**Appendix 2**

Evaluating the Optimal Value of *dgr*,*sm*(*k*)

Assume (*μj* ) *<sup>j</sup>*∈{1,2,...,*m*}=0 and (*σ<sup>j</sup>* ) *<sup>j</sup>*∈{1,2,...,*m*}=1. Then,

$$\begin{aligned} \Pr\left\{ \left\{ e^{\frac{0.5(T\_{k,g} - \mu)^2}{k}} \ge h \left( d\_{gr, sm}(k) \right) e^{0.5(T\_{k,su})^2} \right\} \right\} &= \\ \Pr\left\{ 0.5(T\_{k,gr}|\_{\text{sw}})^2 \ge \ln \left( h \left( d\_{gr, sm}(k) \right) \right) + 0.5(T\_{k,su})^2 \right\} &= \\ \Pr\left\{ \left( T\_{k,gr}|\_{\text{sw}} \right)^2 - \left( T\_{k,su} \right)^2 \ge 2 \ln \left( h \left( d\_{gr, sm}(k) \right) \right) \right\} \end{aligned} \tag{87}$$

Now since (*Tk* ,*sm*, *Tk* ,*gr*|*sm*) follow a standard normal distribution (*μj* ) *<sup>j</sup>*∈{*gr*,*sm*}=0 and (*σ<sup>j</sup>* ) *<sup>j</sup>*∈{*gr*,*sm*}=1, hence (*Tk* ,*gr*|*sm*)2 and (*Tk* ,*sm*) 2 follow a *χ* <sup>2</sup> distribution with one degree of freedom. Then using an approximation, if we assume that (*Tk* ,*sm*) 2 is approximately equal to its mean, we have

$$\left(T\_{\
u\_{k,m}}\right)^2 \propto E\left(T\_{\
u\_{k,m}}\right) = E\left(T\_{\
u\_{k,m}}\right)^2 + Var\left(T\_{\
u\_{k,m}}\right) = 1\tag{88}$$

Now by solving the equation

*δVi*, *<sup>j</sup>* (*N* )

( ( ) ( ))

a

, ,

*gr sm k gr sm*

*B gr O V N*

\*

\*

<sup>1</sup> ; 1

<sup>1</sup> ; 1

( ( ) ( ))

a

, ,

Now using another approximation, if we assume that (*Tk* ,*gr*)

( ( ) ( ))

a

, ,

*gr sm k gr sm*

*B sm O V N*

, ,

Address all correspondence to: Fallahnezhad@yazd.ac.ir

Associate Professor of Industrial Engineering, Yazd University, Iran

*gr sm k gr sm*

*B gr O V N*


\*

<sup>1</sup> ; 1

<sup>1</sup> ; 1

\*

( ( ) ( ))

The approximate optimal value of *dgr*,*sm*(*k*) is obtained as follows,

a

Finally, the approximate value of *dgr*,*sm*(*k*) say *<sup>d</sup>* <sup>1</sup>

mean, the approximate value of *dgr*,*sm*(*k*) say *<sup>d</sup>* <sup>2</sup>

numerically or by a search algorithm.

equation,

**Author details**

**References**

Mohammad Saber Fallah Nezhad\*

*gr sm k gr sm*

*B sm O V N*


*<sup>δ</sup>dgr*,*sm*(*k*) =0, the following equation is obtained.

( ( ( )) )

ln ( ) 1

,

*rd k*


*gr sm*

( ( ( )) )

ln ( ) 1

+

Using Dynamic Programming Based on Bayesian Inference in Selection Problems

2

+

+

+

*gr*,*sm*(*k*) is determined by solving this equation

*gr*,*sm*(*k*) is determined by solving following

is approximately equal to its

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

(94)

155

(95)

,

*hd k*

( ( ( )) )

ln ( ) 1

,

*rd k*


{ } 1 2

[1] Basseville, M. & Nikiforov, I.V. (1993). Detection of abrupt changes: Theory and ap‐

plication. (Information and System Sciences Series), Prentice-Hall.

*gr sm*

( ( ( )) )

, ,, ( ) max ( ), ( ) *gr sm gr sm gr sm d k d kd k* = (96)

ln ( ) 1

,

*hd k*

*gr sm*

*gr sm*

Thus,

$$\begin{split} &\Pr\left\{ \left( T\_{k,gr[sm]} \right)^2 - \left( T\_{\ \_ {k,sm}} \right)^2 \ge 2 \ln \left( h \left( d\_{gr,sm}(k) \right) \right) \right\} \ll\\ &\Pr\left\{ \left( T\_{k,gr[sm]} \right)^2 - E\left( T\_{\ \_ {k,sm}} \right) \ge 2 \ln \left( h \left( d\_{gr,sm}(k) \right) \right) \right\} \\ &= \Pr\left\{ \left( T\_{k,gr[sm]} \right)^2 - 1 \ge 2 \ln \left( h \left( d\_{gr,sm}(k) \right) \right) \right\} =\\ &\Pr\left\{ \left( T\_{k,gr[sm]} \right)^2 \ge 2 \ln \left( h \left( d\_{gr,sm}(k) \right) \right) + 1 \right\} \end{split} \tag{89}$$

