COMPLEXITY MANAGEMENT: New Challenges And ...cpress/ref/Ontonix Stochastic...© C opyr ig h t2 05, O n x srl.A ev dN p a fcumyb w Ontonix COMPLEXITY MANAGEMENT: New Challenges And

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COMPLEXITY MANAGEMENT:New Challenges And Opportunities in the 21-st Century

J. Marczyk, Ph.D.Chief Technical Officer

Ontonix srlwww.ontonix.com


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CONTENTS

• Some Sources of Uncertainty• Complexity Principles• Fitness Landscapes• Fuzzy Cognitive Maps• Complexity: the Source of Fragility• Examples of Complexity-Based Design and Analysis • Complexity and Sustainability• Conclusions


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Sources of Uncertainty in Engineering

• Physical uncertainty– Material properties– Boundary and initial conditions– Assembly and manufacturing imperfections– Environmental loads– Geometry

• Non-physical uncertainty– Modelling errors (material model, element type, etc.)– Choice of method (linear, non-linear, etc.)– Solver– Computer– Engineer (epistemic uncertainty)


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Complexity Principles

• Principle of Complexity:

When the complexity and uncertainty of an engineering system increase, our ability to predict its behavior diminishes until athreshold is reached beyond which accuracy and significance become almost mutually exclusive.

• Principle of Incompatibility:

High precision is incompatible with high complexity.


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• Problem: Given the statistics of the inputs, determine the statistics of the performance (most likely value, scatter, amount of robustness, quality, probability of failure, etc.).

• Solution: Monte Carlo Simulation (MCS)

Definition of a Stochastic Problem

Input variables:

•Loads

•Material properties

•Dimensions

•Boundary conditions

•Initial conditions

•etc.

Output variables:

•Frequencies

•Stresses

•Displacements

•Temperatures

•Energy

•etc.

.

.

.

.


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Why Stochastic Simulation

• Good reasons to do stochastic simulation:– Determine the most likely performance (differs from nominal)– Determine the existence of outliers (or other pathologies) – Helps understand systems better– Nature is stochastic!

Outlier

Most likelyresponse.


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Launcher Failure Rate

Source: http://www.strategypage.com/dls/articles/20030202.asp


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Outliers = Risk


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From Analysis to Simulation: Discovering New Knowledge

Idealized analysis:excessivelyspecialised (optimal) and “fragile”

Realistic simulation:robust,not optimal!


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Car Crash:Optimality or Robustness?

• Example of bifurcation (clustering) in automotive crash (PAM-Crash, 128 samples 512 CPU Cray T3E, 1997).

• It has been found that the dominating variable in this case was the angle of impact, neither the properties of the structure, nor that of the materials.

• A tiny change in the angle of impact will change dramatically the response.

Courtesy, BMW AG

Is this reallythe optimum?

Initial design


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The Input – Output Mappingand the Fitness Landscape

Input space: uniformlysampled to get FL

Output space (FL)

x1

x2

y2

y1

Design 1 Design 2

x y

Each design will exhibit different behaviour because it is located in a different portion of theFitness Landscape. This leads to different behavioral modes. OntoSpace™ determines all the possible modes given a certain Fitness Landscape. Which mode is best? To answer this question it is necessary to resort to complexity.

Design 1

Design 2


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Examples of Fitness Landscapes

The introduction of minute amounts of scatter into computer models (not their surrogates!) reveals an astonishingly rich, complex and far from intuitive behaviour.


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Fitness Landscape of the Artemis Spacecraft(n=7, m=6)

Courtesy, EADS/CASA

From “Principles of Simulation-Based CAE”, J. Marczyk, FIM Publications, 1999


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1: entropy = 0.0278 2: entropy = 0.232 3: entropy = 1.6488




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From Fitness Landscapes to Knowledge

• A great way to represent the knowledge hidden inside a Fitness Landscape is via Fuzzy Cognitive Maps.

