MADM method Input Output Decision types DM interaction Assumptions MCDM software Attribute scores, Moderate Nonrestrictive Spreadsheets 7eights

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1 Aggregation methods This document is made of excerpts extracted from: Malcze7ski, :. (1>>>) GIS and multicriteria decision analysis. :ohn Diley E Sons Inc. Ne7 Gork These methods are only for attribute aggregation, if necessary, there are more methods for obective aggregation. 1. Method Summary MADM method Input Output Decision types DM interaction Assumptions MCDM software Individual DM, Scoring Ordinal ranking Moderate Non Spreadsheets deterministic Logical decisions, Multiattribute Value functions, Very Cardinal ranking DMs, deterministic, High MATS, value fuzzy Multiattribute utility Ideal point Concordance Ordered weighted averaging Utility functions, pair7ise comparisons, ideal point Fuzzy attribute,, order Cardinal ranking Cardinal ranking (ratio scale) Cardinal ranking Partial or ordinal ranking Cardinal or ordinal ranking DMs, probabilistic, fuzzy DMs, deterministic, probabilistic, fuzzy DMs, deterministic, probabilistic, fuzzy DMs, deterministic, probabilistic, fuzzy DMs, fuzzy High High Very Moderately Logical decisions, HIPRE3+, Expert choice, HIPRE3+, DhichE7hy, Moderate Non AIM, Moderate Moderate Non Moderately ELECTRE III and IV,

2 2. Method references MADM Method Citation Full reference Scoring :anko7ski and Richard (1>>4) Scoring Hey7ood et al. (1>>5) Scoring Lo7ry et al. (1>>5) Multiattribute value :anko7ski and Richard (1>>4) Multiattribute utility ceisler and Sundell (1>>7) Multiattribute utility :anssen and Rietveld (1>>0) :anko7ski, P., Richard, L. (1>>4) Integration of GIS-based suitability analysis and multicriteria evaluation in a spatial decision support system for route selection. Environment and Planning B 21(3): > Hey7ood, I., Oliver, :., Tomlinson, S. (1>>5) Building an exploratory multi-criteria modelling environment for spatial decision support. In: P. Fisher (Ed.), Innovations in GIS 2. London: Taylor E Francis, pp Lo7ry, :. H. :r., Miller, H. :., Hepner, G. F. (1>>5) A GIS-based sensitivity analysis of community vulnerability to hazardous contaminations on the Mexico/U.S. border. Photogrammetric Engineering and Remote Sensing 53(10): :anko7ski, P., Richard, L. (1>>4) Integration of GIS-based suitability analysis and multicriteria evaluation in a spatial decision support system for route selection. Environment and Planning B 21(3): > ceisler, :. M., Sundell, R. C. (1>>7) Combining multi-attribute utility and geographic information for boundary decisions: an application to park planning. :ournal of Geographic Information and Decision Analysis 1(2): :anssen, R., Rietveld, P. (1>>0) Multicriteria analysis and geographical information systems: an application to agricultural land use in the Nethernalds. In: H. :. Scholten, :. C. H. Still7ell (eds.), Geographical information systems for urban and regional planning. Dordrecht, The Netherlands: clu7er Academic, pp. 12>-13> Eastman (1>>3) Siddiqui et al. (1>>6) Eastman, :. R. (1>>3) IDRISI: a grid based geographic analysis system, version 4.1. Dorcester, MA: Graduate School of Geography, Clark University Siddiqui, M. e., Everett, :. D., Vieux, B. E. (1>>6) Landfill sitting using geographic information systems: a demonstration. :ournal of Environmental Engineering 122(6): Hickey and :anko7ski (1>>7) Hickey, R., :anko7ski, P. (1>>7) GIS and environmental decision making to aid smelter reclamation planning. Environment and Planning A 2>(1): 5-1>

3 MADM Method Citation Full reference Banai (1>>3) Ideal Point Tkach and Simonovic (1>>7) Banai, R. (1>>3) Fuzziness in geographic information systems: contributions from the analytic hierarchy. International :ournal of Geographical Information Systems 7(4): > Tkach, R. :., Simonovic, S. P. (1>>7) A ne7 approach to multi-criteria decision making in 7ater resources. :ournal of Geographic Information and Decision Analysis 1(1): URL: Ideal Point Malcze7ski (1>>6) Ideal Point Pereira and Duckstein (1>>3) Ideal Point Carver (1>>1) Ideal Point E7art (1>>4) Concordance Martin et al. (in press) Concordance Can (1>>2) Concordance Carver (1>>1) Malcze7ski, :. (1>>6) A GIS-based approach to multiple criteria group decision making. International :ournal of Geographical Information System 10(8): >55->71 Pereira, :. M. C., Duckstein, L. (1>>3) A multiple criteria decision-making approach to GISbased land suitability evaluation. International :ournal of Geographical Information Systems 7(5): Carver, S. :. (1>>1) Integrating multi-criteria evaluation 7ith geographical information systems. International :ournal of Geographical Information Systems 5(3): > E7art, G. M. (1>>4) Integrating geographic information systems and multicriteria decision making to develop a prototype decision support system for health professionals. Unpublished M. S. thesis, University of Idaho, Mosco7, ID Martin, N. :., Onge, B. St., Daaub, :. P. (in press) An integrated decision aid system for develpment of Saint-Charles river's alluvial plain, Quebec, Canada. International :ournal of Environment and Pollution Can, A. (1>>2) Residential quality assessment: alternative approaches using GIS. Annals of Regional Science 23(1): >7-110 Carver, S. :. (1>>1) Integrating multi-criteria evaluation 7ith geographical information systems. International :ournal of Geographical Information Systems 5(3): > Ordered weighted averaging Eastman and :iang (1>>6) Eastman, :. R., :iang, H. (1>>6)

