MODULE - 9 LECTURE NOTES GENETIC ALGORITHMS INTRODUCTIONnptel.ac.in/courses/105108081/module9/lecture37/lecture… · · 2017-08-04Water Resources Systems Planning and Management:

Water Resources Systems Planning and Management: Advanced Topics – Genetic Algorithms

D Nagesh Kumar, IISc, Bangalore

1

M9L2

MODULE - 9 LECTURE NOTES – 2

GENETIC ALGORITHMS

INTRODUCTION

Most real world optimization problems involve complexities like discrete, continuous or

mixed variables, multiple conflicting objectives, non-linearity, discontinuity and non-convex

region. The search space (design space) may be so large that global optimum cannot be found

in a reasonable time. The existing linear or nonlinear methods may not be efficient or

computationally inexpensive for solving such problems. Various stochastic search methods

like simulated annealing, evolutionary algorithms (EA) or hill climbing can be used in such

situations. EAs have the advantage of being applicable to any combination of complexities

(multi-objective, non-linearity etc) and also can be combined with any existing local search

or other methods. Various techniques which make use of EA approach are Genetic

Algorithms (GA), evolutionary programming, evolution strategy, learning classifier system

etc. All these EA techniques operate mainly on a population search basis. In this lecture

Genetic Algorithms, the most popular EA technique, is explained.

CONCEPT

EAs start from a population of possible solutions (called individuals) and move towards the

optimal one by applying the principle of Darwinian evolution theory i.e., survival of the

fittest. Objects forming possible solution sets to the original problem is called phenotype and

the encoding (representation) of the individuals in the EA is called genotype. The mapping of

phenotype to genotype differs in each EA technique. In GA which is the most popular EA,

the variables are represented as strings of numbers (normally binary). If each design variable

is given a string of length ‘l’, and there are n such variables, then the design vector will have

a total string length of ‘nl’. For example, let there are 3 design variables and the string length

be 4 for each variable. The variables are 17,4 321 xandxx . Then the chromosome

length is 12 as shown in the figure.

0 1 0 0 0 1 1 1 0 0 0 1

1x 2x 3x

An individual consists a genotype and a fitness function. Fitness represents the quality of the

solution (normally called fitness function). It forms the basis for selecting the individuals and

thereby facilitates improvements.



2

M9L2

The pseudo code for a simple EA is given below

i = 0

Initialize population P0

Evaluate initial population

while ( ! termination condition)

{

i = i+1

Perform competitive selection

Create population Pi from Pi-1 by recombination and mutation

Evaluate population Pi

}

A flow chart indicating the steps of a simple genetic algorithm is shown in figure 1.



3

M9L2

Fig. 1

The initial population is usually generated randomly in all EAs. The termination condition

may be a desired fitness function, maximum number of generations etc. In selection,

individuals with better fitness functions from generation ‘i' are taken to generate individuals

of ‘i+1’th

generation. New population (offspring) is created by applying recombination and

mutation to the selected individuals (parents). Recombination creates one or two new

individuals by swaping (crossing over) the genome of a parent with another. Recombined

individual is then mutated by changing a single element (genome) to create a new individual.

Generate Initial Population

Start

Encode Generated Population

Evaluate Fitness Functions

Meets

Optimization

Criteria?

Best

Individuals

Yes

Stop

Selection (select parents)

Mutation (mutate offsprings)

Crossover (selected parents)

No

R

E

G

E

N

E

R

A

T

I

O

N



4

M9L2

Finally, the new population is evaluated and the process is repeated. Each step is described in

more detail below.

PARENT SELECTION

After fitness function evaluation, individuals are distinguished based on their quality.

