CHAPTER 1

Introduction

Process capability is a statistical analysis tool. Process capability indices have been widely used in the manufacturing industry providing numerical measures on process potential and process performance. Capability analysis is used in many facets of industrial processes and has been recently used into business processes. It requires collecting data from the process, constructing a histogram, drawing a curve that fits in the histogram, and then finally finding out what percentage of data goes outside the Upper specification limit (USL) & lower specification limit (LSL).Process capability is measured by the calculating the process capability indices (Cp, Cpk, & k). Quality management efforts are often directed for zero defect production by reduction of variability. If a product is found nonconforming, it is usually claimed that the variability is attributed by the processes & thus improvement actions are implemented to enhance the process capability. Process analysis through statistical process control tools can provide insight view of the nature of problem. If the nature of problem can be known, appropriate measures can be taken to obtain stability of the process. As a result, it is very easy to update the process for better quality product by process capability analysis.

1.1 Motivation

Now a day world product market is highly competitive. Customer want high quality product with reachable price. If the process is incapable it will provide the disqualified product which will cause the market fall. Recently process capability indices have been widely used in manufacturing industries for process performance. By process capability analysis one can easily identify the process is capable or not capable which is very essential for qualified product. To maintain the better quality process capability must be maintained. If this study implement in the manufacturing industries, then it will easy to provide better quality product and maintain process stability. For this why, this work is done.

1.2 Research Objectives

The objectives of this thesis work are to justify the process capabilities where the process is capable or not and try to find out the possible causes of process incapability. And try to find out the probability of rejecting the good lots & probability of accepting the bad lots.

1.3 Problem Statement

In case of process improvement, at first one should know about the process whether the existing process is capable or not. Incapable process produces large number of defective items. For this manufacturing cost of the products are increased. In these case manufacturers should justify the process capabilities. In Kohinoor Chemicals Company Bangladesh Limited, the production of defective item is more. As a result their productivity is in a decreasing trend. Due to lower productivity the product cost is more

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and also the company profit is not in a desired range. The recycling of the product is more as a result of defective item production. The quality of the finished product is not in a desired level. In such manufacturing company the capability test can change the production as well as quality rate.

1.4 Methodology

In this section, the whole process of this work is included. At first the literatures are reviewed from various publications, and then data are collected from the relevant manufacturing company. After the data collection the related problems are analyzed of the reference organization (Kohinoor Chemical Company Bangladesh Limited). Then the process capability indices (Cp, Cpk, & k) are calculated for identifying the process capability. After then proposed a linear relationship by regression analysis among these variables or parameters can help the manufacturers. Type I &Type II errors are calculated for the probability of producer’ risk and the customer risk. Then to find out the main culprit for process incapability Cause-Effect diagram are used. The few quality variables which hamper quality are identified though brainstorming.

1.5 Organization of the Thesis

The structure of this thesis has been organized in the following pattern. In the first chapter one of the thesis contain Introduction and in the second chapter literature review is available. Chapter three contain a theoretical framework of the thesis the case study are available in chapter four. Then there also the data collection (from reference industry) is discussed in chapter four. After then, all calculations are placed in chapter five. In chapter six result analyses are available. Then recommendations and future works are placed in chapter seven. After then conclusion, reference and appendix are given respectively.

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CHAPTER 2

Literature Review

Process capability indices are important topics since (Juran 1974) combined process parameters with product specifications to introduce the idea of process capability indices. (Sinhal 1991) even applied PCIs to a multi process performance analysis chart (MPPAC). Burlikoska (2005) proved that monitoring of production processes is a necessary element of a company’s prequalify policy and has been discussed possibility of usage of statistical techniques of quality estimation process. Pearn & Liao (2005) proposed process capability indices, Cp, Cpk, Cpm and Cpmk in manufacturing and service industries providing numerical measures on whether a process is capable of reproducing items within the specification limits preset in the factory. The index Cpmk combines the merits of Cpk and Cpm, which is more sensitive to departure of process mean from the target value than the other two indices. Due to the simplicity and ease of use, those capability indices have become effective tools for evaluating and improving manufacturing quality and popularly used by the managers and shop floor controllers. Job shop factories are now becoming increasingly widespread as consumer requirements increase and production techniques are improved.

Wang (2005) has developed a procedure for constructing multivariate process capability indices based on the principal component analysis (PCA) and Clements method for short-run production.PCA can identify correlations among multiple characteristics and determine independent components. Clements method can accurately determine the 99.865th and 0.135th percentiles. This paper solves the problem of evaluating process quality performance with multiple quality characteristics in a short run production to help to meet the practical requirements of industry. It employs the Pearson distribution curve and considers the correlations among multiple quality characteristics to develop multivariate process capability indices that can accurately depict the process capability performance for short run production.

Process yield has been a standard criterion used in the manufacturing industry as a common measure on process performance. Process yield is currently defined as the percentage of processed product units passing inspection, that is, the product characteristic falls within the manufacturing tolerance. Chen (2005) has considered using the bootstrap simulation techniques to find four approximate lower confidence limits. The four bootstrap methods including the standard bootstrap (SB), percentile bootstrap (PB), biased corrected percentile bootstrap (BCPB), biased corrected and accelerated (BCa) bootstrap methods are compared based on the coverage fraction. The simulations are conducted using four bootstrap methods to find the lower confidence limit of Spk, and plotted are coverage fraction versus the sample size n for various distribution parameters, (µ,sigma) =(15, 1.0), (15, 1.5), (15,2.0), (µ,sigma) =(16, 1.0), (16, 1.5), (16, 2.0) and (µ,sigma) =(17, 1.0), (17, 1.5), (17, 2.0). For identifying, analyzing & eliminating the variability in product manufacturing statistical tools are effective .Process capability indices are widely used in measuring product quality & also help companies to promote marketing sales. A process capability index (Cpp

T) which takes into account all family members. The index Cpp is a simple transformation from index Cpm & Cpp

T provides

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additional , individual information concerning the precision & accuracy of a process as previously demonstrated (Huang, Chen & Li 2005).For practical application evaluation plots are well established & examples are provided to determine the process capability of predetermined set value. Chen, Yu & Sheu (2006) has selected Cpn to replace Cpa & has reconstructed a process capability monitoring chart (PCMC) .The main purpose was to evaluate the process potentials & performance for an entire product. It consists of smaller-the-better (with Cpu), larger-the-better (with Cpl), nominal the best (with Cpn) specification respectively. After all they have proposed an integrated product capability index & have described the process yield of the entire product. Process capability index control chart not only can be used to monitor the stability of process’s quality but also can be used to monitor the quality of the process. A flow path to evaluate the process capability of an entire product composed of multiple process characteristics (Chen, Hsu & Wu 2006).Based on the Cpu, Cpl, and Cpn the research aim is to develop a multi-process capability analysis chart (MPCAC) to evaluate process capability in the normal distribution. Similarly their research aims to define non-normal multi process capability analysis chart (NMPCAC) to evaluate process capability in a non-normal distribution based on Npu, Npl, & Npn. Chen et al. 2007 has constructed the control chart of unilateral specification index Cpl, & Cpu to monitor and evaluate the stability of process and process capability. This control chart contains upper control limit, lower control limit, upper warning limit, lower warning limit and this chart also support a set of sensitizing rules for user can easily to monitor the quality of process and the stability of the process. The two well known process capability indices Cpl, & Cpu which measure larger-the-better and smaller-the –better process capabilities. The control charts can be adopted as a monitor to find if the quality of the process is stable or not, these control charts cannot be adopted to monitor the quality of process. To overcome this problem (Boyles 1991) & (Spiring 1995) point out that as soon as x bar & S control charts are in statistical control, the control chart of process capability indices can be used to monitor the quality of process.

Chang & Wu (2008) has proposed a generalization called Cnpk with asymmetric tolerance

rather than Cpk for processes with symmetric tolerance. A confidence bound estimates the minimum process capability carrying critical information which is necessary for quality assurance. They have provided an efficient algorithm for the calculation of lower confidence bounds and sample size in order to estimate the Cpk for processes with asymmetric tolerance. They have also developed a Mat Lab computer program method using a binary search.Wu, Pearn & Kotz (2009) researched on the nature of actual process yield in terms of the number of non-conformities. They processes fixed indexed values of Cpk,=Cpm=Cpmk with different degree of process centering. They established relationship of Cpm to square error loss. This paper has also analyzed the comparison among PCIs based on various criteria. Process capability is a statistical measure used in the manufacturing industry to assess the variability of process outcomes in relations to their engineering specifications. A general multivariate process capability index based on the Mahanalobis distance which is very easy to use has proposed by the (Ahmed et al. 2009).They have also approximated the distribution of these distance by the Burr XII distribution & then estimate its parameter using a simulated annealing search algorithm. Finally they have given an example, based on the real manufacturing process data which demonstrates that the proportion of non-conformance (PNC) using the proposed method is very close to the actual PNC value.

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van der Merwe & Chikobvu (2010) has developed a process potential index for average of observations from the new or unknown model. They have derived theoretical and simulation results using Bayesian approach. They have removed the complexities of frequency distributions for measuring process performance by this approach. In Bayesian approach inferences the Monte Carlo method is used sometimes in replace posterior distribution to minimize the complexity of problem. Refaie & Bata (2010) proposed a procedure for assessing a measurement system and manufacturing process capabilities using Gage Repeatability (GR&R) desinged experiments with four quality measures. The gage and part variance components are then estimated by conducting analysis of variance (ANOVA) on the GR&R measurement observations. The statistical process control is applied to many different processes now a day for controlling process. The goals of the statistical process control are to improve and ensure quality and thus, reduce the process cost due to waste as a result of rejects. This paper is related with the process capability analysis in the manufacturing of poly bottles for the packaging of table oil. In an oil production factory the difference in the volume of oil in the filled bottles are considered. The blowing process, volume measurements immediately after blowing is capable & also the blowing process, volume measurements immediately after storage is not capable which is measured by process capability analysis The solution for resolving the problem is setup of new blow molding machine in which the bottles could be filled directly after blowing & the problem of post- shrinkage and filling of unequal oil quantities would disappear (Sokele, Sercer & Godec 2010).

Kaya & Kharaman (2011) has used fuzzy set theory to add more information and flexibility to process capability analysis. Here process mean µ and fuzzy variance are used which are obtained by using fuzzy extension principle. There is also fuzzy specification limit are used with mean µ and variance to produce PCIs. The fuzzy formulation includes the development of Cp, Cpk, Ca, Cpm, Cpmk which are the most traditional used PCIs. Aslam et al. (2012) defined acceptance sampling as the inspection and classification of a sample of units selected at random from a larger batch or lot and the ultimate decision about disposition of the lot, usually occurs at two points, i.e. incoming raw materials or components and final production. Attributes and variables sampling plans are widely used in inspection for the purpose of acceptance or rejection of a product, based on adherence to a standard.

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CHAPTER 3

Theoretical Framework

Process capability is an important concept for Industrial Engineers to understand. The challenge in today’s competitive markets is to be on the leading edge of producing high quality products at minimum costs. This cannot be done without a systematic approach and this approach is contained within what has been called “statistical quality control”. So why is process capability so important? Because it allows one to quantify how well a process can produce acceptable product.

3.1 What is Process Capability?

o Process capability is the long-term performance level of the process after it has been brought under statistical control. In other words, process capability is the range over which the natural variation of the process occurs as determined by the system of common causes.

o Process capability is the ability of the combination of people, machine, methods, material, and measurements to produce a product that will consistently meet the design requirements or customer expectation.

3.2 What is a Process Capability Study?

Process capability study is a scientific and a systematic procedure that uses control charts to detect and eliminate the unnatural causes of variation until a state of statistical control is reached. When the study is completed, you will identify the natural variability of the process. 3.3 Assumptions

There are two critical assumptions to consider when performing process capability analyses with continuous data, these are:

o The process is in statistical control. o The distribution of the process considered is Normal.

If these assumptions are not met, then result of the statistics may be highly unreliable.

3.4 Basic Concepts of Process Capability

Process capability is a statistical quality control tool which requires collecting data from a certain process. Then constructing a histogram, a curve is drawn which fits in the histogram and finally finding out what percentage of data goes outside the Upper Specification Limit (USL) and Lower Specification Limit (LSL). This is shown in figure 3.1. From the knowledge of statistics, outputs from a natural process generally follow a normal distribution. The performance is measured using two statistical parameters- a) mean value of the measurements ( μ orx ) and b) spread of the measurements, expressed in terms of standard deviation (σ or s ).

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LSL USL

Figure 3.1: Curve fitting in the histogram.

Five most commonly used indices to measure process capability. These are:i. C p = Process Potential Index

ii. C pk = Process Performance Index iii. CPU = Upper Process Performance Index iv. CPL = Lower Process Performance Index v. k = Process Centering Index

A process is considered “capable” if the process spread is less than or equal to specification limits (figure 3.2). If it goes sufficiently beyond these specification limits the process is considered “not capable”.

LSL USL

(μ or x)

Actual process spread

Allowable process spreadFigure 3.2: Process spread less then specifications.

To achieve a capable process is to attain “process stability”. Once process stability is achieved, there are two statistical parameters which affect process capability:

a) Location, Mean or Average – The mean of process values is its center of its distribution. When the process mean is shifted on either side the probability of

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large amount of data falling outside the specification limit becomes high. Therefore, the closer the process mean to a specification limit, the higher the chance of losing process capability.

b) Variability – This means the spread of process values. If variability, measured in terms of standard deviation, is high, large areas in two tails of distribution go beyond the specification limits. Therefore, variability affects process capability.

Capability of the process can be judged in two ways:

1) Observation of the graphical distribution - though graphical distribution plot can provide fair idea about process capability, this does not provide exact quantified values of capability.

2) Mathematical measures – there are five basic indices which provide quantified values of process capability.

Some graphical situations are illustrated in Figure 3.3, some are capable, whereas some incapable.

LSL LSL USL LSL

Capable Incapable

(μ or x) (μ or x)

LSL USL LSL Incapable USL

Incapable

(μ or x) (μ or x)

Figure 3.3: Process capability.

A process can be producing parts beyond the specification limits if the mean μ or x is located too close to one of the specification limits, or process variability is too high such that both of the tails go beyond the specification limits, as shown in figure 3.3. If too many parts are produced beyond the specification limits, the process is considered as incapable.

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Two areas are necessary to make a process capable:

a) Adjustment in part characteristics – If the part characteristics are adjustable, it is easy to make the process capable.

b) Adjustment in process characteristics – If the part characteristics are impossible to adjustable, significant process changes may be required.

In some cases, it may be possible that the process mean has shifted to one of the specification limits, still maintaining a capable process. Because the shift is not big enough which does not take the corresponding tail of the distribution beyond the limit. It’s possible when the process spread is less than the width of the specification limits.

Now it is clear that both center and variability of quality characteristics are important making a process capable. Appropriate measures must be taken to keep the process at its mean or near mean, and variability low as much as possible.

3.4.1 Process Potential Index

Process potential index, C p is related to process variability (basic statistical parameter). To quantify process potential index,C p, measurement of allowable process spread & actual process spread are necessary.

Allowable process spread = Upper Specification Limit – Lower Specification Limit.

The actual process spread is taken to be 6σ , considering 3σ quality control, when population standard deviation σ is known. In case of unknownσ , standard deviation of the sampling distribution of sample means (σ ) is considered.

LSL USL

(μ or x)

Actual process spread

Allowable process spread

Figure 3.4: Allowable vs. Actual process spread.

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A ratio between the two values givens process potential indexC p:

. Cp =Allowable processspread

Actual process spread=USL−LSL

6 σ

For a capable process, it would be better if actual process spread values is less than the allowable process spread values. Since for a good process, actual spread must be ≤ allowable spread, C p value must be ≥ 1.

From here, it is now possible to evaluate process performance, by computing the percentage of the specification width actually used by the spread, as following-

Percentage of specification used =Actual process spread

Allowable process spread× 100 = 1

Cp ×100

When process is improved through adjustment in process parameter, then actual spread of the process may narrow down and the continuous improvement makes the actual spread steeper. When this happens, C p value increases, which is desirable.

LSL USL LSL USL

C p=1 C p=1.33

LSL USL LSL USL

C p=2

C p=1.66

Figure 3.5: C p varies with process widths

The above figure shows how decreasing variability of actual process influences the process capability.

3.4.2 Process Performance Index

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Location of the set of data is very important. It is true that variability can help in reducing the chance of out-of-control situation; nevertheless, location is another important matter to be taken into account. Mean is the most common measure of location, which, if shifted many of the parts beyond the specification limits may make the process incapable. The process mean may be shifted to any direction, either towards the upper limit or the lower limit. Process Performance Index C pk is logical to measure the index value only on one side of the given specification range, as performance is affected on only one side, the direction in which it is shifted. The other side remains unaffected, thus does not to be considered. It is also measured as a ratio between allowable and actual values, as given below.

Actual upper process spread = only the upper side from the mean = 3 σ

If the center of the process isX , then, allowable upper process spread = USL - X

If performance of the process is measured on one side (the upper side) of the distribution, then the index can be named the Upper Process Performance Index (CPU).

CPU=Allowable uppper Process spread

Actual uppper Process spread =USL−X3 σ

The following figure shows the relationship of measuring CPU, on upper side: Actual Upper Process spread USL

(μ orx)

Allowable Upper Process spread Figure 3.6: CPU measurement Similarly, it may be measured the Lower Process Performance Index (CPL):

CPL =Allowable lower Process spread

Actuallower Process spread = X−LSL3 σ

The minimum of these two values, CPU and CPL, will provide the idea as to which direction of the mean has shifted.

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Process Performance Index, C pk = minimum (CPU, CPL)

Thus, the Process Performance Index, C pk measures capability of the process of that specification limit, which has the highest chance of a part going beyond the limit. And that side will have the highest chance, to which the process has shifted.

3.4.3 Process Centering Index

Process Centering Index, k is considered as another measurement which quantified the process center or mean. Here, k provides a relationship between the center of the actual process and center of the specification limits.

The center or midpoint of the specification limits, m= USL+LSL

2If the process is not shifted at all, then the process center or mean will exactly coincide with the center of the specification limits, i.e. X = m. If it is shifted, then there will be a difference between the two.

The distance between m and X can be computed as following:

m - X

LSL X m USL

X - m

LSL m X USL

Figure 3.7: Process center shifted in one direction

During quantifying the shifting of center, the direction of movement is not considered, thus only absolute value of shifting is considered. Now The process Centering Index,

k =Amount of shift of processcenter ¿ mod point of specification limits ¿

12(allowable process spread)

= ¿m−X∨ ¿

12(USL−LSL)

¿ = 2∨m−X∨ ¿

(USL−LSL)¿

From the above equations, two important decisions can be taken:

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a. If the center (or mean) of the process (X) falls exactly on the center of the specification limits, then m =X , or k =0. In this case, CPU= CPL, and thus C pk =C p.

b. If the process center X lies exactly on the USL, then [m - X = 12(USL−LSL)], and

hence, k = 1. In this case, (USL -X), or (X - LSL) = 0. Thus C pk=0. The estimator for C pk can also be expressed as C pk = Cp (1-k), When k=0, C pk= Cp.

When k=1,C pk=0, but Cp may not be equal to zero. Therefore, the above relationship implies thatC pk is always less than or equal to Cp.

3.5 Cause –Effect Diagram

There are must some potential reasons, or causes which ultimately lead to create an adverse “Effect”. Here the “Effect” is the quality problem. Cause-Effect (CE) analysis is a tool for analyzing and illustrating a process by showing the main causes and sub-causes leading to an effect. Sometimes it is called the Ishikawa diagram since Kaoru Ishikawa developed it and it looks like a fish skeleton diagram.

3.5.1 Procedure of Constructing a CE Diagram

The following materials are needed, for gathering information from a brainstorming session: a flipchart or whiteboard, marking pens.

The following step-by-step procedure may be followed to construct a CE diagram:

1) Agree on a problem statement (effect). Write it at the center right of the flipchart or whiteboard. Draw a box around it and draw a horizontal arrow running to it.

2) Brainstorming the major categories of causes of the problem. If this is difficult use generic heading:

o Methodso Machines (equipment)o People (manpower)o Materials o Measurement o Management o Environment

3) Write the categories of the causes as branches from the main arrow. These branches are known as Twigs.

4) Brainstorm all the possible causes of the problem. For example, ask: “why does this happen?” As each idea is given, the facilitator writes it as a branch from the appropriate category. Causes can be written in several places if they relate to several categories.

5) Again ask “why does this happen?” about each cause. Write sub-causes branching off the causes. The sub-causes are known as Twiglets. Continue to ask “why?” and generate deeper levels of causes.

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6) When the group runs out of the ideas, focus attention to places on the chart where ideas are few.

For simple example figure 3.8 shows the Cause-Effect diagram.

Figure 3.8: Cause-Effect Diagram.

3.6 Hypothesis Test

A statistical hypothesis is an assertion or conjecture one or more population. The structure of the testing will be formulated with the use of the term null hypothesis and denoted by Ho. The rejection of ho lead to the acceptance of an alternative hypothesis, denoted by H1.Rejecting the null hypothesis (Ho) when it is in fact true is called a Type I Error. Many people decide, before doing a hypothesis test, on a maximum p-value for which they will reject the null hypothesis. This value is often denoted α (alpha) and is also called the significance level. When a hypothesis test results in a p-value that is less than the significance level, the result of the hypothesis test is called statistically significant. Not rejecting the null hypothesis when in fact the alternate hypothesis is true is called a Type II Error. This value is often denoted byβ.

Table 3.1: Summary of errors in sampling for control chart:

God’s view (process reality)

Process in control Process out-of-control

Right signal Type II Error(Failure to detect)

Type I Error(False signal)

Right signal

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Process in control

Process out of control

Effect or Problem

Man

MethodMeasurement Environment

Composition ratio SetupSystem

Skill

UpdateAccuracy

CalibrationTemperature

Humidity

MachineMaterialManagement

Since the hypothesis testing based on the sample data, there is always a chance of committing an error. Two types of errors, as stated earlier, namely type I Error and type II Errors may be committed while testing hypothesis.

1. If the null hypothesis (Ho) is rejected (erroneously because of sampling limitations) when it is actually true, then type I error occurs. Probability of type I Error is expressed using α simbol. This is also called producer’s risk, as it denotes probability of rejecting a good lot, which should be acceptable.

2. If the null hypothesis is not rejected (i.e. accepted) when it is false, then a type II Error occurs. Probability of type II Error is expressed using β simbol. This is also called consumer’s risk, as it denotes probability of accepting a bad lot, which should be rejected.

Thus α = p {Type I error} = P {reject HoI Ho is true}

β=P {Type II Error} = P {fail to reject HoI Ho is false}

−Z α2 +Z α

2

Figure 3.9: Probability of Type I Error.

A common practice in statistical quality control is to specify a value ofα , then control the process to obtain a small value ofβ. Thus measuring β is of importance. The following section shows how to measure β value.

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AcceptanceRejection

Rejection

Process mean

α2

Figure 3.9: Probability of Type II Error.

Probability of Type II Error (β) can be measured in the following way-

β= Φ [UCL−μ1

σ√ n

] - Φ [LCL−μ1

σ√ n

] (1)

Where μ1 is the actual process mean obtained from sample, noticing that it may have shifted from population mean μ0.

Here, UCL= μ0+ Z α2

σ√ n and LCL = μ0−¿ Z α

2

σ√ n (2)

And μ1= μ0+δσ , where δ is the shifting of the process mean measured in terms of number of standard deviation units.So, from equations 1 &2

β= Φ [UCL−μ1

σ√ n

] - Φ [LCL−μ1

σ√ n

]

= Φ [μ0+Z α

2

σ√ n

– (μ0+δσ)

σ√ n

] - Φ [μ0−Z α

2

σ√ n

– (μ0+δσ )

σ√ n

]

= Φ ( Z α2 - δ √ n

σ ) – Φ ( −Z α2 -δ √ n

σ )

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LCL UCL

-Z α2

Z α2

0

μ0μ1

Type II Error

3.7 Linear Regression

Simple linear regression is a technique in parametric statistics that’s commonly used for analyzing mean response of a variable Y which chances according to the magnitude of an intervention variable X. It forms the basis of the one of the more important forms of inferential statistical analysis. In regression analysis there is usually the independent variable and a dependent or response variable.

The relationship between two variables is best observed by means of a scatter plot. Then a straight line is drawn which would provide the best estimate of the observed trend. In other word, the line describes the relationship in the best possible manner. Even then for any given value of X there is variability in the values of Y. this is because of the inherent variability between individuals. The line drawn is therefore the line of means. Thus, it expresses the mean of all values of Y corresponding to a given value of X.

Figure 3.10: The regression line

Where the line of means cuts the Y-axis the intercept is found. The intercept is the value of Y corresponding to X=0. Its units are the units of the Y variable. The line has a slope. The slope measures the change in the value of Y corresponding to a unit change in the value of X. Now it is clear that the line of means is an important parameter, its mathematical representation Y= a + b X is called the regression equation, and a & b are the regression coefficients.

