ProMAA (Probabilistic Multicriteria Acceptability Analysis)


The ProMAA (Probabilistic Multicriteria Acceptability Analysis) method, developed within the DECERNS project, assimilates uncertainties of objective values and subjective judgments within the discrete Multiple Criteria Decision Analysis (MCDA). ProMAA algorithm utilizes probability distributions of criteria values and weight coefficients for assessing probabilities of 'likely rank events' based on pairwise comparison of alternatives in an integrated scale. Realization of ProMAA is based on numerical approximation of functions of random variables and numerical assessment of integrals. Algorithm of ProMAA, comparison of ProMAA with SMAA and other multicriteria methods, and application of ProMAA within a practical case study are presented in (Yatsalo, Tkachuk, et al., 2009).
Within the ProMAA probabilities Pik = P{Sik} of the events Sik are determinated, where Sik={Alternative ai has the rank k}, i,k=1,...,n,
(i.e., k-1 alternatives are better ai in a chosen scale for a subset of a space of elementary events).
For values Pik =P{Sik} the term 'rank acceptability indices' are often used (Lahdelma et al, 1998, 2006).
For aggregation of the indicated probabilities a weighted sum may also be used:
(17)
where wkac are weights of relative importance of ranks (Lahdelma et al, 1998).
Thus, ranking or screening alternatives {ai, i=1,...,n} within ProMAA is based on the analysis of the matrix {Pik}, i,k=1,...,n, or/and on the 'holistic acceptability indices' Ri; i=1,...,n, however, the recommendations concerning implementation of such a secondary ranking (17) are restricted.
Utility based ProMAA method, ProMAA-U, has been implemented within the DECERNS WebSDSS. It is based on (probabilistic) extension of the classical MAUT additive model (5), (6) (Keeney and Raiffa, 1976); within ProMAA-U both utilities Uj(aj) and weights wj may be considered as random variates with the given probability distributions, j=1,...,m.
The user interface of ProMAA-U module and corresponding functions allow the user(s) to:
- specify the distribution of Cj(ai) for criterion Cj, j=1,..., m, and the set of alternatives {ai, i=1,...n};
- specify the utility functions Uj(x) (from the class of linear, exponential, and piecewise-linear functions);
- specify the distribution for weight coefficient wj (from the indicated type of distributions), j=1,...,m, see also below a discussion concerning weights in ProMAA; then:
- the distributions of random variates ηi =U(ai), i=1,...,n, and rank acceptability indices Pik, i,k=1,...,n, are calculated by the system as a numerical implementation of the corresponding math expressions (Yatsalo, Tkachuk, et al., 2009);
- the users analyze graphical and tabular representation of the output results for subsequent decision making; and
- users have the possibility to implement utility functions sensitivity analysis of the output results (through changing one or several selected partial utility functions Uj(x)).

Setting weights within ProMAA

Within MAVT/MAUT and within other classical multicriteria methods, weight coefficients are considered as constant/non-random positive numbers. In this case, for uncertainty analysis, as a rule, one-parameter sensitivity analysis to changing the chosen weight coefficient is used. However, extended uncertainty analysis, when weights are not single-valued and are considered as distributed in the intervals given by experts, is justified for most practical multicriteria problems.
Weight coefficients can be assessed with the use of different weighting methods, including swing method(s) for determination of scaling factors in MAVT/MAUT, voting approach for outranking methods and some others (Keeney and Raiffa, 1976; Belton and Stewart, 2002; Malczewski, 1999). Weights of relative importance or scaling factors have uncertainties which are the result of both experts/stakeholders judgments and the weighting method chosen.
In most cases experts can more easily set a range for a scaling factor as opposed to a precise value. For example, to state "the relative value of a swing from worst to best on the second ranked criterion is between 30-60% from a swing from worst to best on the most highly weighted criterion", is easier to do than state this value is equal exactly 45%. The uncertainties of weight coefficients can be a result of both individual and group implementation of a weighting process.
Within ProMAA, the distributions of weights wj in the given variation intervals [wjmin , wjmax] may be used. Therefore, the approach to setting distributed weight coefficients in ProMAA needs a special discussion.
The recommended approach to setting weight coefficients in ProMAA-U is a natural one and corresponds to the steps for assignment of scaling factors as in swing weighting method for MAVT/MAUT:
- weight coefficient w1=1 is assigned for the most highly weighted criterion (let us denote this criterion as C1), taking into account, according to the method, evaluation of increase in overall value as a result of swing from worst to best for each criterion;
- the variation interval [w2min , w2max], 0 < w2min ≤ w2max ≤ 1, is assigned for the weight coefficient w2 of the second ranked criterion (we denote it as C2) based on evaluation of a range for relative value of a swing from worst to best for this criterion in comparison with the corresponding value of swing for the most highly weighted criterion;
- the previous step is repeated for the third, forth, and subsequent criteria;
- the probability distributions (as subjective probabilities or as a result of statistical analysis of expert judgments) for (independent) weight coefficients wj in the given interval [wjmin, wjmax], j=2,...,m, is assigned by experts.
Within the classical MAVT/MAUT methods, the weights, assigned through the swing procedures, are usually normalized according to (6). This seems often to be useful for several reasons, including an interpretation of the importance of weights in percents, or presenting an integrated value/utility function, etc. (Keeney and Raiffa, 1976; Belton and Stewart, 2002; von Winterfeldt and Edwards, 1986). However, in specific cases experts may find it more intuitive to specify a reference criterion whose units are weighted at 1 and against which all other criteria are compared (Belton and Stewart, 2002).
It is evident that a (forced) proportional change of all weights wj, j=1,...,m, (wj→dwj, where d is any real positive number) does not change ranking of alternatives in MAVT/MAUT methods and in ProMAA-U (rank acceptability matrix {Pik} remains the same for distributed or standard/non-distributed type of weights).
In ProMAA-U, according to the current realization within the DECERNS SDSS, the original swing weight coefficients are then automatically normalized to the sum of their mathematical expectations; thus, the sum of mean values for (distributed) weights equals 1. Although, this is not necessary step for ranking alternatives within ProMAA-U, but this is useful for some comparison of ProMAA weights with weights used for other multicriteria methods, where weight normalization is traditionally implemented.