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Optimization Technique

Lecturer: Assistant Professor Dr.Ghanim.M. Alwan

University of Technology, Iraq

Visitor Scholar, Missouri University of Science and Technology.

What is optimization?

Optimization technique is a powerful tool to obtain the desired design parameters and

best set of operating conditions .This would guide the experimental work and reduce

the risk and cost of design and operating.

Optimization refers to finding the values of decision variables, which correspond to

and provide the maximum or minimum of one or more desired objectives.

Reliability of optimum solutions depends on formulation of objective functions and

selected optimization technique.

Optimization requires a mathematical model that describes and predicts the process

behavior.

In complex non-linear processes, optimization search could help to estimate unknown

parameters.

Robust optimization could determine uncertainty variables in dynamic processes.

Optimization could be implemented as a tool to enable scale-up methodology and

design of multiphase reactors and flow systems.

Without optimization of design and operations, manufacturing and engineering

activities will not be as efficient as they are now.

Cautions Regarding Optimization

Optimization algorithms can powerfully assist the workers in operating and design.

1. The researcher should always carefully and thoroughly validate the engineering model.

Optimization of an inaccurate model is modestly illuminating at best and misleading

and a waste of time at worst. Often optimization algorithms will exploit weaknesses in

the model if they exist. As an example, regarding reaction system, you can get some

very high performance design if a model includes reliable kinetics mechanism!

2. We need to make sure the optimization problem represents the real problem, we need

to solve. Care must be taken to optimize with respect to the objective function, which

most nearly reflects the true goals of the optimization problem.

3. We need to be careful to optimize the whole process system and not just individual parts.

Objectives that could be minimized:cost, weight, distance, energy, loss, waste,

processing time, raw material consumption….etc.

Objectives that could be maximized:profit, conversion, yield, utility, efficiency,

capacity…etc.

Optimization process steps:

1. What is the optimization problem?

2. Selecting of the objective.

3. Selecting of the decision variables

4. Definition of constraints.

5. Formulation of the model: presenting the objective as a function of the variables and the

constraints as functional relationships, equalities and/or inequalities of the variables.

*Static and dynamic problems:

Most optimization problems are based on steady state models that could be formulated from

experimental data. Optimization problems involving dynamic models are more suitable for

"Optimal control".

**Continuous and discrete variables:

Continuous variable is any process variable that has real value such

as:pressure,temperature,concentration…..etc,while discrete variable(design variable) is

integer value only, number of tubes in a heat exchanger and number of distillation trays in a

distillation column ….etc.Optimization problems without discrete variables are far easier to

solve. Reliability of optimization technique depends on capturing of discrete variables.

Engineering Models in Optimization:

Engineering models play a key role in engineering optimization. In this section we will

discuss some further aspects of engineering models. We refer to engineering models as

analysis models.

In a very general sense, analysis models can be viewed as shown in Fig 1 below. A model

requires some inputs in order to make calculations. These inputs are called analysis

variables. Analysis variables include design variables (the variables we can change) plus

other quantities such as material properties, boundary conditions, etc. which typically would

not be design variables. When all values for all the analysis variables have been set, the

analysis model can be evaluated. The analysis model computes outputs called analysis

functions. These functions represent what we need to determine the “goodness” of a design.

For example, analysis functions might be stresses, deflections, cost, efficiency, heat transfer,

pressure drop, etc. It is from the analysis functions that we will select the design functions,

i.e., the objectives and constraints.

Fig. 1. The operation of analysis models

Thus from a very general viewpoint, analysis models require inputs—analysis variables—

and compute outputs—analysis functions. Essentially all analysis models can be viewed this

way.

Models and Optimization by Trial-and-Error:

The analysis model is to compute the values of analysis functions. The designer

specifies values for analysis variables, and the model computes the corresponding

functions.

Note that the analysis software does not make any kind of “judgment” regarding the

goodness of the design. If an engineer is designing a reactor, for example, and has software

to predict conversion and yield, the analysis software merely reports those values—it does

not suggest how to change the reactor design to increase conversion in a particular

location. Determining how to improve the design is the job of the designer.

To improve the design, the designer will often use the model in an iterative fashion, as shown

in Fig. 2 below. The designer specifies a set of inputs, evaluates the model, and examines the

outputs. Suppose, in some respect, the outputs are not satisfactory. Using intuition and

experience, the designer proposes a new set of inputs which he or she feels will result in a

better set of outputs. The model is evaluated again. This process may be repeated many

times.

Fig. 2. Common “trial-and-error” iterative design process.

