Design of Experiment

DOE is a systematic approach for investigation of a system or process. A series of structured tests are designed

in which planned changes are made to the input variables of a process or system. The effects of these changes on

a pre-defined output are then assessed. DOE is important as a formal way of maximizing information gained

while minimizing resources required. It has more to offer than „one change at a time‟ experimental methods,

because it allows a judgment on the significance to the output of input variables acting alone, as well input variables acting in combination with one another.

One change at a time‟ testing always carries the risk that the experimenter may find one input variable to have a

significant effect on the response (output) while failing to discover that changing another variable may alter the

effect of the first (i.e. some kind of dependency or interaction). This is because the temptation is to stop the test

when this first significant effect has been found. In order to reveal an interaction or dependency, „one change at

a time‟ testing relies on the experimenter carrying the tests in the first place, and then prescribes exactly what

data are needed to assess them i.e. whether input variables change the response on their own, when combined, or

not at all. In terms of resource the exact length and size of the experiment are set by the design (i.e. before

testing begins). DOE can be used to find answers in situations such as “what is the main contributing factor to a

problem?”, “how well does the system/process perform in the presence of noise?”, “what is the best

configuration of factor values to minimize variation in a response?” etc. In general, these quations are given

labels as particular types of studies. In the examples given above, these are problem solving, parameter design and robustness studies. In each case, DOE is used to find the answer; the only thing that makes them different is

factors used in the experiment.

The order of tasks to using this tool starts with identifying the input variables and the response (output) that is to

be measured. For each input variable, a number of levels are defined that represent the range for which the effect

of that variable is desired to be known. An experiment plan is produced which tells the experimenter where to

set test parameter for each run of the test. The response is then measured for each run. The method of analysis is

to look for differences between response (output) readings for different groups of the input changes. These

differences are then attributed to the input variables acting alone (called a single effect) or in combination with

another input variable (called an interaction). DOE is team oriented and a variety of backgrounds (e.g. design,

manufacturing, statistics etc.) should be involved when identifying factors and levels and developing the matrix

as this is the most skilled part. Moreover, as this tool is used to answer specific questions, the team should have a clear understanding of the difference between control and noise factors.

It is very important to get the most information from each experiment performed. Well – designed experiments

can produce significantly more information and often require fewer runs than haphazard or unplanned

experiments. In addition, a well-designed experiment will ensure that the evaluation of the effects that had been

identified as important. For example, if there is an interaction between two input variables, both variables should

be included in the design rather than doing a „one factor at a time‟ experiment. An interaction occurs when the

effect of one input variable is influenced by the level of another input variable. Designed experiments are

carried out in four phases: planning, screening (also called process characterization), optimization, and

verification.

3.1.1. Planning :

Careful planning helps to avoid problems that can occur during the execution of the experimental plan. For

example, personnel, equipment availability, funding, and the mechanical aspects of the system may affect the ability to complete the experiment. The preparation required before beginning experimentation depends on the

nature of the problem. The following are some of the steps that may be necessary.

Problem Definition: Developing a good problem statement helps make sure that the correct variables are

studied. At this step, the questions that need to be answered are identified.

Object Definition: A well – defined objective will ensure that the experiment answers the right questions and

yields practical, usable information. At this step the goals of the experiment will be defined.

Develop an experimental plan that will provide meaningful information: At this step it is necessary to make sure

that the relevant background information has been reviewed, such as theoretical principles, and knowledge

gained through observation or previous experimentation. For example, you may need to identify which factors

or process conditions affect process performance and contribute to process variability. Or, if the process is

already established and the influential factors have been identified, it may be necessary to determine the optimal process conditions.

Making sure the process and measured systems are in control: Ideally, both the process and the measurement

should be in statistical control as measured by a functioning statistical process control (SPC) system. Even if it

does not have the process completely in control, it must be able to reproduce process settings. Also, it is

necessary to determine the variability in the measurement system.

3.1.2. Screening

In many process development and manufacturing applications, potentially influential variables are numerous.

Screening reduce the number of variables by identifying the key variables that affect product quality. This

reduction allows process improvement efforts to be focused on the rally important variables, or the “vital few.”

Screening may also suggest the “best” or optimal settings for these factors, and indicate whether or not

curvature exists in the responses. Then, it can use optimization methods to determine the best settings and define

the nature of the curvature. Two – level full and fractional factorial designs are used extensively in industry.

