Modelling and Solution for Assignment Problem

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1 Mdellig ad Slui f Assigme Pblem Liyig Yag Mighg Nie Zhewei Wu ad Yiyg Nie Absac I his pape he mixed-iege liea pgammig (MILP) f mii assigme is fmed ad a slui called Opeais Maix is peseed ad cmpaed wih he sluis f exhausi ad MILP Theeical aalyses ad umeical ess shw ha he peais maix ae efficie well-implied eumeai f bh mii ad glbal-miimum assigme pblems Keywds Assigme pblem mehd f exhausi mixed-iege liea pgammig (MILP) peais maix I INTRODUCTION HE glbal-miimum assigme pblem is descibed see eg [] by laguage as: Thee ae peple ad ass T Each pes cmplees ad ly cmplees a as The payme i each pes f a as is give The pblem is which pes cmplees which as such ha he al payme is miimum? Le x be he 0- decisi vaiables whee x = epeses pes i f as j ad x = 0 hewise The glbal-miimum assigme pblem may be descibed as fllws j= j= mi z = c x s x = ( j = ) x = ( i = ) x =0 i j N { } whee c epeses he payme i pes i f as j c give a eal cs maix The pblem () elemes is a iege liea pgammig (ILP) If he csai cdii x = 0 is eplaced by 0 x i () he i becmes he elaxed liea pgammig (RLP) We may Mauscip eceived Ja 7 008: Revised vesi eceived Jue This w was supped by he Naial High Techlgy Reseach ad Develpme Pgam 863 -Pgam (007AA04Z50-) L Y Yag is wih he Sae Key Labay f Rbics Sheyag Isiue f Aumai (SIA) Chiese Academy f Scieces (CAS) ad he Gaduae Schl CAS (yagliyig@siac) M H Nie is wih Sheyag Digial Chia Limied Sheyag 0004 PRChia (iemh@digialchiacm) Z W Wu is wih he Sae Key Labay f Rbics SIA CAS N4 Naa S Sheyag 006 PRChia (wuzw@siac) YY Nie cespdig auh is wih he Sae Key Labay f Rbics SIA CAS N4 Naa S Sheyag 006 PRChia (phe: ; fax: ; yy@siac) () aually slve he pblem () by ILP mehds f isace by he ismeic suface mehd see [] All he ILP mehds ae deived fm he RLP mehds ad hee ae decisi vaiables i () The flps slve ce RLP f () ae 7 usually O(( ) ) = O( ) s ha fm he pi f view f cmpuaial cmplexiy he ILP mehds f () ae bee ha he Hugaia algihm see eg [] If c epeses he pfi gai by pes i f as j i () he he glbal-miimum assigme pblem may be egaded as he assigme pblem f imum gss pfi Le C = ( c ) be he egaive pfi maix ad slve he glbal-miimum assigme pblem f he egaive pfi maix The egaive value f bjecive fuci is he imum gss pfi If he payme is udesd as cs deplei he he bjecive fuci f he glbal-miimum assigme pblem () is easable Hweve if he payme is udesd as ime he he bjecive fuci f () is always easable The al ime exped i peple f hei ass is ieesed hee I is impa miimize he imal ime exped i sme pes f his as Theefe he bjecive fuci f () shuld be mdified Tha is mi z = { c x } i j s x = ( j = ) j= x = ( i = ) { } x =0 i j N This is he mahemaical mdel f he mii assigme pblem I is a iege pgammig whee he bjecive fuci is liea Thee is evide elai bewee he sluis f () ad () Assigme pblem liea pgammig (LP) ILP MILP ad hei applicais have bee eseached f may yeas see eg [34678] I his pape he mehd f exhausi f he iege pgammig () is aalyzed fis ad fems The he MILP f he mii assigme pblem () is fmed ad he ismeic suface mehd f he MILP is discussed A slui called Opeais Maix is peseed f he mii assigme pblem () is peseed ad applied slvig he glbal-miimum assigme pblem () Fially he aalyses f cmpuaial cmplexiy ad umeical ess shw ha he peais maix ae efficie well-implied () Issue Vlume

