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|O=yvW}B OQm} wQ "3pkvwptL |xv} Ry |R=Uxv}tm u=tO}J p�=Ut QO hOy u} QDsyt xm u}= x@ xHwD =@xm �� Cq}yUD u=tO}J |xr�Ut pL |=Q@ l}DvS sD} Qwor= l} =OD@= 'CU= O=wtu=tO}J u=wvax@ u=tO}J u}= "OwW|t x�=Q= �� OQ}o|t Q_v QO =Q xv} Ry Q=}at \kiO}rwD =@ 'Q}PB Q=QmD Ov}=Qi l} QO TBU "CU= xDiQo Q=Qk 2-opt sD} Qwor= |x}=B=@ =yu; u} QD=Q=m ?=NDv= w '2-opt sD} Qwor= \UwD u=tO}J hrDNt |=yw}D=vQDr;"OQ}o|t Q=Qk Q_v Ot hOy u}OvJ u=tRsy '=yxO=O |WWwB p}rLD OQm} wQ R= xO=iDU=

"OyO|t u=Wv =Q |O=yvW}B OQm} wQ pL=Qt 2 pmW

|O=yvW}B l}DvS sD} Qwor= "1"3|=yQ=mwR=U |x}=B Q@ xm CU= |rt=mD |=ysD} Qwor= R= |Y=N wv 'l}DvS sD} Qwor=14Ovr=y \UwD l}DvS sD} Qwor= |rY= |xO}= "CU= Q=wDU= =yuS Ca}@] w pt=mDu=ty xm �� swRwtwQm x@ swUwt |D}atH =@ l}DvS |=ysD} Qwor= [20]"OW KQ]t|}x}=B |=yOL=w R= swRwtwQm Qy "Ovvm|t Q=m �� Ov=xr�Ut |xOW |Q=PoOm |=y?=wH"CU=yswRwtwQm C}kiwt u=R}t |xOvvmu}}aD VR=Q@ `@=D "OwW|t xDN=U uS s=vx@l}DvS sD} Qwor= u; R= TB w OwW|t O=H}= |iO=YD CQwYx@ ,qwtat x}rw= C}atH'OR=OQB|t =y?=wH R= |r=wDt |=ypUv O=H}= x@ u; QO xm OwW|t |}xkrL OQ=w|=yVwQ =Q} R 'OvW=@ s}Ok |=yC}atH R= QDy@ O}OH |=yC}atH xm O}t= u}= x@p=ta= =yu; uOw@ ?U=vt x@ xHwD =@ 'O}OH |=yC}atH ?=NDv= |=Q@ xO=iDU= OQwtpUv O=H}= "CW=O Ovy=wN pFt O}rwD |=Q@ |QDW}@ Tv=W =yu} QDy@ TB "Ov=xOW

55

"""|WW

wBp}rLD

wl}DvS

sD}Qwor=

R=xO=iD

U=

�maxi = ai � (lmin

i )�2 |x]@=Q R= xO=iDU= =@ �maxi CQwYu}= QO OW=@ xOW

"OwW|t x@U=LtR= '|vOWv |=y?=wH |=Q@ xt} QH uDiQo Q_v QO =@ 'xOWx�=Q= l}DvS sD} Qwor=

%R= CU= CQ=@a xt} QH `@=D "Ovm|t |Q}owrH =yu; ?=NDv=ps =

�Costmin

feas � Costminall

��0@vuutns

1 +

NXi=1

maxf0; �i � �maxi g

!1Ak

(2)

|vOW |=y?=wH x@ \w@ Qt O=wt pkvwptL |xv} Ry u} QDsm Costminfeas u; QO xm

|vOWv =} |vOW |=y?=wH x@ \w@ Qt O=wt pkvwptL |xv} Ry u} QDsm Costminall w

ns w COW QDt=Q=B k u}vJty "Ov=xOt; CUO x@ uwvm =D wHwCUH |] QO xm CU=R= =} xOWs=Hv= C=k}kLD QDW}@ QO "CU= pmW C}OwOLt Zk=v |=yu=tDQ=BO O=OaDxOW xO=iDU= xt} QH `@=D ?} Q[ u=wvax@ PN

i=1 maxf0; �i; �maxi g R= =} ns

xO=iDU= xt} QH `@=D ?} Q[ u=wvax@ =yu; |UOvy u}ov=}t R= Q=DWwv u}= QO "CU="OwW|t |vOWv |=yu=tO}J *hrDNt *Cq=L *QDW}@ R}=tD ?@U Q=m u}= "CU= xOW

%R= CU= CQ=@a |vOWv |=yu=tO}J |=Q@ xt} QH uDiQo Q_v QO =@ xv} Ry `@=DCostps =Costs +

�Costmin

feas � Costminall

��0@vuutns

1 +

NXi=1

maxf0; �i � �maxi g

!1Ak

(3)

?=NDv= VwQ "3"1"3u}O@ "CU= xOW xO=iDU= ?=NDv= |=Q@ �CrwQ |xNQJ� VwQ R= sD} Qwor= u}= QO?=NDv= p=tDL= "CU= QDW}@ �QDq=@ |oOvR=Q@ =@ swRwtwQm� ?=NDv= p=tDL= xm CQwYx@U=Lt4 |x]@=Q j} Q] R= swRwtwQm u; |oOvR=Q@ T=U=Q@ 'swRwtwQm Qy =@ Q_=vDt

%OwW|t

ps = e�� Costps

Costpmax

nPi=1

e�� Costpi

Costpmax

(4)

CUO x@ |xv} Ry u} QDW}@ Costpmax w i swRwtwQm |xv} Ry Costps u; QO xm=ysRwtwQm R= |t}v xm OwW|t s}_vD |Qw] w CU= COW QDt=Q=B R}v � "CU= xOt;

"OvW=@ xDW=O =Q ?=NDv= p=tDL= OYQO 80`]=kD Qorta "4"1"3

`]=kD QO "CU= xDUw}B CN=wvm} `]=kD j}kLD u}= QO xOWxO=iDU= `]=kD Qorta|iO=YD swRwtwQm l} TBU "OwW|t ?=NDv= swRwtwQm wO =OD@= xDUw}B CN=wvm}QO "OwW|t O}rwD 'OwHwt |=yswRwtwQm =@ pw]sy '1 w QiY u}@ CN=wvm} `} RwD =@

%OvwW|t O=H}= u=OvRQi 6 w 5 \@=wQ R= xO=iDU= =@ C}=yvy1i = �ix1i + (1� �i)x2i 8 i = 1; 2; :::; N (5)

y2i = �ix2i + (1� �i)x1i 8 i = 1; 2; :::; N (6)

