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.

Bibliographical references

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Multicriteria Methods for Decision Aiding

Multicriteria methods serve to select, rank, classify or make detailed descriptions of alternatives on which decisions will be made [Pomerol & Barba-Romero (2000); Belton & Stewart (2002)]. These methods can be used in combination or not. Thus, for example, a specific method may be used to classify a set of viable alternatives into four categories: very good, good, mediocre and out of the question; following this, one may, by means of another method – which must however have the same axiomatic base as the method previously used – simply rank only those considered very good, in this way obtaining the best alternative of the set.
The genesis of multicriteria methods is to be found in the history of the allied forces during the Second World War. Basically, up to the first half of the twentieth century, when solving complex decision making problems, expected value was used for making decisions in random conditions; however, in many situations it was observed that the risk associated with this procedure was unacceptable. With the end of the Second World War, though, as a result of the experience gained by the Allied Forces with military logistics problems, a large number of research organisations and university departments began to dedicate their time to a systematic study of the analysis and planning of decisions, using the then recently created Operational Research. From that point there arose within the business environment, the immediate need to optimise costs and maximise profits by means of Operational Research methods. With these objectives in mind, various strictly mathematical methods were developed to find the optimum solution for transport and production problems among many others. These methods, in the main, are currently to be found in classical optimization under restrictions or mathematical programming with a single objective function, and many of them are still used today in a series of applications, such as assignment of flows to networks, establishing a minimum path and inventory optimization etc.
In classical optimization under restrictions or in mathematical programming with a single objective function one seeks the maximum or minimum value of a single objective function, submitted to a set of restrictions to be respected. This means that all the consequences derived from the choice of each one of the alternatives must be able to be reduced or expressed in terms of a single evaluating function. However, in practice, the decision maker generally uses various criteria simultaneously to evaluate the different alternatives, some of them difficult to measure in relation to non-monetary consequences (such as, for example, socio-environmental impact, product image, quality, security, comfort etc.). Although they can be incorporated to the model by means of restrictions, the difficulty of dealing with multiple dimensions at the same time as dealing with monetary ones can be observed.
Nevertheless, in the 1950s and 1960s, statistics were used – by means of the expected value concept already mentioned – to find the best solution for single criterion decision problems, through the decision tree concept. As a result of this, a more traditional view of Decision Theory would indiscriminately use this expression and the expression Decision Analysis to designate the construction and application of decision trees for decision problems with a single criterion [Raiffa (1977); (2002)]. Among the excellent works on the classical focus of decision analysis are the books by Fishburn (1970) and de Souza (2002). Examples of works which present both the classical focus as well as the basics of the multicriteria focus are Kaufman & Thomas (1977), Keeney (1982) and Carrasco & Sánchez (1990). Brown’s book (2005) presents the most modern and realistic view – therefore multicriteria based – of decision analysis.
Although attempts had been made to solve decision making problems in the presence of multiple criteria through the assignment of weights – both for alternatives and criteria –, especially in academic environments, scientifically formulated multicriteria methods oriented to real applications basically arose at the end of the 1960s, initially in Paris [Roy (1968)], and during the following decade in the United States of America [Keeney & Raiffa (1976); Saaty (1977)]. These pioneering works notably reflected dissatisfaction with the project evaluation methodologies available at that time, which were either only ordered in consequences expressed in monetary units – such as cost/benefit analyses [Weimer et al. (2005)] – or were presented as incapable of dealing simultaneously with multiple categories of consequences – monetary and non-monetary –, such as cost-effectiveness analyses [Levin & McEwan (2000)].
This being so, the multicriteria methods which arose around the 1970s were designed to solve decision making problems with the following principal characteristics:
· There were at least two criteria involved in solving the problem, these in conflict with each other.
· Both the criteria and the alternatives were not clearly defined, and the consequences of choosing a specific alternative, in relation to at least one criterion, were not duly understood.
· The criteria and the alternatives could be interlinked in such a way that a given criterion seemed to partially reflect another criterion, while the efficacy of opting for a specific alternative depended on whether another was or was not chosen, in the case of the alternatives not being mutually exclusive.
· The solution of the problem depended on a group of people, each with their own point of view, often conflicting with those of the other people.
· The restrictions of the problem were not well defined, meaning that doubts could exist in respect of what was a criterion and what was a restriction.
· Some of the criteria were quantifiable – for example, in terms of monetary units – while others were only so by means of judgement values made against a scale.
