- Email: [email protected]
Decision making with extended fuzzy linguistic computing, with applications to new product development and survey analysis
Expert Systems with Applications 38 (2011) 14052–14059
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Decision making with extended fuzzy linguistic computing, with applications to new product development and survey analysis Shing-Chung Ngan ⇑ Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong, China
a r t i c l e
i n f o
Keywords: Fuzzy sets Fuzzy linguistic computing Decision making New product development Survey analysis
a b s t r a c t Fuzzy set theory, with its ability to capture and process uncertainties and vagueness inherent in subjective human reasoning, has been under continuous development since its introduction in the 1960s. Recently, the 2-tuple fuzzy linguistic computing has been proposed as a methodology to aggregate fuzzy opinions (Herrera & Martinez, 2000a, 2000b), for example, in the evaluation of new product development performance (Wang, 2009) and in customer satisfactory level survey analysis (Lin & Lee, 2009). The 2tuple fuzzy linguistic approach has the advantage of avoiding information loss that can potentially occur when combining opinions of experts. Given the fuzzy ratings of the evaluators, the computation procedure used in both Wang (2009) and Lin and Lee (2009) returned a single crisp value as an output, representing the average judgment of those evaluators. In this article, we take an alternative view that the result of aggregating fuzzy ratings should be fuzzy itself, and therefore we further develop the 2-tuple fuzzy linguistic methodology so that its output is a fuzzy number describing the aggregation of opinions. We demonstrate the utility of the extended fuzzy linguistic computing methodology by applying it to two data sets: (i) the evaluation of a new product idea in a Taiwanese electronics manufacturing firm and (ii) the evaluation of the investment benefit of a proposed facility site. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The strength of fuzzy set theory, which was first proposed by Zadeh (1965), lies in its ability to represent and process both uncertainties in measurements and vagueness in concepts expressed in the natural language. Unlike the classical set theory in which an element must either belong or not belong to a set of interest, an object in the universe of discourse can instead be a member of a fuzzy set to some degree. This added flexibility allows mathematical representation of non-precise human concepts and enables a proliferation of fuzzy set theory applications in a broad range of industrial engineering and electronics areas today. The pervasiveness of applications of fuzzy set theory in industrial engineering research can in fact be found in the recent issues of Expert Systems with Applications (Chuang, Lin, Kung, & Lin, 2010; Dursun & Karsak, 2010; Galetakis & Vasiliou, 2010; Lin, 2010; Pan, 2010; Sen & Baraçlı, 2010): for example, it was used in aggregating evaluators’ opinions in new product development (Wang, 2009) and in survey analysis (Lin & Lee, 2009). As the capability to introduce marketable new products is essential in ⇑ Address: Rm Y6624, Academic Building, 83 Tat Chee Avenue, City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong. Tel.: +852 3442 8400; fax: +852 3442 0172. E-mail address: [email protected] 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.04.213
maintaining and advancing the competitiveness of a firm relative to its rivals, accurate decision making in the domain of new product development becomes increasingly important, and thus the evaluations of new product ideas nowadays are usually carried out by a committee of experts. As pointed out in Hwang and Yoon (1981) and Wang (2009), a great deal of fuzziness and inhomogeneity can occur across the experts’ subjective perceptions and cognitions, and as a results, the subsequent information integration could lead to evaluation results being incompatible with the experts’ expectations. Hence, Wang (2009) proposed using a 2tuple fuzzy linguistic approach for new product development evaluation that can avoid information loss inherent in other fuzzy approaches (Herrera-Viedma, Herrera, Martinez, Herrera, & Lopez, 2004). Similarly, Lin and Lee (2009) applied fuzzy linguistic computing to analyze customer satisfactory level survey data. Vague concepts like strongly unsatisfactory, unsatisfactory, average, satisfactory and strongly satisfactory were represented by fuzzy linguistic terms. In the methodologies of both Lin and Lee (2009) and Wang (2009), the evaluators were required not only to rate the sufficiency of a product idea/service under consideration with respect to various pre-determined criteria and sub-criteria, but also to rate the importance of the criteria and sub-criteria themselves, in determining the overall viability of a new product idea or satisfaction level of a service. This double rating helped capture more fully the experiential cognition of the evaluators.
S.-C. Ngan / Expert Systems with Applications 38 (2011) 14052–14059
The computation procedure used in the above mentioned papers took the fuzzy ratings given by the individual evaluators as inputs and returned a single crisp value describing the average judgment of the evaluators. In this article, we propose to expand the methodology by outputting not a crisp value, but a fuzzy number to represent the aggregation of opinions. This makes logical sense in that one would expect the result of aggregating fuzzy ratings to be fuzzy itself. Furthermore, the resulting fuzzy number can be viewed as a fuzzy set theory analog of the statistical interval used in classical statistics. For example, in the case of customer satisfactory level analysis, a service under consideration might get an overall average crisp rating of ‘‘satisfactory.’’ The fuzzy number that will be obtained with the proposed methodology represents how confident we are that the service is indeed ‘‘satisfactory,’’ reflected by how much of the support of the associated membership function is contained within the support of the linguistic term ‘‘satisfactory.’’ A procedure to formulate and compute statistical confidence intervals for fuzzy data has actually been proposed recently in Wu (2009). It involved the use of a-cut (denoted as h-level set in that paper) and the extension principles in Zadeh (1975a, 1975b, 1975c), leading to an interesting but mathematically non-trivial optimization problem. In this article, we propose to use an alternative formulation based on sampling distribution, giving an efficient and simple procedure for calculating the fuzzy number. The article is organized as follows: In Section 2, the basic format of the survey questionnaires used in eliciting evaluators’ opinions and the 2-tuple fuzzy linguistic approach for aggregating the opinions used in Lin and Lee (2009) and Wang (2009) will be reviewed. After that, we will describe a computationally efficient framework for calculating a fuzzy number representing the aggregation of opinions. In Section 3, we will apply the proposed methodology on evaluating new product development ideas and on survey analysis. A brief summary will be given in Section 4.
2. Background and methodology 2.1. Format of the survey questionnaires The questionnaires used in Lin and Lee (2009) and Wang (2009) have the following common format: A typical questionnaire, for instance to evaluate a new product idea, consists of several main criteria, such as the research and development capability of the firm in producing a prototype, the organizational capacity in manufacturing the proposed product, and the marketability of the product. Associated with each of these main criteria are several sub-criteria. Under the marketability heading, these sub-criteria could be (i) the present market share of related products, (ii) the potential number of major customers, and so on. As discussed in the Introduction, an individual decision maker will rate the product idea with respect to each sub-criterion, for example, giving an ‘‘excellent’’ rating for the product idea regarding item (i) and an ‘‘average’’ rating regarding item (ii), the ratings being chosen from a list of linguistic terms very poor, poor, average, good, and excellent. This yields a first set of ratings. In addition, each decision maker will rate the criteria and each of the associated sub-criteria themselves in order to express his/her opinion about the relative importance of the criteria and sub-criteria in determining the overall viability of a product idea. For example, he/she could give an ‘‘important’’ rating for item (i) and a ‘‘very important’’ rating for item (ii), selected from a list of linguistic terms negligible, not important, average, important, and very important. This produces a second set of ratings. In a later sub-section, we will review how the two sets of ratings are combined to form the decision makers’ aggregate opinion on the overall prospect of success of a new product idea.
14053
2.2. Fuzzy linguistic computing approach Details of the fuzzy linguistic formulation and representation of imprecise data can be found in Herrera and Martinez (2000a, 2000b), Zimmermann (1991) and the references therein. We summarize the core ideas here for completeness. A triangular fuzzy number L with parameters (p, q, r) can be described by a membership function
8 > < ðx pÞ=ðq pÞ when p 6 x 6 q fL ðxÞ ¼ ðx rÞ=ðq rÞ when q 6 x 6 r > : 0 otherwise
ð1Þ
where x is a real number. Such a fuzzy number is graphically depicted in Fig. 1. Thus, if x represents the height of an individual, and L represents a particular person’s (denoted as person A) concept (termed a fuzzy concept) of an average-height individual, then a given individual with an associated height will be considered by person A as having ‘‘average height’’ to a degree determined by the membership function. For example, an individual with height x = q is definitely a member of the concept ‘‘average-height individual’’ according to A (for fL(q) = 1.0), while an individual taller than r is definitely not (for fL(x) = 0 for any x P r). For the scenarios considered in this article, a fuzzy concept can be described by a fuzzy number. Furthermore, the membership function of a fuzzy number does not need to be constrained in the triangular form – other shapes such as the parabola or trapezoid could also be used. However, the triangular form can capture fuzziness in human concepts with good approximation and is employed extensively in this article. A fuzzy linguistic variable is a variable whose domain is a collection of pre-specified fuzzy concepts. For example, a fuzzy linguistic variable can be used to represent the rating of a particular object given by an evaluator in a survey questionnaire. In this case, the domain of the variable consists of the fuzzy concepts (also denoted as the fuzzy linguistic terms) very poor, poor, average, good and excellent. These fuzzy linguistic terms correspond to the five membership functions illustrated in Fig. 2a. Referring to the figure, to represent the crisp value x = 0.4 within the fuzzy linguistic framework, a 2-tuple (average, 0.1) is used. That is, x = 0.4 is expressed as the sum of the ‘‘q’’ value of the fuzzy linguistic term average (qaverage = 0.5) and the offset of x from the ‘‘q’’ position (0.1). Therefore, we can translate between a crisp value and its 2-tuple fuzzy linguistic form using the formulas
DðxÞ ¼ ðSi ; aÞ
ð2Þ
and
Fig. 1. A triangular fuzzy number L with parameters (p, q, r) representing person A’s fuzzy concept of an ‘‘average-height individual.’’ When x = q, the value of the membership function is 1. When x = (p + q)/2, the value is 0.5.
