yu-cheng tang*

Int. J. Business and Systems Research, Vol. 5, No. 1, 2011 35

Copyright © 2011 Inderscience Enterprises Ltd.

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. 2, Shi-Da Road, Changhua 500, Taiwan E-mail: [email protected] *Corresponding author

Thomas W. Lin Leventhal School of Accounting, University of Southern California, 3660 Trousdale Parkway, ACC 109, Los Angeles, CA 90089-0441, USA E-mail: [email protected]

Abstract: After 1 July 2006, 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; lead-free 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. (2011) ‘Application of the fuzzy analytic hierarchy process to the lead-free equipment selection decision’, Int. J. Business and Systems Research, Vol. 5, No. 1, pp.35–56.

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.

36 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.

1 Introduction

Since 1 July 2006, 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, 2006). 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., 2003). 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., 2004), the logarithmic goal programming method (Demirtas and Ustun, 2007) and data envelope analysis (DEA) (Ramanathan, 2005). 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., 2003; Tofallis, 2008).

This study utilises one of ‘multi-criteria decision making’ (MCDM) method, fuzzy analytic hierarchy process (FAHP) first appeared in van Laarhoven and Pedrycz (1983). Previous studies have evaluated FAHP as applied to the overall issue of selection (e.g. facility, vendor or building (Bozda et al., 2003; Kahraman et al., 2003, 2007), supplier selection (Chan and Kumar, 2007) and project selection (Huang et al., 2008), 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 2 describes fuzzy numbers and FAHP. Sections 3 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 5 presents our conclusions.

Application of the FAHP 37

2 Fuzzy numbers and FAHP

2.1 Triangular fuzzy numbers (TFNs)

This study adopts TFNs as they are convenient to use in applications due to their computational simplicity (Moon and Kang, 2001), and useful in promoting representation and information processing in a fuzzy environment (Liang and Wang, 1993). 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 (1) (Chang, 1996; Zimmermann, 1996):

,

( ) ,

0, otherwise

A

x ll x m

m lu x

x m x uu m

(1)

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 = (l1, m1, u1) and B = (l2, m2, u2). Then:

1 Addition:

1 1 1 2 2 2

1 2 1 2 1 2

(+) = ( , , )(+)( , , ) ( + , + , + )A B l m u l m u

l l m m u u

2 Multiplication:

1 1 1 2 2 2

1 2 1 2 1 2

A B= ( , , ) ( , , )= ( , , )l

l m u l m ul m m u u

Inverse:

11 1 1

1 1 1

1 1 1( , , ) , ,l m uu m l

where represents ‘approximately equal to’.

2.2 Construction of FAHP comparison matrices

This study utilises modified synthetic extent FAHP, which was originally introduced in Chang (1996) and developed in Zhu et al. (1999). 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, 2003). This allowance for incompleteness reflects its suitability in decision problems where uncertainty exists in the decision-making 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


between pairs of alternatives in the same hierarchy. The linguistic variables used to make the pairwise comparisons are those associated with the standard 9-unit scale (Saaty, 1980) (see Table 1).

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, 2002). Using TFNs with pairwise comparisons, the fuzzy comparison matrix X = (xij)n n,where xij is an element of the comparison matrix and n is the number of rows and columns. The reciprocal property of the comparison matrix is 1

ijji xx ; i, j = 1, …, n; and

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 1. The ratio comparison between the relative preference of elements indexed i and jon a criterion can be modelled through a fuzzy scale value associated with a degree of fuzziness. Then, an element of X, xij is a fuzzy number defined as xij = (lij, mij, uij), where lij, mij, uij are the lower bound, modal, and upper bound values for xij, respectively.

Table 1 Scale of relative preference based on Saaty (1980)

Numerical value Definition

1 Equally preferred 3 Moderately preferred 5 Strongly preferred 7 Very strongly preferred 9 Extremely preferred

2, 4, 6, 8 Intermediate values between the two adjacent judgements

2.3 Value of fuzzy synthetic extent

Let C = {C1, C2, …, Cn} be a criteria set, where n is the number of criteria and A = {A1,A2, …, Am} be a decision alternative set, where m is the number of decision alternatives. Let 1

iCM , 2iCM ,…,

i

mCM , i = 1, 2, …, n where all the

i

jCM (j = 1, 2, …, m) are TFNs. To

make use of the algebraic operations described in Section 2.1 on TFNs, the value of fuzzy synthetic extent Si with respect to the ith criteria is defined:

1

1 1 1i i

mm nj j

C Cj i j

Si M M (2)

where represents fuzzy multiplication and the superscript 1 represents the fuzzy inverse. The concepts of synthetic extent are also found in Cheng (1999) and Bozdaet al. (2003).

