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xiOyOvJ C=QP |R=Uxv}y@ sD} Qwor= R= xO=iDU=�OWQ= |U=vWQ� |wHWv=O� |rO=@ T=@a
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O}=@ xm CU= |}x}rkv p}=Uw |=Q@ xv}y@ |=yQ}Ut uDi=} hOy '|@=} Q}Ut |xr�Ut QO|@=} Q}Ut |xr�Ut Q=DWwv u}= QO "Ovvm x�=Q= xv} Ry u} QDsm =@ =Q u=} QDWt R=}v OQwt C=tON
|xv} Ry w uwoty=v pkvwptL |xv} Ry |R=Uxv}tm hOy =@ uwoty=v pkvwptL u=ow=v =@Q=}at =@ xOW |UQQ@ |xr�Ut "CU= xDiQo Q=Qk |UQQ@ OQwt pkvwptL p}=Uw |Q=Oxov
|=yu=tR w QiU u=tR wtHt |R=Uxv}tm w uwoty=v pkvwptL u=ow=v =@ |@=} Q}Ut xiOywOxiOyOvJ C=QP xw@v= |R=Uxv}y@ |Q=mD@==Qi VwQ R= Qw_vt u}= |=Q@ "CU= xOW pL Q}N-=D"CU= xOW pL nQR@ w lJwm |xr�Ut xDUO wO |=R=x@ xr�Ut u}= "CU= xOW xO=iDU=s=Hv= lJwm p�=Ut |=R=x@ w Rto Q=Ri=sQv R= xO=iDU= =@ |O=yvW}B |xr�Ut |HvUQ=@Da=
"CU= |O=yvW}B sD} Qwor= |q=@ |}=Q=m Qou=}@ xar=]t u}= R= pY=L G}=Dv "CU= xOW
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'p} wLD Oawt 'uwoty=v pkvwptL u=ow=v 'x}rkv p}=Uw |@=} Q}Ut %|O}rm u=oS=w"xiOyOvJ C=QP xw@v= |R=Uxv}y@ sD} Qwor= 'xiOywO |R=Uxv}y@
xtOkt "1CU= �|@=} Q}Ut |xr�Ut pL QO u=Qm w xN=W sD} Qwor= OQ@ Q=m� 'xRwL u}= QO71 =@ |}xr�Ut |=Q@ sD} Qwor= u}= R}t;C}kiwt |=QH= x@ 1994 p=U QO QW}i xmw urwm '|v=tR |xQHvB =@ x=Qty |@=} Q}Ut |xr�Ut |=Q@ u}vJty "Ot; p}=v |QDWtu}= |=Q@ =Q u; w OvOW u=Qm w xN=W sD} Qwor= |=QH= x@ jiwt �1987� Vv=Q=mty=@ |r�=Ut |=Q@ =ysD} Qwor= R= xDUO u}= xm CU= QmP x@ sRq "OvDiQo Q=m x@ xr�Utu}= QO =Pr "CU= Q@u=tR Q=}U@ xRwL u}= QO =yv; pL w CU}v ?U=vt QDnQR@ O=a@=|Q=mD@= |=ysD} Qwor= R= O}=@ QD`} QU w xv}y@ ?=wH l} x@ |@=}CUO |=Q@ =yxvwtv
[2]"CiQo xQy@%OvwW|t KQ]t xOta pmW wO x@ |@=} Q}Ut QO |v=tR |=yxQHvB ,qwtat
xR=@ |xOwOLt QO =yvD x}rkv |xr}Uw Cr=L u}= QO %CNU |v=tR |=yxQHvB "1|=yxr=kt QO xr�Ut u}= "OwW |QDWt pLt OQ=w CW=OQ@ =} x}rND |=Q@ Ov=wD|tw |D=vwO 'u=Q=mty w |U}=Q@ '|RDw}ro}i 'QwBO=vk w |Q}Yk '=} Q=v}rw@ w =QH=v
"CU= xOW xDiQo Q_v QO u=Q=mty|=yxQHvB R= GQ=N QO x}rkv |xr}Uw C}a[w u}= QO %sQv |v=tR |=yxQHvB "2|v=UQCtON |=Q@ xm Cw=iD u}= =@ 'Ovm|t |v=UQCtON u=} QDWt x@ |v=tR'|RDw}ro}i Cq=kt "OwW|t xDiQo Q_v QO xt} QH |v=tR |xQHvB R= GQ=N QO
"CU= Q=wDU= =v@t u}= Q@ u=Q=mty w wDwt}W=y 'nv=w w nvJ 'Qrwt"OQ}o|t Q=Qk xO=iDU= OQwt |Y=N \}=QW =@ |v=tR |=yxR=@ |y=o 'u}= Q@ xwqaQO p=Ft u=wvax@ &OwW|t xDiQo Q_v QO xr�Ut |=Q@ xiQ]l} |v=tR |xQHvB R}v |y=o
\UwD �uw}t=m `} RwD� u=wvaCLD |@=} Q}Ut |xr�Ut Q=@ u}rw= |=Q@ 1959 p=U QO=QDUvr '�1979� uwUv=H w |Qo\UwDTBU xr�Ut u}= "OW KQ]t QUt=Q wl} RDv=O�1989� Tm =HO w xDQwBq w '�1987� nv}Dlt w wv=} '�1981� u=m |wv} Q wNP-hard p�=Ut |xOQ QO xr�Ut u}= xm OW xO=O u=Wv w CiQo Q=Qk |UQQ@ OQwt|v=tR |oO}J}B l} =@ Ov=wD@ xm O=O x�=Q= |tD} Qwor= u=wD|tv 'wQu}= R= w OQ}o|t Q=Qkw |Q=mD@= |=ysD} Qwor= Cra u}tyx@ "Ovm pL =Q xr�Ut |}xrtHOvJ |xHQO R=
