Probabilistic Solution Discovery for Network Reliability - - PowerPoint PPT Presentation

Probabilistic Solution Discovery for Network Reliability Optimization Jose E. Ramirez-Marquez Assistant Professor SSE Stevens Institute of Tech. Hoboken, NJ

Short Academic Bio

• Graduated from Universidad Nacional

Autonoma de Mexico with Bachelor in Actuarial Science

• Obtained Ph. D., M.Sc. in Industrial &

Systems Engineering and M.Sc. in Statistics from Rutgers University

• Started working at Stevens Institute of

technology Fall 2004

• Project 1: New Methods for network

Reliability Analysis and Optimization

New Methods for Network Reliability Analysis & Optimization

• ARMY Research & Development Center,

Picatinny Arsenal, NJ

• PM: “we need tools that can be

incorporated into our process to measure network operational effectiveness”

• ACADEMIC: research on machine

learning techniques to analyze network reliability

Quick background on Network reliability analysis

• Reliability: Probability a network

(collection of components) works for a specified period of time under specified

• perating conditions
• Analysis:
• understand how network components

interact (build reliability block diagram or network graph)

• obtain component reliability data
• use technique to obtain network

reliability

Tool development- actual research

• TOOL- employee with minimal

background on reliability must be able to analyze network.

• existing methods- computationally

expensive & not easily applicable without background.

• proposed approach- provide interval

for network reliability via a data classification method.

Data Classification Technique

• CART- classification and regression

technique developed by salford networks

• customer profiling, fraud detection,

credit card scoring, etc...

• METHOD- sifts through data and

isolates significant patterns and relationships:

• determines a complete tree with

minimal misclassification, Assigns terminal nodes to a class outcome

Reliability Via Data Classification Methods (1)

• Step 1: Monte Carlo Simulation of

Network

• generate component states
• determine network state
• Step 2: Data Analysis and Rule

Transformation

• set arc states as predictor variables

& network state as target variable

• rules indicate component interaction

related to network behavior

Reliability Via Data Classification Methods (2)

• Step 3: Analysis of Generated Rules
• Extracted rules are analyzed for

validity using a Monte-Carlo simulation

• Step 4: Reliability Approximation
• Valid rules are used to approximate

the reliability of the network via:

Bl ≤ R ≤ Bu

Results

• Results obtained showed a narrow

bound for the network reliability

• With increased arc reliability it is not

necessary to obtain all of the minimal cut and path vectors

• New method is very practical for

medium sized and large networks

• Extendable to many network

configurations & behavior

Network Reliability Optimization

• Problem-network design via strategic

allocation of components & redundancy to:

• determine types & copies of

components assigned to subnetworks to maximize a network function

• Selection of components among

different choices

• Known reliability, cost, weight, etc…
• Constraints should be satisfied

Techniques for Reliability Optimization

• many approaches to generate very good

solutions for most component allocation problems

• integer & dynamic programming,

simulated annealing, tabu search

• genetic algorithms, ant colony,…
• very good results tend to depend on

algorithms with much parameter tuning

• few methods address diverse network

structures or behavior

Probabilistic Solution Discovery Optimization (1)

• Machine learning algorithm with three

main iteration steps:

• Step 1: random generation of network

configuration via vector of component appearance probabilities (two parameters: # of configurations, probabilities)

• appearance probability defines the

frequency of a component appearing in the final design

• equivalent to step 1 in slide 7

Probabilistic Solution Discovery Optimization (2)

• Step 2: (two parameters: Simulation

runs, penalty factor)

• reliability analysis of network

configuration (via method presented in slide 7)

• computation of penalty for deviation

from constraining target.

• Step 3 : (one parameter: sample size)
• Update component appearance

probabilities using a sample of the configurations generated in steps 1& 2.

Example- cost minimization with reliability constraint

1 2 3 4 5

Arc x12 x13 x14 x15 x23 x24 x25 x34 x35 x45 Cost 3 2 5 4 6 2 2 5 3 4 5 8 4 5 3 6 5 2 2 9 Rel . 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 u= 1 x12u x13u x14u x15u x23u x24u x25u x34u x35u x45u

ˆ R xu

h

( )

C xu

h

( )

C xu

h

( )

1 1 1 1 1 1 1 1 0.8984 262 264.1722 2 1 1 1 1 1 1 1 0.8992 282 283.2015 3 1 1 1 1 1 1 1 0.9173 279 283.9117 4 1 1 1 1 1 1 1 0.8957 281 287.1202 5 1 1 1 1 1 1 1 0.8932 281 290.6788

