A fuzzy analysis method for new-product development

A Fuzzy Analysis Method for New-Product

Development

C-C

Lo*

and

P W a g +

Institute

of

Information

Management,

National Chiao Tung

Wniversiv,

Taiwan

K-M Chao,

DSM Research Group

School

of

MS,

Coventry Universiv,

UK

*cclo@faculity.nctu.

edu. tw;

k.chao@coventry.ac.

uk

.

‘ping.

wang88@msa, hinet. net

Abstract

This paper reports a new idea-screening method for new product development

(NPD)

with a group of

decision makers having imprecise, inconsistent and uncertain preferences. The traditional NPD analysis method determines the solution using the membership function of ihefuz.r set which cannot treat the negative evidence. The advantage of vague sets with the capabilig of representing negative evidences supports the decision makers with abiliq of modeling uncertain objects. In this paper? we present a new method for new-product screening in the

NPD

process by relaxing a mmber of assumpiions so that imprecise, inconsistent and uncertain ratings can be considered In addition, a

new simiiariiy measure of vague sets is introduced to proceed with the ratings aggregation for a group of decision makers. From numerical iIlustrations, the proposed model can outperform conventional jaq

methods. It is able to provide decision makers (DMs) with consistent information and to model the simation where vague and ill-defined informalion exist in the decision process.

Keywords: New Product Development, Idea Screening, Vague Sets, Similarity, MPDM

1.

Introduction

New-product development is one of the most critical tasks in business process. Every company develops new products to increase sales, profits, and competitiveness; however

NPD

is

a

complex process and is linked to substantial risks. The objective of NPD is to search for possible products for the target markets. In NPD process, decision makers have to screen new-product ideas according to a number of criteria. Consequently, they recommend the ideas to R&D engineers, marketers, and sales managers in every stage of development. The decision makers’ preferences have a significant impact on the selection of new products and the outcome

of

decision-making. How to reach the consistent group preference on each new-product is an important issue and is notoriously difficult to achieve. in most cases,

NPD is risky due to lacking of sufficient information with consumers’ preferences for making decisions. The information is often imprecise, inconsistent and uncertain. Recent studies [la] report the failure rate of new consumer products at 95% in the United States and 90% in Europe. The failures lead to substantial monetary and non-monetary loses. For example, Ford lost $250 million on i t s Edsel; RCA lost $500 million on its videodisk player etc. Many reasons result in the faiture of new products. Some of important factors in high technology NPD can be summarized as folIows

[Z,

6,101:

1) In idea-screening phase, it is impossible to acquire precise and consistent information regarding customers’ preferences, but it is possible to obtain imprecise, inconsistent and uncertain information; 2) In concept development and testing phase, the criteria

for new-product screening are not always quantifiable or comparable;

3 ) In product development phase, the choice of enabling technologies for developing new products is a challenging issue as the technologies evolve rapidly. In addition, it is often the case that development costs are higher than ‘expected;

4) In commercialization phase, participating competitors will use tactics or other means to contend.

Many methods [2,5,6]and tools [l] are used to control NPD process in an attempt to assist product managers in making better screening decisions. For example, 3M, Hewlett-Packard, Lego, and other companies use the stage-gate system to manage the innovation process [5]. However, the traditional technique [5,6] is likely to use quantitative methods, such

as

optimal techniques, mathematical programming, and utility models etc, which neglect the human behavior and only can be applied to the case if the required data are sufficient. Since new-product screening process must involve the judgments of decision makers and the expression of human judgments often lacks precision. In addition, the confidence levels on the judgments contribute to various degrees of uncertainty. A human-consistent approach is likely to adopt imprecise linguistic terms instead of numerical values in the expression of preference. The issue is compounded when a decision-

The 9th International Conference on Computer Supported Cooperative Work in Design Proceedings

making process involves a group of decision-makers who have inconsistent preferences over the ratings on new products,

This research sets out to tackle more human- consistent by adding the assumptions (i.e., "I am not sure") ofien prohibited by other existing approaches [2,4,8,9]. In this paper, we propose a new method, which extends the traditional NPD methods to a fuzzy environment, uses the simitarity measures of vague sets [3,13] to aggregate the ratings of all decision makers including the negative evidence, and supports the decision on the priority among alternatives through the use of fuzzy synthetic evaluation method [ 121.

