Application of the fuzzy analytic hierarchy process to the lead-free equipment selection decision

Transcription

1 Int. J. Business and Systems Research, Vol., No., 0 Application of the fuzzy analytic hierarchy process to the lead-free equipment selection decision Yu-Cheng Tang* Department of Accounting, National Changhua University of Education, No., Shi-Da Road, Changhua 00, Taiwan loistang888@gmail.com *Corresponding author Thomas W. Lin Leventhal School of Accounting, University of Southern California, 660 Trousdale Parkway, ACC 0, Los Angeles, CA , USA wtlin@marshall.usc.edu Abstract: After July 006, a major challenge that the manufacturing industry has to confront now is the effect of the lead-free equipment system selection process on companies capital expenditure decision. With capital investment, the criteria may be financial (e.g. expected cash flows) and non-financial (e.g. product quality). We use a systems approach with the fuzzy analytic hierarchy process (FAHP) method as the decision support system to help decision makers making better choices both in relation to tangible criteria and intangible criteria. Fuzzy set theory will be utilised to provide an effective way of dealing with the uncertainty of human subjective interpretation of tangible and intangible criteria. Keywords: multi-criteria decision making; systems; capital investment; leadfree equipment; FAHP; fuzzy analytic hierarchy process; uncertainty; imprecision; fuzzy synthetic extent; sensitivity analysis. Reference to this paper should be made as follows: Tang, Y-C. and Lin, T.W. (0) Application of the fuzzy analytic hierarchy process to the lead-free equipment selection decision, Int. J. Business and Systems Research, Vol., No., pp. 6. Biographical notes: Yu-Cheng Tang received her PhD in Accounting from the University of Cardiff (Wales, UK). Currently, she is an Assistant Professor at National Changhua University of Education (Taiwan). Her research interests are in the general area of financial management, in particular in the capital investment, human perceptions on decision making, green accounting, ethic position and budgetary system, etc. Specific methodologies investigates include fuzzy set theory, analytical hierarchy process and balanced scorecards. Her study is at the theoretical development and application-based level, including business and other topics. Copyright 0 Inderscience Enterprises Ltd.

2 6 Y-C. Tang and T.W. Lin Thomas W. Lin is a Professor of Accounting in the Marshall School of Business, University of Southern California, USA. He received his PhD in Accounting from the Ohio State University, MS in Accounting and Information Systems from UCLA and BA in Business Administration from National Taiwan University. His research interests include management accounting, management control systems and business information systems. Introduction Since July 006, electronics companies have been required to conform to the European Union s ban on the use of lead and five other hazardous substances in equipment sold there (Oresjo and Ling, 006). This new lead-free requirement will have a major impact on many companies manufacturing strategies. In particular, a decision to invest in new manufacturing enabling technologies that support the lead-free equipment system must take into account non-quantifiable, intangible benefits to the organisation in meeting its strategic goals (Bozda et al., 00). This study offers a system approach to this unique equipment system selection problem. Equipment system selection has been a topic for research for the last three decades. Some of the research efforts relevant to equipment system selection include integrated approaches (Wang et al., 004), the logarithmic goal programming method (Demirtas and Ustun, 00) and data envelope analysis (DEA) (Ramanathan, 00). These approaches are limited, because they can only provide a set of systematic steps for problem solving without considering the system approach to show relationships between decision factors globally (Kahraman et al., 00; Tofallis, 008). This study utilises one of multi-criteria decision making (MCDM) method, fuzzy analytic hierarchy process (FAHP) first appeared in van Laarhoven and Pedrycz (8). Previous studies have evaluated FAHP as applied to the overall issue of selection (e.g. facility, vendor or building (Bozda et al., 00; Kahraman et al., 00, 00), supplier selection (Chan and Kumar, 00) and project selection (Huang et al., 008), etc.). These different selection processes have all benefited from FAHP and have a common characteristic: the degree of fuzziness in human decision making is fixed. They do not take into account the fact that the degree of fuzziness can vary depending on the criteria being considered. Hence, the question arises, What happens when the degree of fuzziness varies? The proposed FAHP method in this study addresses this question. The method s main advantage is a more general definition of the degree of fuzziness in the scale values used to model pairwise comparisons made by DMs. More importantly, its new developments are general to any FAHP approach. Lastly, it is particularly suited to aid the lead-free equipment system selection process, as the case study will later illustrate. The remainder of this paper is organised as follows. Section describes fuzzy numbers and FAHP. Sections and 4 explore the case study of a Taiwanese manufacturing company and illustrate the use of sensitivity analysis to determine the degrees of fuzziness. Section presents our conclusions.

