1. What is the evaluation function proposed in the paper?
The evaluation function proposed in the paper consists of a weighted sum of the average number of people who consider that an alternative precedes another, within each dimension, when compared pairwise. The weights correspond to the relative importance of those dimensions in the problem. This function is easy to compute and understand, and possesses properties such as independence, symmetry, uniformity, neutrality, and scale. It can also be formulated in terms of 'wins and losses' in pairwise confrontations, and is a multidimensional version of an extended Borda score. The function is applicable to a wide family of problems and provides a cardinal assessment for multidimensional scenarios.
read more
2. What is the precedence function in Proposition 1?
The precedence function in Proposition 1, denoted as ph, is a vector function that represents the average number of individuals who consider one alternative to precede another in a given dimension. It is defined as p i j (k) = 1 n n k i j + e k i j 2, where i and j are alternatives, k is the dimension, and e k i j represents the number of individuals who consider the quality of both alternatives to match. The precedence score ranges from 0 to 1, indicating the degree of precedence between alternatives in a specific dimension.
read more
3. What properties do evaluation functions (i) violate?
Evaluation function (i) violates the property of independence. This function is given by ph i (P) = q k=1 a k 1 2 j =i p i j (k) + p m j (k), where m is a given value. The function (i) satisfies all properties except independence. Independence refers to the property where the value of one variable does not affect the value of another variable. In this case, the function (i) does not satisfy independence as the value of 'p i j (k)' and 'p m j (k)' are added together, indicating a dependency between these variables. This violation of independence can impact the interpretation of the evaluation function and its application in research, as it may introduce bias or confounding factors that can affect the results and conclusions drawn from the data.
read more