A fuzzy group-preferences analysis method for
new-product development
Chin-Chun Lo
a,*, Ping Wang
a, Kuo-Ming Chao
baInstitute of Information Management, National Chiao Tung University, Taiwan bDSM Research Group, Faculty of Engineering and Computing, Coventry University, UK
Abstract
This paper reports a new idea-screening method for new product development (NPD) with a group of decision makers having impre-cise, inconsistent and uncertain preferences. The traditional NPD analysis method determines the solution using the membership func-tion of fuzzy sets which cannot treat negative evidence. The advantage of vague sets, with the capability of representing negative evidence, is that they support the decision makers with the ability of modeling uncertain opinions. In this paper, we present a new method for new-product screening in the NPD process by relaxing a number of assumptions so that imprecise, inconsistent and uncertain ratings can be considered. In addition, a new similarity measure for vague sets is introduced to produce a ratings aggregation for a group of decision makers. Numerical illustrations show that the proposed model can outperform conventional fuzzy methods. It is able to provide decision makers (DMs) with consistent information and to model situations where vague and ill-defined information exist in the decision process.
Ó 2006 Elsevier Ltd. All rights reserved.
Keywords: New product development; Idea screening; Vague sets; Similarity measure; Group decision making
1. Introduction
New-product development is one of the most critical tasks in the business process. Every company develops new products to increase sales, profits, and competitive-ness; however NPD is a complex process and is linked to substantial risks. The objective of NPD is to search for pos-sible products for the target markets. InCopper (1998), the process for NPD is divided into eight phases as follows: (1) idea generation phase; (2) idea screening phase; (3) concept development and testing phase; (4) marketing strategy development phase (5) business analysis phase (6) prod-uct development phase; (7) market testing phase; (8) com-mercialization phase. In the NPD process, decision makers have to screen new-product ideas according to a number of criteria. Subsequently, they recommend the ideas
to R&D engineers, marketers, and sales managers in every stage of development. Idea screening is a concept-level eval-uation process that begins when the collection of new prod-uct ideas is complete. It uses technical, commercial, and financial information to weed out impractical ideas, so that only appropriate ideas can be selected into development and testing (Hart & Hultink, 2002). Idea screening can avoid both the ‘drop-error’ and the ‘go-error’. The former occurs when the company dismisses a viable idea; the latter takes place when the company permits an inferior idea to move into product development and market testing. A wrong decision in idea screening will lose resources, time to market, business opportunity etc. Hence idea screening is perhaps the most critical phase in NPD process. During the idea screening process, the decision makers’ preferences have a significant impact on the selection of new products and the result of the decision making. The method of obtaining the group preference of the decision makers on each new-product in a committee is an important issue which causes many difficulties. In most cases, NPD is risky 0957-4174/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.eswa.2006.01.005
*
Corresponding author.
E-mail addresses: cclo@faculity.nctu.edu.tw (C.-C. Lo), ping.
wang88@msa.hinet.net(P. Wang),k.chao@coventry.ac.uk(K.-M. Chao).
www.elsevier.com/locate/eswa Expert Systems with Applications 31 (2006) 826–834
Expert Systems with Applications
due to the lack of sufficient information about imprecise, inconsistent and uncertain customer preferences. Recent studies (Kim & Kim, 1991; Kotler, 2003) report the failure rate of new consumer products at 95% in the United States and 90% in Europe. The failures lead to substantial mone-tary and non-monemone-tary losses. For example, Ford lost $250 million on its Edsel; RCA lost $500 million on its vid-eodisk player etc. There are many reasons for the failure of a new product. Some of the important factors in high tech-nology NPD can be summarized as follows:
(1) In an idea-screening phase, it is impossible to acquire precise and consistent information regarding custom-ers’ preferences, but it is possible to obtain imprecise, inconsistent and uncertain information.
(2) In a concept development and testing phase, the cri-teria for new-product screening are not always quan-tifiable or comparable.
(3) In a product development phase, the choice of enabling technologies for developing new products is a challenging issue as the technologies evolve rap-idly. In addition, it is often the case that development costs are higher than expected.
(4) In a commercialization phase, participating competi-tors will use a variety of means to contend.
This research sets out to provide more human-consis-tency by including the assumptions (i.e., ‘‘I am not sure’’) often prohibited by other existing approaches (Kao & Liu, 1999; Kessler & Chakrabarti, 1997; Lin & Chen, 2004). In this paper, we propose a new method, which extends the tra-ditional NPD methods to the early product development and evaluation, uses the similarity measures of vague sets (Gau & Buehrer, 1993; Hong & Kim, 1999; Li & Cheng, 2002) to aggregate the ratings of all decision makers includ-ing the negative evidence. It supports decisions on the prior-ity among alternatives through the use of a fuzzy synthetic evaluation method (Chen & Hwang, 1992) for phase.