3 Application of the FAHP Fuzzy numbers and FAHP. Triangular fuzzy numbers (TFNs) This study adopts TFNs as they are convenient to use in applications due to their computational simplicity (Moon and Kang, 00), and useful in promoting representation and information processing in a fuzzy environment (Liang and Wang, ). The definitions and algebraic operations are described as follows. A TFN A can be defined by a triplet (l, m, u) and its membership function A( x ) can be defined by Equation () (Chang, 6; Zimmermann, 6): x l, l x m m l u x A( x), m x u () u m 0, otherwise where x is the mean value of A and l, m, u are real numbers. Define two TFNs A and B by the triplets A = (l, m, u ) and B = (l, m, u ). Then: Addition: A(+) B= ( l, m, u)(+)( l, m, u) ( l + l, m + m, u + u ) Multiplication: AB=( l, m, u) ( l, m, u) =( l, m m, u u ) Inverse: l u m l ( l, m, u ),, where represents approximately equal to.. Construction of FAHP comparison matrices This study utilises modified synthetic extent FAHP, which was originally introduced in Chang (6) and developed in Zhu et al. (). One advantage of the modified synthetic extent FAHP method is that it allows for incompleteness of the pairwise judgements made; though it is not the only FAHP approach that allows this feature (see Interval Probability Theory in Davis and Hall, 00). This allowance for incompleteness reflects its suitability in decision problems where uncertainty exists in the decisionmaking process. The aim of any FAHP method is to priorise ranking of alternatives. Central to this method is a series of pairwise comparisons, indicating the DMs relative preferences

4 8 Y-C. Tang and T.W. Lin between pairs of alternatives in the same hierarchy. The linguistic variables used to make the pairwise comparisons are those associated with the standard -unit scale (Saaty, 80) (see Table ). It is difficult to map qualitative preferences to point estimates, hence a degree of uncertainty exists with some or all pairwise comparison values in an FAHP problem (Yu, 00). Using TFNs with pairwise comparisons, the fuzzy comparison matrix X = (x ij ) nn, where x ij is an element of the comparison matrix and n is the number of rows and columns. The reciprocal property of the comparison matrix is x ji ; i, j =,, n; and x ij the subscripts i and j refer to the row and column, respectively. The pairwise comparisons are described by values taken from a pre-defined set of ratio scale values as presented in Table. The ratio comparison between the relative preference of elements indexed i and j on a criterion can be modelled through a fuzzy scale value associated with a degree of fuzziness. Then, an element of X, x ij is a fuzzy number defined as x ij = (l ij, m ij, u ij ), where l ij, m ij, u ij are the lower bound, modal, and upper bound values for x ij, respectively. Table Scale of relative preference based on Saaty (80) Numerical value Definition Equally preferred Moderately preferred Strongly preferred Very strongly preferred Extremely preferred, 4, 6, 8 Intermediate values between the two adjacent judgements. Value of fuzzy synthetic extent Let C = {C, C,, C n } be a criteria set, where n is the number of criteria and A = {A, A,, A m } be a decision alternative set, where m is the number of decision alternatives. m j Let M C i, M C i,, M C, i =,,, n where all the M (j =,,, m) are TFNs. To i Ci make use of the algebraic operations described in Section. on TFNs, the value of fuzzy synthetic extent S i with respect to the ith criteria is defined: m n m j Si M C M i j i j j C () i where represents fuzzy multiplication and the superscript represents the fuzzy inverse. The concepts of synthetic extent are also found in Cheng () and Bozda et al. (00)..4 Calculating sets of weighted values of FAHP To obtain estimates for sets of weight values under each criterion, one must consider a principle of comparison for fuzzy numbers (Chang, 6). For example, for two fuzzy