· The scale for a specific criterion could be cardinal (that is, numerical), verbal (or possible to be expressed in ordinary language) or ordinal (by establishing relations of ranking), depending on the data available and the nature of the criteria.
Other complications could arise in a real context of decision making, but these seven aspects mentioned characterised the essence of this complexity. In general, decision making problems with these characteristics were considered – and are still considered today – badly formulated problems.
At the same time, a consciousness developed that one should not intend the use of these new methods to lead necessarily to an optimum solution – in the sense of a solution which would ideally be the best possible according to all the points of view relevant to the problem –, but instead, at least to a solution which represented a satisfactory compromise between these points of view. In this way, something was perceived which has become a consensus today: multicriteria methods are heuristic, conceived to deal with decision making problems which have a finite (or countable) number of possible alternative solutions.
As has already been mentioned here, multicriteria methods are used in the analysis which precedes the decision making. Nevertheless, one cannot ignore the fact that the field of application of these methods also includes evaluating at which point a decision already taken has met or not the objectives of the problem. Therefore, it is said that multicriteria methods can be used before or after an implementation. In the first case, one talks of an analysis (of the decision) ex ante, that is, before the decision is made, serving to generate recommendations for the decision making itself. In the second case, it is said that the analysis (of the decision) is ex post, in other words, after the decision has been made, seeking through this analysis to learn from the decisions already made.
Although there are, in the decision analyst’s tool-box, a much greater number of multicriteria methods than the number of problems the analyst is called upon to solve, normally the choice of one particular method, as opposed to the others, is guided by a solid knowledge of a reasonably large number of methods on the part of this professional. This knowledge includes the adequacy of applying each method to the problem, considering here the following principal aspects: (i) the nature of the problem to be solved (that is, selection, ranking, classification and description); (ii) the possible means of collecting and compiling the data; (iii) the structure of the relationships among the objectives of the problem; and (iv) the type of communication to be expected between the analyst and the decision maker, chiefly during the decision analysis stages.
The main characteristics of the most commonly used methods shall now be examined. For a panoramic view of the relatively large number of multicriteria methods available today to practitioners of Decision Theory, see, for example, Schärlig (1990), Vinke (1992), Bana and Costa (1990), Clímaco (1997) and Triantaphyllou (2000).
A very common error committed by novice decision analysts is to try to resolve a specific decision problem by means of commercially available software. However, the software must never be used without the decision analyst having an understanding, principally from a technical point of view, of the analytical method which is embedded in it. On the other hand, the application of any decision aiding method must start from the premise that the problem to be solved has already been adequately structured. Consequently, the stage called problem structuring is crucial and must never be omitted [Montibeller Neto et al. (2008); Bana and Costa et al. (1999); Rosenhead and Mingers (2008); Belton, Ackermann & Sheperd (1997)]. Thus, although the emphasis underlying the use of a multicriteria method is essentially in the modelling and the analysis of the problem, as well the subsequent calculations, the significant need to give due attention to the structuring process is recognised. Even simple techniques, such as Pros and Cons Analysis [Baker et al. (2001)] and the Kepner-Tregoe Decision Analysis [Kepner & Tregoe (1981)] are useful as a first exercise in organising ideas around a decision making problem in the presence of multiple criteria.
Having said this, a listing is made here of some among the main multicriteria decision aiding methods, with important sources of information on them. The following methods are listed in alphabetical order.
· AHP & ANP [Saaty (1988; 1990; 1994; 2001)]
· Dutch Methods of Multicriteria Decision Aiding [Lootsma (1993; 1994a; 1994b); Ancot (1988); Delft & Nijkamp (1977); Jansen (1994); Nijkamp (1977); Nijkamp, Rietveld & Voogd (1990); Paelink & Nijkamp (1976); Rietveld (1980); Voogd (1989)]
· ELECTRE [Roy & Bouyssou (1993)]
· MACBETH [Bana and Costa & Vansnick (1999; 2000)]
· MAUT [Clemen & Reilly (2001); Keeney & Raiffa (1976)]
· PROMÉTHÉE [Brans & Mareschal (2002)]
· Rough Sets Theory [Pawlak (1982); Slowinski (1992)]
· TODIM [Gomes & Lima (1992a; 1992b); Trotta, Nobre & Gomes (1999); Gomes & Rangel (2007)]
· Verbal Decision Analysis [Larichev & Moshkovich (1997); Larichev & Olson (2001)]
In the following section we shall explain one of these methods and show how it can be applied to the practice of decision making aiding.

