contractor selection ahp

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Construction Management and Economics (December 2004) 22, 10211032

Construction Management and Economics

ISSN 0144-6193 print/ISSN 1466-433X online 2004 Taylor & Francis Ltd

http://www.tandf.co.uk/journals

DOI: 10.1080/0144619042000202852

*Author for correspondence. E-mail: [email protected]

Contractor selection using the analytic network

process

EDDIE W. L. CHENG and HENG LI*

Department of Building and Real Estate, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong

Received 30 January 2003; accepted 9 January 2004

Contractor selection is one of the main activities of clients. Without a proper and accurate method for selecting

the most appropriate contractor, the performance of the project will be affected. The multi-criteria decision-

making (MCDM) is suggested to be a viable method for contractor selection. The analytic hierarchy process

(AHP) has been used as a tool for MCDM. However, AHP can only be employed in hierarchical decisionmodels. For complicated decision problems, the analytic network process (ANP) is highly recommended since

ANP allows interdependent influences specified in the model. An example is demonstrated to illustrate how this

method is conducted, including the formation of supermatrix and the limit matrix.

Keywords: Analytic network process, analytic hierarchy process, multi-criteria decision making, contractor

selection

Introduction

Contractor selection is one of the main decisions made

by the clients. In order to ensure that the project can becompleted successfully, the client must select the most

appropriate contractor. This involves a procurement

system that comprises five common process elements:

project packaging, invitation, pre-qualification, short-

listing and bid evaluation (Hatush, 1996; Hatush and

Skitmore, 1997). Moreover, there are methods attemp-

ting to estimate the values of contractors by using various

selection criteria (e.g. Samuelson and Levitt, 1982;

Jaselskis and Russell, 1990). These methods include

multi-criteria decision-making (MCDM), multi-

attribute analysis (MAA), multi-attribute utility theory

(MAUT), multiple regression (MR), cluster analysis

(CA), bespoke approaches (BA), fuzzy set theory (FST)and multivariate discriminant analysis (MDA) (Hatush

and Skitmore, 1997; Holt, 1998; Mahdi et al., 2002).

Selection criteria on the other hand can be classified as

pre-qualification and project-specific (Alarcon and

Mourgues, 2002).

Among those well-known methods, MCDM is

relatively new to be employed to select contractors.

MCDM aims at using a set of criteria for a decision

problem. Since these criteria may vary in the degree of

importance, the analytic hierarchy process (AHP) tech-

nique is employed to prioritize the selection criteria (i.e.assign weights to the criteria). In the existing literature

of contractor selection, studies have utilized AHP to set

up a hierarchical skeleton within which multi-attribute

decision problems can be structured (e.g. Fong and

Choi, 2000; Mahdi et al., 2002). Conceptually, AHP

is only applicable to a hierarchy that assumes a uni-

directional relation between decision levels. The top

level of the hierarchy (apex) is the overall goal for the

decision model, which decomposes to a more specific

level of elements until a level of manageable decision

criteria is met (Meade and Sarkis, 1999). Yet, the strict

hierarchical structure may need to be relaxed whenmodelling a more complicated decision problem that

involves interdependencies between elements of the

same cluster or different clusters. This requires the

generic analytic method the analytic network process

(ANP) that can evaluate multidirectional relationship

among decision elements (Saaty, 1988; Meade and

Sarkis, 1998).

In most studies of contractor selection, selection

criteria are assumed to be independent of each other.


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1022 Cheng and Li

Apparently, these criteria are likely to affect each other.

For example, Fong and Choi (2000) used a sample

of 13 respondents to identify and prioritize eight un-

correlated criteria (tender price, financial capability,

past performance, past experience, resources, current

workload, past relationship and safety performance) for

contractor selection. In fact, the eight criteria are

interrelated to a certain extent. For example, good past

experience may lead to good safety performance if the

past experience includes good safety records. Good past

performance and experience are good evidence of suc-

cessful projects, which in turn results in strong financial

capability. Resources and financial capability may be

positively correlated. Tender price may be negatively

related to other criteria. Therefore, ANP is more

favourable to be employed in this interdependent

relationship framework. Since ANP is new to the

construction field, this study will demonstrate how to

apply ANP to improve the prioritization of contractor

selection criteria. It is expected that by using ANP,clients are able to establish a complete decision model

without sacrificing the validity due to limitations of the

analytical tool.