2 eumeai see [5] f bh mii ad glbal-miimum assigme pblems II THE METHOD OF EXHAUSTION FOR () The basic idea f he mehd f exhausi is cmpae all he bjecive fucis f feasible sluis diecly idiecly Afe he cmpais is cmpleed he pimal slui is fud aually Ufuaely umbe f he feasible sluis f geeal iege pgammig mixed-iege pgammig iceases wih scale f pblem fas f ay cmpue The feasible slui f he mii assigme pblem () f example is a assigme Γ cespdig ad 0 f 0- maix whee all he elemes ae lcaed i diffee ws ad clums f maix The bjecive fuci z f each assigme Γ is he imal eleme cespdig i he cs maix C = ( c ) The assigme Γ miimizig he bjecive fuci z is a slui f () The slui is uique pssibly bu he value f he pimal bjecive fuci is uique Le { j j j } be a pemuai f { } Se a j = a j = a = i j 0- maix Thus a feasible assigme Γ is fmed whee he bjecive fuci z = { cj c j c } Γ j Each pemuai { j j j } cespds a assigme Γ s ha he sum al f assigmes is! Whe is big eugh i Silig s facial fmula! π ( / e) This shws ha he sum al f assigmes iceases i expe wih I is cveie slve () by he mehd f exhausi PC whe exceeds by 0 III AN MILP MODEL Iducig a eal vaiable y + we ca impve he mii assigme pblem () i MILP The MILP equivale () is as fllws mi z = y + s x = ( j = ) j= x = ( i = ) y+ cx 0 ( i j = ) x =0 i j N { } The MILP (3) is equivale he pblem () clealy whee hee is ly e eal vaiable y + bu hee ae me csai iequaliies i cmpais wih () The ismeic suface mehd f ILP see [] may be als applied MILP see [3] I de apply he ismeic suface mehd f he MILP (3) (3) shuld be chaged i he fllwig caical fm (3) z = y + j= j= s x 085 x -5 ( j = ) (4) x 085 x -5 ( ) y+ cx 0 ( i j = ) x =0 i j N { } The ismeic suface mehd f he MILP (4) is deived fm he slui f he fllwig RLP z = y + j= j= s x 085 x -5 ( j = ) (5) x 085 x -5 ( ) y+ cx 0 ( i j = ) x 0 x -0 ( i j = ) Thee ae + vaiables ad csai iequaliies i he RLP (5) The flps slve ce RLP (5) ae me ha O ( 7 ) I de ge a pimal slui f he MILP (4) geeally we eed slve he RLP (5) f imes Theefe he flps slve he MILP (4) ae geeally O ( 9 ) usig he ismeic suface mehd Wha is me he + hypeplaes deemie pimal slui f he RLP (5) csiue ill-cdii liea equais Geeally we ca ge he slui f (5) whaeve usig he ismeic plae mehd see [4] MATLAB simplex mehd Thus i ca be see ha sme ILP MILP f decisi pblem ae difficul slve IV THE OPERATIONS ON MATRIX FOR () The peais maix f he mii assigme pblem () fid he slui diecly fm he cs maix C = ( c ) ad he bjecive fuci hee he subscip f cs maix C epeses he de f maix We aage he paymes c i de fm small lage whee sme paymes ae pssibly equal each he S ha hese paymes ae aaged i de fm small lage i he ( ) eal umbes wih diffee levels R R R ( R < R < < R ) whee hee ae s paymes c equal R ( ) Clealy s ad s = = Le R be a eal umbe lage ha equal R R may be als egaded as he sig If = amely all he c = R he hee ae! sluis f () he bjecive fuci value f evey slui is R Thee is ham i suppsig > I is easy see R = { c } = { c } ad i j j i R c i j j i i j R = mimi{ c } = mimi{ c } Bu mi = mi{ } ad Issue Vlume 008

3 c Rmi = mi{ c } ae equal each he pssibly Le j i c R = { Rmi Rmi} We have Ppsii The pimal bjecive fuci value f he * mii assigme pblem () z is lage ha equal R [Pf] Each pes cmplees ad ly cmplees a as because f he csai cdiis f () The payme exped i he las pes f his as is lage ha equal c R ad lage ha equal mi R mi as well Theefe he payme z * R All he elemes lage ha R i he cs maix C = c ) ae cveed by R This cveig is called basic ( cveig Via Ppsii afe he basic cveig f C R is he lages ucveig eleme ad hee is a ucveig eleme i each w ad each clum a leas If hee exiss a feasible assigme Γ i all he ucveig elemes he Γ is a pimal slui f () Ppsii Assume ha afe