50

CHAPTER III

EXPERIMENTAL DESIGN AND ANALYSIS

Conducting experiments and drawing correct inferences from experimental

observations are two most important prerequisites of any scientific study aimed at

product or process development. Clarity of objectives and proper planning of

experiments is very crucial for assimilating accurate and meaningful conclusions

from the experimental results. Design of experiment is a very powerful tool for

accomplishing these objectives. Adopting robust design of experiments adds value

and reliability to the experimental results besides certain other advantages, like cost

cutting by reducing the number of experimental runs and trials; and determination

and reduction of experimental errors. It is therefore imperative upon researchers to

properly plan and conduct experiments to obtain adequate relevant data which

facilitates interpreting maximum knowledge from the experimental data thereby

developing better insight of the process or subject. The most widely used

experimental approaches are briefly discussed as under,

(1) Trial-and-Error/One-Factor at-a-Time Approach:

Performing a series of experiments arbitrarily or by varying one factor at-a-time

approach wherein each experiment gives some understanding of the basic

phenomena and the effect of individual parameters. This approach usually requires

very large number of experiments, is labour intensive and time consuming and may

be expensive, and at times does not depict the correct behaviour of the process

parameters, and does not give the interaction effect of parameters.

(2) Full Factorial Experiments:

A well planned set of experiments, in which all parameters of interest are varied over

a specified range, is a much better approach to obtain systematic data.

Mathematically speaking, such a complete set of experiments ought to give desired

results. Usually the number of experiments and resources (materials and time)

required are prohibitively large. The analysis is at times tedious and thus effects of

various parameters on the observed data may not be readily apparent. In many

cases, particularly those in which some optimization is required, the method does

not directly point to the BEST settings of parameters.

(3) TAGUCHI Method:

Taguchi of Nippon Telephones and Telegraph Company, Japan has developed a

method based on "ORTHOGONAL ARRAY" experiments which gives much reduced

"variance" for the experiment with "optimum settings" of control parameters. Thus

51

the marriage of Design of Experiments with optimization of control parameters to

obtain BEST results is achieved in the Taguchi Method. "Orthogonal Arrays" (OA)

provide a set of well balanced (minimum) experiments and Taguchi's Signal-to-Noise

ratios (S/N), which are log functions of desired output, serve as objective functions

for optimization, help in data analysis and prediction of optimum results.

The TAGUCHI design of experiment together with the statistical techniques applied

for analysis and validation of results viz. Monte Carlo Simulation, Analytical Hierarchy

Process, Technique for Order Preference by Similarity to Ideal solution, Utility

Concept, Grey Relational Analysis and Response Surface Methodology Modelling

have been discussed in the following sections.

3.1 TAGUCHI EXPERIMENTAL DESIGN AND ANALYSIS

Taguchi’s comprehensive system of quality engineering is one of the greatest

milestones of the 20th century. His methods focus on the effective application of

engineering strategies rather than advanced statistical techniques. It includes both

upstream and shop-floor quality engineering. Upstream methods efficiently use

small-scale experiments to reduce variability and remain cost-effective, and robust

designs for large-scale production and market place. Shop-floor techniques provide

cost-based, real time methods for monitoring and maintaining quality in production.

The farther upstream a quality method is applied, the greater leverages it produces

on the improvement, and the more it reduces the cost and time. Taguchi’s

philosophy is founded on the following three very simple and fundamental concepts

[200,201]:

Quality should be designed into the product and not inspected into it.

Quality is best achieved by minimizing the deviations from the target. The

product or process should be so designed that it is immune to uncontrollable

environmental variables.

The cost of quality should be measured as a function of deviation from the

standard and the losses should be measured system-wide.

Taguchi proposes an “off-line” strategy for quality improvement as an alternative to

an attempt to inspect quality into a product on the production line. He observed that

poor quality cannot be improved by the process of inspection, screening and

salvaging. No amount of inspection can put quality back into the product. Taguchi

recommended a three-stage process: system design, parameter design and

tolerance design [200,201]. In the present work Taguchi’s parameter design

approach is used to study the effect of process parameters on the quality

characteristics.

52

Taguchi recommended orthogonal array (OA) for laying out the design of

experiments. These OA’s are generalized Graeco-Latin squares. Designing an

experiment requires selection of the most suitable OA and assignment of parameters

and interactions of interest to the appropriate columns. The use of linear graphs and

triangular tables suggested by Taguchi makes the assignment of parameters simple

[201].

In the Taguchi method the results of the experiments are analyzed to achieve one or

more of the following objectives [200]:

To establish the best or the optimum condition for a product or process

To estimate the contribution of individual parameters and interactions

To estimate the response under the optimum condition

The optimum conditions are identified by studying the main effects of each of the

parameters. The main effects indicate the general trend of influence of each

parameter. The knowledge of contribution of individual parameter plays a key role in

deciding the nature of control to be established on a production process. The

analysis of variance (ANOVA) is the most commonly used statistical treatment

applied to the results obtained from the experiments in determining the significance

and percent contribution of each parameter against a stated level of confidence.

Study of ANOVA table for a given analysis helps to determine which of the

parameters need control [200].

Taguchi suggested two different routes to carry out the complete analysis, as

reported by Roy[201]. In the first approach, the results of a single run or the average

of repetitive runs are processed through main effect and ANOVA analysis (Raw data

analysis). The second approach is for multiple runs where signal- to- noise ratio (S/N)

is used. The S/N ratio is a concurrent quality metric linked to the loss function as

suggested by Barker [202]. By maximizing the S/N ratio, the loss associated with a

product or process can be minimized. The S/N ratio determines the most robust set

of operating conditions from variation within the results. The S/N ratio is treated as a

response parameter (transform of raw data). Taguchi recommended the use of outer

OA to force the noise variation into the experiment i.e. the noise is intentionally

introduced into experiment, as reported by Ross [200]. Generally, the processes are

subjected to many noise factors that in combination strongly influence the variation

of the response. Most often, it is sufficient to generate repetitions at each

experimental condition of the controllable parameters and analyze them using an

appropriate S/N ratio, as reported by Byrne and Taguchi [203].

53

3.1.1 TAGUCHI LOSS FUNCTION

The heart of Taguchi method is his definition of the nebulous and elusive term

“quality‟ as the characteristic that avoids loss to the society from the time the

product is shipped, Braker [204]. Loss is measured in terms of monetary units and is

related to quantifiable product characteristics. Financial loss is united with the

functional specifications through a quadratic relationship that comes from a Taylor

series expansion, as reported by Roy [201].

