Selection operator and minimizing the fitness function in Genetic Algorithms - matlab

Question

I'm developing a nurse rostering tool im matlab using genetic algorithms, without using GA toolbox.
The individual is a weekly schedule and is represented as a 2-d array with rows equals to the number of nurses and seven columns because it is a weekly schedule.
The fitness function takes the entire population and returns an array with size equal to the population size containing fitness values.
The fitness function should be minimized so the best schedule is the one having the lowest fitness value.
My fitness function is:
function fitness_values =Fitness_Function( thePopulation)
%UNTITLED Summary of this function goes here
% Detailed explanation goes here
[Ar1 Ar2 popsize num_nur] = Return_Data( 0,0,0,0 );
[prev_sched OffArr]=Return_Data1(0,0);
constraints=cell(popsize,1);
fitness_values=zeros(popsize,1);
size=[1 7];
c1=zeros(popsize,1);
c1values=cell(popsize,1);
W1=0.25; %hard
W2=0.25; %hard
W3=0.25; %hard
W4=0.125; %soft
W5=0.125; %soft
for i=1: popsize
c1values{i}=zeros(size);
end
% Checking Constraint c1 (the difference between night and day shifts in
% each day of the schedule
for i=1:popsize
for j=1:7
day_sum=0;
night_sum=0;
for k=1:num_nur
if thePopulation{i}(k,j)==1
day_sum=day_sum+1;
elseif thePopulation{i}(k,j)==2
night_sum=night_sum+1;
end
end
abs_diff=abs(day_sum-night_sum);
c1values{i}(1,j)=abs_diff.^2;
end
c1(i)=sum(c1values{i}(1,:));
%celldisp(c1values);
%defining the array that will hold the result of multiplying the number of
%violations with the correspondig weight,a cell array where each cell
%containts num_nur rows and 4 columns for c2, c3,c4 and c5.
nurse_fitness=zeros(num_nur,1);
for in=1:popsize
constraints{in}=zeros(num_nur,4);
end
for j=1:num_nur
v2=0;
v3=0;
v4=0;
%check violations with the previous schedule(the last day of the
%previous schedule with the first day of the evaluated schedule
% c2
if prev_sched(j,7)==2 && thePopulation{i}(j,1)==1
v2=v2+1;
end
% c3
%check the last day of the previous schedule
if prev_sched(j,7)==1 && thePopulation{i}(j,1)==1 && thePopulation{i}(j,2)~=3
v3=v3+1;
%check the last 2 days of the previous schedule
elseif prev_sched(j,6)==1 &&prev_sched(j,7)==1 && thePopulation{i}(j,2)~=3
v3=v3+1;
end
%c4
%check the last day of the previous schedule
if prev_sched(j,7)==2 && thePopulation{i}(j,1)==3 &&thePopulation{i}(j,2)==1
v4=v4+1;
%check the last 2 days of the previous schedule
elseif prev_sched(j,6)==2 &&prev_sched(j,7)==3 && thePopulation{i}(j,2)==1
v4=v4+1;
end
%check violations of constraints c2,c3 and c4 in the
%evaluated schedule
for k=1:6
%check violations of c2 N->N or N->O (hard)
if thePopulation{i}(j,k)==2 && thePopulation{i}(j,k+1)==1
v2=v2+1;
end
end
%check violations of c3 D->D->O (hard)
for k=1:5
if thePopulation{i}(j,k)==1 && thePopulation{i}(j,k+1)==1 && thePopulation{i}(j,k+2)~=3
v3=v3+1;
end
%check violations of c4 N->O->N or N->O->O (soft)
if thePopulation{i}(j,k)==2 && thePopulation{i}(j,k+1)==3 && thePopulation{i}(j,k+2)==1
v4=v4+1;
end
end
constraints{i}(j,1)=v2*W2;
constraints{i}(j,2)=v3*W3;
constraints{i}(j,3)=v4*W4;
%check violations of c5 (perefrences of each nurse)
offdays=find(thePopulation{i}(j,:)==3);
%disp(offdays);
%disp(OffArr(j,:));
%find intersection between the perefreces and the days off in the
%schedule of each nurse
inters=intersect(offdays,OffArr(j,:));
num_inters=length(inters);
if(length(offdays)==1)
%for head nurse
if num_inters==1
constraints{i}(j,4)=0;
else
constraints{i}(j,4)=3*W5;
end
else
penalty=3-num_inters;
constraints{i}(j,4)=penalty*W5;
end
nurse_fitness(j)=sum(constraints{i}(j,:));
end
%calculating the fitness value for the whole schedule
fitness_values(i)=W1*c1(i)+sum(nurse_fitness);
end
end
I'll summarize how it works: it takes a cell array (the population) each cell contains a schedule represented as matrix having rows =number of nurses and 7 columns (weekly schedule),,the problem has 3 hard constraints and 2 soft constraints, so the fitness will check the violation of these constraints in each schedule,,the violation is penalized by multiplying the number of violations in each nurse with the corresponding wheight of the constraint so the final fitness value is the sum of penalty values of each nurse. finally the fitness value of the the evaluated schedule is saved in an array of fitness values (the same index where the evaluated scheule is stored in the population array).
My question is what is the suitable selection operator to select parents for crossover and mutation operators?

Answer 1

Points missing from your question:
what is the meaning of a value in the array of an "individual"?
what are the constraints that can be violated?
Does an "individual" mean both a Genotype and a Phenotype?
I think you could also illustrate these with a simple example, and for others' best understanding can you please use the GA terminology?
Up to this point I can only give a general answer. Generally I think it is better to search through non-violating individuals. What I would do is not to use a complicated fitting function. I would rather have the phenotype always a non-violating solution that can be quickly calculated from a (possibly violating) genotype. Maybe the genotype should not grasp the whole problem, just give a starting point for a simple allocating algorithm that does the phenotype.
If you have non-violating chromosomes, the mutations should be lightly affecting, leading to similar solutions. Your chromosomes will be potentially some sort of permutations and the mutations could be some transpositions on these. The crossover-born children should jump-away from the parent solutions, preserving some of their characteristics. For permutation-type chromosomes you can find standard crossover operators.