$\mathbf{r}\mathbf{i}$. 
$$\left({\bar{5}}\right)$$
$$\mathbf{r}$$
$$(6)$$
Genetic Crossover Operators 195 The capacity constraint must be met, so q(rk) ≤ d max ∀rk ∈ R (5)
w(vi) measures the time window violation of customer vi, vi ∈ V .

w(vi) = max **max(0;** avi − t
a
vi
); max(0;t
$$t_{v_{i}}^{a}-b_{v_{i}}))$$
vi − bvi)(6)
The fact, that the time window of customer vi**is violated is indicated by** u(vi):
$$u(i)=\left\{\begin{array}{l l l}{{1}}&{{:}}&{{w(i)>0}}\\ {{0}}&{{:}}&{{w(i)=0}}\end{array}\right.$$
0 : w(i) = 0 (7)
The objective functions are given by:

$$g_{1}(R)=\sum_{k=1}^{|R|}t(r_{k})$$
$$\left(7\right)$$
$$({\boldsymbol{\delta}})$$
$$\left({\mathfrak{g}}\right)$$
t(rk**) (8)**
$$g_{2}(R)=\mid R\mid$$
g2(R) =| R | (9)
$$g_{3}(R)=\sum_{i=1}^{N}w(v_{i})$$ $$g_{4}(R)=\sum_{i=1}^{N}u(v_{i})$$
$$(10)$$
$$(11)$$

The objective functions (8) and (9) minimize the costs associated with the solution. Function (8) minimizes the length of the routes, travelled by the vehicles. In addition to that, function (9) minimizes the number of used vehicles. To maximize the provided service it is desired to minimize occurring time window violations. The objective functions (10) and (11) express this circumstance by the measured deviations in time units and by the number of violated time windows. According to the definitions (1) and (2) given in section 2.1, the goal **is to**
find all R ∈ P.