54
R. Bruggemann and L. Carlsen
1. the selection of g, the starting weight tuple and
2. the manner how s is increasing to model the uncertainty about the values of the
weights.
(a) In an m = 2-system one could for example start with g1 = 0, then the
intervals of increasing [gmin, gmax] have always the same lower boundary
and increasing s influences only gmax.
(b) In an m = 2-system where we start with a g, with g1 = 0.5 increasing values
of s affect both boundaries of [gmin, gmax].
3 Results
3.1 Three Fictitious Systems of Crucial Weights
A distribution of crucial weights may be derived from a MIS. Nevertheless, it is also
possible to suppose a certain gc-distribution as an archetype. The latter strategy is
followed within this subsection.
(a) h(gc(1j,k)) as function of k in the following form:
h (gc(1), k) = 1 − k k = 0, .., 1
(once again: without referring to a specific, empirical MIS).
h(gc(1),k) = 1 − k means that the number of realizations of gc s: ts linear
decreasing (just by construction).l.
Beside a normalization constant the behavior of h(gc(k)) as a function of k can
be described by h = f(k), where k is assumed to vary continuously. Then based on
Eq. 11 the following expression holds:
U(k) =
k
0
f
k ’
dk ’
(12)
With f(k ) = 1 – k , Eq. 12 leads to.
U(k) = N*k*(2 − k), where N is a scaling factor and k’ is the integration variable
of an integral with the limit 0 and k.
Beside the factor N the resulting function for U is as shown in Fig. 4:
(b) A maximum for the gc-distribution is assumed at gc = 0.5 and h(gc(1),k) is
supposed to decrease symmetrically like a parabola. This situation is a more
realistic with respect to decision support. Here the maximum of gc-realizations
is thought of as being in the range around g1 = 0.5. A model for that is
h
gc(1), k = 4 ∗ k ∗ (1 − k) , k = 0, .., 1.
R. Bruggemann and L. Carlsen
1. the selection of g, the starting weight tuple and
2. the manner how s is increasing to model the uncertainty about the values of the
weights.
(a) In an m = 2-system one could for example start with g1 = 0, then the
intervals of increasing [gmin, gmax] have always the same lower boundary
and increasing s influences only gmax.
(b) In an m = 2-system where we start with a g, with g1 = 0.5 increasing values
of s affect both boundaries of [gmin, gmax].
3 Results
3.1 Three Fictitious Systems of Crucial Weights
A distribution of crucial weights may be derived from a MIS. Nevertheless, it is also
possible to suppose a certain gc-distribution as an archetype. The latter strategy is
followed within this subsection.
(a) h(gc(1j,k)) as function of k in the following form:
h (gc(1), k) = 1 − k k = 0, .., 1
(once again: without referring to a specific, empirical MIS).
h(gc(1),k) = 1 − k means that the number of realizations of gc s: ts linear
decreasing (just by construction).l.
Beside a normalization constant the behavior of h(gc(k)) as a function of k can
be described by h = f(k), where k is assumed to vary continuously. Then based on
Eq. 11 the following expression holds:
U(k) =
k
0
f
k ’
dk ’
(12)
With f(k ) = 1 – k , Eq. 12 leads to.
U(k) = N*k*(2 − k), where N is a scaling factor and k’ is the integration variable
of an integral with the limit 0 and k.
Beside the factor N the resulting function for U is as shown in Fig. 4:
(b) A maximum for the gc-distribution is assumed at gc = 0.5 and h(gc(1),k) is
supposed to decrease symmetrically like a parabola. This situation is a more
realistic with respect to decision support. Here the maximum of gc-realizations
is thought of as being in the range around g1 = 0.5. A model for that is
h
gc(1), k = 4 ∗ k ∗ (1 − k) , k = 0, .., 1.