The **Interactive Evolutionary Algorithm Approach** is based on
the classical evolutionary algorithm
first described by I. Rechenberg [5], [6],
and H.P. Schwefel [8]. This model of the evolution
is suitable for optimization problems in general, but
is not directly suitable for simulating an imprecise ordinary differential
equation. The
algorithm is called interactive because the objective function assumes
that the fuzzy values are interactive during the evaluation of the
chromosomes.

One possibility to decode the problem by evolutionary algorithms is by expanding the chromosome to represent the problem set more then once, theoretically infinite size. Then the differential equation solutions represented by these sets build the overall minimum/maximum curve. The drawback of this approach is the increasing number of calculations necessary.

The **Interactive Evolutionary Algorithm Approach** runs each
-cut level and codes
the parameters and the initial values of the differential equation
just once by the chromosome.
For each chromosome the differential equation is simulated and
the solution space represented by y-t value pairs is stored in a database.
The first time
the empty database is filled with the first calculated y-t values.
Later the calculated values are saved
when they are smaller/bigger than the actual minimum/maximum value
stored in the database.
When no single problem set of initial values and differential
equation parameters exists, the *Interactive
Evolutionary Algorithm Approach* jumps from
one possible problem set to the other recording the
overall minimum/maximum value in the database.

Fig. 1 visualizes the principle of the approach taken in this work using the graphical notation developed by Rechenberg [6]. From an initial population; parent population 1..

Both the minimum and the maximum should be optimized. Therefore the evolutionary process must run twice for every -cut level. Contrary to the classic evolutionary algorithm is the addition of the database which records the absolute minimum/maximum values of the evaluation process.

- Representation of the ODE Parameter Set
- Mutation of Chromosomes
- Objective Function
- Detailed Example: 1st-Order Imprecise Differential Equation