Some theoretical comments regarding the run-length properties of the synthetic and runs-rules X¯ monitoring schemes – part 1: zero-state. Issue 2 (4th March 2019)

• Record Type: Journal Article
• Authors: Shongwe, Sandile C.; Graham, Marien A.
• Is Part Of: Quality technology & quantitative management. Volume 16:Issue 2(2019)
• Page Start: 170
• Page End: 189
• Publication Date: 2019-03-04
• Subjects: Average run-length (ARL) -- overall performance -- runs-rules charts -- synthetic charts -- transition probability matrix (TPM) -- zero-state; Quality control -- Periodicals; Quality control -- Statistical methods -- Periodicals; Industrial management -- Periodicals; Industrial management; Management -- Research -- Methodology -- Periodicals; Qualitative research -- Periodicals; Management; Quality control; Quality control -- Statistical methods; Periodicals; 658.00721
• DOI: 10.1080/16843703.2017.1389141
• Languages: English
• ISSNs: 1684-3703

Abstract

Abstract: In this paper, we discuss the short-term (also known as zero-state mode) run-length theoretical properties of the four different types of synthetic and runs-rules X ¯ monitoring schemes that were empirically analyzed in another paper. That is, we provide and point out how each corresponding type of the 2 - of -( H + 1 ) runs-rules and synthetic charts' transition probabilities matrices (TPMs) differ from each other in zero-state, for any positive integer H . Next, using these general TPMs and the standard Markov chain formulae, we derive the general form of the average run-length ( ARL ) vectors and the corresponding zero-state ARL expressions for any shift value for each of the four different types of the synthetic and runs-rules X ¯ monitoring schemes. Finally, we provide expressions to calculate the overall run-length performance for each of the schemes. While there is lots of literature available on empirical analysis of zero-state synthetic and runs-rules charts, there is very little on the corresponding theoretical analysis. We believe this paper will, in some part, fill this gap and encourage more research in this area.