ORIGINAL ARTICLE

A median run length-based double-sampling X chart with estimated parameters for minimizing the average sample size

W. L. Teoh & Michael B. C. Khoo & Philippe Castagliola &

S. Chakraborti

Received: 7 May 2013 /Accepted: 23 February 2015 # Springer-Verlag London 2015

Abstract The existing control charts with estimated parameters have been widely studied from the perspective of the average run length (ARL). However, when parameters are estimated, the shape and the skewness of the run length distribution change with the magnitude of the mean shift, the number of phase I samples and sample sizes. Therefore, in this paper, we argue that the median run length (MRL) and the average sample size (ASS) have several advantages over the traditional ARL to effectively evaluate the performance of the double-sampling (DS) X chart with estimated parameters.

Precisely, by correctly accounting for parameter estimation and using the expectation by conditioning approach, we establish a theoretical method for the run length of the DS X chart in phase II process monitoring. Also, theMRL-based DS

X chart with estimated parameters is optimally designed using an optimization algorithm, aiming at minimizing the incontrol ASS by subjecting to both the desired in-control and out-of-control MRLs. Most importantly, the proposed optimal

MRL-based chart with estimated parameters not only employs a smaller sample size on average, when the process is in-control, but also has a lower false-alarm rate and provides a clearer interpretation to practitioners. The proposed optimal chart with estimated parameters is illustrated with some real data from a tape-and-reel packing process used in a manufacturing company.

Keywords Average run length . Average sample size .

Double-samplingX chart . Estimated parameters .Median run length . Optimal design 1 Introduction

A control chart is one of the most effective tools in statistical process control (SPC) for maintaining and achieving process stability over time. In recent years, many researchers, such as

Castagliola et al. [1], Kazemzadeh et al. [2], and Zhang et al. [3], have contributed to the area of control charts. A doublesampling (DS) X chart proposed by Daudin [4], which is a counterpart to double-sampling plans, allows two successive samples to be taken from the same population without any intervening time. The second sample will only be observed if the first sample falls in the warning regions of the firstsample stage of the DS X chart.

Daudin [4] and Costa [5] demonstrated that some of the DS chart’s properties are better than those of the Shewhart,

EWMA, CUSUM, variable sampling interval (VSI), and variable sample size (VSS) charts. Additionally, it is well known that the DS X chart can lead to a notable gain in statistical efficiency, in terms of sensitizing the detection of small and moderate mean shifts while reducing the sample size; hence,

W. L. Teoh (*)

Department of Physical and Mathematical Science, Faculty of

Science, Universiti Tunku Abdul Rahman, Jalan Universiti, Bandar

Barat, 31900 Kampar, Perak, Malaysia e-mail: weilin.teoh@gmail.com

M. B. C. Khoo

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia e-mail: mkbc@usm.my

P. Castagliola

Department of Quality and Logistics, LUNAM Université,

Université de Nantes and IRCCyN UMR CNRS 6597,

Carquefou, France e-mail: philippe.castagliola@univ-nantes.fr

S. Chakraborti

Department of Information Systems, Statistics, and Management

Science, University of Alabama, Tuscaloosa, AL 35487, USA e-mail: schakrab@cba.ua.edu

Int J Adv Manuf Technol

DOI 10.1007/s00170-015-6949-x the inspection and sampling costs can be effectively reduced (see Daudin [4]; He and Grigoryan [6]; Torng et al. [7]). For instance, the in-control sample size of the DS X chart dramatically decreases to nearly 50 % compared to that of the

Shewhart X chart (He et al. [8]). When the time required to collect and measure the samples is negligible, Daudin [4] concluded that the DS X chart outperforms the VSI X chart.

Moreover, Daudin [4] showed that the DS X chart is superior to both the EWMA and CUSUM charts in detecting large mean shifts. Keep in mind that although the CUSUM and

EWMA charts can be effective in detecting small shifts, the control procedures of these two charts are not as easy as that of the DS X chart (Torng et al. [9]). When the goal is to detect small and moderate mean shifts, Costa [5] stated that the necessary sample sizes for the DS X chart may be more economical than those for the VSS X chart. In view of this, Torng et al. [7] claimed that the DS type chart is a good choice for process monitoring with high inspection costs or destructive testing.

Due to the favorable properties of the DS scheme, works on the DS X type and DS S-type charts for monitoring the process mean and variance, respectively, are receiving growing interest among researchers. The DS X -type charts were discussed by Costa and Machado [10], Khoo et al. [11] and

Lee et al. [12]; while the DS S-type charts were studied by He and Grigoryan [6, 13] and Lee et al. [14, 15].

In practice, SPC is generally divided into two distinct phases. In phase I, control charts are used retrospectively to determine the in-control state of a process from a historical reference sample. In phase II, control charts are used prospectively to detect process changes after an unknown time point.

Often in practice, the process parameters are seldom, if ever, known with certainty; hence, they are estimated from an incontrol reference phase I dataset. Nevertheless, many researchers including Calzada and Scariano [16], Testik [17] and Capizzi and Masarotto [18], to name a few, showed that the process parameters which are estimated from a small number of phase I samples give rise to some undesired control chart performances. Bischak and Trietsch [19] found that the control charts based on parameter estimation produce a greater number of false signals when the process is in-control. To circumvent this problem, some researchers proposed a new design approach, i.e. by obtaining the chart parameters of the average run length (ARL)-based chart with estimated parameters, subjected to some specific constraints, such as a fixed in-control ARL (ARL0) value. Some of the recent works include the EWMA S2 chart (Maravelakis and Castagliola [20]), the CUSUM S2 chart (Castagliola and Maravelakis [21]), the synthetic X chart (Zhang et al. [22]), and the VSS