This article was downloaded by: [North Carolina State University]On: 21 October 2014, At: 20:39Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Engineering OptimizationPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/geno20

CATALOG DESIGN: SELECTION USING AVAILABLE ASSETSCATALOG DESIGN: SELECTION USING AVAILABLE ASSETSS. VADDE a , J. K. ALLEN a & F. MISTREE aa Systems Realization Laboratory , The George W. Woodruff School of MechanicalEngineering, Georgia Institute of Technology , Atlanta, Georgia, 30332-0405, USAPublished online: 27 Apr 2007.

To cite this article: S. VADDE , J. K. ALLEN & F. MISTREE (1995) CATALOG DESIGN: SELECTION USING AVAILABLEASSETS CATALOG DESIGN: SELECTION USING AVAILABLE ASSETS, Engineering Optimization, 25:1, 45-64, DOI:10.1080/03052159508941254

To link to this article: http://dx.doi.org/10.1080/03052159508941254

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Eng. Opr., 1995 Val. 25, pp. 45-64 0 1995 OPA (Overseas Publishers Association) Reprints available directly from the publisher Amsterdam B.V. Published under license by Photocopying permitted by license only Gordon and Breach Science Publishers SA

Printed in Malaysia

CATALOG DESIGN: SELECTION USING AVAILABLE ASSETS

S. VADDE, J. K. ALLEN and F. MISTREE

Systems Realization Laboratory, The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta,

Georgia 30332-0405 U S A

(Received 2 August 1994)

Catalog design is a procedure in which a system is assembled by selecting standard components from catalog of available components. Selection in design involves making a choice among a number of alternatives based on several attributes. The information available to a designer during the early stages of project initiatiori may be uncertain. A designer then has to balance limited resources against the quality of solution. This complex task becomes formidable when dealing with coupled selection problems, that is problems which must be solved simultaneously. Coupled selection problems share a number of coupling attributes. An earlier paper has shown how selection problems, both coupled and uncoupled can be reformulated as a single compromise Decision Support Problem (DSP) using a dererminisric model. This paper shows how to extend this to a nondeterministic case. Fuzzy set theory is used to model imprecision and Bayesian statistics to model stochastic information. Formulations with solution schemes are pres- ented to handle both fuzzy and stochastic i n fo rndon in the standard framework of a compromise DSP. The approaches are illustrated by an example involving the coupled selection o i a heat exchanger concept and a cooling fluid to be used on a marine drilling rig. The emphasis in this paper is placed on the method rather than the results per se.

K E Y WORDS: catalog design, uncertainty, compromise Decision Support Problems, cou&ed selection

NOTATION

D S P I

Decision Support Problem Normalized relative importance of attributes in the selection DS P Normalized ratings of alternatives with respect to noncoupling attributes in the selection DSP Normalized ratings of alternatives with respect to coupling attributes in the coupled selection-selection DSP Indicates a fuzzified quantity Indicates a scattered quantity Range of fuzziness in the fuzzy D S P Range of scatter in the Bayesian DSP Crisp merit function Fuzzy merit functions (L-R fuzzy number) Bayesian merit function Non-negative left spreads of L-R fuzzy numbers Non-negative right spreads of L-R fuzzy numbers

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

S. VADDE er 01.

Fuzzy membership function Mean value in a Gaussian distribution Standard deviation in a Gaussian distribution Possibility indicator in the fuzzy DSP Probability indicator in the Bayesian DSP Boolean variable representing an alternative in the selection DSP Probability density function Underachievement deviation variables associated with goals (generally denoted by d - ) Overachievement deviation variables associated with goals (generally denoted by (1')

1 FRAME O F REFERENCE

Selection is the process of making a choice among a number of possibilities taking into account several measures of merit or attributes (Bascaran et a/.'). These multiple measures of merit for judging the goodness of the design may not all be equally important. The key issues in selection are that there are a number of feasible possibilities with a number of attributes and sometimes these are quantified using uncertain information.

Crtrnlog design is a procedure in which a system is assembled by selecting standard components from catalogs of available components. Catalog design is a form of ourirrnc design6. It is generally cheaper to adapt an off-the-shelf component to satisfy a set of requirements than it is to custom-make it. However, selecting an appropriate component is not a simple task because the performance of the compo- nent generally depends upon the other components integrated in the final design solution; such situations are called coupled selection problems. As the procedure of accounting for component interactions is tedious, it can be tempting to accept the first satisfactory option that is found. Methods including, but not limited to, quali- tative optimization, interval approaches, fuzzy set theory, stochastic optimization, subjective design evaluation, and heuristic approaches have been developed to handle various kinds of information in selection design problems. A review of these methods is presented in De Boer2 and Vadde et u13. All of these methods are applicable to specific selection problems and are useful in specific situations. Further, they are either dererministic in the sense that they exploit the notion of dererrninisric dorninrtnce (as put forward by Bradley and Agogino4), in which one option is inferior to some other option for all possible values of random parameters or are applicable only for a specific application. In using these methods there is always a chance that a designer might be left with more than one option when dealing with a large problem with several alternatives and attributes.

Wariant design-the size and/or arrangement of parts or subsystems or the chosen system are varied. The desired tasks and solution principle me not changed. The other two types or design in this classification scheme are original design and adaptive design.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

CATALOG DESIGN 47

Uncertainty can be introduced in the problem formulation itself. A customer is frequently unable to quantify rankings and weightings precisely. In addition, a designer may have available assets, e.g. a trained labor force, a heavy capital invest- ment in specific equipment, a computer program, o r an inventory (or catalog) of already assembled components. The presence of these, or the desire to use them, may give added constraints and introduce additional uncertainty.

