VNU JOURNAL OF SCIENCE, Nat A F IN IT E Sci , t.xv n“ l - 1999 A L G O R IT H M N O N L IN E A R FOR A CLASS O P T IM IZ A T IO N OF PROBLEM S V o V a n T iia ii D u n g Hanoi University o f Technology T ran Vu T h ie u ỉhìiiui Iiistitritc o f Miitììeinntics A b s t r a c t In this paper a finite algonthm IS pvt sent ed for solving a class of nonhn- Ẽữr optiniizatwii prohlevis with special slvuctuvf It IS ha.'icd on 1}umbering IẼchĩììque s to improve feasible solutions commonly used in sobnrig problems of travsportalion type I PR O B L E M STA TEM EN T Given an w x n m atrix = p (0 < p < ») where' a,, € {0, 1}, and given positive nuniboois i = ,2 in Coiisidor the following optimi/sation pioblein: ĨỈÌ iP) -rri — ^ max Ĩ= subject to = /V ^ = ,2 w int('g(TS, i = 1, 2, ĨII < (2 ) J = I '2 II (3 ) SilUT t h e o h j r c t i v o fviiictioii (1) is ro nvo x a m i till' c o n s t r a i n t s (2), (:ị) a r c liiiii'iu and iiitegor, probloiii ( P ) is a Iioiiliiioai integer p i o g i a n u t i i i i g probU'in H o w e v e r, as slicown bolow, (P) may be mlucocl to a linc'ar inU'Rcr proi)l('ni with special stuK tuK ' Th(' constraint (2) may also bo loplacod with iiioquality constraint (2‘) vvilluuiit changing tho solution of (P): > /'m ' = 2, w (2' ;= P r o b l e m ( P ) m a y 1)0 p x p l a i n r c l a s f o l l o w s : t h o l o a r e 111 s t i u l o n t s a i u l ÌI s u b j w t s i foi t h o r n Th(> I i u m b o r o f s i i h j o c t s I('qviir('(l f o r t l i P tho a g m ' a b l e i i o s s o f stu(l('iit i t o s ub j ect I (a, s t i u l e n t is p , C o o f f i c i p i i f s (I, , I('p.r(' >s(' iit I = if s t u d e n t Í is agn>eabl(' t o s iu h x jw t / and a,I = if not) T he quostioii is how to a n a n g e the HtiulentH to learn fho s;iiiì)ị(Ị('( t- A F'.ni.e A l g o r i t h m f o r a Class of , so thit ^acli oi stu dents li'ai'ii ('onipl(‘tf‘ly th(' ĩiiiniboi of suhjiTts roq u im l for liirn and so th a t lu nunihi*r of st udents for each siihjoct is as similar as possihk' It is ('Hsily S(*(‘11 that (P) is ('quivaleiit to th(* following - intoger progranuning probfnu ÌÌ ỈÌÌ > />,, V/; ^ Vj: 7',^ G {(), 1}, 7=1 < r/,^, V7,j} 7-1 Tic model of probli'in (P) was studied ill [1] and [2] In [2] the authors suggested a polyionial time algorithm for problem (P) by solving a finite num ber of m axim um flow probpin.' Eiploiting th r sprcial stnic'tiiip of thí' problrni, in the soquol we shall develop an impnved algorithm for solving (P) which has the following features: (i) it is finite; (ii) it is ba:ed on Iiuinborhig tociiniqui's to improve' feasible solutioiis coninioiily used ill solving probiMn.’ of transportatioii type II FOUNDATION OF T H E SO LU TIO N M E T P O D Ai usually for the convniience \\v a g n v th a t a m atrix /■ = {-Ỉ*;/} whoso entrios satisv (2Ì and (3) is called a fcnsii)Ic soỉỉitioỉi of (P), a frasihle solution ac‘hi(‘\-iĩig the mininuii of (1) is callod an opfiiiirii solution of (P) L ( t VIS cl('iiot(' ỈÌI lit ^ j = //, p ^ /.= (bị H pl > /:=! lit ^ I h e i i t u i i l n ' i u i ill l u h ' i i l ^ a ^ ỉ r f a i f l f t o Ml W jtH I y H i n l p l i e I lil h l U i m b c i (.