(iii) optimization and simulation introduction and basic components d. nagesh kumar, iisc water...

(iii) Optimization and Simulation

Introduction and Basic Components

D. Nagesh Kumar, IIScWater Resources Systems Planning and Management: M1L3

Objectives


Understand the concept of optimization and simulation

Classification of optimization and simulation problems

Economic criteria in water resources

Learn about the challenges in water resources

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Modelling Techniques


Modelling or system analysis techniques - Developed during the Second World War to deploy limited resources in an optimum manner

These techniques were aided for military operations - known as operation research techniques

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.

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Optimization


Science of choosing the best amongst a number of possible alternatives Identify the best through evaluation from a number of possible solutions Driving force in the optimization is the objective function (or functions) Optimal solution is the one which gives the best (either maximum or

minimum) solution under all assumptions and constraints An optimization model can be stated as:

Objective function: Maximize (or Minimize) f(X)

Subject to the constraints

gj(X) ≥ 0, j = 1,2,..,m hj(X) = 0, j = m+1, m+2,.., p

X is the vector of decision variables; g(X) are the inequality constraints; h(X) are the equality constraints.

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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 value of the design variables

Deterministic/ Stochastic nature of the variables

Separability of the functions and number of objective functions

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Classification based on existence of constraints


Constrained optimization problems: Subject to one or more

constraints

Unconstrained optimization problems: No constraints exist

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Classification based on physical structure of the problem


Optimization problems are classified as optimal control and non-optimal control problems.

(i)Optimal control problems

Problem involving a number of stages

Each stage evolves from the preceding stage in a prescribed manner.

Defined by two types of variables: the control or design and state variables.

Control variables define the system and controls how one stage evolves into the next

State variables describe the behavior or status of the system at any stage

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.

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Classification based on the physical structure of the problem…


An OC problem can be stated as follows:

Find X which minimizes

Subject to the constraints

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.

(ii) Problems which are not optimal control problems are called non-optimal control problems.

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Classification based on the nature of the equations involved


Optimization problems can be classified as

(i)Linear programming: Objective function and all the constraints are ‘linear’

functions of the design variables

(ii)Nonlinear programming : Any of the functions among the objectives and

constraint functions is nonlinear

(iii)Geometric programming : Objective function and constraints are

expressed as polynomials

(iv)Quadratic programming: Best behaved nonlinear programming problem

with a quadratic objective function and linear constraints and is concave (for

maximization problems)

* For details refer lecture notes9

Classification based on the permissible values of the decision variables


Objective functions can be classified as integer and real-valued

programming

(i)Integer programming problem: Some or all of the design variables of an

optimization problem are restricted to take only integer (or discrete) values

(ii)Real-valued programming problem: 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

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Classification based on deterministic/ Stochastic nature of the variables


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 problem, some or all

the design variables are expressed probabilistically (non-deterministic or

stochastic).

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Classification based on separability of the functions


Optimization problems can be classified as separable and non-separable

programming problems

(i) Separable programming problems: In this type of problem, the objective

function and the constraints are separable.

Function is said to be separable if it can be expressed as the sum of n

single-variable functions,

,

i.e.

(ii) Non-separable programming problems: Objective function is not

separables

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nni xfxfxf ,..., 221

n

iii xfXf

1

)(

Classification based on the number of objective functions


Objective functions can be classified as single-objective and multi-objective

programming problems.

(i) Single-objective programming: There is only a single objective function.

(ii) Multi-objective programming: A multiobjective programming problem

can be stated as follows:

Find X which maximizes/ minimizes

Subject to gj(X) ≤ 0 , j = 1, 2, . . . , m

where f1, f2, . . . fk denote the objective functions to be maximized/

minimized simultaneously

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XfXfXf k,..., 21

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

Main advantage of simulation is its ability to accurately describe the reality

Operating policies can be tested through simulation before implementing in actual situations

Water resources systems are too complex to be expressed in any analytical expression.

Simulation model duplicates the system’s operation with a defined operational policy, parameters, time series of flows, demands etc

Design parameters and the operation policy are evaluated through the objective function or some reliability measures.

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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.

5. Validation: Test the model and apply the model to the problem

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Classification of Simulation models


Simulation models can be

i. Physical (e.g. a scale model of a spillway)

ii. Analog (system of electrical components such as resistors or capacitors arranged

to act as an analog to the hydrological components) or

iii. Mathematical (action of a system expressed as equations or logical statements.

Simulation models can be

i. Static (fixed parameters and operational policy) or

ii. Dynamic (takes into account the change in the parameters of the system and the

operational policy with time) in nature.

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Classification of Simulation models…


Simulation models can be deterministic or stochastic

Simulation models can be statistical or process oriented, or a mixture of both.

Pure statistical models are based solely on data (field measurements). Regressions

and artificial neural networks are examples

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.