?}DQDx@ =yy2i w y1i w 'swO w pw= u}Or=w |=yuS ?}DQDx@ =yx2i w x1i u; QO xm|xwLv "CU= |iO=YD swRwtwQm uS �i u}vJty "OvDUy swO w pw= u=OvRQi |=yuS

"CU= xOW xO=O u=Wv 5 pmW QO `]=kD C=}rta s=Hv=

"u; x@ \w@ Qt u=tO}J w xr�Ut swRwtwQm R= |}xvwtv "4 pmW

Qw_vtx@ Q=wv Qy QO =yu=tDQ=BO |Q}oQ=Qk ?}DQD R}v w 'OQ}o|t Q=Qk Q=wv Qy QO xmK} QWD |O=yvW}Bl}DvS sD} Qwor= pL=Qt xt=O= QO "O};|tCUO x@ xv} Ry uOQmxv}tm

"OwW|t|Q=PoOm "1"1"3

xr�Ut |=y?=wH V}=tv 'l}DvS sD} Qwor= |R=UxO=}B w |Q}oQ=mx@ QO s=o u}rw=R= sRwtwQm Qy j}kLD u}= QO xOWxO=iDU= u=oRtQ QO "CU= swRwtwQm l} CQwYx@CtUk "CU= B + 1 w 1 u}@ |iO=YD |OOa uS Qy xm xOW p}mWD uS N|Q}oQ=Qk C} wrw= Qou=}=tv u; |Q=Wa= CtUk w Q=wv xQ=tW hQat OOa u}= K}LY4 pmW "�OvQ=OQwNQ@ |QDq=@ C} wrw= R= QDlJwm O=Oa=� CU= Q=wv QO u=tDQ=BO

"Ov=xOW |}=tv=H Q=wv 3 QO u=tDQ=BO 6 u; QO xm OyO|t u=Wv =Q |LQ]swRwtwQm uS u}tWW� sWW u=tDQ=BO =@ \@DQt uS jwi KQ] QO p=Ft u=wvax@uwJ w OQ}o|t Q=Qk swU Q=wv QO u=tDQ=BO u}= |va} "CU= 3 66 Q=Okt |=Q=O �?=wH5 u=tDQ=BO w 3 01 Q=Okt =@ 2 u=tDQ=BO� Q=wv u}= |=yu=tDQ=BO Q}=U =@ xU}=kt QO'swU Q=wv QO =yu=tDQ=BO u=tO}J KQ] QO 'CU= swO |x@DQ |=Q=O �3 96 Q=Okt =@

"CU= xO=O X=YDN= OwN x@ =Q swO C} wrw=

|oDU}=W `@=D "2"1"3|xv} Ry u}=Q@=v@ "OQ}o|t CQwY O=wt pkvwptL |xv} Ry |=v@tQ@ =y?=wH |@=} RQ=1 |x]@=Q j} Q] R= O=wt pkvwptL |xv} Ry "Ow@ Oy=wN QDy@ |oDU}=W hQat QDsm

%OwW|t ?=UL

Costs =NXi=1

NXj 6=i

CijFijdij (1)

"CU= j w i |=yu=tDQ=BO u}@ O=wt u=} QH Fij '=yu=tDQ=BO O=OaD N u; QO xmi |=yu=tDQ=BO u}@ xrY=i OL=w Qy QO O=wt OL=w Qy ptL |xv} Ry Cij u}vJtyCW=O xHwD O}=@ "CU= j wi |=yu=tDQ=BO RmQt u}@ |r}]DUt |xrY=i dij w 'j w=@ |}=yu=tDQ=BO |=Q=O =y?=wH |NQ@ CU= umtt '=y?=wH O}rwD s=ovy QO xm|r}N =} w uyB |r}N =} =yu=tDQ=BO |va} �� OvW=@ �|vOWv� |rtaQ}e |=ypmWu}vJ O}rwD R= |Q}owrH Qw_vtx@ h]avt |Q=wv Q=DN=U =@ p�=Ut QO "OvW=@ l} Q=@Qo = "OwW|t xO=iDU= (�max

i ) �ZQa x@ pw] C@Uv xv}W}@� Q=}at R= |}=y?=wH'OW=@ �u=tO}J� ?=wH l} R= i u=tDQ=BO |xOt; CUO x@ ZQa x@ pw] C@Uv �iQo = "OW=@ i u=tDQ=BO Qy |=R= x@ �i � �max

i Qo = \ki w Qo = CU= |vOW ?=wH u}=xO=O (lmin

i ) R=Ht |Q=vm pw] xv}tm C=aq]= '�maxi |=Hx@ xr�Ut |=yxO=O QO

56

|v=v}Rt

|i]Yt

wxO=RO@=

a|i]Y

t |DU=wNQO CQw=Ht u=R}t �hr=lQDWt x=oDUO =} QwD=QB= |=Q=O u=tDQ=BO wO CU= umtt |O}rwD C=Hv=NQ=m QOu=R}t Qou=Wv |m}ORv MQv "OvW=@ sy Qw=Ht =yu=tDQ=BO u}= CU= QDy@ u}=Q@=v@ &OvW=@xHwD =@ Q=}at u}= |xR=Ov= "CU= Qo}Om} CQw=Ht QO =yu=tDQ=BO u}= Q=QkDU= CQwQ[

%OwW|t x@U=Lt lQDWt RQt pw] w |m}ORv MQv x@

|DU=wNQO CQw=Ht u=R}t =NXi=1

NXj 6=i

Dijcoij (8)

RQt pw] R}v coij w CU= j w i |=yu=tDQ=BO |m}ORv MQv |xOvyOu=Wv Dij"OyO|t u=Wv =Q u=tDQ=BO wO u}@ lQDWt

|DU=wNQO |xrY=i u=R}t �?Qo}Om} R= =yu=tDQ=BO |[a@ CU= sRq 'VRQr =} =OYwQU pFt |r}qO x@ |y=o

%OvW=@ xDW=O |YNWt |xrY=i

|DU=wNQO |xrY=i u=R}t =NXi=1

NXj 6=i

Eijd0ij (9)