According to Darwin's evolution theory the best ones should survive and create new offspring

for the next generation. There are many methods to select the best chromosomes, for example

roulette wheel selection, Boltzmann selection, tournament selection, rank selection, steady

state selection and others. Two of these are briefly described, namely, roulette wheel

selection and rank selection:

Roulette Wheel Selection:

Parents are selected according to their fitness i.e., each individual is selected with a

probability proportional to its fitness value. In other words, depending on the percentage

contribution to the total population fitness, string is selected for mating to form the next

generation. This way, weak solutions are eliminated and strong solutions survive to form the

next generation. For example, consider a population containing four strings shown in the

Table 1. Each string is formed by concatenating four substrings which represents variables

a,b,c and d. Length of each string is taken as four bits. The first column represents the

possible solution in binary form. The second column gives the fitness values of the decoded

strings. The third column gives the percentage contribution of each string to the total fitness

of the population. Then by "Roulette Wheel" method, the probability of candidate 1 being

selected as a parent of the next generation is 28.09%. Similarly, the probability that the

candidates 2, 3, 4 will be chosen for the next generation are 19.59, 12.89 and 39.43

respectively. These probabilities are represented on a pie chart, and then four numbers are

randomly generated between 1 and 100. Then, the likeliness that the numbers generated

would fall in the region of candidate 2 might be once, whereas for candidate 4 it might be

twice and candidate 1 more than once and for candidate 3 it may not fall at all. Thus, the

strings are chosen to form the parents of the next generation.



5

M9L2

Table 1

Candidate Fitness value Percentage of total fitness

1011 0110 1101 1001 109 28.09

0101 0011 1110 1101 76 19.59

0001 0001 1111 1011 50 12.89

1011 1111 1011 1100 153 39.43

Total 388 100

Rank Selection:

The previous type of selection may have problems when the fitnesses differ very much. For

example, if the best chromosome fitness is 90% of the entire roulette wheel then the other

chromosomes will have very few chances to be selected. Rank selection first ranks the

population and then every chromosome receives fitness from this ranking. The worst will

have fitness 1, second worst 2 etc. and the best will have fitness N (number of chromosomes

in population). By this, all the chromosomes will have a chance to be selected. But this

method can lead to slower convergence, because the best chromosomes may not differ much

from the others.

CROSSOVER

Selection alone cannot introduce any new individuals into the population, i.e., it cannot find

new points in the search space. These are generated by genetically-inspired operators, of

which the most well known are crossover and mutation.

Crossover can be of either one-point or two-point scheme. In one point crossover, selected

pair of strings is cut at some random position and their segments are swapped to form new

pair of strings. In two-point scheme, there will be two break points in the strings that are

randomly chosen. At the break-point, the segments of the two strings are swapped so that

new set of strings are formed. For example, let us consider two 8-bit strings given by

'10011101' and '10101011'.

Then according to one-point crossover, if a random crossover point is chosen after 3 bits from

left and segments are cut as shown below:

100 | 11101

101 | 01011

and the segments are swapped to form



6

M9L2

10001011

10111101

According to two-point crossover, if two crossover points are selected as

100 | 11 | 101

101 | 01 | 011

Then after swapping both the extreme segments, the resulting strings formed are

10001101

10111011

Crossover is not usually applied to all pairs of individuals selected for mating. A random

choice is made, where the probability of crossover being applied is typically between 0.6 and

0.9.

MUTATION

Mutation is applied to each child individually after crossover. It randomly alters each gene

with a small probability (generally not greater than 0.01). It injects a new genetic character

into the chromosome by changing at random a bit in a string depending on the probability of

mutation.

Example: 10111011

is mutated as 10111111

It is seen in the above example that the sixth bit '0' is changed to '1'. Thus, in mutation

process, bits are changed from '1' to '0' or '0' to '1' at the randomly chosen position of

randomly selected strings.

REAL-CODED GAs

As explained earlier, GAs work with a coding of variables i.e., with a discrete search space.

GAs have also been developed to work directly with continuous variables. In these cases,

binary strings are not used. Instead, the variables are directly used. After the creation of

population of random variables, a reproduction operator can be used to select good strings in

the population.

AREAS OF APPLICATION IN WATER RESOURCES

Water distribution systems

Hydrological modeling

Watershed Management



7

M9L2

Groundwater modeling

Reservoir Operation

ADVANTAGES AND DISADVANTAGES OF EA:

EA can be efficiently used for highly complex problems with multi-objectivity, non-linearity

etc. It provides not only a single best solution, but the 2nd

best, 3rd

best and so on as required.

It gives quick approximate solutions. EA methods can very well incorporate with other local

search algorithms.

There are some drawbacks also in using EA techniques. An optimal solution cannot be

ensured on using EA methods, which are usually known as heuristic search methods.

