The document discusses using a genetic algorithm to optimize the mass design of a single-stage helical gear unit. The objective is to minimize the total mass of the gear unit, which is calculated based on the volumes and densities of its various components. The design must satisfy 38 constraints related to gear ratios, stresses, clearances, manufacturability, and component life. A genetic algorithm is applied to search for the design variable values that minimize mass subject to all constraints.

1. Genetic Algorithm
in Engineering optimization
Bhushan R Mohite
MIS: 111010025
B.Tech (Mech)
COEP

2. What is Optimization
Optimization is a process that finds a best, or optimal solution for a problem from
a number of alternative solutions in a search space.
The optimization problems are centred around three factors.
1. An objective function which is to be minimized or maximized:
1) In manufacturing, we want to maximise the profit or minimize the cost.
2)In designing an automobile component, we want to minimize the weight
2. A set of unknowns or variables that affect the objective function examples
1)In manufacturing,the variables are amount of resources used or the time spent.
2)In designing component the variables are shape and dimensions of the panel
3.A set of constraints that allow the unknowns to take on only certain values;
1)In manufacturing one constrain is that all time variables to be non negative
2)In component design,we want to constrain the maximum stress
An optimisation problem is defined as finding values of variables that maximise
or minimise the objective function while satisfying the constraints

3. Statement of Optimization problem
• An optimization or a mathematical programming problem can be stated as
follows :
Find X =
𝑥1
𝑥2
⋮
𝑥 𝑛
which maximises/minimises f(X)
subject to constraints
gj(X) ≤ 0 j=1,2,…m
lj X = 0 j=1,2,…p
where, f (X) is termed the objective function;
X is called as design vector
𝑥1,𝑥2 … . . 𝑥 𝑛 are design variables collectively represented as design vector X
and g(X) and l(X) are constraints

4. Design space

5. Applications of Optimization
 Optimization, in its broadest sense, can be applied to solve any engineering
problem.
 Some typical applications from different engineering disciplines are:
1. Design of aircraft and aerospace structures for minimum weight
2. Design of civil engineering structures such as frames, foundations, bridges,
towers, chimneys, and dams for minimum cost
3. Shortest route taken by a salesperson visiting various cities during one tour
4. Minimum-weight or minimum stress design for automobile components.
5. Design of pumps, turbines, and heat transfer equipment for maximum efficiency
6. Allocation of resources among several activities to maximize the benefit
7. Design of optimum pipeline networks for process industries
8. Selection of machining conditions in metal-cutting processes for minimum
production cost
9. Optimum design of linkages, cams, gears, machine tools, and other mechanical
components

6. Optimized components

7. Different Important Optimization
Techniques
• There is no single method available for solving all optimization problems
efficiently. Hence a number of optimization methods have been developed
for solving different types of optimization problems
1. Genetic Algorithms (GA)

8. GENETIC ALGORITHM
“The Gene is by far the most
sophisticated program around.”
-Bill Gates, Business Week,
June 27, 1994

9. INTRODUCTION
• Each species tries to adapt itself with the gradually changing environment
on the earth.
• The knowledge that each species gains is encoded in its chromosomes
automatically, which undergoes transformations due to genetic
recombinations.
• Over a period of time, these changes to the chromosomes give rise to
more fit species that are more likely to survive, and so have a greater
chance of passing their improved characteristics on to future generations.
• Otherwise the species may get extinct.
• Genetic Algorithm is search technique that mimics the process of natural
evolution and helps us to find out the fittest solution of a problem exactly
the same way simulating selection, crossover, mutation etc.
• Often it is used in optimization and search problems.

