MULTI-ATTRIBUTE OPTIMIZATION

BASED ON CONJOINT ANALYSIS

A thesis submitted in partial ful�llmentof the requirements for the degree of

Master of Science

By

HEMANTH AMARCHINTA

B.E., Osmania University, India, 2004

2006

Wright State University

WRIGHT STATE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

December 28, 2006

I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MYSUPERVISION BY Hemanth Amarchinta ENTITLEDMulti-Attribute OptimizationBased on Conjoint Analysis BE ACCEPTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMaster Of Science in Engineering

Ramana V. Grandhi, Ph.D.Thesis Director

George P. Huang, Ph.D.Department Chair

Committe onFinal Examination

Ramana V. Grandhi, Ph.D.

Kenneth C. Cornelius, Ph.D.

Ravi C. Penmetsa, Ph.D.

Joseph F. Thomas, Jr., Ph.D.Dean, School of Graduate Studies

Abstract

Amarchinta, Hemanth, M. S. Engineering., Department of Mechanical and Ma-terials Engineering, Wright State University, 2006. Multi-Attribute Optimizationbased on Conjoint Analysis.

Over the last thirty years, there were tremendous advances in multidisciplinary

design optimization in reducing computational cost, developing algorithms for

e�cient sensitivity analysis in reaching an optimum. Most of these e�orts as-

sumed a single objective (attribute) function and a multitude of constraints. Very

little work has been done in including the designer's preferences as part of the

optimization scheme and in addressing the ability to handle multiple attributes si-

multaneously. This need to develop a systematic method for including designer's

preferences is the main focus of this research work. The concept of modeling

preferences among multi-attribute alternatives is prevalent in consumer product

marketing, and the current work adopts a widely used marketing method known

as conjoint analysis. This method is often implemented to assess the individual

part-worth of the attributes, which provides insightful knowledge of the products

and is then further used to create new products in the market. Conjoint analysis

can be integrated with optimization techniques for engineering applications. In

this work, a novel method of combining advances from management science and

engineering disciplines is implemented. Details about the conjoint analysis algo-

rithm is discussed with an example of designing a pizza. Furthermore, the method

is applied to structural engineering applications such as a cantilever beam, a �xed

plate, and a composite lightweight torpedo demonstrating the advantages of the

method.

iii

Contents

1 Introduction 1

1.1 Pareto Frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Multi-Attribute Optimization . . . . . . . . . . . . . . . . . . . . . 3

2 Multi-Attribute Optimization Approaches 5

2.1 Constraint Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Weighted Sum Method . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Global Criterion Method . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Goal Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.5 GUESS Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.6 Physical Programming . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Conjoint Analysis 12

3.1 Conjoint Value Analysis . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Adaptive Conjoint Analysis . . . . . . . . . . . . . . . . . . . . . . 13

3.3 Choice-Based Conjoint Analysis . . . . . . . . . . . . . . . . . . . . 14

3.4 Algorithm of Conjoint Value Analysis . . . . . . . . . . . . . . . . . 14

3.5 Details of Algorithm With Examples . . . . . . . . . . . . . . . . . 16

3.5.1 Attribute Selection . . . . . . . . . . . . . . . . . . . . . . . 16

3.5.2 Determine Attribute Levels . . . . . . . . . . . . . . . . . . 17

3.5.3 Determine Attribute Combinations . . . . . . . . . . . . . . 18

iv

3.5.4 Select the Presentation Form for the Respondent and Nature

of Judgment . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.5.5 Decide on Aggregation of Judgments . . . . . . . . . . . . . 18

3.5.6 Dummy-Variable Regression Technique . . . . . . . . . . . . 19

3.6 Designing a Pizza . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Combining Marketing and Engineering Tools 26

4.1 Conjoint Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Interpolation and Extrapolation . . . . . . . . . . . . . . . . . . . . 27

4.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.4 Sequential Quadratic Programming Method . . . . . . . . . . . . . 29

5 Engineering Applications 31

5.1 Cantilevered Beam Example . . . . . . . . . . . . . . . . . . . . . . 31

5.1.1 Beam Case Study I . . . . . . . . . . . . . . . . . . . . . . 32

5.1.2 Beam Case Study II . . . . . . . . . . . . . . . . . . . . . . 36

5.2 Fixed Plate Example . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.2.1 Plate Case Study I . . . . . . . . . . . . . . . . . . . . . . . 39

5.2.2 Plate Case Study II . . . . . . . . . . . . . . . . . . . . . . . 43

5.2.3 Plate Case Study III . . . . . . . . . . . . . . . . . . . . . . 48

5.3 Composite Lightweight Torpedo . . . . . . . . . . . . . . . . . . . . 51

5.3.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.3.2 Attributes Considered . . . . . . . . . . . . . . . . . . . . . 53

5.3.3 Optimization of Torpedo Hull . . . . . . . . . . . . . . . . . 54

5.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6 Conclusions and Future Work 59

6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

A Matlab code for optimization of Cantilever Beam 62

A.1 Constraint optimization for displacement and stress as constraint . 62

v

A.2 Conjoint based optimization with stress and displacement as attributes 62

B Matlab Codes and python scripts for Fixed Plate 65

B.1 Matlab code for Constraint optimization for frequency and displacement

as constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

B.2 Python script for output of first natural frequency . . . . . . . . . 66

B.3 Python script for maximum von mises stress . . . . . . . . . . . . . . 67

B.4 Matlab code for Conjoint optimization with displacement and first natural

frequency as attributes . . . . . . . . . . . . . . . . . . . . . . . . . 67

C MATLAB Code for Optimization for Composite Lightweight Torpedo 69

C.1 Matlab Code for Constraint Optimization . . . . . . . . . . . . . . . . 69

C.2 Matlab Code for Conjoint based Optimization . . . . . . . . . . . . . 71

Bibliography 73

vi

List of Figures

1.1 Pareto frontier in two-attribute problem . . . . . . . . . . . . . . . 2

1.2 Schematic representation of multi-attribute optimization . . . . . . 3

2.1 Drawback of constraint method . . . . . . . . . . . . . . . . . . . . 6

2.2 Constant trade-o�s in weighted sum method . . . . . . . . . . . . . 8

3.1 Flow-chart for conjoint analysis . . . . . . . . . . . . . . . . . . . . 15

3.2 Cantilever beam for demonstration . . . . . . . . . . . . . . . . . . 19

4.1 Multi-attribute optimization based on conjoint analysis . . . . . . . 27

5.1 Cantilever beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2 Preference curves for mass and displacement . . . . . . . . . . . . . 35

5.3 Fixed plate with uniformly distributed load . . . . . . . . . . . . . 39

5.4 Finite element model with element numbering . . . . . . . . . . . . 39

5.5 Preference curves for mass, displacement, and stress . . . . . . . . . 42

5.6 Preference curves for attributes - mass, displacement, and frequency 46

5.7 Preference curves for mass and frequency . . . . . . . . . . . . . . . 50

5.8 Finite element mesh of composite lightweight torpedo . . . . . . . . 52

5.9 Cross-section of the composite shell . . . . . . . . . . . . . . . . . . 52

5.10 Preference curves for composite torpedo . . . . . . . . . . . . . . . 56

6.1 Integrating uncertainty with conjoint analysis and optimization . . 60

vii

List of Tables

3.1 Ranking wise preferences for cantilever beam . . . . . . . . . . . . . 20

3.2 Dummy-variable approach . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Logit recode transformation of dependent variable, ranking . . . . . 22

3.4 Zero-centered part-worths before scaling . . . . . . . . . . . . . . . 23

3.5 Part-worths for mass and stress . . . . . . . . . . . . . . . . . . . . 23

3.6 Attribute levels for pizza . . . . . . . . . . . . . . . . . . . . . . . . 24

3.7 Preferences for pizza . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.8 Part-worths for each attribute of a pizza . . . . . . . . . . . . . . . 25

5.1 Levels for mass and displacement . . . . . . . . . . . . . . . . . . . 33

5.2 Preferences chosen for combinations of attribute levels . . . . . . . 33

5.3 Part-worths obtained for each level . . . . . . . . . . . . . . . . . . 34

5.4 Comparision of results of case study I for cantilever beam . . . . . . 36

5.5 Levels for mass, displacement, and stress . . . . . . . . . . . . . . . 37

5.6 Preferences for attributes of mass, displacement and stress . . . . . 37

5.7 Part-worths for mass, displacement, and stress . . . . . . . . . . . . 37

5.8 Comparision of results of case study II for cantilever beam . . . . . 38

5.9 Optimum thickness for displacement and stress constraint . . . . . . 40

5.10 Levels for attributes of mass, displacement, and stress . . . . . . . . 40

5.11 Preference of ranks for mass, displacement, and stress . . . . . . . . 41

5.12 Part-worth for each level for mass, displacement, and stress . . . . 41

viii

5.13 Optimum variables for mass, displacement, and stress using

the conjoint approach . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.14 Comparision of results of case study I for �xed plate . . . . . . . . . 43

5.15 Optimum thickness for displacement and frequency constraints . . . 44

5.16 Attribute levels for mass, displacement, and frequency . . . . . . . . 44

5.17 Preferences for mass, displacement, and frequency attributes . . . . 45

5.18 Part-worth for mass, displacement, and frequency . . . . . . . . . . 45

5.19 Optimum thickness using conjoint approach . . . . . . . . . . . . . 47

5.20 Comparision of results of case study II for �xed plate . . . . . . . . 47

5.21 Optimum design variables con�guration with displacement and

frequency as constraints . . . . . . . . . . . . . . . . . . . . . . . . 48

5.22 Levels for mass and frequency attributes . . . . . . . . . . . . . . . 49

5.23 Preference rankings for mass and frequency attributes . . . . . . . . 49

5.24 Part-worths of attributes mass and frequency for each level . . . . . 49

5.25 Optimum design variables with trade-o�s between mass and fre-

quency with displacement as constraint . . . . . . . . . . . . . . . . 51

5.26 Comparision of results of case study III for �xed plate . . . . . . . . 51

5.27 Material properties of carbon/epoxy . . . . . . . . . . . . . . . . . 53

5.28 Design variables for composite torpedo . . . . . . . . . . . . . . . . 54

5.29 Optimum thickness using constraint approach . . . . . . . . . . . . 54

5.30 Attributes and levels for composite toredo . . . . . . . . . . . . . . 55

5.31 Ranking-based preferences for composite torpedo . . . . . . . . . . 55

5.32 Part-worths obtained for composite torpedo . . . . . . . . . . . . . 55

5.33 Optimum thickness of composite torpedo

using conjoint approach . . . . . . . . . . . . . . . . . . . . . . . . 57

5.34 Comparision of results for composite torpedo . . . . . . . . . . . . . 57

ix

Acknowledgements

I would �rst like to thank my advisor, Dr. Ramana V. Grandhi, for his guid-

ance and support throughout my graduate studies. The knowledge he has shared

with me will remain a tremendous resource for my professional career. I hope to

continue to enjoy a proli�c relationship in future. I would also like to extend my

gratitude to Dr. Kenneth C. Cornelius, and Dr. Ravi C. Penmetsa for serving on

my thesis committee. A special thanks to Dr. Pola Gupta from the Department

of Marketing for providing the software for conjont analysis. I would like to thank

my family for their continuous support during my graduate studies. I would also

like to thank my roomates Rajesh, Hemanth, and Deepak, and CDOC friends

Muthu, Todd, Gulshan, Sangki, Randy, and Arif for providing me support and

strength. I gratefully thank Alysoun for her editorial assistance.

x

1 Introduction

Several design optimization problems could make use of multiple objective func-

tions, but most problems are formulated with a single objective function. Multiob-

jective optimization stayed as a separate and an advanced technique in structural

design optimization. In the �eld of management it is referred to as multiple criteria

decision making (MCDM). In this work it is called as multi-attribute optimiza-

tion because the idea of including preferences in optimization is borrowed from

marketing, and functions are referred as attributes. Formulating a structural op-

timization problem consists of making a mathematical model that describes the

behavior of the physical system [4]. After determining the design variables, objec-

tives, and constraints, which are in�uential in the optimization process, we often

con�ne ourselves to a single objective and allow other performance measures or

attributes to be constraints, even though this is not what we really intend. In

the recent past, many optimization techniques have been developed that break

this tradition of the single objective and include multiple objectives in a problem.

Two of the requirements for an optimization technique are that it must be able to

(i) handle multiple competing attributes, and (ii) incorporate the designer's pref-

erences in solving an optimization problem. In the following section, the pareto

frontier, which is a building block of optimization is discussed followed by basic

understanding of multi-attribute optimization.

1

1.1 Pareto Frontier

A general multi-attribute optimization involves �nding the vector of design vari-

ables X = [x1, x2, x3, . . . , xn]T that minimizes a vector of the objective functions

f(X) = [f1(X), f2(X), f3(X), . . . , fk(X)]T (1.1)

In optimization, the optimal solution is the one that attains minimum values of

each attribute when considered simultaneously. Thus for a multiobjective problem,

the solution x∗ is optimal if and only if x∗ ∈ s where s is a feasible design and

fi(x∗) ≤ fi(x) for all i and for all x ∈ s in a minimization problem [2]. However,

if the attributes are con�icting in nature, there is no unique optimal solution. For

con�icting attributes one may obtain at best a series of e�cient or non-dominant

solutions as shown in Figure 1.1 for two attributes f1(x) and f2(x).

Figure 1.1: Pareto frontier in two-attribute problem

The non-dominated solutions form a set of solutions in which no decrease can

be obtained in any of the attributes without simultaneously increasing in at least

one of the remaining attributes.