Bibliographical references

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BANA E COSTA, C.A. (ed.) (1990) Readings in Multiple Criteria Decision Aid. Berlin: Springer.
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Bana e Costa, C.A.; Ensslin, L.; Corrêa, E.C. & Vansnick, J. (1999) “Decision support systems in action: integrated application in a multicriteria decision aid process”. European Journal of Operational Research, 113: 315-335.
Belton, V.; Ackermann, F. & Sheperd, I. (1997) “Integrated support for problem structuring through an alternative evaluation using COPE and V×I×S×A”. Journal of Multi-Criteria Decision Analysis, 6: 115-130.
BELTON, V. & STEWART, T.J. (2002) Multiple Criteria Decision Analysis An Integrated Approach. Dordrecht: Kluwer.
BRANS, J.-P. & MARESCHAL, B. (2002) Prométhée-Gaia Une Méthodologie d’Aide à la Décision en Présence de Critères Multiples. Bruxelles : Université de Bruxelles/Ellipses.
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CLEMEN, R.T. & REILLY, T. (2001) Making Hard Decisions with DecisionTools. 2a ed. Pacific Grove: Duxbury/ Thomson Learning.
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GOMES, L.F.A.M. & LIMA, M.M.P.P. (1992a) “Todim: Basics and Application to Multicriteria Ranking of Projects with Environmental Impacts”. Foundations of Computing and Decision Sciences, vol. 16, no. 4, p. 113-127.
GOMES, L. F. A. M. & LIMA, M. M. P. P. (1992b) “From Modelling Individual Prefe­rences to Multicriteria Ranking of Discrete Alternatives: A Look at Prospect Theory and the Additive Difference Model”. Foundations of Computing and Decision ­Sciences, vol. 17, no. 3, p. 171-184.
Gomes, L.F.A.M. & Rangel, L.A.D. (2007) “An Application of the TODIM Method to the Multicriteria Rental Evaluation of Residential Properties”. Accepted for publication in the European Journal of Operational Research in 2007, available online at www.sciencedirect.com, doi: 10.1016/j.ejor.2007.10.046.
Janssen, R. (1994) Multiobjective Decision Support for Environmental Management. Dordrecht: Kluwer.
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KEENEY, R.L. & RAIFFA, H. (1976) Decisions with Multiple Objectives: Preferences and Value Tradeoffs. New York: Wiley.
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LARICHEV, O. & MOSHKOVICH, H. (1997) Verbal Decision Analysis for Unstructured Problems. Boston: Kluwer
LARICHEV, O. & OLSON, L. (2001) Multiple Criteria Analysis in Strategic Siting Problems. Boston: Kluwer.
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Pawlak, Z. (1982) “Rough sets”. International Journal of Information and Computer Sciences, vol. 11, no. 5, p. 341-356.
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Decision Aiding for Managers

The study of the paradigms underlying decisions and their analytical bases encompasses what is known as Decision Aiding. The main aim of this text is to present these paradigms and fundamentals to the reader in a non-technical way. It seeks to satisfy this objective by offering a frame of reference which permits the reader to view a wide range of possibilities concerning the use of Decision Aiding in complex decision making processes, notably in various fields of human experience, in human sciences, applied to society or health, in technology etc., as well as in our personal lives and in our civic activities.
In the next step, some basic questions will be proposed and then answers attempted to give an understanding of what Decision Aiding is.
The first question to be put is the following: What is to be understood by decision? Decision is the process which leads - directly or indirectly – to the choice of at least one from among different alternatives, all of which are candidates to solve a specific problem. In this way, one makes a decision when selecting one candidate from several for a job vacancy. One also makes a decision when classifying these candidates as good, average or not satisfactory. While in the first example there is a direct relationship between the process and the act of choosing itself, in the second example the classification obtained is nothing more than the preamble to the act of choosing – which, however, may or may not materialize.
The second question to be put is the following: Why study decisions? Because (i) the way of thinking implicit in the process which constitutes the decision is part of the daily routine of human beings; and (ii) frequently the practical results of that way of thinking are considered extremely important by them. These two aspects mean that special attention needs to be paid to the decision as the subject of study, justifying it as a field of scientific interest.