Plackett – Burman designs have low resolution, but their usefulness in some screening experimentation and

robustness testing is widely recognized. General full factorial designs (designs with more than two – levels) may also be useful for small screening experiments.

3.1.3. Optimization

Next step after identified the “vital few” by screening, the “best” or optimal values for these experimental

factors needed to be determine. Optimal factor values depend on the process objective. For example, maximize

the welding speed and minimize the laser power.

3.1.4. Verification

Verification involves performing a follow – up experiment at the predicted “best” processing conditions to

confirm the optimization results.

3.2. Taguchi Design

3.2.1. Overview

Dr. Genichi Taguchi is regarded as the foremost proponent of robust parameter design, which is an engineering

method for product or process design that focuses on minimizing variation and/ or sensitivity to noise. When used properly, Taguchi designs provide a powerful and efficient method for designing products that operate

consistently and optimally over a variety of conditions. In robust parameter design, the primary goal is to find

factor settings that minimize response variation, while adjusting (or keeping) the process on target. When the

factors affecting variation have been determined, it could be used to find settings for controllable factors that

will either reduce the variation, make the product insensitive to changes in uncontrollable (noise) factors, or

both. A process designed with this goal will deliver more consistent performance regardless of the environment

in which it is used. Engineering knowledge should guide the selection of factors and responses.

3.2.2 The fundamental Terms Used in Taguchi Design

3.2.2.1. Orthogonal arrays

The taguchi method utilizes orthogonal arrays from design of experiments theory to study a large number of

variables with a small number of experiments. Using orthogonal arrays significantly reduces the number of experimental configurations to be studied. Furthermore, the conclusions drawn from small scale experiments are

valid over the entire experimental region spanned by the control factors and their settings.

Orthogonal arrays are not unique to Taguchi. They were discovered considerably earlier. However, Taguchi has

simplified their use by providing tabulated sets of standard orthogonal arrays and corresponding linear graphs to

fit specific projects.

Examples of standard orthogonal arrays:

L-4, L-8, L-12, L-16, L-32 and L-64 all at 2 levels

L-9, L-18 and L-27 at 3 & 2 levels

L-16 and L-32 modified at 4 levels

L-25 at 5 levels

Standard notations for orthogonal arrays:

L-16 (3 5), 16 = Number of experiments 3 = Number of level 5 = Number of factors

To select an appropriate orthogonal array for the experiments, the total degrees of freedom need to be computed.

The degree of freedom are defined as the number of comparisons between process parameters that need to be

made to determine which level is better and specifically how much better it is. For example, a two – level

process parameter counts for one degree of freedom. The degrees of freedom associated with interaction

between two process parameters are given by the product of the degrees of freedom for the two process

parameters. In the present study, the interaction between the laser welding parameters is considered.

Once the degrees of freedom are known, the next step is selecting an appropriate orthogonal array to fit the

specific task. The Degrees of freedom for the orthogonal array should be greater than or at least equal to those

for the process parameters. The tabulations of the typical L16 & L18 orthogonal arrays used in this research

with coded values are shown in tables 3.1. and 3.2. Table 3.1: Typical L16 orthogonal array with coded value