We refer to this process as “optimization by design trial-and-error.” This is the way most

analysis software is used. Often the design process ends when time and/or money run out.

Note the mismatch of technology in Fig.2. On the right hand side, the model may be

evaluated with sophisticated software and the latest high-speed computers. On the left hand

side, design decisions are made by trial-and-error. The analysis is high tech; the decision

making is low tech.

Optimization with Computer Algorithms:

Computer-based optimization is an attempt to bring some high-tech help to the decision

making side of Fig. 2. With this approach, the designer is taken out of the trial-and-error

loop. The computer is now used to both evaluate the model and search for a better

operating conditions. This process is illustrated in Fig.3.

Fig..3. Moving the designer out of the trial-and-error loop with computer-based

optimization software.

The researcher now operates at a higher level. Instead of adjusting variables and

interpreting function values, the designer is specifying goals for the design problem and

interpreting optimization results. Usually a better design can be found in a shorter time.

Specifying an Optimization Problem:

1.Variables, Objectives, Constraints

2.Example: Specifying the Optimization Set-up of a Steam Condenser

It takes some experience to be able to take the description of a design problem and abstract

out of it the underlying optimization problem. In this example, a design problem for a steam

condenser is given. Can you identify the appropriate analysis/design variables? Can you

identify the appropriate analysis/design functions?

Description:

Fig. 4 below shows a steam condenser. The designer needs to design a condenser that will

cost a minimum amount and condense a specified amount of steam, mmin. Steam flow rate,

ms, steam condition, x, water temperature, Tw, water pressure Pw, and materials are

specified. Variables under the designer’s control include the outside diameter of the shell, D;

tube wall thickness, t; length of tubes, L; number of passes, N; number of tubes per pass, n;

water flow rate, mw; baffle spacing, B, tube diameter, d. The model calculates the actual

steam condensed, mcond, the corrosion potential, CP, condenser pressure drop, Pcond, cost,

Ccond, and overall size, Vcond. The overall size must be less than Vmax and the designer would

like to limit overall pressure drop to be less than Pmax.

Discussion:

This problem description is somewhat typical of many design problems, in that we have to

infer some aspects of the problem. The wording, “Steam flow rate, ms, steam condition, x,

water temperature, Tw, water pressure Pw, and materials are specified,” indicates these are

either analysis variables that are not design variables (“unmapped analysis variables”) or

constraint right hand sides. The key words, “Variables under the designer’s control…”

indicate that what follows are design variables.

Statements such as “The model calculates…” and “The designer would also like to limit”

indicate analysis or design functions. The phrase, “The overall size must be less than…”

clearly indicates a constraint.

Fig.4. Schematic of steam condenser.

Thus from this description it appears we have the following,

Process Variables:

Steam flow rate, ms

Steam condition, x

Water temperature, Tw

Water pressure, Pw

Material properties

Design Variables:

Outside diameter of the shell, D

Tube wall thickness, t

Length of tubes, L Number

of passes, N Number of

tubes per pass, n Water flow

rate, mw

Baffle spacing, B,

Tube diameter, d.

Objective Functions:

Minimize Cost

Overall size, Vcond

Steam condensed, mcond

Condenser pressure drop, Pcond

corrosion potential, CP

Constraints:

Vcond ≤Vmax

mcond ≥mmin

Pcond ≤Pmax

Formulating of objective function:

The objective is correlated with decision variables by using an available experimental data.

Several advanced nonlinear regression used that are; Newton-Quasi, Hook-Jeevs pattern

moves and the nonlinear least squares model estimation (Levenberg marquardt method) with

aid of the software (Statistica version10).

Examples:

Max Y=exp(-(x1-4)^2-(x2-4)^2)+exp(-(x1+4)^2-(x2-4)^2+2*exp(-x1^2-

(x2+4)^2)+2*exp(-x1^2-x2^2)

-5

0

5

-5

0

5

0

0.5

1

1.5

2

2.5

Fig. 5.Mesh plot which shows constrained optimum.

Y=4+4.5x1-4x2+x1^2+2x2^2-2x1x2+x1^4-2x1^2x2

Fig.6. Contour plot which shows unconstrained optimum.

2-Those based on Energy-mass balances, thermodynamics, chemical reaction

kinetics….etc.

*Tendency Model or gray model depends on inferential technique.

** Hybrid Model consists frame-work of heat/mass balance equations supplemented with

sub-models of of kinetic parameters and thermodynamic equations.

***Multilayer Model: The first layer is hybrid model and the optimum problem

represents the second modeling layer.

