Page 180 - Special Topic Session (STS) - Volume 3

STS539 Chenglong Li
Li et al. (2009), Ou et al. (2011), Haridy et al. (2013), Ou et al. (2015), among
others. These studies have greatly promoted the theoretical development and
practical application of SPRT charts.
However, all of them carry out the design of a SPRT chart from a statistical
point of view only. It is well known that purely statistical model is not certainly
optimal from the economic point of view. To bridge the research gap, in this
paper, we employ a Markov chain approach and render an economic model
for designing the SPRT chart (for a long-run production).
The remainder of the paper is structured as follows. Section 2 introduces
the operation of a SPRT chart. Section 3 is devoted to the model development
for the SPRT chart. The performance comparison is conducted with several
competing control charts in Section 4. Finally, Section 5 ends with conclusions.

2. SPRT control scheme
A SPRT control scheme has four parameters: the sampling interval (), the
lower limit (), and the upper limit (ℎ), the reference value (). A sample point
or a SPRT is taken at the end of each fixed sampling interval of items, which
are produced in succession from the production line. Within a sample, when
the th observation has been taken, it is used to update the test statistic
. For an upper one-sided SPRT chart (where −∞ < < ℎ < ∞),
= −1 + − (1)

= − 0 (2)

0
where 0 = 0 denotes the starting value of the control statistic; 0 and 0 are
the incontrol (IC) values of the mean and standard deviation of the quality
characteristic. The lower one-sided SPRT chart for detecting a decrease in the
mean operates in a similar manner. In this work, we focus on the upper one-
sided SPRT chart for detecting an increase in the mean. Inside a sample point
or SPRT, if > ℎ, the process is considered to be out-of-control (OC) and an
alert signal is triggered. If ≤ ≤ ℎ , sampling is continued sequentially.
Finally, if < , the process is considered to be IC and the current SPRT is
terminated.

3. Model development
3.1 Probabilistic model

Assume that the process failure mechanism can be described by a
geometric distribution with the parameter and the assignable cause results
in a shift in the mean from 0 to 1 = 0 + 0 ( > 0) but with no change in
the standard deviation 0. Once the process is operating in the OC state, it
remains so until stopped for investigation following a signal on the control
chart. Then, the process resumes in the IC state after the negative effect is

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