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�O=DU=� � |qwHRQ@} Qiu=QyD x=oWv=O |vi |=yxOmWv=O T}OQB '`}=vY |UOvyt |xOmWv=O

�|QDmO� Q=@}wH u=L@U�|QDmO |wHWv=O� xH} QwW Q=R |raOtLt

u=QyD x=oWv=O 'C} Q}Ot |xOmWv=O

|OQ@ Q=m |=yx@vH x@ QDtm ?re= u=v}t]= C}r@=k |}=}=B |R=Uxv}y@ =@ \@DQt C=k}kLD QOQO |}=}=B |R=Uxv}y@ |xr�Ut R= xO=iDU= xr=kt u}= |x�=Q= R= hOy "CU= xOW xHwD u;OOa |]NQ}e |xiOyOvJ |R} Qxt=vQ@ pOt l} xr=kt u}= QO "CU= O}rwD \N |L=Q]

|rwYLt Q=QkDU= Q@ |vD@t O}rwD \N |L=Q] |xr�Ut |=Q@ |}xrY=i |=yQDt=Q=B =@ K}LYxOW x�=Q= C}a]k sOa \}=QW QO |}=}=B uDiQo Q_v QO =@ =yC}OwOLt |x} Q_v OQ@y=Q wO}rwD |=Q@ R=}v OQwt O}rwD \w]N |xv}y@ O=OaD u}}aD 'pOt u}= |x�=Q= R= hOy "CU=

CqwYLt |=[=kD w C}iQ_ '=[i 'xHOw@ |=yC}OwOLt uDiQo Q_v QO =@ CqwYLtOw@tm w |Q=Oyov pm |xv} Ry w xv}W}@ sDU}U pm u=v}t]= C}r@=k xm |wLv x@ &CU=

?wr]t 'G}=Dv xm CU= xOW x�=Q= |OOa p=Ft xU 'pOt uwtR; |=Q@ "OwW xv}tm |OwHwt"OyO|t u=Wv =Q |O=yvW}B pOt uOw@

fjolai@ut.ac.irsobhan.jooybar@ut.ac.irmaz.shourijeh@ut.ac.ir "C}a]k sOa '|}=}=B 'xiOyOvJ |R=Uxv}y@ 'O}rwD \N |L=Q] %|O}rm u=oS=w

xtOkt "1|=ysDU}U `=wv= QO |}=}=B |R=Uxv}y@ "CU= |}=}=B |R=Uxv}y@ |xr�Ut x@ swUwtxDiQo Q=Qk |Q=}U@ u=kkLt xHwD OQwt |R=wt - |QU w |QU - |R=wt '|R=wt '|QUu=wD|tv u; pL |=Q@ xm [6]CU= NP-Hard p�=Ut TvH R= xr�Ut u}= "CU=|x�=Q= x@ [7]OwN |xr=kt QO 2x} ,qFt "OQm xO=iDU= 1j}kO |R=Uxv}y@ |=yVwQ R=p�=Ut `=wv= pL |=Q@ 3C=QP s=LOR= |R=Uxv}y@ Q@ |vD@t |Q=mD@==Qi sD} Qwor= l}|WywSB |xv}tR u}= |xv}W}B |rm Qw] x@ "CU= xDN=OQB 4C}a]k \}=QW QO |}=}=Bs}y=wN xQ=W= \@DQt C=k}kLD R= |NQ@ x@ xt=O= QO 'u}= OwHw =@ "CU}v |ve u=OvJ|L=Q] w |}=}=B s}y=it ?}mQD |=Q@ u=oOvU} wv p=@kDU= sOa p}qO R= |m} O}=W "OQm

"OW=@ j}kO |=yVwQ =@ xiOyOvJ p�=Ut wv u}= pL |Q=wWO O}rwD \NQO xOW xQ=W= |DvU |=ypt=a hqNQ@ xm OvQ=O|t Q=y_= [4]5|vH w Cv@"CU= xOW |}=}=B uwJ |}=ypt=a uOQm xv}W}@ x@ |tm ,=D@Uv xHwD u}W}B h=Qo =Q=B|=yp}rLD w|}=}=B '|@=QNMQv s}y=it uOQm OQ=w =@ Ovvm|t|aU OwNVywSB QO =yu;=@ R}v 6u=Q=mty w nv=J "OvQ=OQ@ s=o =DU=Q u}= QO =ysDU}U COtOvr@ Q=DiQ |Q=t;pL x@ wvtt |wHDUH sD} Qwor= l} |xr}Uw x@ |}=}=B C=_Lqt uDiQo Q_v QO[9]`HQt QO |Qo}O u=kkLt [8]"OvO} RQw CQO=@t |Q=m |=ypwrU O}rwD |xr�Ut|R=Uxv}tm Q@ ptDWt xiOyOvJ \rDNt K}LY OOa |R} Qxt=vQ@ R= xO=iDU= =@x�=Q= |rwrU O}rwD sDU}U |L=Q] |=Q@ |rOt |}=}=B |R=Uxv}W}@ w sDU}U |=yxv} Ry