'Ci=Ut OL=wl} |=R=x@ j w i u=tDQ=BO wO u}@ |DU=wNQO |xrY=i MQv Qou=}@Eij"CU= j w i |=yu=tDQ=BO RmQt u}@ |UO}rk= |xrY=i d0ij R}v w

pmW C@Uv �GCL=Ut 'u=tDQ=BO Qy |=Q@ u=tO}J |L=Q] |xr�Ut pL |=Q@ OwHwt |=yOQm} wQ QOC@Uv u} QDW}@ QwDm =i R= w OwW|t xDiQo Q_v QO xr�Ut �|OwQw� u=wvax@ R=}v OQwt?r=k QO p�=Ut QDW}@ QO QwDm =i u}= "OwW|t xO=iDU= �ZQa x@ pw] C@Uv� pmWQy |=Q@ 'C=R}yHD w Q=m wv x@ xHwD =@ |ak=w \}=QW QO "OwW|t u=}@ C}OwOLtpmW C@Uv hqDN= R= 'pmW C@Uv Q=}at "OQ=O OwHw xv}y@ pmW l} u=tDQ=BO

"OwW|t pY=L ���� p;xO}= pmW C@Uv Q=Okt =@ ��� xOt; CUO x@

pmW C@Uv =NXi=1j�i � ��i j (10)

|xv} Ry uOQmxv}tm 'Cq}yUD u=tO}J p�=Ut QO hOy u} QDsyt OW xQ=W= xm u=vJxOW xO=O |QDW}@ VRQ= hOy u}= x@ R}v xr=kt u}= QO "CU= O=wt pkvwptLxv} Rysm |v=tO}J xm 'l}DvS sD} Qwor= R= xOt; CUO x@ u=tO}J Qw_vtu}O@ "CU=hrDNt |=yw}D=vQDr; w xDiQo Q=Qk 2-opt sD} Qwor= |x}=B u=tO}J u=wvax@ 'CU=pkvwptL |xv} Ry pt=W Q=}at Q=yJ O}Ov=m u=tO}J Qy |=Q@ "OwW|t O=H}= u=tO}Jx@U=Lt pmW C@Uv w |DU=wNQO |xrY=i u=R}t '|DU=wNQO CQw=Ht u=R}t 'O=wtO}Ov=m u=tO}J u} QD=Q=m =yxO=O |WWwB p}rLD |xw}W R= xO=iDU= =@ TBU "OwW|t,=OOHt O}Ov=m |=yu=tO}J u=}t R= u=tO}J u} QD=Q=m |}=U=vW R= Oa@ "OwW|t ?=NDv="OwW|t Q=QmD x} wQ u}= w xDiQo Q_v QO 2-opt sD} Qwor= |x}=B u=tO}J u=wvax@ =Q u;"O@=}|t xtD=N sD} Qwor= 'Ovmv Q}}eD =Q=m u=tO}J |r=wDt |xrLQt wO QO xJv=vJ

"OwW|t O}rwD O}Ov=m u=tO}J

N

2!

O=OaD xrLQt Qy QO xm OwW|t Qw;O=}p}rLD |xw}W QO xm CU= u}= OwW xQ=W= u; x@ O}=@ xm |Qo}O |xDmvCU= ?U=vt=v u=WuOw@sm w ?U=vt u=WuOw@O=} R xm |}=yQ=}at '=yxO=O |WWwBu=wvax@ �|DU=wNQO |xrY=i u=R}t w |DU=wNQO CQw=Ht u=R}t |=yQ=}at pt=W�|xv} Ry |=yQ=}at pt=W� CU= ?wr]t u=WuOw@sm xm |}=yQ=}at w '�|HwQN�xDiQo Q_v QO =yxO=O|WWwB p}rLD �|OwQw� u=wvax@ �pmWC@Uv w O=wt pkvwptL

"`]=kD C=}rta s=Hv= |ovwoJ V}=tv "5 pmW

VyH Qorta "5"1"3%CU= xOW xO=iDU= hrDNt VyH Qorta xU R= xr=kt u}= QO

w 1 u}@ |i=YD OOa l} =@ TBU w OwW|t ?=NDv= |iO=YD Qw]x@ uS l} "1"OwW|t u} Ro}=H 'OwW|t O}rwD 7 |x]@=Q T=U=Q@ xm B + 1

x0i =

8<:xi + ((B + 1)� xi) tanh(kz) tanh(kz) � 0xi + (xi � 1) tanh(kz) tanh(kz) < 0

(7)

|iO=YD OOal} z 'xOWu} Ro}=H Q=Okt x0i 'xOW?=NDv=|iO=YD uS Q=Okt xiOwW|t ?@U jwi VwQ R= xO=iDU= "CU= u; ?} Q[ k w OQ=Ov=DU= p=tQvC=Q}}eD R= w OQ}o CQwY |QDW}@ p=tDL= =@ w |r@k uS |o}=Uty QO =yVyH

"OwW |Q}owrH O=} R |r}Nw |}=DwO |=y|}=Hx@=H u}@ R= TBU w OwW|t ?=NDv= |iO=YD Qw]x@ uS xU "2"OQ}PB|t CQwY 'OwW|t xv} Ry u} QDsm x@ QHvt xm|}=Hx@=H '=yuS u}=|}=DxUswRwtwQm l} |=yuS |}=DwO |=y|}=Hx@=H s=tD 2-opt sD} Qwor= R= xO=iDU= =@ "3?=NDv= OwW|t xv} Ry u} QDsm x@ QHvt xm|}=Hx@=H =yu; u}@ R= w 'OwW|t s=Hv=

"OW Oy=wNO}OH pUv ?=NDv= "6"1"3

?ULQ@ u=oDi=}VyH C}atH w u=OvRQi C}atH '|r@k C}atH Q=DWwv u}= QO|=Q@ ?}DQDx@ =yu} QDy@ 'x}rw= pUv |=yswRwtwQm O=OaD x@ w OwW|t ?DQt |oDU}=W

"OvwW|t ?=NDv= O}OH pUvhkwD \QW "7"1"3

VR=Q@ u} QDy@ =@ ?=wH |YNWt O=OaD Ow@y@ uwO@ O}rwD 'hkwD \QW Q=DWwv u}= QO"CU= xOW xDiQo Q_vQO C}atH

O}Ov=m |=yu=tO}J |}=Q=m |@=} RQ= w u=tO}J |=yw}D=vQDr; O=H}= "2"3|xv} Ry uOQmxv}tm Q@ Cq}yUD u=tO}J |xv}tR QO xDiQos=Hv= C=k}kLD *QDW}@p=L 'OvyO Q=Qk Q_v Ot R}v =Q h=Oy= Qo}O xm u; uwO@ Ov=xDW=O RmQtD O=wt pkvwptL|=ypt=a O}=@ w OvQ=O xQw_vtOvJ C}y=t Cq}yUD u=tO}J x@ \w@ Qt p�=Ut xm u;=@ u}kkLt "OQ}o Q=Qk Q_v Ot u=tO}J |=yKQ] |@=} RQ= |=Q@ xv} Ry RHx@ |Qo}O?U=vt w |rm |@ wJQ=J Ow@v xm OvO}UQ xH}Dv u}= x@ xQw_vtOvJ Cq=kt |UQQ@C=Q_v x@ xHwD =@ u=QoVywSB xm xOW Ea=@ hrDNt |=yu=tO}J |@=} RQ= |=Q@Q=}at xU =yu; wQu}= R= [10]"OvR=OQB@ u=tO}J |=yKQ] |@=} RQ= x@ OwN |YNWu=tO}J p�=Ut |@=} RQ= |=Q@ �� O=wt pkvwptL |xv} Ry Q@ xwqa �� ?U=vt |:tm

%R= Ov=CQ=@a xm xOW xO=iDU= x@=Wt |=yQ=}at R= Q=DWwv u}= QO "OvO=O O=yvW}B

57

"""|WW

wBp}rLD

wl}DvS

sD}Qwor=

R=xO=iD

U=sPr=1

uryrj � mPi=1

wixij � 0 j = 1; :::; n (14)

wi � " � 0 i = 1; :::;m (15)

ur � " � 0 r = 1; :::; s (16)

"CU=y|HwQNO=OaD s w '=y|OwQw O=OaDm '=yDMU O=OaDn 'pw=|xrLQt pOt QOOL=w |HwQN u}t=r Qou=Wv yrj 's=j OL=w |OwQw u}t=i Qou=Wv xij u}vJtyx@ xOWxO=O X}YND uRw ur w 's=i |OwQw x@ xOWxO=O X}YND uRw wi 's=jQO Q=Okt u}= 'pw= |xrLQt pOt R= "� Q=Okt |x@U=Lt R= TB "CU= s=r |HwQN'xOt; xt=O= QO xm swO |xrLQt pOt pL =@ w xOW |Q=Po}=H swO |xrLQt pOt

"OwW|t |}=U=vW DMU u} QD=Q=m"DMU u} QD=Q=m |}=U=vW |=Q@ swO |xrLQt pOt

minZ = M (17)

Subject to :

M � dj � 0 j = 1; :::; n (18)mPi=1

wixij � 1 j = 1; :::; n (19)sPr=1

uryrj � mPi=1

wixij + dj = 0 j = 1; :::; n (20)nPj=1

�j = n� 1 (21)

�j � dj�j = 0 j = 1; :::; n (22)

wi � "� i = 1; :::;m (23)

uj � "� r = 1; :::; s (24)

dj � 0; �j � 1; �j 2 f0; 1g j = 1; :::; n (25)

u}t=j |}=Q=m sOa u=R}t dj w |}=Q=m sOa u=R}t u} QDW}@ M 'swO |xrLQt pOt QOu}t= pOt uOW |]NQ}e Ea=@ xm |rt=a =t= "CU= �DMU� |Q}os}tYD OL=wu=wD|t ' �j � 1 xm u}= x@ xHwD =@ "CU= �j � dj�j = 0 C}OwOLt 'OwW|t

"O}UQ Q} R C}OwOLt x@ �j Q@ xrO=at u}iQ] s}UkD =@�j�j� dj = 0 j = 1; 2; :::; n (26)

%OwW|t pY=L Q} R C}OwOLt j = 1�j

Q}eDt Q}}eD =@ p=L�j j � dj = 0 j = 1; 2; :::; n (27)

%s} Q=O u}vJty�j � 1) 0 < 1

�j� 1) 0 < j � 1 j = 1; 2; :::; n (28)

pOt u=vJty "OQm xO=iDU= 21 C}OwOLt |=Hx@ 27 C}OwOLt R= u=wD|t u}=Q@=v@p}O@D |]N sQi x@ u=wD|t =Q �j j � dj = 0 C}OwOLt =t= 'CU= |]NQ}exm |Qw]x@ (xy) OvW=@ xOW ?Q[ sy QO Q}eDt wO |D}OwOLt QO xJv=vJ "OQmu=wD|t 'OW=@ |Qv}=@ Q}eDt (y) |Qo}O w (� 0) O}kt |Q}eDt (x) =yu; R= |m}

%OQm |]N Q} R VwQ j@=]t =Q C}OwOLt u;x0 �My (29)

x0 � x+M(1� y) (30)

x0 � x�M(1� y) (31)

x; x0 � 0; y = f0; 1g (32)

j}kLD u}= QO xm u}= x@ xHwD =@ =t= "s} Q=O |HwQN wO w |OwQw wO u}=Q@=v@ "OvwW|t|HwQN l} x@ pmW C@Uv Q=}at 'OwW xO=O |QDW}@ uRw xv} Ry Q=}at x@ xOW |aU%OwW|t xO=iDU= 11 |x]@=Q R= pmW C@Uv Q=}at |=Q@ wQ u}= R= "OwW|t p}O@D

pmW C@Uv =NXi=1

1j�i � ��i j (11)

u=tO}J u} QD=Q=m |x@U=Lt |=Q@ =yxO=O |WWwB p}rLD pOt "3"3O}=@ xm CU= u}= =yxO=O |WWwB p}rLD |=ypOt R= |Q=}U@ QO xHwD p@=k |xDmvTBU "OwW pL |]N |R} Qxt=vQ@ pOt l} |@=} RQ= OQwt |=yxv} Ro O=OaD =@ Q@=Q@=yxv} Ro |Ov@x@DQ Q=}at '�|}=Q=m � xOWpL |=ypOt R= pY=L *hOy `@=D Q=Okt�pw= |x@DQ� xv} Ro u} QDy@ uOQm =O}B 'xv} Ro n OwHw CQwY QO u}=Q@=v@ "Ow@ Ovy=wNl}Uqm |=ypOt QO u}vJty "Ow@ Oy=wN |]N |R} Qxt=vQ@ pOt n pL sRrDUt R}vx@ xHwD =@ "OvwW =Q=m DMU |O=} R O=OaD CU= umtt '=yxO=O |WWwB p}rLD�x}=B u=tO}J� u=wvax@ u=tO}J l} O}=@ |O=yvW}B sD} Qwor= |}xrLQtOvJ OQm} wQ|=yu=tO}J u}@ R= u=tO}J u} QD=Q=m ?=NDv= u}=Q@=v@ "OwW ?=NDv= Oa@ |xrLQt |=Q@