LP, linear programming problem: f(x) and ci(x) are linear.

NLP, nonlinear programming problem: f(x) is linear/nonlinear, ci(x):is

nonlinear/linear i.e,one of them is nonlinear.

MATLAB Tool box and the modeling language and optimizer (LINGO15)

Tool box are used throughout this workshop as computational tools for

implementing for cases study.

-Stochastic optimization:Deterministic algorithms for function optimization are

generally limited to convex regular functions. However, many functions are either not

differentiable or need a lot of difficult mathematical treatment (discretization, sensitivity

computation….etc) for differentiating. Therefore, stochastic sampling methods have

been found suitable for optimize such functions.

Genetic Algorithm, GA

Pattern search, PS

Firefly Algorithm,FA

Genetic Algorithm:

Genetic algorithms are search algorithms based on mechanics of natural selection

and natural genetics. Philosophically GAs are based on Darwins′theory.Genetic

algorithm have the following advantages over traditional methods:

GAs search from a population of points, not a single point. Hence GAs are said

to be Global optimization techniques.

GAs use only the value of convex (minimize) objective function. The

derivatives are not used in the search process.

GAs use probabilistic transition rules, not deterministic rules.

Genetic algorithms are the most popular form of evolutionary algorithms.

A population of chromosomes represents a set of possible solution. These solutions are

Classified by an evaluation function, giving better values, or fitness to better solutions.

The simplest representation is a value representation where the chromosome consists

of the values of the design variables placed side by side. For example, suppose we

have 6 discrete design variables whose values are integer values ranging from 1 to 5.

Suppose we also have 4 continuous design variables whose values are real numbers

ranging from 3.000 to 9.000. A possible chromosome is shown in Fig.7:

4

3

1

3

2

5

3.572

6.594

5.893

8.157

Fig. 7: Chromosome

The chromosome in Fig. 7 consists of ten genes, one for each design variable. The

value of each gene is the value of the corresponding design variable. Thus, a

chromosome represents particular design since values are specified for each of the

decision variables.

Fig.8 Process of GA for objective functions.

Genetic Formulation:

A genetic algorithm is search procedure modeled on the mechanics of natural

selection rather than simulated process. Domain knowledge is embedded in the

abstract representation of a candidate solution termed an organism.

Organisms are grouped into sets called populations.Succesive populations are called

Generations. Any string in the search space is evaluated for its fitness, depended on

which the particular string is either dropped or taken into the next generation. In

general a fitness function Ω(x) is first derived from the objective function f(x) and is

used in successive genetic operations.

The fitness function is a blend of objective & constraint function:

Ω(x)=f(x)*(1.0+

µ

(x)) (1)

where the penalty multiplier

µ

(x) that penalizes an infeasible string is given by

linear function:

µ

(x)=

Σ

Wj*Vj (2)

Where the constraint violation is measured by :

Vj=[0,Gm-Bm] (3)

(3)

(3)

Where ;

Gm: mth inequality constraint

Bm:bound on constraint

Wj:penalty weight

The penalty weight at Tth generation is calculated as given

by: Wj=log(1+T) (4)

Figure (3) shows the flow chart of the GA

To capture discrete (integer) decision variables, it is best GA is implemented with

hybrid functions, as pattern search. First does global search using GA and then switches

to Hooke-Jeeves algorithm to refine the decision variables.

Fig.10.Flow chart of GA.

Case study: Spouted bed

The objectives: UI (uniformity index) of solid particles and PD (pressure drop)across the bed are

correlated with the three decision variables(Vg, ρs and dp). Two-advanced nonlinear regression used which

are; Newton-Quasi and Hook-Jeevs pattern moves with the aid of the computer program (Statistica

version10). The conflicted optimization problems are:

UI=0.184Vg-0.214ρs0.12dp-0.267 (1)

PD=0.037Vg0.38 ρs0.407dp-0.221 (2)

Subject to inequality constraints:

0.74 ≤Vg ≤1.0

2400.0 ≤ ρs ≤ 7400.0 (3)

1.09 ≤dp ≤2.18

However, the spouted bed is highly nonlinear interacted process. GA is the best global search for solving

the optimization problem of the process.

Application of GA:

1. Single objective function:

Max UI= -0.184Vg-0.214ρs0.12dp-0.267 (1)

Subject to inequality constraints:

(3)

Adapted parameters of GA

Parameter

Type/value

Population type

Population size

Creation function

Scaling function

Selection function

Crossover function

Crossover fraction

Mutation function

Migration direction

Migration fraction

Hybrid function

Number of generation

Function tolerance

Double vector

80

Feasible population

Rank

Roulette

Scattered

0.8

Adaptive feasible

Forward

0.1

Pattern search

51

1.0E-6

Figure:Results/solution of genetic algorithm.

