1. Multi-objective Optimization
Introduction
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc

2. Objectives
To discuss and formulate Multi-objective optimization problems
To define pareto-optimal or non-inferior solutions
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc

3. Introduction
In a real world problem single objective and multiple constraints problems are not common
The rigidity provided by the General Problem (GP) is, many a times, far away from the practical design problems.
In many of the water resources optimization problems maximizing aggregated net benefits is a common objective.
At the same time maximizing water quality, regional development, resource utilization, and various social issues are other objectives which are to be maximized.
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc

4. Introduction…
There may be conflicting objectives along with the main objective like irrigation, hydropower, recreation etc.
Multiple objectives or parameters have to be met before any acceptable solution can be obtained.
The word “acceptable” implicates that there is normally no single solution to the problems of the above type
Methods of multi-criteria or multi-objective analysis are not designed to identify the best solution, but only to provide information on the tradeoffs between given sets of quantitative performance criteria.
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc

5. Multi-objective Problem
A multi-objective analysis is a vector optimization
The objective function is a vector
A multi-objective optimization problem with inequality (or equality) constraints may be formulated as:
Find (1)
which minimizes f1(X), f2(X), … , fk(X) (2)
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc

6. Multi-objective Problem…
subject to j= 1, 2 , … , m (3)
where k denotes the number of objective functions to be minimized and
m is the number of constraints.
The objective functions and constraints need not be linear
Linear objective functions and constraints - Multi-objective Linear Programming (MOLP).
For the problems of the type mentioned above the very notion of optimization changes
Good trade-offs are to be found rather than a single solution.
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc

7. Multi-objective Problem…
The most commonly adopted notion of the “optimum” proposed by Pareto is
In other words a solution vector X is called optimal if there is no other vector Y that reduces some objective functions without causing simultaneous increase in at least one other objective function.
A vector of the decision variable X is called Pareto Optimal (efficient) if there does not exist another Y such that for i = 1, 2, … , k with for at least one j.
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc

8. Non-inferior Solutions
The various objectives of a multi-objective problem are often
non-commensurate and
may be in conflict.
Trade-offs are necessary to reach consensus or compromise solutions
If there are two solutions, one solution may achieve higher levels of certain objectives and lower levels of other objectives
For example, a multipurpose reservoir serving for irrigation and hydropower generation can have two conflicting objectives:
maximize benefits from power generation and
maximize irrigation benefits.
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc

9. Non-inferior Solutions…
There are three objectives
i, j, k.
Direction of their increment is also indicated.
Surface (which is formed based on constraints) is efficient because no objective can be reduced without a simultaneous increase in at least one of the other objectives k i j
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc

10. Non-inferior Solutions…
f1(x)
f2(x) A B C F E
Feasible region
Indifference curves D
Infeasible region
Transformation curve
Values of objective functions and decision variables can be visualized by plotting them an enveloping curve
Solution X’ is considered non-inferior if and only if there does not exist another solution X” such that fi(x”) ≥ fi(x’) where i = 1,2,…, no:of objective functions.
The collection of all non-inferior solutions is termed as the set of non-inferior solutions.
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc

11. Non-inferior Solutions…
f1(x)
f2(x) A B C F E
Feasible region
Indifference curves D
Infeasible region
Transformation curve
Each point in the feasible region gives a particular set of values for the decision variable, satisfying all the constraints
Each point in the transformation curve gives the maximum value of objective function given a particular value to the other objective function.
Point F should not be considered, since one can get a better value of f1(x) by moving towards B, or a better value of f2(x) by moving towards C.
Portion BECD is the efficient one since a further increase in any objective function is not possible without the decrease of the other.
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc

12. Non-inferior Solutions…
f1(x)
f2(x) A B C F E
Feasible region
Indifference curves D
Infeasible region
Transformation curve
Solutions on AB portion are inferior in the current case.
These solutions are of interest only when some objectives are to be minimized.
Final goal is to identify the plans which lie in the transformation curve which is the set of efficient trade-offs
Curve corresponds to a specific trade-off or marginal rate of substitution between the objectives. The points B, E and C are points with three different marginal rate of substitution between the two objective functions.
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc

13. Non-inferior Solutions…
Indifference curves are the curves of equal preference of a decision-maker.
Optimal plan or the ‘best compromise solution’ is the one which achieves the highest level of preference of the decision-maker, i.e., point E in the figure.
Slope of the transformation curve at point E gives the optimal trade-off.
In order to identify the non-inferior solutions in the decision space as well the objective space in which the best compromise solution lie, methods used are
(i) weighing method
(ii) constraint method
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc

14. Example Maximize subject to
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc

15. Example… Solution: Two decision variables and two objective functions
Feasible region can be found by plotting all the extreme points and the objective function values at each point
Feasible region in the decision space and the set of noninferior solutions
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc

16. Example…
Feasible region in the objective space and the set of noninferior solutions
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc

17. Example…
Set of noninferior solutions: Four extreme points, Z(x2), Z(x3), Z(x4) and Z(x5)
Among these the identification of any particular noninferior solution as the best compromise solution is not possible without any preference information
If any preference is available as represented by an indifference surface (as shown in the figure), then one of the point (here point ] can be identified as the best compromise solution.
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc

18. Thank You Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc