An Analytic Hierarchy Process (AHP) Approach to Port Selection Decisions Empirical Evidence from Nigerian Ports

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1 , 2006, 8, ( ) r 2006 Palgrave Macmillan Ltd All rights reserved /06 $ An Analytic Hierarchy Process (AHP) Approach to Port Selection Decisions Empirical Evidence from Nigerian Ports CHINONYE UGBOMA 1, OGOCHUKWU UGBOMA 2, INNOCENT C OGWUDE 2 1 Department of Maritime Management Technology, Federal University of Technology, P. M. B. 1526, Owerri, Imo State, Nigeria. ccugboma@yahoo.com; 2 Department of Transport Management Technology, Federal University of Technology, P.M.B. 1526, Owerri, Imo State, Nigeria, West Africa. ogougboma23@yahoo.com and icogwude@yahoo.com This study presents the findings of a survey to determine the service characteristics that shippers consider important when selecting a port and the way these characteristics are prioritised according to their importance. Seven criteria for the port selection decision and four ports were identified, and the decision problem was structured into a three-level hierarchy using the Analytic Hierarchy Process. The findings suggest that shippers place high emphasis on efficiency, frequency of ship visits and adequate infrastructure, while quick response to port users needs was insignificant to them. Results from the study are of interest to Port managers because they provide essential information on the key factors that come into the decision process of port users, thus, identifying the strengths and weaknesses of the ports. (2006) 8, doi: /palgrave.mel Keywords: Nigerian ports; shipper s decisions; analytic hierarchy process. INTRODUCTION In today s competitive environment, it is imperative that port managers develop the ability to determine the critical port selection factors their users desire. Rapid environmental changes will necessitate continued attention and emphasis in formulating strategies on the part of port management. In essence,

2 252 transportation decisions of shippers are crucial for policy formulation in ports. The essential element in matching the needs of shippers and the services of ports is in the development of a positioning strategy. Positioning involves the perceptual differentiation of the port and its services from its competitors. Thus, by tailoring their marketing strategies to stress the satisfaction of their customers, port managers may be able to influence the decisions of their customers. Shippers choose those ports, which act as transshipment ports for their cargoes or as origin/destination ports, where their cargoes are handled reliably, efficiently and economically. Shippers can be grouped into three types: those who have long-term contracts with shipping lines, those who are using freight forwarders and those that are independent shippers. The first category of shippers are committed to a particular carrier for a number of years and are therefore dependent on the shipping lines chosen port of call, while the second group of shippers delegate the responsibility for port selection to the freight forwarders. Thus, only the freight forwarders and the independent shippers are engaged in regular port selection. This study examines the predominant factors that have a bearing on shippers port selection decisions by employing the Analytical Hierarchy Process (AHP). The AHP is a multi-objective, multi-criteria theory of measurement. It addresses the issue of how to structure a complex decision problem, identify its criteria (tangible or intangible), measure the interaction among them and finally synthesise all the information to arrive at priorities, which depict preferences. The AHP is able to assist port managers in obtaining a detailed understanding of the criteria that shippers deem important in port selection decisions and the strength of their preferences. Other multi-criteria decision making techniques such as the Multi-Attribute Utility Theory (MAUT) and the Multi-Attribute Value Theory (MAVT) could provide alternative methodologies to AHP. However, a study by Lai and Hopkins (1995) shows that the MAUT is not significantly different from AHP in effectiveness. Although the usefulness of the AHP methodology has been demonstrated in studies conducted in other parts of the world (Song and Yeo, 2004; Dey, 2004) and in a variety of industries (Beynon, 2002; Chwolka and Raith, 2001; Lirn et al., 2004), it has not been much researched upon in Nigeria and to date, no research has been conducted on shippers port selection decisions using AHP in Nigeria. Through this study, we wish to: contribute to the literature by using AHP to model shippers port selection decisions in Nigeria; highlight the usefulness of the AHP methodology in shippers port selection decisions in Nigeria; and

3 253 to bring awareness to the existence of this methodology to shippers who are looking for alternative means of decision making. OBJECTIVES OF THE STUDY The objectives of this study are as follows: To determine the service characteristics that shippers consider important when selecting a port and how these characteristics are prioritised according to their importance; To determine shippers relative preferences of Lagos Port (LPC) and the other major ports in Nigeria: Tincan Island Port (TCIPC), RO-RO Port (RRP) and Port Harcourt Port (PPPC) complexes; with respect to each characteristic considered in the decision process. To determine shippers overall preferences of the ports For this study, the decision problem was structured into a three-level hierarchy. The top-most level (Level 1) is the macro decision to select a port. The middle level (Level 2) contains the seven selection attributes and the final level (Level 3) represents the four ports. LITERATURE REVIEW There have been few studies carried out on the factors determining port choice. Slack (1985) provided interesting information on port choice. He surveyed port end users and freight forwarders engaged in trans-atlantic container trade between the American mid-west and Europe to identify port selection criteria. A study by Bird (1988) on the perceptions of European freight forwarders again indicated that the frequency of service is the main reason for port choice. Tongzon (1995, 2002) confirmed that time is of essence in freight forwarding and that frequency of service is a major determinant of that time. He also showed that port efficiency is the most important factor in port choice and performance and a certain mode of decision-making. Branch (1986) gave a general list of factors accounting for shippers choice of port. Other studies attempting to identify and explain the various factors in port choice include Willingale (1984) and Murphy et al (1991, 1992). There are numerous studies on freight transport choice by shippers, but they have centred on modal choice and carrier selection, rather than addressing the more specific question of choice between competing ports, for example, Saleh and Lalonde (1972), Bardi (1973), Brooks (1984, 1985), and D Este and Meyrick (1992). However, it seems reasonable to assume that modal choice and

4 254 the question of port choice are ultimately related and that the results from modal choice studies can provide some basis for formulating certain hypotheses about port choice factors. Investigations of decision factors, for example; McGinnis (1979), Ogden and Rattray (1982), Gilmour (1976), Wilson et al (1986), and D Este and Meyrick (1992), have identified three categories of factors: route factors (which include frequency, capacity, convenience, directness, flexibility and transit time), cost factors (freight rate and other costs) and service factors (delays, reliability and urgency, avoidance of damage, loss and theft, fast response to problems, cooperation between shipper and carrier, documentation and tracing capability). They have found that shippers are generally risk averse and thus have limited options. Service factors, particularly service frequency, take precedence over price. The predominance of service factors is not unusual and has been previously reported by Cook (1967), Bayliss and Edwards (1970), Brooks (1984), Wilson et al (1986), and Meyrick and D Este (1989). These studies have shown that there are many potential determinants of port choice, which may be quantitative or qualitative in nature. Quantitative factors are those that can be potentially measured and compared in an objective manner. Qualitative factors include subjective influences such as flexibility and ease of use, the port s marketing efforts, tradition, personal contacts and the level of cooperation that may be developed between the shipper and the port. In practice, the distinction between quantitative and qualitative factors is blurred because a shipper s perception of the level of port performance may not be a fair reflection of the actual performance. In many cases, perceptions can take precedence over actual performances (D Este and Meyrick, 1992). They have also shown that there exists a perception gap between the port users and port operators. However, the above studies have focused mainly on port users based in North America, Europe and Australia and may not be applicable universally particularly to Africa. Based on the related literature survey, the following factors are postulated to have a significant impact on the choice of ports among shippers located at the industrial centers of Lagos and Port Harcourt: frequency of ship visits, efficiency, adequacy of port infrastructure, location, competitive port charges, quick response to port users needs and port s reputation for cargo damage. RESEARCH METHODOLOGY The AHP methodology is a flexible tool that can be applied to any hierarchy of performance measures (Rangone, 1996). The success of the AHP in research in a number of areas supports its use to solve transport problems, as for example