Contractor selection

Existing literature on contractor selection mainly deals

with how to identify and assess the criteria to make the

most appropriate decisions (Holt, 1997). A more pro-

mising approach to classifying the contractor selection

criteria has been provided by Hatush and Skitmore

(1997), who focused on two of the five-stage process ofcontractor selection: (1) pre-qualification, and (2) bid

evaluation. Holt (1998) and Valentine (1995) referred

to this as a two-stage procedure: (1) pre-qualification,

and (2) evaluation of tenderers. Figure 1 illustrates a

typical bespoke approach that shows where these stages

are located.

Pre-qualification is the process that compares the key

contractor-organizational criteria among a group of

contractors desirous to tender. Such criteria can be past

performance, past experience, and financial stability. In

order to identify the contractor-organizational criteria,

researchers have proposed useful methods, such as

MAA (e.g. Russell and Skibniewski, 1987; Russell et al.,

1992; Holt et al., 1994).

Evaluation of contractors on the other hand considers

specific criteria that can measure the suitability of con-

tractor for the proposed project (Holt, 1998). Contrac-

tor evaluation is not equivalent to contractor selection.

Specifically, contractor evaluation is the process of

investigating or measuring project-specific attributes,

while contractor selection refers to as the process of

aggregating the results of evaluation to identify

optimum choice. In practice, these two processes are

always grouped together to represent a single procedure

to prioritize the contractors according to the project spe-

cific criteria, which can be office location with respect to

the project, experience in the geographical region, and

experience of the proposed construction methods.

There are methods apt to identify project-specific

criteria. For example, MAUT is one of the current

available techniques (Alarcon and Mourgues, 2002).

MAA, MAUT and AHP are comparable methods

that assign weights to selection criteria (Holt, 1998;

Alarcon and Mourgues 2002). Table 1 illustrates their

respective formula to show why their functions are alike

(Holt, 1998). As shown in Table 1, these contractor

evaluation methods are known to calculate an aggregate

(or composite) score for each criterion. The differences

between these methods are that: (1) MAA and AHP use

Figure 1 A simplified bespoke approach (note: this simpli-

fied bespoke approach (BA) is typically run in a large client in

Hong Kong. It may be different from other BAs (e.g. Holt,

1998)


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1023Contractor selection using analytic network process

simple scoring for rating the criteria, while MAUT

makes use of utility value; and (2) AHP employs pair-

wise comparison for determining the weights, while

MAA and MAUT use simple scoring. Yet, Mahdi et al.(2002) suggested that AHP could be the weighting

method incorporated into MAA or MAUT. When con-

sidering that the criteria are somewhat related and what

Holt (1997) mentioned about the rationalization,

resource saving and objectivity, ANP (analytic network

process) would be a more reliable method to assign

weights to correlated attributes. This paper is not

intended to compare existing contractor evaluation

methods. Those who are interested in knowing more

about other methods may refer to Holt (1998).

AHP and ANP

AHP and ANP are two separate concepts introduced by

Saaty (1980, 1996). Saaty (1980) first developed the

AHP, which helps to establish decision models through

a process that contains both qualitative and quantitative

components. Qualitatively, it helps to decompose a

decision problem from the top overall goal to a set of

manageable clusters, sub-clusters, and so on down to

the final level that usually contains scenarios or alterna-

tives. The clusters or sub-clusters can be forces,

attributes, criteria, activities, objectives, etc. Quantita-

tively, it uses pair-wise comparison to assign weights to

the elements at the cluster and sub-cluster levels and

finally calculates global weights for assessment taking

place at the final level. Each pair-wise comparison

measures the relative importance or strength of the

elements within a cluster by using a ratio scale. One of

the main functions of AHP is to calculate the consis-

tency ratio to ascertain that the matrices are appropriate

for analysis (Saaty, 1980). Nevertheless, AHP models

assume that there are unidirectional relationships

between clusters of different decision levels along the

hierarchy and uncorrelated elements within each cluster

as well as between clusters. It is not appropriate for

models that specify interdependent relationships

in AHP. ANP is then developed to enhance the toolsanalytical power.

