Economic design of time-between-events control chart system

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Abstract

This article presents the economic design of the control chart system consisting of several individual control charts based on time-between-events (TBE) data for monitoring multistage manufacturing processes. The design algorithm considers all the TBE charts within a system in an integrative and optimal manner. Numerical studies show that the proposed design algorithm improves the performance characteristics (in terms of profit) considerably. The proposed control chart system is easy to understand and operate, and thus the floor operators can utilize and understand it as easily as for the traditional system.

Introduction

The economic design of control chart aims at maximizing the profit (or minimizing the cost) associated with the implementation of the Statistical Process Control (SPC) scheme in a process. Many research works (Al-Oraini and Rahim, 2002, Castillo and Montgomery, 1996, Ho and Case, 1994, Montgomery, 1986, Saniga, 1989, Wu et al., 2006, Yang and Rahim, 2005) have been carried out in order to improve the effectiveness and economic performance of the Shewhart chart since the pioneering work of Duncan (1956). However, in a low defect environment, quality control based on traditionally used Shewhart chart results in high false alarm rates and inability to detect further process improvements (Chan et al., 2000, Goh and Xie, 2003).

In the past few years, several approaches have been investigated for the monitoring of low defect processes (Calvin, 1983, Chan et al., 2000, Liu et al., 2006, Ohta et al., 2001, Rassoul et al., 2007, Xie and Goh, 1993). The main focus of these approaches is to employ Cumulative Count of Conforming (CCC) charts that monitor the quantity of conforming or non-defective products produced between successive occurrences of nonconformities or defects. The TBE charts use the same concept and monitor the time between successive events (Xie et al., 2002, Zhang et al., 2005), where, the time-between-events follows an exponential distribution. The terms “time” and “events” have been interpreted in different ways. In this article, the term “events” is defined as the production of defects and the term “time” refers to the time between the occurrences of the events. The advantages of TBE charts are that it can be used in any environment irrespective of whether the environment is of low or high defect; it does not need any subjective sample size or sampling interval; and it can detect further process improvement (Chan et al., 2000, Xie et al., 2002). However, the previously proposed TBE models consider different single processes and develop algorithms for the design of individual TBE control charts.

The fabrication of a product usually goes through several process stages in series. The integration of all these stages constitutes a multistage manufacturing system. For example, in the manufacturing of a mechanical part, each stage usually pertains to the machining of a dimension. Some of the dimensions are critical to the overall quality of the product and the corresponding process stages have to be monitored by control charts. A TBE control chart system is the combination of all the TBE charts that are used to monitor the time between successive events at different stages of a manufacturing system. Due to the difference between the production rates and other factors, some of the process stages may have more than one parallel streams or machines. In some applications, a single chart or a group chart is used to monitor the outputs from all the streams of a stage. However, in this article, a separate chart will be applied to the output of each individual stream. This scenario helps to detect and diagnose the out-of-control stream (Montgomery, 2005). Typically, the quality characteristics in parallel streams in a single stage have the same mean, standard deviation and target (Runger, Alt, & Montgomery, 1996). Therefore, in this article, without loss of generality, a group of identical TBE charts (with the same rate of occurrences of the events) are used to monitor the quality characteristics of the parallel streams in a stage.

Even though many manufacturing systems consist of a series of process stages, the literature on the design of the control chart systems that monitor multistage manufacturing systems is still limited. Nelson, 1986, Mortell and Runger, 1995, Runger et al., 1996 developed the group control chart for monitoring the output from multiple streams of a single stage. Peters and Williams (1987) developed a control scheme for a three-stage manufacturing system based on a lost-cost model. Williams and Peters (1989) presented an np-control scheme for a multistage production process. Wade and Woodall, 1993, Ding et al., 2002, Zantek et al., 2002, Zantek et al., 2006, Sun and Tsung, 2004, Zhou et al., 2004, Kong et al., 2008 published several papers studying the multistage processes and the diagnosis problems. However, none of these approaches considered the multistage manufacturing processes as a whole and designed the charting parameters in an integrative and optimal manner.

In traditional chart system designs, the type I error probability α is identical for every individual chart, typically being set at 0.0027 (i.e. same power is indiscriminately allocated to each chart in the chart system). However, if all charts in a system are designed in an integrative and optimal manner, the overall effectiveness of the system may be improved significantly. The optimization design of the integrated control chart system may result in different α values or allocate a different power to different charts in a system based on the values of certain parameters such as rate of occurrences of the events and magnitudes of shifts, which would affect the performance of the chart system. Wu et al., 2004, Lam et al., 2005, Shamsuzzaman et al., 2005, Wu and Shamsuzzaman, 2005, Wu et al., 2007 developed the integrated X¯ and X¯&S control chart systems for monitoring multistage manufacturing systems. Recently, Shamsuzzaman, Xie, Goh, and Zhang (2008) developed a statistical model for the optimization design of the integrated control chart system based on TBE data. However, the implementation of the TBE chart system has significant economic impact as it involves various costs, such as the cost incurred by the occurrence of the event, cost of false alarms, cost of locating and removing the assignable cause and cost of allowing the system to operate in an out-of-control state. Therefore, it is quite reasonable to take into account the economic issues in designing the TBE chart system.

This article presents an economic design of the integrated control chart system for monitoring multistage manufacturing processes based on TBE data. It designs control limits of all the TBE charts in the system in an integrative and optimal manner in order to maximize the profit associated with the implementation of the SPC system. The false alarm rate is used as the constraint.

The remainder of the paper is organized as follows. Section 2 gives assumptions, parameter definition and estimation methods for the design of the proposed chart system. In addition, the optimization model and the required optimization algorithm are also discussed in this section. Section 3 conducts numerical studies to evaluate the performance of the chart system. Finally, Section 4 draws conclusions.

Section snippets

Assumptions

A few assumptions are adopted in this article.

  • (1)

    Since the processes will often operate in the in-control condition for most of the time or relatively long periods, it is assumed that only one process is out of control at any moment in a manufacturing system (Wu et al., 2004).

  • (2)

    There is a single assignable cause shifting the system from an in-control state to the out-of-control state. The occurrence rate of the assignable cause, λa,i in each stage is independent and known (Zhang et al., 2005).

  • (3)

    The

Performance analysis

This section studies the effects of different input parameters on the performance of the control chart system. Firstly, the effects of the process parameters (λa,i, λ0,i, λδ,i, gi, pi, di) are investigated, and then, the errors associated with the estimation of the cost parameters (B0, B1, A0, Ai, C) are analyzed. Two different control chart systems, i.e., a traditional chart system and an integrated chart system are designed. For the traditional chart system, the αi values of the control

Conclusions

This article proposes the economic design of the control chart system based on time-between-events (TBE) data. The design algorithm optimizes the control limits of each chart in the chart system in order to maximize the profit associated with the monitoring of a multistage manufacturing system. It is found that, by properly allocating the power among the individual charts based on the values of the influential parameters, the profit of the SPC system can be significantly improved.

The economic

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    This manuscript was processed by Area Editor E.A. Elsayed.

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