10. History
• 1859:Natural selection, genetic inheritance and evolution was first
described by Charles Darwin
• 1954:Computer simulations of evolution started as early as in 1954 with
the work of Nils Aall Barricelli at the Institute of Advanced Study,
Princeton.
• 1957:Fraser developed first Genetic Algorithm.
• 1960:John Holland developed GA on his own way.
• 1966:The idea of ‘Evolutionary computation’, as a part of artificial
intelligence first introduced by Lawrence J. Fogel.
• 1970-75:Genetic Algorithm is Developed by John Holland and his students,
University of Michigan (1970’s) almost in the form today we are using it.
Thus called the father of ‘GA’.
• 1980 -. In the late 1980s, General Electric started selling the world's first
genetic algorithm product for industrial processes

11. Motivation & Necessity
MOTIVATION:
Nature is the motivation. Nature automatically finds out the best, fittest
individual from a species those who are likely to survive more easily than
others. Exactly the same way we can find out the best solutions among a
no of solutions by using GA from a given search space for a specific
problem.
NECESSITY:
Often we need a search or an optimization process which-
1) Can deal with complex multidimensional discontinuous problem..
2) Is easy and efficient to find global maxima or minima.
3) Faster.
4) Can be implemented by computer.
5) Can be used in Huge search space defined for those variables.
GA satisfies the following criteria.

12. Why would we use genetic algorithms?
Isn’t there a simple solution we learned in
Calculus?
• Solving engineering problems can be complex and a time consuming
process when there are large numbers of design variables and constraints.
Hence, there is a need for more efficient and reliable algorithms that solve
such problems. The improvement of faster computer has given chance for
more robust and efficient optimization methods. Genetic algorithm is one
of these methods.
• Newton-Raphson and it’s many relatives and variants are based on the use
of local information
Perfect for a parabola

13. Where Newton-Raphson Fails
• A local method will only find local
extrema If we start our search here
We’ll end up here

14. What is a Genetic Algorithm
• A genetic algorithm is a search and optimization method developed by
mimicking the evolutionary principles and chromosomal processing in natural
genetics.
• A GA begins its search with a random set of solutions usually coded in binary
string structures.
• Each possible solution is tested against the problem by using a fitness function
which is directly related to objective function of optimization problem.
• Best solutions are retained and are modified to new population of solutions by
applying three operators similar to natural genetic operators- selection,
crossover and mutation
• A GA works iteratively by successively applying these three operators in each
generation till best solution is found or maximum number of generations are
produced.

15. Biological background
• Chromosome(individual): A set of genes. Chromosome is a solution in
form of genes.
• Gene: A part of chromosome. A gene contains a part of solution. It
determines the solution. E.g. 16743 is a chromosome and 1, 6, 7, 4 and 3
are its genes.
• Population: Total number of chromosomes present
• Fitness: Fitness is the value assigned to an individual chromosome. It is
based on how far or close a individual is from the solution. Greater the
fitness value better the solution it contains.
• Fitness function: Fitness function is a function which assigns fitness value
to the chromosome. It is problem specific.

16. Simple Genetic Algorithm
1. [Start] Generate random population of n chromosomes(i.e suitable
solutions for the problem)
2. [Fitness] Evaluate the fitness f(x) of each chromosome x in the population.
3. [New population] Create a new population by repeating following steps
until the new population is complete.
(a) [Selection] Select two parent chromosomes from a population
according to their fitness(better fitness, greater chance to get selected)
(b) [Crossover]Perform crossover to form new offspring chromosomes.
(c) [Accepting]place new offspring in new population
4. [Replace] Use new generated population for a further run of the algorithm
5. [Test]If a solution is found that satisfies minimum criteria, stop, and return
the best solution in current population
6. [Loop]Go to step 2

17. Encoding solutions into
chromosomes
101011101001010111010000100110
698 349 350
x y z
Before a genetic algorithm can be put to work on any problem,a method is needed
To encode potential solutions to that problem in a form so that a computer can process.
-One common approach is to encode solutions as binary strings
For example
A chromosome looks like: 1100010100011100011101011001
Gene1 Gene2 Gene3 Gene4
A chromosome in some way should contain information about solution which it represents
It thus requires encoding.

