51 Partial Order Scoring

51.1  Special Terminology and Definitions

a)  Binary relation. A binary relation R in V is such that for any two elements a, b Î V

a R b

For every binary relation R in V there exists a converse relation R⁺ in V such that

b R+ a

b)  Reflexive and anti-reflexive relation. A relation R is reflexive when

a R a      for all a Î V

otherwise R is anti-reflexive.

c)  Symmetric and anti-symmetric relation. A relation R is symmetric when R = R⁺, that is when

a R b Û b R a      for all a, b Î V

otherwise R is anti-symmetric.

d)  Transitive relation. A relation R is transitive when

a R b Ù b R c Þ a R c      for all a, b, c Î V

e)  Equivalence relation. A relation R defined on a set of elements V is an equivalence relation when it is:

• reflexive,
• symmetric, and
• transitive.

f)  Strict partial order relation. A relation R is called a strict partial order when it satisfies the conditions:

• a R b and b R a cannot hold simultaneously, and
• R is transitive.

g)  Partially ordered set. A set V is called a partially ordered set if a strict partial order relation "«" is defined on it. The fundamental properties of a partially ordered set are:

• a « b Ù b « c Þ a « c      for all a, b, c Î V
• a « b and b « a cannot hold simultaneously.

h)  Ordered set. A set V is called an ordered set if there are two relations " » " and "«" defined on it and they satisfy the axioms of ordering:

• for any two elements a, b Î V, one and only one of the relations a » b, a « b, b « a holds,
• " » " is an equivalence relation, and
• "«" is a transitive relation.

i)  Subset of elements dominating an element a.

G (a) = {   g   |   g Î V ;   a « g   }

j)  Subset of elements dominated by an element a.

L (a) = {   l   |   l Î V ;   l « a   }

k)  Subset of comparable elements.

C (a) = G (a) ÈL (a)

Note that G (a) ÇL (a) = Æ.

l)  Strict dominance. An element b strictly dominates an element a if

a « b    and    not (b « a)

It can also be said that "b is strictly better than a", or that "a is strictly worse than b".

51.2  Calculation of Scores

{ x1, x2, ..., xi, ..., xv }

and a priority list associated to them by

{ p1, p2, ..., pi, ..., pv } .

The "partial order relation" constructed on the basis of this collection of variables,

a « b    for any cases a and b

is equivalent to the condition

x1 (a) £ x1 (b),   x2 (a) £ x2 (b),   ...,   xv (a) £ xv (b)

where x_i (a) and x_i (b) denote values of the i^th variable for cases a and b respectively.

When comparing two cases, the variables of highest priority (lowest LEVEL value) are considered first. If they unambiguously determine the relation, the comparison procedure ends. In the situation of equality, the comparison is continued using variables of the next priority level. This procedure is repeated until the relation is determined at one of the priority levels, or the end of the variable list is reached.

For each case a from the analyzed set, the program calculates:

N   (a) = the number of cases strictly dominating the case a N (a) = the number of cases equivalent to the case a N (a) = the number of cases strictly dominated by the case a

and then one (or two) of the following scores:

s1 (a) = S  N (a)  /   æ è N (a) + N (a) + N   (a) ö ø r1 (a) = S - s1 (a) s2 (a) = S   æ è N (a) + N (a) ö ø  /   æ è N (a) + N (a) + N   (a) ö ø r2 (a) = S - s2 (a) s3 (a) = S  N (a) /  N r3 (a) = S   æ è N   (a) + N (a) ö ø  /  N s4 (a) = S   æ è N (a) + N (a) ö ø  /  N r4 (a) = S   N   (a)  /  N

where

N = total number of cases in the analyzed set S = the value of the scale factor (see the SCALE parameter).

The values of the ORDER parameter select the score(s) as follows:

ASEA : r3 (a) DEEA : s4 (a) ASCA : r4 (a) DESA : s3 (a) ASER : s1 (a), r1 (a) DESR : s1 (a), r1 (a) ASCR : s2 (a), r2 (a) DEER : s2 (a), r2 (a) .

51.3  References

Hunya, P., A Ranking Procedure Based on Partially Ordered Sets, Internal paper, JATE, Szeged, 1976.