Books on the An I ic Hier rch Proce. Qroup decision making and conflict resolution. Saaty is Distinguished University Professor at the University of Pittsburgh, before that was a professor at the Wharton School, University of Pennsylvania for 10 years and before that was for seven years in the Arms Control and Disarmament Agency at the U. State Department.

## Thomas Saaty - Google Scholar Citations

He is also the author of numerous other books on mathematics and mathematical modeling. A nce 0 ic. Therefore, this paper seeks to identify the application of AHP in property sectors and how it can be an important instrument in property sectors for a decision making tool.

It has been used as a tool to identify the importance of criteria in decision making or problem solving to achieve a goal. AHP bringing the qualitative and quantitative approach in research and combines it into the context as a sole empirical question.

AHP applies the qualitative approach to restructure problems into hierarchy which is more systematic. On the other hand, based on a quantitative approach, it uses more of the comparison method of pair-wise to obtain responses and reliability that are more consistent through questionnaire forms [1]. Figure 1 below reveals the hierarchy towards the application of the AHP method. This is a measurement theory to discuss each criterion that can be quantified and make known the differences. This method has been applied in various situations involving the result theory and problem solving [2].

Practically, AHP functions to test the weightage among the related elements. The weightage of each element or criteria has two main functions, which are firstly, to give priority ranking to every element so that the importance of each element can be determined. With this method, performance of each element or criteria can be evaluated.

Secondly, apart from determining the weightage, this process can make a more precise decision regarding each criterion. This process is more to strategic planning to solve problems. This paper has discovered and summarised the application of AHP that includes four main steps [5], [6], [7] which are; 1. Diagnose the problem, and determine the objective. Comparing among the criteria via pairwise comparison method. Identify the relative weightage of each level, criterion and sub-criteria to get the importance of each element.

## Thomas L. Saaty

In addition, the most crucial part in the AHP method is to determine the relative weightage for each criterion. Every criterion has a priority and each comparison among the criteria has its own importance or priority among one another. The indicator that states the relative importance of each criterion is in the scale as shown in Table 1. TABLE i. The comparison scale of pairwise Saaty [5] The Fundamental Scale For Pairwise Comparisons Intensity of Importance Definition Explanation 1 Equal importance Two elements contribute equally to the objective 3 Moderate importance Experience and judgment slightly favor one element over another 5 Strong importance Experience and judgment strongly favor one element over another 7 Very strong importance One element is favored very strongly over another, its dominance is demonstrated in practice 9 Extreme importance The evidence favoring one element over another is of the highest possible order of affirmation Intensities of 2, 4, 6 and 8 can be used to express intermediate values.

Intensities 1. Once all the elements are compared, with the importance's scale among the criteria, the pair for each comparison will be considered via the metrics method [7]. Then, the total relative score for each criterion will be gathered and combined along with the regarded weightage value to produce one absolute sum. This process must be presented by applying the metrics method in pairwise comparison via normalized eigenvector to evaluate the metrics comparison [8].

### Fundamentals of Decision Making and Priority Theory

It has already been widely applied as a tool to solve a problem. The application of AHP has also been shown to give an impact in property sectors. However, due to the growth of AHP applications, there have been modified by researchers based on its purpose and suitability in the property sectors. In the property sectors, AHP has been used as a tool for decision making including investment, building quality, planning and deciding the best alternative such as contractors, property managers, and tenants.

In the previous phase, AHP application is a sole and profound tool. Realising the need in problem solving and decision making in property sectors, researchers then changed the AHP versions via merging with other instruments [7]. According to [9], AHP can be used in the project management field to decide on the best contractor. He has developed one hierarchy structure that covers the difficult criteria in choosing experienced contractors for the project.

Some of the most widely used are: the scatterplots matrix, the correlation matrix, variance inflation factors, condition numbers [24,25] and the Gleason-Staelin indicator. The scatterplots matrix is a visual tool that plots each attribute against the other in matrix form; this allows the observation of patterns or linear trends that helps to determine the dependence between attributes. Values of VIF greater than 10 indicate strong dependence in the data [26]. If any of the condition numbers calculated for the set of attributes is greater than , it indicates that there is no independence between the attributes.

The Gleason-Staelin redundancy measure Phi is given by [28]:. If this indicator is higher than 0. To deal with the problem of dependence among the attributes, Vega et al. Using the Mahalanobis distance is not necessary to normalize the initial data. The value of the Mahalanobis distance is the same, apart from the used normalization mode that is used. The value is also the same without normalization.

