# Introduction

In the Scheduling on Unrelated Parallel Machines problem, the goal is to find an jobs/machines assignment to minimize the overall makespan. In other words, the goal is to have the best balance between machines.

# Problem data

In our problem, we’ll consider n jobs to be assigned on m machines.

## Processing time

The jobs processing time will be manage as follow :

## Job assignment

The jobs/machines assignment will be manage as follow : If the job j is schedule on machine i then Xij = 1, else Xij = 0.

# Genetic algorithms

## Introduction

In a genetic algorithm, a population of chromosomes is evolved toward better solutions. Each chromosome is defined by its genes. For each chromosome, you should be able to calculate it’s score, also called fitness.

To find better solutions, the process is: 1- Evaluation: Sort the population based on chromosomes scores (fitness). 2- Selection: Choose the best chromosomes to generate the next population (natural selection). 3- Crossover: Mate the chromosomes between them by mixing their genome. 4- Mutation: As in a natural environment, some genes are changed arbitrarily.

## Example

The goal is to give a practical idea of the genetic algorithm operations. We’ll consider a problem with 2 machines (m=2) and 4 jobs (n=4).

### Population

Let’s generate 4 chromosomes randomly :

### Evaluation

Evaluation of the generated chromosomes :

### Selection

Select only the bests chromosomes, here we’ll choose to keep 75% of the sorted population :

### Crossover

1 – Choose two random chromosomes in the selected ones (the best ones). 2 – Merge these two chromosomes by mixing their genome. 3 – Store the new generated chromosome in the new population. 4 – Repeat the crossover operation until the new population is fully generated.

### Mutation

The mutation operation is not systematic. Usually, around 1% of the crossover chromosomes will go through a mutation. During this operation, a random gene is arbitrarily changed:

# Code example

`````` __author__ = 'rfontenay'
__description__ = 'Genetic algorithm to solve a scheduling problem of N jobs on M parallel machines'
import random
import time
# ******************* Parameters ******************* #
# Jobs processing times
jobsProcessingTime = [543, 545, 854, 766, 599, 657, 556, 568, 242, 371, 5, 569, 9, 614, 464, 557, 460, 970, 772, 886,
69, 423, 181, 98, 25, 642, 222, 842, 328, 799, 651, 197, 213, 666, 112, 136, 150, 810, 37, 620,
139, 721, 823, 351, 953, 765, 204, 800, 840, 132, 764, 336, 587, 514, 948, 134, 203, 766, 954,
537, 933, 458, 936, 835, 335, 690, 307, 102, 639, 635, 923, 699, 71, 913, 465, 664, 49, 198, 747,
931, 124, 41, 214, 246, 954, 676, 811, 295, 977, 100, 316, 453, 903, 50, 120, 320, 517, 441, 874,
842]
# Number of jobs
n = len(jobsProcessingTime)
# Number of machines
m = 2
# Genetic Algorithm : Population size
GA_POPSIZE = 256
# Genetic Algorithm : Elite rate
GA_ELITRATE = 0.1
# Genetic Algorithm : Mutation rate
GA_MUTATIONRATE = 0.25
# Genetic Algorithm : Iterations number
GA_ITERATIONS = 1000
# ******************* Functions ******************* #
def random_chromosome():
"""
Description :Generate a chromosome with a random genome (for each job, assign a random machine).
Input : -Line 2 of comment...
Output : Random chromosome.
"""
# Jobs assignment : Boolean matrix with 1 line by job, 1 column by machine
new_chromosome = [[0 for i in range(m)] for j in range(n)]
# For each job, assign a random machine
for i in range(n):
new_chromosome[i][random.randint(0, m - 1)] = 1
return new_chromosome
def fitness(chromosome):
"""
Description : Calculate the score of the specified chromosome.
The score is the longest processing time among all the machines processing times.
Input : A chromosome.
Output : The score/fitness.
"""
max_processing_time = -1
for i in range(m):
machine_processing_time = 0
for j in range(n):
machine_processing_time += chromosome[j][i] * jobsProcessingTime[j]
# Save the maximum processing time found
if machine_processing_time > max_processing_time:
max_processing_time = machine_processing_time
return max_processing_time
def crossover(chromosome1, chromosome2):
"""
Description : Crossover two chromosomes by alternative genes picking.
Input : Two chromosome.
Output : One chromosome.
"""
new_chromosome = [[0 for i in range(m)] for j in range(n)]
for i in range(n):
# Alternate the pickup between the two selected solutions
if not i % 2:
new_chromosome[i] = chromosome1[i]
else:
new_chromosome[i] = chromosome2[i]
return new_chromosome
def evolve(population):
"""
Description : Create a new population based on the previous population.
The new population is generated by mixing the best chromosomes of the previous population.
Input : Old population.
Output : New population.
"""
new_population = [[] for i in range(GA_POPSIZE)]
# First : Keep elites untouched
elites_size = int(GA_POPSIZE * GA_ELITRATE)
for i in xrange(elites_size): # Elitism
new_population[i] = population[i]
# Then generate the new population
for i in range(elites_size, GA_POPSIZE):
# Generate new chromosome by crossing over two from the previous population
new_population[i] = crossover(population[random.randint(0, GA_POPSIZE / 2)],
population[random.randint(0, GA_POPSIZE / 2)])
# Mutate
if random.random() < GA_MUTATIONRATE:
random_job = random.randint(0, n - 1)
# Reset assignment
new_population[i][random_job] = [0 for j in range(m)]
# Random re-assignment
new_population[i][random_job][random.randint(0, m - 1)] = 1
return new_population

# ******************* Program ******************* #
# Measure execution time
start = time.time()

# Generate an initial random population
population = [[] for i in range(GA_POPSIZE)]
for i in range(GA_POPSIZE):
population[i] = random_chromosome()

# Sort the population based on the fitness of chromosomes
population = sorted(population, key=lambda c: fitness(c))
# Print initial best makespan
print “Starting makespan = %03d” % (fitness(population))
#Iterate
for i in range(GA_ITERATIONS):
# Sort the population : order by chromosone’s scores.
population = sorted(population, key=lambda c: fitness(c))
#Generate the following generation (new population)
population = evolve(population)

# Print the best fitness and the execution time after iterations
print "Ending makespan = %03d" % (fitness(population))
print "Execution time = %02d s" % (time.time() - start)``````