Modern trends in optimization techniques

Modern Trends in
Optimization Techniques
24th Nov 2017
Mr. Prasad P. Prabhu
Assistant Professor
Department of Civil Engineering,
Ashokrao Mane Group of Institutions,
Vathar tarf vadgaon
Website: https://sites.google.com/a/amgoi.edu.in/ppp
Email: ppp@amgoi.edu.in

Objectives of Today’s Session
• Share my views and experiences on optimization
techniques
• Build up confidence among UG/PG students to handle
optimization problems
02-02-2018 AMGOI, vathar 2
Outcomes of this talk
At the end of this talk, the audience will be able to …
1) Recognize the importance of formulation of
optimization problems
2) Solve the optimization problem using excel solver

Outline of todays session :
Part – I Overview
• Formulation of optimization problem
• Multi objective optimization
• Multi level optimization
• Application in civil Engineering
Part – II Hands-on Session on Optimization and Solver (add-ins Excel)
• Problem I- Simple Function (minima)
• Problem II- Simple Function (maxima)
• Problem III – Fit experimental data to given model
• Practice Problems & solution/ presentation by audience

• Optimization is….
 the act of obtaining the best result under given circumstances.
 process of finding the best way of using the existing resources while taking
in to the account of all the factors that influences decisions in any
experiment.
• Why Optimization ???
 It is future, because the investment cost, efficiency, energy saving are
crucial aspects.
 Various disciplines need to develop mathematical formulation improve
efficiency.
• It is not a screening technique.

The problem formulation of any optimization problem
can be thought of as a sequence of steps and they are:
1. Choosing design variables
2. Formulating constraints
3. Formulating objective functions
4. Setting up variable limits
5. Choosing an algorithm to solve the problem
6. Solving the problem to obtain the optimal solution

• There is to date no universal method for solving all
the optimization problems
• Many difficulties  when case applied to real-world
problems.
• Typical optimization difficulties
o the functions are often very expensive to
evaluate.
o The existence of noise in the objective and
constraint functions
o the presence of discontinuities in the functions,
constitute further obstacles in the application of
standard and established methods.

Classical Methods Evolutionary Methods
Liner Programming, Non Linear
Programming, Integer Programming
& so on
Fuzzy Logic, Artificial Neural Network,
Genetic Algorithm etc.
Advanced Topics in Optimization:
• Multi Objective Optimization
• Multi Level Optimization

MULTI OBJECTIVE
OPTIMIZATION

Multi Objective Optimization
In real world examples: situation of single objective and
multiple constraints more often than not.
e.g.- water resources optimization problems
common objective.
- maximizing water quality
- regional development
- resource utilization
- various social issues
conflicting objectives.
- irrigation
- hydropower
- recreation
There is normally no single
solution to the problems of
the this type !
(But… have acceptable
solution)

Multi-objective Problem
A multi-objective optimization problem with inequality (or
equality) constraints may be formulated as
Here k denotes the number of objective functions to be minimized and
m is the number of constraints.
*Objective functions and constraints need not be linear but when they are, it is called
Multi-objective Linear Programming

MULTI LEVEL
OPTIMIZATION

Multi level Optimization
• Some optimization problem involve  large number of variables and
constraints.
• Solving such a problem will be quite cumbersome.
• Such large sized problems are decomposed into smaller independent
problems
• Then overall optimum solution can be obtained by solving each sub-problem
independently.
Model Coordination Method
Consider a minimization optimization problem F(x) consisting of ‘n’ variables, x1,
x2, … xn
Min F( x1, x2, x3,… xn)
subjected to constraints
gj(x1, x2, x3,… xn) <=0 j= 1,2,3…m
lxi <= xi <= uxi i= 1,2,3…n
where lxi and uxi represents the lower and upper bounds of the decision variable
xi.

Some applications in Civil Engineering Filed
Reservoir Operation
The goals of a multipurpose reservoir operation problem can be:
A) Flood control
B) Hydropower generation
C) Meeting irrigation demand
D) Maintaining downstream water quality
Water Distribution Systems
The typical goals of water distribution systems problem in designing urban pipe
system can be:
A) Meeting the household demands.
B) Minimizing cost of pipe system.
C) Meeting the required water pressure at all nodes of the distribution system.
D) Optimal positioning of valves.
Transportation Engineering
Vehicle Routing Problem
Structural Engineering
Design of reinforced concrete frame members
3D Steel Structure Frame (shape, design, connections)

hands-on session

Consider a transport company which has to supply 4 units of materials from
each of the place S1 and S2 to three cities. The material is to be supplied to D1,
D2 and D3 with demands of 4, 1 and 3 units respectively. Cost of transportation
per unit of supply (cij) is indicated below in the figure. Decide the pattern of
transportation that minimizes the cost.
S1
S2
D1
D2
D3