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Slope

Intercept

X- Axis

Y- Axis

CHAPTER 4

Case Study and Data Collection

If you like to measure the process capability indices, you must have certain related data from a related manufacturing industry. This thesis paper shows the data related to a manufacturing industry. Kohinoor Chemical Company (Bangladesh) limited (KCCL) was a fully government owned industry under the direct control of Bangladesh Chemical Industries Corporation (BCIC) up to 05 may, 1988. Among various products like Sandalina Sandal Soap, Tibet Toilet & Laundry Soap, Tibet Ball, Tibet detergent, Tibet coconut oil, Tibet snow, Tibet glycerin, Tibet petroleum jelly etc. Sandalina Sandal Soap becomes very popular soap. During production of Sandalina Sandal Soap some major parameter should be maintained in a certain range. The major parameters of the Sandalina Sandal Soap are given below:

o Total Fatty Metal (TFM)o Free Alkali (as NaOH)o Chloride (as NaCl)o Moisture

Soap production involves many processing steps.The flow diagram of the soap production are given as following:

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Figure 4.1: Soap Production.

In flaked soap unit the following steps are-

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Bleaching tank

Air cooling 24-28 hoursCutting into blocks

Soap flakes unitsSoap flakes packaging

Equipment & floor wash

MucilageNaOHIndirect

StreamSodium Hypochlorit

e

Waste water

Chlorite

Solid WasteSolid

waste(soap) Solid waste

Waste water

Inputs Processing step

Pollution source

Figure 4.2: Flaked Soap Unit

The following table shows the parameter and minimum and maximum values:

Table 4.1: Parameter and their range values of neat Sandalina.

Parameter Maximum value Minimum value% Free Alkali (FA) 0.06 0.03

% Chloride (as NaCl) 0.6 0.4

During production the data of the neat soap lump of the process parameter of %FA & %Chloride are collected which are shown in the appendix A.

After production, the data of the major parameters are collected; these are based on finished products which are shown in the Appendix B.

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Saponification

Separation

Sweet water to glycerol recovery

Vacuum dryingSoap paste

compressorSoap barring

Soap Cutting

Stamping

FatNaOHSteam

Additive

Additive

Additive

FumesWaste waterHeat

stress(workplace)

Solid waste

Inputs Processing steps

Pollution source

CHAPTER 5

Calculation

5.1 Calculation of Process Capability Indices for the Neat (Toilet Sandalina Sandal Soap) Lump

The process mean µ = ∑i=1

n

xi

n

Standard deviation, δ =√∑i=1

n

¿¿¿¿

From appendix A: Mean of the collected data of the % FA (as NaOH), µ = 0.042Standard deviation of the collected data of the % FA (as NaOH), δ= 0.0133

Process Potential Index, Cp =USL−LSL

6 σ = 0.06−0.036 × 0.0133 = 0.03759 which is less than 1 so

that the process is not capable.

Now Process Performance Index, Cpk = min (CPU, CPL)

Upper Process Performance Index, CPU = allowable upper proces spread

actual upper process spread

= USL−X

3σ = 0.06−0.0423 × 0.0133 = 0.451

Lower Process Performance Index, CPL = allowable lower proces spread

actual lower process spread

= X−LSL3 σ = 0.042−0.03

3× 0.0133 = 0.301

Therefore Cpk = min (0.451, 0.301) = 0.301, So that the process mean is shifted in left side

Process Centering Index, k = ¿m−X∨ ¿

12(USL−LSL)

¿

The center or midpoint of specification limits, m = USL+LSL

2

= 0.06+0.032

= 0.045

21

Therefore k = ¿0.045−0.042∨ ¿

12(0.06−0.03)

¿ = 0.2, So that the process mean is

shifted in left side.

The process mean µ = ∑i=1

n

xi

n

Mean of the collected data of the % Chloride (as NaCl),µ = 0.47

Standard deviation δ =√∑i=1

n

¿¿¿¿

Standard deviation of the collected data of the % Chloride (as NaCl), δ= 0.028

Process Potential Index, Cp =USL−LSL

6 σ = 0.6−0.46×0.028 = 1.19

Process Performance Index, Cpk = min (CPU, CPL)

Upper Process Performance Index, CPU = allowable upper proces spread

actual upper process spread

= USL−X

3σ = 0.6−0.473× 0.028 = 1.58

Lower Process Performance Index, CPL = allowable lower proces spread

actual lower process spread

= X−LSL3 σ = 0.47−0.4

3× 0.028 = 0.833

Therefore Cpk = min (1.58, 0.833) = 0.833, So that the process mean is shifted in left side slightly.

Process Centering Index, k = ¿m−X∨ ¿

12(USL−LSL)

¿

The center or midpoint of specification limits, m = USL+LSL

2

= 0.6+0.42

= 0.5

Therefore k = ¿0.5−0.47∨ ¿

12(0.6−0.4)

¿ = 0.3, So that the process mean is shifted in

left side slightly.

22

5.2 Linear Regression Analysis

The simple linear relationship between dependent & independent variable is Y= a + b X

Where X = independent variable, Y= dependent variable, n= Sample size

b =n ∑ XY−∑ X ∑Yn ∑ X2−(∑ X )2 , a = ∑Y−b ∑ X

n

5.2.1 Relationship between %TFM & %Moisture

Considering %TFM is the dependent variable Y & % moisture is the independent variable X.

Then from Appendix B:

∑ X2 ∑ X ∑Y ∑ XY n

7882.7 485.27 2293.02 37062.64 30

b = 30× 37062.64−485.27 ×2293.02

30× 7882.7−(485.27)2 = −¿0.859

a = 2293.02−(−0.859× 485.27)

30 = 90.34

14 14.5 15 15.5 16 16.5 17 17.5 18 18.5 197273747576777879

f(x) = − 0.859738477443898 x + 90.34084303164R² = 0.839454748933901

Relationship between % TFM & % Moisture

% Moisture

% T

FM

Figure 5.1: Scatter chart relation between % TFM & % Moisture.

Table 5.1: Relationship between %TFM & %Moisture.

Dependent variables (Y) Independent variables (X) Linear equation

23

% TFM % Moisture Y= 90.34 – 0.859X

5.2.2 Relationship between %TFM & %FA

Considering %TFM is the dependent variable Y & %FA is the independent variable X.