[1]"CU= xOW xO=O xaUwD xr�Ut u}= pL |=Q@ |Q=}U@ |Q=mD@==Qiw xDQwBqO '�1999� Vv=Q=mty w w} QOvo '�2003� xDQwBq w wvOQwm |=yQ=m QOx@ |@=} Q}Ut |xr�Ut pL |=ysD} Qwor= '�2001� w} QOvo w |U}=Q@ '�1999� CtUw 'l}Uqm |Q=mD@= |=ysD} Qwor= 'j}kO |=ysD} Qwor= %CU= xOW s}UkD xDUO xUKQ]t =Q |@=} Q}Ut |=Q@ |Q=mD@= |=ypOt �2003� pR}y "|Q=mD@==Qi |=ysD} Qwor=KQ]t |@=} Q}Ut |=Q@ =Q |twta |=yVwQ R}v �2005� xmB wQ w Qov}UB w 'OQm|@=} Q}Ut |xr�Ut pL |=ysD} Qwor= T}U=w w K@U R}v 2002 p=U QO "OvOQm|=ysD} Qwor= uwUvwQ; 1995 p=U QO 'xt=O= QO "OvOQm u=}@ =Q |v=tR |xQHvB =@ x=Qtyj}kO |=ysD} Qwor= x=oO}O "OQm u=wva `t=H |xar=]t l} ?r=k QO =Q xr�Ut pLVv=Q=mty w RO}=iwDU} Qm '�1995� |Rwov}t w RO}=iwDU} Qm |=yQ=m QO u=wD|t =QhwQat j}kO |=ysD} Qwor= R= |m} "OQm xOy=Wt �2002� wo} w w TwD w �1981�
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xr�Ut u=}@ "2|yOT} wQU |=Q@ x}rkv |xr}Uw w Q}Ut ?=NDv= |xr�Ut |UQQ@ =t Q_v OQwt pOtpkvwptL u=ow=v "Ov=xOW xOvm =QB wBO l} pwL xm CU= |v=} QDWt |xawtHt x@|=yxv} Ry =@ Cw=iDt |xr}Uw wvK 'CU= uwoty=v u=} QDWt x@ |yOT} wQU |=Q@OwOLt wv Qy R= OwHwt p}=Uw O=OaD w 'TQDUO QO hrDNt QiU u=tR w |Q=OxovOawt u; R= TB x}rkv |xr}Uw Qo = xm CU= QQkt Oawt l} |=Q=O |QDWt Qy "CU=|yOT} wQU u=tR |QDWt Qy u}vJty "CU= xO=O MQ Q}N-=D 'OUQ@ |QDWt x@ QQkt|va} &CU= uQ=kDt=v wv R= =yxQo u}@ QiU u=tR T} QD=t "OQ=O =Q OwN x@ XwYNt|QDWt x@ j |QDWt R= QiU u=tR R= Cw=iDt j |QDWt x@ i |QDWt R= QiU u=tR\}=QW R= |Q=}U@ w =yu=@=}N R= |Q=}U@ uOw@ xiQ]l} p}rOx@ ZQi u}= "CU= i"CU= xOW p=ta= |ak=w \}=QW x@ xr�Ut uOW QDl}ORv xJQy |=Q@ w |ak=w |=}vO'|Q=Oxov |xv} Ry u} QDsm xm CU= |}xvwox@ pkvwptL u=ow=v w Q}Ut uDi=} hOypY=L u=} QDWt u}@ QiU u=tR u} QDsm R}v w '|yOT} wQU QO Q}N-=D u=tR u} QDsm
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xr�Ut |=yQDt=Q=B "1"2%OwW|t h} QaD Q} R |=yQDt=Q=B =OD@= pOt u=}@ Qw_vtx@
&f1; :::; Ng u=} QDWt |xOvyOu=Wv %i; j&k |xr}Uw \UwD j |QDWt x@ i |QDWt R= QiU |=Q@ R=}v OQwt u=tR %tijk
&k |x}rkv |xr}Uw |Q=Oxov |xv} Ry %hk&k |x}rkv |xr}Uw C}iQ_ %Vk
&i |QDWt |=[=kD %di&i |QDWt x@ k |xr}Uw uO}UQ u=tR %aik
&=[=kD Ci=} QO |=Q@ i |QDWt QQkt Oawt %mi
&i |QDWt x@ |yOT} wQU u=tR %Si&k wv R= OwHwt |xr}Uw O=OaD %zk
QO w '1 Q@=Q@ s}tYD Q}eDt 'Ovm QiU j |QDWt x@ i |QDWt R= k |xr}Uw Qo = %xijk&CU= QiY Q@=Q@ CQwYu}= Q}e
&OUQ|t i |QDWt x@ xm |t=ovy k xr}Uw Q}N-=D u=R}t %yik&xr�Ut pw= hOy `@=D %OB1"xr�Ut swO hOy `@=D %OB2
|[=} Q pOt "2"2|xr�Ut |=Q@ |O=yvW}B |[=} Q pOt p@kCtUk QO xOW x�=Q= |=yQDt=Q=B x@ xHwD =@
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OB1 =NXi=1