5 255 in Vreeker et al (2002), Lirn et al (2003), Chang and Yeh (2001), Poh and Ang (1999), Tzeng and Wang (1994) and Frankel (1992). For this study, a survey method earlier used by Ta and Har (2000) was modified and adopted. Chou and Liang (2001) used the AHP to create a model capable of evaluating the performance of shipping companies. Lirn et al (2004) used the AHP to study and evaluate the criteria shippers use for transshipment port selection. In the Analytic Hierarchy Process, a decision problem is decomposed into a hierarchy. In the context of ports, decomposing a shippers choice problem involves the structuring of a hierarchy in terms of the overall objective (decision to select a port), the selection criteria and the decision alternatives (ports). The first step in the research design is to identify the criteria that shippers deem important when selecting a port. A literature review on port selection decisions of shippers carried out generated an extensive list of criteria. From the list, it was found that shippers ranked several factors such as port efficiency, adequate infrastructure, frequency of ship visits, quick response to port users needs, location, port charges and port reputation for cargo damage, high in importance. To identify the relevant criteria of shippers port selection decisions in Nigeria, a focus group interview was conducted. The group comprised 10 freight forwarders each patronising each port. In Nigeria, shippers are represented by freight forwarders who make independent decisions on port choice on their (shippers) behalf. From the discussion, a list of factors that were important for shippers port selection decisions was generated. Most of the factors were similar to those discussed in past literature. Based on the literature review and the focus group discussion, a total of seven criteria were identified for this study. They are; port efficiency, adequate infrastructure, frequency of ships visits, quick response to port users needs, location, port charges and ports reputation for cargo damage. The decision hierarchy for the selection of a port, based on seven and four decision alternatives, is illustrated in Figure 1. DATA COLLECTION Primary data for this survey were collected through the use of questionnaires. The questionnaire was constructed from information gathered from the focus group interviewed and the literature. The questionnaire was divided into three sections. The first section required the respondents to make pairwise comparisons of the seven criteria. The second section asked the respondents to make pairwise comparisons of the four ports with respect to each of the

6 256 Decision to Select a Port. Port Efficiency Adequate Infrastruc ture Frequency of Ship Visits Quick Response to Port Users Needs Location Port Charges Ports Reputation for Cargo Damage Lagos Port Complex (LPC) Tincan Island Port Complex (TCIPC) Port Harcourt Port Complex (PHPC) Ro-Ro Port (RRP) Figure 1: Decision Hierarchy to Select a Port. seven criteria. The last section focused on the profile and port use behaviour of the respondents. The nine-point fundamental scale (Saaty and Vargas, 1994) was used for the pairwise comparisons (Table 1). Detailed instructions on how to use the scale to complete a pairwise comparison of the criteria with respect to the objective of selecting a port and the explanation of the major criteria used were given on the introductory page of the questionnaire. This was important in order to familiarise the respondents with pairwise comparisons of the AHP survey and minimise inconsistent replies. When answers were found to be inconsistent, the researcher followedup respondents asking them to revise their responses as suggested by Selly and Forman (2002). In this study, a total of 280 questionnaires were given out. Of this number, 190 questionnaires were returned, yielding a response rate of 67.9 per cent. Of these responses, 155 questionnaires (55.4 per cent) provided complete information used for the analysis in this survey. Such a sample was considered most satisfactory especially when compared with earlier research works on port choice selection decisions (Tongzon, 2002; Lirn et al., 2004). The analysis was carried out using the Expert Choice Software (1994). The AHP methodology The AHP is a theory of measurement for dealing with quantifiable and/or intangible criteria. It is a multi criteria decision-making approach that employs

