Economic-Statistical Design of X-bar and R or X-bar and S Charts - ASQ

Economic-Statistical Design of and R or and S Charts

In this paper we present a comprehensive FORTRAN program that can be used to jointly determine the parameters of charts used in combination with either R or S charts. The parameters are determined using economic, statistical, or economic- statistical design criteria. The general cost model due to Lorenzen and Vance (1986) is used in economic and economic-statistical designs.

Keywords: Economic Control Charts, Economic Design, Process Control, Variables Control Charts.

by Thomas P. Mcwilliams, Drexel University, Philadelphia, PA 19104, Erwin M. Saniga and Darwin J. Davis, University of Delaware, Newark, DE 19716


The most common control charts used to monitor a process when the quality variable of interest is continuous are charts used in combination with R or S charts (Saniga and Shirland (1977)). The chart is used to control the mean of the process and either the R or S chart is used to control the variance of the process, thus providing control over the two common types of shifts.

To employ control charts, the user must specify the control limit widths for both charts, the sample size, and the sampling frequency. Collectively, these four variables constitute the design of the control charts. Perhaps the most common design in practice is a heuristic design where samples of size four or five are taken every hour and three sigma limits are used on both charts. While this design is simple to implement, it is easy to show that it can have undesirable statistical and economic characteristics. Three improvements to this type of heuristic design have been developed. These are statistical design, economic design, and economic-statistical design.

Statistical designs meet constraints on average run length (ARL), or equivalently Type I and Type II error probabilities, or the average time to signal (ATS) a particular shift (see, e.g., Saniga (1991)). An advantage of these designs is that they require no cost or system parameter estimation other than a specification of ARL's for particular shifts against which protection is desired. Furthermore, by assuring that shifts are signaled rapidly and false searches or improper adjustments are avoided, one can guarantee high quality products or services.

Economic designs were first proposed by Duncan (1956). In the economic approach, the objective is to find a design that balances the costs of using a control chart with the costs of allowing the process to operate when a shift has occurred. Economic designs require that the user estimate a number of cost and system parameters but yield a design that is economically optimal in that it minimizes the expected total relevant cost. Economic designs for and R charts used jointly were developed by Saniga (1977, 1979).

Economic-statistical design is a method proposed by Saniga (1989) in which an economic design is found that meets certain constraints on ARL's (or equivalently Type I and Type II error probabilities or ATS's). These designs require the estimation of the same cost and system parameters as in economic design as well as the specification of desired ARL's and shifts against which protection is desired. While more expensive than economic designs, economic-statistical designs can, like statistical designs, guarantee high quality products or services while being more flexible and robust (Saniga (1992)). In addition, they are at least as good as statistical designs in terms of their statistical properties, but they can be more economical to use (Saniga (1989)).

In practice, one must evaluate all types of designs to ensure that the competing objectives of the organization, such as maintenance of quality, minimization of incorrect shut downs, minimization of cost, etc., are met. The methods to do so have been addressed by Saniga (1989) and Collani, Saniga and Weigand (1994). They argue that control chart design is a problem where alternate designs must be compared and solutions chosen that best fit the needs of the organization.

The determination of joint control chart parameters and the comparison of the resulting plans, for the types of designs discussed above, is a difficult task which existing published programs are too narrow in scope to perform. For example, papers by Rahim (1989) and Saniga (1991) provide the user with the ability to generate joint and R designs, but Rahim's program only deals with economic designs and lacks the flexibility of the Lorenzen-Vance (1986) model to cope with different production related assumptions, while Saniga's program is limited to determining statistical designs. A program by McWilliams (1994) can generate all three designs for charts but cannot determine joint and R chart or joint and S chart designs. Its value is therefore limited to situations where only shifts in the process mean are of interest, while in practical applications users are much more likely to be concerned about shifts in the process mean and/or process variability. Finally, a program by Saniga, Davis and McWilliams (1995) does provide a comprehensive and flexible tool for generating the three design types when process monitoring is based on attribute data, but this program cannot be used for continuous (variables) data.

In this paper, we present a user-friendly, stand alone program that can be used to determine economic, statistical, or economic-statistical designs for an chart used jointly with an R or with an S chart. This program uses the Lorenzen and Vance model, a general model that can be configured to represent most actual production situations.

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