2. Preliminary Description of Vague Set

Theory

The Vague Set (VS), which

is

a generalization of the concept of a

fuzzy

set, has been introduced by Gau and Buehrer [ 131 as follows:

A vague set

A ' ( x )

in

X,

x={x,,x,

,...,

x,,}, is characterized by the truth-membership 6, and a fafse- membership function

f,

of the element

xk

E X

to

A ' ( x )

E

X

,

(k=1,2

,...,

n); t,:X+Ql] and f, :X+[0,1], where the functions t,(x,)and

f , ( x , )

areconstrainedby OIt,(x,)+ f A ( x k ) I 1 , (1) where t , ( x k ) is a lower bound on the grade of membership of the evidence for X t , f , ( x k ) is a lower bound on the negation of xk derived from the evidence against xk . The grade of membership of X k in the vague set A' is bounded to a subinterval [ f A ( x k ) , l

-

f , ( x , ) ]

of [0,1]. In other words, the exact grade of membership [ f A ( x k ) , l -

f A ( x k ) ]

of

xk may

be unknown, but it is bounded by

Fig 1. shows a vague set in the universe of discourse t"(Jc&) 5

WJ

l-fA(%).

X .

Fig. 1. A vague set

When Xis continuous, a vague set

A'

(X) can be written as

&)= hIf.(x).X'-fi(Jc.c)l~.~, Xk

EX.

(2)

4 x 1 = C [ f A ( X t ) , 1 - j A G k ) I / X k , X k E

x.

(3) When X is discrete, a vague

set

A ' ( x )

can be written

as

"

&=l

The vague value is simply defined as unique element of a vague set. For example, we assume that

x

={1,2,

...

J O ] ,

a

vague value "Beuuryl' o f

X

may be defined by &@~=[0,5,0.4]/5 . It implies the positive preference is 0.5 and the negative preference is 0.6 (i.e.,

In the sequel, we wilI redefine

A'

(x)

is a vague set,

A'

is a vague value, and omit the argument

xk

of

f A (x,) and

f ,

(xk)

throughout unless they are needed for cIarity.

Definition 1. The intersection of two vague sets

A ' ( x )

and

B'(x)

is a vague set C'(x), written as

d(x)

=A(X)Ad(X), truth-membership function and false-membership function are

t ,

and

fc

,

respectively, where

t ,

=

Min(f,,

le), and

1 - f ,

=Minll-f,,l-S,).Thatis, 1-0.4).

rw

-

f,i

= [ f A

,I

- A I

,,

[ t , ~

-

f,

I

=

WW,,

t,

1,

Mi41

-

f"

,I-

f,

11.

(4) Definition 2. The union of vague set

A'(x)

and

B'(x)

is a vague set ~ ' ( x )

,

written as

C'(x) =

A'

( x ) v

B'(x),

where truth-membership function and false-membership function are

t,

and

fc

, respectively, where t ,

= M c c E ( ~ ~ , ~ ~ ) ~

and 1- f , = M m ( l - f , , l - f , > . That is, Pc 9 1

-

sc

1

=

[ t A J

-

f,

1

v

P,

7 1

-

f,

1

=

I M 4 , , t , ) , M W - f,,l-f,>l- (5) Further, let us define the similarity measures between two vague values in order to represent the preference agreement between experts' ratings as follows: [3] Let A' = [ t A ( x L ) , l

-

f , ( x , )] be a vague value, where

r,4 b k

1

E

[OJl,

fA (x,

1

E

[OJI

I and

0 5 t A ( X L ) f f " ( X , ) S I .

Definition 3. Let

A'

be a v a g u e v a i u e i n

X,

x=&,

...,xn}

,

A' = [ f A ( x k ) , 1 - f A ( x k ) ]

.

The median valueof

A'

is [6]

2

Definition 4. For two vague values

A'and

B'

in

X,

X

= { . ] I ~ , . , . , X , ) , S ( A ' , B ' ) is the degree of similarity between

A'

and B'which preserves the properties (P1)-

(P4).

cpl) 0 I S ( A ' , B ' )

s

1;

(E)

s ( A ' , B ' ) = ~

if

A ' = B * ;

(7)

(p3) S( A ' , B ' ) = S(B', A ' ) .

(P4)

S(A',C')<S(A',B') and $(A'$') 5 S(E',C') if

A'

~3

cc,

C'is

avague set inX

3. The'Proposed Method

In NPD process, decision makers including marketers, customers, managers, and

R&D

members, have to form a new-product committee. Each decision maker has to evaluate and screen new-products according to fie well- defined criteria, and then assign performance ratings to the alternatives for each criterion individually. The decision-makers allocate ratings based on their own preferences and subjective judgments. The explicit representation of their preference and judgment with precise numerical values may not be simple, whereas the use of linguistic terms is more natural to human decision makers. This formulation is imprecise, ambiguous and often leads to an increase in the complexity of the decision making process. To simplify the evaluation process of group decision-making, the evaluation criteria are pre-defined here. Hence the new-product screening activity of NPD can be regarded

as

a

fuzzy

MPDM problem. A fuzzy MPDM problem [4,S], however, can be formulated as a generic decision making matrix.

