The TODIM Method

This section presents the theory and calculation procedures of one of the multicriteria methods mentioned in the previous section: the TODIM method.
The TODIM (acronym of Tomada de Decisão Interativa e Multicritério – Interactive and Multicriteria Decision Making) multicriteria method, conceived in the early 1990s and the subject of two articles published in a European scientific journal at the same time (Gomes & Lima, 1992a; Gomes & Lima, 1992b), is a discrete multicriteria method based on Prospect Theory (Kahneman & Tversky, 1979). In this way, while practically all the other multicriteria methods start from the premise that the decision maker makes a decision always seeking the solution corresponding to the maximum of a global measurement of value – for example, the greatest possible value of a multiattribute utility function, in the case of MAUT (Keeney & Raiffa, 1993; Belton & Stewart, 2002) ¾, the TODIM method makes use of the notion of a global measurement of value calculable by the application of the paradigm of Prospect Theory. Hence, the method is based on a description, proved by empirical evidence, of how people effectively make decisions in the face of risk (He & Huang, 2008; Schmidt & Starmer, 2008).
As regards the structuring of the decision problem, the TODIM method essentially consists of a multicriteria method for the ranking and selection of the alternatives. As such, it is a method to evaluate, from a multicriteria viewpoint, a given set of alternatives, without the intention of supporting the decision maker in assuming a position in the face of a specific context. Taking this line, the TODIM method follows the tradition of authors such as Brans & Mareschal, 2002; Roy, 1996; Keeney, 1992; and Von Winterfeldt & Edwards, 1986. In this way, it can be coupled to a problem structuring process (Montibeller Neto et al., 2008; Ensslin, Montibeller Neto & Noronha., 2001; Bana e Costa et al, 1999; Rosenhead, 1989; Belton, Ackermann & Sheperd, 1997). As a consequence, although the emphasis underlying the practical use of the TODIM method is in the modelling of the problem itself, as well as in the subsequent calculations, it recognises the need to give due attention to this structuring process. It is expected that, even when not using a particular technique to structure the problem, the organisation where the study is developed should reach a consensually accepted understanding of the problem through a process of intense discussions in which both executives and technical analysts participate. This understanding characterises what is understood as a required model for the problem (Phillips, 1984; 1990). Included in this required model are the definition of the objectives – which can be broken down into decision criteria and attributes of viable alternatives – and the identification of the same. The subsequent option to use the TODIM method is normally based on the combination of the relative simplicity of its use, possible through the means of a simple Excel® spreadsheet, and the opportunity provided by its foundations in Prospect Theory, thus giving a dimension of practicality and realism to the results obtained (Kumar & Lim, 2008; Huber, Viscusi & Bell, 2008; Jou et al., 2008).
In order to be able to apply the Prospect Theory paradigm to a database arising from calculations and judgement values, the TODIM method must test specific forms of the losses and gains functions. These, once validated empirically, will serve to construct the additive difference function of the method, which supplies measurements of dominance of each alternative over each other alternative. Although it may seem complicated to have to test the validity of the application of the paradigm to the database – which could on occasions oblige the decision analyst to use other forms of losses and gains functions -, in reality it is not, as, since the first practical uses of the TODIM method published in the literature on Multicriteria Decision Support at the beginning of the 1990s, the same two mathematical forms have been used with success, having been validated empirically in different applications (Rangel & Gomes, 2007; Gomes & Rangel, 2007; Passos & Gomes, 2005; Costa, Almeida & Gomes, 2003; Passos & Gomes, 2002; Gomes et al., 2001; Trotta, Nobre & Gomes, 1999; Gomes, Duarte & Moraes, 1999). A complete presentation of the theory of the TODIM method can be found in Gomes, Araya & Carignano, p. 137-157 (2004).
From the construction of the aforementioned TODIM method additive difference function – which functions as a multiattribute value function and, as such, must also have its use validated by the verification of the condition of preferential mutual independence (Keeney & Raiffa, 1993; Clemen & Reilly, 2001) –, the method leads to a global ranking of the alternatives. It is observed that the multiattribute value function – or additive difference function – of the TODIM method is constructed from a projection of the differences between the values of any two alternatives (perceived in relation to each criterion) over a reference criterion or referential criterion. The concept of the additive difference function used by the TODIM method is based on Tversky’s research on the analytical treatment of the multidimensionality of a value function (Tversky, 1969).