2.4 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, 1996). For example, for two fuzzy


numbers, M1 and M2, the degree of possibility that M1 M2 is defined as:

1 21 2 sup min ( ), ( )M Mx y

V M M x y

where sup represents supremum, it follows that V(M1 M2) = 1. Since M1 and M2 are convex fuzzy numbers defined by the TFNs (l1, m1, u1) and (l2, m2, u2), respectively, it follows:

1 2 1 21 iff V M M m m

12 1 1 2hgt M dV M M M M x (3)

where iff represents ‘if and only if’, d is the ordinate of the highest intersection point between the

1M and 2M TFNs (see Figure 1), and xd is the point in the domain of

1M

and 2M where the ordinate d is found. The term hgt is the height of fuzzy numbers on

the intersection of M1 and M2. For M1 = (l1, m1, u1) and M2 = (l2, m2, u2), the possible ordinate of their intersection is given by Equation (3). The degree of possibility for a convex fuzzy number can be obtained from Equation (4):

1 22 1 1 2

2 2 1 1hgt

l uV M M M M d

m u m l (4)

The degree of possibility for a convex fuzzy number M to be greater than the number of kfuzzy numbers Mi (i = 1, 2, …, k) is given by the use of the operations max and min (Dubois and Prade, 1980) and is defined by:

1 2 1 2, , , and and and

min , 1, 2, ,k k

i

V M M M M V M M M M M M

V M M i k

Assume that d (Ai) = min V(Si Sk), where k = 1, 2, …, n, k i, and n is the number of criteria as described previously. Then, a weight vector is given by:

1 2, , , mW d A d A d A (5)

where Ai (i = 1, 2, …, m) are the m decision alternatives. Hence, each d (Ai) value represents the relative preference of each decision alternative and the vector W is normalised and denoted:

1 2, , , mW d A d A d A (6)

Figure 1 The comparison of two fuzzy numbers M1 and M2


If two fuzzy numbers, say M1 = (l1, m1, u1) and M2 = (l2, m2, u2), in a fuzzy comparison matrix satisfy l1 u2 > 0, then V(M2 M1) = hgt(M1 M2) =

2( )M dx , where )(2 dM x

is given by (Zhu et al., 1999):

2

1 21 2

2 2 1 1,

( ) ( )0, otherwise

M d

l ul u

m u m lx (7)

2.5 Degree of fuzziness

Referring back to fuzzy numbers, for example, an element xij in a fuzzy comparison matrix, if DA i is preferred to DA j then mij takes an integer value from two to nine (from the 1–9 scale). More formally, given the entry mij in the fuzzy comparison matrix has the kth scale value vk, then lij and uij have values either side of the vk scale value. It follows the values lij and uij directly describe the fuzziness of the judgement given in xij. In Zhu et al. (1999) this fuzziness is influenced by a (degree of fuzziness) value, wheremij – lij = uij – mij = . That is, the value of is a constant and is considered an absolute distance from the lower bound value (lij) to the modal value (mij) or the modal value (mij)to the upper bound value (uij) (see Figure 2).

Given the modal value mij(vk), the fuzzy number representing the fuzzy judgement made is defined by (mij , mij, mij + ), with its associated inverse fuzzy number subsequently described by (1/( )ijm , 1/ ijm , 1/( )ijm ).

In Figure 2, the definition of the fuzzy scale value given in Zhu et al. (1999) is that the distance from mij(= vk) to vk 1 is equal to the distance from mij to vk+1 ( distance). In the case of mij given a value of one (mij = 1) off the leading diagonal (i j), the general form of its associated fuzzy scale value is defined as (1 / (1 ) , 1, 1 + ). For example, given mij = 1, the fuzzy number will be (0.6667, 1, 1.5) when = 0.5.