3.1 Problem Formulation

Suppose that a decision group contains g decision makers who have to give linguistic ratings on m

alternatives according to n evaluation criteria, then a

hzzy MPDM problem can be expressed concisely in preference-agreement matrix [ 121 as follows: Suppose that a decision group has m decision makers who have to give linguistic ratings on n evaluated targets,

I

m

~ = [ q y

...

w.1, m d X w i

= I ,

where D is a decision matrix of the group,

d , E { d , , d z ,

...,

d , } are a set of decision makers, and t , ~ ( f l , f 2 ,

...,

t , ) are a finite set of possible targets(i.e., new-products) from which decision makers have to select, X Y (iQ=l, ..., m) is the Iinguistic rating on

,=I

*

target

t,

by di

,

and

w,

is the importance weights of

4

.

These linguistic terms can be transformed into a vague value

A ' ,

A

= [ t A ( x ~ ) , 1 - f A ( x ~ > l / x , ~ ~ E X .

In the following, we use the similarity measure of vague sets to aggregate linguistic ratings of a group's preferences in order to obtain their preferences on each new-product.

(9)

3.2. SimiIarity Measure

We present a new similarity measure

between

two

vague sets which may be continuous or discrete form. We give corresponding proofs of these similarity measures as follows.

According to Def. 3, we use the median value of A'and B' to represent the mean of truth-membership and false-membership function. The preference agreement between two experts can be represented by the proportion of the consistent area to the total area, as shown in Figure 2 [ll].

$44

%(4

PBW

Fig. 2. Preference agreement between two DMs'

linguistic ratings expressed by median of vague sets

Definition 5. Using median of vague value, S " ( A ' , B ' ) is defined

as

the similarity measure between two vague values

3.3 Preferences Aggregation

We calculate the preference-agreement degree of two experts' ratings expressed

by

Eq.(9)

and

denote

S"(1,i) as

aii'

,

i,i' = 1

,...,

m

,

where two vague sets

I , and I' represents the linguistic rating of decision maker di,di . The preference-agreement matrix A for evaluated targets tl

..

1, is

The 9th International Conference on Computer Supported Cooperative Work in Design Proceedings

Remark. For ai; = S m ( I , I ' ) if

I#i

,

and =i if

I=i;

It means that if two decision makers fully agree to

an

evaluated target,

and

they have ari' = I ; it implies:

t,(X)=t,(x),

l - ~ ( x ) = l - f B ( x ) .

By contrast, if they have completely different estimates, then we get

-

a..,

= 0,

After all the preference-agreement degrees between two decision makers are measured, we then aggregate those pairs of vectors using the average aggregation rule to obtain the preference of the

group

on each new- product.

By applying simple additive aggregation rule, we have the group preference of all the decision makers on an evaluated target as

' i = l ,

3.4 Group preference on New-Product

In order to synthesize the preference degree of group, a general compensation operator proposed by Zimmermann and Zysno (1983) is adopted as the group-preference operator in this paper [7]. This index synthesizes confidence level of preference of all experts on an evaluated target r j , A global measure of

preference on each evaluated targets ( t l , ..., t , ) is obtained as

As

the compensation parameter y varied &om 0 to 1, the operator describes the aggregation properties of "AND" and "OR", that is,

max ( r j ) 2 F(t,,

...,

t B ) 2 min ( t , ) .

]=I, ..., n j = l , ..., n

where F is an aggregation b c t i o n of Eq.( 15).

The compensation parameter y indicates the confidence level of preference of decision maker. A small y implies the higher degree of confidence. Finally, the moderator can estimate the degree of confidence and decide whether the group preference has been reached throught the use of C(t) and y. If the group consensus

has

not been reached,

then

the decision

makers

have to modify their ratings according to the Delphi iterative procedures.

3.5 Fuzzy Synthetic Evaluation Method

Once the group preference for all decision makers on each new-product has reached, the fuzzy synthetic evaluation method is employed to attain the priorities of new products. The

fuzzy

simple weighting additive rule is adopted to derive the synthetic evaluations of alternatives by multiplying the importance weight o f

each decision maker ( W , ) with fizzy rating of

alternatives ( x y ). The formulation of synthetic

evaluations of new products which is shown as follows.

v=[v,]

= @x,

,

i=1,2,. ;.,m,j=1,2,;.

.,.

