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Planning of Barus & Holley Addition

Planning of Barus & Holley Addition. Activity Duration, Days Predecessors. P rocurement. 215. S ite P reparation. 60. P our F oundation. 31. SP. E rect S teel . 15. P, PF. R oof, E xt. W all. 40. ES. F abricate G lass. 50. ES. I nt W alls & G l. 85. REW.

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Planning of Barus & Holley Addition

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  1. Planning of Barus & Holley Addition Activity Duration, Days Predecessors Procurement 215 Site Preparation 60 Pour Foundation 31 SP Erect Steel 15 P, PF Roof, Ext. Wall 40 ES Fabricate Glass 50 ES Int Walls & Gl. 85 REW Landscaping 55 REW Acquire Furn. 280 Install Furn. 15 IWG, AF, FG

  2. ES(I) = Earliest Start of Activities emanating from node I Forward Pass: Find ES(I) 60 270 2 5 PF L REW SP IWG P ES FG 9 3 4 1 6 7 370 215 230 280 355 0 AF IF 8 355

  3. LC(I) = Latest Completion of Activities terminating at Node I. Backward Pass: Find LC(I) 184 270 60 270 2 5 PF L REW SP IWG P ES FG 9 3 4 1 6 7 370 215 230 280 355 0 215 230 370 355 0 355 AF IF 8 355 355

  4. 8 Identify Critical Path 184 270 60 270 2 5 PF L REW SP IWG P ES FG 9 3 4 1 6 7 370 215 230 355 280 0 215 370 230 355 355 0 AF IF 355 355

  5. 8 Determine Slack Times 184 270 60 270 2 5 L (45) SP(124) PF(124) REW IWG ES P P FG(75) 9 3 4 1 6 7 370 215 230 355 280 0 215 370 230 355 355 0 AF(75) IF 355 355

  6. Two-Person, Zero-Sum Game: The Campers Matrix of Payoffs to Row Player: Column Player: Carol (j) (1) (2) (3) (4) (1) (2) (3) (4) 7 2 5 1 2 2 3 4 5 3 4 4 3 2 1 6 Row Player: Ray (i)

  7. Two-Person, Zero-Sum Game: The Campers Column Player: Carol (j) Matrix of Payoffs to Row Player: Row Minima: (1) (2) (3) (4) (1) (2) (3) (4) 7 2 5 1 2 2 3 4 5 3 4 4 3 2 1 6 1 2 3 1 Row Player: Ray (i) Column Maxima: 7 3 5 6

  8. Two-Person, Zero-Sum Game: The Campers Column Player: Carol (j) Matrix of Payoffs to Row Player: Row Minima: (1) (2) (3) (4) (1) (2) (3) (4) 7 2 5 1 2 2 3 4 5 3 4 4 3 2 1 6 1 2 3 1 Row Player: Ray (i) MaxiMin Column Maxima: 7 3 5 6 Game has a saddle point! MiniMax

  9. Two-Person, Zero-Sum Game: Advertising Matrix of Payoffs to Row Player: Column Player: 0 TV N TVN 0 TV N TVN 0 -.6 -.4 -1 .6 0 .2 -.4 .4 -.2 0 -.6 1 .4 .6 0 Row Player:

  10. Two-Person, Zero-Sum Game: Advertising Matrix of Payoffs to Row Player: Column Player: Row Minima: 0 TV N TVN 0 TV N TVN 0 -.6 -.4 -1 .6 0 .2 -.4 .4 -.2 0 -.6 1 .4 .6 0 -.6 -.4 -.6 0 Row Player: Column Maxima: 1 .4 .6 0

  11. Two-Person, Zero-Sum Game: Advertising Matrix of Payoffs to Row Player: Column Player: Row Minima: 0 TV N TVN 0 TV N TVN 0 -.6 -.4 -1 .6 0 .2 -.4 .4 -.2 0 -.6 1 .4 .6 0 -.6 -.4 -.6 0 Row Player: MaxiMin Column Maxima: 1 .4 .6 0 MiniMax Game has a saddle point!

  12. Two-Person, Zero-Sum Game: Advertising Matrix of Payoffs to Row Player: Column Player: Row Minima: 0 TV N TVN 0 TV N TVN 0 .2 -.4 -.2 0 -.6 .4 .6 0 -.4 -.6 0 Row Player: MaxiMin Column Maxima: .4 .6 0 Game has a saddle point! MiniMax

  13. Two-Person, Zero-Sum Game: Advertising Matrix of Payoffs to Row Player: Column Player: Row Minima: 0 TV N TVN 0 TV N TVN 0 -.6 .6 0 -.6 0 Row Player: MaxiMin Column Maxima: .6 0 Game has a saddle point! MiniMax

  14. Two-Person, Zero-Sum Game: Advertising Matrix of Payoffs to Row Player: Column Player: Row Minima: 0 TV N TVN 0 TV N TVN 0 0 Row Player: MaxiMin Column Maxima: 0 Game has a saddle point! MiniMax

  15. Two-Person, Zero-Sum Game: Mixed Strategies Column Player: Matrix of Payoffs to Row Player: Row Minima: C1 C2 R1 R2 0 5 10 -2 0 -2 Row Player: Column Maxima: 10 5

  16. Two-Person, Zero-Sum Game: Mixed Strategies Column Player: Matrix of Payoffs to Row Player: Row Minima: C1 C2 MaxiMin R1 R2 0 5 10 -2 0 -2 Row Player: Column Maxima: 10 5 MiniMax VC VR No Saddle Point!

  17. Two-Person, Zero-Sum Game: Mixed Strategies Column Player: Matrix of Payoffs to Row Player: Row Minima: Y1 Y2 C1 C2 MaxiMin X1 R1 X2 R2 0 5 10 -2 0 -2 Row Player: Column Maxima: 10 5 MiniMax MiniMax MaxiMin No Saddle Point!

  18. Graphical Solution VR 10 VR < 10(1-X1) VR < -2 +7X1 50/17 Optimal Solution: X1=12/17, X2=5/17 VRMAX=50/17 0 1 12/17 X1

  19. Graphical Solution VR 10 VR < 10(1-X1) Y1=1 Y1=.75 Y1=0 VR < -2 +7X1 Y1=.5 50/17 Y1=.25 Optimal Solution: X1=12/17, X2=5/17 VRMAX=50/17 0 1 12/17 X1

  20. Two-Person, Zero-Sum Games: Summary • Represent outcomes as payoffs to row player • Evaluate row minima and column maxima • If maximin=minimax, players adopt pure strategy corresponding to saddle point; choices are in stable equilibrium -- secrecy not required • If maximin minimax, use linear programming to find optimal mixed strategy; secrecy essential • Number of options to consider can be reduced by using iterative dominance procedure

  21. The Minimax Theorem “Every finite, two-person, zero-sum game has a rational solution in the form of a pure or mixed strategy.” John Von Neumann, 1926

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