(3.1)

Where

L = Loss in monetary units

m = value at which the characteristic should be set

y = actual value of the characteristic

k = constant depending on the magnitude of the characteristic and the monetary

unit involved

The following two observations can be made from figure 3.1 which represents the

difference between the traditional and the Taguchi loss function concept.

The farther the product’s characteristic from the target value, the greater is

the loss. The loss must be zero when the quality characteristic of a product

meets its target value.

The loss is a continuous function and not a sudden step as in the case of

traditional approach. This consequence of the continuous loss function

illustrates the point that merely making a product within the specification

limits does not necessarily mean that product is of good quality.

In a mass production process, the average loss per unit can be expressed by eqn.3.2

as given by Roy[239]:

(3.2)

where

y1, y2,….…,yn = Actual value of the characteristic for unit 1, 2,…n respectively

n = Number of units in a given sample

k = Constant depending on the magnitude of the characteristic and the monetary

unit involved

m = Target value at which the characteristic should be set

54

The Eq. 4.2 can also be expressed as:

(3.3)

Where MSDNB = Mean squared deviation and represents the average of squares of all

deviations from the target or nominal value

NB = “Nominal is Best”

Figure 3.1(a, b): The Taguchi Loss-Function and The Traditional Approach [205]

55

(a)

(b)

Figure 3.2(a, b): The Taguchi Loss-Function for LB and HB Characteristics, Barker[202]

56

The loss-function can also be applied to product characteristics other than the

situation where the nominal value is the best value (m). The loss-function for a

“smaller is better‟ type of product characteristic (LB) is shown in Figure 4.2a. The loss

function is identical to the “nominal-is-best‟ type of situation when m=0, which is

the best value for “smaller is better‟ characteristic (no negative value). The loss

function for a “larger-is-better‟ type of product characteristic (HB) is also shown in

Figure 4.2b, where also m=03.1.2.

3.1.2 SIGNAL TO NOISE RATIO

Taguchi transformed the loss function into a concurrent statistic called S/N ratio,

which combines both the mean level of the quality characteristic and variance

around this mean into a single metric [202,205]. The S/N ratio consolidates several

repetitions (at least two data points are required) into one value. A high value of S/N

ratio indicates optimum value of quality with minimum variation.

The equation for calculating S/N ratios for “smaller is better‟ (LB); “larger is better‟

(HB); and “nominal is best‟ (NB) types of characteristics are as follows[203]:

1. Higher the Better:

(3.4)

2. Lower the Better:

(3.5)

57

3. Nominal the Best

(3.6)

The mean squared deviation (MSD) is a statistical quantity that reflects the deviation

from the target value. The expressions for MSD are different for different quality

characteristics. For “nominal the better‟ type of characteristic, the standard

definition of MSD has been used. For the other two characteristics the definition is

slightly modified. For “Lower the better‟, type the target value is zero. For “Higher

the better‟ type, the inverse of each large value becomes a small value and again the

target value is zero. Thus for all the three expressions, the smallest magnitude of

MSD is being sought. The constant 10 has been purposely used to magnify S/N

number for easy analysis and negative sign is used to set S/N ratio of “higher the

better” relative to the square deviation of the “lower the better”.

3.1.3 TAGUCHI PROCEDURE FOR EXPERIMENTAL DESIGN AND ANALYSIS

The stepwise procedure for Taguchi experimental design and analysis, as illustrated

in the flow diagram shown in Figure 3.3, has been described in the following

sections.

3.1.3.1 SELECTION OF ORTHOGONAL ARRAY (OA)

In selecting an appropriate OA, the following pre-requisites as reported by Ross[200]

and Roy[201] are to be taken care of:

Selection of process parameters and/or interactions to be evaluated

Selection of number of levels for the selected parameters

The determination of parameters to be investigate hinges upon the product or

process performance characteristics or responses of interest. Several methods are

58

Fig. 3.3 Taguchi Experimental Design and Analysis Flow Diagram

59

suggested by Taguchi for determining which parameters to include in an experiment.

These include [200]:

a) Brainstorming

b) Flow charting

c) Cause-Effect diagrams

The total Degrees of Freedom (DOF) of an experiment is a direct function of total

number of trials. If the number of levels of a parameter increases, the DOF of the

parameter also increases because the DOF of a parameter is the number of levels

minus one. Thus, increasing the number of levels for a parameter increases the total

degrees of freedom in the experiment which in turn increases the total number of

trials. Thus, two levels for each parameter are recommended to minimize the size of

the experiment. However, if curved or higher order polynomial relationship between

the parameters under study and the response is expected, at least three levels for

each parameter should be considered [202]. The standard two level and three level

arrays reported by Taguchi and Wu[206] are:

Two Level Arrays: L4, L8, L12, L16, L32

Three Level Arrays: L9, L18, L27

The number as subscript in the array designation indicates the number of trials in

that array. The total degrees of freedom (DOF) available in an OA are equal to the

number of trials minus one :

(3.7)

where

fLN = Total degrees of freedom of an OA

LN = OA designation

N = Number of trials

When a particular OA is selected for an experiment, the following inequality must be

satisfied (Ross, 1988):

fLN ≥ Total degree of freedom required for parameters and interactions (3.9)

Depending on the number of levels of the parameters and total DOF required for the

experiment, a suitable OA is selected.

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3.1.3.2 ASSIGNMENT OF PARAMETERS AND INTERACTIONS TO OA

An OA has several columns to which various parameters and their interactions are

assigned. Taguchi has provided two tools to aid in the assignment of parameters and

their interactions in the columns of OA, viz. linear graphs and triangular tables.

Each OA has a particular set of linear graphs and a triangular table associated with it.

The linear graphs indicate various columns to which parameters may be assigned

and the columns subsequently evaluate the interaction of these parameters. The

triangular tables contain all the possible interactions between parameters (columns).

Using the linear graphs and /or the triangular table of the selected OA, the

parameters and interactions are assigned to the columns of the OA. The linear graph

of L27 OA is given in Figure C.1 (Appendix C).

3.1.3.3 SELECTION OF OUTER ARRAY

Taguchi separates factors (parameters) into two main groups: controllable factors

and uncontrollable factors (noise factors). Controllable factors are factors that can

easily be controlled. Noise factors, on the other hand, are nuisance variables that are

difficult, impossible, or expensive to control. The noise factors are responsible for

the performance variation of a process. Taguchi recommends the use of outer array

for the noise factors and inner arrays for controllable factors. If an outer array is

used, the noise variation is forced into the experiment. However, experiments

against the trial conditions of the inner array (the OA used for the controllable

factors) may be repeated and in this case the noise variation is unforced into the

experiment [203]. The outer array, if used, will have same assignment

considerations. However, the outer array should not be complex as the inner array

because the outer array is noise only which is controlled only in the experiment

[200]. An example of inner and outer array combination is shown in Table C.1

(Appendix C).

