29 Acceptance Sampling

Kajal Kiran

29.1 OBJECTIVES

This chapter will enable the students to understand:

The concept of acceptance sampling
Situations where acceptance sampling is used and the risks associated with it
Operating characteristics of different sampling plans: single sampling plan, double sampling plan and multiple sampling plan
Advantages and limitations of acceptance sampling

29.2 INTRODUCTION

Every manufacturing organization strives to ensure that the finished product meet the standard specifications defined either by the manufacturer or the consumer. The extent to which the product manufactured meets the standard specifications describes the quality of the product. For a business organization to become successful, it is must on its part to offer quality products to its customers. It necessitates the need of effective quality control system in the manufacturing concerns to ensure that the products conform to the specifications and meet the customer’s requirements.

However the inspection method which includes inspection of each and every unit of production to ensure quality of the product is a costly affair in terms of money, time and labor. Moreover, the repetitive inspection process causes boredom and fatigue and as a result there is always a possibility that even the competent and efficient inspections may overlook some defective items. Inspection method is also not suitable in the cases where the products manufactured are destructive in nature e.g. life of a bulb, explosive strength of crackers or life of a candle.

Thus, in the cases where the inspection method is not suitable, statistical sampling inspection method can be employed to determine the quality of the products manufactured. Acceptance Sampling is a statistical technique which involves decisions for accepting or rejecting lots (batches) based on inspection of sample(s)

29.3 MEANING

Acceptance sampling is one of the important techniques of statistical quality control. This technique encompasses decision for accepting or rejecting a lot (or batch) of product that has already been produced or is in final stage of production on the basis of a sample of the product. The quality of the whole lot is presumed to be good if the sample meets the specified standards of quality. In other words, it is a process to measure the quality of a batch with a specified degree of statistical certainty without inspecting every unit of product. Since this method is based on samples, thus it is also known as ‘Sampling Inspection Plan’. The basic assumption underlying acceptance sampling is that the sample is randomly selected and includes the products of varying qualities from the lot. However, there is always a probability that the sample is accepted but the quality of the batch is bad or the sample is rejected but the quality of the batch is good. But by increasing the batch size, increasing the sample size or by inspecting more than one sample, more reliable conclusions can be made and better decisions for accepting or rejecting the lot can be taken.

Acceptance sampling should not be confused with statistical process control. Statistical process control aims at improving the process so that future product can be improved. On the other hand acceptance sampling relates to evaluation of products after their production to ensure their quality.

Three basic determinations involved in acceptance sampling are:

(i) batch size

(ii) sample size and

(iii) maximum number of defects to be uncovered for rejecting the entire lot.

The basic procedure involved in acceptance sampling is:

A random sample is taken from a large quantity of items and its quality characteristics such as height, radius etc. are measured.
If the selected sample passes the test then entire quantity is accepted.
If the sample fails the test then either

(a) entire quantity is subjected to 100% inspection and then all defective items are found which are then either reworked or rejected. This is a costly and time consuming procedure. Another way round this is that quality manager resort to multiple sampling plans which involves taking another sample from entire quantity and its quality characteristics are checked. Such sampling plans would be discussed in next section.

(b) The entire quantity is rejected and returned.

29.4 DECISIONS INVOLVED IN SETTING UP ACCEPTANCE SAMPLING PLANS

Acceptance sampling involves decisions from both consumer and manufacturer. The producer would like its lot to be accepted by the consumer without being reworked or rejected. The consumer would only accept if the lot given by producer is defect free. Consumer would like to reduce the risk of rejecting good quality lot or accepting bad quality lot. As the entire lot when items are in large numbers rarely be 100% defect free so there would always be some likelihood of having few defective parts. Thus in this scenario both consumer and producer reach an agreement in deciding acceptable number of defective parts in one lot. Following definitions or concepts need to be understood while making such decisions:

Acceptance Quality Level (AQL): It is the maximum percent defectives that can be considered satisfactory for the purpose of sampling inspection. This level represents the maximum proportion of defectives that can be accepted without any major effect on further processing or customer relation. It can be considered as producer’s safe point. For example, an agreement between producer and consumer might call for acceptable level of one defective item per 10,000 items indicating an acceptable level of 0.0001. it is pertinent to mention here that acceptable quality level indicates number of defective items per lot and number of non-defective items. But in case even with such low level of acceptance level the lot gets rejected or not accepted by consumer due to faulty sampling plan then it would lead to loss of producer. Such a risk is termed as producer’s risk and probability of happening of such case i.e. rejection of good lot (also called as Type-I error) is denoted by alpha (α). Generally most of sampling plans work with 1% of AQL implying that probability of occurrence of Type-I error or producers’ risk in a sample of items is 1%.
Lot Tolerance Proportion Defective (LTPD): The LTPD also known as Rejectable Quality Level (RQL) represents a designated high defect level which would not be acceptable to the consumer. At this level, the consumer wants to be quite sure that the lot will not be passed. It represents the worst level of quality that consumer will tolerate. LTPD is the definition of bad quality that consumer would like to reject. But in case such a lot with unacceptable level of defective items is passed or accepted by consumer due to faulty sampling plan then that would lead to loss of consumer. Such a risk is known as consumers’ risk and probability of happening of such case i.e acceptance of bad lot (also called as Type-II error) is denoted by beta (β). Generally most of sampling plans work with 10% of LTPD level implying that in a sample of items the chance of occurrence of Type-II error or of consumers’ risk is 10%.

The above mentioned concepts are based on two important criteria’s:

Firstly, selection of proper sampling plans. Next section discusses three different types of sampling plans.
Secondly, performance of a particular sampling plans in analyzing discrimination between good and bad lots. This is done by calculating probabilities of Type-I and Type-II error for different sample size and acceptance number. This is illustrated graphically through operating characteristic curve and also by discussing few numerical.

29.5 TYPES OF SAMPLING PLANS

There are different types of sampling plans depending upon the number of samples drawn from the lot.

Single Sampling Plan :-

Under this plan, a single sample is drawn from the lot. Each and every unit of the sample is inspected and the number of defectives (d) in the sample is noted. The number of defectives (d) is then compared with the acceptance number (a). The lot is accepted only if the number of defectives in the sample are less than or equal to acceptance number. For instance an acceptance number (a) of 3 in a lot size (n) of 50 would mean that if the number of defectives (d) in the sample are equal to 0,1,2 or 3, the lot will be accepted otherwise rejected.

Double Sampling Plan :-

Double sampling plan is used when the first sample drawn is neither very good nor very bad. This plan involves decision for accepting or rejecting the lot on the basis of combined result of first and second sample.

Under this plan firstly a sample of size n₁ is drawn and number of defectives (d₁) in the sample are noted. If number of defectives (d₁) in the sample are less than or equal to acceptable number (a₁), the lot is accepted and if the number of defectives (d₁) are more than a₂, the lot is rejected. If the number of defectives (d₁) are in between a₁ and a₂ then another sample of a different size n₂ is drawn. If the number of defectives (d₁ and d₂) in the combined sample of n₁ and n₂ is equal to or less than a₂, the lot is accepted, otherwise rejected.

Multiple Sampling Plan :-

Multiple Sampling plan is characterized by drawing several samples until a decision to accept or reject a lot is taken. It can be stopped at a particular stage or can be continued till the whole lot is exhausted.

Since multiple sampling plan is difficult to administer, hence it is not frequently applied.

29.6 OPERATING CHARACTERISTIC CURVE

An OC curve is a graphical representation between proportion of defective items and probability of acceptance of a specific lot. There would always be some number of defective items in a lot. Two things that should be emphasized here are firstly number of defective items and secondly lot size. With change in lot size number of defective items also change. For instance, there could be only 1 defective item in a lot size of 100 but with increase in size of lot to 1000 number of defective items might also increase to 10.

OC curve indicates probability of acceptance with change in lot size and number of defective items in that batch of items. X-axis of the curve represents proportion of defective items and Y-axis represents corresponding probability of acceptance (Fig. 1).
From the figure it can be clearly deduced that if proportion defective is zero then probability of acceptance would be 100%. This probability would keep falling with increase in proportion of defective items.
The curve also indicates alpha and beta values. In the following figure if the acceptable quality level is 1% then probability of acceptance suppose is 90%. But if even with 1% AQL the batch size is rejected it would indicate Type-I error shown by alpha in the figure. Similarly if LTPD is 10% then probability of acceptance suppose falls to 20%. But even with such low level of acceptance (the batch should be rejected) if it gets accepted it would imply occurrence of Type-II error shown by beta in the figure.

Fig 1: Operating Characteristic Curve

Following cases illustrate the application of OC curve by calculating values of alpha and beta for different acceptance number (a) and sample size (n)

Calculating the probability of acceptance involves decision of either item is defective or non-defective. Thus, binomial probability distribution is followed in calculation procedures. Following formula would be used:

This implies that with lot size remaining same and increase in acceptance number probability of occurrence of Type-I error i.e producers’ risk decreases. As more defe ctive items in the same lot size are now acceptable so chances of rejecting the lot which is termed as good decreases. But this would lead to acceptance of lot with higher number of defective items increasing the chances of occurrence of Type-II error or cons umers’ risk.

But due to faulty sampling plan if the lot is still rejected then probability of rejection of good lot indicated by alpha would be 1 – 0.8786 = 0.1214.

29.7 PERFORMANCE OF SAMPLING PLANS

Example of Single Sampling Plan:

A manufacturer receives large batches of items daily and decides to apply acceptance sampling scheme. Three possible plans are considered each of which requires a sample of 30 items. The agreement between producer and consumer indicates AQL = 2%.

Plan A: accept the batch if no defective items are found otherwise reject.

Plan B: accept the batch if not more than one defective item is found otherwise reject Plan C: accept the batch if two or fewer defective items are found otherwise reject Find probability of acceptance for each single sampling plan.

Example of Double Sampling Plan:

Find the probability of acceptance for following plan with acceptance number as 1%. Take a sample of 40 and accept if no defective item is found. Reject the batch if 2 or more defective items are found. If one defective is found then take a further sample of 40 items. If total of 2 or fewer defective items out of 80 are found accept the batch otherwise reject it.

Solution: Accept the batch if 0 defective items are found in batch of 40 items. Accept if 1 in first sample and 0 in second sample are found defective. Accept if 1 in first sample and 1 insecond sample are found to be defective.

Thus Probability of acceptance = P(0) + P(1)*P(0) + P(1)*P(1)

= 0.923

29.8 ADVANTAGES OF ACCETANCE SAMPLING

The advantages of acceptance sampling can be listed as follows:-

Less cost and time is required for acceptance sampling as compared to 100% inspection.
Smaller inspection staff is required.
Problem of boredom and fatigue which arises in 100% inspection is eliminated.
The lot is disposed of in shorter time and hence scheduling and delivery are improved.
The items subjected to destructive test can be inspected by sampling inspection only.
Less damage to products as only a few items are subjected to handing during inspection.
Quality of product is improved as the rejection of entire lot on the basis of sample puts much pressure on quality improvement than the rejection of individual article.

29.9 LIMITATIONS OF ACCEPTANCE SAMPLING

Since the decisions of accepting or rejecting the lot are based on sample, so there is always some likelihood of making wrong decisions regarding the quality of lot resulting in consumer’s risk or producer’s risk.
The effective outcomes of acceptance sampling technique depend on the randomness of samples, quality characteristics to be tested, lot size, acceptance critical etc.
It provides less information as compared to 100% inspection.

29.10 SUMMARY

Acceptance sampling helps in assuring the quality on one hand and overcomes the limitations involved in 100% inspection. Thus acceptance sampling allows the manufacturing organization to determine the quality of the lot with a specified degree of statistical certainty without inspecting each and every unit of product. It helps in deciding whether to accept or reject a particular lot of material purchased or whether to pass or not to pass a particular lot of materials in process or whether to pass or not to pass a particular lot of finished product for dispatch to customers and such decisions are taken by inspecting samples only.

29.11 GLOSSARY

Defectives:- A unit of product having one or more defect.
Sample:- Number of items taken out of the lot.
Acceptance Number :- The maximum number of defects or defective units in the sample which can be tolerated for acceptance of the lot.

29.12 EFERENCES/ SUGGESTED READINGS

Nair N G, Production and Operation Management, Tata McGraw- Hill Publishing Company Limited, New Delhi
Adam and Eben, Production and Operations, 5th Ed Prentice Hall
Robert Fetter B. , Quality Control Systems, Richard D. Irwin, Illinois, USA
Chary, Production and Operations Management, McGraw Hill