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|[a@ QO "CU= xOW xO=iDU= xr�Ut |=Q@ p} wLD Oawt R= u=Q=mty w nv=m |xr=ktl} R= |v=tR |=yxQHvB ,qwtat C}a[w u}= QO &CU= sQvxt}v |v=tR |xQHvB \}=QW|xDi=}s}taD xm CU= |R=i |v=tR |xQHvB 'Qo}O Cr=L "OvQ=O Rw=HD u=mt= CtU
[3]"CU= QDOwOLt u; x@ C@Uv xm CU= R=@ |v=tR |=yxQHvBwQW |x]kv u}OvJ =@ |k}@]D |x_i=L |R} Qxt=vQ@ VwQ l} u=Q=mty w |rO}rwD |=Q@ |Q=mD@= VwQl} |k}@]D |x_i=L |x} wQ '=yu; VwQ QO "OvOQm O=yvW}B|wHwCUH sD} Qwor= l} 'x} wQ u}= uOQm p=@vO =@ "OQ}o|t Q=m x@ p}=Uw |=yQ}UtOvJ pOt QO "CU= xOW |L=Q] =y?=wH C}i}m Ow@y@ |=Q@ xOWKqY= wvttC}=yv QO "OW Oy=wN Q=QmD '=yQ=QmD R= |v}at O=OaD |=Q@ x} wQ u}= wQW x]kv=Q 3-Opt w 2-Opt |=y|}=Hx@=H R= |QU l} xm 1|R=Uxv}y@=UB |x} wQ l}Q@ xwqa "CU= xDiQo Q=Qk xO=iDU= OQwt ?=wH u} QDy@ uDi=} |=Q@ 'OOv@|t Q=m x@|xaUwD |=Q@ 2Q}Ut OOHt p=YD= |Q=mD@==Qi VwQ l} R= u=Q=mty w |r 'u}=l} [4]"Ov=xDiQo xQy@ u=Q=mty w Qwwro C=ar=]t QO |O=yvW}B sD} Qwor= OQmrtap}=Uw |@=} Q}Ut |xr�Ut pL |=Q@ w=Ov=Q@ \UwD wvtt wHwCUH |Q=mD@==Qi VwQ|tD} Qwor= 'VwQ u}= QO "CU= xOW xO=O xaUwD 'uwoty=v pkvwptL u=ow=v =@ x}rkvx@ QHvt =yCr=L |NQ@ QO xm wvtt wHwCUH |rY= <=RH= |t=tD Q@ |vD@tu=Q=mty w =vB [5]"OW xO=O O=yvW}B 'OW|t |r@k G}=Dv u} QDy@ QO QDW}@ |=yOw@y@w Q}eDt xDU=Q; |o}=Uty |x} wQ =@ xOW ?}mQD |Q=QmD |rLt |wHwCUH l}|@=} Q}Ut `=wv= pL |=Q@ =Q 3�ILS-RVND� |iO=YD |o}=Uty |R=U?DQt|Qq=U w |t}_a |H=v Q=m QO [6]"Ov=xOQ@ Q=m x@ uwoty=v u=ow=v =@ x}rkv p}=Uw|xr�Ut pL |=Q@ K}LY OOa |]N |R} Qxt=vQ@ |x}=B Q@ |Q=mD@= OQm} wQ l}u; R= u=wD|t xm OW O=yvW}B uwoty=v pkvwptL u=ow=v =@ x}rkv |xr}Uw |@=} Q}Ut'p�=Ut u}= pL QO OwHwt |=yVwQ OQmrta Ow@y@ |=Q@ |r}tmD |Q=R@= u=wvax@|@}mQD |rt=mD |rLt |wHwCUH VwQ l} u=Q=mty w pt=yv=O [7]"CiQo xQy@pkvwptL u=ow=v =@ x}rkv |xr}Uw |@=} Q}Ut |=Q@ OwHwt |=yxvwtv |t=tD Q@ ptDWtQwWm QO =yQyW u}@ |ak=w pY=wi R= OwN Q=m QO =yu; "OvO=O O=yvW}B uwoty=v|@=} Q}Ut p�=Ut R= |}xvwo u=Q=mty w |r [8]"OvOQm xO=iDU= QDtwr}m T=U=Q@ xUv=QiwQW wBO R= x}rkv |xr}Uw u; QO xm OvOQm |UQQ@ =Q uwoty=v u=ow=v =@ x}rkv p}=UwwBO x@� Ov=UQ|t u=}=B x@ =Q OwN Q=m u=} QDWt \=kv R= |m} QO w Ovm|t CmQL x@|x_i=L |R} Qxt=vQ@ |Q=mD@==Qi VwQ R= xr�Ut u}= pL |=Q@ =yv; "�CWo Oy=wNvR=@KqY= wvtt |wHwCUH sD} Qwor= R= xO=iDU= =@ ' wQW x]kv u}OvJ =@ |k}@]D