In such cases, a designer often may be able to reduce the uncertainty by perform- ing analysis, experimentation, etc. Acquiring the required information nevertheless has an associated cost. A decision-analytic approach for catalog selection which complements some of the previously mentioned methods was developed by Bradley and Agogino4. They propose a utility function as a means of measuring the expected value of perfect information, EVPI which is computed for individual parameters. Their method is based on the assumption that an explicit utility function is available. Bradlev and Agogino4 mention that. "such a function is often unknown or - - prohibitively expensive to determine, and may contain a great deal of expensive and unnecessary information describing the utility of many designs that are infeasible or obviously inferior."

The approach herein is to model catalog design by formulating a coupled selec- tion Decision Support Problem (DSP). Motivated by the paper of Bradley and Agogino4 the present authors' earlier work3 is extended by assuming that uncertain- ty will be present either in the form of imprecision or stochasticity in the informa- tion about attributes and alternatives. This is a unified and domain independent approach which is capable of handling any kind of information without a known utility function for catalog design. The approach consists of representing the various component characteristics in "databases" or "catalogs" (with one selection DSP querying each database). These selection DSPs are coupled and solved as compro- mise D S P S ' . ~ . ~ . The efficiency and effectiveness of the solution process is enhanced by formulating the problem as a coupled fuzzy o r Bayesian selection-selection DSP. Uncertain interactions between components are modelled using fuzzy set theory and Bayesian statistics, which is incorporated in the formulation allowing a solution to the problem to be obtained without iterative effort. The present approach is illus- trated through the formulation of a coupled DSP (Sections 3-5) and the solution of a coupled pr'oblem of selecting a heat exchanger and a cooling fluid (Sections 6 and 7). The example is based on the work of Bascaran et al.' and is representative of an early stage in a variant design process.

2 SELECTION AND COUPLED SELECTION DECISION SUPPORT PROBLEMS

A selection DSP is of the form7:

Given A set of alternatives. Identify The principal attributes influencing selection.

The relative importance of attributes. The candidate alternatives.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

48 S. VADDE rr al.

Rate The alternatives with respect to their attributes. Rank The feasible alternatives in order of preference based on the computed

merit function values.

A coupled selection-selection DSP arises whenever there are several inter- dependent subsystems to be selected. This is prevalent in catalog design. Consider a svstem that is to be assembled using three components. Assume - further that there is coupling between components 1 and 2 and components 2 and 3. Figure I. The coupling between the alternatives is taken into account by - . - modelling the interaction between the appropriate attributes. A selection DSP can be formulated as a compromise DSP, using the multiple goal, linear compro- mise DSP formulation1. The DSlDES program incorporating the ALP nlgorithms provides an efficient solution method. ALP is based on a modified sequential linear programming approach with an added integer programming capability and i t provides a vertex solution. All formulations in this paper are solved with the DSlDES software on a SUN 41260 workstation.

In this paper, a fuzzy coupled selection DSP and a Bayesian coupled selection DSP are developed in Sections 3 and 4, respectively. Then both models are applied to the solution of a catalog design problem-the simultaneous selection of a heat exchanger concept and a cooling fluid. This is done because the design, at different points on the design time line, may be most suitably modelled by imprecise (fuzzy) information at one time and stochastic information at another time. As the quality of uncertain information changes, naturally the mathematical representation of the design must change, also. However, a great deal of effort can go into creating the mathematical model of the design. Therefore, it would be ideal to have one

1 Rank: I Alternative A1 Alternative A2

. . .

. . . Alternative A3

I With respect to: Attributes

Alternative B1 Alternative %

Alternative 83

Alternative C2 . . .

~l temahve C3

ith respect to:

Attributes

Atmbute cl

ribute

Atmbute c,

Figure 1 Illustration of coupled selection-selection DSPs'

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

CATALOG DESIGN 49

formalism available which can be modified appropriately depending on the quality of information available.

3 T H E FUZZY COUPLED SELECTION DSP

Assume that a designer has previously determined that a fuzzy representation of the problem is appropriate and the data available are suitably represented by fuzzy numbers. Fuzzy numbers and fuzzy arithmetic are used to formulate the fuzzy selection DSP. Dubois and Prade9 developed a deterministic method of ranking fuzzy numbers based on four indices of grades of the possibility of donlinance (PD) . Roubens and Teghemto developed a method for fuzzy goal programming which is ideal for nondeterministic cases. The criteria of P D may be used to transform a nondeterministic case into a deterministic case by converting fuzzy inequalities and equalities to their equivalent crisp forms. This approach is extended to reformulate a fuzzy selection D S P in the form of a compromise DSP.

What is a fuzzy number? A fuzzy number is characterized by a possibility dis- tribution (or a membership function) o r is a fuzzy subset of real numbers. Zadeh's extension principle" is used to extend the algebraic operations. The basic binary operations of addition and multiplication required in the present formulation are derived from this principle in the max-min form.

For demonstration purposes, L-R (left reference-right reference) membership functions are used to represent positive fuzzy number^^.^^^'^ . Th ey are simple to handle and they satisfy the definitions set forth by Dubois and Pradet2. Consider an L-R fuzzy number, A:

where m is the mean value a and /3 are the non-negative left and right spreads of A respectively and define the

membership function, p,(x), i.e.