ANP is a generic form of AHP and allows for more

complex interdependent relationships among elements

(Saaty, 1996). It is also known as the systems-with-

feedback approach (Meade and Sarkis, 1998). Inter-

dependence can occur in several ways: (1) uncorrelated

elements are connected, (2) uncorrelated levels are con-

nected and (3) dependence of two levels is two-way (i.e.

bi-directional). Figure 2 illustrates examples of these

interdependencies. By incorporating interdependencies

Table 1 Comparison of MAA, MAUT and AHP

Method Formula Description

Multi-attribute n ACrj is the aggregate score for contractorj;

analysis (MAA) ACrj= SVijWi Vijis the attribute iscore with respect to contractorj;

i = 1 n is the number of attributes considered in the analysis;

Wi is the weighting indices to Vi

Multi-attribute utility n ACrj is the aggregate score for contractorj;theory (MAUT) ACrj= SUijWi Uijis the attribute iscore with respect to contractorj;

i = 1 n is the number of attributes considered in the analysis;

Wi is the weighting indices to UiAnalytic hierarchy n Cri is the composite score for contractor i;

process (AHP) Cri= SciVij ciis the relative weight for Viwith respect to contractorj;

i = 1 and Vijis the selection criterion iwith respect to contractorj

Note: Partially adapted from Holt (1998).

Figure 2 Examples of interdependence (notes: (1) uncorre-

lated elements are connected; (2) uncorrelated levels are

connected; (3) dependence of two levels is two-way (i.e.

bi-directional)


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1024 Cheng and Li

Figure3

Hierarchicalstructureofcontractorselection(source:FongandChoi,2000)


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(i.e. addition of the feedback loops in the model), a

supermatrix will be developed. The supermatrix

adjusts the relative importance weights in individual

matrices to form a new overall matrix with the eigen-

vectors of the adjusted relative importance weights

(Meade and Sarkis, 1998). According to Sarkis (1999),

ANP comprises four main steps:

(1) Conducting pair-wise comparisons on the

elements at the cluster and sub-cluster levels;

(2) Placing the resulting relative importance

weights (eigenvectors) in submatrices within the

supermatrix;

(3) Adjusting the values in the supermatrix so that

the supermatrix can achieve column stochastic;

and

(4) Raising the supermatrix to limiting powers until

the weights have converged and remain stable.

Methodology

The current study revises the hierarchical model of

Fong and Choi (2000) by adding interdependent influ-

ences at the selection criteria level. Figure 3 illustrates

the original model being composed of four levels. At the

top level is the decision problem itself, while the bottom

level comprises the three decision alternatives (i.e. con-

tractor candidates). The criteria and sub-criteria repre-

sent the middle two levels. Figure 4 illustrates a general

view of the new decision network model. In this model,

the main difference from the original model is that there

Figure 4 The ANP network component

is a feedback loop in the selection criteria level. It is

assumed that the eight selection criteria are interdepen-

dent. Figure 4 also illustrates a clearer view of the inter-

relationship structure by the callout box. Moreover,

only four of the eight criteria have sub-criteria (see

Figure 3). It is worth noting that sub-criteria decom-

posed from their respective criterion are not assumed tobe interdependent.

Pair-wise comparisons

The normal procedure of a pair-wise comparison is to

invite experts to compare two sub-clusters elements

with respect to their respective clusters element. Saaty

(1980) has developed a 9-point priority scale of mea-

surement, with a score of 1 representing equal impor-

tance of the two compared elements and 9 being

overwhelming dominance of one element (row element)

over another element (column element). When there is

overwhelming dominance of a column element over arow element, a score of 1/9 is given. Figures 5 and 6

provide an illustration of the use of the scale to represent

the judgments generated in this study.

After having consulted with five construction pro-

fessionals, the pair-wise comparisons in this paper are

of three bases. First, this paper adopts the original

pair-wise comparison results in Fong and Choi (2000)

who compared the criteria and sub-criteria for the three

candidates, which had fifteen sets of judgment matrices.

Second, this paper adjusted part of the original relative

weights of the criteria with respect to the top goal and


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1026 Cheng and Li

those of the sub-criteria with respect to their respective

criteria. Figure 5 presents these five sets of judgment

matrices. Third, for synthesizing the relative weights

among the criteria, other pair-wise comparisons have

to be made for this study. This is to compare two criteria

with respect to a selected criterion. This requires

establishing eight additional sets of judgment matrices

for analysis. Figure 6 presents these eight paired

comparison matrices. For example, with respect to the

criterion tender price, financial capability is relatively

Figure 5 Relative weights of criteria and sub-criteria

more important than safety performance. Noteworthy,

clients should develop their own set of scores for

the criteria and sub-criteria matching their project

requirements.