he basic cveig f C ) ucveig elemes i each w ad each hee ae s ( s clum a leas Remvig he w ad clum whee sme ucveig eleme is lcaed a he css pi we ge he esidual maix f de C whee hee ae s ucveig elemes i each w ad each clum a leas [Pf] I is sufficie pve f s = Assume ha afe he basic cveig f C hee ae w ucveig elemes c ad c ( c c R ) i he i -h w Sice hee ae w ucveig elemes i he j -h clum he j -h clum f C a leas hee is a ucveig eleme less ha equal R i each w ad each clum f C a leas afe he i -h w ad he j -h clum he i -h w ad he j -h clum f C ae emved Similaly assume ha afe he basic cveig f C hee ae w ucveig elemes c ad c i he j -h clum i j Thee is a ucveig eleme less ha equal R i each w ad each clum f C a leas afe he j -h clum ad he i -h w he j -h clum ad he i -h w f C ae emved Se : = C : = C i he begiig ad mae he basic cveig f he maix C = ( c ) Fid he sequece umbe f R i seies R R R ad le R = R The elemes lage ha equal R + = R i he cs maix + C have bee cveed by R The algihm peais maix f he mii assigme pblem () csiss f he fllwig seps: Ivesigae he ucveig elemes i evey w ad clum f C Suppse hee exiss w clum whee ly e eleme is ucveed ad he ucveig eleme is c pq Recd he miimal c (pimal assigme i sigle ey) pq ad emve he p -h w ad he q -h clum f he maix C The esidual maix f de is C Se : = C : = C ad u Sep Ivesigae he ucveig elemes i evey w ad clum f C Suppse ha each w clum has w ucveig elemes a leas Tu Sep 3 3 The pimal bjecive fuci value f C is R Each w clum f C has w ucveig elemes a leas Deflae C wih e de accdig Ppsii Thee ae may selecis f ecded ucveig eleme c pq duig he deflai The selecive sequece is as fllws I) Remved he p -h w he q -h clum cais he leas ucveig elemes II) The umbe f elemes caied i he p -h w ad he q -h clum is miimum III) Recd he miimal c pq (pimal assigme i sigle ey) Se : = C : = C afe deflai Tu Sep if If { c pq = } is lage ha R = R he he R cveig des mae a feasible assigme Cside he R cveig f : = + ad pefm agai he ex cveig cicle f Sep - 3 uil { c pq = } is equal R (he pimal bjecive fuci value) Ppsii 3 Assume ha he paymes f cs maix C ae aaged i de fm small lage i he eal umbes wih diffee levels R R R whee hee ae s i paymes c equal R i ( i = ) ad ha afe he basic cveig f C he sequece umbe f R i seies R R R is ( ) The usig + cveig cicles a ms he peais maix fid a slui f he mii assigme pblem () [Pf] Sice f he fis cveig cicle hee is a ucveig eleme i each w ad each clum a leas w ucveig elemes ae ecded a leas i he fis cveig cicle If ucveig elemes ae ecded i he fis cveig cicle he he peais maix fid a pimal slui wih he basic cveig Ohewise iceasig a cveig cicle f example : = + he level f ucveig elemes is iceased R + ad he umbe f ucveig + elemes is iceased s Sice i si = he peais maix fid a slui f () usig + cveig cicles a ms We shall pve ha he peais maix give he pimal slui f he pblem () i well-implied eumeai A eumeaive algihm is called well-implied eumeai see [5] ha meas i ca bai he pimal Issue Vlume

4 slui i fac wih ly a few eumeaed feasible sluis whee ly a few is udesd as he umbe f eumeaed feasible sluis is a plymial fuci f dimesi f slui space ad umbe f csais ad i fac is udesd as sme pbabiliy bai he pimal slui is equal The peais maix via Ppsii 3 bai a slui f he pblem () wih ly a few eumeaed feasible sluis I is ecessay pve ha sme pbabiliy give he pimal slui wih peais maix is equal The fllwig hypheses cceig pbabiliy calculai ae cfm ealiy U) All he elemes f cs maix C ae equally liely disibui; U) Afe R cveig f C all he elemes ae divided i w classes cveig elemes ad ucveig elemes hee is diffeece bewee he elemes f each class Ppsii 4 Assume ha he umbe f ucveig elemes f cs maix C whee hee is a ucveig eleme i each w ad each clum a leas is m The m ad he pbabiliy p ( Z m) bai he pimal slui wih peais maix f C is equal whe m= m [Pf] Sice f he -de cs maix C hee is a ucveig eleme i each w ad each clum a leas clealy m If m = he p ( Z ) = because he ucveig elemes ae disibued diffee w ad clum f maix Sep f peais maix ca give he pimal slui If m he p ( Z m) = as well This is easily pve by iducive mehd I fac pz ( m m ) = hlds clealy f he 3-de cs maix wih 6 ucveig elemes a leas Suppse ha he cclusi hlds f he -de cs maix Le = + If hee is ly e ucveig eleme i sme w clum f cs maix he afe deflai f maix wih peais maix hee is e cveig eleme a ms i he -de cs maix ad he cclusi hlds clealy f he -de maix If hee ae w ucveig elemes i each w ad each clum f cs maix a leas he afe deflai f maix wih peais maix hee ae ( + ) 3( + ) + = ucveig elemes a leas f he -de cs maix ad he cclusi hlds as well I is pssible ha hee is feasible slui f he assigme pblems f he basic cveig f he cs maix C whe < m < Theefe i is pssible ha p ( Z m) < f < m < Ppsii 5 Assume ha he umbe m f ucveig elemes f cs maix C saisfies m ( + )/ ad ha duig he deflai f each -h de maix ( = ) wih peais maix ucveig elemes ae emved a ms he he pbabiliy bai he pimal slui by peais maix p ( Z m) = [Pf] I is easy pve he ppsii by iducive mehd I fac he equaliy pz ( m m ( + )/) = hlds f = clealy Suppse ha he ppsii hlds f = s amely pz ( ms m s ss ( )/) = whee ms epeses he umbe f ucveig elemes f he ( s )-h de maix Le = s ad he umbe f ucveig elemes f he s -h de maix ms s( s+ )/ Sice s ucveig elemes ae emved a ms whe he s -h de maix is deflaed wih peais maix he umbe f ucveig elemes f he ( s )-h de deflaed maix ms ss ( + )/ s= ss ( )/ Theefe p( Z m s ms s( s+ )/) = I is easy see ha f ay cveig f cs maix C he exi f peais maix is always Sep whehe he cveig has feasible slui f pblem () Assume ha hee ae s ucveig elemes i each w ad each clum f sme cveig a leas hus he umbe f ucveig elemes m s whee s If s = hlds fm sa fiish duig he deflais f maix wih peais maix he he slui is clealy cec Hweve if s > ccued duig he deflais he he slui is always cec F example C = whee epeses ucveig eleme ad 0 cveig eleme Thee is a feasible slui a leas f maix C ad hee ae 4 ucveig elemes s = Fu ucveig elemes ae emved whe he maix is deflaed wih Sep 3 f peais maix ad hee ae 4 selecis f ecded ucveig eleme If w ad clum 5 ae emved he he peais maix ca give feasible slui Wih he excepi f his he peais maix fid he feasible slui Theefe whe s > he peais maix misae pssibly he cveig ha has slui f he cveig ha has slui I de avid pssibly misae abu his case he peais maix shuld deal specially wih he cs maix Tha is he ecded eleme is mdified aificially as he sig he he peais maix ae applied he mdified cs maix agai This special eame guaaees ha he peais maix give always a cec slui f he pblem () Ppsii 6 Assume ha hee ae s ucveig elemes i each w ad each clum f cs maix C a leas whee s If he cs maix is mdified aificially whe he cclusi f feasible slui is give by peais maix f s > he he pbabiliy bai he cec slui f pblem () by peais maix p ( Z m s) = [Pf] Sice he ecded eleme becmes i ad is cveed f each aificial mdificai hee ae s Issue Vlume