5 Application of the FAHP numbers, M and M, the degree of possibility that M M is defined as: sup min M ( ), M ( ) V M M x y x y where sup represents supremum, it follows that V(M M ) =. Since M and M are convex fuzzy numbers defined by the TFNs (l, m, u ) and (l, m, u ), respectively, it follows: V M M iff m m V M M hgt MM M x d () where iff represents if and only if, d is the ordinate of the highest intersection point between the M and M TFNs (see Figure ), and x d is the point in the domain of M and M where the ordinate d is found. The term hgt is the height of fuzzy numbers on the intersection of M and M. For M = (l, m, u ) and M = (l, m, u ), the possible ordinate of their intersection is given by Equation (). The degree of possibility for a convex fuzzy number can be obtained from Equation (4): l u V M M M M d hgt m u m l The degree of possibility for a convex fuzzy number M to be greater than the number of k fuzzy numbers M i (i =,,, k) is given by the use of the operations max and min (Dubois and Prade, 80) and is defined by: VM M, M,, MkV M M and M M andand M Mk min V M M, i,,, k Assume that d(a i ) = min V(S i S k ), where k =,,, n, k i, and n is the number of criteria as described previously. Then, a weight vector is given by:,,, m W d A d A d A () where A i (i =,,, m) are the m decision alternatives. Hence, each d(a i ) value represents the relative preference of each decision alternative and the vector W is normalised and denoted:,,, m W d A d A d A (6) i (4) Figure The comparison of two fuzzy numbers M and M

6 40 Y-C. Tang and T.W. Lin If two fuzzy numbers, say M = (l, m, u ) and M = (l, m, u ), in a fuzzy comparison matrix satisfy l u > 0, then V(M M ) = hgt(m M ) = M ( x ) d, where M ( x ) d is given by (Zhu et al., ): M x d. Degree of fuzziness l u, l u ( m u) ( ml) () 0, otherwise Referring back to fuzzy numbers, for example, an element x ij in a fuzzy comparison matrix, if DA i is preferred to DA j then m ij takes an integer value from two to nine (from the scale). More formally, given the entry m ij in the fuzzy comparison matrix has the kth scale value v k, then l ij and u ij have values either side of the v k scale value. It follows the values l ij and u ij directly describe the fuzziness of the judgement given in x ij. In Zhu et al. () this fuzziness is influenced by a (degree of fuzziness) value, where m ij l ij = u ij m ij =. That is, the value of is a constant and is considered an absolute distance from the lower bound value (l ij ) to the modal value (m ij ) or the modal value (m ij ) to the upper bound value (u ij ) (see Figure ). Given the modal value m ij (v k ), the fuzzy number representing the fuzzy judgement made is defined by (m ij, m ij, m ij + ), with its associated inverse fuzzy number subsequently described by (/( mij ), / m ij, /( mij )). In Figure, the definition of the fuzzy scale value given in Zhu et al. () is that the distance from m ij (= v k ) to v k is equal to the distance from m ij to v k+ ( distance). In the case of m ij given a value of one (m ij = ) off the leading diagonal (i j), the general form of its associated fuzzy scale value is defined as (/( ),, + ). For example, given m ij =, the fuzzy number will be (0.666,,.) when = 0.. Figure Description of the degree of fuzziness according to Zhu et al. () 0 v k v k v k + l ij m ij u ij

7 Application of the FAHP 4 One restriction of the method described by Zhu et al. () is that it assumes equal unit distances between successive scale values. However, with respect to the traditional AHP there has been a growing debate on the actual appropriateness of the Saaty scale, with a number of alternative sets of scales being proposed (see Beynon (00) and references contained therein). Here, is defined as a proportion (relative) of the distance between successive scale values. Hence, the associated fuzzy scale value for the case of m ij given scale value v k is defined as:,, v v v v v v v (8) k k k k k k k Therefore, m ij = v k, l ij = v k (v k v k ) and u ij = v k + (v k+ v k ). When the maximum scale value v is used, consideration has to be given to its associated upper bound values. That is given m ij = v k then it is not possible to use the previously defined expressed, instead of u ij = u = v + ( v v8) /( v8 v). The reason is that there is no v 0 (v + ) value to use, so instead the new expression takes into account the difference between successive scale values (for the details of degree of fuzziness, see Tang and Beynon (00))..6 Sensitivity analysis of resultant weight values Sensitivity analysis is a fundamental concept for the effective use and implementation of quantitative decision models (Dantzig, 6). The objective of sensitivity analysis here is to find out when the input data (preference judgements and degrees of fuzziness) are changed into new values, how the ranking of the DAs will change. This study will utilise sensitivity analysis to measure degrees of fuzziness and will explain it in Section 4. Application of FAHP to a lead-free equipment system selection problem This section presents the case study, electronics company (EL), including the details of its capital investment problem and the solution proposed by FAHP.. Description of EL EL is a listed company in Taiwan. It aims to create a safe, convenient and obstacle-free environment, provide better life quality, safety and comfort for customers and constantly devote itself to research and development of high-quality, low-price, competitive products. To follow all the procedures in ISO 400 and OHSAS 800 regulations, to prevent calamities, and to control air and waste pollution, EL s top management decided to implement a new lead-free equipment system. EL s capital investment decisions are normally made by three senior managers: the finance department manager, the engineering department manager and the manufacturing department manager (hereafter referred to as DMs). In this particular decision, only three well-known suppliers, A, A and A, provide price quotes. The three types of lead-free equipment system they provide, Equipment A, Equipment A and Equipment A, are the decision alternatives in this case study.