[9]"OvOQm xO=iDU= 'xOWQO =Q uwoty=v u=ow=v =@ x}rkv p}=Uw |@=} Q}Ut |xr�Ut u=Q=mty w nvw}|rQiU |xv} Ry |R=Uxv}tm xr�Ut u}= hOy "OvOQm |UQQ@ |Oa@ wO |Q}oQ=@ Cr=L|O} Q@D |R=Ux}@W VwQ R= xr�Ut u}= pL |=Q@ =yu; "CU= XNWt |=yQ}UtuO=O CaQU Qw_vtx@ "OvOQm xO=iDU= |rLt |wHwCUH |Q=mD@= sD} Qwor= =@ x=Qty|Q}PBu=mt= x@ \w@ Qt C=aq]= C@F |=Q@ xO=O Q=DN=U l} R= wHwCUH Ov}=Qi x@|Q=Oxov |xv} Ry Q}F-=D p=L@ =D w[wt C=}@O= x@ xHwD =@ [10]"OvOQm xO=iDU= |Q}oQ=@p} wLD QQkt Oawt u=tRty uDiQo Q_v QO w 'x}rkv |xr}Uw w Q}Ut ?=NDv= |xr�Ut Q@'|OQ@ Q=m |=ypOt x@ QDW}@ xJQy |m}ORv |=Q@ Q=DWwv u}= QO =Pr "CU= xOWv |UQQ@w 'xOW xDiQo Q_v QO uwoty=v u=ow=v =@ x}rkv |xr}Uw w Q}Ut ?=NDv= |xr�Utu}vJty "CU= xOW p=ta= QiU u=tR w Cw=iDt |Q=Oxov |=yxv} Ry Q@ u; Q}F-=D|QDWt Qy |=Q@ w xOW |UQQ@ xr�Ut ?=wH Q@ |Q=Oxov |=yxv} Ry C=Q}}eD Q}F-=DsD} Qwor= R= xO=iDU= =@ xr�Ut u}= "CU= xOW xDiQo Q_v QO p} wLD QQkt Oawt l}
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VWwB =Q =ypLQo}O xm OvDUy |}=ypL ?wret=v |=ypL Qo}O CQ=@ax@QO p=Ft u=wvax@ "OvwW|tv xO=O VWwB =ypL u; \UwD u=WOwN |rw OvyO|txOW xO=O u=Wv ?wretQ}e |=ypL \UwD =ypL Q}=U VWwB |ovwoJ 1 pmW
"CU=xO=O u=Wv Q_v OQwt |x}rkv p}=Uw |@=} Q}Ut |xr�Ut pOt 'p@k CtUk QOp�=Ut <RH x}rkv p}=Uw |@=} Q}Ut |xr�Ut xm xOW u=}@ u}vJty "CU= xOWnQR@ O=a@= QO p�=Ut u}= pL x@ QO=k j}kO pL |=yVwQ w CU= NP-Hard
xr�Ut u}= pL |=Q@ Qw_vt u}tyx@ "OvDU}v |ak=w |=}vO p�=Ut pL u}vJty w|xiOyOvJ sD} Qwor= R= Q=DWwv u}= QO "OQm xO=iDU= O}=@ |Q=mD@==Qi |=ysD} Qwor= R=xiOyOvJ C=QP |R=Uxv}y@ sD} Qwor= "CU= xOW xO=iDU= C=QP xw@v= |R=Uxv}y@wDQ=B OQm} wQ |=v@t Q@ 1999 p=U QO 4utB=J w Qwt \UwD Q=@u}rw= �MOPSO�"Ow@ C=QP |OQi w |ywQo |wHwCUH Q@ u=v; sD} Qwor= |rY= O}m -=D "CU= xOW =v@?}mQD R= xm OvO=O xaUwD C=QP sD} Qwor= x@ x}@W |tD} Qwor= 5w}r w |Q xt=O= QO
"OQm xO=iDU= C=QP xw@v= =@ 6|rt=aD |=yl}vmD swyit w wDQ=B OQm} wQl} w CaQU Q=OQ@ l} =@ Ov=wD|t xwQo R= s=i |xQP '|Oa@ M |=[i l} QOw C}akwt Q=DN=U QO Q}}eD =@ xQP Qy C}akwt Q}}eD "OwW xO=O u=Wv C}akwt Q=OQ@CU= |Q=Okt u} QDy@ pt=W |D=aq]= |xQP Qy "CU= Q}PBu=mt= V=|r@k CaQUC=aq]= u}= "OQ=O =Q Xt
i C}akwt w �7|YNW |xv}y@� xO}UQ u; x@ uwvm =D xm"OyO|t s=Hv= ?=wH u} QDy@ uDi=} |=Q@ xQP Qy xmCU=|}=yVqD |xU}=kt pY=L|xU}=kt R= 'xOt; CUOx@ xwQo pm QO uwvm =D xm =Q |@=wH u} QDy@ xQP Qy u}vJtyx@ uO}UQ |=Q@ xQP Qy "�8Q}o =Qi |xv}y@� OU=vW|t hrDNt C=QP |xv}y@ Q}O=kt%OyO Q}}eD Q} R C=aq]= R= xO=iDU= =@ =Q OwN C}akwt Ovm|t |aU ?=wH u} QDy@
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OB2 =KXk=1
hkzk (1)
NXi=0
KXk=1
xijk = 1 j = 1; 2; :::; N (2)
NXj=0
KXk=1
xijk = 1 i = 1; 2; :::; N (3)
NXj=1
x0jk � zk k = 1; 2; :::;K (4)
NXi=0
xipk =NXj=0
xpjk p = 0; 1; 2; :::; N k = 1; 2; :::;K (5)
NXi=1
diNXj=0
xijk � vkzk k = 1; 2; :::;K (6)
yik =NXj=1
xijk(aik �mi) j = 1; 2; :::; N k = 1; 2; :::;K (7)
a0k = 1 (8)
ajk =NXi=1
xijk(aik + si + tijk)
j = 1; 2; :::; N k = 1; 2; :::;K (9)NXj=1
x0jk � r(s)
xijk 2 f0; 1g (10)