14054
S.-C. Ngan / Expert Systems with Applications 38 (2011) 14052–14059
ij Þ ¼ D 1 ½D1 ðW ij1 ; 0Þ þ þ D1 ðW ijK ; 0Þ ðW ij ; b K
ð5Þ
where Wijk e {W0, . . . , W4} is the importance rating of the subcriterion Di,j given by the kth expert. Furthermore, let us denote Wik e {W0, . . . , W4} as the importance rating of the criterion Ci given by the kth expert. Then, the aggregate importance rating of Ci is determined by the formula
ðW i ; bi Þ ¼ D
1 1 ½D ðW i1 ; 0Þ þ þ D1 ðW iK ; 0Þ K
ð6Þ
Subsequently, the aggregate opinion of the product idea with respect to criterion Ci across the K experts is
(PN i ðSi ; ai Þ ¼ D
j¼1 D
ij Þ ij Þ D1 ðW ij ; b ðSij ; a PNi 1 j¼1 D ðW ij ; bij Þ
1
) ð7Þ
and the overall aggregate opinion of the product idea across the K experts and across all criteria is calculated using
Þ ¼ D ðS; a
Fig. 2. (a) depicts the membership functions corresponding to the linguistic terms very poor, poor, average, good and excellent. For example, ‘‘average’’ is represented by the membership function with paverage = 0.25, qaverage = 0.5 and raverage = 0.75, and (b) depicts the five membership functions corresponding to the linguistic terms negligible, not important, average, important and very important. For example, the p, q and r values for W2 are 0.25, 0.5 and 0.75, respectively.
D1 ðSi ; aÞ ¼ x ¼
i þa g
ð3Þ
where i = round(x g), a = x i/g, g = N 1 and N is the total number of linguistic terms in the domain. 2.3. Aggregating the evaluators’ opinions Let us consider the scenario in which a total of K experts have rated a new product idea based on M criteria {C1, . . . , CM} and for each criterion Ci, Ni sub-criteria fDi;1 ; . . . ; Di;Ni g. As discussed above, let us assume that the linguistic variable describing the rating of the product idea with respect to a sub-criterion Di,j is endowed with the domain {very poor (S0), poor (S1), average (S2), good (S3) and excellent (S4)}, and meanwhile, the linguistic variable representing the importance rating of a criterion/sub-criterion is endowed with the domain {negligible (W0), not important (W1), average (W2), important (W3), and very important (W4)} (see Fig. 2b). A fuzzy linguistic approach for aggregating the opinions of the experts on the product idea with respect to the sub-criterion Di,j is achieved through the computation
ij Þ ¼ D ðSij ; a
1 1 ½D ðSij1 ; 0Þ þ þ D1 ðSijK ; 0Þ K
ð4Þ
where Sijk e {S0, . . . , S4} is the product idea rating with respect to the sub-criterion Di,j given by the kth expert. The aggregate importance rating of the sub-criterion Di,j is similarly calculated:
(P M
i Þ i Þ D1 ðW i ; b ðSi ; a PM 1 i¼1 D ðW i ; bi Þ
i¼1 D
1
) ð8Þ
The computation procedure delineated in Eqs. (4)–(8) is the methodology taken in Wang (2009). The approach developed in Lin and Lee (2009) is largely similar, with the exception that an evaluator can select more than one linguistic term when he/she gives a rating, such as choosing ‘‘average’’ with a weight of 0.8 and ‘‘good’’ with a weight of 0.2. The choosing of more than one linguistic term represents the evaluator’s uncertainty in his/her rating and the weights reflect that the evaluator leans more towards ‘‘average’’ than ‘‘good’’ when a product idea/service under consideration is rated against some sub-criterion. This additional flexibility can be accommodated by modifying Eq. (4) appropriately. The approaches used in both papers return a single crisp value (from the result of Eq. (8)) as a final output. 2.4. An alternative formulation of fuzzy linguistic computing based on sampling distribution As discussed in the Introduction, given the fuzzy ratings by the evaluators, our goal is to output a fuzzy number instead of a crisp number characterizing the aggregation of opinions. The fuzzy number allows us to capture the degree of confidence that the aggregate opinion indeed belongs to a particular overall rating. Instead of calculating the fuzzy number based on the a-cut and extension principles, which can lead to mathematically non-trivial optimization problems, the present formulation is based on sampling distribution. For concreteness, let us reconsider the ‘‘average-height individual’’ fuzzy concept delineated in Fig. 1. We will assume a probabilistic interpretation of the fuzzy concept as follows (see Bilgiç & Turksen, 1999 for a survey of various interpretations): Suppose that in an occasion (denoted as occasion 1), a friend B of person A gathers one hundred individuals, each with an actual height of q. Without informing A of their actual heights, B presents those one hundred individuals to A and asks A to visually judge them one by one. Person A, following his subjective concept of ‘‘average-height individual’’, will end up judging all one hundred of these individuals to be of ‘‘average height,’’ since the value of the membership function of his concept of ‘‘average height’’ at x = q is 1. On the other hand, if the above experiment is repeated (denoted as occasion 2) with the difference that B instead gathers one hundred individuals each with an actual height of (p + q)/2. Then A will classify only about half of them as having ‘‘average height’’ according to his subjective concept, because the value of the membership function at x = (p + q)/2 is 0.5. In general, the value of the membership function at the height of
S.-C. Ngan / Expert Systems with Applications 38 (2011) 14052–14059
14055
interest gives the probability that an individual with that height will be classified as being ‘‘average height.’’ Following conventional conditional probability, we have the relation
Pð\average height"jactual heightÞ Pðactual heightÞ ¼ Pðactual heightj\average height"Þ Pð\average height"Þ
ð9Þ
Given an individual with a particular actual height (known by B but not by A), the term P(‘‘average height’’|actual height) refers to the probability that A will classify that individual as having ‘‘average height.’’ Based on the probabilistic interpretation of fuzzy concepts discussed in the previous paragraph, this term is fully specified by the membership function of the fuzzy concept ‘‘average-height individual.’’ P(actual height|‘‘average height’’) gives the probability of an individual possessing a particular actual height given that the individual has been classified by A as having ‘‘average height.’’ On the other hand, P(actual height) represents the prior probability of encountering an individual with a particular actual height in our universe of discourse. Note that the prior probability can be further specified if one possesses sufficient knowledge about the actual height distribution of individuals in the universe of discourse. Otherwise, following conventional practices in Bayesian statistics, we set it as a non-informative prior. For example, we let P(actual height) = C, where C is a constant, for a broad enough range of actual heights. Combining with Eq. (9), this then yields
Pðactual heightj\average height"Þ /
Pð\average height"jactual heightÞ Pð\average height"Þ
ð10Þ
Let us consider yet another occasion (denoted as occasion 3), in which A has just finished judging a number of individuals and concluded that one hundred of them have ‘‘average height.’’ Then, we can infer that the actual height distribution of the one hundred individuals follows P(actual height|‘‘average height’’). Furthermore, from Eq. (10), we obtain for this occasion
Pðactual heightj\average height"Þ / Pð\average height"jactual heightÞ
ð11Þ
As discussed above, the right hand side of Eq. (11) corresponds to the membership function of the fuzzy concept ‘‘average-height individual.’’ Therefore, by properly normalizing the membership function (such that the area under the curve equals 1), the actual height distribution P(actual height|‘‘average height’’) can be determined:
Pðxj\average height"Þ ¼ ^f L ðxÞ
ð12Þ
with
8 > < Fðx pÞ=ðq pÞ when p 6 x 6 q ^f L ðxÞ ¼ Fðx rÞ=ðq rÞ when q 6 x 6 r > : 0 otherwise
and F ¼
2 ðr pÞ
In the preceding example, the concept ‘‘average-height individual’’ is fuzzy, while the crisp values ‘‘actual heights’’ are objective measurements. In general, the fuzzy linguistic framework can also be applied to describe fuzzy concepts together with subjective crisp values. For example, consider occasion 4 in which person A is to give his subjective rating (on the scale of 0–1) for the tastes of sandwiches. Instead of committing to a crisp value to represent his subjective rating on a sandwich under consideration, A may use a linguistic term ‘‘average’’ instead (see Fig. 3). In other words, the linguistic term ‘‘average’’ enables A to express his opinion while
Fig. 3. Three linguistic terms taste bad, average and taste good representing person A’s fuzzy concepts of taste of sandwiches.
relieving him from the responsibility of needing to specify a particular crisp value to describe his subjective rating. In analogy with Eq. (12), we have
Pðxj\average"Þ ¼ ^f average ðxÞ
ð13Þ
where
8 when 0 6 x 6 0:5 > < 4x ^f average ðxÞ ¼ 4ðx 1Þ when 0:5 6 x 6 1 > : 0 otherwise is the normalized form of the membership function for ‘‘average.’’ In the present context, the term P(x|‘‘average’’) can be viewed as describing the range and distribution of the crisp values person A would have given if he were pressed to give his subjective crisp ratings for say the N sandwiches that he considers average taste. 2.5. Application of the alternative formulation to aggregating opinions Recall the basic questionnaire setup consisting of K experts rating a new product idea/service based on M criteria {C1, . . . , CM} and Ni sub-criteria fDi;1 ; . . . ; Di;Ni g for each criterion Ci, with the linguistic variables Sijk representing the fuzzy ratings of the product idea/ service and Wik and Wijk representing the fuzzy ratings for the relative importance of the criteria and sub-criteria. We aggregate these ratings with a sampling approach: One run of sampling and aggregation consists of first randomly generating a crisp value from each linguistic variable based on the fuzzy ratings given by the experts. For instance, suppose that the kth expert rates the product idea under consideration with respect to sub-criterion Di,j as Sijk = S3 based on his/her subjective opinion, i.e. corresponding to the linguistic term ‘‘good’’ in Fig. 2a. In analogy with Eq. (13), a crisp value x will be generated based on the distribution
pðxjS3 Þ ¼ ^f S3 ðxÞ
ð14Þ
with
8 > < 16ðx 0:5Þ when 0:5 6 x 6 0:75 ^f 3 ðxÞ ¼ 16ð1 xÞ when 0:75 6 x 6 1 S > : 0 otherwise where ^f S3 ðxÞ is the normalized version of the membership function fS3 ðxÞ. After a crisp value has been generated for each linguistic variable, these values are numerically combined stage-wise:
b S ik ¼
PNi b c j¼1 S ijk W ijk PN i c j¼1 W ijk
ð15Þ
14056
b Sk ¼
S.-C. Ngan / Expert Systems with Applications 38 (2011) 14052–14059
PM b c i¼1 S ik W ik PM c i¼1 W ik
ð16Þ
and K 1X b b S¼ Sk K k¼1
ð17Þ
c ijk and W c ik represent the randomly sampled crisp where b S ijk ; W values for the linguistic variables Sijk, Wijk and Wik. Note that if historical data or background knowledge were available, specific c ijk and W c ik could conditional probabilistic relation among b S ijk ; W be formulated and estimated. Lacking these data and knowledge, we take the simplest assumption of conditional independence among these variables, and generate the samples independently. Now, Eq. (15) computes the weighted rating of a product idea/ service with respect to criterion Ci according to the opinion of the kth expert. Eq. (16) determines the overall weighted rating of the product idea/service according to the kth expert across
all criteria. Finally, Eq. (17) averages these overall weighted rating across the K experts, yielding a single sampling of the aggregate rating b S. The above-mentioned sampling and aggregation run is repeated N times (e.g. say N = 10,000). In analogy with occasions 3 and 4 in the previous example, the N sampling and aggregation runs can be viewed as representing the evaluations of N highly similar product ideas/services. (These product ideas/services are highly similar in the sense that they receive the same fuzzy ratings from the K experts, much like in occasion 3, in which the one hundred individuals were highly similar in that they were all identified as of ‘‘average height’’, or in occasion 4, in which the N sandwiches were highly similar in that they were all identified as of ‘‘average’’ taste.) In occasion 3, according to Eq. (12), the actual height distribution of the one hundred individuals can be used to recover the membership function for the fuzzy concept ‘‘average-height individual.’’ In the present context, the resulting collection of the b S values from the N runs gives a distribution, which can then be properly scaled to obtain a fuzzy number
Table 1 The fuzzy ratings given by the four decision makers D1, D2, D3 and D4 for a new product idea under consideration with respect to various sub-criteria. There are altogether four criteria. Under each criterion are 3–5 sub-criteria. Each decision maker specifies his/her ratings by choosing from the five linguistic terms {S0, S1, S2, S3, S4}. For example, D1 selects S3 as his/her rating for the product idea with respect to the sub-criterion ‘‘quality and speed to market,’’ leading to the ‘‘3’’ entry in the table. Criteria
Sub-criteria
D1
D2
D3
D4
Market
Quality and speed to market Number of major customer Market share rate Widening customer choice and expectation Delivery
3 4 3 4 2
4 2 4 3 2
2 4 3 2 4
4 3 3 4 4
Innovation
Flexibility Number of new products or processes Number of patent Fee of research/fee of total
4 4 2 3
2 3 3 4
4 2 4 4
2 3 3 2
Organization
Competitive priorities of responsiveness Trademark Information system Index of productivity
4 3 4 2
4 4 2 3
2 3 4 4
4 3 3 4
Employee
Capability of employees Output merit of employees Skill training of employees
4 2 3
3 4 4
3 2 4
4 4 3
Table 2 The fuzzy ratings given by the four decision makers D1, D2, D3 and D4 to express their views regarding the relative importance of the criteria and sub-criteria in determining the overall viability of a new product idea. Each decision maker specifies his/her ratings by choosing from the five linguistic terms {W0, W1, W2, W3, W4}. For example, D1 selects W4 as his/her importance rating for the sub-criterion ‘‘quality and speed to market,’’ leading to the ‘‘4’’ entry in the table. Criteria
Sub-criteria
D1
D2
D3
D4
Quality and speed to market Number of major customer Market share rate Widening customer choice and expectation Delivery
4 4 2 3 3 4
4 4 4 3 3 4
4 4 2 2 4 4
3 3 2 2 3 4
Flexibility Number of new products or processes Number of patent Fee of research/fee of total
3 3 3 2 4
3 4 2 3 4
4 4 4 2 4
3 4 4 3 3
Competitive priorities of responsiveness Trademark Information system Index of productivity
4 4 2 4 4
3 3 3 4 2
4 3 2 3 4
3 4 2 4 3
Capability of employees Output merit of employees Skill training of employees
3 3 3 2
2 4 3 4
3 4 3 2
3 3 3 3
Market
Innovation
Organization
Employee
14057
S.-C. Ngan / Expert Systems with Applications 38 (2011) 14052–14059
Fig. 4. Fuzzy number V representing the aggregate opinion of the four decision makers on the overall viability of a new product idea under consideration.
representing the aggregate opinion of the experts regarding the product idea/service under consideration.
are reproduced in Tables 1 and 2. The five linguistic terms {S0, S1, S2, S3, S4} are available for the product ratings, and the five linguistic terms {W0, W1, W2, W3, W4} are available for the importance ratings. By applying the sampling approach to combine ratings given by the decision makers, we obtain a fuzzy number V as displayed in Fig. 4, representing the aggregate opinion of these experts on the overall viability of the new product idea. The membership function of V peaks at x = 0.8 (agreeing closely with the average crisp value of 0.8056 reported in Wang (2009), and resembles a bell-shaped curve, but with a finite support. For any x e (0.77, 0.83), the corresponding value of the membership function is larger than 0.05. (The range (0.77, 0.83) is denoted as the 0.05+-cut of the fuzzy number V.) Referring to Fig. 2a, we note that the membership function of S3 (the ‘‘good’’ rating) intersects with that of S2 (the ‘‘average’’ rating) at x = 0.625, and intersects with that of S4 (the ‘‘excellent’’ rating) at x = 0.875. The range (0.625, 0.875) is incidentally the 0.5+-cut of the linguistic term ‘‘good,’’ since the value of the membership function of S3 corresponding to any x e (0.625, 0.875) is larger than 0.5. We can conclude that the aggregate opinion of the new product idea as captured by the fuzzy number V is well described by an overall ‘‘good’’ rating, in the sense that the 0.05+-cut of V (a large majority of the support of V) is within the 0.5+-cut of S3.
