Probability Collectives

1
Probability Collectives: A Distributed
Optimization for Multi-Agent Systems
Anand J. Kulkarni, Tai Kang
Optimization and Agent Technology Research (OAT Research) Lab
www.oatresearch.org

2
Outline
 Introduction
 Motivation and Objectives
 Probability Collectives (PC)
 Unconstrained PC Formulation
 Validation of the Unconstrained PC
 Constrained Handling Techniques
Heuristic Approach
Penalty Function Approach
Feasibility-based Rule I
Feasibility-based Rule II
 Conclusions
 Future Recommendations

3
Introduction- What are Complex Systems?
 Complex systems: a broad term encompassing a research approach to
problems in the diverse areas such as Social Structures, earthquake
prediction, climate change and weather forecasting, counter-terrorism,
financial systems, project rescheduling, molecular biology,
cybernetics, etc.
 Complex systems generally have many (interconnected) components that
not only interact but also compete with one another to deliver the best they
can to reach the desired system objective.
 Any move by a component affects the moves by other components and so
on. So it is difficult to understand the behavior of the entire system simply by
knowing the individual components and their behavior
Complex Systems in Engineering:
1) Internet Search
2) Manufacturing and Scheduling
3) Supply Chain
4) Sensor Networks
5) Aerospace Systems
6) Telecommunication Infrastructure

4
Introduction- Solving Complex Systems- Centralized
System
Limitations:
1. Communication Overload
2. Computational Overload
3. Large Storage Space
4. Processing Bottleneck
5. Adds Latency (delay)
6. Limited Scalability
7. Reduced Robustness
 A Single/Central Agent is supposed to have all the capabilities such as
problem solving in order to alleviate user’s cognitive load.
 The Agent is provided with general knowledge, storage space, etc. to deal
with wide variety of tasks/computations.
Central
Agent
Tasks/Sensors
Centralized
System

5
Introduction- Solving Complex Systems- Distributed System
Advantages
1. Reduced Risk of Bottleneck
2. Reduced Risk of Latency
3. Robustness
4. Highly Scalable
5. Easy to Maintain & Debug
 In a Decentralized and Distributed System, the total work is decomposed into
different expert modules. Each expert module is an autonomous
agent, i.e. having local control, decision. All Agents achieve their
individual goals contributing towards the system objective.
 Local cooperation is to avoid the duplication of the work.
Challenges
1. Coordination
2. Handling Constraints

Probability Collectives (PC): Motivation and Objectives
• GA, PSO, ACO, Wasp Colony System, Swarm-bot, etc. have been
used for solving complex problems
• As the complexity of the problem domain grew these problems
became quite tedious to be solved using above algorithms.
• Probability Collectives is an emerging AI tool in the framework of
COllective INtelligence (COIN) for modeling and controlling
distributed MAS. Proposed by Dr. David Wolpert in 1999 in a
technical report presented to NASA and further elaborated by S.R.
Bieniawski in 2005.
• It is an obvious tool to deal with the increasing complexity as it
decomposes the problem into sub-problems.
6

State-of-the-Art - Probability Collectives (PC)
• Joint Routing and Resource Allocation in Wireless Sensor Networks
--- Choosing the optimal number of nodes in a cluster and the cluster head
(Ryder et al. 2005, Mohammed et al. 2007)
• Solving the Benchmark Problems
– Multimodality, non-separability, non-linearity, etc. (Huang et al. 2005)
– Robustness, rate of descent, trapping in false minima, etc.
• University Course Scheduling (Autry et al. 2008)
7

State-of-the-Art - Probability Collectives (PC)
8
Mechanical Design
10 bar truss problem
(Bieniawski et al. 2004)
Conflict Resolution
Airplanes Collision Avoidance
(Sislak et al. 2011)
Airplane fleet assignment
(Wolpert et al. 2004)

9
Objectives: Probability Collectives (PC)
 Develop a more generic and powerful approach of PC by
incorporating constraint handling techniques necessary for solving
constrained optimization problems and further test these techniques
by solving a variety of challenging constrained problems
 Solve the path planning of Multiple Unmanned Aerial Vehicles
(MUAVs) by modeling it as a MTSP and solving by the PC approach
 Modify the PC approach to make it more efficient and faster
- inherent and desirable characteristics
- key benefits of being a distributed, decentralized and
cooperative approach

Characteristics of PC
PC works through the COllective INtelligence (COIN) framework
exploiting the advantages of Decentralized, Distributed & Cooperative
approach.
• Deep connections to Game Theory, Statistical Physics &
Optimization
• Successfully exploits the important concept of “Nash Equilibrium”
• PC can be applied to continuous, discrete or mixed variables, etc.,
• Works on Probability Distribution directly incorporating Uncertainty
10

Characteristics of PC
• The Homotopy function for each agent (variable) helps the
algorithm to jump out of the local minima and further reach the
global minima.
• It can successfully avoid the tragedy of commons, skipping the
local minima and further reach the true global minimum.
• It can efficiently handle problems with a large number of variables
i.e. scalable.
• It is robust and can accommodate the agent failure case.
11

Formulation of Unconstrained PC
• Consider a general unconstrained problem (in minimization sense)
comprising variables
• Variables Agents/Players of a game being played iteratively.
• Initially, every agents is given a sampling interval/space
• Every agent randomly samples strategies from within the
corresponding sampling interval .
12
( ) ( )1 2 1
, ,..., ,..., ,i N N
G f X X X X X−
=X
,lower upper
i i i
 Ψ ∈ Ψ Ψ 
N
[ ][1] [2] [ ]
{ , ,..., ,..., } , 1,2,...,imr
i i i i iX X X X i N and= =X
1 2 1... ...i N Nm m m m m−= = = = = =
i im
iΨ

13
{ }[1] [?] [?] [1] [?] [?]
1 2 1, ,..., ,..., ,i i N NX X X X X−=Y
Agent selects its first strategy and samples randomly from other
agents’ strategies as well.
( )[1]
iG Y
1 [ ] [ ][ ][1] [2] [1] [2] [1] [2]
1 1 1 1{ , ,..., } ,..., { , ,..., } ,..., { , ,..., }i Nm mm
i i i i N N N NX X X X X X X X X= = =X X X
{ } ( )
{ } ( )
{ } ( )
{ } ( )
[2] [?] [?] [2] [?] [2]
1 2
[3] [?] [?] [3] [?] [3]
1 2
[ ] [?] [?] [ ] [?] [ ]
1 2
[ ] [ ] [ ][?] [?] [?]
1 2
, ,..., ,...,
, ,..., ,...,
, ,..., ,...,
, ,..., ,...,i i i
i i N i
i i N i
r r r
i i N i
m m m
i i N i
X X X X G
X X X X G
X X X X G
X X X X G
= ⇒
= ⇒
= ⇒
= ⇒
Y Y
Y Y
Y Y
Y Y
M
M
( )[ ]
1
im
r
i
r
G
=
⇒ ∑ Y
i