The rest of the paper is structured as follows. Section2
reviews important NPD literature. Section 3 introduces basic concepts and definitions in vague sets and their oper-ations. Section 4 formulates the problem of new-product screening and describes the proposed algorithms and asso-ciated methods. Proofs for four resulting properties from the proposed algorithms are also included. In Section 5, an example of evaluating new ideas is shown, to illustrate the proposed method. Section 6 compares the outcomes with other approaches. Section 7 offers the conclusion on this work.
2. Literature review
Many methods (Calantone, Benedetto, & Schmidt, 1999; Copper, 1981, 1993, 1998; Copper & Kleinschmidit, 1986; Kessler & Chakrabarti, 1997; Lin & Chen, 2004) and tools (Henriksen & Traynor, 1999; Rangaswamy & Lilien, 1997) 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 com-panies use the stage-gate system to manage the innovation process (Kotler, 2003). Rangaswamy and Lilien (1997)
comprehensively classified these methods into three main classes: (1) factor-weighting techniques (Kao & Liu, 1999); (2) eigenvector method, e.g., analytic hierarchy pro-cess (AHP) for NPD (Calantone et al., 1999); (3) screening regression methods. The factor-weighting method decides the importance of critical successful factors (CSF) of NPD using the weighted distance function (Kao & Liu, 1999). The AHP method (Satty, 1980) determines the weights of CSF of NPD by solving for the eigenvalues of a rating matrix (Liberatore, 1987; Calantone et al., 1999; Zimmermann & Zysno, 1983). Screening regression meth-ods use a set of variables to analyze the importance weight of factors and to predict the success or failure of a NPD project using regression and statistics techniques (Copper, 1993). Other well-known techniques for NPD include beta-testing, conjoint analysis, quality function deployment (QFD), break-even analysis (Hart & Hultink, 2002).
However, the traditional technique (Calantone et al., 1999; Copper, 1981; Hart & Hultink, 2002; Kao & Liu, 1999; Kessler & Chakrabarti, 1997; Liberatore, 1987; Satty, 1980) is likely to use quantitative methods, such as optimal techniques, mathematical programming, AHP, and multi-ple regression models etc., which can only be applied to the case of performance evaluation of the product develop-ment phase when the required data are in numeric format. Since the early phase of new-product screening most often operates in an uncertain situation with incomplete information, it must involve the judgements of decision makers. The expression of human judgment often lacks precision and the confidence levels on the judgment con-tribute to various degrees of uncertainty. A human-consis-tent approach is likely to adopt imprecise linguistic terms instead of numerical values in the expression of preference. The issue is compounded when a decision-making process involves a group of decision-makers who have inconsistent preferences.
In the next section, we use vague sets to represent the imprecise linguistic ratings of the group, and define three similarity measures based on mean value of vague sets. These allow the accumulation of the ratings of all the deci-sion makers in order to make an appropriate decideci-sion on the priority among alternatives.
3. 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 (1993)as follows:
A vague set A0(x) in X, X = {x
1, x2, . . . , xn}, is
character-ized by a truth-membership function, tA, and a
false-mem-bership function, fA, for the elements xk2 X to A0(x)2 X,
(k = 1, 2, . . . , n); tA: X! [0, 1] and fA: X! [0, 1], where
condition 0 6 tA(xk) + fA(xk) 6 1. tA(xk) is a lower bound
on the grade of membership of the evidence for xk, fA(xk)
is a lower bound on the negation of xkderived from the
evi-dence against xk. The grade of membership of xk in the
vague set A0is bounded to a subinterval [t
A(xk), 1 fA(xk)]
of [0, 1]. In other words, the exact grade of membership of xk may be unknown, but it is bounded by tA(xk) 6
uA(xk) 6 1 fA(xk).
Fig. 1shows a vague set in the universe of discourse X. Let X be the universe of discourse, X = {x1, . . . , xn},
xk2 X, a vague set A0 of the universe of discourse X can
be represented byChen (1997) A0ðxÞ ¼½tAðx1Þ; 1 fAðx1Þ x1 þ þ½tAðxnÞ; 1 fAðxnÞ xn : ð1Þ
(1) can be represented as the following formula: A0ðxÞ ¼X n k¼1 ½tAðxkÞ; 1 fAðxkÞ xk ; xk 2 X : ð2Þ
The vague value is simply defined as a unique element of a vague set. For example, X = {Number of friends} the vague set Likeable could then have vague values associated with each number [0.1, 0.0]/0, [0.2, 0.1]/2,. . .
In the sequel, we will refer to A0(x) as a vague set, A0as a
vague value, and omit the argument xkof tA(xk) and fA(xk)
throughout unless they are needed for clarity.