18. Representation
Chromosomes could be:
– Bit strings (0101 ... 1100)
– Real numbers (43.2 -33.1 ... 0.0 89.2)
– Permutations of element (E11 E3 E7 ... E1 E15)
– Lists of rules (R1 R2 R3 ... R22 R23)
– Program elements (genetic programming)
– ... any data structure ...

19. 13SELECTION
 Selection operator selects good chromosomes from the
population as parents to crossover and produce
offspring
 Parents are selected according to their fitness
 There are many methods to select the best
chromosomes
1. Roulette Wheel Selection
2. Rank Selection
3. Tournament Selection
4. Steady State Selection
5. Elitism
 The better the chromosomes are, the more chances to
be selected they have

20. Evaluate the fitness
• Each individual created in initial population is assigned a fitness value
which is related to the objective function value.
• Particular solution is ranked against all solutions in the population
• Measures the quality of the particular solution
• A larger fitness value will give the individual a higher probability of being
selected as parent chromosome

21. Roulette-Wheel Selection
 Probability of any individual to be
selected is exactly proportional to
its fitness
Here, fitness of 3rd individual is higher
than others so the wheel will choose
3rd individual more than others
No. String Fitness % of Total
1 111110 62 31
2 01010 10 5
3 1001100 76 38
4 110000 24 12
5 11100 28 14
Total 200 100 Roulette wheel game
 Roulette wheel selection also known as fitness proportionate selection is
a genetic operator used for selecting fitter solutions for recombination

22. Crossover
 Crossover is a genetic operator that combines two chromosomes
(parents) to produce a new chromosome(offspring)
 Applied to create a better string
 A random pair of chromosomes is selected as parents; then a cross site
is selected at random on string length; then the genes are swapped
between the two strings at crosspoint to form offspring.
 MAY NOT ALWAYS yield a better or good solution
 Since parents are good, probability of the child being good is high
 If offspring is not good, it will be removed in the next iteration during
“Selection”

23. 13
1. Single Point Crossover:
 A cross-site is selected randomly along the length of
the mated strings
 Bits next to the cross-site are exchanged
 If good strings are not created by crossover, they will
not survive beyond next generation
Offspring1
Offspring2Parent2
Parent1
Strings before Mating
Contd..
0 1 0 1 0 0 0 1
1 0 1 1 1 1 1 1
0 1 0 1 0 1 1 1
1 0 1 1 1 0 0 1
Strings after Mating
Single Point Crossover
 A cross-site is selected randomly along the length of the mated
strings
 Then it copies everything before this point from first parent and
then everything after this point from second parent
 If good strings are not created by crossover, they will not
survive beyond next generation
Offspring1
Offspring2
Strings before
Mating
0 1 0 1 0 1 1 1
1 0 1 1 1 0 0 1
Strings after Mating
Parent2
Parent1
0 1 0 1 0 0 0 1
1 0 1 1 1 1 1 1

24. 13
Two-point Point Crossover
 It randomly selects two crossover points within parent
chromosomes
 Then the contents bracketed by these two points are
interchanged to form offsprings
Offspring1
Offspring2
Strings before
Mating
0 1 1 1 0 1 0 1
1 0 0 1 0 0 1 1
Strings after
Mating
Parent2
Parent1
0 1 1 1 0 0 0 1
1 0 0 1 0 1 1 1

25. Mutation
• Mutation operator alters one or more gene values in a chromosome
• Helps to maintain diversity in population.
• Providing new genetic materials it helps to find out global maxima
rather than local maxima
• Mutation of a bit involves flipping it, changing 0 to 1 and vice versa
1011011111
1000000000Offspring2
Offspring1 Offspring1
Offspring2
1011001111
1010000000
mutation
Mutated offspringOriginal offspring

26. Genetic algorithm overview

27. GA Softwares
• GAOT- Genetic Algorithm Optimization Toolbox in Matlab
• JGAP is a Genetic Algorithms and Genetic Programming component
provided as a Java framework
• Generator is another popular and powerful software running on
Microsoft Excel

