Mei C., Zhou J., Peng X. Simulation and Optimization of Furnaces and Kilns for Nonferrous Metallurgical Engineering

Подождите немного. Документ загружается.

10 Multi-objective Systematic Optimization of FKNME 

of the first class center are the desired optimization parameters. The sample should

be classified again if the difference between the two target values is small.

To assure the reliability of the optimized class center, the self organizing feature

mapping (SOFM) raised by Keinan is applied to find the class center clusters

(CCC). The optimization result can be assured further if the mapping results can

distinguish the two samples clearly.

The optimization result of the characteristic variables that is centered in the

class can be made by applying PCA, PLS or ODP, which can be verified further

with the result gotten by SOFM.

üThe optimization using GA and BP: the GA and BPN algorithms are applied

to get better patterns. The fitness function is the trained neural network. The

decimalization coding is used, the value scopes of the character variables are

limited to ±5%, by which the optimum pattern could be obtained. Finally, the

artificial neural network mathematical model is constructed with multi targets, in

which the original character variables are the input parameters, the multi target

variables are the output parameters and have a number of hidden nodes. The

network structure can be obtained after being trained, and it can describe the

training samples variables and the targets. The sample patterns of the two class

centers and the performance indices of the optimum samples can be forecasted.

c) Intelligent decision system (Wang and Dai, 1990):

A key optimum parameter

(variable A) that can be controlled strictly should be found out from the

optimization industry parameter. As mentioned above, if necessary, an expert

system can be constructed to forecast the optimum parameter according to other

component parameters.

üKnowledge express:

Expressing of network knowledge: the target variables and other characteristic

variables are the input parameters. A number of hidden nodes are constructed. The

variable A is regarded as the output parameter. The neural network is trained by

the collected sample data until the fitting precision reaches ±5%. Then the target

variables are assured and the input variables is also fixed by the users. The value

of A can be given when the users input the character variables which can be

transferred to the neural network in graphical user interface (GUI).

Knowledge expressing assisted by experience:

the above neural network has

such shortcomings as: the learning knowledge is too limited, as the learning

sample numbers are not large enough; the industry parameter of the training

samples has a value limit, if the value of the input parameters exceeds the scope,

the neural network will give a less credible forecasting value because the network

cannot extrapolate. Considering such a condition, the knowledge of the expert

system depends on human beings. The rules summarized by people should be

added into the rule base.

Knowledge expressing by pattern recognition: whether the target parameter

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

forecasting by neural network is correct or not can be judged qualitatively by

pattern recognition of classification. The force of the sample is calculated by

regarding variable A and other character variables as the character parameters. The

sample belongs to a good sample if its force is less than zero and an output

parameter is gotten when the sample is considered as the input parameter, by

which the sample can be tested whether it is good.

üRunning process: the input parameters must be given when the system is

running. The input parameters and sample character parameters are compared by

the system. Variable A will be estimated with the experience rule if the input

parameters exceed the scope of the later. The value of variable A will be gotten if

the input parameters are in the scope of the later. The value of A is tested by the

classification pattern recognition. The value of A is estimated by experience if A

belongs to a bad sample. Otherwise it is directly showed in GUI if it was a good

sample.

10.3.2



Consistent target approach

The essence of the consistent target approach is that the target function f

(x),

(x),…, f

(x)of Eq. 10.1 is united into a overall consistent target function f (x), i.e.

(Liu, 1994)

f (

x)=f{f

(x), f

(x),

, f

(x) } (10.27)

The problem of Eq. 10.1 is translated into such a pattern:

min f (

(x)

0 (u =1, 2,

, m) (10.28)

The optimum problem of a multi-object function can then be solved by

converting it into a single target function.

Such approaches can be taken in the process of minimizing the consistent target

function f

(x), so that each of the partial target function can approach its optimum

value:

a) The weight combined approach, also called linear combined approach or weight

factor approach, i.e. a weight factor is introduced into the consistent target function in

the process of combining the partial target function into the overall consistent target

function, so that the difference of importance among the partial target functions, and

the difference in the magnitudes and dimensions are taken into account. Therefore, Eq.

10.27 can be rewritten as.

() ()

fwf

∑

(10.29)

where w

is the weight factor of the jth partial target function f

(

) and it is larger

than zero. Its value is decided by the magnitude and the importance of each target.

