STATISTICAL QUALITY CONTROL WITH SAMPLING

2.1 Statistical Quality Control with Sampling

Sampling by attributes is a widely applied quality control method. The procedure is intended to determine whether or not a particular group of materials or work products is acceptable. In the literature of statistical quality control, a group of materials or work items to be tested is called a lot or batch. An assumption in the procedure is that each item in a batch can be tested and classified as either acceptable or deficient based upon mutually acceptable testing procedures and acceptance criteria. Each lot is tested to determine if it satisfies a minimum acceptable quality level (AQL) expressed as the maximum percentage of defective items in a lot or process.
In its basic form, sampling by attributes is applied by testing a pre-defined number of sample items from a lot. If the number of defective items is greater than a trigger level, then the lot is rejected as being likely to be of unacceptable quality. Otherwise, the lot is accepted. Developing this type of sampling plan requires consideration of probability, statistics and acceptable risk levels on the part of the supplier and consumer of the lot. Refinements to this basic application procedure are also possible. For example, if the number of defectives is greater than some pre-defined number, then additional sampling may be started rather than immediate rejection of the lot. In many cases, the trigger level is a single defective item in the sample. In the remainder of this section, the mathematical basis for interpreting this type of sampling plan is developed.
More formally, a lot is defined as acceptable if it contains a fraction p₁ or less defective items. Similarly, a lot is defined as unacceptable if it contains a fraction p₂ or more defective units. Generally, the acceptance fraction is less than or equal to the rejection fraction, p₁ p₂, and the two fractions are often equal so that there is no ambiguous range of lot acceptability between p₁ and p₂. Given a sample size and a trigger level for lot rejection or acceptance, we would like to determine the probabilities that acceptable lots might be incorrectly rejected (termed producer's risk) or that deficient lots might be incorrectly accepted (termed consumer's risk).
Consider a lot of finite number N, in which m items are defective (bad) and the remaining (N-m) items are non-defective (good). If a random sample of n items is taken from this lot, then we can determine the probability of having different numbers of defective items in the sample. With a pre-defined acceptable number of defective items, we can then develop the probability of accepting a lot as a function of the sample size, the allowable number of defective items, and the actual fraction of defective items. This derivation appears below.
The number of different samples of size n that can be selected from a finite population N is termed a mathematical combination .

For any combination of n and r, we can read off the value of g(p) for a given p from the corresponding OC curve. For example, n = 15 is specified in Figure 13-1. Then, for various values of r, we find:

r=0 r=0 r=1 r=1

p=24% p=4% p=24% p=4%

g(p) 2% g(p) 54% g(p) 10% g(p) 88%

The producer's and consumer's risk can be related to various points on an operating characteristic curve. Producer's risk is the chance that otherwise acceptable lots fail the sampling plan (ie. have more than the allowable number of defective items in the sample) solely due to random fluctuations in the selection of the sample. In contrast, consumer's risk is the chance that an unacceptable lot is acceptable (ie. has less than the allowable number of defective items in the sample) due to a better than average quality in the sample. For example, suppose that a sample size of 15 is chosen with a trigger level for rejection of one item. With a four percent acceptable level and a greater than four percent defective fraction, the consumer's risk is at most eighty-eight percent. In contrast, with a four percent acceptable level and a four percent defective fraction, the producer's risk is at most 1 - 0.88 = 0.12 or twelve percent.
In specifying the sampling plan implicit in the operating characteristic curve, the supplier and consumer of materials or work must agree on the levels of risk acceptable to themselves. If the lot is of acceptable quality, the supplier would like to minimize the chance or risk that a lot is rejected solely on the basis of a lower than average quality sample. Similarly, the consumer would like to minimize the risk of accepting under the sampling plan a deficient lot. In addition, both parties presumably would like to minimize the costs and delays associated with testing. Devising an acceptable sampling plan requires trade off the objectives of risk minimization among the parties involved and the cost of testing.