Convergence of EA techniques are problem oriented. Sensitivity analysis should be carried

out to find out the range in which the model is efficient. Also, the implementation of these

techniques requires good programming skill.

MULTI-OBJECTIVE EVOLUTIONARY ALGORITHMS

Genetic algorithms are efficient in solving multiobjective problems. Considerations in Multi-

Objective Evolutionary Algorithms (MOEAs) implementation are

1. Preserve non-dominated points – elitism

2. Progress towards points on Pareto front

3. Maintain diversity of points on Pareto Front (phenotype) and/or Pareto Optimal

solutions (genotype)

4. Provide decision maker a limited number of Pareto Front (PF) points.

Non-dominated solutions are always better than 1st-level dominated solutions, which are

always better than 2nd-level dominated solutions, etc. Within the same level of dominance,

solutions which are isolated are better than solutions that are clumped together.



8

M9L2

Flow chart of Multi-objective Genetic Algorithm with Elitism

Fig. 2

Multi Objective Genetic Algorithms for Optimal Reservoir Operation – Case Study

Bhadra Reservoir – A multi-purpose reservoir located in the district of Chickmangalur,

Karnataka state, India; 75o38

’20

’’ E longitude and 13

o42

’ N

latitude.



9

M9L2

Fig. 3 Schematic diagram of Bhadra reservoir Project

Multi Objective Reservoir Operation Model

Objective functions:

1. Minimize irrigation deficit (f1)

212

1

212

1 t

trtr

t

tltl IRDIRDSQDV ,,,,

2. Maximize hydropower production (f2)

12

1

321

t

tbtbtrtrtltl HRkHRkHRkE ,,,,,,

Subject to constraints in

(i) Reservoir storage continuity constraint

tttttttt OERRRISS ,,, 3211

(ii) Storage bounds

maxmin SSS t

(iii)Turbine capacity limits

max,,,

max,,,

max,,,

333

222

111

EHpR

EHpR

EHpR

tt

tt

tt

(iv) Canal capacity limits



10

M9L2

max,,

max,,

22

11

CR

CR

t

t

(v) Irrigation demands

ttt

ttt

DRD

DRD

,,,

,,,

maxmin

maxmin

222

111

(vi) Water quality requirements

tt MDTR ,3

Pareto optimal solution for reservoir operation

0 2 4 6 8 10 12

x 104

150

160

170

180

190

200

210

220

230

gen=50

gen=200

gen=500

Fig. 4 Improvement in Pareto optimal front over the iterations. f1 is annual squared irrigation

deficit; f2 is hydropower generated MkWh

Model Application

MOGA model is solved for three different inflow scenarios into the reservoir

Scenario 1: Mean monthly inflows – 0.5 * SD

Scenario 2: Mean monthly inflows

Scenario 3: Mean monthly inflows + 0.5 * SD

where SD is the standard deviation of monthly flows

f1

f2



11

M9L2

Fig. 5 Pareto optimal front, showing the trade-off between irrigation ( f1) and hydropower (

f2) for different inflow scenarios. f1 = sum of squared irrigation deficits, (Mm3)2; f2 =

hydropower generated, (MkWh)



12

M9L2

Fig. 6 Reservoir operating policies for different inflow scenarios, showing the initial storages

for different situations, viz., equal priority case, irrigation only priority case and hydropower

only priority case.



13

M9L2

Fig. 7 Optimal release policy obtained for equal priority case, showing releases in Mm3 for

Left bank canal (R1), Right bank canal (R2) and River bed (R3) for different inflow

scenarios.

Advantages of MOEAs:

MOEAs are easy to adopt and can provide efficient solutions for multi-objective problems.

They are capable of handling nonlinear objectives/ constraints, disconnected Pareto-fronts,



14

M9L2

non-convex decision space. They can find solutions to extremely complex and high

dimensional real-world applications in reasonable computation time. They have high

potential for multi-objective optimization of hydrological and water resources problems.

Documents

MODULE - 9 LECTURE NOTES GENETIC ALGORITHMS INTRODUCTIONnptel.ac.in/courses/105108081/module9/lecture37/lecture… · · 2017-08-04Water Resources Systems Planning and Management: