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"OvwW|t |iQat |]N |R} Qxt=vQ@pOt C=[wQit "4"1"5

p}yUD w |QDWt xm 'xQo Qy QO xm CU= pL p@=k |DQwY QO pOt u}= %pw= ZQiQ}PBu=mt= |QDWt x@ p}yUD \UwD |yOT} wQU 'OvwW|t OQ=w u; x@ u=tRtyxOW?=UL w |k]vt w j}kO |}x]@=Q xm u; Qot Ow@ Oy=wNv umtt Qt= u}= "OW=@u}vJty w xQo l} x@ xOW OQ=w u=} QDWt |=y=[=kD =} |QDWt l} |=[=kD u}@x=Q= pOt QO x]@=Q u}= OW=@ xDW=O OwHw xQo u; QO OwHwt Cq}yUD C}iQ_

%CU= u}vJ xOWXj2F

capj �Xi2F

Di

|=yC}iQ_ ẁtHt O}=@ pOt uOw@ Q}PBpL |=Q@ xm OyO|t u=Wv x]@=Q u}= `k=w QO"OW=@ QDW}@ v |@=NDv= xQo Qy QO OwHwt u=} QDWt |=y=[=kD ẁtHt R= Cq}yUD"CU= xm@W QO =yxQo pm |xawtHt Q@=Q@ V |xawtHt jwi pOt QO %swO ZQixQo Qy x@ i xQo Qy xm swyit u}= x@ xOw@ xDUw}B xOW xDiQo Q_v QO xm@W =Hv}= QO"OQ=Ov OwHw xm@W u}= QO |}xrwR}= w =yvD T-=Q I}y w CU= pYDt xm@W QO j"CU= xm@W |=yxQo pm Q@=Q@ V |xawtHt OW xQ=W= xm u=vJ %swU ZQiQ_v QO Cq}yUD Q@=Q@ F |xawtHt w u=} QDWt |xawtHt =@ Q@=Q@ C |xawtHtxawtHt pm R= |}=yxawtHtQ} R Cq}yUD w u=} QDWt |xawtHt "OwW|t xDiQoQO "OvW=@ Q@=Q@ R}v =yxQo xawtHt pm =@ Ovv=wD|t O=OaD ^=Lr R= xm OvDUy =yxQo=} p}yUD l} Q=QkDU= pLt =} |QDWt l} Q=QkDU= pLt Ov=wD|t xQo Qy jwi pOtCq}yUD w =y|QDWt R= s=OmQy Ok=i Ov=wD|t Qo}O Cr=L QO =} 'OW=@ =yu; wO Qy

"OW=@ xm@W

80

u=}QoD=Q

=Uwu=}U

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t

"xQo 10 R= pmWDt |i=Qo "1 pmW

"pOt pLR= TB Cq}yUD w u=} QDWt Q=QkDU= pLt "2 pmW

"pOt pL R= TB Cq}yUD w u=} QDWt Q=QkDU= pLt "3 pmW

Q=QkDU= uwO@ '=yvD xm@W pw= |xrLQt QO "s} R=OQB|t p=Ft u}= K}[wD x@ xrLQt"OwW|t xO=O u=Wv 1 pmW QO u=} QDWt w Cq}yUD

Cq}yUD w u=} QDWt |=ypLt |iO=YD CQwYx@ xm@W |rm |UQQ@ R= TB|mWt nvQ =@ xm@W |O=a |=yxQo pmW u}= QO "�2 pmW� OvwW|t XNWtOv=xOWXNWt Cq}yUD |=Q@ |iO=YD CQwYx@ xm |}=yxQo =t= Ov=xOW xO=O u=WvQO xt=O= QO "CU= xOW XNWt |@; nvQ =@ u=} QDWt |=ypLt w RtQk nvQ =@"CU= xOW XNWt pOt pL R= TB Cq}yUD w u=} QDWt Q=QkDU= pLt 3 pmWx@ CtON Ci=} QO |=Q@ 2 |QDWt p=Ft u=wvax@ OwW|t xOy=Wt xm u=vJ|=[=kD"CU= xOQm CmQL xQo u=ty CtU x@ R}v 2 p}yUD xm OwQ|t 9 xQo CtUw u=} QDWt x}k@ |}=Hx@=H "CU= xOt; 3 pwOH QO Cq}yUD C}iQ_ w u=} QDWt

"OwW|t s=Hv= jwi K}[wD x@ xHwD =@ w 4 pwOH j@] R}v Cq}yUD'Qo}O |QDt=Q=B uDiQo Q_v QO =@ |twO pOt xOWx=Q= pOt |xaUwD Qw_vtx@

"CU= xOW xO=O xaUwD w |L=Q]

"pw= pOt |OOa p=Ft C}iQ_ w =[=kD Q}O=kt "1 pwOH10 9 8 7 6 5 4 3 2 1 V �C�F40 0 0 10 30 10 0 0 20 10 D(i)0 0 30 40 0 0 40 30 0 0 cap(j)

"d(i; j) = d(j; i) Q} R T} QD=t j@=]t %d(i; j)"pw= pOt |OOa p=FtQO xQo wO u}@ |=yCi=Ut "2 pwOH

i

10 9 8 7 6 5 4 3 2 1 d(i; j)

j

22 75 28 77 12 9 45 98 4 0 149 23 2 88 91 63 99 94 0 236 97 62 14 80 27 60 0 382 86 50 28 58 15 0 426 11 79 92 94 0 570 52 42 24 0 683 33 84 0 737 97 0 86 0 90 10

QO xQo wO u}@ xrY=i xR=Ov= u=wvax@ =Q =yu; w xOQw; CUO x@ EXCEL Q=Ri=sQv QOwO Qy |=Q@ &OwW|t h} QaD d(i; j) O=tv =@ i; j xQo wO u}@ xrY=i "s} Q}o|t Q_vl} QO 2 pwOH j@] xOt; CUO x@ Q}O=kt "d(i; j) = d(j; i) s} Q=O i; j xQoXNWt xQo wO Qy u}@ pY=wi |DQ=@ax@ w =yCi=Ut =D OvwW|t xO=O Q=Qk T} QD=t

"OwWV = f1; 2; 3; : : : ; 10gC = f1; 2; 5; 6; 7; 10gD = f10; 20; 10; 30; 10; 40gF = f3; 4; 7; 8gcap = f30; 40; 40; 30g

wO Qy =} 'p}yUD =} |QDWt CU= umtt xQo l} QO OwW|t xOy=Wt xm Qw]v=typ=Ft "OvwWv xO=O Q=Qk xQo l} QO =yu; R= s=OmI}y CU= umtt u}vJty "OvQ}o Q=Qk

%OwW|t xDiQo Q_v QO Q} R pmW x@ GAMS |U} wvxt=vQ@ QO jwi|OOa p=Ft 'xt=vQ@ |=y|OwQw u=wvax@ jwi C=[wQit |t=tD uDiQo Q_v QO =@|=R=x@ hOy `@=D Q=Okt "s}vm|t pL GAMS Q=Ri=sQv R= xO=iDU= =@ =Q xOW KQ]tQ@=Q@ cplex xOvvmpL R= xO=iDU= =@ xrUt pL u=tR w xOw@ 354 Q@=Q@ Q}O=kt u}= pLu=tR Q_v R= GAMS Q=Ri=sQv |=yxOvvmpL Qo}O p@=kt QO xm CU= x}v=F 0 45=Q pOt |=yQDt=Q=B 'hrDNt |=yCr=L |=Q@ xt=O= QO p=L "CU= xHwD p@=k pL

"OwW|t p}rLD w x} RHD u; G}=Dv w xO=O Q=Qk |UQQ@ OQwt

|t}UQD p=Ft "3"5xm@W u}= QO xm CU= xOW xDiQo Q_v QO xQo 10 R= pmWDt |}xm@W CtUk u}= QOQy =@ \@DQt p=mW= sUQ =@ w xrLQt xU |] "Ov=xDiQo Q=Qk |QDWt 7 w p}yUD 3

81

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}=tv=HQ@=Q@ |}=[=kD |=Q=O w C}iQ_ uwO@ xQo u}= pOt |Q}PBpL Qw_vtx@ "Ovvm CmQL

CU= u; xHwD p@=k |xDmv "OQ=Ov OwHw xQo u}= QO |QDWt I}y `k=w QO &CU= QiYu; |=Q@ OwWv Q=OxWON pOt pL C}r@=k xm u}= |=Q@ =yvD w xOw@ |R=Ht xQo u}= xm

"OQ=Ov |HQ=N OwHw |}xQo u}vJ |ak=w Cr=L QO &OwW|t xDiQo Q_v QO

swO pOt |[=} Q |Ov@pwtQi "1"6xOW h} QaD CQwY u=ty x@ R}v swO pOt QO pw= pOt |=yQDt=Q=B w =yT}Ov= x}rmw xOwRi= =yQDt=Q=B Qo}O Q@ =Q w uRw QDt=Q=B swO pOt QO =t= OvwW|t xDiQo Q=mx@ w