"CU= Q=OQwNQ@ |}q=@ C}ty= R= =Q=m Qy=_x@|}=U=vW |}=v=wD xm Ov=xOW KQ]t =yxO=O |WWwB p}rLD O}OH |=ypOt ,=Q}N=w |DQ= Tm =t}v}t pOt x@ u=wD|t '=ypOt u}= |xrtH R= "OQ=O =Q OL=w u} QD=Q=m=Q DMU u} QD=Q=m =]N w |aU VwQ R= xO=iDU= =@ pOt u}= "OQm xQ=W= [18]u=Q=mtyu}= `iQ |=Q@ "O};|t ?=UL x@ pOt u}= ha[ |x]kv OwN u}= xm Ovm|t |}=U=vW[18]'Vv=Q=mty w |DQ= |O=yvW}B Tm =t}v}t pOt T=U=Q@ [19] wr] w u}t= pmWtpOt l} pL =@ u=wD|t u; ltmx@ xm Ov=xO=O x�=Q= |}xrLQt wO DEA pOtxv} Ro u} QDy@ 'K}LY OOa \rDNt |]N |R} Qxt=vQ@ pOt l} w |]N |R} Qxt=vQ@'xv} Ro n u=}t R= xm Ow@ u=vJ pOt u}= OQmrta "OQm u}}aD =Q �DMU u} QD=Q=m �xQ=wty xm Ow@ u}= pOt u}=ha[ "OW|t ?=NDv= �u} QDQ@� u} QD=Q=m u=wvax@ |}xv} Ro|=yDMU u}@ |va} "Ow@ 1 Q=Okt =yDMUx}k@ w QiY Q=Okt DMU l} |}=Q=m =vpOt QO |mOv= C=Q}}eD =@ 2009 p=U QO [21]u}t= "CiQo|tv CQwY |Ov@x@DQ =Q=m =v"OWN@ Ow@y@ =Q |O=yvW}B OQm} wQ w Ovm hQ]Q@ =Q ha[ u}= CUv=wD swO |xrLQtuOQmhQ]Q@ =@ xr=kt u}= QO "OW swO |xrLQt pOt uOW |]NQ}e ?@U Qt= u}= =t=\rDNt |]N |R} Qxt=vQ@ pOt l} x@ swO |xrLQt |]NQ}e pOt 'ha[ u}=OQm} wQ pL=Qt QO u=tO}J u} QD=Q=m |}=U=vW |=Q@ u; R= w xOW p}O@D K}LY OOa|}=yQ=Ri=sQv xm OwW|t ?@U u}t= pOt uOQm|]N "CU= xOW xO=iDU= |O=yvW}BpL u=tR 'Ovvm pL =Q pOt u}= Ovv=wD@ OvDU}v |]NQ}e |=ypOt pL x@ QO=k xm|xwLv w u}t= |O=yvW}B pOt x@ xt=O= QO "O@=} Vy=m |yHwD p@=k u=R}t x@ R}vxm "�=OD@= 'u}t= |}xrLQtwO pOt QO "s}=xDN=OQB swO |xrLQt pOt |R=U|]Nx@ sRq "OwW|t x@U=Lt pw= |xrLQt pOt R= xO=iDU= =@ 'CU= " Q=Okt xv}W}@=yDMU *QDy@ R}=tD Qw_vtx@ [22]u=Q=mty w 15lwm =Q "� R= xO=iDU= xm CU= QmPQO xm CU= u=Q=mty w lwm pOt |xDi=}s}taD 'pw= |xrLQt pOt "OvOQm O=yvW}B

"CU= xOt; xt=O=""� |}=U=vW |=Q@ pw= |xrLQt pOt

max "� = " (12)

Subject to :mPi=1

wjxij � 1 j = 1; :::; n (13)

58

|v=v}Rt

|i]Yt

wxO=RO@=

a|i]Y

t "|}xvwtv p�=Ut "1 pwOH`HQt C}OwOLt KQ] O=a@= O=OaD xr=Ut=yxO=O pmW Q=QkDU= =yu=tDQ=BO[23] lmin = 5 51�25 10 VC10[23] lmin = 5 51�25 10 VC10Rs[24] �max = 5 51�25 10 VC10Ra[25] lmin = 1 10�6 12 BA12[25] lmin = 1 9�7 14 BA14[26] �max = 4 4�3 12 NUG12[26] �max = 4 3�5 15 NUG15[27] �max =Q}eDt 30�20 20 AB20

C@Uv |O=yvW}B l}DvS sD} Qwor= OQmrta =yp=Ft u}= QO "Ov=xOW pL 7 RwOv} w|R=i |@}mQD pOt [4]'C}tU= w C}D l}DvS sD} Qwor= Q}_v Qo}O |=yOQm} wQ x@|=ysD} Qwor= w [8]u=Q=mty w l =vwm MILP OQm} wQ [7]'u=Q=mty w =}v= l}DvS wxU}=kt [14]Vv=Q=mty w l =vwm� pQwDrm w [13]'u}OwQ=tm w nv=w u=oJQwt |vwrmxOQ@ Q=m x@ |Owta |=yQ=wv =yvD Cq=kt u}= =@ |R=Uu=Uty Qw_vtx@ "CU= xOW

"OwW|t|xr�Ut l} u=wvax@ Rwvy 'OW x�=Q= [27]=iw@ w QwtQ; \UwD xm AB20 |xr�Ut'OwW pL xv}y@ CQwYx@ |k]vt u=tR COt QO Ov=wD|tv xm nQR@ ,=D@Uv u=tO}Jh]avt |Q=wv Q=DN=U |=v@tQ@ =Q AB20 [4]'C}tU= w C}D "OwW|t xDiQo Q_v QO|UQQ@ 1000 =D 1 70677 u}@ hrDNt |=yZQa x@ pw] C@Uv xv}W}@ w|R=i |@}mQD pOt R= xO=iDU= =@ =Q |y@=Wt p�=Ut [7]u=Q=mty w =}v= ,=Oa@ "OvOQmw C}D l}DvS sD} Qwor= R= xOt; CUO x@ G}=Dv R= |NQ@ w OvOQm pL l}DvS ww l =vwm� pQwDrm w [13]u}OwQ=tm w nv=w 2010 p=U QO "OvO}WN@ Ow@y@ =Q C}tU=QDW}@ =Q |r@k C=k}kLD G}=Dv u=oJQwt |vwrm sD} Qwor= R= xO=iDU= =@ [14]VQ=mty|=yxO=O 'CU= FBS Q@ |vD@t xDWPo |=yOQm} wQ xm u; p}rO x@ "OvO=O Ow@y@|O=yvW}B l}DvS sD} Qwor= |@=} RQ= |=Q@ =Q |@ wN |xU}=kt Ovv=wD|t p�=Ut u}=xOt; CUO x@ G}=Dv 2 pwOH QO "OyO x�=Q= hrDNt ZQa x@ pw] |=yC@Uv =@u} QDy@ "CU= xOW x�=Q= u; |=QH= Q=@ GvB QO |O=yvW}B l}DvS sD} Qwor= \UwDQO "CU= |r@k C=k}kLD R= pY=L |=y?=wH *|@ wNx@ xOt; CUO x@ ?=wHu=Wv nvQQB CQwYx@ xOWs=Hv= C=k}kLD pm QO xr�Ut Qy ?=wH u} QDy@ 2 pwOHR}t;C}kiwt Q=}U@ |O=yvW}B l}DvS sD} Qwor= xm CU= K[=w "CU= xOW xO=Opw] C@Uv Cr=L 12 R= Cr=L 11 QO xDWPo |=yOQm} wQ =@ xU}=kt QO w xOw@|=y?=wH u} QDy@ 'O}OH ?=wH 11 u}= "CU= xDW=O |QDy@ OQmrta ZQa x@=Q hrDNt |=yZQa x@ pw] C@Uv xv}W}@ =@ AB20 |xr�Ut |=Q@ xOWxDN=vW

