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Under the Guidance of

Mr.M.JagadeeshAssistant Professor

CSE department

ByM.Praveen Kumar1221010121M.Tech-SE-IV sem


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Introduction

Determination of a single solution for multi-objectiveproblems is performed using methods .

Traditional methods require that the decision-maker hasbroad knowledge about the underlying problem.

The other general approach is the determination of aset of solutions, i.e., a Pareto-optimal set.


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Post-Pareto optimality analysis

Obtain a small sub-set of preferred solutions from thelarge Pareto-optimal set.

The results obtained from any optimization method.

Decision-maker identify the most preferred solution inmulti-objective optimization problems.

Pareto-optimal set methods to reduce solution set

Pruning by using non-numerical objective functionranking preferences method.

Pruning by using data clustering.


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Required characteristics for Reliability

A SRGM can be viewed as product of positive constant.

The fault detection rate must be finite

The SRGM can have limitations

Some SRGMs:

Weibull SRGMs

Gamma SRGMs

Lognormal and Inverse Weibull SRGMs

A SRGM with decreasing fault detection rate


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Weibull SRGMs Fault detection rate per fault at any testing time is a

constant, i.e., d(t) =b

H(t) = a[1exp(bt)] , (10)with

G(t) = 1exp(bt) .

The Weibull distribution is given by

G(t) = 1exp[(t / ) ]

It includes the exponential distribution as its special

case (i.e., when = 1 ).


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Inverse Weibull distributionThe inverse Weibull distribution, given by

G(t) = exp[( /t) ]

It is also close to the exponential distribution when

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Standard Approach :Weighted

Sum of Objective Functions Conventional approach :

Combine the objectives into a single weight formula:

E.g., Fitness = W1 * f1(U) + W2 * f2(U)

and Normalizing the weights W1 + W2 = 1 Limitations:

- Result depends on weights.

- Some solutions cannot be reached.

- Multiple runs of the algorithm are required in order toget the whole picture.

- Difficult to select weights cost functions may be indifferent scales.


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The algorithm used to prune Pareto-optimal solutions :

1.Rank Objectives

2.Scale Objectives

3.Randomly generate weights based on ranks using the

weight function

4. Sum weighted objectives to form a single function


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5. Find the solution that yields the max (optimal) value

6. Increment the counter corresponding to that solutionby a value of one

7. Repeat Steps 2 to 5 numerous (several thousand) times

8. Determine the pruned Pareto optimal set i.e. the

solutions that have non zero counter values (counter > 0)


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Pruning by Using Data Clustering

In this decision-maker is not required to specify anyobjective function preference.

k-means clustering algorithm is used to group thesolutions into clusters.

Determine the optimal number of clusters.

Data into k clusters. The members within a cluster aresimilar to one another.


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The k-means algorithm :

1. Place k points into the space represented by the objects

that are being clustered. These points represent initial

group centroids.

2. Assign each object to the group that has the closest

centroid.

3. When all objects have been assigned, recalculate the

positions of the k centroids.

4. Repeat Steps 2 and 3 until the centroids stabilize (no

longer move).


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