3. Results and discussion 3.2. Application of the sampling approach to survey analysis 3.1. Application of the sampling approach to new product development In Wang (2009), four decision makers from a Taiwanese electronic manufacturer evaluated a new product idea based on a number of criteria and sub-criteria (see Table 1). Their fuzzy ratings of the product idea with respect to the various sub-criteria and of the importance of the criteria and sub-criteria themselves
In Lin and Lee (2009), two experts evaluated the investment benefit of a proposed facility site based on five criteria and ten sub-criteria (see Tables 3a and 3b). Their fuzzy ratings of the facility site as well as of the importance of the criteria and sub-criteria are detailed in Tables 3a, 3b and 4. The seven linguistic terms {S0, . . . , S6} and the eleven linguistic terms {W0, . . . , W10} are
Table 3a The fuzzy ratings given by an expert D1 for the investment benefit of a proposed facility site with respect to various sub-criteria. There are altogether five criteria. Under each criterion are 2 sub-criteria. D1 specifies a rating by choosing a weight for each of the seven linguistic terms {S0, S1, S2, S3, S4, S5, S6}. For example, D1 puts a weight of 0.17 for S0, and a weight of 0.83 for S1 (total weight summed up to 1.0) as his/her rating of the proposed site with respect to the sub-criterion ‘‘salary level.’’ This reflects both D1’s uncertainty and that D1 leans more towards S1 than S0 in his/her rating. Criteria
Sub-criteria
S0
S1
S2
S3
S4
S5
S6
Labor
Salary level Manpower level
0.17 0
0.83 0.80
0 0.20
0 0
0 0
0 0
0 0
Geography
Usage condition level of factory place Nearing market level of delivery system
0 0
0.60 0.89
0.40 0.11
0 0
0 0
0 0
0 0
Economic
Index of industry production growth Index of industry modern times
0.15 0
0.85 0.75
0 0.25
0 0
0 0
0 0
0 0
Reward
Reward obtain level Institution perform level
0.25 0
0.75 0.80
0 0.20
0 0
0 0
0 0
0 0
Politics
Regulatory restrictions level Investment subsidy level
0 0
0.75 0.60
0.25 0.40
0 0
0 0
0 0
0 0
Table 3b Analogous to Table 3a, this table shows the fuzzy ratings given by an expert D2 for the investment benefit of a proposed facility site with respect to various sub-criteria. Criteria
Sub-criteria
S0
S1
S2
S3
S4
S5
S6
Labor
Salary level Manpower level
0 0.15
0.80 0.85
0.20 0
0 0
0 0
0 0
0 0
Geography
Usage condition level of factory place Nearing market level of delivery system
0 0.20
0.70 0.80
0.30 0
0 0
0 0
0 0
0 0
Economic
Index of industry production growth Index of industry modern times
0.10 0
0.90 0.70
0 0.20
0 0.10
0 0
0 0
0 0
Reward
Reward obtain level Institution perform level
0.10 0.20
0.20 0.80
0.70 0
0 0
0 0
0 0
0 0
Politics
Regulatory restrictions level Investment subsidy level
0 0
0.60 0.60
0.25 0.30
0.15 0.10
0 0
0 0
0 0
14058
S.-C. Ngan / Expert Systems with Applications 38 (2011) 14052–14059
Table 4 The fuzzy ratings given by the two experts D1 and D2 to express their views regarding the relative importance of the criteria and sub-criteria used in determining the overall investment benefit of a proposed facility site. Each expert specifies his/her ratings by choosing from the eleven linguistic terms {W0, W1, . . . , W10}. For example, D1 selects W5 as his/her importance rating for the sub-criterion ‘‘salary level,’’ leading to the ‘‘5’’ entry in the table. Criteria
Sub-criteria
D1
D2
Salary level Manpower level
1 5 3
3 3 4
Usage condition level of factory place Nearing market level of delivery system
3 2 5
2 8 4
Index of industry production growth Index of industry modern times
4 3 7
3 4 6
Reward obtain level Institution perform level
4 6 8
2 6 3
Regulatory restrictions level Investment subsidy level
6 4 2
3 7 3
Labor
Geography
Economic
Reward
Politics
Fig. 5. Fuzzy number I representing the aggregate opinion of the two experts on the investment benefit of a proposed facility site.
available for the site ratings and the importance ratings respectively. Application of the sampling approach to combine opinions yields the fuzzy number I delineated in Fig. 5, representing the aggregate view of the two experts on the investment benefit of the proposed site. The membership function peaks at x = 0.2, in close agreement with the average crisp value of 0.204 reported in Lin and Lee (2009). For any x e (0.14, 0.27), the corresponding value of the membership function is larger than 0.05. Thus, (0.14, 0.27) is the 0.05+-cut of the fuzzy number I. Referring to Fig. 6, the 0.5+-cut of S1 is (0.0833, 0.25), while that of S2 is (0.25, 0.417). Thus, the aggregate opinion of the investment benefit of the proposed site can be viewed as straddling across the S1 and S2 ratings, leaning heavily towards S1, in the sense that the 0.05+cut of I is mostly contained within the 0.5+-cut of S1, but with overlap with the 0.5+-cut of S2. 4. Conclusion Fuzzy set theory, with its ability to represent and process uncertainties and vagueness inherent in subjective human reasoning, has been applied to a wide variety of industrial domains and under continuous development since its introduction in the 1960s. Recently, the 2-tuple fuzzy linguistic computing (Herrera & Martinez, 2000a, 2000b) has been proposed as a methodology to aggregate fuzzy opinions, for example, in the evaluation of new product development performance (Wang, 2009) and in customer satisfactory level survey analysis (Lin & Lee, 2009). The 2-tuple fuzzy linguistic approach has the advantage of avoiding information loss that can potentially occur when combining opinions of experts, due to the fuzziness and inhomogeneity of the experts’ subjective cognitions (Herrera-Viedma et al., 2004). Given the fuzzy ratings of the evaluators, the computation procedure used in both Wang (2009) and Lin and Lee (2009) returned a single crisp value as an output, representing the average judgment of those evaluators. In this article, we took an alternative view that the result of aggregating fuzzy ratings should be fuzzy itself, and therefore we further developed the 2-tuple fuzzy linguistic methodology so that its output is a fuzzy number describing the aggregation of opinions. Instead of relying on the a-cut and the extension principles, which could lead to interesting but mathematically non-trivial optimization problems as in Wu (2009), our approach was based on sampling distribution, giving an efficient and simple procedure. We demonstrated the utility of the extended fuzzy linguistic computing methodology on aggregating fuzzy ratings by applying it to two data sets: (i) the evaluation of a new product idea in a Taiwanese electronics manufacturing firm and (ii) the evaluation of the investment benefit of a proposed facility site. For each of these two studies, the peak location of the membership function representing the aggregate opinion of the evaluators agreed closely with the average crisp value reported in the respective publication. Moreover, the fuzzy numbers obtained from the present methodology gave additional information beyond the crisp values about the aggregate opinions. In the first study, we concluded that the aggregate opinion of the new product idea was well described by an overall S3 rating (i.e. a ‘‘good’’ rating), while in the second study, the aggregate opinion of the investment benefit of the proposed site actually straddled across the S1 and S2 ratings. References
Fig. 6. The seven linguistic terms S0, S1, S2, S3, S4, S5 and S6 available for the experts to rate the proposed facility site with respect to various sub-criteria. For example, the p, q and r values of S2 are 0.167, 0.333 and 0.5, respectively. Note that S0 and S1 intersect at x = 0.0833, S1 and S2 at x = 0.25, S2 and S3 at x = 0.417.
Bilgiç, T., & Turksen, I. B. (1999). Measurement of membership functions: Theoretical and empirical work. In Handbook of fuzzy sets and systems. In D. Dubois & H. Prade (Eds.). Fundamentals of fuzzy sets (Vol. 1, pp. 195–232). Kluwer. Chuang, T.-N., Lin, C.-T., Kung, J.-Y., & Lin, M.-D. (2010). Planning the route of container ships: A fuzzy genetic approach. Expert Systems with Applications, 37, 2948–2956.
S.-C. Ngan / Expert Systems with Applications 38 (2011) 14052–14059 Dursun, M., & Karsak, E. E. (2010). A fuzzy MCDM approach for personnel selection. Expert Systems with Applications, 37, 4324–4330. Galetakis, M., & Vasiliou, A. (2010). Selective mining of multiple-layer lignite deposits. A fuzzy approach. Expert Systems with Applications, 37, 4266–4275. Herrera-Viedma, E., Herrera, F., Martinez, L., Herrera, J. C., & Lopez, A. G. (2004). Incorporating filtering techniques in a fuzzy linguistic multi-agent model for information gathering on the web. Fuzzy Sets and Systems, 148, 61–83. Herrera, F., & Martinez, L. (2000a). A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Transactions on Fuzzy Systems, 8, 746–752. Herrera, F., & Martinez, L. (2000b). An approach for combining linguistic and numerical information based on 2-tuple fuzzy representation model in decision making. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 8, 539–562. Hwang, C. L., & Yoon, K. (1981). Multiple attributes decision making methods and applications. Springer-Verlag. Lin, L.-Z. (2010). A perceptual measure of trade shows using fuzzy quality deployment development. Expert Systems with Applications, 37, 3921–3933. Lin, L., & Lee, H.-M. (2009). Fuzzy assessment method on sampling survey analysis. Expert Systems with Applications, 36, 5955–5961.
14059
Pan, W.-T. (2010). Combining fuzzy Sammon mapping and fuzzy clustering approach to perform clustering effect analysis: Take the banking service satisfaction as an example. Expert Systems with Applications, 37, 4139–4145. Sen, C. G., & Baraçlı, H. (2010). Fuzzy quality function deployment based methodology for acquiring enterprise software selection requirements. Expert Systems with Applications, 37, 3415–3426. Wang, W.-P. (2009). Evaluating new product development performance by fuzzy linguistic computing. Expert Systems with Applications, 36, 9759–9766. Wu, H.-C. (2009). Statistical confidence intervals for fuzzy data. Expert Systems with Applications, 36, 2670–2676. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353. Zadeh, L. A. (1975a). The concept of linguistic variable and its application to approximate reasoning I. Information Sciences, 8, 199–249. Zadeh, L. A. (1975b). The concept of linguistic variable and its application to approximate reasoning II. Information Sciences, 8, 301–357. Zadeh, L. A. (1975c). The concept of linguistic variable and its application to approximate reasoning III. Information Sciences, 9, 43–80. Zimmermann, H. J. (1991). Fuzzy set theory and its applications (2nd ed.). Kluwer Academic Publishers.
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Decision making with extended fuzzy linguistic computing, with applications to new product development and survey analysis Shing-Chung Ngan ⇑ Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong, China
a r t i c l e
i n f o
Keywords: Fuzzy sets Fuzzy linguistic computing Decision making New product development Survey analysis
a b s t r a c t Fuzzy set theory, with its ability to capture and process uncertainties and vagueness inherent in subjective human reasoning, has been under continuous development since its introduction in the 1960s. Recently, the 2-tuple fuzzy linguistic computing has been proposed as a methodology to aggregate fuzzy opinions (Herrera & Martinez, 2000a, 2000b), for example, in the evaluation of new product development performance (Wang, 2009) and in customer satisfactory level survey analysis (Lin & Lee, 2009). The 2tuple fuzzy linguistic approach has the advantage of avoiding information loss that can potentially occur when combining opinions of experts. Given the fuzzy ratings of the evaluators, the computation procedure used in both Wang (2009) and Lin and Lee (2009) returned a single crisp value as an output, representing the average judgment of those evaluators. In this article, we take an alternative view that the result of aggregating fuzzy ratings should be fuzzy itself, and therefore we further develop the 2-tuple fuzzy linguistic methodology so that its output is a fuzzy number describing the aggregation of opinions. We demonstrate the utility of the extended fuzzy linguistic computing methodology by applying it to two data sets: (i) the evaluation of a new product idea in a Taiwanese electronics manufacturing firm and (ii) the evaluation of the investment benefit of a proposed facility site. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The strength of fuzzy set theory, which was first proposed by Zadeh (1965), lies in its ability to represent and process both uncertainties in measurements and vagueness in concepts expressed in the natural language. Unlike the classical set theory in which an element must either belong or not belong to a set of interest, an object in the universe of discourse can instead be a member of a fuzzy set to some degree. This added flexibility allows mathematical representation of non-precise human concepts and enables a proliferation of fuzzy set theory applications in a broad range of industrial engineering and electronics areas today. The pervasiveness of applications of fuzzy set theory in industrial engineering research can in fact be found in the recent issues of Expert Systems with Applications (Chuang, Lin, Kung, & Lin, 2010; Dursun & Karsak, 2010; Galetakis & Vasiliou, 2010; Lin, 2010; Pan, 2010; Sen & Baraçlı, 2010): for example, it was used in aggregating evaluators’ opinions in new product development (Wang, 2009) and in survey analysis (Lin & Lee, 2009). As the capability to introduce marketable new products is essential in ⇑ Address: Rm Y6624, Academic Building, 83 Tat Chee Avenue, City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong. Tel.: +852 3442 8400; fax: +852 3442 0172. E-mail address: [email protected] 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.04.213
maintaining and advancing the competitiveness of a firm relative to its rivals, accurate decision making in the domain of new product development becomes increasingly important, and thus the evaluations of new product ideas nowadays are usually carried out by a committee of experts. As pointed out in Hwang and Yoon (1981) and Wang (2009), a great deal of fuzziness and inhomogeneity can occur across the experts’ subjective perceptions and cognitions, and as a results, the subsequent information integration could lead to evaluation results being incompatible with the experts’ expectations. Hence, Wang (2009) proposed using a 2tuple fuzzy linguistic approach for new product development evaluation that can avoid information loss inherent in other fuzzy approaches (Herrera-Viedma, Herrera, Martinez, Herrera, & Lopez, 2004). Similarly, Lin and Lee (2009) applied fuzzy linguistic computing to analyze customer satisfactory level survey data. Vague concepts like strongly unsatisfactory, unsatisfactory, average, satisfactory and strongly satisfactory were represented by fuzzy linguistic terms. In the methodologies of both Lin and Lee (2009) and Wang (2009), the evaluators were required not only to rate the sufficiency of a product idea/service under consideration with respect to various pre-determined criteria and sub-criteria, but also to rate the importance of the criteria and sub-criteria themselves, in determining the overall viability of a new product idea or satisfaction level of a service. This double rating helped capture more fully the experiential cognition of the evaluators.
S.-C. Ngan / Expert Systems with Applications 38 (2011) 14052–14059
The computation procedure used in the above mentioned papers took the fuzzy ratings given by the individual evaluators as inputs and returned a single crisp value describing the average judgment of the evaluators. In this article, we propose to expand the methodology by outputting not a crisp value, but a fuzzy number to represent the aggregation of opinions. This makes logical sense in that one would expect the result of aggregating fuzzy ratings to be fuzzy itself. Furthermore, the resulting fuzzy number can be viewed as a fuzzy set theory analog of the statistical interval used in classical statistics. For example, in the case of customer satisfactory level analysis, a service under consideration might get an overall average crisp rating of ‘‘satisfactory.’’ The fuzzy number that will be obtained with the proposed methodology represents how confident we are that the service is indeed ‘‘satisfactory,’’ reflected by how much of the support of the associated membership function is contained within the support of the linguistic term ‘‘satisfactory.’’ A procedure to formulate and compute statistical confidence intervals for fuzzy data has actually been proposed recently in Wu (2009). It involved the use of a-cut (denoted as h-level set in that paper) and the extension principles in Zadeh (1975a, 1975b, 1975c), leading to an interesting but mathematically non-trivial optimization problem. In this article, we propose to use an alternative formulation based on sampling distribution, giving an efficient and simple procedure for calculating the fuzzy number. The article is organized as follows: In Section 2, the basic format of the survey questionnaires used in eliciting evaluators’ opinions and the 2-tuple fuzzy linguistic approach for aggregating the opinions used in Lin and Lee (2009) and Wang (2009) will be reviewed. After that, we will describe a computationally efficient framework for calculating a fuzzy number representing the aggregation of opinions. In Section 3, we will apply the proposed methodology on evaluating new product development ideas and on survey analysis. A brief summary will be given in Section 4.
2. Background and methodology 2.1. Format of the survey questionnaires The questionnaires used in Lin and Lee (2009) and Wang (2009) have the following common format: A typical questionnaire, for instance to evaluate a new product idea, consists of several main criteria, such as the research and development capability of the firm in producing a prototype, the organizational capacity in manufacturing the proposed product, and the marketability of the product. Associated with each of these main criteria are several sub-criteria. Under the marketability heading, these sub-criteria could be (i) the present market share of related products, (ii) the potential number of major customers, and so on. As discussed in the Introduction, an individual decision maker will rate the product idea with respect to each sub-criterion, for example, giving an ‘‘excellent’’ rating for the product idea regarding item (i) and an ‘‘average’’ rating regarding item (ii), the ratings being chosen from a list of linguistic terms very poor, poor, average, good, and excellent. This yields a first set of ratings. In addition, each decision maker will rate the criteria and each of the associated sub-criteria themselves in order to express his/her opinion about the relative importance of the criteria and sub-criteria in determining the overall viability of a product idea. For example, he/she could give an ‘‘important’’ rating for item (i) and a ‘‘very important’’ rating for item (ii), selected from a list of linguistic terms negligible, not important, average, important, and very important. This produces a second set of ratings. In a later sub-section, we will review how the two sets of ratings are combined to form the decision makers’ aggregate opinion on the overall prospect of success of a new product idea.
14053
2.2. Fuzzy linguistic computing approach Details of the fuzzy linguistic formulation and representation of imprecise data can be found in Herrera and Martinez (2000a, 2000b), Zimmermann (1991) and the references therein. We summarize the core ideas here for completeness. A triangular fuzzy number L with parameters (p, q, r) can be described by a membership function
8 > < ðx pÞ=ðq pÞ when p 6 x 6 q fL ðxÞ ¼ ðx rÞ=ðq rÞ when q 6 x 6 r > : 0 otherwise
ð1Þ
where x is a real number. Such a fuzzy number is graphically depicted in Fig. 1. Thus, if x represents the height of an individual, and L represents a particular person’s (denoted as person A) concept (termed a fuzzy concept) of an average-height individual, then a given individual with an associated height will be considered by person A as having ‘‘average height’’ to a degree determined by the membership function. For example, an individual with height x = q is definitely a member of the concept ‘‘average-height individual’’ according to A (for fL(q) = 1.0), while an individual taller than r is definitely not (for fL(x) = 0 for any x P r). For the scenarios considered in this article, a fuzzy concept can be described by a fuzzy number. Furthermore, the membership function of a fuzzy number does not need to be constrained in the triangular form – other shapes such as the parabola or trapezoid could also be used. However, the triangular form can capture fuzziness in human concepts with good approximation and is employed extensively in this article. A fuzzy linguistic variable is a variable whose domain is a collection of pre-specified fuzzy concepts. For example, a fuzzy linguistic variable can be used to represent the rating of a particular object given by an evaluator in a survey questionnaire. In this case, the domain of the variable consists of the fuzzy concepts (also denoted as the fuzzy linguistic terms) very poor, poor, average, good and excellent. These fuzzy linguistic terms correspond to the five membership functions illustrated in Fig. 2a. Referring to the figure, to represent the crisp value x = 0.4 within the fuzzy linguistic framework, a 2-tuple (average, 0.1) is used. That is, x = 0.4 is expressed as the sum of the ‘‘q’’ value of the fuzzy linguistic term average (qaverage = 0.5) and the offset of x from the ‘‘q’’ position (0.1). Therefore, we can translate between a crisp value and its 2-tuple fuzzy linguistic form using the formulas
DðxÞ ¼ ðSi ; aÞ
ð2Þ
and
Fig. 1. A triangular fuzzy number L with parameters (p, q, r) representing person A’s fuzzy concept of an ‘‘average-height individual.’’ When x = q, the value of the membership function is 1. When x = (p + q)/2, the value is 0.5.