"CU= s=i |QDWt �C}ty=� uRw Q@=Q@ wi %s}=xOQm h} QaD u}vJ =Q u;

pOt C=[wQit "1"1"6pOt X=N \}=QW Crax@ 'pw= pOt C=[wQit Q@ xwqa swO pOt C=[wQit

%CU= h}YwD p@=k Q} R CQwYx@p=tDL= u}= 'OwW|t xO=O X}YND uRw l} pOt QO |QDWt Qy x@ xm =Hv; R= "1|x]}L QO u=} QDWt R= |[a@ xm CW=O Oy=wN OwHw |a]k O=ONQ O}=W =}'t |R=Ht xQol} uDiQo Q_v QO =@\}=QW u}= QO "OvwWv xDiQo Q=Qk |v=UQCtONpLt R= |}=Hx@=H s=ovy OvQ}o|tv Q=Qk |v=UQCtON |x]}L QO xm |v=} QDWt|v=UQCtON Qw_vtx@ xrUtC}rm R= uDU=m uwO@ w |R=Ht CQwYx@ u=W|rai|Dr=L QO s=N + 1 xQo u=wvax@ =Q xQo u}= "Ovvm|t CmQL xQo u}= CtU x@

"s} Q}o|t Q_v QO 'OW=@ N Q@=Q@ |R=Ht xQo uwO@ =yxQo pm O=OaD xm|DQ=@ax@ =} Dt = 0 |va} 'OQ=Ov OwHw |QDWt xOW h} QaD |R=Ht xQo QO "2|ak=w Cr=L QO xm OyO|t u=Wv ZQi u}= "CU=QiY Q@=Q@ QwmPt xQo |=[=kD

"OQ=Ov OwHw Q_v OQwt xm@W QO |}xQo u}vJ|DQ=@ax@ "CU=yxQo pm |=[=kD Q@=Q@ xOW xDiQo Q_v QO |R=Ht xQo C}iQ_ "3xty uOW xOQw;Q@ Qw_vtx@ ZQi u}= "capt = Pi6=tDi %CW=O s}y=wN

"OwW|t xDiQo Q_v QO u=} QDWt |=y=[=kD&CU= QiY Q@=Q@ jwi |R=Ht xQo R= p}yUD xQo Qy =} |QDWt xQo Qy |xrY=i "4QO =Q ZQi u}= "djt = 0; j 2 F; dit = 0; i 2 V %s} Q=O Qo}O CQ=@ax@

"s} Q}o|t Q_v QO t xQo uOw@ |R=Ht |=DU=Q"CU= 0 �Wi � 1 xm CW=O s}y=wN =yDv= QO "5

pOt |=yC}OwOLt w hOy `@=D "2"1"6%OvwW|t u=}@ Q} R CQwYx@ jwi pOt hOy `@=wD

Minimize f1 =Xi2C

Xv2V

xivDidiv +Xj2F

Xv2V

yjvdjv

Minimize f2 = � Xi2V;i6=t

Xv2V;v 6=t

xivDiwi

pm uOQmxv}tm |=DU=Q QO �� pw= pOt QO Q_v OQwt hOy u=ty �� pw= hOy `@=D'OW u=}@ pw= pOt KQW QO xm u=vJ w CU= Cq}yUD w u=} QDWt |=y|}=Hx@=H|r=L QO CU= u=} QDWt |=y|}=Hx@=H ẁtHt Qou=Wv f1 hOy `@=D R= pw= CtUk

"CU= Cq}yUD |=y|}=Hx@=H ẁtHt Qou=Wv u; swO CtUk xmCyH u; R= ẁtHt u}= "CU= |ivt Ctqa =@ C=R=}Dt= ẁtHt swO hOy `@=DpL p@=k xrUt pm hOy `@=D uOQmp=tQv xrLQt QO xm xOW xDiQo Q_v QO |ivt

"pw= pOt |t}UQD p=Ft Cq}yUD C}iQ_ u=R}t w u=} QDWt |=[=kD u=R}t "3 pwOHC}iQ_ u =R}t p}yUD pLt =[=kD u=R}t |QDWt u=mt

230 7 199 9121 6 88 10250 2 10 2

128 448 138 535 7

"pw= pOt |t}UQD p=Ft Cq}yUD w u=} QDWt OY=kt "4 pwOHxm xm@W xQo xQ=tW xQ=tW xm@W xQo xQ=tW xQ=tWu; CtU x@ p}yUD xQo CtU x@ |QDWt xm xQoOvm|t CmQL p}yUD Ovm|t CmQL u; |QDWt