u=R}t uwJ |tyt |=yQDt=Q=B xQ=wty |O}rwD |=ysDU}U |L=Q] QO |DvU Qw] x@|=yx=oDU}= wCq;u}W=t Q=QkDU=|=Q@ R=}v OQwt|=[i 'pwYLt Qy|=Q@ Q=R=@ |=[=kD|wU R= [1]"OwW|t xDiQo Q_v QO O}rwDC}iQ_ w u=Q}Ot TQDUO QO |r=t `@=vt '|Q=mw |QDWt uO=O CUO R= x@ CU= umtt Q=R=@ |=[=kD x@ MU=B sOa xm s}v=O|t Qo}Ou} QDtyt R= |m} "OwW xDiQ CUO R= |=yCYQi |xv} Ry OWQ u; `@D x@ w Q=R=@ syU|=ysDU}U |oO=Di= Q=m R= 'xwkr=@ u=} QDWt uDiQ CUO R= w O}rwD Vy=m p}qOR}v w O}rwD Vy=m "CU= O}rwD QO xikw w O}rwD |=RH= |=y|@=QN Cra x@ |O}rwDxOw@ u=oOvvmO}rwD xHwD OQwt xQ=wty |O}rwD |=ysDU}U |@=QN R= pY=L C=a}=["CU= sDU}U u=v}t]= C}r@=k uOQ@ q=@ hOy u}= x@ p}v |=yx=Q R= |m} [3w2]"CU=sDU}U |}=}=B |R=Uxv}W}@ u=tRsy xOW QmP |DvU |=yQDt=Q=B Q@ xwqa O}=@ 'u}=Q@=v@[4]"OQ}o Q=Qk u=Q}Ot |OH xHwD OQwt �sDU}U |oO=Di= Q=m R= uOQm xv}tm � |O}rwD|=Q@ xm |iOy jkLD |=Q@ sDU}U u; |}=v=wD sDU}U l} |}=}=B |rm Qw] x@l} |L=Q] |xr�Ut [5]"CU= XNWt |v=tR |xQwO l} QO xOW |L=Q] u;xQ}e w xR=Ov= 'uRw 'xHOw@ uwJ `@=vt C}OwOLt uDiQo Q_v QO =@ q=@ |}=}=B =@ sDU}U

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DOI:10.24200/J65.2019.7214.1758

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=@ =Q Cq}yUD u=tO}J w |Q=m |=[i C=t=Rr= s}y=it 8u=Q=mty w uwOwU=wQO [11]"OvOQm |UQQ@ |@}mQD Qw] x@ Oqwi |xv=NQ=m l} |QwxQy@ w |}=}=B s}y=itpwtat OQm} wQ hqN Q@ u=oOvU} wv '|}=Pe O=wt O}rwD \N |xQ=@ QO |Qo}O j}kLD|=yxikw uOQm xv}tm |=Q@ 9x=owro R= ?=vDH= T=U= Q@ |Q=m |=yx=oDU}= |L=Q]u=oOvU} wv \UwD xm |QwQt =@ ' wtHt QO [3]"OvOQ@ xQy@ |}=}=B s}y=it R= |O}rwDu}= QDW}@ xm OW xOy=Wt 'CiQo CQwY |WywSB |xv}tR u}= OwOLt |xv}W}B Q@QO pwrU CN=U x@ \w@ Qt R}v �|}=}=B w O}rwD \w]N |L=Q] ?}mQD� =yVywSB

[17�12'9]"CU= |rwrU O}rwD |=ysDU}Us=tD xm OwW|t ZQi |}=}=B |xQ=@ QO xOW s=Hv= C=k}kLD R= |Q=}U@ QOx@ Cq=tDL= R= l} Qy xm |vat u}O@ "OvDUy XNWt j}kO pmW x@ Cq=tDL=sDU}U |xQ=@ QO |rt=m |r=tDL= C=aq]= w OvDUy |Q}oxR=Ov= p@=k |@ wN wLvQ=QkQ@ O}=@ \QW wO |r=tDL= C=aq]= p}tmD |=Q@ "OQ=O OwHw u; |=RH= Q=DiQ w|Q}oxR=Ov= p@=k |@ wN pmW x@ =} XNWt Cq=tDL= |=y`} RwD s=tD "1 %OW=@|xOvvm h}YwD |iO=YD |=yQ}eDt s=tD |va} 'Ov=pkDUt sDU}U |=RH= "2 w OvW=@u=wD|t OW=@v Q=QkQ@ \QW l} pk=OL xm|DQwY QO "Ov=pkDUt |=RH= |}=}=B Q=DiQQO [18]"OQm xO=iDU= �|R=i =} |}xrY=i O=Oa= 'p=Ft u=wva x@� |a]kQ}e |}=}=B R=xDWPov sDU}U |R=Ov= x=Q R= |v=OvJ u=tR xm OQ=wt R= |Q=}U@ QO |ak=w |=}vO&OQ=Ov OwHw |k}kO |Q=t; |=yxO=O 'Ovm|t Q=m xSwQB l} ?r=k QO \ki =} CU=|xr}Uw x@ Ov=wD|t xm CU= |k}kO=v OOa sDU}U R= <RH l} |}=}=B u}=Q@=v@x@U=Lt |a]kQ}e Cq=tDL= Oa=wk ltm x@ w h} QaD [19]|R=i =} |}xrY=i |OOa

"OwWQ=Qk u=kkLt xHwD OQwt R}v C}a]k sOa \}=QW QO |}=}=B |R=Uxv}y@ |xr�UtCq;u}W=t |}=}=B |=yxO=O C}a]k sOa QF= [20]`HQt QO p=Ft u=wva x@ "CU= xDiQou}= [18]10u=Q=mty w =}vw@ "CU= xOW|UQQ@ |O}rwD |=ysDU}U C=}rta w|L=Q] Q@|=yC}OwOLt ZQi w|}xrY=i |=yxO=O =@ |R=wt - |QU sDU}Ul} |=Q@ =Q xr�Ut11u=Q=mty w wy=W u}vJty "Ov=xOQm pL l}DvS sD} Qwor= =@ w xO=O xaUwD |iO=YDpmW x@ |R=wt - |QU sDU}U l} |=Q@ =Q |}=}=B |R=Uxv}y@ |xr�Ut [21]`HQt QOl}DvS sD} Qwor= Q@ |vD@t |}xrLQt wO |WwQ =@ w |a]k |=yC}OwOLt =@ xiOyOvJ|R=Uxv}y@ p�=Ut pL |=Q@ O} Q@D |R=Ux}@W sD} Qwor= [22]`HQtQO "Ov=xOQm pLp=tDL= `} RwD |=yQDt=Q=B uOw@ |R=i ZQi =@ |R=wt - |QU |=ysDU}U |=Q@ |}=}=B