14
• The ultimate goal of every agent is to identify its strategy value
which contributes the most towards the minimization of the sum
(collection) of these system objectives i.e. .
• Possibly many local minima
• Directly minimizing may require excessive computational efforts
• Homotopy Method: modify the function by converting it into
another topological space by constructing a related and easier
function . This forms the Homotopy function:
( )[ ]
1
im
r
i
r
G
=
∑ Y
i
( )( ) ( ) [ )[ ]
1
, ( ) , 0,
im
r
i i i i
r
J q T G T f T
=
= − ∈ ∞∑X Y X
( )if X

• Analogy to Helmholtz free energy
One of the ways to achieve the thermal equilibrium and hence minimize
the energy to do work is actually minimizing the internal energy
through an annealing schedule, i.e. stepwise drop the temperature of
the system from to achieving the equilibrium
in every step.
15
( )( ) ( ) [ )[ ]
1
, ( ) , 0,
im
r
i i i i
r
J q T G T f T
=
= − ∈ ∞∑X Y X
L D T S= −
Energy available to do work Internal energy Spontaneous (Random) energy
initialT T= 0 finalT or T T→ →

Deterministic Annealing
• It suggests conversion of the variables into random real valued
probabilities which converts the into .
16
( )( ) ( ) [ )[ ] [ ] [ ]
2
1 1
, ( ) ( )log ( ) , 0,
i im m
r r r
i i i i i
r r
J q T E G T q X q X T
= =
 
= − − ∈ ∞ ÷
 
∑ ∑X Y
[ ]
1
( )
im
r
i
r
G
=
∑ Y ( )[ ]
1
( )
im
r
i
r
E G
=
∑ Y
( )( ) ( ) [ )[ ]
1
, ( ) , 0,
im
r
i i i i
r
J q T G T f T
=
= − ∈ ∞∑X Y X
( )( ) ( ) [ )[ ]
1
, ( ) , 0,
im
r
i i i i
r
J q T E G T S T
=
= − ∈ ∞∑X Y

17
0
0.05
0.1
0.15
1 2 3 4 5 6 7 8 9 10
0
0.05
0.1
0.15
1 2 3 4 5 6 7 8 9 10
[ ]
( ) [ ]
( )1
1 1 1... 1/im
q X q X m= = = [ ]
( ) [ ]
( )1
... 1/im
N N Nq X q X m= = =
L
0
0.05
0.1
0.15
1 2 3 4 5 6 7 8 9 10
L
[ ]
( ) [ ]
( )1
... 1/im
i i iq X q X m= = =
Agent 1 Agent i Agent N
{ }
{ }
{ }
[1] [?] [1] [?]
1
[ ] [?] [ ] [?]
1
[ ] [ ][?] [?]
1
,..., ,...,
,..., ,...,
,..., ,...,i i
i i N
r r
i i N
m m
i i N
X X X
X X X
X X X
=
=
=
Y
Y
Y
M
M
( )( )[ ]
1
im
r
i
r
E G
=
∑ Y
( ) ( ) ( ){ } [ ]
( ) [ ]
( ) ( )
[ ]
( )( )
( )( )
( ) ( ) ( ){ } [ ]
( ) [ ]
( ) ( )
[ ]
( )( )
( )( )
( ) ( ) ( ){ } [ ]
( ) [ ]
( ) ( )
[ ]
( )( )
( )( )
1 1 ?[?] [1] [?] [1]
1
?[?] [ ] [?] [ ]
1
?[ ] [ ][?] [?]
1
,..., ,..., Y
,..., ,..., Y
,..., ,..., Y i ii i
i N i i ii
i
r rr r
i N i i ii
i
m mm m
i N i i ii
i
q X q X q X G q X q X E G
⇒ =
⇒ =
⇒ =
∏
∏
∏
Y
Y
Y
M
M
Strategies Strategies Strategies

• The minimization of the Homotopy function can be carried out using
a suitable second order optimization approach such Nearest
Newton Descent Scheme as well as Broyden-Fletcher-Goldfarb-
Shanno (BFGS) scheme.
18
0
0.05
0.1
0.15
1 2 3 4 5 6 7 8 9 10
0
0.05
0.1
0.15
1 2 3 4 5 6 7 8 9 10
0
0.05
0.1
0.15
1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1 2 3 4 5 6 7 8 9 10
[ ] [ ] [ ] [ ] [ ] [ ]
{ } [ ]
( )1 2 1, ,..., ,..., ,fav fav fav fav fav fav fav
i N NX X X X X G−= ⇒Y Y
Favorable Strategy Favorable Strategy Favorable Strategy

• Updating of the Sampling Interval (Neighboring Method)
• Convergence and Final Solution
 If
 If there is no significant change in the system objectives for
successive considerable number of iterations
19
[ ]
( ) [ ]
( ), , 0 1
fav favupper lower upper lower
i i down i i i down i i downX Xλ λ λ Ψ ∈ − Ψ − Ψ + Ψ − Ψ < ≤
 
[ ] [ ] [ ] [ ] [ ]
{ } [ ], , , , , ,
1 2 1, ,..., , ( )
fav final fav final fav final fav final fav final fav final
N NX X X X G−= ⇒Y Y
0finalT T or T= →
[ ] [ ], , 1
( ) ( )
fav n fav n
G G ε−
− ≤Y Y

20
Nash Equilibrium (Necessary Properties):
Rationality: Select the best possible strategy by guessing other agents’
strategies
Convergence: Same class policy of selecting the best possible strategy and
guessing other agents’ strategies (guaranteed: policy does not change)
Nash Equilibrium in PC
: by guessing other agents’ strategies
and : is communicated to every other agent
[ ]fav
iX
[ ]fav
iX [ ]
( )
fav
G Y

21
Solution to Rosenbrock Function using PC
( ) ( ) ( )
1 2 22
1
1
100 1
N
i i i
i
f x x x
−
+
=
 = − + −
  ∑X
where [ ]1 2 3....... Nx x x x=X
lower limit upper limit
1,2,...,
ix
i N
≤ ≤
=
Agents/
(Variables)
Strategy Values Selected with maximum Probability
Trial-1 Trial-2 Trial-3 Trial-4 Trial-5 Range of Values
Agent-1 1.0000 0.9999 1.0002 1.0001 0.9997 -1.0 to 1.0
Agent-2 1.0000 0.9998 1.0001 1.0001 0.9994 -5.0 to 5.0
Agent-3 1.0001 0.9998 1.0000 0.9999 0.9986 -3.0 to 3.0
Agent-4 0.9998 0.9998 0.9998 0.9995 0.9967 -3.0 to 8.0
Agent-5 0.9998 0.9999 0.9998 0.9992 0.9937 1.0 to 10.0
Fun. Value 2 x 10-5
1 x 10-5
2 x 10-5
2 x 10-5
5 x 10-5
Fun. Evals. 288100 223600 359050 204750 242950
Results