Definition 1. The intersection of two vague sets A0(x) and B0(x) is a vague set C0(x), written as C0(x) = A0(x)^ B0(x), truth-membership and false-membership functions are tC
and fC, respectively, where tC= min(tA, tB), and 1 fC=
min(1 fA, 1 fB). That is,
½tC;1 fC ¼ ½tA;1 fA ^ ½tB;1 fB
¼ ½minðtA; tBÞ; minð1 fA;1 fBÞ: ð3Þ
Definition 2. The union of vague set A0(x) and B0(x) is a
vague set C0(x), written as C0(x) = A0(x)_ B0(x), where
truth-membership function and false-membership function are tC and fC, respectively, where tC= max(tA, tB), and
1 fC= max(1 fA, 1 fB). That is,
½tC;1 fC ¼ ½tA;1 fA _ ½tB;1 fB
¼ ½maxðtA; tBÞ; maxð1 fA;1 fBÞ: ð4Þ
Further, let us define the similarity measures between two vague values in order to represent the preference agree-ment between experts’ ratings as follows:
Let A0= [t
A(xk), 1 fA(xk)] be a vague value, where
tA(xk)2 [0, 1], fA(xk)2 [0, 1], and 0 6 tA(xk) + fA(xk) 6 1
(xk2 X).
Definition 3. Let A0be a vague value in X, X = {x1, . . . , xn},
A0= [t
A(xk), 1 fA(xk)]. The mean value of A0(Li & Cheng,
2002) is uAðxkÞ ¼
tAðxkÞ þ 1 fAðxkÞ
2 : ð5Þ
Definition 4. If a vague set A0is a subset of a vague set B0,
we denote as A0 B0.
Proposition 1. For two vague sets A0, B0, u
A(xk) 6 uB(xk)
holds, if A0 B0.
If A0 B0, then each subinterval [t
A(xk),1 fA(xk)] is
contained inside [tB(xk),1 fB(xk)]. According toDefinition
3, it implies that the mean values of A0are smaller than those of B0, which can be expressed as uA(xk) 6 uB(xk) for all xk.
Definition 5. For two vague values A0 and B0 in X,
X = {x1, . . . , xn}, S(A0, B0) (Li & Cheng, 2002) is a degree
of similarity between vague values if it preserves the prop-erties (P1)–(P4). Let D be the set of vague values in X = {x1, x2, . . . , xn} then S(a, b) is a degree of similarity
for D if it preserves the properties (P1)–(P4). ðP1Þ For all A0; B02 D 0 6 SðA0; B0Þ 6 1;
ðP2Þ SðA0; B0Þ ¼ 1 if A0¼ B0;
ðP3Þ For all A0; B02 D SðA0; B0Þ ¼ SðB0; A0Þ;
ðP4Þ For all A0; B0; C02 D such that A0 B0 C0;
SðA0; C0Þ 6 SðA0; B0Þ and SðA0; C0Þ 6 SðB0; C0Þ: ð6Þ 4. The proposed method
In a NPD process, decision makers including marketers, customers, managers, and R&D members, have to form a new-product committee. Each decision maker has to eval-uate and screen new-products according to some 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 eval-uation criteria are pre-defined here. Hence the new-product screening activity of NPD can be regarded as a fuzzy
1–fA(xk) 1–fA(xk) tA(xk) xk 0 1
MPDM problem. A fuzzy MPDM problem (Chen & Hwang, 1992; Hwang & Lin, 1987), however, can be for-mulated as a generic decision making matrix.
4.1. Problem formulation
Suppose that a decision group contains m decision mak-ers who have to give linguistic ratings on n alternatives according to q evaluation criteria, then a fuzzy MPDM problem can be expressed concisely in preference-agree-ment matrix (Chen & Hwang, 1992) as follows:
DðtjÞ ¼ ~ x11 ~x12 ~x1n ~ x21 ~x22 ~x2n .. . .. . .. . .. . ~ xm1 ~xm2 ~xmn 2 6 6 6 6 6 4 3 7 7 7 7 7 5 ; ð7Þ W ¼ ½w1w2 wm; and Xm i¼1 wi¼ 1;
where D is a decision matrix of the group, di2 {d1,
d2, . . . , dm} are a set of decision makers. tj2 {t1, t2, . . . , tn}
are a finite set of possible targets (i.e., new-products) from which decision makers have to select, ~xij ði ¼ 1; . . . ; m;
j¼ 1; . . . ; nÞ is the linguistic rating on target tj by di, and
wi is the importance weight of di. These linguistic terms
can be transformed into a vague value A0 according to
Table 1,
A0¼ ½tAðxkÞ; 1 fAðxkÞ=xk; xk 2 X : ð8Þ
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.
4.2. Similarity measure
We present a new similarity measure between two vague sets with discrete form. We give corresponding proofs of these similarity measures as follows.
The preference agreement between two experts can be represented by the proportion of the interception to the union. Based on this idea, we use theDefinition 6to repre-sent the similarity between two vague values.
Definition 6. Using mean of vague value, Sm(A0, B0) is defined as the similarity measure between two vague values according toZwick, Carlstein, and Budescu (1987)
SmðA0; B0Þ ¼ Pn i¼1ðuAðxiÞ ^ uBðxiÞÞ Pn i¼1ðuAðxiÞ _ uBðxiÞÞ ¼ Pn i¼1minðuAðxiÞ; uBðxiÞÞ Pn i¼1maxðuAðxiÞ; uBðxiÞÞ : ð9Þ
According toDefinition 3, we use the mean value of A0and
B0 to represent the mean of truth-membership and
false-membership function.
Theorem 1. Sm(A0, B0) preserves the four important
proper-ties (P1)–(P4) of the similarity measure of vague value. Proof. It is obvious thatTheorem 1satisfies the properties (P1)–(P3) of Definition 6. In the following, Sm(A0, B0) will
be proved to satisfy (P4) as follows. For any C0= [t C(x), 1 fC(x)]/x and A0 B0 C0, we have A0 B0, as A0 B0 implies u A0ðxÞ 6 uB0ðxÞ. SmðA0; C0Þ ¼ Pn i¼1minðuAðxiÞ; uCðxiÞÞ Pn i¼1maxðuAðxiÞ; uCðxiÞÞ ¼ Pn i¼1uAðxiÞ Pn i¼1uCðxiÞ 6 Pn i¼1uBðxiÞ Pn i¼1uCðxiÞ ¼ Pn i¼1minðuBðxiÞ; uCðxiÞÞ Pn i¼1maxðuBðxiÞ; uCðxiÞÞ ¼ SmðB0; C0Þ: Since A0 B0, we have Sm (A0, C0) 6 Sm (B0, C0). Similarly,
we can prove that Sm(A0, C0) 6 Sm
(A0, B0) if A0 B0
C0. h
In the following, we introduce the explicit form of Sm(A0, B0), called Mean Similarity.
In some cases, the weight of the element x2 X might be considered. Then, we present the following weighted mea-sure between vague sets.
Assume that the weight of x2 X = {x1,. . ., xn} is
wk(k = 1, 2, . . . , n), where 0 6 wk61, andPnk¼1wk ¼ 1. We denote SwðA0; B0Þ ¼ Pn i¼1wðxiÞ: minfuAðxiÞ; uBðxiÞg Pn i¼1wðxiÞ: maxfuAðxiÞ; uBðxiÞg : ð10Þ
Theorem 2. Sw(A0, B0) is a degree of similarity between the
two vague sets A0and B0in X.
Proof. This proof is similar to that of Theorem 1
(omitted). h
Obviously, if wk= 1/(b a) (k = 1, 2, . . . , n), Eq. (12)
becomes Eq.(11). So Eq.(12)is a general form of Eq.(11). Definition 7. Sw(A0, B0) is the weighted similarity between vague sets A0 and B0.
4.3. Preferences aggregation
We calculate the preference-agreement degree of two experts’ ratings expressed by Eq. (9) and denote Sm(i, i0)
as aii0, i, i0= 1, . . . , m, where two vague sets i, and i0
Table 1
Linguistic variables for the rating of new product
Very low/very poor [tA(1), 1 fA(1)]/1
Low/poor [tA(2), 1 fA(2)]/2
Medium [tA(3), 1 fA(3)]/3
High/good [tA(4), 1 fA(4)]/4
represents the linguistic rating of decision maker di, di0. The preference-agreement matrix A(t) for evaluated targets t = t1. . ., tnis (need to show dependence on t in the matrix.)
AðtÞ ¼ 1 a12ðtÞ a1mðtÞ a21ðtÞ 1 a2mðtÞ .. . .. . .. . .. . am1ðtÞ am2ðtÞ 1 2 6 6 6 6 4 3 7 7 7 7 5 ð11Þ
Remark. For aii0 ¼ Smði; i0Þ if i 5 i0, and aii0 ¼ 1 if i = i0. Two decision makers fully agree to an evaluated target, if they have aii0 ¼ 1; it implies: tA(x) = tB(x), 1 fA(x) = 1 fB(x). By contrast, if they have completely different
estimates, then we get aii0 ¼ 0.
After all the preference-agreement degrees between two decision makers have been 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 (not sure that this is what it is, you are adding up similarities.) of all the decision makers on an evaluated target as CðtjÞ ¼ 2 mðm 1Þ Xm1 i¼1 Xm i0¼iþ1 aii0ðtjÞ: ð12Þ
(There should be a definition here. The quantity appears to be an agreement average on a given target. It might be use-ful to call it that.)
4.4. Group preference on new-product
In order to synthesize the preference degree of group, a general compensation operator proposed byZimmermann and Zysno (1983)is adopted as the group-preference oper-ator in this paper (Kacprzyk & Fedrizzi, 1989; Zimmer-mann & Zysno, 1983). This index synthesizes a confidence level of preference for all experts on an evalu-ated target tj. A global measure of preference on each
eval-uated targets (t1, . . . , tn) is obtained as
CðtÞ ¼ Y n j¼1 CðtjÞ !1r 1Y n j¼1 ð1 CðtjÞÞ !r : ð13Þ
(The above formula needs correcting, there is no defini-tion of Cs and product is over values of ‘j’ which is not
mentioned). As the compensation parameter c varied from 0 to 1, the operator describes the aggregation properties of ‘‘AND’’ and ‘‘OR’’, that is,
max
j¼1;...;nCðtjÞ P CðtÞ P minj¼1;...;nCðtjÞ; ð14Þ
where F (so is F the same thing as C?) is an aggregation function of Eq.(15)(This does not make sense, the tihave
not yet been defined numerically so how can we have a max and min).
The compensation parameter c indicates the comple-ment level of decision maker. A small c implies the higher degree of complement. Finally, the moderator can estimate the degree of consensus depending on c and decide whether group consensus has been reached using CQ1nEnQ2ðtÞ (some explanation of Q1nEnQ2 would be helpful) and c. If the
group consensus has not been reached, then the decision makers have to modify their ratings according to the Del-phi iterative procedures.
4.5. Fuzzy synthetic evaluation method
Once the group preference for all decision makers on each new-product has reached, the fuzzy synthetic evalua-tion 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 of each decision maker (wi) with fuzzy rating of alternatives (~xij). The
for-mulation of synthetic evaluations of new products which is shown as follows: e V ¼ ½~vj ¼ Xn j¼1 wi ~xij; i¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n: ð15Þ However, the aggregation results eV are still vague val-ues, which cannot be applied directly to decision making. The use of fuzzy ranking method and a-cuts of fuzzy num-ber is to rank the order of alternatives and to transform them into numerical values, according to the synthetic eval-uation results.
Based onDefinition 3, the synthetic evaluation values eV can be represented as e V ¼X n i¼1 ½tAðxkÞ; 1 fAðxkÞ xk ¼X n k¼1 uAðxkÞ xk : ð16Þ
Finally, the fuzzy ranking method proposed by Yager (1981) is adopted to determine the ranking of results of synthetic evaluation as follows (Chen & Hwang, 1992):
Given a fuzzy number eV, Yager’s index is defined as Fð eVÞ ¼
Z amax
0
Xð eVaÞda; ð17Þ
where amax¼ supxuV~ðxÞ and X ð eVaÞ represents the average
value of the elements having at least a degree of membership.
In summary, the solution algorithm can be summarized as follows:
Step 1. Form a new-product committee and identify the appropriate criteria and importance weights for each decision maker.
Step 2. Select the appropriate linguistic terms for repre-senting the rating of new products and perform the idea screening process using vague value according to confidence level of decision maker.
Step 3. Calculate the preference-agreement vector between two decision makers using Eq.(9).
Step 4. Construct the preference-agreement matrixes for all decision makers using Eq.(11).
Step 5. Aggregate the preference-agreement vectors to obtain the group preference of each new product using Eq.(12).
Step 6. Calculate the group-preference index on all prod-ucts using Eq.(13).
Step 7. The new-product manager judges whether group preference on each new-product has been reached according to the index. If it has not reached, then decision maker has to modify his/her rating according to the Delphi iterative procedures. Step 8. Repeat steps (2)–(6) until group-preference index is
reached the accepted level by all decision makers. If group preference has been reached, then go to step 9, else go to step 2.
Step 9. The new-product manager determines the ranking of new products using Eq. (19) and make one of four decisions: go, kill, hold, or recycle according to the company’s screening policy of NPD.
5. 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 pro-posed method in a realistic scenario, as well as a validation of the effectiveness of the method. The evaluation process of products screening is specified asFig. 2.
Suppose that there is a new-product committee consist-ing of six decision makers, {R&D manager, quality man-ager, sales manman-ager, engineering manman-ager, accounting manager, customer} has to screen new-product ideas as
Table 1according to the five criteria: (c1) project resource compatibility (c2) product superiority and unique, (c3) technology complexity and magnitude, (c4) market need,
growth and size (c5) maintenance of market share and sunk cost (Balachandra & Friar, 1997; Copper, 1993; Kim & Kim, 1991). A set of four concept models has been built. Four concept models must be selected through idea-screen-ing process and be sent to mass product and market test-ing. The committee has to perform the screening process and select the best target from the four candidates accord-ing to the defined criteria. The proposed method is applied to solve this problem according to the following computa-tional procedure:
Step 1: Form a working group d = {d1, d2, d3, d4, d5, d6},
and possible targets t = {t1, t2, t3, t4}. In the
follow-ing, we have the priori information to determine the weighting vectors of each decision maker by his/her relative importance, wi¼ wi=
Pn
i¼1wi, that
is,
W ¼ ½wi ¼ f0:15; 0:2; 0:25; 0:15; 0:15; 0:1g:
Step 2: Let a vague set A0 in X = {VL, L, M, H, VH}
pre-sents linguistic variables of sales price as Table 1. For example, ‘‘High’’ may be represented as A0= (0.7, 0.8)/4, where t
A(4) = 0.7, fA(4) = 0.2.
We use the linguistic variables, shown in Table 1, to assess the ratings of new products using vague value asTable 2.
Step 3: For evaluated target t1, we calculate the preference
agreement vectors between d1, d2using Eq.(9)as
1. Idea
generation 2. Idea screening
3. Concept development & testing 4. Marketing strategy development 7. Product development & marketing
5. Business analysis 6. New products screening Group preferences reached ? Rating modifications
Fig. 2. The evaluation process of LCD-TV new products screening.
Table 2
Ratings of evaluated targets using vague sets DMs Targets t1 t2 C1 C2 C3 C4 C5 C1 C2 C3 C4 C5 d1 (0.7, 0.8)/2 (0.8, 0.8)/3 (0.7, 0.7)/4 (0.6, 0.7)/3 (0.7, 0.9)/4 (0.6, 0.6)/4 (0.7, 0.9)/4 (0.7, 0.9)/3 (0.7, 0.7)/3 (0.7, 0.8)/4 d2 (0.8, 0.9)/2 (0.6, 0.7)/4 (0.8, 0.8)/4 (0.7, 0.7)/3 (0.8, 0.9)/4 (0.7, 0.7)/3 (0.8, 0.8)/4 (0.6, 0.8)/3 (0.8, 0.9)/3 (0.8, 0.9)/3 d3 (0.6, 0.8)/2 (0.6, 0.7)/3 (0.5, 0.7)/4 (0.8, 0.9)/3 (0.8, 0.8)/3 (0.6, 0.8)/4 (0.8, 0.9)/4 (0.6, 0.7)/3 (0.9, 0.9)/4 (0.7, 0.8)/4 d4 (0.5, 0.6)/3 (0.5, 0.8)/3 (0.6, 0.7)/3 (0.6, 0.6)/4 (0.6, 0.7)/4 (0.5, 0.6)/4 (0.7, 0.9)/4 (0.6, 0.9)/3 (0.6, 0.7)/3 (0.8, 0.8)/4 d5 (0.9, 0.9)/2 (0.9, 0.9)/3 (0.6, 0.7)/4 (0.8, 0.8)/3 (0.7, 0.7)/4 (0.6, 0.6)/4 (0.6, 0.6)/4 (0.8, 0.9)/3 (0.6, 0.8)/3 (0.9, 0.9)/4 d6 (0.6, 0.7)/2 (0.9, 0.9)/3 (0.9, 0.9)/4 (0.8, 0.9)/3 (0.8, 0.8)/4 (0.6, 0.7)/4 (0.6, 0.7)/4 (0.7, 0.9)/3 (0.6, 0.7)/3 (0.8, 0.8)/4 t3 t4 d1 (0.7, 0.7)/5 (0.7, 0.8)/4 (0.7, 0.8)/5 (0.6, 0.8)/2 (0.8, ,0.8)/4 (0.6, 0.6)/3 (0.9, 0.9)/3 (0.6, 0.8)/5 (0.6, 0.8)/3 (0.7, 0.8)/5 d2 (0.6, 0.6)/4 (0.8, 0.9)/3 (0.8, 0.9)/5 (0.7, 0.7)/3 (0.7, 0.9)/4 (0.6, 0.7)/3 (0.8, 0.9)/3 (0.5, 0.6)/4 (0.7, 0.9)/4 (0.7, 0.7)/4 d3 (0.6, 0.7)/4 (0.7, 0.7)/4 (0.8, 0.8)/4 (0.8, 0.9)/3 (0.8, 0.9)/5 (0.7, 0.7)/3 (0.8, 0.8)/3 (0.7, 0.8)/4 (0.8, 0.9)/3 (0.8, 0.8)/4 d4 (0.9,0.9)/4 (0.7, 0.7)/4 (0.9, 0.9)/5 (0.6, 0.6)/3 (0.9, 1.0)/4 (0.6, 0.6)/3 (0.5, 0.8)/3 (0.5, 0.6)/4 (0.5, 0.6)/3 (0.4, 0.6)/4 d5 (0.7, 0.8)/4 (0.8, 0.9)/4 (0.8, 0.9)/5 (0.8, 0.8)/3 (0.9, 0.9)/4 (0.8, 0.9)/3 (0.8, 0.9)/3 (0.5, 0.6)/4 (0.8, 0.9)/3 (0.6, 0.8)/4 d6 (0.8, 0.9)/4 (0.7, 0.7)/4 (0.7, 0.7)/5 (0.7, 0.8)/3 (0.8, 0.8)/4 (0.6, 0.8)/3 (0.7, 0.8)/3 (0.6, 0.7)/4 (0.7, 0.8)/3 (0.7, 0.8)/4
a12¼ R3 2½minft11; t21g; minf1 f11;1 f21gdx R3 2½maxft11; t21g; maxf1 f11;1 f21gdx ¼ R3 2½0:7; 0:8dx R3 2½0:8; 0:9dx ¼ R3 20:75dx R3 20:85dx ¼0:75 0:85¼ 0:882:
Following the same way, we can obtain the others elements a13, a14, . . . , a65for targets t1, t2, t3and t4.