2.2 Safety

Construction is a relatively hazardous undertaking. As Table 13-1 illustrates, there are significantly more injuries and lost workdays due to injuries or illnesses in construction than in virtually any other industry. These work related injuries and illnesses are exceedingly costly. The Construction Industry Cost Effectiveness Project estimated that accidents cost $8.9 billion or nearly seven percent of the $137 billion (in 1979 dollars) spent annually for industrial, utility and commercial construction in the United States. [3] Included in this total are direct costs (medical costs, premiums for workers' compensation benefits, liability and property losses) as well as indirect costs (reduced worker productivity, delays in projects, administrative time, and damage to equipment and the facility). In contrast to most industrial accidents, innocent bystanders may also be injuried by construction accidents. Several crane collapses from high rise buildings under construction have resulted in fatalities to passerbys. Prudent project managers and owners would like to reduce accidents, injuries and illnesses as much as possible.

TABLE 2-1  Nonfatal Occupational Injury and Illness Incidence Rates Industry 1996 2006 Agriculture, forestry, fishing Mining Construction Manufacturing Trade,Transportation and utilities Financial activities Professional and business services 8.7 5.4 9.9 10.6 8.7 2.4 6.0 6 3.5 5.9 6 5 1.5 1.2

Note: Data represent total number of cases per 100 full-time employees

Source: U.S. Bureau of Labor Statistics, Occupational injuries and Illnesses in the United States by Industry, annual

As with all the other costs of construction, it is a mistake for owners to ignore a significant category of costs such as injury and illnesses. While contractors may pay insurance premiums directly, these costs are reflected in bid prices or contract amounts. Delays caused by injuries and illnesses can present significant opportunity costs to owners. In the long run, the owners of constructed facilities must pay all the costs of construction. For the case of injuries and illnesses, this general principle might be slightly qualified since significant costs are borne by workers themselves or society at large. However, court judgements and insurance payments compensate for individual losses and are ultimately borne by the owners.
The causes of injuries in construction are numerous. Table 13-2 lists the reported causes of accidents in the US construction industry in 1997 and 2004. A similar catalogue of causes would exist for other countries. The largest single category for both injuries and fatalities are individual falls. Handling goods and transportation are also a significant cause of injuries. From a management perspective, however, these reported causes do not really provide a useful prescription for safety policies. An individual fall may be caused by a series of coincidences: a railing might not be secure, a worker might be inattentive, the footing may be slippery, etc. Removing any one of these compound causes might serve to prevent any particular accident. However, it is clear that conditions such as unsecured railings will normally increase the risk of accidents. Table 13-3 provides a more detailed list of causes of fatalities for construction sites alone, but again each fatality may have multiple causes.

TABLE 2-2  Fatal Occupational Injuries in Construction, 1997 and 2004
Year	1997	2004
Total fatalities Falls Transportation incidents Contact with objects & equipment Exposure to harmful substances and environments	1,107 376 288 199 188	1,234 445 287 267 170

Source: Bureau of Labor Statistics

TABLE 2-3  Fatality Causes in Construction, 1996/1997 and 2006/2007 Year 96/97 06/07 Total accidents Falls from a height Struck by a moving vehicle Struck by moving/falling object Trapped by something overturning/collapsing Drowning/asphyxiation 287 88 43 57 16 9 241 45 30 40 19 16

Source: Bureau of Labor Statistics

Various measures are available to improve jobsite safety in construction. Several of the most important occur before construction is undertaken. These include design, choice of technology and education. By altering facility designs, particular structures can be safer or more hazardous to construct. For example, parapets can be designed to appropriate heights for construction worker safety, rather than the minimum height required by building codes.
Choice of technology can also be critical in determining the safety of a jobsite. Safeguards built into machinery can notify operators of problems or prevent injuries. For example, simple switches can prevent equipment from being operating when protective shields are not in place. With the availability of on-board electronics (including computer chips) and sensors, the possibilities for sophisticated machine controllers and monitors has greatly expanded for construction equipment and tools. Materials and work process choices also influence the safety of construction. For example, substitution of alternative materials for asbestos can reduce or eliminate the prospects of long term illnesses such as asbestiosis.
Educating workers and managers in proper procedures and hazards can have a direct impact on jobsite safety. The realization of the large costs involved in construction injuries and illnesses provides a considerable motivation for awareness and education. Regular safety inspections and safety meetings have become standard practices on most job sites.
Pre-qualification of contractors and sub-contractors with regard to safety is another important avenue for safety improvement. If contractors are only invitied to bid or enter negotiations if they have an acceptable record of safety (as well as quality performance), then a direct incentive is provided to insure adequate safety on the part of contractors.
During the construction process itself, the most important safety related measures are to insure vigilance and cooperation on the part of managers, inspectors and workers. Vigilance involves considering the risks of different working practices. In also involves maintaining temporary physical safeguards such as barricades, braces, guylines, railings, toeboards and the like. Sets of standard practices are also important, such as: [4]

• requiring hard hats on site.
• requiring eye protection on site.
• requiring hearing protection near loud equipment.
• insuring safety shoes for workers.
• providing first-aid supplies and trained personnel on site

While eliminating accidents and work related illnesses is a worthwhile goal, it will never be attained. Construction has a number of characteristics making it inherently hazardous. Large forces are involved in many operations. The jobsite is continually changing as construction proceeds. Workers do not have fixed worksites and must move around a structure under construction. The tenure of a worker on a site is short, so the worker's familiarity and the employer-employee relationship are less settled than in manufacturing settings. Despite these peculiarities and as a result of exactly these special problems, improving worksite safety is a very important project management concern.
Example 2-1: Trench collapse  

To replace 1,200 feet of a sewer line, a trench of between 12.5 and 18 feet deep was required down the center of a four lane street. The contractor chose to begin excavation of the trench from the shallower end, requiring a 12.5 deep trench. Initially, the contractor used a nine foot high, four foot wide steel trench box for soil support. A trench box is a rigid steel frame consisting of two walls supported by welded struts with open sides and ends. This method had the advantage that traffic could be maintained in at least two lanes during the reconstruction work.

In the shallow parts of the trench, the trench box seemed to adequately support the excavation. However, as the trench got deeper, more soil was unsupported below the trench box. Intermittent soil collapses in the trench began to occur. Eventually, an old parallel six inch water main collapsed, thereby saturating the soil and leading to massive soil collapse at the bottom of the trench. Replacement of the water main was added to the initial contract. At this point, the contractor began sloping the sides of the trench, thereby requiring the closure of the entire street.
The initial use of the trench box was convenient, but it was clearly inadequate and unsafe. Workers in the trench were in continuing danger of accidents stemming from soil collapse. Disruption to surrounding facilities such as the parallel water main was highly likely. Adoption of a tongue and groove vertical sheeting system over the full height of the trench or, alternatively, the sloping excavation eventually adopted are clearly preferable.

 References

1. Ang, A.H.S. and W.H. Tang, Probability Concepts in Engineering Planning and Design: Volume I - Basic Principles, John Wiley and Sons, Inc., New York, 1975.
2. Au, T., R.M. Shane, and L.A. Hoel, Fundamentals of Systems Engineering: Probabilistic Models, Addison-Wesley Publishing Co., Reading MA, 1972
3. Bowker, A.H. and Liebermann, G. J., Engineering Statistics, Prentice-Hall, 1972.
4. Fox, A.J. and Cornell, H.A., (eds), Quality in the Constructed Project, American Society of Civil Engineers, New York, 1984.
5. International Organization for Standardization, "Sampling Procedures and Charts for Inspection by Variables for Percent Defective, ISO 3951-1981 (E)", Statistical Methods, ISO Standard Handbook 3, International Organization for Standardization, Paris, France, 1981.
6. Skibniewski, M. and Hendrickson, C., Methods to Improve the Safety Performance of the U.S. Construction Industry, Technical Report, Department of Civil Engineering, Carnegie Mellon University, 1983.
7. United States Department of Defense, Sampling Procedures and Tables for Inspection by Variables, (Military Standard 414), Washington D.C.: U.S. Government Printing Office, 1957.
8. United States Department of Defense, Sampling Procedures and Tables for Inspection by Attributes, (Military Standard 105D), Washington D.C.: U.S. Government Printing Office, 1963.

Problems

1. Consider the following specification. Would you consider it to be a process or performance specification? Why?

"Water used in mixing or curing shall be reasonably clean and free of oil, salt, acid, alkali, sugar, vegetable, or other substance injurious to the finished product...Water known to be potable quality may be used without test. Where the source of water is relatively shallow, the intake shall be so enclosed as to exclude silt, mud, grass, or other foreign materials." [6]

2. Suppose that a sampling plan calls for a sample of size n = 50. To be acceptable, only three or fewer samples can be defective. Estimate the probability of accepting the lot if the average defective percentage is (a) 15%, (b) 5% or (c) 2%. Do not use an approximation in this calculation.
3. Repeat Problem 2 using the binomial approximation.
4. Suppose that a project manager tested the strength of one tile out of a batch of 3,000 to be used on a building. This one sample measurement was compared with the design specification and, in this case, the sampled tile's strength exceeded that of the specification. On this basis, the project manager accepted the tile shipment. If the sampled tile was defective (with a strength less than the specification), the project manager would have rejected the lot.

a. What is the probability that ninety percent of the tiles are substandard, even though the project manager's sample gave a satisfactory result?
b. Sketch out the operating characteristic curve for this sampling plan as a function of the actual fraction of defective tiles.

5. Repeat Problem 4 for sample sizes of (a) 5, (b) 10 and (c) 20.
6. Suppose that a sampling-by-attributes plan is specified in which ten samples are taken at random from a large lot (N=100) and at most one sample item is allowed to be defective for the lot to be acceptable.

a. If the actual percentage defective is five percent, what is the probability of lot acceptance? (Note: you may use relevant approximations in this calculation.)
b. What is the consumer's risk if an acceptable quality level is fifteen percent defective and the actual fraction defective is five percent?
c. What is the producer's risk with this sampling plan and an eight percent defective percentage?

7. The yield stress of a random sample of 25 pieces of steel was measured, yielding a mean of 52,800 psi. and an estimated standard deviation of s = 4,600 psi.

a. What is the probability that the population mean is less than 50,000 psi?
b. What is the estimated fraction of pieces with yield strength less than 50,000 psi?
c. Is this sampling procedure sampling-by-attributes or sampling-by-variable?

8. Suppose that a contract specifies a sampling-by-attributes plan in which ten samples are taken at random from a large lot (N=100) and at most one sample is allowed to be defective for the lot to be acceptable.

a. If the actual percentage defective is five percent, what is the probability of lot acceptance? (Note: you may use relevant approximations in this calculation).
b. What is the consumer's risk if an acceptable quality level is fifteen percent defective and the actual fraction defective is 0.05?
c. What is the producer's risk with this sampling plan and a 8% defective percentage?

9. In a random sample of 40 blocks chosen from a production line, the mean length was 10.63 inches and the estimated standard deviation was 0.4 inch. Between what lengths can it be said that 98% of block lengths will lie?

 Footnotes

1. This illustrative pay factor schedule is adapted from R.M. Weed, "Development of Multicharacteristic Acceptance Procedures for Rigid Pavement," Transportation Research Record 885, 1982, pp. 25-36. Back
2. B.A. Gilly, A. Touran, and T. Asai, "Quality Control Circles in Construction," ASCE Journal of Construction Engineering and Management, Vol. 113, No. 3, 1987, pg 432. Back
3. See Improving Construction Safety Performance, Report A-3, The Business Roundtable, New York, NY, January 1982. Back
4. Hinze, Jimmie W., Construction Safety,, Prentice-Hall, 1997. Back
5. This example was adapted from E. Elinski, External Impacts of Reconstruction and Rehabilitation Projects with Implications for Project Management, Unpublished MS Thesis, Department of Civil Engineering, Carnegie Mellon University, 1985. Back

6. American Association of State Highway and Transportation Officials, Guide Specifications for Highway Construction, Washington, D.C., Section 714.01, pg. 244. Back