22
Solution to Rosenbrock using PC (Comparison)
Method No. of Var./
Agents
Function
Value
Function
Evaluations
Variable Range(s)/
Strategy Sets
CGA 2 0.000145 250 -2.048 to 2.048
PAL 2
5
≈ 0.01
≈ 2.5
5250
100000
-2.048 to 2.048
-2.048 to 2.048
Modified DE 2
5
1 × 10-6
1 × 10-6
1089
11413
-5 to 10
-5 to 10
LCGA 2 ≈ 0.00003 -- -2.12 to 2.12
PC 5 0.00001 223600 -1.0 to 1.0
-5.0 to 5.0
-3.0 to 3.0
-3.0 to 8.0
1.0 to 10.0

Unconstrained Test Problems
1. Ackley Function
2. Beale Function
3. Bohachevsky Function
4. Booth Function
5. Branin Function
6. Colville Function
7. Dixon & Price Function
8. Easom Function
9. Goldstein & Price Function
10. Griewank Function
11. Hartmann Functions
12. Hump Function
13. Levy Function
14. Matyas Function
15. Michalewicz Function
16. Perm Functions
17. Powell Function
18. Power Sum Function
19. Rastrigin Function
20. Rosenbrock Function
21. Schwefel Function
22. Shekel Function
23. Shubert Function
24. Sphere Function
25. Sum Squares Function
26. Trid Function
27. Zakharov Function
23

Constrained PC
• Approach 1: Heuristic Approach
Two variations of the MDMTSP and several cases of
the SDMTSP
• Approach 2: Penalty Function Approach
Three Test Problems
• Approach 3: Feasibility-based Rule I
Two cases of the Circle Packing Problem
Feasibility-based Rule II
Two cases and associated cases of the Sensor
Network Coverage Problem
24

Constrained PC Approach 1: Heuristic Approach
• Explicitly uses the problem specific information and combines them
with the unconstrained optimization technique to push the objective
function into the feasible region.
• Validated by solving two cases of the Multiple Depot Multiple
Traveling Salesmen Problem (MDMTSP) and several cases of the
Single Depot Multiple Traveling Salesmen Problem (SDMTSP)
– Solve the path planning of Multiple Unmanned Aerial Vehicles
(MUAVs) by modeling it as a MTSP
25

26
Multiple Traveling Salesmen Problem (MTSP)
-40 -30 -20 -10 0 10 20 30 40
-40
-30
-20
-10
0
10
20
30
40
2
13
15
D1
1
3
4
5
6
7
8
9
10
11
12
14
D2
D3
-40 -30 -20 -10 0 10 20 30 40
-40
-30
-20
-10
0
10
20
30
40
2
13
15
D1
1
3
4
5
6
7
8
9
10
11
12
14
D2
D3
400
450
500
550
600
650
1 3 5 7 9 11 13 15 17 19 21 23 25 27
Iterations
TotalTravelingCost
(a) Test Case 1 (b) Solution to Test Case 1
Convergence Plot for Test Case 1

27
Multiple Traveling Salesmen Problem (MTSP)
-10 -5 0 5 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
10
14
1
3
7
12
13
1
2
3
4
5
6
8
9
11
15
D1
D2
D3
(a) Test Case 2 (b) Solution to Test Case 2
60
110
160
210
260
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37
Iterations
TotalTravelingCost
Convergence Plot for Test Case 2
-10 -5 0 5 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
10
14
1
3
7
12
13
1
2
3
4
5
6
8
9
11
15
D1
D2
D3

28
MTSP (Randomly Located Nodes)
10 20 30 40 50 60 70 80 90 100
1
2
34
5
6
7
8
9
1
2
34
5
6
7
8
9
1
2
3
4
5
6
7
8 9
10
11 12
13
14
15
D
Randomly Located Nodes (Sample Case 1) Randomly Located Nodes (Sample Case 2)
20 30 40 50 60 70 80 90 100
0
10
20
30
40
50
60
70
80
90
100
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
2
3
4
5
6
7 8
9
10
11
12
13
14
15
D

29
(MTSP) Comparison
Method Nodes Vehicles Avg. CPU Time
(Minutes)
PDP*
MTSP to std. TSP 20
20
20
2
3
4
2.05
2.47
2.95
--
Cutting Plane 20
20
20
2
3
4
1.71
1.50
1.44
--
--
Elastic Net+
22
22
22
2
3
4
12.03
13.10
12.73
28.71
74.18
33.33
Branch on an Arc with
LB0
15 3 0.56 --
Branch on an Arc with
LB2
15
20
3
4
1.01
2.32
--
Branch on an Route with
LB0
15 3 0.44 --
PC (MDMTSP)
PC (SDMTSP)
15 (Case 1)
15 (Case 2)
15
3
3
2.09
1.27
3.34
0.00
0.00
2.94
PDP*: percent deviation from the average solution
+: unable to reach the optimum

30
MTSP (Heuristic Techniques)
Insertion Heuristic
-20 -15 -10 -5 0 5 10 15 20 25
15
10
-5
0
5
10
15
20
1
2
3
11
2
6
78
9
8
14
15
1
2
D1
10
1
6
78
9
7
14
15
1
2
D2
4
5
6
78
9
6
9
14
15
Vehicle 1: D1-5-6-7-8-9-10-D1 Vehicle 2: D2-5-1-2-3-4-10-D2
Vehicle 1: D1-5-6-7-8-6-9-10-D1
Vehicle 2: D2-5-1-2-3-4-11-D2
Vehicle 1: D1-5-6-7-8-6-9-10-D1
Vehicle 2: D2-5-1-2-3-4-10-D2
Inter-vehicle Repetition
-20 -15 -10 -5 0 5 10 15 20 25
-15
-10
-5
0
5
10
15
20
1
2
3
11
2
6
78
9
8
14
15
1
2
D1
10
1
6
78
9
7
14
15
1
2
D2
4
5
6
78
9
6
9
14
15
13
Vehicle 1: D1-5-6-7-8-6-9-10-D1 Vehicle 2: D2-5-1-2-3-4-11-D2
Elimination Heuristic

31
MTSP (Heuristic Techniques)
Elimination Heuristic
Swapping
Heuristic
-20 -15 -10 -5 0 5 10 15 20 25
-15
-10
-5
0
5
10
15
20
2
3
11
2
6
78
9
8
2
D1
10
1
6
78
9
7
2
D2
4
5
6
78
9
6
9
13
Vehicle 1: D1-5-6-7-8-6-9-10-D1 Vehicle 2: D2-1-2-3-4-11-D2
-20 -15 -10 -5 0 5 10 15 20 25
-15
-10
-5
0
5
10
15
20
1
2
3
11
2
6
78
9
8
14 1
2
D1
10
1
6
78
9
7
14 1
2
D2
4
5
6
78
9
6
9
14
13
Vehicle 2: D2-1-2-3-4-11-D2Vehicle 1: D1-5-6-7-8-9-10-D1
-20 -15 -10 -5 0 5 10 15 20 25
-15
-10
-5
0
5
10
15
20
1
2
3
11
2
8
14
15
1
2
D1
10
1
7
14
15
1
2
D2
4
5
6
9
14
15
13
Vehicle 2: D2-1-2-3-4-11-D2Vehicle 1: D1-5-6-7-8-9-10-D1
-20 -15 -10 -5 0 5 10 15 20 25
5
0
5
0
5
0
5
0
1
2
3
11
2
8
14
15
1
2
D1
10
1
7
14
15
1
2
D2
4
5
6
9
14
15
Vehicle 2: D2-5-6-7-8-9-10-D2Vehicle 1: D1-1-2-3-4-11-D1 Vehicle 1: D1-1-2-3-4-11-D1
Vehicle 2: D2-5-6-7-8-9-10-D2
Vehicle 1: D1-5-6-7-8-9-10-D1
Vehicle 2: D2-1-2-3-4-11-D2
Vehicle 1: D1-5-6-7-8-6-9-10-D1
Vehicle 2: D2-1-2-3-4-11-D2
Vehicle 1: D1-5-6-7-8-9-10-D1
Vehicle 2: D2-1-2-3-4-11-D2
Intra-vehicle
Repetition