7 257 Table 1: The fundamental scale Intensity of importance Definition Explanation 1 Equal importance Two activities contribute equally to the objective 2 Weak 3 Moderate importance Experience and judgement slightly favour one activity over another 4 Moderate plus 5 Strong importance Experience and judgement strongly favour one activity over another 6 Strong plus 7 Very strong or demonstrated importance An activity is favoured very strongly over another; its dominance demonstrated in practice 8 Very, very strong 9 Extreme importance The evidence favouring one activity over another is of the highest possible order of affirmation 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 Source: Saaty and Vargas (1994). pairwise comparisons to arrive at a scale of preferences among a set of alternatives. Using the AHP methodology in a decision problem involves four steps (Zahedi, 1986): 1. Structuring the decision hierarchy by breaking down the decision problem into a hierarchy of interrelated decision elements (criteria, decision alternatives); 2. Collecting input data, depicted by matrices of pairwise comparisons, of decision elements (Appendix I provides a detailed description of the input matrix); 3. Using the eigenvalue method to estimate the relative weights of the decision elements (Appendix II presents the mathematics of the eigenvalue method); 4. Aggregating the relative weights of decision elements to arrive at a set of ratings for decision alternatives. Using the scale mentioned earlier, the respondents judgments were first obtained. For example, if a respondent compares two attributes, port efficiency (A 1 ) and quick response to port users needs (A 2 ), and indicates that A 1 is stronger than A 2, then a value of a 12 ¼ 5 is assigned to this pairwise comparison.

8 258 By definition, comparison of an attribute with itself results in a relative importance value of 1. In addition, the value of reciprocal comparison is the reciprocal of the relative importance value. Thus, with respect to the decision of selecting a port, the pairwise comparison A matrix has two rows and two columns, and its elements are a 11 ¼1, a 12 ¼ 4, a 21 ¼1/5, a 22 ¼1. When applied to this study, the above process requires all pairwise comparisons of the seven attributes to be made. The result is a 7 7 pairwise comparison matrix for Level 2. Next, for each of the seven attributes, the same procedure is used for pairwise comparisons of the four ports. The result is seven matrices of size 4 4 for Level 3. The pairwise comparison matrix for Level 2 is then used to obtain the priorities of the seven attributes. Let A be the matrix of a respondent s relative preference, and W ¼ (w 1, w 2, y, w 7 ) be the vector of priorities of Level 2. An estimate of the vector of priorities, Ŵ is obtained from the equation ÂŴ ¼ l max Ŵ, where l max is the largest eigenvalue of Â. For each of the seven attributes, the same procedure is used to obtain Level 3 priorities. For example, consider the attribute quick response to port users needs. Using the pairwise comparison matrix of a respondent s relative preference on quick response to port users needs with respect to the four ports, we obtain the vector of priorities, say, V ¼ (v 1, v 2,y, v 4 ) for quick response to port users needs. To obtain the priorities of Level 2 and Level 3 for the sample, the geometric means of the pairwise comparisons given by the individual respondents were computed. These geometric means were then used as inputs in the pairwise comparison matrix of Level 2 and in the seven pairwise comparison matrices of Level 3. Another important consideration in the AHP is the notion of consistency. Consistency is the degree to which the perceived relationship between elements in the pairwise comparison is maintained. It is important because comparisons lacking consistency may indicate that the respondents did not understand the differences in the choices presented or were unable to assess accurately the relative importance of the elements compared. On the other hand, a lack of information about the criteria being compared or a lack of concentration during the judgement process can also cause inconsistency. The AHP is able to provide a measure of consistency and to reduce the inconsistency inherent in the judgement process. This measure is known as the consistence ratio (CR). A CR of 0.10 or less is evidence of informed judgments. According to Saaty (1988), the value of the consistency ratio should be 10 per cent or less. If it is more than 10 per cent, the judgments may be somewhat random and should perhaps be revised. The AHP is able to show one by one, in sequential order, which judgments are the most inconsistent. The AHP also

