below consists in replacing the original MMP (3) by a search for a non-dominated solu- tion belonging t o a curve g which lies inside the demanded set Q. If Q is defined by ( I ) , g begins a t an attainable reference point ql and ends a t an unattainable dominating one, 92 i.e. g ( 0 ) = q l and g ( l ) = q 2 . The solution thus found belongs t o t h e intersection of
FP( U,O)
and g*:= g(
[ O ; l ] ) , and is non-dominated provided t h a t the setFP(
U,O) divides the demanded set into two parts. The latter condition is fulfilled e.g. when F ( U ) is convex and (6) is satisfied.The algorithm of the search.
The choice of t h e curve g is based on the analysis of the specific properties of ele- ments of g*. Consequently we will consider the curves which satisfy the maximal safety principle, i.e. those for which the probability t h a t the compromise solution chosen will remain within the randomly perturbed demanded set is maximal.
This may be achieved by selecting the curve maximizing the mean value of distance from the boundary of the demanded set. Considering moreover the fact t h a t some criteria may turn o u t t o be redundant leads us t o choosing the so-called ordinal skeleton curve (Gorecki (1981), Wiecek (1984)) as the curve the search should be expected on.
The general algorithm of the search on a curve g may be sketched a s follows:
H . Gorecki, A . Skulirnowski - 66
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Sajety principle...
Step 0 : selection of g , the choice of the algorithm A of detection of a non-dominated point p on g*,
Step 1 :
f;=A (fi-l)ri-l),
Step 2 : check whether
f,
is attainable; set r;:=l iff,
is attainable, otherwise r;:=O,Step 9 :
e , :=
II
fi-fi-1I1
i j ri<ri-, and e , < e o then
p:= fi+ fi- 1
2 stop
else i:=i+l, go to 1.
The result of an execution of the algorithm is a non-dominated solution p. The Pareto- optimality of p is an immediate consequence of the assumption t h a t ql and q2 are separated by the non-dominated surface
F P ( U , O ) .
The uniqueness is assured if g is a linearly ordered subset ofQ
which will be assumed further on. T h e maximal safety of p will be discussed in the following section.In selecting a curve g so t h a t safety principle is satisfied, a crucial role is played by the norm in the criteria space since it determines the value of the distance of the solution chosen from t h e exterior (or, equivalently, boundary) of
Q .
On the other hand, choice of distance influences t h e properties of the probability distribution of finding a non- dominated point along a curve. T h e justification of the choice of the normsIl
or 1, in the criteria space is contained in the following subsection.The algorithm is assumed t o possess the following properties a) A ( f , r ) € g t whenever f ~ g *
b)
I
1 - fI < I f
f - 1 for 1>
1c) the assumed number of iterations of A depends only on the value of
1 I
ql-q2( 1,
noton t h e shape of g*.
T o check whether a point
F,
belonging t o g* is attainable one should examine the existence of solutions t o the equationIn convex cases this may be done a s proposed by Wiecek (1984).
The value of eo must be sufficiently small t o assure the accuracy of the method. T h e recommended value which proved t o be adequate in numerical experiments is
H. Gorecki, A. Skulimowski
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67 - Safety principle. . .
where p , ( Q ) is the diameter of the projection of Q on the i-th axis in the criteria space.The choice of a distance in the criteria space
We will s t a r t this section from the following definition:
Definition 1 : A curve g : ( ~ , l ] + E is linearly ordered iff
where is the partial order in
E.
The set of all linearly ordered curves linking the points x and y will be denoted by L ( x , y).Further on we will require t h a t the following property of the line-search for a non- dominated solution, imposed by the choice of the class searching algorithms, is satisfied.
Assumption 1
.
Let x and y be two elements of the demanded set Q such t h a t x L e y . Then the probability of finding a non-dominated point on a linearly ordered curve con- necting z and y is constant and does not depend on the choice of this curve.On the other hand we may require t h a t the search on a curve gives better results when the curve is longer which can be formalized as
Assumption 2. T h e probability of detecting a non-dominated point on a curve linking two points is proportional t o the length of this curve.
Consequently, the Assumptions 1 and 2 imply t h a t all linearly ordered curves linking two fixed points in the criteria space should have the same length. T h e above require- ments imply the limitations in the choice of the distance and the derived differential form (element of distance) which defines the length of the curve.
It is easy t o see t h a t the following statement is true.
Proposition 1 : T h e Assumptions 1 and 2 are fulfilled by the length of the curve gen- erated by the ll or
I,
norm, i.e. bywhere g= (gl ,...,gN) is the curve considered, and z ( l l ) is the element of length associated t o the L1 norm. T h e length of g for
I,
norm is defined similarly t o (10).Proof of the Proposition 1 is given in Gorecki and Skulimowski (1986b), i.e. we prove t h a t
jor each z , ~ E Q , a , b € L ( z , y ) : h l ( a ) = h l ( b ) (11) Observe t h a t only certain distances in
R~
satisfy the above requirement ( l l ) , e.g. i t is not fulfilled by the Euclidean distance.The Assumptions 1 and 2, and the subsequent distance in the criteria space are in compliance with the assumption about the class of algorithms applied for looking for a non-dominated point on a curve, namely we will assume t h a t these algorithms satisfy:
Assumption 3 . T h e a priori imposed maximal number of steps of an algorithm of detect- ing a non-dominated point on a curve g connecting the elements z and y of the criteria space does not depend on the choice of g but on the differences between coordinates of x and y. In particular, i t may be defined a s
H . Gorecki, A . Skulimowski