Water Resources Systems Planning and Management: Introduction and Basic Concepts: Optimization and
Simulation
1
MODULE – 1 LECTURE NOTES – 3
OPTIMIZATION AND SIMULATION
INTRODUCTION
In the previous lecture we studied the basics of an optimization problem and its formulation
as a mathematical programming problem. In this lecture we look at the various criteria for
classification of optimization problems, economic considerations and challenges in water
resources.
MODELLING TECHNIQUES
The modelling or system analysis techniques were developed during the Second World War
to deploy limited resources in an optimum manner. Since, these techniques were aided for
military operations, these were known as operation research techniques. The popular
operations research techniques include optimization methods, simulation, game theory,
queuing theory etc. Among, these, the popular ones in water resources field are optimization
and simulation.
OPTIMIZATION
Optimization is the science of choosing the best amongst a number of possible alternatives.
There may be number of possible solutions for many engineering problems. It is required to
identify the best through evaluation. The driving force in the optimization is the objective
function (or functions). The optimal solution is the one which gives the best (either maximum
or minimum) solution under all assumptions and constraints. Optimization theory is defined
as the branch of mathematics dealing with techniques for maximizing or minimizing an
objective function subject to linear, non-linear and integer constraints.
An optimization model can be stated as:
Objective function: Maximize (or Minimize)
Subject to the constraints
f(X)
gj(X) ≥ 0,
j = 1,2,..,m
hj(X) = 0,
j = m+1, m+2,.., p
where X is the vector of decision variables, g(X) are the inequality constraints and h(X) are
the equality constraints.
Classification of Optimization Techniques
Optimization problems can be classified based on the type of constraints, nature of design
variables, physical structure of the problem, nature of the equations involved, permissible
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Water Resources Systems Planning and Management: Introduction and Basic Concepts: Optimization and
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value of the design variables, deterministic/ stochastic nature of the variables, separability of
the functions and number of objective functions. These methods are briefly discussed below.
Classification based on existence of constraints
Under this category optimization problems can be classified into two groups as follows:
Constrained optimization problems: which are subject to one or more constraints.
Unconstrained optimization problems: in which no constraints exist.
Classification based on the physical structure of the problem
Based on the physical structure, optimization problems are classified as optimal control and
non-optimal control problems.
(i) Optimal control problems
An Optimal control (OC) problem is a mathematical programming problem involving a
number of stages, where each stage evolves from the preceding stage in a prescribed manner.
It is defined by two types of variables: the control or design and state variables. The control
variables define the system and controls how one stage evolves into the next. The state
variables describe the behavior or status of the system at any stage. The problem is to find a
set of control variables such that the total objective function (also known as the performance
index, PI) over all stages is minimized, subject to a set of constraints on the control and state
variables. An OC problem can be stated as follows:
l
Find X which minimizes f(X) =
f i ( xi , y i )
i 1
Subject to the constraints
qi ( xi , yi )
yi
yi
1
i = 1, 2, …., l
g j (x j )
0,
j = 1, 2, …., l
hk ( y k )
0,
k = 1, 2, …., l
Where xi is the ith control variable, yi is the ith state variable, and fi is the contribution of the
ith stage to the total objective function. gj, hk, and qi are the functions of xj, yj ; xk, yk and xi, yi
respectively, and l is the total number of states. The control and state variables xi and yi can
be vectors in some cases.
(ii) Problems which are not optimal control problems are called non-optimal control
problems.
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Water Resources Systems Planning and Management: Introduction and Basic Concepts: Optimization and
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Classification based on the nature of the equations involved
Based on the nature of equations for the objective function and the constraints, optimization
problems can be classified as linear, non-linear, geometric or quadratic programming
problems. This classification is much useful from a computational point of view since many
predefined special methods are available for effective solution of a particular type of
problem.
(i) Linear programming problem
If the objective function and all the constraints are ‘linear’ functions of the design variables,
the optimization problem is called a Linear Programming Problem (LPP). A linear
programming problem is often stated in the standard form:
x1
x2
Find X = .
