This document discusses multi-objective optimization and Pareto multi-objective optimization. It provides examples of multi-objective optimization problems with two or more competing objectives that must be optimized simultaneously. The key concepts covered include Pareto optimal solutions, which define the best trade-offs between objectives and are non-dominated by other solutions. Methods for solving multi-objective optimization problems include traditional approaches that aggregate objectives and Pareto techniques using genetic algorithms and multi-objective evolutionary algorithms.

1. CHAPTER-5
MULTI OBJECTIVE OPTIMIZATION
AND
PARETO MULTI OBJECTIVE OPTIMIZATION
OPTIMIZATION TECHNIQUES
AND
OPERATIONS RESEARCH
PRESENTED BY:
ADITYA DESHPANDE (S-42)
GUIDED BY:
PROF. G. N. KOTWAL SIR

2.  Involve more than one objective function that are to be minimized
or maximized
 Answer is set of solutions that define the best trade off between
competing objectives
 Basic difference is in the single-objective optimization problem,
the superiority of a solution over other solutions easily determined
by comparing their objective function values.
2
CONCEPT OF MOOP

3.  In multi-objective optimization problem, the goodness of a solution is
determined by the dominance
 Real-world problems have more than one objective function,
each of which may have a different individual optimal solution.
 Different in the optimal solutions corresponding to different
objectives because the objective functions are often conflicting
(competing) to each other.
 Set of trade-off optimal solutions instead of one optimal solution,
generally known as “Pareto-Optimal” solutions (named after
Italian economist Vilfredo Pareto (1906)).
 No one solution can be considered to be better than any other with
respect to all objective functions. The non-dominant solution
concept.
3
CONCEPT OF MOOP(cont…)

4. 4
MOOP EXAMPLES

5.  Other example fields:
1. Economics
2. Optimal control and design
3. Process optimization
4. Radio resource management
5. Electric power systems
5
MOOP EXAMPLES

6.  Non-dominated - given two objectives, a non-dominated solution
is when none of both solution are better than the other with
respect to two objectives. Both objectives are equally important.
e.g. speed and price.
 Dominated: when solution a is no worse than b in all objectives,
and solution a is strictly better than b in at least one objective,
then solution a dominate solution b.
6
Non Dominated & Dominated Solutions

7.  F(x) = [F1(x), F2(x),...,Fm(x)],
 Min F(x), subject to Gi(x)=0, i=1,...,ke; Gi(x)≤0, i=ke+1,...,k; l≤x≤u.
 Simple car design example: two objectives - cost and accident rate –
both of which are to be minimized.
7
MOOP MATHS
A, B, D - One objective
can only be improved at
the expense of at least
one other objective!

8.  Need to save money
 Or
 Have shortest flying time
 If we compare tickets A & B, we can’t say that
either is superior without knowing the relative
importance of Travel Time vs. Price.
 comparing tickets B & C shows that C is
better than B in both objectives, so we can say
that C “dominates” B.
 as long as C is a feasible option, there is no
reason we would choose B. 8
Ticket Travel
Time
(hrs)
Ticket
Price
(Rs)
A 10 17000
B 9 20000
C 8 18000
D 7.5 23000
E 6 22000
Flying Example

9.  D is dominated by E
 rest of the options (A, C, & E) have a trade-off associated with
Time vs. Price, so none is clearly superior to the others
9
Flying Example
Plane Ticket Options
0
1000
2000
3000
4000
5000
0 5 10 15 20 25
Flight Time (hrs)
Price($)
AE
D
C
B
Feasible Region

10.  Solutions that lie along the line are non-dominated solutions
 those lie inside the line are dominated because there is always another
solution on the line that has at least one objective that is better.
 The line is called- Pareto front and solutions on it are Pareto-optimal.
 All Pareto-optimal solutions are non-dominated.
 Imp to find the solutions as close as possible to the Pareto front & as
far along it as possible.
10
f1
f2
Feasible
RegionPareto
Front

11. 1. Traditional Approach
A.Aggregating approaches
i. Weighting Method
ii. Constraint Method
iii.Goal Programming
iv.Minmax Approach
B. VEGA (Vector Evaluated Genetic Algorithm)
2. Pareto Techniques
B. Genetic Algorithm (GA)
C. Multi-Objective Evolutionary Algorithm (MOEA)
11
Two approaches for MOOP

12. 12
RESPONSE SURFACE OPTIMIZATION
MOOP IN ANSYS

13. 13
MOOP IN ANSYS

14. 14
MOOP IN MATLAB

15. 15
The gamultiobj solver attempts to create a set of Pareto optima for a
multiobjective minimization.
 You may optionally set bounds or other constraints on variables.
 gamultiobjuses the genetic algorithm for finding local Pareto optima.
 As in the ga function, you may specify an initial population, or have the solver
generate one automatically.
Multiobjective optimization is, therefore, concerned with the generation and
selection of noninferior solution points.
Noninferior solutions are called Pareto optima. A general goal in multiobjective
optimization is constructing the Pareto optima using GAMULTIOBJ
MOOP IN MATLAB

16. 16

17. 17

18.  The function f(x) is called the objective function. This is the
function you wish to minimize
 The inequality x1
2+x2
2≤1 is called a constraint. Constraints limit
the set of x over which you may search for a minimum.
 Non linear problem only using inequality
 So need to add equality by ourselves 18
MOOP IN MATLAB (ROSEN BROCK FUNCTION)

19.  STEPS
1. Define Rosen brock function in editor…….. save
2. Define constraint in other editor window……..save
Here, use [ ] blank to add equality
3. Use optimization app>>>>fmincon-constrained nonlinear minimization
4. Give objective function, starting point, constraint function
START………………
FIND FINAL POINT WITH MINIMUM OB. FUNCTION
19
MOOP IN MATLAB (ROSEN BROCK FUNCTION)

20. THANK YOU20