MCom I Semester Analysis Statistical Quality Control Study Material Notes ( Part 2 )

MCom I Semester Analysis Statistical Quality Control Study Material Notes ( Part 2 ): Control Charts for Attributes Types of Acceptance Sampling Plans Product Control or Acceptance Sampling Quality Control Factor Table Control Charts at a Glance Examinations Questions Long Answer Questions  Short Answer Questions Objectives Questions ( Most Important Notes for MCom I Semester Students )

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Control Charts for Attributes

The control chart for variables can be used whenever the quality characteristic can be measured and expressed as a quality. Sometimes the quality characteristics of a product are not amenable to measurement. In such cases, we usually classify each item inspected as either confirmation to the specification on the quality characteristics or non-conforming to those specifications. The terminology ‘defective or non-defective’ or ‘acceptable or non-acceptable’ is often used to identify these two classifications of product. Quality characteristics of this type are called attributes. For example, we may say that plastic is cracked or not cracked, whether the bottles that have been manufactured contain holes or not. In such a case the defects cannot be measured as such, but number of defective items or number of defects per item can be ascertained. Hence, control charts for attributes may be constructed to reflect the pattern of variation and stability in terms of the proportion or fraction of items produced that are defective. Charts may also be based on the number of defects per unit of output. Generally, in case of attributes the following control charts are used:

(a) Control chart for fraction or proportion defectives (p-chart)

(b) Control chart for the actual number of defectives (np-chart)

(c) Control chart for the number of defects per unit (C-chart)

(a) Control Chart for Fraction or Proportion Defectives (p-chart) : The control chart for fraction (proportion) defectives, known as p-chart is used whenever the quality characteristic observed in the classification of items as defective or not-defective is the result of inspection of castings, go and not-go gauge test results, etc. The objective of this chart is to evaluate the quality of the items that is the average fraction defective or percent defective) and to note the changes in quality over a period of time.

Hence, the p-chart is designed to control the percentage or proportion of defectives per sample. A control chart for fraction defective, or p-chart, is based on the distribution of sample proportions. It is assumed that the items are produced by Bernoulli process. This assumption implies that (1) there are only two possible outcomes (acceptable or defective), (2) the outcomes occur randomly, and (3) the probability of either outcome remain unchanged for each trial. The steps in constructing the p-chart are as follows:

(1) Compute the fraction defective for each of the samples. This can be expressed as

Number of defective units in the sample np ” Total number of units inspected (sample size) n

(ii) Obtain the average fraction defective from all the samples combined. This is given by

(iv) Draw the central line at D as a dark horizontal line and the control limits as dotted horizontal lines against the corresponding sample numbers as points (dots).

(V) Plot the individual fraction defectives (or percent defective) values. the process is judged for control in the same way as is done for X or arts. It all the points fall within the control limits without giving

(c) Control Chart for the Number of Defects per unit (C-chart): The C-chart Is used when we count the number of defects per unit rather than classifying a unit as good or bad i.e., defective or non-defective. In a number of manufacturing processes we come across situations where :

(1) The opportunities for the occurrence of a defect in any unit are very large.

(ii) The actual occurrence of a defect is rare i.e., the probability of occurrence of a defect in any spot is very small. For example, we count the number of imperfections in a piece of cloth, the number of air bubbles in a piece of glass, the number of blemishes in a sheet of paper, etc.

From statistical theory we know that the theoretical distribution to be used in such situations is Poisson probability distribution. Accordingly 3.0 control limits for C-chart are based on the Poisson probability model.

In short, the C-chart is designed to control the number of defects per unit and is based on the Poisson distribution. The C-chart applies to the number of defects in sub-groups of constant size; the variable ‘C’ consists of the number of defects observed in one article. The limits of the C-chart are based on Poisson distribution and in the case of Poisson distribution the average number of defects and the square of the standard deviation are the same.

The construction of control chart for number of defects where the sample size is constant is as follows:

(i) The defects in a sample counted individually.

(ii) Compute the average number of defects C.

Number of defects in all samples It may be noted that when LCL is negative, it is taken as zero.

(iv) The sample points C, C), …,Care plotted as points (dots) by taking the sample statistic C along the vertical scale and the sample number along the horizontal scale. The central line (CL) is drawn as bold horizontal line at C and UCLc and LCLc are plotted as dotted lines at the computed values given. The interpretation of C-chart is similar to the charts for fraction defective and number of defectives.

Illustration 15.

The Number of defeats found in each one of the 15 Pieces of 2*2 meters of a synthetic fiber cloth is given below construct and appropriate control chart and state our conclusion also show three lines in the graph paper :

3, 7 12, 5, 21, 4, 3, 20, 0, 8, 10, 20, 9, 7, 6

Illustration 18

A 36′ x 20′ carpet produced on a large scale is found on an average to have 16 defects in its texture. What is the maximum number of defects likely to be encountered in a carpet? (Use Poisson distribution.) Give the upper and lower control limits for the number of defects. Solution.