Figure 2 Description of the degree of fuzziness according to Zhu et al. (1999)

1

0 vk 1 vk +1vk

lijmij uij


One restriction of the method described by Zhu et al. (1999) 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 1–9 scale, with a number of alternative sets of scales being proposed (see Beynon (2002) 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 mij given scale value vk is defined as:

1 1, ,k k k k k k kv v v v v v v (8)

Therefore, mij = vk, lij = vk (vk vk 1) and uij = vk + (vk+1 vk). When the maximum scale value v9 is used, consideration has to be given to its associated upper bound values. That is given mij = vk then it is not possible to use the previously defined expressed, instead of uij = u9 = v9 + 2

9 8 8 7( ) / ( )v v v v . The reason is that there is no v10(v9+1) 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 (2009)).

2.6 Sensitivity analysis of resultant weight values

Sensitivity analysis is a fundamental concept for the effective use and implementation of quantitative decision models (Dantzig, 1963). 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.

3 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.

3.1 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 14001 and OHSAS 18001 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, A1, A2 and A3, provide price quotes. The three types of lead-free equipment system they provide, Equipment A1, Equipment A2 and Equipment A3, are the decision alternatives in this case study.


3.2 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 2. Table 2 Information table of the lead-free equipment

Equipment

Criteria Sub-criteria Equipment A1 Equipment A2 Equipment A3

C11 1,500,000 1,900,000 1,300,000 C12 Moderate Expensive Relatively cheap

C1

C13 Convenient Inconvenient Convenient C21 High compatible Low compatible Middle compatible

Already space reserved

Additional augmentation needed

Additional augmentation needed

C31 Domestic/abroad have service centres

Need agents/difficult to maintain

Company made/easy to maintain

C32 2–4 hr/easy difficult 5 hr/easy

C3

C33 3 hr 6–8 hr 2–4 hr C41 Excellent Excellent Excellent C4

Training days C42 7 days 4–5 days 5 days C51 Small Large Middle C52

Air pollution Low Low Low Noise pollution Low Low Low

C5

Water pollution No No No C61 Capable Capable Capable Augmentation Yes Yes Yes Easy to upgrade Yes Yes Yes

C6

Reserved the space Yes Yes Yes C71 Good Medium Relatively low C7

C72 Above 15 Below 10 Between 10 and 15 C73 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:

C1: Acquisition cost of equipment and parts – C11: price of the equipment (NT$); C12:price of parts; C13: 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 A2 is the most expensive equipment in both acquisition cost and replacement parts cost.


C2: Compatibility – C21: The DMs prefer the new equipment to be downwardly compatible. Equipment A1 is highly compatible with EL’s existing equipment.

C3: Response and maintenance time – C31: service ability (The numbers of distributor service centres and the distance of distributor service centres); C32: maintenance ability (maintenance time – by hours); C33: time to arrive.

For example, in Table 2, C32, Equipment A1 needs 2–4 hr for maintenance, while Equipment A3 needs 5 hr for maintenance. Equipment A2 is difficult to maintain, and the supplier has difficulty arriving at EL in a short period of time.

C4: Education and training – C41: install; C42: 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.

C5: Equipment size and pollution control – C51: space of the equipment; C52:environmental assessment.

The DMs prefer equipment with less air pollution, noise pollution and water pollution.

C6: Upgrades and expansibility – C61: 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.

C7: Supplier and brand reputation – C71: brand; C72: quantity of customers of the supplier at present; C73: financial situation of the supplier. A1 has a good reputation and already supplies more than 15 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 3 and 4 illustrate the results of the pairwise comparisons between the seven criteria and the sub-criteria, respectively. Table 5(a)–(o) shows 15 further fuzzy comparison matrices for pairwise comparisons between equipment alternatives on each of the criteria and sub-criteria. For example, in Table 3, the three DMs made judgements on C1 compared to C4, with the pairwise comparisons of (3, 5, and 5). The fuzzy scale values 3 and 5 represent ‘moderately preferred’ and ‘strongly preferred’, respectively, as shown in Table 1. For the comparisons between the sub-criteria shown in Table 4, the criteria C2and C6 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 1 with the degree of fuzziness, the distances between successive scale values are equal, that is, vk – vk 1 = vk+1 – vk.


