Run Length Distribution of CUSUM Control Schemes for Negative Binominal Processes

Abstract

The CUSUM control scheme is an important tool in detecting the quality of goods or services. It is used in many industries for rapid detection of many industrial processes. The average run length (ARL) and distribution of run length and standard deviation of run length (SDRL) are the essential characteristics in constructing a control chart and are also important indexes for comparing performance with other control charts. Until now, no theoretic derivation for the run length and its standard deviation underlying negative binominal distribution for the CUSUM scheme have been developed. In this paper, an analytic method for steady state average run length as well as SDRL and two-sided ARL and SDRL with different control statistics is developed for negative binomial CUSUM (NB CUSUM). To decrease computer memory requirements and computing time, a sparse matrix operation method is also proposed to obtain the run length when conducting the Markov chain method. This analytic method makes the computations of run length for one-sided and two-sided NB CUSUM schemes easier. A real numerical example also shows the analytic results are consistent with the simulation results.

Key Words:

• Markov processes
• quality control
• run length
• SDRL
• sparse matrix.

Additional information

Notes on contributors

Sheng Shu Cheng

Sheng Shu Cheng is an Assistant Professor at the Department of Business Administration, Yu-Da University, Taiwan. He received his Ph.D. in 2002 from Yuan Ze University and Master in 1994 from National Central University, Taiwan, and gained his Bachelor degree in 1991 from Tamkang University. His research focuses on Statistical computing, SPC and Quality related fields.

Fong-Jung Yu

Fong-Jung Yu is a Professor at the Department of Industrial Engineering and Management, Da-Yeh University, Taiwan. He received his Ph.D. in 2002 and Master in 1992 from Yuan Ze University, Taiwan, and gained his Bachelor degree in 1977 from Chung-Yuan college of Science and Engineering. His research focuses on SPC, TQM and Quality related fields. He worked for a fighter program at logistic engineering department of Chung Shan institute of Science and Technology for over 12 years before he came to Da-Yeh University and also worked at the industrial engineering related field of several factories from June 1977 to October 1984. He also gained the individual award of quality award from Chinese Society for quality in 2010.