# 1.1 and Riaz (2006) used a probability

1.1

An Introduction to Statistical Quality Control and its Brief Review of

Literature:

Many

people used the word “quality” in different contexts. The quality of a service

or product is recognized as very important by various companies. However, it is

very difficult to frame a definition that differentiates between product and

services of bad quality from products and services of high quality. Shewhart

(1931) states that there are two aspects of quality. Firstly there is an

objective concept of quality, resulting in quantatively measurable physical

characteristics, which are independent of second, subjective, aspect of

quality. He recognized that the subjective side of quality is commercially

interesting, but that it is necessary to establish standards of quality in a

quantitative manner. Quality can be defined only in terms of the agent (Deming

(1982)). Juran and Gryna (1988) defines quality as “quality is fitness for use.” In recent

years, the importance of quality has become increasingly apparent. Harder competition,

complex environmental and safety regulations, and abruptly changing economic

conditions have been key factors in tightening plant product quality.

used a synthetic

control charting approach, Muhammad and

Riaz (2006) used a probability weighted moments based approach, He and

Grigoryan (2006) used a double sampling approach, Riaz and Saghir (2007) used a

gini mean difference based approach. Riaz (2008a) proposed a process variance

chart and claimed its superiority over the well known

chart, the cause-selecting and regression adjusting

control charts.

1.2

An Approach to Acceptance Sampling and its brief review of literature:

Acceptance sampling has become typical to work

with suppliers to improve their process performance by using of statistical

process control (SPC) with designed experiments. It has been focused on when the inspection required is destructive,

testing 100% inspection is not feasible

due to the cost or time so an acceptance sampling plan is created to define how

many samples must be taken to verify the lot. Juran and Gryna (1988) defined

acceptance sampling as an inspection procedure applied in SQC. It is a method

of measuring random samples of population called “lots” of materials or

products against predetermined standards. Acceptance sampling is a part of

operations management and service quality supervision. Acceptance sampling

seems very beneficial for industrial and business purposes as its helps in

decision making process. Sampling plans are hypothesis tests regarding product

that has been submitted for an appraisal and subsequent acceptance and rejection.

The products may be grouped into lots or may be single pieces from a continuous

operation. A random sample is selected and could be checked for various

characteristics. Accepting or rejecting a lot is similar to not rejecting or

rejecting the null hypothesis in a hypothesis test. The acceptance methods are

used for attributes and variables. The attribute sampling is a simple

statistical method utilizes representative samples to analyze traits of a large

body of a data and decision based on number of defectives in a lot. In variable

sampling plans, one or more samples of items are drawn from a given lot,

measurement of quality characteristic in each sampled item is recorded, and the

decision of acceptance or rejection of the lot is made as a function of such

measurements. The variable sampling plans are used in the situations where the

quality characteristic of sampled item is measurable on continuous scale and

the functional form of the probability distribution is assumed to be known. A

variable sampling plan is advantageous to attribute sampling plan in the sense

that it generates more information from each item inspected, requires small

sample and provides same protection as provided by attribute sampling plan. Studies relating to sampling plans when the

assumptions of normality and independence of the quality characteristic fails

or the functional form of the underlying distribution deviates from normal or

the form of the distribution is not known are found in the literature of acceptance

sampling. Some of the early works on

variable sampling includes Liberman and Resineff (1955), Schilling (1982), Owen

(1966, 1967) and Hamaker (1979). Saveral authors tried to design the sampling

plans in case the assumptions of normality and independence are not

fulfilled. Srivastava (1961) studied

variable sampling inspection for non-normal samples. Das and Mitra (1964)

examined the effect of non-normality on plans for sampling inspection by

variables. Geetha and Vijayaragharvan (2011) studied studied the selection of

single sampling plan by variables based on Logistic distribution. Geetha and Vijayaraghavan

(2013) examined the procedure for the selection of single sampling plan by

variables based on Pareto distributions. For non-normal distributions, the

designing of unknown sigma plans is much complicated. Takagi (1972) attempted

to provide solution to such problems and proposed a methodology for determining

the parameters of variable sampling plans under the non-normality population by

introducing an expansion factor in terms of measure of skewness and kurtosis. The performance of an acceptance sampling plan

is based on the operating characteristic (OC) curve. The OC curve plots the

probability of accepting the lot against the actual product fraction defective,

which displays the discriminating power of the sampling plan. Pukar et al.

(2011) designed an OC curve for acceptance sampling plan to minimize a

consumer’s risk. Khandwawala (2012) constructed OC curve for acceptance

sampling plan by using MATLAB software.

1.3

A Brief Literature Review On Economic Acceptance Sampling

Plans: Economically

designed plans guarantee the lowest cost, but typically they have poor

statistical performance as they ignore statistical properties. Type I error

rate of the economic design may be too high for many situations and will cause

a large number of false alarms. This has infact became a major limitation of

the economic design. A statistically designed sampling plan is a structured

method in which the Type I error probability and power are generally fixed at

the desired levels. Statistically designed plans may yield high power and low

Type I error rate but they may cost more than economic designs. Saniga (1989)

was the first to introduce economic statistical design to combine the benefits

of both pure statistical and economic designs while minimizing their weakness.

The objective of both economic and statistical designs is to minimize the

expected total cost per unit via a non-linear constrained optimization. The

main difference between the two is that economic statistical designs are

subject to constraints on the type I error rate and power. Various authors have

tried to study sampling plans from economic view point. Some of the works

already done on economic designs includes Watherill and Chou (1975) have

compiled a through bibliography of papers dealing with acceptance sampling

schemes with emphasis on the economic aspects. Champernowni (1953) considers

the problem of deriving sequential sampling plans that minimize the sum of

decision and inspection costs. Beta distribution was used by him as the prior

distribution of lot quality. His plans are based on critical fraction

defective, p0, where decision costs are zero. Pukar et al. (2011)

designed an OC curve for acceptance sampling plan to minimize a consumer’s

risk. Farrel and Chhoker (2010) developed economically optimal acceptance

sampling plans in a two stage supply chain and tried to minimize the producer’s

and consumer’s total quality cost while

satisfying both the producer’s and

consumer’s quality and risk requirements. Vispute and Singh (2014) examined

economic effect of variable sampling plan for autocorrelated data. Narayanan

and Rajarathinam (2013) provided the procedure for selection of single sampling

plans by variables for Pareto distributions.

1.4

An approach to Correlation and its brief review of literature: Correlation means

association – more precisely it is a measure of the extent to which two

variables are related. If an increase in one variable tends to be associated

with an increase in the other then this is known as a positive correlation. A correlational study determines whether or not two

variables are correlated. This means to study whether an increase or decrease in one

variable corresponds to an increase or decrease in the other variable.

Correlational Research is also known as Associational Research. Relationships

among two or more variables are studied without any attempt to influence them

and investigated the possibility of relationships between two variables.

One purpose for doing correlational research is to determine the degree to which a

relationship exists between two or more variables. Guo and

Manatunga (2007) proposed to estimate the concordance correlation coefficient

(CCC) non-parametrically through the bivariate survival function. They proved

the presented estimator of the CCC to be strongly consistent and asymptotically

normal, with consistent bootstrap variance estimator. Additionally, they

developed a non-parametric estimator for the time-dependent agreement

coefficient. It has the same asymptotic properties as the estimator of the CCC.

Previously, Liu et al. (2005) have also worked in this field of CCC and they

studied inter-rater agreement in

measurements of time to event, usually not observed with perfect consistency

between raters. As a function of the first two moments of rating measures, the

CCC can be estimated with data subject to censoring, using a likelihood-based

estimation method employed under the assumptions of random censoring and

parametric distribution models for the ratings of time to event.

In traditional

quality control charts, fixed sampling interval (FSI) schemes are used where

the time between samples has fixed intervals. More efficient methods called VSI

schemes have been developed where one takes the next observation sooner than

usual if there is an indication that the process is operating off the target

value. Another traditional assumption behind most statistical process control

charts is that the sequential observations are independent. However, there are

many situations where the sequential observations should not to be treated as

independent. Rather, a time series model, in particular the first order

autoregressive (AR (1)) model, is appropriate. Baik (1991) used Markov chain

representation study the properties of the FSI and VSI Shewhart X control

charts. They showed that if the process variance is properly estimated and if

traditional control limits are used in the FSI control charts, then the

detection time is shorter when the consecutive observations are negatively

correlated than when they are positively correlated. If they are positively

correlated, then the false alarm rate decreases as the correlation between

consecutive observations increases. Consecutively, the detection time increases

as the correlation increases. In VSI control charts with traditional control

limits, if the process mean is near the target, then the average time to signal

(ATS) and average number of samples to signal (ANSS) tend to decrease as the

correlation increases until the correlation becomes rather moderate. Then, for

more highly correlated data, the ATS and ANSS tend to increase as the

correlation increases. Even under the AR (1) process, the VSI chart is more

efficient than the FSI chart in terms of ATS. In contrast, the VSI chart is

less efficient than the FSI chart in terms of ANSS. The inefficiency (efficiency)

of ATS (ANSS) tends to increase (decrease) as the correlation between the

consecutive observations becomes stronger.1.1

An Introduction to Statistical Quality Control and its Brief Review of

Literature:

Many

people used the word “quality” in different contexts. The quality of a service

or product is recognized as very important by various companies. However, it is

very difficult to frame a definition that differentiates between product and

services of bad quality from products and services of high quality. Shewhart

(1931) states that there are two aspects of quality. Firstly there is an

objective concept of quality, resulting in quantatively measurable physical

characteristics, which are independent of second, subjective, aspect of

quality. He recognized that the subjective side of quality is commercially

interesting, but that it is necessary to establish standards of quality in a

quantitative manner. Quality can be defined only in terms of the agent (Deming

(1982)). Juran and Gryna (1988) defines quality as “quality is fitness for use.” In recent

years, the importance of quality has become increasingly apparent. Harder competition,

complex environmental and safety regulations, and abruptly changing economic

conditions have been key factors in tightening plant product quality.

used a synthetic

control charting approach, Muhammad and

Riaz (2006) used a probability weighted moments based approach, He and

Grigoryan (2006) used a double sampling approach, Riaz and Saghir (2007) used a

gini mean difference based approach. Riaz (2008a) proposed a process variance

chart and claimed its superiority over the well known

chart, the cause-selecting and regression adjusting

control charts.

1.2

An Approach to Acceptance Sampling and its brief review of literature:

Acceptance sampling has become typical to work

with suppliers to improve their process performance by using of statistical

process control (SPC) with designed experiments. It has been focused on when the inspection required is destructive,

testing 100% inspection is not feasible

due to the cost or time so an acceptance sampling plan is created to define how

many samples must be taken to verify the lot. Juran and Gryna (1988) defined

acceptance sampling as an inspection procedure applied in SQC. It is a method

of measuring random samples of population called “lots” of materials or

products against predetermined standards. Acceptance sampling is a part of

operations management and service quality supervision. Acceptance sampling

seems very beneficial for industrial and business purposes as its helps in

decision making process. Sampling plans are hypothesis tests regarding product

that has been submitted for an appraisal and subsequent acceptance and rejection.

The products may be grouped into lots or may be single pieces from a continuous

operation. A random sample is selected and could be checked for various

characteristics. Accepting or rejecting a lot is similar to not rejecting or

rejecting the null hypothesis in a hypothesis test. The acceptance methods are

used for attributes and variables. The attribute sampling is a simple

statistical method utilizes representative samples to analyze traits of a large

body of a data and decision based on number of defectives in a lot. In variable

sampling plans, one or more samples of items are drawn from a given lot,

measurement of quality characteristic in each sampled item is recorded, and the

decision of acceptance or rejection of the lot is made as a function of such

measurements. The variable sampling plans are used in the situations where the

quality characteristic of sampled item is measurable on continuous scale and

the functional form of the probability distribution is assumed to be known. A

variable sampling plan is advantageous to attribute sampling plan in the sense

that it generates more information from each item inspected, requires small

sample and provides same protection as provided by attribute sampling plan. Studies relating to sampling plans when the

assumptions of normality and independence of the quality characteristic fails

or the functional form of the underlying distribution deviates from normal or

the form of the distribution is not known are found in the literature of acceptance

sampling. Some of the early works on

variable sampling includes Liberman and Resineff (1955), Schilling (1982), Owen

(1966, 1967) and Hamaker (1979). Saveral authors tried to design the sampling

plans in case the assumptions of normality and independence are not

fulfilled. Srivastava (1961) studied

variable sampling inspection for non-normal samples. Das and Mitra (1964)

examined the effect of non-normality on plans for sampling inspection by

variables. Geetha and Vijayaragharvan (2011) studied studied the selection of

single sampling plan by variables based on Logistic distribution. Geetha and Vijayaraghavan

(2013) examined the procedure for the selection of single sampling plan by

variables based on Pareto distributions. For non-normal distributions, the

designing of unknown sigma plans is much complicated. Takagi (1972) attempted

to provide solution to such problems and proposed a methodology for determining

the parameters of variable sampling plans under the non-normality population by

introducing an expansion factor in terms of measure of skewness and kurtosis. The performance of an acceptance sampling plan

is based on the operating characteristic (OC) curve. The OC curve plots the

probability of accepting the lot against the actual product fraction defective,

which displays the discriminating power of the sampling plan. Pukar et al.

(2011) designed an OC curve for acceptance sampling plan to minimize a

consumer’s risk. Khandwawala (2012) constructed OC curve for acceptance

sampling plan by using MATLAB software.