2

1.2 Multi-Attribute Optimization

The basics of multi-attribute optimization are discussed in this section. As men-

tioned in the previous section, we can generate a series of non-dominant solutions.

The problem is, in real world applications, engineers may not be interested in a

series of solutions but are interested in one preferred solution. Also, an approach

that optimizes a single attribute is not realistic in most applications. It is a com-

plex task to pick one solution from the set of optimal solutions. Figure 1.2 shows

a schematic diagram of a multi-attribute optimization (MAO) [7].

Figure 1.2: Schematic representation of multi-attribute optimization

MAO is a two-step process. The �rst step optimizes multiple attributes and

generates a pareto frontier. The second step is selecting the best trade-o� solution.

In choosing one solution from this series of solutions, higher level information is

needed, which could include the designer's preferences. The best trade-o� solution

can be chosen accordingly.

The main focus of this research is to (i) develop an optimization technique

which has the ability to handle multiple attributes and (ii) incorporate designer's

preferences in the optimization to obtain the best trade-o� design. This thesis is

divided into 6 chapters. Chapter 2 discusses in brief some of the methods used

in multi-attribute optimization, their advantages and disadvantages. Chapter 3

is dedicated to conjoint analysis, basic types of conjoint analysis, an algorithm

3

used, and a few examples showing the implementation of the method. Chap-

ter 4 combines the tools of marketing and optimization with development of the

preference-based multi-attribute optimization method based on conjoint analysis.

This chapter is the heart and soul of this thesis. In Chapter 5, engineering ex-

amples of a cantilever beam, a �xed plate, and a composite lightweight torpedo

are solved to demonstrate the method. This is followed by conclusions and future

directions in Chapter 6.

4

2 Multi-Attribute Optimization

Approaches

As mentioned in Chapter 1, it is a complex task to select a design from non-

dominated set or for that matter, even from a dominated set. The optimization

approach should include the capability to optimize simultaneously for multiple

competing attributes, it must be an e�cient method especially if the attribute

calculation comes from a black box such as Finite Element Analysis (FEA) or

Computational Fluid Dynamics (CFD) and it must also include the designer's

preferences. Several algorithms have been applied to tackle this problem [12, 8].

A survey of the methods for solving multiobjective optimiztion is presented by

Marler and Arora in [18], which o�ers a broad review of the methods. This chapter

discusses some of the approaches available to solve the multi-attribute optimization

problem. The �rst two methods are presented with a simple example of cost and

comfort level.

2.1 Constraint Method

The constraint method is the most basic method. One of the attributes is con-

verted to an objective function and the other attributes are converted to con-

5

straints. The standard formulation is shown in Equation 2.1:

Minimize f1(x) (2.1)

subject to

f2(x) ≤ 0 h1(x) = 0

......

Design Variables : xL ≤ x ≤ xU

The advantages of this method are, easy set up of the formulation and its single

objective function nature. But the disadvantage of this method is that there

are no trade-o�s involved in the optimization. Consider an example involving

attribute f1 as cost and attribute f2 as comfort level. The problem is formulated

as Minimize f1 and a bound is set on comfort level as fmax2 = 50%. We might

obtain a design of $100 cost for a comfort level of 50%, but this method will discard

a design with cost of $105 for a comfort level of 95%, which may be a preferred

design, because there are no designer preferences involved in the optimization.

The iterative optimization is stopped when any of the constraints are active. This

drawback is shown graphically in Figure 2.1:

Figure 2.1: Drawback of constraint method

6

2.2 Weighted Sum Method

The weighted sum method is also quite commonly used in optimization. Each

attribute is given a weight, and the objective function is formed by the weighted

sum of the attributes. The constraints from the previous method are now incorpo-

rated in the objective function by corresponding weighting factors. The standard

formulation is shown in Equation 2.2:

Minimize w1f1(x) + w2f2(x) + · · · (2.2)

where

w1 + w2 + · · · = 1

0 ≤ w1 ≤ 1, 0 ≤ w2 ≤ 1

Design Variables xL ≤ x ≤ xU

The weights are selected based on the importance of the attributes relative to each

other. These weights now represent the trade-o�s between the attributes. But the

drawback in this method is that these weights are always the same throughout the

optimization, which means a constant trade-o�. To illustrate this, using the same

example of cost and comfort level as discussed before, consider f1 as the cost and

f2 as the comfort level. If the weights are selected as w1 = 0.75 and w2 = 0.25,

this means that the designer is willing to trade-o� 0.750.25

= 3 units of cost for every

level of comfort level, regardless of the current value of the comfort level, which

may not be true, because it seems intuitive that the designer will tend to trade-o�

less at a higher comfort level. The constant trade-o�s are shown in Figure 2.2:

7

Figure 2.2: Constant trade-o�s in weighted sum method

2.3 Global Criterion Method

The global criterion method [25] develops a global objective function that is derived

from the sum of the deviations of the values of the individual objective functions

fj(x) from their respective ideal values fj(x∗) as a ratio to that of the ideal values.

Thus, a single objective function is formulated. The modi�ed problem is shown

in Equation 2.3:

Minimize F =∑k

j=1

[fj(x

∗)−fj(x)

fj(x∗)

]p

(2.3)

subject to

g1(x) ≤ 0 h1(x) = 0

......

Design Variables xL ≤ x ≤ xU

where p is the integer valued exponent that re�ects the importance of the ob-

jectives. The trade-o� information is embedded in the p value; setting p = 1

implies that equal importance is given to all deviations, while p = 2 implies that

these deviations are weighted proportionately, with the largest deviation having

the largest value.

8

2.4 Goal Programming

In goal programming [4] each attribute is given a target value. The objective

function is formulated as deviations from these target values and it is minimized.

Target values are considered as additional constraints in which new variables are

added to represent the deviations. The goal programming problem is formulated

as shown in Equation 2.4:

Minimize [∑k

j=1(d+j + d−j )p]

1p , p ≥ 1 (2.4)

subject to

gj(X) ≥ 0, j = 1, 2, ...,m

fj(X)− d+j + d−j = bj, j = 1, 2, ..., k

d+j ≥ 0, j = 1, 2, ...k

d−j ≥ 0 j = 1, 2, ...k

d+j d−j = 0, j = 1, 2, ...k

(2.5)

where bj is the target set by the designer for the jth attribute; d+j and d−j are the

under- and over- achievement of the jth target, respectively. Goal programming

brings designer preferences into the optimization by allowing the designer to set

the target values. The target values control the optimal solution. Generally, p = 2

which is an euclidean metric value. The main disadvantage of this method is that

if the target values are not placed properly, the optimization leads to a dominated

design which means any one of the attributes can be improved without worsening

others.

9

2.5 GUESS Method

The GUESS method is an interactive solution method based on a class of methods

called reference methods and is used with continuous multiple criteria decision

problems [5]. The maximum and minimum values of each attribute based on

the criterion are calculated and are referred to as Uj and Mj, (j = 1, 2, ..., k),

respectively. The vector of Uj is called the ideal point and the vector of Mj is

called the nadir point. The criterion vector is normalized,

dj(x) =fj(x)−Mj

Uj −Mj

, j = 1, 2, ...k (2.6)

and the optimization problem is formulated as

Maximize y (2.7)

subject to fj(x)− (Aj −Mj)y ≥ Mj, j = 1, 2, 3, ...k

where Ak is the initial guess of the attributes. At each iteration, this method

�nds a solution whereby proportional achievement of each criterion is maintained.

Demonstration of this method for linear attributes can be seen in [5]. This method

also involves designer preferences by setting ideal values for each vector, but the

drawback is that it is di�cult to implement if there is a lack of knowledge about

the performance of attributes. Also other than the solutions which result from the

guesses, no guidance is given to the decision maker.

2.6 Physical Programming

Physical programming [20] divides individual attributes into ranges based on de-

signer preferences. These ranges are indicated as highly desirable, desirable, tol-

10

erable, undesirable, highly undesirable, and unacceptable. These priorities are

converted into class functions using a physical programming lexicon and di�erent

curve �tting methods depending on the ranges of attributes. The main advantage

of physical programming as addressed by Messac [20] is that when compared to a

weighted sum method, this method eliminates an extra optimization loop that is

prevalent in the weighted sum method. Once the individual preference functions

are obtained, the objective function is created by summation of log10 of the indi-

vidual functions. The disadvantage of this method is that the assumptions must

be made for the individual functions formation that can incorporate preferences

of the designer. The goal of this method is to get all attributes in the desirable

range; when this is not possible, the individual function formulation determines

what will be the mix of ranges that are stated above in the optimal solution.

2.7 Summary

A review of some of the methods used to optimize multiple competing attributes

with advantages and disadvantages are presented. These disadvantages give the

need to develop an e�cient way of incorporating designer preferences in the op-

timization. Conjoint analysis which is described in the following chapter tries to

address this issue.

11

3 Conjoint Analysis

In marketing science the concept of modeling consumer preferences among multi-

attribute alternatives has received much attention. The idea is to characterize

a product into a bundle of attributes and assign levels for each attribute. A

technique known as conjoint analysis is used to obtain the numerical values of

the product. Conjoint analysis is a technique that breaks down attributes to

derive the part-worth associated with each level of a product based on the overall

preferences of choice alternatives by a group of respondents [22]. Conjoint analysis

can be best understood by an example. The product under consideration is a car,

and the attributes that are in�uencing customers to make judgments for buying

the car are (i) make, (ii) price, (iii) seating capacity, etc. Yet many respondents

�nd it very di�cult to estimate the contribution of each attribute to the �nal

decision. Conjoint analysis attempts to handle this problem by estimating the

values of each attribute by calculating the part-worths on the basis of preferences

respondents make along product concepts that are varied in a systematic way. The

dictionary de�nes conjoint as joined together or combined. The word conjoint

is used here because the relative values of attributes are considered jointly and

can be determined when they might not be measurable taken one at a time. In

the marketing discipline, conjoint analysis can be applied in health care, energy

policies, and public policy decisions just to name a few applications. Further

details are provided by Green and Srinivasan [9] and in engineering it is applied

for acoustic design by Grissom, et al [11]

According to Orme [21] conjoint analysis can be divided into three types: (i)

12

conjoint value analysis, (ii) adaptive conjoint analysis, and (iii) choice-based con-

joint analysis. The above mentioned types of analysis are discussed brie�y in the

next section.

3.1 Conjoint Value Analysis

The traditional full pro�le conjoint analysis is also called Conjoint Value Analysis

(CVA). Full pro�le conjoint has been the mainstay of the conjoint community for

decades. Green and Srinivasan[9] have suggested that the full pro�le approach

is useful for measuring up to six attributes. This number varies depending on

the project and respondent familiarity with the category. This method can be

implemented with paper and pencil and, these days with computers. The main

disadvantage to CVA is that as the number of attributes grow it becomes a big

burden for the respondents to make their preferences and errors creep in the results

3.2 Adaptive Conjoint Analysis

Adaptive Conjoint Analysis (ACA) was introduced in the late 1980s and was a very

popular method through the 1990s [14, 26]. ACA's main advantage is its ability

to handle more attributes than CVA. In ACA respondents do not evaluate all

attributes at the same time and thus are not overloaded. ACA is a hybrid approach

combining state evaluations of attributes and levels with pairwise comparisions.

Given two products how much one would prefer product 1 over product 2. One of

the main drawback of this method is that it can be administered only via computer

as the interview (combinations of attributes) adapts to respondents' answers as the

survey progresses, which cannot be done with pencil and paper. Other limitation

is when price is included as an attribute, its importance is likely to be understated

[26].

13

3.3 Choice-Based Conjoint Analysis

Choice-Based Conjoint (CBC) analysis gained popularity in the early 1990s and

has now become the most widely used conjoint technique in the world, according

to Sawtooth Software Company [23] which markets the software named Sawtooth

Software SMRT. CBC closely mimics the products in the competing market. In-

stead of rating or ranking scales, respondents are shown a set of products and

asked which one they would like to purchase. As in the real world the respondents

can decline to purchase an item in a CBC interview by choosing none.

Conjoint Value Analysis addresses the needs of the problems faced in engineering

world. It is easy to implement and hence used in this research.

3.4 Algorithm of Conjoint Value Analysis

In this research work, Conjoint Value Analysis (CVA) is used to perform conjoint

analysis. This section describes the step by step detailed procedure used. Figure

3.1 shows a �ow chart [6] of the CVA procedure:

14

Figure 3.1: Flow-chart for conjoint analysis

The �rst step in the process of conjoint analysis is to decide the attributes (or

functions), fi, i = 1, 2, · · · k, where k is the number of attributes, that are relevant

functions in the design of the system. Next, for each of these selected attributes,

levels must be chosen, fij, j = 1, 2, · · ·m , where m denotes the number of levels for

each attribute, and i is ranging from 1, 2, · · · k as stated above. The third major

decision for the designer is to choose the number of combinations of attribute

levels to consider for including preferences. For example, the designer must decide

whether to consider a full factorial design with all the combinations. But if there

are too many combinations, it is di�cult to consider a full factorial design as it

will be a burden to come up with preferences. Therefore in these circumstances a

fractional factorial designs must be considered. The fourth step in the process is

to make a description of the above considered combinations such that the designer

has some insight into the combinations to make judgments about the preferences.

Also nature of judgments can be a rating-wise where the designer can rate the

combinations on a 1 to 10 rating scale. The judgments can be made ranking-wise,

15

such as ranking the products in the increasing order of preference. The next step

is situation dependent: if there is a group that is involved in the design of the

system, then the judgments of preferences of these people have to be aggregated.

