# SPC: Acceptance sampling: sampling criteria

Title: SPC: Acceptance Sampling: sampling criteria

Authors: Anwar Stephens, Chris Garcia, Meng Yang Ng, Winardi Kusumaatmaja

Date Presented: 11/28/06 /Date Revised:

## Introduction

Thus far, we have learned how to design, implement, and optimize control schemes for processes. We also learned that most control strategies cannot always handle all types of disturbances that they face. In other words, we should expect some level of deviation from what is desired - like fulfilling a product specification. In order to combat this inherent control issue, many chemical engineers use statistical process control, or SPC. SPC is a method that utilizes process data and very basic statistical analyses to determine process stability. This method gives chemical engineers an opportunity to see how well a process is being controlled by the control system. SPC is comprised of a group of charts and diagram that can determine, with respect to time, process efficiency, number and frequency of deviant products, and boundaries of chosen process variables. Some of the main analytical tools are flow charts, control charts (Basic Control Charts), and acceptance sampling plans. There are quite a few more useful tools in the SPC toolbox but this wiki will focus on acceptance sampling.

## Acceptance Sampling

Acceptance sampling is a method used to determine if you accept or reject a particular lot of products. This is done by first establishing an acceptance plan or sampling plan which sets the product acceptability criteria. This is known as the decision rule. The decision rule is an rule that explicitly states how many out-of-specification items (in a batch of arbitrary size) can be shipped to a customer. Depending on the type of sampling plan being utilized by the company, there could be more than one decision rule in effect (for double or multiple plans).

Sampling plans utlizing can be single, double, or multiple. Single sampling plan consists of a sample size n and an acceptance number c. The lot is rejected if there are more than c defective for the sample size. Single sampling plan is the most common plan to use although not the most efficient in terms of average number of samples needed. Double sampling plan involves taking a second sample if no decision can be made regarding the first sampling and combining the information obtained from both sampling to make a final decision. But sometimes both tests might be lead to contradictory results, therefore a multiple sampling plan would be used instead. Below is an example of a double sampling plan:

If a situation occurs when a sampling technique yields a result of "no decision", then the company may still ship. This, of course, depends on the industry and the customers' sensitivity to product quality. The decision to ship can be a risky one. For example, if the customer becomes totally debilitated by the slip in quality, then chances are the manufacturer would scrap the lot. The losses incurred here are much less than the losses of trust and continued business from the consumer!

Multiple sampling plan is just an extension of the double sampling plan whereby more than two samples are needed to reach a conclusion. This plan would definitely lead to a more conclusive result as compared to a single or double sampling plan. The advantage of multiple sampling is smaller sample sizes as compared to 100% sampling which actually analyzes the whole lot rather than just a few samples.

This entire concept is implicitly based on the assumption that each sample is representative of the entire product lot and also that each lot is highly variable. There in lies the limitation of this method as the customer cannot be assured of the consistency of each batch. Although this method would be appropriate for chemical plants whereby the products are usually homogenous therefore a single sample would actually be representative of the entire batch.

### Conditions for using Acceptance Sampling

There are situations when 100% sampling is not practical. Some of these situations are listed below:

• When the testing results in the destruction of the material.
• When there are high cost involved for the inspection.
• When there are time or technology constraints.
• When the size of the lot is large and the chances of making inspection error is high.
• When the supplier has been very reliable in producing goods that are within the inspection criteria.

### Steps for Acceptance Sampling

These are the basic steps followed for acceptance sampling:

For Batch processes

• Lot is received from the production line.
• A random sample is taken from the batch.
• The sample is analyzed and checked whether it meets the acceptance criteria.
• If the sample meets the acceptance criteria, the batch is accepted and sent for further processing.
• If the sample fails to meet the criteria, the batch will be removed and evaluated (determining next step for that particular batch).

For Continuous Processes

• Sample is taken from process line at a certain frequency
• The sample is analyzed and checked whether it meets the acceptance criteria.
• If the sample meets the acceptance criteria, then continue to accept products.
• If the sample fails to meet the criteria, then that particular group of out-of-spec products will be separated and evaluated (determining next step for that particular group of products).