The TODIM method makes use of pair comparisons between decision criteria, and has technically simple and correct resources to eliminate eventual inconsistencies arising from these comparisons. It also allows judgement values to be carried out on a verbal scale, the use of criteria hierarchy, fuzzy judgement values and the use of relations of interdependence between alternatives (Gomes, Araya & Carignano, 2004).
Roy & Bouyssou (1993), writing on the TODIM method, state that it is: “a method based on the French School and the American School. It combines aspects from the Theory of Multiattribute Utility, the AHP method and the Electre methods.” (p. 638).
The idea, contained in the formulation of the TODIM method, of introducing expressions of gains and losses in the same multiattribute value function gives this method some similarity with the PROMÉTHÉE methods (Brans & Mareschal, 2002; 1990), which make use of the notion of net outranking flow. Barba-Romero & Pomerol (2000) understood this, stating the following in respect of the TODIM method: “it is based on a notion which is extremely similar to net flow, in the sense of PROMÉTHÉE” (p. 229).
It considers a set of n alternatives to be ranked in the presence of m quantitative and qualitative criteria, and allows one of these criteria to be considered as the reference criterion. After the definition of these elements, the specialists are asked to estimate, for each of the qualitative criteria c, the contribution of each alternative i to the objective associated with the criterion. This method requires the values of the evaluations, of the alternatives in relation to the criteria, to be numerical and to be normalized. With this, the quantitative criteria evaluated on a verbal or nominal scale are transformed into a cardinal scale. The evaluations of the quantitative criteria are obtained through the performance of the alternatives in relation to the criteria, such as, for example, noise level in decibels, the power of a motor in horsepower, the grade of a student in a subject etc.
After the evaluation of the alternatives in relation to all the criteria, the evaluation matrix is obtained, where the values are all numerical. Then normalization is carried out, using, for each criterion, the division of the value of an alternative by the sum of all the alternatives. This normalization is carried out for each criterion, thus obtaining a matrix, where all the values are between zero and one, called the matrix of partial desirabilities W = [Wnm], where n indicates the number of alternatives and m the number of criteria. Once the scale giving the reading of the measurement of estimated performance of each alternative in relation to each criterion is determined from the set of alternatives itself, by means of the normalization cited above, the occasional occurrence of a reversal of the ranking can be minimised by the two following paths: (i) the addition of a new alternative which contributes to expanding the variation interval of the normalised values; or, alternatively, (ii) the weighting of each alternative in relation to a criterion with the value of one unit in the scale in which the criterion is measured (Belton & Stewart, 2002, p. 159; Belton & Gear, 1983, 1985; Tallarico, 1990).
After attributing weights to the criteria and their normalization, the partial dominance matrixes and the final dominance matrix must be calculated. The decision makers must indicate the reference criterion r, which may initially be the criterion with the greatest weight. However, it is easy to prove algebraically that, whatever the reference chosen, the result obtained will be the same. Thus, arc represents the substitution rate of the criterion under analysis c in relation to the reference criterion r. The measure of dominance of each alternative i over each alternative j, now incorporated to Prospect Theory, is given by a mathematical formula. In this formula gains and losses are represented as the differences between Wic and Wjc, the weights of the alternatives i and j in relation to c. An attenuation factor of the losses q takes care of the shape of the value function in the negative quadrant. The construction of the value function Fc(i,j) permits the adjusting of the data of the problem to the value function of Prospect Theory, thus explaining aversion and propensity to risk. This function takes the form of an “S” but is not symmetrical with respect to the origin. Above the horizontal axis, considered as the reference in this analysis, there is a concave curve representing the gains, and, below the horizontal axis, there is a convex curve representing losses. The concave part reflects the aversion to risk in the face of gains, and the convex part, in turn, symbolises the propensity to risk when dealing with losses.
After calculating the various partial dominance matrices, one for each criterion, the final dominance matrix [d(i,j)] is obtained, through the sum of the elements of the various matrices. The final dominance matrix is then normalised, in order to obtain the global value of each alternative. Each number calculated must be interpreted as the measurement of desirability or global utility, or, simply, as the global value of a specific alternative. The ranking of the alternatives originates from the ranking of their respective values.
