Application and Development of a Fuzzy Analytic Hierarchy Process within a Capital Investment Study

Transcription

1 經 濟 與 管 理 論 叢 (Journal of Economics and Management), 2005, Vol., No. 2, Application and Development of a Fuzzy Analytic Hierarchy Process within a Capital Investment Study Yu-Cheng Tang and Malcolm J. Beynon * Capital budgeting as a decision process is among the most important of all management decisions. The importance (consequents) of the subsequent outcome may bring a level of uncertainty to the judgement-making process by the decision maker(s) in the form of doubt, hesitancy, and procrastination. This study considers one such problem, namely the selection of the type of fleet car to be adopted by a small car rental company, a selection that accounts for a large proportion of the company s working capital. With a number of criteria to consider, a fuzzy analytic hierarchy process (FAHP) analysis is undertaken to accommodate the inherent uncertainty. Developments are made to the FAHP method utilised to consider the preference results with differing levels of precision in the pairwise judgements made. Keywords: capital budgeting, degree of fuzziness, fuzzy analytic hierarchy process, sensitivity analysis, uncertainty Introduction Capital budgeting is the decision process relating to long-term capital investment programmes. The decisions made are among the most important of all management decisions (Smith, 994). However, numerous capital budgeting decision-making methods only take into account financial criteria and fail to analyse political and market risks (see Ye and Tiong, 2000). A number of research models take risks into account, but each of these focuses on different factors and has its limitations (ibid.). Received August 3, 2004, revised November 3, 2004, accepted January 3, * Authors are respectively at: Department of Accounting, National Changhua University of Education; Accounting and Finance Section, Cardiff Business School, University of Wales Cardiff.

2 208 Yu-Cheng Tang and Malcolm J. Beynon Two of the major problems that decision-makers (DMs) encounter in making capital budgeting decisions involve both the uncertainty and ambiguity surrounding the different criteria, including financial and non-financial criteria (e.g., Kaplan, 988; Weber, 993; Bromwich and Bhimani, 994; Dompere, 995; Perego and Rangone, 996; Abdel-Kader et al., 999; Abdel-Kader and Dugdale, 200; Tang, 2003). In the case of investment in new technologies, the problems of uncertainty and ambiguity assume even greater proportions because of the difficulty in estimating the impact of unexpected changes on cash flows (Franz et al., 995; Sutardi et al., 995). Moreover, it is difficult to measure the positive impact on cash flows brought about by the increase in quality and quicker reactions to changes in the market (Kaplan, 986; Franz et al., 995). Apart from uncertainty in quantitative (objective) data, the other problem of uncertainty in capital budgeting investment comes from DMs subjective opinions. These uncertainties involve incomplete information, inadequate understanding, and undifferentiated alternatives (Lipshitz and Strauss, 997; Ahn, 2000; Ahn et al., 2000). Here we focus on eliciting subjective opinions from DMs with the ultimate objective the selection of a best course of action from a set of available alternatives. Specifically, the problem considered here relates to the selection of the type of fleet car to be adopted by a small car rental company with the selection made by one of its directors. Even for this company, the decision affects a large proportion of its working capital. Hence, the importance (consequents) of the decision may bring a level of doubt and hesitancy to the decision-making process by the DM (ibid.). To operationalize this decision-making process, there exist a number of methods to elicit the DMs subjective opinions (e.g., the CW method in Wang, 997; the AHP in Saaty, 980; the Java AHP in Zhu and Dale, 200; the Randomized Expert Panel Opinion Marginalizing Procedure in Tenekedjiev et al., 2004). Among the most well known is the analytic hierarchy process (AHP) in Saaty (980). Here, to accommodate the acknowledged possible uncertainty in the subjective judgements to be made, a Fuzzy AHP (FAHP) approach is adopted. The earliest work in the FAHP appeared in van Laarhoven and Pedrycz (983), which utilised triangular fuzzy numbers (TFNs) to model the pairwise comparisons made in order to elicit weights of preference of the decision alternatives considered. Since then, FAHP-related developments have been repeatedly reported in the concomitant