8 4 Y-C. Tang and T.W. Lin. Details of equipment system selection problem The lead-free equipment system has the following components: equipment, parts, service support, education and training support, and pollution control function. First, we identified which selection criteria should be considered through a semi-structured interview with the DMs. The DMs decided to restrict the criteria to seven areas based on EL s requirements. Details of these criteria and their sub-criteria are shown in Table. Table Information table of the lead-free equipment Equipment Criteria Sub-criteria Equipment A Equipment A Equipment A C C,00,000,00,000,00,000 C Moderate Expensive Relatively cheap C C Convenient Inconvenient Convenient C High compatible Low compatible Middle compatible C Already space reserved Domestic/abroad have service centres Additional augmentation needed Need agents/difficult to maintain C 4 hr/easy difficult hr/easy C hr 6 8 hr 4 hr C 4 C 4 Excellent Excellent Excellent Training days C 4 days 4 days days Additional augmentation needed Company made/easy to maintain C C Small Large Middle C 6 C Air pollution Low Low Low Noise pollution Low Low Low Water pollution No No No C 6 Capable Capable Capable Augmentation Yes Yes Yes Easy to upgrade Yes Yes Yes Reserved the space Yes Yes Yes C C Good Medium Relatively low C Above Below 0 Between 0 and C Good Medium Medium Brief descriptions of the seven criteria and some DMs opinions of how well the equipment alternatives meet the criteria are listed below: C : Acquisition cost of equipment and parts C : price of the equipment (NT$); C : price of parts; C : convenience to get parts. The DMs want to minimise the price of the equipment and the price of its parts, and they want accessibility of replacement parts. Equipment A is the most expensive equipment in both acquisition cost and replacement parts cost.

9 Application of the FAHP 4 C : Compatibility C : The DMs prefer the new equipment to be downwardly compatible. Equipment A is highly compatible with EL s existing equipment. C : Response and maintenance time C : service ability (The numbers of distributor service centres and the distance of distributor service centres); C : maintenance ability (maintenance time by hours); C : time to arrive. For example, in Table, C, Equipment A needs 4 hr for maintenance, while Equipment A needs hr for maintenance. Equipment A is difficult to maintain, and the supplier has difficulty arriving at EL in a short period of time. C 4 : Education and training C 4 : install; C 4 : education and training. The DMs are concerned about how much training is necessary for the installation and testing of the equipment. They also care about the quantity and quality of education and training that suppliers are willing to provide. C : Equipment size and pollution control C : space of the equipment; C : environmental assessment. The DMs prefer equipment with less air pollution, noise pollution and water pollution. C 6 : Upgrades and expansibility C 6 : research and development ability. The DMs want to know the extent of suppliers research and development facilities, relatively easy to upgrade to high-level products and reserved the space to expand. C : Supplier and brand reputation C : brand; C : quantity of customers of the supplier at present; C : financial situation of the supplier. A has a good reputation and already supplies more than companies. Using a systems approach with the structured questionnaire, the DMs first indicated their preferences between pairs of criteria. This study allows DMs to leave blank any comparison for which they had no opinion or preference. Thus, by allowing for incomplete responses, the questionnaire avoided pressuring the DMs into an inappropriate decision. Tables and 4 illustrate the results of the pairwise comparisons between the seven criteria and the sub-criteria, respectively. Table (a) (o) shows further fuzzy comparison matrices for pairwise comparisons between equipment alternatives on each of the criteria and sub-criteria. For example, in Table, the three DMs made judgements on C compared to C 4, with the pairwise comparisons of (,, and ). The fuzzy scale values and represent moderately preferred and strongly preferred, respectively, as shown in Table. For the comparisons between the sub-criteria shown in Table 4, the criteria C and C 6 are left out, since they have only one sub-criterion. Table 6 shows the fuzzy preference comparison matrix for each pair of the seven criteria from Table with the degree of fuzziness, the distances between successive scale values are equal, that is, v k v k = v k+ v k.

10 44 Y-C. Tang and T.W. Lin Table Pairwise comparisons between criteria based on three DMs opinions C C C C 4 C C 6 C C / / /8 / /6 / /6 / / / / C 8 / / C 6 / / / /6 C 4 / / / / / / / / / / / / / C / 6 / / / / / / / C 6 /6 / / 6 / / / C / /8 / /8 / / / / / / / / / / Table 4 (a) (e) Comparisons between sub-criteria a) C C C C b) C C C C c) C 4 C 4 C 4 d) C C C e) C C C C C / / C / / C 4 C C / / / C / / C / / C / / C 4 / / / / / / / / C / C / C / / C / / / /

11 Application of the FAHP 4 Table (a) (o) Comparisons between decision alternatives over the different sub-criteria A A h) C4 C4 A A A / / / / A 6 / A / / /6 / A / / / / / g) C C A A o) C C A A A / / / A / / / / / / A / / / / / / A / / / f) C C A A n) C C A A A / / / / / A / / /4 A / / / / A / / / / /6 /6 e) C C A A m) C C A A A / / / A 4 / / / 6 A / / / / / / A / / /6 / / /4 d) C C A A l) C6 C6 A A A / / / / A / / / / A / / / A / / / c) C C A A k) C C A A A / / / / /6 / A / / / 6 6 A / / / A / / /6 / / / b) C C A A j) C C A A A / /6 / / /8 /4 A / / 8 4 / / / A / / / 6 A / / a) C C A A i) C4 C4 A A

12 46 Y-C. Tang and T.W. Lin Table 6 The fuzzy comparison matrix version of comparisons between criteria C C C C 4 C C 6 C C (,, ) C (,, + ) (,, + ) (8, 8, 8 + ) C (,, + ) (/( + ),, + ) (6, 6, 6 + ) C 4 (/( + ), /, /( )) (/( + ), /, /( )) (/( + ), /, /( )) C (,, + ) (/( + ), /, /( )) (6, 6, 6 + ) C 6 (,, + ) (,, + ) (,, + ) C (/( + ), /, /( )) (/( + ),, + ) (,, + ) (/( + ), /, /( )) (/( + ), /, /( )) (/(8 + ), /8, /(8 )) (,, ) (,, + ) (/( + ), /, /( )) (/( + ), /, /( )) (/( + ), /, /( )) (/( + ), /, /( )) (/( + ), /, /( )) (/( + ), /, /( )) (/( + ), /, /( )) (/( + ), /, /( )) (,, + ) (/( + ),, + ) (/(6 + ), /6, /(6 )) (/(8 + ), /8, /(8 )) (/( + ), /, /( )) (/(8 + ), /8, /(8 )) (/( + ), /, /( )) (/( + ),, + ) (/(6 + ), /6, /(6 )) (/( + ), /, /( )) (,, + ) (,, + ) (,, ) (/( + ), /, /( )) (/( + ), /, /( )) (/( + ), /, /( )) (/( + ), /, /( )) (/( + ), /, /( )) (,, + ) (/( + ), /, /( )) (/( + ), /, /( )) (6, 6, 6 + ) (/( + ), /, /( )) (/( + ), /, /( )) (/( + ), /, /( )) (,, + ) (,, + ) (,, + ) (,, + ) (,, + ) (,, + ) (,, + ) (,, + ) (,, + ) (,, ) (,, + ) (/( + ), /, /( )) (,, + ) (,, + ) (/( + ), /, /( )) (,, + ) (/( + ), /, /( )) (/( + ), /, /( )) (/( + ),, + ) (/( + ), /, /( )) (,, + ) (/(6 + ), /6, /(6 )) (,, + ) (,, + ) (,, + ) (/( + ), /, /( )) (/( + ), /, /( )) (/( + ), /, /( )) (/( + ), /, /( )) (/( + ),, + ) (6, 6, 6 + ) (,, + ) (,, + ) (,, + ) (,, + ) (/( + ), /, (/(6 + ), /6, /( )) /(6 )) (/( + ), /, /( )) (,, + ) (/( + ), /, /( )) (,, ) (,, + ) (/( + ), /, /( )) (/( + ), /, /( )) (/( + ), /, /( )) (/( + ),, + ) (/( + ), /, /( )) (/( + ), /, /( )) (,, + ) (/( + ), /, /( )) (/( + ), /, /( )) (,, + ) (,, + ) (,, ) (/( + ), /, /( )) (/( + ), /, /( )) (/( + ), /, /( )) (,, + ) (/( + ),, + ) (/( + ), /, /( )) (8, 8, 8 + ) (,, + ) (8, 8, 8 + ) (,, + ) (,, + ) (,, + ) (,, + ) (,, + ) (/( + ),, + ) (,, + ) (/( + ),, + ) (,, + ) (,, + ) (,, + ) (,, + ) (,, )