"OyO|t x�=Q=l}DvS sD} Qwor= \UwD xOt; CUO x@ |=y?=wH u} QDy@ C=}�RH 3 pwOH QOsD} Qwor= |=y?=wH pwOH u}= QO "CU= xOW x�=Q= AB20 |xr�Ut |=Q@ |O=yvW}Bw Ov=xOW =OH Qo}Om} R= �j� O=tv =@ =yQ=wv "Ov=xOW |Q=PoOm |O=yvW}B l}DvSxHwD xDmv u}= x@ O}=@ "CU= q=@ x@ u}}=B R= Q=wv Qy uwQO =yu=tDQ=BO ?}DQDCU= |O=aQ}e 25 w 50 '1000 ZQa x@ pw] C@Uv xv}W}@ Q}O=kt xm CW=Oxm u}= C=@F= |=Q@ p�=Ut xvwou}= p=L Qy x@ "OvO=} R u=tO}J |xr�Ut |=Q@ wC}ty= 'CU= |awvDt Q=}U@ |=yu=tO}J |wHwCUH x@ QO=k l}DvS sD} Qwor=QO "CU= Q=wv l} pt=W ?=wH u} QDy@ �max = 1000 |=Q@ p=Ft |=Q@ "OQ=O

?Q[pY=L =@ u=wD|t =Q x0 Q}eDt u}=Q@=v@ "CU= nQR@ |OOa M u; QO xmxJv=vJ xm OwW|t xH}Dv jwi |=yC}OwOLt R= "OQm u} Ro}=H y w x |=yQ}eDtQ=Okt y Q}eDt xJv=vJ w OwW|t QiY x0 Q}eDt OQ}o@ OwN x@ QiY Q=Okt y Q}eDtx@ xHwD =@ xr�Ut pOt QO "Ovm|t Q=}DN= x Q}eDt Q@=Q@ |Q=Okt x0 Q}eDt 'OQ}o@ 1ZQi 1 Q@=Q@ M Q=Okt 'Ovm Q=}DN= 1 R= QDW}@ |Q=Okt Ov=wD|tv j Q}eDt xm u}=

%OQm u=}@ u}vJ u=wD|t xr�Ut |=Q@ =Q jwi |=yC}OwOLt "CU= xOW 0j � dj = 0 j = 1; 2; :::; n (33)

0j � �j j = 1; 2; :::; n (34)

0j � j + (1� �j) j = 1; 2; :::; n (35)

0j � j � (1� �j) j = 1; 2; :::; n (36)

0j � 0; l � j � 1 j = 1; 2; :::; n (37)

xr�Ut pOt u}=Q@=v@ "CU= |mJwm Q=}U@ C@Ft Q=Okt l '37 |x]@=Q QO CU= |vDio%OQm |U} wvR=@ K}LY OOa \rDNt |]N |R} Qxt=vQ@ Cr=L QO u=wD|t =Q

"DMU u} QD=Q=m |}=U=vW |=Q@ |O=yvW}B MILP pOtminZ = M (38)

Subject to :

M � dj � 0 j = 1; :::; n (39)mPi=1

wixij � 1 j = 1; :::; n (40)sPr=1

uryrj � mPi=1

wixij + dj = 0 j = 1; :::; n (41)nPj=1

�j = n� 1 (42)

0j � dj = 0 j = 1; 2; :::; n (43)

0j � �j j = 1; 2; :::; n (44)

0j � j + (1� �j) j = 1; 2; :::; n (45)

0j � j � (1� �j) j = 1; 2; :::; n (46)

wi � "� i = 1; :::;m (47)

ur � "� r = 1; :::; s (48)

dj ; 0j � 0; l < j � 1; � 2 f0; 1g j = 1; 2; :::; n (49)

G}=Dv |xU}=kt w |OOa |=yp=Ft "4=@ "OwW|t |UQQ@ |O=yvW}B OQm} wQ |}=Q=m 'p=Ft u}OvJ pL =@ VN@ u}= QO\UwD ,q@k �h]avt |Q=wv Q=DN=U =@ Cq}yUD u=tO}J� |xr�Ut xm u}= x@ xHwDQO OwHwt hwQat |=yp=Ft w =yxO=O R= 'xDiQo Q=Qk |UQQ@ OQwt |Qo}O u}kkLtpY=L G}=Dv w 'xOW xO=iDU= |O=yvW}B sD} Qwor= |@=} RQ= Qw_vtx@ w[wt C=}@O=p�=Ut "CU= xOW xU}=kt |r@k C=k}kLD R= pY=L G}=Dv =@ |O=yvW}B OQm} wQ R='xR=Ov= Q}_v �� =yu; |=y|oS} w xYqN w Q=DWwv u}= QO xO=iDU= OQwt |}xvwtv"CU= xOW xO=O u=Wv 1 pwOH QO �� x}rw= |=yxO=O `HQt w 'pmW C}OwOLt|=yw}D=vQDr; |x@U=Lt |=Q@ 2-opt sD} Qwor= w |O=yvW}B l}DvS sD} Qwor= u}vJty|=Q@ 23 3 Rto Q=Ri=sQv R= w '|U} wvxt=vQ@ 2009 MATLAB Q=Ri=sQv QO u=tO}J=@ |}xv=}=Q QO |}xvwtv p�=Ut |t=tD "CU= xOW xO=iDU= |O=yvW}B DEA pOt pLpt=a sDU}U QO C}=@=o}o 4 |rY= |x_i=L w RDQy=o}o 2 2 |}xDUywO |xOvR=OQB