"CU= xDiQ Q=m x@ =RH= Qtaxm OvDUy |QU - |R=wt sDU}U l} |L=Q] p=@vO x@ u=oOvU} wv xr=kt u}= QO\N l} u:}@t h}OQ Qy w hrDNt `=wv= R= |}=yu}W=t hQat =RH= R= l} Qy u; QOuDi=} hOy "Ovvm|t O}rwD u=Um} |rwYLt 'O}rwD \w]N u}= s=tD xm CU= O}rwDQ=R=@ |=[=kD w =[i 'xHOw@ C}OwOLt uDiQo Q_v QO =@ O}rwD |xv}y@ \w]N O=OaD|=yQDt=Q=B =@ |}=tv =yu}W=t Qta `} RwD w CU= |r=tDL= =[=kD "CU= pwYLt |=Q@x=Qty x@ |[=} Q pOt TBU w |}xrY=i C=}[=} Q =OD@= xt=O= QO "CU= |}xrY=i

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"CU="CU= XNWt `} RwD `@=D l} |=Q=O w |iO=YD |}=yv pwYLt |=[=kD "7

"CU= |rwYLt Q=QkDU= T=U= Q@ O}rwD sDU}U "8"CU=yC}OwOLt |x} Q_v T=U= Q@ O}rwD |R} Qxt=vQ@ OQ@y=Q "9

|Q=PoO=tv&Q=@v= QO |}=yv pwYLt Qy |Q=Oyov |xv} Ry %h

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s=j wv R= u}W=t Qy O}it Qta |=Q@ |}=tv `} RwD QDt=Q=B %�j = [�jkL; �jkR]

&s=k sDU}UQ} R QO (j = 1; 2; :::; n)

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&s=k sDU}UQ} R QO O}rwD \N Qy |R=Ov=x=Q |=Q@ R=}v OQwt O}it |=[i %sk

"|QU - |R=wt sDU}U "1 pmW

|}xrY=i O=Oa= �|w=Ut w QDnQR@ 'QDlJwm � \@=wQ "2"2%CW=O Ovy=wN =Q Q} RCr=L xUR=|m} OOa wOu}= 'OvW=@|}xrY=i OOa wOB wA Qo =

"Ov=sy R= =OH B w A %1 wv"OvQ=O l =QDW= Qo}Om} =@ B w A %2 wv

"B � A =} CU= A � B %3 wvxOW KQ]t |}xrY=i O=Oa= u=}t \@=wQ |=Q@ |vwo =vwo h} Q=aD [18]`@vt QO"xv=v}@O@ w xv=v}@VwN |Q}os}tYD "OQ=O OwHw |Q}os}tYD wv wO |rm Qw] x@ "CU=u=v}t]= sOa uDiQo Q_v QO uwO@ =Q xv} Ro u} QDy@ xOvQ}os}tYD 'xv=v}@VwN wv QOsOa |tm =@ =Q xv} Ro u} QDy@ xOvQ}os}tYD 'xv=v}@O@ wv QO xm|r=L QO "Ovm|t ^=Lr

"Ovm|t ?=NDv= u=v}t]=A = [aL; aR] = haC ; aW i|}xrY=i OOa wO|=Q@�0min|x]@=Q "1h} QaD|R=Uxv}tm |xr�Ut |=Q@ xv=v}@VwN Cr=L QO B = [bL; bR] = hbC ; bW i w

%OwW|t h} QaD Q} R CQwY x@ (0min)

A �0min B , aL � bL; A <0min B ^A 6= B

"CU}v 12uQ=kDt�0min |x]@=Q "OQ}PB|t =QA xOvQ}os}tYD 'h} QaD u}= T=U= Q@"�CU= �|k]vt AND� �|k]vt w� |vat x@ ^�

A = [aL; aR] = haC ; aW i|}xrY=i OOa wOu}@�0max |x]@=Q "2h} QaD|R=Uxv}W}@ |xr�Ut |=Q@ xv=v}@VwN Cr=L QO B = [bL; bR] = hbC ; bW i w

%OwW|t h} QaD Q} R CQwY x@ (0max)

A �0max B , aR � bR; A >max B ^A 6= B

|x]@=Q "Ovm|t ?=NDv= =Q A u}@VwN |xOvQ}os}tYD 'h} QaD u}= T=U= Q@"CU}v uQ=kDt =t= 'CU= 14|OaD w 13|U=mav= �omax

|}xrY=i Oa wO u=}t <pmin |x]@=Q "3 h} QaDxv=v}@O@ Cr=L QO B[bL; bR] = hbC ; bW i w A = [aL; aR] = haC ; aW i

%Ow@ Oy=wN Q} R CQwY x@ (pmin) |R=Uxv}tm w xr�Ut |=Q@A <pmin B , ac < bc

2w1 wv |}xrY=i O=Oa= |=Q@A <pmin B , aC � bC ^ aW < bW

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"aC < bC ^ aW > bWA = [aL; aR] = haC ; aW i|}xrY=i OOa wOu}@>pmax|x]@=Q "4h} QaD|R=Uxv}W}@ |xr�Ut |=Q@ xv=v}@O@ Cr=L QO B = [bL; bR] = hbC ; bW i w

%OwW|t h} QaD Q} R CQwY x@ (pmax)

A >pmax B , aC > bC

2w 1 wv |}xrY=i O=Oa= |=Q@A >pmax B , aC � bC ^ aW < bW

3 wv |}xrY=i OOa |=Q@'ac < bC ^ aW < bW xm|v=tR xv=v}@O@ s}tYD 3 wv |}xrY=i OOa |=Q@ '=t=

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41

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(dk �Qk)f(dk)ddi (4)