(15)

However, the aggregation results V are still vague values, which cannot be applied directly to decisian- making. The use of

fkzzy

ranking method

and

a -cuts of fuzzy number is to

rank

the order

of

alternatives

and

to transform them into numerical values, according to the synthetic evaluation results.

Based on Def.3, the synthetic evaluation values can be represented as

-

-

"

-

J=1

-

= -f

lf,

( X t

Finally, the fizzy ranking method proposed by Yager(l981) is adopted to determine the ranking of results of synthetic evaluation as follows [ 121:

-

SA

( x k

11

=

2

P A ( x k

1.

₍₁₆₎

,=1 X t k = l

Given a fi~zzy number

,

Yager's index is defined as

F ( Y )

=

c-

X , ( f , ) d a ,

(17)

where amax = SUP U - and

i(f,)

represents the

average value of the elements having at least

a

degree of membership.

r v

4.

Numerical Example: New-products

Screening

In this section an example

for

a

LCD

TV development is used as a demonstration of the application of the proposed method in a realistic

scenario, as well as a validation of the effectiveness of the method.

The evaluation

process

of products

screening is specified as Figure 3,

4 M . * m I m *r@=.1 I I&. F m L I -

1

5- -m 7 &I h r r + A d 4 q

Fig. 3. The evaluation process of LCD-TV products

screening.

Suppose that there i s a new-product committee consisting of six decision makers, {R&D manager, marketer, sales manager, sales, accounting manager, customer] and a set of four different models with various choices such as colors, shapes, and prices, which must be selected through product-screening process. The resulting selection will be sent to product

LOO a88 a93 aoo a83 a87

a88 1.00 Q82 a00 a94 a77

4‘1)-

aoo aoo aoo 1.00 000 am

a87 a77 a93 aoo a72 1.00

a00 e80 a87 0.83 1.00 Q88

1

QOO a72 0.77 a94 aas 1.00 a93 ~ 8 2 too a00

a n

a93

083 Q94 a78 000 100 a72

1.00 0.00 aoo 0.00 0.00 0.06

a00 1.00 0.92 067 0.80 0.72

QOO a92 1.00 a72 Q87 CL77

I

44)-

aoo a67 a n 1.00 Q83 0.94 A f 4 ) =

development and market testing. The committee has to perform the screening process and select the best target from the four candidates according to the defied criteria.

The proposed method is applied to solve this problem according to the following computational procedure:

and possible targets t = { t , , t 2 , f , , t 4 ) , In the following, we have the priori information to determine the weighting vectors of each decision maker by hidher relative importance,

-

,Pw,

,

that is,

Step 1: Form a working W U p d ={dl,d2,d3,d,,d~.ds}.

I - 1

i=l

W

= [w, J = {O.l5,0.2,0.25,0.15,0.15,0.1}.

Step 2: Let

a

vague set

A’

in

X=

{VL, L, M, H, VH) presents linguistic variables of sales price a~ Table 2. For example, “High” may be represented as

A ’ = (0.7,0.2) / 4,

where tA(4) = 0.7, f A ( 4 ) = 0.8.

Table 2. Linguistic variables far the rating of new-product

Very low/ Very Poor [ t A ( . r ) , l - f A ( x ) ] / l Lowt Poor ( x ) d - f A ( x ) 1 ’ 2

Medium

r ~ A ( x > J - - f R ( x ) l / 3

High! Good

[~,tx)~-f,(x)il4

Vely high/ Very Good _{it, (x)J} - fA (x)]/ 5

1.00 aoo 0.8s 0.69 0.75 as9

0.00 LOO 0.00 0.00 0.00 aoo

0.69 aoo 0.79 1.00 a92 0.61

0.89 aoo 0.78 a61 a67 LOO

088 a00 1.00 0.79 Q86 a78

Q75 000 a66 a92 LOO 067

1.00 0.92 0.86 0.67 0.71 0.86 0.92 1.00 0.93 0.72 0.77 0.93 0.86 0.93 1.00 0.78 0.82 0.87 0.71 0.77 0.82 0.94 1.00 o a2 0.86 0.93 as7 0.78 o.sz l a q 0.67 0.72 0.78 1.00 0.94 0.78 We use the linguistic variables, shown in Table 2, to assess the ratings of new products using vague value as Table3.

Step 3: For -evaluated target

t, ,

we calculate the preference agreement vectors between d , , d , using Eq.(8)

as

matrixes are also constructed.