9 259 suggests the value that best improves inconsistency. The decision maker then could refine the information on the criteria. DATA ANALYSIS In order to understand the prioritisation of the service characteristics that shippers deem important in port selection decisions, the respondents judgments on the seven criteria were examined. Table 2 reports the priority vectors for Level 2 and Level 3 of the decision hierarchy. The priorities of Level 2 provide the relative importance of the seven criteria when choosing a port. The priorities of Level 3 give the relative preference of the four ports with respect to each of the seven criteria in Level 2. DISCUSSION OF FINDINGS Interpretation of level 2 priorities selection criteria Table 2 shows the priority vectors for the decision hierarchy. Based on the priorities of Level 2, the three most important criteria affecting shippers port selection decisions in Nigeria are; efficiency, frequency of ship visits and adequate infrastructure. They are followed by location, port charges, port s reputation for cargo damage and quick response to port users needs. Efficiency is the most important factor in shippers selection decisions for a port while frequency of ship visits and adequate infrastructure ranked second and third respectively in importance. Also, location, port charges and ports reputation for cargo damage ranked fourth, fifth and sixth respectively while quick response to port users needs was the least in importance. In other words, respondents considered efficiency (weight ¼ 0.197) more than twice as important as quick response to port users needs (0.081). The consistency ratio for this level was 0.02 within the acceptable range of 0.10 (Saaty, 1988). These findings are consistent with previous studies and observations in many freight transport markets. For example, in evaluating shippers port choice decisions among Malaysian and Thailand shippers, Tongzon (2002) found that shippers ranked (for Malaysian shippers) efficiency, shipping frequency and adequate infrastructure as the most important and port charges, quick response to port users needs and reputation for cargo damage as the least important while (for Thailand shippers) efficiency, location, shipping frequency were ranked as the most important and quick response to port users needs and reputation to cargo damage as the least important. Also, in evaluating the importance of port charges in the US, Malchow and Kanafani (2001) have also

10 260 Table 2: Priority vectors for the decision hierarchy Attributes Level 2 Priorities Ports Level 3 Findings Priorities C.R 1. Efficiency PHPC LPC TCIPC RRP Frequency of Ship visits LPC TCIPC RRP PHPC Adequate Infrastructure TCIPC LPC RRP PHPC Location LPC TCIPC RRP PHPC Port Charges PHPC LPC TCIPC RRP Ports Reputation for Cargo Damage PHPC TCIPC RRP LPC Quick Response to Port Users Needs PHPC TCIPC LPC RRP Final Synthesis Global Priorities LPC PHPC TCIPC RRP Consistency Ratio 0.02 found that port charges relative to other factors are not significant in port selection. The less significance of port charges and greater emphasis on qualitative service factors seem to be consistent with the global trend attributed to changes in commodity patterns involving greater proportions of highvalue added products and the adoption of logistical approaches (such as

11 261 inter-modality and just-in-time inventory) to freight management in response to greater competition between producers (D Este and Meyrick, 1992). Interpretation of level 3 priorities decision alternatives To address the second objective of determining shippers preferences of the ports with respect to each criterion considered, the Level 3 findings were examined. From the priorities of the ports in Table 2, Port Harcourt Port Complex (PHPC) is the most preferred port for four out of the seven selection criteria. These are efficiency (weight ¼ 0.257), port charges (0.313), port reputation for cargo damage (0.422) and quick response to port users needs (0.275). All these could be attributed to low number of vessels visiting the port and the government s efforts in decongesting Lagos Port Complex (LPC) and Tincan Island Port Complex (TCIPC) thereby granting certain concessions to PHPC port users in the form of low port charges, provision of modern cargo handling equipment and dredging of the channel and berths to accommodate larger vessels. On the other hand, LPC is most preferred on two criteria: frequency of ships visits (0.434) and location (0.248) while TCIPC is mostly preferred on the remaining criterion of adequate infrastructure (0.266). For the factor frequency of ship visits, LPC (0.434) is more than twice preferred to TCIPC (0.159) and RORO Port (0.154) and more than thrice preferred to PHPC (0.129). This shows that although the port (PHPC) is located in the eastern part of the country, shippers (mostly businessmen from the major markets in Nigeria (Onitsha, Nnewi and Aba markets), situated in the eastern part of the country and industrialists in this region) mostly patronise LPC and TCIPC because of this factor. The same applies to the location factor where LPC (0.248) and TCIPC (0.224) are mostly preferred to PHPC (0.156). This implies that majority of cargo destined for the eastern part of the country is handled in the ports located in Lagos; LPC and TCIPC. Another significance of this is that location factor is not particularly very important to these shippers thus supporting result of an earlier studies by Tongzon (2002) among Malaysian shippers, Willingale (1984) and Murphy et al (1991). Overall, the priorities in Level 3 show that shippers relative preferences for the ports vary over the seven attributes. The results provide the specific attribute on which each port is the most or least preferred. This information is of significant importance to port managers who need to improve and modify the service attributes so as to increase shippers preferences of the ports. Final synthesis global priorities To address the third objective of determining the shippers overall preferences of the ports, the priorities were synthesised. The global priorities of the ports are

