Yuen 2010 975 Multicriteria & Prioritization

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Applied Soft Computing 10 (2010) 975989

Contents lists available at ScienceDirect

Applied Soft Computing

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a s o c

Analytic hierarchy prioritization process in the AHP application development:

A prioritization operator selection approach

Kevin Kam Fung Yuen

Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

a r t i c l e i n f o

Article history:

Received 1 May 2008

Received in revised form 20 August 2009

Accepted 30 August 2009

Available online 25 October 2009

Keywords:

AHP

Prioritization methods

Prioritization operators

Prioritization method measurement

Multicriteria decision making

a b s t r a c t

In the analytic hierarchyprocess,prioritizationof the reciprocalmatrix is a coreissue to influence thefinal

decision choice. Various prioritization methods have been proposed, but none of prioritization methods

performs better than others in every inconsistent case. To address the prioritation operator selection

problem, thispaper proposes theanalytichierarchy prioritizationprocess,which is anobjectivehierarchy

model (without using subjective pairwise comparisons) to approximate the real priority vectors with

selection of the most appropriate prioritization operator among the various prioritization candidates,

for a reciprocal matrix, and on the basis of a list of measurement criteria. Nine important prioritization

operators and seven measurement criteria are illustratedin AHPP. Two previous applications are revised

andillustratethe validity andusabilityof the proposed model. Theresults showthat the mostappropriate

prioritization operator is dependent of the content of the reciprocal matrix and AHPP is an appropriate

method to address the prioritization problem to make better decisions.

2009 Elsevier B.V. All rights reserved.

1. Introduction

The pairwise comparison method is originated from psycholog-

ical research [1,2]. Saaty developed the concept in a mathematical

way, and applied such a concept in the analytic hierarchy/network

process [37]. AHP has been widely studied and applied in mul-

ticriteria decision making (MCDM) domains [8,9]. However, there

are criticisms on this method [1015].

In AHP, verbal judgments are given for pairwise comparison

by decision makers. The reciprocal matrices are formed by trans-

forming the linguistic labels to numerical values. Next, the priority

vectors are generated from the reciprocal matrices by a prioriti-

zation operator. Finally, these priority vectors are aggregated as

a global priority vector in the synthesis stage. This induces three

fundamental problems: selection of numerical scales in stage one,

selection of prioritization operators (or methods) in stage two, and

selection of aggregation operators in stage three.This research typically is interested in the selection of prior-

itization operators. The prioritization operator (PO) refers to the

algorithms of deriving a priority vector from the reciprocal matrix.

Various prioritization methods in the AHP models have been stud-

ied in the literature. Each method is claimed to overperform some

of the existing methods. Usually, the authors of POs claimed that

their proposed prioritization operator is superior by the way of

Tel.: +852 60112169.

E-mail address: [email protected].

finding the better results (e.g. less value in the average of mean

absolute deviation) to their limited opponents (other prioritization

operators) in their proposed test scenarios on the basis of some cri-

teria, such as Euclidean distance, root mean squared error, mean

absolute deviation, or/and worst absolute deviation [e.g. 1619].

Such tests are similar to statistical methods. It is true that the sta-

tistical summarized performance of an operator A is better than

an operator B. This does not follow that A is better than B when

comparison is on a case-by-case basis.In addition,the fact that lim-

ited opponents, which arechosenfor comparison, likelymeans less

objective.

Actually, the best prioritization operator relies on the content

of a pairwise matrix, and none of prioritization methods performs

better than others in every inconsistent case. Two applications

in this study and [20,21] also verify this issue. Thus, it is most

appropriate to propose a framework to select the most appropriate

prioritization operator for each reciprocal matrix among suffi-cient candidates with more objective measurement criteria and

methods.

The related research is in [21], which proposed a Multicrite-

ria Prioritization Synthesis (MPS). Seven prioritization operators

and two evaluation criteria (minimum violations and Euclidean

distance) were used in [21]. Srdjevic [21] also showed that the pri-

oritization method was dependentof reciprocalmatrix,and none of

prioritization methods was thebest.The limitations of the research

are that the measurement criteria are insufficient and aggregation

of the results of the measurement criteria is linear and subjective

as weights of measurement criteria are not justified. The results

1568-4946/$ see front matter 2009 Elsevier B.V. All rights reserved.

doi:10.1016/j.asoc.2009.08.041
http://localhost/var/www/apps/conversion/tmp/scratch_3/dx.doi.org/10.1016/j.asoc.2009.08.041http://www.sciencedirect.com/science/journal/15684946http://www.elsevier.com/locate/asocmailto:[email protected]:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_3/dx.doi.org/10.1016/j.asoc.2009.08.041http://localhost/var/www/apps/conversion/tmp/scratch_3/dx.doi.org/10.1016/j.asoc.2009.08.041mailto:[email protected]://www.elsevier.com/locate/asochttp://www.sciencedirect.com/science/journal/15684946http://localhost/var/www/apps/conversion/tmp/scratch_3/dx.doi.org/10.1016/j.asoc.2009.08.041


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976 K.K.F. Yuen / Applied Soft Computing 10 (2010) 975989

Fig. 1. Compound analytic hierarchy process model.

of the appropriate prioritization operator may have bias. In this

study, two more prioritization operators are chosen as candidates,

five more measurement criteria are chosen, and the algorithms of

AHPP are hierarchical and objective. The details of the algorithms

of AHPP can be found in Section 2.The details of the comparisons

can be found in Section 5.As an accurate method to derive the priorities of the criteria is

critical to enable decision makers to make the correct choice, this

paper proposes an AHPP for evaluating the prioritization methods

and selecting the most appropriate one in the development of AHP

applications. This study selects nine important prioritization oper-

ators: Eigenvector [3], normalization of the row sum method [3],

normalization of reciprocals of column summethod [3], arithmetic

mean of normalized columns method [3] (which is the same as

additive normalization [21]), normalization of geometric means of

rows [3] (or logarithm least squares [e.g. 16, 19,2228], direct least

squares [30], weighted least squares [30], fuzzy programming [17],

and enhanced goal programming [19,29]). The details can be found

in Section 3.

The structure of this paper is organized as follows. The ideasof Compound AHP, which comprises of AHPP, are presented in the

next section. In Section 3, several important prioritization methods

are reviewed.The measurementcriteria and methodsare presented

in Section 4. In Section 5, two applications are selected for discus-

sion to illustrate the validity and usability of the proposed model.

Section 6 is the conclusion. The notation summary is in Appendix

C.

2. Compound AHP

The Compound AHP can be summarized as five processes ( D, E,

, S, M) where D is the definition process, E is the evaluation andassessment process, is the analytic hierarchy prioritization pro-

cess, Sis the synthesis process, and Mis the measurement process.If the AHP applies AHPP for its prioritization method, it is named as

CAHP (Fig. 1).

2.1. Definition process

The definition process D consists of two parts: the AHP problem

definition process and the AHPP definition process, i.e. D = (D1, D2).

In theAHP problem definition process D1 = (O,C, T, ), a hierarchymodel is defined by an objective O, a set ofcriteria C={c1, c2, . . ., ci,. . ., cn}, a set of alternatives T = {t1, t2, . . . , t j, . . . , t m}, and a ratingscale schema = {1, 2, . . . , i, . . . , p}. If the data type of the

rating scales is the fuzzy number, then the AHP model is extended

as fuzzy AHP problem. If the data type is the crisp number, then

the AHP model is the crisp AHP model or the AHP model. The crisp

AHP problem is the special case of the fuzzy AHP problems as the

crispAHP model does notneedthe interval computing, butrelies on

the modal value of the fuzzy number. This paper only discuss crisp

AHP model. For the typical AHP model, nine point rating scale is

applied, = {1/9, . . . , 1/2, 1, 2, . . . , 9} and the definition is shown

in Table 2.In the AHPPdefinition process D2 = (S, S, P), S= {s1, s2, . . ., sq, . . .,

sQ} is a set of measurement criteria, P = {p1, p2, . . . , pk, . . . , pK} is

a set of prioritization operators, which is discussed in Section 3,

and Sq S = {S1, . . . , SQ} is a measurement function to measure Punder a measurement criterion sq, which is derived by Sq(P). Thedefinitions for the measurement functions Sqs (or the set S) can be

found in Section 4.

2.2. Evaluation and assessment process

In the evaluation and assessment process E =

((C),(ci, T)ni=1), the decision makers or raters assess apairwise matrix of all criteria AC by assessment function (C),and the set {AT,i } of the pairwise comparison matrices of allalternatives for each criterion ci by the set of assessment functions,

(C, T) =

(ci, T)

ni=1

ni=1

= {(c1, T), . . . , (cn, T)}. AC is the

form:

AC = (C) =

c1c2...

cn

c1 c2 . .. c n

a11 a12 . .. a1na21 a22 . .. a2n

......

. . ....

an1 an2 ann

(1)

A pairwise matrix AT,i of all alternativesT for a criterion ci bythe pairwise assessmentfunction,andAT,i AT = {AT,1, . . . , AT,n }.

AT,i is the form:

AT,i = (ci, T) =

t1t2...

tm

t1 t2 . .. t m

a11 a12 . . . a1ma21 a22 . . . a2m

..

....

. . ....

am1 am2 amm

, for all i (2)

To interpret the pairwise matrix, let a set of the real (ideal)

relative weights (or a priority vector) be W = {w1, . . . , wn}, andthe comparison score is aij = wi/wj. The ideal pairwise matrix

A = [wi/wj] can be representedby a subjectivejudgmental pairwise


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K.K.F. Yuen / Applied Soft Computing 10 (2010) 975989 977

Table 1

Random consistency index (RI) ].

