Fuzzy Multiple-Criteria Decision-Making Approach
for Industrial Green Engineering
HUA-KAI CHIOU
Department of Statistics, College of Management National Defense University
P.O. Box 90046-15 Chungho Taipei 235, Taiwan
GWO-HSHIUNG TZENG*
Institute of Technology Management, and Energy and Environmental Research Group
College of Management, National Chiao Tung University 1001, Ta-Hsuch Rd.
Hsinchu 300, Taiwan
ABSTRACT / This paper describes a fuzzy hierarchical analytic approach to determine the weighting of subjective judgments. In addition, it presents a nonadditive fuzzy integral technique to evaluate a green engineering industry case as a fuzzy multi-criteria decision-making (FMCDM) problem. When the
invest-ment strategies are evaluated from various aspects, such as economic effectiveness, technical feasibility, and environmen-tal regulation, it can be regarded as an FMCDM problem. Since stakeholders cannot clearly estimate each considered criterion in terms of numerical values for the anticipated alter-natives/strategies, fuzziness is considered to be applicable. Consequently, this paper uses triangular fuzzy numbers to establish weights and anticipated achievement values. By ranking fuzzy weights and fuzzy synthetic utility values, we can determine the relative importance of criteria and decide the best strategies. This paper applies what is called a fuzzy measure and nonadditive fuzzy integral technique to evaluate the synthetic performance of green engineering strategies for aquatic products processors in Taiwan. In addition, we dem-onstrate that the nonadditive fuzzy integral is an effective eval-uation and appears to be appropriate, especially when the criteria are not independent.
Along with technological and economic develop-ment, mass production has resulted in increasing waste, including hazardous emissions and toxic waste from manufacturing process. According to United States En-vironmental Protection Agency statistics, in 2000, over 400 million tons of hazardous waste emissions and in-dustrial waste is processed annually worldwide. Further-more, about 480 million tons of municipal waste is produced in daily life. Preserving the planet on which we live is an urgent challenge for our time.
Green engineering aims to reclaim industrial or mu-nicipal waste and is an increasingly important view-point, which also provides the opportunity for sustain-able development of enterprise. In 1992, the United Nations Environmental Planning Board (UNEP) pre-sented Agenda 21 of the Rio Declaration on Environ-ment and DevelopEnviron-ment as a guideline to improve sus-tainable development. In addition, in 1996 UNEP proposed the structure and approaches of sustainable development index. The United States developed 10 goals and a related sustainable development index for their country in the same year. The United Kingdom
declared 120 sustainable development indices for their country in 1992. They then integrated these into 13 major indices to evaluate the performance of economic development, social investment, climate change, envi-ronmental quality and ecological conservation for their country in 1996 (Mendoza and Prabhu 2000).
Environmental planning and decision-making in green engineering industries are essentially conflict analysis characterized by sociopolitical, environmental, and economic value judgments. Several alternatives/ strategies have to be considered and evaluated in terms of many different criteria resulting in a vast body of data that are often inaccurate or uncertain.
In real world systems, the decision-making problems are very often uncertain or vague in a number of ways. Due to lack of information, the future state of the system might not be known completely. This type of uncertainty has long been handled appropriately by probability the-ory and statistics. However, in many areas of daily life, such as engineering, medicine, meteorology, manufactur-ing, and others, human judgment, evaluation, and deci-sions often employ natural language to express thinking and subjective perception. In these natural languages the meaning of words is often vague. The meaning of a word might be well defined, but when using the word as a label for a set, the boundaries within which objects do or do not belong to the set become fuzzy or vague.
KEY WORDS: Green engineering industry; Nonadditive fuzzy integral; Fuzzy multicriteria decision-making; fuzzy measure
*Author to whom correspondence should be addressed; email: ghtzeng@cc.nctu.edu.tw
Furthermore, human judgment of events may be significantly different based on individuals’ subjective perceptions or personality, even using the same words. Fuzzy numbers are introduced to appropriately express linguistic variables. We will provide a more clear de-scription of linguistic expression with fuzzy scale in a later section.
In this paper the fuzzy hierarchical analytic ap-proach was used to determine the weights of criteria from subjective judgment, and a nonadditive integral technique was utilized to evaluate the performance of green engineering strategies for aquatic products pro-cessors in Taiwan. Traditionally, researchers have used additive techniques to evaluate the synthetic utilities of each criterion. In this article, we demonstrate that the nonadditive fuzzy integral is a good means of evalua-tion and appears to be more appropriate, especially when the criteria are not independent situations.
The conceptual development of green engineering is discussed in the next section, and the fuzzy hierar-chical analytic approach and nonadditive fuzzy integral evaluation process for multicriteria decision-making (MCDM) problem are derived in the subsequent sec-tion. Then an illustrative example is presented, apply-ing the MCDM methods for aquatic products proces-sors in Taiwan, after which we discuss and show how the MCDM methods in this paper are effective. Finally, the conclusions are presented.
Concept Development of Green Engineering
Thinking
Recently, environmental concerns have raised pub-lic awareness of environmental issues and are driving forces for regulation. The impact of regulation on the cost of production is expected to become an important determinant for the international competitiveness of industries. In response to cost pressures, industries have launched a number of initiatives aimed at improv-ing efficiency and reducimprov-ing environmental impact; re-claiming techniques are effective and economic ap-proaches to enable enterprises to achieve goals of sustainable development.
When a consumer no longer wants to keep a prod-uct, any of the following options may be possible (Lave and others 1994, 1999), of which options 1– 4 are kinds of green engineering (Simon 1992): (1) reuse (as with old furniture); (2) remanufacture (as with copier ma-chines or automobile alternators); (3) recycle for the same use in a “closed loop” (as with asphalt pavement); (4) recycle into a lower valued use (as with plastics formed into park benches); (5) incinerate (as with burning paper to reclaim energy); (6) landfill (as with
most municipal solid waste); and (7) discard directly to the environment (as with littering or dumping into the ocean).
Since the United Nations General Assembly proposed “Our Common Future” in 1987, the international social system began to take account of environmental and sus-tainable development issues. There have been many bilat-eral, multilatbilat-eral, regional, and global agreements to pro-vide environmental protection, and some of the important regulations are described in Appendix 1.