12 262 presented in the last section of Table 2. The global priority of LPC for instance, is the sum of the products of Level 2 priorities and the LPC s priorities in Level 3. In other words, the global priority of LPC ¼ (0.197)(0.216)+(0.124) (0.434)+(0.120)(0.264)+(0.117)(0.248)+(0.115)(0.247)+(0.110)(0.131)+(0.081) (0.191) ¼ This means that LPC (0.215) is the most preferred port among Nigerian shippers, whereas PHPC is the second most preferred (0.214). the next port preferred is TCIPC (0.170) while RORO Port trailed behind (0.138). CONCLUSION AND MANAGERIAL IMPLICATIONS The study has used AHP to evaluate shippers selection decisions criteria among four ports in Nigeria namely; Lagos Port Complex (LPC), Port Harcourt Port Complex (PHPC), Tincan Island Port Complex (TCIPC) and Ro-Ro Port (RRP), to determine the criteria that are more important to these shippers. The study revealed that for the efficiency, port charges, quick response to port users needs and reputation for cargo damage criteria, PHPC is the most preferred while RRP is the least preferred in these criteria. Location and frequency of ship visits criteria have LPC mostly preferred while adequate infrastructure criterion has TCIPC mostly preferred. These findings are important because in an increasingly competitive port environment, it is essential to know the key factors that come into the decision process of major port users and their mode of port selection. In particular, these findings provide an empirical support that port efficiency is the most important factor in port selection and throughput determination from the perspective of the shippers. It is, therefore, essential that port operators and policy makers give top priority to improving their overall level of efficiency relative to other factors in order to attract more shippers to use their ports. The findings of this study is also of interest to port managers since the strengths and weaknesses of the ports could be identified. With this understanding, the ports could better position themselves and formulate strategies to gain competitive advantages. Thus, if any of the ports aim to surpass their competitors, they must attempt to be more competent in the important criteria. For example, if they were already preferred in that criterion, they should try to maintain status quo. They could also try to improve on that criterion since they already have a competitive advantage over their competitors. However, if they are not preferred in that criterion, they could adopt other strategies that may enhance their standing.