Std Run Factor 1 Factor 2 Factor Factor 4 Factor Response 1

1 1 1 1 1 1 1

6 2 2 2 1 4 3

8 3 2 4 3 2 1

2 4 1 2 2 2 2

5 5 2 1 2 3 4

4 6 1 4 4 4 4

10 7 3 2 4 3 1

15 8 4 3 2 4 1

16 9 4 4 1 3 2

14 10 4 2 3 1 4

13 11 4 1 4 2 3

7 12 2 3 4 1 2

12 13 3 4 2 1 3

11 14 3 3 1 2 4

3 15 1 3 3 3 3

9 16 3 1 3 4 2

Table 3.2: Typical L18 orthogonal array with coded value

Control Factors

Expt. No. A B C D E F G H

1 1 1 1 1 1 1 1 1

2 1 1 2 2 2 2 2 2

3 1 1 3 3 3 3 3 3

4 1 2 1 1 2 2 3 3

5 1 2 2 2 3 3 1 1

6 1 2 3 3 1 1 2 2

7 1 3 1 2 1 3 2 3

8 1 3 2 3 2 1 3 1

9 1 3 3 1 3 2 1 2

10 2 1 1 3 3 2 2 1

11 2 1 2 1 1 3 3 2

12 2 1 3 2 2 1 1 3

13 2 2 1 2 3 1 3 2

14 2 2 2 3 1 2 1 3

15 2 2 3 1 2 3 2 1

16 2 3 1 3 2 3 1 2

17 2 3 2 1 3 1 2 3

18 2 3 3 2 1 2 3 1

3.2.2.2. S/N rations and MSD analysis

Taguchi recommends the use of signal to noise (S/N) as opposed to simple process optimizing process

parameters. The rationale is that while there is a need to maximizing the mean (signal) in the sense of its

proximity to nominal value, it is also desirable to minimize the process variations (noise). The use of S/N accomplishes both objectives simultaneously.

In order to evaluate the influence of each selected factor on the responses: The S/N for each control factor

should be calculated. The signals have indicated that the effect on the average responses, which would indicate

the sensitiveness of the experiment output to the noise factors. The appropriate S/N ratio must be chosen using

previous knowledge, expertise, absent signal factor (Static design), it is possible to choose the S/N ratio

depending on the goal of the design. S/N ratio selection is based on Mean Squared Deviation (MSD) for analysis

of repeated results. MSD expression combines variation around the given target and is consistent with Taguchi‟s

quality objective. The relationships among observed results, MSD and S/N rations are follows (3.1 to 3.4):

(( ̅) ( ) ( ̅) ) …………. For nominal is better ……………. (3.1)

(

) …………. For smaller is better ……………. (3.2)

(

)

…………. For bigger is better ……………. (3.3)

S/N = - 10Log (MSD) …………. For all characteristic ……………. (3.4)

3.2.2.3. Analysis of variance (Anova) Analysis of variance (analysis of variance) is a general method for studying sampled – data relationships. The

method enables the difference between two or more sample means to be analyzed, achieved by subdividing the

total sum of squares. One way anova is the simplest case. The purpose is to test for significant differences

between class means, and this is done by analyzing the variances. Analysis of variance (anova) is similar to

regression in that it is used to investigate and model the relationship between a response variable and one or

more independent variables. In effect, analysis of variance extends the two sample t – test for testing the

equality of two population means to a more general null hypothesis of comparing the equality of more than two

means, versus those that are not all equal. Table 3.3 is a sample of the Anova table used for analysis of the

models developed in this work. Sum of squares and mean square errors are calculated using Eq. 3.5 to 3.8.

Table 3.3: Sample anova table for a model

Source SS df MS FV – Value Prob.>Fv

Model SSM p Each SS Divided by Its df Each MS Divided by MSE From Table or

automatically

from the

software

P SSI

S SS2

F SS3

PS SS12

PF SS13

SF SS23

P2 SS11

S2 SS22

F2 SS33

Residual SSE n – p – 1

Cor. Total SSt n – l - - -

Where, p : Number of coefficients in the model.

df : Degree of freedom,

SS : Sum of squares, MS : Mean squares,

n: Total number of runs (For this work n = 16 or 25) Cor. Total : Sum of squares total corrected for the mean.

∑( ̂ ̅) ( )

∑( ̂ )

( )

∑( ̅) ( )

( )

3.3 Optimization

The optimization will allow the industrial user to achieve the optimum welding composition and process

parameter to achieve the desired weld pool shape and mechanical properties. All independent variables are

measurable and can be repeated with negligible error. The objective function can be represented by :

Objective = f (x1, x2, …………… , xn) ………………… (3.9)

Where : n is number of independent variables.

3.3.1. Determination of optimal condition(s). With time, complexity in welding process dynamics has increased and as a consequence, problems related to

determination of optimal or near – optimal welding condition(s) are faced with discrete and continuous

parameter spaces with multimodel, differentiable as well as non differentiable objective function or response (s).

search for optimal or acceptable near optimal solution (s) by a suitable optimization technique based on input –

output and in – process parameter relationship or objective function formulated from model(s) with or without

constraint (s), is a critical and difficult task for researchers and practitioners. A large number of techniques have

been developed by researchers to solve these types of parameter optimization problems, and my be classified as

conventional and nonconventional optimization techniques. Fig. 1 provided a general classification of

parameters relationships modeling and optimization techniques in welding.

Whereas conventional techniques attempt to provide a local optimal solution, non – conventional techniques

based on extrinsic model or objective function development, are only an approximation, and attempt to provide

near – optimal welding condition (s). Conventional techniques may be broadly classified into two categories. In the first category, experimental techniques that include statistical design of experiment.

Such as taguchi method, and response surface design methodology (RSM) are referred to. In the second

category, iterative mathematical search techniques, such as linear programming (LP), non-linear programming

(NLP), and dynamic programming (DP) algorithms are included. Non – conventional meta- heuristic search –

based techniques, which are sufficiently general and extensively used by researchers in recent times are based

on genetic on genetic algorithm (GA), tabu search (TS), and simulated annealing (SA).

3.4. Experimental Procedure

The taguchi method is used to improve the quality of welded components improved quality results when a

higher level of performance is consistently obtained. The highest possible performance is obtained by

determining the optimum combination of design factors. The consistency of performance is obtained by making

the process insensitive to the influence of the uncontrollable factor. In Taguchi‟s approach, optimum design is

determined by using design of experiment principles, and consistency of performance is achieved by carrying

out the trial conditions under the influence of the noise factors. The following steps are performed in order to develop and optimizing a mathematical model in case of

dissimilar laser welding.

3.4.1. Planning Experiments (Brainstorming)

This is a first step in any application. The session should include individuals with firsthand knowledge of the

project. The literature review covers this step.

- Determine the vital process factors in this study; the laser welding variables were determined fro the literature

review.

- Identify all influencing factors and those to be included in the study. The selected welding parameters for this

study are: welding power, welding speed, focus point position and gap between the plates to be jointed in some

butt welding experiments.

Determine the factor levels. Before determining the factor levels the operating range has been determined

through a pilot experiment which is carried out by changing one factor at time. Visual inspections were carried

out and the criteria selected for accepting the applicable range were; the absence of welding defects, a

continuous, smooth the uniform welding line and in some experiments a full depth penetration were decided.

Once the operating range was determined, Design-Expert 7 software was used to divide the operating range into

levels according to the selected design. Three and five levels were chosen depending on a select orthogonal

array.

Optimizing tools and techniques

Conventional techniques (Optimal Solution) Non - Conventional techniques [Near Optimal Solution(s)]

Design of Experiment (DOE) Mathematical Iterative search Meta Heuristic Search Problem specific Heuristic Search

Dynamic

Programming

(DP) – based

algorithm

Non – linear

Programming

(NLP) – based

algorithm

Linear

Programming

(LP) – based

algorithm Genetic

algorithm Simulated

Annealing

Tabu

Search

Taguchi

Method -

Based

Factorial

Design

based

Response surface

Design Methodology

(RSM) - based

3.1.1. Classification of modeling and optimization techniques in welding problems

3.4.2 Designing Experiments

Using the factors and levels determined in the previous step, the experiments now can be designed and the

method carrying them out established. To design the experiment, implement the following:

-Select the appropriate orthogonal array.

In the present study, the interaction between the welding parameters is considered. Therefore, degrees of

freedom owing to the three level welding process parameters were evaluated. Tht degrees of freedom for the

orthogonal array should be greater than or at least equal to those for the process parameters. In this study, L 18

orthogonal arrays were used.

3.4.3 Running Experiment

All the experiments of laser welding were carried out (during joining process only) in random order of the

developed matrices by the software to avoid any systematic error during the experiments. After the joining

process the responses, mentioned earlier in this work, were tested and measured in sequential order following

the standard procedures when available for each response. An average of at least three (in most cases) recorded

measurements in calculated and considered for more analysis.

3.4.4 Analyzing Results

Before analysis, the raw experimental data might have to be combined into an overall evaluation criterion. This

is particularly true when there are multiple criteria of evaluation.

Analysis is performed to determine the following:

The optimum design.

Influence of individual factors.

Performance at the optimum condition.

Relative influence of individual factors.

The steps in this analyzing stage are following in this sequence:

3.4.4.1 Developing the mathematical model

Design expert software develops and exhibits the possible modules which can fit input data and suggest the

model that best fits the experiment data.

3.4.4.2 Estimating of the coefficients of the model independent factors

Regression analysis is carried out by software to estimate the coefficients for all factorsin each experiment.