3 Application and Development of a Fuzzy Analytic Hierarchy Process within a Capital Investment Study 209 literature; e.g., spatial allocation within FAHP (Wu et al., 2004), the method of FAHP and fuzzy multiple-criteria decision-making (Hsieh et al., 2004), deriving priorities from FAHP (Mikhailov, 2003), and revisiting the original FAHP (Buckley et al., 200). The utilisation of the FAHP presented in this paper brings together a number of advantageous aspects of group decision-making in a fuzzy environment. That is, while many of these aspects are present in other techniques, they are most salient in this FAHP method. Several studies have argued that the role of group decisionmaking is increasingly important (Belton and Pictet, 997; Armacost et al., 999; Ahn, 2000). The method used in this paper takes into account group decisionmaking; see Equation (2). Each matrix can involve all the DMs judgements. However, in this case study only one of the directors from the car hire company was willing to answer the questionnaire. A major consequence of the incorporation of decision-making in a fuzzy environment is the acknowledgement of and allowance for imprecision in the judgements made. Imprecision refers to the contents of the considered judgements and depends on the granularity of the language used in those judgements (Smets, 99). The method in this paper allows judgements in the judgement matrices to be given a measure of imprecision by using the degree of fuzziness δ as the quantifiable allowance for a level of imprecision in the judgement(s) made. One aspect of the FAHP method in this paper is the prevalence of and allowance for incompleteness in the judgements made by DMs. For example, if a DM is unwilling or unable to specify preference judgements in the detailed way required by the method, then a DM is able to not make a judgement in the form of a pairwise comparison between two decision alternatives (DAs). The problem of DMs being unable to provide complete information in the above circumstances is addressed by the allowance for incompleteness in the FAHP. In this study, an attempt is made to further the understanding of the FAHP method introduced in Chang (996) and developed by Zhu et al. (999) which includes the utilisation of the Extent Analysis method and the use of group decisionmaking in the FAHP. A redefining of the measure of the degree of fuzziness in the pairwise comparison judgements is made which removes the restriction on the scale values able to be utilised in future studies. A sensitivity analysis is undertaken to

4 20 Yu-Cheng Tang and Malcolm J. Beynon observe the influence of changes in the value of the degree of fuzziness on the sets of weights. Throughout this paper, there is a balance between the size of the problem considered and the developments of the existing FAHP technique used, and as such there is a level of an expositional approach to the application investigated. The structure of the rest of the paper is as follows. In Section 2, the details of the car rental selection problem are described. In Section 3, the synthetic extent method of the FAHP is presented, including the new developments. In Section 4, the results of the FAHP analysis of the car rental selection problem are illustrated. In Section 5, conclusions are given as well as directions for future research. 2 Identification of the rental car selection problem This section presents the details of the capital investment problem investigated throughout this study. The problem concerns a small car rental company and their selection of the type of fleet car to be adopted. This selection is an important investment decision, with a large proportion of working capital tied up in the final choice. Here, one of the three directors of the company agreed to make the necessary judgements to be used in a FAHP analysis of this decision problem. However a number of a priori decisions were made on the specific structure of the rental car selection problem. The first stage was the identification of the necessary criteria to be considered, which here was a consequence of a semi-structured interview with the director (hereafter DM). Following discussion with the DM concerning the nature of the application, it was decided to restrict the number of criteria to five areas: equipment, comfort, safety, image, and price (hereafter C, C 2, C 3, C 4, and C 5 ); see Appendix. Apart from the five criteria, the initial interview also identified five types of cars. This is because initial discussions with the DM indicated that five model cars are the most commonly used by them for rental purposes; see Appendix. The five types of cars are Proton Persona, Honda New Civic, Vauxhall Merit, Volkswagen Polo, and Daewoo Lanos (hereafter A, A 2, A 3, A 4, and A 5 ). These are the decision alternatives (DAs) in this case study. Given the necessary details of the criteria and DAs, the DM was asked to indicate preferences between pairs of criteria and then between pairs of alternatives