N 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

RI 0 0 .58 .90 1.12 1.24 1.32 1.41 1.45 1.49 1.51 1.48 1.56 1.57 1.59

matrix A = [aij] formed as follows:

A = [aij] =

w1

w1

w1

w2

. . .w1

wnw2w1

w2w2

. . .w2wn

..

....

. . ....

wnw1

wnw2

wnwn

=

a11 a12 . .. a1na21 a22 . .. a2n

......

. . ....

an1 an2 ann

= [aij] = A (3)

where aij = a1

ji, and aii = 1, i, j = 1, . . ., n. A is the pair matrix from

expert judgment. A can be AC or AT,i .A is an ideal consistent judg-

ment matrix generated from W, which is generated from A. Wcan

be WC or WT,i, which is generated from AC or AT,i .

2.3. Measurement process

Before the calculation of the priority vectors of the pairwise

matrices, it is necessary to evaluate the validity of the input data.

Measurement process M= (CR) is to determine a consistency ratio

CR, which is obtained by a consistency index CIanda randomindex

RI, which is an average random consistency index derived from a

sample of randomly generated reciprocal matrices using the nine

point scales (Table 2). CR has the form:

CR(CI,RI) =CI

RI(4)

where RIcan be found in Table 1, and

CI = max nn 1

(5)

max is a principal eigenvalue of a pairwise matrix. Saaty [3] alsoproved that max n. max can also be derived by PerronFrobenius

theorem [7]:

max =1

n

ni=1

wi

wi, where [wi] = A W and wi W (6)

As the random index associated with Eigenvector method com-

prises many studies, this study applies Eigenvector method to

evaluate the consistency index of reciprocal matrix. To determine

the validity, if CR > 0.1, the pairwise matrix is not consistent, then

the comparisons should be revised. Otherwise, the pairwise matrixis accepted.

2.4. Analytic hierarchy prioritization process

AHPP returns the most appropriate priority vector by selecting

the bestprioritization methods. An analytic hierarchy prioritization

process is theform = (A, (D2, MP,Agg, EP),W). An inconsistent pair-wise matrix A is prioritized to the priority vector W, i.e. :A W. consists of four core processes (D2, MP, Agg, EP) illustrated asfollows.

2.4.1. AHPP definition process (D2)

D2

= (S, S, P) is introduced in Section 2.1.

2.4.2. Measurement and prioritization process (MP = (, VP))

VP = (vs1 , vs2 , . . . , vsq , . . . , vsQ)T

= (vqk)QK is a measurement pri-

ority matrix, which is written explicitly:

VP =

s1s2...

sq...

sQ

p1 p2 . .. pk . .. pK

v11 v12 . . . v1k . . . v1Kv21 v22 . . . v2k . . . v2K

......

. . ....

......

vq1 vq2 vqk vqK

......

......

......

vQ1 vQ2 vQk vQK

(7)

In the above matrix, VP = [(Sq(P))]T

= [(S{1,...Q}(P))]T

=

[(S1(P)), . . . , (SQ(P))]T

, and (Sq(P)) VP. A measurementpriority vector, vsq = (Sq(P)) = {vq1, . . . , vqk, . . . , vqK}, where

Kk=1

vqk = 1, q = 1, . . . Q , is produced by the transformation

function (Sq(P)) which is to convert a set of measurement values

by the measurement function Sq(P) into a measurement priorityvector vsq of measurement criterion sq. In Section 4, a normalization

of inversion function NI() (from Eq. (35)) is defined for (), i.e.(Sq(P)) = NI(Sq(P)).

2.4.3. Aggregation process (Agg = ())In this process, a Maximum Individual Interest Aggregation

Operator () is proposed. It consists of two parts: column normal-

ization and ordering function, which are shown as follows.

Column normalization of VP is to generate the related weightsof the measurement criteria for each prioritization operator, and is

the form:

CN(VP) =

vqk

=vqknq=1

vqk

| q = 1, 2, . . . , Q, k = 1, 2, . . . , K

=VPQ

q=1vqk, k = 1, 2, . . . , K

(8)which is written explicitly,

V = CN(VP) = (vqk)QK

=

s1s2...

sq...

sQ

p1 p2 . .. pk . .. pK

v11

v12

. . . v1k

. . . v1K

v21

v22

. . . v2k

. . . v2K

..

....

. . ....

..

....

vq1 v

q2

vqk

vqK

.

.....

.

.....

.

.....

vQ1

vQ2

vQk

vQK

,

(9)

where vqk

= vqk/n

q=1vqk, k = 1, 2, . . . , K .

From the matrix VP (Eq. (7)), a row vector is vsq ={vq1, . . . , vqk, . . . , vqK}, q = 1, . . . , Q . Ordering function of vsqreturns a set of rank positions of vsq in ascending order, and has

the form:ordering(vsq ) = {Iq(j)|j = 1, . . . , K }, q = 1, . . . , Q where

Iq(j) =K

k=1rj(vqk), and

rj(vqk) =1, vqj > vqk&j /= k0, otherwise

(10)


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Table 2

The fundamental scale of absolute numbers ].

Intensity of Importance Definition Explanation

1 Equal importanceTwo activities contribute equally to the objective

2 Weak or slight

3 Moderate importanceExperience and judgment slightly favor one activity over another

4 Moderate plus

5 Strong importanceExperience and judgment strongly favor one activity over another

6 Strong plus

7 Very strong or demonstrated

importance

An activity is favored very strongly over another; its dominance

demonstrated in practice

8 Very, very strong The evidence favoring one activity over another is of the highest

possible order of affirmation9 Extreme importance

Reciprocals of above If activity i has one of the

above nonzero numbers

assigned to it when compared

with activity j, then j has the

reciprocal value when

compared with i

A reasonable assumption

Rationals Ratios arising from the scale If consistency were to be forced by obtaining n numerical values to

span the matrix

If the matrix VP is taken as parameter, the matrix I = ordering(VP),which is written explicitly

I = ordering(VP) = (iqk)QK

=

s1s2...

sq...

sQ

p1 p2 . .. pk . .. pK

i11

i12

. .. i1k

. .. i1K

i21

i22

. .. i2k

. .. i2K

.

.....

. . ....

.

.....

iq1

iq2

iqk

iqK

......

......

......

iQ1 iQ2 i

Qk

iQK

, iqk {1, . . . , K } (11)

Finally, the aggregation agg(V, I) of V and Iis to take a column

mean of non-scalar product V I, which is the form:

(V, I) =

uk |uk =

1

Q

Qq=1

iqk vqk, k = 1, . . . , K

= {u1, . . . uk, . . . , uK} = (12)

Thus, the Mean Individual Interest Aggregation Operator isthe form:

(VP) = (V, I) = (CN(VP), ordering(VP))

= {u1, . . . uk, . . . , uK} = (13)

is a set of preference values of prioritization operators by theMean Individual Interest Aggregation Operator taking VP as aparameter. The advantage of may help to address the problem to

set a weight for each measurement criterion. No universal weights

are given to the measurement criteria as the relative weights are

based on CN(VP). Thehigherrelativeweightlikely obtainthe higherrank score in ordering(vsq ). The aggregation of the weights and cor-responding rank scores is based on each prioritization operators

optimum interest with considering other operations. Such allo-

cation of is more reasonable than those allocation of a simplemean, max, or min operator as is tail-made for the prioritizationoperators.

2.4.4. Exploitation process (EP = (p, f(P, ), W))The prioritization selection function f(P, ) returns the best

prioritization method p* in P with respect to = (VP). The

best prioritization method is determined by the highest score

* = Max(), and its position is returned by the argument of themaximum function argmax shown as follows:

p = f(P, ) = p,

where = argmaxj {1,2,...,m}

({u1, . . . uq, . . . , uQ}), p P (14)

An ideal priority vector W of a reciprocal matrix A derived by

the best prioritization method P* in AHPP model is the form:

W = (A) = P(A) (15)

W can be a priority vector WC of criteria or a priority vector

WT,i of all alternatives T under a criterion i, from the reciprocalpairwise matrices (AC or AT,i ) by AHPP function : AC WC or

: AT,i WT,i . In CAHP model, AHPP returns WTC

and WT, whichare as follows:

WTC = [(AC)]T

= P(AC) = (1, 2, . . . , i, . . . , n)T,

n

i=1

i = 1

(16)

The upper subscript Tis the transposition function.

WT = (WT,i )T

= (WT,1, WT,2, . . . , W T,i , . . . , W T,n )T

= [wij]nm

where WT,i = (AT,i ) = {wi1,, . . . , wim} and

m

j=1wij = 1 , i = 1, . . . , n (17)


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WT is written explicitly as the form:

WT =

c1c2...

ci...

cn

t1 t2 . . . t j . .. t m

w11 w12 . . . w1j . .. w1m

w21 w22 . . . w2j . .. w2m

......

. . ....

......

wi1 wi2 wij wim

......

......

......

wn1 wn2 wnj wnm

(18)

WTC

and WT are used in synthesis process discussed in Section2.5.

2.5. Synthesis process

Synthesis process is 3-tuple, i.e. S = ((WC, WT), t, f). Synthesis

process is to selectthe best alternative t* froma matrix WT = {WT,i }

of the priority vectors of all alternatives for all criteria, and a crite-

ria priority vector WC, by a selection function f(WC, WT), which isillustrated as follows.