3.1.3.4 EXPERIMENTATION AND DATA COLLECTION

The experiment is performed against each of the trial conditions of the inner array.

Each experiment at a trial condition is repeated simply (if outer array is not used) or

according to the outer array (if used). Randomization should be carried to reduce

bias in the experiment. The data (raw data) are recorded against each trial condition

and S/N ratios of the repeated data points are calculated and recorded against each

trial condition.

3.1.3.5 ANALYZING EXPERIMENTAL DATA

A number of methods have been suggested by Taguchi for analyzing the data viz.

observation method, ranking method, column effect method, ANOVA, S/N ANOVA,

61

plot of average response curves, interaction graphs etc. as reported by Ross[200].

However, in the present investigation the following methods have been used:

Plot of average response curves

Plot of S/N response graphs

ANOVA for S/N data

The plot of average responses at each level of a parameter indicates the trend. It is a

pictorial representation of the effect of parameter on the response. The change in

the response characteristic with the change in levels of a parameter can easily be

visualized from these curves. Typically, ANOVA for OA’s are conducted in the same

manner as other structured experiments.

The S/N ratio is treated as a response of the experiment, which is a measure of the

variation within a trial when noise factors are present. A standard ANOVA can be

conducted on S/N ratio which will identify the significant parameters (mean and

variation)[207].

3.1.3.6 PARAMETER CLASSIFICATION AND SELECTION OF OPTIMAL LEVELS

Average response curves and ANOVA of the SN ratio identifies the control factors,

which affect the average response and the variation in the response respectively.

The control factors are classified into four groups, Ross[200]:

Group I: Parameters which affect both average and variation

Group II: Parameters which affect variation only

Group III: Parameters which affect average only

Group IV: Parameters which affect nothing

The parameter design strategy is to select the suitable levels of group I and group II

parameters to reduce variation and group III parameters to adjust the average values

to the target value. The group IV parameters may be set at the most economical

levels.

3.1.3.7 PREDICTION OF MEANS

After determination of the optimum condition, the mean of the response (μ) at the

optimum condition is predicted. The mean is estimated only from the significant

parameters as identified by ANOVA. Suppose, parameters A and B are significant and

A2B2 (second level of A=A2, second level of B=B2) is the optimal treatment

condition. Then, the mean at the optimal condition (optimal value of the response

characteristic) is estimated, Ross[205]:

62

(3.8)

It may so happen that the predicted combination of parameter levels (optimal

treatment condition) is identical to one of those in the experiment. If this situation

exists, then the most direct way to estimate the mean for that treatment condition is

to average out all the results for the trials which are set at those particular

levels[205].

3.1.3.8 DETERMINATION OF CONFIDENCE INTERVAL

The estimate of the mean (μ) is only a point estimate based on the average of results

obtained from the experiment. Statistically this provides a 50% chance of the true

average being greater than μ. It is therefore customary to represent the values of a

statistical parameter as a range within which it is likely to fall, for a given level of

confidence as reported by Ross[200]. This range is termed as the confidence interval

(CI). In other words, the confidence interval is a maximum and minimum value

between which the true average should fall at some stated confidence interval[200].

The following two types of confidence interval are suggested by Taguchi for

estimated mean of the optimal treatment condition.

1. Around the estimated average of a treatment condition predicted from the

experiment. This type of confidence interval is designated as CIPOP (confidence

interval for the population).

2. Around the estimated average of a treatment condition used in a confirmation

experiment to verify predictions. This type of confidence interval is designated as

CICE (confidence interval for a sample group).

The difference between CIPOP and CICE is that CIPOP is for the entire population i.e., all

parts ever made under the specified conditions, and CICE is for only a sample group

made under the specified conditions. Because of the smaller size (in confirmation

experiments) relative to entire population, CICE must slightly be wider. Ross [200]

and Roy [208] have given the expressions for computing the confidence intervals as

under.

63

(3.9)

(3.10)

In Eq. 3.10, as R approaches infinity, i.e., the entire population, the value 1/R

approaches zero and CICE = CIPOP. As R approaches 1, the CICE becomes wider.

3.1.3.9 CONFIRMATION EXPERIMENTS

The confirmation experiment is a crucial step and is highly recommended to verify

the experimental conclusion. The suggested optimum levels are set for significant

parameters while the economic levels are selected for the insignificant parameters

and a selected number of test runs are conducted. The average values of the

responses obtained from confirmation experiments are compared with the predicted

values. The average values of the response characteristics obtained through the

confirmation experiments should lie within the 95% confidence interval, CICE.

However, these may or may not lie within 95% confidence interval, CIPOP [205].

3.2 MONTE CARLO SIMULATION BASICS

Monte Carlo method is a technique that involves using random numbers and

probability to solve problems. The term Monte Carlo Method was coined by S. Ulam

and Nicholas Metropolis[209] in reference to games of chance, a popular attraction

in Monte Carlo, Monaco. Hoffman[210]

Computer simulation has to do with using computer models to imitate real life or

make predictions. When you create a model with a spread sheet like Excel, you have

a certain number of input parameters and a few equations that use those inputs to

give you a set of outputs (or response variables)[211]. This type of model is usually

deterministic, meaning that you get the same results no matter how many times you

64

re-calculate. Fig. 3.4 shows a deterministic model mapping a set of input variables to

a set of output variables.

Figure 3.4 A parametric deterministic model.

Monte Carlo simulation is a method for iteratively evaluating a deterministic model

using sets of random numbers as inputs. Weisstein [212] suggested employing this

method when the model is complex, nonlinear, or involves more than just a couple

of uncertain parameters. A simulation can typically involve over 10,000 evaluations

of the model, a task which in the past was only practical using super computers.

Coddington[213] elucidates that by using random inputs, we are essentially turning

the deterministic model into a stochastic model. The Monte Carlo method is just one

of many methods for analyzing uncertainty propagation, where the goal is to

determine how random variation, lack of knowledge, or error affects the sensitivity,

performance, or reliability of the system that is being modeled. Monte Carlo

simulation is categorized as a sampling method because the inputs are randomly

generated from probability distributions to simulate the process of sampling from an

actual population. So, we try to choose a distribution for the inputs that most closely

matches data we already have, or best represents our current state of knowledge.

The data generated from the simulation can be represented as probability

distributions (or histograms) or converted to error bars, reliability predictions,

tolerance zones, and confidence intervals. Wittwer [214] demonstrated the basic

principle of stochastic uncertainty propagation behind Monte Carlo simulation as

shown in Fig. 3.5.

The steps in Monte Carlo simulation corresponding to the uncertainty propagation

shown in Figure 3.3 are fairly simple, and can be easily implemented in Excel for

simple models. All we need to do is follow the five simple steps listed below:

Step 1: Create a parametric model, y = f(x1, x2, ..., xq).

Step 2: Generate a set of random inputs, xi1, xi2, ..., xiq.

Step 3: Evaluate the model and store the results as yi.

Step 4: Repeat steps 2 and 3 for i = 1 to n.

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Step 5: Analyze the results using histograms, summary statistics, confidence

intervals, etc.

Figure 3.5 Schematic showing the basic principle of stochastic uncertainty

propagation

3.3 ANALYTICAL HIERARCHY PROCESS

The Analytic Hierarchy Process (AHP) is a structured technique for dealing

with complex decisions. Rather than prescribing a "correct" decision, the AHP helps

decision makers find one that best suits their goal and their understanding of the

problem—it is a process of organizing decisions that people are already dealing with,

but trying to do in their heads.

Based on mathematics and psychology, the AHP was developed by Thomas L.

Saaty[215] in the 1970s and has been extensively studied and refined since then. It

provides a comprehensive and rational framework for structuring a decision

problem, for representing and quantifying its elements, for relating those elements

to overall goals, and for evaluating alternative solutions.

Users of the AHP first decompose their decision problem into a hierarchy of more

easily comprehended sub-problems, each of which can be analysed independently.

66

The elements of the hierarchy can relate to any aspect of the decision problem—

tangible or intangible, carefully measured or roughly estimated, well- or poorly-

understood—anything at all that applies to the decision at hand.

Once the hierarchy is built, the decision makers systematically evaluate its various

elements by comparing them to one another two at a time, with respect to their

impact on an element above them in the hierarchy. In making the comparisons, the

decision makers can use concrete data about the elements, or they can use their

judgments about the elements' relative meaning and importance. It is the essence of

the AHP that human judgments, and not just the underlying information, can be

used in performing the evaluations[216].

The AHP converts these evaluations to numerical values that can be processed and

compared over the entire range of the problem. A numerical weight or priority is

derived for each element of the hierarchy, allowing diverse and often

incommensurable elements to be compared to one another in a rational and

consistent way. This capability distinguishes the AHP from other decision making

techniques.

In the final step of the process, numerical priorities are calculated for each of the

decision alternatives. These numbers represent the alternatives' relative ability to

achieve the decision goal, so they allow a straightforward consideration of the

various courses of action.

3.3.1 PROCEDURE FOR USING THE AHP

1. Model the problem as a hierarchy containing the decision goal, the

alternatives for reaching it, and the criteria for evaluating the alternatives.

2. Establish priorities among the elements of the hierarchy by making a series of

judgments based on pairwise comparisons of the elements. For example,

when comparing potential real-estate purchases, the investors might say

they prefer location over price and price over timing.

3. Synthesize these judgments to yield a set of overall priorities for the

hierarchy. This would combine the investors' judgments about location, price

and timing for properties A, B, C, and D into overall priorities for each

property.

4. Check the consistency of the judgments.

5. Come to a final decision based on the results of this process.

These steps are more fully described below.

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3.3.2 MODEL THE PROBLEM AS A HIERARCHY

The first step in the Analytic Hierarchy Process is to model the problem as

a hierarchy. In doing this, participants explore the aspects of the problem at levels

from general to detailed, then express it in the multileveled way that the AHP

requires. As they work to build the hierarchy, they increase their understanding of

the problem, of its context, and of each other's thoughts and feelings about both

[217].

An AHP hierarchy is a structured means of modeling the decision at hand. It consists

of an overall goal, a group of options or alternatives for reaching the goal, and a

group of factors or criteria that relate the alternatives to the goal. The criteria can be

further broken down into subcriteria, sub-subcriteria, and so on, in as many levels as

the problem requires. A criterion may not apply uniformly, but may have graded

differences like a little sweetness is enjoyable but too much sweetness can be

harmful. In that case the criterion is divided into subcriteria indicating different

intensities of the criterion, like: little, medium, high and these intensities are

prioritized through comparisons under the parent criterion, sweetness.

The design of any AHP hierarchy will depend not only on the nature of the problem

at hand, but also on the knowledge, judgments, values, opinions, needs, wants, etc.

of the participants in the decision making process. Constructing a hierarchy typically

involves significant discussion, research, and discovery by those involved. Even after

its initial construction, it can be changed to accommodate newly-thought-of criteria

or criteria not originally considered to be important; alternatives can also be added,

deleted, or changed[218].

To better understand AHP hierarchies, consider a decision problem with a goal to be

reached, three alternative ways of reaching the goal, and four criteria against which

the alternatives need to be measured.

Such a hierarchy can be visualized as a diagram as shown in fig.3.6 below, with the

goal at the top, the three alternatives at the bottom, and the four criteria in

between. There are useful terms for describing the parts of such diagrams: Each box

is called a node. A node that is connected to one or more nodes in a level below it is

called a parent node. The nodes to which it is so connected are called its children.

68

Fig. 3.6 A simple AHP hierarchy.

3.3.3 Evaluate the hierarchy

Once the hierarchy has been constructed, we analyze it through a series of pairwise

comparisons that derive numerical scales of measurement for the nodes. The criteria

are pairwise compared against the goal for importance. The alternatives are pairwise

compared against each of the criteria for preference. The comparisons are processed

mathematically, and priorities are derived for each node.

3.3.4 Establish priorities

Priorities are numbers associated with the nodes of an AHP hierarchy. They

represent the relative weights of the nodes in any group.

Like probabilities, priorities are absolute numbers between zero and one, without

units or dimensions. Depending on the problem at hand, "weight" can refer to

importance, or preference, or likelihood, or whatever factor is being considered by

the decision makers.

Priorities are distributed over a hierarchy according to its architecture, and their

values depend on the information entered by users of the process. Priorities of the

Goal, the Criteria, and the Alternatives are intimately related, but need to be

considered separately.

By definition, the priority of the Goal is 1.000. The priorities of the Alternatives

always add up to 1.000. Things can become complicated with multiple levels of

Criteria, but if there is only one level, their priorities also add to 1.000. All this is

illustrated by the priorities in the Fig. 3.7 below.

