The lexicographic decision function

In this paper the lexicographic decision process is presented in a unified way. We construct a lexicographic decision function using a universal preference function and a unary function. This construction incorporates the different outranking approaches,the lexicographic decision process and the utility based decision making models. Finally we consider the connection of the lexicographic decision method and the Arrow paradox. 1.Introduction In this section we describe the concept of the lexicographic method. We use the terminology of P.C. Fishburn, see [6]. The general concept of a finite lexicographic order involves a set I = {1, 2, ..., n} and an order relation ≺i on a nonempty set Xi for each i ∈ I. We let ∼i denote the symmetric complement of ≺i so that xi ∼i yi if and only if (xi ≺i yi or yi ≺i xi) does not hold. With x = (x1, x2, ..., xn) and y = (y1, y2, ..., yn) , y precedes x lexicographically under the natural order < on I and with respect to the ≺i or x < L y for short, iff {i : i ∈ I and (xi ≺i yi or yi ≺i xi)} is nonempty, and xi ≺i yi for the first (smallest) i in this set. For this reason a lexicographic order <is also referred to as an order by first difference. An example of a lexicographic order arises from the alphabetical order of words in a dictionary or lexicon. To show this let I = {1, 2, ..., n} , let Xi = A = {∅, a, b, ..., z} with ∅ ≺i a ≺i b ≺i ... ≺i z for each i , take n as large as the longest listed word , and let the English word α1α2...αm withm ≤ n correspond to (α1, α2, ..., αm, ∅, ..., ∅) in A . Then <on the subset of A which corresponds to the ”legitimate ” words orders these words in their natural alphabetical order. For example, ”as” precedes ”ask” since (a, s, ∅, ..., ∅) < (a, s, k, ∅, ..., ∅) which is to say that a ∼1 a, s ∼2 s, ∅ ≺3 k. In multicriteria decision making the idea of the lexicographic decision consists of a hierarchy or ordered set of attributes or criteria. Decision alternatives are examined initially on the basis of the first or most important criterion, If more than one alternative is ”best” or ”satisfactory” on this basis, then these are compared under the second most important criterion and so forth. The principle of order by first difference says that one alternative is ”better” than another iff the first is ”better” than the second on the most important criterion on which they differ. So let x and y be two alternatives (actions) and c1, c2, ..., cn be different criteria, xi and yi are the utilities (evaluations) of x and y. We identify x and y with their evaluation vector x = (x1, x2, ..., xn) , y = (y1, y2, ..., yn). Then ≺i is the order relation according to ci on the set of alternatives. We say that xi ∼i yi iff the alternatives x and y are indifferent according to the ci, and we say that x < y iff xi ∼i yi for i ∈ {1, 2, ..., k − 1} where 0 ≤ k − 1 ≤ n− 1 and xk ≺k yk in other words the alternative y is preferred to the alternative x , according to the criteria ck. The lexicographic decision method is a well adaptable method. It can arrange data of arbitrary scales, and it is suitable to evaluate a set of considerable alternatives. This method does not require the weight of criteria and in spite of its simlicity always arranges the alternatives, Rapcsák [18]. Some decision procedures have lexicographic decision rules to prevent ties, Temesi [20]. Sequential screening procedures illustrate another common application of the lexicographic idea. Candidates or alternatives are first screened under a given criterion (perhaps with the use of a test or an interview) and separated into ”rejects” and ”others”. In terms of ≺1 of the set of candidates, x ∼ y whenever both x and y are ”rejects”