The left and right spreads, a and P, of a fuzzy number are represented in terms of the ratio between the extent of fuzziness and the main value, c. c may also be expressed as a percentage. a = m ( 1 - c) and / = m (1 + c).

Addition of L - R fuzzy numbers: The addition two L-R fuzzy numbers results in an L-R fuzzy numbert4, if

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

50 S. VADDE er ol

then

M~tltiplicciriori of an L-R firzzy nurnher by a real number: The extended product operation also results in an L-R fuzzy number. For a fuzzy number given by (111, a, b),, and a real number (n,0,0); 11 > 0,14,

A designer can incorporate imprecise information into the selection process by converting either or both l j (the relative normalized importance of attribute j ) or Rkj (the normalized rating of alternative k with respect to attribute j) to fuzzy numbers then the fuzzy merit function is

N N

l jdk j or i j ~ k j (for positive integers k and j ) j = 1 j = 1

and i t is calculated using the extension principle. A crisp coupled selection problem can be transformed to the fuzzy form using

Roubens and Teghem's approachlo. The question that arises is: 'What happens to the crisp inequality for a merit function rnX < 1 when i t is transformed into an - equality of the form riiX % I? This is answered as follows:

Given the fuzzy merit function = r i i (which has a maximum value which 'almost cquals 1' in a fuzzy sense), then: -

rii = (rn, a, p),, and 1 = ( 1 , c l , E ~ ) ~ . ,

I f we call ht ( i n f i n sup rii) the non-negative height of the intersection of the increasing left-hand side for pi and the decreasing right-hand side for pfi (4, or

h t ( i n r i n sup 1;) = max ( ~ ; ~ + 1 , 0 } < 1 , - w h e n r n c l

we obtain the grade of possibility of dominance (PD) of Dubois and Pradeg which represents the fuzzy extension for rii > 1.

Applying this concept for some aspiration level+ 0, leads to the statement that at - asprintion level, 0, rii is approximately equal to I .

f i z O T iff PD(rii, 7) 2 0 and P D ( ~ , ni)> 0

-

' ~ h c crisp relation So, where a Soh iR2 $ ,b (a = I I I , and b = 1 for our case), presents a total interval order structure for each OE[O, I ] and the fuzzy relation is a fuzzy complete Ferrers relation".

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

CATALOG DESIGN 5 1

The set of conditions in Eq. (I) and the extension principle are used to solve for a - system of fuzzy equalities of the form 1 by transformation into the crisp nonlinear programming problem:

K

max C 0, k = 1

For very small values of E~ and e2 (<< I ) , i approaches its crisp value of 1 . Since we are interested in the maximum~values of the merit functions, the associated equa- tions are transformed into the following set of goals of a compromise DSP that are to be satisfied:

where I;, I:, r;, r:, t ; , and c,+ are deviation' variables associated with various goals. The objective is to minimize the sum of these deivations.

K

min [I ( I ; + 1: + r; + r: + r; + t : ) k = 1 I

Thus 0, is maximized to its greatest possible value of one. This is interpreted as maximizing the possibility of a suitable alternative being chosen in the presence of imprecision. I t should also be noted that 0, varies and that the above formulation corresponds that first proposed by Zimmermannl' and now extended to nonlinear goal programming. In the selection DSP formulated as a compromise DSP1, X is a dimensian Kx1 vector and contains boolean variables of which only one, the most

-

'The system and deviation variables in a compromise DSP are always non-negative. To effect solution, one of the following conditions must hold; [(d; = 0) and ( d l = 0) or (d; = 0) and ( d l O)] or (d; = 0) and ( i l l = O)].

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

52 S. VADDE et al.

promising alternative, has a value of one and the others are all zero. T o ensure that the variables X , take only hooleun values and that only one alternative is selected, there are two additional equality constraints:

2 X k = 1 and 2 (X , ) ( I - X , ) = 0 k = 1 k = l

By including these constraints the variables X , are treated as continuous real t~utnbers but are forced to take boolean values. Based on the approach presented and the assumptions made, the fuzzy selection DSP is represented in the form of a compromise DSP in Figure 2.

In this case, an Archimedean f ~ r m u l a t i o n ~ ~ ~ " is used. The objective is to mini- mize the deviation function. The goals have been modelled to be equally important as it is assumed that uncertainty is predominant and there is no way of determining the value of the information available. Thus weights for all the deviations are equal. Further, the Archimedean formulation is sensitive to the change in extent of fuzziness and is found ideal for the study of varying uncertainty. Although here, maximizing the possibility (directly related to 0,) and selecting an alternative, X , , are considered equally important, this is not a necessary condition. This form of the compromise DSP is found to be convenient to represent stochastic uncertainty too, as discussed in the next section.

4 BAYESIAN SELECTION DSP ,

Assume that a designer has previously determined that a Bayesian representation of the problem is appropriate and the data available are suitably represented by Bayesian statistics. T o illustrate the probability approach, Gaussian distributions are used to model stochastic uncertainties in the design parameters. These par- ameters are the normalized relative importance of attributes, I, and the normalized ratings of alternatives with respect to attributes, R. This method is based on the use of the quantification of the designer's preferences based on his/her knowledge of priors and likelihoods. This can be contrasted with the concept of imprecision in fuzzy numbers where a designer chooses preference levels by using desirable values. Further, the concept of probability is viewed as a logic of inductive inference and not in'the traditional frequency sense. The probability calculus approach of Wood et d l 8 , using the axioms of probability logiclg can be used to perform elementary binary operations of multiplication and addition of stochastically uncertain par- ameters which are of interest. Here an input design parameter is represented by a probability density function, pdf or p( ) with unit area, and the ordinate of the pdf reflects the subjective character of the input parameter. The ordinate p( ) is maxi- mum for those values of the input parameter that a designer rates as highest in terms of preference o r desirability and zero for those a designer rates as least preferred. If I stands for prior information and two independent input parameters x