13 Application of the FAHP 4 4 Using sensitivity analysis to determine degrees of fuzziness This section explains how to use sensitivity analysis to measure degrees of fuzziness. Figure shows the sensitivity of the varying degrees of fuzziness for Table judgement data of the pairwise comparisons between seven criteria. Variable represents the degree of fuzziness. There are seven lines in Figure that represent the weighted values of the different criteria. The numbers (with criteria) on the -axis represent the degrees of fuzziness with respect to each criterion. For example, the degrees of fuzziness up to 0.46 (in Figure, -axis) shows that C has the absolute dominant preference (and hence, the least amount of fuzziness). This result means that C is an important criterion to be considered when the DMs make decisions, so the weight value is. After reaches 0.46, the criterion C has the next priority weight. The next criterion is C 6, which has a priority weight as approaches 0.8, etc. The values of at which the criteria have positive weight values (non-zero) are hereafter referred to as appearance points. For judgements between criteria, all seven criteria have positive weights when is greater than. This means that if is less than, some criteria will have no positive weights. Zahir () discusses this aspect within traditional AHP, suggesting that DMs do not favour one criterion and ignore all others, but rather place criteria at various scales. In addition, when pairwise comparisons are made between criteria, it is expected that all weights should have positive values. Therefore, it is useful to choose a minimum workable degree of fuzziness. The expression minimum workable degree of fuzziness is defined as the largest of the values of at the various appearance points of criteria on the -axis. In this case, the minimum workable degree of fuzziness for decisions between criteria is. In general, when considering the final results, the domain of workable is expressed as and is defined by the maximum of the various minimum workable degrees of T fuzziness throughout the problem; that is T = max(, T, C T,, T T n, ), where the - Tn subscript T is the maximum of the minimum workable values in the n + (T c matrix) fuzzy comparison matrices. Figure Set of weight values from judgements on seven criteria over 0 C 0. 0 C C 4 C C C 6 C

14 48 Y-C. Tang and T.W. Lin In the comparisons between sub-criteria (C, C, C 4, C and C ), their minimum workable values are = 0.4, T =.0, T =.04, T =.8 and T =., respectively (see 4 T Figure 4(a) (e)). For the comparisons between equipment alternatives with respect to individual sub-criteria, the resulting fuzzy comparison matrices (on C, C, C, and C ) reveal their minimum workable values are 4.0,.6,,.4,., 4,, 0, 0.6,., 0.6,.,.,.8 and.8, respectively (see Figure (a) (o)). For the maximum of the minimum workable values is = max(, 0.4,.0,.04, T.8,., 4.0,.6,,.4,., 4,, 0, 0.6,., 0.6,.,.,.8,.8) = 4.0. The weights results should possibly be considered only in the workable region of > 4.0. The minimum workable degree of fuzziness excludes values of at which there are no positive weights for the three equipment alternatives. When = 4.0, the weight values for the seven criteria are 0.6, 0.4, 0., 0., 0.8, 0.6 and 0.08, respectively (see Table ). Subsequently, the weight values for the sub-criteria based on each criterion are derived from Table 4, and the weight values are listed in Table. For example, for C, the weight values for the comparisons between C, C and C are 0.80, 0.0 and 0.4, respectively. The last column in Table shows the weight values for equipment alternatives over the different sub-criteria. For instance, for C, the weight values for the comparisons by the three DMs are 0.4, and 0.66, respectively. The final results of this case study reveal two clear decisions made by the DMs of EL, which are the most preferred decision alternative and the most important criterion. The most preferred lead-free alternative is Equipment A ; Equipment A is the next preferred alternative, while Equipment A is the least preferred alternative (see Table ). This preference for Equipment A is found in the weight values for not only the criteria, but also the sub-criteria shown in Table. In those sub-criteria, apart from C, C, C 4, C and C, Equipment A has greater weight values than the other two alternatives. The most preferred criterion is C, that is, the compatibility between new and old equipment (see Table ). This result means that the DMs care more about the compatibility between new and old equipment than any other single criterion. Figure 4 (a) (e) Comparisons between sub-criteria over 0 (a)

