Featured Product
This Week in Quality Digest Live
Statistics Features
Douglas C. Fair
Part 3 of our series on SPC in a digital era
Scott A. Hindle
Part 2 of our series on SPC in a digital era
Donald J. Wheeler
Part 2: By trying to do better, we can make things worse
Douglas C. Fair
Introducing our series on SPC in a digital era
Donald J. Wheeler
Part 1: Process-hyphen-control illustrated
Statistics News
How to use Minitab statistical functions to improve business processes
New capability delivers deeper productivity insights to help manufacturers meet labor challenges
Day and a half workshop to learn, retain, and transfer GD&T knowledge across an organization
Elsmar Cove is a leading forum for quality and standards compliance
InfinityQS’ quality solutions have helped cold food and beverage manufacturers around the world optimize quality and safety
User friendly graphical user interface makes the R-based statistical engine easily accessible to anyone
Collect measurements, visual defect information, simple Go/No-Go situations from any online device
Good quality is adding an average of 11 percent to organizations’ revenue growth
Ability to subscribe with single-user minimum, floating license, and no long-term commitment
Statistics

## Does C = 0 Sampling Really Save Money?

### Only if you count just the cost of the customer’s inspection

Published: Monday, August 5, 2013 - 15:32

The claim is made and widely believed that C = 0 sampling plans are more cost effective than classic sampling plans such as ANSI/ASQ Z1.4. Below is a preliminary analysis of the cost difference between the two sampling plans using the hypergeometric probability distribution to compare a Squeglia C = 0 sampling plan with the ANSI/ASQ Z1.4 Single Normal sampling plan.

However, before starting our investigation we need to define some of the important terms used in acceptance sampling.

1. A lot is a collection of size N of units that are required to meet the same specifications for each quality characteristic.
2. A sample is a subset of size n of the N units in the lot.
3. The letter C is the acceptance number for the sample plan. If the number of nonconformities in the sample is less or equal to the acceptance number, then the lot is accepted without further inspection. If it is greater than the acceptance number, then 100-percent inspection and rectification are required for the lot.
4. The acceptable quality level (AQL) is a level of quality routinely accepted by the sampling plan. It is generally defined as that level of quality (i.e., fraction nonconforming) that the sampling plan will accept a high percent of the time (typically 90 to 99%). This means lots at or better than the AQL percentage nonconforming are accepted at least about 90 to 99 percent of the time.
5. The operating characteristic curve (OC) is a plot showing the probability of lot acceptance for a stated incoming quality level as measured by the fraction nonconforming for a given sampling plan.
6. The average outgoing quality (AOQ) is the average nonconformance rate in released lots assuming rejected lots are 100-percent inspected and all nonconformities are removed. The outgoing quality is better than the incoming quality as a result of the 100-percent inspection of rejected lots. AOQ(p) = p × OC(p) where p is the nonconformance rate.
7. The average total inspected (ATI) is the number of units that can be expected to be inspected under the sampling plan at the given incoming fraction nonconforming.
8. The average run length (ARL) is the number of lots that will be accepted before a lot is rejected under the given sampling plan with a given incoming fraction nonconforming.

For example, from Applied Technology’s Sampling and Inspection software:
ANSI/ASQ Z1.4, Single Normal
Using Z1.4, N = 500, n = 50, AQL = 1.0, C = 1
Yields Pa = 0.99, ATI = 54.4, ARL = 101.8 @ p = 0.004

Squeglia’s C = 0, Single Normal
Using “C = 0,” N = 500, n = 29, AQL = 1.0, C = 0
Yields Pa = 0.88, ATI = 82.1, ARL = 8.9 @ p = 0.004

In figures 1–4 below are graphs of the various sample plan performance metrics OC, AOQ, ATI, and ARL for different acceptance numbers C = 0 and C = 1.

Figure 1: The operating characteristic curve, N = 500, n = 50. Click here for larger image.

Figure 2:
The average outgoing quality, N = 500, n = 50. Click here for larger image.

Figure 3: The average total inspected, N = 500, n = 50. Click here for larger image.

Figure 4: The average run length, N = 500, n = 50. Click here for larger image.

The cost analysis for rejected lots includes the following categories:
• Hold-time cost (impacts customer)
• MRB cost (customer)
• Shipping cost (supplier)
• Unpacking cost (supplier)
• Inspection/test cost (supplier)
• Rework cost (supplier)
• Inspection cost (supplier)
• Packaging cost (supplier)
• Return shipping cost (supplier)
• Reinspection cost (customer)

There is about an 11-percent increase in the number of lots rejected using the C = 0 plan vs. the C = 1 plan for the same AQL = 1.0 @ p = 0.004. Each time a lot is rejected, it goes through this costly rectification process. Cost-wise there is a 64-percent reduction in customer sampling cost, and only an 11-percent increase in supplier sampling cost. However, the supplier is doing 100-percent inspection, and the customer is only doing sampling inspection for a net inspection cost increase of \$3,700 in this example (see Appendix below). However, the real impact is in the cycle time and the noninspection activities listed above.