^1 subji ’ts irqiiii('(l for all stu(l(‘iits) 7) ; = 1,2 iti , = /-1 it is 110M'1 in [2] that in order to (P) has an optim al solution A Iif‘('(‘ssarv and sufficii'iit condtioi for the oxistoiKT of an optimal solution of (P) is > Pi for all / = ,2 ÌĨỈ (4) Ccadition (4) is very simple and easily to be checked So we assum e th at (P) satisfies this ondLon Furtherm ore, without loss of generality ve may assum e th at the students and sibjfcts are n um bered so th at ~ P\ ^ ^^2 — P2 ^ ■• • ầ “ Pnt find i n > h\ > h’2 ^ > hjt > Vo Van Tiian Dĩing, Tran Vu T h i e i u It is Iiatiual to suppose that Ị)j > for all ; = 1, 2, » ÌỈ, hocauso if bj = for soMifu' ; thou tho subject J must he deleted (there is no studoiit who wants to loarii the suhjel(' solution r wo c n ‘atp a tahlí' consist ÌMíip, of ĨÌỈ rows and 71 columns, in which oach of rows (‘oriospoiuls to a student and c a c h oof cohunns corrosponds to a subject The cell lying at tho intersoction of row / and (olunu.n J is (lonoted by A feasible solution /• — {-i'/j} of (P) will C'oirespoiul to a rabltk' consisting of zeros and ones in its colls A coll (?, j ) is called ỉ)ỈHck if a,Ị — (blark celhls will bo forbicldon to use, hocause studoiit i is not agreeable to suhjoct ý, HO that :i\i = ())) The lem aining colls will bo divided into two classes; white cells if :r,y = (stiulenr ? i is agreeable to subject j , but ho is not allocated for this subject) and hliic cells if :r,, ~ (student / is allocated for subject j) Denote Vi = 1.2, n '5')) 7=1 - - ( f is the num ber of stu d en ts allocatod for siibjoct J and is tho objectiv(' function valu'H' of :r) For any feasible solution r ĨÌ ;= ;=1 of (P), according to (5) we liave 7Ì -/ rn Til X J a Ì1Ì Í. ề Í. ^ J = \ 1=1 Colum n J is called full if f n Í - 1=1 7= n d ,= l (loficieiit if tj < t'' - It should be iK)tot'(l ĩhat the notions of blue coll, white cell, full column and deficient column arc c o n c fn io t’cl with a given feasible solution The following proposition gives a simple criterion for an optim al solution of ( P ) P ro p o s itio n I f n foafíi/)/ọ R o /ijfio ij = tìieĩi r is r /jaK n o r i o / i r j n n f r i ) i i i i n n f- > f ~ u y j i o (T7) optiiiiHl soììition of (P) Proof: From (6 ) and (7) it follows p = '£f'', -D - m Suppose the contrary that there oxists a foasihlo solution Ụ th at is Ix'ttoi til a n T i.e t'j < - 1, Vj - ,2 ,7 /, Combining (G) and (9) yiolds 7-1 (Í9) A F'.fiie A l g o r i t h m f o r a Class of whica s (’ontrarv to (8 ) Tims, T is an optim al solution □ } of (P) Let c ho a sequenrc of a ltm ia tin g \)i)si(lci now a foa.sihlc soluf ion r = whit< aid !)hic cells wi th n'spoc't to r Joiniiii’ roluniii Jo and coluniii JA- : y o ) ( ' o , y i ( A - > t = 0,1 , , , wher- 1)}, ( 10) are whitocplls ( r „ „ = 0), while ( i i j i +i), f = 1, bluocclls (■(',, ,,^ | = 1) We iiứro(luc(' tho following tiansfoniiation of r TrarỉS ioriĩiatioii A On tho spqiK'iiro c ipplaco all the forniPi' whit(‘ cells by blue oiiPS a n d clltho foniior blur rolls by whitp ones Thi s moans t h a t wo set •'■^7, = 1- =0, / = 1, rV = T , „ y o , j ) Ệ C "’ho following li'ninia sliows that this transforination does not change t h r objective functoi value of r L e m m i Assiiine tìiiìt r ' is uhtHÌned from r Ị)y