Step 4: Construct the preference-agreement matrixes for color criterion for all targets as
Aðt1Þ ¼ 1:00 0:88 0:93 0:00 0:83 0:87 0:88 1:00 0:82 0:00 0:94 0:77 0:93 0:82 1:00 0:00 0:78 0:93 0:00 0:00 0:00 1:00 0:00 0:00 0:83 0:94 0:78 0:00 1:00 0:72 0:87 0:77 0:93 0:00 0:72 1:00 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 ; Aðt2Þ ¼ 1:00 0:00 0:88 0:69 0:75 0:89 0:00 1:00 0:00 0:00 0:00 0:00 0:88 0:00 1:00 0:79 0:86 0:78 0:69 0:00 0:79 1:00 0:92 0:61 0:75 0:00 0:86 0:92 1:00 0:67 0:89 0:00 0:78 0:61 0:67 1:00 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 ; Aðt3Þ ¼ 1:00 0:00 0:00 0:00 0:00 0:00 0:00 1:00 0:92 0:67 0:80 0:72 0:00 0:92 1:00 0:72 0:87 0:77 0:00 0:67 0:72 1:00 0:83 0:94 0:00 0:80 0:87 0:83 1:00 0:88 0:00 0:72 0:77 0:94 0:88 1:00 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 ; Aðt4Þ ¼ 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 0:67 0:72 0:78 1:00 0:94 0:78 0:71 0:77 0:82 0:94 1:00 0:82 0:86 0:93 0:87 0:78 0:82 1:00 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 :
Similarly, c2, c3, c4 and c5 of the preference-agree-ment matrixes are also constructed.
Step 5: Aggregate the preference-agreement vectors to obtain the group preference of each new product using Eq.(12)as
t1 t2 t3 t4
CðtjÞ 0:564 0:715 0:575 0:676
Step 6: Calculate the group-preference index on all targets for c = 0, c = 0.5, c = 1, respectively
c¼ 0 c¼ 0:5 c¼ 1 CðtÞ 0:157 0:393 0:983
Step 7: The new-product manager averages new-product with three different levels of confidences: low, mod-erate, and high, C(t) = 0.511 to judge that group preferences have been reached due to the fact C(t) = 0.511 P 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:
(9.1) The weighted fuzzy rating is obtained using Eq. (15) as shown in Table 3 and synthetic results for four target is obtained by integrat-ing Xð eVaÞ at a = 0.05, 0.10, 0.15–1 through
Eqs.(16) and (17).
For example, the mean form of eV for eVð1; 1Þ (i.e., rating on t1 evaluated by d1) is
e
Vð1; 1Þ ¼ 0:11=2 þ 012=3 þ 011=4. The various a level sets are e Va¼ f4; 3; 2g; 0 < a 6 0:05; e Va¼ f4; 3; 2g; 0:05 < a 6 0:1; e Va¼ f0g; 0:10 < a 6 0:15:
From this set of eVa, we can compute Xð eVaÞ
as
Xð eVaÞ ¼ ð4 þ 3 þ 2Þ=3 ¼ 3; 0:00 < a 6 0:05;
Xð eVaÞ ¼ ð4 þ 3 þ 2Þ=3 ¼ 3; 0:05 < a 6 0:1;
Xð eVaÞ ¼ 0; 0:10 < a 6 0:15;
Xð eVaÞ ¼ 0; 0:15 < a 6 1:00:
Since the synthetic evaluation is a discrete form, Fð eVÞ index is computed by
Fð eVÞ ¼ Z 1 0 Xð eVaÞda ¼ Z 0:05 0 3daþ Z 0:10 0:05 3da¼ 0:30: Table 3
Weighted ratings of evaluated targets using vague sets
DMs Targets t1 t2 t3 t4 d1 0.11/2 + 0.22/3 + 0.23/4 0.23/3 + 0.32/4 0.11/2 + 0.23/4 + 0.22/5 0.34/3 + 0.22/5 d2 0.16/2 + 0.14/3 + 0.48/4 0.45/3 + 0.33/4 0.31/3 + 0.28/4 + 0.17/5 0.48/3 + 0.25/4 d3 0.18/2 + 0.37/3 + 0.35/4 0.39/3 + 0.59/4 0.37/3 + 0.59/4 0.59/3 + 0.39/4 d4 0.37/3 + 0.10/4 0.24/3 + 0.32/4 0.28/3 + 0.25/4 + 0.14/5 0.35/3 + 0.17/4 d5 0.14/2 + 0.26/3 + 0.21/4 0.24/3 + 0.33/4 0.25/3 + 0.25/4 + 0.13/5 0.39/3 + 0.19/4 d6 0.07/2 + 0.18/3 + 0.17/4 0.15/3 + 0.21/4 0.08/3 + 0.24/4 + 0.07/5 0.30/3 + 0.08/4
Similarly, we can obtain the other elements for all decision makers. We, then, average the rat-ing derived from six decision makers with re-spect to t1, t2, t3and t4are
t1 t2 t3 t4
VðtiÞ 0:455 0:592 0:620 0:524
(9.2) The order of the preferences of the decision makers on four models can be stated as t3 t2 t4 t1.