There is much evidence that environmental issues may affect business profits. In addition, all enterprises must take responsibility to value our resources by com-plying with regulations. Reclaiming of resources is an ecoefficient strategy and a paragon of sustainable de-velopment. According to our survey of the literature, several multicriteria analytic methods have been used to deal with environmental problems. The main ap-proaches can be classified based on the type of decision model they used (Lahdelma and others 2000): (1) value or utility function-based methods, such as multi-attribute utility theory (Keeney and Raiffa 1976, Merk-hofer and Keeney 1987, Teng and Tzeng 1994, Tzeng and others 1996), AHP (Saaty 1980), DEA (Oral and others 1991), and the stochastic multiobjective accept-ability analysis methods (Lahdelma and others 1998, Roy and others 1986); or (2) outranking methods such as ELECTRE (Siskos and Hubert 1983, Grassin 1986, Roy and Bouyssou 1986, Roy 1991, Hokkanen and oth-ers 1995, Hokkanen and Salminen 1997a, b, Salminen and others 1998), PROMETHEE I and II (Brans and Vincke 1985, Briggs and others 1990), and GFD (Ca-ruso and others 1993).
Hierarchical Analytic Process and Evaluation
Methods
In real MCDM problems, it is necessary to divide the process into distinct stages. First, based on a general problem statement, the various stakeholders are de-fined, typically including decision-makers, various inter-est groups affected by the decision, experts in the ap-propriate fields, as well as planners and analysts responsible for the preparations and managing the process. The overall objective will be set up in this stage. Second, based on various points of view from stakehold-ers, the problems can be categorized into distinct as-pects. Third, defining alternatives/strategies and crite-ria, a discrete MCDM problem consisting of a finite set of alternatives/strategies can be evaluated in terms of multicriteria. Finally, choosing a suitable method to measure the criteria can help the evaluators and ana-lysts to process the evaluating cases.
Building a Hierarchical System for Green Engineering Industry
First of all, we establish a hierarchy system of green engineering industry for analysis and evaluation through scenario writing and brainstorming, as shown in Figure 1. Phase 1 includes our overall objectives. Second, we consider three aspects for achieving goals in phase 2, including business activities, government roles and socioeconomic effects. Third, we consider four criteria in business activities, five criteria in government roles, and three criteria in socioeconomic effects with respect to our consideration aspects that are evaluated and selected outranking listed in phase 3. All criteria considered are measured by evaluators, consisting of individuals with different viewpoints. Finally, the strat-egies of green engineering to carry on the business of participating companies are listed in phase 4. The pos-tuse process of products with eight strategies from source materials is considered to meet green engineer-ing concepts. Each enterprise will choose the strategies based on technical feasibility, financial status, manage-rial ability, relevant business situation, etc. The
defini-tions of relevant criteria and strategies are listed in Table 1.
Determination of Evaluation Criteria Weights
Because the evaluation of criteria entails diverse meanings, we cannot assume that each evaluation cri-terion is of equal importance. There are many methods that can be employed to determine weights (Hwang and Yoon 1981), such as the eigenvector method, weighted least square method, entropy method, AHP, as well as linear programming techniques for multidi-mension of analysis preference (LINMAP). The selec-tion of method depends on the nature of the problems. We use the fuzzy geometric mean method to determine the criteria weights in this paper.
Saaty (1980) originally introduced the Analytic Hi-erarchy Process (AHP) to systematically cope with com-plex problems in social system. He used the principal eigenvector of the comparison matrix to find the com-parative weight among the criteria of the hierarchy systems. If we wish to compare a set of n criteria pair-wise according to their relative importance (weights), then denote the criteria by C1, C2, . . . ,Cn and their
weights by w1, w2, . . . ,wn. If w ⫽ (w1, w2, . . . , wn) T
is given, the pairwise comparisons may be represented by matrix A of the following formulation:
共A ⫺ maxI兲w ⫽ 0 (1)
Equation 1 denotes that A is the matrix of pairwise comparison values derived by intuitive judgment for ranking order. The procedure for AHP can be summa-rized in four steps, as follows:
Step 1. Set up the decision system by decomposing the problem into a hierarchy of interrelated ele-ments.
Step 2. Generate input data consisting of pairwise com-parative judge of decision elements.
Step 3. Synthesize the judgment and estimate the rela-tive weight.
Step 4. Determine the aggregating weights of the deci-sion elements to arrive at a set of ratings for the alternatives/ strategies.
Obtaining Synthetic Utility Value
The evaluators choose a performance value for each participating company based on their subjective judg-ments. This way of estimating the achievement level of each criterion in each strategy can use the methods of fuzzy theory for treating the fuzzy environment.
Fuzzy number. Since Zadeh (1965) proposed the
fuzzy set theory and Bellman and Zadeh (1970) subse-quently described the decision-making methods in Table 1. Definitions of criteria and strategies in green engineering industry
Description Criterion
C11. Technical feasibility To measure the degree of reclaim technique
C12. Benefit–Cost effectiveness To measure the benefit–cost effectiveness from leading reclaim technique, including the value-increasing of new products and reduction of power expenditure and waste treatment costs, etc.
C13. Managerial ability To measure who possesses the managerial ability in technique of waste treatment and reclamation.
C14. New technology acceptance To measure the degree of acceptance of all inner members about reclaim technique in waste treatment and recovery that leads to company. C21. Financial support and
preferential taxes
To encourage business to engage in reclaiming the waste from process or material.
C22. Technique support and training To measure the degree of government to provide the reclaim technique and knowledge in waste that will enhance business competence.
C23. Regulation completeness To indirectly encourage business to develop and lead in reclamation techniques; it also gives protection to the legitimate companies.
C24. Knowledge providing To hold technical seminars and publish (by government or organization) to provide knowledge of reclamation techniques in waste.
C25. Waste treatment network To provide the channel of waste treatment that will prevent and reduce environment damage to ensure sustainable development.
C31. Environmental loading To measure the degree of loading from enterprise or municipal waste, including water waste, waste liquid, viscera, mud, fishbone, shell, in addition to the offensive smell of fish in aquatic products processing.
C32. Job creation and protection To measure contributions to the community from enterprise.
C33. Interest groups impacts To include the protest by civil organizations, or residents of the impact area for pollution accident.
Strategies
S1. Source reduction Material and source reduction in the early part of product manufacturing. S2. Life cycle product design Expand product lifecycles in design stage.
S3. Reducing emission and waste in manufacturing
Emission and waste reduction in manufacturing process. S4. Volume reduction, recyclable
package material
Volume reduction, using recyclable package material. S5. Green labeling product and
green image in marketing
Produce green labeling of product and establish green image in marketing will encourage consumer to buy and use it.
S6. Consumer education in PR/education
Green label products will help consumer, to value resources
S7. Collecting partnerships Establish good collecting partnerships and complete recycling network. S8. Recycling composting energy in
postuse processing
Develop new reclaiming technology transfer the waste that from produce and post-used process to new product, it will create new value to originally products and also might bring new niche to industry.
fuzzy environments, an increasing number of studies have dealt with uncertain fuzzy problems by applying fuzzy set theory. Similarly, this study includes fuzzy decision-making theory, considering the possible fuzzy subjective judgment during evaluation process.