Theresults ofWCand WT

are aggregatedto obtain global priority

of each alternative Wby synthesis aggregation function Saggwhichis defined as the Scalar Dot Product sdp of WT

Cand WT shown as

follows:

W = Sagg(WC, WT) = sdp(WC, WT) = WTC WT

= {w1, w2, . . . , wj, . . . , wm} (19)

Usually, the best alternative is determined by the highest score

w = Max(W), and its position is returned by the argument of themaximum function argmax as follows:

t = (W) = t, where = argmaxj {1,2,...,m}

({w1, w2, . . . , wj, . . . , wm})

(20)

(In rare cases, if lowest score is applied, the argument of the

minimum function argmin is used.)

Finally, the selection function is the form:

t = f(WC, WT) = (W) = (sdp(WC, WT)) (21)

3. Prioritization operators

3.1. Eigenvector (EV)

Eigenvector operator for the intuitive justification is proposed

by [3]. EV is to derive the principal Eigenvector max of A as thepriority vector w by solving following the Eigen system.

Aw = maxw, andn

i=1

wi = 1 (22)

A is consistent if and only If max = n, and is not consistent if andonly ifmax > n where max n.

3.2. Normalization operators

Normalization operator was introduced in [3]. The following

methods are named according to their calculation steps.

3.2.1. Normalization of the row sum (NRS)

NRS is to sum the elements in each row and normalize by divid-

ing each sum by the total of all the sums, thus the results now add

up to unity. NRS has the form:

ai

=

nj=1

aij i = 1, 2, . . . , n

wi =a

ini=1

ai

i = 1, 2, . . . , n

(23)

3.2.2. Normalization of reciprocals of column sum (NRCS)

NRCS is to take the sum of the elements in each column, form

the reciprocals of these sums, and then normalize so that these

numbers add to unity, e.g. to divide each reciprocal by the sum of

the reciprocals. It is the form:

ai

=1n

i=1aij

j = 1, 2, . . . , n

wi =a

ini=1

ai

i = 1, 2, . . . , n

(24)

3.2.3. Arithmetic mean of normalized columns (AMNC)

AMNC is also named as additive normalization methodin [20] as

the proposed name is relatively clear about its calculation process.

Each element in A is divided by the sum of each column in A, andthen the mean of each row is taken as the priority wi . It has the

form:

aij

=aijni=1

aiji, j = 1, 2, . . . , and

wi =1

n

nj=1

aij i = 1, 2, . . . , n

(25)

The AHP applications applied this method due to the simplicity

of its calculation process.

3.2.4. Normalization of geometric means of rows (NGMR)

NGMR is to multiply the n elements in each row and take the

nth root, and then normalize so that these numbers add to unity. It

is the form:

wi

=

nj=1

a1/nij

i = 1, 2, . . . , n

wi =w

ini=1

wi

i = 1, 2, . . . , n

(26)

Although it is more complex than other three normalization

methods, it is recommended by some authors [e.g. 16, 18, 31] as

this method produces the same result as LLS.

3.3. Direct least squares/weighted least squares (DLS/WLS)

This method is used to minimize the sum of errors of the differ-

ences between the judgments and their derived values. The direct

least squares, which was proposed in [30], is the form:

Min

ni=1

nj=1

aij

wiwj

2

Subject to

ni=1

wi = 1, wi > 0, i = 1, 2, . . . , n

(27)

The above non-linear optimization problem has no special

tractable form or closed form and is very difficult to be solved [30].

However, there is no clear evidence (e.g. no formal mathematical


6/15


proof can be found in the literature) to show it has multiple solu-

tions although [26] claimed that it may have multiple solutions,

and [17,20] indicated that it generally has multiple solutions

using the incorrectcitations of[26,30]. At least, this research pro-

duces the same LLS results of the two examples (Section 5) as [21]

does.

For efficient computation with closed form, [30] modified the

objective function and proposed the weighted least squares (WLS)

in the form:

Min

ni=1

nj=1

(wi aijwj)2

Subject to

ni=1

wi = 1, wi > 0, i = 1, 2, . . . , n

(28)

Although this method provides the closed form for the answer,

which is shown in [20], the reliability is likely less than DLSM (refer

to Example 2).

3.4. Logarithm least squares (LLS)

LLS has a long history and has been intensively studied by many

authors [e.g. 16, 19, 2228]. The LLS is of the form:

Min

ni=1

nj>i

(ln aij (ln w

i ln wj))

2

Subject to

ni=1

wi

= 1, wi > 0, i = 1, 2, . . . , n

(29)

The finalresult(wi) is derived from normalization of (wi). Craw-

ford and Williams [16] indicated that the solution is unique, and is

equivalent to NGMR, which is preferable due to its simplicity.

3.5. Fuzzy programming (FP)

The FP is proposed by Mikhailov [17], which has the form:

max

Subject to d+j

+ RjWT d+

j,

dj

RjWT d

j, j = 1, 2, . . . , m , 1 0

ni=1

wi = 1, wi > 0, j = 1, 2, . . . , n

(30)

Rj

Rmn = {aij

} is the row vector. The values ofthe leftand right

tolerance parametersdj

and d+j

representthe admissible interval of

approximate satisfaction of the crispequalityRjWT= 0.The measure

of intersection of is a natural consistency index of theFP. Its valuehowever depends on the tolerance parameters. For the practical

implementation of the FP it is reasonable all these parameters to

be set equal. Limitation of this method is that parameters dj

and

d+j

are undermined in [17]. This leads to infinite candidate values

for the parameters. Mikhailov [17] set dj

= d+j

= 1 in his example.

3.6. Enhanced goal programming (EGP)

Bryson [29] proposed goal programming operator (GP), which

uses relative deviations

+

ij /

ij to measure the relationship between

wi/wj and aij. The relationship has the form:+

ij

ij

wiwj

= aij, where (

+

ij 1&

ij= 1) or (

ij 1&

ij= 0)

(31)

In Brysons method, the aim is to minimize

ij>1(+

ij

ij). To

solve Eq. (31), the non-linear programming problem is translated

into the linear goal programming problem with the form:

Min ln =

ni=1

nj>i

(ln +ij

+ ln ij

)

Subject to ln wi ln wj + ln +

ij ln

ij= ln aij, (i, j) IJ

(32)

where IJ = {(i, j) : 1 i < j n}; ln +ij

and ln ij

are non-negative.

Ideally the objective value should be 0 when ln +ij

= ln ij

= 0,

i.e. +ij

= ij

= 1. The computer tries to minimize the value as low

as possible for the solution by many loops. In the simulation of

this research, GP does not provide the unique results of the prior-

ities, even if the same objective value is achieved by the functions

FindMinimum[] and NMinimize[] in Mathematica. Also Lingo and

Mathematica provide differentresultswhen objective values arethesame.Lin [19] illustratedan example toadd a wi > 0 asa constraint;the priority is differentbut the objective values arethe same. These

facts can be concluded that GP leads to various priority vectors for

thesameobjective value.(The mathematicalproof is left forreaders

as it is beyond the research purpose.)

To address this issue, one approach is to modify GP as LLS. If

the objective function is modified asn

i=1

nj>i

(ln +ij

+ ln ij

)2

, it

will become another form of the LLS model, which is equivalent to

Eq. (29). Thus, there are two forms of LLS which provide the same

result as NGMR:

Min ln =

n

i=1

n

j>i

(ln +ij

+ ln ij

)2

Subject to ln wi ln wj + ln +

ij ln

ij= ln aij, (i, j) IJ

(33)

where IJ = {(i, j) : 1 i < j n}; ln +ij

0 and ln ij

0. The objec-

tive function is also equivalent ton

i=1

nj>i

((ln +ij

)2

+ (ln ij

)2

).

As NGMR is relatively easy for calculation, it is more preferable.

However, there exists a tradeoff. The GP, which gives various solu-

tions, performs better than the LLS when outliers exist but suffers

from the problem of alternative optimal solutions while the LLS

gives a unique solution but is sensitive to outliers [18,19].

Another approach is to use the enhanced goal programming

model [19], which is the combination of GP and LLS, and has the

form:

Min ln +

Subject to ln wi ln wj + ln +ij ln ij = ln aij, (i, j) IJ

ln =

ni=1

nj>i

(ln +ij

+ ln ij

)

=

ni=1

nj>i

((ln +ij

)2

+ (ln ij

)2

)

(34)

where IJ = {(i, j) : 1 i < j n}, ln +ij

0, ln ij

0 and is a suf-

ficient small positive number. The term sufficient small means that

any increase of will cause ln to lose its optimality. In his paper, is set to 1010 as an example, which is approximate to 0. When

ln reaches its optimum, a tradeoff rate exists between ln and. The decrease of leads to the increase of ln . The effect of

is to depress ln to increase. When is sufficient small, any sacri-


7/15


fice ofln fir reducing would be fruitless. Thus, themodelforcesthesolutionto minimize ln before is minimized.However, thisoptimization method induces more computational effort than LLS

and GP.

4. Measurement priority vectors

If the matrix is consistent, any above prioritization methods can

produce real priority vectors. This usually happens in the matrixof size 3 3 and 4 4. A matrix of size of 5 5 or above is likely

an inconsistent matrix. For the inconsistent matrix, different pri-

oritization operators produce different results, which possibly lead

to different priority orders. Criteria to measure the fitness of the

prioritization operators are needed. Fitness presents the measure-

ment of how fit a priority vector of a prioritization operator can

represent a reciprocal matrix. In Section 2, a measurement priority

vector of a measurement criterion sq of prioritization operator P isthe form vsq = (Sq(P)). Sq(P) is a measurement method or func-

tion ofPfor sq. This study defines seven measurement methods onthe basis of variances. Less variance leads to more fitness of a pri-

oritization operator. The normalization of inversion NI() is defined

for the transformation function () which is to convert a set of

measurement values by Sq(P) into a vector of the measurementpriorities vsq . Let a set of variances (or measurement values) be

= {1, . . . , k, . . . , K}, K is the cardinal number of the set of pri-oritization operators, thus inversion of is 1 =

1

1, . . . , 1K

.

The normalization of inversion (NI()) is of the form:

NI() =

11zi=1

1, . . . ,

1kz

i=11

, . . . ,1zzi=1

1

(35)

can be determined by Sq(P), i.e. = Sq(P). Seven measurementmethods S1,...,7(P) propagating to measurement priorityvectors areintroduced as follows.

4.1. Normalization of inversion of mean absolute variance(NI(MAV))

The mean absolute variance is of the form:

MAV(A, W) =1

n n

ni=1

nj=1

aij wiwj (36)

whereA is a pairwise matrix (aij), Wis a priorities vector of a prior-

itization operator, and wi, wj W {W1, . . . , W K} = {W}. Then thepriorities vector of the set of the prioritization candidates is of the

form:

MAV(A, {W}) = {MAV1, . . . , M A V K}

Less variance reflects better fitness. Higherinversion of variance

leads to higher fitness of a prioritization operator. Then, inversion

ofMAVis of the form:

MAV1(A, {W}) = {MAV11, . . . , M A V K

1}

Finally the above result is normalized, and then NI(MAV) is of

the form:

NI(MAV(A, {W}))=

MAV1

1

MAV1(A, {W})

, . . . ,MAVK

1

MAV1(A, {W})

(37)

4.2. Normalization of inversion of root mean square variance

(NI(RMS))

The root mean square variance is of the form:

RMSV(A, W) =

1

n n

ni=1

nj=1

aij

wiwj

2(38)

The Euclidean distance is of the form:

ED(A, W) =

ni=1

nj=1

aij

wiwj

2(39)

Similar to the calculation approach of NI(MAV), then NI(RMSV)

or NI(ED) is of the form:

NI(RMSV(A, {W}))

=

RMSV1

1RMSV1(A, {W})

, . . . ,RMSVK

1RMSV1(A, {W})

(40)

Although the form ofRMSVis different from ED, the final results

are the same after NIis taken.

4.3. Normalization of inversion of worst absolute variance

(NI(WAV))

The worst absolute variance is the form:

WAV(A, W) = Maxi,j

aij wiwj

(41)

And NI(WAV) is the form:

NI(WAV(A, {W}))

= WAV11

WAV1(A, {W}) , . . . ,

WAVK1

WAV1(A, {W})

(42)

4.4. Normalization of inversion of consistency variance (NI(CV))

Consistence index has two purposes in this paper. One is used

for testing the validity of user inputs for the pairwise matrices.This

belongs to the topic of consistence ratio. The second one is that

CI is as a measurement criterion to measure the relative fitness

among candidates. To distinguish these two purposes, consistency

variance is the name used for the second purpose.

The CI(Eq. (6)) proposed by Saaty for the EVM can be expressed

as the average of the differences between the errors and the unity,

and is of the form [31].

CV = CI =1

n(n 1)

ni /= j

aijwj

wi 1

(43)

And NI(CV(A,{W})) is the form:

NI(CV(A, {W})) =

CV1

1CI1(A, {W})

, . . . ,CVK

1CI1(A, {W})

(44)

4.5. Normalization of inversion of geometric consistency variance

(NI(GCV))

Analogous to Eq. (43), [31] proposed geometric consistency

index, which can be seen as an average of the square difference


8/15


between the logarithmof the errors and the logarithm of unity, i.e.

GCV = GCI =1

n(n 1)

i 1wi = wj&aji /= 1wi /= wj&aji = 1otherwise

There is a critical error of theaboveform. Iij mustbe equal to1 in

the condition (wi < wj & aji < 1). Also as the value of MVdependson the size (n2) of the matrix (usually the larger size of the matrix

leads to the higher value of MV), the mean value of MV(MMV) is

a more appropriate method to measure the prioritization method

in the reciprocal matrix. Thus, the revised MMVhas the following

form:

MMV(A, W) =1

n2

1 +

i

j

Iij

Iij =

1, (wi > wj & aji > 1) or (wi < wj & aji < 1)0.5, (wi = wj & aji /= 1) or (wi /= wj & aji = 1)

0, otherwise

(47)

An ideal prioritization method means MV=0. The zero value

cannot be used by NI(). To use NI() properly, 1 is added into sum-

mations ofIij which is taken in Eq. (47). NI(MMV(A,{W})) is of the

form:

NI(MWV(A, {W})) =

(MWV1)1zi=1

(MWVi)1

, . . . , (MWVK)1z

i=1(MWVi)

1

(48)

4.7. Normalization of inversion of weighted distance variance

(NI(WDV))

This study proposes the weighted distance variance method,

which is expressed as

WDV =1

ni jYij

where

Yij =

1

aij wiwj , wi wj & aij 1

or wi wj & aij 1

2

aij

wiwj

, wi = wj & aij /= 1, 1 = 1 2 3

or wi /= wj&aij = 1

3aij wiwj , otherwise

(49)

WAV is the special case of WDV if 1 = 2 = 3 = 1. By default,1 = 1, 2 = 1.5, 3 = 12 are defined. And NI(MDV(A,{W})) is of theform:

NI(WDV(A, {W}))

=

WDV1

1WDV1(A, {W})

, . . . ,WDVK

1WDV1(A, {W})

(50)

How the most appropriate prioritization operator is selected on

the basis of above measurement priority vectors is discussed in the

next section.

5. Applications

Two decision problems, which are also discussed in [21], are

selected to illustrate the AHPP concept: (1) allocating the reservoir

storage to multiple purposes, which is taken from [32], (2) choos-

ing the high school, which is taken from [3, pp. 2628]. These two

cases shown in [21] are worthy to be reused since this study also

compares the results between the Multicriteria Prioritization Syn-

thesis (MPS) [21] and the proposed AHPP. Example 1 is illustrated

for details of the calculation approach. And Example 2 is illustrated

for the essentialsof the approach.Various prevailingsoftwarepack-

ages such as Excel, Mathlab, and Lindo can conveniently compute

themodels.Thisstudyuses Mathematica to perform the calculation.

For the comparison with [21], six prioritization operators arechosen in [21]. Only Euclidean distance and minimum violations

with the relative weights of 0.8 and 0.2respectively are considered

in [21]. The calculation of the measurement methodis rather rough

and subjective. In this article, seven measurement criteria and nine

prioritization operators are used to form an objective AHPP model.

The calculation of AHPP is relatively comprehensive and objective

as this is based on the measure priority matrix VP. The details arediscussed as follows.

5.1. Example 1

5.1.1. Definition process

The AHP problem is to allocate the surface water reservoir stor-

ageto multiple uses. A globaleconomical goal is definedas themostprofitable use of reservoir.Six purposes, T = {t1, t2, . . . , t 6},arecon-sidered as decision alternatives. Five economical criteria, C= {c1,

c2, . . ., t5} of different metrics are used as alternatives. Details areshown in Appendix A.

In the AHPPdefinition process, a set of sevenmeasurement crite-

ria S={s1, s2, . . ., s7} is measurement by a setof seven measurementfunctions, S = {S1, . . . , S7}, which is discussedin Section 6, andasetof nineprioritizationoperators P = {p1, p2, . . . , p9} areusedtoform

an AHPP model. The function AMNC used in this paper is the same

as AN using in [21]. The AHPP structure is shown in Fig. 2.

5.1.2. Evaluation and assessment, and measurement process

The details of the pairwise matrices and consistence ratio are

shown in Appendix A. There are one 5 5 matrix AC at the second


9/15


Fig. 2. The structure of AHPP.

level, and a set of five 6 6 matrices AT at the third level. They aredenoted as A = {AC, AT} = {A1, A2, . . . , A6}.

5.1.3. Analytic hierarchy prioritization process

In Table 3, it can be concluded that different prioritization

operators produce different values of priorities. Whist many

studies argue which one is the best, AHPP addresses this prob-

lem and is used for selecting the best prioritization operator

among several candidates by some measurement criteria for an

individual reciprocal matrix. The detailed steps are shown as fol-

lows.

In the definition process, the AHPP structure is shown in Fig. 2.

Table 3

Priority vectors and synthesis results with nine prioritization operators (Example 1).

