Evidence based Uncertainty Models and Particles Swarm Optimization for Multiobjective Optimization of Engineering Systems

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The present work develops several methodologies for solving engineering analysis and design  problems involving uncertainties and evidences from multiple sources. The influence of uncertainties on the safety/failure of the system and on the warranty costs (to the  manufacturer) are also investigated. Both single and multiple objective optimization problems  are considered.

A methodology is developed to combine the evidences available from single or  multiple sources in the presence (or absence) of credibility information of the sources using  modified Dempster Shafer Theory (DST) and Fuzzy Theory in the design of uncertain engineering  systems.

To optimally design a system, multiple objectives, such as to maximize the belief  for the overall safety of the system, minimize the deflection, maximize the natural frequency and minimize the weight of an engineering structure under both deterministic and uncertain parameters, and subjected to multiple constraints are considered.

We also study the various  combination rules like Dempster’s rule, Yager’s rule, Inagaki’s extreme rule, Zhang’s center  combination rule and Murphy’s average combination rule for combining evidences from multiple sources. These rules are compared and a selection procedure was developed to assist the analyst in selecting the most suitable combination rule to combine various evidences obtained from multiple sources based on the nature of evidence sets. A weighted Dempster Shafer  theory for interval-valued data (WDSTI) and weighted fuzzy theory for intervals (WFTI) were proposed for combining evidence when different credibilities are associated with the  various sources of evidence.

For solving optimization problems which cannot be solved using  traditional gradient-based methods (such as those involving nonconvex functions and discontinuities), a modified Particle Swarm Optimization (PSO) algorithm is  developed to include dynamic maximum velocity function and bounce method to solve both deterministic multi-objective problems and uncertain multi-objective problems (vertex method is used in  addition to the modified PSO algorithm for uncertain parameters).

A modified game theory approach (MGT) is coupled with the modified PSO algorithm to solve multi-objective  optimization problems. In case of problems with multiple evidences, belief is calculated for a safe design (satisfying all constraints) using the vertex method and the modified PSO algorithm is used to solve the multi-objective optimization problems. The multiobjective  problem related to the design of a composite laminate simply supported beam with center load  is also considered to minimize the weight and maximize buckling load using modified game  theory.

A comparison of different warranty policies for both repairable and non repairable  products and an automobile warranty optimization problem is considered to minimize the total  warranty cost of the automobile with a constraint on the total failure probability of the system. To illustrate the methodologies presented in this work, several numerical design  examples are solved. We finally present the conclusions along with a brief discussion of the future scope of the research.


Dempster Shafer Theory:

Many generalized models of uncertainty have been developed to treat different situations; including possibility theory and fuzzy sets, Dempster-Shafer theory of evidence, imprecise probabilities, convex models, random sets and others. These generalized models of uncertainty have a variety of mathematical descriptions. However, they all can typically be described as either random or fuzzy. If the uncertain parameters are treated as random variables, they are described by suitable probability  distributions and the response of the structure, such as  displacement, strains and stresses, can be computed using probability principles.

Fuzzy Set Theory:

When the parameters of a system contain information and features that are vague, qualitative and linguistic, a fuzzy approach can be used to predict the response. Various attempts are made to apply fuzzy set theory to solve structural optimization problems since there exists a vast amount of fuzzy information in both the objective and constraint functions for the design optimization of structures. Many papers have discussed the application of fuzzy set theory to structural design and in particular in structural  optimization. The theory of fuzzy sets was developed for a domain in which descriptions of activities and observations are not well defined.

Multiobjective Optimization:

During the past decade the, subject of structural multi-objective optimization has been explored extensively. Some investigators have treated structures subject to static constraints, e.g., maximum stress limit or minimum deflection, while others have considered structures subject to dynamic constraints, e.g. natural frequency. The literature on multi-objective or multi-criteria decision making has grown tremendously in the last decade.


Particle Swarm Optimization (PSO):

The PSO algorithm is based on the swarm intelligence techniques. The concept of swarm  intelligence was inspired by the social behavior of groups of animals such as a flock of birds, ants, or a school of fish. It is also a population based algorithm in which  individuals are called particles and the population is called swarm. The PSO is similar to evolutionary algorithms (EA) in the sense that both approaches are population-based  and  each individual has a fitness function. Based on the bounds or limits on the design variables,  randomly generated values are taken as initial population in the algorithm.

