TOPSIS (Technique for Order Preference by Similarity to the Ideal Solution)


The PROMETHEE method, developed by Brans and Vincke (Brans 1985), belongs to so called outranking (ORT, Outranking Relation Theory) methods (Roy 1996).
ORT approaches imply forming an ordered relation of a given set of alternatives. Outranking methods are based on a pairwise comparison of alternatives for each criterion under consideration with subsequent integration of obtained preferences according to a chosen algorithm. Among outranking approaches, the ELECTRE family of methods developed by Roy (Roy 1996) and the PROMETHEE method developed by Brans and Vincke (Brans 1985) are most used (Belton and Stewart, 2002; Figueira et al, 2005).
PROMETHEE is based on utilization of a performance matrix {zi(a)} (where zi(a) is an evaluation of alternative a against criterion i) and a chosen preference function fi(x), 0 ≤ fi(x) ≤ 1, with specified indifference and preference thresholds (all the main types of preference functions are used in DecernsSDSS). PROMETHEE method determines the intensity of preference for alternative a over alternative b, Pi(a,b)= fi(zi(a) - zi(b)), and the preference index, P(a,b),
(10)
where weights wi reflect the relative importance of criteria and meet the requirements of (3). According to the features of preference functions fi(x), if Pi(a,b) > 0, then Pi(b,a)=0. Preference indices are used for determination of positive outranking flow Q+(a):
(11)
and negative outranking flow Q-(a):
(12)
summed over all alternatives ba.
According to the PROMETHEE 1 method, a outranks b if Q+(a) ≥ Q+(b) and Q-(a) ≤ Q-(b); a is indifferent to b if Q+(a) = Q+(b) and Q-(a)=Q-(b); a and b are incomparable if Q+(a) > Q+(b) and Q-(b) < Q-(a), or Q+(b) > Q+(a) andQ-(a) < Q-(b). Thus, PROMETHEE 1, like some other outranking methods, does not presuppose that a single best alternative can be identified, since some alternatives may be incomparable. The PROMETHEE 2 method is based on the "net flow" criteria Q(a):
(13)
and it may be used for a complete ranking of alternatives: alternative a outranks b if Q(a) > Q(b).
PROMETHEE, like other outranking methods, is considered an attractive and transparent method, although both positive and negative flows depend on the complete set of alternatives under consideration (Belton 2002; Brans 1985). However, a drawback of outranking is that "indifference" and "preference" thresholds - though often based on expert knowledge - are essentially arbitrary, and the relationship representing which alternatives outrank depends on selection of those thresholds. One way to analyze the robustness and check consistency between thresholds is to manipulate the thresholds.
Outranking techniques allow inferior performance on some criteria to be compensated for by superior performance on others (Belton 2002; Roy 1996). They do not necessarily, however, take into account the magnitude of relative underperformance in a criterion versus the magnitude of over-performance in another criterion. Therefore, outranking models are known as "partially compensatory".