Step 5: Aggregate the preference-agreement vectors to obtain the group preference of each new product using Eq.(12) as

11 (2

t,

t 4

c(t,)

0.564

0.715 0.575

0.676

Step 6: Calculate the group-preference index

on

aII targets for y 4, 7 4 . 5 , 7 =1, respectively

r=O

r=0.5 r = l

c(r)

0.157 0.393 0.983

Step 7: The new-product manager averages new-product with three different levels of confidences: low, moderate, and high, C(t) = 0.5 11 to judge that group preferences have been reached due to the fact

c(r)

= 0.51 1 2 0.5. Step 8: If a group has been reached a consensus over the preferences, then go to step 9. If not, it goes

back

to step

1. Step 9:

1.The weighted fuzzy rating is obtained using Eq.(15) and synthetic results for

four

target is obtained by integrating

i(ca)

at a =0.05,0.10,0.15-1 through Eqs.(16)+17)

’ For example, the median form of

rating on

t ,

evaluated by

dl

) is

i(1.1)

= 0.1112 + 01213 + 0 1 ~ 4

.

The various

a

level sets are for ?(i,l) (i.e.,

~ , = ( 4 , 3 , 2 ) , O < n ~ 0 . 0 5 ; Y.a(4,3,2},0.05<aS0.1;

Y, ={O}, O . I O i a < O . l S :

From this set of

im

,

we

can

compute

i(;*)

as .k(?‘.)~(4+3+2)/3=3, 0.00<U50.05;

Xv.)

_ _

-(4+3+2)/3 = 3, 0.05 < a 5 0.k

x@.)

= 0, 0.10 < a 5 0.15;

k(i.)=

0, 0.15 < a

<

1-00;

Since the synthetic evaluation is a discrete form,

F(Y)

index is computed by

F ( Y ) = jX(G,)dcr = 9as3d@+f~503da=0,30.

Similarly, we can obtain the other elements for all decision makers, We, then, average the rating derived from six decision makers with respect t o t ,

,

t,

, t,

and

t ,

are

4

t 2 t 3 1 4

V ( r , ) 0.455 0.592 0.620 0.524

2. The order of the preferences of the decision makers

on

four models can be stated

as

t,

+

t,

+

t,

+

f,

.

3. The new-product manager makes the decision according to new-product screening rule

of

company as

f ,

t,

(3 I, Decision kill go go kill ,

The 9th International Conference on Computer Supported Cooperative Work in Design Proceedings

5.

Discussion

Without any comparison of the proposed method with other well-established methods, the resuiting decision may be questionable. In this section, we will compare the new-product ranking procedures, developed by Lin and Chen’s approach

[Z],

to treat the

same

problem.

From Eq.(l8), the synthetic evaluation o f traditional fuzzy approach can be obtained when it is true that t{x)=I-f(x) for vague sets (i.e., ignore uncertainty) as Table 3.

Then, the average value of rating alI decision makers

is given

by

- 1 ” -

-

v, = - C [ v ; @ v ; @

...

@.;;I n in1

have attained. In addition, the synthetic evaluation is neither flexible nor can it illustrate the degree of confidence level of the decision makers. We can conclude that the proposed method is more effective than the traditional fuzzy method on

NPD

process.

6.

Conclusion

This paper presents a new fuzzy approach to solve NPD screening problems. The proposed method allows the decision makers to express their preferences

in

linguistic terms and explicitly represent their uncertainty of their judgments using vague sets. The experimental results indicate that our approach can not only effectively reveal the uncertainty of decision makers’ subjective judgments, but also is applicable to

NPD screening problems. From a numerical illustration, the usefulness

and

effectiveness of the proposed .model

has

been demonstrated. T

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R&D

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[3] D. H., Hong, and C. Kim, A note on similarity measures between vague sets and between elements, Fupv Sets undSystems, vol. 115, 1999, pp.83 -96.

[4] F. Herrera, et a], “A Rational Consensus Modef in Group

Decision Making Using Linmistic Assessments”, Fuzzy set In the following, the left-and-right

fuzzy

ranking

method is applied to synthesize the fuzzy ratings [I21

YR

= SUP [U - ( X ) A U, (X)],

vr.

= SUP [U - (XI

*

U,,, ( X I ] ,

(20) (21)

x “ I

r v i

The synthetic evaluation on each target is given by

The synthetic value on

each

target

is

calculated

using

Eqs.(19)-(22) or geometric graphics described as [13]

tl I, 13 t 4 .

Y ( t , )

0.47 0.51 0.55 0.49

Obviously, the target 3 is the best choice and

the

ranking order is

t3

+

t,

+

t4

r

tr

.

The solution of Lin and Chen’s method concludes the same result as our proposed model.

From Table 3 and Eq. (22), the traditional fuzzy method on

NFD

process is not able to dynamically analyze the group preferences of decision makers and does not specify whether the group consensus on targets

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