3.4.4.3 The Signal-to-noise (S/N) ratio analysis

A signal to noise ratio in the ANOVA Table is presents as an Adequate Precision. Equations 3.15 and 3.16 are

applied to the model to compares the range of the predicted values at the design points to the average prediction

error. Ratios greater than 4 indicate adequate model discrimination.

Adequate Precision max(Y) min(Y)

4V(Y)

…(3.15)

2n

f 1

1 PV(Y) V(Y)

n n

…(316)

P = number of model parameters, 2 = residual MS from ANOVA table, n = number of experiments.

3.4.4.4 ANOVA Outputs

The analyses of variances (ANOVA) were applied to test adequacy of the developed models. Each term in

developed models was examined by the following statistical significance tools using Eq. 3.15-3.20 [140]:

VF value: Test for comparing model variance with residual (error) variance. When the variances are close to

each other, the ratio will be close to one and it is less likely that any of the factors have a significant effort on the

response. Model VF =Value and associated probability value (Prob.> VF ) to confirm model significance. VF

value is calculated by term mean square divided by residual mean square.

Prob.> VF : Probability of seeing the observed VF value if the null hypothesis is true (there is no factor effect).

If the Prob.> VF of the model and/or of each term in the model does not exceed the level of significance (for

chosen a = 0.05 in this work) then the model can be considered adequate within the confidence interval (1-a).

Precision of a parameter estimate is based on the number of independent samples of information which can be

determined by degree of freedom f(d ).

Degree of Freedom f(d ) : the degree of freedom equals to the number of experiments minus the number of

experiments minus the number of additional parameters estimated for that calculation.

The same tables show also the other adequacy measures 2R , adjusted

2R and adequacy precision 2R for each

response. In this study, all adequacy measures were close to 1, which indicates adequate models.

The Adequate Precision compares the range of the predicated value at the design points to the average predicted

error. The adequate precision ratio above 4 indicates adequate model discrimination. In this study, the values of

adequate precision are significantly greater than 4.

2 r

M r

SSR 1

SS SS

…(3.17)

2 2n 1Adj. R (1 R )

n p

…(3.18)

2

r M

PRESSPredicted R 1

SS SS

…(3.19)

1

n2

f i ,f 1

PRESS (Y Y )

…(3.20)

3.4.4.5 Model reduction

Model reduction consists of eliminating those terms that are not desired or which are statistically insignificant.

In this case it was done automatically by the software used. For each response regression the starting model can

be edited by specifying fewerecandidate terms than the full model would contain. In the three automatic

regression variations, those terms which are forced into the model regardless of their entry/exit a value could be

controlled. There are three basic types of automatic model regression: Step-Wise: A term is added, eliminated or

exchanged at each step. Step-wise regression is a combination of forward and backward regressions. Backward

elimination: A term is eliminated at each step. The backward method may be the most robust choice since all

model terms will be given a chance of inclusion in the model. Conversely, the forward selection procedure starts

with a minimal core model, thus some terms vever get included. Forward selection: A term is added at each

step.

3.4.4.6 Development of final model form

The program automatically defaults to the “Suggested” polynomial model which best fits the criteria discussed

in the Fit Summary section. The responses could be predicted at any midpoints using the adequate model. Also,

essential plots, such as Contour, 3D surface, and perturbation plots of the desirability function at each optimum

can be used to explore the function in the factor space. Also, any individual response

3.4.5 Running Confirmation Experiments

The final step is to predict and verify the improvement of the response using the optimal level of the welding

process parameters. In addition, to verify the satisfactoriness of the developed models, at least three

confirmation experiments were carried out using new test conditions at optimal parameters conditions, obtained

using the Design Expert software.

3.5 Gray system theory

3.5.1 Overview

The multi-criteria decision-making problem must be determined not with the exact criteria values, but with

fuzzy values or with values taken from some intervals. Deng (1982) developed the Grey system theory.

According to him, the Grey relational analysis has some advantages: it involves simple calculations and required

a smaller number of samples; a typical distribution of samples is not needed; the quantified outcomes from the

Grey relational grade do not result in contradictory conclusions from the qualitative analysis; the Grey relational

grade model is a transfer functional model that is effective in dealing with discreate4 data (Deng 1988).