The key to the approach is the choice of the weight factor.

10 Multi-objective Systematic Optimization of FKNME 

b) The target layout approach. The optimization value f

(

) of each target

function is obtained respectively. It is then adjusted properly according to the

overall demands of the optimization design of the multi-object function, and

obtain the ideal optimization value

)0(

f . The consistent target function can be

constructed as follows:

() (0)

(0)

()

−

⎡⎤

⎢⎥

⎣⎦

∑

(10.30)

By Eq. 10.30, it means that the consistent target function f (

x) gets the minimum

value

)0(

f when each of the partial target function reaches its ideal optimization

value. The key lies in choosing the proper value of the

()

qjf

,...,2,1

)0(

= .

c) The efficacy coefficient approach. Here, each of the partial target function f

()

1, 2, ,

q= K is expressed in a proper efficacy coefficient

()

,...,2,1=

, 0

İ1 (by

=0, it means the worst;

=1 is the best). The overall efficacy

coefficient is the average value of the coefficients

...

ηη η

•=

That is, it can be used to evaluate the quality of the design scheme. Therefore, the

optimum designing scheme is:

...

ηη η

•=

max→

Thus, when the coefficient

=1, it means an ideal scheme is obtained. Otherwise,

when

=0, it means the scheme can not be accepted. In the latter, there must be

one efficacy coefficient of the partial target that equals to zero. Usually, 0.7<



stands for good conditions.

The overall efficacy coefficient

can be regarded as the consistent target function

()

...

ηη

•==x

max→ (10.31)

The approach is very effective as it is intuitionistic and can be easily adjusted,

although its calculation is a little complicated. Moreover, no matter what the

magnitudes and the dimensions of the individual partial targets are, their values

can be converted into the numerical values between zero and one. If there is any

partial target function that is not good enough, its overall efficacy coefficient



will be surely zero, which indicates the designing scheme cannot be accepted and

the restricted condition or the critical value of the target function should be

adjusted. Such an approach is especially applicable for conditions when the target

functions are neither too big nor too small.

d)The relative cost approach. With this method, all the q targets in the

optimization problem of the multi-targets are divided into two kinds. The first

includs variables such as consumption of raw materials, working hours, cost and

weight, for which, the less the variables, the better the target value. The second are

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

variables including turnout, production values, profits and benefits. For such

variables, however, the more the variables, the better the target value. If for the

first kind target, there are s terms

⎥

⎦

⎤

⎢

⎣

⎡

∑

)(x , and for the second kind there are

˄q̢s˅terms

⎥

⎦

⎤

⎢

⎣

⎡

∑

)(x , the consistent target function can be expressed as

∑

)(

(10.32)

It shows that the minimum value of f (

) is then the optimization value.

10.3.3



The main target approach

The importance of individual targets is different in the multi-target optimization.

As the key idea of the main target approach, the main targets should be considered

first and the secondary targets must be considered simultaneously. With this

approach, firstly, all the target functions of the multi-target optimization are

rearranged according to their importance. The most important one should be put in

the first. Secondly, the restriction optimization value of each target function is

solved. The estimated proper values of all the other target functions, which are

considered as accessorial restrictions, should be given based on the primary design.

In this way, the restriction optimization of multi target functions is transformed

into that of single target functions. The relative optimization values of entire

problem can then be achieved (Liu, 1994).

The mathematical model of restriction optimization value of the kth target

function for the Eq. 10.1 is

)()(min

ff =

(1İkİq)

)(.. x

gts İ

1, 2,..., )um=

(10.33)

)()()(

xxx

jjjm

ffg −=

İ

( j =1, 2, …, k̢1, k, k+1, …, q)

where, )(

f is the estimated optimization value of other target functions

besides minimizing, before getting the optimization value; it is the optimization

value of the target function, after getting the optimization value. s.t.g

(

) is the

accessorial restriction condition.

In the situation with many target functions, when the target functions located in

the front of the multi-target function arrangement have been reached the

10 Multi-objective Systematic Optimization of FKNME 

optimization values, the optimization process may be interrupted because no

optimization value can be obtained for the next target functions. In such a case, an

excessive function

)(

fΔ

could be added to the optimization value )(

f ,

by which the optimization precision will be decrease, to avoid the interruption.

