### 20110523_A two

```A two-stage approach for multiobjective decision making with
applications to system reliability
optimization
Zhaojun Li, Haitao Liao, David W. Coit
Reliability Engineering and System Safety
Hui-Yu, Chung
Agenda


Introduction
Multi-objective optimization problem
◦ Mathematical formulation
◦ Non-dominated sorting genetic algorithm

Statistical classification methods
◦ Unsupervised and supervised data classification
◦ Self-organizing map

Reduction of Pareto optimal solutions
◦ Data envelopment analysis


Application to multi-objective RAP
Conclusions
Introduction
Though there are multiple design
objectives, a decision-maker must
ultimately select one or a small set of
solutions to consider.
 In this approach, prospective solutions are
clustered, pruned for the decision-maker
to consider only a small subset of the
promising solutions.

Introduction

problem (RAP) is to maximize the system
reliability under various constraints
◦ Single-objective integer programming problem,
which is a NP hard

Mathematical programming approaches for
RAP usually restrict the solution space by
considering only one component choice for
each subsystem
◦ Without allowing the mixture of those
functionally equivalent components
Introduction

Component mixing in system redundancy
increases the problem solution space
◦ May result in higher system reliability values
◦ Need to employ heuristic algorithms such as
GA or Tabu search

Mixing functionally equivalent
components may potentially reduce the
variance of system reliability estimate and
minimizes the likelihood of common
cause failures
Introduction


problem, a new two-stage approach is
proposed in this paper.
First Stage:
◦ A multiple objective evolutionary algorithm
(MOEA) is applied to identify a representative
Pareto optimal solution set

Second Stage:
◦ Classify the Pareto optimal solutions by selforganizing map (SOM) method
◦ Eliminate the non-efficient solutions using data
envelopment analysis (DEA) method
Two-stage method for solving multiobjective RAP
Agenda


Introduction
Multi-objective optimization problem
◦ Mathematical formulation
◦ Non-dominated sorting genetic algorithm

Statistical classification methods
◦ Unsupervised and supervised data classification
◦ Self-organizing map

Reduction of Pareto optimal solutions
◦ Data envelopment analysis


Application to multi-objective RAP
Conclusions
Multi-objective optimization
problem
Mathematical formulations:
 Let x be a vector containing p decision
variables

◦ The optimization problem with n objective
functions is expressed as:
What is “Pareto Optimal” Solutions?

In multi-objective optimization problems,
we cannot expect that every objective
being satisfied in the result solutions
◦ The Pareto optimal Solution
If we want to improve one of the subobjective, we have to worse some other
sub-objectives.
 Pareto optimal solutions are often
continuous, and we can find infinite
number of that kind of solutions.

What is “Pareto Optimal” Solutions?
Mathematical formulations

Some approaches to solve the problem:
◦ Transform the original problem into a singleobjective problem
◦ Using Pareto optimal concept based on nondominance

Pareto dominance & non-dominance
◦ Determined by multiple pair-wise vector
comparison
Mathematical formulations
x is non-dominated in a p-dimensional set X
if there is no other y  x in X such
that f (y )  f ( x ) .
 If N is a set containing all the non-dominated
solutions in X, then the set N is called the
Pareto optimal set. (Pareto frontier in multiobjective optimization problem)
 The number of solutions in the Pareto
optimal solution set is large as the number
of conflicting objectives increases

Non-dominated sorting genetic
algorithm

To identify the Pareto optimum solution
set, some kinds of MOEA Genetic
Algorithms can be applied.