59

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wBp}rLD

wl}DvS

sD}Qwor=

R=xO=iD

U="AB20 |xr�Ut |xU}=kt "2 pwOHu=tR OYQO Q=QmD O=OaD u}ov=}t u} QDy@ w nv=w l =vwm�pQwDrm w l =vwm w =}v= w C}D

�maxpL Ow@y@ u} QDy@ GA ?=wH GA ?=wH u}OwQ=tm l =vwm w u=Q=mty u=Q=mty C}tU=�x}v=F� ��� ?=wH |O=yvW}B |O=yvW}B [13] [14] [8] [7] [4]

313 3 09 5 1587 91 1587 91 � 1638 5 � 1638 5 1638 5 1000329 11 99 5 2381 86 2381 86 � 2706 5 � 3009 5 3009 5 50336 3 82 5 3391 95 3391 95 � 3526 5 � 3535 1 3535 1 25334 1 84 5 4043 91 4043 91 � 4119 8 � 4140 5 4296 1 15372 1 71 4 4396 23 4364 74 � 4440 7 � 4440 7 4633 3 10356 1 58 2 4829 52 4717 53 � 4793 5 � 4793 5 5255 7383 2 15 4 5192 13 5183 52 � 5297 6 � 5397 6 5524 7 5374 3 31 3 5206 71 5183 52 � 5360 8 � 5370 9 5743 1 4392 4 02 3 5386 08 5369 3 � 5594 3 � 5594 3 5832 6 3413 2 87 2 5723 92 5677 83 � 5845 3 � 6023 2 6171 1 2417 0 2 5742 37 5677 83 5677 83 5845 3 6890 82 6453 1 7205 4 1 75452 2 87 1 5863 46 5677 83 � 5845 3 � 6029 3 6662 9 1 70667

1000� xtD=N \QW '70� x}rw= C}atH '0 8� `]=kD p=tDL= '0 2� VyH p=tDL=

'5 pwOH "CU= xOW xO=O u=Wv nvQQB CQwYx@ xr�Ut Qy |=Q@ FBS ?=wH"OyO|t u=Wv =Q xOt; CUO x@ |=y?=wH u} QDy@ R= pY=L u=tO}J V}=tv |xwLv=@ AB20 |xr�Ut R= '|OOa p=Ft l} |=Q@ u=tO}J u} QD=Q=m |x@U=Lt Qw_vtx@|D=aq]= x@ R=}v 'Q=m u}= s=Hv= |=Q@ "OwW|t xO=iDU= 2 ZQa x@ pw] C@Uv xv}W}@|m}ORv MQv T} QD=t Qw_vtu}O@ "CU= p;xO}= pmW C@Uv w |m}ORv MQv |=Q@u=wvax@ (��i = 1 66) |}q] OOa u}vJty "OwW|t O}rwD |iO=YD CQwYx@"CU= xOW xDiQo Q_v QO =yu=tDQ=BO |xty |=Q@ ZQa x@ pw] C@Uv p;xO}= Q=Okt=Q AB20 |xr�Ut |=Q@ xOWO}rwD |iO=YD |=yC}r=ai |x]@=Q Q=Owtv '6 pmW

"OyO|t u=WvCUO x@ x}rw= u=tO}J |wQ 2-opt sD} Qwor=\UwD|}=DwO |=yZ} waD s=Hv= =@

"|i}m \@=wQ VRQ= u=R}t pwOH w |m}ORv MQv T} QD=t "6 pmW

"AB20 |xr�Ut ?=wH "3 pwOH|O=yvW}B l}DvS sD} Qwor= R= xOt; CUO x@ ?=wH u} QDy@ �max

1'18'5'20'8'7'6'2'4'19'3'10'14'9'15'12'17'13'16'11 100011j16j13 j17j12j15j9j14j10j3j19j4j2j6j7j8j20j5j18j1 501j11j16 j17j12j13j15j10j14j3j19j4j2j5'9'8'7'6j20j18 2511j16j17j12j15j20'8'7'2'4'19'9'10'14'3'13j5'6'1j18 1511j16j17j12j15'13'14'10'9'19'3'1j5'20'8'7'2'4'6'18 1011j16j17j13'12j15'14'10'9'3'1j5'19j20'8'7'6'2'4'18 711'16j15'17j13'14'10'9'12j3'19j20'8'7'2'4'6j1'5'18 4�511'16j15'13'17j14'10'9'12j3'19'5j1'2'4'7'8'6j18'20 3

16'17j11'15'12 j13'14'9'10j1'3'19j5'8'7'2'4j20'6'18 1 70667�1 75�2

pL G}=Dv "CU= Q=wv 20 pt=W ?=wH u} QDy@ �max = 50 |=Q@ xm |r=LQO=k |QF-wt Qw]x@ '|O=yvW}B l}DvS sD} Qwor= xm OyO|t u=Wv AB20 |xr�UtuOQm =O}B |}=v=wD w xOw@ |irDNt Q=}U@ |=yQ=DN=U =@ =y?=wH |wHwCUH x@=Q 1 70667 |mJwm x@ w u}}=B ZQa x@ pw] C@Uv xv}W}@ =@ |vOW |=y?=wH

"OQ=Oxm |Qo}O p�=Ut |=Q@ |O=yvW}B l}DvS sD} Qwor= R= pY=L G}=Dv 4 pwOH QOp�=Ut u}= QDW}@ "CU= xOW x�=Q= Ov=xDiQo Q=Qk xO=iDU= OQwt FBS Cq=kt QO ,q@k?=wH w Ov=xOW pL j}kO CQwYx@ [8]u=Q=mty w l =vwm MILP pOt \UwDsD} Qwor= xm OyO|t u=Wv 4 pwOH R= pY=L G}=Dv "CU= OwHwt =yu; |xv}y@u} QDy@ 4 pwOH QO "Ovm =O}B =Q FBS |xv}y@ |=y?=wH Ov=wD|t |O=yvW}B l}DvS

60

|v=v}Rt

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t "Qo}O |}xvwtv p�=Ut "4 pwOHu=tR OYQO Q=QmD O=OaD u}ov=}t u} QDy@ w nv=w l =vwm�pQwDrm w l =vwm

xr�UtpL Ow@y@ u} QDy@ GA ?=wH GA ?=wH u}OwQ=tm l =vwm w u=Q=mty�x}v=F� ��� ?=wH |O=yvW}B |O=yvW}B [13] [14] [8]

87 0 5 20320 52 20320 52 � 20320 52 � VC1083 0 5 22899 65 22899 65 22899 65 22899 65 22899 65 VC10Rs

94 0 5 21463 07 21463 07 21463 07 21463 07 21463 07 VC10Ra

218 0 3 8892 52 8786 8786 8801 33 8801 33 BA12254 0 4 5037 47 5004 55 5004 55 5004 55 5004 55 BA14107 0 5 262 003 262 003 262 003 262 003 265 5 Nug12139 0 5 524 75 524 75 536 75 524 75 526 75 Nug15