Relative weights of elements and consistency

ratio of matrices

After the pair-wise comparison matrices are developed,

a vector of priorities (i.e. a proper or eigen vector) in


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Figure 6 Criteria interdependency comparisons


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1028 Cheng and Li

Figure 6 Continued


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each matrix is calculated and is then normalized to

sum to 1.0 or 100 per cent. This is done by dividing

the elements of each column of the matrix by the sum

of that column (i.e. normalizing the column); then,

obtaining the eigen vector (eVector) by adding the

elements in each resulting row (to obtain a row sum)

and dividing this sum by the number of elements in

the row (to obtain priority or relative weight)(Cheng and Li, 2002). Moreover, for ascertaining the

consistency of the judgment matrices, Saaty (1994)

suggested three threshold levels: (1) 0.05 for 3-by-3

matrix; (2) 0.08 for 4-by-4 matrix; and (3) 0.1 for all

other matrices. Those who want to know the algo-

rithm for computing consistency ratio may refer to

Saaty (1980) and Cheng and Li (2001). Figures 5 and

6 present the relative weights (priorities) and the CR

values for the comparison matrices.

Supermatrix and the limit matrix

With interdependent influences, the system that con-

sists of cluster and sub-cluster matrices must translate

to a supermatrix. This can be achieved by entering the

local priority vectors in the supermatrix, which in turn

obtains global priorities. Table 2 shows the super-

matrix for the ANP decision model. It contains the

weights (or priorities) for the judgment matrices.

After entering the sub-matrices into the super-

matrix and completing the column stochastic, the

supermatrix is then raised to sufficient large power

until convergence occurs (Saaty, 1996; Meade and

Sarkis, 1998). Table 3 presents the final limit matrix.

Each column is the same and provides the localrelative weights of individual sets of elements.

Discussions

The limit matrix shows the local relative weights for

all the elements in the supermatrix. In order to ascer-

tain the value of ANP, results of the normalized rela-

tive weights of the candidates obtained from ANP and

AHP are compared. Table 4 presents the local relative

weights of the three candidates based on the results

from ANP, as well as the local relative weights from

AHP. In the demonstrated example, candidate Ashould be chosen because it has the largest relative

weights (= 0.473, from ANP in Table 4). However, if

the decision model does not specify the interdepen-

dent relationships (i.e. only AHP model), candidate

B should have been selected since it had the largest

relative weights (= 0.448, from AHP in Table 4).

Candidate A was even the worst among all candi-

dates. It is because the ANP decision model has taken

into account the interdependencies among selection

criteria that exert extra influences on the model.

AHP has its limitation because it can only be applied in

simple hierarchical structures, while ANP provides pow-

erful capability in solving nowadays construction manage-

ment issues that involve more complicated decision

problems. It is not to say that results from AHP would be

different from those of ANP. It depends on the subjective

and/or objective ratings given to the judgment matrices.

However, when there are interdependent influences, ANP

is a viable method for prioritization. In this study, the

ANP method is applied in contractor selection, and ANP

enhances the increasingly popular multi-criteria decision-

making (MCDM) approach to criteria prioritization.

When researchers and practitioners have realized that

lowest-price is not the promising approach to attain the

overall lowest project cost upon project completion,

multi-criteria selection becomes more popular (Wong

et al., 2001). There are various methods used for multi-

criteria contractor evaluation. Multivariate statistical ana-

lytic methods are more quantitative in nature. Wong et al.

(2001) refer to them as the objective tender evaluationmethods. Yet, these methods need a sufficiently large

amount of respondents in order to ensure the objective

quality. Although researchers tend to believe the need for

identifying a set of general selection criteria using empiri-

cal surveys (Holt et al., 1995; Fong and Choi, 2000; Wong

et al., 2001), assigning weights to these general criteria is

however the internal business decisions made by indi-

vidual clients. In other words, the clients evaluate contrac-

tors according to the requirements of individual projects.

Basically, there are two types of projects: public and pri-

vate. There may be necessary to develop individual sets of

criteria for these project types. For example, there is an

expanding view that long-term or strategic partneringbecomes more appropriate in private projects procure-

ment. Hence, the relationship between the client and

the bidding contractors may be an important selection

criterion. On the other hand, the competitive tendering

process is still the current practice for public projects

although some may argue that it is becoming less popular.