◦ In this paper, non-dominated sorting genetic
algorithms (NSGA) or NSGA-II is used

NSGA v.s. Simple GA:
◦ The same crossover & mutation as GA
◦ Different selection operator
◦ Ranking method
NSGA-II Algorithm
Agenda


Introduction
Multi-objective optimization problem
◦ Mathematical formulation
◦ Non-dominated sorting genetic algorithm

Statistical classification methods
◦ Unsupervised and supervised data classification
◦ Self-organizing map

Reduction of Pareto optimal solutions
◦ Data envelopment analysis


Application to multi-objective RAP
Conclusions
Statistical classification methods

Unsupervised data classification
◦ e.g. k-means and Self-organizing map (SOM)
◦ No or few prior information is available
◦ Labels are not specified beforehand

Supervised data classification
◦ e.g. Artificial neural network and SVM
◦ Relationship between the data and its
corresponding cluster is known
◦ The label for each input vector needs to be
specified first (by training process)
Self-organizing map
An unsupervised classification method
 Generates a set of representations for
multi-dimensional input vector while
preserving the topological properties of
similarity
 Dimensional reduction process

Self-organizing map
(Training Process)

Best Matching Unit (BMU):
◦ The Euclidean distance to all weight vectors is
computed
◦ The neuron with the weight vector most
similar to the input

◦ The weight of the input vectors is adjusted
according to the distances of the BMUs
◦ The adjustment decreases with time
Self-organizing map
(Training Process)

The weight w(t) is updated iteratively:
w(t  1)  w(t )   (v , t ) (t )[I(t )  w(t )]
◦ w(t  1) : the weight vector at step t+1
◦ I(t ) : the input vector
◦  (t ) : the learning coefficient (monotonically
decreasing with time)
◦  (v , t ) : the neighborhood function
 Gaussian neighborhood function is often used
 (v ,t )  exp(v 2 /  2 )
Self-organizing map
Eventually, output nodes are associated
with groups or patterns corresponding to
the input vectors
 The input vector is mapped to a specific
location on the lattice based on its
similarity to the weight vector for a
specific neuron

Self-organizing map

SOM measures the similarity by the Euclidian distance
as well as the angle between the input vectors by
updating the weight vectors iteratively
Agenda


Introduction
Multi-objective optimization problem
◦ Mathematical formulation
◦ Non-dominated sorting genetic algorithm

Statistical classification methods
◦ Unsupervised and supervised data classification
◦ Self-organizing map

Reduction of Pareto optimal solutions
◦ Data envelopment analysis


Application to multi-objective RAP
Conclusions
Reduction of Pareto optimal
solutions
Selecting representative solutions from
each cluster can be regarded as a multiobjective solution optimization problem
(MOSO)
 Data Envelopment Analysis

◦ A special MOSO method
◦ Is able to eliminate non-efficient Pareto
optimal solution from each cluster
Data envelopment analysis

A linear programming-based technique
for measuring relative performance of
decision making units (DMUs)
◦ A unit whose performance can be measured
in terms of input-output analysis

For MOSO, each alternative solution is
treated as a DMU in the DEA method
◦ The DMUs are assumed to be
homogeneously comparable (to make the
result efficiency meaningful)
Data envelopment analysis

Relative Efficiency (RE)
weighted sumof outputs
RE 
weighted sumof inputs

Considering a problem involving l DMUs,
each has m inputs and n outputs, the RE
of the kth DMU is:
weights
Data envelopment analysis

The RE of a specific DMU k0 can be
obtained by:


is a small positive quantity
Data envelopment analysis

Normalized programming problem:
Data envelopment analysis
When applying DEA, all DMUs are
attempting to select their most favorable
weights
 There may be more than one efficient
unit whose relative efficiency has the
value of one

◦ Efficient frontier
Data envelopment analysis


In the MOSO formulation for the RAP, all the Pareto
Optimal solutions in each cluster can be considered as
DMUs
A higher relative efficiency value indicates a higher
output value (ex. system reliability)
Data envelopment analysis
In this paper, method are presented when
the decision-makers have not expressed
any objective function preferences
 Ordinal ranking of objective function

◦ Used to prune the Pareto optimal set
◦ Weight sets adhering to the stated
preferences are randomly and repeatedly
elected to identify the best solution
Agenda


Introduction
Multi-objective optimization problem
◦ Mathematical formulation
◦ Non-dominated sorting genetic algorithm

Statistical classification methods
◦ Unsupervised and supervised data classification
◦ Self-organizing map

Reduction of Pareto optimal solutions
◦ Data envelopment analysis


Application to multi-objective RAP
Conclusions
Application to multi-objective RAP

In the Pareto optimal solution
identification stage, MOGA is initially
applied
◦ 75 Pareto optimal solutions by Taboada and
Coit using NSGA-II method

Each Pareto optimal solution has three
dimensions (input vectors)
◦ System reliability, total cost, system weight

A 10  10 output lattice are employed to
get the SOM clustering results
Application to multi-objective RAP

Consider a system consisting of
◦ 3 subsystems (with 5 options)
◦ 4 or 5 types of components in each subsystem
◦ Maximum # of components is 8 per subsystem
Application to multi-objective RAP
Application to multi-objective RAP


Each cluster has its own characteristics
The solutions in a specific cluster are topologically similar to each
other
Application to multi-objective RAP
Application to multi-objective RAP

Results:
◦ 3 solution achieve the RE to 90%
◦ 2 of the above solutions’ RE is equal to 1
Agenda


Introduction
Multi-objective optimization problem
◦ Mathematical formulation
◦ Non-dominated sorting genetic algorithm

Statistical classification methods
◦ Unsupervised and supervised data classification
◦ Self-organizing map

Reduction of Pareto optimal solutions
◦ Data envelopment analysis


Application to multi-objective RAP
Conclusions
Conclusions


This paper introduces a two-stage method
to get Pareto optimal solutions and classify
them to reduce the solution set.
In the Solutions pruning stage, SOM is first
applied in classification