5 ucveig elemes a leas i each w ad each clum f cs maix mdified aificially If s > ad he cclusi f feasible slui is sill give by peais maix he he cs maix is aificially mdified agai Theefe he esul pefm peais maix f sme cs maix mdified aificially is ha a feasible slui is baied ha s = hlds fm sa fiish duig he deflais ad feasible slui is baied The peais maix give a cec slui always amely p ( Z m s) = Ppsii 7 The peais maix wih mdifyig aificially cs maix f he mii assigme pblem () ae well-implied eumeai Usig + cveig cicles a ms he peais maix fid he pimal slui whee ad ae specified as i Ppsii 3 [Pf] If hee is feasible slui f he pblem () f he basic cveig f cs maix C he via Ppsiis 4-6 he pimal slui is fud by he fis cveig cicle Ohewise he ex cveig cicle is pefmed Iceasig a cveig cicle e level f ucveig elemes is iceased Usig + cveig cicles a ms via Ppsii 3 a feasible slui is fud Suppse ha he feasible slui is fud a he ealies j -h cveig cicle The hee is feasible slui befe he j -h cveig cicle Tha is i de fid feasible slui j levels f R cveig f cs maix shuld be iceased fm he level R f basic cveig Theefe he pimal bjecive fuci value equals R + j whee R = R The slui fud by peais maix i he j -h cveig cicle is he pimal slui T pefm Sep - 3 f he -de maix he flps ae O ( ) ; deceases fm s ha pefm a cveig 3 cicle he flps ae O ( ) T pefm + cveig cicles a ms he pimal slui is fud Theefe eglec he imes f aificial mdificais he flps slve he mii assigme pblem () by peais maix ae O ( 5 ) a ms The cmpuaial cmplexiy f he peais maix is gealy lwe ha he cmpuaial cmplexiy slve he MILP (4) V TO APPLY OPERATIONS ON MATRIX TO SOLUTION OF () Sice he miimal c pq i Sep ad Sep 3 is ecded amely he pimal assigme i sigle ey is adped i he algihm he slui f he pblem () baied wih peais maix is fe he slui f he pblem () Hweve he peais maix may ecessaily slve he glbal-miimum assigme pblem () if he cs maix is asfmed i he elaive cs maix befe R cveig I he cs maix C = ( c ) evey eleme f each w subacs he leas eleme f he w (w cveial umbe) ad evey eleme f each clum subacs he leas eleme f he clum (clum cveial umbe) The ew cs maix B = ( b ) afe such a asfmai is a egaive maix ad hee is e eleme 0 a leas i each w ad each clum Cespdig he eleme b = 0 he sum f he i -h w cveial umbe ad he j -h clum cveial umbe equals he payme c Via he Hugaia algihm see eg [] he cs maix B is equivale C f he assigme pblem () The eleme f B epeses he elaive payme If all he 0-elemes f B ca csiue a feasible assigme he he sum f he elemes f C cespdig he assigme amely he sum f w ad clum cveial umbes is he pimal bjecive fuci value Nw he peais maix wih he pimal assigme i sigle ey ae applied he elaive cs maix B The basic cveig is he cveig f ze elemes If a feasible assigme ca be fud by he basic cveig f B he he assigme is a slui f he pblem () Ohewise usig psiive eleme cveig f he lwes level he peais maix give a slui f he pblem () f B whee he leas ucveig elemes ae caied as may as pssible Via he pimal assigme i sigle ey his slui is geeally he slui f he pblem () The bjecive fuci value f he pblem () equals he sum f cespdig elemes f B plus he sum f w ad clum cveial umbes Nice ha he slui f he pblem () give by he peais maix wih he pimal assigme i sigle ey f he elaive cs maix B is always he slui f he pblem () F example he deflaed elaive cs maix f sme cveig cicle is as fllws B 3 0 = B3 = The peais maix fid he cec sluis f he mii assigme pblem () f B B ad bh bjecive 3 fuci values ae equal Bu hese sluis ae he sluis f he glbal-miimum assigme pblem () Sice he peais maix wih he pimal assigme i sigle ey will be applied -de deflaed elaive cs maix ulimaely he abslue payme sum f he las w assiged elemes shuld be veified afe he glbal-miimum assigme pblem () is slved by he peais maix Thee is small pbabiliy mae a misae eve if he veificai has bee de I de esimae a uppe bud f he pbabiliy p (E) whee he eve E epeses mae a misae by he peais maix f he pblem () a hyphesis is added besides he hypheses U) ad U) U3) Pvided ha he slui f peais maix f he pblem () is baied wih basic cveig f elaive cs maix hee ae icmplee ucveig elemes i he elaive cs maix such ha he sum f hem is less ha he sum f slui elemes f he pblem () Issue Vlume