The private relationship criterion would be inappropri-

ate when selecting contractors in the process of public

tendering.

In normal practices of contractor selection, only a small

group of experts (mainly the top management of the

client) is responsible for evaluating the contractor candi-

dates. In such circumstances, using the MCDM approach

to contractor selection is more plausible (Mahdi et al.,

2002). ANP and AHP can help assign weights to selection

criteria so as to increase the accuracy of judgments made

by experts. They are not only the decision tools appropri-

ate for contractor evaluation at both the pre-qualification

and project-specific stages but can also act as the quantita-

tive tools for assigning weights to criteria in other methods

(e.g. MAA and MAUT). These decision tools set the new

horizon for contractor selection by raising key processes of


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1030 Cheng and Li

Table2

Supermatrix(major

components)

Go

al

Selectioncriteria

Selectionsub-criteria

F.C.

P.P.

P.E.

R.

T.P.

F.C.

P.P.

P.E.

R.

C.W.

C.R.S.P.

F.S.

F.R.C

.C.C.O.

D.

A.Q.S.C.T.C.

E.A.

P.R.

H.R.

Goal

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

Selection

T.P.0.38

0.00

0.56

0.33

0.31

0.21

0.20

0.17

0.18

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

criteria

F.C.0.28

0.14

0.00

0.37

0.32

0.24

0.26

0.17

0.14

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

P.P.

0.13

0.20

0.06

0.00

0.20

0.21

0.18

0.17

0.17

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

P.E.

0.08

0.11

0.06

0.06

0.00

0.18

0.18

0.09

0.16

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

R.

0.04

0.20

0.06

0.06

0.04

0.00

0.07

0.17

0.23

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

C.W.0.03

0.11

0.06

0.06

0.04

0.09

0.00

0.17

0.03

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

C.R.0.04

0.11

0.06

0.06

0.06

0.04

0.03

0.00

0.10

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

S.P.

0.02

0.11

0.12

0.06

0.03

0.04

0.07

0.04

0.00

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

Selection

F.C.F.S.

0.00

0.00

0.90

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

sub-criteria

F.R.0.00

0.00

0.10

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

P.P.C.C.0.00

0.00

0.00

0.71

0.00

0.00

0.00

0.00

0.00

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

C.O.0.00

0.00

0.00

0.11

0.00

0.00

0.00

0.00

0.00

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

D.

0.00

0.00

0.00

0.11

0.00

0.00

0.00

0.00

0.00

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

A.Q.0.00

0.00

0.00

0.07

0.00

0.00

0.00

0.00

0.00

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

P.E.S.C.0.00

0.00

0.00

0.00

0.48

0.00

0.00

0.00

0.00

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

T.C.0.00

0.00

0.00

0.00

0.41

0.00

0.00

0.00

0.00

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

E.A.0.00

0.00

0.00

0.00

0.11

0.00

0.00

0.00

0.00

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

R.

P.R.0.00

0.00

0.00

0.00

0.00

0.50

0.00

0.00

0.00

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

H.R.0.00

0.00

0.00

0.00

0.00

0.50

0.00

0.00

0.00

0.00

0.000

.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

Contractor

A

0.00

0.07

0.00

0.00

0.00

0.00

0.47

0.81

0.18

0.20

0.750

.18

0.80

0.69

0.77

0.73

0.40

0.12

0.75

0.69

candidates

B

0.00

0.65

0.00

0.00

0.00

0.00

0.08

0.07

0.59

0.40

0.060

.59

0.12

0.22

0.07

0.08

0.20

0.42

0.18

0.09

C

0.00

0.28

0.00

0.00

0.00

0.00

0.45

0.12

0.23

0.40

0.190

.23

0.08

0.09

0.16

0.19

0.40

0.46

0.07

0.22

Note:Allfiguresareroundedtotw

odecimalplaces.


11/12


Table3

Thelimitmatrix(lo

calprioritiesformajorcomponents)

G

oal

Selectioncrite

ria

Selectionsub-criteria

F.C.

P.P.

P.E.

R.

T.P.

F.C.

P.P.

P.E.

R.

C.W.

C.R.S.P.

F.S.F.R.C.C.C.O.

D.

A.Q.

S.C.

T.C.