The accessorial restriction condition is then changed into

() () [ ( ) ( )]

mj j j j

gfff

=− +Δ

xx xİ

( j =1, 2, …, k-1, k, k+1, …, q) (10.34)

In the practical optimization design, a right estimation and decision of all the

designing indices can be made according to the basic requirements, and be

rearranged in the light of importance. It is therefore not very difficult to apply the

main target approach to practice.

10.3.4



The coordination curve approach

In this method, the optimization scheme of the multi-target function optimization

is searched in the whole designing space according to the coordination relation

between the isoline and the restriction surface of the individual target functions.

In solving the n-dimensional optimization of over two targets, a winding surface of

coordination can be formed according to the coordination relations among the isolines

of all the target functions. However, it is impossible to express the super winding

surface by graphs. Only the variation scope of the target function can be given by the

following mathematical model (Liu, 1994).

)(min x

(j =1, 2, …, q)

.. () ()

vvv

sth f f=−xx

İ

(v=1, 2, …, q

̢1; vĮj) (10.35)

)(

g İ (u =1, 2, …, m)

where

is the expectation value of ),...,2,1)(( qvf

=x .

The coordination curve approach is very useful for solving the target function

with two contradictive target functions. The relation between the targets and the

overall design scheme can be analyzed clearly by the coordination curve, by

which the ideas to improve the design can be got, and the satisfactory result can be

obtained.

10.3.5



The partition layer solving approach

Differing from the minimum mathematical model of the common multi-target

optimization, each target in Eq. 10.1 is not at the equal level; on the contrary, they

has different preferential layer. As the model has the characteristics by considering

target in each layer, the optimization is carried out by obtaining the optimized

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

solution for each layer according to the turns of preferential layer given in the

model. The solution of the last layer is then the solution of the problem. This

approach is called the complete partition approach.

The simplest complete partition approach is (Hu, 1993): the target function

)(

xf in the first preferential layer is minimized in the feasible solution field x,

then the target function

)(

xf in the second preferential layer is minimized in

the optimization solution collection of the first preferential layer, and so on.

Usually, the target function

)(

f in the s+1 preferential layer is minimized

in the optimization solution collection of the sth

preferential layer.

10.3.6



Fuzzy optimization of the multi targets

The restriction conditions and the target functions of the usual optimization

problems are very clear. However, they may be fuzzy in many practical problems.

for example, the restriction conditions may be flexible and the restriction target

functions are of a fuzzy data collection etc. The mathematical optimization

problem under the fuzzy conditions, i.e., fuzzy restriction conditions or fuzzy

targets is called fuzzy optimization (Kunii and Levenspied, 1969).

It is impossible to get the maximum or minimum values for all the target

functions at one point when there are many target functions. Hence, a compromise

scheme has been adopted to get either the maximum or the minimum value of each

target function as much as possible. To achieve this, the target function can be made

fuzzy with the fuzzy mathematical approach. The details of the approach are as

follows.

Firstly, to solve the maximum value

z of the single target z

(i=1, 2,

, r)

under the restriction condition A

xİb xı

⎪

⎩

⎪

⎨

⎧

∑

jijiii

xczzz

,|max Axİb xı

⎭

⎬

⎫

i=1, 2, …, r (10.36)

This is the typical linear optimization problem of the single target.

Secondly, the flexible index d

is given for each single target z

, i=1, 2, Ă, r,

>0. The more important is the target, the smaller is the flexible index. In this way,

each target will be made fuzzy. Supposing the fuzzy target G

corresponds to the

ith target z

, the subject function of G

is defined as

⎟

⎠

⎞

⎜

⎝

⎛

∑

jijii

xcgxG

)(

(10.37)

10 Multi-objective Systematic Optimization of FKNME 

ij j i i

jj j

ij j

cx z d

zcxzd

⎧

<−

⎪

⎛⎞

⎪

=− − −

⎨⎜ ⎟

⎝⎠

⎪

≤

⎪

⎩

∑

jij

zxc <

∑

(10.38)

i =1, 2, …, r

Let

, which is the fuzzy target of the typical linear problem; and D={ x|Axİ

 xı}, which is the typical collection, i.e., the possible solution collection that

meets the restriction conditions, or called the feasible solution field. Thus, the

fuzzy optimization solution of the typical linear optimization of the multi-targets

can be obtained by the fuzzy discriminant.