We will construct the c-chart to control the number of defects. An average number of defects (C) has already been given in the question. Hence, Central Line = c = 16

UCL = c + 3Vc = 16 + 3 V16 = 16+ (3 X 4) = 28

LCL = 7 – 3Vc = 16 – 3 V16 = 16 – (3 x 4) = 4

(II) Product Control or Acceptance

Sampling So far we confined our attention to Process Control’ which is achieved through the control charts. In this case, the producer had the complete control of the process which made the product. But when it comes to the marketing of the product, the problem is different. Here, we have to take into consideration the requirements of the customers and the firms or companies who receive the end products from the process. For this we need what is termed as product control in which the producer wants to ensure himself that the manufactured goods are according to the specifications of the customers or the receiving firms or companies and do not contain a large number of defectives. For this, it is imperative that he should have his product examined at strategic points, which is designated as receiving inspection.

The need of acceptance sampling arises from the fact that when shipments (or lots of product are transferred from one firm to another, or say from seller to buyer, the manager (or the buyer) receiving the lot wants to be reasonably sure that the lot meets whatever standards had been already agreed upon with regard to the quality of the product. The supplier being aware of these standards must have worked to meet them but the buyer may still find it desirable to subject the product lot to further inspection. Certainly the supplier is more likely to adhere to standards when he knows the buyer is inspecting incoming lots. As a result acceptance sampling is used.

Thus, product control is that phase of statistical quality control where in the purpose is to accept or reject a lot of a given product once it has been manufactured. The decision to accept or to reject a given lot is taken on the basis of evidence provided by inspection of samples drawn at random from the lot. There are two reasons in favour of sampling inspection as opposed to 100% inspection. They are (i) the cost of total inspection is very low, (ii) when inspection is destructive i.e., the inspected items gets destroyed), only sampling inspection is possible. Hence, sampling inspection is usually done in practice. If on the basis of sampling inspection plan, the lot meets the Acceptable Quality Level (AQL) it is accepted but if it does not meet the AQL, then it is rejected. Accordingly the product control is also known as Lot Acceptance Sampling or simply Acceptance Sampling.

Types of Acceptance Sampling Plans

The following three types of acceptance sampling plans are commonly used :

(1) Single Sampling Plan: When the decision whether to accept a lot or reject a lot is made on the basis of only one sample, the acceptance plan is described as a Single sampling plan. This is the simplest type of sampling plan.

In any plan for single sampling usually three numbers are specified, one is the total number of articles N in the lot from which the sample is to be drawn. The Second is the number of articles in the random sample drawn from the lot. The is the acceptance number c. This acceptance number is the maximum allowable number of defective articles in the sample. More than this will cause the rejection of the lot. For instance, single sampling plan may be as follows:

N = 300, n = 30, c = 3

These numbers may be interpreted as saying, “Take a random sample of 30 from a lot of 300. If the sample contains more than 3 defective, reject the lot; otherwise accept the lot.”

(2) Double Sampling Plan : Double sampling involves the possibility of putting off the decision on the lot until a second sample has been taken. A lot may be accepted at once if the first sample is good enough or rejected at once if the sample is bad enough. If the first sample is neither good enough nor bad enough, the decision is based on the evidence of the first and second samples combined.

Thus, double sampling inspection plan provides for taking a second sample if we are not in a position to arrive at a decision about accepting or rejecting a lot on the basis of a single sample.

EXAMINATION QUESTIONS

Long Answer Theoretical Questions

1 What do you understand by statistical quality control ? Explain its main techniques.

2. What do you understand by ‘statistical quality control’ ? Discuss its aspects and advantages.

3. Distinguish between Process control and Product control. State the different types of Acceptance sampling plans and give their merits and demerits.

4. Write a short note on Statistical Quality Control

5. Explain the need and importance of statistical quality control in an industry.

6. Discuss the role of control charts in the manufacturing process.

7. “Statistical quality control cannot be so effective as the cent per-cent inspection”. Discuss.

8. What is control chart ? Explain clearly the difference between p-chart and c-chart. For what purpose are they used ? Explain by giving a numerical example.

9. Discuss Process Control’ and ‘Acceptance Inspection with reference to statistical quality control.

Short Answer Theoretical Questions

1 What do you understand by “Statistical Quality Control?

2. Distinguish between product control and process control.

3. What is Control Chart?

4. Explain clearly the difference between p-chart and c-chart.

5. Write the steps in the construction of the p-chart with necessary formulae for the upper control limit and lower control limit.

6. What are X and R-Charts.

7. What do you understand by ‘Acceptance Sampling’ ? Explain its utility.

8. Discuss the need and utility of statistical quality control in the industry.

Objective Questions Indicate whether the following statements are ‘True’ or ‘False’:

1 Under the sampling inspection method, every unit of finished goods is inspected. (False)

2. The use of statistical methods to control the quality of finished goods is known as statistical quality control. (True)

3. Generally, upper control of standard error from the process mean.

4. To control the attributes, a mean control chart range control chart and a standard deviation chart can be prepared. (True)