Minimize f = 5 x11 + 3 x12 + 8 x13 + 4 x21 + x22 + 7 x23
Let the amount of material
supplied from S to D be xij.
Total Supply = 8 Units and
Demand = 4 + 1 + 3 = 8
Hence, Balanced Problem
Now Objective is to minimize the
total cost of transportation from
all combinations:
subject to the constraints as explained below:
(1) The total amount of material supplied from each source city should be equal to 4.
(1) x11 + x12 + x13 = 4
(2) x21 + x22 + x23 = 4
Consider a transport company which has to supply 4
units of materials from each of the place S1 and S2 to
three cities. The material is to be supplied to D1, D2 and
D3 with demands of 4, 1 and 3 units respectively. Cost
of transportation per unit of supply (cij) is indicated
below in the figure. Decide the pattern of

(2) The total amount of material
received by each destination city
should be equal to the
corresponding demand.
x11 + x21 = 4
x12 + x22 = 1
x13 + x23 = 3
(3) Non – negativity constraints
xij ≥ 0 or
X11,x12,x13,x21,x22,x23 ≥ 0
Thus, the optimization problem has 6
decision variables and 5 constraints.
Excel Solver
Solution
Consider a transport company which has to supply 4
units of materials from each of the place S1 and S2 to
three cities. The material is to be supplied to D1, D2 and
D3 with demands of 4, 1 and 3 units respectively. Cost
of transportation per unit of supply (cij) is indicated
below in the figure. Decide the pattern of

Then  Excel Solver ( available in data or add-ins)

Typical Example – Water Resources Engineering
Consider two crops 1 and 2. One unit of crop 1 produces four units of profit and one
unit of crop 2 brings five units of profit. The demand of production of crop 1 is A units
and that of crop 2 is B units. Let x be the amount of water required for A units of crop 1
and y be the same for B units of crop 2. The amount of production and the amount of
water required can be expressed as a linear relation as shown below
A = 0.5(x - 2) + 2
B = 0.6(y - 3) + 3
Minimum amount of water that must be provided to 1 and 2 to meet their demand is
two and three units respectively. Maximum availability of water is ten units. Find out the
optimum pattern of irrigation.
(http://nptel.ac.in/courses/105108127/pdf/Module_4/M4L4_LN.pdf)
The objective is to maximize the profit from crop 1 and 2, which can be represented
as
Maximize f = 4A + 5B
Expressing as a function of the amount of water,
Maximize f = 4[0.5(x - 2) + 2] + 5[0.6(y - 3) + 3]
= 2x + 3y + 10

Example 2:
Typical Example – Water Resources Engineering
Consider two crops 1 and 2. One unit of crop 1 produces four units of profit and one
unit of crop 2 brings five units of profit. The demand of production of crop 1 is A units
and that of crop 2 is B units. Let x be the amount of water required for A units of crop 1
and y be the same for B units of crop 2. The amount of production and the amount of
water required can be expressed as a linear relation as shown below
A = 0.5(x - 2) + 2
B = 0.6(y - 3) + 3
Minimum amount of water that must be provided to 1 and 2 to meet their demand is
two and three units respectively. Maximum availability of water is ten units. Find out the
optimum pattern of irrigation.
(http://nptel.ac.in/courses/105108127/pdf/Module_4/M4L4_LN.pdf)
x + y ≤ 10 ; Maximum availability of water
x ≥ 2 ; Minimum amount of water required for crop 1
y ≥ 3 ; Minimum amount of water required for crop 2
Solution : After 3rd iteration,
x = 2; y = 8; Therefore, f = 4 + 24 + 10 = 38
Maximize f = 2x + 3y + 10

Excel Solver

• Organize the data for your problem in the spreadsheet in a
logical manner.
• Choose a spreadsheet cell to hold the value of each decision
variable in your model.
• Create a spreadsheet formula in a cell that calculates the
objective function for your model.
• Create a formulas in cells to calculate the left hand sides of
each constraint.
• Use the dialogs in Excel to tell the Solver about your decision
variables, the objective, constraints, and desired bounds on
constraints and variables.
• Run the Solver to find the optimal solution.
Summarizing…..

EXCEL Solver to fit experimental data to a model
In this exercise we will use the solver option to fit a nonlinear
equation to an experimental dataset.
Problem: soil contamination test is performed. The data is as
below.
Water
concen
tration
(c )
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7
Soil
concen
tration
(S)
0 0.5 1.1 1.25 1.2 1.3 1.4 1.5 1.5
5
1.6 1.7 1.7 1.7
2
1.7
3
1.8

Excel Solver

Minimize
f = ( x1 -2)2 + ( x2 – 1)2
Subject to,
2 ≥ x1 + x2
x2 ≥ (x1)2
Maximize
f = - x1 - x2
Subject to,
(x1)2 + x2 ≥ 2
4 ≤ (x1 + 3 x2)
30 ≥ (x1 + (x2)4)

EXCEL Solver to fit experimental data to a model
The problem involves understanding of how dry density changes
with voids ratio.
A typical data set of experiment is given below.
Voids
Ratio (e)
0.65 0.62 0.60 0.59 0.59 0.60 0.64 0.66 0.68
Dry
density
(𝛾 𝑑)
1.58 1.60 1.62 1.63 1.64 1.62 1.59 1.57 1.55
Find (single) constants for best curve fitting

Thank you

Modern trends in optimization techniques

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Modern trends in optimization techniques