28. Optimal mass design of single stage
Helical gear unit using Genetic algorithm
The problem of minimum mass design of simple and multi-stage spur gear
trains has been a subject of considerable interest, since many high-
performance power transmission applications (e.g. automotive,aerospace,
machine tools, etc.) require low mass. For this reason, in this study,
minimization of single-stage helical gear unit mass is the objective function
and it is defined as shown in Eq. (1).
F(x) = M1 + M2 + M3
=
where: V11i is the volume of the reducer’s housing body, V12i represents the
volume of the reducer’s housing cover, V21j is the volume of the pinion, V22j is
the volume of the wheel (2), V31k is the volume of the input shaft, V32k is the
volume of the output shaft, ρ1 is the density of cast iron (i.e. 7.2·10-6
mm3/kg), ρ2 is the density of steel (i.e. 7.85·10-6 mm3/kg).

29. a) Completely assembled reducer; b) –exploded view of the reducer’s housing body; c) –sub-assembly of
the input/output reducer’s shafts; d) –exploded view of the reducer’s housing cover

30. Constraints
C1–The relative difference between the required and the actual gearing ratio must be within the range of [-
2.5% … +2.5%]. C2–The Hertzian contact pressure on the teeth of gears must not exceed a specified value. C3,
4–The bending stress on the teeth of gears must not exceed a specified value. C5, 6–The teeth on pinion (1)
and wheel (2) must not be undercut. C7, 8–The top land of the teeth on pinion (1) and wheel (2) must not
vanish. C9. The contact ratio of the gearing must be greater than a specified value. C10–The addendum
coefficient of the wheel (2) should be in the range of [-0.6, 1]. C11-16 A set of measurability constraints of the
gears. C17, 18–Constraints related to gears manufacture. C19–The numbers of teeth of both gears must be
relative primes. C20, 21–The input and output shaft ends must have sufficient diameter
step to allow the mounting of a belt wheel. C22, 23–The inside diameter of the tapered roller bearings on the
input and output shafts must be less than the mounting diameter of the seal. C24, 25–Geometrical constraint
related to the space required by the outside ring of the tapered roller bearings on the input and output shafts.
C26–The minimum distance between adjacent tapered roller bearings must be greater than 15 mm. C27, 28
Equivalent von Misses maximum stresses experienced by the input and output shafts should be less then the
allowable bending stresses. C29–31 The bending strains on the input and output shaft must be below certain
threshold values to enable the correct functioning of the gearings and the bearings. C32, 33–The fatigue life
safety factors on the two shafts must not fall below a specified value. C34, 35–The torsional strains in the shafts
must be below a threshold value. C36, 37–The service life of the tapered roller bearings must exceed a
specified value. C38–41 The shearing and crushing stresses must not exceed a specified value on the keys and
keyways of the input and output shafts. C42, 43–The shearing and crushing stresses must not exceed a
specified value on the key and keyway for mounting the wheel (2) on the output the shaft. C44. The operating
temperature of the reducer must not exceed a specified value. C45–Lubrication constraint—the margin
between the minimum and maximum allowable lubricant levels should be no less than 10 mm.

31. Let now us consider the following optimal design problem. A 6.3 kW single-
stage helical gear unit) is to be designed for minimum mass and a service life
of 8000 h, given an input speed of 750 RPM and a transmission ratio of 3.15;
when the pinion (1) and the wheel (2) were made of quenched and tempered
alloy steel 42CrMo4 and 41Cr4 respectively
• Initial population: 50;
• Crossover probability: 0.75;
• Mutation probability: 0.15;
• Total trials: 10000000
Running the algorithm led to a single-stage helical gear unit weighing
32.969 kg.
Inputs