Then from the Appendix B:

∑ X2 ∑ X ∑Y ∑ XY n

4.5×10−3 0.35 2293.02 26.7863 30

b = 30× 26.7863−0.35 × 2293.02

30 × 4.5 ×10−3−(0.35)2 = 82.56

a = 2293.02−82.56 ×0.35

30 = 75.47

0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.0227273747576777879

f(x) = 82.56 x + 75.4708R² = 0.0973434074634661

Linear relationship between %TFM & %FA

%FA

%T

FM

Figure 5.2: Scatter Chart relation between %TFM & %FA.

Table 5.2: Relationship between %TFM & %FA.

Dependent variables (Y) Independent variables (X) Linear equation% TFM % FA Y= 75.47 + 82.56X

5.2.3 Relationship between %TFM & %Chloride

Considering %TFM is the dependent variable Y & % Chloride is the independent variable X.

Then from the Appendix B:

24

∑ X2 ∑ X ∑Y ∑ XY n

7.36 14.86 2293.02 1135.81 30

b = 30× 1135.8072−14.86 × 2293.02

30 ×7.3644−(14.86)2 = −¿0.544

a = 2293.02−(−0.544 ×14.86)

30 = 76.70

0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.537273747576777879

f(x) = − 0.544483985765094 x + 76.7037010676156R² = 3.8070948408464E-05

Relationship between %TFM &% Chloride

% Chloride

%T

FM

Figure 5.3: Scatter chart relation between %TFM & % Chloride

Table 5.3: Relationship between %TFM & %Chloride.

Dependent variables (Y) Independent variables (X) Linear equation% TFM %Chloride (as NaCl) Y= 76.70 – 0.544X

5.2.4 Relationship between %TFM, %Moisture, %FA & %Chloride

Considering %TFM is the dependent variable Y & the independent variables are %Moisture X1, %FA X2, % Chloride X3.

Table 5.4: Relationship between %TFM %Moisture, %FA & %Chloride.

Dependent variables (Y)

Independent variables (X)

%TFMY

%MoistureX1

%FA (as NaOH)X2

%Chloride (as NaCl)X3

Overall relationship

Y = 80.83667 – 0.2863X1 +27.52X2 – 0.18133X3

25

5.3 Calculation of Type І & Type ІІ Error of the Process Parameter % FA

From Appendix A:Mean of the collected data of the process parameter % FA, X = 0.042Sample size of the collected data, n = 30 Standard deviation of the collected data of the process parameter % FA, σ = 0.0133

Consider, Null hypotheses H0: µ= 0.045 Alternative hypotheses H1: µ≠ 0.045Probability of Type І error (or level of significance, α) is specified as 0.05.This is a two tail test, thus level of significance on both side is α /2 = 0.05/2= 0.025

−Z α2 +Z α

2

Z =X−μ

σ√n

= 0.042−0.045

0.0133√ 30

= −1.18 ¿ Z critical (−¿1.96)

Since Z (= −¿1.18) is greater than Z critical (=−¿1.96), the null hypothesis is rejected. That means, the process mean has really shifted.

Probability of type ІІ error, β = Φ ( Z α2 −¿

δ √ nσ ) – Φ ( −Z α

2 −¿

δ √ nσ )

= Φ (1.96 –0.003 √ 30

0.0133 ) – Φ (−¿ 1.96 −¿ 0.003 √ 30

0.0133 )

= Φ (1.96 – 1.1804) −¿ Φ (−¿1.96 – 1.1804) = Φ (0.7796) – Φ (−¿3.1404) = 0.78219 – 0.00084 =0.78135Now power of this process,Power = 1 – β = 1−¿ 0.78135 = 0.21865

26

Process mean

0.475

0.025

−¿1.96

1.96

=21.865% Therefore, probability of detecting the shift of mean, or probability of not accepting the bad lot is 21.865%.

5.4 Calculation of Type І & Type ІІ error of the Process Parameter % Chloride

From Appendix A:Mean of the collected data of the process parameter % Chloride, X = 0.47Sample size of the collected data, n = 30 Standard deviation of the collected data of process parameter %Chloride, σ = 0.028

Consider, Null hypotheses H0: µ= 0.50 Alternative hypotheses H1: µ≠ 0.50Probability of Type І error (or level of significance, α) is specified as 0.05.This is a two tail test, thus level of significance on both side is α /2 = 0.05/2= 0.025

0.475

−Z α2 +Z α

2

-1.96 1.96

Z =X−μ

σ√n

= 0.47−0.50

0.028√ 30

= −¿ 5.868¿ Z critical (−¿1.96)

Since Z (= −¿5.868) is less than Z critical (=−¿1.96), the null hypothesis is not rejected. That means, the process mean has not shifted.

Probability of type ІІ error, β = Φ ( Z α2 −¿ δ √ n

σ ) – Φ ( −Z α2 −δ √ n

σ )

= Φ (1.96 –0.03 √ 300.028 ) – Φ (−¿1.96 −¿ 0.03 √ 30

0.028 ) = 0.00005Therefore, Power = 1 – β = 1 – 0.00005 = 0.99995

27

0.025

= 99.995% Therefore, probability of detecting the shift of mean, or probability of not accepting the bad lot is 21.865%.

5.5 Cause-Effect Diagram

Cause-Effect diagram indentifies one-by-one all the possible causes from brainstorming sessions and then classifies into groups. During production the common type of causes or effects are occurred. These types of causes or effects hamper the productivity greatly. In this study try to find out the major causes or sub causes by cause-effect diagram.

5.5 .1 Cause-Effect Diagram of Improper Saponification

For the effect of improper saponification the several causes are responsible. These are man, machine, material, management, method, measurement and environment. And each cause has some sub-causes. These are given below:

Figure 5.4: Cause-Effect Diagram of Improper Saponification.

After brainstorming, the major causes or sub-causes are type of row materials, composition ratio during mixing, skill of the operator and SOP. If the types of raw materials are not good in quality then improper saponification occurred. Composition ratio also falls bad in condition if the quality of raw material is not good. So the raw material quality should be good in quality. The skill full operator plays a vital rule in

28

Improper Saponification

MaterialMachineManagement

Man

Method Measurement Environment

Composition ratio Type

HardnessSetup

Accuracy

PowerCommitment

RuleSystem

Skill

Motivation

SOPUpdate

Accuracy

Calibration

Noise

Temperature

Humidity

saponification process. If possible operator should be trained. And Standard Operating Procedure (SOP) also affects the effect of improper saponificaton.