On the other hand, although decisions are a constant presence in human activity, at times, people generally considered highly intelligent make decisions which lead to consequences which are very different from those expected. In fact, someone’s performance as a decision maker does not only depend on intelligence: it also depends on the degree of adjustment within the organizational culture – company, political party, family etc. – where the decision is made and the psychological style in decision making. As regards this last aspect, it can be observed that there are, for example, people who prefer to make important decisions alone, after hearing all of the relevant points of view, while others always prefer to work in a group.
A decision is seen, in turn, through the following three dimensions: (i) importance, in terms of satisfaction of values; (ii) the speed required; and (iii) the degree of individuality.
Practically no-one questions the importance of an excellent performance for a decision maker in skills such as leadership, negotiation, entrepreneurship and administration in general. In management, for example, decisions are made because important problems need to be resolved; in this type of decision, decisions are frequently made in groups. In general, but not always, there must be or should be sufficient time to organize the ideas well before the decision itself. This organization of ideas is normally the most important element of the decision making process, called problem structuring.
Next comes the basic question of Decision Aiding: How to make a good decision? It was precisely in the efforts to answer this question that Decision Aiding established itself as a field of scientific knowledge. Even so, it is not difficult to see that a decision which may seem excellent today may tomorrow be revealed as catastrophic. This suggests that the notion of what is a good decision is only valid for a particular scenario, including here the values of the client of the decision, be it a person, a group of people or an organization.
As a result, it can be stated that Decision Aiding arose from the need to support that human activity which consists of making good decisions. There is a consensus among students of Decision Aiding that the path to a good decision usually includes the following stages, not necessarily in sequence:
· Be sure that one is trying to solve the right problem.
· Think sufficiently about the problem, seeking to maintain a distance from possible emotional involvements, never taking as truths the opinions of others and thus avoiding the so-called psychological traps.
· Seek out all the relevant information.
· Clearly identify what effectively matters, in other words, the crux of the decision.
· Consider explicitly the commitments of a moral and ethical nature.
· Generate the widest possible set of viable alternatives.
· List the objectives of the decision making process, both quantitative – to find the solution to the lowest possible annual cost, or minimize the total annual cost – and qualitative – to find the best solution from an aesthetic point of view, for example, or maximize the aesthetic; note that the objectives are always formulated using the infinitive of the verbs.
· For each of the objectives listed, make the decision criteria clear – thus, an objective such as maximize the social importance of the project can be covered in the criteria (i) meeting the most urgent necessities of the population in need and (ii) promotion of social mobility; the criteria are always formulated as nouns.
· Make the consequences of each alternative clear in relation to each of the decision criteria, together with an estimate of the probability of each one of these consequences in fact materializing – the best way to do this is by the construction of a table, in which the lines are associated with the alternatives and the columns correspond to the criteria; the information contained in the intersection of each line with each column will provide the calculations, judgment values, and/or consultations with experts.
· Starting from the nine stages above, although feeding new information as necessary, use one of the various analytical methods available in the literature on Decision Aiding – the multicriteria methods – to select, rank, classify or describe in detail the alternatives on which the decision will be made – the feeding of new information is necessary due to the fact that, during this technical analysis, some aspect of the problem might arise which had not been considered during the previous nine stages, thus generating, for example, new alternatives or new criteria.
· Carry out a criticism of the results obtained in this tenth stage, attempting to place oneself in the position of the decision maker as well as those who will live with the direct and indirect consequences of the decision – on occasions, as a result of this criticism, it may be necessary to redo the tenth stage.
· Produce objective recommendations for the decision maker, here including a proposal of the decision itself and the best way to implement it, guaranteeing the transparent documentation of all the stages, with a view to organizational learning. The perception of the viability of implementing each of the alternative candidates should, in other words, permeate all of the process described above, which may, in many cases, constitute one of the decision criteria.
The first nine of these twelve stages constitute what is usually called problem structuring. The tenth and the eleventh of the stages constitute the decision analysis while the last stage is the synthesis. Throughout this twelve stage process values, alternatives, criteria, consequences, possible risks and trade-offs between alternatives and between criteria are elicited. As a general principle, equal attention should be paid to each of these twelve stages. One of the most common errors – and one which frequently leads to bad results – is to give less attention to the structuring of the problem than to the decision analysis and synthesis. In addition, one should not naively believe that this is a purely rational process, as intuition is always present when performing it. Other difficulties which may arise in carrying this out are related to the presence of different points of view, even among experts; to the usual lack of perfect and complete information; to ever-present uncertainty and imprecision; and, without doubt, to the size of the problem – that is, the range of relationships which may exist among elements of what seems to be the decision problem – which together, are called the decision system – and other components – although outside that system – of the context in which the decision should be made.
