**Author details**

( ) ( ) ( ) , ,, , <sup>2</sup> 2 2 ( ) <sup>1</sup> *k sm k sm k sm k sm*

{ ( ( ))}


,

*k sm*

2 2 , , 2 2 , ,

2

2 , ,

Pr ( ) 2ln ( ) 1

154 Dynamic Programming and Bayesian Inference, Concepts and Applications

Pr ( ) ( ) 2ln ( )

1 2 2

Pr ; ( ) Pr

*t*

2

<sup>1</sup> <sup>2</sup> 2

æ ö Gç ÷ è ø

Replacing the above equations in equation (53) results in

a

() ; 1

*i j gr sm k i j rd k*

a

*e t dt* - -

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

( ( ))

2ln ( ) 1 1

+

,

*rd k*

¥

ò

*gr sm*

{ ( ( ))}

³ +

2 2 <sup>2</sup> , , 2ln ( ) 1 1

2 2 2 2 , , 2ln ( ) 1 1

¥

ì ü ³= ³ í ý

<sup>2</sup> <sup>2</sup> , ,

*k gr sm k sm*

0.5( ) 0.5( )

î þ

,

*gr sm*

*t*


1

+

2

<sup>1</sup> <sup>2</sup> 2

æ ö Gç ÷ è ø

*T T*

*k sm gr sm*

<sup>+</sup> - ³

*e t T T hd k dt*

¥

*e t <sup>T</sup> hd k dt*

<sup>+</sup> <sup>³</sup> + =

*gr sm*

{ ( ( ))}

*k gr sm gr sm*

*T ET h d k*

{ ( ( )) }

Pr ( ) 2ln ( ) 1

*k gr sm gr sm*

*T hd k*

{ ( ( )) } ( ( , ))

{ ( ( ))} ( ( )) , ,

*k gr sm gr sm hd k*

{ ( ) } ( ( ))

*B gr O d k e r d k e*

( ( ) ( )) ( ( ( )) )

2 2 \* \* ,, , 2ln ( ) 1 1

*e t V N B gr O V N dt*

( ( ) ( )) ( ( ( )) ) ( )

, , 1 , 2ln ( ) 1

*e t B sm O V N dt V N*

*gr sm k i j i j hd k*

¥

ò

,

µ -- +

*gr sm*

2 2 \* \*


+

¥

ò

; 1 1

, , ,

*gr sm k gr sm gr sm*

*k gr sm gr sm hd k*

, ,

*k gr sm gr sm*

= - ³ =

*T hd k*

Pr ( ) 1 2ln ( )

,


*k sm*

Pr ( ) ( ) 2ln ( )

Pr ( ) ( ) 2ln ( )

*k gr sm gr sm*

*T T hd k*

Thus,

Hence,

Similarly

*T E T E T Var T* µ=+ = (88)

1

2

ò (90)

1

2

;

ò ; (91)

<sup>1</sup> <sup>2</sup> 2

æ ö Gç ÷ è ø

1

2

<sup>1</sup> <sup>2</sup> 2

 a

æ ö Gç ÷ è ø

*t*


*t*


<sup>1</sup> <sup>2</sup> 2

æ ö Gç ÷ è ø

*t*


(89)

(92)

(93)

Mohammad Saber Fallah Nezhad\*

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

Associate Professor of Industrial Engineering, Yazd University, Iran