E.A.

P.R.H.R.

Selection

T.P.0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.250.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

criteria

F.C.0.19

0.19

0.19

0.19

0.19

0.19

0.19

0.19

0.19

0.19

0.190.19

0.19

0.19

0.19

0.19

0.19

0.19

0.19

0.19

P.P.0.14

0.14

0.14

0.14

0.14

0.14

0.14

0.14

0.14

0.14

0.140.14

0.14

0.14

0.14

0.14

0.14

0.14

0.14

0.14

P.E.0.10

0.10

0.10

0.10

0.10

0.10

0.10

0.10

0.10

0.10

0.100.10

0.10

0.10

0.10

0.10

0.10

0.10

0.10

0.10

R.

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.110.11

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.11

C.W.0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.070.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

C.R.0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.070.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

S.P.0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.070.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

Selection

F.C.F.S.0.90

0.90

0.90

0.90

0.90

0.90

0.90

0.90

0.90

0.90

0.900.90

0.90

0.90

0.90

0.90

0.90

0.90

0.90

0.05

sub-criteria

F.R.0.10

0.10

0.10

0.10

0.10

0.10

0.10

0.10

0.10

0.10

0.100.10

0.10

0.10

0.10

0.10

0.10

0.10

0.10

0.01

P.P.C.C.0.71

0.71

0.71

0.71

0.71

0.71

0.71

0.71

0.71

0.71

0.710.71

0.71

0.71

0.71

0.71

0.71

0.71

0.71

0.03

C.O.0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.110.11

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.01

D.

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.110.11

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.01

A.Q.0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.070.07

0.07

0.07

0.07

0.07

0.07

0.07

0.07

0.00

P.E.S.C.0.48

0.48

0.48

0.48

0.48

0.48

0.48

0.48

0.48

0.48

0.480.48

0.48

0.48

0.48

0.48

0.48

0.48

0.48

0.02

T.C.0.41

0.41

0.41

0.41

0.41

0.41

0.41

0.41

0.41

0.41

0.410.41

0.41

0.41

0.41

0.41

0.41

0.41

0.41

0.01

E.A.0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.110.11

0.11

0.11

0.11

0.11

0.11

0.11

0.11

0.00

R.

P.R.0.50

0.50

0.50

0.50

0.50

0.50

0.50

0.50

0.50

0.50

0.500.50

0.50

0.50

0.50

0.50

0.50

0.50

0.50

0.02

H.R.0.50

0.50

0.50

0.50

0.50

0.50

0.50

0.50

0.50

0.50

0.500.50

0.50

0.50

0.50

0.50

0.50

0.50

0.50

0.02

Contractor

A

0.47

0.47

0.47

0.47

0.47

0.47

0.47

0.47

0.47

0.47

0.470.47

0.47

0.47

0.47

0.47

0.47

0.47

0.47

0.47

candidates

B

0.27

0.27

0.27

0.27

0.27

0.27

0.27

0.27

0.27

0.27

0.270.27

0.27

0.27

0.27

0.27

0.27

0.27

0.27

0.27

C

0.26

0.26

0.26

0.26

0.26

0.26

0.26

0.26

0.26

0.26

0.260.26

0.26

0.26

0.26

0.26

0.26

0.26

0.26

0.26

Notes:(1)Eachcolumnisthesam

e.(2)Allfiguresareroundedtotwodecimalplaces.


12/12

1032 Cheng and Li

decomposing a complex problem to a manageable

network or hierarchical structure, eliciting accurate

rating by employing pair-wise comparison and consis-

tency measure, and obtaining overall priority vector by

dependent and/or interdependent matrix computations.

Conclusions

ANP extends the function of AHP and is a viable

method for multi-criteria decision problems that involve

interdependent relationships. In order to highlight the

possible difference between the use of AHP and ANP,the results obtained from both supermatrix and limit

matrix are compared. The mathematics performed in

this research may not be familiar to every reader. Yet,

Saaty is now developing a software tool for conducting

ANP. Once the software tool is available for sale, a

much faster growing use of ANP can be anticipated.

Acknowledgement

This paper was financially supported by The Hong

Kong Polytechnic University under grant number

G-YW72.

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Table 4 Comparing the results from ANP and AHP

Candidate ANP AHP

A 0.473 0.262

B 0.271 0.448

C 0.255 0.290

contractor selection ahp

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