%OQm xO=U 5 |x]@=Q CQwY x@ u=wD|t =Q 4 |x]@=Q

TC =�kXk=1

�h

qxkZqIxk

(qkxk � dx)f(dk)ddk + :

q:UxkZqxk

(dk � qkxk)f(dk)ddk�

(5)

'pw= |xQ=Ro x]@=Q u}= QO "OwW xv}tm O}=@ xm CU= xr�Ut swO hOy `@=D 5 |x]@=Q"CU= T |xQwO QO s=k pwYLt |=[=kD Q@ O=R=t |q=m |Q=Oyov |xv} Ry Qou=}@O=R=t Q=Okt QO ?Q[ q=m OL=w l} |Q=Oyov |xv} Ry wtHt =@CU= Q@=Q@ xv} Ry u}=(qkxk � dk) Q=R=@ |=[=kD w xOW O}rwD pwYLt u=}t Cw=iD "T |xQwO QO q=mOw@tm |xv} Ry hQat swO |xQ=Ro "CU= O=R=t |q=m |OwHwt K]U |xOvvm u}}aDOw@tm Q=Okt QO ?Q[ q=m OL=w l} Ow@tm |xv} Ry wtHt =@ Q@=Q@ xm CU= pwYLtO}rwD pwYLt w Q=R=@ |=[=kD u=}t Cw=iD =@ Q@=Q@ xm CU= T |xQwO QO pwYLtx@ xr=kt u}= QO |O=yvW}B |[=} Q pOt T=U= u}= Q@ "CU= (dk � qkxk) xOW

%Ow@ Oy=wN Q} R CQwYMaximize

Rs(X) =�kXk=1

�1�

xkYi=1

(1�nYj=1

e��ijkRT ); 1�xkYi=1

(1�nYj=1

e��ijkLT )�

(6)

Minimize

TC =�kXk=1

�h

qkxkZqkLxk

(qkxk � dk)f(dk)ddk + :

qkUxkZqkxk

(dk � qkxk)f(dk)ddk (7)

St:2Xk=1

ckxk � B (8)

2Xk=1

skxk � S (9)

prob[Dk � qkxk] � 1� � 8k = 1; 2; :::; �k (10)

Lxk � xk � Uxk; int 8k = 1; 2; :::; �k (11)

|=Q@ O}rwD \N O=OaD xJ xmu}= Q@ |v@t CU= xHOw@ C}OwOLt %pw= C}OwOLt"OQm |R=Ov=x=Q u=wD|t TQDUO QO |xHOw@ x@ xHwD =@ pwYLt Qy O}rwD

s=tD |=Q@ O}rwD \w]N |=[i wtHt "CU=[i C}OwOLt %swO C}OwOLt"OW=@ TQDUO QO O}it |=[i pm =@ |w=Ut =} QDlJwm O}=@ CqwYLt

&O}rwD \N |R=Ov=x=Q |=Q@ Q=}DN= QO |=[i pm %Swv R= u}W=t \UwD CN=U u=} QH QO |q=m OL=w l} O}rwD |=Q@ sRq u=tR %tjk

&s=k sDU}UQ} R QO (j = 1; 2; :::; n) s=j&|R} Qxt=vQ@ ji= %T

|iO=YD |Q}eDt xm [0; T ] |v=tR |xQwO QO s=k |}=yv pwYLt |=Q@ =[=kD %Dk&Dk = f(dk)w CU= XNWt `} RwD =@

&Dk |iO=YD Q}eDt Q}O=kt %dkQ_v QO =@ s=k sDU}UQ} R QO O}rwD \N Qy QO xOW O}rwD |}=yv pwYLt Q=Okt %qk

&[0; T ] |R} Qxt=vQ@ |xQwO QO xO=O MQ |=y|@=QN uDiQo&|@=QN Qy R= TB Q}taD u=tR |[=} Q O}t= %MRT

&s=k pwYLt O}rwD pm Q=Okt %QkpwYLt O}rwD |=Q@ sRq O}rwD \w]N O=OaD |xOvyO u=Wv xm s}tYD Q}eDt %xk

&CU= s=k&(Lxk � xk � Uxk) s}tYD Q}eDt |q=@ u=Qm w u}}=B u=Qm %Uxk w Lxk

&(LQ = qk:Lxk ) Ovm O}rwD O}=@ s=k sDU}UQ} R xm|rwYLt Q=Oktu} QDtm %LQkOQ=O =Q u; O}rwD C}r@=k s=k sDU}UQ} R xm |rwYLt O=OaD u} QDW}@ %UQk

"(UQk = qk:Uxk)

Q} R |x]@=Q pmW x@ |}=}=B 'sDU}UQ} R k =@ |QU � |R=wt sDU}U l} QO%CU=

Rs(x) =nYj=1

[RSL(X); RSR(X)]

%xmRSL(X) =

�kXk=1

�1�

xYi=1

(1�nYj=1

rijkL)�;

RSR(X) =�kXk=1

�1�

xYi=1

(1�nYj=1

rijR)�

(1)

Qta uOw@ |}=tv x@ xHwD =@ "OwW|t h} QaD u}W=t Qy |}=}=B T=U= Q@ x]@=Q u}=sOa x@ xHwD =@ s=k sDU}UQ} R QO s=j wv R= u}W=t Qy |}=}=B 'u}W=t Qy O}it

%CU= 2 x]@=Q CQwY x@ w |}xrY=i |Q=Okt `} RwD QDt=Q=B C}a]krijk = [e��jkrT ; e��jkLT ] 8 i = 1; 2; :::; xk (2)

