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

2. employees jobs projects managers jobs machines service teams cars doctors night shifts Linear Programming Applications Assignment Problem Assignment „1 to 1“ Objective: maximize the effect of assignment ___________________________________________________________________________ Operations Research  Jan Fábry

3. Linear Programming Applications Assignment Problem Example – Prague Build, Inc. • 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 Objective: minimize total distance necessary for all movements ___________________________________________________________________________ Operations Research  Jan Fábry

4. ___________________________________________________________________________ Operations Research  Jan Fábry

5. Linear Programming Applications Assignment Problem Example – Prague Build, Inc. Distances ___________________________________________________________________________ Operations Research  Jan Fábry

6. Linear Programming Applications Assignment Problem Example – Prague Build, Inc. Decision variables ___________________________________________________________________________ Operations Research  Jan Fábry

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

8. Linear Programming Applications Assignment Problem Example – Prague Build, Inc. Optimal solution ___________________________________________________________________________ Operations Research  Jan Fábry

9. Linear Programming Applications Assignment Problem Example – Prague Build, Inc. Optimal solution ___________________________________________________________________________ Operations Research  Jan Fábry

10. ___________________________________________________________________________ Operations Research  Jan Fábry

11. Network Models ___________________________________________________________________________ Operations Research  Jan Fábry

12. UNDIRECTED DIRECTED j i j i UNDIRECTED NETWORK DIRECTED NETWORK Network Models • Network • Nodes • Arcs ___________________________________________________________________________ Operations Research  Jan Fábry

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

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

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

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

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

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

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

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

21. yij Values Arcs j j i i Nodes yi yj - distance - time - cost - capacity Network Models • Evaluated Network ___________________________________________________________________________ Operations Research  Jan Fábry

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

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

24. 18 2 5 14 23 12 30 10 1 15 4 15 25 6 3 16 2 4 5 6 1 3 Network Models Shortest Path Problem Shortest path between 2 nodes ___________________________________________________________________________ Operations Research  Jan Fábry

25. Network Models Shortest Path Problem Shortest Paths Between All Pairs of Nodes ___________________________________________________________________________ Operations Research  Jan Fábry

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

27. 18 2 5 14 Homecity 23 12 30 1 10 1 15 4 15 25 6 3 16 1 5 4 2 6 3 Network Models Traveling Salesperson Problem (TSP) ___________________________________________________________________________ Operations Research  Jan Fábry

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

29. Network Models 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 ___________________________________________________________________________ Operations Research  Jan Fábry

30. 85 2 7 61 88 Power 76 63 9 70 1 90 1 5 55 40 43 54 75 80 8 3 60 74 68 35 52 10 4 120 6 71 7 6 9 5 2 8 10 3 4 Network Models Minimal Spanning Tree Example - Exhibition ___________________________________________________________________________ Operations Research  Jan Fábry

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

32. Capacitednetwork Input Output 7 9 3 Network Models Maximum Flow Problem Source GasFluidTrafficInformationPeople Sink ___________________________________________________________________________ Operations Research  Jan Fábry

33. UNDIRECTEDARC DIRECTEDARC j j i i Network Models Maximum Flow Problem Flow Flow  Capacity ___________________________________________________________________________ Operations Research  Jan Fábry

34. Network Models 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 ___________________________________________________________________________ OperationsResearch Jan Fábry

35. Network Models 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 ___________________________________________________________________________ OperationsResearch Jan Fábry

36. 740 780 300 900 1050 270 800 370 410 470 520 230 1100 700 280 370 720 800 1400 840 800 550 400 300 1510 250 660 290 130 220 700 420 8 2 3 5 7 8 4 6 2 3 7 6 5 4 Northern Channel Reservoir Lake Southern Channel ___________________________________________________________________________ Operations Research  Jan Fábry

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

38. 780 2 6 1050 780 470 3 470 9 200 1 800 1400 840 4 8 400 250 210 5 360 150 7 3 8 7 6 4 2 5 Southern Channel ___________________________________________________________________________ Operations Research  Jan Fábry