C1 C2 C3 C4 C5 W C1 C2 C3 C4 C5 W

P1: EV P2: NRS

C 0.358 0.306 0.041 0.171 0.123 0.288 0.321 0.040 0.192 0.159

t1 0.440 0.365 0.409 0.416 0.046 0.363 6 0.389 0.344 0.364 0.359 0.048 0.314 6

t2 0.077 0.110 0.180 0.150 0.209 0.120 2 0.084 0.103 0.179 0.168 0.227 0.133 2

t3 0.286 0.078 0.032 0.036 0.187 0.157 5 0.322 0.076 0.031 0.032 0.168 0.151 4

t4 0.065 0.040 0.106 0.040 0.352 0.090 1 0.069 0.039 0.124 0.048 0.335 0.100 1

t5 0.078 0.211 0.153 0.125 0.162 0.140 4 0.086 0.227 0.192 0.163 0.176 0.164 5

t6 0.054 0.197 0.120 0.233 0.043 0.130 3 0.050 0.211 0.110 0.230 0.047 0.138 3

P3: NRCS P 4: AMNC

C 0.418 0.278 0.048 0.140 0.115 0.352 0.300 0.043 0.172 0.133

t1 0.514 0.406 0.479 0.478 0.061 0.425 6 0.432 0.364 0.403 0.413 0.048 0.356 6t2 0.066 0.104 0.152 0.130 0.155 0.100 2 0.082 0.108 0.178 0.151 0.223 0.125 2

t3 0.216 0.063 0.039 0.043 0.223 0.141 5 0.281 0.079 0.034 0.036 0.192 0.156 5

t4 0.060 0.043 0.091 0.039 0.329 0.085 1 0.066 0.040 0.107 0.043 0.325 0.091 1

t5 0.075 0.194 0.131 0.110 0.177 0.127 4 0.081 0.206 0.159 0.129 0.166 0.142 4

t6 0.070 0.190 0.108 0.200 0.055 0.122 3 0.057 0.203 0.119 0.228 0.045 0.131 3

P5: NGMR/LLS P 6: DLS

C 0.356 0.313 0.042 0.166 0.122 0.276 0.345 0.047 0.166 0.166

t1 0.448 0 .3 84 0 .4 32 0.423 0.053 0.375 6 0.384 0.335 0.353 0.346 0.053 0.305 6

t2 0.073 0 .1 08 0 .1 67 0.147 0.212 0.117 2 0.063 0.098 0.182 0.17 0.204 0.122 2

t3 0.282 0 .0 71 0 .0 34 0.036 0.193 0.154 5 0.348 0.06 0.041 0.042 0.169 0.154 4

t4 0.063 0 .0 41 0 .1 06 0.039 0.326 0.086 1 0.057 0.043 0.087 0.04 0.334 0.097 1

t5 0.079 0 .2 04 0 .1 54 0.125 0.167 0.140 4 0.073 0.239 0.23 0.173 0.192 0.174 5

t6 0.056 0 .1 92 0 .1 08 0.231 0.049 0.129 3 0.075 0.224 0.107 0.229 0.047 0.149 3

P7: WLS P 8: FP

C 0.411 0.291 0.047 0.140 0.111 0.391 0.283 0.065 0.152 0.109

t1 0.498 0.394 0.469 0.464 0.057 0.412 6 0.453 0.389 0.442 0.423 0.17 0.399 6

t2 0.060 0.099 0.160 0.131 0.126 0.093 2 0.121 0.117 0.184 0.165 0.188 0.138 5

t3 0.233 0 .056 0 .040 0.046 0.220 0.145 5 0.201 0.078 0.063 0.059 0 .16 0.131 3

t4 0.057 0 .043 0 .087 0 .039 0.369 0.087 1 0.05 0.067 0.095 0.045 0 .279 0.082 1

t5 0.075 0.205 0.132 0.117 0.177 0.133 4 0.086 0.194 0.104 0.132 0.146 0.132 4

t6 0.077 0.203 0.112 0.202 0.051 0.130 3 0.088 0.156 0.112 0.176 0.056 0.119 2

P9: EGP AHPP

C 0.356 0.313 0.042 0.166 0.122 0.352 0.300 0.043 0.172 0.133

t1 0.426 0.397 0.465 0.436 0.053 0.375 6 0.432 0.384 0.432 0.413 0.053 0.364 6

t2 0.085 0.099 0.155 0.145 0.263 0.124 2 0.082 0.108 0.167 0.151 0.212 0.123 2

t3 0.273 0.066 0.031 0.045 0.158 0.146 5 0.281 0.071 0.034 0.036 0.193 0.154 5

t4 0.063 0 .04 0.085 0 .037 0.316 0.083 1 0.066 0.041 0.106 0.043 0 .326 0.091 1

t5 0.091 0.199 0.171 0.112 0.158 0.140 4 0.081 0.204 0.154 0.129 0.167 0.141 4

t6 0.063 0.199 0.093 0.224 0.053 0.132 3 0.057 0.192 0.108 0.228 0.049 0.128 3


10/15


Table 4

Measurement variances (S(P)s) of nine prioritization operators for six pairwise matrices (Example 1).

P1 P2 P3 P4 P5 P6 P7 P8 P9 P1 P2 P3 P4 P5 P6 P7 P8 P9

A1 A2s1 0.551 0.566 0.628 0.517 0.536 0.482 0.605 0.652 0.536 0.624 0.654 0.698 0.595 0.624 0.547 0.698 0.750 0.554

s2 0.992 0.854 1.113 0.917 0.963 0.733 1.102 1.089 0.962 1.043 1.112 1.216 1.035 1.020 0.789 1.156 1.338 1.026

s3 3.644 2.251 3.684 3.125 3.407 1.997 3.759 3.333 3.407 3.272 3.178 3.714 3.588 3.117 1.896 3.305 5.333 3.794

s4 0.104 0.141 0.128 0.107 0.104 0.153 0.124 0.164 0.104 0.112 0.130 0.143 0.115 0.111 0.151 0.153 0.257 0.121

s5 0.101 0.135 0.123 0.103 0.101 0.143 0.119 0.155 0.101 0.108 0.124 0.137 0.110 0.107 0.143 0.146 0.225 0.116

s6 0.04 0.12 0.04 0.04 0.04 0.12 0.04 0.04 0.04 0.167 0.167 0.222 0.111 0.167 0.278 0.278 0.167 0.139s7 0.551 0.635 0.628 0.517 0.536 0.56 0.605 0.652 0.536 0.718 0.75 0.839 0.649 0.72 0.738 0.896 0.889 0.659

A3 A4s1 0.491 0.500 0.464 0.477 0.455 0.445 0.451 0.604 0.425 0.638 0.599 0.645 0.611 0.578 0.574 0.610 0.686 0.568

s2 0.893 0.846 0.899 0.884 0.881 0.764 0.867 1.151 0.904 1.230 1.110 1.231 1.133 1.221 0.879 1.186 1.220 1.482

s3 3.295 3.000 2.925 3.385 3.146 2.711 2.653 4.667 3.000 5.786 4.600 5.128 4.963 5.823 2.210 4.768 5.333 8.000

s4 0.095 0.102 0.099 0.096 0.093 0.121 0.112 0.150 0.097 0.142 0.163 0.157 0.141 0.138 0.203 0.158 0.263 0.159

s5 0.091 0.097 0.093 0.092 0.089 0.110 0.102 0.142 0.091 0.134 0.150 0.148 0.132 0.129 0.184 0.149 0.239 0.145

s6 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.139 0.139 0.139 0.139 0.139 0.194 0.139 0.194 0.194

s7 0.491 0.500 0.464 0.477 0.455 0.445 0.451 0.604 0.425 0.752 0.747 0.757 0.723 0.688 0.787 0.728 0.853 0.765

A5 A6s1 0.605 0.693 0.643 0.599 0.601 0.537 0.608 0.694 0.519 0.525 0.442 0.555 0.464 0.435 0.424 0.572 0.926 0.389

s2 1.075 1.088 1.264 1.018 1.104 0.817 1.176 1.106 1.104 0.843 0.775 0.885 0.798 0.774 0.747 0.888 1.457 0.831

s3 3.432 2.637 5.208 2.997 3.940 1.994 4.829 3.075 4.662 3.315 3.521 2.877 3.541 3.464 3.367 2.763 4.361 3.800

s4 0.083 0.117 0.103 0.084 0.082 0.115 0.104 0.146 0.095 0.122 0.126 0.149 0.121 0.117 0.126 0.186 0.439 0.144

s5 0.081 0.114 0.099 0.082 0.080 0.111 0.100 0.139 0.092 0.116 0.117 0.141 0.113 0.110 0.118 0.172 0.390 0.129

s6 0.028 0.028 0.083 0.028 0.028 0.139 0.083 0.083 0.083 0.111 0.167 0.222 0.111 0.111 0.167 0.222 0.222 0.111

s7 0.605 0.693 0.690 0.599 0.601 0.623 0.659 0.751 0.571 0.525 0.486 0.666 0.464 0.435 0.472 0.707 1.086 0.389

Table 5

The measurement priority matrices (VP = NI(S(P))) for six pairwise matrices (Example 1).


A1 A2s1 0.113 0.110 0.099 0.120 0.116 0.129 0.103 0.095 0.116 0.112 0.107 0.101 0.118 0.112 0.128 0.101 0.094 0.127

s2 0.107 0.124 0.095 0.116 0.110 0.145 0.096 0.097 0.110 0.113 0.106 0.097 0.114 0.116 0.149 0.102 0.088 0.115

s3 0.093 0.150 0.092 0.108 0.099 0.169 0.090 0.101 0.099 0.110 0.114 0.097 0.101 0.116 0.190 0.109 0.068 0.095

s4 0.130 0.096 0.106 0.127 0.130 0.089 0.110 0.083 0.130 0.134 0.116 0.105 0.131 0.135 0.100 0.098 0.058 0.124

s5 0.128 0.096 0.106 0.126 0.129 0.091 0.110 0.084 0.129 0.132 0.115 0.104 0.130 0.134 0.100 0.098 0.064 0.124

s6 0.130 0.043 0.130 0.130 0.130 0.043 0.130 0.130 0.130 0.116 0.116 0.087 0.173 0.116 0.069 0.069 0.116 0.139

s7 0.116 0.101 0.102 0.124 0.119 0.114 0.106 0.098 0.119 0.116 0.112 0.100 0.129 0.116 0.113 0.093 0.094 0.127