• The reason the maps are fuzzy: the Principle of Incompatibility.• A few examples of FCMs are shown in the following slides.• Most of these are subjective and are generated manually.• OntoSpace generates the FCMs automatically.


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Fuzzy Cognitive Maps in Biology (Dolphin)


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Fuzzy Cognitive Maps in Politics


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Fuzzy Cognitive Maps and Highway Traffic


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Cognitive Maps and Socio-economics


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Fuzzy Cognitive Map of Metabolic Network


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Monte CarloSimulation

ResultsData-base

Results:• System behavioral modes (maps)• Complexity measures• Robustness measures• Risk & vulnerability assessment• Dynamic Sets of Rules• Better problem understanding• All What-if? scenarios

STEP 1:Establish model

STEP 2:Run Monte Carlo Simulation

STEP 3:Run OntoSpace™

OntoSpace™

OntoSpace: Transforming Data Into NewKnowlege

OntoSpace™ can also process data originating from tests, historical records, sensors, etc.


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Fuzzy Cognitive Maps: Fitness LandscapeAttractors

As the Fitness Landscape is navigated (the design is moved to different locations) the topology of the FCM changes. Each stable topology is called a mode. A mode is an attractor in the Fitness Landscape.


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FCMs: Understanding How Systems Function

Hub

- Input

- Output

- Connector: significant relationship


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Hubs – Critical Elements in Networks

• Hubs are important since they determine the level of robustness of a system. A targetedattack at the single hub of a particular system can cause the collapse of the whole.

• Multi-hub systems are inherently more robust (e.g. Internet, certain ecosystems)


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Connectivity Histogram and System Vulnerability

Robust system Vulnerable system


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Complexity x Uncertainty = Fragility

• Today, we think that uncertainty is the biggest concern when designing and making decisions.

• However, the problem is not uncertainty, it is complexity.• When uncertainty meets high complexity, the result is fragility.

Simple systems can cope better with uncertainty than highly complex systems. They are less vulnerable.

• Highly complex systems are more exposed to the effects of uncertainty because of the countless ways in which they process information. They can fail in many ways, often due to apparently innocent causes.

• Uncertainty/scatter in the environment, cannot be avoided. We must learn to live with it. Hence the need to manage complexity.


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Complexity and Controllability

Input Input Input

Output Output Output

CASE 1 CASE 2 CASE 3


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Complex or Complicated?

• A system may be complicated, but still have low complexity.

• A large number of parts doesn’t generally imply high complexity. It does, in general, imply a complicated system.


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Classical Graph Complexity Measures

No. Nodes: 6No. Links: 15Density: 0.5N/L : 0.4

No. Nodes: 5No. Links: 10Density: 0.5N/L = 0.5

One way to define a map’s complexityis to compute its Density:

D=L/(N(N-1)),

where N – No. NodesL – No. Links

This definition implies that nodesdo not have causal effects on themselves.

Two examples, with 6 and 5 nodesare shown, with all the possible links,15 and 10 respectively.

Other simple measures of complexity:

C=L/Lmax, where Lmax is the maximumpossible No. of links

C=N/L – node to link ratio.


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Nature Increases Complexity (Functionality): There is a Price to Pay!

Time

Func

tiona

lity


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Autoimmunity: Complexity and Fragility

•Nervous System:•Multiple sclerosis•Myasthenia gravis•Autoimmune neuropathies such as Guillain-Barré•Autoimmune uveitis•Blood:•Autoimmune hemolytic anemia•Pernicious anemia•Grave's Disease•Autoimmune thrombocytopenia•Blood Vessels:•Temporal arteritis•Anti-phospholipid syndrome•Vasculitides such as Wegener's granulomatosis•Skin:•Psoriasis•Dermatitis herpetiformis•Pemphigus vulgaris•Vitiligo

•Gastrointestinal System:•Crohn's Disease•Ulcerative colitis•Primary biliary cirrhosis•Autoimmune hepatitis•Endocrine Glands:•Type 1 or immune-mediated diabetes mellitus•Grave's Disease•Hashimoto's thyroiditis•Autoimmune oophoritis and orchitis•Temporal arteritis•Autoimmune disease of the adrenal gland•Multiple Organs :•Rheumatoid arthritis•Systemic lupus erythematosus•Scleroderma•Polymyositis, dermatomyositis•Spondyloarthropathies such as ankylosing•spondylitis•Sjogren's syndrome

Autoimmunity is a prominent illustration of the fragility (in somecases cascadingfailure) that is the consequence of an exceedingly complex immune system.