3.5.2 The Meaning of ‘Grey’ in Grey System

The cognition of our natural and/or artificial universe has been a tedious and a progressive process. The

formulations of natural and artificial laws are certainly not overnight happenings. Nature to us is not white (full

of precise information), but on the other hand, it is not black (completely lack of information) either, and it is

mostly grey (a mixture of black and white). Our thinking, no matter how analytical, is grey, while our action and

reaction, no matter how practical, is also grey. In fact, since the beginning of our existence, we are confined in a

high dimensional grey information relational space.

Natural phenomena have given us numerous difficult problems. We are confronted with numerous such grey

systems: social system, environmental system, economic system, human anatomical system, and our own human

race relational system, just to name a few. To insure continuation of our very existence, it is imperative that we

investigate and understand these systems. However, given our present knowledge or scientific information, we

have to simplify much of the complex embodiment of these systems. During this process, we have to delete

information left and right. After such an endeavor, we have a system that only possesses bone but no flesh and

blood. Such a model can only be at best homomorphic to, or vaguely resemble the original system. As a result,

we can only command partial information, that can be extracted from the system, the color we can obtain from a

system is grey. Therefore, the grey of a system is absolute, and the black and white of a system is grey.

Therefore, the grey of a system is absolute, and the black and white of a system is relative. Confronting such

truths, in 1982, Professor Deng Dulong of Huazhong University of Science and Technology, P.R.C, wrote the

first landmark article Control Problem of Grey Systems, the hence started the theory of grey system. This

inaugural article enunciated the concepts and numerical methods of treating systems wherein only partial

information was known. It was recognized as a great breakthrough contribution in the in depth study of system

theory.

What is a characteristic of a grey system? The incompleteness of information is the basic characteristic, and it

serves as the fundamental starting point of the investigation of grey systems. The emphasis is to discover the

true properties of these systems under poorly informed situation. The main melody of grey system theory is t

supply information so that we can within the greyness. Incomplete information follows from the limited

availability of data. Therefore, incomplete data analysis is really the theory of scarce (or few) data analysis. The

central problem of grey system theory is to seek only the intrinsic structure of the system given such limitation

of data. In other words, we need to devise a methodology to achieve an early understanding of the system under

this predicament. Out of whatever complexity of a given system, information in still its basic elements.

What is information ? Most people identify information as numerical data. In grey system theory, we consider

such a concept to be narrow. In reality, data is only part of the total information. Information should consist of

two types. The first is the qualitative elements; that is, the type that cannot be measured, and it exemplifies the

information‟s qualitative appearance. The second type is the quantitative data elements, exemplifying its

measurable property. In real life, we may be faced with a system, knowing only part of its informational

qualitative elements and no more. At the same time, we may know only certain variation intervals of its

informational quantitative data elements, with their precise numerical values unknown. No doubt such a system

has only provided us with information that are grey. Furthermore, such grey information in the system may be

constraining each other, and they may be very interdependent with each other. So such intrinsic relational

behavior may differentiate one grey system from another. Therefore, relations between grey information

constitute another central study of grey system theory.

Facing the challenge of understanding our nature and ourselves, we have built many classical system, and have

devoted a long period of time in investigating them. Unfortunately, in keeping up with our high scientific and

technological development the system‟s complexity and the technological quest for rigor and precision have

become paradoxically uncompromising. At this serious juncture, in 1965, L. A. Zadeh enunciated the famous

Fuzzy Logic (Fuzzy Set) Theory, and thus created the Fuzzy System. As we know, the theory of classical

system based itself on the classical Cantor set theory. In that, element x has only two exemplification‟s x A or

x A (Boolean Logic). Fuzzy systems based on the fuzzy set theoretic membership function A (x), takes

values from 0 to 1, instead of 0 or 1. Therefore, the difference between classical and fuzzy systems is merely the

unclear boundaries of the systems and the imprecise intrinsic attribute of the systems. However, the incomplete

information (or how much do we know of the system) still eludes the attention of the classical or fuzzy set

theorists. This objective characteristic of the system seems to be forgotten in the development of classical or

fuzzy system theory. Grey system, in addition, focus keenly on what partial or limited information the system

can provide, and try to paint its total picture from this. In fact, the theory or grey system bases itself on Grey