4 MADM Method Citation Full reference Ordered weighted averaging Eastman (1>>7) Eastman, :. R. (1>>7) IDRISI for Dindo7s, version 2.0: tutorial exercises, Dorcester, MA: Graduate School of Geography, Clark University 3. Method description Scoring They are based on the concept of a 7eighted average. The decision maker directly assigns of relative importancek to each attribute. A total score is then obtained for each alternative by multiplying the importance 7eight assigned for each attribute by the scaled value given to the alternative on that attribute, and summing the products over all attributes. Dhen the overall scores are calculated for all the alternatives, the alternative 7ith the highest overall score is chosen. The decision rule evaluates each alternative, A i, by the follo7ing formula: A i " # w! x 7here x i is the score of the ith alternative 7ith respect to the th attribute, and the 7eight w is a normalized 7eight, so that! w = 1. Multiattribute value The value function approach is applicable in the decision situations under certainty (deterministic approach). This approach assumes that the decision maker is relatively risk neutralk or that the attributes are kno7n 7ith certainty. Formally, the value function model is similar to scoring methodk, except that the score x i is replaced by a value v i derived from the value function. The value function model can be 7ritten: V i " # w! v 7here V i is the overall value of the ith alternative, v i is the value of the ith alternative 7ith respect to the th attribute measured by means of the value function, and the 7eight w is a normalized 7eight or scaling constant for attribute, so that! w = 1. Multiattribute utility In the utility function procedure, the decisionms maker attitude to7ard risk is incorporated into assessment of a single-attribute utility function (ceeney, 1>80). Thus utility is a convenient method of including uncertainty (risk preference) into decision-making. The concept of a utility function is i i

5 inherently probabilistic in nature. Formally, the utility function model is similar to scoring methodk, except that the score x i is replaced by a utility u i derived from the utility function. The utility function model can be 7ritten: U i " # w! u 7here U i is the overall value of the ith alternative, u i is the utility of the ith alternative 7ith respect to the th attribute measured by means of the utility function, and the 7eight w is a normalized 7eight or scaling constant for attribute, so that! w = 1. The analytical hierarchy (AHP) method, developed by Saaty (1>80), is based on tree principles: decomposition, comparative udgment and synthesis of priorities. The decomposition principle requires that the decision problem be decomposed into a hierarchy that captures the essential elements of the problem, the principle of comparative udgment requires assessment of pair7ise comparisons of the elements 7ithin a given level of the hierarchical structure, 7ith respect to their parent in the next-higher level, and the synthesis principle takes each of the derived ratio-scale local priorities in the various levels of the hierarchy and constructs a composite set of priorities for the elements at the lo7est level of the hierarchy. In this final step, the goal is aggregate the relative of the levels obtained in the previous step to produce composite, this is done by means of a sequence of multiplications of the matrices of relative at each level of the hierarchy. Ideal point methods Ideal point methods order a set of alternatives on the basis of their separation from the ideal point. This point represents a hypothetical alternative that consists of the most deliverable 7eighted standardized levels of each criterion across the alternatives under consideration. The alternative that is closed to the ideal point is the best alternative. The separation is measured in terms of a distance metric. The ideal point decision rule is: + p ( p si- " )# w $ vi, v- % & * ' 7here s i+ is the separation of the ith alternative from the ideal point, w is a 7eight assigned to the criterion, v i is the standardized criterion value of the ith alternative, v + is the ideal value for the th criterion, and p is a po7er parameter ranging from 1 to ". Concordance methods i 1 p

6 Concordance methods are based on a pair7ise comparison of alternatives. They provide an ordinal ranking of the alternatives; that is, 7hen t7o alternatives are compared, these methods can only express that alternative A is preferred to alternative B, but cannot indicate by ho7 much. The most kno7n concordance approach is the ELECTRE method and its modifications. Ordered weighted averaging Ordered 7eighted averaging is an aggregation technique based on the generalization of three basic types of aggregation functions, 7hich are: (1) operators for the intersection of fuzzy set, (2) operators for the union of fuzzy sets, and (3) averaging operators. It provides continuous fuzzy aggregation operations bet7een the fuzzy intersection and union, 7ith a 7eighted-average combination falling mid7ay in bet7een. Emilio Ortega E Santiago Mancebo TRANSyT UPM Spain