%Ow@ Oy=wN 3 |x]@=Q CQwY x@ CUNv hOy `@=D 'u}=Q@=v@Maximize

Rs(X) =�kXk=1�

1�xkYi=1

(1�nYj=1

e��ijkRT ); 1�xkYi=1

(1�nYj=1

e��ijkLT )�

(3)

pwYLt |=Q@ (Dk) =[=kD R= QDW}@ =} QDtm Ov=wD|t (Qk) O}rwD Q=Okt xm =Hu; R=Oy=wN p}tLD sDU}U Q@ pwYLt |Q=Oyov w pwYLt Ow@tm |=yxv} Ry 'OW=@ s=ks=yr= =@ xH}Dv QO ' LQk � Qk � UQk w Qk = qkxk xm s}v=O|t "OWR= |m} =@ uOW xH=wt CQwY QO sDU}U x@ xOW p}tLD pm |xv} Ry [23]`HQtR=

%=@ Ow@ Oy=wNQ@=Q@ pwYLt O=R=t =} Ow@tm Cr=L wO

42

1 2|xQ

=tW'35

�1|xQwO

'�139

8u=DU@

=D�h}QW

C}Q}Ot

w`}=vY

|UOvyt C=L}[wD� OyO|t u=Wv =Q |R} Qxt=vQ@ |xQwO |=yDv= O=R=t|v=tR |xR=@ 14 |x]@=Q

'uwU=wB w |}=tv `} RwD X=wN j@] Qo}O hQ] R= "�O}v}@@ 2 CUw}B QO =Q QDW}@|=y|@=QN |va} "CU= 1

� =@ Q@=Q@ |@=QN u}rw= O=ONQ =D |v=tR |xrY=i |[=} Q O}t="OvwW|t `} RwD CN=wvm} CQwY x@ 1

� pY=wi =@ T |v=tR ji= pw] QO |r=wDtO=R=t |v=tR |xR=@ RH x@� CU= 1

� sDU}U CtqU |v=tR |=yxR=@ Q} R pw] u}=Q@=v@12 |x]@=Q j@] xH}Dv QO "�OQ=O 1

� R= QDtm |rw] xm � u=ty =} xQwO |=yDv=O}rwD O=OaD |=vat x@ xm � s=k pwYLt |=Q@ xOW s=Hv= O}rwD pt=m |=yxNQJ O=OaD

%=@ CU= Q@=Q@ |v=tR |=yxR=@ Q} R u}= R= l} Qy QO �CU= s=k |}=yv pwYLt

�m =� 1� � � +

�8k = 1; 2; :::; �k (15)

|=yDv= O=R=t |xR=@ QO s=k pwYLt |=Q@ xOW s=Hv= O}rwD pt=m |=yxNQJ O=OaD w%=@ CU= Q@=Q@ xQwO

��m�

�� � +

�(16)

%=@ CU= Q@=Q@ T |v=tR ji= pw] QO |}=yv s=k pwYLt O}rwD O=OaD u}=Q@=v@qk = #: �m+ ��m 8k = 1; 2; :::; �k (17)

Xk=v |=yxNQJ QOCU= umtt xm|tr=U|}=yv CqwYLt O=OaD R= 17 |x]@=Q QOxv=v}@O@ Cr=L QO |}=yv pwYLt O=OaD qk '|va} &CU= xOW Q_v hQY 'OwW O}rwD"CU= |}xrY=i |OOa R}v qk '14w 13 \@=wQ QO � uOw@ |}xrY=i p}rO x@ "CU=

xiOyOvJ |R=Uxv}y@ "4xiOywO |]NQ}e |R} Qxt=vQ@ pOt l} xr=kt u}= QO |O=yvW}B |R} Qxt=vQ@ pOt|=yVwQ xiOyOvJ |R} Qxt=vQ@ p�=Ut pL |=Q@ "CU= |]N |=yC}OwOLt =@"OvQ=O =Q OwN X=N ?}=at w =}=Rt =yVwQ u}= R= s=Om Qy xm OQ=O OwHw |vwo =vwo=Q} R &CU}v =yVwQ Qo}O =@ Q@=Q@ ,=twRr VwQ Qy R= xOt; CUO x@ ?=wH xmu}= ut[Cw=iDt pL Ov}=Qi QO xOvQ}os}tYD CmQ=Wt u=R}t u}vJty w VwQ Qy C=[wQitxOW xO=iDU= |}=yv pOt pL |=Q@ `t=H Q=}at VwQ R= j}kLD u}= QO [24]"CU=

"CU=

`t=H Q=}at VwQ "1"4|Ov@ C} wrw= x@ |R=}v xiOyOvJ p�=Ut pL Qo}O |=yVwQ hqN Q@ VwQ u}= QOxDU@ '15`t=H Q=}at VwQ "CU}v C}OwOLt x@ h=Oy= p}O@D =} '|yO uRw 'h=Oy=xv}tm =Q u=Wxv}y@ Q=Okt R= h=Oy= |@Uv C=i=QLv=: : :'swO 'pw= u=wD wtHt 'OQwt x@x@ 'CU= xHwD OQwt u; |R=Uxv}tm xQ=wty xm hOy `@=D 'VwQ u}= QO "Ovm|t

%OwW|t h} QaD Q} R CQwY

min Z =kXi=1

�f�i � fif�i

�p(18)

"CU= �Qo}O h=Oy= uDiQo Q_v QO uwO@� s=i hOy `@=D |xv}y@ Q=Okt f�i u; QO xm|va}� Ovv=O|t ?U=vt =Q p = 1 "OQ=O OwHw p Q=Okt |=Q@ |irDNt |=yO=yvW}B|va}� Ovv=O|t QD ?U=vt =Q p = 2 R}v |NQ@ w �OwW xv}tm C=i=QLv= |@Uv wtHt

[24]�"OwW xv}tm C=i=QLv= swO u=wD wtHt

=} QDnQR@ p=tDL= OyO|t u=Wv xm CU= |r=tDL= C}OwOLt l} %swU C}OwOLt(�) u=v}t]= K]U |=yvt 1 R= QDW}@ O}=@ =[=kD x@ C@Uv x[Qa =@ uOw@ |w=UtC}OwOLt l} x@ Q} R h} QaD T=U= Q@ u=wD|t =Q |r=tDL= C}OwOLt u}= "OW=@