6 The umbe f hese eves is less ha equal + a imum f cveig cicles give by Ppsii 3 Hyphesis U3) amplifies he pbabiliy f eve E bu simplifies he esimai f p (E) Ppsii 8 Wih he elaive cs maix he peais maix f he glbal-miimum assigme pblem () ae a well-implied eumeai he pbabiliy fid he pimal slui wih + cveig cicles a ms eds wih iceasig [Pf] The peais maix ae applied he elaive cs maix via Ppsii 7 fid he pimal slui f he mii assigme pblem () wih + cveig cicles a ms Whe he cveig cicles ae lage ha i is pssible mae a misae sice he pimal slui f pblem () is egaded as he pimal slui f pblem () Accdig Hyphesis U3) pvided ha he slui f pblem () is baied wih icmplee 0-elemes f elaive cs maix hee ae icmplee ucveig elemes such ha he sum f hem is less ha he sum f slui elemes f he pblem () I he + eves a ms he imal pbabiliy eve is ha hee ae elemes equal 0 excep e cveig eleme We w calculae he pbabiliy fm a feasible slui f pblem () wih hese elemes Accdig Hypheses U) ad U) hee ae C disibuis f e psiive cveig eleme ad 0-elemes i B whee! disibuis ae diffee feasible sluis f he pblem () S ha he pbabiliy fm feasible slui f he glbal-miimum assigme pblem () wih hese elemes is τ =! /( C ) = (( )!) ( )!/( )! The pbabiliy f eve E saisfies p ( E) ( + ) τ Wih Silig s facial fmula esimae ( + ) τ igh side f p (E) Whe is big eugh ( + ) τ π ( ) /( + )(( ) /( + )) ( ) ( /( + )) / e Theefe wih he peais elaive cs maix B f pblem () he pbabiliy mae a misae p (E) eds 0 wih iceasig VI NUMERICAL TESTS Via cmpais f cmpuaial cmplexiy he peais maix f he mii assigme pblem () he glbal-miimum assigme pblem () ae fa efficiecy i cmpais wih mehd f exhausi ad MILP mdel We eed ly es he peais maix f pblems () ad () wih highe de cs maix The cs maix C is geeaed by cmpue i he umeical ess The ess f he peais maix f pblems () ad () ae cmpleed a PC The pgam is u usig MATLAB70 ude WidwsXP Icludig he ime geeae cs maix he cmpuaial CPU ime equied a PC is give i secds Example The maix elemes c = 0 + 5i+ 5 j ( i j) cii = i; (6) = The pimal bjecive fuci value f pblem () is clealy Thee ae w pimal sluis (f eve umbe ) a ms The diagal elemes fm lef lwe igh uppe ae always a pimal assigme The sluis bewee pblems () ad () ae ally diffee f he cs maix wih elemes (6) Via he elaive cs maix he uique slui f () is he diagal elemes fm lef uppe igh lwe The pimal bjecive fuci value is Example The maix elemes c = 0 + 5i+ 5 j ( i j) cii = i; (7) = The pimal bjecive fuci value f pblem () is (f eve umbe ) (f dd umbe ) Thee is uique pimal slui f eve umbe Thee ae 4 pimal sluis f dd umbe The diagal elemes fm lef lwe igh uppe ae always a pimal assigme Via he elaive cs maix hee ae may sluis f pblem () f he cs maix wih elemes (7) May feasible assigmes wihu diagal elemes ae all pimal sluis ad he pimal bjecive fuci value is 5+ 5 Example 3 The maix elemes c = 0 + 5i+ 5 j; (8) = The pimal bjecive fuci value f pblem () is The pimal slui is uique amely he diagal elemes fm lef lwe igh uppe Via he elaive cs maix hee ae! sluis f pblem () f he cs maix wih elemes (8) Ay feasible assigmes ae all pimal sluis f () ad he