Constrained PC (Approach 2): Penalty Function Approach
32
• Penalty based methods are the most generalized constraint handling
methods: simplicity, ability to handle non linear constraints and
compatibility with most of the unconstrained optimization methods
• Converts constrained optimization problem into unconstrained one.
[ ]
( ) ( ) ( ) ( )
2 2
[ ] [ ] [ ]
1 1
s t
r r r r
i i j i j i
j j
G g hφ θ +
= =
     = + +    
  
∑ ∑Y Y Y Y
( ) ( )( )[ ] [ ]
max 0,r r
j i j iwhere g g and is scalar parameterθ+
=Y Y

Every agent obtains the probability distribution
identifying its favorable strategy
START
Every agent sets up a strategy set. Initialize ‘n’, ‘T’
Every agent forms a combined strategy set for its every
strategy and computes system objectives and
constraints, and corresponding collection of pseudo
system objectives
Every agent assigns uniform probabilities to its
strategies and computes expected collection of system
objectives
Every agent forms a modified Homotopy function
Every agent minimizes the Homotopy function using
Nearest Newton Method/BFGS Method
Compute the global objective function and associated
constraints
1
2
33

Accept current objective function and
related favorable strategies
N
Discard current and retain previous objective
function with related favorable strategies
STOP
Accept final values
Convergence ?
Y
YN
Maximum constraint
value ≤
1
2
PC…
34
Every agent updates its sampling interval and
forms corresponding updated strategy set, and
Update the Penalty Parameter
µ
A
B

Spring Design
35
( ) ( )
( )
( )
( )
( )
( )
2
3 2 1
3
2 3
1 4
1
2
2 1 2
2 23 4
12 1 1
1
3 2
2 3
1 2
4
1 2 3
Minimize 2
Subject to 1 0
71785
4 1
1 0
510812566
140.45
1 0
1 0
1.5
where 0.05 2, 0.25 1.3, 2 15
f x x x
x x
g
x
x x x
g
xx x x
x
g
x x
x x
g
x x x
= +
= − ≤
−
= + − ≤
−
= − ≤
+
= − ≤
≤ ≤ ≤ ≤ ≤ ≤
X
X
X
X
X

Spring Design
No. of runs Avg. CPU time Best Sol. Mean Sol. Worst Sol. % with Best Sol.
10 24.5 Sec 0013500 0.02607 0.05270 6.63
36
Design
variables &
Constrains
Best Solutions Found
Cultural
algorithm
Constraint
correction
algorithm
Self-adaptive
penalty
app.
Multi-obj.
app.
GA
HPSO Proposed
PC
0.050000 0.053390 0.051480 0.051980 0.051700 0.050600
0.317390 0.399180 0.351660 0.363960 0.357120 0.327810
14.031790 9.185400 11.632200 10.890520 11.265080 14.056700
0.000000 0.000010 -0.003300 -0.001900 -0.000000 -0.052900
-0.000070 -0.000010 -0.000100 0.000400 0.000000 -0.007400
-3.967960 -4.123830 -4.026300 -4.060600 -4.054600 -3.704400
-0.755070 -0.698280 -0.731200 -0.722700 -0.727400 -0.747690
0.012720 0.012730 0.012700 0.012680 0.012660 0.013500
Fun. Evals 80000 5214
2
x
3
x
1
g
2
g
3
g
4
g
f
0 100 200 300 400 500
0
10
20
30
40
50
60
70
Iterations
f(X)
1
x

Himmelblau Function
No of runs 10
Avg. CPU time 11 Mins
Best Sol. -30641
Mean Sol. -30635
Worst Sol. -30626
% with Best Sol 0.078
37
( )
( )
( )
( )
2
3 1 5 1
1 2 5 1 4 3 5
2 2 5 1 4 3 5
3
Minimize 5.3578547 0.8356891 37.293239 40792.141
Subject to 85.334407 0.0056858 0.0006262 0.0022053 92 0
85.334407 0.0056858 0.0006262 0.0022053 0
80.51249 0
f x x x x
g x x x x x x
g x x x x x x
g
= + + −
= + + − − ≤
= − − − + ≤
= +
X
X
X
X
( )
( )
( )
2
2 5 1 2 3
2
4 2 5 1 2 3
5 3 5 1 3 3 4
6 3 5
.0071317 0.0029955 0.0021813 110 0
80.51249 0.0071317 0.0029955 0.0021813 90 0
9.300961 0.0047026 0.0012547 0.0019085 25 0
9.300961 0.0047026 0.0012547
x x x x x
g x x x x x
g x x x x x x
g x x
+ + − ≤
= − − − − + ≤
= + + + − ≤
= − − −
X
X
X
( )
1 3 3 4
1 2
0.0019085 20 0
where 78 102, 33 45, 27 45, 3,4,5i
x x x x
x x x i
− + ≤
≤ ≤ ≤ ≤ ≤ ≤ =
0 500 1000 1500 2000 2500 3000 3500
-3.1
-3
-2.9
-2.8
-2.7
-2.6
-2.5
-2.4
-2.3
-2.2
x 10
4
Iterations
f(X)

Algorithm Best Mean Worst Std Dev Average FE
Cultural Algorithm -30665.5000 -30662.5000 -30636.2000 9.3
Cultural Differential
Evolution
-30665.5386 -30665.5386 -30665.5386 0.00000
Homomorphous
Mapping
-30664.0000 -30655.0000 -30645.0000 - 1400000
Filter SA -30665.5380 -30665.4665 -30664.6880 0.173218 86154
Self Adaptive
Penalty Approach
-31020.8590 -30984.2400 -30792.4070 73.633
Gradient Repair
Method
-30665.5386 -30665.3538 -30665.5386 0.000000 26981
Multiobjective
Approach
-30665.5386 -30665.3539 -30665.5386 0.000000 66400
Bubble Sort
Algorithm
-30665.5390 -30665.5390 -30665.5390 1.1E-11 350000
PSO (Zahara et al.
2008)
-30665.5386 -30665.35386 -30665.5386 0.000000 19658
Feasibility-based
Rule (Deb (2000))
-30665.5370 -- -29846.6540 --
Dynamic Penalty
Scheme
-30665.5000 -30665.2000 -30663.3000 4.85E-01 --
PSO (Hu et al. 2002) -30665.5000 -30665.5000 -30665.5000 -- --
PSO (Dong et al.
(2005))
-30664.7000 -30662.8000 -30656.1000 -- --
Multi-criteria
Approach
-30651.6620 -30647.1050 -30619.0470 35408
HPSO -30665.5390 -30665.5390 -30665.5390 1.7E-06
Stratum Approach -30373.9500 -- -30175.8040 -- --
PC -30641.5702 -30635.4157 -30626.7492 7.5455 2780449
38