5 Application and Development of a Fuzzy Analytic Hierarchy Process within a Capital Investment Study 2 over the different criteria through the structured interview. One characteristic of this questionnaire was that it stated that the DM could leave blank any rows corresponding to comparisons for which the DM had no opinion. The questionnaire was designed to avoid pressuring the DM into an inappropriate decision by allowing for incomplete responses. This advantage is one facet of the utilization of the Chang (996) method. The linguistic variables used to make the pairwise comparisons were those associated with the standard 9-unit scale (Saaty, 980, 986, and 988); see Table. The results of the pairwise comparisons made by the DM are illustrated in Tables 2 for the five criteria and 3 for the five DAs. Table. Scale of relative preference based on Saaty (980) Intensity of preference (Numerical Value) Definition (Verbal Scale) Explanation Equally preferred; equal preference Two elements contribute equally to the objective 3 Moderately preferred; weak Experience and judgement preference of one over other slightly favour one element over another 5 Strongly preferred; essential or strong Experience and judgement favour preference 7 Very strongly preferred; demonstrated preference 9 Extremely preferred; absolute preference 2, 4, 6, 8 Intermediate values between the two adjacent judgements Reciprocals of above nonzero If an element i has one of the above numbers assigned to it when compared with element j, then j has the reciprocal value when compared with i. one element over another An element is very strongly favoured and its dominance is demonstrated in practice The evidence favouring one element over another is of the highest possible order of affirmation When compromise is needed Ratios Ratios arising from the scale If consistency were to be forced by obtaining n numerical values to span the matrix. Table 2. Pairwise comparisons between criteria based on the DM s opinions C C 2 C 3 C 4 C 5 C 3-3 /5 C 2 /3 /9 - /2 C C 4 /3 - /7 /5 C

6 22 Yu-Cheng Tang and Malcolm J. Beynon Table 3. Pairwise comparisons between alternatives over the different criteria a) C A A 2 A 3 A 4 A 5 b) C 2 A A 2 A 3 A 4 A 5 A A /8 /8 /7 - A 2-7 /3 5 A A 3 - /7 /5 5 A /8 5 A A A 5 /5 /5 /5 /5 A 5 - /5 /5 /5 c) C 3 A A 2 A 3 A 4 A 5 d) C 4 A A 2 A 3 A 4 A 5 A /8 - /7 - A /7 /5 /9 - A A /6 A /8 3 A 3 5 /7 /8 6 A A A 5 - /3 /3 /3 A 5-6 /6 /6 e) C 5 A A 2 A 3 A 4 A 5 A /3 A 2 /3 /3 3 /5 A 3 /2 3 4 /3 A 4 /5 /3 /4 /9 A Presentation of the synthetic extent FAHP method In this study the modified synthetic extent FAHP is utilised, which was originally introduced in Chang (996), developed in Zhu et al. (999), and recently applied to the selection of computer integrated manufacturing systems (Bozdağ et al., 2003). One reason for its employment is that it allows for incompleteness of the pairwise judgements made, though it is not the only FAHP approach to allow this (see Interval Probability Theory in Davis and Hall, 2003). This feature reflects its suitability in decision problems where uncertainty exists in the judgement-making process. A brief exposition of triangular fuzzy numbers and the FAHP method are given next. 3. Triangular fuzzy numbers (TFNs) In applications it is often convenient to work with TFNs because of their computational simplicity (Giachetti and Young, 997; Moon and Kang, 200), and they are useful in promoting representation and information processing in a fuzzy environment (Liang and Wang, 993). In addition, TFNs are the most utilized in FAHP studies (e.g., Kuo et al., 2002; Chiou and Tzeng, 200; Zhu et al., 999; Chang, 996). This paper adopts TFNs in the FAHP and describes their algebraic operations in the following subsection.

7 Application and Development of a Fuzzy Analytic Hierarchy Process within a Capital Investment Study 23 A TFN can be defined by a triplet (l, m, u) and the membership function can be defined by Equation () (Kaufmann and Gupta, 988; Chang, 996): 0 x < l; x l l x m; μ ( ) = m l A x m x () m x u; u m 0 x > u. 3.. Algebraic Operations on TFNs There are various operations on TFNs (see Kaufmann and Gupta, 988); in this subsection, three important operations used in this paper are illustrated. Define two TFNs A and B by the triplets A = (l, m, u ) and B = (l 2, m 2, u 2 ). Then (i) Addition: A (+) B = (l, m, u ) (+) (l 2, m 2, u 2 ) = (l + l 2, m + m 2, u + u 2 ), (ii) Multiplication: A B = (l, m, u ) (l 2, m 2, u 2 ) = (l l 2, m m 2, u u 2 ), (iii) Inverse: (l, m, u ) - (/u, /m, /l ), where represents approximately equal to. 3.2 Construction of the FAHP comparison matrices The aim of any FAHP method is to elucidate an order of preference on a number of DAs, i.e., a prioritised ranking of DAs. Central to this method is a series of pairwise comparisons, indicating the relative preferences between pairs of DAs in the same hierarchy. It is difficult to map qualitative preferences to point estimates, and hence a degree of uncertainty is associated with some or all pairwise comparison values in an FAHP problem (Yu, 2002). Using triangular fuzzy numbers with the pairwise comparisons made, the fuzzy comparison matrix X = (x ij ) n m is constructed.