Therefore, the TODIM method determines a choice, in ranking all of the alternatives, based on the preferences expressed by the decision makers. Changing this set of preferences may occasionally arrive at a new result, by means of a sensitivity analysis.
Thus for its application in a decision making process in a company, the following steps are followed: selection of the criteria; (ii) comparison in pairs of the criteria, using the Saaty scale (1991); (iii) the attribution of a score between 0 and 1 to each of the alternatives, relative to each of the criteria; (iv) a sensitivity analysis; and (v) a comparison with the status quo, with a mathematical treatment related to the method in the relevant stages. Step (ii) produces a positive reciprocal matrix, with its elements in the interval [1, 9], which is essentially used to obtain the weights of the criteria. These weights are obtained by following the steps as follow: (1) add the elements of the matrix down each column and form the reciprocals of the totals thus obtained; (2) divide each reciprocal by the sum of the reciprocals, thus obtaining a first ordering of the weights – having done this, the generic criterion c will have its weight V0c. These steps (1) and (2), according to Saaty (1991, p. 24), consist of the best estimate of the normalized principal eigenvector, corresponding to the positive reciprocal matrix. If this positive reciprocal matrix which produced the set of generic criteria weights V0c is not absolutely consistent – in other words, if the property of transitivity (Vansnick, 1990, p. 83) is not obeyed -, it will still be possible to correct it. For this purpose, in each cell of the revised matrix, corresponding to the intersection of criterion c with criterion c’, the quotient V0c / V0c’ is introduced. The new matrix of pair comparisons between the criteria will then be absolutely consistent, provided that the property of strict transitivity is respected. This technique to arrive at a consistent positive reciprocal matrix respects the pair comparisons initially carried out by the decision maker, without causing the possible embarrassment of asking the decision maker to review some of the value judgements (Gomes, 1993).
The score between 0 and 1 attributed to each of the alternatives, in relation to each of the criteria, can be obtained in two distinct ways. When dealing with a quantitative criterion, divide the measurement – for example, the estimate of the financial cost of an alternative – by the sum of the financial cost of all of the alternatives. When dealing with a qualitative criterion – such as the estimate of the social importance of an alternative -, attribute to each of them a score between 1 and 9 and, then, divide each of these scores by the sum of them. In this way, the partial desirabilities matrix denominated W = [Wnm] is formed, mentioned earlier in this article.
Based on the technique of Heuristic Minimisation of Interdependence between Criteria (HMIC), interdependence among the initial set of criteria can be minimised. Basically, what the HMIC technique does, through the interpretation of conceptual interactions among the twelve criteria initially established, is to associate to each intersection of these a level of strength of interaction, read in a pre-defined verbal scale, of increasing levels of interdependence. This allows the criteria to be progressively aggregated until a final set of criteria of a minimal size is reached. The heuristics in which the HMIC technique is based thus permit a reduction in the initial family of criteria (Gomes, Damázio & Araújo, 1992).
Even after this reduction it is desirable to present the decision makers with the resulting set of criteria, with a view to testing its potential functional validity. This allows them to have an active involvement in the modelling of the problem, included here, and, with due emphasis, the obtaining of the final list of evaluation criteria. Next comes the passing of the values read in the verbal or nominal scales from 0 to 1 by means of the association of a reading of 0 to the worst possible value and a reading of 100 to the best possible value. From this point, the decision makers are asked to position each reading in those scales inside the interval [0, 1]. The values of the normalised weights of the criteria are then determined, using for this the calculation procedure previously explained in this article. The attribution of weights to the criteria, performed by the decision analyst, generates the matrix of desirabilities. In applications of the TODIM method it is also desirable to carry out a sensitivity analysis of the results and variations in  and, in the case of obtaining different final rankings, it will be the interaction between the analyst and the decision makers which will determine which value of  will be used in the calculations. The desirability of each alternative indicates its global value. The ranking of the alternatives originates from the ranking of their respective values.
In essence, the TODIM method is a multicriteria method which can be expected to be well-received due to its theoretical foundations, based on Prospect Theory, to the opportunity its interactive focus presents and, without doubt, to the practicality of its application.
In the next section we shall illustrate the use of the TODIM method through a real application.

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