

9. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6) 1 2 3 4 5 6 (6,8) (3,3) (4,4) (4,10) Maximal Flow 1.Capacity is only relevant parameter. 2.Find maximal flow from source to sink.

10. Maximum Flow Models (Flow, Capacity) (0,3) (2,2) (5,7) (0,8) (3,6) 1 2 3 4 5 6 (6,8) (3,3) (4,4) (4,10) Maximal Flow 1.Capacity is only relevant parameter. 2.Find maximal flow from source to sink.

11. Maximum Flow Models (Flow, Capacity) [External Flow] (0,3) (2,2) (5,7) (0,8) (3,6) 1 2 3 4 5 6 (6,8) (3,3) (4,4) (4,10) Maximal Flow 1.Capacity is only relevant parameter. 2.Find maximal flow from source to sink. S S [M] [-M]

12. Network with Gains 2 1 3 4 [3][-3] (Flow, capacity, gain, cost) [External Flow] (1,2,.5,3) (2,2,.5,2) (0,2,1,1) (1,2,1,-1) (0,4,2,5) (1.5,4,2,1)

13. Relationships Less GeneralMore General Assignment Transpor- tation Shortest Path Pure Min Cost Flow Maximal Flow General Min Cost Flow Linear Program

14. Network with Slack External Flow 2 1 3 4 [2,2,1][-1] [External Flow, max slack external flow, slack cost] [0,-2,1] [-1,2,-2]

15. Pure Min Cost Flow Consider K-Chair Corp. Plant Cost / ChairMax ProductionMin Production 1 $5 500 0 2 7 750 400 3 3 1000 500 4 4 250 250 Wood comes from 1 of 2 suppliers and K-Chair agrees to buy 8 tons (800 chairs at 20 lbs / chair) from each supplier. Cost is $0.10 per lbs from supplier one and $0.075 per lbs from supplier 2. Transportation costs follow. P1P2P3P4 Supplier 10.010.020.040.04 Supplier 20.040.030.020.02

16. Pure Min Cost Flow Chairs are sold in NY, Houston, San Francisco, and Chicago. Transportation costs From each plant to each city follows NYHSFC P11120 P23673 P33153 P48214 Selling price, maximum demand, and minimum demand follow SPMaxMin NY$202000500 H 15 400100 SF 201500500 C 181500500

17. K-Chair (Supplier) 1 2 3 4 5 6 [800,M,2] [800,M,1.5] (lower, upper, cost) [Fixed, slack, cost] (0,M,.2)

18. K-Chair (Production) 1 2 3 4 5 6 [800,M,2] [800,M,1.5] (lower, upper, cost) [Fixed, slack, cost] (0,M,.2) 7 8 9 10 (400,750,7)

19. K-Chair (Shipping) 1 2 3 4 5 6 [800,M,2] [800,M,1.5] (lower, upper, cost) [Fixed, slack, cost] (0,M,.2) 7 8 9 10 (400,750,7) NY H SF C (0,M,1) (0,M,2) (0,M,0)

20. K-Chair (Sales) 1 2 3 4 5 6 [800,M,2] [800,M,1.5] (lower, upper, cost) [Fixed, slack, cost] (0,M,.2) 7 8 9 10 (400,750,7) NY H SF C (0,M,1) (0,M,2) (0,M,0) [-500,-1500,-20] [-100,-300,-15] [-500,-1000,-20] [-500,-1000,-18] (0,500,5) (500,1000,3) (250,250,4)