8 24 Yu-Cheng Tang and Malcolm J. Beynon 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 (i.e., a comparison of the ith DA with the jth DA with respect to a specific criterion) is a fuzzy number defined as x ij = (l ij, m ij, u ij ), where m ij, u ij, and l ij are the modal, upper bound, and lower bound values for x ij, respectively. 3.3 Value of fuzzy synthetic extent Let C = {C, C 2,, C n } be a criteria set, where n is the number of criteria and A = {A, A 2,, A m } is a DA set with m the number of DAs. Let M, M C i 2 C i m,., M be values of extent analysis of the ith criteria for m DAs. Here i =, 2,, n and all the M j C i (j =, 2,, m) are triangular fuzzy numbers (TFNs). To make use of the algebraic operations on TFNs as described in subsection 3.., the value of fuzzy synthetic extent S i with respect to the ith criteria is defined as: C i S i = m j= M j C i n m i= j= j MC i (2) where represents fuzzy multiplication and the superscript represents the fuzzy inverse. The concepts of synthetic extent can also be found in Cheng (999), Kwiesielewicz (998), and Bozdağ et al. (2003). 3.4 Calculation of the sets of weight values of the FAHP To obtain the estimates for the sets of weight values under each criterion, it is necessary to consider a principle of comparison for fuzzy numbers (Chang, 996). For example, for two fuzzy numbers M and M 2, the degree of possibility of M M 2 is defined as: V(M M 2 ) = sup x y [min ( μ ( ), μ ( ) )], M x M 2 y where sup represents supremum (i.e., the least upper bound of a set) and when a pair (x, y) exists such that x y and μ ( ) = μ ( ) =, it follows that V(M M 2 ) = M x M 2 y

9 Application and Development of a Fuzzy Analytic Hierarchy Process within a Capital Investment Study 25 and V(M 2 M ) = 0. Since M and M 2 are convex fuzzy numbers defined by the TFNs (l, m, u ) and (l 2, m 2, u 2 ) respectively, it follows that V(M M 2 ) = iff m m 2 ; V(M 2 M ) = hgt (M M 2 ) = μ ( x ), M d (3) where iff represents if and only if and d is the ordinate of the highest intersection point between the μ and μ TFNs (see Figure ) and x M M 2 d is the point on the domain of μ and μ where the ordinate d is found. The term hgt is the height of fuzzy M M 2 numbers on the intersection of M and M 2. For M = (l, m, u ) and M 2 = (l 2, m 2, u 2 ), the possible ordinate of their intersection is given by Equation (3). The degree of possibility for a convex fuzzy number can be obtained from the use of Equation (4): V(M 2 M ) = hgt (M M 2 ) = l u 2 ( m u ) ( m l ) 2 2 = d. (4) M 2 M VM ( > = M) 2 d 0 l 2 m 2 l x d u 2 m u x Figure : The comparison of two fuzzy numbers M and M 2 The degree of possibility for a convex fuzzy number M to be greater than the number of k convex fuzzy numbers M i (i =, 2,, k) can be given by the use of the operations max and min (Dubois and Prade, 980) and can be defined by: V(M M, M 2,, M k ) = V[(M M ) and (M M 2 ) and and (M M k )] = min V(M M i ), i =, 2,, k. Assume that d (A i ) = min V(S i S k ), where k =, 2,, n, k i, and n is the number of criteria as described previously. Then a weight vector is given by: W = (d (A ), d (A 2 ),, d (A m )), (5)