1.3

A Brief Literature Review On Economic Acceptance Sampling

Plans: Economically

designed plans guarantee the lowest cost, but typically they have poor

statistical performance as they ignore statistical properties. Type I error

rate of the economic design may be too high for many situations and will cause

a large number of false alarms. This has infact became a major limitation of

the economic design. A statistically designed sampling plan is a structured

method in which the Type I error probability and power are generally fixed at

the desired levels. Statistically designed plans may yield high power and low

Type I error rate but they may cost more than economic designs. Saniga (1989)

was the first to introduce economic statistical design to combine the benefits

of both pure statistical and economic designs while minimizing their weakness.

The objective of both economic and statistical designs is to minimize the

expected total cost per unit via a non-linear constrained optimization. The

main difference between the two is that economic statistical designs are

subject to constraints on the type I error rate and power. Various authors have

tried to study sampling plans from economic view point. Some of the works

already done on economic designs includes Watherill and Chou (1975) have

compiled a through bibliography of papers dealing with acceptance sampling

schemes with emphasis on the economic aspects. Champernowni (1953) considers

the problem of deriving sequential sampling plans that minimize the sum of

decision and inspection costs. Beta distribution was used by him as the prior

distribution of lot quality. His plans are based on critical fraction

defective, p0, where decision costs are zero. Pukar et al. (2011)

designed an OC curve for acceptance sampling plan to minimize a consumer’s

risk. Farrel and Chhoker (2010) developed economically optimal acceptance

sampling plans in a two stage supply chain and tried to minimize the producer’s

and consumer’s total quality cost while

satisfying both the producer’s and

consumer’s quality and risk requirements. Vispute and Singh (2014) examined

economic effect of variable sampling plan for autocorrelated data. Narayanan

and Rajarathinam (2013) provided the procedure for selection of single sampling

plans by variables for Pareto distributions.

1.4

An approach to Correlation and its brief review of literature: Correlation means

association – more precisely it is a measure of the extent to which two

variables are related. If an increase in one variable tends to be associated

with an increase in the other then this is known as a positive correlation. A correlational study determines whether or not two

variables are correlated. This means to study whether an increase or decrease in one

variable corresponds to an increase or decrease in the other variable.

Correlational Research is also known as Associational Research. Relationships

among two or more variables are studied without any attempt to influence them

and investigated the possibility of relationships between two variables.

One purpose for doing correlational research is to determine the degree to which a

relationship exists between two or more variables. Guo and

Manatunga (2007) proposed to estimate the concordance correlation coefficient

(CCC) non-parametrically through the bivariate survival function. They proved

the presented estimator of the CCC to be strongly consistent and asymptotically

normal, with consistent bootstrap variance estimator. Additionally, they

developed a non-parametric estimator for the time-dependent agreement

coefficient. It has the same asymptotic properties as the estimator of the CCC.

Previously, Liu et al. (2005) have also worked in this field of CCC and they

studied inter-rater agreement in

measurements of time to event, usually not observed with perfect consistency

between raters. As a function of the first two moments of rating measures, the

CCC can be estimated with data subject to censoring, using a likelihood-based

estimation method employed under the assumptions of random censoring and

parametric distribution models for the ratings of time to event.

In traditional

quality control charts, fixed sampling interval (FSI) schemes are used where

the time between samples has fixed intervals. More efficient methods called VSI

schemes have been developed where one takes the next observation sooner than

usual if there is an indication that the process is operating off the target

value. Another traditional assumption behind most statistical process control

charts is that the sequential observations are independent. However, there are

many situations where the sequential observations should not to be treated as

independent. Rather, a time series model, in particular the first order

autoregressive (AR (1)) model, is appropriate. Baik (1991) used Markov chain

representation study the properties of the FSI and VSI Shewhart X control

charts. They showed that if the process variance is properly estimated and if

traditional control limits are used in the FSI control charts, then the

detection time is shorter when the consecutive observations are negatively

correlated than when they are positively correlated. If they are positively

correlated, then the false alarm rate decreases as the correlation between

consecutive observations increases. Consecutively, the detection time increases

as the correlation increases. In VSI control charts with traditional control

limits, if the process mean is near the target, then the average time to signal

(ATS) and average number of samples to signal (ANSS) tend to decrease as the

correlation increases until the correlation becomes rather moderate. Then, for

more highly correlated data, the ATS and ANSS tend to increase as the

correlation increases. Even under the AR (1) process, the VSI chart is more

efficient than the FSI chart in terms of ATS. In contrast, the VSI chart is

less efficient than the FSI chart in terms of ANSS. The inefficiency (efficiency)

of ATS (ANSS) tends to increase (decrease) as the correlation between the

consecutive observations becomes stronger.