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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. 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 demand an enormous computational effort.

Optimization will determine the best decision; but the solution is often based on many limiting assumptions.

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.

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Economics in Water Resources


Economics in engineering deals with applying economic criteria to select the

best solution from a group of feasible alternatives

Evolving the best economic policy for planning and management of an

engineering project

Ranking and selection of alternatives are done based on the principles of

engineering economics

Magnitude of consequences expected from employing each alternative are

assessed and converted into commensurable units for comparison.

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Concepts used in economic analysis


Cash flow diagram: Assess the consequences of each alternative and assign a monetary value for each

consequence Graphic representation of each monetary value with time is called a cash flow

diagram. Benefits are represented as upward arrows and costs as downward arrows. 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.

At the beginning, a large expenditure is made.

Benefits are received thereafter every year.

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Benefits

Costs500

50 50 50 50 50 50 50

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Cash flow diagram

Concepts used in economic analysis…


Discount factors:

Amounts at different times have different values.

All monetary values are converted into equivalent amounts at some definite time using discount factors, for comparison.

Many discount factors are used:

Compound amount factor – An amount P invested at the beginning of first year grows to Q at the 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 – Amount X that will be received at the end of each year to get Q at the end of n years, X = Q i / [(1+i)n -1]

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Concepts used in economic analysis…


Discount factors:

Capital recovery factor – Amount X that should be invested at the end of each year, if amount P is invested at the beginning of first year, X = P i (1+i)n / [(1+i)n -1]

Series compound factor – Amount Q that will be received at the end of nth year, if an amount X is invested at the end of each year, Q = X [(1+i)n -1] / i

Series present worth factor – Present value of P if an amount X is invested at the end of each year, P = X [(1+i)n -1] / i (1+i)n

Other terms

Sunk cost: Money spent already which has no economic relevance in deciding future alternatives

Salvage value: Value of the unused life of an element at the end of the period of analysis. Salvage value, S = I (1 – U/L), where I = initial value, U = unused life and L = total life.

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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

4. Annual cost method

The first two methods are explained

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Benefit – Cost ratio method


BC ratio: Ratio of the present worth of benefits and the present worth of cost

where

Bt and Ct are the monetary values of benefits and costs incurred at time t

respectively

i is the discount rate

n is the life of the project in time steps (years or months or weeks).

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n

tt

t

n

tt

t

iC

iB

CBR

0

0

1

1

Benefit – Cost ratio method…


Steps for choosing the best alternative are:

1. Calculate the BC ratio for each alternative

2. Retain all alternatives with BC>1 and reject the rest. If sets of mutually exclusive alternatives are involved then go to steps 3, 4 and 5.

3. 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.

4. Choose the more costly alternative of the incremental BC >1. Otherwise choose the less costly alternative.

5. Repeat the analysis for all alternatives in the order of rank.

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BC ratio: 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 3 and 4, 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.

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Present worth method


Net worth (benefit – cost ) for each year is computed and discounted to the present with using the present worth factor – Net Present Value (NPV).

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:

1. Determine the NPV of each alternative.

2. Retain those alternatives with NPV > 0 and reject the rest. If there is any mutually exclusive alternative, then proceed to steps 3 and 4. Otherwise, stop.

3. Choose the one with greatest NPV from the set of mutually exclusive alternatives.

4. 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.

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nnn

iCB

iCB

iCBNPV

1

.....11 1

11000

Challenges in Water Sector


The major challenges in water sector as per World Water Forum and are listed below:

Meeting basic needs

Protecting ecosystems

Securing the food supply

Sharing water resources

Dealing with hazards

Valuing water

Governing water wisely

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Challenges in Water Sector…


Main actions to be taken to solve problems in water sector are:

Integrated management of river basins

Participation of community in the management

Improved agricultural practices

Extensive and reliable database of information dissemination

Proper valuation of water

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BIBLIOGRAPHY/ FURTHER READING:


Global Water Partnership (GWP), Integrated Water Resources Management, Background Papers No. 4, Technical Advisory Committee (TAC), 2000.

Jain, S.K. and V.P. Singh, Water Resources Systems Planning and Management, Vol. 51, Elsevier Science, 2003.

Haimes, Hierarchical Analyses of Water Resources Systems: Modeling and Optimization of Largescale systems, McGraw-Hill, New York, 1977.

Loucks D.P. and van Beek E., ‘Water Resources Systems Planning and Management’, UNESCO Publishing, The Netherlands, 2005.

Loucks, D.P., J.R. Stedinger, and D.A. Haith, Water Resources Systems Planning and Analysis, Prentice-Hall, N.J., 1981.

Mays, L.W. and K. Tung, Hydrosystems Engineering and Management, McGraw-Hill Inc., New York, 1992.

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Thank You

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