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

Copyright 2011 Inderscience Enterprises Ltd.

Appl ication of the fuzzy analytic hierarchy process tothe 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 industryhas 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 hierarchyprocess (FAHP) method as the decision support system to help decision makersmaking 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 intangiblecriteria.

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-freeequipment selection decision, Int. J. Business and Systems Research, Vol. 5,

No. 1, pp.3556.

Biographical notes:Yu-Cheng Tang received her PhD in Accounting from theUniversity of Cardiff (Wales, UK). Currently, she is an Assistant Professor at

National Changhua University of Education (Taiwan). Her research interestsare in the general area of financial management, in particular in the capitalinvestment, human perceptions on decision making, green accounting, ethic

position and budgetary system, etc. Specific methodologies investigates includefuzzy set theory, analytical hierarchy process and balanced scorecards. Herstudy is at the theoretical development and application-based level, including

business and other topics.

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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 TFNAcan 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 l

u xx m x u

u m

(1)

wherexis the mean value ofAand l, m, uare real numbers. Define two TFNsAandBby

the tripletsA= (l1, m1, u1) andB= (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 u

l m m u u

Inverse:

11 1 1

1 1 1

1 1 1( , , ) , ,l m u

u 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 (seeInterval Probability Theoryin 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

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38 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 9-unit scale (Saaty,

1980) (see Table 1).

It is difficult to map qualitative preferences to point estimates, hence a degree ofuncertainty 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)nn,

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 1ij

ji xx ; i,j= 1, , n; and

the subscripts iandjrefer 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 iandj

on a criterion can be modelled through a fuzzy scale value associated with a degree of

fuzziness. Then, an element ofX,xijis a fuzzy number defined asxij= (lij, mij, uij), where

lij, mij, uijare the lower bound, modal, and upper bound values forxij, 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 adjacentjudgements

2.3 Value of fuzzy synthetic extent

Let C= {C1, C2, , Cn} be a criteria set, where nis the number of criteria and A= {A1,

A2, ,Am} be a decision alternative set, where mis the number of decision alternatives.Let 1

iCM , 2

iCM ,,

i

mCM , i= 1, 2, , nwhere all the

i

jC

M (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 Siwith respect to the ith criteria is defined:

1

1 1 1i i

mm nj jC C

j i j

Si M M

(2)

where represents fuzzy multiplication and the superscript 1 represents the fuzzyinverse. The concepts of synthetic extent are also found in Cheng (1999) and Bozda

et 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

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numbers,M1andM2, the degree of possibility thatM1 M2is 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 areconvex fuzzy numbers defined by the TFNs (l1, m1, u1) and (l2, m2, u2), respectively, it

follows:

1 2 1 21 iffV M M m m

12 1 1 2

hgt M dV M M M M x (3)

where iff represents if and only if, d is the ordinate of the highest intersection pointbetween the

1M and

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

1M

and2M

where the ordinate dis 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 aconvex fuzzy number can be obtained from Equation (4):

1 22 1 1 2

2 2 1 1

hgtl u

V M M M M d m u m l

(4)

The degree of possibility for a convex fuzzy numberMto be greater than the number of k

fuzzy 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 nis 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 numbersM1andM2

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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(M2M1) = hgt(M1M2) =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 iis preferred to DAjthen mijtakes an integer value from two to nine (from

the 19 scale). More formally, given the entry mijin the fuzzy comparison matrix has the

kth scale value vk, then lijand uijhave values either side of the vkscale value. It follows

the values lijand uijdirectly 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 numbersubsequently 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 vk1is equal to the distance from mijto vk+1(distance). In

the case of mijgiven a value of one (mij= 1) off the leading diagonal (ij), the generalform 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 +1vklij mij uij

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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 19 scale,

with a number of alternative sets of scales being proposed (see Beynon (2002) andreferences 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 mijgiven scale value vkis

defined as:

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

Therefore, mij= vk, lij= vk (vk vk1) and uij= vk+ (vk+1 vk). When the maximumscale value v9is 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, ELs top management decided

to implement a new lead-free equipment system.