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

Bov

esio

n S

elec

tion

U

SP

<

rirg

S

drc

rio

n

DS

P

irrn

K

ca

ndn!

alea

llem

ativ

es

N

allr

ibur

n

I, no

rmal

iscd

rela

tive

impm

ance

, j*

attr

ibut

e

RkJ

lh

e no

rmal

ircd

rat

ing

of !

he k

*

allc

mat

ivc

wilh

res

psl

lo l

hc j*

al

trib

~le

Fu

ry

S

elec

tion

D

SP

G

ive

n

K

cand

idal

e al

lem

ativ

es

N at

uibu

tes

I, no

rmal

iwd

rela

live

imp

om

ce. j"

at

uibu

le

Ekj

the

fuzz

y na

mal

ized

rat

ing

of !

he k

" al

tern

ativ

e w

ith r

espe

ct to

the

jLha

urbi

ua

mk

he

cris

p m

ertl

func

tion

ol l

he

N k"

al

tcm

awc

= 1 lj

R

j 1

'1

otis

fy

Sclc

cuon

Svs

tem

Con

svai

ns

Lk

the

furr

y m

erit

func

tion

of l

he

N k"

al

tern

ativ

e= 1 Ij

Rk

j

j=

I

ind V

ecto

r of

Sys

tem

Var

iabl

es X

k (I

;=).

.... K

)

Dev

iatio

n V

aria

bles

ek,

and

e;

Selc

ctio

n Sy

rtcm

Goa

ls

(k=

l. ..

.. K

)

Fin

d

Vec

ta o

f Sy

slem

Var

iabl

es X

k, t&

(k=

I. ..

.. K:

Dev

iatio

n V

aria

bles

t.

it,

r;. ri,

f;, and

5:

Bou

nds

(k=

I. ...

. K)

0 5

Xk,

e;.

and

el

5 1

'in

imiz

e

Sat

isfy

F

uu

y S

elec

tion

Syst

em C

onsl

rain

ls

K K

EX

k =

l a

nd 1 Xk

(1-X

k)

= 0

k

=l

k= l

Fu

uy

Sel

cclio

n Sy

slcm

Goa

ls

(k =

I. ....

K)

1 mk

[I+

(1-

ek

)(l+

~k

)]X

k +

1-

-I*

= 1

k

k

~~

II

-(

I-

B~

)(

I-

C~

)~

X~

+

I-

-I

+=

I

k k

ek +

f; . f

; =

I

Bou

nds

(k=

I. ..

.. K

)

0 5

Xk.

r;.

r:. 1;.

I;. l;.

and

f; 5

I

Min

imiz

e

K Z =

L

(r

;+r

;+lk

~+

l~+

I;+

l;)

k

=l

;ive

n

K

cand

idal

e al

tern

ativ

es

N at

uibu

les

2 no

rmal

iscd

rela

tive

impo

nanc

c. jh

atu

ibul

e

Rkj

sc

atte

red

norm

aliz

ed ra

ting

of th

e k"

alte

rnat

ive

wilh

res

pect

to th

e jl

hatv

ibut

e

r;lk

the

Bay

esia

n m

erit

func

tion

of t

he

rin

d

Vec

tor

of S

ysle

m V

aria

bles

Xk,

pi

(k=

1. ..

.. K)

Dev

iatio

n V

aria

bles

I*'.

I:, r;,

r;. b;

. an

d b+

k

Sat

isfy

B

ayes

ian

Sele

ctio

n S

yaem

Con

stra

ints

K

K

xx

k =

1 a

nd 1 Xk

(l-X

k )=

0

k=l

k=

l B

ayes

ian

Sele

ctio

n Sy

stem

Goa

ls

(k =

I. ...

. K)

my

[I+

$J 0

.22

1n

~1

1p

~l I ~

k +

r' L

- r+

I. =

I

mk

[I- $

J 0.

221n

111p

k1 1x

1; +

I '

- I +

=

I k

k

Bou

nds

@=I

. ....

K)

) ~

~k

.r'.

r+

.l~

an

dl;

<l

kk

k

and0

.011

5

bi.

b:f

l

Min

imiz

e

K Z=

1 r;

+ r;

+ +

I,+ +

b; .b

i )

k=l

Figu

re 2

T

he c

risp

. lu

uy

an

d B

ayes

ian

sele

ctio

n D

SP

s

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

54 S . VADDE er 01.

and y are represented by probability density functions, then:

z = f (s, y) is also a pdfwith unit area. I f the stochastic nature of input parameters has been obtained from the knowl-

edge of priors and likelihoods, then Bayesian statistics can be used effectively. Bayesian statistics oRers a convenient way of modelling the likelihood that a given point within the uncertain region satisfies the goal or constraint. A Gaussian distribution is employed to define the range in which a variable is believed to exist. In this distribution / I is the most likely value and u is the standard deviation. Although the Gaussian distribution is continuous from - m to + m, the probabil- ity density falls oRalmost to zero by the time the value of .u in the Gaussian function rcaches / r + 30. Vadde er ( 1 1 . ~ ~ , have shown that the problem of infinite limits of the Gaussian distribution is avoided by representing finite regions on the real line. The inverse Gaussian distribution is defined for m in terms of .u. This is P-'(p) where p is the probability indicator. Then stochastic uncertainty may be introduced into a variable by using an inverse Gaussian distribution. In this paper it shall be used to modify a crisp merit function to include uncertain information. The resulting merit function is referred to as the Bayesian merit function which is:

whcre L, is the scatter of the merit function and is the stochastic analog of fuzziness. The scatter is expressed as a percentage of the mean value of a stochastic parameter. The interpretation of the probability indicator p is analogous to the aspiration 0 (or possibility indicator) considered in the fuzzy selection problem. By maximizing the probability indicators a designer tries to select an alternative which has the highest likelihood of having the maximum utility.

This formulation is also Archimedean and all deviations are given equal weights for the same reasons as mentioned in the previous section. Maximizing theprobabil- ity indicators (p,) and selecting an alternative (X, ) are considered equally important. For the case of zero scatter this reduces to the formulation in Bascaran, et u1.I.

5 REPRESENTATION O F COUPLING ATTRIBUTES

Independent fuzzy and Bayesian selection DSPs are represented in terms of a compromise DSP by means of system goals as shown in the preceeding sections. Assume that there are two coupled selection DSPs, the first with p alternatives, the second with q alternatives and a number of coupling attributes. In this case, the ratings corresponding to each of the coupling attributes cannot be expressed in vectorial I 'O~ITI. They must be formulated as a (p x q) matrix or two dimensional

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

CATALOG DESIGN

array as follows:

CR(X1, X2) = ... for the crisp formulation.

The matrix becomes ( X I , X,j for the fuzzy selection DSP and TR ( X I , X,) for the Bayesian selection DSP. X I is a vector of dimension p representing the alterna- tives of the first selection DSP; X, is a similar vector of dimension q representing the alternatives of the second DSP. If more than two problems are coupled, the ratings are represented by an S dimensional array, where S is the number of selection problems coupled by that attribute. Since the selection DSP can be defined in terms of compromise descriptors the mathematical formulation of the crisp coupled selec- tion-selection DSP can then be written as follows1.

Given W number of selection DSPs involved in the coupled problem For w = 1,. . . , W :

Xkw vector of alternatives k for selection problem w K ,, number of alternatives for selection problem k Nw number of noncoupling attributes for selection problem k Cw number of coupling attributes for selection problem k

'j number of selection problems coupled by attribute j Rk j w normalized ratings of alternative k with respect to noncoupling

attribute j for selection problem rv CRkj,,(X,) S dimensional aray of normalized ratings of alternative i for

coupling attribute j, selection problem w

I jw relative importance of attribute j for selection problem ,tl MFk w merit function for the kth alternative of selection problem 1s

Satisfy Selection System Constraints

Selection System Goals (k = I , . .. , K,), (w = 1, . . . , W )

Independent part Coupling part

Find Selection Variables: Xkw (w = I , . . . , W )

Bounds

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

56 S. VADDE el al.

Minimize The Deviation Function:

There are two parts of the merit function, namely, the independent and the coupling parts. This merit function can easily be modified to include uncertainty by trans- formation procedures described earlier, Figure 2. The use of the compromise formu- lation makes i t computationally efficient to solve coupled problems whose size grows exponentially with the number and size of the coupled DSPs considered'.

6 AN EXAMPLE O F A C O U P L E D SELECTION-SELECTION DSP: T H E SIMULTANEOUS SELECTION O F A HEAT EXCHANGER AND COOLING FLUID

The Problem: During the operation of an open sea marine platform, it is necess- ary to cool a stream of lubricant a t low pressure used in drilling systems. The problem involves the selection of the best combination of a heat exchanger design and the cooling fluid to be used from a catalog ofalternatives. This is more than just a data base search, we have to decide which data bases (or catalogs) to use. The crisp formulation of this problem is presented in Bascaran e t a / . ' .

Heat exchangers, Figure 3, are used in every situation in which it is necessary to transfer energy between two fluid streams at different temperatures. The shell and tube configuration is the most popular due to its well established design procedure and adaptability, however other heat exchanger configurations are available and any one of these might be a better choice for a specific situation. Although the primary attribute in determining the feasibility of an exchanger concept is the ability to transfer the required heat duty within the allowed pressure drop, other attributes such as maintenance and future expandability must be considered. In addition, some of the required attributes pertain to characteristics of the fluids involved in the energy exchange, hence, a coupled problem arises.

The two selection problems are formulated and solved as a coupled selection- selection DSP. The heat exchanger selection DSP involves five candidate alternatives and six attributes while the cooling fluid selection DSP involves three alternatives and three attributes. Two of the attributes are common to both DSPs; hence the coupled selection-selection DSP. This example, though simplified, is representative of the complexities inherent in multiple discrete option selection problems under uncer- tainty, where the alternatives and configuration of coupling attributes are specified vaguely and a system is assembled by selecting suitable alternatives of each type. It is emphasized that the example is presented principally to demonstrate a method.

Alternatives and attributes are described in detail in Bascaran et ul.', for each of the selection DSPs. The same data is used as that of Bascaran but the merit functions are fuzzifiedJscattered to study the effect of uncertainty on the solution obtained. The data is presented in Tables 1-5. Some results and comments with respect to the effects of uncertainty in catalog design are then presented.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

CATALOG DESIGN

(a) Shell and Tube Heat Exchanger

(b) Double Pipe Heat Exchanger

(d) Spiral Heat Exchanger

(c) Gasketed Heat Exchanger

(e) Welded Plate Exchanger

Figure 3 Heat Exchanger Concepts

6.1 Heat Exchanger Concept Selection DSP

Five alternatives are considered for this problem. They are representative of the numerous design concepts available in the market. The alternatives are Shell and Tube Heat Exchanger, Double Pipe Heat Exchanger, Gasketed Plate Heat Exchanger, Spiral Heat Exchanger, and Welded Plate Heat Exchanger.

