
## 3 A Genetic Approach 3.1 Encoding Technique, Initialization

We use a string of N **genes to encode the solutions of the problem. Each gene**
represents a customer of the problem and the sequence of them corresponds to the sequence of visiting the customers. A possible string, e.g. "5 2 1 3 7 9 8 4 6 10" is then partitioned to a set of tours, e.g. r1 = [v5, v2, v1], r2 = [v3, v7, v9, v8], r3 = [v4, v6, v10**], assuring feasibility with respect to the defined**
side constraints (3) and (5) in section 2.3. The decoding technique inserts the next stored customer of the representation in the current tour **as long as the** side constraints are not being violated.

The initial population inherits n pop individuals i**, consisting of randomly generated strings. To assure a sufficient potential of genetic information and avoid**
premature convergence, n pop **is chosen equal to 500.**

## 3.2 Fitness Function

Most investigations on genetic algorithms for multi-objective optimization are based on examining the effectiveness of certain approaches of **fitness functions. As we have a different focus, we restrict ourselves to the use of a single**
function, similar to the the well known fitness function proposed by **Fonseca** and Fleming [14].

For individual i **of the current population, the number of individuals whose**
corresponding objective vectors dominate the objective vector of i is indicated by ξi. All currently not being dominated alternatives receive ξi **= 0.**
The fitness f(i) of individual i **is derived using a linear transformation with**
ξ max **= max**
i(ξi**) and two external given parameters** f min and f max:

 $$f(i)=f^{max}-\frac{\xi_i(f^{max}-f^{min})}{\xi^{max}}\tag{12}$$  The parameters $f^{min}$ and $f^{max}$ are set to $f^{min}=1$ and $f^{max}=100$. 

## 3.3 Genetic Recombination Operators

As we want to study the effectiveness of different crossover operators, several of them are implemented and tested. From the first approaches of **genetic**
crossover operators, improvements were made by incorporating **problem specific knowledge into the recombination process. This is in the case of the**
travelling salesman problem as well as in the case of the vehicle routing problem e.g. done by considering the distances of the customers to each **other and**
selecting between two possible genes the route minimizing part. As we deal with more than one objective, problem specific orientation during the crossover process is not as obvious. We therefore restrict the investigation