.
xn
n
Which maximizes f(X) =
ci xi
i 1
Subject to the constraints
n
aij xi
bj ,
j = 1, 2, . . . , m
i 1
xi
0,
j = 1, 2, . . . , m
where ci, aij, and bj are constants.
(ii) Non-linear programming problem
If any of the functions among the objectives and constraint functions is non-linear, the
problem is called a Non-Linear Programming (NLP) problem. This is the most general form
of a programming problem and all other problems can be considered as special cases of the
NLP problem.
(iii) Geometric programming problem
A Geometric Programming Problem (GPP) is one in which the objective function and
constraints are expressed as polynomials in X. A function h(X) is called a polynomial (with
m terms) if h can be expressed as
h( X )
c1 x1a11 x2a21  xnan1
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c2 x1a12 x2a22  x nan 2
 cm x1a1m x2a2 m  x nanm
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Water Resources Systems Planning and Management: Introduction and Basic Concepts: Optimization and
Simulation
where cj ( j 1,  , m ) and aij ( i 1,, n and j 1,  , m ) are constants with c j
xi
4
0 and
0.
Thus GPP problems can be posed as follows:
Find X which minimizes
N0
n
cj
f(X) =
a
x i ij , cj > 0,
j 1
xi > 0
i 1
subject to
Nk
gk(X) =
n
a jk
j 1
q
xi ijk
0,
ajk > 0,
qijk > 0, xi > 0, k = 1,2,…..,m
i 1
where N0 and Nk denote the number of terms in the objective function and in the kth constraint
function, respectively.
(iv) Quadratic programming problem
A quadratic programming problem is the best behaved non-linear programming problem with
a quadratic objective function and linear constraints and is concave (for maximization
problems). It can be solved by suitably modifying the linear programming techniques. It is
usually formulated as follows:
n
f(X) = c
n
n
q i xi
i 1
Qij xi x j
i 1 j 1
Subject to
n
aij xi
bj ,
j = 1,2,….,m
i 1
xi
0,
i = 1,2,….,n
where c, qi, Qij, aij, and bj are constants.
Classification based on the permissible values of the decision variables
Under this classification, objective functions can be classified as integer and real-valued
programming problems.
(i) Integer programming problem
If some or all of the design variables of an optimization problem are restricted to take only
integer (or discrete) values, the problem is called an integer programming problem. For
example, the optimization problem is to find number of articles needed for an operation with
D Nagesh Kumar, IISc, Bangalore
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Water Resources Systems Planning and Management: Introduction and Basic Concepts: Optimization and
Simulation
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least effort. Thus, minimization of the effort required for the operation being the objective,
the decision variables, i.e., the number of articles used can take only integer values. Other
restrictions on minimum and maximum number of usable resources may be imposed.
(ii) Real-valued programming problem
A real-valued problem is that in which it is sought to minimize (or maximize) a real function
by systematically choosing the values of real variables from within an allowed set. When the
allowed set contains only real values, it is called a real-valued programming problem.
Classification based on deterministic/ stochastic nature of the variables
Under this classification, optimization problems can be classified as deterministic or
stochastic programming problems.
(i) Deterministic programming problem
In a deterministic system, for the same input, the system will produce the same output
always. In this type of problems all the design variables are deterministic.
(ii) Stochastic programming problem
In this type of an optimization problem, some or all the design variables are expressed
probabilistically (non-deterministic or stochastic).
Classification based on separability of the functions
Based on this classification, optimization problems can be classified as separable and nonseparable programming problems based on the separability of the objective and constraint
functions.
(i) Separable programming problems
In this type of problem, the objective function and the constraints are separable. A function is
said to be separable if it can be expressed as the sum of n single-variable functions,
f1 xi , f 2 x2 ,... f n xn , i.e.
n
f (X )
f i xi
i 1
and separable programming problem can be expressed in standard form as :
n
Find X which minimizes f ( X )
f i xi
i 1
n
subject to
g j (X )
g ij xi
bj ,
j = 1,2,. . . , m
where bj is a constant.
i 1
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Water Resources Systems Planning and Management: Introduction and Basic Concepts: Optimization and
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Classification based on the number of objective functions
Under this classification, objective functions can be classified as single-objective and multiobjective programming problems.
