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Optimal Multilevel System Design under Uncertainty NSF Workshop on Reliable Engineering Computing Savannah, Georgia, 16 September 2004. M. Kokkolaras and P.Y Papalambros University of Michigan. Z. Mourelatos Oakland University. Outline. Design by Decomposition

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Optimal Multilevel SystemDesign under UncertaintyNSF Workshop on Reliable Engineering ComputingSavannah, Georgia, 16 September 2004

M. Kokkolaras and P.Y Papalambros

University of Michigan

Z. Mourelatos

Oakland University

  • Design by Decomposition
  • Hierarchical Multilevel Systems
  • Analytical Target Cascading
    • Deterministic Formulation
    • Nondeterministic Formulations
  • Propagation of Uncertainty
  • Practical Issues
  • Example

Design Target Problem

Optimal System Design

design by decomposition
Design by Decomposition
  • When dealing with large and complex engineering systems, an “all-at-once” formulation of the optimal design problem is often impossible to solve
  • Original problem is decomposed into a set of linked subproblems
  • Typically, the partitioning reflects the hierarchical structure of the organization (different design teams are assigned with different subproblems according to expertise)
decomposition example
Decomposition Example












multilevel system design


subsystem 1

subsystem 2

subsystem n

component m

component 1

component 2

Multilevel System Design
  • Multilevel hierarchy of single-level (sub)problems
  • Responses of higher-level elements are depend on responses of lower-level elements in the hierarchy
  • Need to assign design targets for the subproblems to the design teams
  • Design teams may focus on own goals without taking into consideration interactions with other subproblems; this will compromise design consistency and optimality of the original problem
analytical target cascading
Analytical Target Cascading
  • Operates by formulating and solving deviation minimization problems to coordinate what higher-level elements “want” and what lower-level elements “can”
  • Parent responses rp are functions of
    • Children response variables rc1,rc2, …,rcn, (required)
    • Local design variables xp (optional)
    • Shared design variables yp(optional)
  • In the following formulations:
    • Subscript index pairs denote level and element
    • Superscript indices denote computation “location”

optimization inputs

optimization outputs

response and shared

variable values cascaded

down from the parent

response and shared

variable values passed

up to the parent

response and shared

variable values passed

up from the children

response and shared

variable values cascaded

down to the children

Information Exchange

element optimization problem pij, where rij is provided by the analysis/simulation model

multilevel system design under uncertainty


subsystem 1

subsystem 2

subsystem n

component m

component 1

component 2

Multilevel System Designunder Uncertainty
  • Multilevel hierarchy of single-level (sub)problems
  • Outputs of lower-level problems are inputs to higher-level problems: need to obtain statistical properties of responses
nondeterministic formulations
Nondeterministic Formulations
  • For simplicity, and without loss of generalization, assume uncertainty in all design variables only
  • Introduce random variables (and functions of random variables)
  • Identify (assume) distributions
  • Use means as design variables assuming known variance
  • “Hard” and “soft” inequalities
  • “Hard” and “soft” equalities
  • Typically, a target reliability of satisfying constraints is desired
propagation of uncertainty
Propagation of Uncertainty

State of the Art (?):

Since functions are generally nonlinear, use first-order approximation

(Taylor series expansion around the means of the random variables)

validity of linearization
Validity of Linearization





consistency constraints in ATC formulation secure validity


* 1,000,000 samples

moment approximation using advanced mean value method
Moment Approximation UsingAdvanced Mean Value Method
  • Consider Z=g(X)
  • Discretize “b-range” (from b = 4 (Pf = 0.003%) to b = -4 (Pf = 99.997%))
  • Find MPP for P[g(X)>0]<F(-bi) for all i
  • Evaluate Z=g(XMPP), i.e., generate CDF of Z
  • Derive PDF of Z by differentiating CDF numerically
  • Integrate PDF numerically to estimate moments

* 1,000,000 samples

example piston ring liner subassembly
Example:Piston Ring/Liner Subassembly

Brake-specific fuel consumption (BSFC)

GT Power

Power loss due to friction

Oil consumption


Liner wear rate


Ring and liner surface roughness

Liner material properties

results and reliability assessment
Results and Reliability Assessment

0.03% less reliable than assumed

* 1,000,000 samples

statistical properties of power loss
Statistical Properties of Power Loss

MCS – PDF (1,000,000 samples)


probability distribution of bsfc
Probability Distribution of BSFC


MCS with 1,000,000 samples

practical issues
Practical Issues
  • Computational cost
  • Noise/accuracy in the model vs. magnitude of uncertainty in inputs
  • Convergence of multilevel approach
concluding remarks
Concluding Remarks
  • Practical yet rational decision-making support
    • Value of optimization results is in trends not in numbers
    • Strategies should involve a mix of deterministic optimization and stochastic “refinement”
  • Need for accurate uncertainty quantification (and propagation)
error issues
Error Issues
  • y=f(x) + emodel + emetamodel + edata

+ enum + eunc. prop.

  • Need to keep ALL errors relatively low

Optimum Symmetric Latin Hypercube (OSLH) Sampling

OSLH Samples

Partitioned Group #2

Partitioned Group #1


a : vector of constants


Global Least Squares :


Moving Least Squares :

where :

Cross-Validated Moving Least Squares (CVMLS) Method

  • Polynomial Regression using Moving Least Squares (MLS) Method
    • In MLS, sample points are weighted so that nearby samples have more influence on the prediction.
metamodel errors
Metamodel Errors
  • Optimal symmetric Latin hypercube sampling (200 train points and 150 trial points for Ringpak, 45 train points and 40 trial points for GT-power)
  • Moving least squares approximations