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 decision models. 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 be completed 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 criteria decision-making (MCDM), multi-attribute analysis (MAA), multi-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 when modelling 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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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 of contractor 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 contracContrac-tor 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)
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 tool’s analytical 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 contractor j;
analysis (MAA) ACrj=SVijWi Vij is the attribute i score with respect to contractor j;
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 contractor j;
theory (MAUT) ACrj=SUijWi Uij is the attribute i score with respect to contractor j;
i =1 n is the number of attributes considered in the analysis;
Wi is the weighting indices to Ui Analytic hierarchy n Cri is the composite score for contractor i;
process (AHP) Cri=SciVij ci is the relative weight for Vi with respect to contractor j; i =1 and Vij is the selection criterion i with respect to contractor j
Note: Partially adapted from Holt (1998).
Figure 2 Examples of interdependence (notes: (1) uncorre-lated elements are connected; (2) uncorreuncorre-lated levels are connected; (3) dependence of two levels is two-way (i.e. bi-directional)
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Figure 3
(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 to be interdependent.
Pair-wise comparisons
The normal procedure of a pair-wise comparison is to invite experts to compare two sub-cluster’s elements with respect to their respective cluster’s 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 a row 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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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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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 local relative 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 A should 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 evaluation methods. 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 partnering becomes 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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Table 2
Supermatrix (major components)
Goal Selection criteria Selection sub-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 0.00 0.00 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 0.00 0.00 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 0.00 0.00 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 0.00 0.00 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 0.00 0.00 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 0.00 0.00 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 0.00 0.00 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 0.00 0.00 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 0.00 0.00 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 0.00 0.00 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 0.00 0.00 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 0.00 0.00 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 0.00 0.00 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 0.00 0.00 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 0.00 0.00 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 0.00 0.00 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 0.00 0.00 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 0.00 0.00 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 0.00 0.00 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 0.00 0.00 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 0.75 0.18 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 0.06 0.59 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 0.19 0.23 0.08 0.09 0.16 0.19 0.40 0.46 0.07 0.22
Table 3
The limit matrix (local priorities for major components)
Goal Selection criteria Selection sub-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 0.25 0.25 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 0.19 0.19 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 0.14 0.14 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 0.10 0.10 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 0.11 0.11 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 0.07 0.07 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 0.07 0.07 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 0.07 0.07 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 0.90 0.90 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 0.10 0.10 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 0.71 0.71 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 0.11 0.11 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 0.11 0.11 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 0.07 0.07 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 0.48 0.48 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 0.41 0.41 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 0.11 0.11 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 0.50 0.50 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 0.50 0.50 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 0.47 0.47 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 0.27 0.27 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 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26
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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
