**•** Positive shifts in the process mean

The values of *B* <sup>+</sup> (*Ok* ), showing the probability of occurring a positive shift in the process mean, will be calculated applying equation (61) recursively. Pr{*xk* | *In* −*control*} is defined by the following equation,

$$\Pr\left\{\mathbf{x}\_k \middle| \text{In}-control\right\} = \mathbf{0}.5\tag{62}$$

**4.2. A decision on beliefs approach**

**•** A positive shift is occurred in the process mean

**•** No positive shift is occurred in the process mean

**•** Calculate the posterior Beliefs in terms of prior Beliefs.

Decision making framework is as follows:

**•** Gather a new observation.

negative shift)

value of *V* (*n*, *d* +(*n*)) thus,

We present a decision making approach in terms of Stochastic Dynamic Programming

Using Dynamic Programming Based on Bayesian Inference in Selection Problems

http://dx.doi.org/10.5772/57423

147

**•** Determine the value of the minimum acceptable belief (*d* +(*n*) is the minimum acceptable belief for detecting the positive shift and *d* <sup>−</sup>(*n*) is the least acceptable belief for detecting the

**•** If the maximum Belief was more than the minimum acceptable belief, *d* +(*n*), select the belief

**•** In terms of above algorithm, the belief with maximum value is chosen and if this belief was more than a control threshold like *d* +(*n*), the candidate of that Belief will be selected as optimal candidate else the sampling process is continued. The objective of this model is to determine the optimal values of *d* +(*n*). The result of this process is the optimal strategy with

Suppose new observation *xk* is gathered. (*k* is the number of gathered observations so far). *V* (*n*, *d* +(*n*)) is defined as the probability of correct selection when *n* decision making stages are remained and we follow *d* +(*n*) strategy explained above also *V* (*n*) denotes the maximum

> { ( , ( ))} *d n V n Max V n d n* <sup>+</sup>

CS is defined as the event of correct selection. *S1* is defined as selecting the out-of-control condition (positive shift) as an optimal solution and *S2* is defined as selecting the in-control condition as an optimal decision and *NS* is defined as not selecting any candidate in this stage.

<sup>+</sup> <sup>=</sup> (68)

(69)

*n* decision making stages that maximize the probability of correct selection.

( )

( ( )) 11 22 , x{Pr{ }} Pr{ }Pr{ } Pr{ }Pr{ }

<sup>+</sup> == + +

*V n d n Ma CS CS S S CS S S*

Suppose n stages for decision making is remained and two decisions are available.

**•** Order the current Beliefs as an ascending form and choose the maximum.

candidate with maximum value as a solution else go to step 1.

( )

Hence, using the total probability law, it is concluded that:

Pr{ }Pr{ }

*CS NS NS*

approach. Presented approach is like an optimal stopping problem.

For positive shift, the probability of being a positive shift in the process at iteration *k*, Pr{*xk* |*Out* −*of* −*control*}, is calculated using equation (63).

$$\Pr\{\mathbf{x}\_k \Big| Out - of -control\} = \wp\{\mathbf{x}\_k\} \tag{63}$$

where *φ*(*xk* ) is the cumulative probability distribution function for the normal distribution with mean μ and variance σ<sup>2</sup> . Above probabilities are not exact probabilities and they are a kind of belief function to ascertain good properties for *B* <sup>+</sup> (*Ok* )

Therefore *B* <sup>+</sup> (*Ok* ) is determined by the following equation,

$$B^{+}\left(O\_{k}\right) = \frac{B^{+}\left(O\_{k-1}\right)\phi\left(\mathbf{x}\_{k}\right)}{B^{+}\left(O\_{k-1}\right)\phi\left(\mathbf{x}\_{k}\right) + 0.5\left(1 - B^{+}\left(O\_{k-1}\right)\right)}\tag{64}$$

**•** Negative shifts in the process mean

The values of *B* <sup>−</sup> (*Ok* ) denotes the probability of being a negative shift in the process mean that is calculated using equation (61) recursively. In this case, Pr{*xk* | *In* −*control*} is defined by the following equation,

$$\Pr\left\{\mathbf{x}\_k \middle| In -control\right\} = 0.5\tag{65}$$

Also is Pr{*xk* |*Out* −*of* −*control*} calculated using equation (66).

$$\Pr\{\mathbf{x}\_k \Big| Out-of-control \} = 1 - \varphi\{\mathbf{x}\_k\} \tag{66}$$

Thus *B* <sup>−</sup> (*Ok* ) is determined by the following equation,

$$B^-\left(O\_k\right) = \frac{B^-\left(O\_{k-1}\right)\left(1-\phi\left(\mathbf{x}\_k\right)\right)}{B^-\left(O\_{k-1}\right)\left(1-\phi\left(\mathbf{x}\_k\right)\right) + 0.5\left(1-B^-\left(O\_{k-1}\right)\right)}\tag{67}$$