Chemical Equilibrium Problem
39
( )
( )
( )
( )
10
1 1 2 10
1 1 2 3 6 10
2 4 5 6 7
3 3 7 8 9 10
1 2 3 4 5
Minimize ln
...
Subject to 2 2 2 0
2 1 0
2 1 0
0.000001, 1,2,...,10
where 6.089 17.164 34.054 5.914 24.721
j
j j
j
i
x
f x c
x x x
h x x x x x
h x x x x
h x x x x x
x i
c c c c c
=
 
= + ÷
+ + + 
= + + + + − =
= + + + − =
= + + + + − =
≥ =
= − = − = − = − = −
∑X
X
X
X
6 7 8 9 1014.986 24.100 10.708 26.662 22.179c c c c c= − = − = − = − = −

Chemical Equilibrium Problem
40
Best Solutions Found
Design
Variables
Hock et al.
(1981)
GENOCOP PC
0.01773548 0.04034785 0.0308207485
0.08200180 0.15386976 0.2084261218
0.88256460 0.77497089 0.6708869580
0.0007233256 0.00167479 0.0371668767
0.4907851 0.48468539 0.3510055351
0.0004335469 0.00068965 0.1302810195
0.01727298 0.02826479 0.1214712339
0.007765639 0.01849179 0.0343070642
0.01984929 0.03849563 0.0486302636
0.05269826 0.10128126 0.0486302636
8.6900E-08 6.0000E-08 -0.0089160590
0.0141 1.0000E-08 -0.0090697995
5.9000E-08 -1.0000E-08 -0.0047181958
-47.707579 -47.760765 -46.7080572120
Average FE -- -- 389546
1x
2x
3x
4x
5x
6x
7x
8x
9x
10x
( )1h X
( )2h X
( )3h X
( )f X
0 200 400 600 800 1000
-7
-6
-5
-4
-3
-2
-1
0
x 10
4
Iterations
f(X)
8000 8200 8400 8600 8800 9000
-7000
-6000
-5000
-4000
-3000
-2000
-1000
Iterations
No of runs 10
Avg. CPU time 21.60 Mins
Best Sol. -46.7080572120
Mean Sol. -45.6522267370
Worst Sol. -44.4459333503
% with Best Sol 2.20

Constrained PC (Approach 3): Feasibility-based Rule I
• Feasibility-based rule allows the objective and constraint information
to be considered separately.
• The constraint violation tolerance is tightened iteratively to obtain
the fitter solution and further drive the solution towards the
feasibility.
• Convert the equality constraint into inequality constraints
41
Minimize
Subject to 0 , 1,2,...,
0, 1,2,...,
j
j
G
g j s
h j t
≤ =
= =
0 1,2,...,
0
0
Minimize
Subject to 0 , 1,2,...,
s j j
j
s w j j
j
g h j w
h
g h
G
g j t
δ
δ
+
+ +
= − ≤ =
= ⇒ 
= − − ≤
≤ =

Feasibility-based Rule I:
• Any feasible solution is preferred over any infeasible solution.
• Between two feasible solutions, the one with better objective is
preferred.
• Between two infeasible solutions, the one with fewer violated
constraints is preferred.
42

• Updating of the Sampling Space and Perturbation Approach
In order to jump out of this possible local minimum, every agent
perturbs its current feasible strategy
The value of and +/- sign are selected based on preliminary trials.
Every agent expands the sampling space as follows:
43
i
[ ] [ ] [ ]
( )
( ) [ ]
( ) [ ]
1 1
2 2
1
,
1
,
fav fav fav
i i i i
lower upper
fav
i
i
lower upper
fav
i
X X X fact
randomvalue if
X
where fact
randomvalue if
X
σ σ γ
σ σ γ
= ± ×

∈ ≤

= 
 ∈ >

1 1 2 20 1lower upper lower upper
σ σ σ σ< < ≤ < <
γ
( ) ( ), , 0 1lower upper lower upper upper lower
i i up i i i up i i upλ λ λ Ψ ∈ Ψ − Ψ − Ψ Ψ + Ψ − Ψ < ≤
 

44
( ) ( )
2 2
1
2 2
Minimize
Subject to
0.001
2
, 1,2,...,
z
i
i
i j i j i j
i i l
i i u
i i l
i i u
i
f L r
x x y y r r
x r x
x r x
y r y
y r y
Lr
i j z i j
π
=
= −
− + − ≥ +
− ≥
+ ≤
− ≥
+ ≤
≤ ≤
= ≠
∑
Circle Packing Problem Formulation
Tragedy of Commons
4 5 6 7 8 9 10 11
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
Shipping, Apparel,
Automobile,
Aerospace, Food
Industry, etc.

Circle Packing Problem (Case 1): Solution History
45
4 5 6 7 8 9 10 11
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
X-Coordinates
Y-Coordinates
2
5
4
1
3
4 5 6 7 8 9 10 11
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
X-Coordinates
Y-Coordinates
1
2
3
4
5
4 5 6 7 8 9 10 11
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
X-Coordinates
Y-Coordinates
1
2
3
5
4
4 5 6 7 8 9 10 11
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
X-Coordinates
Y-Coordinates
12
4
53
4 5 6 7 8 9 10 11
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
X-Coordinates
Y-Coordinates
1
3
2
4
5
Randomly Generated Initial Solution Iteration 401 Iteration 901
Iteration 1001 Stable Solution at iteration 1055
No. of Circles Avg CPU time Avg F.E. No. of runs
5 14.05 Mins 17515 30

Circle Packing Problem (Case 2): Solution History
46
3 4 5 6 7 8 9 10 11 12
3
4
5
6
7
8
9
10
11
12
X-Coordinates
Y-Coordinates
5
1
3
2
4
4 5 6 7 8 9 10 11
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
X-Coordinates
Y-Coordinates
1
4
3
2
5
4 5 6 7 8 9 10 11
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
X-Coordinates
Y-Coordinates
2
3
4
1
5
4 5 6 7 8 9 10 11
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
X-Coordinates
Y-Cooridnates
1
2
3
4
5
4 5 6 7 8 9 10 11
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
X-Coordinates
Y-Coordinates
1
23
4
5
Randomly Generated Initial Solution Iteration 301 Iteration 401
Stable Solution at iteration 955Iteration 801Iteration 601
No. of Circles Avg CPU time Avg F.E. No. of runs
5 15.70 Min 68406 30
4 5 6 7 8 9 10 11
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
X-Coordinates
Y-Coordinates
1
2
3
4
5