The Mahalanobis distance [5,6] determines the similarity between two multi-dimensional random variables as well as considering the existent correlation between them is required to obtain. The Mahalanobis distance between two random variables with the same and probability distribution and with S variance-covariance matrix is formally defined as:.

This value coincides with the Euclidean distance if the covariance matrix is the identity matrix, i. In order to deal with this contradiction, in the next section, we advance a new synthesis procedure Step 5 of the TOPSIS algorithm , based on the Analytic Hierarchy Process AHP , that allows the consideration of the relative importance of the distances from the ideal and anti-ideal alternatives and provides results that are closer for the distances employed than those obtained with the traditional TOPSIS approach, regardless of the normalization model used.

When the distance employed to measure proximity is the Euclidean Step4 , data are previously normalised using the Euclidean mode If the distance employed is the Mahalanobis distance 14 , then it is not necessary to normalize the original data the results obtained when normalizing with any norm and not normalizing are the same. As already mentioned, the results obtained for Ri, and therefore for the associated rankings, using both distances Euclidean and Mahalanobis can be clearly different if there is any dependence, even if it is very small.

The new, AHP-based, procedure deals with this drawback, as well as the problem of selecting the method for normalizing the data; it further allows the assignation of a different relative importance for both distances.

## The Evolution of Analytical Hierarchy Process (AHP) as a Decision Making Tool in Property Sectors

The rest of this section presents the new, AHP-based, procedure. In order to derive the priorities of the proximities to the ideal and to the anti-ideal solutions for a particular distance Euclidean, Mahalanobis etc. By means of any of the existing prioritization procedures the Row Geometric Mean in this case , we derive for each alternative the priorities of its distances to the ideal and anti-ideal solutions and , respectively.

This procedure has been applied to the example Profiles of Graduate Fellowship Applicants used in Vega et al. The data corresponding to the distances to the ideal and the anti-ideal as well as the relative proximity 12 and the resulting rankings for the two distances Euclidean and Mahalanobis can be seen in Table 3.

This is also true for very small dependencies [1]. In order to deal with this conflict and after ranking the distances to the ideal and the anti-ideal from the minimum to the maximum, we ask the decision maker to evaluate the relative importance of the distances Euclidean and Mahalanobis to the ideal and the anti-ideal solutions.

The pairwise comparisons matrices provided by the decision maker are given in Tables 4a and 4b for the Euclidean distance and in Tables 5a and 5b for the Mahalanobis distance. Using the Row Geometric Mean method as the prioritization procedure, the local priorities for the two distances Euclidean and Mahalanobis are obtained in five different scenarios, which depend on the weights assigned to the priorities of the distances to the ideal and the anti-ideal see Tables 6 and 7.

With respect to the judgment of Saaty's fundamental scale assigned by the decision maker to each comparison of distances, it should be mentioned that these judgments capture the holistic vision of the reality and are given in accordance with the decision maker's experience and culture. It is not easy to assign these judgments in a systematic way because of the diversity of the distances for both metrics. In order to solve this problem and capture the dependence among the attributes, [1] proposed the use of the Mahalanobis distance no need to normalize the data instead of the Euclidean.

To deal with this problem, this paper proposes a new synthesis procedure for the distances of the alternatives from the ideal and the anti-ideal ones. The new proposal aims to be a stepping-stone in the process of obtaining a synthesis procedure for the distances to the ideal and the anti-ideal that allows us to reduce the gap between the results obtained with dependent and independent attributes. The results, that appear to be justified by the relative importance captured by the AHP, should be tested with some other examples.

This work was partially financed by:. Procedia Computer Science, 31, pp. DOI: Expert Systems with Applications, 39, pp. New York. NY: Springer-Verlag.

Decision Support Systems, 35, pp. Chemo AC. Pharmaceutical Institute. Department of Pharmacology and Biomedical Analysis, 50, pp. Proceedings National Institute of Science, 2 1 , pp. International Statistical Review, 73 3 , pp. DYNA, 81 , pp. USA: Penguin Books. Bolton: KluwerAcademic Pres. Journal of Optimization Theory and Applications, 13, pp. In Multicriteria Decision Making. Cochrane and M. Zeleny, South Carolina. Management Science, 1, pp. Computer-Aided Civil and Infrastructure Engineering, 17, pp.

Berlin: Springer-Verlag. Thousand Oaks. CA: Sage Publication.