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 operating 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: B l ≤ R ≤ B u

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 Arc x 12 x 13 x 14 x 15 x 23 x 24 x 25 x 34 x 35 x 45 2 5 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 1 4 3 appearance u 12u 13u 14u 15u 23u 24u 25u 34u 35u 45u probabilities 1 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 ˆ ( ) ( ) ( ) h h h u = 1 x 12u x 13u x 14u x 15u x 23u x 24u x 25u x 34u x 35u x 45u R x u C x u C x u generated 1 1 1 0 1 1 0 0 1 1 1 0.8984 262 264.1722 2 1 0 0 1 1 1 1 1 1 0 0.8992 282 283.2015 solutions 3 1 1 0 1 0 1 1 1 0 1 0.9173 279 283.9117 4 1 1 1 1 1 0 1 0 0 1 0.8957 281 287.1202 5 1 1 1 1 1 0 1 0 0 1 0.8932 281 290.6788

Example- cost minimization with reliability constraint 1 Arc x 12 x 13 x 14 x 15 x 23 x 24 x 25 x 34 x 35 x 45 2 5 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 4 3 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 0 1 1 0.35 0 . 2 1 0.95 1 8 1 0 . 6 0 1 1 0 0 . 4 1 1 1 9 1 0.15 0 1 1 0 0.85 1 1 1 1 0 1 0 0 1 1 0 1 1 1 1

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

Results for networks with unknown optima Worst Best Cost Mean Cost Problem n l r ij R 0 Cost GA ANN ANN SDA GA [9] SDA SDA [9] [10] [10] 344.6 1 15 105 0.90 0.95 317 304 268 307.6 281.4 301 956.0 2 20 190 0.95 0.95 926 270 200 281 233.1 262 1651.3 3 25 300 0.95 0.90 1606 402 331 421.8 348.1 367 ˆ ( h ) R x u ˆ ( h ) Problem Worst Cost Configuration R x u Best Cost Configuration 4,5,8,10,13,15,22,26,37,39,46, 4,5,8,15,22,25,26,38,46,47,53, 1 47,53,58,65,69,72,74,77,79,84, 0.9570 58,65,69,72,73,77,79,84,88, 0.9505 88,92 92,101 4,8,29,33,36,42,43,58,66,79, 4,8,11,33,35,42,44,56,66,79,80, 80,92,96,102,106,119,131, 2 92,102,106,123,131,140,146, 0.9525 0.9550 137,140,146,154,160,165,177, 151,154,168,172,188,189 188,189 6,14,32,37,40,51,57,63,70,71, 3,17,32,40,51,54,63,73,97, 87,97,129,137,149,156,174,176, 102,114,119,137,141,160,176, 3 0.9027 0.9003 187,193,206,207,209,215,225, 177,195,207,209,210,225,229, 233,235,242,246,268,274,300 233,242,243,257,269,274,300 Solutions Searche d Fraction Problem Search Spac e SAMPL E out of [9] GA [9] SDA 1.02 × 10 31 1.40 × 10 5 4.50 × 10 3 1 0.032 3 0 0 7 1.02 × 10 57 2.00 × 10 5 6.00 × 10 3 2 0.030 3 0 0 7 2.10 × 10 90 4.00 × 10 5 1.20 × 10 4 3 0.025 400 7

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 S 2 S 1 S K 1 1 1 2 2 2 . . . . . . . . . . . . n n n

Step 1 Arc States Probabilities 4 S i1 S i2 P i1 P i2 i 1 8 1 0 9 0.3 0.7 2 0 6 0.3 0.7 3 5 3 0 4 0.3 0.7 7 4 0 5 0.3 0.7 2 9 5 0 8 0.3 0.7 6 0 4 0.3 0.7 6 7 0 2 0.3 0.7 8 0 8 0.3 0.7 Figure 1: Demand = 6 units 9 0 7 0.3 0.7 ϕ u x u x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 Max Flow x 1 9 6 4 5 8 4 0 8 7 12 1 x 2 0 0 0 0 0 6 0 5 2 8 7 x 3 0 0 0 0 4 0 0 4 2 8 7 x 4 9 6 4 0 8 4 2 0 7 6 1 x 5 9 6 4 5 0 4 2 0 7 6 1 x 6 9 6 0 5 0 4 2 8 0 5 0 x 7 9 6 4 5 8 0 2 8 7 10 1 x 8 0 0 0 0 9 0 4 5 8 0 2 x 9 4 2 0 6 1 9 6 0 5 8 7 x 10 0 2 0 2 0 9 6 4 5 8 7