Due to their technical importance in the process, the seven – non-sequential, although interactive – phases of the decision analysis must be clearly identified. They are as follows:
· Phase 1 – Identification of the decision agents and the decision maker.
· Phase 2 – Listing of the alternatives, all them being viable candidates to solve the problem in question. In some cases, it will be easy to identify the alternatives; in others, however, it will be necessary to define them progressively. There will also be cases in which it may be necessary to reduce a long list of alternatives to a smaller list simpler to administrate; in principle, this can be done in various ways, such as, for example, eliminating the alternatives which do not satisfy some of the criteria in any way, in this way selecting a basic and representative set of the alternatives or, even determining a relatively small number of critical criteria for the evaluation and selection of those alternatives which perform better according to these criteria. Although using one of these techniques, there is no theoretical limit for the number of alternatives to be evaluated. It is considered that the collection of information for a large number of alternatives may be an exhausting task, especially if the number of criteria is relatively large.
· Phase 3 – Definition of the effectively relevant criteria. The definition of the alternatives and criteria, as seen above, will usually be an interactive process, in which new alternatives can suggest new criteria and vice-versa. Eventually a criteria hierarchy is formed; the criteria hierarchy most often used is linear and takes the form of a tree, in which each criterion is progressively decomposed, starting from the highest branch (or criterion) to those located beneath – through this technique, a family of criteria is formed starting from a father-criterion (the highest branch in the hierarchy). It is observed that there are few formal procedures which help in the structuring of a criteria hierarchy; this is a skill which is acquired with practice. In other words, there is no “correct” hierarchy for any problem in particular, and it is possible to develop alternative criteria structures. However, soon after the construction of a tree (or hierarchy) of criteria, it can be judged whether this representation is useful to the decision analyst, using the five factors suggested by Keeney and Raiffa (1976): (i) Completeness: if the tree is complete, all the criteria which interest the decision maker will be included in it. (ii) Operationality: all of the criteria at the lowest level of the tree should be sufficiently specific to permit the decision analyst to use them in the process of solving the problem (iii) Decomposability: it must be possible to evaluate the performance of an alternative in relation to a criterion, independent of its performance in relation to the other criteria. (iv) Absence of redundancy: if two criteria – partially or totally – reflect the same reality (for example, environmental impact and sound pollution) then one of them is clearly redundant; the danger of redundancy lies in the fact that it may cause double counting with the result that the recommendation generally tends to be spurious. One practical manner of identifying redundancy consists of establishing if it is possible to modify in some way the recommendation reached – by using a multicriteria method - if a specific criterion is eliminated from the tree. If the elimination of the criterion does not alter the choice of the best alternative then it will not be necessary to include it in the analysis. (v) Minimum size: if the tree is relatively large, any decision analysis will be impossible from a practical point of view. To ensure that this does not occur, criteria should not be decomposed beyond the level at which they can be evaluated. Common sense should always prevail. At times, the size of the tree can be reduced through the elimination of criteria which do not permit distinctions to be established between the alternatives.
· Phase 4 – Evaluation of the alternatives in relation to the criteria. There are various ways to carry out this phase, depending on the multicriteria method employed. In this phase, scales will be used to represent the consequences of each alternative in relation to each of the criteria, whether quantitative (such as, for a project, the value of the internal rate of return) or not (such as the importance of the point of view of mitigation of environmental impact).
· Phase 5 – To determine the relative importance of the criteria. This phase of the decision analysis consists of giving the criteria weights. As in Phase 4, there are many ways to carry out this weighting process, depending on the multicriteria method selected. It is important that the measurements of the relative weightings of the criteria are expressions of trade-offs between criteria: for example, one criterion may be considered twice as important as any other, which would bring with it consequences for the calculations to be carried out (by a multicriteria method). These weightings reflect, from the point of view of the decision maker, how much one is prepared to compromise regarding losses in terms of one criterion provided that there is a gain in another criterion, thus providing the idea of a trade-off relationship.