"500� xtD=N \QW '40� x}rw= C}atH '0 8� `]=kD p=tDL= '0 2� VyH p=tDL=

"?=wH u} QD=Q=m w u} QDxv} Ry sm "7 pwOH|O=yvW}B l}DvS sD} Qwor= R= xOt; CUO x@ ?=wH u} QDy@

18'6'20 j4'2'7'8'5j19'3'1j10'9'14'13j12'15'11j17'16 x}=B u=tO}J�x}rw=�

18'6'20 j4'2'7'8'5j19'3'15j10'9'14'13j12'1'11j17'16 =Q=m u=tO}J�|}=yv�

xm u=vJ "CU= xOW xO=O u=Wv �=Q=m � |}=yv w x}rw= ?=wH V}=tv |ovwoJ 7|}=Hx@=H =@ Q=QmD xrLQt l} R= TB 2-opt Q}PB Q=QmD sD} Qwor= 'OwW|t xOy=Wt"O=O Ow@y@ =Q u; u=wD|tv Qo}O xm xO}UQ |}=Q=m u=tO}J x@ 15 w 1 |=yu=tDQ=BO

|Q}oxH}Dv "5|OQ@y=Q C=t}tYD wv R= |}=tv=H p�=Ut OQwtQO |Q}os}tYD xm xDmv u}= x@ xHwD =@Q_v QO u=tRsy =Q Q=}at u}OvJ xm |v=tO}J p�=Ut |UQQ@ =Pr 'OwW|t ?wULtu=tO}J p�=Ut QO Q=}at u} QDsyt xm u}= x@ xHwD =@ "OUQ|t Q_v x@ |QwQ[ OQ}o@|QDW}@ xHwD OQwt Q=}at u}= xm CU= |k]vt 'CU= �O=wt pkvwptL |xv} Ry�x@ QHvt Ov=wD|t xv} Ry Q=}at Q@ |vD@t |x}rw= u=tO}J ?=NDv= u}=Q@=v@ "OQ}o Q=Qku=tO}J uOQw;CUO x@ |=Q@l}DvS sD} Qwor=l} Qw_vtu}O@ "OwW |QDy@ |=y?=wHp�=Ut |=y?=wH Ov=wD|t |O=yvW}B l}DvS sD} Qwor= "CU= xOW O=yvW}B x}rw==Q =yu; OQ=wt |NQ@ QO |DL w OQw; CUO x@ |r@k |=ysD} Qwor= *|@ wNx@ =Q |}xvwtv|}=U=vW x@ QO=k xm =yxO=O |WWwB p}rLD pOt uOQm|]N =@ u}vJty "OyO Ow@y@=Q |Qo}O |=yQ=}at �xv} Ry� Q@ xwqa xm u=tO}J u} QD=Q=m 'CU= DMU u} QD=Q=m

"OwW|t u}}aD 'OyO|t Q=Qk Q_v Ot R}v|=ysD} Qwor= R= xr�Ut pL |=Q@ xm OwW|t O=yvW}B |D; C=k}kLD u=wvax@'OvQ@|t Q=m x@ VR=Q@ `@=D w ?=NDv= R=i QO =Q =yxO=O |WWwB p}rLD xm |Q=mD@==Qi|=ysD} Qwor= R= xO=iDU= =@ xQw_vtOvJ Cr=L QO p�=Ut u}= |UQQ@ "OwW xO=iDU=C=k}kLD Qo}O R= MOPSO w NSGA-II'NRGA Q}_v xQw_vtOvJ |Q=mD@==Qi

"CU= |D;

"p�=Ut |=y?=wH u} QDy@ "5 pwOHsD} Qwor= R= xOt; CUO x@ ?=wH u} QDy@ xr�Ut|O=yvW}B l}DvS

1j2'6j9j10'7j8'4j5'3 VC103'5j9'10'8j2'4j6'7j1 VC10Rs

3'5'8'10j9'4'7j2'6j1 VC10Ra

4j10'5'9j11'8'12j3'2'1j6'7 BA127'8'14j4j2j9'6'12'13j3j1j11'5'10 BA14

3'2'10'6'1'5j12'9'11'7'8'4 Nug121'4'3'2'14'13'7'8'9'11j10'15'6'5'12 Nug15

"u=tO}J KQ] |=Q@ =yQ=}at Q=Okt "6 pwOHC@Uv xrY=i CQw=Ht ptL |xv} RypmW |DU=wNQO |DU=wNQO O=wt pkv w

4 53 74 24 196 45 5677 83|xv} Rysm ?=wH\UwD xOWO}rwDl}DvS sD} Qwor=

6 22 96 08 712 61 5872

u=tO}J u} QD=Q=mR= xOt; CUOx@2-opt sD} Qwor=

DEA w

Q=yJ TBU "O};|t CUO x@ u=tO}J w}D=vQDr; 190 O=OaD 'l}DvS sD} Qwor= R= xOt;xOt; CUO x@ |=yxO=O C}=yv QO w 'x@U=Lt =yw}D=vQDr; |t=tD |=Q@ xOWQmP Q=}atQ=Okt pw= |xrLQt pOt |=QH= =@ "OwW|t |O=yvW}B *|}xrLQtwO DEA pOt OQ=wu} QD=Q=m w OQ=w swO |xrLQt pOt QO Q=Okt u}= TBU w O};|t CUO x@ uwr}UB=w |}=OD@= ?=wH |=yQ=}at R= l} Qy Q=Okt 6 pwOH QO "OwW|t |}=U=vW Z} waD|=yQ=}at =Q=m u=tO}J QO xmCU= u; Qou=}@ pwOH u}= C=aq]= "CU= xOW x�=Q= =Q=m"Ov=xDW=O |Q}osWJ Ow@y@ |DU=wNQO |xrY=i u=R}t w |DU=wNQO CQw=Ht u=R}tpwOH QO "CU= xOW xv} Ry |QDW}@ Q=Okt hQY ?@U =yQ=}at u}= V}=Ri= xD@r=

61

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wBp}rLD

wl}DvS

sD}Qwor=

R=xO=iD

U==yCWwv=B1. facility layout problem2. exible bay structure3. genetic algorithm4. data envelopment analysis5. multiple criteria decision making6. mixed integer linear programming7. degenerate solution8. ELECTRE9. facility relayout problem10. decision making unit11. analytic hierarchy process12. Viz Factory13. Minimax14. Holland15. Cook

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