# Sequential Updating of Decisions and GamesPlans under Uncertainty

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```					 Sequential Updating of Decisions
and Games/Plans under Uncertainty

Berc Rustem

Imperial College London
Department of Computing
SEAS DTC Conference, Edinburgh, June 2008

1
Agenda
n Multiple asset allocation to targets
n Central planning without complete
information at the center
n Decisions in a Dynamic Setup

2
Assignment of multiple assets
n   Allow multiple assets to visit targets
n   Higher probability of mission success
n   Framework : target score and MILP

3
Assignment/routing example
3           5               7   target scores

Targets

2   2   2       4       4   6      Sub-assets needed
to obtain max
target score

Assets

3               7           sub-assets

4
Assignment / routing

5
Target score maximisation
n Objective : sum of target score for each
target visited
n Constraints : do not exceed capacity

6
Assignment/Routing Constraints
3           5                    7

2   2   2       4       4       6

•Target 1 reachable by assets 1,2
•Target 2 reachable by assets 1,2
•Target 3 reachable by asset 2
3               7
•Maximum score achieved : 12

7
Allow multiple assets to visit a
target
3                      5                     7

1/2          1/2       2/4       2/4   3/6

3                      7
v       t         mvt/tmtv           Completion         Score
1        2           1/2                  50%             2.5
1        3           2/4                  50%             3.5
2        1           1/2                  50%             1.5
2        2           2/4                  50%             2.5
2        3           3/6                  50%             3.5
Total Score    15     8
Vehicle routing
n Assignment decides which vehicle visits
which target.
n Routing decides the order with which a
vehicle visits a target.

9
Routing example
3                         5               7

2/4                   3/6

1/2       2/4
1/2

3                     7

10
Routing
n Routing demands a larger number of
constraints
n Problem is more difficult to solve due to
dimensionality
n Problem has an exploitable pattern

11
Dantzig Wolfe Decomposition
n Parts of the overall problem are
independent
n Problem is split into
¨ Subproblems  : independent part of problem
¨ Master problem : ties together subproblems

12
Central Planning
without complete information
at the center
Evaluate plans
se
ns
it   ivi
ty
it ivi                                                                                 ty
s
s   en                                                              Su
bm
la n                                            it
it p             Submit plan                                pl
an
bm

sensitivity
Su

13
Dantzig Wolfe Decomposition
&
Routing
n subproblem : calculate best route for a
vehicle. Objective contains sensitivity
information from master problem.
n Master problem : decides which routes
(subproblem solution) will be used. Also
calculates sensitivity information.

14
Central Planning
without complete information
at the center
Master problem
se
ns
it   ivi
ty
it ivi                                                                 ty
s
s   en
R
1                               ou
e                                        te
ut            Route 2                            3
Ro

sensitivity

15
Decomposition Stopping
criterion

n   Newly generated routes do not contribute
to optimality.

16
Independence of subproblems
n Block Angular Form
n Li,Ai,bi,ci refer to vehicle i (i=1,2,3)

C1        C2        C3
L1        L2        L3        b0
A1                            b1
A2                  b2
A3        b3

17
Decisions in a Dynamic Setup
n   Updating Decisions for Setup Changes (Repair)
n   Anticipating Opponent’s Strategy (Games)

18
Updating Decisions for Setup Changes
(Repair)

19
Updating Decisions for Setup Changes
(Repair)

20
Updating Decisions for Setup Changes
(Repair)

21
Updating Decisions for Setup Changes
(Repair)          Solution method:
Find the difference between the
initial problem and the current
one and use a similar solution
technique on the difference

22
Anticipating Opponent’s Strategy
(Games)

Opponent                    Asset

Defended
Area

Asset   23
Anticipating Opponent’s Strategy
(Games)

24
Anticipating Opponent’s Strategy
(Games)

25
Anticipating Opponent’s Strategy
(Games)

26
Anticipating Opponent’s Strategy
(Games)

27
Anticipating Opponent’s Strategy
(Games)

28
Anticipating Opponent’s Strategy
(Games)

29
Anticipating Opponent’s Strategy
(Games)

30
Anticipating Opponent’s Strategy
(Games)

31
Anticipating Opponent’s Strategy
(Games)

Choose the Best Strategy amongst
All Possible Strategies

32
Dynamic Decisions

Strategy       .
Static         Dynamic
Static    Integrated       Games
Setup
Dynamic    Repair      Repair of Games

33
Re-optimisation and Repair
n   Respond to dynamic changes to the
environment
malfunction
¨ Vehicle
¨ New observation, better knowledge of the
opponent
¨…

34
Re-optimisation and repair
techniques
n Sensitivity analysis
n Robust Linear Programming (not
analyzed)
n Warm starts
n Constraint Programming

35
Summary
n   Integrated Task Assignment & Routing
•   Multi Asset Assignment
•   Routing Calculus
n   Decisions in a Dynamic Setup
•   Anticipating Opponent’s Strategy (Games)
•   Updating Decisions for Setup Changes (Repair)
•   Sensitivity Analysis
•   Warm Starts

36

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