[18]%OQm p}O@D |a]k' Dk � N(�dk ; �

2dk ) 'Cr=L u}= QO "CU= p=tQv `} RwD |=Q=O Dk |Dkw

R= Tv=} Q=w Qou=}@ �2Dk w Dk = E(Dk) R= u}ov=}t hQat �dk u; QO xm

x@ Ov=wD|t prob[Dk � qkxk] � 1� � 'u}=Q@=v@ "CU= Dk = V ar(Dk)

CU= OQ=Ov=DU= p=tQv Q=Okt xm OwW xDWwv qkxk � �Dk + :�Dk CQwY=Hu}= QO O}; |t CUO x@ �( ) = � |x]@=Q R= xm

jwi C}OwOLt |=H x@ u}=Q@=v@ "CU= �(z) = 1p2�+1Rzexp�� t2

2

�dt

CN=wvm} `} RwD |=Q=O Dk |Dkw "OwW|t u} Ro}=H qkxk � �Dk + :�Dkprob[Dk � qk:xk] � |r=tDL= C}OwOLt TB Dk � U(lk; uk) 'CU=u; QO xm OwW xDWwv �k � qkxk |a]k C}OwOLt CQwY x@ Ov=wD|t 1 � �"CU= � = 1� � u; QO xm � = uk� + lk� =} �kR

lk

1uk�lk ddk = 1� �

"OwW|t u} Ro}=H qkxk � uk� + lk� jwi C}OwOLt |=H x@ Cr=L u}= QOxOW |L=Q] k pwYLt CN=U |=Q@ xm =Q |QU O}rwD \N l} =@ sDU}U l}xvwoI}y uwO@ |}=yv pwYLt u}rw= |QU O}rwD sDU}U QO "O} Q}o@ Q_v QO 'CU=�\N QO u}W=t n \UwD VR=OQB u=tR wtHt� � =

nPj=1

tjk u=tR |] QO w xikwQ=m u=tR COt 'CU= hrDNt xOW |QU |=yu}W=t wv xm =Hu; R= "OwW|t O}rwDxm |v}W=t 'C}a[w u}= QO "CU= Cw=iDt xDN=Uxt}v pwYLt |wQ Q@ u}W=t QyQo = "OwW|t ?wULt O}rwD \N x=owro 'CU= |QDW}@ VR=OQB u=tR COt |=Q=OpwYLt OwQw w pw= u}W=t uOW |r=N ZLt x@ swO |}=yv pwYLt O}rwD |=Q@CqwYLt R= |iY |y=owro u}W=t CWB QO 'OwW s=Ok= swO u}W=t x@ xDN=Uxt}vuO=O MQ R= ?=vDH= |=Q@ 'u}=Q@=v@ "OW Oy=wN O=H}= VR=OQB Q=_Dv= QO xDN=Uxt}vpwYLt x@ C@Uv |v=tR |xikw l} =@ O}=@ |r=wDt CqwYLt O}rwD '|D}a[w u}vJu}W=t VR=OQB u=tR COt =@ CU= Q@=Q@ xikw u}= u=tR COt "OwW wQW u}W}Bu=tR COt uOw@ Q@=Q@ CQwY QO xm CU= |y}O@ " = max

j;k(tjk) |va} |y=owro

"CU= tjk =} =yVR=OQB u=tR COt Q@=Q@ xikw u}= u=tR COt =yu}W=t |xty VR=OQBu=Wv I =@ =Q u; w s}t=v@ xDUw}B |v=tR |xR=@ l} =Q |@=QN uwO@ |v=tR |xR=@ l} Qo =|xNQJ l} u=tR COt w () |y=owro u}W=t VR=OQB u=tR x@ xHwD =@ 's}yOO}rwD |=vat x@ xm s=k pwYLt |=Q@ xO=O MQ O}rwD pt=m |=yxNQJ O=OaD '(�) O}rwD

%O};|t CUO x@ Q} R |x]@=Q R= 1 CUw}B j@] 'CU= s=k |}=yv pwYLt

m =�1� � +

�8k = 1; 2; :::; �k (12)

Q=JO sDU}UT |v=tRji=l} QO,qwtat xmCU=u}= 'OQ=O OwHwC}ak=w QO xJu; =t=|=yxR=@ O=OaD O}=@ u}=Q@=v@ "Ow@ Oy=wNv xDUw}B |xR=@l} T 'wQ u}= R= &OwW|t|@=QN|=yx]@=Q j} Q] R= syt u}= "OQw; CUO x@ =Q T |R} Qxt=vQ@ ji= QO xDUw}B |v=tR%OwW|t pY=L �CU= xOW K} QWD =yv; |x@U=Lt |xwLv 2 CUw}B QO xm � Q} R# =

�T +MRT1� +MRT

�(13)

� = T � #(1�

+MRT ) (14)

w 'CU= 1� Q@=Q@ ,=k}kO =yv; pw] xm =Q sDU}U CtqU R= |}=yxR=@ O=OaD 13 |x]@=Q

43

"""O}rwD\

N|L=Q]

|=Q@|[

=}Q|rO

t"=yp=Ft pL R= pY=L |}=yv |=yxDi=} "1 pwOHxt=vQ@ |=QH= u=tR sDU}U pm |xv} Ry \N |}=}=B �|}xrY=i� \N |}=}=B

(Xk) \w]N O=OaD O}rwD \N wv p=Ft�x}v=F� �Ow@tm � |Q=Oyov� CU=Q OL AJ OL0 284 118628462 0 83280855 0 839594793 0 8260223 4 1 10 69865527 0 717029406 0 68028114 5 2