9 2 9 93 7 3 101 6 9 2

3 41 11 51 7

u=} QDWt C}ty= uDiQo Q_v QO =@ pOt |xaUwD "6w =yxYNWt |t=tD "CU= xOW x=Q= |x}rw= pOt R= |}xaUwD `k=w QO swO pOtu; |=yC}OwOLt w =yQDt=Q=B w =yQ}eDt |t=tD R}v w 'pw= pOt QO xOW u=}@ \}=QWpOt QO xm CU= xDmv u}= QO pOt wO u}= Cw=iD =t= "CU= Q=QkQ@ R}v swO pOt |=Q@`k=w QO xm OwW|t xi=[= pOt x@ C}ty= =} uRw QDt=Q=B s=vx@ |O}OH QDt=Q=B swO|yOT} wQU |=Q@ |QDWt Qy Qo}O u=}@ x@ "CU= xm@W QO OwHwt u=} QDWt XDNtxm Ow@ Oy=wN |vRw |DQ=@ax@ =} C}ty= |=Q=O xm@W QO OwHwt Cq}yUD Q_v R=|[a@ hPL 'pOt QO uRw QDt=Q=B uDiQo Q_v QO "O@=}|t X}YND |QDWt u; x@u; R= C}ak=w u}= "CW=O Oy=wN p=@vO x@ |v=UQCtON |x]}L R= =Q u=} QDWt R=Q=Qk p}yUD l} |v=UQCtON |xOwOLt QO xm |v=} QDWt u=}t R= xm CU= CyH'OvwW|t ?=NDv= OvQ=O Q_v OQwt p}yUD |=Q@ |QDW}@ C}ty= xm |}=yu; OvQ}o|ts}kDUt \=@DQ= u; |=[=kD u=R}t =@ |QDWt l} x@ |yOT} wQU xm =Hv; R= =t=u=wvax@ 'u; C}ty= u=R}t QO |QDWt Qy |=[=kD ?Q[pY=L swO pOt QO OQ=OQU}t |yOT} wQU Qw_vtx@ =Q |QDWt l} Vv} Ro xm OwW|t ?=NDv= |QDt=Q=B'u}}=B |=[=kD w q=@ C}ty= |=Q=O |QDWt l} CU= umtt Cr=L u}= QO "OR=U|t|QDWt x@ C@Uv QDq=@ |=[=kD w u}}=B C}ty= |=Q=O |Qo}O |QDWt p@=kt QO w|xv} Ro QDt=Q=B wO u}= ?Q[pY=L Q}O=kt x@ xHwD =@ Cr=L u}= QO xm 'OW=@ |r@k|m} hPL ?=NDv= u}= xm =Hv; R= "OQ}o|t Q=Qk |v=UQCtON ZQat QO |@=NDv=C}rm QO pqDN= sOa |=Q@ 'OQ=O p=@vO x@ =Q |v=UQCtON |x]}L R= u=} QDWt R='CU= |Q}oT} wQU Qw_vtx@ p}yUD l} CtU x@ |QDWt Qy CmQL Qou=}@ xm pOtR= Vv} Ro |xrLQt QO xm |v=} QDWt =D xDiQo Q_v QO xm@W |=Q@ |R=Ht xQo l}xQo u}= CtU x@ u=W}=[=kD jkLD |=Q@ 'OvwW|t hPL |v=UQCtON |xOwOLt

82

u=}QoD=Q

=Uwu=}U

=@Qm|Oy

t

"xQo 15 R= pmWDt |}xm@W "4 pmW

"|}xQo 15 |xm@W QO Cq}yUD w u=} QDWt |iO=YD Q=QkDU= pLt "5 pmW"swO pOt |t}UQD p=Ft QO Cq}yUD C}iQ_w u=} QDWt |=[=kDQ}O=kt "5 pwOH

C}iQ_ u=R}t Cq}yUD u=mt =[=kD u =R}t u=} QDWt u=mt121 1 81 10120 2 10 4179 3 58 831 7 88 14136 8 44 12537 �|R=Ht xQo �16 29 2

23 570 1187 747 13

u=Wh=Oy= x@ |@=}CUO Qw_vtx@ =Q u=} QDWt w Cq}yUD OY=kt xm 6 pmW QOxm@W |rY= |=yxQo Q@ xwqa pmW u}= QO "CU= xOW xO=O u=Wv Ovm|t XNWtxOW xDiQo Q_v QO 'CU= |} woMU=B x@ QO=k u; |x]U=wx@ pOt xm '|R=Ht xQo l}|=yu=mt =} OY=kt xm xOW xDW=Po V}=tv x@ |rwOH pmW u}= |xt=O= QO "CU="Ovm|t XNWt u=W|rai pLt R= |}=Hx@=H R= Oa@ =Q Cq}yUD w u=} QDWt O}OHQw_vtx@ u=} QDWt w Cq}yUD |}=yv OY=kt Qou=}=tv 6 pmW OW xQ=W= xm u=vJxQo l} QO xm CU= XNWt pmW u}= QO "CU= CtON Ci=} QO w |v=UQCtONj@] "OW=@ xDW=O OwHw |Q}oT} wQU Qw_vtx@ |QDWt l} R= V}@ CU= umtt

C@Ft Ctqa =@ C=R=}Dt= ẁtHt uOQmxv}W}@ u=ty swO |rY= hOy `@=D =t= "OW=@|=Q@ �QDW}@ C}ty=� QDq=@ C} wrw= =@ |QDWt ?=NDv= f2 uOQmxv}tm R= hOy "CU=u}vJ w CU= pw= pOt |=yC}OwOLt uwJty =yC}OwOLt "CU= |v=UQCtON

%OvwW|t h} QaDXv2V

xiv = 1 8 i 2 CXv2V

yjv = 1 8 j 2 FXj

capj :yjv �Xi2C

Dixi 8 v 2 V

w 'xOW pL uOQmxv}tm hOy =@ f1 hOy `@=D |=Q@ pOt Q=@l} jwi pOt pL QOl} f2 hOy `@=D |=Q@ jwi pOt TBU "OwW|t pL uOQmxv}W}@ hOy =@ Qo}O Q=@=@ pOt TBU "OwW|t pL uOQmxv}W}@ hOy =@ Qo}O Q=@ w uOQmxv}tm hOy =@ Q=@

"OwW|t p=tQv CU= |@}mQD hOy `@=D l} xm Q} R hOy `@=Dmin(1(f1 f1 min)=(f1 max f1 min) + 2(f2 f2 min)=

(f2 max f2 min))

"OvQ}o|t Q=Okt l} w QiY u}@ xm OvDUy hOy `@=D Qy |=yuRw 1; 2x]@=Q u}= QOxDU@ u}vJty w xrUt QO OwHwt \}=QW x@ xDU@ xOWp=tQv hOy `@=D QO ?}=Q[ u}=\}=QW xOvQ}os}tYD OQi Q_v QO CU= umtt "OvwW|t u}}aD xOvQ}o s}tYD OQi x@QDq=@ C}ty= =@ u=} QDWt ?=NDv= x@ C@Uv =y|}=Hx@=H uOQmxv}tm xm OW=@ |}xvwox@|QDW}@ Q=Okt |=Q=O 1 ?} Q[ Cr=L u}= QO "OW=@ xDiQo Q=Qk |QDq=@ C} wrw= QO

1; 2 Q}O=kt u}=Q@=v@ "Ow@ Oy=wN jwi xOWp=tQv CQ=@a QO 2 ?} Q[ x@ C@UvQO w OwW|t u}}aD xOvQ}os}tYD OQi Q_v w xrUt \}=QW x@ xDU@ jwi CQ=@aQO

"OW Oy=wNv xDiQo Q_v QO |D}OwOLt OQwt u}=swO pOt |t}UQD p=Ft "3"1"6

p}yUD 5 u; QO xm CU= xOW xDiQo Q_v QO xQo 15 R= pmWDt |}xm@W p=Ft u}= QOx@ xrLQt Qy =@ \@DQt p=mW= sUQ =@ w xrLQt xU |] QO "OQ=O Q=Qk |QDWt 10 w

"s} R=OQB|t p=Ft u}= K}[wDQO OwHwt Cq}yUD w u=} QDWt uDiQo Q_vQO uwO@ =yvD xm@W pw= xrLQt QOQ=QkDU= pLt 5 pmW QO xrLQt u}= R= TB "OwW|t xO=O u=Wv 4 pmW QO xm@WCq}yUD w u=} QDWt Q=QkDU= pLt "OwW|t XNWt Cq}yUD w u=} QDWt |iO=YDu=} QDWt |=[=kD Q}O=kt "CU= xOW ?=NDv=|iO=YD CQwYx@|}xQo 15 |xm@W QOCW=O xHwD O}=@ "�5 pwOH� Ov=xOW u}at |iO=YD CQwYx@ R}v Cq}yUDC}iQ_ wxm CU= swyit u}= x@ CU= 10 Q@=Q@ |QDWt xQo xQ=tW p=Ft u=wvax@ xm |v=tR