|R=Uxv}tm x@ pw=VN@ QO xmCU= xr�Ut hOy `@=D Qou=Wv '1 |x]@=Q q=@ pOt QO|QDWt x@ x}rkv |xr}Uw xm |t=ovy �� Q}N-=D pm u=tR w x}rkv p}=Uw QiU pm u=tR"OR=U|t xv}tm =Q x}rkv p}=Uw |Q=Oxov |xv} Ry R}v swOVN@ w OR=OQB|t �� OUQ|t|xr}Uw l} \UwD \ki O}=@ |QDWt Qy x@ xm CU= xOW u=}@ 3 w 2 |=yx]@=Q QO"CU= xOW u=}@ x}rkv p}=Uw O=OaD C}OwOLt 4 |x]@=Q QO "OwW xO=O T} wQU x}rkvQo = 'CU= xOW xO=O u=Wv 5 |x]@=Q \UwD u=} QDWt |xm@W QO u=} QH ^iLC}iQ_ 6 |x]@=Q \UwD "OW Oy=wN GQ=N u; R= ,=tDL OwW OQ=w |yQo x@ u}W=t7 |x]@=Q \UwD |QDWt Q}N-=D u=R}t |x@U=Lt "OW Oy=wN |UQQ@ x}rkv |xr}UwxO=O u=Wv 8 |x]@=Q =@ R}v wBO R= p}=Uw GwQN |=Q@ x}rw= Q=Okt w 'OwW|t x@U=LtQO =yQwD Q} R hPL '9 |x]@=Q QO |QDWt x@ x}rkv |xr}Uw OwQw u=tR "CU= xOW
"CU= xOW u=wva 10 C}OwOLt
xr�Ut pL |Q=mD@==Qi VwQ "3"Ovvm|t xO=iDU= wHwCUH QO x@re swyit R= xiOyOvJ|@=}xv}y@ |=ysD} Qwor=?re=
"s} R=OQB|t x@re swyit h} QaD x@ =t =Hv}= QOwO x1; x2 2 F w OW=@ xr�Ut xHwt |=[i pm F O}vm ZQi %?wret w ?r=e�CU= x1 ?wret x2 =}� CU= ?r=e x2 Q@ x1 s}} wo|t "OW=@ xr�Ut u}= R= ?=wH
%OW=@ Q=QkQ@ Q} R \QW wO Qo = \ki w Qo =|va} "�CU= Q@=Q@ =} QDy@� CU}v QDO@ h=Oy= R= s=OmI}y QO x2 x@ C@Uv x1 "1
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QO w 2 |QDWt x@ TBU 5 |QDWt x@ =OD@= �pw= Q]U� pw= u}W=t p=Ft Qw]@w 1 |QDWt x@ =OD@= �swU Q]U� 3 u}W=t u}vJty "OwQ|t 3 |QDWt x@ C}=yv
"OwW|tv |}xO=iDU= sy swO u}W=t R= &OwQ |t 4 |QDWt x@ TBU
|O=yvW}B sD} Qwor= |=yQDt=Q=B s}_vD "2"3u}}aD =]N w |aU =@ C=W}=tR; |=QH= =@ |O=yvW}B VwQ |=yQDt=Q=B xr=kt u}= QO
"CU= xOW xO=O V}=tv 2 pwOH QO =yQDt=Q=B u}= Q}O=kt "CU= xOW
|D=@U=Lt G}=Dv "4|xwLv |UQQ@ x@ uwvm = 'u; QDt=Q=B s}_vD w |O=yvW}B sD} Qwor= Q=m |xwLv KQW R= Oa@"s} R=OQB|t Q=DWwv u}= QO |UQQ@ OQwt |xr�Ut =@ xyH=wt QO sD} Qwor= u}= OQmrta|=y?=wH uOw@ l}ORv w uOw@ CUQO u=R}t |UQQ@ VN@ u}= hOy `k=w QOu}vJty &CU= |UQQ@ OQwt p�=Ut |rm |xv}y@ ?=wH x@ sD} Qwor= u}= R= pY=L|R=Uxv}y@ |Q=mD@==Qi sD} Qwor= ?=wH =@ Gams Q=Ri=sQv R= xOt;CUO x@ |=y?=wH|=Q@ LP-Metric VwQ R= xr�Ut j}kO pL VwQ QO "OwW|t xU}=kt C=QP xw@v=xiOywO |xr�Ut l} VwQ u}= T=U=Q@ "CU= xOW xO=iDU= xr�Ut uOQm xiOylDxHwD =@ TBU w OwW|t pL =OH CQwYx@ `@=wD R= |m} uDiQo Q_v QO uwO@ =OD@=OQ=Ov=DU= wtHt xm OwW|t xrwtQi |R=Uxv}tm hOy `@=D l} xv}y@ Q}O=kt u}= x@pOt OQwtQO "OR=U|t xv}tm =Q u=Wxv}y@ Q}O=kt w hOy `@=wD u}@ hqDN= |xOWs=OmQy |xv}y@ Q}O=kt w OB2 w OB1 =@ hOy `@=wD Q}O=kt 'xr=kt u}= |O=yvW}B`@=D p=L "OwW|t xO}t=v OB�2 w OB�1 'Qo}O hOy `@=D ?=}e QO hOy `@=wD R=
%OwW|t |Ov@pwtQi u}vJ xr�Ut LP-Metric hOy
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+W2:OB2 �OB�2
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�(14)
|=Q@ hOy `@=wD R= l} Qy xm |D}ty= x@ xHwD =@ w x=wNrO x@ W2 w W1 Q}O=kt"OwW 1 =yv; wtHt xm |}xvwox@ 'OW Ovy=wN ?=NDv= 'OQ=O xOvQ}os}tYD