Warranty Policies:

In today’s competitive market, product warranties have become an important consideration for  both the manufacturer (or seller) and the customer (or consumer). A warranty is an assurance  given by the manufacturer to the buyer at the time of sale that the product will perform its  intended functions satisfactorily for a specified length of warranty.


Combination Rules Based on DST:

The versatility of DST is the motivation for selecting DST to represent and combine different  types of evidence obtained from multiple sources. The various combination rules to combine  evidence obtained from multiple sources are discussed in this section.

Proposed Selection Procedure:

The following guidelines are proposed to provide an insight in to the use of different combination rules depending on the nature of evidence available from different sources.

Engineering Application – A Welded Beam Problem:

The application of the various models for combining evidence is illustrated by considering  the failure analysis of a welded beam in the presence of different bodies of evidence in the various evidence sets. Consider a beam of length L and cross-sectional dimensions t and b that is welded to a fixed support. The weld length is l on both the top and bottom surfaces and the beam is required to support a load P. The weld is in the form of a triangle with a depth of h.


 Fuzzy Approach for Combining Evidences:

When evidences from multiple sources are available regarding the uncertainty of a system, Dempster-Shafer theory has traditionally been used to combine the evidences. This theory was studied and described in detail by Dempster and Shafer. Dempster-Shafer  theory is considered as a generalization of probability theory where probabilities are  assigned to sets instead of mutually exclusive events.

In Dempster-Shafer theory, evidence can be associated with multiple or sets of events. By combining evidence from multiple sources, Dempster-Shafer theory provides the lower and upper bounds, in the form of belief and  plausibility, for the probability of occurrence of an event.


Modified Particle Swarm Optimization:

Particle swarm optimization algorithm, described, is considered along with modification to  include dynamic velocity function for maximum adaptable velocity for the particles, bounce method and dynamic penalty function to the optimization algorithm.

Applications with Single Objective Function:

Two engineering design problems with single objective function are solved to validate the  proposed approaches using modified PSO. The design of a welded beam, with continuous design  variables, is considered by applying the dynamic velocity function to limit the maximum  velocity of the design variables in the algorithm. The design of a pressure vessel with mixed discrete design variables is considered to illustrate the MDNLP where CDA is used to handle the discrete design variables.


This work studies various combination rules to combine evidences from multiple sources  to  understand the procedure of combining evidences in depth and how the conflict among the  evidences can be treated. The solutions of different optimization problems, which are framed and solved for combining two sources of evidences, indicate the similarities and distinctions among various combination rules.

The proposed selection procedure guides the user or analyst to select the most suitable combination rule for combining various evidences obtained from multiple sources based on the nature of evidence sets. At the same time, the user or analyst  is free to use other rules for combining the evidences. Evidence sets given are constructed  in such a way that the applied combination rule gives more satisfactory/logical results compared to other combination rules in each of the five cases.

For each of the cases  described, if any combination rule other than the one suggested is used for combining the evidences we may get misleading results which may not convergence to actual reality when more and more evidence/information becomes available. We also considered an example, in which data is available in various forms of evidence namely deterministic, probabilistic, fuzzy and Dempster’s bpa to combine evidence.


Using a procedure similar to the one , other rules of combining evidences can be explored and  suitable selection procedures can be developed to apply these rules. A Weighted Dempster Shafer Theory for Interval-valued data (WDSTI) and Weighted Fuzzy Theory for Interval-valued  Data (WFTI) are proposed to combine evidences from multiple sources, to evaluate the safety/failure of any uncertain system in the presence of evidence on the uncertain  parameters, when the multiple sources of evidence have different credibilities.

These  approached can be explored for other engineering applications. The scope of the automobile  example considered can be expanded to include more sub-assemblies (or design variables) of the automobile to obtain more realistic optimum total warranty costs. This work can be extended to include the actual data available to the real world auto manufacturers instead of the data assumed. The proposed method using modified particle swarm optimization (PSO) based algorithm  to solve multi-objective optimization of uncertain engineering problems involving different  types of design variables (continuous, discrete and/or mixed).

We can apply this algorithm to solve many optimization problems in the area  of composite structures,  aerospace  structures and civil engineering. We can also explore design of composite structures by expanding the problem solved in chapter-8 for multi-objective optimization using  uncertain  parameters. We can also solve real world automobile optimization examples with inherent uncertainty present in the formulation of the problem. We can expand the example problem in chapter 4 to combine various forms of evidences available from different sources with  different credibilities for complex engineering applications.

Source: University of Miami
Author: Kiran Kumar Kishore Annamdas

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