58 98 264607959

0 85992699 0 920257227 0 79959675 6 1

20 45233555 0 574807252 0 32986384 7 20 61860443 0 709749954 0 5274589 4 30 73067398 0 803386049 0 65796191 4 40 78001406 0 855185827 0 70484229 5 5

19784 467868052

0 73613995 0 814734641 0 65754527 4 1

3

0 49945006 0 564779755 0 43412036 4 20 69758571 0 786957931 0 60821348 5 30 73067398 0 803386049 0 65796191 4 40 78001406 0 855185827 0 70484229 5 50 64185176 0 716917974 0 56678554 4 60 91194183 0 930244909 0 89363874 5 70 68237598 0 772600593 0 59215138 5 80 69513626 0 715632438 0 67464007 4 90 84678891 0 871979462 0 82159837 4 10

x@ xHwD =@ xt=O= QO "OwW|t x@U=Lt 17 |x]@=Q ltm x@ O}rwD \N Qy |=Q@x@ \w@ Qt C}OwOLt ltm x@ s}tYD |=yQ}eDt u}}=B OL Q=Okt xr�Ut |=yxO=Ou; Q=Okt |=H x@ Q}eDt l} u}}=B OL uO=O Q=Qk =@ "CU= xOW x@U=Lt 10 |x]@=Q|q=@ OL Q=Okt '=yv; l =QDW= u}}aD w 9 w 8 \@=wQ x@ \w@ Qt |=yC}OwOLt QO'CU= K}LY OOa s}tYD |=yQ}eDt TvH xm =Hu; R= "O};|t CUO x@ Qo}O Q}eDt|=Q@ 's}OW xH=wt |Q=Wa= O=Oa= =@ xJv=vJ u}}=B w q=@ OwOL |x@U=Lt s=ovy QOOOa u} QDoQR@ x@ q=@ OL |=Q@ w �q=@ x@� QDnQR@ K}LY OOa u}rw= x@ u}}=B OL|]NQ}e |R=Uxv}y@ TvH R= xr�Ut u}= "s}vm|t OQo �u}}=B x@� QDlJwm K}LYVwQ R= xr�Ut u}= pL |=Q@ "CU= |}xrY=i |=yQDt=Q=B =@ K}LY OOa xiOyOvJVwQ u}= j@] Q@ "CU= xOW xO=iDU= 'OW xQ=W= u=O@ p@k VN@ QO xm `t=H Q=}atQO uO=O Q=Qk =@ w OW x@U=Lt xv=o =OH Qw] x@ hOy `@=wD R= l} Qy |xv}y@ Q=Okt|xv=oQ=yJ |=yC}OwOLt x@ xHwD =@ xiOywO |xr�Ut u}= |}=yv ?=wH 18 |x]@=Q

"CU= xOW xO=O V}=tv 1 pwOH QO xm OW x@U=Lt u;

|Q}oxH}Dv "6=@ K}LY OOa |xiOyOvJ |]NQ}e |R} Qxt=vQ@ pOt l} u=oOvU} wv xr=kt u}= QO|x} Q_v uDiQo Q_v QO =@ O}rwD \N |L=Q] |xr�Ut pL |=Q@ |}xrY=i |=yQDt=Q=B\w]N |R=Ov=x=Q |xQ=@ QO =D Ovm|t ltm u=Q}Ot x@ pOt u}= "Ov=xOQm x�=Q= |}=}=B

|OOa p=Ft "5|L=Q]hrDNt |=yxR=Ov= QO |OOa p=Ft xU '|O=yvW}B pOt Q=@Da= u}}aD Qw_vt x@xOW pL K} QY VQ=tW sD} Qwor= R= xO=iDU= =@ w MATLAB Q=Ri=sQv \UwD wxO=O u=Wv 3 CUw}B w 5 w 4'3 |=ypwOH QO =yp=Ft u}= x@ \w@ Qt C=aq]= "CU=OvDUy O}rwD \N wv 10 w 5 '2 |=Q=O ?}DQD x@ 3 w 2 '1 |=yp=Ft "CU= xOWVR=OQB u=tR "CU= xOW u}}aD hrDNt \w]N |=Q@ u}W=t 4 xv}W}@ w 2 xv}tm xm|}xR=@ CQwY x@ |@=QN MQv w CU= xOW xDiQo Q_v QO xk}kO x@ =yu}W=t |=Q@|=Q@ R=}v OQwt |xHOw@ w =[i =yp=Ft R= l} Qy QO O}rwD \N Qy |=Q@ "Ow@ Oy=wN|=yQDt=Q=B Qo}O R= R}v =[i w xHOw@ pm "CU= xOW xDiQo Q_v QO l} Qy |R=Ov=x=QhrDNt CqwYLt |=Q@ =[=kD `} RwD xmu}= ut[ "OvDUy p�=Ut u}= QO ZwQitp@=k 5 w 4 |=ypwOH QO xm CU= xOW xDiQo Q_v QO p=tQv =} CN=wvm} `} RwD|R} Qxt=vQ@ |xQwO 'u=v}t]= K]U '=yq=m Ow@tm w |Q=Oyov |xv} Ry "CU= xOy=Wt"CU= xOW xDiQo Q_v QO u=Um} xr�Ut xU QO |@=QN wkw R= TB \N Q}taD u=tR wQ@ "s}DUy CqwYLt O}rwD p=@vO x@ QtDUt Qw] x@ Q_v OQwt |R} Qxt=vQ@ |xQwO QOuOQm xv}tm hOy =@ wv Qy R= =Q xv}y@ \w]N O=OaD s}y=wN|t =yxO=O u}= T=U=CUO x@ sDU}UQ} R Qy |}=}=B uOQm xv}W}@ w |Q=Oyov w Ow@tm |=yxv} Ry wtHt

"s} Qw;x@U=Lt s}tYD |=yQ}eDt |=Q@ =Q u}}=B w q=@ OL O}=@ =OD@= 'p�=Ut pL |=Q@Q=Okt u}= "OQm x@U=Lt s=k O}rwD \N |=Q@ =Q qk Q=Okt O}=@ =OD@= Qw_vt u}O@ "OQm

