On the Analysis and Design of CUSUM-Shewhart Control Schemes

In recent years cumulative sum (CUSUM) control charts have become increasingly popular as an alternative to Shewhart's control charts. These charts use sequentially accumulated information in order to detect out-of-control conditions. They are philosophically related to procedures of sequential hypothesis testing (the relation being similar to that existing between Shewhart's charts and classical procedures for hypothesis testing). In the present paper we present a new approach to design of CUSUM-Shewhart control schemes and analysis of the associated run length distributions (under the assumption that the observations correspond to a sequence of independent and identically distributed random variables). This approach is based on the theory of Markov chains and it enables one to analyze the ARL (Average Run Length), the distribution function of the run length, and other quantities associated with a CUSUM-Shewhart scheme. In addition, it enables one to analyze situations in which out-of-target conditions are not present initially, but rather appear after a substantial period of time during which the process has operated in on-target mode (steady state analysis). The paper also introduces an APL package, DARCS, for design, analysis, and running of both one- and two-sided CUSUM-Shewhart control schemes and gives several examples of its application.