69

Fig.3.7 Simple AHP hierarchy with associated default priorities.

The priorities shown are those that exist before any information has been entered

about weights of the criteria or alternatives, so the priorities within each level are all

equal. They are called the hierarchy’s default priorities. If a fifth Criterion were

added to this hierarchy, the default priority for each Criterion would be .200. If there

were only two Alternatives, each would have a default priority of .500.

We now calculate the geometric mean of ith row and normalize the geometric means

of each row in the comparison matrix to obtain the normalized weight (Wj) of each

factor by using eqns.3.11 and 3.12. The normalized weaights are expressed in the

form of A25x1 matrix,as under.

Two additional concepts apply when a hierarchy has more than one level of

criteria: local priorities and global priorities. Consider the hierarchy shown below in

fig.3.8, which has several Sub-criteria under each Criterion.

Fig.3.8 A more complex AHP hierarchy, with local and global default priorities.

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3.3.4 MATHEMATICAL EXPRESSIONS USED FOR AHP PROCEDURE

Once our objectives are clearly defined we construct a pair-wise comparison matrix,

using Satty’s scale of relative importance, comparing factor i with factor j which

yields a square matrix A15x5 where rij denotes the comparative importance of factor i

with respect to factor j. In the matrix rij = 1 where i = j and rij =1/rij. Based on our

experience and the reviewed literature we allocate intensities to the compared

factors and obtain the square comparison matrix A15x5.

Table 3.2 Saaty’s intensities of importance[219]

Intensity of importance

Definition Explanation

1

Equal importance Two activities contribute equally to the objective

3 Weak Importance The judgment is to favor one activity over Another, but it is not conclusive

5 Essential or Strong Importance

The judgment is strongly in the favor of one activity over another

7 Demonstration importance

The Conclusive judgment as to the Important of one activity over another

9 Absolute

The judgment in the favor of one activity over another is of the highest possible order of affirmation

2,4,6,8 Intermediate values between the two adjacent judgments

When compromise is needed

We now calculate the geometric mean of ith row and normalize the geometric means

of each row in the comparison matrix to obtain the normalized weight (Wj) of each

factor by using eqns.3.11 and 3.12. The normalized weaights are expressed in the

form of A25x1matrix.

1/

1

NN

i ij

j

GM a

(3.11)

N

i

i

i

j

GM

GMW

1

(3.12)

Matrix A3 5x1 is calculated as A3 5 x1 = A15x5 x A25 x 1

λmax is worked out which is nothing but the average of matrix A45x1 which is given by

(A45x1 = A35x1 / A25x1) and can be expressed by eqn. 3.13 as

71

5 1

1

13 /

5

N

avg iii

A W

(3.13)

If the value of λavg is closer to the number of attributes n, the result is more

consistent. The deviation from consistency is represented by Consistency Index (CI)

and is obtained by eqn. 3.14.

1

max

N

NCI

(3.14)

Random Index or correction for random error is denoted by RI and their values for

different values of attributes (n) are given by Saaty, as shown in table 3.3.

Table 3.3 RI Values of different values of n

N 1 2 3 4 5 6 7 8 9

RI 0 0 0.58 0.90 1.12 1.24 1.32 1.41 1.45

Finally we calculate the Consistency Ratio, CR, which is the ratio of Consistency Index

to the Random Index, and is ratio given by,

CR = CI/RI

CR value of less than 0.1 indicates minimal deviation from accuracy which validates

our choice of comparison matrix and confirms good consistency in choices of relative

importance values assigned to the process parameters.

3.4 TECHNIQUE OF ORDER PREFERENCE BY SIMILARITY TO IDEAL SOLUTION

(TOPSIS)

Multi-attribute decision-making (MADM) techniques are employed to help decision-

makers to identify the best alternative from a finite set. MADM techniques have

been successfully applied in the selection of work materials [220, 221], rapid

prototyping processes [222], thermal power plants [223], industrial robots [224],

evaluation of projects [225], mobile phones [226], product design [227], flexible

manufacturing systems[228], performance measurement models for manufacturing

organizations [229], plant layout design [230], and so on. Hwang and Yoon [231]

developed TOPSIS to assess the alternatives while simultaneously considering the

distance to the ideal solution and negative-ideal solution, regarding each alternative,

and selecting the closest relative to the ideal solution as the best alternative.

72

The selection of an alternative from amongst a list of possible alternatives on the

basis of several attributes is clearly a multiple attribute decision making problem for

which the TECHNIQUE of ORDER PREFERENCE by SIMILARITY to IDEAL SOLUTION

(TOPSIS) provides a simple yet powerful decision making tool. TOPSIS method is

based on the concept that the chosen alternative should not only have the shortest

Euclidean distance from the ideal solution but also have the farthest Euclidean

distance from the negative ideal solution. TOPSIS thus provides a solution that is not

only closest to the hypothetical best, but is also the farthest from the hypothetically

worst. Combined multi-attribute decision-making is aimed at integrating different

measures into a single global index which facilitates ranking alternatives on the basis

of their suitability.

If each attribute has a monotone increasing (or decreasing) function, the ideal

solution, which is composed of the best attribute values, and the negative ideal

solution, which is composed of the worst, are calculated. From the viewpoint of

geometry, an alternative with the shortest Euclidean distance from the ideal solution

is chosen, i.e. the best alternative is the nearest one to the ideal solution and the

farthest one from the negative ideal solution [232, 233]. The AHP can efficiently deal

with tangible and non-tangible attributes in the light of subjective judgements of

different individuals in the process of decision-making [234]. However, in some

cases, an unmanageable number of pair-wise comparisons of attributes and

alternatives with respect to each of the attributes may result. TOPSIS is more

efficient in dealing with the tangible attributes and the number of alternatives to be

assessed. However, the TOPSIS method needs a powerful procedure to determine

the relative importance of different attributes with respect to the objective; AHP

provides such a procedure. Hence, to take advantage of both the methods, a

combined MADM (using TOPSIS and AHP) approach is adopted to select the most

suitable alternative from amongst the available alternatives. The procedures for

implementing this combined TOPSIS–AHP method are described below.

Step 1:

Model the problem as a hierarchy containing the decision goal, the alternatives for

reaching it, and the criteria for evaluating the alternatives. The goal presents the

optimum solution of the decision problem. It can be selection of the best alternative

among many feasible alternatives. Also, the ranking of all alternatives can be

performed, by obtaining the priorities. Criteria (attributes) are the quantitative or

qualitative data (judgments) for evaluating the alternatives.