10 26 Yu-Cheng Tang and Malcolm J. Beynon where A i (i =, 2,, m) are the m DAs. Hence each d (A i ) value represents the relative preference of each DA. To allow the values in the vector to be analogous to weights defined from the AHP type methods, the vector W is normalised and denoted: W = (d(a ), d(a 2 ),, d(a m )). (6) One point of concern, highlighted in this paper, is when two elements (fuzzy numbers, say M = (l, m, u ) and M 2 = (l 2, m 2, u 2 ) in a fuzzy comparison matrix satisfy l u 2 > 0 then V(M 2 M ) = hgt(m M 2 ) = μ ( x ), with μ ( x ) given by (Zhu et al., 999): M 2 d M 2 d μ ( x ) M2 d = l u2 ( m u ) ( m l ) 2 2 l u ; 2 0 otherwise. (7) 3.5 The modified synthetic extent FAHP: Redefinition of the proportional distance 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 -9 scale). It follows that the values l ij and u ij directly describe the fuzziness of the judgement given in x ij. In Zhu et al. (999) this fuzziness is influenced by δ (the degree of fuzziness), where m ij l ij = u ij m ij = δ. That is, δ 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 ). Given the modal value (scale value) m ij, the fuzzy number representing the fuzzy judgement made is defined by (m ij δ, m ij, m ij + δ), with its associated inverse fuzzy number subsequently described by ( m ij + δ, m, m δ ij ij ). 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 (/( + δ),, + δ) (Escobar and Moreno-Jiménez, 2000). One restriction of the method described by Zhu et al. (999) 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

11 Application and Development of a Fuzzy Analytic Hierarchy Process within a Capital Investment Study 27 Saaty -9 scale, with a number of alternative sets of scales being proposed (see Ma and Zheng, 99; Mon et al., 994; Triantaphyllou et al., 994; Beynon, 2002; Tang, 2003, and references contained therein). Here the effect of the δ value on a fuzzy number (l ij, m ij, u ij ) will be elucidated. For example, around this scale value, the domain of the fuzzy scale value measured is between 0 and (for a discussion on the notion of an unbounded scale see Jensen, 984). With the sub-domain 0 to associated with one direction of preference (e.g., j preferred to i) and to the reverse preference (e.g., i preferred to j). In the case of fuzzy scale values, there is still a need for the strict partition of the scale value domain. That is, the support of any fuzzy scale value should be in either the 0 to or the to sub-domains of δ (Moreno-Jiménez and Vargas, 993). To illustrate, using the fuzzy scale value m ij = v k = 2, following Zhu et al. (999) if δ =.5 the associated fuzzy number is (0.5, 2, 3.5). (More formally, given the entry m ij in the fuzzy comparison matrix has the k th scale value v k, then l ij and u ij have values either side of the v k scale value.) It follows that l ij = 0.5 < and implies that a sub-domain of the support (0.5, ) is meaningless with the fuzzy scale value m ij = 2. One further example shows that when δ = 2.5, the associated fuzzy number is ( 0.5, 2, 4.5), and the value 0.5 has no meaning as part of a fuzzy judgement (ratio scale measure). To remove this potential of conflict, a restraint on the l ij value needs to be constructed. (This argument assumes that the fuzziness in a preference judgement m ij does not include the possibility of reversal in the direction of the preference initially implied.) Expressed more formally, if m ij is given a fuzzy scale value such that m ij = v k then l ij is bound by l ij m ij, whose value depends on the value of δ, and is given by: vk δ ( vk lij = v k ) vk δ ; v v k k vk δ >. v v k k (8) This expression for l ij ensures that irrespective of the value of δ, the support associated with a fuzzy scale value includes no conflicting sub-domain. There is no limit on the upper bound of the fuzzy scale value, and hence the value of u ij remains u ij = v k + δ(v k+ v k ). A similar discussion and subsequent expression can be given for the inverse case (when m ij < ).