· Phase 6 – Determining satisfactory solutions. These, as seen above, will be the results of a selection procedure (of at least one best alternative for the final choice by the decision maker, or, possibly, a subset of the better alternatives), of a ranking (in which the set of viable alternatives is ranked from best to worst), of a classification (in which the alternatives are classified in pre-established categories) or, simply, of a detailed description of the alternatives (this description, frequently expressed in logical rules, may be used as a preliminary for a selection, a ranking or a classification).
· Phase 7 – Sensitivity Analysis. In this last stage of the decision analysis, as well as playing the role of devil’s advocate, the analyst seeks to introduce realistic modifications (that is, ones which may in fact materialize) in the variables and parameters used by the multicriteria method used, so as to test how robust the results obtained are. Occasionally the analyst may introduce these modifications in order to simulate possible changes in the decision maker’s preferences.
In other words, the preferences are nothing more than binary relationships between two objects – the alternatives. There are four main categories of preferences, which are:
· Indifference – when there are clear and positive reasons which justify equivalence between two alternatives.
· Strong preference (or strict) – when there are clear and positive reasons which justify a significant preference in favor of one of the two alternatives as opposed to the other.
· Weak preference – when there are clear and positive reasons which do not imply a strict preference in favor of one of the two alternatives as opposed to the other, but these reasons are insufficient to deduce if it is a strong preference or indifference between these two alternatives (thus the reasons do not permit one of the two preceding situations – indifference and strict preference - to be isolated as being the only appropriate one).
· Incomparability A – when there are no clear and positive reasons justifying one of the three preceding situations.
These four main categories of preference relationships are present in the carrying out of decision analysis. There are multicriteria methods – the ELECTRE methods – which make intensive use of refinements of them (Roy and Bouyssou, 1993). The twelve stage process is called, as a whole, Decision Aiding. It is fitting here to make a consideration about the terminology. As one makes use of at least two conflicting criteria to resolve all and any decision problems, Decision Aiding is appropriately called Multicriteria Decision Aiding (or Decision Making with Multiple Objectives).
Consequently, it is said that Multicriteria Decision Aiding is Decision Aiding put into practice. This practice, however, although it should be carried out in the best possible way, cannot always guarantee reaching a good decision. It is easy to understand the reason for this: the context – or scenario – in which the decision is made can change with time. Therefore, a new cast of values might arise, making obsolete the initial set of values on which the practice of Multicriteria Decision Aiding was based. In addition to this, new information might arise, as time passes, which may, by means of the introduction of new parameters, invalidate the recommendations which were reached at the end of the initial process.
It can now be stated, recognizing that the decision generally occurs in the presence of a dynamic scenario, that is one which evolves with time, that a good decision is one which solves a problem based on Multicriteria Decision Aiding; as the scenario changes, better decisions, founded on that same base, may materialize.
Therefore, Multicriteria Decision Aiding plays a crucial role, of an extremely technical nature, in the making of the decision concerning complex decision making processes. It illuminates, by means of ample structuring of the problem and analytical focus, through the application of methods, the search for a good solution of the problem. As one is dealing simultaneously with multiple – and conflicting – decision criteria, it can be imagined that this good solution which is sought will meet in different degrees the various objectives which characterize the decision making problem. Thus, based on the ideas of Simon (1982), it is said that one is seeking a satisfactory solution which represents the best possible compromise among multiple decision criteria. According to this last author it is recognized that rationality in decision making is always limited by three main factors, inherent to the participants in making the decision: (i) their cognitive capacities are not infinite; (ii) their personal values and motivations do not always coincide with those of the organization in which they are placed as decision makers; and (iii) their knowledge of the problem which they are attempting to solve is usually partial. In this way, it can be understood why one does not work towards an ideally best possible solution according to the decision criteria, but instead towards at least a satisfactory solution.
At this point it becomes indispensable to define some of the principal participants involved in the practice of Decision Aiding:
· The decision maker – is the last person responsible for the decision to be made, or, simply, the decider; this may be one single person or a group of people, the individual/s for whom the recommendation is produced on which decision should be made.
· The decision agent – is the individual or group or individuals, who, directly or indirectly, carry out calculations, generate estimates and elicit preferences and value judgments which are used during the decision analysis.
· The decision aider (or decision analyst) – is the professional versed in the principles and methods of Decision Aiding to whom is attributed the tasks of administrating and structuring the problem, its analysis and the production of recommendations for the decision maker; it can also be said that the modeling and the solution of the problem are the essential activities of the decision analyst, who constantly interacts with the decision agents and with the decision maker himself/herself. Therefore, the functions performed by the decision maker and the decision analyst are complementary, even though, in the end, the direct responsibility for the decision is assumed by the first and not the second.