0 50 100

-0.55

-0.5

-0.45

Generation

Fitness value

Best: -0.53484 Mean: -0.53369

123

0

5000

10000

Number of variables (3)

Current best individual

Current Best Individual

-0.54 -0.535 -0.53 -0.525

0

5

Raw scores

Expectation

Fitness Scaling

20 40 60 80 100

-0.6

-0.5

-0.4

Generation

Best, Worst, and Mean Scores

-0.54 -0.535 -0.53 -0.525

0

10

20

Score Histogram

Score (range)

Number of individuals

0 5 10 15 20

0

5

10

Selection Function

Individual

Number of children

Best fitness

Mean fitness

0.538 0.5382 0.5384 0.5386 0.5388 0.539 0.5392 0.5394 0.5396 0.5398

0.7405

0.741

0.7415

0.742

0.7425

0.743

Uniformity Index

Gas Velocity,(m/s )

0.538 0.5382 0.5384 0.5386 0.5388 0.539 0.5392 0.5394 0.5396 0.5398

6646

6646.2

6646.4

6646.6

6646.8

6647

6647.2

6647.4

6647.6

6647.8

6648

Uniformity Index

Solid density ,(Kg/m3)

Mutation of decision variables corresponding to objective change.

Optimal values by single GA

Decision variables

Optimum value

Gas velocity(m/s)

0.741

Density of solid (Kg/m3)

6648

Diameter particle(mm)

1.09

0.538 0.5382 0.5384 0.5386 0.5388 0.539 0.5392 0.5394 0.5396 0.5398

1.09

1.091

1.092

1.093

1.094

1.095

1.096

1.097

1.098

1.099

Uniformity Index

Solid diameter,(mm)

2. Multi-objective optimization:

Multi-objective optimization technique is used to solve the conflicted optimization problems:

Max UI= - 0.184Vg-0.214ρs0.12dp-0.267 (1)

Min PD= 0.037Vg0.38 ρs0.407dp-0.221 (2)

Subject to inequality constraints:

0.74 ≤Vg ≤1.0

2400.0 ≤ ρs ≤ 7400.0 (3)

1.09 ≤dp ≤2.18

Adapted parameters of GA

Parameter

Type/Values

Population type

Population size

Creation function

Scaling function

Selection function

Crossover function

Crossover fraction

Mutation function

Migration direction

Migration fraction

Hybrid function

Number of generation

Function tolerance

Double vector

80

Feasible population

Rank

Roulette

Scattered

0.8

Adaptive feasible

Forward

0.1

Pattern search

107

1.0E-6

The objective functions are conflicted, so that 14 chromosomes were created. Each

chromosome represents the possible solution to the optimization problem, and each

bit(gene) represents the value of the variables of the problem.This means that the

optimization search created 26 sets for the process design and operating conditions.

-0.54 -0.52 -0.5 -0.48 -0.46 -0.44 -0.42 -0.4 -0.38 -0.36

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

Objective 1

Objective 2

Pareto front

Decision variables

Min. .

Max.

Gas velocity (m/s)

0.740

0.756

Density of solid (KG/M3)

2400

6259

Diameter of particle(mm)

1.109

2.178

Value of objective

UI = 0.39 – 0.53

PD = 0.66 – 1.19

100 200 300 400 500 600

1000

2000

3000

Generation

Avergae Distance

Average Distance Between Individuals

-1 0 1 2

0

10

20

Score Histogram

Score (range)

Number of individuals

5 91317212529333741

0

10

20

Selection Function

Individual

Number of children

0 10 20 30 40

0

0.1

0.2

Distance of individuals

Individuals

Distance

1 2 3 4 5 6

0

10

20

Rank histogram

Rank

Number of individuals

0 200 400 600

0

0.5

Generation

Average Spread

Average Spread: 0.07348

fun1 [-0.537319 -0.399723]

fun2 [0.663763 1.19368]

Mutation change of decision variables corresponding to objectives.

0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56

0.74

0.742

0.744

0.746

0.748

0.75

0.752

0.754

0.756

0.758

0.76

Uniformity Index

Gas veloc ity,(m/s )

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1. 1 1.15

0.74

0.742

0.744

0.746

0.748

0.75

0.752

0.754

0.756

0.758

0.76

Pressure drop,(Kpa)

Gas veloc ity,(m/s )