If there is only one designer who has the �nal word, then this step is not relevant.

The �nal step is to systematically convert these preferences of input data into

part-worth or utilities for all levels of each attribute. Several methods exist [10]

for this step, and the method's selection is dependent on the nature of judgments.

The dummy-variable regression technique[24] is used in this work. This is a simple

technique that converts the preferences, either ranking or rating, systematically

using regression analysis.

3.5 Details of Algorithm With Examples

This section presents the details of each step discussed in the previous section. The

marketing perspective is presented �rst, followed by the engineering application.

The analysis technique of dummy-variable regression [24] that is used to calculate

part-worths is discussed with an example.

3.5.1 Attribute Selection

The �rst step in the process is to select the attribute for analysis. In marketing,

this is a di�cult task, as there are many attributes present, and it is important to

know which ones to consider. For example, for an automobile (car) the attributes

can be brand, color of the car, number of doors, car size, car power, fuel e�ciency,

tires used, brake system, car warranty, engine life , cost and so on. For engineering

applications this is not a problem because if we design a structure, there are certain

things that we look for such as mass, stress, strain, frequency, displacement etc.

which become the attributes.

16

3.5.2 Determine Attribute Levels

In the above considered example of car and in most marketing products, the levels

are discrete. For example, the brand of the car can be Ford, GM, or Toyota and

the number of doors can be 2 or 4. Some of the attributes are continuous like

cost and engine life. For discrete attributes, selecting the levels is easy because we

know which levels are available, but for continuous attributes, selecting the levels

is a di�cult task. For example, cost is dependent on many factors such as (a)

vehicle class (mid-size, luxury, etc.) (b) competition and competitor's pricing, (c)

local market factors, such as which part of the world they are in and the prevailing

standard of living, and so on. For engineering applications, selecting levels is a

di�cult task because most of the attributes are continuous. We might have an

idea of one level for one or two attributes, but selecting the correct number of

levels, with values for each level, requires some experience and intuition by the

designer. For example,

If we are designing a beam which is used in construction. Assuming the beam

is made up of aluminum. We know that the stress in the beam has to be less

than 30 ksi because aluminum's yield strenght is 30 ksi. So, one level for stress

can be 30 ksi. The next questions are how many levels should be there and what

should their values be. The designer needs to have some experience about the

stress range of the beam in order to properly select the number of levels and their

values. One approach is, the designer must decide the lowest value that will be

considered in the trade-o� and assign it as a level for stress. In this case, we know

that if we set the stress value too low, for example, 10 ksi, the result will be an

unacceptable mass attribute. A simple approach is to select the range for stress

and then linearly increase the stress value from the lowest. This approach reveals

the drawback of this method: the designer should have some knowledge of the

expected response to select the number of levels and the values for each level.

17

3.5.3 Determine Attribute Combinations

The third major decision the designer makes to conduct a conjoint analysis involves

deciding the speci�c number of combinations of attributes that will be used. Ac-

cording to Green and Srinivasan [9] at most �ve or six attributes can be used for a

full factorial design. With a higher number, it is di�cult for respondents to make

their preferences correctly because there are so many combinations to consider,

requiring the use of fractional factorial designs. In this work, Sawtooth Software

is used to select the number of combinations based on the number of attributes

and the levels for each.

3.5.4 Select the Presentation Form for the Respondent and

Nature of Judgment

Step 4 involves selecting the form of presentation for the respondent to evaluate.

The three basic approaches are verbal description, paragraph description, and

pictorial description. For engineering applications, it is di�cult to get descriptions

about the response because the designer does not have a clear idea of how the

response behaves. Therefore, based on the combinations of the attributes available

the designer has to make the trade-o� decisions. The two most popular nature

of judgment options are rank wise or rating wise. Ranking method ranks all the

given combinations in increasing or decreasing order of preference. Ranking-based

preferences are used in this work. Rating scale of 1 to 10 can be used and then

judge each combination.

3.5.5 Decide on Aggregation of Judgments

Managers �nd it very di�cult to develop marketing strategies based on part-worths

calculated from the response of single individual. Step 5 involves deciding if the

responses from individuals will be aggregated and if so, how? At one extreme, we

can pool all the responses across all individuals and then estimate the part-worth.

18

This option fails to recognize any heterogeneity in preferences that might exist

among individuals. Making average of individual responses is another option. In

engineering, if only one designer is working and has the �nal word, then there

is no need for aggregating, but if the decision depends on many individuals then

aggregation is required.

3.5.6 Dummy-Variable Regression Technique

The dummy-variable regression technique [24] is used to estimate the part-worths

and can be best understood by an example. An engineering example is shown

below. Consider a cantilever beam as shown in Figure 3.2 . Stress and mass are

of importance, so these are the two attributes.

Figure 3.2: Cantilever beam for demonstration

The levels for each attribute are chosen as Mass → 0.2 lb, 0.3 lb, 0.4 lb, and

0.5 lb and for Stress → 27 ksi, 28 ksi , 29 ksi, and 30 ksi. The levels are chosen

independently; one way to choose is that the designer knows the stress has to be

less than 30 ksi, so based on this, the other levels for stress are chosen. Lower

stress values are preferred, hence all the levels are lower than 30 ksi. Stress levels

are chosen decreasing linearly because the designer has no real experience on how

the stress varies; therefore linear variation is the safe bet. Similarly, for mass,

the range of mass is expected based on the stress range; the levels for mass are

chosen appropriately. There are 4 × 4 = 16 combinations in all. Ranking scale

preferences are used, where rank 1 is least preferred and rank 16 is most preferred.

The preferences given are shown in Table 3.1:

19

Table 3.1: Ranking wise preferences for cantilever beam

Mass (lb) Stress (ksi) Ranking

0.2 27 160.2 28 150.3 27 140.2 29 130.3 28 120.2 30 110.3 29 100.4 27 9

Mass (lb) Stress (ksi) Ranking

0.3 30 80.4 28 70.5 27 60.4 29 50.5 28 40.4 30 30.5 29 20.5 30 1

The �rst design of 0.2 lb and 27 ksi, (and possibly other designs) is not practical,

but is given a rank of 16 as this is the ideal design, and based on this, other ranking

values are made. These rankings are the designer's preferences. The design of

0.2 lb mass and 28 ksi is ranked second, and 0.3 lb mass and 27 ksi is ranked

third. This shows the designer is willing to have higher stress levels, as he is

more concerned about mass. In this manner the trade-o�s are considered in the

ranking. The next step is to convert the trade-o� rankings in a systematic way

into part-worths for all levels of each attribute. The dummy-variable approach is

used to convert this data to �t into a model. In this method, for each ranking if

the level for an attribute is present it is substituted by 1 otherwise 0 is substituted.

The above rankings are now converted as shown in Table 3.2:

20

Table 3.2: Dummy-variable approach

0.2(x1) 0.3(x2) 0.4(x3) 0.5(x4) 27(x5) 28(x6) 29(x7) 30(x8) Rank(y)

1 0 0 0 1 0 0 0 161 0 0 0 0 1 0 0 150 1 0 0 1 0 0 0 141 0 0 0 0 0 1 0 130 1 0 0 0 1 0 0 121 0 0 0 0 0 0 1 110 1 0 0 0 0 1 0 100 0 1 0 1 0 0 0 90 1 0 0 0 0 0 1 80 0 1 0 0 1 0 0 70 0 0 1 1 0 0 0 60 0 1 0 0 0 1 0 50 0 0 1 0 1 0 0 40 0 1 0 0 0 0 1 30 0 0 1 0 0 1 0 20 0 0 1 0 0 0 1 1

The variables x1, x2, · · ·x8 are assigned for each level. The �rst four columns

are for mass and the next four are for stress. Consider the �rst row of the Table

3.2, the data of rank 16 preference is converted by placing a 1 in the column of

0.2 lb mass and 27 ksi stress and rest of columns are substitued by zeros, because

for this rank the levels that are present are only 0.2 lb mass and 27 ksi. Similarly

other ranking data are now converted systematically. Until now the coding has

been straightforward, but there is one complication that must be resolved: in

regression analysis, no independent variable may be perfectly predictable based

on the state of any other independent variable or combinations of independent

variables because the regression analysis procedure cannot separate the e�ects of

confounded variables. There is a situation of linear dependency, knowing data

about three levels of each attribute gives the information about the fourth level.

To resolve this linear dependency, one column from each attribute is omitted.

Any column from each attribute can be omitted. In this case �rst column of each

attribute is omitted, encompassing variables x1 and x5. The model to �t the data

is selected as yL = β0 + β2x2 + β3x3 + β4x4 + β6x6 + β7x7 + β8x8, where β2, β3,

and β4 are the part-worths that are to be estimated, indicating the e�ect of each

21

attribute level of mass on the overall preference. Similarly β6, β7, and β8 are the

part-worths indicating the e�ect of each attribute level of stress on the overall

preference. β0 is an intercept term, and yL is the logit recoding [24] of the the

dependent variable, ranking. This is a simple transformation of the rankings used.

The equation for converting is shown in equation 3.1:

p =y −min + 1

max−min + 2(3.1)

where y is ranking, p is probability and min andmax are minimum and maximum

ranking respectively. After calculating the probability p logit recode transforma-

tion is calculated using equation 3.2:

Logit recode = yL = lnp

1− p(3.2)

We can use no transformation, that is directly use ranking as the output variable.

We can also use zero-centered transformation, where we subtract the mean from

every ranking. The dependent variable vector after transformation is shown in

Table 3.3:

Table 3.3: Logit recode transformation of dependent variable, ranking

Ranking (y) Logit recode (yL)

16 2.772615 2.014914 1.540413 1.178712 0.875511 0.606110 0.35679 0.11788 -0.11787 -0.35676 -0.60615 -0.87554 -1.17873 -1.54042 -2.01491 -2.7726

22

The coe�cients, βs are solved using regression analysis, hence the name dummy-

variable regression method [24]. These βs represent the part-worths associated

with each level of every attribute. The part-worths obtained after regression anal-

ysis are β0 = 2.59, β2 = −0.98, β3 = −2.31, β4 = −3.29, β6 = −0.62, β7 =

−1.29, β8 = −1.91. The intercept is divided by the number of attributes and the

quotient is added to each β including those which are previously assumed zeros and

scaled. In this work, the part-worths are scaled using the zero-centered di�erences

method [24]. The details of the method is documented in the following discussion.

After adding the quotient and subtracting the mean for each attribute, (hence the

name zero-centered di�erences) we obtain part-worths as shown in Table 3.4.

Table 3.4: Zero-centered part-worths before scaling

Mass (lb) Part-worth

0.2 1.64310.3 0.66370.4 -0.66370.5 -1.6431

Stress (ksi) Part-worth

27 0.956228 0.338829 -0.338830 -0.9562

This di�erences method rescales part-worths so that for each individual or re-

spondent, the total sum of the part-worth di�erences between the worst and best

levels for each attribute across attributes is equal to the number of attributes times

100. This basically means for each attribute the sum between best and worst levels

is calculated, for attribute mass it is 3.2861 and for attribute stress it is 1.9123

and the total sum of di�erences is 5.1985. Now each zero-centered part-worth is

scaled by 200/5.1985. Here 200 is used since there are only two attributes, if there

are three attributes 300 would have been used. The part-worths obtained for mass

and stress are shown in Table 3.5 and represent the preferences of the designer.

Table 3.5: Part-worths for mass and stress

Mass (lb) Part-worth

0.2 63.210.3 25.530.4 -63.210.5 -25.53

Stress (ksi) Part-worth

27 36.7928 13.0329 -13.0330 -36.79

23

From the above table the part-worths associated with 0.4 lb, and 0.5 lb mass

and similarly for stress are negative. This is only due to the scaling, it means it is

least preferred. The scaling can also be on a 0 to 100 scale where 100 means most

preferred and 0 means least preferred. These part-worths are now representative

of the preferences for selected levels for mass and stress.

3.6 Designing a Pizza

This is a marketing example of a pizza and is a variation from the problem shown

in [17]. The manager of a pizza store is interested in knowing which of the factors

are contributing towards pizza sales. The attributes are assumed to be type of

crust, type of toppings, price, and type of cheese. Each attribute has three levels

as shown in Table 3.6.

Table 3.6: Attribute levels for pizza

Crust

ThinThickPan

Topping

VeggiePepporoniSausage

Price ($)

7.998.999.99

Cheese

MozzarelloRomanoMixed

There are total 3× 3× 3× 3 = 81 combinations; sawtooth software designs the

analysis using 27 combinations. Ranking-based preferences for these combinations

are used and shown in Table 3.7. These rankings are average of customer rankings

for the given combinations. Customers are given the questionaire to rank the

combinations and then average of these preferences are made.