Hazy set. Grey huzy set exemplify itself in several stages: the embryonic state, the hazy state, the4 white4ning

state, and the verifiable state. Grey hazy set possesses several properties: co-existence (co-habitationality),

verifiability, time effectiveness, informationality, and constructability. It can be seen therefore, that grey hazy

set is completely different from the classical Cantor set and Zadeh‟s fuzzy set. Nevertheless, the Cantor set

(crisp) is the transparent nuclear state of the grey hazy set, and classical and/or fuzzy systems are special cases

of grey systems when the degree of greyness is zero. The mathematical foundation of grey systems and fuzzy

systems are rather different in their formulation. We have to point out, after studying the two systems, grey hazy

set possesses excellent characteristics in treating natural dynamical and static systems.

3.5.3 Grey Information Relation vs Fuzzy Relations

Let us recall our classical ordinary (white) relations. For x, y A, let xRy denote the relation between x and y,

and xRy denote that x and y are not related. Here R is a relation between the elements of set A. Using numerical

values to describe the above relation we have: xRy => R = 1 and xRy => R=0. Note the use of {0,1} to describe

this Boolean relation. Fuzzy system generalizes this extrems relational value to the full interval [0,1], and thus

gives us a large convenience in studying the system. This is no doubt a big step forward in system theory. As

fuzzy relation give: x y A => (xRy) (x, y) [0,1], and this replaces the classical relation: x y A => xRy or

xRy. However, neither classical relation nor fuzzy relation focuses on a profound concept of the property of

self-characteristics of the elements x and y of A. For example, x y A contain how much information, or what

are the degree of greyness in x (or y) itself. Questions like these were never addressed by either classical or Fuzy

theory. Grey system theory treats this fundamental academic perspective and emphasizes the incompleteness of

system information and the greyness of elements themselves. These are the foundations of the research of grey

information relation theory. In such an investigation, one has to start by paying attention to the greyness and/or

whiteness of the elements under focus, and these elements pervade our information decision space. In fact, these

relation between the grey-white elements further relate4 to the degree of greyness (grey relational grade) of the

elements themselves. In addition not all the elements in the dynamical system are grey, for in time, some grey

elements, through the whitening process, would become white elements. Further, the grey information relations

and the white information relations in the system often mingle to form certain grey-white information relations,

which define the transparency grade of information decision making. Therefore the creation of the grey system

theory, as it breaks through the investigation of such relation, is indeed a great leap forward for the system

theory research. So we see grey information relation is built on the foundation of grey hazy set theory, which is

dynamical in nature, where fuzzy relation is built on fuzzy set theory, which is a mere convenient extension of

the classical set theory. Is rather static in nature. From this point of view, and because of their conceptual

foundations, problems described by grey information relation, and by fuzzy relation are in fact different.

3.5.4 Grey Relational Model

3.5.4..1 Existence of Grey Relation

Objective observation of many existing systems shows they consist of a number of subsystems, and the relations

between these subsystems are extremely complex. In particular, the different states of appearances and the

randomness of changes (chaotic system), cause great confusion in the cognition of the true nature of the

systems. But the very essence of grey system theory is to provide an analytic concept of the grey relational

degree of these subsystems. Here the central methodology is to seek out the relations (including the numerical

relations) between subsyste3ms and sub causalities. We find, in the course of grey systems research, that if the

basic states of causal changes of two subsystems are similar, their synchronized degree of changes is high, and

hence their grey relational grade is high; otherwise their grey relational grade is low. Therefore, we can provide

a quantitative measure in grey relational analysis of systems during the course of its dynamic. There are

differences between grey relational analysis and the regression analysis of statistics. In that:

1. They are different in their theoretical foundations. Grey relational analysis is based on the grey process

of the grey system theory, whereas regression analysis is based on the random process of the

probability theory;

2. Grey relational analysis compares and computes the dynamic causalities of the subsystems of the given

system, whereas regression analysis focuses on the grouped values of the random variables;

3. Grey relational analysis requires very minimal raw data (as few as 4 in cardinality), whereas regression

analysis require sufficiently large set of sample data; and

4. Grey relational analysis mainly investigates and dynamic process of the system, whereas regression

analysis mainly studies the static behavior of the system.

