## Numerical Results

The GA described previously is used to search for best feeding polices for the system outlined in Section 2.2. The final culture time and volume were fixed to be 10 days and two liters, respectively. For the above problem, suppose p(g) = [pi(g) P2(g) ■■■ pq(g)]T (2.6)

is the feed rate matrix for the reactor,

Pi(g) = [Pii(g) Pi2(g) ■■■ Pim(g)] (i = 1,2,■■■,q) (2.7)

is the feed rate vector, where g is the sequence number of a generation, q is the number of the individuals of a generation, and m is the number of intervals within 10 days. The matrix of p forms a population in the GA.

Each individual feed rate vector is constrained by the following conditions:

0 < Pij (g) < 0.5 L/day 2 > 10 j Pij (g) (i = 1,2,

>¿=1 Pij(g) c = 12r• •,q; j = 1 ^■■■,m) The performance is measured by the index given as follows: J(0,10) = MAb(10) ■ V(10)

Then the application of the GA to search for the best feed rate profile can be described as follows:

(1) An initial population of feed rate matrix p(0) = [p1(0) p2(0) ■ ■ ■ pq(0)] is formed with randomly selected individuals.

(2) Each individual pi(g) is used to calculate the performance index J(0,10) by solving the non-linear differential equation of the kinetic models. To have the final volume V(10) = 2 L, the initial volume is chosen to be V(0) = 2 - £127=1 Pij(g). If V(0) < 0, we set J(0,10) = 0.

(3) A new generation p(g + 1), with the same individual number of p(g) is formed by means of reproduction, crossover and mutation based on p(g).

(4) The process will stop if g = maximum generation number. Otherwise, g = g +1, go back to Step 2 to continue.