According to Dubois and Prade (1978), a fuzzy num-ber A˜ is a fuzzy subset of a real number, and its mem-bership function is A˜( x): R3 [0,1], where x
repre-sents the criterion and is described by the following characteristics: (1)A˜( x) is a continuous mapping from
R to the closed interval [0,1]; (2)A˜(x) is a convex fuzzy
subset; and (3) A˜( x) is the normalization of a fuzzy
subset, which means that there exists a number x0such
thatA˜( x0) ⫽ 1.
According to the characteristics of triangular fuzzy numbers and the extension principle put forward by Zadeh (1975), the operational laws of two triangular fuzzy numbers, A˜⫽ (a1, a2, a3) and B˜⫽ (b1, b2, b3), are
as follows:
1. Addition of two fuzzy numbers Q.
共a1, a2, a3兲 丣 共b1, b2, b3兲 ⫽ 共a1⫹ b1, a2⫹ b2, a3⫹ b3兲
(2) 2. Subtraction of two fuzzy numbers⌰
共a1, a2a3兲 ⌰ 共b1, b2, b3兲 ⫽ 共a1⫺ b3, a2⫺ b2, a3⫺ b1兲
(3) 3. Multiplication of two fuzzy numbers R
共a1, a2, a3兲 丢 共b1, b2, b3兲 ⬵ 共a1b1, a2b2, a3b3兲 (4)
4. Multiplication of any real number k and a fuzzy number R
k 丢 共a1, a2, a3兲 ⫽ 共ka1, ka2, ka3兲 (5)
5. Division of two fuzzy numbers⭋
共a1, a2, a3兲 ⭋ 共b1, b2, b3兲 ⬵ 共a1/b3, a2/b2, a3/b1兲
(6)
Linguistic variables. According to Zadeh (1975), it is
very difficult for conventional quantification to express reasonably those situations that are overtly complex or hard to define; thus the notion of a linguistic variable is necessary in such situations. A linguistic variable is a variable whose values are words or sentences in a nat-ural or artificial language, and we use this kind of expression to compare two green engineering criteria by linguistic variables in a fuzzy environment as “abso-lutely important,” “very strongly important,” “essentially important,” “weakly important,” and “equally important” with respect to a fuzzy five-level scale. The use of linguistic
variables is currently widespread, and the linguistic effect values of strategies found in this paper are primarily used to assess the linguistic ratings given by evaluators. Further-more, linguistic variables are used as a way to measure the performance value of green engineering strategies for each criterion as “very low,” “low,” “fair,” “high,” and “very high.” In this paper we employ the triangular fuzzy num-bers to express the fuzzy scale as above.
Fuzzy weights for the hierarchy process. Buckley (1985)
was the first to investigate fuzzy weights and the fuzzy utility for the AHP technique, extending AHP by the geometric mean method to derive the fuzzy weights. In Saaty (1980), if A⫽ [aij] is a positive reciprocal matrix,
then the geometric mean of each row rican be
calcu-lated as ri⫽
冉
j⫽ 1写
maij
冊
1/m
. Here Saaty definedmaxas the
largest eigenvalue of A and the weight wias the
com-ponent of the normalized eigenvector corresponding tomax, where wi⫽ ri/(r1⫹ . . . ⫹rm).
Buckley (1985) considered a fuzzy positive recipro-cal matrix A˜ ⫽ [a˜ij], extending the geometric mean
technique to define the fuzzy geometric mean of each row r˜iand fuzzy weight w˜icorresponding to each
crite-rion as follows:
r˜i⫽ 共a˜i1 丢 · · · 丢 a˜im兲1/m;
w˜i⫽ r˜i 丢 共r˜1 丣 · · · 丣 r˜m兲⫺1 (7)
Ranking the fuzzy measure and aggregation. Sugeno
(1974) introduced the concepts of fuzzy measure and fuzzy integral, generalizing the usual definition of a measure by replacing the usual additive property with a weaker requirement, i.e., the monotonicity property with respect to set inclusion. In this section, we give a short introduction to some notions from the theory of fuzzy measure and fuzzy integral. For a more detailed account, refer to Dubois and Prade (1980), Grabisch (1995), Hougaard and Keiding (1996), among others.
Definition 1. Let X be a measurable set that is endowed
with properties of -algebra, where Ꭽ is all subsets of X. A fuzzy measure g, defined on the measurable space (X, Ꭽ), is a set function g: Ꭽ3[0,1], which satisfies the following properties: (1) g() ⫽ 0, g(X) ⫽ 1 (boundary conditions); (2) @A, B⑀ Ꭽ, if A 債 B then g(A) ⱕ g(B) (monotonicity); (3) for every sequence of subsets of X, if either A1 債 A2 債 . . . or A1 傶 A2 傶 . . ., then
limi3⬁g共Ai)⫽ g(limi3⬁Ai) (continuity).
As in the above definition, (X,Ꭽ, g) is said to be a fuzzy measure space. Furthermore, as a consequence of the monotonicity condition, we can obtain:
再
g共 A 艛 B兲 ⱖ max兵 g共 A兲, g共B兲其while the two strict cases of measure g as
再
g共 A 艛 B兲 ⫽ max兵 g共 A兲, g共B兲其g共 A 艚 B兲 ⫽ min兵 g共 A兲, g共B兲其 (9)
are called possibility measure and necessity measure, respectively. We have summarized some definitions and properties of these topics in Appendix 2.
Definition 2. Let (X, Ꭽ, g) be a fuzzy measure space.
Then the Choquet integral of a fuzzy measure g: Ꭽ3[0,1] with respect to a simple function h is defined by
冕
h共 x兲 䡠 dg ⬵冘
i⫽ 1 n 关h共 xi兲 ⫺ h共 xi⫺ 1兲兴 䡠 g共 Ai兲 (10) with the same notions as above, and h(x(0))⫽ 0.From the beginning of the application of fuzzy mea-sures and fuzzy integrals to multicriteria evaluation problems, it has been thought there was dependence between criteria. Keeney and Raiffa (1976) advocated the multiattribute multiplicative utility function, called the nonadditive multicriteria evaluation technique, to refine situations that do not conform to the assumption of independence between criteria (Ralescu and Adams 1980, Chen and Tzeng 2001, Chen and others 2000). In this paper, we apply Keeney’s nonadditive multicriteria evaluation technique using Choquet integrals to derive the fuzzy synthetic utilities of each strategy for criteria as follows.