Q=Ri=sQv 23 5 1 |xNUv QO MINLP |xOvvmpL R= xO=iDU= =@ xvwtv p�=Ut|}xv=}=Q QO 'xOW |U} wvxt=vQ@ 2012b ?rDt Q=Ri=sQv QO |O=yvW}B sD} Qwor= w RtosDU}U QO C}=@=o}o 4 |rY= x_i=L w RDQy =o}o 2 40 w |}xDUyGvB |xOvR=OQB =@|=y?=wH |xU}=kt |D=@U=Lt G}=Dv "Ov=xDiQo Q=Qk V}=tR; OQwt 7 RwOv} w pt=a
"CU= xOt; 3 pwOH QO C=QP xw@v= |R=Uxv}y@ sD} Qwor= w Rto
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12 w 11 \@=wQ j@=]t O}OH C}akwt u; `@D x@ w xQP Qy CaQU ?}DQD u}O@%Ovm|t Q}}eD
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Q=QmD QO i |xQP CaQU V ti 'O}OH Q=QmD QO i |xQP CaQU V t+1i u; QO xm
u} QDy@ pbesti 'O}OH Q=QmD QO xQP C}akwtXt+1i 'xQP |vwvm C}akwtXt
i '|rais=tD xm CU= |D}akwt u} QDy@ gbestt w 'xOQm Q=}DN= uwvm =D i |xQP xm |D}akwtCU= 1 w QiY u}@ |iO=YD OOa l} Rand(0; 1) "Ov=xOQm Q=}DN= uwvm =D C=QP?}DQDx@ c2 w c1 |=yQDt=Q=B "OwQ|t Q=m x@ xwQo |vwo =vwo w wvD ^iL |=Q@ xm
"CU= |atH w |OQi |Q}oO=} ?} Q[xaUwD |} wDQ=B ?=wH O}rwD |=v@tQ@ C=QP xw@v= |R=Uxv}y@ |xiOyOvJ sD} Qwor=u} QDy@ u=wvax@ |}xQP xJ xm CU= u; sD} Qwor= u}= QO xr�Ut u}rw= "CU= xDi=}|}xawtHt VwQ u}= QO Qw_vt u}= |=Q@ "OwW ?=NDv= C=QP |x}k@ C}=Oy |=Q@ xQPx@ ?r=e |=y?=wH |t=tD xm CU= xDi=} VQDUo �REP� `HQt |xawtHt s=v x@s=ovy QO "OwW|t |Q=Oxov u; QO '?=wH |=[i |wHwCUH u}L QO xOt; CUO`@=wD |=[i pm '?r=e |=y?=wH u}@ R= ?=wH u} QDy@ ?=NDv= Qw_vtx@ |v=UQRwQx@Q}O=kt x@ xHwD =@ ?r=e |=y?=wH w OwW|t s}UkD |w=Ut |=yCtUk x@ hOy
"OvQ}o|t |=H =yCtUk R= l} Qy QO hOy `@=wD|=Q@ "OwW|t |v=UQRwQx@ xawtHt ?r=e |=y?=wH sD} Qwor= R= Q=QmD Qy QO
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pi = e��:niPnj=1 e��:ni
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"CU= xOW xO=iDU= VwQ u}= R= `HQt |xawtHtu} QDy@ |v=UQRwQx@ |xwLv xiOylD Cr=L =@ xiOyOvJ sD} Qwor= u}= Cw=iDO}OH |x]kv xJv=vJ |v=UQRwQx@ s=ovy QO "CU= pbest |xQP Qy X=N ?=wH|x]kv u} QDy@ u} Ro}=H x]kv u}= OwW ?r=e |r@k u=mt u} QDy@ Q@ xOW Ci=}QO |v=UQRwQx@ I}y OwW ?wret O}OH ?=wH Qo = TmaQ@ "OwW|t xOW Ci=}CQwYx@ OvwWv ?r=e =y?=wH R= l}I}y xJv=vJ =t= OwW|tv s=Hv= pbest OQwt2 pmW QO "OwW|t |r@k ?=wH u} Ro}=H O}OH ?=wH |w=Ut p=tDL= =@ w |iO=YD
"CU= xOW x�=Q= Q_v OQwt |O=yvW}B sD} Qwor= OQmrta |xwLv
C=QP V}=tv |xwLv "1"3umtt pLx=Q l} w wHwCUH |=[i QO x]kv l} Qou=Wv xQP Qy sD} Qwor= u}= QOKxm CU= T} QD=t l} xQP Qy 1 pwOH x@ xHwD =@ "CU= Q_v OQwt |xr�Ut |=Q@
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"�CU= xOW xO=iDU= x}rkv p}=Uw |t=tD R= ?=wH u}= QO�|xr}Uw "OyO|t T} wQU 9 xQ=tW |QDWt x@ =yvD 1 xQ=tW |x}rkv |xr}UwxQ=tW |x}rkv |xr}Uw "OyO|t T} wQU 16 xQ=tW |QDWt x@ =yvD 2 xQ=tW |x}rkv|QDWt x@ =yvD 4 xQ=tW |x}rkv |xr}Uw "OyO|tT} wQU 7 xQ=tW |QDWt x@ =yvD 3
