Handling Conflicts in Aspectual Requirements Compositions - Transactions on Aspect-Oriented Software Development - page 150

multiple-objective decision making (MODM) and multiple attribute decision making
(MADM) [27]. In our work we focus on MADM but we will use the more general
notation of criteria instead of attribute, i.e., MCDM. MCDM methods use
mathematical techniques to help decision makers to choose among a discrete set of
alternative decisions. These methods do not try to compute an optimal solution, but to
determine, via various ranking procedures, either a ranking of the relevant actions
(decision alternatives) that is “optimal” with respect to several criteria, or the
“optimal” actions amongst the existing solutions (decisions alternatives) [21].
Two phases are usually needed to rank the alternatives or to select the most
desirable one: (i) the aggregation of the degree of satisfaction for all criteria, per
decision alternative (rating) and (ii) the ranking of the alternatives with respect to the
global aggregated degree of satisfaction [27].
Triantaphyllou [21] warns that there may never be a single MCDM method that
guaranties that a solution (derived ranking of alternatives) is the correct one because
of the subjective assignment of alternative classifications and weights for criteria.
Even within the fuzzy MCDM domain [10] this type of problem remains ill-defined
by nature. It is a hard problem to know the best solution, even under perfect
knowledge of the input data of a MCDM problem.
There are three parameters in an MCDM problem: the set of alternatives to be ranked;
the set of criteria that will be used for classifying (rating) each alternative; and the
weights (importance) attributed to each criterion. The weights represent the relative
importance of that criterion in relation to others in a decision scenario. The higher the
weight, the higher the importance of the criteria is for the decision maker. Usually,
decision problems are represented in a decision matrix, as depicted in Table 2.

Table 2. Decision Matrix

Weight 1 Weight 2 … Weight j

Criteria 1 Criteria 2 … Criteria j

Alternative 1 x 11 x 12 … X 1j

… … … … …

Alternative i x i1 x i2 ... x ij

The normalized mathematical formulation of a MCDM problem is:

⎡ ⎤

D ( A ) = ⊕ ( x ⊗ w ) / w

∑

(1)

⎢ ⎣ ⎥ ⎦

i ij j j

j

j

w is relative importance of criteria, i x rating of the alternative for the

where

respective criteria and are appropriate (to be selected) aggregation operators.

When we are dealing with a simple weighted average aggregation, the

⊕ ⊗

,

⊗

is the

⊕

operator multiplication and is the summation. The best alternative A i is the one

with the highest ranking.
To solve MCDM problems many methods have been proposed [10, 21, 26]: direct
scoring and outranking methods, trade-off schemes, distance based methods, value
and utility functions, interactive methods. Direct scoring techniques are widely used,