An adaptive Bayesian scheme for joint monitoring of process mean and variance George Nenes a,n , Sofia Panagiotidou b,1 a University of Western Macedonia, Department of Mechanical Engineering, Bakola & Sialvera, 50100 Kozani, Greece b Aristotle University of Thessaloniki, Department of Mechanical Engineering, 54124 Thessaloniki, Greece article info Available online 7 June 2013 Keywords: Quality control Bayes theorem Economic optimization abstract This paper presents a new model for the economic optimization of a process operation where two assignable causes may occur, one affecting the mean and the other the variance. The process may thus operate in statistical control, under the effect of either one of the assignable causes or under the effect of both assignable causes. The model employed uses the Bayes theorem to determine the probabilities of operating under the effect of each assignable cause. Based on these probabilities, and following an economic optimization criterion, decisions are made on the necessary actions (stop the process for investigation or not) as well as on the time of the next sampling instance and the size of the next sample. The superiority of the proposed model is estimated by comparing its economic outcome against the outcome of simpler approaches such as Fp (Fixed-parameter) and adaptive Vp (Variable-parameter) Shewhart charts for a number of cases. The numerical investigation indicates that the economic improvement of the new model may be significant. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction In the last decades there has been a massive production of scientific articles proposing numerous economically optimized statistical quality control charts. The pioneering work of Duncan [14], who was the first who associated the use of a simple Shewhart control chart with the economic outcome, and optimized its use, has inspired many scientists for over half a century. An indicative list of early approaches in the field of economically designed control charts would include the works of Bather [2], Goel et al. [17], Knappenberger and Grandage [18], von Collani [40,41] and Duncan [15] who extended his early model to the case of multiple assign- able causes. Lorenzen and Vance [19] also proposed an economic- ally designed model that is flexible enough to be used either for Shewhart charts or for Cumulative Sum (CUSUM) and Exponentially Weighted Moving Average (EWMA) charts. The common characteristic of all aforementioned approaches is that (a) they all assume fixed design parameters and (b) they all consider assignable causes that affect the process mean. Since those early approaches it soon became evident that the design of control charts with adaptive design parameters could significantly improve the economic and statistical behavior of the charts. To this effect, Reynolds et al. [27] were the first to introduce adaptive control charts, and in particular VSI (Variable Sampling Interval) charts. Typical economically designed VSI control charts were later proposed by Das et al. [10] and Bai and Lee [1]. Similar adaptive control charts where the sample size (instead of the sampling interval) is allowed to vary are called VSS (Variable Sample Size) charts and their economic design was first introduced by Park and Reynolds [25]. The economic design of VSSI (Variable Sample Size and sampling Interval) control charts has been analyzed in the works of Das et al. [10] and Park and Reynolds [25] while De Magalhães et al. [12], Costa and Rahim [9], Nenes [21] and Celano et al. [5] present the economic design of fully adaptive control charts (Vp-Variable parameter control charts). A different stream of economically designed control charts uses the Bayes theorem in order to determine the optimum design parameters. To this end, instead of using just the last sample's outcome, these charts use all information available to reach to the optimum decisions. Bayesian charts have their origins in the first theoretical approaches of Girshick and Rubin [16], Bather [2] and Taylor [38,39]. However, the economic design of Bayesian control charts has not received much attention from the academics, mainly because of the increased modeling complexity. Tagaras [33,34] was the first to introduce a Bayesian one-sided X control scheme with adaptive parameters, while, at about the same time, Calabrese [3] developed a one-sided Bayesian p chart for monitor- ing the fraction nonconforming. Porteus and Angelus [26] identify opportunities for cost reduction in SPC through the use of Bayesian Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/caor Computers & Operations Research 0305-0548/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cor.2013.05.018 n Corresponding author. Tel.: +30 24610 56665; fax: +30 24610 56601. E-mail addresses: gnenes@uowm.gr, gnenes@auth.gr (G. Nenes), span@uowm.gr (S. Panagiotidou). 1 Tel.: +30 2310 995914; fax: +30 2310 996018. Computers & Operations Research 40 (2013) 2801–2815