It is also said that there are two possible focuses to Multicriteria Decision Aiding: the constructive focus (or constructivist) and the prescriptive focus. According to the constructive focus, the structuring of the problem advances in an interactive way – that is, by means of interaction between the decision analyst (versed in Decision Aiding) and the other participants in the decision making process (in other words, the decision agents) – in a way which is coherent with the values, objectives, consequent criteria and preferences of these agents and of the decision maker himself/herself. The prescriptive focus, meanwhile, consists of starting from a description of all the elements relevant to the problem, including here a description of the preferences of the decision maker, proposing – via the decision analyst – prescriptions to the decision maker, based on normative hypotheses. In the prescriptive focus, the involvement of the actors (or decision agents) in the process is restricted to the structuring of the problem. Due to the greater facility of adapting it to constantly evolving scenarios, the constructive focus to Multicriteria Decision Aiding has increased considerably in relative importance in recent years which, at the same time, has relegated the prescriptive focus to a second plane.
Multicriteria Decision Aiding therefore does not seek an optimum solution to a specific problem, as occurs in traditional Operations Research, but instead a compromise solution, where a consensus must preferentially prevail among the parties involved. From this viewpoint, the criteria used, as well as the importance attributed to them, have an essential role in the results obtained. This type of analysis allows the decision making process to be dealt with in a more transparent way thus increasing its credibility. However, it must be noted that this approach to the decision problem, from the Multicriteria Decision Aiding perspective, does not seek to present the decision maker with a definitive solution to the problem, electing a single truth represented by the selected alternative. This approach instead seeks to support the decision making process with a recommendation for actions which are in line with the preferences expressed by the many decision agents.
Therefore, a multicriteria approach applied to a complex decision making process generally results in the following advantages: (i) the construction of a base for the dialogue between the different decision agents; (ii) a concrete possibility of working with the subjectivities, uncertainties and imprecision always present in such a process; (iii) a visualization of each potential satisfactory solution as a compromise between the distinct points of view in conflict.
While Expected Utility Theory (VON NEUMANN and MORGENSTERN, 1944) reflects a normative vision of the decision, Prospect Theory (KAHNEMAN and TVERSKY, 1979) describes how decisions are made in the presence of risk. As all and any decision implies running some type of risk – at least the risk of not meeting, at a minimally adequate level, the objectives of the problem -, it can be said that both of the theories are decisional paradigms in competition with one another: while the first establishes the norm according to which the rational being seeks, when deciding, to maximize a measure of utility expected by him/her, the second, based on empirical observations, hopes that the rationality of the decision maker, confronted by risk, will reflect on the relative gains and losses, always defined in relation to a point of reference. Prospect Theory provides indeed a wider framework than Expected Utility Theory, based on a description of how to make decisions effectively in the face of risk. However, a decision always occurs in the presence of some type of uncertainty and, therefore, risk. When studying decisions one cannot therefore set out to ignore human behavior in the face of risk. Thus, it is natural that a decision paradigm such as Prospect Theory has established itself as a base for Multicriteria Decision Aiding. At least one multicriteria method exists which has in its base Prospect Theory: the TODIM method (Gomes and Rangel, 2007).

Bibliographical references

GOMES, L.F.A.M.; RANGEL, L.A.D. (2007) “An application of the TODIM method to the multicriteria rental evaluation of residential properties”, doi:10.1016/j.ejor.2007.10.046, paper available through www.sciencedirect.com. To be published in European Journal of Operational Research.
KAHNEMAN, D., TVERSKY, A. (1979) “Prospect Aiding: An Analysis of Decision Under Risk”. Econometrica, vol. 47, p.263-292.
KEENEY, R.L.; RAIFFA, H. (1976) Decisions with Multiple Objectives: Preferences and Value Tradeoffs. New York: Wiley.
ROY, B.; BOUYSSOU, D. (1993) Aide Multicritère à la Décision: Methods et Cas. Paris: Economica.
SIMON, H. (1982) Models of Bounded Rationality. 3 volumes. Cambridge: The MIT Press.
VON NEUMANN, J.; MORGENSTERN, O. (1944) Aiding of Games and Economic Behavior. 3rd ed. 1953 edition. Princeton: Princeton University Press.