Table 3 Pairwise comparisons between criteria based on three DMs’ opinions

C1 C2 C3 C4 C5 C6 C7

C1 1 1/51/31/8

1/71

1/6

355

1/35

1/6

1/71/31/5

31

1/5C2 5

38

1 1/737

759

355

1/516

878

C3 7 16

71/31/7

1 7 53

77

1/3

55

1/6

733

C4 1/3 1/51/5

1/71/51/9

1/71/51/3

1 1/3 3

1/3

1/55

1/5

531

C5 3 1/56

1/31/51/5

1/71/73

31/33

1 1/5 33

512

C6 7 35

51

1/6

1/51/56

51/55

51/31/3

1 5 25

C7 1/3 15

1/81/71/8

1/71/31/3

1/51/31

1/51

1/2

1/51/21/5

1

Table 4 (a)–(e) Comparisons between sub-criteria

a) C1 C11 C12 C13 b) C3 C31 C32 C33 c) C4 C41 C42 d) C5 C51 C52 e) C7 C71 C72 C73

C11 1 1/325

1/323

C31 1 51/55

51/51

C41 1 535

C51 1 519

C71 1 1/5 3

1/5

1/515

C12 3 1/21/5

1 51/31/3

C32 1/5 5

1/5

1 1/5 1

1/5

C42 1/51/31/5

1 C52 1/51

1/9

1 C72 5 1/35

1 5 25

C13 3 1/21/3

1/533

1 C33 1/5 51

515

1 C73 5 1

1/5

1/51/21/5

1


Table 5 (a)–(o) Comparisons between decision alternatives over the different sub-criteria

A 3 1 1 1 1 1 1 1

A 2 1 1 1 1 1 1 1

A 1 1 1 1 1 1 1 1

h)C

4C

41

A 1 A 2 A 3

A 3 1/5 1 2 1/9

1/3

1/3 1 A 3 9 3 5 1 1 1 1

A 2 9 3 6 1 9 3 3 A 2 9 1/3 5 1 1 1 1

A 1 1 1/9

1/3

1/6 5 1 1/2 A 1 1 1/9 3 1/5

1/9

1/3

1/5

g)C

3C

33

A 1 A 2 A 3 o)C

7 C73

A 1 A 2 A 3

A 3 9 3 5 1/9

1/5

1/3 1 A 3 3 3 2 3 5 3 1

A 2 9 7 7 1 9 5 3 A 2 1/3

1/2

1/3 1 1/3

1/5

1/3

A 1 1 1/9

1/7

1/7

1/9

1/3

1/5 A 1 1 3 2 3 1/3

1/3

1/2

f)C

3C

32

A 1 A 2 A 3 n)C

7 C72

A 1 A 2 A 3

A 3 1/9

1/2 5 1/9

1/3

1/3 1 A 3 9 6 6 9 3 4 1

A 2 9 5 7 1 9 3 3 A 2 9 5 5 1 1/9

1/3

1/4

A 1 1 1/9

1/5

1/7 9 2 1/5 A 1 1 1/9

1/5

1/5

1/9

1/6

1/6

e)C

3C

31

A 1 A 2 A 3 m)