"OwW|t QkDUt xm@W 10 xQ=tW xQo QO |rY= |xm@W QO |QDWt?=NDv= |iO=YD CQwYx@ xm Cq}yUD w u=} QDWt Q=QkDU= pLt 5 pwOH R=C}iQ_ w u=} QDWt |=[=kD Q}O=kt |t=tD u}vJty "OwW|t XNWt Ov=xOW"CU= xOW xOQw; pwOH u}= QO Ov=xOW ?=NDv= |iO=YD CQwYx@ xm R}v Cq}yUDw =y|oS} w Q@=v@ swO p=Ft QO "CU= xDiQo CQwY p=Ft QDy@ lQO Qw_vtx@ Qt= u}=xQo Qw_vt u}= x@ "CU= R=}v |R=Ht xQo l} x@ pOt pL |=Q@ 'pOt |=yxYNWtCU= pOt ZQi ?@U x@ R}v u}= w CU= xOW xDiQo Q_v QO |R=Ht xQo u=wvax@ 16xm@W |=yxQo O=OaD N � CU= N + 1 Q@=Q@ |R=Ht xQo xQ=tW xm u}= Q@ |vD@t

"�CU=

83

"""xm@Wl

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}=tv=H"Q=Ri=s Qv R= xO=iDU= =@ swO pOt R= |OOa p=Ft OvJ pL "7 pwOH

C=R=}Dt= ẁtHt |=yCi=Ut ẁtHt hOy `@=D Q=Okt u=tR xrY=i ?} Q[ xv}W}@ O=OaD O=OaD O=OaDu=} QDWt �|=yuRw� u=} QDWt \UwD xOW|] |=R= x@ xrUt pL R= =yxQo C}iQ_ u=R}t |=yxQo |=yxQo =yxQo|ivt Ctqa =@ Cq}yUD w xv}y@ pL x}v=F x@ Qo}Om} p}yUD =[=kD u=} QDWt Cq}yUD�2177 813 9610 88 1 9 524 84 100 1 1 100 1 9 524 84 110�2596 72 7342 1 065 581 005 100 1 1 100 98 22 120�2602 45 7679 5 1 075 683 935 100 1 1 100 105 25 130�2919 145 5573 5 0 78 960 795 100 1 1 100 115 25 1402461 80 6870 67 1 373 1107 605 100 1 1 100 125 25 150�3201 74 6734 66 0 6 1762 93 100 1 1 100 135 35 170�3773 775 5699 5 0 45 2393 31 100 1 1 100 145 45 190�4197 663 6049 0 16 3518 61 100 1 1 100 150 50 200�4611 69 5984 0 41 3809 23 100 1 1 100 185 65 250

x@ xHwD =@ u; G}=Dv w xOW pL Gams Q=Ri=sQv R= xO=iDU= =@ xOW x=Q= pOt"CU= xOW xOQw; 7 pwOH QO OvwW|t ?=NDv= |iO=YD CQwYx@ xm |Q}O=kt

|Q}oxH}Dv "7|}=tv=H QO =y|}=Hx@=H |R=Uxv}tm VywSB |rY= hOy 'OW xQ=W= xm u=vJQw_vtx@ "Ow@ lQLDt Cq}yUD w u=} QDWt R= xDUw}B |}xm@W QO lQLDt Cq}yUDswO pOt xm OW x=Q= |]N |R} Qxt=vQ@ pOt wO |rY= hOy u}= x@ |@=}CUOuO}UQ Qw_vtx@ pOt wO u}= QO "CiQo Q=Qk xar=]t OQwt pw= pOt |xaUwD u=wvax@hOy x@ pOt uO}UQ Q}Ut QO xm |O}rm |=yC}OwOLt w =yQDt=Q=B '|rY= hOy x@