3.5.4.2 Grey Relational of Primitive Data

3.5.4.2.1 The Processing of Primitive Data

The physical meanings of the causal elements in a system could be different. As a result there are

differences in the system‟s data index (catalog), and during the process of analytic comparison, we find

difficulty in reaching a proper are correct conclusion. Therefore, we use:

1. Mean value processing. We first compute the mean values of all the primitive sequences x1,

X2,…-, Xp (data space of the dynamic). Then we use these mean values to divide values of the

corresponding sequences to obtain a collection of new sequences, which is now called the mean

valued sequences –Xi, X2,….., Xp.

2. Initial value processing. We use the first value of each sequence to divide each succeeding value of

the corresponding sequence to form a collection of quotient sequences, which are now called the

initialized sequences, Xi, X2,…., Xp.

In general when analyzing the dynamic process of certain stable socio-economic systems, we often

employ this initial valued process.

3.5.4.2.2 Grey Relational Coefficient

Let X = {Xi I I 1} be a space sequence where I = {1, 2, …., n}. If we denote the numerically

proessed parent sequence 0X by 0X (1), 0X (2),….. 0X (p), the generated sequence iX { by x; (1),

iX (2),….. iX (p), then the grey relational coefficient 0 , i(k) of iX (k) is defined to be:

min max0,i

i max

(k)(k)

Where i 0 i(k) X (k) X (k)

Is the absolute difference of the two comparing sequences 0X and iX ,

max jj I k

min min (k)

min jj I k

min min (k)

Are respectively the maximum and minimum values of the absolute differences of all comparing

sequences, and [0, 1] is a distinguishing coefficient, the purpose of which is to weaken the

effect of maxA when it gets too big, and thus enlarges the difference significance of the relational

coefficient, 0 , i(k) reflect the degree of closeness between the two comparing sequences at k. At

minA , 0 , i=1, that is, the relational coefficient attains its largest value. While at maxA , ^0 attains

the smallest value. Hence 0, i i0 1 .

3.5.4.2.3 Grey Relational Grade

In reality, grey relational analysis compares relations of sequences in their appropriate metric

spaces. If two sequences agree at all points, then their grey relational coefficient is 1 everywhire,

and therefore, their grey relational grade should be 1. In view of this, the relational grade of two

comparing sequence can be quantified by the mean value of their grey relational coefficients; i.e.

Here 70 is designated as the grey relational grade between X, and 0X , and p is the length of the

two comparing sequences.

p

0, i 0,ik 1

1(k)

p

3.5.4.2.4 Grey Relational Ordering

In relational analysis, the practical meaning of the numerical values of grey relational grades

between elements is not absolutely important, while the grey relational ordering between them

yields more subtle information. Here, being primary or secondary form the bases of decision

making.

1. If o o, , we say X to 0X is better than X to , 0X and we denote this

0 0X X X X .

2. o o, , we say X to 0X is worse than X to 0X , and we denote this

0 0X X X X

3. o o, , we say X to 0X is worth equally this X to 0X , and we denote this

0 0X X X X

3.5.4.2.5 Relational Matrix

If we have n parental sequences Yi, Y2, …., Yn, n 1, and m offspring (generated)

sequences 1 2 m,X ,X ,.....X m 1 , then the relational grades of the parental sequences

1 2 nY ,Y ,..... Y to each offspring sequences are:

0 1,2, , 1,1. ,m

0,2 0,2,2 , 2. ,m

n,2. n,2,2-, n, m

When arranged properly we have either

1,1 1,2 1,m 1,1 1,2 1,n

2,1 2 ,2 2 ,m 2,1 2 ,2 2 ,n

n ,1 n ,2 n ,m m,1 m,2 m,n

... ...

... ...R or R =

... ...

Matrices of grey relational grade, which from the bases of decision making. Given a relational matrix, if for all

I, the columns satisfy :

1, j1,i

2 , j2 ,i

m , jm ,i

Where j = 1, 2, ……. , n, j I, we say yi is optimally better than Yi, j . In order words, the relational grade of

Xi, Yi, is optimumly the best columns of the system, and we write

Yi>>Yj, j = 1,2,….,n,j i

If n n

k,i k , jk 1 k 1

1 1,i, j 1,2,...,m, i j,

n n we say Yi relative to Yj in respect to relational

grade of Xi is pseudo optimumly the best in the system, and we denote as

Yi > Yj, I, j = 1,2,…,n,j i