44

1 2|xQ

=tW'35

�1|xQwO

'�139

8u=DU@

=D�h}QW

C}Q}Ot

w`}=vY

|UOvyt Oy=wN Ot;Q=m =v xr�Ut O=a@=V}=Ri= CQwY QO xmCU= xOW xO=iDU= K} QY VQ=tW

xaUwD |Q=mD@==Qi |=yVwQ R= u=wD|t |D; C=k}kLD |=Q@ |O=yvW}B u=wva x@ w Ow@u}vJty "OQm xO=iDU= : : :w C=QP s=LOR= |R=Uxv}y@ 'l}DvS sD} Qwor= Ovv=t xDi=}

"OwW xO=O xaUwD R}v O}rwD \w]N C=a}=[ uDiQo Q_v QO =@ Ov=wD|t pOt u}=

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=yCWwv=B1. exact optimization methods2. Wei-Chang Yeh3. particle swarm optimization4. deterministic5. Bennett & Jenny6. Chung, Wu & Chang7. Kumral8. Vasudevan et al9. bottleneck10. Bhunia, et al11. Sahoo, et al12. symmetric13. re exive14. transitive15. global criterion method

�References� `@=vt1. Stevenson, W. J. Production/operations management,

Tom Casson, USA (1996).

2. Lin, Y.-K. and et al. \Reliability evaluation for a waste-reduction parallel-line manufacturing system", Journalof Cleaner Production, 35, pp. 93-101 (2012).

3. Liberopoulos, G. and Tsarouhas, P. \Reliability analysisof an automated pizza production line", Journal of FoodEngineering, 69, pp. 79-96 (2005).

4. Bennett, D. and Jenney, B. \Reliability: its implicationssystems design in production ",OMEGA, 8(4), pp. 433-440 (1980).

5. Thoft-Christensen, P. and Murotsu, Y. Application ofStructural Systems Reliability Theory Springer, Berlin,Heidelberg (1986).

6. Garg, H. and et al. \Intuitionistic fuzzy optimizationtechnique for solving multi-objective reliability optimiza-tion problems in interval environment", Expert Systemswith Applications, 41, pp. 3157{3167 (2014).

7. Yeh, W.-C. \A two-stage discrete particle swarm opti-mization for the problem of multiple multi-level redun-dancy allocation in series systems", Expert Systems withApplications, 36, pp. 9192-9200 (2009).

8. Chung, S.-H. and et al. \An e�cient tabu search al-gorithm to the cell formation problem with alternativeroutings and machine reliability considerations", Com-puters & Industrial Engineering, 60, pp. 7-15 (2011).

9. Das, K. and et al. \Reliability consideration in the designand analysis of cellular manufacturing systems", Int. J.Production Economics, 105, pp. 243-262 (2007).

10. Kumral, M. \Reliability-based optimisation of a mineproduction system using genetic algorithms", Journal ofLoss Prevention in the Process Industries, 18, pp. 186-189 (2005).

11. Vasudevan, K. and et al. \Application of simulation todesign and operation of steel mill devoted to manufac-ture of line pipes", 2010 Second International Confer-ence on Advances in System Simulation, pp. 1-6, IEEE(2010).

12. Diallo, M. and et al. \ Manufacturing cell design with exible routing capability in presence of unreliable ma-chines, International Journal of Production Economics,74(1-3), pp. 175-182 (2001).

13. JabalAmeli, M. S. and Arkat, J. \Cell formationwith-alternative process routings and machine reliability con-sideration", International Journal of Advanced Manufac-turing Technology, 35, pp. 761-768 (2008).

14. JabalAmeli, M. S. and et al. \Modelling the e�ects ofmachine breakdowns in the generalized cell formationproblem", International Journal of Advanced Manufac-turing Technology, 39, pp. 838-850 (2008).

15. Savsar, M. \Reliability analysis of a exible manufactur-ing cell", Reliability Engineering & System Safety, 67,pp. 147-152 (2000).

16. Zakarian, A. and Kusiak, A. \Modeling manufacturingdependability", IEEE Transactions Robotics and Au-tomation, 13(2), pp. 161-168 (1997).

17. Logendran, R. Talkington, D. \Analysis of cellular andfunctional manufacturing system in the presence of ma-chine breakdown", International Journal of ProductionEconomics, 53(3), pp. 239-256 (1997).

18. Bhunia, A. K. and et al. \Reliability stochastic optimiza-tion for a series system with interval component relia-bility via genetic algorithm", Applied Mathematics andComputation, 216, pp. 929-939 (2010).

45

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19. Gupta, R. K. and et al. \ A GA based penalty functiontechnique for solving constrained redundancy allocationproblem of series system with interval valued reliabilitiesof components", Journal of Computational and AppliedMathematics, 232, pp. 275-284 (2009).

20. Colledani, M. and Yemane, A. \Impact of machine reli-ability data uncertainty on the design and operation ofmanufacturing systems", Procedia CIRP, 7, pp. 557-562(2013).

21. Sahoo, L. and et al. \Genetic algorithm based multi-objective reliability optimization in interval environ-

ment", Computers & Industrial Engineering, 62, pp.152-160 (2012).

22. SeyedEsfahani, M. and et al. \A simulated annealingalgorithm for fuzzy reliability optimization model inseries-parallel and standby systems", International Jour-nal of Industrial Engineering & production Management,24(4), pp. 413-422 (2014).

23. Sadjadi, S. J. and et al. \A new nonlinear stochasticsta� scheduling model", Scientia Iranica TransactionsE: Industrial Engineering, 18(3), pp. 699-710 (2011).

24. Momeni, M., New topics of Operations Research, 2th edi-tion, Tehran University Press, Tehran (2008).

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