A3 A4s1 0.107 0.106 0.114 0.111 0.116 0.119 0.117 0.087 0.124 0.106 0.113 0.105 0.111 0.117 0.118 0.111 0.099 0.119

s2 0.111 0.117 0.110 0.112 0.112 0.129 0.114 0.086 0.109 0.106 0.117 0.105 0.115 0.106 0.148 0.109 0.106 0.088

s3 0.105 0.116 0.118 0.102 0.110 0.128 0.131 0.074 0.116 0.089 0.112 0.101 0.104 0.089 0.234 0.108 0.097 0.065

s4 0.122 0.114 0.118 0.121 0.125 0.097 0.104 0.078 0.120 0.128 0.111 0.116 0.128 0.131 0.089 0.114 0.069 0.114

s5 0.120 0.113 0.118 0.120 0.124 0.100 0.107 0.077 0.120 0.126 0.112 0.114 0.127 0.130 0.091 0.113 0.070 0.116

s6 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.123 0.123 0.123 0.123 0.123 0.088 0.123 0.088 0.088

s7 0.107 0.106 0.114 0.111 0.116 0.119 0.117 0.087 0.124 0.111 0.112 0.111 0.116 0.122 0.106 0.115 0.098 0.109

A5 A6s1 0.111 0.097 0.105 0.112 0.112 0.125 0.111 0.097 0.130 0.105 0.124 0.099 0.118 0.126 0.130 0.096 0.059 0.141

s2 0.111 0.109 0.094 0.117 0.108 0.145 0.101 0.107 0.108 0.113 0.123 0.108 0.119 0.123 0.127 0.107 0.065 0.115

s3 0.108 0.141 0.071 0.124 0.094 0.186 0.077 0.121 0.079 0.114 0.107 0.131 0.106 0.109 0.112 0.136 0.086 0.099

s4 0.135 0.095 0.108 0.131 0.135 0.096 0.107 0.076 0.117 0.132 0.128 0.108 0.133 0.137 0.128 0.087 0.037 0.112

s5 0.134 0.095 0.108 0.131 0.134 0.097 0.107 0.077 0.117 0.129 0.128 0.106 0.133 0.136 0.127 0.087 0.038 0.116

s6 0.181 0.181 0.060 0.181 0.181 0.036 0.060 0.060 0.060 0.146 0.098 0.073 0.146 0.146 0.098 0.073 0.073 0.146

s7 0.117 0.102 0.103 0.119 0.118 0.114 0.108 0.095 0.124 0.112 0.121 0.089 0.127 0.136 0.125 0.084 0.054 0.152

Table 6

Preference values of nine prioritization operators for A1 (Example 1).

A1 V = CN(VP) I = ordering(VP)


s1 0.138 0.152 0.136 0.141 0.139 0.165 0 .138 0.138 0.139 5 4 2 8 7 9 3 1 6

s2 0.131 0.172 0.130 0.136 0.132 0.185 0 .129 0.141 0.132 4 8 1 7 5 9 2 3 6

s3 0.113 0.208 0.126 0.127 0.119 0.217 0 .121 0.147 0.119 3 8 2 7 4 9 1 6 5

s4 0.159 0.133 0.145 0.149 0.156 0.114 0 .147 0.120 0.156 7 3 4 6 9 2 5 1 8

s5 0.157 0.134 0.145 0.148 0.155 0.117 0 .147 0.122 0.155 7 3 4 6 9 2 5 1 8

s6 0.160 0.060 0.179 0.153 0.156 0.056 0 .175 0.189 0.156 3 1 3 3 3 1 3 3 3

s7 0.142 0.140 0.140 0.145 0.143 0.147 0 .142 0.142 0.143 6 2 3 9 8 5 4 1 7

0.728 0.685 0.395 0.932 0.931 0.908 0.48 0.342 0.883 = 4, p*(A1) =p4(A1 )


11/15


Table 7




s1 0.135 0.137 0.146 0.132 0.133 0.151 0 .150 0.161 0.149 6 4 2 7 5 9 3 1 8

s2 0.136 0.135 0.140 0.127 0.137 0.176 0 .152 0.152 0.135 5 4 2 6 8 9 3 1 7

s3 0.132 0.145 0.141 0.112 0.137 0.224 0 .163 0.116 0.112 6 7 3 4 8 9 5 1 2

s4 0.160 0.147 0.152 0.146 0.160 0.117 0 .146 0.100 0.145 8 5 4 7 9 3 2 1 6

s5 0.158 0.147 0.151 0.145 0.158 0.118 0 .146 0.110 0.146 8 5 4 7 9 3 2 1 6

s6 0.139 0.147 0.126 0.194 0.137 0.082 0 .103 0.199 0.163 4 4 3 9 4 1 1 4 8

s7 0.140 0.142 0.144 0.144 0.137 0.133 0 .139 0.162 0.149 7 4 3 9 6 5 1 2 8

0.909 0.675 0.431 1.03 1.014 0.915 0.364 0.251 0.944 = 4, p*(A2) =p4(A2)

Table 8




s1 0.137 0.135 0.142 0.140 0.142 0.148 0.146 0 .145 0.150 3 2 5 4 6 8 7 1 9

s2 0.141 0.149 0.137 0.142 0.138 0.161 0.142 0 .143 0.133 4 8 3 5 6 9 7 1 2

s3 0.134 0.148 0.148 0.130 0.135 0.159 0.163 0 .124 0.140 3 5 7 2 4 8 9 1 6

s4 0.156 0.146 0.147 0.154 0.154 0.120 0.130 0.129 0.146 8 4 5 7 9 2 3 1 6

s5 0.153 0.145 0.147 0.152 0.152 0.125 0.134 0 .129 0.146 8 4 5 6 9 2 3 1 7

s6 0.142 0.142 0.138 0.141 0.136 0.139 0.139 0 .185 0.135 1 1 1 1 1 1 1 1 1

s7 0.137 0.135 0.142 0.140 0.142 0.148 0.146 0 .145 0.150 3 2 5 4 6 8 7 1 9 0 .6 29 0 .54 0 .638 0 .60 4 0.852 0.817 0.777 0.143 0.835 = 5, p*(A3) =p5(A3)

In the measurement and prioritization processes, the variances

of P which are measured by S(P) with respect to each pairwisematrix is shown in Table 4. The relative weights of the seven

measurement variances (or measurement priorities) of nine pri-

oritization operators of six pairwise matrices are determined by

(S(P)) which is to take the function NI() of the measurementvariances shown in Table 4, and then the measurement priority

matrices VP

s of the six pairwise matrices are shown in Table 5.

In the aggregation and exploitation processes, a Mean Indi-

vidual Interest Aggregation Operator is used to aggregatethe measurement priority matrices V

P

s of the six pairwise

matrices (in Table 5). (Eq. (13)) returns the set of mean val-

ues of V and I, where V = CN(VP), and I = ordering(VP). The

set of the mean values are the set of the preference val-

ues of prioritization operators. The most appropriate operator

is located in the argument of the maximum of the prefer-

ence values, i.e. p*() =p(), = argmaxj {1,2,...,m}

({u1, . . . uq, . . . , uQ}).

These results are shown in Tables 611. The highest prefer-

ence values of the selected POs are bolded in Tables 611,

and the values of selected POs are bolded and underlined in

Table 3.

5.1.4. Synthesis process

The results of the synthesis process with various prioritizationoperators are shown in Table 3. Unlike other prioritization oper-

ators, an AHPP method is used to select the best prioritization

Table 9




s1 0.135 0.141 0.136 0.135 0.143 0.135 0.140 0.157 0.171 3 6 2 4 7 8 5 1 9

s2 0.134 0.146 0.136 0.139 0.130 0.169 0.138 0 .170 0.125 3 8 2 7 4 9 6 5 1

s3 0.113 0.140 0.130 0.127 0.109 0.268 0.137 0 .155 0.093 3 8 5 6 2 9 7 4 1

s4 0.162 0.139 0.149 0.156 0.160 0.102 0.144 0 .110 0.163 7 3 6 8 9 2 5 1 4

s5 0.160 0.140 0.147 0.154 0.159 0.105 0.142 0 .112 0.166 7 3 5 8 9 2 4 1 6

s6 0.156 0.153 0.159 0.149 0.150 0.100 0.155 0 .140 0.126 4 4 4 4 4 1 4 1 1

s7 0.141 0.140 0.143 0.140 0.149 0.122 0.145 0 .156 0.157 5 6 4 8 9 2 7 1 3

0.675 0.776 0.576 0.925 0.936 0.824 0.772 0.306 0.571 = 5, p*(A5) =p5(A5)

Table 10

Preference values of prioritization operators for A5 (Example 1).


PI P2 P3 P4 P5 P6 P7 P8 P9 P1 P2 P3 P4 P5 P6 P7 P8 P9

s1 0.124 0.118 0.161 0.123 0.127 0.157 0 .165 0.153 0.176 5 2 3 7 6 8 4 1 9

s2 0.123 0.133 0.145 0.128 0.122 0.182 0 .151 0.170 0.146 7 6 1 8 4 9 2 3 5

s3 0.121 0.171 0.110 0.135 0.107 0.232 0 .114 0.190 0.108 5 8 1 7 4 9 2 6 3

s4 0.150 0.116 0.166 0.144 0.153 0.120 0.159 0.120 0.159 8 2 5 7 9 3 4 1 6

s5 0.149 0.115 0.167 0.143 0.152 0.121 0 .160 0.122 0.160 8 2 5 7 9 3 4 1 6

s6 0.202 0.221 0.093 0.198 0.205 0.045 0 .090 0.095 0.082 6 6 2 6 6 1 2 2 2

s7 0.131 0.125 0.158 0.130 0.134 0.142 0 .161 0.149 0.169 6 2 3 8 7 5 4 1 9

0.925 0.635 0.438 1.009 0.942 0.923 0.470 0.341 0.891 = 4, p*(A5) =p4(A5)


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Table 11




s1 0.123 0.150 0.139 0.134 0.138 0.153 0.144 0 .144 0.161 4 6 3 5 7 8 2 1 9

s2 0.133 0.148 0.151 0.135 0.135 0.151 0.160 0.158 0.130 4 7 3 6 8 9 2 1 5

s3 0.133 0.129 0.183 0.120 0.119 0.132 0.203 0 .209 0.113 7 4 8 3 5 6 9 1 2

s4 0.155 0.154 0.151 0.150 0.150 0.151 0.129 0 .088 0.127 7 5 3 8 9 6 2 1 4

s5 0.152 0.154 0.149 0.150 0.149 0.150 0.130 0.093 0.132 7 6 3 8 9 5 2 1 4

s6 0.172 0.118 0.103 0.166 0.160 0.115 0.109 0 .177 0.166 6 4 1 6 6 4 1 1 6

s7 0.132 0.146 0.124 0.144 0.149 0.148 0.125 0 .132 0.172 4 5 3 7 8 6 2 1 9

0.809 0.765 0.530 0.893 1.069 0.911 0.474 0.143 0.843 = 5, p*(A6) =p5(A6 )

Table 12

Comparison of AHPP and MPS [21] (Example 1).

