1. ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Assignment Problem Linear Programming

2. ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Assignment Problem Assignment „1 to 1“ employeesjobs machinesjobs projectsmanagers service teams cars doctors night shifts Objective: maximize the effect of assignment

3. Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Example – Prague Build, Inc. Objective: minimize total distance necessary for all movements Assignment Problem  Excavating shafts for basements (Michle, Prosek, Radlice, Trója)  Each excavation takes 5 days  4 excavators stored in 4 separated garages (everyday‘s movement)  One excavator to one destination  Distances between garages and destinations

4. ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry

5. Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Example – Prague Build, Inc. Assignment Problem MichleProsekRadliceTrója Garage 1 5221218 Garage 2 1517610 Garage 3 825520 Garage 4 10121912 Distances

6. Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Example – Prague Build, Inc. Assignment Problem MichleProsekRadliceTrója Garage 1 x 11 x 12 x 13 x 14 Garage 2 x 21 x 22 x 23 x 24 Garage 3 x 31 x 32 x 33 x 34 Garage 4 x 41 x 42 x 43 x 44 Decision variables

7. Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Example – Prague Build, Inc. Assignment Problem Decision variables x ij = 1 if the excavator from the garage i goes to the destination j 0 otherwise Binary variable

8. Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Example – Prague Build, Inc. Assignment Problem Optimal solution MichleProsekRadliceTrója Garage 1 1000 Garage 2 0001 Garage 3 0010 Garage 4 0100

9. Linear Programming ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Applications Example – Prague Build, Inc. Assignment Problem Optimal solution 1 movement MichleProsekRadliceTrója Garage 1 5 km --- Garage 2 --- 10 km Garage 3 -- 5 km - Garage 4 - 12 km -- Minimal total distance 320 km

10. ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry

11. ___________________________________________________________________________ Operations Research  Jan Fábry Network Models

12. ___________________________________________________________________________ Operations Research  Jan Fábry  Nodes  Arcs j jj j i j jj j iUNDIRECTEDDIRECTED UNDIRECTED NETWORK DIRECTED NETWORK  Network

13. Network Models ___________________________________________________________________________ Operations Research  Jan Fábry  Path Sequence of arcs in which the initial node of each arc is identical with the terminal node of the preceding arc. 3 7 5

14. Network Models ___________________________________________________________________________ Operations Research  Jan Fábry  Path 1 3 2 4 5 6 1 2 3 4 5 6 Open Path 1 6 

15. Network Models ___________________________________________________________________________ Operations Research  Jan Fábry  Circuit (Cycle) Path starting and ending in the same node (closed path). 1 3 2 4 5 6 1 2 3 4 5 6 1 

16. Network Models ___________________________________________________________________________ Operations Research  Jan Fábry  Connected Network There is a path connecting every pair of nodes in the network. 1 3 2 4 5 6 1 2 3 4 5 6

17. Network Models ___________________________________________________________________________ Operations Research  Jan Fábry  Unconnected Network 1 3 2 4 5 6 1 2 3 4 5 6

18. Network Models ___________________________________________________________________________ Operations Research  Jan Fábry  Tree Connected network without any circuit. Exactly 6 arcs (n-1) Removing 1 arc Unconnected network 2 4 1 4 3 2 3 3 4 5 4 7 6 Adding 1 arc Circuit in the network

19. Network Models ___________________________________________________________________________ Operations Research  Jan Fábry  Tree STAR „CHRISTMAS“ TREE SNAKE

20. Network Models ___________________________________________________________________________ Operations Research  Jan Fábry  Spanning Tree Tree including all the nodes from the original network. 1 3 2 4 5 6 1 2 3 4 5 6

21. Network Models ___________________________________________________________________________ Operations Research  Jan Fábry  Evaluated Network - distance - time - cost - capacity ValuesArcs Nodes j i ii i j i ii i y ij yiyiyiyi yjyjyjyj

22. Network Models ___________________________________________________________________________ Operations Research  Jan Fábry Basic Network Applications Project Management  Shortest Path Problem  Traveling Salesperson Problem (TSP)  Minimal Spanning Tree  Critical Path Method (CPM)  Maximum Flow Problem  Program Evaluation Review Technique (PERT)

23. ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Shortest Path Problem Network Models

24. ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Shortest Path Problem 14 14 25 10 12 15 16 15 23 3018 1 3 2 4 5 6 1 2 3 4 5 6 Shortest path between 2 nodes

25. Network Models ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Shortest Path Problem Shortest Paths Between All Pairs of Nodes 123456 1-1424263240 214-10121826 32410-152816 4261215-2315 532182823-30 64026161530-

26. ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Traveling Salesperson Problem Network Models

27. ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Traveling Salesperson Problem (TSP) 14 14 25 10 12 15 16 15 23 3018 1 3 2 4 5 6 1 2 3 4 5 6 1 Home city Shortest tour 110 km

28. ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Minimal Spanning Tree Network Models

29. ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Minimal Spanning Tree Example - Exhibition  Exhibition area with 9 locations that need electricity power  Use cable for extensions  Price of cable = 10 CZK / 1 m Objective: minimize the cost of all the extensions

30. Network Models ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Minimal Spanning Tree 88 60 90 76 40 80 55 63 708575 52 71 74 61 68 43 120 35 54 Example - Exhibition 3 2 5 7 8 4 9 10 6 1 3 2 4 5 6 7 8 9 10 1Power

31. Network Models ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Minimal Spanning Tree Example - Exhibition 60 76 40 55 52 61 68 43 35 3 2 5 7 8 4 9 10 6 1 3 2 4 5 6 7 8 9 10 1 Power Optimum 490 m 4 900 CZK

32. Network Models ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Maximum Flow Problem 3 7 9 Input Output Capacited network Gas Fluid Traffic Information People Source Sink

33. Network Models ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Maximum Flow Problem j i i j UNDIRECTED ARC DIRECTED ARC Flow Flow Capacity

34. Network Models ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Maximum Flow Problem Mathematical Model  Flow through each arc  Capacity of the arc  Quantity flowing out = Quantity flowing into (except the source and the sink)  Total flow into the source = 0  Total flow out of the sink = 0  Total flow out of the source = Total flow into the sink

35. Maximum Flow Problem Example – White Lake City  The city is situated on the edge of a small lake  To minimize disruptive effects of possible flood  Reconstruction of drain system  2 alternatives - Northern Channel & Southern Channel Objective: maximizing the quantity of water being pumped in one hour Network Models ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry

36. ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry 2 3 5 7 900 800 370 270 410740 4 8 6 1100 220 400 300 300 280 130 1510 720 550 700 2 4 6 8 800 660 370 230 1050780 5 7 800 420 290 1400 700 250 840 3 520 470 Lake Reservoir Northern Channel Southern Channel

37. ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Northern Channel 2 3 5 7 610 220610 4 8 6 1100 220 390 300 300 130 1500 300 700 100 410 1 2 3 4 5 6 7 8 9 Optimum 1 930 m 3

38. ___________________________________________________________________________ ___________________________________________________________________________ Operations Research  Jan Fábry Operations Research  Jan Fábry Southern Channel Optimum 2 450 m 3 2 4 6 8 780 400 200 1050780 5 7 800 150 1400 360 840 3 470 470 210210210210 250 1 2 3 4 5 6 7 8 9