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14 264 Ogden, KW and Rattray, AL. 1982: Analysis of freight mode choice. Proceedings of the 7th Australian Transport Research Forum Hobart, Australia, pp Poh, KL and Ang, BW. 1999: Transportation of fuels and policy for Singapore: an AHP planning approach. Computers & Industrial Engineering 37: Rangone, A. 1996: An analytical hierarchy process framework for comparing the overall performance of manufacturing departments. International Journal of Operations & Production Management 16: Saaty, TL. 1980: The Analytic Hierarchy Process,. McGraw-Hill Publications: New York, NY. Saaty, TL. 1988: Decision Making for Leaders,. RWS Publications: Pittsburgh, PA. Saaty, TL and Vargas, LG. 1994: Decision Making in Economic, Political, Social and Technological Environments with the Analytic Hierarchy Process. RWS Publications: Pittsburgh, PA. Saleh, F and Lalonde, BJ. 1972: Industrial buying behaviour and the motor carrier Selection decision. Journal of Purchasing 8: Selly, MA and Forman, EH. 2002: Decision by objectives 1st edn, World Scientific Pub Co: Petersburg. Slack, B. 1985: Containerisation and inter-port competition. Maritime Policy and Management 12: Song, DW and Yeo, KT. 2004: A competitive analysis of Chinese container ports using the Analytic Hierarchy Process. Maritime Economics and Logistics 6: Ta, HP and Har, KY. 2000: A study of bank selection decision in Singapore using the Analytic Hierarchy Process. International Journal of Bank Marketing 18: Tongzon, J. 1995: Determinants of port performance and efficiency. Transportation Research 29A: Tongzon, J. 2002: Port choice determinants in a competitive environment. The International Association of Maritime Economists Annual Conference 2002 Conference Proceedings Panama. Tzeng, GH and Wang, RT. 1994: Application of AHP and Fuzzy MADM to the evaluation of a bus system s performance in Taipei City. Third International Symposium on the Analytical Hierarchy Process George Washington University, Washington, DC, July Vreeker, R, Nijkamp, P and Welle, CT. 2002: A multicriteria decision support methodology for evaluating airport expansion plans. Transportation Research: Part D 7: Willingale, MC. 1984: Ship-operator port-routeing behaviour and the development process. In: Hoyle, B.S. and Hilling, D. (eds) Seaport Systems and Spatial Change. John Wiley & Sons: New York. pp Wilson, FR, Bisson, BJ and Kobia, BK. 1986: Factors that determine mode choice in the transportation of general freight. Transportation Research Record 1061: Zahedi, F. 1986: The Analytic Process: survey of the method and its applications. Interfaces 16: APPENDIX I INPUT MATRIX When employing the AHP methodology, the input data for the decision problem consists of matrices of pairwise comparisons of elements of one level that contribute to achieving the objectives of the immediate preceding level. Thus, the Level 2 attributes are compared pairwise with one another, in relation to their importance to the Level 1 objective. If there are n attributes in Level 2 of the hierarchy, a total of n(n 1)/2 comparisons are required. This results in a n n matrix. Similarly, the Level 3 attributes are pairwise compared with one

15 265 another, in relation to their preference with regard to each of the nm m matrices. For this study, the input matrix of the respondents judgments would look like the following: Attribute 1 Attribute Attribute 7 Attribute 1 1 a k Attribute 2 1=a Attribute 7 1=k The matrix shows that attribute 1 is a times more important than attribute 2 and is k times more important than attribute 7. the matrix has the property that its principal diagonal elements are all unity because when compared with itself, each elements has equal importance. The lower triangle elements of the matrix are the reciprocal of upper triangle elements. Thus, pairwise comparisons are collected for only half of the matrix elements. APPENDIX II THE EIGENVALUE METHOD The eigenvalue method of the AHP takes in as inputs the pairwise comparisons of the respondents and judgements and produces the relative weights of the elements at each level of the decision hierarchy. Following Zahedi (1986), if the evaluator could know the actual relative weights of n elements (at one level of the hierarchy with respect to the level above), the matrix of the pairwise comparisons would be 2 w 1 =w 1 w 1 =w A¼ w n =w 1 w n =w 1 In this case, the relative weights could be obtained from each of the n rows of matrix A. In other words, matrix A has rank 1 and the following holds: A W ¼ n W Where W is the vector of actual relative weights and n is the number of elements. In matrix algebra, n and W are called the eigenvalue and the right eigenvector of A respectively.

16 266 The AHP posits that the evaluator does not know W and therefore, is not able to produce the pairwise relative weights of matrix A accurately. Thus, the observed matrix A contains inconsistencies. The estimation of W, denoted by Ŵ could be obtained from ^A ^W ¼ l max ^W where  is the observed matrix of pairwise comparisons, l max is the largest eigenvalue of Â, and is its right eigenvector. Saaty (1980) has shown that l max can be considered an estimation of n and that l max is always greater than or equal to n. Furthermore, when the observed values of  are consistent, the value of computed l max is very close to n. this property allows the construction of the consistency index (C.I) as C:I ¼ðl max nþ=ðn 1Þ and the construction of the consistency ratio (C.R) as C:R ¼ðCI=ACIÞ100 where ACI is the average index of randomly generated weights. The computational algorithm is available in Expert Choice Software. In summary, the eigenvalue method in the AHP is one of the widely used methods for estimating the relative weights of W from the matrix of pairwise comparisons.

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