Appendix S1
This section explains the Markov chain approach for computing the ANOS of the EWMA-T
chart, presented in Pehlivan and Testik [31] and Zhang et al. [53]. For a one-sided EWMA-T
chart with design parameters  , LCL E T and B , let the interval between the boundary B and the
LCL E T , i.e.  LCLE T , B  of the in-control region be divided into n subintervals, each of width
2 (Figure 5 ) such that

B  LCL E T
2n
(A1)
FIGURE 5 HERE
Each subinterval represents a state and there are n transient states altogether. Since the TBE
observation is always a positive value, the out-of-control region  0, LCLET  represent the
absorbing state, which is the (n+1)th state. The statistic Z t can fall in any of the n transient states
when the process is in-control and fall in the (n+1)th absorbing state when the process is out-ofcontrol. The Z t is said to be in a transient state j at time t when it falls in the jth subinterval
 B  2 j, B  2  j 1 for
Hj 
j  1, 2,..., n. The midpoint of jth subinterval is represented by
B  2 j  B  2  j  1
2
 B  2 j    B   2 j  1  .
(A2)
The run length of the EWMA-T chart is defined as the number of steps taken from an initial state
of Z 0 until Z t falls in the absorbing state, which can be determined by a transition probability
matrix. Let Pij be the transition probability among transient states that Z t transitions from state i
(for i  1, 2, , n at time t 1 ) to state j (for j  1, 2, , n at time t). Assume that Z t 1 is equal to
H i in state i at time t 1 . Since the largest value of Zt is set as B, the largest Zt always fall in
state 1 of interval  B  2 , B  . Thus, when j  1 , the transition from state i to the 1st state is
given by
Pi1  P  B  2  Z t  B Z t 1  H i 
 P  H1    Z t  B Z t 1  H i 
 P  Z t  H1   Z t 1  H i 
 P  X t  (1   ) Z t 1  H1   Z t 1  H i 
(A3)
H    (1   ) H i 

 P  Xt  1




 H    (1   ) H i 
 1  FT  1
 , i  1,..., n



The Pij for state j  2,3,, n is given as
Pij  P  B  2 j  Z t  B  2  j  1 Z t 1  H i 
 P  B  2 j   X t  (1   ) Z t 1  B  2  j  1 Z t 1  H i 
 P  B  2 j   X t  (1   ) H i  B  2  j  1 
  B  2 j   (1   ) H i
 B  2 j  2   (1   ) H i 
 P
 Xt 





 H j    (1   ) H i
 H j     (1   ) H i 
 P
 Xt 




  H j     (1   ) H i
 FT 




 H    (1   ) H i
  FT  j




The zero-state ANOS for the EWMA-T chart is

 , i  1,..., n; j  2,..., n

(A4)
ANOSEWMA T  S T  I  R  1 ,
1
(A5)
where R is the n  n transition probability matrix of the transient states with elements Pij ; I is a
n  n identity matrix; 1 is a n1 row vector of ones and; S is a n1 initial probability vector
which determines the starting state of the EWMA-T chart. Meanwhile, the out-of-control steadystate ANOS for the EWMA-T chart is [54]


ssANOSEWMAT  S0T  I  R1  1  0.5 ,
1
(A6)
where matrix I and vector 1 are as defined previously. Here, R1 is the out-of-control transition
T
probability matrix of the transient states; S 0 is the steady-state probability vector, obtained by
first normalizing the in-control transition probability matrix R (based on   1 ) and then solving
the equation S 0  R T S 0 subject to 1T S 0  1 . For an accurate approximation of ANOSE-T using the
Markov chain method, n  201 is considered in this study. The ANOS computed have been
verified with simulation. The ANOS performance is similar to the ARL performance, for the
EWMA-T chart as r  1 .