Assuming that all the heat exchanger alternatives can adequately handle the required heat duty, six attributes are considered. Thev are Maximum Pressure. Maximum Temperature Range, Compactness, Versatility, Fouling and Corrosion. Fouling and Corrosion are coupling attributes. Several case studies or scenarios are created by modifying the relative importance of these six attributes, Table 5. An illustration of the use of the reciprocal matrix comparison method for obtaining the relative importance of attributes is given in Bascaran et ul. ' .

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

58 S. VADDE er a/ .

'l'uble I Ratings of noncoupling attributes for the heat exchanger selection problem - - -

Alrerr~t~ric~s pmnr Tma, Cornparrncs Versariliry Fou l iq & [MP(J] p] [-I [-1 Currosiotr [ - I

Shell :~nd Tube 30.7 600 3 8 50 600 I 9 1)ouble Pipe ' Coupled

G:~sketed Plnte 1.6 175 8 7 Attributes Spiral 1.8 400 7 5 Wcldcd I'htc 3.0 400 8 5 Scale Ratio Ratio Interval Interval Interval Kmge 0.5-50.0 100-700 1-9 1-9 1-9 I'rcfcrred Value High High High High Low

Table 2 Kotings of noncoupling attributes for the coolant selection problem

Cosr Fouliny & Corrusiun

[$I c - I Sen Water 2 Regular Water 5 Coupling Purified Watcr 8 Attributes Scalc Interval Interval Range 1-9 1-9 I'relerred Value Low Low

Tuble 3 Ratings for the coupling attribute fouling

Fouli~ng [-] Sea Worer Tap Warcr Purified Water

Shcll and Tube 8 5 3 Double Pipc 4 2 2 Gaskcted Plate 5 3 I Spir:d 5 3 I Welded I'late 8 7 4

Table 4 Ratings for the coupling attribute corrosion

Corrosiorn [ - I Sen Wnter Trip Water Purified Water

Shell xnd Tube 7 5 3 Double Pipe 8 7 6 Gasketed Plate 5 3 2 Spiral 4 2 I Welded Plate 4 2 I

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

CATALOG DESIGN

Table 5 Scenarios based on the relative importance of attributes

Heor Eschorryur Scenorio EX1 Scenario E X 2 Scerrrrrio E X 3 Scenario E X 4 Selecrion

pmax 0.166 0.1 0.2 0.16 Tmnx 0.166 0.1 0.2 0.16 Compactness 0.166 0.1 0.2 0.20 Versatility 0.166 0.1 0.2 0.11 Fouling 0.166 0.3 0.1 0.19 Corrosion 0.166 0.3 0.1 0.18

Coolant Selection Scer~nrio C L l Scerrario CL2 Scenario CL3

Initial Cost 0.333 0.2 0.8 Fouling 0.333 0.4 0.1 Corrosion 0.333 0.4 0.1

6.2 Cooling Fluid Selection DSP

Tap water is the alternative of choice for most industrial cooling applications because of its low cost, availability and safety. In a marine platform, however, several alternatives need to be considered because of its isolated location. Only three alternatives are evaluated in this example, they are Sea Water, Regular Water and Purified Water and the three attributes are C o s t , ' ~ o u l i n ~ and Corrosion. Similarly to the heat exchanger problem, variations of relative importance among the coolant attributes results in a set of scenarios, Table 5.

6.3 Scenurios Based on Relative Importnnce of Attributes

From the twelve possible combinations that can be formed with the proposed scenarios of Table 5, solutions for the two of the most interesting ones are included in this paper. Two scenarios with fuzziness/scatter are presented to demonstrate the application of the various formulations and to get an insight into their use. The scenarios also facilitate the study of the effects of uncertainty in the information available to a designer during the selection process. The first scenario is EXI, CLl where all the attributes for each of the two selection problems are rated equally. This is a hypothetical situation and is introduced for purposes of comparison. The second scenario is one in which the ratings of the attributes are generally different within each problem. The only ratings that are the same are those for Pressure and Tem- perature and Fouling and Corrosion which indicates a designer's preference. These two scenarios are presented in Table 5.

7 IMPLEMENTATION, VALIDATION, OBSERVATIONS AND FUTURE WORK

Implementation: Various cases were run from different starting points and for four different values of fuzziness/scatter. All cases resulted in a solution, that is, only

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

60 S. VADDE cr (11.

one alrernative was identified as most promising for each selection problem. The values of the possibility and probability indicators were very close to one (0.998 + 0.002) which indicates that the solutions are reliable. The three different skirting points are classified as low infeasible (infeasible and close to the lower bounds), high infeasible (infeasible and close to the upper bounds), and infeasible (infeasible and midway between the bounds). Convergence plots for both formula- tions for different situations are included. The solution does not depend on the starting point. Plots of the deviation function for goals and constraint violation are shown in Figures 4 and 5.

Both fuzzy and Bayesian formulations converged in 17 o r less analysis/synthesis cycles. The time taken for convergence was always less than 2 seconds of CPU time. The time For convergence for Bayesian formulations was at least 13%less than fuzzy formulations.