"OW xDiQo Q_v QO OvQ=PoQ}F-=DlQLDt Cq}yUD w u=} QDWt R= xDUw}B |}xm@W QO |]N |R} Qxt=vQ@ pOt wOQyl} QDt |=[i QO h=Qol} CQwYx@ xm@W u}= xm |Qw]x@ 'CU= xOW xDiQo Q_v QOQo}O CQ=@ax@ "OW |@=} RQ= 'xOw@ |UO}rk= xrY=i u; QO OwHwt pY=wi |t=tD xmxDiQo Q=Qk xHwD OQwt |UO}rk= sQv =@ l} QDt |=[i l} u=wvax@ Q_v OQwt |=[iu}= \=kv |t=tD QO Ovv=wD|t xm@W u}= QO OwHwt Cq}yUD w u=} QDWt x}rm "CU=QO =yvD u=v; hkwD |=ypLt =t= Ovvm CmQL �� =yT-=Q xJ w =yp=} xJ �� h=QoxQo l} p}yUD w |QDWt Qy |=Q@ xm@W u}= QO "OW xDiQo Q_v QO xm@W |=yxQo|yOT} wQU =} |Q}oT} wQU |=Q@ p}yUD w |QDWt xm OW xDiQo Q_v QO OYktw |QDWt Qy CmQL V}=tv |=Q@ "Ovvm CmQL xQo u}= CtU x@ �p}yUD \UwD�xDiQo Q_v QO pOt s}tYD |=yQ}eDt u=wvax@ xm xOW h} QaD |}=yQ}eDt p}yUD

"Ov=xOWu}@ xrY=i w u=} QDWt |=[=kD u=wvax@ |}=yQDt=Q=B pOt wO Qy QO u}vJtyu}at xm |}=yC}OwOLt Q@ xwqa 'pw= pOt |O}rm C}OwOLt "OW h} QaD =yxQoCtU x@ =yvD R}v p}yUD Qy w xQo l} CtU x@ =yvD |QDWt Qy xm CU= u; 'OvOQmẁtHt R= Cq}yUD |=yC}iQ_ ẁtHt Rw=HD sOaC}OwOLt w 'OvmCmQL xQol}QDt=Q=B uDiQo Q_v QO |=DU=Q QO C}OwOLt u}= xm OW h} QaD u=} QDWt |=y=[=kD'=yC}OwOLt w =yQDt=Q=B w =yQ}eDth} QaD R=TB "CiQo pmW Cq}yUD |=Q@C}iQ_

"swO pOt |t}UQD p=Ft pL R= TB Cq}yUD w u=} QDWt Q=QkDU= pLt "6 pmW"swO pOt |t}UQD p=Ft u=} QDWt w C q}yUD OY=kt "6 pwOHxm@W xQo xQ=tW xQ=tW xm@W xQo xQ=tW xQ=tWCtU x@ p}yUD xm xQo CtU x@ |QDWt xm xQoOvm|t CmQL u; p}yUD Ovm|t CmQL u; |QDWt

14 1 2 211 2 14 47 3 16 52 7 7 78 8 8 8

16 1011 1116 128 1314 14

"Ov=xOW XNWt RtQk |=yErFt =@ Cq}yUD w |@; Q}=wO =@ u=} QDWt h=Qo |=tvy=Q|rY= 8 xQo QO p=Ft u=wvax@ "OvyO|t u=Wv =Q xm@W |rY= |=yxQo |mWt Q}=wO|QDWt OYkt xrUt pL R= Oa@ Ow@ xDiQo |=H u; QO |QDWt l} w p}yUD l} xm

"OwW|t QDu=U; Q=}U@ 6 pmW lQO 6 pwOH R= xO=iDU= =@ "CUy R}v 13

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=Uwu=}U

=@Qm|Oy

t QO OwHwt \}=QW Qo}O x@ xHwD =@ u=wD|t =Q Q}eDt u}= "CU= EL@ OQwt ẁ[wt =@"OQm xi=[= pOt x@ xrUt

