A twostage approach for multiobjective decision making with

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 Advisor: Frank, Yeong-Sung Lin

Agenda �Introduction �Multi-objective optimization problem ◦ Mathematical formulation ◦ Non-dominated sorting genetic algorithm �Statistical classification methods �Reduction of Pareto optimal solutions ◦ Unsupervised and supervised data classification ◦ Self-organizing map ◦ Data envelopment analysis �Application to �Conclusions multi-objective RAP

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 decisionmaker to consider only a small subset of the promising solutions.

Introduction �Traditionally, 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

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 �To 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 selforganizing map (SOM) method ◦ Eliminate the non-efficient solutions using data envelopment analysis (DEA) method

Two-stage method for solving multi-objective RAP

Agenda �Introduction �Multi-objective optimization problem ◦ Mathematical formulation ◦ Non-dominated sorting genetic algorithm �Statistical classification methods �Reduction of Pareto optimal solutions ◦ Unsupervised and supervised data classification ◦ Self-organizing map ◦ Data envelopment analysis �Application to �Conclusions multi-objective RAP

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 single-objective problem ◦ Using Pareto optimal concept based on non-dominance �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 in X such that. �If N is a set containing all the nondominated 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

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 �Reduction of Pareto optimal solutions ◦ Unsupervised and supervised data classification ◦ Self-organizing map ◦ Data envelopment analysis �Application to �Conclusions multi-objective RAP

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 �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

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 �Conclusions multi-objective RAP

Reduction of Pareto optimal solutions �Selecting 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 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) �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 obtained by: � is a small positive quantity can be

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 �Reduction of Pareto optimal solutions ◦ Unsupervised and supervised data classification ◦ Self-organizing map ◦ Data envelopment analysis �Application �Conclusions to multi-objective RAP

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 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 �Reduction of Pareto optimal solutions ◦ Unsupervised and supervised data classification ◦ Self-organizing map ◦ Data envelopment analysis �Application to �Conclusions multi-objective RAP

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 ◦ Basic trade-offs information about the solution set can be observed �DEA method is further used to reduce the original solutions ◦ Which makes multi-objective decision making

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