Circle Packing Problem with Agent Failure: Solution History
47
4 5 6 7 8 9 10 11
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
X-Coordinates
Y-Coordinates
1
2
3
4
5
4 5 6 7 8 9 10 11
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
X-Coordinates
Y-Coordinates
2
5
4
1
3
4 5 6 7 8 9 10 11
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
X-Coordinates
Y-Coordinates
2
5
4
1
3
4 5 6 7 8 9 10
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
X-Coordinates
Y-Coordinates
2
4
5
1
3
4 5 6 7 8 9 10 11
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
X-Coordinates
Y-Coordinates
2
5
4
1
3
Randomly Generated Initial Solution Iteration 124 Iteration 231 Iteration 377
Iteration 561 Iteration 723
Stable Solution at iteration
901
Stable Solution at iteration
1051
4 5 6 7 8 9 10 11
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
X-Coordinates
Y-Coordinates
2
5
4 3
1
4 5 6 7 8 9 10 11
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
X-Coordinates
Y-Coordinates
2
5
4
1
3
4 5 6 7 8 9 10
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
X-Coordinates
Y-Coordinates
2
5
4 3
1
No. of Circles Avg CPU time Avg F.E. No. of Trials
5 -- 17365 30

Constrained PC (Approach 3): Feasibility-based Rule II
• Feasibility-based rule II allows the objective and constraint
information to be considered separately.
• In addition to the iterative tightening of the constraint violation
obtaining the fitter solution and further drive the solution towards the
feasibility, the rule helps the solution jump out of possible local
minima.
• Procedure starts with initializing the number of constraints improved
initialized to , i.e. . The value of is updated iteratively.
48
µ
0 0µ = µ

Feasibility-based Rule II:
• Any feasible solution is preferred over any infeasible solution.
• Between two feasible solutions, the one with better objective is preferred.
• Between two infeasible solutions, the one with more number of improved
constraint violations is preferred.
• If the solution remains feasible and unchanged for successive number of
iterations, and current feasible system objective is worse than the
previous feasible solution, accept the current solution.
49

50
Sensor Network Coverage Problem
• Strategic Applications of Sensor Network
Natural disaster relief, Hostile and Hazardous environment monitoring, critical
infrastructure monitoring and protection, Habitat exploration and surveillance,
Situational awareness in battlefield and target detection, Industrial sensing and
diagnosis, Biomedical health monitoring, Seismic sensing, etc.
• How to best deploy/position the sensors over a FoI to achieve best possible
Coverage and Detection capability, connectivity, etc.
• Coverage directly affects the quality and effectiveness of the surveillance/
monitoring provided by the sensor network

51
(Sweep) Barrier coverageBlanket Coverage
Point Set CoverageComplete coverage
Coverage Classification
Deterministic
Static and systematic deployment of the
sensors over certain (or weighted) FoI.
Sensor Network Coverage Problem
Stochastic
Sensor positions are selected based
on some distributions such as
uniform, Gaussian, Poission, etc.

52
( )( ) ( )( )( )
( )( ) ( )( )( )
( )
( )
1 2 1 2
1 2 1 2
Minimize
max , ,..., ,..., min , ,..., ,...,
max , ,..., ,..., min , ,..., ,...,
Subject to
, 2 , 1,2,..., ,
1,2,...,
, ~
i z s i z s
i z s i z s
s
i s l
i s u
i s l
i s u
A x x x x r x x x x r
y y y y r y y y y r
d i j r i j z i j
x r x
x r x
y r y
y r y
i z
d i j i jγ
= + − −
× + − −
≥ = ≠
− ≥
+ ≤
− ≥
+ ≤
=
≤
X
, , , 1,2,...,i j i j z≠ =
Sensor Network Coverage Problem Formulation
3
2
, ,
1 1
3
z
collective c i c i
i i
A A A rπ
= =
= = =∑ ∑
Deploy a set of
Homogeneous sensors
over a certain FoI to
achieve the maximum
possible Deterministic,
Connected Blanket
Coverage

Sensor Network Coverage Problem (Variation 1)
Solution History and Convergence
53
5 6 7 8 9 10 11
4
5
6
7
8
9
10
11
X-Coordinates
Y-Coordinates
0 500 1000 1500 2000 2500 3000
5
10
15
20
25
30
35
40
45
Iterations
AreaoftheEnclosingRectangle
0 500 1000 1500 2000 2500 3000
0
0.5
1
1.5
2
2.5
3
3.5
4
Iterations
CollectiveCoverage

Sensor Network Coverage Problem (Variation 2- Case 1)
54
2 4 6 8 10 12 14
2
4
6
8
10
12
14
X-Coordinates
Y-Coordinates
0 1000 2000 3000 4000 5000 6000 7000 8000
20
40
60
80
100
120
140
160
180
200
220
Iterations
0 1000 2000 3000 4000 5000 6000 7000 8000
0
2
4
6
8
10
12
14
16
18
Iterations
CollectiveCoverage

55
2 4 6 8 10 12 14
0
2
4
6
8
10
12
14
X-Coordinates
Y-Coordinates
0 2 4 6 8 10 12
0
2
4
6
8
10
12
14
X-Coordinates
Y-Coordinates

Summary of Sensor Network Coverage Problem Results
SN Particulars Variation 1 Variation 2
1 Cases -- Case 1 Case 2 Case 3
2 Number of Sensors ( ) 5 5 10 20
3 The Sensing Range ( ) 0.5 1.2 1 0.6
4 Average Collective Coverage 3.927 18.5237 19.4856 16.3631
5 Minimum and Maximum
Collective Coverage
3.9270, 3.9270 18.0920, 18.7552 17.5427, 20.8797 15.5347, 17.3377
6 Standard Deviation associated
with Collective Coverage
0.0000 0.1687 1.1837 1.2217
7 Average area of the Enclosing
Rectangle
5.8311 34.3014 49.0938 39.3480
8 Minimum and Maximum area
of the Enclosing Rectangle
5.7046, 5.9750 33.0448, 39.7099 44.7135, 52.6277 34.1334, 43.8683
9 Standard Deviation associated
with the area of Enclosing
Rectangle
0.1040 1.9899 2.6995 2.8829
10 Average CPU time (Approx.) 20 Mins 1 Hr 2Hrs 3.5 Hrs
11 Average number of Function
Evaluations
90417 315063 1172759 3555493
56
z
sr

57
• Discrete Problems
Variables GA
[31]
HS
[34]
PSO
[36]
PSOPC
[36]
HPSO
[36]
DHPSACO
[37]
Proposed
PC
0.4 0.01 0.01 0.01 0.01 0.01 0.01
2.0 2.0 2.6 2.0 2.0 1.6 0.4
3.6 3.6 3.6 3.6 3.6 3.2 3.6
0.01 0.01 00.01 0.01 0.01 0.01 0.01
0.01 0.01 0.4 0.01 0.01 0.01 2
0.8 0.8 0.8 0.8 0.8 0.8 0.8
2.0 1.6 1.6 1.6 1.6 2.0 0.01
2.4 2.4 2.4 2.4 2.4 2.4 4
563.52 560.59 566.44 560.59 560.59 551.61 477.16684
Variables GA
[31]
PSO
[36]
PSOPC
[36]
HPSO
[36]
DHPSACO
[37]
Proposed
PC
0.307 1.000 0.111 0.111 0.111 0.111
1.990 2.620 1.563 2.130 2.130 0.563
3.130 2.620 3.380 3.380 3.380 3.13
0.111 0.250 0.111 0.111 0.111 0.141
0.141 0.307 0.111 0.111 0.111 1.8
0.766 0.602 0.766 0.766 0.766 0.766
1.620 1.457 1.990 1.620 1.620 0.111
2.620 2.880 2.380 2.620 2.620 3.88
556.49 567.49 567.49 551.14 551.14 464.14708
Case 1: The discrete variables are selected from
the set
{0.01, 0.4, 0.8, 1.2, 1.6, 2.0, 2.4, 2.8, 3.2, 3.6,
4.0, 4.4, 4.8, 5.2, 5.6, 6.0}
.

Conclusions and Original Contributions
60
Improvements to the original PC approach:
• The original PC approach was improved with a reduction in the
computational complexity.
- A neighboring scheme developed for updating the solution space
was developed which contributed to faster convergence and
improved efficiency of the overall algorithm.
- the modified PC was successfully validated optimizing
Rosenbrock Function
- Nash Equilibrium successfully formalized and demonstrated

Constraint Handling Techniques
• A number of constraint handling techniques were developed . This
allowed PC to solve practical problems which inevitably are
constrained problems.
• Problem specific heuristics were developed and incorporated into
the PC algorithm for solving the NP-hard problem such as MTSP.
• True optimum solution was achieved for two specially developed
cases of the MDMTSP, several cases of the SDMTSP were also
solved.
• For the first time, the MTSP was solved using a distributed,
decentralized and cooperative approach such as PC.
61

• Penalty function approach was successfully incorporated and tested
by solving variety of test problems with in/equality constraints.
• Feasibility-based rule I was successfully formalized and
demonstrated solving two specially developed cases of the Circle
Packing Problem (CPP).
• In order to make the solution jump out of possible local minima, a
perturbation approach and voting heuristic were developed.
• Demonstrate the desirable and key characteristic of a distributed
approach to avoid the tragedy of commons.
• Important ability of PC to deal with the practically significant agent
failure problem was demonstrated solving the CPP.
62

• Feasibility-based rule II was successfully formalized and
demonstrated solving two variations and associated cases of the
Sensor Network Coverage Problem (SNCP).
• Two variations and associated cases produced sufficiently robust
results.
• BFGS method was successfully used as an alternative to the
Nearest Newton Descent Scheme.
• CPP and SNCP were first time solved using a distributed,
decentralized approach such as PC.
63

Recommendations for Future Work
• Make the approach more generalized and increase the efficiency of
the PC algorithm by developing a self adaptive scheme for the
parameters, improving diversification of sampling, etc.
• More realistic path planning problems of the Multiple Unmanned
Vehicles (MUVs) can then be solved with the MTSP and VRP
approaches.
• Multi-Objective Probability Collectives (MOPC)
64

Recommendations for Future Work
Solve the Traffic Control Problem using PC
• Distributed, decentralized approach
• Every intersection represents an independent agent dynamically
optimizing the signal durations, cycle time, phase sequence, etc.
• Local traffic optimization → Network traffic optimization
• Traffic simulator will be used to set up the traffic scenario
• Flow rate will be measured at intersections (agents)
• PC will optimize the variables such as signal durations, cycle time,
phase sequence, etc.
• Optimized variables will be fed back to evaluate the performance.
65

66
Thanks for your attendance
Q & A

68
Solution to Rosenbrock using PC

69
Nash Equilibrium
The basic concept states that when a social game is being played iteratively
by number of agents, if a state comes when any agent changes its
strategy/state unilaterally without taking into consideration the other agents’
strategy/state, it does not benefit that agent and also does not benefit the
entire game output. If the game is in such state then the agents are assumed
to be in Nash Equilibrium.
It is worth to mention that Nash Equilibrium does not necessarily gives
best payoffs to agents but as a social system best collective / global /
system objective can be achieved.
n i

70
Probability Collectives (PC) Comparison
Sampling, Convergence criterion and Neighboring makes the PC presented
here different than the originally proposed by Dr. David Wolpert.
Proposed PC Original PC
Sampling
 Pseudorandom scalar
values drawn from uniform
distribution
 Fewer number of samples
 Monte Carlo sampling
 Computationally
expensive and slower
Convergence criterion
 Predefined number of
iterations and/or there is no
change in the final goal value for
considerable number of
iterations.
 No change in the
probability values for
considerable number of iterations

71
Probability Collectives (PC) Comparison
Proposed PC Original PC [1, 3]
Neighboring
 Sample around the
‘favorable strategy values’
and continue from the beginning.
 Narrows down the sampling
options of Agents forcing
them to sample only from the
neighbored range.
 Increases convergence
speed.
 Computationally cheaper
 Regression
 Data-aging
 Computationally
expensive/Large
memory

• Procedure starts with initializing the constraint violation tolerance
where is the cardinality of .
Feasibility-based Rule I
• Any feasible solution is preferred over any infeasible solution
If the current system objective as well as the previous
solution are infeasible, accept the current system objective
and corresponding as the current solution if the number of
constraints violated is less than or equal to , i.e. ,
and then the value of is updated to , i.e. .
73
µ = C [ ]1 2 ... tg g g=CC
[ ]
( )fav
G Y
[ ]fav
Y
[ ]
( )fav
G Y
violatedC µ violatedC µ≤
µ violatedC violatedCµ =

preferred
If the current system objective is feasible, and the previous
solution is infeasible, accept the current system objective
and corresponding as the current solution and then the value of
is updated to , i.e. .
74
[ ]
( )fav
G Y
[ ]fav
Y
[ ]
( )fav
G Y
0
µ
0violatedCµ = =

• Between two infeasible solutions, the one with fewer violated
constraints is preferred.
If the current system objective is feasible, i.e. and
is not worse than the previous feasible solution, accept the current
system objective and corresponding as the current
solution.
• If all the above conditions are not met, then discard current system
objective and corresponding , and retain the previous
iteration solution.
75
[ ]
( )fav
G Y
[ ]fav
Y
[ ]
( )fav
G Y
0violatedC =
[ ]
( )fav
G Y [ ]fav
Y

Updating of the Sampling Space and Perturbation Approach
• On completion of pre-specified iterations,
• If then shrink the sampling intervals:
• If and are feasible and
the system objective is referred to as .
76
[ ] [ ], ,
( ) ( )testfav n fav n n
G G
−
≤Y Y
testn
[ ]
( ) [ ]
( ), , 0 1
fav favupper lower upper lower
i i down i i i down i i downX Xλ λ λ Ψ ∈ − Ψ − Ψ + Ψ − Ψ < ≤
 
[ ],
( )
fav n
G Y [ ],
( )testfav n n
G
−
Y
[ ] [ ], ,
( ) ( )testfav n fav n n
G G ε−
− ≤Y Y
[ ],
( )
fav n
G Y [ ],
( )
fav s
G Y

• Updating of the Sampling Space and Perturbation Approach
In order to jump out of this possible local minimum, every agent
perturbs its current feasible strategy
The value of and +/- sign are selected based on preliminary trials.
Every agent expands the sampling space as follows:
77
i
[ ] [ ] [ ]
( )
( ) [ ]
( ) [ ]
1 1
2 2
1
,
1
,
fav fav fav
i i i i
lower upper
fav
i
i
lower upper
fav
i
X X X fact
randomvalue if
X
where fact
randomvalue if
X
σ σ γ
σ σ γ
= ± ×

∈ ≤

= 
 ∈ >

1 1 2 20 1lower upper lower upper
σ σ σ σ< < ≤ < <
γ
( ) ( ), , 0 1lower upper lower upper upper lower
i i up i i i up i i upλ λ λ Ψ ∈ Ψ − Ψ − Ψ Ψ + Ψ − Ψ < ≤
 

• How about the convergence or the stable solution acceptance
78

Feasibility-based Rule II:
• Any feasible solution is preferred over any infeasible solution
If the current system objective as well as the previous
solution are infeasible, accept the current system objective
and corresponding as the current solution if the number of
improved constraints is greater than or equal to , i.e. ,
and then the value of is updated to , i.e. .
80
[ ]
( )fav
G Y
[ ]fav
Y
[ ]
( )fav
G Y
µ improvedC µ≥
µ improvedC improvedCµ =

preferred
If the current system objective is feasible, and the previous
solution is infeasible, accept the current system objective
and corresponding as the current solution and then the value of
is updated to , i.e. .
• Between two infeasible solutions, the one with more number of
improved constraint violations is preferred.
If the current system objective is feasible, and is not worse
than the previous feasible solution, accept the current system
objective and corresponding as the current solution.
81
[ ]
( )fav
G Y
[ ]fav
Y
[ ]
( )fav
G Y
0
µ
0improvedCµ = =
[ ]
( )fav
G Y
[ ]fav
Y
[ ]
( )fav
G Y

• If the solution remains feasible and unchanged for successive
predefined number of iterations, and current feasible system
objective is worse than the previous iteration feasible solution,
accept the current solution.
If the solution remains feasible and unchanged for successive pre-
specified iterations i.e. and are feasible and ,
and the current feasible system objective is worse than the previous
iteration feasible solution, accept the current system objective
and corresponding as the current solution.
82
( )[ ],fav n
G Y
[ ]fav
Y
testn ( )[ ], testfav n n
G −
Y
( )[ ]fav
G Y

84
{ }[1] [?] [?] [1] [?] [?]
1 2 1, ,..., ,..., ,i i N NX X X X X−=Y
Agent selects its first strategy and samples randomly from other
agents’ strategies as well.
( )[1]
iG Y
[ ] [ ] [ ][1] [2] [1] [2] [1] [2]
1 1 1 1{ , ,..., } ,..., { , ,..., } ,..., { , ,..., }N i Nm m m
i i i i N N N NX X X X X X X X X= = =X X X
{ } ( )
{ } ( )
{ } ( )
{ } ( )
[2] [?] [?] [2] [?] [2]
1 2
[3] [?] [?] [3] [?] [3]
1 2
[ ] [?] [?] [ ] [?] [ ]
1 2
[ ] [ ] [ ][?] [?] [?]
1 2
, ,..., ,...,
, ,..., ,...,
, ,..., ,...,
, ,..., ,...,i i i
i i N i
i i N i
r r r
i i N i
m m m
i i N i
X X X X G
X X X X G
X X X X G
X X X X G
= ⇒
= ⇒
= ⇒
= ⇒
Y Y
Y Y
Y Y
Y Y
M
M
( )[ ]
1
im
r
i
r
G
=
⇒ ∑ Y
i

85
0
0.05
0.1
0.15
1 2 3 4 5 6 7 8 9 10
0
0.05
0.1
0.15
1 2 3 4 5 6 7 8 9 10
[ ]
( ) [ ]
( )1
1 1 1... 1/im
q X q X m= = = [ ]
( ) [ ]
( )1
... 1/im
N N Nq X q X m= = =
L
0
0.05
0.1
0.15
1 2 3 4 5 6 7 8 9 10
L
[ ]
( ) [ ]
( )1
... 1/im
i i iq X q X m= = =
{ }
{ }
{ }
[2] [?] [2] [?]
1
[ ] [?] [ ] [?]
1
[ ] [ ][?] [?]
1
,..., ,...,
,..., ,...,
,..., ,...,i i
i i N
r r
i i N
m m
i i N
X X X
X X X
X X X
=
=
=
Y
Y
Y
M
M
( )( )[ ]
1
im
r
i
r
E G
=
∑ Y
( ) ( ) ( ){ } [ ]
( ) [ ]
( ) ( )
[ ]
( )( )
( )( )
( ) ( ) ( ){ } [ ]
( ) [ ]
( ) ( )
[ ]
( )( )
( )( )
( ) ( ) ( ){ } [ ]
( ) [ ]
( ) ( )
[ ]
( )( )
( )( )
1 1 1[?] [2] [?] [2]
1
[?] [ ] [?] [ ]
1
[ ] [ ][?] [?]
1
,..., ,..., Y
,..., ,..., Y
,..., ,..., Y i i ii i
i N i i ii
i
r r rr r
i N i i ii
i
m m mm m
i N i i ii
i
⇒ =
⇒ =
⇒ =
∏
∏
∏
Y
Y
Y
M
M

86
[ ]
( )( ) [ ]
( ) [ ]
( ) ( )
[ ]
( )( )
Y
r r r r
i i i i
i
E G G q X q X= ∏Y
[ ]
( ) [ ] [ ]
( )
[ ]
( )
?
1 1
( ) (Y ) ( ) ( )
i im m
r r r
i i i i
r r i
E G G q X q X
= =
=∑ ∑ ∏Y

88
Multiple Unmanned Aerial Vehicles (MUAVs) Path Planning
Related Work
 Probabilistic Map Approach
- real-time and local updating of the map
 Flock formation
- collision avoidance, obstacle avoidance, formation keeping
- single objective function Vs individual objective function
 Gyroscope force
- real-time change in the path avoiding the collision
 Magnetic forces
- Attraction and Repulsion
 Concept of Auto-pilot – the airplanes with conflicting trajectories change
their ways with local communication avoiding latency in decision making

90
2 4 6 8 10 12 14
0
2
4
6
8
10
12
14
X-Coordinates
Y-Coordinates
0 0.5 1 1.5 2 2.5 3
x 10
4
40
60
80
100
120
140
160
180
200
Iterations
0 0.5 1 1.5 2 2.5 3
x 10
4
0
5
10
15
20
25
Iterations
CollectiveCoverage

91
0 2 4 6 8 10 12
0
2
4
6
8
10
12
14
X-Coordinates
Y-Coordinates
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 10
4
20
40
60
80
100
120
140
160
180
200
Iterations
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 10
4
0
2
4
6
8
10
12
14
16
18
Iterations
CollectiveCoverage

Probability Collectives

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Editor's Notes