(i) Single-objective programming problem in which there is only one objective function.
(ii) Multi-objective programming problem
A multiobjective programming problem can be stated as follows:
Find X which maximizes/minimizes f1 X , f 2 X ,... f k X
Subject to
gj(X)
0,
j = 1, 2, . . . , m
where f1, f2, . . . fk denote the objective functions to be maximized/minimized simultaneously.
SIMULATION
Simulation process duplicates the system’s behaviour by designing a model of the system and
conducting experiments for a better understanding of the system functioning in various
probable scenarios. Simulation reproduces the response of the system to any imposed future
conditions. The main advantage of simulation is its ability to accurately describe the reality.
The operating policies can be tested through simulation before implementing in actual
situations. Often, water resources systems are too complex to be expressed in any analytical
expression. A simulation model duplicates the system’s operation with a defined operational
policy, parameters, time series of flows, demands etc. The design parameters or operation
policy are usually evaluated through the objective function or some reliability measures.
Steps in Simulation
1.
Problem definition: Define the goals of the study
2.
System definition: Identify the water resources system components and its
hydrological aspects. Identify the performance measures to be analysed.
3.
Model design: Understand the behavior of actual system. Decide the model structure
by determining the variables describing the system, its interaction and various
parameters of structures. Decide the inputs (time series of flows, demands of the
system, operation policies etc) and outputs (hydrological variables and design
variables).
4.
Data Collection: Determine the type of data to be collected. New/ Old data is
collected/ gathered.
D Nagesh Kumar, IISc, Bangalore
M1L3
Water Resources Systems Planning and Management: Introduction and Basic Concepts: Optimization and
Simulation
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Validation: Test the model and apply the model to the problem
Classification of Simulation models
Simulation models can be Physical (e.g. a scale model of a spillway), analog (system of
electrical components such as resistors or capacitors arranged to act as an analog to the
hydrological components) or mathematical (action of a system expressed as equations or
logical statements.
Simulation models can either be static (fixed parameters and operational policy) or be
dynamic (takes into account the change in the parameters of the system and the operational
policy with time) in nature.
Since many hydrological variables are stochastic in character, simulation models can be
deterministic or stochastic depending on the way this stochasticity is accounted for.
Simulation models can be statistical or process oriented, or a mixture of both. Pure statistical
models are based solely on data (field measurements). Regression and artificial neural
networks are examples of purely statistical models. Pure process oriented models are based
on knowledge of the fundamental processes that are taking place. In this, calibration using
field data is required to estimate the parameter values in the process relationships. Hybrid
models incorporate some process relationships into regression models or neural networks.
Comparison between Optimization and Simulation
Optimization models eliminate the worst solutions. Simulation tools evaluate the
performance for various configurations of the system; but they are not effective for choosing
the best configuration. Simulation simply addresses ‘what-if’ scenarios – what may happen if
a particular scenario is assumed or if a particular decision is made. The users have to specify
the value of design or decision variables for conducting simulation. Simulation is not feasible
when there are too many alternatives for decision variables, which demands an enormous
computational effort. On the other hand, optimization will determine the best decision; but
the solution is often based on many limiting assumptions. Hence, to take full advantage of
systems techniques, optimization should be used to define a relatively small number of good
alternatives that can later be tested, evaluated and improved by means of simulation.
ECONOMICS IN WATER RESOURCES
Economics in engineering deals with applying economic criteria to select the best solution
from a group of feasible alternatives or evolving the best economic policy for planning and
D Nagesh Kumar, IISc, Bangalore
M1L3
Water Resources Systems Planning and Management: Introduction and Basic Concepts: Optimization and
Simulation
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management of an engineering project. The ranking and selection of alternatives are done
based on the principles of engineering economics. The magnitude of consequences expected
from employing each alternative are assessed and converted into commensurable units for
comparison.