Let g be a fuzzy measure that is defined on a power set P(x) and satisfies definition 1 above. The following characteristic is evidently,
᭙ A, B 僆 P共X兲, A 艚 B ⫽ f g共 A 艛 B兲 ⫽ g共 A兲
⫹ g共B兲 ⫹ g共 A兲 g共B兲 for ⫺1 ⱕ ⬍ ⬁ (11)
where set X⫽ {x1, x2, . . . , xn}, and the density of fuzzy
measure gi⫽ g({xi}) can be formulated as follows:
g共兵x1,x2,· · ·,xn其兲 ⫽
冘
i⫽ 1 n gi⫹ 冘
i1⫽ 1 n⫺ 1冘
i2⫽ i1⫹ 1 n gi1 䡠 gi2 ⫹ · · · ⫹ n⫺ 1 䡠 g 1 䡠 g2· · ·gn⫽ 1 冏
i写
⫽ 1 n 共1 ⫹ 䡠 gi兲 ⫺ 1冏
for ⫺1 ⱕ ⬍ ⬁ (12)For an evaluation case with two criteria, A and B, one of three cases as following will be sustained, based on the above properties:
Case 1: if ⬎ 0, i.e., g(A艛 B) ⬎ g(A)⫹ g(B),
then this implies A and B have multiplicative effect. Case 2: if ⫽ 0, i.e., g(A艛B)⫽g(A)⫹g(B), then this implies A and B have additive effect.
Case 3: if ⬍ 0, i.e., g(A艛B)⬍g(A)⫹g(B), then
this implies A and B have substitutive effect.
Let h be a measurable set function defined on the fuzzy measurable space (X,Ꭽ) and suppose that h(x1)ⱖ
h(x2)ⱖ. . .ⱖh(xn), then the fuzzy integral of fuzzy
mea-sure g(䡠) with respect to h(䡠) can be defined as follows (Ishii and Sugeno 1985).
冕
h 䡠 dg ⫽ h共 xn兲 䡠 g共Hn兲 ⫹ 关h共 xn⫺ 1兲⫺ h共 xn兲兴 䡠 g共Hn⫺ 1兲 ⫹ · · · ⫹ 关h共 x1兲 ⫺ h共 x2兲兴 䡠 g共H1兲
⫽ h共 xn兲 䡠 关 g共Hn兲 ⫺ g共Hn⫺ 1兲兴 ⫹ h共 xn⫺ 1兲 䡠 关 g共Hn⫺ 1兲
⫺ g共Hn⫺ 2兲兴 ⫹ · · · ⫹ h共 x1兲 䡠 g共H1兲 (13)
where H1⫽ {x1}, H2⫽ {x1, x2},. . .,Hn⫽ {x1,x2,. . .,xn}⫽
X. In addition, if ⫽ 0 and g1 ⫽ g2 ⫽ . . . ⫽ gnthen
h(x1)ⱖ h(x2)ⱖ ⱖ h(xn) is not necessary.
In order to clarify the operation of the fuzzy integral technique, we give numerical example in Appendix 3. On the other hand, the result of fuzzy synthetic decisions reached by each alternative is a fuzzy number. Therefore, it is necessary that the nonfuzzy ranking method for fuzzy numbers be employed during the comparison of the strategies. In previous work, the procedure of defuzzification has been to locate the best nonfuzzy performance (BNP) value. Methods of such defuzzified fuzzy ranking generally include the mean of maximal, center of area (COA), and␣-cut (Zhao and Govind 1991, Tsaur and others 1997, Tang and others 1999). Utilizing the COA method to determine the BNP is simple and practical, and there is no need to introduce the preferences of any evaluators. The BNP value of the triangular fuzzy number (LRi, MRi, URi)
can be found by the following equation:
BNPi⫽ 关共URi⫺ LRi兲 ⫹ 共MRi⫺ LRi兲兴/3 ⫹ LRi, ᭙ i
(14) For those reasons, the COA method is used in this paper to rank the order of importance of each criterion. Accord-ing to the value of the derived BNP, the evaluation of each green engineering strategy can then proceed.
In this paper when the criteria are not necessary mutually independent, we use factor analysis and the nonadditive fuzzy integral technique to find the syn-thetic utilities of green engineering strategies, and to observe the order of the synthetic utilities in different values.
Illustrative Example
In this section we take an illustrative example for evaluating the green engineering industry to
demon-strate that these methods of fuzzy measure and nonad-ditive fuzzy integral provide a good evaluation and appear to be more appropriate, especially when the criteria are not independent situations in a fuzzy envi-ronment. This section is divided into five subsections: (1) problem description, (2) determining of evaluation criteria weights, (3) determining the performance ma-trix, (4) calculating the nonadditive fuzzy synthetic util-ities, and (5) discussions.
Problem Description
The aquatic products industry is a branch of the food products industry. There are abundant fishery resources in Taiwan because of its geographical fea-tures, and aquatic products are an important dietary resource in daily life. However, for example, about 50% of harvested fish material is not edible, and how to reclaim this waste is an important challenge. In Japan, special techniques are used to process the waste from aquatic products for extracts such as fish oil, fish meal, and fish solution, which are used to make health food, forage additives, and so on, in addition to uses in agriculture and medical science.
There are about 600 aquatic products processors in Taiwan based on the Fishery Annual Report in 1998, the majority of which are small-sized enterprises. Only some of them have engaged in reclaiming waste from processing aquatic products such as fish, shrimp, and shellfish. In this study, we apply the fuzzy AHP ap-proach and the nonadditive fuzzy integral technique to evaluate the performance of green engineering strate-gies, reviewing ten companies as samples of aquatic products processors in this island.
Determining of Evaluation Criteria Weights
First, we establish the green engineering decision hierarchy frame shown in Figure 1, where the prelimi-nary classification is comprised of aspects involving business, government, and socioeconomic dimensions, with 12 criteria selected. Secondly, we have 15 evalua-tors, including staff from the government sector who are in charge of sustainable development, academic experts, company executives of aquatic products pro-cessors, members of environmental interest groups, and residents. We integrate their subjective judgments to develop the fuzzy criteria weights with respect to aspects by the fuzzy geometric mean method as in equation 7. We then derive the final fuzzy weights and nonfuzzy BNP values corresponding to each criterion, as shown in Table 2.
Determining the Performance Matrix
To determine the performance value of each strat-egy, the evaluators can define their own individual range for the linguistic variables employed in this paper according to their subjective judgments within a fuzzy scale. Under future uncertainties, the anticipated per-formance values of unquantifiable criteria cannot be specified with qualitative numerical data in qualitative evaluation pertaining to the possible achievement value of each strategy.
