Morton Klein

Two Alternatives to the Shewhart X̄ Control Chart

Average run length (ARL) values are calculated for two X̄ control chart schemes and compared with those of a standard Shewhart chart. Both control charts are based on runs rules and are easily implemented. An out-of-control condition for one of the charts is a run of two of two successive points beyond a special control limit. The other chart uses a run of two of three successive points beyond a different control limit. Both schemes are shown to have better, that is, lower, ARL values than the standard Shewhart chart for process average shifts as large as 2.6 standard deviations from the mean.

Composite Shewhart-EWMA statistical control schemes

A group of composite Shewhart-EWMA statistical control schemes were evaluated with simulations. The main results are given in terms of average run-length profiles for each scheme. Results in terms of suggested secondary criteria, percentile points of the run-length distribution of false out-of-control signals, are also given. It is shown that Shewhart-EWMA control schemes using either time-dependent or constant control limits can be found that have better average run-length characteristics than standard Shewhart-Runs Rules schemes. When the secondary criteria are used, only the constant control limit schemes are better than the Shewhart-Runs Rules schemes.

Examination schedules for breast cancer

Annual breast cancer examination programs are known to yield favorable survival statistics, presumably because of early disease detection. Earlier detection may be possible if examinations are scheduled more frequently. Since empirical evaluation of many such schedules is impractical, a mathematical model is used to obtain theoretical evaluations. Calculations using this model indicate that a semi‐annual examination program may detect cancer significantly earlier than an annual examination program. Also, for mass screening programs, there appears to be a slight economic advantage if examination frequency is a function of age rather than fixed throughout life.

How often should patients be sigmoidoscoped? A mathematical perspective

Abstract A simple mathematical model is used to help determine suitable intervals for routine periodic sigmoidoscopy. Although annual examinations have been recommended in the past, our calculations indicate that such frequent examinations may only be needed for high-risk patients or very cautious examiners. Our calculations suggest examinations every 2 or 3 years for asymptomatic persons. These recommendations are intended as guides for physicians in their daily practice. They may not apply to large scale screening programs where costs and logistics factors require consideration.

Prospective evaluation of periodic breast examination programs.

A mathematical model is developed to obtain prospective estimates of average tumor sizes and the expected proportion of positive regional lymph node cases for periodic breast cancer examination programs. Semiannual, annual, and biannual programs, with or without mammography, are evaluated and compared with results from the National Breast Project. Calculations indicate that a semiannual clinical‐mammographic program can be expected to result in about 37% fewer positive node cases than those observed in the National Breast Project and substantially smaller tumors. The expected reduction in positive node cases from a similar annual program is about 30%, and from a biannual program, even without mammography, about 20%. Calculations also indicate the important role of diligent patient self‐examination in such programs, especially for the detection of fast‐growing tumors.

Modified s-charts for controlling process variability

To help in the detection of variance increases and decreases, three modified versions of traditional Shewhart S-charts are evaluated in terms of their average run length values. One scheme uses control limits based on equal tail chi-square distribution probabilities. The second uses control limits based on unequal tail probabilities. The third uses warning limits based on equal tail probabilities, but requires two successive points beyond the warning limit to give an out-of-control signal. They all result in better average run length values than the traditional S-chart. Also, if the only concern is the detection of variance increases, then both S-charts and warning limit charts without lower control limits are shown to have better average run length values than those of the traditional charts.

Modified Shewhart-exponentially weighted moving average control charts

A modified Shewhart-exponentially weighted moving average X-bar statistical control scheme is evaluated. It is characterized by the use of a Shewhart component requiring two (or more) plotted points beyond reduced width (i.e., less than three-sigma) control limits to give an out-of-control signal. Schemes with both better average run length profiles and better early false-signal characteristics than those associated with standard Shewhart-runs rules schemes are suggested.

Prospective evaluation of periodic breast examination programs: interval cases.

A mathematical model is developed and used to obtain estimates of the expected proportion of breast cancer cases which will be detected at scheduled examinations and between them in various periodic examination programs; the latter are called interval cases. Another model is used to get estimates of the proportion of women in such programs who get cancer and are expected to have axillary node metastases at the time of first treatment. Programs with or without patient self‐examinations are also evaluated. The purpose of both models is to enable evaluations of different periodic schedules. Although interval cases have been reported to have poorer prognostic characteristics than those found at scheduled examinations, calculations indicate that large reductions in these cases do not imply an equivalent reduction in positive axillary node cases. Thus, interval case counts are a relatively weak measure of the success of a periodic program. However, they may still be valuable since programs which can be expected to have low proportions of interval cases are less dependent on efficient patient self‐examinations. Cost‐benefit calculations show how the relative costs of mammographic and physical examinations can be considered in the development of mass screening programs. For example, if a mammographic examination costs three times as much as a physical, then a physical examination program may be preferred to one involving both modalities.

A Transportation Model for Production Planning with Convex Costs

Abstract This expository note reviews a somewhat neglected, but elementary, method for reformulating linear and convex cost production planning problems with inventory and back-order constraints as un-capacitated transportation-type problems.

Solving the Shortest Route Problem

Abstract This note presents a simple and intuitive graphical method for finding the shortest route between two specified nodes in a network. The approach is similar to the Hungarian Method for solving assignment problems. E. A. Silver