14054
S.-C. Ngan / Expert Systems with Applications 38 (2011) 14052–14059
ij Þ ¼ D 1 ½D1 ðW ij1 ; 0Þ þ þ D1 ðW ijK ; 0Þ ðW ij ; b K
ð5Þ
where Wijk e {W0, . . . , W4} is the importance rating of the subcriterion Di,j given by the kth expert. Furthermore, let us denote Wik e {W0, . . . , W4} as the importance rating of the criterion Ci given by the kth expert. Then, the aggregate importance rating of Ci is determined by the formula
ðW i ; bi Þ ¼ D
1 1 ½D ðW i1 ; 0Þ þ þ D1 ðW iK ; 0Þ K
ð6Þ
Subsequently, the aggregate opinion of the product idea with respect to criterion Ci across the K experts is
(PN i ðSi ; ai Þ ¼ D
j¼1 D
ij Þ ij Þ D1 ðW ij ; b ðSij ; a PNi 1 j¼1 D ðW ij ; bij Þ
1
) ð7Þ
and the overall aggregate opinion of the product idea across the K experts and across all criteria is calculated using
Þ ¼ D ðS; a
Fig. 2. (a) depicts the membership functions corresponding to the linguistic terms very poor, poor, average, good and excellent. For example, ‘‘average’’ is represented by the membership function with paverage = 0.25, qaverage = 0.5 and raverage = 0.75, and (b) depicts the five membership functions corresponding to the linguistic terms negligible, not important, average, important and very important. For example, the p, q and r values for W2 are 0.25, 0.5 and 0.75, respectively.
D1 ðSi ; aÞ ¼ x ¼
i þa g
ð3Þ
where i = round(x g), a = x i/g, g = N 1 and N is the total number of linguistic terms in the domain. 2.3. Aggregating the evaluators’ opinions Let us consider the scenario in which a total of K experts have rated a new product idea based on M criteria {C1, . . . , CM} and for each criterion Ci, Ni sub-criteria fDi;1 ; . . . ; Di;Ni g. As discussed above, let us assume that the linguistic variable describing the rating of the product idea with respect to a sub-criterion Di,j is endowed with the domain {very poor (S0), poor (S1), average (S2), good (S3) and excellent (S4)}, and meanwhile, the linguistic variable representing the importance rating of a criterion/sub-criterion is endowed with the domain {negligible (W0), not important (W1), average (W2), important (W3), and very important (W4)} (see Fig. 2b). A fuzzy linguistic approach for aggregating the opinions of the experts on the product idea with respect to the sub-criterion Di,j is achieved through the computation
ij Þ ¼ D ðSij ; a
1 1 ½D ðSij1 ; 0Þ þ þ D1 ðSijK ; 0Þ K
ð4Þ
where Sijk e {S0, . . . , S4} is the product idea rating with respect to the sub-criterion Di,j given by the kth expert. The aggregate importance rating of the sub-criterion Di,j is similarly calculated:
(P M
i Þ i Þ D1 ðW i ; b ðSi ; a PM 1 i¼1 D ðW i ; bi Þ
i¼1 D
1
) ð8Þ
The computation procedure delineated in Eqs. (4)–(8) is the methodology taken in Wang (2009). The approach developed in Lin and Lee (2009) is largely similar, with the exception that an evaluator can select more than one linguistic term when he/she gives a rating, such as choosing ‘‘average’’ with a weight of 0.8 and ‘‘good’’ with a weight of 0.2. The choosing of more than one linguistic term represents the evaluator’s uncertainty in his/her rating and the weights reflect that the evaluator leans more towards ‘‘average’’ than ‘‘good’’ when a product idea/service under consideration is rated against some sub-criterion. This additional flexibility can be accommodated by modifying Eq. (4) appropriately. The approaches used in both papers return a single crisp value (from the result of Eq. (8)) as a final output. 2.4. An alternative formulation of fuzzy linguistic computing based on sampling distribution As discussed in the Introduction, given the fuzzy ratings by the evaluators, our goal is to output a fuzzy number instead of a crisp number characterizing the aggregation of opinions. The fuzzy number allows us to capture the degree of confidence that the aggregate opinion indeed belongs to a particular overall rating. Instead of calculating the fuzzy number based on the a-cut and extension principles, which can lead to mathematically non-trivial optimization problems, the present formulation is based on sampling distribution. For concreteness, let us reconsider the ‘‘average-height individual’’ fuzzy concept delineated in Fig. 1. We will assume a probabilistic interpretation of the fuzzy concept as follows (see Bilgiç & Turksen, 1999 for a survey of various interpretations): Suppose that in an occasion (denoted as occasion 1), a friend B of person A gathers one hundred individuals, each with an actual height of q. Without informing A of their actual heights, B presents those one hundred individuals to A and asks A to visually judge them one by one. Person A, following his subjective concept of ‘‘average-height individual’’, will end up judging all one hundred of these individuals to be of ‘‘average height,’’ since the value of the membership function of his concept of ‘‘average height’’ at x = q is 1. On the other hand, if the above experiment is repeated (denoted as occasion 2) with the difference that B instead gathers one hundred individuals each with an actual height of (p + q)/2. Then A will classify only about half of them as having ‘‘average height’’ according to his subjective concept, because the value of the membership function at x = (p + q)/2 is 0.5. In general, the value of the membership function at the height of
S.-C. Ngan / Expert Systems with Applications 38 (2011) 14052–14059
14055
interest gives the probability that an individual with that height will be classified as being ‘‘average height.’’ Following conventional conditional probability, we have the relation
Pð\average height"jactual heightÞ Pðactual heightÞ ¼ Pðactual heightj\average height"Þ Pð\average height"Þ
ð9Þ
Given an individual with a particular actual height (known by B but not by A), the term P(‘‘average height’’|actual height) refers to the probability that A will classify that individual as having ‘‘average height.’’ Based on the probabilistic interpretation of fuzzy concepts discussed in the previous paragraph, this term is fully specified by the membership function of the fuzzy concept ‘‘average-height individual.’’ P(actual height|‘‘average height’’) gives the probability of an individual possessing a particular actual height given that the individual has been classified by A as having ‘‘average height.’’ On the other hand, P(actual height) represents the prior probability of encountering an individual with a particular actual height in our universe of discourse. Note that the prior probability can be further specified if one possesses sufficient knowledge about the actual height distribution of individuals in the universe of discourse. Otherwise, following conventional practices in Bayesian statistics, we set it as a non-informative prior. For example, we let P(actual height) = C, where C is a constant, for a broad enough range of actual heights. Combining with Eq. (9), this then yields
Pðactual heightj\average height"Þ /
Pð\average height"jactual heightÞ Pð\average height"Þ
ð10Þ
Let us consider yet another occasion (denoted as occasion 3), in which A has just finished judging a number of individuals and concluded that one hundred of them have ‘‘average height.’’ Then, we can infer that the actual height distribution of the one hundred individuals follows P(actual height|‘‘average height’’). Furthermore, from Eq. (10), we obtain for this occasion
Pðactual heightj\average height"Þ / Pð\average height"jactual heightÞ
ð11Þ
As discussed above, the right hand side of Eq. (11) corresponds to the membership function of the fuzzy concept ‘‘average-height individual.’’ Therefore, by properly normalizing the membership function (such that the area under the curve equals 1), the actual height distribution P(actual height|‘‘average height’’) can be determined:
Pðxj\average height"Þ ¼ ^f L ðxÞ
ð12Þ
with
8 > < Fðx pÞ=ðq pÞ when p 6 x 6 q ^f L ðxÞ ¼ Fðx rÞ=ðq rÞ when q 6 x 6 r > : 0 otherwise
and F ¼
2 ðr pÞ
In the preceding example, the concept ‘‘average-height individual’’ is fuzzy, while the crisp values ‘‘actual heights’’ are objective measurements. In general, the fuzzy linguistic framework can also be applied to describe fuzzy concepts together with subjective crisp values. For example, consider occasion 4 in which person A is to give his subjective rating (on the scale of 0–1) for the tastes of sandwiches. Instead of committing to a crisp value to represent his subjective rating on a sandwich under consideration, A may use a linguistic term ‘‘average’’ instead (see Fig. 3). In other words, the linguistic term ‘‘average’’ enables A to express his opinion while
Fig. 3. Three linguistic terms taste bad, average and taste good representing person A’s fuzzy concepts of taste of sandwiches.
relieving him from the responsibility of needing to specify a particular crisp value to describe his subjective rating. In analogy with Eq. (12), we have
Pðxj\average"Þ ¼ ^f average ðxÞ
ð13Þ
where
8 when 0 6 x 6 0:5 > < 4x ^f average ðxÞ ¼ 4ðx 1Þ when 0:5 6 x 6 1 > : 0 otherwise is the normalized form of the membership function for ‘‘average.’’ In the present context, the term P(x|‘‘average’’) can be viewed as describing the range and distribution of the crisp values person A would have given if he were pressed to give his subjective crisp ratings for say the N sandwiches that he considers average taste. 2.5. Application of the alternative formulation to aggregating opinions Recall the basic questionnaire setup consisting of K experts rating a new product idea/service based on M criteria {C1, . . . , CM} and Ni sub-criteria fDi;1 ; . . . ; Di;Ni g for each criterion Ci, with the linguistic variables Sijk representing the fuzzy ratings of the product idea/ service and Wik and Wijk representing the fuzzy ratings for the relative importance of the criteria and sub-criteria. We aggregate these ratings with a sampling approach: One run of sampling and aggregation consists of first randomly generating a crisp value from each linguistic variable based on the fuzzy ratings given by the experts. For instance, suppose that the kth expert rates the product idea under consideration with respect to sub-criterion Di,j as Sijk = S3 based on his/her subjective opinion, i.e. corresponding to the linguistic term ‘‘good’’ in Fig. 2a. In analogy with Eq. (13), a crisp value x will be generated based on the distribution
pðxjS3 Þ ¼ ^f S3 ðxÞ
ð14Þ
with
8 > < 16ðx 0:5Þ when 0:5 6 x 6 0:75 ^f 3 ðxÞ ¼ 16ð1 xÞ when 0:75 6 x 6 1 S > : 0 otherwise where ^f S3 ðxÞ is the normalized version of the membership function fS3 ðxÞ. After a crisp value has been generated for each linguistic variable, these values are numerically combined stage-wise:
b S ik ¼
PNi b c j¼1 S ijk W ijk PN i c j¼1 W ijk
ð15Þ
14056
b Sk ¼
S.-C. Ngan / Expert Systems with Applications 38 (2011) 14052–14059
PM b c i¼1 S ik W ik PM c i¼1 W ik
ð16Þ
and K 1X b b S¼ Sk K k¼1
ð17Þ
c ijk and W c ik represent the randomly sampled crisp where b S ijk ; W values for the linguistic variables Sijk, Wijk and Wik. Note that if historical data or background knowledge were available, specific c ijk and W c ik could conditional probabilistic relation among b S ijk ; W be formulated and estimated. Lacking these data and knowledge, we take the simplest assumption of conditional independence among these variables, and generate the samples independently. Now, Eq. (15) computes the weighted rating of a product idea/ service with respect to criterion Ci according to the opinion of the kth expert. Eq. (16) determines the overall weighted rating of the product idea/service according to the kth expert across
all criteria. Finally, Eq. (17) averages these overall weighted rating across the K experts, yielding a single sampling of the aggregate rating b S. The above-mentioned sampling and aggregation run is repeated N times (e.g. say N = 10,000). In analogy with occasions 3 and 4 in the previous example, the N sampling and aggregation runs can be viewed as representing the evaluations of N highly similar product ideas/services. (These product ideas/services are highly similar in the sense that they receive the same fuzzy ratings from the K experts, much like in occasion 3, in which the one hundred individuals were highly similar in that they were all identified as of ‘‘average height’’, or in occasion 4, in which the N sandwiches were highly similar in that they were all identified as of ‘‘average’’ taste.) In occasion 3, according to Eq. (12), the actual height distribution of the one hundred individuals can be used to recover the membership function for the fuzzy concept ‘‘average-height individual.’’ In the present context, the resulting collection of the b S values from the N runs gives a distribution, which can then be properly scaled to obtain a fuzzy number
Table 1 The fuzzy ratings given by the four decision makers D1, D2, D3 and D4 for a new product idea under consideration with respect to various sub-criteria. There are altogether four criteria. Under each criterion are 3–5 sub-criteria. Each decision maker specifies his/her ratings by choosing from the five linguistic terms {S0, S1, S2, S3, S4}. For example, D1 selects S3 as his/her rating for the product idea with respect to the sub-criterion ‘‘quality and speed to market,’’ leading to the ‘‘3’’ entry in the table. Criteria
Sub-criteria
D1
D2
D3
D4
Market
Quality and speed to market Number of major customer Market share rate Widening customer choice and expectation Delivery
3 4 3 4 2
4 2 4 3 2
2 4 3 2 4
4 3 3 4 4
Innovation
Flexibility Number of new products or processes Number of patent Fee of research/fee of total
4 4 2 3
2 3 3 4
4 2 4 4
2 3 3 2
Organization
Competitive priorities of responsiveness Trademark Information system Index of productivity
4 3 4 2
4 4 2 3
2 3 4 4
4 3 3 4
Employee
Capability of employees Output merit of employees Skill training of employees
4 2 3
3 4 4
3 2 4
4 4 3
Table 2 The fuzzy ratings given by the four decision makers D1, D2, D3 and D4 to express their views regarding the relative importance of the criteria and sub-criteria in determining the overall viability of a new product idea. Each decision maker specifies his/her ratings by choosing from the five linguistic terms {W0, W1, W2, W3, W4}. For example, D1 selects W4 as his/her importance rating for the sub-criterion ‘‘quality and speed to market,’’ leading to the ‘‘4’’ entry in the table. Criteria
Sub-criteria
D1
D2
D3
D4
Quality and speed to market Number of major customer Market share rate Widening customer choice and expectation Delivery
4 4 2 3 3 4
4 4 4 3 3 4
4 4 2 2 4 4
3 3 2 2 3 4
Flexibility Number of new products or processes Number of patent Fee of research/fee of total
3 3 3 2 4
3 4 2 3 4
4 4 4 2 4
3 4 4 3 3
Competitive priorities of responsiveness Trademark Information system Index of productivity
4 4 2 4 4
3 3 3 4 2
4 3 2 3 4
3 4 2 4 3
Capability of employees Output merit of employees Skill training of employees
3 3 3 2
2 4 3 4
3 4 3 2
3 3 3 3
Market
Innovation
Organization
Employee
14057
S.-C. Ngan / Expert Systems with Applications 38 (2011) 14052–14059
Fig. 4. Fuzzy number V representing the aggregate opinion of the four decision makers on the overall viability of a new product idea under consideration.
representing the aggregate opinion of the experts regarding the product idea/service under consideration.
are reproduced in Tables 1 and 2. The five linguistic terms {S0, S1, S2, S3, S4} are available for the product ratings, and the five linguistic terms {W0, W1, W2, W3, W4} are available for the importance ratings. By applying the sampling approach to combine ratings given by the decision makers, we obtain a fuzzy number V as displayed in Fig. 4, representing the aggregate opinion of these experts on the overall viability of the new product idea. The membership function of V peaks at x = 0.8 (agreeing closely with the average crisp value of 0.8056 reported in Wang (2009), and resembles a bell-shaped curve, but with a finite support. For any x e (0.77, 0.83), the corresponding value of the membership function is larger than 0.05. (The range (0.77, 0.83) is denoted as the 0.05+-cut of the fuzzy number V.) Referring to Fig. 2a, we note that the membership function of S3 (the ‘‘good’’ rating) intersects with that of S2 (the ‘‘average’’ rating) at x = 0.625, and intersects with that of S4 (the ‘‘excellent’’ rating) at x = 0.875. The range (0.625, 0.875) is incidentally the 0.5+-cut of the linguistic term ‘‘good,’’ since the value of the membership function of S3 corresponding to any x e (0.625, 0.875) is larger than 0.5. We can conclude that the aggregate opinion of the new product idea as captured by the fuzzy number V is well described by an overall ‘‘good’’ rating, in the sense that the 0.05+-cut of V (a large majority of the support of V) is within the 0.5+-cut of S3.
3. Results and discussion 3.2. Application of the sampling approach to survey analysis 3.1. Application of the sampling approach to new product development In Wang (2009), four decision makers from a Taiwanese electronic manufacturer evaluated a new product idea based on a number of criteria and sub-criteria (see Table 1). Their fuzzy ratings of the product idea with respect to the various sub-criteria and of the importance of the criteria and sub-criteria themselves
In Lin and Lee (2009), two experts evaluated the investment benefit of a proposed facility site based on five criteria and ten sub-criteria (see Tables 3a and 3b). Their fuzzy ratings of the facility site as well as of the importance of the criteria and sub-criteria are detailed in Tables 3a, 3b and 4. The seven linguistic terms {S0, . . . , S6} and the eleven linguistic terms {W0, . . . , W10} are
Table 3a The fuzzy ratings given by an expert D1 for the investment benefit of a proposed facility site with respect to various sub-criteria. There are altogether five criteria. Under each criterion are 2 sub-criteria. D1 specifies a rating by choosing a weight for each of the seven linguistic terms {S0, S1, S2, S3, S4, S5, S6}. For example, D1 puts a weight of 0.17 for S0, and a weight of 0.83 for S1 (total weight summed up to 1.0) as his/her rating of the proposed site with respect to the sub-criterion ‘‘salary level.’’ This reflects both D1’s uncertainty and that D1 leans more towards S1 than S0 in his/her rating. Criteria
Sub-criteria
S0
S1
S2
S3
S4
S5
S6
Labor
Salary level Manpower level
0.17 0
0.83 0.80
0 0.20
0 0
0 0
0 0
0 0
Geography
Usage condition level of factory place Nearing market level of delivery system
0 0
0.60 0.89
0.40 0.11
0 0
0 0
0 0
0 0
Economic
Index of industry production growth Index of industry modern times
0.15 0
0.85 0.75
0 0.25
0 0
0 0
0 0
0 0
Reward
Reward obtain level Institution perform level
0.25 0
0.75 0.80
0 0.20
0 0
0 0
0 0
0 0
Politics
Regulatory restrictions level Investment subsidy level
0 0
0.75 0.60
0.25 0.40
0 0
0 0
0 0
0 0
Table 3b Analogous to Table 3a, this table shows the fuzzy ratings given by an expert D2 for the investment benefit of a proposed facility site with respect to various sub-criteria. Criteria
Sub-criteria
S0
S1
S2
S3
S4
S5
S6
Labor
Salary level Manpower level
0 0.15
0.80 0.85
0.20 0
0 0
0 0
0 0
0 0
Geography
Usage condition level of factory place Nearing market level of delivery system
0 0.20
0.70 0.80
0.30 0
0 0
0 0
0 0
0 0
Economic
Index of industry production growth Index of industry modern times
0.10 0
0.90 0.70
0 0.20
0 0.10
0 0
0 0
0 0
Reward
Reward obtain level Institution perform level
0.10 0.20
0.20 0.80
0.70 0
0 0
0 0
0 0
0 0
Politics
Regulatory restrictions level Investment subsidy level
0 0
0.60 0.60
0.25 0.30
0.15 0.10
0 0
0 0
0 0
14058
S.-C. Ngan / Expert Systems with Applications 38 (2011) 14052–14059
Table 4 The fuzzy ratings given by the two experts D1 and D2 to express their views regarding the relative importance of the criteria and sub-criteria used in determining the overall investment benefit of a proposed facility site. Each expert specifies his/her ratings by choosing from the eleven linguistic terms {W0, W1, . . . , W10}. For example, D1 selects W5 as his/her importance rating for the sub-criterion ‘‘salary level,’’ leading to the ‘‘5’’ entry in the table. Criteria
Sub-criteria
D1
D2
Salary level Manpower level
1 5 3
3 3 4
Usage condition level of factory place Nearing market level of delivery system
3 2 5
2 8 4
Index of industry production growth Index of industry modern times
4 3 7
3 4 6
Reward obtain level Institution perform level
4 6 8
2 6 3
Regulatory restrictions level Investment subsidy level
6 4 2
3 7 3
Labor
Geography
Economic
Reward
Politics
Fig. 5. Fuzzy number I representing the aggregate opinion of the two experts on the investment benefit of a proposed facility site.
available for the site ratings and the importance ratings respectively. Application of the sampling approach to combine opinions yields the fuzzy number I delineated in Fig. 5, representing the aggregate view of the two experts on the investment benefit of the proposed site. The membership function peaks at x = 0.2, in close agreement with the average crisp value of 0.204 reported in Lin and Lee (2009). For any x e (0.14, 0.27), the corresponding value of the membership function is larger than 0.05. Thus, (0.14, 0.27) is the 0.05+-cut of the fuzzy number I. Referring to Fig. 6, the 0.5+-cut of S1 is (0.0833, 0.25), while that of S2 is (0.25, 0.417). Thus, the aggregate opinion of the investment benefit of the proposed site can be viewed as straddling across the S1 and S2 ratings, leaning heavily towards S1, in the sense that the 0.05+cut of I is mostly contained within the 0.5+-cut of S1, but with overlap with the 0.5+-cut of S2. 4. Conclusion Fuzzy set theory, with its ability to represent and process uncertainties and vagueness inherent in subjective human reasoning, has been applied to a wide variety of industrial domains and under continuous development since its introduction in the 1960s. Recently, the 2-tuple fuzzy linguistic computing (Herrera & Martinez, 2000a, 2000b) has been proposed as a methodology to aggregate fuzzy opinions, for example, in the evaluation of new product development performance (Wang, 2009) and in customer satisfactory level survey analysis (Lin & Lee, 2009). The 2-tuple fuzzy linguistic approach has the advantage of avoiding information loss that can potentially occur when combining opinions of experts, due to the fuzziness and inhomogeneity of the experts’ subjective cognitions (Herrera-Viedma et al., 2004). Given the fuzzy ratings of the evaluators, the computation procedure used in both Wang (2009) and Lin and Lee (2009) returned a single crisp value as an output, representing the average judgment of those evaluators. In this article, we took an alternative view that the result of aggregating fuzzy ratings should be fuzzy itself, and therefore we further developed the 2-tuple fuzzy linguistic methodology so that its output is a fuzzy number describing the aggregation of opinions. Instead of relying on the a-cut and the extension principles, which could lead to interesting but mathematically non-trivial optimization problems as in Wu (2009), our approach was based on sampling distribution, giving an efficient and simple procedure. We demonstrated the utility of the extended fuzzy linguistic computing methodology on aggregating fuzzy ratings by applying it to two data sets: (i) the evaluation of a new product idea in a Taiwanese electronics manufacturing firm and (ii) the evaluation of the investment benefit of a proposed facility site. For each of these two studies, the peak location of the membership function representing the aggregate opinion of the evaluators agreed closely with the average crisp value reported in the respective publication. Moreover, the fuzzy numbers obtained from the present methodology gave additional information beyond the crisp values about the aggregate opinions. In the first study, we concluded that the aggregate opinion of the new product idea was well described by an overall S3 rating (i.e. a ‘‘good’’ rating), while in the second study, the aggregate opinion of the investment benefit of the proposed site actually straddled across the S1 and S2 ratings. References
Fig. 6. The seven linguistic terms S0, S1, S2, S3, S4, S5 and S6 available for the experts to rate the proposed facility site with respect to various sub-criteria. For example, the p, q and r values of S2 are 0.167, 0.333 and 0.5, respectively. Note that S0 and S1 intersect at x = 0.0833, S1 and S2 at x = 0.25, S2 and S3 at x = 0.417.
Bilgiç, T., & Turksen, I. B. (1999). Measurement of membership functions: Theoretical and empirical work. In Handbook of fuzzy sets and systems. In D. Dubois & H. Prade (Eds.). Fundamentals of fuzzy sets (Vol. 1, pp. 195–232). Kluwer. Chuang, T.-N., Lin, C.-T., Kung, J.-Y., & Lin, M.-D. (2010). Planning the route of container ships: A fuzzy genetic approach. Expert Systems with Applications, 37, 2948–2956.
S.-C. Ngan / Expert Systems with Applications 38 (2011) 14052–14059 Dursun, M., & Karsak, E. E. (2010). A fuzzy MCDM approach for personnel selection. Expert Systems with Applications, 37, 4324–4330. Galetakis, M., & Vasiliou, A. (2010). Selective mining of multiple-layer lignite deposits. A fuzzy approach. Expert Systems with Applications, 37, 4266–4275. Herrera-Viedma, E., Herrera, F., Martinez, L., Herrera, J. C., & Lopez, A. G. (2004). Incorporating filtering techniques in a fuzzy linguistic multi-agent model for information gathering on the web. Fuzzy Sets and Systems, 148, 61–83. Herrera, F., & Martinez, L. (2000a). A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Transactions on Fuzzy Systems, 8, 746–752. Herrera, F., & Martinez, L. (2000b). An approach for combining linguistic and numerical information based on 2-tuple fuzzy representation model in decision making. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 8, 539–562. Hwang, C. L., & Yoon, K. (1981). Multiple attributes decision making methods and applications. Springer-Verlag. Lin, L.-Z. (2010). A perceptual measure of trade shows using fuzzy quality deployment development. Expert Systems with Applications, 37, 3921–3933. Lin, L., & Lee, H.-M. (2009). Fuzzy assessment method on sampling survey analysis. Expert Systems with Applications, 36, 5955–5961.
14059
Pan, W.-T. (2010). Combining fuzzy Sammon mapping and fuzzy clustering approach to perform clustering effect analysis: Take the banking service satisfaction as an example. Expert Systems with Applications, 37, 4139–4145. Sen, C. G., & Baraçlı, H. (2010). Fuzzy quality function deployment based methodology for acquiring enterprise software selection requirements. Expert Systems with Applications, 37, 3415–3426. Wang, W.-P. (2009). Evaluating new product development performance by fuzzy linguistic computing. Expert Systems with Applications, 36, 9759–9766. Wu, H.-C. (2009). Statistical confidence intervals for fuzzy data. Expert Systems with Applications, 36, 2670–2676. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353. Zadeh, L. A. (1975a). The concept of linguistic variable and its application to approximate reasoning I. Information Sciences, 8, 199–249. Zadeh, L. A. (1975b). The concept of linguistic variable and its application to approximate reasoning II. Information Sciences, 8, 301–357. Zadeh, L. A. (1975c). The concept of linguistic variable and its application to approximate reasoning III. Information Sciences, 9, 43–80. Zimmermann, H. J. (1991). Fuzzy set theory and its applications (2nd ed.). Kluwer Academic Publishers.