The run sum t control chart for monitoring process mean changes in manufacturing

Abstract

The $\overline{X}$ type charts are not robust against estimation errors or changes in process standard deviation. Thus, the t type charts, like the t and exponentially weighted moving average (EWMA) t charts, are introduced to overcome this weakness. In this paper, a run sum t chart is proposed, and its optimal scores and parameters are determined. The Markov chain method is used to characterize the run length distribution of the run sum t chart. The statistical design for minimizing the out-of-control average run length (ARL₁) and the economic statistical design for minimizing the cost function are studied. Numerical results show that the t type charts are more robust than the $\overline{X}$ type charts for small shifts, in terms of ARL and cost criteria, with respect to changes in the standard deviation. Among the t type charts, the run sum t chart outperforms the EWMA t chart for medium to large shifts by having smaller ARL₁ and lower minimum cost. The run sum t chart surpasses the $\overline{X}$ type charts by having lower ARL₁ when the charts are optimally designed for large shifts but the run sum $\overline{X}$ and EWMA $\overline{X}$ prevail for small shifts. In terms of minimum cost, the $\overline{X}$ type charts are superior to the t type charts. As occurrence of estimation errors is unpredictable in real process monitoring situations, the run sum t chart is an important and useful tool for practitioners to handle such situations.

Appendix

In this section, the Markov Chain representation of the run sum t chart RS _t (k,K,{S ₁,S ₂, …,S _k }) is discussed. For illustration, consider the RS _t (4,K,{0,2,3,6}) chart. Table 9 provides the state to which the Markov chain moves, given the ordered pair cumulative score (U _r ,L _r ) of the state at time r and (U _r + 1,L _r + 1) at time r + 1. The charting procedure is assumed to start with an initial score of U ₀ = + 0 and L ₀ = − 0, i.e., (U ₀,L ₀) = (+0, − 0). Then at time r, all the possible in-control ordered pair cumulative scores are (U _r ,L _r ) = (+0, − 0), (+0, − 5), (+0, − 4), (+0, − 3), (+0, − 2), (+2, − 0), (+3, − 0), (+4, − 0), and (+5, − 0). Let these ordered pairs correspond with the transient states 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively, and let state 10 be the absorbing state. Note that if T _r  < CL, then U _r is reset to zero, and if T _r  > CL, L _r is reset to zero.

Table 9 State-next-state transitions by regions at time r and the region that contains T _r+1 at time r + 1, for the RS _t (4, K, {0, 2, 3, 6}) chart

Using Table 9, the transition probabilities from the current state (at time r) to the next state (at time r + 1) can be easily obtained. For example, the transition probability from state 1 (current state) to state 1 (next state) (see Table 9) can be obtained as

$$ {P}_{11}= \Pr\;\left({T}_{r+1}\in \left({\mathrm{LCL}}_1,\mathrm{CL}\right]\cup {T}_{r+1}\in \left[\mathrm{CL},{\mathrm{UCL}}_1\right)\right)={p}_{-0}+{p}_0. $$

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The complete transition probabilities of the RS _t (4, K,{0, 2, 3, 6}) chart are shown in Table 10.

Table 10 Transition probabilities with nine transient states, based on Table 9

From Table 10, the transition probability matrix (with the absorbing state), P, for this RS _t (4, K,{0, 2, 3, 6}) chart with (U ₀,L ₀) = (+0, − 0) is a 10 × 10 matrix whose entries are the transition probabilities P _ij , for i, j = 1, 2, …, 10. The last row and last column of this matrix correspond to state 10, i.e., the absorbing state. The transition probability matrix, P, for the RS _t (k, K, {S ₁,S ₂, …, S _k }) chart does not have a general form, where its dimension depends on the choice of the scores S ₁, S ₂,..., S _k .

The number of samples drawn until the RS _t (k, K,{S ₁, S ₂, …, S _k }) chart issues, an out-of-control signal is a random variable known as the run length. The ARL is an important parameter of the run length distribution. The zero state ARL is obtained using the following equation [5]:

$$ \mathrm{ARL}={s}^T{\left(I-Q\right)}^{-1}1, $$

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where s ^T = (1, 0,..., 0) is the initial probability vector having a unity in the first element and zeros elsewhere, I is the identity matrix and Q is the transition probability matrix, for the transient states. For the RS _t (4,K,{0, 2, 3, 6}) chart, Q comprises the remaining elements in P, after removing the last row and last column of P. Note that 1 is a vector having all elements unity.

For the steady state case, the cyclical steady state probability vector, s ₀ presented in Crosier [30] is employed. The method for obtaining s ₀ which was simplified by Champ [31] is briefly explained as follows: First, s ₀ is computed by solving r = P ^T r subject to 1 ^T r = 1, where P is the transition probability matrix with the absorbing states. Then s ₀ = (1 ^T q)^− 1 q, where q is a vector whose length is similar to the number of rows (or columns) of Q. Here, q is obtained from r by deleting the entry corresponding to the absorbing state. The steady state ARL is computed using Eq. 19, but by replacing the initial probability vector s in this equation with the steady state probability vector s ₀.