Let h˜ij k
represent the fuzzy performance score by the
kth evaluator of the ith strategy under the jth criterion.
Since the perception of each evaluator varies according to individual experience and knowledge, and the defi-nitions of linguistic variables also vary, we employ the fuzzy geometric mean method to integrate the fuzzy Table 2. Criteria weights for evaluating green engineering industrya
Aspects and criteria Local weights Overall weights BNP
Business activities (0.103,0.311,0.917)
Technical feasibility (0.102,0.337,1.178) (0.011,0.105,1.080) 0.398 (2)
Benefit/Cost effectiveness (0.086,0.307,1.032) (0.009,0.096,0.946) 0.350 (3)
Managerial ability (0.050,0.185,0.731) (0.005,0.058,0.670) 0.244 (8)
New technology acceptance (0.040,0.171,0.653) (0.004,0.053,0.598) 0.219 (10)
Government roles (0.128,0.373,1.080)
Financial support and preferential taxes (0.036,0.133,0.444) (0.005,0.049,0.480) 0.178 (12)
Technique support and training (0.049,0.169,0.537) (0.006,0.063,0.580) 0.216 (11)
Regulation completeness (0.087,0.251,0.738) (0.011,0.094,0.797) 0.301 (5)
Knowledge providing (0.066,0.201,0.639) (0.008,0.075,0.690) 0.258 (7)
Waste treatment network (0.085,0.246,0.735) (0.011,0.092,0.793) 0.299 (6)
Social economics effects (0.109,0.316,0.945)
Environmental loading (0.162,0.454,1.288) (0.018,0.143,1.218) 0.460 (1)
Job creation and protection (0.072,0.206,0.687) (0.008,0.065,0.649) 0.241 (9)
Interest groups impacts (0.108,0.340,0.954) (0.012,0.107,0.902) 0.340 (4)
performance score h˜ij for m evaluators, as shown in
Table 3. This is,
h˜ij⫽ 共h˜ij
1 丢 · · · 丢 h˜
ijm兲
1/m
(15) Furthermore, we employ the COA defuzzification pro-cedure to compute the BNP values of fuzzy perfor-mance score h˜ij, as shown in Table 4.
Calculating the Nonadditive Fuzzy Synthetic Utilities When the criteria are not necessarily mutually inde-pendent, in order to drive the synthetic utilities, we first exploit the factor analysis technique to extract the cri-teria in four common factors. The first factor includes five criteria: technical feasibility (C11), benefit– cost effectiveness (C12), financial support and preferential taxes (C21), technique support and training (C22), and environmental loading (C31). The second factor includes three criteria: managerial ability (C13), new technology acceptance (C14), and knowledge provid-ing (C24). The third factor also includes three criteria: waste treatment network (C25), job creation and pro-tection (C32), and interest groups impacts (C33). The final factor includes only one criterion, regulation com-pleteness (C23). The criteria within the same factor are not independent; rather they are a nonadditive mea-surement case, so we utilize the nonadditive fuzzy inte-gral technique to find the synthetic utilities of each strategy within the same factor. On the other hand, there is mutual independence between factors, and the
measurement is an additive case, so we utilize the ad-ditive aggregate method to conduct the synthetic utili-ties (see Figure 2). A more explicit procedure for con-ducting final synthetic utilities is summarized in Appendix 4.
Futhermore, we have conducted the synthetic utili-ties of each strategy using different values, with the results as shown in Table 5.
Discussions
Earlier we introduced the value representing the properties of substitution between criteria, where values range from ⫺1 to a positive infinite value (⬁). We can find the variation of synthetic utilities in differ-ent value is given. For each strategy, the synthetic utilities decrease with respect to and rapidly decrease in ⫽ 0. Furthermore, situations where ⬍ 0 are substitutive effect cases, for example, where ⫽ ⫺1. In this case we outrank the fuzzy synthetic utilities, as follows: S7ⱭS3ⱭS1ⱭS4ⱭS5ⱭS6ⱭS8ⱭS2. Moreover, when ⫽ 0, it is an additive effect case, and we out-rank the fuzzy synthetic utilities, as follows: S7ⱭS5ⱭS3ⱭS6ⱭS1ⱭS4ⱭS8ⱭS2. Finally, when ⬎ 0, there are multiplicative effect cases, for example, where ⫽ 5. Then we have different outranking fuzzy synthetic utilities, as follows: S5ⱭS7ⱭS3ⱭS6ⱭS2ⱭS8ⱭS1ⱭS4, where AⱭB means A outranks B (see Table 5).
In addition, if the criteria are independent in a fuzzy environment, conducting the fuzzy synthetic utilities by Table 3. Fuzzy performance score of green engineering strategies with respect to criteria
Strategy
Criteria
C11 C12 C13 C14 C21
S1. Source reduction (1.55,2.65,4.91) (1.63,3.06,5.30) (2.83,4.91,6.94) (2.27,4.44,6.49) (1.12,3.16,5.17) S2. Life cycle product
design (2.81,4.99,7.06) (3.71,5.91,7.67) (2.03,4.22,6.27) (2.95,5.16,7.24) (1.12,2.03,4.22) S3. Reducing emission and waste in manufacturing (2.39,4.59,6.65) (1.25,2.39,4.59) (1.25,2.67,4.83) (1.25,2.67,4.83) (1.00,1.93,4.08) S4. Volume reduction, recyclable package material (2.95,5.16,7.24) (2.27,4.44,6.49) (2.79,5.08,7.18) (4.44,6.49,8.35) (1.31,2.21,4.47) S5. Green labeling product and green image in marketing (3.11,5.34,7.42) (3.78,5.96,7.87) (2.14,4.36,6.43) (1.73,3.47,5.62) (1.00,1.25,3.32) S6. Consumer education in PR/ Education (2.67,4.83,6.88) (3.30,5.44,7.48) (3.41,5.57,7.48) (2.98,5.08,7.12) (1.46,2.90,5.12) S7. Collecting partnerships (2.67,4.83,6.88) (1.82,4.01,6.07) (2.25,4.51,6.60) (1.93,4.08,6.12) (1.25,2.39,4.59) S8. Recycling composting energy in postuse processing (3.78,5.96,7.87) (3.13,5.26,7.30) (3.30,5.44,7.48) (3.68,5.72,7.74) (1.82,4.01,6.07)
the simple additive weight method is traditionally used. This method is especially appropriate to employ in independent criteria situations. In this paper we also compute the fuzzy synthetic utilities by the simple ad-ditive weight method and obtain a different outrank-ing, as follows: S5ⱭS7ⱭS1ⱭS3ⱭS2ⱭS8ⱭS4ⱭS6.