R= pY=L |=y?=wH |m}ORv u}vJty w pOt |DUQO xOt; CUO x@ G}=Dv"Ovm|t O}}-=D Rto |=y?=wH =@ =Q |Q=mD@==Qi sD} Qwor=
u=tR 'xrY=i Q=}at '|oOvm =QB Q=}at 'wDQ=B |=y?=wH O=OaD xr�Ut Qy pL |=R=x@u} QDO@ 'swO hOy `@=D Q=Okt u} QDy@ 'pw= hOy `@=D Q=Okt u} QDy@ 'xr�Ut |=QH=QO hOy `@=wD Q}O=kt u}ov=}t w swO hOy `@=D Q=Okt u} QDO@ 'pw= hOy `@=D Q=OktQO |Q=Oxov |=yxv} Ry u=R}t C}ty= x@ xHwD =@ "CU= xOt; G}=Dv QO wDQ=B xawtHt"CU= xOW pL Cw=iDt |Q=Oxov |xv} Ry xawtHt wO |=R=x@ xr�Ut Qy xr=kt u}=xOy=Wt xr�Ut Qy |=y?=wH QO =Q |Q=Oxov |xv} Ry Q}F-=D u=wD|t G}=Dv u}= T=U=Q@pY=L G}=Dv &CU= xOW pL H2 w H1 hrDNt Q=Okt wO |=R=x@ xr�Ut Qy "OQm=@ =Q Q}Ut u} QDy@ w x}rkv p}=Uw u} QDy@ wDQ=B ?=wH Qy |=R=x@ sD} Qwor= |=QH= R=?=NDv= Q}N-=D u=tR w QiU u=tR |R=Uxv}tm w |Q=Oxov |xv} Ry |R=Uxv}tm x@ xHwDxQ}NP Q}eDt l} QO R}v |QDWt Qy |=R=x@ x}rkv |xr}Uw Qy Q}N-=D Q}O=kt "OQm Oy=wN
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"CU=u; |=Q@ swO hOy `@=D Q=Okt w 44 1 ?=wH u}= |=R=x@ pw= hOy `@=D Q=OktOvQ=O Q=Qk wDQ=B ?=wH xawtHt u}= QO xm Qo}O ?=wH 6 |t=tD "CU= 10913w Q}N-=D Q}O=kt 'Q}Ut wv 'p}=Uw wv xm Cw=iD u}= =@ 'OvDUy C=aq]= u}= pt=W
"CU= Cw=iDt hOy `@=wD Q}O=ktQou=Wv |vLvt u}= "CU= xOt; 3 pmW QO ?=wH u}= |=R=x@ wDQ=B \=kv V}=tvQo}O ?=wH Qy &CU= 11 xQ=tW |xr�Ut QO |vOWv ?wret wDQ=B ?=wH xawtHt=y?=wH |xU}=kt Qw_vtx@ "Ow@ Oy=wNv =y?=wH u}= R= QDy@ =y?=wH u}= R= Q}e
"MOPSOVwQ11xQ=tW xvwtv|xr�Ut<=R=x@ wvQ=@\=kv xawtHt Q=Owtv "3pmW
w 3 xQ=tW |QDWt x@ =OD@= 5 xQ=tW |x}rkv |xr}Uw "OyO|t T} wQU 11 xQ=tW|xr}Uw "OyO|t T} wQU 15 xQ=tW |QDWt x@ ,=D}=yv w 5 xQ=tW |QDWt x@ TBUT} wQU 8 xQ=tW |QDWt x@ TBU w 10 xQ=tW |QDWt x@ =OD@= 6 xQ=tW |x}rkvxQ=tW|QDWt x@TBU w 13 xQ=tW|QDWt x@ =OD@= 7 xQ=tW|x}rkv |xr}Uw "OyO|tT} wQU 4 xQ=tW |QDWt x@ \ki 8 xQ=tW |x}rkv |xr}Uw "OyO|t T} wQU 14|QDWt x@ TBU w 12 xQ=tW |QDWt x@ =OD@= 9 xQ=tW |x}rkv |xr}Uw "OyO|t|x}rkv |xr}Uw C}=yv QO w "OyO|t T} wQU 2 xQ=tW |QDWt x@ QN; QO w 1 xQ=tW"OyO|t T} wQU 6 xQ=tW |QDWt x@ TBU w 17 xQ=tW |QDWt x@ =OD@= 10 xQ=tW
"MOPSO w Rto |=y?=wH |xU}=kt "3 pwOHpL u=tR C=QP u}ov=}t pL u=tR
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415 9 2 241 8 OB1 12279 16 2235 OB2
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485 26 61 312 25 5 OB1 55229 14 5153 OB2
511 25 35 312 23 7 OB1 76506 3 6493 2 OB2
543 32 45 611 31 1 OB1 97806 54 7795 3 OB2
"lJwm p�=Ut |=Q@ MOPSO VwQ R= xOt; CUO x@ G}=Dv "4 pwOHhOy `@=D |=y?=wH u}ov=}t hOy `@=D ?=wH u} QDO@ hOy `@=D ?=wH u} QDy@ Q=}at Q=}at =QH= u=tR ?=wH O=OaD xQ=tW
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swO pw= swO pw= swO pw= xrY=i |oOvm =QB �x}v=F� wDQ=B p=Ft14694 21 65 14790 5 25 1 14616 5 20 3 0 5159 24 8807 91 0988 3 14