C7 C

71

A 1 A 2 A 3

A 3 9 2 5 1/9

1/2

1/3 1 A 3 5 3 4 1/9

1/3

1/3 1

A 2 9 3 7 1 9 2 3 A 2 9 5 6 1 9 3 3

A 1 1 1/9

1/3

1/7

1/9

1/2

1/5 A 1 1 1/9

1/5

1/6

1/5

1/3

1/4

d)C

2C

21

A 1 A 2 A 3 l)C

6 C61

A 1 A 2 A 3

A 3 1 1/2 1 1/9

1/5

1/3 1 A 3 9 1/5

1/3 1 1 1/5 1

A 2 9 5 3 1 9 5 3 A 2 9 1/5 3 1 1 1 5

A 1 1 1/9

1/5

1/3 1 2 1 A 1 1 1/9 5 1/3

1/9 5 3

c)C

1C

13

A 1 A 2 A 3 k)C

5 C52

A 1 A 2 A 3

A 3 1/9

1/3

1/3

1/9

1/6

1/5 1 A 3 9 2 5 1/9

1/2

1/3 1

A 2 9 7 3 1 9 6 5 A 2 9 3 6 1 9 2 3

A 1 1 1/9

1/7

1/3 9 3 3 A 1 1 1/9

1/3

1/6

1/9

1/2

1/5

b)C

1C

12

A 1 A 2 A 3 j)C

5 C51

A 1 A 2 A 3

A 3 1/9

1/6

1/3

1/9

1/8

1/4 1 A 3 1/9

1/2 3 1 1 2 1

A 2 9 7 3 1 9 8 4 A 2 1/9

1/3 5 1 1 1 1/2

A 1 1 1/9

1/7

1/3 9 6 3 A 1 1 9 3 1/5 9 2 1/3

a) C1

C11

A 1 A 2 A 3 i) C4 C

42

A 1 A 2 A 3


Table 6 The fuzzy comparison matrix version of comparisons between criteria

C1 C2 C3 C4 C5 C6 C7

C1

(1, 1, 1)

(1/(5 + ), 1/5, 1/(5 ))

(1/(3 + ), 1/3, 1/(3 ))

(1/(8 + ), 1/8, 1/(8 ))

(1/(7 + ), 1/7, 1/(7 ))

(1/(1 + ), 1, 1 + )

(1/(6 + ), 1/6, 1/(6 ))

(3 , 3, 3 + )

(5 , 5, 5 + )

(5 , 5, 5 + )

(1/(3 + ), 1/3, 1/(3 ))

(5 , 5, 5 + )

(1/(6 + ), 1/6, 1/(6 ))

(1/(7 + ), 1/7, 1/(7 ))

(1/(3 + ), 1/3, 1/(3 ))

(1/(5 + ), 1/5, 1/(5 ))

(3 , 3, 3 + )

(1/(1 + ), 1, 1 + )

(1/(5 + ), 1/5, 1/(5 ))

C2 (5 , 5, 5 + )

(3 , 3, 3 + )

(8 , 8, 8 + )

(1, 1, 1)

(1/(7 + ), 1/7, 1/(7 ))

(3 , 3, 3 + )

(7 , 7, 7 + )

(7 , 7, 7 + )

(5 , 5, 5 + )

(9 , 9, 9 + )

(3 , 3, 3 + )

(5 , 5, 5 + )

(5 , 5, 5 + )

(1/(5 + ), 1/5, 1/(5 ))

(1/(1 + ), 1, 1 + )

(6 , 6, 6 + )

(8 , 8, 8 + )

(7 , 7, 7 + )

(8 , 8, 8 + )

C3 (7 , 7, 7 + )

(1/(1 + ), 1, 1 + )

(6 , 6, 6 + )

(7 , 7, 7 + )

(1/(3 + ), 1/3, 1/(3 ))

(1/(7 + ), 1/7, 1/(7 ))

(1, 1, 1)

(7 , 7, 7 + )

(5 , 5, 5 + )

(3 , 3, 3 + )

(7 , 7, 7 + )

(7 , 7, 7 + )

(1/(3 + ), 1/3, 1/(3 ))

(5 , 5, 5 + )

(5 , 5, 5 + )

(1/(6 + ), 1/6, 1/(6 ))

(7 , 7, 7 + )

(3 , 3, 3 + )

(3 , 3, 3 + )

C4 (1/(3 + ), 1/3, 1/(3 ))

(1/(5 + ), 1/5, 1/(5 ))

(1/(5 + ), 1/5, 1/(5 ))

(1/(7 + ), 1/7, 1/(7 ))

(1/(5 + ), 1/5, 1/(5 ))

(1/(9 + ), 1/9, 1/(9 ))

(1/(7 + ), 1/7, 1/(7 ))

(1/(5 + ), 1/5, 1/(5 ))

(1/(3 + ), 1/3, 1/(3 ))

(1, 1, 1)

(1/(3 + ), 1/3, 1/(3 ))

(3 , 3, 3 + )

(1/(3 + ), 1/3, 1/(3 ))

(1/(5 + ), 1/5, 1/(5 ))

(5 , 5, 5 + )

(1/(5 + ), 1/5, 1/(5 ))

(5 , 5, 5 + )

(3 , 3, 3 + )

(1/(1 + ), 1, 1 + )

C5 (3 , 3, 3 + )

(1/(5 + ), 1/5, 1/(5 ))

(6 , 6, 6 + )

(1/(3 + ), 1/3, 1/(3 ))

(1/(5 + ), 1/5, 1/(5 ))

(1/(5 + ), 1/5, 1/(5 ))

(1/(7 + ), 1/7, 1/(7 ))

(1/(7 + ), 1/7, 1/(7 ))

(3 , 3, 3 + )

(3 , 3, 3 + )

(1/(3 + ), 1/3, 1/(3 ))

(3 , 3, 3 + )

(1, 1, 1)

(1/(5 + ), 1/5, 1/(5 ))

(3 , 3, 3 + )

(3 , 3, 3 + )

(5 , 5, 5 + )

(1/(1 + ), 1, 1 + )

(2 , 2, 2 + )

C6 (7 , 7, 7 + )

(3 , 3, 3 + )

(5 , 5, 5 + )

(5 , 5, 5 + )

(1/(1 + ), 1, 1 + )

(1/(6 + ), 1/6, 1/(6 ))

(1/(5 + ), 1/5, 1/(5 ))

(1/(5 + ), 1/5, 1/(5 ))

(6 , 6, 6 + )

(5 , 5, 5 + )

(1/(5 + ), 1/5, 1/(5 ))

(5 , 5, 5 + )

(5 , 5, 5 + )

(1/(3 + ), 1/3, 1/(3 ))

(1/(3 + ), 1/3, 1/(3 ))

(1, 1, 1)

(5 , 5, 5 + )

(2 , 2, 2 + )

(5 , 5, 5 + )

C7 (1/(3 + ), 1/3, 1/(3 ))

(1/(1 + ), 1, 1 + )

(5 , 5, 5 + )

(1/(8 + ), 1/8, 1/(8 ))

(1/(7 + ), 1/7, 1/(7 ))

(1/(8 + ), 1/8, 1/(8 ))

(1/(7 + ), 1/7, 1/(7 ))

(1/(3 + ), 1/3, 1/(3 ))

(1/(3 + ), 1/3, 1/(3 ))

(1/(5 + ), 1/5, 1/(5 ))

(1/(3 + ), 1/3, 1/(3 ))

(1/(1 + ), 1, 1 + )

(1/(5 + ), 1/5, 1/(5 ))

(1/(1 + ), 1, 1 + )

(1/(2 + ), 1/2, 1/(2 ))

(1/(5 + ), 1/5, 1/(5 ))

(1/(2 + ), 1/2, 1/(2 ))

(1/(5 + ), 1/5, 1/(5 ))

(1, 1, 1)


4 Using sensitivity analysis to determine degrees of fuzziness

This section explains how to use sensitivity analysis to measure degrees of fuzziness. Figure 3 shows the sensitivity of the varying degrees of fuzziness for Table 3 judgement data of the pairwise comparisons between seven criteria. Variable represents the degree of fuzziness. There are seven lines in Figure 3 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 3, -axis) shows that C2 has the absolute dominant preference (and hence, the least amount of fuzziness). This result means that C2 is an important criterion to be considered when the DMs make decisions, so the weight value is 1. After reaches 0.46, the criterion C3 has the next priority weight. The next criterion is C6, 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 2. This means that if is less than 2, some criteria will have no positive weights. Zahir (1999) 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 2. In general, when considering the final results, the domain of workable is expressed

as T and is defined by the maximum of the various minimum workable degrees of fuzziness throughout the problem; that is T = max(

CT ,1T ,

2T , …, -1nT ,

nT ), where the subscript T is the maximum of the minimum workable values in the n + 1(Tc matrix) fuzzy comparison matrices.

Figure 3 Set of weight values from judgements on seven criteria over 0 5

0

0.5

1

0.46 0.8 1.31.46

1.7 2 3 4 5

C2

C3 C5C6

C1 C4 C7


In the comparisons between sub-criteria (C1, C3, C4, C5 and C7), their minimum workable values are

1T = 0.43, 2T = 1.07,

3T = 2.04, 4T = 1.83 and

5T = 1.2, respectively (see Figure 4(a)–(e)). For the comparisons between equipment alternatives with respect to individual sub-criteria, the resulting fuzzy comparison matrices (on C11, C12, C13, … andC73) reveal their minimum workable values are 4.05, 3.56, 2, 3.4, 3.1, 4, 2, 0, 0.65, 3.25, 0.6, 3.1, 3.7, 1.38 and 1.28, respectively (see Figure 5(a)–(o)).