Evaluating and planning the strategies and criteria in the green engineering industry or in another real MCDM problem can result in a vast body of data that are often inaccurate or uncertain and come from the subjective judgment by various stakeholders who are
the evaluators. Moreover, despite the correlation be-tween different criteria, the conventional MCDM meth-ods are based on the assumption of independence among criteria within the evaluating system, with the subsequent decision-making activities being performed in an additive process. However, in such complex MCDM problems, we can show through a factor analysis statistical approach that the criteria are not indepen-dent. Therefore, we demonstrate that the nonadditive fuzzy integral is more appropriate for real-world MCDM problems.
Table 4. BNP values of fuzzy performance score with respect to criteria
Strategy
BNP values of criteria
C11 C12 C13 C14 C21 C22 C23 C24 C25 C31 C32 C33
S1. Source reduction 3.035 3.328 4.894 4.399 3.148 5.406 2.914 5.309 5.406 6.321 1.758 2.077 S2. Life cycle product design 4.953 5.766 4.175 5.118 2.455 3.473 3.620 4.175 2.884 5.118 2.077 2.608 S3. Reducing emission and waste
in manufacturing
4.544 2.741 2.914 2.914 2.336 4.544 5.589 4.312 6.210 6.311 4.638 4.312 S4. Volume reduction, recyclable
package material
5.118 4.399 5.012 6.424 2.665 2.741 2.608 5.714 6.987 2.884 3.505 3.260 S5. Green labeling product and
green image in marketing
5.290 5.870 4.312 3.608 1.856 4.953 5.406 5.676 5.963 3.265 2.077 4.054 S6. Consumer education in PR/
Education
4.793 5.406 5.489 5.059 3.162 4.247 5.870 5.779 5.779 2.436 1.962 2.065 S7. Collecting partnerships 4.793 3.965 4.453 4.043 2.741 5.290 6.111 5.909 7.232 6.111 3.162 5.118 S8. Recycling composting energy
in postuse processing 5.870 5.229 5.406 5.714 3.965 6.424 6.992 6.424 5.290 4.953 3.620 3.654 Table 3. (Continued) Criteria C22 C23 C24 C25 C31 C32 C33 (3.30,5.44,7.48) (1.25,2.67,4.83) (3.24,5.39,7.30) (3.30,5.44,7.48) (4.29,6.33,8.35) (1.00,1.12,3.16) (1.00,1.55,3.68) (1.39,3.50,5.53) (1.92,3.33,5.62) (2.03,4.22,6.27) (1.39,2.52,4.75) (2.95,5.16,7.24) (1.00,1.55,3.68) (1.12,2.27,4.44) (2.39,4.59,6.65) (3.47,5.62,7.67) (2.14,4.36,6.43) (4.22,6.27,8.14) (4.36,6.43,8.14) (2.53,4.67,6.71) (2.14,4.36,6.43) (1.25,2.39,4.59) (1.12,2.27,4.44) (3.68,5.72,7.74) (5.16,7.24,8.56) (1.39,2.52,4.75) (1.82,3.22,5.48) (1.39,3.13,5.26) (2.81,4.99,7.06) (3.30,5.44,7.48) (3.59,5.76,7.67) (3.91,6.11,7.87) (1.54,3.00,5.25) (1.00,1.55,3.68) (2.14,3.91,6.11) (2.33,4.14,6.27) (3.78,5.96,7.87) (3.65,5.81,7.87) (3.65,5.81,7.87) (1.25,1.92,4.14) (1.00,1.39,3.50) (1.12,1.46,3.62) (3.11,5.34,7.42) (4.08,6.12,8.14) (3.87,5.92,7.94) (5.44,7.48,8.78) (4.08,6.12,8.14) (1.46,2.90,5.12) (2.95,5.16,7.24) (4.44,6.49,8.35) (5.08,7.12,8.78) (4.44,6.49,8.35) (3.11,5.34,7.42) (2.81,4.99,7.06) (1.92,3.33,5.62) (1.72,3.53,5.72)
Conclusions
Generally, the green engineering industry provides en-vironmental planning and decision-making problems that are essentially conflict analyses characterized by sociopo-litical, environmental, and economic value judgments. Several alternatives/strategies have to be considered and evaluated in terms of many different criteria, resulting in a vast body of data that are often inaccurate or uncertain. We introduce fuzzy numbers to express linguistic variables that consider the possible fuzzy subjective judgment of the evaluators. Furthermore, the fuzzy geometric mean tech-nique is an effective method to obtain the final fuzzy weights of each criterion.
In this study, we successfully demonstrate the non-additive fuzzy integral technique to deal with the deci-sion-making problem if the criteria are not indepen-dent. Actually, in real MCDM problems, where the criteria are not necessarily mutually independent, if we employ the simple additive aggregate method (also called the weighted mean method) to derive the final
synthetic utility, it will overestimate when the criteria have substitutive properties, or underestimate when the criteria have multiplicative properties. We provided two examples earlier to illustrate the value for the prop-erty of the criteria, which will result in a different ranking order.
In this paper, we employ fuzzy synthetic utilities to rank green engineering strategies. The strategy called “establish good collecting partnerships and complete recycling network (S7)” is the best strategy when enter-prise would like to engage in green engineering if the criteria considered are substitutive and independent. On the other hand, the strategy called “produce green labeling product and establish green image exhibiting in marketing (S5)” is the best strategy when the criteria considered are multiplicative. This is a useful informa-tion for new businesses in this industry. Furthermore, if we want to evaluate the individual synthetic utility of participating companies, the nonadditive fuzzy integral technique is an effective method.
Figure 2. Synthetic utilities with addi-tive and nonaddiaddi-tive measurements.
Acknowledgments
The authors are grateful to Dr. Virginia H. Dale, Editor-in Chief, and the two anonymous referees for providing valuable comments and suggestions, which helped to improve the presentation of this article. This work was supported in part by the National Science Council of Taiwan under grant NSC 90-2416-4-009-002.
Appendix 1
We describe some of important environmental reg-ulations as follows:
1. Basel Convention—including 52 nations, the ma-jority of the Organization of Economic Corpora-tion and Development (OECD) naCorpora-tions, signed in 1989 and taking effect in 1992, to prohibit OECD nations from exporting waste for final disposal or recycling treatment by non-OECD nations. 2. Rio Declaration—the majority of nations who
par-ticipated in the United Nations Conference on En-vironment and Development (UNCED) signed in 1992. This declaration clearly expressed the princi-ple of rights and responsibilities for environmental issues.
