Planning of Barus & Holley Addition

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Presentation transcript:

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

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

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

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

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

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)

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 Row Player: Ray (i) Column Maxima: 7 3 5 6

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 Row Player: Ray (i) MaxiMin Column Maxima: 7 3 5 6 Game has a saddle point! MiniMax

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

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

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

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

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

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

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 -2 Row Player: Column Maxima: 10 5

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 -2 MaxiMin Row Player: Column Maxima: 10 5 MiniMax VR VC No Saddle Point!

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

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

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

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

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