**Step I**

Since the exact value of the out-of-control mean vector *μ***1***i* is not known a priori, two approx‐

determine the exact probability. Instead, the aim is to have an approximate probability (a belief) on each characteristic being out-of-control. In the first approximation method, define *ICi*

of observing *xk* given all characteristics are in-control. Further, let

Pr{*xk* | *I Ci*

<sup>0</sup> <sup>1</sup> Pr{ } <sup>1</sup> exp ; 1,2,..., , Pr{ } <sup>2</sup>

*i i <sup>x</sup> OOC <sup>R</sup> i m*

æ ö æ ö - µ = ç ÷ ç ÷ <sup>=</sup> ç ÷ è ø è ø

where *R* is a sufficiently big constant number to ensure the above definition is less than one.

**•** The determination of a threshold for the decision-making process (derived later) will be

Niaki and Fallahnezhad [8] defined another equation for the above conditional probability and

proposed a novel method of detection and classification and used simulation to compare its performances with that of existing methods in terms of the average run length for different mean shifts. The results of the simulation study were in favor of their proposed method in almost all shift scenarios. Besides using a different equation, the main difference between the current research and Niaki and Fallahnezhad [8] is that the current work develops a novel heuristic threshold policy, in which to save sampling cost and time or when these factors are

} can be approximately evaluated using Pr{*xk* | *ICi*

(*xk* **'**

, *μ*0*i*+1, ..., *μ*0*<sup>m</sup> <sup>T</sup>* in the aforementioned multivariate normal density, so that

), and since it is assumed that characteristic *i* is under control, no matter the

} and also we do not know the exact value of out-of-control mean vector,

2

*ki i*

s

m

*<sup>x</sup> <sup>x</sup>* (37)

th characteristic is to its in-control mean the smaller is Pr{*xk* |*OOCi*

} in equation (37) has the following two properties:

*Bi*

(*x<sup>k</sup>* , *O<sup>k</sup>* <sup>−</sup>1)= *Bi* =1. They

}. Note that we do not want to

} be the conditional probability

*<sup>k</sup>* | *ICi*

} =Pr{*x*'

−*μ***0**)). Note that this evaluation is proportional to

}<sup>∝</sup> <sup>1</sup> Pr{*xk* | *I Ci*

} , since we only need a belief function to

}. Moreover, it can be easily seen that

to be

}, where

} . Although it is

}

imations are used in this research to determine Pr{*xk* |*OOCi*

138 Dynamic Programming and Bayesian Inference, Concepts and Applications

the event that all characteristics are in-control, and let Pr{*xk* | *ICi*

<sup>−</sup>*μ***0**)*<sup>T</sup> <sup>Σ</sup>* **-1**

In the second approximation method, we assume Pr{*xk* |*OOCi*

*IC*

**•** It does not require the value of out-of-control means to be known.

showed that if a shift occurs in the mean of variable *i*, then*Limk*→*<sup>∞</sup>*

constrained, the number of the data gathering stages is limited.

*k*

this approximation is just used to determinePr{*xk* |*OOCi*

*i*

*k*

The approximation to Pr{*xk* |*OOCi*

condition of the other characteristics, this approximation is justifiable.

} is not equal to <sup>1</sup>

*xk* **'**

Pr{*x*'

exp( −<sup>1</sup> 2 ( *xki* <sup>−</sup> *<sup>μ</sup>*0*<sup>i</sup> σi* ) 2

= *μ*01, ..., *xki*

*<sup>k</sup>* | *ICi*

} <sup>=</sup> *<sup>A</sup>* exp( <sup>−</sup><sup>1</sup>

obvious that Pr{*xk* |*OOCi*

the closer the value of the *i*

as expected. We thus let

easier.

evaluate Pr{*xk* |*OOCi*

2 (*xk* **'**

Pr{*xk* | *ICi*

Define *i* =1, 2, ..., *m* as the set of indices for the characteristics, all of which having the potential of being out-of-control.