For the maximum of the minimum workable values is T = max(2, 0.43, 1.07, 2.04, 1.83, 1.2, 4.05, 3.56, 2, 3.4, 3.1, 4, 2, 0, 0.65, 3.25, 0.6, 3.1, 3.7, 1.38, 1.28) = 4.05. The weights results should possibly be considered only in the workable region of > 4.05. The minimum workable degree of fuzziness excludes values of at which there are no positive weights for the three equipment alternatives.

When = 4.05, the weight values for the seven criteria are 0.1265, 0.1941, 0.1757, 0.1129, 0.1387, 0.1621 and 0.0899, respectively (see Table 7). 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 7. For example, for C1, the weight values for the comparisons between C11, C12 and C13 are 0.3580, 0.3077 and 0.3343, respectively. The last column in Table 7 shows the weight values for equipment alternatives over the different sub-criteria. For instance, for C11, the weight values for the comparisons by the three DMs are 0.4319, 0.0016 and 0.5665, 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 A1; Equipment A3 is the next preferred alternative, while Equipment A1 is the least preferred alternative (see Table 7). This preference for Equipment A1 is found in the weight values for not only the criteria, but also the sub-criteria shown in Table 7. In those 15 sub-criteria, apart from C12, C13,C42, C52 and C72, Equipment A1 has greater weight values than the other two alternatives.

The most preferred criterion is C2, that is, the compatibility between new and old equipment (see Table 7). 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 5

(a)


Figure 4 (a)–(e) Comparisons between sub-criteria over 0 5 (continued)

(b)

(c)

(d)

(e)


Figure 5 (a)–(o) Graphs of weight values between the decision alternatives on sub-criteria

(a)

(b)

(c)

(d)


Figure 5 (a)–(o) Graphs of weight values between the decision alternatives on sub-criteria (continued)

(e)

(f)

(g)

(h)



(i)

(j)

(k)

(l)



(m)

(n)

(o)

Table 7 The sets of weight values for all fuzzy comparison matrices and the final results obtained where = 4.05 based on the DM’s opinions

Weight values for criteria Weight values for sub-

criteria Weight values for decision

alternatives

C11 0.3580 [0.4319, 0.0016, 0.5665] C12 0.3077 [0.4280, 0.4824, 0.5238]

C1 0.1265

C13 0.3343 [0.4213, 0.1504, 0.4283] C2 0.1941 C21 1 [0.5521, 0.0502, 0.3977]

C31 0.3477 [0.4654, 0.0966, 0.4380] C32 0.2813 [0.5776, 0.0558, 0.4168]

C3 0.1757

C33 0.3709 [0.4412, 0.1265, 0.4323] C4 0.1129 C41 0.6995 [0.3333, 0.3333, 0.3333]


Table 7 The sets of weight values for all fuzzy comparison matrices and the final results obtained where = 4.05 based on the DM’s opinions (continued)

Weight values for criteriaWeight values for sub-

criteriaWeight values for decision

alternatives

C42 0.3005 [0.2668, 0.3721, 0.3611] C51 0.6868 [0.5370, 0.0676, 0.3954] C5 0.1387 C52 0.3132 [0.3483, 0.2955, 0.3561]

C6 0.1621 C61 1 [0.5067, 0.0939, 0.3994] C71 0.3051 [0.5534, 0.3995, 0.4719] C72 0.4075 [0.3647, 0.4372, 0.1981]

C7 0.0899

C73 0.2874 [0.4271, 0.2922, 0.2807] Final results [0.4709, 0.1324, 0.3967] Final ranking [A1, A3, A2]

5 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 decision-making process. This study’s contribution to FAHP methodology is the demonstration that fuzziness should not be fixed only at 0.5. DMs from manufacturing companies faced


with selecting lead-free equipment system can establish their own evaluation procedure for their company’s capital investments based on their subjective opinions. The lead-free equipment system selection problem is a typical capital expenditure problem. Manufacturing companies can also apply FAHP to a variety of other capital expenditure decisions.

Acknowledgements

We thank helpful comments and suggestions from Ruben Davila, Margaret Palisoc and Amber Sturdivant. This work was supported in part by the National Science Council under the Grants NSC 93-2416-H-025-005.

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yu-cheng tang*

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