Now, if you only count the cost of the customer’s inspection, it may appear that C = 0 sampling saves the customer time and money. In addition the argument is made that it also sends a message to the supplier, that they better not send any nonconformities to you because they will be punished financially if they do. However, the reality is significantly different because there are many costs that are hidden or ignored in this superficial analysis.

First, it is true that the supplier is hit with higher cost do to the high level of rejected lots, but this is just viewed as part of the cost of manufacturing by the supplier and so it is rolled into the price that the customer eventually pays for the product.

Second, the rejected lots are shipped back, reworked, inspected, tested, and returned to the customer, where they are reinspected by the customer. So while all this is going on, the customer may be without material to operate its factory at full capacity.

Third, all this shipping, handling, testing, return shipping, and reinspection may actually create additional nonconformities, which, as luck might have it, could result in the lot being rejected a second time.

Fourth, and most important, all this cost, cycle time increase, potential production stoppages, and product handling damage did not improve the quality of the product reaching the production line one bit.

So contrary to popular belief, C = 0 sampling plans do not save money. In fact, they are actually a very expensive and disruptive approach when you combine the cost to both the customer and the supplier, and recognize that this is the cost of the system. The belief that it is a good program stems from the notion that customer-supplier relations are a zero-sum game (i.e., one wins and the other loses, whereas the reality is that it is a win-win, or lose-lose game). I think most rational people would expect that this additional cost of doing business incurred by the supplier when the customer enforces a C = 0 sampling will be passed on as a cost of production, and it will inhibit the supplier’s ability to provide cost reductions to the customer in the long run. As W. Edwards Deming advocated years ago, “Buyers and suppliers need to work together in a partnership relationship based on loyalty and trust” to achieve system optimization and real cost reduction.

### Appendix

System inspection cost model
Supplier (100%)Customer (sample)<Ac/Re>Supplier (100%)Customer (sample)

Customer sampling inspection
Since Pa (Z1.4) – Pa (C = 0) = .99 – .88 = .11, so in this case there will be an 11-percent increase in the number of rejected lots.

Customer inspection cost
100 lots of size N = 500, inspection cost \$1/unit
Z1.4 result: 99 (50) + 1 (50 + 50) = 5,050
“C = 0” result: 88 (29) + 12 (29 + 29) = 3,248
A reduction in customer sampling cost of 64 percent, or \$1,800

Supplier inspection cost
100 lots of size N = 500, inspection cost \$1/unit
Z1.4 result: 100 (500) + 1 (500) = 101*500 = 50,500
“C = 0” result: 100 (500) + 12 (500) = 112*500 = 56,000
An increase in supplier sampling cost of 11 percent, or \$5,500. This results in a net inspection system cost increase of \$3,700.

### References

1. Squeglia, N. L. Zero Acceptance Number Sampling Plans, Fifth Edition (ASQ Quality Press, 2008)
2. ANSI/ASQ Z1.4—“Sampling procedures and tables for inspection by attributes” (ASQ Quality Press, 2008)
3. Flaig, J. J. Sampling and Inspection software, Applied Technology, San Jose, CA

### John Flaig

John J. Flaig, Ph.D., is a fellow of the American Society for Quality and is managing director of Applied Technology at www.e-at-usa.com, a training and consulting company. Flaig has given lectures and seminars in Europe, Asia, and throughout the United States. His special interests are in statistical process control, process capability analysis, supplier management, design of experiments, and process optimization. He was formerly a member of the Editorial Board of Quality Engineering, a journal of the ASQ, and associate editor of Quality Technology and Quantitative Management, a journal of the International Chinese Association of Quantitative Management.

### But what are the system costs?

You make some very good points about looking at the system costs, but it seems to me that if you are considering system costs you need to look at the whole system. For example, what is the cost to the customer of a defective part getting into their process? How about 2% or 4%? And what if it makes it out the door to the end user? Wouldn't these costs be relevant also?

### C = 0 vs Z1.4

Beth, The answer to your question is yes these customer costs are certainly important. However, we are comparing two sampling procedures which have essentially the same AQL So the quality of the product accepted by the customer should be very similar. In which case the costs that you are referring to should be approximately the same. So for comparison purposes we can ignore them. Regards, John

### Sampling economics and customer protection

Dear sirs:

Can I know what are the major differences of sampling plan metircs (OC curve, AOQ, ATI) among the classic c=0 sampling plan (since 1963), c=0 in ANSI/ASQ Z1.4 2008 and c=0 in MIL-STD 1916 (since 1996)? Thanks a lot.

Tom