12 28 Yu-Cheng Tang and Malcolm J. Beynon Following the exposition of the FAHP extent analysis, the subjective opinions in Tables 2 and 3 need to be transferred into the fuzzy comparison matrix. An example demonstrates the transformation for δ = 0.5 as shown in Table 4. Table 4. The fuzzy comparison matrix over different criteria when δ = 0.5 C C 2 C 3 C 4 C 5 C (,, ) (2.5, 3, 3.5) - (2.5, 3, 3.5) (0.88, /5, ) C 2 (0.2857, /3, 0.4) (,, ) (/9, /9, 0.76) - (0.4, 0.5, ) C 3 - (8.5, 9, 9.5) (,, ) (6.5, 7, 7.5) (0.6667,,.5) C 4 (0.2857, /3, 0.4) - (0.333, /7, 0.538) (,, ) (0.88, /5, ) C 5 (4.5, 5, 5.5) (.5, 2, 2.5) (0.6667,,.5) (4.5, 5, 5.5) (,, ) 4 Results of the FAHP analysis in the car rental selection problem In this section, the concepts presented above are applied to the data from the car rental selection case study. The redefinition of the proportional distance between lower bound and upper bound values associated with fuzzy numbers in the FAHP in subsection 3.5 is now applied in a practical environment to reach a decision on capital investment. The application of the FAHP to the data from the car rental selection case study is described as follows. 4. The process of weight evaluation Utilising the expression given in Section 3, we apply the modified FAHP extent analysis method to the data on capital budgeting case study described above. The following stages demonstrate how to obtain the weight values for DAs. In this demonstration, the degree of fuzziness is set at 0.5. (The degree of fuzziness is not necessarily 0.5; it can be any number as explained later.) 4.. Weights evaluation for criteria In this rental car selection case study, only the judgements between criteria obtained from the DM are demonstrated. Subsequently, the judgements between DAs over different criteria are dealt with in an identical manner.

13 Application and Development of a Fuzzy Analytic Hierarchy Process within a Capital Investment Study 29 The first stage of the weight evaluation process is the aggregation of l ij, m ij, and u ij values present in the pairwise comparison matrix for the judgements between criteria. Following the fuzzy synthetic extent concept shown in Equation (2), the evaluation with respect to the five criteria in terms of the -9 scale from Saaty (980) based on δ = 0.5 can be illustrated as shown in Table 5. Table 5. Sum of rows and columns based on different criteria Row Sums Column Sums C (6.88, , ) (6.074, , ) C 2 (.7968,.9444, 2.843) (3.500, 5.000, 6.000) C 3 (6.6667, 8.000, 9.000) (.9, , 2.775) C 4 (.600,.6762,.776) (4.500, 6.000, 7.500) C 5 (2.667, 4.000, 6.000) (2.4303, , 3.6) Sum of column sums (38.428, , ) The associated S i values can be found as follows: S = (6.88, , ) (,, ) = (0.30, 0.68, 0.240); S 2 = (.7968,.9444, 2.843) ( S 3 = (6.6667, 8.000, 9.000) ( S 4 = (.600,.6762,.776) ( S 5 = (2.667, 4.000, 6.000) ( , , , , , , , , Using Equations (3) and (4) described in Section 3, one obtains: ) = (0.038, , ); ) = (0.3532, , ); ) = (0.0339, 0.039, ); ) = (0.2579, , 0.465); V (S S 2 ) = ; V (S S 3 ) = = = 0; ( ) ( ) V (S S 4 ) = ; V (S S 5 ) = 0; V (S 2 S ) = 0; V (S 2 S 3 ) = 0; V (S 2 S 4 ) = ; V (S 2 S 5 ) = 0; V (S 3 S ) = ; V (S 3 S 2 ) = ; V (S 3 S 4 ) = ; V (S 3 S 5 ) = ;V (S 4 S ) = 0;

14 220 Yu-Cheng Tang and Malcolm J. Beynon V (S 4 S 2 ) = ( ) ( ) = ; V (S 4 S 3 ) = 0; V (S 4 S 5 ) = 0; V (S 5 S ) = ; V (S 5 S 2 ) = ; V (S 5 S 3 ) = = ; ( ) ( ) V (S 5 S 4 ) =. Finally, using Equation (5) in Section 3, it follows that: d (C ) = V(S S 2, S 3, S 4, S 5 ) = min(, 0,, 0) = 0, d (C 2 ) = V(S 2 S, S 3, S 4, S 5 ) = min(0, 0,, 0) = 0, d (C 3 ) = V(S 3 S, S 2, S 4, S 5 ) = min(,,, ) =, d (C 4 ) = V(S 4 S, S 2, S 3, S 5 ) = min(0, , 0, 0) = 0, d (C 5 ) = V(S 5 S, S 2, S 3, S 4 ) = min(,, , ) = Therefore, W = (0, 0,, 0, ). Through normalization, the weight vectors are obtained with respect to the decision criteria C, C 2, C 3, C 4 and C 5 to yield W = (0, 0, 0.723, 0, ). Similarly, the transformation procedures for comparisons between criteria based on other alternative scales can be found, and the final results based on δ = 0.5 are shown below. Table 6 shows that A 4 has the largest weight while δ = 0.5. It reveals that DM prefers the Volkswagen over the others with the Daewoo second. However, the preferences for A, A 2, and A 3 as obtained from Table 6 are not known since there is no weight with these three DAs when δ = 0.5. Table 6. The sets of weight values for all fuzzy comparison matrices and the final results obtained when δ = 0.5 based on the DM s opinions Weight values for Das Weight values for criteria DAs A A 2 A 3 A 4 A 5 C C C C C Final Results Ranking orders [A 4, A 5, A = A 2 = A 3 ]