C1 C2 C3 C4 C5 Global priorities

AHPP MPS AHPP MPS AHPP MPS AHPP MPS AHPP MPS AHPP MPS

(AMNC,AMNC) 0.352 0.352 0.300 0.300 0.043 0.043 0.172 0.172 0.133 0.133

p* AMNC AMNC LLS LGP LLS AMNC AMNC AMNC LLS LLS

t1 0.432 0.432 0.384 0.391 0.432 0.403 0.413 0.413 0.053 0.053 0.3639(6) 0.3648(6)

t2 0.082 0.082 0.108 0.098 0.167 0.178 0.151 0.151 0.212 0.212 0.1228(2) 0.1201(2)

t3 0.281 0.281 0.071 0.065 0.034 0.034 0.036 0.036 0.193 0.193 0.1538(5) 0.1517(5)

t4 0.066 0.066 0.041 0.056 0.106 0.107 0.043 0.043 0.326 0.326 0.0909(1) 0.0954(1)

t5 0.081 0.081 0.204 0.195 0.154 0.159 0.129 0.129 0.167 0.167 0.1407(4) 0.1382(4)

t6 0.057 0.057 0.192 0.195 0.108 0.119 0.228 0.228 0.049 0.049 0.1279(3) 0.1294(3)

Italic form indicates the notation representing functions.

method among the candidates to prioritize the pairwise matrix

or reciprocal matrix since the best prioritization method is case

dependent. The case means the content of the pairwise matrix.

A comparison between AHPP and MPS [21] for Example 1

is shown in Table 12. Although the best prioritization operators

with respect to C2 and C3 are not the same, their ranks are the

same, but final utilities are different. Regarding the methodology,

AHPP is regarded to be superior to MPS because of the following

reasons:

1. AHPP uses more measurement criteria to evaluate the perfor-

mance of each individual prioritization operator. Appling more

related measurement criteria means more objective or less

biased to reflect the actual performance of the prioritization

operator.

Table 13

Priority vectors and synthesis results with various prioritization operators (Example 2).

C1 C2 C3 C4 C5 C6 W C1 C2 C3 C4 C5 C6 W

P1: EV P2: NRS

C 0.321 0.140 0.035 0.128 0.237 0.139 0.242 0.188 0.031 0.131 0.232 0.175

t1 0.157 0.333 0.455 0.772 0.250 0.691 0.367 2 0.151 0.333 0.455 0.695 0.250 0.657 0.378 3

t2 0.594 0.333 0.091 0.055 0.500 0.091 0.378 3 0.575 0.333 0.091 0.054 0.500 0.090 0.344 2

t3 0.249 0.333 0.455 0.173 0.250 0.218 0.254 1 0.274 0.333 0.455 0.251 0.250 0.254 0.279 1

P3: NRCS P 4: AMNC

C 0.381 0.105 0.045 0.131 0.211 0.127 0.305 0.149 0.038 0.141 0.221 0.146

t1 0.169 0.333 0.455 0.809 0.250 0.711 0.369 2 0.159 0.333 0.455 0.750 0.250 0.685 0.377 3

t2 0.607 0.333 0.091 0.068 0.500 0.101 0.397 3 0.589 0.333 0.091 0.060 0.500 0.093 0.365 2

t3 0.225 0.333 0.455 0.124 0.250 0.189 0.234 1 0.252 0.333 0.455 0.190 0.250 0.221 0.258 1

P5: NGMR/LLS P 6: DLS

C 0.316 0.139 0.036 0.125 0.236 0.148 0.184 0.220 0.037 0.150 0.210 0.197t1 0.157 0 .333 0 .455 0 .772 0.250 0.691 0.370 2 0.178 0.333 0.455 0.788 0.250 0.687 0.430 3

t2 0.594 0 .333 0 .091 0 .055 0.500 0.091 0.376 3 0.592 0.333 0.091 0.082 0.500 0.108 0.325 2

t3 0.249 0 .333 0 .455 0 .173 0.250 0.218 0.254 1 0.230 0.333 0.455 0.130 0.250 0.205 0.245 1

P7: WLS P 8: FP

C 0.415 0.094 0.035 0.112 0.219 0.125 0.349 0.144 0.053 0.123 0.192 0.139

t1 0.174 0.333 0.455 0.804 0.250 0.707 0.353 2 0.159 0.333 0.455 0.796 0.250 0.703 0.371 2

t2 0.606 0.333 0.091 0.074 0.500 0.107 0.417 3 0.619 0.333 0.091 0.082 0.500 0.109 0.390 3

t3 0.221 0.333 0.455 0.122 0.250 0.187 0.230 1 0.222 0.333 0.455 0.122 0.250 0.188 0.239 1

P9: EGP AHPP

C 0.431 0.131 0.027 0.137 0.144 0.131 0.305 0.149 0.038 0.141 0.221 0.146

t1 0.157 0.333 0.455 0.772 0.250 0.691 0.355 2 0.178 0.333 0.455 0.788 0.250 0.687 0.388 3

t2 0.594 0.333 0.091 0.055 0.500 0.091 0.393 3 0.592 0.333 0.091 0.082 0.500 0.108 0.371 2

t3 0.249 0.333 0.455 0.173 0.250 0.218 0.251 1 0.230 0.333 0.455 0.130 0.250 0.205 0.241 1


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Table 14

Comparison of AHPP and MPS [21] (Example 2).

C1 C2 C3 C4 C5 C6 Global priorities

AHPP MPS AHPP MPS AHPP MPS AHPP MPS AHPP MPS AHPP MPS AHPP MPS

(AN,AN) 0.305 0.149 0.038 0.141 0.221 0.146

p* DLS WLSM Any Any DLS FPM Any DLS FPM

t1 0.178 0.174 0.333 0.455 0.788 0.796 0.250 0.687 0.703 0.388(3) 0.390(3)

t2 0.592 0.606 0.333 0.091 0.082 0.082 0.500 0.108 0.109 0.371(2) 0.376(2)t3 0.230 0.221 0.333 0.455 0.130 0.122 0.250 0.205 0.188 0.241(1) 0.234(1)

Italic form indicates the notation representing functions.

2. AHPP uses objective relative weights for the measurement cri-

teria rather than subjective weights for them, which are used by

MPS.

3. In AHPP, the prioritization operators selected are calculated

by Mean Individual Interest based Aggregation Operator. This

method cares the individual interest of each prioritization oper-

ator. However, MPSjust applies a universalvaluewithsubjective

judgment. If more criteria are considered in MPS, it is difficult to

determine the weights as coefficients of the linear equation.

4. Measurement criteria of MPS are very limited and less flexible.

Minimum violations (MV) criterion [26] used by MPS [21] is ill-defined, and the correct form is discussed in Section 4.6. MPS

used the ill-defined MV and likely produced incorrect results.

Measurement criteria of AHPPS are determined by analysts. This

research introduces seven criteria for illustration.

5.2. Example 2

Example 2 is to select the high school [3, pp. 2628]. The def-

initions of the five criteria, four alternatives and seven pairwise

matrices are illustrated in Appendix B. The reciprocal matrices

include three consistent matrices, i.e. A3, A4, A6, and four non-

consistent matrices, e.g. A1, A2, A5, A7. For the consistent matrices,

any prioritization operator can be used as the value of the priority

vector is the same. For the inconsistent matrices, AHPP is applied.The local priorities and global priorities with nine prioritization

operators are summarized in Table 13. It can be observed that dif-

ferent prioritization operators produce different priorities, which

likelylead to different preference orders. This research defines that

higher score means higher preference. To address this problem,

similar to Example 1, the AHPP is applied to select the best prior-

itization operator which is tailor-made for each pairwise matrix.

For the non-consistent matrices, in Table 13, the values bolded and

underlined are selected as the priority vectors in columns WT,i orin row WC.

The results of using AHPP are illustrated in Table 14. The AHPP

solution is the combination of prioritization operators: AMNC/AN

(p4), DLS (p6), any, any, DLS (p6), any, DLS (p6), corresponding to

pairwise matrices A1

, A2

,. . .

, A7

. The comparison of AHPP and MPS

[21] is also shown in Table 14. Although the global priorities of

the two methods are not the same, interestingly, both AHPP and

MPS [21] produce the same preference orders, which is however

different from ones obtained by EVM in [3].