73

Fig.3.9 TOPSIS model with “n” criteria and “m” alternatives

Step 2:

Construct a decision matrix such that each row of this matrix is allocated to one

alternative, and each column to one attribute. Therefore, an element dij of the

decision matrix D gives the value of jth attribute, in original real values and units, for

the ith alternative. Thus, if the number of alternatives is M and the number of

attributes is N, then the decision matrix is an MxN matrix can be represented as

follows,

[D]MxN = [D]4x4 = [M]ij

a11 a12 a13 a14

a21 a22 a23 a24

[D] = a31 a32 a33 a34 (3.15)

a41 a42 a43 a44

Step 3:

Obtain the normalized decision matrix, Rij by using eqn.3.16.

2/1

1

2/

M

j

ijijij mmR (3.16)

Step 4:

Determine the relative importance of different attributes with respect to the

objective for assignment of weightage to different attributes for logical decision-

making by application of Analytical Hierarchy Process (AHP), as already described in

the last section 3.3.

74

Step 5:

The weighted normalized matrix Vij is obtained by the multiplication of each element

of the column of the matrix Rij with its associated weight wj , obtained from AHP

procedure, using Eqn. 3.17.

Vij = wj Rij (3.17)

Step 6:

Obtain the ideal (best) and negative ideal (worst) solutions in this step. The ideal

(best) and negative ideal (worst) solutions can be expressed as:

V+ =

max min

/ , / ' / 1,2,.....i i

Vij j J Vij j J i M

V+ 1 2 3, , , ..., NV V V V (3.18)

V- =

min max

/ , / ' / 1,2,.....,i i

Vij j J Vij j J i M

V- 1 2 3, , ,..., NV V V V (3.19)

Where J = (j = 1, 2… N) / j is associated with the beneficial attributes and J’ = (j = 1,

2… N) / j is associated with the non-beneficial attributes. Vj indicates the ideal (best)

value of the attribute for different alternatives. In case of beneficial attributes (i.e.

whose higher values are desirable for the given application), Vj indicates the higher

value of the attribute. In case of non-beneficial attributes (i.e. whose lower values

are desired for the given application), Vj indicates the lower value of the attribute. Vj

indicates the negative ideal (worst) value of the attribute for different alternatives.

In case of beneficial attributes (i.e. whose higher values are desirable for the given

application), Vj indicates the lower value of the attribute. In case of non-beneficial

attributes (i.e. whose lower values are desired for the given application), Vj indicates

the higher value of the attribute.

Step 7:

Obtain the separation measures which indicate the separation of each alternative

from the ideal solution as given by the Euclidean distance using eqns. 3.20 and 3.21

Si+

0.5

2

1

N

ij j

j

V V

Ni ....,,2,1 (3.20)

75

Si-

0.5

2

1

,N

ij j

j

V V

Ni ....,,2,1 (3.21)

Step 8:

The relative closeness of a particular alternative to the ideal solution, expressed as Pi

is calculated using eqn.3.22.

Pi = Si- / (Si

+ + Si

-) (3.22)

The preferred feasible solutions, Pi may also be called as overall or composite

performance score of alternative. This relative closeness to ideal solution can be

considered as the “Global Index (GI).”

Step 9:

A set of alternatives are arranged in descending order of their composite

performance score, Pi values, indicating the most preferred solution for highest Pi

value and the least preferred solution for the lowest Pi value.

3.5 MULTI-PERFORMANCE OPTIMIZATION TECHNIQUES

It requires a consideration of diverse quality characteristics for evaluating any

product or service in order so as to be able to make a rational choice. These

evaluations of diverse quality characteristics should be combined to arrive at a

composite index which may suitably represent the overall utility of a product or

service. The overall utility of a product measures the usefulness of that product from

the evaluators perspective. Whereas, the utility of a product based on a particular

characteristic measures the usefulness of that particular characteristic only. The

utility concept proposes that the overall utility of a product is the sum of utilities of

each of the quality characteristics.

3.5.1 UTILITY CONCEPT

Kumar et al.[235] suggested that in accordance to the utility theory, if Xi is the

measure of effectiveness of an attribute (quality characteristics) i and there are n

attributes evaluating the outcome space, then the overall utility function is given by:

U (X1, X2, X3…, Xn) = f (U1 (X1), U2 (X2)… Un (Xn)) (3.23)

where, Ui (Xi) is the utility of the ith attribute.

The overall utility function is the sum of individual utilities if the attributes are

independent, and is given by:

76

∑

where, Ui(Xi) is the utility of the ith attribute. The attributes may be assigned

weights depending upon the relative importance or priorities of the characteristics.

The overall utility function after assigning weights to the attributes can be written as:

∑

where Wi, is the weight assigned to attribute i and the sum of the weights for all

attributes is equal to 1.

To determine the utility value for a number of quality characteristics, a preference

scale for each quality characteristic is constructed and later these scales are

weighted to obtain a composite number (overall utility). The Preference Scale may

be linear, exponential or logarithmic. The minimum acceptable quality level for each

quality characteristic is set at a preference number of 0 and the best available quality

is assigned a preference number of 9 ( the preference numbers for minimum or best

values of characteristics is optimal). Gupta and Murthy [236] suggested that if a log

scale is chosen, the preference number, Pi, is given as eqn.3.26.

Pi = A log

(3.26)

where Xi is the value of the quality characteristic or attribute i, Xi” is the minimum

acceptable value of the quality characteristic or attribute i and A is a constant.

Arbitrarily, we may choose A such that Pi = 9 at Xi = X*, where X* is the optimum

value of Xi assuming that such a number exists.

The next step is to assign weights or relative importance to the quality characteristic.

Bosser [237] suggested a number of methods for the assignment of weights (AHP,

Conjoint Analysis, etc). The weights should be assigned such that the following

condition holds:

∑

The overall utility can be calculated as:

∑

77

PROCEDURE FOR MULTI-CHARACTERISTIC OPTIMIZATION USING TAGUCHI’S

PHILOSOPHY AND UTILITY CONCEPT

1. Find optimal values of the selected quality characteristics separately using

Taguchi’s experimental design and analysis (parameter design).

2. Using the optimal values and the minimum quality levels, construct

preference scales for each quality characteristic using eqn.3.26.

3. Assign weights Wi, i = 1,2,…,n, to various quality characteristics based on

experience and the end use of the product such that the sum of weights is

equal to 1.

4. Find utility values of each product against each trial condition of the

experiment using eqn. (4.28).