94314 3 23 97633 25 4 93613 5 19 3 1 98644 155 217 198 644 4 1514765 1 23 3167 14843 2 26 9 14711 3 21 6 0 61377 25 6279 95 9988 3 1696366 7 23 1 97829 6 25 5 93695 1 21 8 1 89541 109 444 209 153 3 1715015 2 28 2714 15271 2 32 5 14859 9 26 9 0 75103 42 0812 101 325 4 1893901 1 31 5667 93980 3 36 9 93814 7 27 3 1 21933 13 6773 234 675 3 1915139 4 34 4143 15398 8 38 15014 4 32 2 1 22874 40 6268 109 346 4 2094814 1 32 22 98029 4 35 4 93981 9 31 5 1 90703 140 667 259 149 4 2115835 6 39 5667 16478 7 43 1 15470 4 37 5 1 97159 55 9261 121 327 3 22110916 36 9 124228 39 1 104227 35 6 1 98796 244 464 303 761 3 2316842 6 30 2333 30991 3 36 6 14790 6 25 2 1 87626 429 225 146 13 8 24116667 28 1 192825 34 8 94034 9 24 6 1 87215 875 663 365 121 5 2516878 1 31 5556 18942 3 35 6 15269 3 27 2 0 26184 180 526 172 675 6 26104557 33 425 117799 36 8 94453 2 29 3 0 34714 416 536 401 284 6 2717742 2 35 4714 19648 9 40 6 15941 28 5 1 92609 316 755 211 099 4 28
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"OQm Oy=wN x�=Q= =Q G}=Dv u=ty|Q=Oxov |xv} Ry Q}O=kt Q}}eD =@ 'lJwm p�=Ut |=Q@ xm OyO|t u=Wv G}=Dvp�=Ut |=Q@ "OQm Ovy=wN Q}}eD |@=NDv= |=yQ}Ut u; `@Dx@ w pkvwptL u=ow=v ?=NDv=
"O@=}|t V}=Ri= |OwaY CQwYx@ MOPSO sD} Qwor= pL u=tR 'nQR@pL |=Q@ |Q=mD@==Qi |=yVwQ Q}=U R= 'OwW|t O=yvW}B |D; C=k}kLD |=Q@QO =Q xr�Ut CU= QDy@ "OwW xU}=kt Qo}Om} =@ =yu; G}=Dv w OwW xO=iDU= xr�Ut|xr�Ut x@ =Q C}a]k sOa w |r=tDL= \}=QW u}vJty w 'pL |} wBOOvJ Cr=L
"OwW pL TBU w xOwRi= OwHwt
xrY=i XN=W w |oOvm =QB XN=W 'wDQ=B |=y?=wH O=OaD |=yXN=W R= u=wD|tQDW}@ xJQy |oOvm =QB XN=W w wDQ=B |=y?=wH O=OaD |=yXN=W "OQm xO=iDU=QDsm ?=wH l} QO xrY=i XN=W Q=Okt xJQy TmaQ@ &CU= QDy@ ?=wH OvW=@
%xrY=L G}=Dv T=U=Q@ "CU= QDy@ ?=wH u; 'OW=@u}=Q@=v@ "OQm Oy=wN =O}B V}=Ri= CaQUx@ R}v u; pL u=tR xr�Ut O=a@= V}=Ri= =@ �u}= QO |O=yvW}B sD} Qwor= R= OwW|t x}YwD |ak=w |=}vO O=a@= =@ p�=Ut |=R=x@
"OwW xO=iDU= xr=kt|=Q@ |@=NDv= |x}rkv p}=Uw wv 'x}rkv p}=Uw |Q=Oxov |xv} Ry Q}O=kt Q}}eD =@ �
"OQm Oy=wN Q}}eD |@=NDv= |=yQ}Ut u; `@Dx@ w u=} QDWt x@ |yOT} wQU"O=O Oy=wN |QDy@ |=y?=wH xr�Ut QD|ak=w |Q=Oxov |=yxv} Ry |=R=x@ �
QDy@ Q=}U@ |oOvm =QB w xrY=i |=yXN=W 'QD|ak=w |Q=Oxov |=yxv} Ry |=R=x@ �"Ow@ Oy=wN
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"CU= ?U=vt Q=}U@ C=QP xw@v= VwQ |=R=x@ xr�Ut pL u=tR �
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=yCWwv=B1. post optimization2. path-relinking3. iterated local search-random variable neighborhood decent4. J Moore and R Chapman5. T Ray and KM Liew6. evolutionary techniques7. personal best8. global best
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