Concepts used in economic analysis:
Cash flow diagram:
Assess the consequences of each alternative and assign a monetary value for each
consequence. The graphic representation of each monetary value with time is called a cash
flow diagram. The benefits are represented as upward arrows and costs as downward arrows.
It is drawn to convert the time stream of monetary value into an equivalent single number.
All cash flows are combined into an equivalent single lump sum at the end of a period. An
example cash flow diagram is shown below. At the beginning, a large expenditure is made.
Benefits are received thereafter every year.
50
50
50
50
50
50
50
Benefits
Costs
40
500
Cash flow diagram
Discount factors:
The amounts at different times have different values. In order to compare them, all monetary
values are converted into equivalent amounts at some definite time using discount factors.
Many discount factors are used as given below.
Compound amount
An amount P invested at the beginning of first year grows to Q at the
factor
end of n years,
Q = P(1+i)n
Present worth factor
Inverse of the above, gives the present value of a future amount,
P = Q/(1+i)n
Sinking fund factor
The amount X that will be received at the end of each year to get Q at
the end of n years,
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Water Resources Systems Planning and Management: Introduction and Basic Concepts: Optimization and
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X = Q i / [(1+i)n -1]
Capital recovery
The amount X that should be invested at the end of each year, if
factor
amount
P
is
invested
at
the
beginning
of
first
year,
X = P i (1+i)n / [(1+i)n -1]
Series compound
The amount Q that will be received at the end of nth year, if an
factor
amount X is invested at the end of each year,
Q = X [(1+i)n -1] / i
Series present worth
The present value of P if an amount X is invested at the end of each
factor
year,
P = X [(1+i)n -1] / i (1+i)n
Sunk cost: The money spent already which has no economic relevance in deciding future
alternatives
Salvage value: The value of the unused life of an element at the end of the period of analysis.
The salvage value, S = I (1 – U/L), where I = initial value, U = unused life and L = total life.
Discounting techniques
Discounting techniques are used to find the feasible one among various alternatives.
Commonly used discounting techniques are (1) Benefit-cost ratio method (2) Present worth
method (3) Rate of return method and (4) Annual cost method. The first two methods are
explained here.
Benefit – Cost (BC) ratio method
BC ratio, R is defined as the ratio of the present worth of benefits and the present worth of
cost. It can be expressed as
n
R
B
C
t 0
n
t 0
Bt
1 i
Ct
t
1 i
t
where Bt and Ct are the monetary values of benefits and costs incurred at time t respectively, i
is the discount rate and n is the life of the project in time steps (years or months or weeks).
The steps to be followed for choosing the best alternative are:
(i)
Calculate the BC value for each alternative
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Water Resources Systems Planning and Management: Introduction and Basic Concepts: Optimization and
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(ii)
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Retain all alternatives with BC>1 and reject the rest. If sets of mutually exclusive
alternatives are involved then go to steps (iii), (iv) and (v).
(iii) Rank the set of mutually exclusive alternatives in the order of increasing cost.
Calculate the BC ratio using incremental cost and incremental benefit of the next
alternative above the least costly alternative.
(iv) Choose the more costly alternative of the incremental BC >1. Otherwise choose the
less costly alternative.
(v)
Repeat the analysis for all alternatives in the order of rank.
Example
Two alternative plans are feasible. The estimated cost of 1st plan is 70 lakhs and the
corresponding benefit is 80 lakhs. The cost for the 2nd plan is 85 lakhs and benefit is 100
lakhs. Which plan should be selected?
Solution
BC ratio for 1st plan = 80/70 = 1.14
BC ratio for 2nd plan = 100/85 = 1.176
Since BC ratios are >1, according to steps (iii) and (iv), rank the plans based on cost.
Plan 1 is ranked 1 and plan 2 is ranked 2.
Now, incremental cost (2nd plan over 1st plan) = 85 – 70 = 15
Incremental benefit (2nd plan over 1st plan) = 100 – 80 =20
Incremental BC ratio = 20/15 = 1.33.
Since Incremental BC ratio > 1, the more costly alternative should be selected.
Therefore, plan 2 is chosen.