Now, since (*Tk* ,*gr*|*sm*)2∝*<sup>χ</sup>* 2(1), we have

$$\Pr\left\{(T\_{k,gr|sm})^2 \ge 2\ln\left(h\left(d\_{gr,sm}(k)\right)\right) + 1\right\} = \int\_{2\ln\left(h\left(d\_{gr,sm}(k)\right)\right)+1}^{\infty} \frac{e^{-\frac{t}{2}}t^{-\frac{1}{2}}}{\Gamma\left(\frac{1}{2}\right)2^{\frac{1}{2}}}dt\tag{90}$$

Hence,

$$\left| \Pr \left\{ \left( T\_{k,gr|m} \right)^2 - \left( T\_{\
u,m} \right)^2 \ge 2 \ln \left( h \left( d\_{gr,sm}(k) \right) \right) \right\} \right| \simeq \int\_{2\ln \left\{ h \left( d\_{gr,m}(k) \right) \right\} + 1}^n \frac{e^{-\frac{t}{2}} t^{-\frac{1}{2}}}{\Gamma \left( \frac{1}{2} \right) 2^{\frac{1}{2}}} dt \tag{91}$$

Similarly

$$\Pr\left\{B\_{gr,sm}\left(gr;\mathcal{O}\_k\right) \ge d\_{gr,sm}(k)\right\} = \Pr\left\{e^{\frac{0.5\left(T\_{k,pl}\right)^2}{2}} \ge r\left(d\_{gr,sm}\left(k\right)\right)e^{\frac{0.5\left(T\_{k,sm}\right)^2}{2}}\right\} \simeq \prod\_{l=2\left(r\left(d\_{gr,sm}(k)\right)\right)+1}^{\infty} \frac{e^{-\frac{l}{2}}\left(-\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}\right)2^{\frac{1}{2}}}dt\tag{92}$$

Replacing the above equations in equation (53) results in

$$\begin{aligned} \left(V\_{i,j}"\left(\mathbf{N}\right)\propto\left(\mathbf{B}\_{gr,sm}\left(gr;\mathbf{O}\_k\right)-aV\_{i,j}^\*\left(\mathbf{N}-1\right)\right)\right)\_{\left(2\ln\left(r\left(d\_{g,sm}\left(k\right)\right)\right)+1\right)}^{\left\lfloor\frac{t}{2}\frac{1}{2}-\frac{1}{2}}dt+\\ \left(\mathbf{B}\_{gr,sm}\left(sm;\mathbf{O}\_k\right)-aV\_{i,j}^\*\left(\mathbf{N}-1\right)\right)\right)\_{\left(2\ln\left(d\_{g,sm}\left(k\right)\right)\right)+1}^{\left\lfloor\frac{t}{2}-\frac{1}{2}}\frac{1}{2}dt+aV\_{i,j}^\*\left(\mathbf{N}-1\right) \end{aligned} \tag{93}$$

Now by solving the equation *δVi*, *<sup>j</sup>* (*N* ) *<sup>δ</sup>dgr*,*sm*(*k*) =0, the following equation is obtained.

$$\begin{split} \left( \operatorname{B}\_{gr,sm} \left( \operatorname{gr}; \operatorname{O}\_{k} \right) - \alpha \operatorname{V}\_{gr,sm}^{\star} \left( \operatorname{N} - 1 \right) \right) \frac{1}{\sqrt{\left( \ln \left( r \left( \operatorname{d}\_{gr,sm} \left( k \right) \right) \right) + 1 \right)}} &= \\ - \left( \operatorname{B}\_{gr,sm} \left( \operatorname{sn}; \operatorname{O}\_{k} \right) - \alpha \operatorname{V}\_{gr,sm}^{\star} \left( \operatorname{N} - 1 \right) \right) \frac{1}{\sqrt{\left( \ln \left( h \left( \operatorname{d}\_{gr,sm} \left( k \right) \right) \right) + 1 \right)}} \end{split} \tag{94}$$

Finally, the approximate value of *dgr*,*sm*(*k*) say *<sup>d</sup>* <sup>1</sup> *gr*,*sm*(*k*) is determined by solving this equation numerically or by a search algorithm.

Now using another approximation, if we assume that (*Tk* ,*gr*) 2 is approximately equal to its mean, the approximate value of *dgr*,*sm*(*k*) say *<sup>d</sup>* <sup>2</sup> *gr*,*sm*(*k*) is determined by solving following equation,

$$\begin{split} & \left( B\_{gr,sm} \left( sm; \mathcal{O}\_k \right) - \alpha \boldsymbol{V}\_{gr,sm}^\* \left( \boldsymbol{N} - 1 \right) \right) \frac{1}{\sqrt{\left( \ln \left( r \left( \boldsymbol{d}\_{gr,sm}(k) \right) \right) + 1 \right)}} = \\ & - \left( B\_{gr,sm} \left( gr; \mathcal{O}\_k \right) - \alpha \boldsymbol{V}\_{gr,sm}^\* \left( \boldsymbol{N} - 1 \right) \right) \frac{1}{\sqrt{\left( \ln \left( h \left( \boldsymbol{d}\_{gr,sm}(k) \right) \right) + 1 \right)}} \end{split} \tag{95}$$

The approximate optimal value of *dgr*,*sm*(*k*) is obtained as follows,

$$d\_{\mathcal{g}^r, \text{sm}}(k) = \max \left\{ d^1\_{\mathcal{g}^r, \text{sm}}(k), d^2\_{\mathcal{g}^r, \text{sm}}(k) \right\} \tag{96}$$