32. Results
The main characteristics of the single-stage helical gear unit (traditional
design and optimal solutions for both cases) are shown side-by-side
symbol Traditional design Optimal design
Gear characteristics Gear characteristics
z1 33 21
z2 104 67
i12 3.1515 3.1904
mn (mm) 1.75 2.25
aw (mm) 125 100
xn1 0.669 0.15
xn2 0.8238 -0.2531
b1 (mm) 54 57
b2 (mm) 50 53
df1 (mm) 57.00666 42.8889
df2 (mm) 184.5745 145.865
dw1 (mm) 60.2189 47.7272
dw2 (mm) 189.5745 152.2727
da1 (mm) 64.5504 53.0099
da2 (mm) 192.1183 155.986
σh (Mpa) 558.0433 744.3064
σhp(Mpa) 593.9152 745.8103
σf1(Mpa) 134.9747 180.1192
σf2(Mpa) 149.297 185.4118
σfp1(Mpa) 479.9631 569.0124
σfp2(Mpa) 467.9097 493.7047

33. Results
Shaft characteristics Shaft characteristics
des1(mm) 25 25
des2(mm) 35 38
d1es1(mm) 30 26
d1es2(mm) 40 40
dtrb1(mm) 30 30
dtrb2(mm) 40 40
σes1(Mpa) 36.6887 41.3545
σes2 (Mpa) 37.9119 0.0019
δ11(mm) 0.0209 0.0014
δ12(mm) 0.0006 4.226
δ21(mm) 0.0009 5.6873
Overall dimensions of reducer Overall dimensions of reducer
L(mm) 388 347
W(mm) 326 308
H(mm) 132 203
Objective function Objective function
Obj(kg) 37.6619 32.9693
As it can be observed from Table, the optimal design solution offers a mass reduction of
12.2% . If a large series of production is considered, the advantages are obviously and
manufacturing costs are significantly diminished. For example at 10 reducers
produced, 1 is for free taking into account the material

34. Conclusion
• The results obtained show that the genetic algorithm to provide better
solution than those obtained from traditional design method. It can be
concluded that the genetic algorithm that can be successfully and
efficiently used for design optimisation.
• Genetic Algorithms can be applied to virtually any problem that has a
large search space.
• It is a useful search and optimization algorithm. However, several
improvements can be made in order that Genetic Algorithm could be
more generally applicable.

35. References:
1. Charles Darwin “Origin of Species by Means of Natural Selection”
2. Genetic algorithm in search and optimization:The technique and applications by Kalyanmoy Deb(IITK)
3. DEB K., JAIN S., Multi-speed gearbox design using multi-objective evolutionary algorithms, ASME
Journal of Mechanical Design
4. An Introduction to Genetic Algorithm by Melanie Michell
5. An Introduction to Genetic Algorithm by S.N.Shivanandan and S.N. Deepa,Springer
6. Genetic Algorithms and Evolutionary Computation by Adam Marczyk
7. Comparison of genetic algorithm amd particle swarm optimisation” by Dr. Karl O. Jones
8. Neural Networks, Fuzzy Logic, And Genetic Algorithm:by S.Rajasekaran and G.A.Vijayalakshmi Pai
9. Genetic Algorithm and Engineering Optimization by Mitsuo Gen and Runwei Cheng
10. Genetic Algorithms-in search, optimisation and machine learning by David E. Goldberg
11. Engineering Optimization by Singiresu S. Rao
12. Overview: http://en.wikipedia.org/wiki/Genetic_algorithm
13. Roulette-Wheel Selection at a glance: http://www.edc.ncl.ac.uk/highlight/rhjanuary2007g02.php/
14. Methods of Selection and Crossover : http://en.wikipedia.org/wiki/Crossover_(genetic_algorithm)
15. Vose, Michael (1999). The Simple Genetic Algorithm: Foundations and Theory. Cambridge
16. Optimal mass design of single stage helical gear unit using genetic algorithms-Ovidiu buiga, Claudiu-
Ovidiu popa(Proceedings of romanian academy)
17. Renner G Ekarta , Genetic algorithms in computer aided design, Computer-Aided Design

36. QUESTIONS???

37. THANK YOU