pt=aQ}e Ovi=OB p=Ut OQ@ Q=m 'lQLDt Cq}yUD |}=tv=H p=Ut |xRwL QO=D OwW|t s=Hv= xm CU= |D=t=Ok= xawtHt pt=aQ}e Ovi=OB "CU= Q}otWJ Q=}U@|=QH= R= hOy "OUQ@ OwN u=R}t u} QDsm x@ |r=tDL= C=Q=UN 'nvH RwQ@ CQwYQOC=R}yHD w x}v@= w |v=Uv= |wQ}v |Q}PB?}U; R= uDU=m pt=a Q}e Ovi=OB |=yKQ]CqtL seQ|ra QwWm =} xk]vt O=Qi= |x}LwQ ^iL 'syt w T=UL w |D=}L|=yR=}v u}t-=D w |}=v@ Q} R C=tON w =yC}r=ai Q=QtDU= w utWO ?QNt w xv=tYN|r=Ft u=wvax@ "CU= nvH R= |W=v |v=QL@ \}=QW QO QwWm xQ=O= sw=OD w |D=}LC=U}U-=D |=Q@ x=vB QU O=H}= w =iDN= 'Q=DDU= x@ u=wD|t pt=aQ}e Ovi=OB R= xO=UuDiQo Q_v QO Q_v OQwt Q=DWwv =@ xRwL u}= \=@DQ= "OQm xQ=W= l} SD=QDU= w sytu=} QDWt u=wvax@ |ovH |xk]vt l} QO xm CU= |ovH |=y=t}B=wy R= |}xawtHtQ_v QO |v=UQCNwU |=y=t}B=wy u=wD|t =Q pOt Cq}yUD "OvR=wQB p=LQO pOt|v=UQCNwU |ovH |=y=t}B=wy x@CNwU Ow@tm CQwY QO R=wQB pw] QO xmCiQo'OQ=O |v=w=Qi OQ@ Q=m |=y|OwHwt pQDvm EL@t 'Q}N= |xRwL QO xm =Hv; R= "Ovvm|tQO u; |xaUwD x@ xOWx=Q= pOt QO |OwHwt |=yxv} Ry uDiQo Q_vQO =@ u=wD|t|xOvQ=OQ@ QO OwN |OwHwt |=yxv} Ry u}= "CN=OQB |R} Qxt=vQ@ |=ypOt |=[ixi=[= pOt x@ =Q =yu; u=wD|t xm CU= |Q=Oxov |=yxv} Ry w VQ=iU |=yxv} Ry'Q}N= xOW x=Q= |=ypOt x@ |OwHwt pQDvm EL@t uOwRi= |=yOQ@ Q=m Qo}O R= "OQm\UwD |}xO=H |xm@W QO CNwU OvtR=}v lQLDt |=ywQOwN CNwU R=}v `iQ

"CU= u=UQCNwU lQLDt |=ywQOwN

\UwD CU= Cq}yUD w u=} QDWt |=y|}=Hx@=H ẁtHt xm |iOy `@=D =@ pw= pOt"OW |HvUQ=@Da= w 'pL Gams Q=Ri=sQv

xaUwD u=} QDWt |=Q@ �C}ty=� uRw QDt=Q=B uDiQo Q_v QO =@ pw= pOt xt=O= QO|=yC}OwOLt w =yQDt=Q=B '=yQ}eDt |t=tD "OW x=Q= swO pOt u; R= TB w OW xO=Ou=} QDWt |=Q@ uRw QDt=Q=B swO pOt QO xwqax@ "CU= Q=QkQ@ R}v swO pOt |=Q@ pw= pOtw u=} QDWt |=y|}=Hx@=H pm uOQmxv}tm Q@ xwqa u; hOy `@=D w OW xDiQo Q_v QOQO "CU= |v=UQCtON |=Q@ QDq=@ C} wrw= =@ u=} QDWt ?=NDv= OOYQO Cq}yUDu=} QDWt R= |NQ@ hPL swyit x@ u}= w OW xi=[= pOt x@ C}ty= QDt=Q=B swO pOtZk=vD QO pOt |rY= C=}[Qi =@ ẁ[wt u}= =t= 'CU= |v=UQCtON x]}L R=w xOW xi=[= xm@W x@ |R=Ht xQo l} Zk=vD u}= `iQ |=Q@ ?@U u}tyx@ CU=

"OW pL xm@W QO |R=Ht xQo u}= OwHw ZQi =@ pOt

xOWx=Q= pOt |xaUwD |=Q@ |}=yO=yvW}B "8pL |=Q@ |Q=mD@==Qi |=ysD} Qwor= R= xO=iDU= xOWs=Hv= VywSB QO |U=U= O=yvW}B|=ypOt xm =Hv; R= "CU= Q=Ri=sQv \UwD pL p@=k Q}e w O=} R |=yxQo O=OaD QO pOt'Ov=|]N lQLDt Cq}yUD |}=tv=H QO =y|}=Hx@=H uOQmxv}tm hOy =@ xOW x=Q=xaUwD Qw_vtx@ |UQQ@ OQwt EL=@t |=DU=Q QO Qo}O |}=yQDt=Q=B w =yQ}eDt uOwRi=

"OwW|t O=yvW}B '|]N Q}e Cr=L x@ |]N Cr=L R= uOQmp}O@D |DL w pOt\@DQt |=yQ}eDt xrtHR= u=} QDWt x@ |v=UQCtON u=tR COt uDiQo Q_v QO

=yCWwv=B1. Basch2. kinetic data structure3. certi�cate4. responsive5. e�ciency6. compactness7. locality8. un Capacitated facility Location problem

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