ELs 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 A2and EquipmentA3,are the

decision alternatives in this case study.

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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

ELs 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 spacereserved

Additionalaugmentation needed

Additionalaugmentation needed

C31 Domestic/abroadhave service centres

Need agents/difficultto maintain

Company made/easyto maintain

C32 24 hr/easy difficult 5 hr/easy

C3

C33 3 hr 68 hr 24 hr

C41 Excellent Excellent ExcellentC4

Training days C42 7 days 45 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 lowC7

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. EquipmentA2is the most expensive equipment in

both acquisition cost and replacement parts cost.

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C2: Compatibility C21: The DMs prefer the new equipment to be downwardly

compatible. EquipmentA1is highly compatible with ELs existing equipment.

C3: Response and maintenance time C31: service ability (The numbers of distributor

service centres and the distance of distributor service centres); C32: maintenanceability (maintenance time by hours); C33: time to arrive.

For example, in Table 2, C32, Equipment A1 needs 24 hr for maintenance, while

EquipmentA3needs 5 hr for maintenance. Equipment A2is 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.A1has 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 aninappropriate 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

C1compared 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 vk1= vk+1 vk.

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Table 3 Pairwise comparisons between criteria based on three DMs opinions

C1 C2 C3 C4 C5 C6 C7

C1 1 1/5

1/3

1/8

1/7

1

1/6

3

5

5

1/3

5

1/6

1/7

1/3

1/5

3

1

1/5

C2 5

3

8

1 1/7

3

7

7

5

9

3

5

5

1/5

1

6

8

7

8

C3 7

1

6

7

1/3

1/7

1 7

5

3

7

7

1/3

5

5

1/6

7

3

3

C4 1/3

1/5

1/5

1/7

1/5

1/9

1/7

1/5

1/3

1 1/3

3

1/3

1/5

5

1/5

5

3

1

C5 3

1/5

6

1/3

1/5

1/5

1/7

1/7

3

3

1/3

3

1 1/5

3

3

5

1

2

C6 7

3

5

5

1

1/6

1/5

1/5

6

5

1/5

5

5

1/3

1/3

1 5

2

5

C7 1/3

1

5

1/8

1/7

1/8

1/7

1/3

1/3

1/5

1/3

1

1/5

1

1/2

1/5

1/2

1/5

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 1/3

2

5

1/3

2

3

C31 1 5

1/5

5

5

1/5

1

C41 1 5

3

5

C51 1 5

1

9

C71 1 1/5

3

1/5

1/5

1

5

C12 3

1/2

1/5

1 5

1/3

1/3

C32 1/5

5

1/5

1 1/5

1

1/5

C42 1/5

1/3

1/5

1 C52 1/5

1

1/9

1 C72 5

1/3

5

1 5

2

5

C13 3

1/2

1/3

1/5

3

3

1 C33 1/5

5

1

5

1

5

1 C73 5

1

1/5

1/5

1/2

1/5

1

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

A3

1 1 1 1 1 1 1

A2

1 1 1 1 1 1 1

A1 1 1 1 1 1 1 1

h)

C4

C41

A1

A2

A3

A3

1/5

1 21/9

1/3

1/3

1 A3

9 3 5 1 1 1 1

A2

9 3 6 1 9 3 3 A2

91/3

5 1 1 1 1

A1

11/9

1/3

1/6

5 11/2

A1

11/9

31/5

1/9

1/3

1/5

g)

C3

C33

A1

A2

A3

o)

C7C

73

A1

A2

A3

A3

9 3 51/9

1/5

1/3

1 A3

3 3 2 3 5 3 1

A2

9 7 7 1 9 5 3 A2

1/3

1/2

1/3

11/3

1/5

1/3

A1

11/9

1/7

1/7

1/9

1/3

1/5

A1

1 3 2 31/3

1/3

1/2

f)

C3C

32

A1

A2

A3

n)