15 Application of the FAHP 4 Figure 4 (a) (e) Comparisons between sub-criteria over 0 (continued) (b) (c) (d) (e)

16 0 Y-C. Tang and T.W. Lin Figure (a) (o) Graphs of weight values between the decision alternatives on sub-criteria (a) (b) (c) (d)

17 Application of the FAHP Figure (a) (o) Graphs of weight values between the decision alternatives on sub-criteria (continued) (e) (f) (g) (h)

18 Y-C. Tang and T.W. Lin Figure (a) (o) Graphs of weight values between the decision alternatives on sub-criteria (continued) (i) (j) (k) (l)

19 Application of the FAHP Figure (a) (o) Graphs of weight values between the decision alternatives on sub-criteria (continued) (m) (n) (o) Table The sets of weight values for all fuzzy comparison matrices and the final results obtained where = 4.0 based on the DM s opinions Weight values for criteria C 0.6 Weight values for subcriteria Weight values for decision alternatives C 0.80 [0.4, 0.006, 0.66] C 0.0 [0.480, 0.484, 0.8] C 0.4 [0.4, 0.04, 0.48] C 0.4 C [0., 0.00, 0.] C 0. C 0.4 [0.464, 0.066, 0.480] C 0.8 [0.6, 0.08, 0.468] C 0.0 [0.44, 0.6, 0.4] C 4 0. C [0., 0., 0.]

20 4 Y-C. Tang and T.W. Lin Table The sets of weight values for all fuzzy comparison matrices and the final results obtained where = 4.0 based on the DM s opinions (continued) Weight values for subcriteria Weight values for decision alternatives Weight values for criteria C [0.668, 0., 0.6] C 0.8 C [0.0, 0.066, 0.4] C 0. [0.48, 0., 0.6] C C 6 [0.06, 0.0, 0.4] C 0.08 C 0.0 [0.4, 0., 0.4] C 0.40 [0.64, 0.4, 0.8] Final results [0.40, 0.4, 0.6] Final ranking [A, A, A ] C 0.84 [0.4, 0., 0.80] Conclusions This study has shown that FAHP has the potential to benefit the manufacturing industry by minimising any negative effects of being forced to invest in the lead-free equipment system by new regulations. Decisions about capital expenditures required by new laws, like the lead-free requirement system, can be particularly complex for DMs. Due to the uncertain and fuzzy nature of such complex problems, FAHP allows for imprecision in judgement. This study takes FAHP even further by including more allowances for imprecision in its model. Most importantly, it allows for variations in degrees of fuzziness. Previous studies assumed fixed fuzziness. Fuzziness of a decision can change depending on the criteria being considered by the DM. This study uses sensitivity analysis to find the degree of fuzziness appropriate to the weight values of decision alternatives and also allows for imprecision by not forcing DMs to choose between alternatives or criteria when they have no preference. In our case study, we used criteria based completely on subjective opinions elicited directly from the DMs and found that the DMs successfully made judgements regarding which lead-free equipment system to purchase utilising FAHP. The results of this case study suggest that a suitable degree of fuzziness, that is, the maximum of the minimum workable values of, is necessary to obtain the sets of weights. Moreover, where there are different maximums of the minimum workable values of for different scales or different models of aggregation, as in the comparisons in this study, we suggest that the highest of the maximums of the minimum workable values of should be chosen. In summary, we provide a real-world example to illustrate a new MCDM method with the systems approach for selecting lead-free equipment system when company confronts the different policies from government. This suggested FAHP method adequately addresses the inherent uncertainty and imprecision of the human decisionmaking process. This study s contribution to FAHP methodology is the demonstration that fuzziness should not be fixed only at 0.. DMs from manufacturing companies faced

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