24

Table 3.7: Preferences for pizza

Combination Rank

Pan, pepporoni, 7.99, mixed 27Thin, pepporoni, 7.99, mozzarello 26Thin, pepporoni, 8.99, romano 25Pan, pepporoni, 7.99, mozzarello 24Thin, pepporoni, 8.99, mozzarello 23Thin, pepporoni, 9.99, romano 22Pan, pepporoni, 9.99, mixed 21Thin, sausage, 9.99, romano 20

Thick, pepporoni, 8.99, romano 19Pan, sausage, 8.99, mixed 18

Thick, pepporoni, 9.99, mixed 17Thin, sausage, 7.99, mozarello 16Pan, sausage, 7.99, romano 15Thick, sausage, 7.99, romano 14

Combination Rank

Thin, sausage, 8.99, mixed 13Pan, sausage, 8.99, romano 12

Thick, sausage, 9.99, mozzarello 11Pan, veggie, 8.99, mozzarello 10Thin, veggie, 7.99, mixed 9Thin, veggie, 7.99, romano 8Pan, veggie, 9.99, mozzarello 7Thick, veggie, 7.99, mixed 6Thick, veggie, 8.99, mixed 5

Thick, veggie, 8.99, mozarello 4Thin, sausage, 9.99, mixed 3Pan, veggie, 9.99, romano 2Thin, veggie, 9.99, romano 1

part-worths are obtained using Sawtooth Software. The calculated part-worths

are shown in Table 3.8

Table 3.8: Part-worths for each attribute of a pizza

Crust Part-worth

Thick -12.18Thin -7.51Pan 19.69

Topping Part-worth

Veggie -106.21Sausage -6.50Pepporoni 112.71

Price ($) Part-worth

7.99 47.238.99 -3.809.99 -43.43

Cheese Part-worth

Romano -29.74Mozzarello 28.81Mixed 0.93

From the Table 3.8, the ideal pizza combines Pan + Pepporoni + 7.99 + Moz-

zarello. From this study, the manager knows that pepporoni topping is given more

importance and therefore introducing a new pizza with pepporoni at a higher price

may be a better option than introducing a veggie pizza, even at a lower price.

25

4 Combining Marketing and

Engineering Tools

As the title implies, in this chapter the concepts of marketing such as conjoint

analysis are applied in the optimization of structural engineering problems. Con-

joint analysis, which is discussed in Chapter 3, is a popular method in marketing

for obtaining individual part-worths by including preferences. Taking advantage

of this idea of including preferences of the designer in optimization is the key in

this chapter. As mentioned in Chapter 2 there is a need for including preferences

in optimization iterations. Also, the optimization algorithm should be able to han-

dle multiple attributes. In this optimization, �rstly conjoint analysis is performed

which includes designer preferences. This acts as a front end to for multi-attribute

optimization, it generates part-worths which are optimized with respect to design

variables. There are three steps to perform the optimization, which are discussed

in the sections below and can be expressed pictorially as shown in Figure 4.1.

From Figure 4.1 it can seen that individual continuous part-worths p1, p2, . . . , pk

are generated using cojoint analysis and piece-wise linear approximation, until this

stage the concepts from marketing are used. Nextly an objective function is made

up of these part-worths, that are function of attributes, is developed and opti-

mzed. These attributes which may be mass, displacement, stress, etc, are inturn

function of design variables. The second stage of optimization is performed using

engineering tools such as Finite Element Analysis (FEA) or Computational Fluid

Dynamics (CFD).

26

Figure 4.1: Multi-attribute optimization based on conjoint analysis

4.1 Conjoint Analysis

The �rst step is to perform conjoint analysis by selecting attributes that are in�u-

ential in the output, which is the response and levels for each attribute. Preferences

of the designer are included in this step. Appropriate conjoint analysis has to be

selected based on number of attributes and levels. The output from this step is

the part-worth for each attribute at every level.

4.2 Interpolation and Extrapolation

The part-worths obtained from the above step are discrete at selected levels. To

perform optimization, part-worth values are needed for all combinations of design

variables. Piece-wise linear interpolation and extrapolation are used in this work

to generate continuous part-worths with respect to each attribute. Preference

curves are generated for each attribute separately. These curves show the part-

worth for every value of attribute. Linear approximation, even though not an

exact prediction, but it is a fairly good approximation because in engineering

applications several of the attributes have monotonic preferences. For example,

27

lowering the stress always the better, and higher fundamental natural frequency is

always good. Therefore there are no ups and downs in the preferences and linear

approximation is able to capture this e�ect approximately.

4.3 Optimization

The �nal step is to set up the formulation for optimization. From the above

step in Section 4.2, di�erent continuous part-worths, which can be obtained from

preference curves are available, such as p1, p2, · · · pk where k is the number of

attributes. The optimization is formulated as shown in Equation 4.1. The actual

problem of optimization is now solved in part-worths. For a given combination of

design variables, there are attributes associated and for every attribute value there

is a corresponding part-worth value from the preference curve and the objective

is sum of these part-worths.

Minimize− {p1(f1(x)) + p2(f2(x)) + · · · pk(fk(x))}

xL ≤ x ≤ xU (4.1)

Additive model of part-worths is assumed and is valid under certain assumptions

[11]. A well known assumption is that the attributes are mutually preferentially

independent (MPI), which basically says that the trade-o�s between any pair of

attributes fi and fj, keeping the levels of other attributes �xed, does not depend

on the �xed levels. For example, if there are three attributes, then the trade-

o�s between any two are independent of the third. De�nition of MPI and the

existence of additive model are presented in detail by Keeney and Rai�a [15].

Continuous part-worths are obtained by piece-wise linear interpolation and ex-

trapolation of discrete part-worth. These discrete part-worths are a function of

attributes f1,f2, . . . fk based on the preferences (rankings or ratings) by the de-

signer and there is one-to-one mapping between attributes and part-worths. The

28

part-worths p1, p2, · · · pk need to be maximized to obtain the optimal solution,

and by having a negative sign before the objective function, the objective function

is being maximized. MATLAB, which uses Sequential Quadratic Programming

(SQP) method is used for optimization in this work.

4.4 Sequential Quadratic Programming Method

MATLAB uses the SQP method for optimization, which is discussed in this sec-

tion [3] . SQP is a gradient-based method. The advantages of the SQP method

are that the starting point can be in an infeasible region, gradients of only active

constraints are required, and equality constraints can also be handled. The task

of getting a new point is divided into two parts. The �rst part is getting the

direction, dk and the second part is step size, αk. The new point in the design

space is obtained as xk+1 = xk + αkdk. In this method, the direction vector, dk is

obtained by solving the QP sub problem which is shown in Equation 4.2:

Minimize 12dT d +5fT d

subject to5 gTi d + gk

i ≤ 0

5hTi d + gk

i ≤ 0 (4.2)

xL ≤ d + xk ≤ xU

where d is the design variable. 5f, 5g, and5h are the gradients of the objective,

inequality constraints and equality constraints respectively. Thus dk, the direction

vector is obtained. Step size, αk is chosen equal to (0.5)J where J is the �rst of

the integers q = 1, 2, · · · for which the following inequality holds:

θ(xk + (0.5)qdk) ≤ θ(xk)− γ(0.5)q‖d‖2 (4.3)

29

where θ(x) can be used as a descent function. Further details of the method can

be seen in [3] and [19]

30

5 Engineering Applications

This Chapter applies the conjoint analysis concepts to engineering problems. The

procedure addressed in Chapter 4 is implemented here. Engineering examples are

considered, including a cantilevered beam, �xed plate and a composite lightweight

torpedo. Each example is �rst solved using the constraint method and then fol-

lowed by multi-attribute optimization, based on preferences obtained from conjoint

analysis.

5.1 Cantilevered Beam Example

The cantilever beam shown in Figure 5.1 is used in the analysis. It is subjected

to a tip load P = 100 lb. Mass, tip displacement, and stress are considered as

attributes. The breadth and height of the beam are design variables. The closed-

form equations are given in Equations 5.1 and 5.2 for displacement and stress,

respectively. Mass is calculated as product of density and volume. The length

of the beam = 20 in, density = 0.1 lb/in3, and Young's modulus, E = 107 psi.

MATLAB is used for optimization; analytical gradients are supplied since closed-

form equations are available

31

Figure 5.1: Cantilever beam

Tip displacement =4PL3

Ebh2(5.1)

Stress =12PL

bh2(5.2)

5.1.1 Beam Case Study I

In this case study, only mass and displacement are considered as the attributes. For

solving the problem using the constraint approach, the optimization is formulated

as shown in Equation 5.3:

Minimize mass

displacement ≤ 0.15 in (5.3)

0.5 in ≤ breadth, height ≤ 5 in

The design variables, as mentioned before, are breadth and height of the beam.

The optimum result is obtained as mass = 1.6219 lb and displacement = 0.15 in,

with breadth = 0.5 in and height = 1.6219 in. Displacement is at the bound and

the lowest possible mass is achieved by this method. The same problem will now be

solved using the conjoint analysis approach. The levels for mass and displacement

are chosen as shown in Table 5.1:

32

Table 5.1: Levels for mass and displacement

# Mass (lb)

1 1.452 1.553 1.654 1.75

# Displacement (in)

5 0.126 0.137 0.148 0.15

The levels are chosen based on designer's experience, since the designer knows

the displacement has to be less than 0.15 in, based on this other levels for dis-

placement are chosen. In this way the four levels for displacement are chosen.

Displacements are linearly decreased because the designer does have enough ex-

perience to judge which levels between the range are in�uential in the decision

making process. More levels can be chosen to accurately cover the displacement

range, but the designer will be burdened to make preferences for more number of

combinations. For mass levels the designer has enough experience to know where

the mass values will be and based on this levels for mass are chosen. Same ar-

gument is made for other case studies of �xed plate and composite lightweight

torpedo. Because there are only 16 combinations, a full factorial design is consid-

ered for including preferences, but if there are large number of combinations, then

it is not feasible to consider all of them, and fractional factorial designs are used.

Sawtooth Software's SMRT is used to perform conjoint analysis. The preferences

are chosen as shown in Table 5.2.

Table 5.2: Preferences chosen for combinations of attribute levels

Rank Combination

16 1,515 1,614 1,713 2,512 2,611 1,810 2,79 3,5

Rank Combination

8 3,67 4,56 2,85 3,74 4,63 3,82 4,71 4,8

The preferences are based on the designer's priorities for mass and displace-

33

ment. The higher the rank, the more the product is preferred. Hence the ideal

combination is 1,5, which is 1.45 lb mass and 0.12 in displacement. Second best

is 1,6 which is 1.45 lb mass and 0.13 in, which means the designer is ready to in-

crease the displacement but not the mass, giving more importance to mass. Other

combinations are chosen in a similar fashion. In this way preferences are included.

The part-worths obtained for each level are shown in Table 5.3:

Table 5.3: Part-worths obtained for each level

Mass (lb) Part-worth

1.45 65.161.55 16.961.65 -22.701.75 -59.42

Displacement (in) Part-worth

0.12 34.890.13 14.980.14 -9.330.15 -40.53

The part-worths obtained are discrete and preferences are interpolated and ex-

trapolated piecewise linearly and preference curves are shown in Figure 5.2:

34

1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8−60

−40

−20

0

20

40

60

80

Mass (lb)P

artw

orth

(a) Preference curve for mass (p1)

0.12 0.125 0.13 0.135 0.14 0.145 0.15−50

−40

−30

−20

−10

0

10

20

30

40

Displacement (in)

Par

twor

th

(b) Preference Curve for Displacement (p2)

Figure 5.2: Preference curves for mass and displacement

These curves are one-to-one mapping between attributes and part-worths. Hence

for every mass or displacement obtained during function evaluation (generally

comes from FEA or CFD) a corresponding part-worth value is considered. The

constraints in the previous optimization in Equation 5.3 are now converted into

objective function and the problem is formulated as shown in Equation 5.4:

Minimize − (p1 + p2)

where p1 is the part-worth of mass (5.4)

p2 is the part-worth of displacement

0.5 in ≤ breadth, height ≤ 5 in

35

The optimization results are mass = 1.7963 lb and displacement = 0.1104 in, with

breadth = 0.5 in and height = 1.7963 in. Comparing both results, shown in Table

5.4

Table 5.4: Comparision of results of case study I for cantilever beam

Attribute Constraint Method Conjoint Method % di�erence

Mass (lb) 1.6219 1.7963 10.75↑Displacement (in) 0.15 0.1104 26.4↓

Therefore for an increase of 10.75% in mass a decrease of 26.4% in the displace-

ment can be seen. This is obtained by a small increase in height of the beam with

breadth remaining same.

5.1.2 Beam Case Study II

In this study mass, displacement and stress are considered as attributes. The

optimization problem using the constraint approach is shown in Equation 5.5:

Minimize mass

displacement ≤ 0.15 in

stress ≤ 104 psi

0.5 ≤ breadth, height ≤ 5in (5.5)

The optimization results are mass = 2.1909 lb, Displacement = 0.0609 in and

Stress = 10,000 psi, with breadth = 0.5 in and width = 2.1909 in. Stress is

the active constraint. Solving the same problem using conjoint analysis approach.

The levels for mass, displacement and stress are shown in Table 5.5

36

Table 5.5: Levels for mass, displacement, and stress

# Mass(lb)

1 1.02 1.53 2.04 2.5

# Displacement(in)

5 0.056 0.087 0.18 0.15

# Stress(psi)

9 800010 900011 950012 10000

The preferences for the above attributes are chosen as shown in Table 5.6. Since

there are 64 total combinations. A fractional factorial design of 30 combinations

is considered. The preferences are based on the designer's priorities for mass,

displacement and stress, such as how much importance is given to individual

attributes, how much more can be spent on mass, and how willing the designer is

to decrease the other two attributes.