pimal bjecive fuci value is 5+ 5 Le =5 ad =6 i de ivesigae he sluis f peais maix f Examples -3 Table gives he w ad clum assigmes ad he paymes slve he pblem () Nice ha hee ae w sluis f =6 ad Example ad ha hee ae fu sluis f =5 ad Example The sluis give by Table ae diagal assigmes fm lef lwe igh uppe while hey accd wih he pimal assigme i sigle ey The pimal assigme i sigle ey f cuse is pssible ly f muliple sluis f () Usually hee ae muliple sluis f he mii assigme pblem () The sluis f Examples ad 3 fud diecly by he peais maix f he pblem () ae als f he pblem () bu he siuai f Example is s Table gives all he cmpuai ime slve he pblem () f Examples -3 f = 0 50 icludig he ime geeae cs maix ad display esuls The imal expeded ime is abu 05s f = ad Example sice hee is e cveig cicle f R = me pefmed I is easy see fm Table ha efficie peais maix ae Issue Vlume

7 E 5 E 5 E Table : Sluis f he pblem () =5 = Table : Time slve he pblem () E E E E 5 E 5 E Table 3: Sluis f he pblem () =5 = E - 5 E -5 E Table 4: Sluis f he pblem () f egaive pfi maix =5 = cmpleely fi slve lie bh mii ad glbal-miimum assigme pblems The ess f Example f =00 ad 000 have bee de ad he ime is 050s ad 688s especively The ess f Example f =0 ad 00 have bee de as well ad he ime is 04s ad 53750s especively The ime slve he 000-de pblem () wih he peais maix des exceed 5 miues Hweve he ime slve PC he 000-de MILP pblem (4) wih he ismeic suface mehd is ve 0 miues see [3] while he 000-de MILP pblem cespds ly he 3-de assigme pblem ( 000 3) Table 3 gives he w ad clum assigmes ad he paymes slve he pblem () via he elaive cs maix wih he peais maix Ad Table 4 gives he w ad clum assigmes ad he paymes slve he pblem () f egaive pfi maix The pimal bjecive fuci value equals he sum f evey payme The esuls cfm cmpleely he es f Examples -3 I Table 4 he imum gss pfi equals he egaive pimal bjecive fuci value REFERENCES [] HW Kuh The Hugaia Mehd f he Assigme Pblem Naval Reseach Lgisics Qua Vl (9) [] YYNie LJSu ad CLi A Ismeic Suface Mehd f Iege Liea Pgammig Ie J Cmpue Mah Vl80 (003) N [3] LYYag JDHa LJSu ad YYNie Ismeic Suface Mehd ad Is Numeical Tess f Mixed-Iege Liea Pgammig ISAST Tasacis Cmpues ad Sfwae Egieeig appea [4] YYNie ad SRXu A Ismeic Plae Mehd f Liea Pgammig J Cmpuaial Mah Vl9 (99) N3 6-7 [5] YYNie XSg LJSu JYu MZYua Well-Implied ad Nea-Implied Eumeais Ifmai ad Cl Vl34 (005) N (i Chiese) [6] Musapha Diaby The avelig salesma pblem: A liea pgammig WSEAS Tasacis Mahemaics v 6 6 Jue 007 p [7] Rabih A Jab A biay iege pgammig appach wse-case liea cicui leace aalysis WSEAS Tasacis Cicuis ad Sysems v 6 8 Augus 007 p -53 [8] Xili Xu Yu Tg Xiagli Wag Shiya Yig Waliag Wag Pduci pla pimizai assigme based muli-age WSEAS Tasacis Sysems v 5 6 Jue 006 p LY Yag is pesely pusuig he Dcal Degee a he Sae Key Labay f Rbics Sheyag Isiue f Aumai Chiese Academy f Scieces Sheyag Chia He eseach ieess iclude aumus plaig f muli-mblie bs i dyamic evime mixed iege liea pgammig Issue Vlume

8 MH Nie is a geeal maage f Sheyag Digial Chia Techlgy Develpme Cmpay His eseach ieess iclude maageme ifmai sysem (MIS) ad lgisics sysem ZW Wu is a pfess a he Sae Key Labay f Rbics Sheyag Isiue f Aumai Chiese Academy f Scieces Sheyag Chia His eseach ieess iclude b cl ad iellige cl He has published me ha 30 papes i vaius juals YY Nie is a pfess a he Sae Key Labay f Rbics Sheyag Isiue f Aumai Chiese Academy f Scieces Sheyag Chia His eseach ieess iclude applied mahemaics ad cmpue aided egieeig He has published me ha 70 papes i vaius juals Issue Vlume 008 5