Esr~rhli.shi~~g the j t z z y crnd Boyesirrn fortnu1rrtiun.s: At 0 % fuzziness/scatter, both the fuzzy and Bayesian formulations identified the same alternative. For scenario EXI, CLI, the Shell and Tube heat exchanger concept is selected with Purified Water as the cooling fluid for both formulations (see Table 6) . This is reasonable since the coupling ratings are high for this combination and also the noncoupling attribute ratings for the Shell and Tube concept are high for Pressure, Temperature and Versatility. Likewise, for scenario EX4, CL3, the solution is a Shell and Tube heat exchanger with Sea Water ior both formulations. This is justified since this scenario has a high rating for the Cost attribute which forces Sea Water into the solution.

Cori~prrriso~l betwee11 rhe fuzzy rrnd Brryesim DSPs: Although further development is needed to augment the versatility of the formulations presented, preliminary

Iteration No. Iteration No.

Figure 4 Convergence of the fuzzy DSP for scenario EXI, CLI and 25% fuzziness.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

CATALOG DESIGN

- Infeasible point High Infeasible point - Low Infeasible ~ o i n t

Iteration No. Iteration No.

Figure 5 Convergence of the Bayesian DSP lor scenario EX4. CL3 and 25%scatter.

indications shown in the form of data and plots suggest numerous advantages of incorporating uncertainty in coupled selection problems. The Bayesian DSP yields a stable solution at a higher percentage of uncertainty than the fuzzy DSP. This is evident from the results for the two scenarios presented. For scenario E X I , CLI, as shown in Table 6, the Bayesian DSP gives the same solution up to a scatter value of 50%. For the same scenario, in the case of the fuzzy DSP, the solution is

Table6 Solution lor scenario EXI. C L I

Percentage Fuzz). Selecrion D S P Bayesian Selerrion D S P

F u z z i n [ w / Alt~~rnoriues CPU rime Alternatives C P U rime Scurler Selecred ( i n x c o n d s ) Selected ( i n seconds)

0% Shell and Tube 1.89 Heat Exchanger. Purified Water

25% Shell and Tube 1.63 Heat Exchanger. Purified Water

50% Double Pipe 1.53 Heat Exchanger, Tap Water

75% Double Pipe 1.39 Heat Exchanger, Tap Water

Shell and Tube 1.63 Heat Exchanger. Purified Water Shell and Tube 1.25 Heat Exchanger. Purified Water Shell and Tube 0.85 Heat Exchanger, Purified Water Double Pipe 0.71 Heat Exchanger, Tap Water

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

62 S. VADDE er (11

the same only to a value of 25%. For scenario EX4, CL3, (Table 7) the Bayesian DSI' gives a stable solution only to a scatter value of 50% whereas the fuzzy DSI' gives dinerent solutions for different percentages. Prior knowledge incorpo- rated in the case of Bayesian DSP is believed to be the essential cause for its stable solutions.

Ejfect Oj i n c ~ w ~ i n g f trzzines~Js~utter on the fuzzy ond Bugesiun DSPs: The etlect of uncertainty is not predominant for scenario EX1, CLl since i t is a neutral scenario depicting a hypothetical case where a designer has no preference for any attributes. Nevertheless, the results for this scenario gives an indication of the stability of solution. The effect of increasing fuzziness/scatter is studied on the basis of the results prescnted for scenario EX4, CL3 in Table 7. The effect of uncertainty on the results is pronounced in this scenario. For the crisp case with 0% fuzzinessJscatter the solutions obtained by both the DSPs is the same. For 25% fuzziness, the so- lution for the fuzzy DSP is the Double Pipe Heat ExchangerJSea Water. The Double Pipe heat exchanger has the highest ratings for Pressure, Temperature, and Vcrsatility. For the Bayesian DSP the results do not change for 25% increase in scatter. The Double Pipe Heat ExchangerJSea Water is the solution for the Ihyesinn DSP for both 50%and 75%values of scatter. In general, the fuzzy DSP is found to be sensitive to changes in the extent of uncertainty whereas the Bayesian DSP is not.

The eJJicuc~1 ($using the fuzzy und Buyesiun DSPs to model uncertainty in cutulog desigrr prohlerns: Both the fuzzy and the Bayesian DSPs solve the problem and also provide a means of modelling varying types of uncertain information. incorporating the fuzzinessJscatter negates the deleterious etlect of an initial infeasible design on the number of iterations required for convergence. This observation is of particular importance during the early stages of project initiation when little is known of the system and uncertainties associated are very high.

Table 7 Solution for scen:~rio EX4, CL3

Percmlroge Furzg Selection DSP Bnyesiun Selecfiort DSP

F~~mirlesa/ Alferrtnriues CPU rime Alfernofiorcs CPU rww Scoffer Selecred (in seconds) Selecrcd toke11

0 % Shell and Tube 1.91 Shell and Tube 1.42 Heat Exchanger, Heat Exchanger. - Sea Water

25% Double Pipe I .76 Heat Exchaneer. - . Sen Water

50% Spiral Heat 1.08 Exchanger, Purified Water

75% Welded plate 0.82 Heat Exchanger, Tap Water

Sea Water Shell and Tube 1 . 1 1 Heat Exchanger. - . Sea Water Double Pipe 0.75 Heat Exchanger, Sea Water Double Pipe 0.68 Heat Exchanger, Sea Water

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

CATALOG DESIGN 63

Etnployiny the fuzzy und Buyesiun DSPs increases the speed with which a solution is obtained: The convergence for the fuzzy and Bayesian DSPs for the catalog design problem is much faster than the corresponding crisp formulation. The C P U time taken for convergence for all formulations was less than 2 seconds. The time for convergence for the Bayesian formulations was found to be less by 13% to 48% than the fuzzy formulations.