Table 5.6: Preferences for attributes of mass, displacement and stress

Rank Combination

30 1,5,1029 1,6,1028 1,5,1227 1,7,926 2,5,925 1,6,1124 1,8,1123 1,8,1222 2,6,921 2,5,11

Rank Combination

20 2,6,1119 2,7,1018 2,7,1217 2,8,1016 3,5,1015 2,8,1214 3,6,913 3,5,1212 3,7,911 3,6,12

Rank Combination

10 3,7,119 4,5,98 3,8,107 3,8,116 4,5,115 4,6,104 4,7,103 4,6,122 4,7,121 4,8,9

The part-worths obtained from the above preferences are shown in Table 5.7:

Table 5.7: Part-worths for mass, displacement, and stress

Mass(lb) Part-worth

1.0 101.351.5 32.392.0 -27.692.5 -106.05

Displacement(in) Part-worth

0.05 33.970.08 7.440.1 -8.200.15 -33.21

Stress(psi) Part-worth

8000 12.289000 11.259500 -10.3910000 -13.14

These part-worths are linearly interpolated and extrapolated in the same way as

before. The optimization problem is now formulated which rede�nes the objective

function as shown in Equation 5.6:

37

Minimize − (p1 + p2 + p3)

where p1 is the part-worth of mass

p2 is the part-worth of displacement (5.6)

p3 is the part-worth of stress

0.5 in ≤ breadth, height ≤ 5 in

The optimum result ismass=2.3094 lb, displacement = 0.052 in, and stress=9000.008 psi,

with breadth = 0.5 inand height = 2.3094 in. Comparing both the results, the

di�erences are shown in Table 5.8:

Table 5.8: Comparision of results of case study II for cantilever beam

Attribute Constraint Method Conjoint Method % di�erence

Mass (lb) 2.1909 2.3094 5.41↑Displacement (in) 0.0609 0.052 14.61↓

Stress (psi) 10,000 9000.08 9.99↓

Therefore, as shown, for a small percentage increase in mass, there is a consid-

erable decrease in displacement, and stress also decreases, but not considerable to

displacement. Again here a small increase in mass is obtained due to a little in-

crease in height, breadth remaining same and due to this attributes- displacement,

and stress have decreased considerably.

5.2 Fixed Plate Example

A plate that is �xed at both ends is considered in this example. The dimensions

of the plate are 10 in ×10 in. A uniformly distributed load of 100 psi is applied.

The material properties are Young's modulus, E=107psi, Density, ρ = 0.1lb/in3,

Poisson's ratio, ν=0.3. Figure 5.3 shows the details:

38

Figure 5.3: Fixed plate with uniformly distributed load

The plate is modeled using quadrilateral �nite elements; ABAQUS software is

used for �nite element analysis. Due to double symmetry, a quarter model is

modeled using 10 × 10 elements and symmetric boundary conditions are placed.

The �nite element mesh and element numbering are shown in Figure 5.4. The

design variables are thickness along one column of the plate and this is shown by

di�erent colors in the �gure. There are 10 design variables.

Figure 5.4: Finite element model with element numbering

5.2.1 Plate Case Study I

In this study mass, maximum displacement, and maximum Von mises stress are

considered as attributes. The problem is �rst solved using the constraint method.

The optimization problem is formulated as shown in Equation 5.7

39

Minimize mass

displacement ≤ 0.05 in

stress ≤ 25, 000 psi

0.1 in ≤ xi, i = 1, . . . , 10 ≤ 5 in (5.7)

The optimum results are mass = 0.9912 lb, stress = 25,000 psi and displacement

= 0.0483 in. The stress constraint is active and the displacement constraint is

almost active. The optimum design variables obtained are shown in Table 5.9:

Table 5.9: Optimum thickness for displacement and stress constraint

Thickness # Value(in)

1 0.12 0.37043 0.43344 0.49405 0.49446 0.49407 0.49298 0.19 0.492910 0.4929

The same problem is now solved using the conjoint analysis approach. The

levels for each attribute are shown in Table 5.10. The levels for displacement and

stress are chosen based on the requirements, and the levels for mass are chosen

based on the designer's experience.

Table 5.10: Levels for attributes of mass, displacement, and stress

# Mass(lb)

1 0.52 0.83 1.24 1.5

# Displacement(in)

5 0.026 0.037 0.048 0.05

# Stress(psi)

9 23,00010 25,00011 26,00012 27,000

40

The preferences of the designer are based on how much of an increase in the

mass and decrease in stress and displacement are allowable. These ranks are

shown in Table 5.11. Again, since there are three attributes and four levels for

each, Sawtooth Software's SMRT provides 30 combinations of the total number

of combinations, which is 4× 4× 4 = 64.

Table 5.11: Preference of ranks for mass, displacement, and stress

Rank Combination

30 1,6,1029 2,5,1028 2,5,927 1,7,1026 1,7,925 1,6,1124 1,5,1223 2,6,1122 2,7,921 2,6,12

Rank Combination

20 1,8,1119 2,8,1218 1,8,1217 2,8,1116 2,7,1015 3,5,1014 3,6,913 4,5,912 4,6,1011 3,8,9

Rank Combination

10 3,5,119 3,8,108 4,5,117 4,6,126 3,7,115 3,7,124 4,8,93 4,7,112 4,8,101 4,7,12

The fractional factorial design is based on D-e�ciency[16], given the conditions

de�ned by the designer. If the design is orthogonal and balanced then it has

optimum e�ciency. The D-e�ciency measures the goodness of the design relative

to the hypothetical orthogonal design. The part-worths obtained based on the

dummy variable and least squares approach are shown in Table 5.12

Table 5.12: Part-worth for each level for mass, displacement, and stress

Mass(lb) Part-worth

0.5 82.320.8 49.301.2 -40.991.5 -90.62

Displacement(in) Part-worth

0.02 34.410.03 28.420.04 -27.350.05 -35.48

Stress(psi) Part-worth

23,000 26.8425,000 17.3226,000 -13.8427,000 -30.32

These part-worths are linearly interpolated and extrapolated to make them

continuous, and the preference curves for each attribute is shown in Figure 5.5.

These preference curves are piecewise linear.

41

0.5 1 1.5−100

−80

−60

−40

−20

0

20

40

60

80

100

Mass(lb)

Par

twor

th

(a) Preference curve for mass (p1)

0.02 0.025 0.03 0.035 0.04 0.045 0.05−40

−30

−20

−10

0

10

20

30

40

Displacement(in)

Par

twor

th

(b) Preference curve for displacement (p2)

23,000 23,500 24,000 24,500 25,000 25,500 26,000 26,500 27,000−40

−30

−20

−10

0

10

20

30

Stress(psi)

Par

twor

th

(c) Preference curve for stress (p3)

Figure 5.5: Preference curves for mass, displacement, and stress

The optimization problem is now formulated as shown in Equation 5.8

Minimize − (p1 + p2 + p3)

where p1 is the part-worth of mass

p2 is the part-worth of displacement (5.8)

p3 is the part-worth of stress

0.1 in ≤ xi, i = 1, . . . , 10 ≤ 5 in

The optimum results obtained are mass = 1.0338 lb, displacement = 0.0165 in,

and stress=13,631 psi. The optimum design variables are shown in Table 5.13

42

Table 5.13: Optimum variables for mass, displacement, and stress usingthe conjoint approach

Thickness# Value (in)

1 0.12 0.86483 0.14 0.99335 0.72166 0.17 0.18 0.10099 0.103910 0.9506

A comparision on both the methods and results are shown in Table 5.8:

Table 5.14: Comparision of results of case study I for �xed plate

Attribute Constraint Method Conjoint Method % di�erence

Mass (lb) 0.9912 1.0338 4.29 ↑Displacement (in) 0.0483 0.0165 65.84 ↓

Stress (psi) 25,000 13,621 45.48 ↓

We can see the designer can now achieve huge decrease in displacement and

stress for a nominal increase in mass. This is achieved by redistribution of the

thickness di�erently when compared to constraint method. Four thicknesses are

having the lower bound in conjoint approach. Using the constraint approach the

designer would have never come up with this solution in the design space, because

the mass is a slightly higher than the conjoint approach.

5.2.2 Plate Case Study II

In this study mass, maximum displacement, and �rst fundamental frequency are

considered as attributes. A non-structural mass of 0.5 lb is added to the structure,

the quarter model, which is distributed equally at each node. The problem is �rst

solved using the constraint approach, and the optimization is formulated as shown

in Equation 5.11

43

Minimize mass

displacement ≤ 0.05 in

fundamental frequency ≥ 8 Hz

0.1 in ≤ xi, i = 1, . . . , 10 ≤ 5 in (5.9)

The optimum results are mass = 2.32689 lb, displacement = 0.00688 in, and

frequency = 8.0024 Hz. Frequency is the driving constraint in the optimization,

and the optimum design variables are shown in Table 5.15:

Table 5.15: Optimum thickness for displacement and frequency constraints

Thickness # Value (in)

1 0.78862 0.76683 0.80334 0.77525 0.80196 0.17567 0.65538 1.22819 0.656310 0.6563

Solving the same problem using the conjoint approach, the levels for each at-

tribute are shown in Table 5.16:

Table 5.16: Attribute levels for mass, displacement, and frequency

# Mass (lb)

1 2.252 2.503 2.754 3.00

# Displacement (in)

5 0.026 0.037 0.048 0.05

# Frequency (Hz)

9 810 911 1012 11

The preferences for the above-chosen levels are shown in Table 5.17 and are

based on the priorities of the designer.

44

Table 5.17: Preferences for mass, displacement, and frequency attributes

Rank Combination

30 1,6,1229 2,5,1228 1,5,1127 2,5,1126 1,7,1225 3,5,1224 2,7,1223 1,6,1122 1,5,1021 3,8,12

Rank Combination

20 2,8,1119 1,8,1018 3,6,1117 3,7,1116 2,8,1015 3,6,1014 1,7,913 4,6,1212 3,5,911 3,7,10

Rank Combination

10 2,6,99 4,7,118 2,7,97 1,8,96 4,5,105 4,8,124 4,8,113 4,6,102 4,5,91 3,8,9

The part-worths obtained for each level after calculations are shown below in

Table 5.18:

Table 5.18: Part-worth for mass, displacement, and frequency

Mass (lb) Part-worth

2.25 47.452.50 33.682.75 -4.143.00 -76.99

Displacement (in) Part-worth

0.02 29.210.03 5.680.04 -6.610.05 -28.27

Frequency (Hz) Part-worth

8 -65.729 -10.7510 24.1011 52.36

These part-worths are linearly interpolated and extrapolated to make them con-

tinuous for optimization. The preference curves for attributes mass, displacement,

and frequency are shown in Figure 5.6. These part-worths represent designer's

choice for mass, displacement and frequency. Hence for a particular value of at-

tributes there is a corresponding value for part-worth.

45

2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3−80

−60

−40

−20

0

20

40

60

Mass (lb)

Par

twor

th

(a) Preference curve for mass (p1)

0.02 0.025 0.03 0.035 0.04 0.045 0.05−30

−20

−10

0

10

20

30

Displacement (in)

Par

twor

th

(b) Preference curve for displacement (p2)

8 8.5 9 9.5 10 10.5 11−80

−60

−40

−20

0

20

40

60

Frequency (Hz)

Par

twor

th

(c) Preference curve for frequency (p3)

Figure 5.6: Preference curves for attributes - mass, displacement, and frequency

The optimization is now formulated as shown in Equation 5.10 by converting

the constraints in the constraint method into an objective function.

Minimize − (p1 + p2 + p3)

where p1 is the part-worth of mass

p2 is the part-worth of displacement (5.10)

p3 is the part-worth of frequency

0.1 in ≤ xi, i = 1, . . . , 10 ≤ 5 in

The optimum results are mass=2.50005 lb, displacement = 0.003801 in, and fre-

quency = 12.096 Hz. The optimum design variables are shown in Table 5.19:

46

Table 5.19: Optimum thickness using conjoint approach

Thickness # Value (in)

1 0.75482 1.91243 0.15214 0.16465 0.31246 0.04577 1.81258 0.66859 0.638610 1.5685

A comparision of the results from both methods is shown in Table 5.8:

Table 5.20: Comparision of results of case study II for �xed plate

Attribute Constraint Method Conjoint Method % di�erence

Mass (lb) 2.3269 2.50005 7.44 ↑Displacement (in) 0.0069 0.0038 44.78 ↓

Fundamental frequency (hz) 8.0024 12.096 51.15 ↑

As seen before, the same trend of a little increase in mass resulting in a consider-

able decrease displacement and a considerable increase in fundamental frequency

is observed. This is achieved by di�erent distribution of thickness when compared

to constraint method. Based on the result of conjoint approach if we pose the

constraint optimization problem as follows

Minimize mass

displacement ≤ 0.004 in

fundamental frequency ≥ 12 Hz

0.1 in ≤ xi, i = 1, . . . , 10 ≤ 5 in (5.11)

The optimum results obtained are mass=2.4345 lb, displacement = 0.003797 in,

and frequency = 12.0 Hz which is almost the same optimum as the conjoint

approach of Table5.8.

47

5.2.3 Plate Case Study III

This problem is a modi�cation of Case Study II. The attributes considered are

the same as above - mass, displacement, and frequency, but for conjoint approach

trade-o� is considered only for mass and frequency and displacement is posed as

a constraint. The optimization formulation using constraint method is shown in

Equation 5.12

Minimize mass

displacement ≤ 0.02 in

fundamental frequency ≥ 5 Hz (5.12)

0.1 in ≤ xi, i = 1, . . . , 10 ≤ 5 in

The optimum results are mass = 1.91563 lb, displacement = 0.01814 in, and

frequency = 5.0007 Hz. The optimum variables are shown in Table 5.21. It can

be seen that the frequency constraint is active and the displacement is almost near

the boundary.