(9.3) The new-product manager makes the deci-sion according to new-product screening rule of company as
t1 t2 t3 t4
Decision kill go go kill:
6. Discussion
Without any comparison of the proposed method with other well-established methods, the resulting decision may be questionable. In this section, we will compare the new-product ranking procedures, developed by Lin and Chen’s approach (Lin & Chen, 2004), to treat the same problem.
From Eq. (17), the synthetic evaluation of traditional fuzzy approach can be obtained when it is true that t(x) = 1 f(x) for vague sets (i.e., ignore uncertainty) as
Table 4.
Then, the average value of rating all decision makers is given by
In the following, the left-and-right fuzzy ranking method is applied to synthesize the fuzzy ratings
VR¼ supx½u~vjðxÞ ^ umaxðxÞ; ð19Þ
VL¼ supx½u~vjðxÞ ^ uminðxÞ; ð20Þ
where umaxðxÞ ¼ x; 0 6 x 6 1; 0; otherwise; uminðxÞ ¼ 1 x; 0 6 x 6 1; 0; otherwise:
The synthetic evaluation on each target is given by V ¼jVRþ ð1 VLÞj
2 : ð21Þ
The synthetic value on each target is calculated using Eqs. (18)–(21) or geometric graphics described as (Chen & Hwang, 1992)
t1 t2 t3 t4
VðtiÞ 0:47 0:51 0:55 0:49
:
Obviously, the target 3 is the best choice and the ranking order is t3 t2 t4 t1. The solution of Lin and Chen’s
method concludes the same result as our proposed model. FromTable 4and Eq.(21), the rational outcomes can be obtained using either our method or Lin and Chen’s method. Furthermore, our method is capable of revealing the positive and negative preference degree associated with DM’s subject judgements and assisting the DM to make a normal decision based on group consensus. We believed that this method is complimentary to Lin and Chen (2004) as it introduces another dimension to new product development based on group preference.
7. Conclusion
This paper presents a new fuzzy approach to solve NPD screening problems considering the group consensus. 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 during the conceptual design phase. From a numerical illustration for early evaluation of LCD-TV new products screening, it can assist the manager to make the screening decision based on the proposed model. The experimental results ~vj¼ 1 n Xn i¼1 fv1 ij fv2ij . . . fvnij h i ; ð18Þ t1 t2 t3 t4 e VðtiÞ 0:72=2 þ 0:67=3 þ 0:68=4 0:686=3þ 0:634=4 0:78=3 þ 0:7=4 þ 0:78=5 0:74=3 þ 0:55=4 þ 0:6=5 : Table 4
Rating of evaluated targets using fuzzy sets
DMs Targets t1 t2 t3 t4 d1 0.7/2 + 0.8/3 + 0.7/4 0.6/4 + 0.7/4 + 0.7/3 0.7/5 + 0.7/4 + 0.7/4 0.6/3 + 0.9/3 + 0.6/5 d2 0.8/2 + 0.6/4 + 0.8/4 0.7/3 + 0.8/4 + 0.6/3 0.6/4 + 0.8/3 + 0.8/5 0.6/3 + 0.8/3 + 0.5/4 d3 0.6/2 + 0.6/3 + 0.5/4 0.6/4 + 0.8/4 + 0.6/3 0.6/3 + 0.7/4 + 0.8/4 0.7/3 + 0.8/3 + 0.7/4 d4 0.5/3 + 0.5/3 + 0.6/3 0.5/4 + 0.7/4 + 0.6/3 0.9/3 + 0.7/4 + 0.9/5 0.9/3 + 0.8/3 + 0.5/4 d5 0.9/2 + 0.9/3 + 0.6/4 0.6/4 + 0.6/4 + 0.8/3 0.7/4 + 0.8/3 + 0.8/5 0.8/3 + 0.8/3 + 0.5/4 d6 0.6/2 + 0.6/3 + 0.9/4 0.6/4 + 0.7/4 + 0.8/3 0.8/4 + 0.7/4 + 0.7/5 0.6/3 + 0.7/3 + 0.6/3
indicate that our approach not only effectively reveals the uncertainty of decision makers’ subjective judgments, but also is applicable to analyze the consensus degree of group during the NPD screening process.
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