15 Application and Development of a Fuzzy Analytic Hierarchy Process within a Capital Investment Study 22 The results from Table 6 cannot fully represent the preferences for DAs. That is, since pairwise comparisons are made between criteria (or DAs), it is expected that all weights should have positive values. Zahir (999) discussed this aspect within the traditional AHP, suggesting that the DM does not favour one criterion (or DA) and ignore all others but rather places the criteria (or DAs) at various levels. Furthermore, it is suggested the -9 scale forces the concentration of the weight values, whereas only with an unbounded scale range would it be possible for the weights to overwhelmingly prefer one criterion (or DA). In addition, in accordance with the aspects of Oskamp (982), people become confident enough to make a decision (i.e., people who don t know how much they don t know). Hence the degree of fuzziness should be enlarged beyond 0.5 rather than only within certain domains. Therefore, there has been an attempt made using sensitivity analysis to observe the weight values changing on different criteria and discussed below. 4.2 Sensitivity analysis of resultant weight values The objective of a typical sensitivity analysis is to explore how the ranking of the DAs change when the input data (preference judgements and degrees of fuzziness) are changed. To illustrate this, we first consider the judgements made between the criteria shown in Table 2. Figure 2. The weights between criteria based on the DM s opinions for 0 δ 5 The five lines in Figure 2 represent the weight values associated with the different criteria. In Figure 2 the numbers (with criteria) on the δ-axis represent the degree of fuzziness appearance points with respect to each criterion. For example the degree of fuzziness δ up to 0.3 shows that C 3 has the absolute dominant

16 222 Yu-Cheng Tang and Malcolm J. Beynon preference. This means that safety is an important criterion to be considered when the DM makes decisions and the weight value is. After δ reaches 0.3, the criterion C 5 has weight. The next criterion is C which has weight as δ approaches.4, etc. The values of δ at which the criteria (or DAs) have positive weight values (non-zero) are hereafter referred to as appearance points. There is one cross point, of C 2 and C 4 where δ = 3.85: C 4 has greater preference over C 2 after δ > Hence the ranking orders become clear after δ > The final ranking order shown on the right hand of Figure 2 is C 3, C 5, C, C 4, C 2. For the fuzzy comparison matrix with judgements between criteria (see Figure 2), all five criteria have positive weights when δ is greater than 3.5. This means that if δ is less than 3.5, some criteria have no positive weights. It is suggested therefore that 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 δ at the various appearance points of criteria (or DAs) on the δ-axis. For example in Figure 2, the weights of the criteria C 5, C, C 2, and C 4 at the appearance points are 0.3,.4, 3.36, and 3.5, respectively. The largest of these appearance points is 3.5. Hence for this fuzzy comparison matrix a minimum workable δ value can be expressed as δ T C = 3.5 where the subscript T C represents the comparisons between different criteria. Figure 2 can also be compared with Table 2. In Table 2, safety (C 3 ) has most of the preferences when comparing it with other criteria apart from when it has no opinion on comparison with C. Price (C 5 ) also has most of the preferences when compared with other criteria. The least preference shown in Figure 2 (where δ < 3.85) is image (C 4 ). This can also be verified from Table 2. In the C 4 row shown in Table 2, there is no preference made by the DM between criteria. It seems as if the image (colour) of the DA is not very important when DM is making her/his decisions. Although the order changes between C 2 and C 4 after δ > 3.85, they both are still very close to each other as shown in Figure 2. In Figure 2, where the degree of fuzziness is between 0 and, only two criteria have positive weights. This result goes against the extant research (i.e., Zhu et al., 999), as δ should be in the domain 0.5 to. Referring to the comparisons between DAs on these five criteria, Figures 3a to 3e show the varied movement of the weight values for δ in the domain 0 to 5. For the fuzzy comparison matrix with judgements between DAs on different criteria (see