6. Conclusions

A core issue in AHP is how to derive the priority vector from a

reciprocalmatrix filledby verbal judgment represented by numeri-

calvalues from decisionmakers. There arevariousstudiesto handle

this problem. However, different prioritization operators produce

different values for a priority vector. This is also shown in the two

applications. To address this issue, this paper proposes the AHPP

model which is to select the most appropriate prioritization oper-

ator among a list of the operator candidates according to a list

of measurement criteria. In this study, nine prioritization opera-

tors are selected and seven measurement criteria are proposed for

building the AHPP model. The Compound AHP is the AHP problem

applying AHPP. In general, CAHP applies a combination of prioriti-

zation operators to prioritize all reciprocal matrices in the problem

domains,rather thanto use one singleuniversal prioritizationoper-

ator to all cases.

For the classical validation approach, many studies propose

a large number of reciprocal matrices with attemptions to con-

clude that their proposed prioritization methods (or prioritizationoperator) are superior to others in general. However, as the most

appropriate prioritization operator is in facton a case-by-case basis.

The fact that a prioritization operator is superior to others in gen-

eral does notfollow that every case shouldapplya uniqueoperator.

In other words,it is notnecessary to attempt to conclude which pri-

oritization operator is the best in general (in general means being

similar to statistical conclusion) unless there is a best prioritization

operator with all inconsistent reciprocal matrices.

Rather than the tests of many reciprocal matrices, two appli-

cations in this study are sufficient to illustrate the usability and

validity of AHPP. The illustrated applications show theAHPP model

does not favorite any prioritization operator. For each reciprocal

matrix, only the fittest prioritization operator selected by AHPP is

regarded as the best priority vector for the reciprocal matrix. (Inother words, the best prioritization operator is tailor-made for an

individual reciprocal matrix.) The best vectors of all matrices are

propagated in the synthesis process.

If allpairwisereciprocal matricesof the AHPare perfectlyconsis-

tent, the mixed prioritization operators approach such as AHPP or

MPSis notrecommendedsince prioritization of a consistent matrix

has a single solution only regardless of the prioritization operator.

If a significant number of matrices of the AHP is inconsistent such

that 0 < CR 0.1, the AHPP approach is highly recommended since

both examples show that a pure aggregation operator methodcan-

not guarantee the optimal result. The contribution of the AHPP is

to provide guidelines for decision makers to compare the strengths

and weaknesses of prioritization operators and choose the most

appropriate operator, especially under the inconsistent matrices of

the AHP.

Acknowledgments

TheauthorwishestothanktheResearchOfficeofTheHongKong

Polytechnic University for support in this research project. Thanks

are also extended to the editors and anonymous referees for their

constructive feedbacks to improve this work.

Appendix A. Example 1

Thedataof this example is taken from [32]. CIand CRare added

in this research.


14/15


Criteria Alternatives

c1: national income t1: electric power generation

c2: foreign exchange t2: irrigation

c3: balance of payment t3: flood protection

c4: import substitution t4: water supply

c5: regional gains t5: tourism and recreation

t6: river traffic

Criteria (A1) National income (A2)

C1 C2 C3 C4 C5 t1 t2 t3 t4 t5 t6

C1 1 2 5 3 2 t1 1 5 3 6 7 5

C2 1 7 3 3 t2 1 1/7 1/2 2 2

C3 1 1/4 1/5 t3 1 7 3 4

C4 1 3 t4 1 1/2 1

C5 1 t5 1 2

t6 1

CI = 0.1043, CR = 0.093 CI = 0.1122, CR = 0.091

Foreign exchange (A3) Balance of payment (A4)

t1 t2 t3 t4 t5 t6 t1 t2 t3 t4 t5 t6

t1 1 4 6 7 2 2 t1 1 3 7 6 3 4

t2 1 2 2 1 1/3 t2 1 5 2 3 1/2

t3 1 2 1/6 1 t3 1 1/4 1/7 1/3

t4 1 1/5 1/7 t4 1 1/2 2

t5 1 1 t5 1 2

t6 1 t6 1

CI = 0.0954, CR = 0.0769 CI = 0.0499, CR = 0.040

Import substitution (A5) Regional gains (A6)

t1 t2 t3 t4 t5 t6 t1 t2 t3 t4 t5 t6

t1 1 3 9 7 4 3 t1 1 1/5 1/3 1/6 1/3 1

t2 1 3 6 2 1/3 t2 1 2 1/5 2 4

t3 1 1/2 1/4 1/5 t3 1 1 2 3

t4 1 1/6 1/6 t4 1 1 7

t5 1 1/2 t5 1 5

t6 1 t6 1

CI = 0.0213, CR = 0.017 CI = 0.1769, CR = 0.143

Appendix B. Example 2

The data of this example is taken from [3, pp. 2628].

Criteria Alternatives

C1: learning t1: school A

C2: friends t2: school B

C3: school life t3: school C

C4: vocational training

C5: college preparation

C6: music classes

Criteria (A1)

C1 C2 C3 C4 C5 C6

C1 1 4 3 1 3 4C2 1/4 1 7 3 1/5 1

C3 1/3 1/7 1 1 1 1/6

C4 1 1/3 5 1 1 1/3

C5 1/3 5 5 1 1 3

C6 1/4 1 6 3 1/3 1

CI=0.3, CR=0.24

Learning (A2) Friends (A3 ) School life (A4)

t1 t2 t3 t1 t2 t3 t1 t2 t3

t1 1 1/3 1/2 t1 1 1 1 t1 1 5 1

t2 3 1 3 t2 1 1 1 t2 1/5 1 1/5

t3 2 1/3 1 t3 1 1 1 t3 1 5 1

CI = 0.025, CR = 0.04 CI = CR = 0 CI = CR = 0

Vocationaltraining (A5 ) Collegepreparation(A6) Music classes (A7)

t1 t2 t3 t1 t2 t3 t1 t2 t3

t1 1 9 7 t1 1 1/2 1 t1 1 6 4

t2 1/9 1 1/5 t2 2 1 2 t2 1/6 1 1/3

t3 1/7 5 1 t3 1 1/2 1 t3 1/4 3 1

CI = 0.105, CR = 0.18 CI = CR = 0 CI = CR = 0.04

Appendix C. Notation summary

(D, E, , S, M) Definition of Compound AHP

D Definition process D = (D1, D2)

D1 The AHP problem definition process D1 = (O,C, T, ).

O An objective

C A set of criteria C={c1, c2, . . ., ci , . . ., cn}

T A set of alternatives T = { t1, t2, . . . , t j, . . . , t m}

A rating scale schema = { 1, 2, . . . , i, . . . , p}

D2 The AHPP definition process D2 = (S, S, P)

S A set of measurement criteria S={s1, s2, . . ., sq , . . ., sQ}

P A set of prioritization operators P = {p1 , p2, . . . , pk , . . . , pK}

Sq A measurement function to measure Punder a measurement

criterion sq such that Sq S = { S1, . . . , SQ}

E The evaluation and assessment process

E = ((C),(ci, T)ni=1)() An assessment function

A A pairwise matrix which basically has two forms depending on

the candidates, e.g. AC = (C), AT,i = (ci, T)A = [aij] An ideal consistent judgment matrix generated from W

W The priority set (or normalized weight set) which generally has

two form WC or WT,i , which is generated from AC or AT,i . In the

AHPP, the ideal Wis determined by W= (A) = P*(A)

M Measurement process

CI Consistency index

RI Random index

CR Consistency ratio

max: A principal eigenvalue

Analytic hierarchy prioritization process = (A,(D2, MP, Agg,

EP),W)

MP Measurement and prioritization process MP = (, VP)

VP A measurement priority matrix

VP = (vs1 , vs2 , . . . , vsq , . . . , vsQ )T

= (vqk)QK

() A tran sf or matio n fu nctionNI() A normalization of inversion function

Agg Aggregation process

A Maximum Individual Interest Aggregation Operator, which is

the form (VP) = (V, I) = (CN(VP), ordering(VP)) =

{u1, . . . uk , . . . , uK} =

CN() Column normalization function

ordering() An ordering function which returns a set of rank positions in

ascending order

I The matrix from I = ordering(VP) = (i

qk)

QK

A set of preference values of prioritization operators = {u1,

. . ., uk , . . ., uK}

* The highest score of {u1, . . ., uk , . . ., uK} such that = Max()

EP The exploitation process EP = (p, f(P, ), W)

f(P, ) The prioritization selection function which returns the best

prioritization method p* in Pwith respect to = (VP)

argmax() The argument of the maximum function

: A position which is returned by argmaxj { 1,2,...,m}({u1 , . . . uq, . . . , uQ})

p* The best PO determined by p = f(P, ) = p

s The synthesis process S = ((WC, WT), t, f), WT =

WT,i

t* The best alternative determined by

t = f(WC, WT) = (W) = (sdp(WC, WT))

f() A PO selection function

(W): (W) = t, where = argmaxj { 1,2,...,m}

({w1 , w2, . . . , wj , . . . , wm})

Sagg A synthesis aggregation function

sdp A Scalar Dot Product function

EV Eigenvector function

NRS Normalization of the row sum function

NRCS Normalization of reciprocals of column sum function

AMNC Arithmetic mean of normalized columns function

NGMR Normalization of geometric means of rows function

DLS/WLS Direct least squares/weighted least squares

LLS Logarithm least squares


15/15


FP Fuzzy programming

EGP Enhanced goal programming

MAV Mean absolute variance

RMSV Root mean square variance

WAV Worst absolute variance

CV Consistency variance

GCV Geometric consistency variance

MMV Mean minimum violation

MV Minimum violation

WDV Weighted distance variance

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Yuen 2010 975 Multicriteria & Prioritization