Multi-objective Optimization
Introduction
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc
Objectives
 To discuss and formulate Multi-objective optimization
problems
 To define pareto-optimal or non-inferior solutions
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Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc
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.
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Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc
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.
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Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc
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
which minimizes
5
 x1 
x 
 2
 . 
X  
.
.
 
 xn 
(1)
f1(X), f2(X), … , fk(X)
(2)
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc
Multi-objective Problem…
subject to
g j ( X )  0,
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.
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Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc
Multi-objective Problem…
 The most commonly adopted notion of the “optimum” proposed by Pareto is
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.
 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.
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Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc
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.
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Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc
Non-inferior Solutions…
 There are three objectives
i, j, k.
k
 Direction of their increment is also indicated.
i
j
 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
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Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc
Non-inferior Solutions…
 Values of objective functions and
Infeasible region
B
decision variables can be visualized by
plotting them an enveloping curve
A
f1(x)
Indifference curves
E
F
C
 Solution X’ is considered non-inferior
if and only if there does not exist
Feasible region
Transformation
curve
another solution X” such that fi(x”) ≥
fi(x’) where i = 1,2,…, no:of objective
D
f2(x)
functions.
 The collection of all non-inferior solutions is termed as the set of non-inferior
solutions.
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Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc
Non-inferior Solutions…
 Each point in the feasible region gives a
Infeasible region
B
particular set of values for the decision
variable, satisfying all the constraints
A
f1(x)
Indifference curves
E
F
C
 Each point in the transformation curve
gives the maximum value of objective
Transformation
curve
Feasible region
function given a particular value to the
other objective function.
D
f2(x)
 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.
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Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc
Non-inferior Solutions…
 Solutions on AB portion are inferior in
Infeasible region
B
the current case.
 These solutions are of interest only
A
f1(x)
Indifference curves
E
F
C
when some objectives are to be
minimized.
Feasible region
 Final goal is to identify the plans
which lie in the transformation curve
Transformation
curve
D
f2(x)
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.
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Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc
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
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Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc
Example
Maximize
subject to
Z1 x   10 x1  3x2
Z 2 x   2 x1  6 x2
g1  x  :  x1  x 2  2  0
g 2 x  :
g 3 x  :
g 4 x  :
g 5 x  :
g 6 x  :
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x1  x 2  7  0
x1  5  0
x2  4  0
x1  0
x2  0
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc
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
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Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc
Example…
Feasible region in the objective space and the set of noninferior
solutions
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Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc
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.
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Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc
Thank You
Water Resources Planning and Management: M5L1
D Nagesh Kumar, IISc