Iteration 1

u

12u 13u 14u 15u 23u 24u 25u 34u 35u 45u

1 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

appearance probabilities generated solutions

Example- cost minimization with reliability constraint

Arc x12 x13 x14 x15 x23 x24 x25 x34 x35 x45 Cost 3 2 5 4 6 2 2 5 3 4 5 8 4 5 3 6 5 2 2 9 Rel . 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8

Iteration 10

appearance probabilities

u

12u 13u 14u 15u 23u 24u 25u 34u 35u 45u

1 0 . 5 0 . 5 0 . 5 0 . 5 0 . 5 0 . 5 0 . 5 0 . 5 0 . 5 0 . 5 2 0.65 0.75 0 . 6 0.75 0.65 0.55 0.75 0 . 5 0 . 6 0.65 3 0.75 0 . 7 0.55 0.85 0.75 0.65 0 . 7 0 . 6 0 . 6 0 . 9 4 0.75 0.55 0.35 1 0.75 0 . 7 0.55 0.65 0.65 0.95 5 0.85 0 . 4 0 . 3 1 0.85 0.65 0.25 0 . 6 0.75 1 6 1 0.35 0.15 1 0.95 0.55 0 . 2 1 0 . 8 1 7 1 0 . 5 1 1 0.35 0 . 2 1 0.95 1 8 1 0 . 6 1 1 0 . 4 1 1 1 9 1 0.15 1 1 0.85 1 1 1 1 0 1 1 1 1 1 1 1 1 2 3 4 5

Results for networks with known optima

Problem n l rij R0 Best Costb Mean Cost Median u Cv Cv [9] 1 5 1 0 0.80 0.90 2 5 5 255.00 2 0.0000 0.0000 2 5 1 0 0.90 0.95 2 0 1 204.60 3 0.0309 0.0000 3 7 2 1 0.90 0.90 7 2 0 746.00 6 0.0276 0.0000 4 7 2 1 0.90 0.90 8 4 5 852.00 6 0.0126 0.0185 5 7 2 1 0.95 0.95 6 3 0 661.00 7 0.0561 0.0344 6 8 2 8 0.90 0.90 2 0 8 215.70 4 0.0315 0.0211 7 8 2 8 0.90 0.90 2 4 7 259.10 6 0.0314 0.0183 8 8 2 8 0.95 0.95 1 7 9 181.70 5 0.0284 0.0228 9 9 3 6 0.90 0.90 2 3 9 246.30 6 0.0356 0.0497 10 9 3 6 0.90 0.90 2 8 1 294.60 7 0.0474 0.0340 11 9 3 6 0.95 0.95 2 0 9 216.20 7 0.0569 0.0839 12a 1 0 4 5 0.90 0.90 1 5 4 167.10 7 0.0791 0.0618 13a 1 0 4 5 0.90 0.90 1 9 7 210.00 7 0.0448 0.0095 14 1 0 4 5 0.95 0.95 1 3 6 145.00 8 0.0618 0.0802

Results for networks with unknown optima

Best Cost Mean Cost Problem n l rij R0 GA [9] ANN [10] SDA GA [9] ANN [10] SDA Worst Cost SDA 1 15 105 0.90 0.95 317 304 268 344.6 307.6 281.4 301 2 20 190 0.95 0.95 926 270 200 956.0 281 233.1 262 3 25 300 0.95 0.90 1606 402 331 1651.3 421.8 348.1 367

Solutions Searche d Problem Search Spac e GA [9] SDA Fraction

• ut of [9]

SAMPL E 1 1.02×1031 1.40×105 4.50×103 0.032 3 0 0 7 2 1.02×1057 2.00×105 6.00×103 0.030 3 0 0 7 3 2.10×1090 4.00×105 1.20×104 0.025 400 7

Problem Worst Cost Configuration

ˆ R xu

h

( )

Best Cost Configuration

ˆ R xu

h

( )

1 4,5,8,10,13,15,22,26,37,39,46, 47,53,58,65,69,72,74,77,79,84, 88,92 0.9570 4,5,8,15,22,25,26,38,46,47,53, 58,65,69,72,73,77,79,84,88, 92,101 0.9505 2 4,8,11,33,35,42,44,56,66,79,80, 92,102,106,123,131,140,146, 151,154,168,172,188,189 0.9525 4,8,29,33,36,42,43,58,66,79, 80,92,96,102,106,119,131, 137,140,146,154,160,165,177, 188,189 0.9550 3 6,14,32,37,40,51,57,63,70,71, 87,97,129,137,149,156,174,176, 187,193,206,207,209,215,225, 233,235,242,246,268,274,300 0.9027 3,17,32,40,51,54,63,73,97, 102,114,119,137,141,160,176, 177,195,207,209,210,225,229, 233,242,243,257,269,274,300 0.9003

Future work & questions

• extend technique to larger reliability

problems

• all terminal reliability with

redundancy

• implement algorithm outside reliability

area:

• orienteering problem
• single & group
• completely characterize algorithm

parameters for any problem size

Reliability Block Diagram & network Graph

1 2 n . . . 1 2 n . . . 1 2 n . . .

. . . S1 SK S2

Step 1

1 2 3 4 5 6 7 8 9

Figure 1: Demand = 6 units

Arc i S i1 S i2 P i1 P i2 1 9 0.3 0.7 2 6 0.3 0.7 3 4 0.3 0.7 4 5 0.3 0.7 5 8 0.3 0.7 6 4 0.3 0.7 7 2 0.3 0.7 8 8 0.3 0.7 9 7 0.3 0.7 States Probabilities

xu x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 Max Flow ϕ u x1 9 6 4 5 8 4 8 7 12 1 x2 6 5 2 8 7 x3 4 4 2 8 7 x4 9 6 4 8 4 2 7 6 1 x5 9 6 4 5 4 2 7 6 1 x6 9 6 5 4 2 8 5 x7 9 6 4 5 8 2 8 7 10 1 x8 9 4 5 8 2 x9 9 6 5 8 4 2 7 6 1 x10 9 6 4 5 8 2 7 2

Step 2

x3 ≥ 4 x2 ≥ 6 x2 ≥ 6 x3 ≥ 4 x7 ≥ 2 x3 ≥ 4 x3 < 4 x1 < 9 x1 ≥ 9 x3 < 4 x2 < 6 x7 < 2

x5 x8 x6 x2 x7 x9 x4

x5 < 8

x1 x7 x8 x3 x3 x2 x9 x6 x2 x4 x3 x1 x3 x1 x4

DFV = 0 DFV = 0 DFV = 0 DFV = 0 DFV = 0 DFV = 0 DFV = 0 DFV = 0 DFV = 0 DFV = 0 DFV = 0 DFV = 0 DFV = 0 DFV = 0 DFV = 0 DFV = 1 DFV = 1 DFV = 1 DFV = 1 DFV = 1 DFV = 1 DFV = 1 x5 ≥ 8 x6 < 4 x6 ≥ 4 x9 ≥ 7 x9 < 7 x4 < 5 x1 ≥ 9 x7 ≥ 2 x1 < 9 x8 < 8 x8 ≥ 8 x3 < 4 x8 ≥ 8 x8 < 8 x4 ≥ 5 x7 < 2 x9 ≥ 7 x9 < 7 x6 ≥ 4 x6 < 4 x2 < 6 x4 ≥ 5 x4 < 5 x2 ≥ 6 x2 < 6 x3 ≥ 4 x3 < 4 x1 ≥ 9 x1 < 9 x4 ≥ 5 x4 < 5

x2

x2 < 6 x2 ≥ 9 DFV = 1 DFV = 0

Step 3

/*Terminal Node 1*/ If (X5 <= 4 && X6 <= 2) { terminalNode = -1; class = 0;

CVT1 = {9,6,4,5,0,0,2,8,7} CVT2 = {9,6,4,5,0,4,2,8,0} /*Terminal Node 23*/ If (X5 > 4 && X8 > 4 && X2 <= 3 && X3 > 2 && X1 > 4.5 && X4 > 2.5 ) {terminalNode = -23; class = 1; PVT23 = {9,0,4,5,8,0,0,8,0} PVT24 = {0,6,0,0,8,0,0,8,0}

1(9) 2(6) 3(4) 4(5) 7(2) 8(8) 9(7)

/*Terminal Node 2*/ If (X5 <= 4 && X6 > 2 && X9 <= 3.5) {terminalNode = -2; class = 0; /*Terminal Node 24*/ If ( X5 > 4 && X8 > 4 && X2 > 3 ) {terminalNode = -24; class = 1;

1(9) 2(6) 3(4) 4(5) 7(2) 8(8) 6(4) 1(9) 3(4) 4(5) 8(8) 5(8) 2(6) 8(8) 5(8)

Results

1 (3) 2 (1) 3 (8) 4 (4) 5 (2) 6 (5) 9 (3) 10 (2) 7 (4) 8 (2) 11 (3) 15 (1) 14 (2) 13 (9) 12 (4) s t

Network Arc Reliability Bl Bu Bound Width 1 0.80 0.82750 0.83361 0.00611 0.85 0.90303 0.90597 0.00294 37 CMCV 0.9 0.95763 0.95850 0.00087 0.95 0.98982 0.98995 0.00013 30 CMPV 0.99 0.99956 0.99957 1E-0 5