Present worth method
In this method, the net worth (benefit – cost ) for each year is computed and discounted to the
present with using the present worth factor. Their sum is the Net Present Value (NPV).
NPV
B0 C0
0
1 i
Bn Cn
B1 C1
.....
1
n
1 i
1 i
where Bt and Ct are the monetary values of benefits and costs incurred at time t respectively, i
is the discount rate and n is the life of the project. The steps for selecting the best alternative
are:
(i)
Determine the NPV of each alternative.
D Nagesh Kumar, IISc, Bangalore
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Water Resources Systems Planning and Management: Introduction and Basic Concepts: Optimization and
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(ii)
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Retain those alternatives with NPV > 0 and reject the rest. If there is any mutually
exclusive alternative, then proceed to steps (iii) and (iv). Otherwise, stop.
(iii) Choose the one with greatest NPV from the set of mutually exclusive alternatives.
(iv) If in a set of mutually exclusive alternatives, some have benefits that cannot be
quantified but are approximately equal, then choose the one with least cost.
CHALLENGES IN WATER SECTOR
The major challenges in water sector as per World Water Forum are listed below:
Meeting basic needs
20% of the world population do not have access to adequate safe drinking water. 3-4 million
people die due to water-borne diseases every year. Access to water is a basic human need.
While the participation of community is essential to ensure secure and sustainable supplies of
water, women as the custodians of health and hygiene hold the key in such effort.
Protecting ecosystems
Aquatic ecosystems are repositories of biodiversity and form or constitute a crucial part of
hydrological cycle. The decline in their area and their quality will turn rivers into open sewers
without any aquatic life and also reduce the biological diversity. A balance between human
needs and the intrinsic value of ecosystems needs to be ensured.
Securing food supply
Agriculture is the largest user of water (around 90%). The demand for food grains is expected
to increase by 30-40 % in the next 25 years due to the increase in population and also change
in the consumption patterns.
Sharing water resources
River basins as the main source of freshwater, need a cooperative management. Upper states
should consider the interests of other riparian states in the same basin.
Dealing with hazards
Too much water and also too little water affects people and property. Floods and droughts are
recurrent phenomena that cannot be prevented. Government is responsible to provide security
from such hazards. Both structural (embankments, dams etc) and non-structural (forecasting
systems, contingency plans etc) methods are to be adopted for a sustainable flood and
drought management.
Valuation of water
It is necessary to understand the economic, social and cultural values of water. Wastage
decreases when the price is high. For effective water management, the concerned agencies
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Water Resources Systems Planning and Management: Introduction and Basic Concepts: Optimization and
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must have adequate resources. The valuation of water resources and charging for water
services need active participation of stakeholders.
Governing water wisely
River basin management requires a suitable institutional framework. Governing bodies
should allocate water effectively and also manage water resources based on legitimate
requirements as per agreed policies and laws. An integration of private and public sectors and
also frequent interactions between stakeholders will create an enabling environment.
The main actions to be taken to solve problems in water sector are:
a) Integrated management of river basins
b) Participation of community in the management
c) Improved agricultural practices
d) Extensive and reliable database of information dissemination
e) Proper valuation of water
BIBLIOGRAPHY/ FURTHER READING:
1. Global Water Partnership (GWP), Integrated Water Resources Management, Background
Papers No. 4, Technical Advisory Committee (TAC), 2000.
2. Jain, S.K. and V.P. Singh, Water Resources Systems Planning and Management, Vol. 51,
Elsevier Science, 2003.
3. Haimes, Hierarchical Analyses of Water Resources Systems: Modeling and Optimization
of Largescale systems, McGraw-Hill, New York, 1977.
4. Loucks D.P. and van Beek E., ‘Water Resources Systems Planning and Management’,
UNESCO Publishing, The Netherlands, 2005.
5. Loucks, D.P., J.R. Stedinger, and D.A. Haith, Water Resources Systems Planning and
Analysis, Prentice-Hall, N.J., 1981.
6. Mays, L.W. and K. Tung, Hydrosystems Engineering and Management, McGraw-Hill
Inc., New York, 1992.
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