For a primer on autoimmune diseases see: http://www.niaid.nih.gov/publications/autoimmune


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Complexity-Based Design – The Concept

Design 1

Design 2

Given two or more equivalent designs (in terms of functionality, performance, cost, etc.), the one with the lower complexity should be chosen as it will result intrinsically more robust.


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Complexity-Based Comupter-Aided Design: Example

Multitude of design parameters:

Height

Thickness factor

Dimension fraction

Spacing factor

Rib spacing

Cut widthCut Depth

Radius

Flange distance


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Geometric parameters (Continued)

Quarter model view:

x

Thickness factor = x/Height

If the thickness factor is increased

Height

Rib Spacing is the amount of holes between ribs

D

T

The dimension fraction is D/T

SThe spacing factor is S/T

Cut depth, width and radius determine the shape of the ribs

t

t is the flange distance


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Complexity-Based CAD – The Concept

Which one is best? What is “best”?


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Complexity-Based Design – The Concept

Allowable design range

x

f(x)

y*

Desired performance

•Five solutions which deliver identical performance are found.•They all possess a characteristic value of complexity.•In solution 3, small changes in x don’t affect performance.

x5x4x3x2x1

C5C4C3C2C1


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Complexity-Based Design: The James Webb Space Telescope

Option 3 Option 4

Option 2Option 1

James Webb Space Telescope payload adapter.Courtesy EADS CASA Espacio.


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Computational Geopolitics and Conflict Anticipation: How Complex is The World Getting?

2003 2004


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Nuclear Power Plant: Accident Analysis

Courtesy, PBMR Ltd.


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Process Plant: Vulnerability AnalysisA certain process plant is given, for which it is desired to perform vulnerability (risk) analysis. Given that vulnerability is related to complexity, the complexity of the plant shall be measured in its different operational modes (e.g. production, maintenance, etc.) using OntoSpace™.

To evaluate the complexity of a process or a system it is first necessary to obtain the equivalent process map(s). These are computed automatically by OntoSpace™. In order to extract the maps, OntoSpace™ requires data from the process sampled with a certain frequency at a set of significant locations or sensor points.


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Business Process Maps & Diagnosis

Inputs (red):•€/$ rate•Fed/ECB rate•Oil price•Raw materials costs•etc.

Outputs (blue):•Revenue•Production volume•Profit margin•Debt•Staff turnover•Quality•Growth•Taxes•etc.

Hubs – criticalvariables


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Mergers & Acquisitions Analysis

Corporation 1 Corporation 2

Total Complexity 772.7

Total Entropy 152946.4

Total Complexity 422.2

Total Entropy 47567.8

Two-third

s of a

ll merg

ers fa

il !


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ATC: Clasifying Airports Using Complexity (Peak Period)

Airport 1 Airport 2 Airport 3


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Genomics: Generation of Genomic and Metabolic Maps


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Patient Health Monitoring (Dialysis, 1)

Complexity = 30.12 Complexity = 15.12

Patient 3 Patient 4


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Patient Health Monitoring (Dialysis, 2)

Complexity = 46.43 Complexity = 20.22

Patient 9 Patient 10


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Insurance: Analysis of Daily Data From Agents – Regional Analysis

Region 1Agents 1 to 200




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Sustainable Development and Complexity

Complexity = 91.4Complexity = 80.1Complexity = 63.0Complexity = 32.1

1980 1995 2006 2009

1980. A corporation is established. An initial strucure is defined, together with a nominal business model.

1995. The corporation has evolved, entered new markets and rolled out new products. Revenue has increased, as well as number of employees, customers, suppliers, etc. More business units are needed. The corporation is nearly twice as complex as it was 15 years ago when it was established.

2006 – present. The company has grown further. In 11 years it has increased in terms of complexity by over 40%. The management of the company is becoming increasingly difficult, as the number of interconnections between the different business units has increased substantially. It is difficult to grow the business as changes are more and more difficult to implement. More and more compromises are necessary.

2009. This is the estimated time when the corporation and its business model will attain critical complexity unless drastic measures are taken. Beyond this value of complexity decline of the business will be inevitable. The corporation has now only 3 years left before it becomes critical and fragile. In 2009 it will be very difficult to further grow the business. Either heavy restructuring takes place or a merger/acquisition is considered.


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Sustainable Development and Complexity

Complexity = 91.4

Complexity = 80.1

Complexity = 63.0

Complexity = 32.1

Time

Com

plex

ity

System is critically complex here and becomes fragile

Time left before system starts to degrade (loss of structure in map)

20092006

1995

1980

2009

2006


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Air Traffic: Saturation Limits

Version 2 of OntoSpace estimates theupper bound on complexity a givensystem may reach before incrementsin entropy will cause loss of structure.

At that point, the system becomesfragile, loses functionality and becomesdifficult to manage.


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Complexity Tracking

• In certain applications (battle management, plant monitoring, airtraffic control, etc.) it is of interest to track the time evolution ofoverall system complexity. Critical levels may be determined(based on historical data) and early warnings may be issuedbased on complexity trends.

• Sudden complexity changescorrespond to “traumatic”events (e.g. extinction spasmsin our biosphere)

Critical upper limit

Critical lower limit


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Does Optimal Mean Best?


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Stochastic Design Improvement (SDI):Beyond Optimization

• Problem: Given an uncertain environment and (engineering) tolerances, determine the values of design variables so that certain outputs have prescribed behavior.

Input variables:

Loads

Material properties

Dimensions

Boundary conditions

Initial conditions

etc.

Output variables:

Frequencies

Stresses

Displacements

Temperatures

Energy

etc.

.

.

.

.


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Courtesy of BMW AG

Stochastic Design Improvement (SDI)Some Successes

Courtesy of Nissan

Problem: reduce mass, maintain safety and stiffnessResult:16 kg mass reduction20% reduction of A-pillar deformation40% reduction of dashboard deformationCost = 60 runs (tolerances in all materials andthicknesses) of PAM-Crash and MSC.Nastran

Problem: reduce mass, maintain safety in two crash conditionsResult:15 kg mass reductionCost = 90 runs (tolerances in all materials and thicknesses) of PAM-Crash


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Conclusions

• Computational Science can become a source of knowledge, but it must first transition from an analysis-based to a simulation-based paradigm. Predictive models cannot generate new knowledge!

• Complexity management is fundamental for rigorous risk anticipation, assessment and management.

• Complexity is not a magical phenomenon on the border between order and chaos – it is a quantity, just like energy, or mass, that can be measured.

• The study of the structure of complexity helps to better understand the way systems really function and evolve.


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Conclusions

• It can be shown that optimal systems/decisions are fragile.• Optimality and robustness are mutually exclusive!• It is very dangerous to seek optimal solutions to very complex

problems without understanding the nature of their complexity.• Sustainable development of complex systems (society, markets,

economy, biosphere, etc.) is impossible if the pursuit of perfection is the underlying philosophy.

• Nature doesn’t optimize – it favors systems fit for the function. Optimal solutions quickly go extinct.

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COMPLEXITY MANAGEMENT: New Challenges And ...cpress/ref/Ontonix Stochastic...© C opyr ig h t2 05, O n x srl.A ev dN p a fcumyb w Ontonix COMPLEXITY MANAGEMENT: New Challenges And