Using SAS® To Control Multistream Binomial Processes

Process control schemes for multistream binomial processes are described. A multistream process is one in which the streams are uncorrelated or very weakly correlated. When each of the J streams are identical (a homogeneous multistream process) one can use a group control chart (a k-sigma p-chart on which only the largest and smallest of the J sample proportions nonconforming are plotted). A test (runs rule) that signals when the same stream produces the largest proportion nonconforming R consecutive times is presented along with two SAS® programs useful for designing runs rule schemes, p-charts, and ChiSquared control charts. INTRODUCTION A multistream process is best introduced by means of an example. Consider an injection molding process that makes a plastic toy. Tubes running from a reservoir carry liquid plastic into molds in an apparatus (a frame) which contains four molds. Suppose for simplicity that one tube is used to fill each toy. Each mold location can be considered to be a “stream” and this will be referred to as a 4-stream process. After cooling and separation from the mold a toy can be inspected and classified as defective (nonconforming) or nondefective (conforming). A process quality control scheme is a methodology for determine whether the process is producing “conforming units” in a stable and predictable manner; that is, one may decide whether the process is in-control or out-of-control. The control schemes described here require that periodically samples of product be inspected. One might be concerned with two ways in which the process could cease to be in-control. 1. A process change that affects all of the streams 2. A change that impacts on one or more streams The first might correspond with to a poor grade of plastic being used; the second to a clog or other problem with an individual tube. Note that identifying which type of out-of-control condition occurs will be useful for bringing the process back under control. PROCESS MODELING AND CONTROL CHARTS Control charts can be used to control processes (see Montgomery). In order to use control charts a statistical model for the process is required. Suppose that n frames are selected for inspection at the end of each epoch. Then each stream will be sampled n times also. If the streams are uncorrelated then the process is called a j-stream multistream process. A plausible process model then is that each stream is a binomial process with n trials and success probability p, where p is the proportion of items in the process (stream) that are nonconforming. When p is the same for all the streams the process is a homogeneous multistream process. (See Jacobs and Wludyka for details). The process is out-of-control if 1. The overall p changes 2. p changes for one or more streams A frequently used tool for detecting changes in a binomial process is the p-chart (see Montgomery). The sample proportion nonconforming is plotted on the chart at each epoch and a signal occurs whenever the sample proportion plots outside the k-sigma control limits. Typically k = 3 since experience has shown this to be a useful control band width. For the multistream process one may 1. construct a single overall p-chart 2. Construct J individual p-charts (one for each stream). 3. Use a Chi-squared chart In (2) it is unwise to set k = 3 since the false alarm rate will usually be much higher than most quality engineers would find reasonable. That is, k should be greater than three. See Jacobs and Wludyka for methods for determining the proper k. Note that is it sufficient to plot only the largest and smallest of the J sample (stream) proportions on a single chart for (2). This chart is called a group p-chart. The Chi-squared chart will not be described in this paper; however, the simulation program provided produces estimated ARLs (average run lengths) for this chart.