a two-stage approach for multi- objective decision making with applications to system reliability...

A two-stage approach for multi-objective decision making with applications to system reliability optimization

Zhaojun Li, Haitao Liao, David W. CoitReliability Engineering and System Safety

Hui-Yu, ChungAdvisor: Frank, Yeong-Sung Lin

AgendaIntroductionMulti-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 RAPConclusions

IntroductionThough 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.

IntroductionTraditionally, the redundancy allocation

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

IntroductionComponent 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 searchMixing functionally equivalent

components may potentially reduce the variance of system reliability estimate and minimizes the likelihood of common cause failures

IntroductionTo address the multi-objective

optimization 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 self-

organizing map (SOM) method ◦Eliminate the non-efficient solutions using

data envelopment analysis (DEA) method

Two-stage method for solving multi-objective RAP

Multi-objective optimization problemMathematical 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 sub-objective, 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

single-objective problem◦Using Pareto optimal concept based

on non-dominancePareto dominance & non-

dominance◦Determined by multiple pair-wise

vector comparison

Mathematical formulationsx is non-dominated in a p-dimensional

set X if there is no other in X such that .

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 multi-objective optimization problem)

The number of solutions in the Pareto optimal solution set is large as the number of conflicting objectives increases

y x( ) ( )f y f x

Non-dominated sorting genetic algorithmTo 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



Statistical classification methods◦Unsupervised and supervised data classification◦Self-organizing map



Statistical classification methodsUnsupervised 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 mapAn unsupervised classification

methodGenerates 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

Adjusted weights◦ 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:

◦ : the weight vector at step t+1

◦ : the input vector◦ : the learning coefficient

(monotonically decreasing with time)◦ : the neighborhood function

Gaussian neighborhood function is often used

( 1) ( ) ( , ) ( )[ ( ) ( )]w t w t v tt I t w t

( 1)w t ( )I t( )t

( , )v t

2 2( , ) exp( / )v t v

Self-organizing mapEventually, 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 mapSOM measures the similarity by the Euclidian

distance as well as the angle between the input vectors by updating the weight vectors iteratively

Reduction of Pareto optimal solutionsSelecting representative

solutions from each cluster can be regarded as a multi-objective 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 analysisA 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 analysisFor 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 analysisRelative Efficiency (RE)

Considering a problem involving l DMUs, each has m inputs and n outputs, the RE of the kth DMU is:

weighted sumof outputsRE

weighted sumofi nputs

weights

Data envelopment analysisThe RE of a specific DMU can

be obtained by:

is a small positive quantity

0k

Data envelopment analysisNormalized programming

problem:

Data envelopment analysisWhen 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 analysisIn 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

Application to multi-objective RAPIn the Pareto optimal solution

identification stage, MOGA is initially applied◦75 Pareto optimal solutions by Taboada

and Coit using NSGA-II methodEach Pareto optimal solution has

three dimensions (input vectors)◦System reliability, total cost, system

weightA output lattice are employed

to get the SOM clustering results

10 10

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 Each cluster has its own characteristics The solutions in a specific cluster are topologically

similar to each other

Application to multi-objective RAPResults:

◦3 solution achieve the RE to 90%◦2 of the above solutions’ RE is equal

to 1

ConclusionsThis 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◦Basic trade-offs information about the

solution set can be observedDEA method is further used to reduce

the original solutions◦Which makes multi-objective decision

making for RAP more easier

THANKS FOR YOUR ATTENTION~!!!

a two-stage approach for multi- objective decision making with applications to system reliability...

Documents

multiobjective decision

n objective functions

promising solutions

kind of solutions

stage approach

prospective solutions

system redundancy

variance of system reliability