New Shewhart-type synthetic $$\bar{X}$$ X ¯ control schemes for non-normal data

Abstract

Abstract In this paper, Burr-type XII $$\bar{X}$$ X ¯ synthetic schemes are proposed as an alternative to the classical $$\bar{X}$$ X ¯ synthetic schemes when the assumption of normality fails to hold. First, the basic design of the Burr-type XII $$\bar{X}$$ X ¯ synthetic scheme is developed and its performance investigated using exact formulae. Secondly, the non-side-sensitive and side-sensitive Burr-type XII $$\bar{X}$$ X ¯ synthetic schemes are introduced and their zero-state and steady-state performances, in terms of the average run-length and expected extra quadratic loss values, are investigated using a Markov chain approach. Thirdly, the proposed schemes are compared to the existing classical runs-rules and synthetic $$\bar{X}$$ X ¯ schemes. It is observed that the proposed schemes have very interesting properties and outperform the competing schemes in many cases under symmetric and skewed underlying process distributions. Finally, an illustrative real-life example is given to demonstrate the design and implementation of the proposed Burr-type XII $$\bar{X}$$ X ¯ synthetic schemes.

Keywords

• Non-side-sensitive synthetic schemes
• Side-sensitive synthetic schemes
• Zero-state mode
• Steady-state mode
• Transition probability matrix (TPM)