Because a sampling plan bases its decision on the random sample taken from the batch, there is a probability of making an incorrect decision in determining whether a particular batch meets the acceptance criteria. There are two types of error that might arise. Type I error (Producer's risk) involves incorrectly rejecting a lot that is actually acceptable. The producer suffers when this occurs because a lot with acceptable quality was rejected. Type II error (Consumer's risk) involves incorrectly accepting a lot that is really unacceptable. The consumer suffers when this occurs, because a lot with unacceptable quality was accepted.

In acceptance sampling, we use the Operating Characteristic or OC curve to estimate the probabilty of making a Type I or Type II error. The OC curve plots the probability of accepting the lot (Y-axis) versus the lot fraction or percent defectives (X-axis). The OC curve is the primary tool for displaying and investigating the properties of an Acceptance Sampling Plan. A sample plot of OC curve is shown below.

The curve is read according to this example

• If the lot quality is 0.02 fraction defective, then the probability of acceptance, Pa, is 0.09 for both plans.
• If the lot quality is 0.12 fraction defective, then the probability of acceptance, Pa, is 0.025 for plan 1 and 0.010 for plan 2.

The operating curve is out of scope for the purpose of this class -- those who use acceptance sampling here should focus on using the decision rule from before.

• There are less damage due to inspection handling.
• It is more economical than doing 100% inspection.
• It takes much lesser time than doing 100% inspection.

• There may errors (Producer's and Consumer's risk) associated with the sampling.
• The sample does not provide 100% accurate information of the condition of the bacth.

## Applications for Chemical engineers

When a sampled product lot is not "in-spec" the company needs a standardized approach for determining the fate of the lot. With regards to acceptance sampling (a tool used in SPC), chemical engineers can identify how out-of-specification lots that should be handled. There is a concept in acceptance sampling known as the decision rule. Given a number of out-of-spec units in a batch, one could have a standard approach for making decisions on how to handle the lot. Handling options vary from process to process but can include shipping, re-using, or scrapping the non-standard batch (known as a “lot”). Shipping slightly off-spec products usually requires additional standards due to certain ethical considerations (e.g. shipping slightly off-spec baby food). Therefore the frequency of performing acceptance sampling would vary from process to process. For instance, inspection for food products and loose bolts within a machinery would be done at a higher frequency as compared to checking the consistency of a chemical product or tires for vehicle. Acceptance sampling is also used extensively by other industries such as the Military, the American National Standards Institute, and the International Standards Organization (ISO) for setting various industry standards that have to be followed when manufacturing a particular product.

## Worked out Example 1

PVC piping is a plastic. It is made in big extrusion machines. Extrusion is the process of melting plastic pellets and then pushing it through an orifice that resembles the shape of the product. For PVC piping, the orifice is circular ring that defines the outer diameter, inner diameter, and wall thickness. This process is kept continuous and the piping is cut to the desired length. Before the PVC pipe can be sent to the distributors, it must be inspected for the outer diameter specification, inner diameter specification, surface defects, flexibility, compression, etc.

1) For this case, we will look at the compression strength of the PVC piping. An Instron machine is used to determine the compressible strength. The Instron machine applies pressure to the top of the PVC pipe, while the base is fixed, until the PVC pipe breaks. The force required to break the PVC pipe is recorded and compared to the minimum desired value. If the recorded value is above the minimum desired value the PVC piping is saleable. Is acceptance sampling or 100% inspection more viable?

2) For this case, we will look at the outer diameter, inner diameter, and surface defects of PVC piping. To test the outer diameter, the PVC pipe is run through a laser micrometer that uses lasers to determing the outer diameter. Since there is no online testing, a quality technician needs to test the pieces by hand. Each technician can only test a certain portion of each lot. Is acceptance sampling or 100% inspection more viable?