5. Use these values as a response of the trial conditions of the selected

experimental plan.

6. Analyse results using the procedure suggested by Taguchi [238]

7. Find the optimal settings of the process parameters for optimum utility

(mean and minimum deviation around the mean).

8. Predict the individual characteristic values considering the optimal significant

parameters determined in step 7.

9. Conduct a confirmation experiment at the optimal setting and compare the

predicted optimal values of the quality characteristics with the actual ones.

3.5.2 GREY RELATIONAL ANALYSIS

Optimization of multiple response characteristics is more complex compared to

optimization of single performance characteristics. In recent years, the theories of

grey relational analysis have attracted the interest of researchers. Deng [239]

proposed application of the principles of grey relational analysis as a method of

measuring degree of approximation among sequences according to the grey

relational grade. In the grey relational analysis, the measured values of the

experimental results are first normalized in the range between zero and one, which

is also called grey relational generation. Next, the grey relational coefficients are

calculated from the normalized experimental results to express the relationship

between the desired and the actual experimental results. The next step involves

assignment of weighting factors to each quality characteristic. Then, the grey

relational grades are computed by averaging the grey relational coefficient

corresponding to each performance characteristic. The overall equation of the multi-

performance characteristic is based on the grey relational grade. As a result,

optimization of the complicated multi-performance characteristics can be converted

into optimization of a single grey relational grade. The optimal level of the process

parameters is the level with the highest grey relational grade. Further, a statistical

student’s t-test was performed to identify the statistically significant parameters. In

78

addition, empirical model was developed for grey relational grade. Hence, an

empirical model for the multi-objective optimization is available. This response

surface model for the grey relational grade was further used for the optimization of

the process parameters. Thus, grey relational analysis coupled with RSM has been

employed to identify the optimum parameter settings of the significant factors.

Finally, a confirmation experiment was conducted to confirm the optimum levels of

the process parameters identified by the optimization method.

Based on the above discussion, the use of the grey relational analysis with Taguchi

design of experiment to optimize the process parameters considering multiple

performance characteristics includes the following steps as suggested by Siddiqui et

al.[240] and Lue et al.[241].

1. Normalize the experimental results by data pre-processing which is basically

a means of transferring the original sequence to a comparable sequence.

2. Perform the grey relational generating and calculate the corresponding grey

relational coefficient.

3. Assignment or Calculation of weighting factors to each quality characteristic.

4. Calculate the grey relational grade by averaging the grey relational

coefficient.

5. Plot the average responses at each level of parameter.

6. Select the optimal levels of process parameters.

7. Conduct confirmation experiments.

Data pre- processing

Data pre-processing is normally required since the range and unit in one data

sequence may differ from the others. Data pre-processing is also necessary when the

sequence scatter range is too large, or when the direction of the target in the

sequence are different. Data pre-processing is a means of transferring the original

sequence to a comparable sequence. Depending on the characteristics of a data

sequence, there are various methodologies of Data pre-processing available for the

gray relational analysis.

If the target value of original sequence is infinite, then it has a characteristic of the

“higher is better” and the original sequence can be normalized by using eqn.3.29.

* ( ) min ( )( )

max ( ) min ( )

o oi i

i o oi i

x k x kk

x k x kx

(3.29)

79

When the “lower is better” is a characteristic of the original sequence, then the

original sequence should be normalized using eqn. 3.30.

* max ( ) ( )( )

max ( ) min ( )

o oi i

i o oi i

x k x kk

x k x kx

(3.30)

However, if there is a definite target value (desired value) to be achieved, the

original sequence may be normalized using eqn.3.31.

* ( )( ) 1

max ( )

o oi

i o oi

x k xk

x k xx

(3.31)

Or, the original sequence can simply be normalized by the most basic methodology,

i.e. let the value of original sequence be divided by the first value of the sequence

using eqn.3.32.

( )( )

(1)

oio

i oi

x kk

xx

(3.32)

Where i = 1… m; k = 1… n. m is the number of experimental data items and n is the

number of parameters. xio

(k) denotes the original sequence, xi*(k) the sequence after

the data pre-processing, max xio(k) the largest value of xi

o(k), min xio (k) the smallest

value of xio(k) and xo

is the desired value.

Gray relational coefficient and gray relational grade

In gray relational analysis, the measure of the relevancy between two systems or

two sequences is defined as the gray relational grade. When only one sequence, xo

(k), is available as the reference sequence, and all other sequence serves as

comparison sequence, it is called a local gray relation measurement. After data pre-

processing is carried out, the gray relation coefficient ξi(k) for the kth performance

characteristics in the ith experiment can be expressed as,

min max( )

( ) maxi k

oi k

(3.33)

Where, Δoi is the deviation sequence of the reference sequence and the

comparability sequence; x0*(k) denotes the reference sequence and xi

*(k) denotes

the comparability sequence. ξ is distinguishing or identification coefficient which is

defined in the range 0 ≤ ξ ≤ 1.

A weighting method is used to integrate the grey relational coefficients of each

experimental run into the grey relational grade, which is a weighting sum of the grey

relational coefficients and we have pressumed equal weightage for both the

80

performance characteristics. Deng [239] reported that it is usual to take the average

value of the grey relational coefficients as the grey relational grade. The grey

relational grade was calculated using eqn. 4.34. The overall evaluation of the

multiple-performance characteristics is based on the grey relational grade.

γi =

n

kn 1

1i(k) (3.34)

However, in a real engineering system, the importance of various factors to the

system varies. In the real condition of unequal weight being carried by the various

factors, the grey relational grade in Eq.3.34 was extended and defined as eqn,3.35.

γi =

n

k

kwn 1

1ξi (k) (3.35)

Where, wk denotes the normalized weight of factor k for the performance

characteristic and n is the number of performance characteristics. The grey relational

grade γi represents the level of correlation between the reference sequence and the

comparability sequence. If the two sequences are identical, then the value of grey

relational grade is equal to 1. The grey relational grade also indicates the degree of

influence that the comparability sequence could exert over the reference sequence,

and then the grey relational grade for that comparability sequence and reference

sequence will be higher than other grey relational grades.

To understand the relationship between the process parameters and the multi-

characteristic grey relation grades, a model has been developed using response

surface methodology as explained in section 3.6. The general expression for the

developed model is given in eqn.3.36.

YGrG = b0 + juiuij

ji1i

2

iujiiui

k

1i

xxbxbxb

(3.36)

Where, Ygrg is the grey relational grade considered as multiple performance

response. Further, the coefficients and constants are denoted by their usual

notations as mentioned and explained in section 3.6.