The eficcrcy ofusing the fuzzy und Bngesiun DSPs over u range o f t h e design rime line: In practice, uncertainties should decrease as one proceeds along a design time- line. The solutions of the fuzzy and Bayesian DSPs for the catalog design problem for a fuzziness/scatter value of zero were the same. Thus, both methods converge to the same solution as uncertainty decreases, hence, either may be used to model the design. It is left to the designer to determine which model accurately reflects the type and quality of the information available at each stage of design- and to switch between models o i use combinations of models, if necessary.

The authors gratefully acknowledge the financial contribution made by their corporate sponsor, The BF Goodrich Company. The NSF Equipment Grant 880681 1 is gratefully acknowledged. The cost of com- puter time was underwritten by the University of Houston. During the course of this research, Srinivas Vadde was suported as a teaching assistant by the University of Houston. This work was completed while all three authors were in Houston.

References

1. Bascaran, E., Bannerot, R. B. and Mistree, F. (1989) Hierarchical selection decision support problems in conceptual design. Engineering Optimizarion, 14, 207-238.

2. De Boer, S. J. (1987) Selection techniques in methodical design. In Proceedings 1987 lnternarional Co~ferenre on Engineerir~y Design. (Eden, E. W., Editor) Boston: The American Society of Mechanical Engineers, 1. 303-310.

3. Vadde, S., Allen, J . K. and Mistree, F. (1992) Catalog design: design using available assets. In Ad- vances in Design Auromation. D. A. Hoeltzel, Editor, ASME, New York, 345-354.

4. Bradley, S. R. and Agogino, A. M. (1991) An intelligent real time design methodology for catalog selection. In Design Theory and Methodologg, ASME, New York. 31, 201-208.

5. Allen, J. K., Simovich, G. and Mistree, F. (1989) Selection under uncertain conditions: a marine application. In Fourth Internationnl Symposium on Prncrical Design of Ships and hlohile Unirs, Varna, Bulgaria; Bulgarian Ship Hydrodynamics Centre. 2, 80.1-80.8.

6. Reddy, R. and Mistree, F. (1992) Modeling uncertainty in selection using exact interval arithmetic. In Design Theory and Metkodoloyj~ 92, L. A. Staufer and D. L. Taylor, Editors, ASME, New York 193-201.

7. Kup~uraiu. N., Ittimakin, P. and Mistree, F. 11985) Design through selection. A method that works. ~ e s . i y n studies, 6,(2), 9 1 - 106.

- -

8. Mistree, F., Hughes, 0. F. and Bras, B. A. (1993) The compromise decision support problem and the adaptive h e a r programming algorithm. In Structural Optimization: Starus and Promise, M. P. Kamat, Editor, AIAA, Washington. D.C.. 247-286.

9. Dubois, D. and ~ r a d e , ~ . (1983) Ranking fuzzy numbers in the setting of possibility theory. Iiforina- tion Sciences. 30. 183-224.

10. Roubens, M. and Teghem, J. Jr. (1991) Comparison of methodologies for fuzzy and stochastic multiobjective programming. Fuzzy Sets and Sysrerns, 42, 119-132.

1 I. Zadeh. L. A. (1975) The concept of a linguistic variable and its application to approximate reasoning- 1. InJorn~ation Sciences, 8, 199-249.

12. Dubois, D. and Prade, H. (1979) Fuzzy real algebra: some results. Fuzzg Sets and Spstenls. 2,(4), 327-348.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014

64 S. VADDE et al.

13. Kaufmonn. A. and Gupta. M. M. (1985) Inrorductio~~ ro Fuzzy Arirhmetic. New York, Van Nostrand Reinhold Compnny.

14. Dubois, D. and Prade. H. (1978) Operations on fuzzy numbers. lntcrnotional Journal of Sptems Science. 9. 613-626.

IS. Zimmermann. H. J. (1978) Fuzzy programing and linear programming with several objective func- tions. Fuzzy Sers and Systems. 1. 45-55.

16. Mislree. F., Patcl. B. and Vadde. S. (1994) On modeling multiple objectives and multi-level decisions in concurrent design. In 1994 Dcsiyr! Automation Cofference. (Gilmore, B. J . , Hoeltzel, D., Dutta, D., Eschenhauer, H.. Eds.) Minneapolis, Minnesota, ASME, 69-2, 151-161.

17. Ignizio, J. P. (1985) lrltroductior: to Linear Coal Programming. Qunntitatioe Applications in the Social Scioiccs. Ed. J. L. Sullivan and R. G. Niemi. Vol. Number 07-056. Beverly Hills, California, Sage University Papers, 17. 187-203.

18. Wood. K. L.. Antonsson. E. K. and Beck, J. L. (1989) Comparing fuzzy and probability calculus for representing iniprecision in engineering design. In Design Theory and Methoduloyg, ASME.

19. Cox, R. T. (1961) T lw Alyehrn ofProhnhle Inference. Baltimore, Maryland, Johns-Hopkins University Press.

20. Vxdde, S.. Allen J. K. and Mislree. F. (1994) The Bayesian compromise decision support problem for multilevel design involving uncertainty. Journal of Mechanical Design, 116,(2), 388-395.

Dow

nloa

ded

by [

Nor

th C

arol

ina

Stat

e U

nive

rsity

] at

20:

39 2

1 O

ctob

er 2

014