5.5 .2 Cause-Effect diagram of Improper Mixing

For the effect (or cause) of improper mixing the several causes are responsible. These are man, machine, material, management, method, measurement and environment. And each cause has some sub-causes. These are given below:

Figure 5.5: Cause-Effect Diagram of Improper Mixing.

After brainstorming, the major causes or sub-causes of the effect improper mixing are type of row materials, composition ratio of the raw materials, skill of the operator and SOP. Raw material quality is very important in this case. If raw material worst in condition than proper mixing occurred. Therefore improper is occurred at this time. Skill-full workers are also needed for operating the process with Standard Operating Procedure (SOP).

29

Improper Mixing

Man

Method Measurement Environment

Composition ratio

Type

Hardness

Setup

Accuracy

Power

Commitment

RuleSystem

Skill

Motivation

SOPUpdate

Accuracy

Calibration

Noise

Temperature

Humidity

Machine MaterialManagement

5.5 .3 Cause-Effect Diagram of Rework

For the effect (or cause) of rework the several causes are responsible. These are man, machine, material, management, method, measurement and environment. And each cause has some sub-causes. These are given below:

Figure 5.6: Cause-Effect Diagram of Rework.

After Cause-Effect diagram of rework, the major causes or sub-causes are the skill of the operator and SOP, type of raw materials, composition ratio, machine set-up, calibration, and temperature etc. Skill of the operator is very much important. Because of lack of skill of the operator, the process can produces much defect full products. Also the process should be operated according to SOP. Raw material grade and composition ratio are also very important in this case.

30

Rework

Man

Method Measurement Environment

Composition ratio

Type

Hardness

Setup

Accuracy

Power

Commitment

RuleSystem

Skill

Motivation

SOPUpdate

Accuracy

Calibration

Noise

Temperature

Humidity

MaterialMachineManagement

CHAPTER 6

Results Analysis

Results analysis is the most important in every thesis work, in results analysis one can easily understand what is happen in the process or in the system from which the data are collected. 6.1 Result Analysis of Process Capability Indices for the Process Parameter % FA

Result of process capability indices for % FA for the neat lump Sandalina Sandal soap is given below:

Table 6.1: Result Analysis of Process Capability Indices for the Process Parameter % FA.

Process capability index Value CommentCp 0.3759 Process is not capableCpk 0.301 Mean is shifted in left sidek 0.2 Mean is shifted in left side

Since the process potential index, Cp is less than 1, the process is not capable and it produces large number of non-conformities products. The value of process performance index, Cpk and process centering index, k indicates than the process mean actually shifted in the left side. The possible cause of the process incapability is the sample size (30) is small. One should consider the large number of the sample size as possible. One when collect the much data per a day then the result becomes accurate. Other possible causes are: When collecting the data-

a. The operator might be not responsible about his duties. b. Machine might be operated without follow the SOP.

If the above correction brings worst in result, new target value should be setup.

6.2 Result Analysis of the Process Capability Indices for the Process Parameter % Chloride

Result of the process capability indices for % Chloride (as NaCl) neat lump Sandalina Sandal is given below:

Table 6.2 Result Analysis of the Process Capability Indices for the Process Parameter % Chloride.

Process capability index Value CommentCp 1.19 Process is capable

31

Cpk 0.833 Mean is shifted in left side slightly

k 0.3 Mean is shifted in left side slightly

Since the process potential index, Cp is greater than 1, the process is capable and it produces large number of conformities products. The value of process performance index, Cpk and process centering index, k indicates than the process mean actually slightly shifted in the left side, but produces less number of non-conformities product. Although the process in capable, the mean of the process is slightly shifted, so that one should collect the more data for the accurate result.

6.3 Result Analysis of Type І & Type ІІ Errors Since the hypothesis test is based on sample data, there is always a chance of committing an error as stated earlier, namely Type І & Type ІІ Errors may be committed while testing hypothesis.

i. If the null hypothesis is rejected (erroneously because of sampling limitations) when it is true, then a Type І error occurs. Probability of Type І is expressed using α symbol. This is also called producer’s risk as it denotes probability of rejecting a good lot, which should be acceptable.

ii. If the null hypothesis is not rejected when it is false, then a Type ІІ error occurs. Probability of Type ІІ error is expressed using β symbol. This is also called customer’s risk as it denotes probability of accepting a bad lot, when it should be rejected.

So that, Type І & Type ІІ Errors are used for measuring probability of rejecting a good lot, which should be acceptable and probability of accepting a bad lot, when it should be rejected.

6.3.1 Result Analysis of Type І & Type ІІ Errors of the Process Parameter % Chloride

Table 6.3: Result analysis of Type І & Type ІІ error of the process parameter % FA.

Parameter Mean Sample size

Standard deviation

Power(1-β )

CommentX μ

%FA 0.042 0.045 30 0.01392 21.865% Mean shifted

Mean of the process parameter % FA is really shifted and produces large no. of non-conformities products. However, a sample size 30 should not be enough to justify the accurate result. It is thus recommended sample size should be increased in all future estimation.

6.3.2 Result Analysis of Type І & Type ІІ Error of the Process Parameter % Chloride

Table 6.4: Result analysis of Type І & Type ІІ error of the process parameter % Chloride.

32

Parameter Mean Sample size

Standard deviation

Power(1-β )

CommentX μ

%Chloride 0.47 0.50 30 0.028 99.995% Mean not shifted

Mean of the process parameter %Chloride is not shifted and produces less no. of non-conformities products. However, a sample size 30 should not be enough to justify the accurate result. It is thus recommended sample size should be increased in all future estimation.

6.4 Regression Analysis

The study uses the simple linear regression relationship between the dependent variable %TFM and the independent variables %FA, %Moisture, %Chloride. These relationships help the manufacturer to find out the %TFM very easily.

6.4.1 Regression Analysis between % TFM & % Moisture

The relationship between % TFM & % Moisture are given below:

Dependent variables (Y)

Independent variables (X) Linear equation

% TFM % Moisture Y= 90.34 – 0.859X

From Appendix B, consider the sample no. 10, the independent variable % Moisture is X= 15.52, then the dependent variable % TFM is Y=90.34 −¿0.859×15.52 = 77.00832One can easily find out the approximate value of % TFM by following above process.