### Solutions

1) This is an example of a destructive test. For destructive tests only acceptance sampling can be used. A predetermined percentage of PVC piping from each lot will be randomly selected for testing. Using 100% inspection would result in no saleable products.

2) To test all the piping a multitude of technicians and laser micrometers are needed. This becomes very costly. And since there is no need for high tolerances, acceptance sampling should be used.

## Worked out Example 2

Imagine a situation where a manufacturing company known as WOMT (We-Only-Make-Tires) ships two batches of their best tire to a consumer. The consumer complained that more than a quarter of one batch was defective! As a result, WOMT has been doing final inspections on entire lots to prevent the complaints from occurring. However, they soon realize that the approach is wildly inefficient. So, they hire some chemical engineers and tell them to develop an acceptance sampling plan to determine which lots should be shipped.

So, the new hires take weeks worth of data and find that with usual operating conditions the mean weight = 50g. A week later, they take two random samples of two 101-part batches made during several production runs. They took the data, used other SPC tools, and found that the upper control limit (UCL) was 80g and the lower control limit (LCL) was 20g. Supposedly, the decision rule in use by some of the semi-conscious operators was usually “we’ll ship even when 250 of 1000 are slightly off-specifications.” Given the decision rule from above, the chemical engineers set 26 parts (in the sample batch) being off-spec as the upper boundary for shipping the lot. The two lot samples are shown below. Which batch would be sent to the consumer?

The following excel plots show the weights of all the units sampled from both lots (2 batches). Using the upper control limit as 80g and the lower limit as 20g, we wanted to see which units were out-of-control. This was done by counting how many units were outside the upper and lower limits.

Should either of the two lots (that these batches represent) be shipped to the customer?

Figure 1: Data sampled from batch 1

Figure 2: Data sampled from batch 2

### Solution

Ship the first batch. The first plot shows no tires produced outside the acceptable range of values discussed earlier. However, the second plot has numerous tires outside of those ranges. Depending on the total number of out-of-spec tires, the lot may still be shipped. Using the Media:showing_acceptance.xls excel spreadsheet, the engineers found 34 deviant tires. Since this number exceeds the decision rule, the second lot should not be shipped. Only the first one should be shipped to the customer.

Initially, this may be best option. But lets review an alternate outcome. On sending batch 1 to the consumer, they return the shipment back to us with more then 25% defects. We can infer that weight is not a useful indicator of product quality and thus we are sampling using an ineffective product characteristic. Diameter and width may be more suitable measurables. Diameter and width will insure that the tires fit the axels and rims onto with the tire will be placed.

## Multiple Choice Question 1

1) Which criterion is NOT one of the conditions for acceptance sampling?

(A) When the testing is destructive.

(B) When there are time and technology limitations.

(C) When the lot sizes are very large and the probability of inspection errors is high.

(D) When the supplier produces a high percentage of bad quality products and is declared a health hazard by the FDA and CDC.

## Multiple Choice Question 2

2) What are the advantages of using Acceptance sampling instead of Statistical Process Control (SPC)?

(A) Less damage incurred from product handling.

(B) Less costs involved compared with 100% inspection.

(C) Possible occurence of Type I or Type II error.

(D) A & B

## Submitting answers to the multiple choice questions

• Everyone else, the deadline for submitting your answers is the start of class on Thursday, 11/30.

You are expected to work on these multiple choice questions under the Honor Code.

https://lessons.ummu.umich.edu/2k/che_466/SPCAcceptance

## References

Wheeler, Donald J. & Chambers, David S. (1986). The Fallacy of Acceptance Sampling. Understanding Statistical Process Control. 2nd Edition. SPC Press. Knoxville, Tennessee.

Steward, Douglas M. Statistical Process Control. The Anderson Schools of Managament. The University of New Mexico.

Woolf, Peter (2006). Acceptance Sampling Plans vs. 100% Inspection. 11.30.2006 notes.

http://www.shsu.edu/~mgt_ves/mgt481/lesson9/, visited 11/25/06