3.6 RESPONSE SURFACE METHODOLOGY

Response surface methodology (RSM) is a collection of mathematical and statistical

techniques useful for analyzing problems in which several independent variables

influence a dependent variable or response, and the goal is to optimize this

response, Cochran and Cox[242]. In statistics, response surface methodology

(RSM) explores the relationships between several explanatory variables and one or

more response variables. The method was introduced by Box and Wilson [243] in

81

1951. The main idea of RSM is to use a sequence of designed experiments to obtain

an optimal response.

Some extensions of response surface methodology deal with the multiple response

problem[244]. Multiple response variables create difficulty because what is optimal

for one response may not be very optimal for other responses. Other extensions are

used to reduce variability in a single response while targeting a specific value, or

attaining a near maximum or minimum while preventing variability in that response

from getting too large.

In many experimental conditions, it is possible to represent independent factors in

quantitative form as given in Equation 3.37. Then these factors can be thought of as

having a functional relationship with response as follows:

(3.37)

This represents the relation between response Y and x1, x2,…… ,xk of k quantitative

factors. The function Φ is called response surface or response function. The residual

er measures the experimental errors[242]. For a given set of independent variables, a

characteristic surface is responded. When the mathematical form of Φ is not known,

it can be approximated satisfactorily within the experimental region by a polynomial.

Higher the degree of polynomial, better is the correlation but at the same time costs

of experimentation become higher.

For the present work, RSM has been applied for developing the mathematical

models in the form of multiple regression equations for the quality characteristic of

machined parts produced by AFM process. In applying the response surface

methodology, the dependent variable is viewed as a surface to which a

mathematical model is fitted. For the development of regression equations related

to various quality characteristics of AFM process, the second order response surface

has been assumed as:

(3.38)

This assumed surface Y contains linear, squared and cross product terms of variables

xi’s. In order to estimate the regression coefficients, a number of experimental

design techniques are available.

Regression analysis is based on some assumptions, the experimental error (residue)

is normally distributed with the constant variance and errors are normally

distributed. The limitation of the regression analysis is that, it cannot be used for

82

extrapolation. This means that the values of independent variables must lie within

the upper and lower limits those are set at the time of testing. These models give the

predicted value with some error, which cannot be extracted. The error in the

prediction is estimated by the coefficient of determination (R2). The value of the R2 is

an important criteria to decide validity of regression model (Montgomery, 2001). If

this value is 0.8 or more, the relationship established by regression model is

acceptable. To know the behavior of population based sample, adjusted R2 value

gives the level of the validity, to use the regression model for population. Standard

error gives the error in the predicted value of Y.

There is a difference between the predicted and actual value of the response for the

same set of independent variables. It is possible to attribute this difference to a set

of independent variables and the difference due to random or experimental errors.

For analysis of the experimental data, checking of goodness of fit of model is very

much required. Model adequacy checking includes test for significance of regression

model and on model coefficients as suggested by Montgomery [245]. Analysis of

variance (ANOVA) is performed for this purpose. A statistical software program,

Mini-Tab-15[246] and Microsoft excel (MS-Office-2007) were used for training

models.

MODELLING OF THE PROCESS CORRELATING QUALITY CHARACTERISTICS WITH

VARIABLE PARAMETERS

The functional relationship between the output response and the input process

parameters can be represented in a general form by the following expression.

MR = c Pa Ab Mc Nd Le ε1 (3.39)

∆Ra = c1 Pa1Ab1 Mc1 Nd1Le1 ε1 (3.40)

The model can be transformed to logarithmic equation as shown below:

y1 = y – ε = lnc1+ a lnP + blnA + clnM + dlnN + dlnL + lnε1 (3.41)

which represent the following linear mathematical equation,

ὴ = b0x0 + b1x1 +b2x2 + b3x3 + b4x4 +b5x5 (3.42)

where ὴ is the true response of the input parameters on a logarithmic scale, x0 = 1

(dummy

variable), x1, x2, x3, x4 and x5 are logarithmic transformation of the input.

The linear model of eqn.3.40 in terms of the estimated response can be represented

as

Ῠ = y-ε = b0 x0 +b1 x1 +b2 x2 +b3 x3 +b4 x4 + b5x5 (3.43)

83

where Ῠ is the estimated response based on first order equation and y is the

measured response of based on experimental results on a logarithmic scale; ε is the

experimental error and ‘b’ values are the estimates of the corresponding

parameters. The constants and exponents c, a, b, c, d & e can be determined by the

method of least squares. The basic formula is given by eqn.

b = (XT. X)-1 XT Y (3.44)

where the calculation matrices are X and the variance matrix (XT.X)-1, hence the b

values can be determined by using eqn. (3.42) (Montgomery, 2005).

If this model is not sufficient to represent the process, then the second order model

will be developed. The general equation for the response model has been

represented as:

Y = b0 + juiuij

ji1i

2

iujiiui

k

1i

xxbxbxb

(3.45)

The general second order model can be represented as:

Y2 = y-ε = b0 x0 +b1 x1 +b2 x2 +b3 x3 +b4 x4 + b12 x1x2 +b23 x2x3+ b14 x1x4

+ b24 x2x4 + b13 x1x3 + b34 x3x4 + b11 x12 + b22 x2

2 + b33 x32 + b44 x4

2 (3.46)

where Y2 is the estimated response based on second order equation. The

parameters, i.e. b0, b1, b2, b3, b4, b12, b23, b14 are to be estimated by the method of

least squares, eqn (3.42). x12, x2

2, x32 and x4

2 are the quadratic effects of these

variables and x1x2, x1x3, x2x3, x1x4 and x3x4 represent the interaction between them;

b0, b1, b2,…,….b14 are the regression coefficient to be estimated. In order to

understand the process, the experimental values are used to develop the

mathematical models using response surface method. In this work, commercially

available mathematical software package MINITAB-15 was used for the computation

of the regression constants and exponents.

The variables are coded by taking into account the capacity and the limiting

condition of the process. The variables are transformed according to natural

logarithmic equation as follows:

84

Lnxn- ln xn0 (3.47)

ln xn1 - ln xn0

where, x is the coded value of any factor corresponding to its natural value xn, xn1 is

the natural value of the factor at the +1 level and xn0 is the natural value of the factor

at the middle level.

Assuming null hypothesis in the form

H0 = 0 (b0 = b1 = b2 = b3 = b4 = b5 = 0) (3.48)

That is to say, none of the parameters have significant influence on output response.

X =