Table 5.21: Optimum design variables con�guration with displacement andfrequency as constraints

Thickness # Value (in)

1 0.56912 0.56913 0.56484 0.56625 0.56506 0.56627 0.56628 0.56629 0.564810 0.5648

When solving this problem using the conjoint approach, since only trade-o�s

between mass and frequency are of importance, the displacement attribute is posed

48

as a constraint. The attributes considered for conjoint analysis are mass and

frequency. The levels are shown in Table 5.22:

Table 5.22: Levels for mass and frequency attributes

# Mass (lb)

1 1.52 1.753 2.04 2.25

# Frequency (Hz)

5 5 Hz6 6 Hz7 7 Hz8 8 Hz

Since there are only 16 total combinations, a full factorial design is considered

for preferences and the rankings are shown in Table 5.23 according to the priorities

of the designer:

Table 5.23: Preference rankings for mass and frequency attributes

Rank Combination

16 1,815 1,714 1,613 2,812 2,711 3,810 3,79 2,6

Rank Combination

8 4,87 1,56 4,75 3,64 2,53 4,62 3,51 4,5

The part-worths obtained after performing conjoint analysis are shown in Table

5.24

Table 5.24: Part-worths of attributes mass and frequency for each level

Part-worth Mass (lb)

1.50 54.861.75 9.122.00 -17.712.25 -46.27

Part-worth Frequency (Hz)

-58.09 5-6.96 624.26 740.79 8

These part-worths are interpolated and plotted as shown in Figure 5.7 The

optimization problem is now formulated as shown in Equation 5.13.

49

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3−60

−40

−20

0

20

40

60

Mass (lb)

Par

twor

th

(a) Preference curve for mass (p1)

5 5.5 6 6.5 7 7.5 8−60

−40

−20

0

20

40

60

Frequency (Hz)

Par

twor

th

(b) Preference curve for frequency (p2)

Figure 5.7: Preference curves for mass and frequency

Minimize − (p1 + p2)

displacement ≤ 0.02 in

where p1 is the part-worth of mass

p2 is the part-worth of frequency (5.13)

0.1 in ≤ xi, i = 1, . . . , 10 ≤ 5 in

The optimum results obtained are mass = 2.0 lb, displacement = 0.01149 in, and

frequency = 6.5429 Hz. The optimum variables are shown in Table 5.25:

50

Table 5.25: Optimum design variables with trade-o�s between mass and frequencywith displacement as constraint

Thickness # Value (in)

1 0.45532 0.50043 0.50044 1.03935 0.94956 0.85267 1.05918 0.21859 0.225510 0.1992

A comparision between the methods is shown in Table 5.17:

Table 5.26: Comparision of results of case study III for �xed plate

Attribute Constraint Method Conjoint Method % di�erence

Mass (lb) 1.9156 2.0 4.4 ↑Fundamental frequency (hz) 5.0007 6.5429 30.84 ↑

As shown, a small increase in mass increased the fundamental frequency sig-

ni�cantly, and displacement value of 0.0115 in is well below the limit of 0.02 in.

Therefore, again, with a little increase in mass, signi�cant gain can be obtained

in frequency. This result is obtained because preferences are included in the opti-

mization.

5.3 Composite Lightweight Torpedo

5.3.1 Modeling

A lightweight torpedo, similar to the MK-44 con�guration, was modeled using

the �nite element method. The overall length and diameter are 2.42 m and 0.32

m, respectively. These dimensions are chosen such that the torpedo �ts in the

present launch infrastructure. The �nite element model was created using 1176

quadrilateral shell elements and 48 triangular shell elements. Figure 5.8 shows the

51

�nite element model:

Figure 5.8: Finite element mesh of composite lightweight torpedo

To represent the total mass of the structural components inside the torpedo,

concentrated mass elements were distributed along the nodes of the torpedo. Lay-

ered composite material properties are assigned to the outer shell. GENESIS, a

�nite element software, was used for modeling and analysis. A conceptual design

modeled by Adduri[1] with a sandwich honeycomb panel was used in this work.

This model consists of honeycomb surrounded by �ber reinforced laminates on the

top and bottom plates of the shell. Figure 5.9 shows the cross-section of the shell.

Figure 5.9: Cross-section of the composite shell

The stacking of the layers used are 0o/±45o/90oon both sides of the honeycomb.

The material properties of the laminates are AS/3501 carbon/epoxy and are given

in Table 5.27:

52

Table 5.27: Material properties of carbon/epoxy

Property Carbon/Epoxy

Longitudinal Modulus, E11 139GPaTransverse Modulus, E22 8.96GPa

In-plane Shear Modulus, G12 7.1GPaPoisson's Ratio, ν12 0.3Laminate Density, ρ 1600 kg/m3

Longitudinal Tensile Strength, F1t 1447MPaLongitudinal Compressive Strength, F1c 1447MPa

Transverse Tensile Strength, F2t 51.6MPaTransverse Compressive Strength, F2c 206MPa

In-plane Shear Strength, F6 93MPa

The honeycomb can only handle transverse shear hence only transverse shear

modulii and transverse shear strength are de�ned for material properties, which

are shown in Equation 5.14:

G13 = 110MPa, G23 = 55MPa, S13 = 0.65MPa, S23 = 0.40MPa (5.14)

5.3.2 Attributes Considered

Torpedoes travel underwater and should have enough strength to withstand pres-

sure at crush depth. Hence the composite shell should be designed for crush depth.

Since the considered torpedo is composite and not isotropic, the Holfman failure

criterion which is described by Gurdal, et al in [13] is used. The failure index

is de�ned as the ratio of applied stress to critical stress. Buckling is the second

attribute. The torpedo traveling at a velocity should not buckle under hydrostatic

pressure loading. The critical buckling load factor is de�ned as the ratio of actual

buckling load to the applied load. Fundamental natural frequency is the third

attribute. The torpedo structure should be away from resonance to avoid any

catostrophy. Mass of the torpedo is the fourth attribute.

53

5.3.3 Optimization of Torpedo Hull

There are four design variables, which are the thickness of the laminates and

honeycomb. Symmetry is maintained on both sides of the honeycomb and hence

the laminates on the top and bottom are linked. Also, the thickness of the+45o

and −45olayers are linked. The design variables are shown in Table 5.28:

Table 5.28: Design variables for composite torpedo

# Thickness

1 Honeycomb2 0olayer3 ±45olayer4 90olayer

The constraint method is used and the optimization is formulated as shown in

Equation 5.15:

Minimize mass

ω1 ≥ 22.2 Hz

Pcr ≥ 1.0 (5.15)

FI ≤ 0.9

0.000001 m ≤ xi, i = 1, · · · , 4 ≤ 0.05 m

where ω1 is fundamental natural frequency, Pcr is critical buckling load factor, and

FI is the failure indices of each layer. The results obtained are mass = 220.1303

kg, ω1= 22.2045 Hz, Pcr= 1.0002, FI = 0.45. The optimum design variables are

shown in Table 5.29.

Table 5.29: Optimum thickness using constraint approach

Design Variable Value (m)

Honeycomb 0.050olayer 0.0014±45olayer 0.000690olayer 0.00003

54

Solving using the conjoint approach, the attributes and levels for each are shown

in Table 5.30:

Table 5.30: Attributes and levels for composite toredo

# Mass (kg)

1 2222 2243 2264 228

# Pcr

5 1.16 1.27 1.38 1.4

# ω1(Hz)

9 22.210 2311 2412 25

# FI

13 0.614 0.715 0.816 0.9

There are 4 × 4 × 4 × 4 = 256 combinations, and it is di�cult to consider all

combinations for making preferences. Sawtooth Software provides 39 combinations

for a fractional factorial design. The preferences are ranking-based and are shown

in Table 5.31:

Table 5.31: Ranking-based preferences for composite torpedo

Rank Combination

39 1,8,11,1538 1,8,12,1637 1,7,12,1436 2,8,11,1535 2,8,10,1434 1,7,10,1433 3,8,12,1632 2,8,9,1331 1,7,9,1330 3,8,10,15

Rank Combination

29 1,6,11,1428 3,8,9,1427 3,7,12,1426 3,7,11,1325 2,7,10,1624 1,6,9,1523 2,5,12,1322 3,7,11,1621 2,7,9,1620 2,6,12,15

Rank Combination

19 1,5,10,1318 2,5,12,1517 4,8,12,1316 3,6,10,1315 1,5,9,1614 2,5,9,1413 2,5,11,1412 3,6,9,1411 3,8,10,1410 3,5,12,15

Rank Combination

9 4,7,10,158 4,5,12,147 4,8,9,166 3,5,10,165 3,5,9,154 4,7,9,153 4,5,11,132 4,6,11,161 4,6,10,16

The part-worths obtained after conjoint analysis are shown in Table 5.32:

Table 5.32: Part-worths obtained for composite torpedo

Part-worth Mass (kg)

78.94 22228.50 224-3.91 226-103.53 228

Part-worth Pcr

-61.77 1.1-35.88 1.223.53 1.374.11 1.4

Part-worth ω1(Hz)

-28.07 22.2-7.59 238.83 2426.83 25

Part-worth FI

9.38 0.68.25 0.7-0.26 0.8-17.37 0.9

Preference curves obtained after linear interpolation of the above part-worths

are shown in Figure 5.10:

55

222 223 224 225 226 227 228−120

−100

−80

−60

−40

−20

0

20

40

60

80

Mass (kg)

Par

twor

th

Preference Curve for Mass

(a) Preference curve for mass

1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45−80

−60

−40

−20

0

20

40

60

80

Critical Buckling Load Factor

Par

twor

th

Preference curve for Buckling Load Factor

(b) Preference curve for BLF

22 22.5 23 23.5 24 24.5 25−30

−20

−10

0

10

20

30

Fundamental natural frequency

Par

twor

th

Preference Curve for Frequency

(c) Preference curve for Frequency

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95−20

−15

−10

−5

0

5

10

Failure Index

Par

twor

th

Preference Curve for Failure Index

(d) Preference curve for FI

Figure 5.10: Preference curves for composite torpedo

The optimization is now formulated as shown in Equation 5.16 for the conjoint

approach:

Minimize − (p1 + p2 + p3 + p4)

where p1 is part-worth for mass

p2 is part-worth for ω1 (5.16)

p3 is part-worth for Pcr

p4 is part-worth for FI

0.000001 m ≤ xi, i = 1, . . . , 4 ≤ 0.05 m

56

The optimum result obtained is mass = 226.0078 kg, ω1= 22.9698 Hz, Pcr=

1.5309, FI = 0.0265, and the optimum design variables are shown in Table 5.33.

Table 5.33: Optimum thickness of composite torpedousing conjoint approach

Design Variable Value (m)

Honeycomb 0.04990olayer 0.00132±45olayer 0.0010490olayer 0.000029

A comparision between both methods are shown in Table 5.34 indicating that

huge dividends can be obtained in the buckling load factor attribute and failure

index for a small trade-o� in mass.

Table 5.34: Comparision of results for composite torpedo

Attribute Constraint Method Conjoint Method % di�erence

Mass (lb) 220.1303 226.0078 2.67 ↑Buckling Load Factorcr) 1.0002 1.5309 53.06 ↑

Failure Index (FI) 0.45 0.00265 94.11 ↓Fundamental frequency (hz) 22.2045 22.9699 3.45 ↑

From Table 5.34 , it can be seen that for a small mass increase of 2.67%, the

critical buckling load factor which was active in the previous constraint optimiza-

tion has met the design goal with a decent margin of safety with 53.06% increase.

Furthermore the failure index has signi�cantly decreased, where as there is a small

increase in fundamental frequency. Comparing the optimum thickness for both the

cases, the ±45o layer thickness is increased from 0.0006 m for constraint approach

to 0.00104 m for conjoint approach and other thickness remained almost same.

Hence this increase in ±45o layer thickness provides the strength for the hull to

withstand the buckling pressure and stresses developed inside the hull.

57

5.4 Chapter Summary

The conjoint approach is able to incorporate designer preferences into the opti-

mization. This approach �nds a solution in the design space that has the best

trade-o�s available. The levels set by the designer and the preferences drive the

optimization. Therefore, the optimum design is a preferred design. The optimum

solution is reached when there are no better trade-o�s available. Based on the so-

lution obtained, the designer can change the attribute levels and preferences if he is

not satis�ed with the current solution. Comparing this method with the constraint

method, in the constraint method there is no trade-o� analysis considered and a

solution is reached when one of the constraints is active. The constraint method

arrives at the trade-o� condition in an arbitrary manner. The only preferences

involed are maximum values on the attributes. Therefore, the constraint method

fails to give the designer complete information about what happens to other at-

tributes when there is a small increase in one of the attributes. Comparing the

conjoint method with the weighted sum method, in the weighted sum method the

weights represent the preferences, but these weights are always constant through

out the optimization iterations, which means there is no preference on the levels

of the attributes. This is not necessarily accurate because the preferences change

depending on the current value of the constraint. In the conjoint approach, the

preferences are piecewise linear, which is a better representation of the preferences

of the designer.

The computational cost of performing the conjoint analysis is negligible. The

only e�ort involved in performing conjoint analysis is to come up with levels for

each attribute and to include preferences which requires designer's experience.

Once part-worths are determined, the e�ort is similar to solving a standard op-

timization problems. Unlike a single objective function, here with the assistance

of conjoint analysis, preferences are incorporated in multi-attribute problem. The

computational cost is involved during the optimization stage where the function

evaluations come from expensive FEA or CFD calculations.