6.4.2 Regression Analysis between % TFM & % FA

The relationship between % TFM & % FA are given below:

Dependent variables (Y)

Independent variables (X) Linear equation

%TFM %FA (as NaOH) Y= 75.47 + 82.56X

From Appendix B, consider the sample no. 10, the independent variable % FA is X= 0.01, then the dependent variable % TFM is Y=75.47 + 82.56×0.01 = 76.2956One can easily find out the approximate value of % TFM by following above process.

6.4.3 Regression Analysis between % TFM & % Chloride

The relationship between % TFM & % FA are given below:

Dependent variables (Y)

Independent variables (X) Linear equation

33

%TFM %Chloride (as NaCl) Y= 76.70 – 0.544X

From Appendix B, consider the sample no. 10, the independent variable % Chloride is X= 0.47, then the dependent variable % TFM is Y=76.70 −¿0.544×0.47 = 76.44432One can easily find out the approximate value of % TFM by following above process

6.4.4 Regression Analysis between % TFM, % Moisture, % FA & % Chloride

The overall relationship is given below:

Dependent variables (Y)

Independent variables (X)

%TFMY

%MoistureX1

%FA (as NaOH)X2

%Chloride (as NaCl)X3

Overall relationship

Y = 80.83667 – 0.2863X1 +27.52X2 – 0.18133X3

From Appendix B, consider the sample no. 10, the independent variables are % Moisture is X1 = 15.52, % FA is X2= 0.01 and % Chloride is X3= 0.47, then the dependent variable % TFM is Y=80.83667 – 0.2863× 15.52 + 27.52× 0.01 – 0.1813×0.47 = 76.578083One can easily find out the approximate value of % TFM by following above process.

34

CHAPTER 7

Recommendations and Future Work

Brainstorming can help to find out the defects and the parameters or quality variables which hamper the quality. Cause effect diagram also can helpful to find out the defects. In order to know the major causes everyone can use the Pareto Chart. Also Control Chart is very useful to control the process within specification limits. This work recommends that after quantify the major parameters or factors anyone can use Design of Experiment technique. Design of Experiment (DoE) is highly structured method of quality control or process control. After identifying the important quality variables or factors that affect quality, DoE can very useful to eliminate those preferably at the design stage. In future this study recommends DoE which is also useful on all new designs. If in a production floor, there are number of machines which produce a particular product, then Randomized Complete Block Diagram (RCBD) is also very useful in such cases. This study consider the sample size only 30, in future one should consider as high sample size as possible.

35

Conclusion

An essential part of implementing any quality improvement process is to use process capability indices to quantify manufacturing process capability and performance. Process capability analysis is an important tool to improve the process quality as well as product quality. This study shows that in soap production the raw material quality is very important which demonstrate the process capability. Standard Operating Procedure is also responsible for process capability. The result analysis can be used to improve the process performance. This paper also shows the cause effect diagram and linear regression which can help to improve the process performance. This study tries to develop a novel procedure of integrated analysis of the process capability for a certain product. This work considers the sample size 30 which is not enough. The data was collected only from one machine which is considered as limitation of this work.

36

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Appendix A:

From neat soap lump during production the data of the %FA & %Chloride are collected.

Sample No. %FA (As NaOH) %Chloride ( As NaCl)1 0.04 0.462 0.05 0.463 0.05 0.524 0.03 0.435 0.06 0.456 0.03 0.467 0.03 0.518 0.01 0.439 0.03 0.5010 0.04 0.4611 0.06 0.5012 0.06 0.4813 0.04 0.4514 0.04 0.4715 0.06 0.4416 0.03 0.4517 0.03 0.4318 0.04 0.5119 0.03 0.4520 0.06 0.4821 0.04 0.5122 0.02 0.4723 0.05 0.4624 0.06 0.4925 0.03 0.4826 0.04 0.4627 0.04 0.4928 0.04 0.5029 0.04 0.5330 0.07 0.50

38

Appendix B:

After production, the data of the major parameters are collected.

Sample No. % TFM % Moisture % FA %NaCl1 74.18 18.63 0.01 0.502 76.77 17.41 0.01 0.503 76.96 15.62 0.01 0.484 75.37 17.35 0.01 0.495 75.68 16.33 0.01 0.506 75.29 17.42 0.01 0.507 74.49 18.30 0.01 0.498 76.69 15.91 0.01 0.499 76.63 15.97 0.01 0.4910 77.06 15.52 0.01 0.4711 76.47 16.14 0.01 0.4912 76.44 16.66 0.01 0.5113 77.15 15.39 0.01 0.5014 77.14 15.38 0.01 0.5215 76.71 15.89 0.01 0.4916 77.37 15.13 0.02 0.5117 77.33 15.21 0.01 0.4918 76.58 15.98 0.02 0.5219 75.79 16.90 0.01 0.4820 76.69 15.89 0.01 0.5021 76.44 16.17 0.01 0.5022 76.66 15.94 0.01 0.4923 78.08 14.38 0.02 0.4924 74.98 16.20 0.01 0.4925 74.27 18.53 0.01 0.5026 77.43 15.10 0.01 0.4927 77.00 15.55 0.02 0.5128 76.94 15.63 0.01 0.5029 77.85 14.64 0.01 0.4930 76.58 16.10 0.02 0.48

39

List of Abbreviations Used in this Thesis Work

ANOVA= Analysis of Variance BCa= Biased Corrected and Accelerated Bootstrap BCIC= Bangladesh Chemical Industries Corporation BCPB = Biased Corrected Percentile BootstrapCE = Cause –Effect CPL = Lower Process Performance Index CPU = Upper Process Performance Index C p= Process Potential Index Cpa = Asymmetric Process Capability Index Cpl = Lower Process Performance IndexC pk = Process Performance Index Cpm = Taguchi IndexCpmk = Asymmetric Taguchi IndexCpu = Upper Process Performance IndexDoE = Design of ExperimentFA = Free Alkali k = Process Centering Index KCCL = Kohinoor Chemical Company (Bangladesh) limited LSL = Lower Specification Limitm= Center or Midpoint of the Specification LimitsMPCAC = Multi-Process Capability Analysis ChartMPPAC= Multi Process Performance Analysis Chart NMPCAC= Non-Normal Multi Process Capability Analysis ChartPCA= Principal Component AnalysisPCMC= Process Capability Monitoring ChartPNC= Proportion of Non-ConformancePCI= Process Capability IndexRCBD= Randomized Complete Block DiagramSB= Standard BootstrapSOP= Standard Operating Procedure TFM= Total Fatty Metal USL= Upper Specification Limitα = Type I Errorβ= Type II Error

40