The fuzzy discriminant is D

=DģG.

The best discriminant is x

, and x

is defined as the fuzzy optimization solution.

satisfies

( ) max( () ()) max ()

fDG

DuuG

≥∈

=∧=

xxxx

(10.39)

It can be concluded that the fuzzy optimization value

is the maximum point

in the feasible solution field D.

The above problem can be transformed into a typical linear optimization

problem. Suppose:

)()(

∧==

The typical linear optimization of multi- targets can be transformed as

max

with the restriction conditions of:

1,1,2,,

,1,2,,

0, 0

iijj

ij j k

zcx i r

ax b k m

⎧

⎛⎞

−− ≥=

⎪

⎜⎟

⎝⎠

⎪

≤=

⎨

⎪

≤

⎪

≥≥

⎪

⎩

∑

(10.40)

Or it could be expressed as

max

with the restriction condition of

ı ı

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

,1,2,,

0, 0

ij j i i i

ij j k

cx d z d i r

ax b k m

⎧

−≥− =

⎪

≤=

⎨

⎪

≤

⎪

≥≥

⎪

⎩

∑

(10.41)

Eq. 10.38 is a typical linear scheme problem to get the optimization solution

** * *

(, , , , )

xx x

Eq. 10.38 is equivalent to the following formula

max

of which, the restriction conditions are

(1 ) , 1, 2, ,

,1,2,,

0, 0

ij j i i

ij j k

cx z d i r

ax b k m

⎧

≥−=

⎪

≤=

⎨

⎪

≤

⎪

≥≥

⎪

⎩

∑

(10.42)

It indicates that max

λ make the individual target functions

∑

jij

),,2,1( ri Λ=

approach

z the most closen.

By solving Eq. 10.38, (

** * *

,,,,

xx x

) can be obtained, in which λ

the maximum value of

, and (

** *

,,,

xxK

) is the fuzzy optimization

solution of the original typical multitarget linear optimization. z

= cx

, is the

optimization solution of the target.

10.4



Technical Carriers of Furnace Integral Optimization

The main forms of the furnace integral optimization technology applied in

engineering are:

a) CAD software for optimum design of engineering.

b) Offline instruction system (or optimum decision support system) for furnace

operation optimization.

c) Online optimum control system or integrated optimum system integrating

monitoring, control and management for furnace.

ı ı

10 Multi-objective Systematic Optimization of FKNME  

10.4.1



Optimum design CAD

In view of a given furnace, hologram simulation software packages could be

developed by means of multifield coupling hologram simulation, which could be

used to execute the simulation test after being verified by production practice or in

situ measured data. Therefore, the best design parameters and design scheme

could be found out by artificial optimizing or auto-optimizing program. As an

example, the content and structure of six-field coupling simulation software

package of aluminium electrolysis cell is shown in Fig. 10.2.

Fig. 10.2 Coupling simulation software package of aluminium electrolysis cell (Mei et al., 1996)

(A, B, C, D: Simulation data and optimization information)

The principle of auto-optimizing or artificial optimizing (selecting artificially

the optimal design scheme) using hologram simulation software is shown in

  Xiaoqi Peng, Yanpo Song, Zhuo Chen and Junfeng Yao

Fig.10.3. This is the operation process of optimum design aided by a computer

(optimum design CAD).

Fig. 10.3 Optimizing by simulation experimentation

10.4.2



Intelligent decision support system for furnace operation

optimization

Intelligent decision support system has advantages such as low developing cost,

small investment, fast effect and high efficiency, so it has been widely applied in

assisted management, optimal decision making and process control. It is especially

fit for multi-variable nonlinear system such as a complex production process with

big random interference and accurate mathematical model being difficult to

establish.

10.4.2.1



Introduction of intelligent decision support system

Decision support system is an interactive compute software system which is used

to help decision makers to solve the unstructured or semistructured problems. An

intelligent decision support system is an integration of expert system and decision

support system, which can accomplish qualitative knowledge reasoning, quantitative

model calculations, mass data processing and so on. It usually includes four

components as shown in Fig. 10.4.

Human computer interface component is used to receive, check and transform