58

6 Conclusions and Future Work

Ideas from marketing science have been used and implemented with optimization

for engineering applications. This method helps to �nd the optimum trade-o�s

that can be achieved in the design space. Drawbacks of the constraint method

and weighted sum method are discussed. The need for an optimization to incor-

porate the designer preferences and include ability to handle multiple compet-

ing attributes is demonstrated. An optimization method that is able to handle

multiple attributes is achieved, and the designer preferences are included in the

optimization routine. Conjoint analysis, which is a systematic way of converting

the priorities into part-worths is applied. Details of the algorithm for performing

the conjoint analysis are discussed from a marketing point of view �rst, followed

by engineering insight. A marketing example of designing a pizza is also shown.

A novel way of combining the conjoint analysis and optimization is discussed

where after obtaining the part-worths, they are piece-wise linearly interpolated,

and an optimization formulation is shown. An additive formulation between the

attributes is assumed during this formulation. One of the disadvantages of the

above method is that the designer should have experience with the product to

determine with attributes and levels for each. If no experience is available, linear

variation between the levels for attributes can be used.

Engineering applications using a cantilevered beam where closed form solution is

available is demonstrated for two case studies with displacement, stress and mass

as attributes, followed by a �xed plate example that is modeled using the �nite

element method (FEM). ABAQUS is used as FEM software and demonstrated

59

for three case studies with mass, displacement, stress and �rst natural frequency

as attributes. In one of the case studies trade-o� between mass and �rst natural

frequency with displacement as a constraint is achieved successfully. A composite

lightweight torpedo example in which the attributes are derived from multiple

disciplines is also discussed.

6.1 Future Work

Future work can address developing methods that do not require for the designer

to have background knowledge of the output but will still allow the designer be

able to incorporate preferences in the optimization. Uncertainty in design has

never been addressed in this work, which opens new horizons. Reliability-based

designs can be integrated with the present optimization routine. Figure 6.1 shows

a potential future model for incorporating uncertainties.

Figure 6.1: Integrating uncertainty with conjoint analysis and optimization

Doing this will make the problem truly multidisciplinary, with disciplines from

marketing, solid mechanics, optimization, and reliability. To address uncertainty

60

apart from above-mentioned FORM, other methods, such as evidence theory or

fast fourier transforms can also be used.

61

A Matlab code for optimization of Cantilever Beam

A.1 Constraint optimization for displacement and stress as constraint

function [] = cbeamclosed()

format short

clear

clc

x0=[1,1]; %x(1) = b; x(2)=h;

lb=[0.5,0.5];

ub=[5.0, 5.0];

options = optimset('LargeScale','off'); %,'display','iter');

options = optimset(options,'GradObj','on','GradConstr','on');

options = optimset(options,'DerivativeCheck','on');

[x,fval,exitflag]= fmincon(@obj_func,x0,[],[],[],[],lb,ub,@con_func,options);

fprintf( ' b and h are %10.4f and %10.4f\n',x(1), x(2));

fprintf('The mass is %10.4f\n',fval);

function [f,Df] = obj_func(x) %objective function

% fprintf( ' b and h are %10.5f and %10.5f\n',x(1), x(2));

rho=0.1; L=20;

f = rho*L*x(1)*x(2);

% fprintf('The mass is %10.5f\n',f);

Dfx1 = rho*L*x(2); % Df is the gradient information

Dfx2 = rho*L*x(1);

Df = [Dfx1,Dfx2];

function [c,ceq,Dc,Dceq]=con_func(x) %constraints no equality constraints

P = 100; L = 20; E = 10^7;

g1 = ((4*P*(L^3))/(E*x(1)*(x(2)^3)))-0.5; % displacement constraint

g2 = ((12*P*L)/(x(1)*x(2)^2))- 10^4; % stress constraint

c=[g1;g2];ceq=[];

% fprintf('The displacement value is %10.5f\n\n' ,g1+0.5);

Dg1x1 = -((4*P*(L^3))/(E*(x(1)^2)*(x(2)^3))); % these are gradient of constraints

Dg1x2 = -((3*4*P*(L^3))/(E*x(1)*(x(2)^4))); % of g1 wrt x1 and x2

Dg2x1 = -((12*P*L)/(E*(x(1)^2)*(x(2)^2)));

Dg2x2 = -((2*12*P*L)/(E*x(1)*(x(2)^3)));

Dc = [Dg1x1 Dg2x1;

Dg1x2 Dg2x2];

Dceq=[];

A.2 Conjoint based optimization with stress and displacement as attributes

function [] = cbeamclosedconjoint()

clear

clc

x0=[1.0,1.0]; %x(1) = b; x(2)=h;

lb=[0.5,0.5];

ub=[5.0,5.0];

options=optimset('LargeScale','off'); %,'MaxFunEvals',2000);

62

options = optimset(options,'GradObj','on');

options = optimset(options,'DerivativeCheck','on');

% [x,fval,exitflag]= fminunc(@obj_func,x0,options);

[x,fval,exitflag]= fmincon(@obj_func,x0,[],[],[],[],lb,ub,@con_func,options);

fprintf( ' b and h are %10.4f and %10.4f\n',x(1), x(2));

function [f,Df] = obj_func(x) %objective function

% fprintf( ' b and h are %10.4f and %10.4f\n',x(1), x(2));

rho=0.1; L=20;P = 100; E = 10^7;

mass = rho*L*x(1)*x(2);

if (mass<=1.2)

f1 = -215.85*mass + 287.23;

elseif (mass>1.2)&&(mass<=1.4)

f1 = -84.3*mass + 129.37;

elseif (mass>1.4)&&(mass<=1.6)

f1 = -154.05*mass+ 227.02;

elseif (mass>1.6)&&(mass<=1.8)

f1 = -64.25*mass + 83.34;

else

f1 = -134.35*mass + 209.52;

end

% fprintf('The mass is %10.4f\n',mass);

disp = ((4*P*(L^3))/(E*x(1)*(x(2)^3)));

% g2 = ((12*P*L)/(x(1)*x(2)^2))- 10^4;

if (disp<=0.11)

g1 = -771*disp + 114.8;

elseif(disp>0.11)&&(disp<=0.12)

g1 = -2641*disp + 320.5;

elseif(disp>0.12)&&(disp<=0.13)

g1 = -1845*disp + 320.5;

elseif(disp>0.13)&&(disp<=0.14)

g1 = -978*disp + 112.27;

else

g1 = -709*disp + 7.461;

end

f = -(f1+g1);

% fprintf('The displacement value is %10.4f\n\n' ,disp);

% now the gradients

% gradients for mass

if (mass<=1.2)

Df1x1 = -215.85*rho*L*x(2);

Df1x2 = -215.85*rho*L*x(1);

elseif (mass>1.2)&&(mass<=1.4)

Df1x1 = -84.3*rho*L*x(2);

Df1x2 = -84.3*rho*L*x(1);

elseif (mass>1.4)&&(mass<=1.6)

Df1x1 = -154.05*rho*L*x(2);

Df1x2 = -154.05*rho*L*x(1);

elseif (mass>1.6)&&(mass<=1.8)

Df1x1 = -64.25*rho*L*x(2);

Df1x2 = -64.25*rho*L*x(1);

else

Df1x1 = -134.35*rho*L*x(2);

Df1x2 = -134.35*rho*L*x(1);

end

%gradients for displacement

if (disp<=0.11)

Dg1x1 = -771* -((4*P*(L^3))/(E*(x(1)^2)*(x(2)^3)));

Dg1x2 = -771*-((3*4*P*(L^3))/(E*x(1)*(x(2)^4)));

63

elseif(disp>0.11)&&(disp<=0.12)

Dg1x1 = -2641*-((4*P*(L^3))/(E*(x(1)^2)*(x(2)^3)));

Dg1x2 = -2641*-((3*4*P*(L^3))/(E*x(1)*(x(2)^4)));

elseif(disp>0.12)&&(disp<=0.13)

Dg1x1 = -1845*-((4*P*(L^3))/(E*(x(1)^2)*(x(2)^3)));

Dg1x2 = -1845*-((3*4*P*(L^3))/(E*x(1)*(x(2)^4)));

elseif(disp>0.13)&&(disp<=0.14)

Dg1x1 = -978*-((4*P*(L^3))/(E*(x(1)^2)*(x(2)^3)));

Dg1x2 = -978*-((3*4*P*(L^3))/(E*x(1)*(x(2)^4)));

else

Dg1x1 = -709*-((4*P*(L^3))/(E*(x(1)^2)*(x(2)^3)));

Dg1x2 = -709*-((3*4*P*(L^3))/(E*x(1)*(x(2)^4)));

end

Dfx1 = -(Df1x1 + Dg1x1);

Dfx2 = -(Df1x2 + Dg1x2);

Df = [Dfx1,Dfx2];

function [c,ceq]=con_func(x) %no constraints

c=[];

ceq=[];

64

B Matlab Codes and python scripts for Fixed Plate

B.1 Matlab code for Constraint optimization for frequency and displacement

as constraints

function [] = platenormaloptstressdispfreq()

format short

clear

clc

x0 = [1,1,1,1,1,1,1,1,1,1];

lb = [0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1];

ub = [5,5,5,5,5,5,5,5,5,5];

options = optimset('largescale','off','DiffMinChange',0.00001,'display','iter');

options = optimset(options,'TolFun',0.00001,'TolX',0.00001);

[x,fval,exitflag] = fmincon(@obj_func, x0,[],[],[],[],lb,ub,@con_func,options)

x'

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [f] = obj_func(x);

filemanagement(x);

[obj]=mass();

f = obj;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function[c,ceq]=con_func(x)

[const1] = displacement();

g1 = const1 - 0.05;

[const3] = freq();

g3 = 30 - const3;

c=[g1;g3];

ceq=[];

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [] = filemanagement(x)

fid1 = fopen('5by5staticmassunits.inp','r');

fid2 = fopen('5by5varythickmassunits.inp','w');

for i = 1:383

tline = fgets(fid1);

fprintf(fid2,tline);

end

for i = 1:10

a = fprintf(fid2,'%7.5f',x(i));

a = fprintf(fid2,',');

a = fprintf(fid2,' ');

a = fprintf(fid2,'5');

tline = fgets(fid1);

fprintf(fid2,'\n');

for i = 1:5

tline = fgets(fid1);

fprintf(fid2,tline);

65

end

end

for i = 1:127

tline = fgets(fid1);

fprintf(fid2,tline);

end

fclose('all');

a = system('abaqus interactive job=5by5varythickmassunits');

a = system('abaqus python von_mises_and_disp_massunits.py');

a = system('abaqus python get_freq_data.py');

function[const1] = displacement()

fid4 = fopen('disp3.dat','r');

% a = fscanf(fid4,'%*s',5);

a = fscanf(fid4,'%s',1);

format long

const1 = eval(a)

fid5 = fopen('displacementdata.dat','a');

fprintf(fid5,'%10.5f \n',const1);

fclose('all');

function [obj] = mass();

total = 'TOTALMASS';

temp1 = 'temp1';

c1=0;

fid3 = fopen('5by5varythick.dat','r');

for i = 1:220

temp = fgetl(fid3);

end

while(c1==0)

temp = fgetl(fid3);

temp1 = sscanf(temp,'%s',2);

c1 = strcmp(total,temp1);

end

for i = 1:2

temp = fgetl(fid3);

end

format long

obj = eval(temp)

fid4 = fopen('massdata.dat','a');

fprintf(fid4,'%10.6f',obj);

fclose('all');

B.2 Python script for output of first natural frequency

For first natural frequency

from odbAccess import *

test=open("freq.dat","w")

test1 = open("freq1.dat","a")

odb = openOdb('5by5varythick.odb')

for i in range(1,2):

freq=odb.steps['Step-1'].frames[i].frequency

test.write('%10.4f \n'%(freq))

test1.write('%10.4f \n'%(freq))

test.close()

test1.close()

odb.close()

66

B.3 Python script for maximum von mises stress

from odbAccess import *

odb = openOdb(path='5by5varythick.odb')

step1 = odb.steps['Step-2']

frame = step1.frames[-1]

maxMises = 0.

maxDisp2 = 0.

maxDisp3 = 0.

v_mises = frame.fieldOutputs['S']

for stressValue in v_mises.values:

if (stressValue.mises > maxMises):

maxMises = stressValue.mises

stressFile = open('stress.dat','w')

stressFile.write('%10.4E \n' % (maxMises))

#stressFile.write('%10.4E ' % (maxMises))

stressFile.close()

disp = frame.fieldOutputs['U']

for dispValue in disp.values:

if (abs(dispValue.data[2]) > maxDisp3):

maxDisp3 = abs(dispValue.data[2])

disp3File = open('disp3.dat','w')

disp3File.write('%10.4E \n' % (maxDisp3))

#disp3File.write('%10.4E ' % (maxDisp3))

disp3File.close()

odb.close()

B.4 Matlab code for Conjoint optimization with displacement and first

natural frequency as attributes

function [] = plateconjointoptdispStressfreq()

format long

clear

clc

x0 = [3,3,3,3,3,3,3,3,3,3];

lb = [0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1];

ub = [5,5,5,5,5,5,5,5,5,5];

options = optimset('largescale','off','DiffMinChange',0.00001,'display','iter');

% options = optimset(options,'TolFun',0.00005,'TolX',0.00005);

[x,fval,exitflag] = fmincon(@obj_func, x0,[],[],[],[],lb,ub,@con_func,options);

x'