17 Application and Development of a Fuzzy Analytic Hierarchy Process within a Capital Investment Study 223 Figures 3a to 3e), all the DAs have positive weights when δ is greater than 4.73 (see Figure 3c). This means that if δ is less than 4.73 some DAs have no positive weights. For the comparisons between DAs with respect to individual criterion fuzzy comparison matrices (on C, C 2, C 3, C 4, and C 5 ) their minimum workable δ values are δ T = 2.75, δ = 4.55, δ = 4.73, δ = 4.55, and δ = 2.86, respectively (see T2 T3 T4 T5 Figures 3a to 3e). When considering the final results, the domain of workable δ is expressed as δ T and is defined by the maximum of the various minimum workable degrees of fuzziness throughout the problem; that is, here δ T = max( δ, δ, δ, δ, δ, δ ) TC T T2 T3 T4 T5 = 4.73 where the subscript T is the maximum of the minimum workable δ values in the six fuzzy comparison matrices. It follows that for this problem the workable region of δ is δ > 4.73 and the results on weights should possibly only be considered in the workable δ region. The use of minimum workable degree of fuzziness is intended to exclude values of δ at which there are no positive weights for the DAs. However the use of a workable value of δ is not to be strictly enforced. Table 7. The sets of weight values for all fuzzy comparison matrices and the final results obtained where δ =4.73 based on the DM s opinions Weight values for Das Weight values for criteria DAs A A 2 A 3 A 4 A 5 C C C C C Final Results Ranking orders [A 2, A 4, A 3, A 5, A ] In Table 7, the final results show that the most preferred car is the Honda (A 2 ), then the Volkswagen (A 4 ), the Vauxhall (A 3 ), the Daewoo (A 5 ) and the Proton (A ), which is the least preferred car in the DM s mind. From the comparison between the criteria in Table 7, the first two preferred criteria out of five criteria are safety and price. This means that the DM cares about the cost (car price) and safety more than other criteria. In this study, the phenomenon of rank reversal occurs when sets of weight values are obtained in the pairwise comparisons between criteria (see Figure 2). Each fuzzy comparison matrix contains uncertain and incomplete information. Therefore when these fuzzy comparison matrices are aggregated together they also involve imperfect information. The increase in degree of fuzziness also increases the uncertainty of the information.

18 224 Yu-Cheng Tang and Malcolm J. Beynon Figure 3: The weights between alternatives based on C to C 5

19 Application and Development of a Fuzzy Analytic Hierarchy Process within a Capital Investment Study 225 One thing which should be highlighted here is that the use of the minimum workable degree of fuzziness to obtain the final aggregation results does not take into account the possibility that the phenomenon of rank reversal might exist when δ becomes larger. This shows the need for more research in future work. 5 Conclusion The aim of this study is to investigate the application of the Fuzzy Analytic Hierarchy Process (FAHP) method of multi-criteria decision-making within a capital budgeting problem. The application problem in question is the selection of the type of fleet car to be adopted by a small car rental company. The important consequences of the choice outcome may confer a level of uncertainty on the decision maker, in the form of doubt, procrastination etc. This is one reason for the utilisation of FAHP, with its allowance for imprecision in the judgements made. The issue of imprecision is reformulated in this study which further allows a sensitivity analysis on the preferences weights evaluated to changes in the levels of imprecision. It is found that the DM (one director) of the car rental company successfully made the necessary judgements. This included an allowance to not make specific pairwise comparisons between all pairs of decision alternatives the incompleteness of another aspect of the possible inherent uncertainty in the decision process. The redefinition of the degree of fuzziness associated with the preference judgements made allows the change of imprecision (fuzziness) to be succinctly reported. Among the criteria in the rental car selection problem was price; from its definition this has an associated value with each alternative and hence it is a tangible criterion. Within AHP and subsequently FAHP, an ongoing question is how to effectively incorporate the tangible with the intangible criteria. Specifically to FAHP, whether the change in the degree of fuzziness may aid in this appropriateness is, again, left for future research. An important development in this study is the notion of a workable degree of fuzziness, possibly specific to the synthetic extent FAHP and it needs adoption in future studies to strengthen its appropriateness.

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