C7C

72

A1

A2

A3

A3

1/9

1/2

51/9

1/3

1/3

1 A3

9 6 6 9 3 4 1

A2

9 5 7 1 9 3 3 A2

9 5 5 11/9

1/3

1/4

A1

11/9

1/5

1/7

9 21/5

A1

11/9

1/5

1/5

1/9

1/6

1/6

e)

C3

C31

A1

A2

A3

m)

C7C

71

A1

A2

A3

A3

9 2 51/9

1/2

1/3

1 A3

5 3 41/9

1/3

1/3

1

A2

9 3 7 1 9 2 3 A2

9 5 6 1 9 3 3

A1

11/9

1/3

1/7

1/9

1/2

1/5

A1

11/9

1/5

1/6

1/5

1/3

1/4

d)

C

2

C21

A1

A2

A3

l)

C

6C

61

A1

A2

A3

A3

11/2

11/9

1/5

1/3

1 A3

91/5

1/3

1 11/5

1

A2

9 5 3 1 9 5 3 A2

91/5

3 1 1 1 5

A1

11/9

1/5

1/3

1 2 1 A1

11/9

51/3

1/9

5 3

c)

C1

C13

A1

A2

A3

k)

C5C

52

A1

A2

A3

A3

1/9

1/3

1/3

1/9

1/6

1/5

1 A3

9 2 51/9

1/2

1/3

1

A2

9 7 3 1 9 6 5 A2

9 3 6 1 9 2 3

A1

11/9

1/7

1/3

9 3 3 A1

11/9

1/3

1/6

1/9

1/2

1/5

b)

C1

C12

A1

A2

A3

j)

C5C

51

A1

A2

A3

A3

1/9

1/6

1/3

1/9

1/8

1/4

1 A3

1/9

1/2

3 1 1 2 1

A2

9 7 3 1 9 8 4 A2

1/9

1/3

5 1 1 11/2

A1

11/9

1/7

1/3

9 6 3 A1

1 9 31/5

9 21/3

a)

C1

C11

A1

A2

A3

i) C4C

42

A1

A2

A3

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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)

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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 judgementdata 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 C2has 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 C3has 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 DMsdo 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 expressedas

T and is defined by the maximum of the various minimum workable degrees of

fuzziness throughout the problem; that isT

= 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. 3

1.46

1.7 2 3 4 5

C2

C3

C5

C6

C1 C4 C

7

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In the comparisons between sub-criteria (C1, C3, C4, C5and C7), their minimum workable

values are1T

= 0.43,2T

= 1.07,3T

= 2.04,4T

= 1.83 and5T

= 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 isT = 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, C12and C13are 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 EquipmentA1is the least preferred alternative (see Table 7).

This preference for Equipment A1is 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, C52and C72, EquipmentA1has 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)

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

(b)

(c)

(d)

(e)

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

(a)

(b)

(c)

(d)

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

(e)

(f)

(g)

(h)

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

(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 DMs opinions

Weight values for criteriaWeight values for sub-

criteriaWeight values for decision

alternatives

C11 0.3580 [0.4319, 0.0016, 0.5665]

C12 0.3077 [0.4280, 0.4824, 0.5238]

C10.1265

C13 0.3343 [0.4213, 0.1504, 0.4283]

C20.1941 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]

C30.1757

C33 0.3709 [0.4412, 0.1265, 0.4323]

C40.1129 C41 0.6995 [0.3333, 0.3333, 0.3333]

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Table 7 The sets of weight values for all fuzzy comparison matrices and the final results

obtained where = 4.05 based on the DMs opinions (continued)

Weight values for criteria

Weight values for sub-

criteria

Weight values for decision

alternatives

C42 0.3005 [0.2668, 0.3721, 0.3611]

C51 0.6868 [0.5370, 0.0676, 0.3954]C50.1387

C52 0.3132 [0.3483, 0.2955, 0.3561]

C60.1621 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]

C70.0899

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 methodwith 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 studys contribution to FAHP methodology is the demonstration

that fuzziness should not be fixed only at 0.5. DMs from manufacturing companies faced

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with selecting lead-free equipment system can establish their own evaluation procedure

for their companys 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 expendituredecisions.

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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