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [f] = obj_func(x);

filemanagement(x); % calling function filemanagement

[obj]=mass(); % calling function mass

if (obj<=1.5)

f1 = -54.08*mass+131.58;

elseif (obj>1.5) && (obj<=2.0)

f1 = -151.42*mass+ 277.59;

else

f1 = -154.94*mass+284.63;

end

%displacements

[const1] = displacement(); % calling function displacement

if (const1<=0.03)

67

f2 = -3791*const1+120.8;

elseif (const1>0.03) && (const1<=0.04)

f2 = -1589*const1+54.74;

else

f2 = -3441*const1+128.82;

end

[const3]=freq();

if (const3<=31.0)

f4 = 1.1*const3-38.16;

elseif (const3>31.0) && (const3<=31.0)

f4 = 8.11*const3-255.47;

else

f4 = 1.12*const3- 31.79;

end

f = -(f1+f2+f4);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function[c,ceq]=con_func(x)

c = [];

ceq=[];

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

68

C MATLAB Code for Optimization for Composite Lightweight Torpedo

C.1 Matlab Code for Constraint Optimization

function [ ]= torpedonormalopt()

% This file is normal optimization of composite torpedo, there are four

% design variables thickness of layers of 0degree, 45degree,90degree,

% honeycomb, and response functions are buckling, frequency, and failure

% index x1 = honeycomb, x2 = zero degree, x3 = 45 degree, x4 = 90degree

% failure index of layer 2 is considered as critical

format long

clear

clc

x0 = [2.9086E-02,1.5E-003,6.0E-004,6.0E-004,2.0E-004,2.0E-004,6.0E-004,6.0E-004,1.5E-003];

lb = [1.0E-010,1.0E-010,1.0E-010,1.0E-010,1.0E-010,1.0E-010,1.0E-010,1.0E-010,1.0E-010];

ub = [0.050,0.050,0.050,0.050,0.050,0.050,0.050,0.050,0.050];

options = optimset('largescale','off','DiffMinChange',1.0e-6,'display','iter');

% options = optimset(options,'TolFun',0.00001,'TolX',0.00001);

[x,fval,exitflag] = fmincon(@obj_func, x0,[],[],[],[],lb,ub,@con_func,options);

fprintf('the thickness variables are \n');

x'

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [f] = obj_func(x);

torpedo_filemanagement(x); % calling function to create input file

[obj]=torpedo_mass(); %calling function to get mass output

f = obj;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function[c,ceq]=con_func(x)

[buck] = torpedo_buck(); % calling function to get BLF output

g1 = 1.0 - buck;

[freq] = torpedo_freq(); % calling function to get freq output

g2 = 22.2 - freq;

[fi] = torpedo_fi(); % calling function to get FI output

g3 = fi - 0.9;

c=[g1;g2;g3];

ceq=[];

% a = system('del torpedo_analysis*.*');

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function[] = torpedo_filemanagement(x);

fid1 = fopen('CopyofTorpedo.dat','r');

fid2 = fopen('torpedo_analysis.dat','w');

for i = 1:17

tline = fgets(fid1);

fprintf(fid2,tline);

end

tline = fgets(fid1);

fprintf(fid2,'DVAR,10,T10,%8.6E,1.0E-010,0.050 \n',x(1));

69

tline = fgets(fid1);

fprintf(fid2,'DVAR,11,T11,%8.6E,1.0E-010,0.050 \n',x(2));

tline = fgets(fid1);

fprintf(fid2,'DVAR,12,T12,%8.6E,1.0E-010,0.050 \n',x(3));

tline = fgets(fid1);

fprintf(fid2,'DVAR,13,T12,%8.6E,1.0E-010,0.050 \n',x(3));

tline = fgets(fid1);

fprintf(fid2,'DVAR,14,T13,%8.6E,1.0E-010,0.050 \n',x(4));

tline = fgets(fid1);

fprintf(fid2,'DVAR,15,T13,%8.6E,1.0E-010,0.050 \n',x(4));

tline = fgets(fid1);

fprintf(fid2,'DVAR,16,T12,%8.6E,1.0E-010,0.050 \n',x(3));

tline = fgets(fid1);

fprintf(fid2,'DVAR,17,T12,%8.6E,1.0E-010,0.050 \n',x(3));

tline = fgets(fid1);

fprintf(fid2,'DVAR,18,T11,%8.6E,1.0E-010,0.050 \n',x(2));

for i = 1:2594

tline = fgets(fid1);

fprintf(fid2,tline);

end

fclose('all');

a = system('genesis torpedo_analysis');

function[obj] = torpedo_mass()

total = 'MASS';

temp1 = 'temp1';

c1=0;

fid3 = fopen('torpedo_analysis.out','r');

for i = 1:50

temp = fgetl(fid3);

end

while(c1==0)

temp = fgetl(fid3);

temp1 = sscanf(temp,'%s',4);

c1 = strcmp(total,temp1);

end

for i =1:2

temp = fgetl(fid3);

end

a = sscanf(temp,'%*s %*s %s',[1 2]);

obj = str2num(a)

fid4 = fopen('massdata.xls','a');

fprintf(fid4,'%10.4f \n',obj);

fclose('all');

function[buck] = torpedo_buck();

total = 'MODEBUCKLING';

temp1 = 'temp1';

c1 = 0;

fid3 = fopen('torpedo_analysis.out','r');

for i = 1:60

temp = fgetl(fid3);

end

while(c1==0)

temp = fgetl(fid3);

temp1 = sscanf(temp,'%s',2);

c1 = strcmp(total,temp1);

end

for i =1:2

temp = fgetl(fid3);

70

end

a = sscanf(temp,'%*s %s',[1 1]);

buck = str2num(a)

fid4 = fopen('buckdata.xls','a');

fprintf(fid4,'%10.4f \n',buck);

fclose('all');

function[fi] = torpedo_fi();

total = 'QUAD4IDLAYER';

temp1 = 'temp1';

c1 = 0;

fid3 = fopen('torpedo_analysis.out','r');

for i = 1:70

temp = fgetl(fid3);

end

while(c1==0)

temp = fgetl(fid3);

temp1 = sscanf(temp,'%s',3);

c1 = strcmp(total,temp1);

end

for i =1:3

temp = fgetl(fid3);

end

a = sscanf(temp,'%*s %*s %*s %*s %*s %*s %*s %*s %*s %s',[1 9]);

fi = str2num(a)

fid4 = fopen('FIdata.xls','a');

fprintf(fid4,'%10.4f \n',fi);

fclose('all');

function[freq] = torpedo_freq();

total = 'MODECYCLES';

temp1 = 'temp1';

c1 = 0;

fid3 = fopen('torpedo_analysis.out','r');

for i = 1:55

temp = fgetl(fid3);

end

while(c1==0)

temp = fgetl(fid3);

temp1 = sscanf(temp,'%s',2);

c1 = strcmp(total,temp1);

end

for i =1:2

temp = fgetl(fid3);

end

a = sscanf(temp,'%*s %s',[1 1]);

freq = str2num(a)

fid4 = fopen('freqdata.xls','a');

fprintf(fid4,'%10.4f \n',freq);

fclose('all');

C.2 Matlab Code for Conjoint based Optimization

function [ ]= torpedoconjointopt()

% This file is conjoint optimization of composite torpedo, there are four

% design variables thickness of layers of 0degree, 45degree,90degree,

% honeycomb, and response functions are buckling, frequency, and failure

% index x1 = honeycomb, x2 = zero degree, x3 = 45 degree, x4 = 90degree

71

% failure index of layer 2 is considered as critical

format long

clear

clc

x0 = [2.9086E-02,1.5E-003,6.0E-004,2.0E-004];

lb = [1.0E-010,1.0E-010,1.0E-010,1.0E-010];

ub = [0.050,0.050,0.050,0.050];

options = optimset('largescale','off','DiffMinChange',1.0e-6,'display','iter');

% options = optimset(options,'TolFun',0.00001,'TolX',0.00001);

[x,fval,exitflag] = fmincon(@obj_func, x0,[],[],[],[],lb,ub,@con_func,options);

fprintf('the thickness variables are \n');

x'

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [f] = obj_func(x);

torpedo_filemanagement4variables(x); % calling function to create input file

%Mass

[mass]=torpedo_mass(); %calling function to get mass output

if (mass<=223.0)

f1 = -25.22*mass + 5652.6;

elseif (mass >223.0) && (mass <=225.0)

f1 = -16.205*mass + 3642.2;

else

f1 = -49.81*mass + 11203.34;

end

%Buckling

[buck] = torpedo_buck(); % calling function to get BLF output

if (buck<=1.2)

f2 = 258.9*buck - 346.56;

elseif (buck > 1.2) && (buck <=1.3)

f2 = 594.1*buck - 748.80;

else

f2 = 505.8*buck - 634.01;

end

%Frequency

[freq] = torpedo_freq(); % calling function to get freq output

if (freq<=23.0)

f3 = 25.60*freq - 596.39;

elseif (freq > 23.0) && (freq <=24.0)

f3 = 16.42*freq - 385.25;

else

f3 = 18.00*freq - 423.17;

end

%Failure Index

[fi] = torpedo_fi(); % calling function to get FI output

if (fi<=0.7)

f4 = -11.3*fi + 16.16;

elseif (fi > 0.7) && (fi <=0.8)

f4 = -8.51*fi + 67.82;

else

f4 = -171.1*fi+136.62;

end

f = -(f1+f2+f3+f4);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function[c,ceq]=con_func(x)

c = [];

ceq=[];

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

72

Bibliography

[1] Adduri, P., Penmetsa, R. C., and Grandhi, R. V. Optimization oflightweight composite torpedo structure with system reliability constraint.Journal of Structural and Multidisciplinary Optimization (Submitted 2005).

[2] Bagchi, T. Multiobjective scheduling by genetic algorithms. Kluwer Aca-demic Publishers, Massachusetts, 1999.

[3] Belugundu, A. D., and Chandraputla, T. R. Optimization Conceptsand Applications in Engineering. Prentice-Hall Inc, New Jersey, 1999.

[4] Bharatram, G. Multiobjective optimization of large scale structures withreliability as decision criterion. Master's thesis, Wright State University, Day-ton, OH, 1991.

[5] Buchanan, J. T. A naive approach for solving MCDM problems: theGUESS approach. Journal of Operational Research Society 48 (1997), 202�206.

[6] Churchill, G. A. Marketing Research: methogological foundations. TheDryden Press, Florida, 1991.

[7] Deb, K. Multi-Objective Optimization using Evolutionary Algorithms. JohnWiley and Sons, Ltd, Chichester, 2001.

[8] Deb, K., and Tiwari, S. Multi objective optimization of a leg mechanismusing genetic algorithms. Engineering Optimization 37 (2005), 325�350.

[9] Green, P. E., and Srinivasan, V. Conjoint analysis in consumer research;issues and outlook. The Journal of Consumer Research 5 (1978), 103�123.

[10] Green, P. E., and Srinivasan, V. Conjoint analysis in marketing : Newdevelopments with implications for research and practice. Journal of Market-ing 54 (1990), 3�19.

[11] Grissom, M. D., Belegundu, A. D., Rangaswamy, A., and Koop-

mann, G. H. Conjoint-analysis-based multiattribute optimization: applica-tion in acoustic design. Struct Multidisc Optim 31 (2006), 8�16.

[12] Gunawan, S., and Azarm, S. Multi-objective robust optimization using asensitivity region concept. Struct Multidisc Optim 29 (2005), 50�60.

[13] Gurdal, Z., Haftka, R. T., and Hajela, P. Design and Optimizationof Laminated Composite Materials. John Wiley & Sons, New York, 1999.

73

[14] Huber, J. What we learned from 20 years of conjoint research: When to useself-explicated, graded pairs, full pro�les or choice experiments. In Proceedingsof Sawtooth Software Conference (pp 243-256, Seattle, 1997).

[15] Keeney, R. L., and Raiffa, H. Decision with Multiple Objectives: Pref-erences and Value Tradeo�s. John Wiley & Sons, New York, 1976.

[16] Kuhfeld, W. H., Tobais, R. D., and Garratt, M. E�cient experi-mental design with marketing research applications. Journal of MarketingResearch 31 (1994), 545�557.

[17] Lilien, G. L., and Rangaswamy, A. Marketing Engineering: Computerassisted marketing analysis and planning. Addison-Wesley Educational Pub-lishers Inc, 1998.

[18] Marler, R. T., and Arora, J. S. Survey of multi-objective optimizationmethods in engineering. Struct Multidisc Optim 26 (2004), 369�395.

[19] The Mathworks Inc. Optimization Toolbox Users Guide. Natick, MA,2006.

[20] Messac, A. Physical programming:e�ective optimization for computionaldesign. AIAA Journal 34 (1996), 149�158.

[21] Orme, B. K. Getting Started With Conjoint Analysis: Strategies for productdesign and pricing research. Research Publishers LLC, Wisconsin, 2006.

[22] Reutterer, T., and Kotzab, H. W. The use of conjoint analysis for mea-suring preferences in supply chain design. Industrial Marketing Management29 (2000), 27�35.

[23] Sawtooth-Software. Summer 2005 customer feedbackresults. Published in Sawtooth Solutions. Available onhttp://www.sawtoothsoftware.com/ssolutions.shtml.

[24] Sawtooth-Software. SMRT Application Help. Sawtooth Software, Se-quim, WA.

[25] Tabucanon, M. T. Multi criteria decision making in industry. Elsevierscience publishers, Amsterdam, 1988.

[26] Williams, P., and Kilroy, D. Calibrating price in ACA: The ACA pricee�ect and how to manage it. In Proceedings of Sawtooth Software Conference(pp 81-95, Sequim, WA, 2000).

74