3. The Framework Convention on Climate Change— the majority of nations who participated in the UNCED signed in 1992. This convention includes 5 principles and 10 commitments for waste emission standards that would contribute to the greenhouse effect, such as carbon dioxide (CO2), methane
(CH4), chlorofluorocarbons (CFC5), nitrous oxide
(N2O), etc.
4. Convention of Biological Diversity—the majority of nations who participated the UNCED signed in
1992 to ensure the sustainable growth of the eco-system.
5. Agenda 21—the majority of nations who partici-pated the UNCED signed in 1992 to establish the global consensus overcoming the environmental impacts and reaching the overall sustainable devel-opment.
6. ISO 14000 — developed by the International Orga-nization for Standardization (ISO) in 1993 and declared after three years. The generic standards provide business management with a structure for managing environmental impacts. The standards comprise a broad range of environmental disci-plines, including basic environmental management system, environmental performance evaluation, au-diting, labeling, life-cycle assessment, and environ-mental aspects in product standards.
Appendix 2
According to Shafer (1976), the mathematical the-ory of evidence is based on complementary belief and
plausibility measures. This was motivated by previous
work on upper and lower probabilities by Dempster (1967).
1. Given a universal set X, assumed here to be finite, a
belief measure is a function
Bel:P共X兲 3 关0, 1兴 such that Bel(⭋) ⫽ 0, Bel(X) ⫽ 1, and Bel共 A1艛 A2艛 · · · 艛 An兲 ⱖ
冘
j Bel共 Aj兲 ⫺冘
j⬍ k Bel共 Aj艚 Ak兲 ⫹ · · · ⫹ 共⫺1兲n⫹ 1Bel共 A 1艚 A2艚 · · · 艚 An兲 (2.1)Table 5. Synthetic utilities with values
S1 S2 S3 S4 S5 S6 S7 S8 ⫺1.0 13.377 8.764 13.497 12.197 11.567 10.667 14.126 9.046 ⫺0.5 4.297 3.461 4.780 3.962 5.210 4.277 5.353 3.644 ⫺0.0 3.679 3.104 4.226 3.410 4.710 3.763 4.755 3.226 0.5 3.331 2.905 3.915 3.097 4.427 3.472 4.421 2.991 1.0 3.091 2.768 3.699 2.881 4.231 3.270 4.191 2.830 3.0 2.560 2.468 3.223 2.401 3.795 2.821 3.686 2.476 5.0 2.284 2.312 2.974 2.150 3.566 2.586 3.425 2.293 10.0 1.923 2.111 2.647 1.821 3.265 2.277 3.086 2.056 20.0 1.604 1.930 2.353 1.528 2.993 2.001 2.787 1.848 40.0 1.341 1.782 2.108 1.285 2.765 1.773 2.544 1.679 100.0 1.078 1.629 1.853 1.040 2.528 1.540 2.299 1.511 150.0 0.990 1.578 1.766 0.957 2.448 1.462 2.218 1.457 200.0 0.935 1.545 1.710 0.905 2.395 1.412 2.167 1.422 SAWa 4.041 2.298 4.020 2.125 4.467 2.073 4.252 2.227
2. Given a universal set X, assumed here to be finite, a
plausibility measure is a function
Pl: P(X) 3 关0, 1兴 such that Pl(⭋) ⫽ 0,Pl(X) ⫽ 1, and Pl(A1艚 A2艚 · · · 艚 An) ⱕ
冘
j Pl(Aj) ⫺冘
j⬍ kPl(Aj艛 Ak)⫹ · · · ⫹ 共⫺1兲n⫹ 1Pl(A 1艛 A2艛 · · · 艛 An) (2.2)3. Let A1⫽ A and A2 ⫽ A for n ⫽ 2, where A is the
complementary set of A, then the following prop-erties of belief measure and plausibility measure are satisfied.
Bel(A)⫹ Bel(A) ⱕ 1 (2.3)
Pl(A)⫹ Pl(A) ⱖ 1 (2.4)
Pl(A)⫽ 1 ⫺ Bel(A) (2.5) Bel(A)⫽ 1 ⫺ Pl(A) (2.6) 4. Belief and plausibility measures can conveniently be characterized by a function m: P(X)3[0,1] such that m(⭋) ⫽ 0 and
冘
A⑀P共X兲m共A兲 ⫽ 1. This functionis called a basic probability assignment.
5. Let a given finite body of evidence具ℑ,m典 be nested. Then the associated belief and plausibility mea-sures have the following properties for all A, B 僆
P(X):
Bel(A艚 B) ⫽ min[Bel(A), Bel(B)] (2.7) Pl(A艛 B) ⫽ max[Pl(A), Pl(B)] (2.8) 6. Let necessity measures and possibility measures be de-noted by the symbols Nec(䡠) and Pos(䡠), respec-tively. Those measures are a special branch of evi-dence theory that deals only with bodies of evidence whose focal elements are nested. There-fore, we have following basic equations of possibil-ity theory, which hold for every A, B僆 P(X)
Nec(A艚 B) ⫽ min[Nec(A), Nec(B)] (2.9) Pos(A艛 B) ⫽ max[Pos(A), Pos(B)] (2.10) 7. Since necessity measures are special belief mea-sures and possibility meamea-sures are special plausibil-ity measures, hence the following properties hold:
共a兲
冦
Nec共 A兲 ⫹ Nec共A ⱕ 1Pos共 A兲 ⫹ Pos共A兲 ⱖ 1 Nec共 A兲 ⫽ 1 ⫺ Pos(A)(2.11)
共b兲
再
min[Nec共 A兲, Nec共A兲] ⫽ 0max[Pos共 A兲, Pos共A兲] ⫽ 1 (2.12) (c)再
Nec共 A兲 ⬎ 0 f Pos共 A兲 ⫽ 1Pos共 A兲 ⬍ 1 f Nec共 A兲 ⫽ 0 (2.13) On the other hand, the concept of a fuzzy measure was introduced by Sugeno (1974). Fuzzy measures are used to assign a value to each crisp subset of the universal set to represent the degree of evidence that a particular ele-ment belongs to the set. The fuzzy measure g must satisfy three axioms as in definition 1 in the section “Ranking the fuzzy measure and aggregation,” in the main text, that is boundry conditions, monotonicity, and continuity.If a fuzzy measure g(䡠) satisfies the additive condition
g(A艛 B) ⫽ g(A) 艛 g(B), for A 艚 B ⫽ ⭋, then g(䡠) is a probability measure. It can be seen that the probability
measure is one of fuzzy measures with additivity. It follows from the above monotonicity that
再
g共 A 艛 B兲 ⱖ max兵 g共 A兲, g共B兲其g共 A 艚 B兲 ⱕ min兵 g共 A兲, g共B兲其 (2.14)
In the strict cases, we have
再
g共 A 艛 B兲 ⫽ max兵 g共 A兲, g共B兲其g共 A 艚 B兲 ⫽ min兵 g共 A兲, g共B兲其 (2.15)
The former is called the possibility measures Pos(䡠), and the later is called the necessity measure Nec(䡠), those have same meaning and properties as above evidence theory.
Furthermore, the relationship among the six types of measures employed can be depicted in Figure 3.
Appendix 3
In this article we utilize nonadditive Choquet inte-grals to aggregate fuzzy performance scores with weights. Here we give an example to compare the results with traditional independent assumption among considered criteria.
Example
Consider an employer who would like to recruit new staff for the company. The recruiting committee set three criteria, skill (C1), professional knowledge (C2) and experience (C3). Three persons, A, B, and C, are interviewed, and the scores from interviewers are summed as follows: Recruit Skill (C1) Knowledge (C2) Experience (C3) A 90 80 50 B 50 60 90 C 70 75 70
In addition, the committee set the weights as follows: 共兵c1其兲 ⫽ 共兵c2其兲 ⫽ 0.45; 共兵c3其兲 ⫽ 0.3; 共兵c1,c2其兲
⫽ 0.5; 共兵c2,c3其兲 ⫽ 共兵c1,c3其兲 ⫽ 0.9,
Applying the Choquet integral with the above fuzzy measure and the traditional weighted mean methods leads to following evaluation:
Recruit Global evaluation (Choquet integral) Global evaluation (weighted mean) A 69.50a 76.25b B 68.00 63.75 C 72.25 71.875
a. Nonindependent case among criteria:
where g(䡠) presents fuzzy measure of criteria, and C1,
C2, and C3are defined as above.
b. Independent case among criteria:
1. Find the criteria weights through normalization:
g共兵C1其兲 ⫽ g共兵C2其兲 ⫽ 0.375; g共兵C3其兲 ⫽ 0.25
2. Global evaluation⫽ 90 * 0.375 ⫹ 80 * 0.375 ⫹ 50 * 0.25⫽ 76.25
Through the above results, we can see the difference between independent and nonindependent cases based on ranking by global evaluation. If the criteria considered have nonindependent relationships (either substitutive or multiplicative), fuzzy integrals might be an appropriate method to evaluation.
Appendix 4
How to conduct the final synthetic utilities is a con-cern for analysts. Here we summarize the procedure of nonadditive fuzzy hierarchical analytic approach as fol-lows.
Figure 3. Relationship among the types of measures discussed.
g({C1})⫽ 0.45; g({C1, C2}) ⫽ 0.50g({C1, C2, C3}) ⫽ 1.0 Global evaluation
1. Setting up the hierarchical system including goals, subobjectives, criteria, alternatives/strategies. 2. Generating the relative important score of
consid-ered criteria and performance score (called hijwith
the ith strategy corresponding to the jth criterion in this article) of alternatives by subjective judg-ment of evaluators. Utilizing statistical factor anal-ysis to extract independent common factors from criteria scores will help the analyst to verify inde-pendent or nonindeinde-pendent relationships among criteria.
3. Establishing pairwise comparison matrix among criteria and then aggregating the relative weights (called wjfor the jth criterion in this article) using
a geometric mean or other appropriate method. 4. Using the fuzzy integral technique to aggregate
performance score with weights in common fac-tors, the evaluation value called uicorresponding
to the ith strategy in this article. Then employing the weighted mean method to gain the final syn-thetic utilities of each alternative. There exist inde-pendent relationships among common factors. 5. Ranking the alternatives based on their final
syn-thetic utilities will provide useful information to decision-maker.
Example
Consider one decision-making case including three independent criteria, C1, C2, C3, and four alternatives,
A1, A2, A3, A4. In addition, define hij to represent the
performance score with the ith alternative correspond-ing to the jth criteria, (a higher performance score is better), and wjto represent the weight with respect to
the j th criteria. If we have an ordinary performance matrix H⫽ [hij], and have driven the ordinary weights
w⫽ [wj] T as follows, H⫽
冤
5 1 9 7 3 5 3 7 1 5 7 9冥
w⫽冋
1 3 1 5 7 15册
T Moreover, we define ui⫽冘
j hij 䡠 wjas representing thefinal utility corresponding to the vth alternative. Then we conduct the final utilities as u⫽ [6.067 5.267 2.867 7.267]T. Finally, the ranking of alternatives based on final utilities as A4Ɑ A1Ɑ A2Ɑ A3, where AⱭ B means
A is preferential to B.
On the other hand, if we define a triangular fuzzy number as in an earlier section:
˜1⫽ 共1, 1, 3兲; ˜3 ⫽ 共1, 3, 5兲; ˜5 ⫽ 共3, 5, 7兲; ˜7 ⫽ 共5, 7, 9兲; ˜9 ⫽ 共7, 9, 9兲
then we can transfer the ordinary performance score matrix and ordinary weighting to fuzzy performance matrix H˜ ⫽ [h˜ij] and fuzzy weights w˜⫽ [w˜j]
T as in the following matrix: H˜ ⫽
冤
3 5 7 1 1 3 7 9 9 5 7 9 1 3 5 3 5 7 1 3 5 5 7 9 1 1 3 3 5 7 5 7 9 7 9 9冥
w˜ ⫽冋
3 15 5 15 7 15 1 15 3 15 5 15 5 15 7 15 9 15册
Twhere we define the fuzzy final utility as u˜i⫽ (h˜i1R w˜1
Q h˜i2R w˜2Q h˜i3R w˜3), where Q and R are addition and
multiplication operators in fuzzy number arithmetic. Then we can intuitively compute the fuzzy final utility u˜
⫽ [u˜i] as follows. u˜⫽
冢
3.000 6.067 9.667 2.067 5.267 10.07 0.867 2.867 7.133 3.267 7.267 11.67冣
Furthermore, utilizing the center-of-area method to conduct the best nonfuzzy performance value of final utility as u⫽ (6.244 5.800 3.622 7.400)T
, the ranking of alternatives based on final utilities as A4 Ɑ A1Ɑ A2 Ɑ
A3, we have the same ranking result as in the case of
crisp ordinary weights. It is important that fuzzy mea-sure and fuzzy synthetic appraisal might be appropri-ately used to evaluate the subjective semantic judg-ments or qualitative methods used in evaluating process for social science research such as in public policy, mass transit system, environmental issues.
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