Tiansfoiinatioii A oil sonic seqtience of íììteriaiug w h ite a n d i>iiie CCỈỈS j o i n i n g t w o c u l u i i i n s w h i c h a r e n o t f i d l t h e n f ’’' = /' ■ Prooj niỊiposí' that thoLc ('xists a secuH'iice of fonii (10) joining two columns jo < f') Since ill oach of rows it {t = 0, I , k whicl ;i(' not full arc just two \vhir(> and hhu' cells of c .r' = {.r' J Katisfic's (2), (3), - 1) there i.e r ' is also a feasible in oacliof colunins J, {t = 1,2 Ả- solutiji of (P) Siiiiihuly, since Jh - 1) th e n ' are jilst tw o v h f c a n d h h i c cells o f c wo h a v e vj^.yo,.yA- ( 11) ( n tli(' otlu'i hand, as coliuuii Jd has only 01U' (('11 of c (wliito cell (/(J, ;q)), wc have + (19) and a- (oluniii 7^ lias oiil\' one cell of c (l)lu(' coll {if, 1, ii, )) W(' ftcf Ik Ỉ' Iiallv, HS c o l u n m s Jo S n i i l a i l v , S U P Ị) ( ) S ( ' t h a t Ik -1 (13) a n d ji, a i r not full, fr o m ( 1) - ( 13) it follows t l i a t f ' = f □ c is a c y c l e o f a l t t ' n i a t i n g w h i t e ' a n d b l u e c pl ls : ('o-Jo)^ ('o ;i) i u ■Jk)A>k- Jo) {>0 , Ji)) (A- > 1) w h o r e (/,./,), / = , and (i,,/o) aro blue cclls a r c w h i t r O 'lls = 7',^ = 0), w h ile (/,, ), ^ = , , _ Ẳ' - = 1) Cunsiclpi' tlio followinp, tiaiisformatioii: T r a n f f d - in a tio n B On the cvcle c roplaro all the foinu'i white cells by blue onos and all th( fiiiiioi' blue rf'lls 1)V wliito OIK'S T h i s m e a n s th at \V(> set ■r' = r „ , V ( K j ) ệ C Vo Van Tuan Dung, Tran Vu Th/.io.u 10 L e m m a Sĩipposc tiicit r ' is uhtniiierl tioin r I)V TiiUisfuiiiintion B on ÍÌ cycle of HÌitoruHtiiig white and liluc cclls then f = T h e r o w s a n d c o l u i u n s n u m b e r i n g T h e p i o m l m e of rows aiul co lu m ns immlx-r iiiR is dpfiiK'd as follows First o f all, VVP assign t o oach c o l u m n J wh ich is full (/;■ = f ) If (‘o l u i u n j is imni'bri'pd, W(' a s s i g n i m in h f i' J t o o a c h r o w i w h i d i h a s n o t b e e n iim n h c 'H ’d = ( Ụ , j ) is a blue cell) Then, if row / is numbPied, wo assign minib.'i to aiul has eac h colvmm J w h ic h h a s n o t b e e n lum iboiecl a n d h a s a , I - T , J = ( th is is equivaloint to a J = r,, = i.(' (/ j ) is a white roll) and so on T h e above piocediire must stop aiftor a t m o s t 111 + V t i m e s o f r o w s a n d c o l i i n i i i s n u n i b o r i n g If a (leficiont column, o.g column Jo with < / ' - 2, is numbeipcl th(>io niusit he a soquencp of a lto in atiag white and blue colls joining a somo full column and Jo- Sluch a sequence of clue e v i l s j Ul l i Ul g H f ul l i ul i i J l i i J i u n l it J f / i c i c i i t c u J i i n i i i t j i c i i r c.'Ui l> imiulK'rs in tliP a b o v e i n e q u a l i t i e s a r e i nt f' gers it f o l l o w s t h a t i.^ a ' v h i t o c e l l w i t h r o s p c r t t o /■) M o i e o v e i a s b o t l i r a n d !J s a t i s f y ( ), w e m u s t have ỈÌ IỈ ^ J-I HU ~ ^ ^ UỉQ.Ì ~ J-1 /^0' This nu'ans That tiuMc is (’t)luiiin /1 such that (/(Ị ]\ ) is a l)lu(' cell; ~ 11 ^ yio.il " ih(’iT' (‘xists row /Ị siicli that (/Ị, /i) is a wliiti' C'('ll: ■’ >ị.ìì and Hso Ỉ)V (2 ) ” imisf ỉ)í' ach row i fioiii to III we write p, o n r s ( t h e i c m a i i i i i i g ( ('Us a i c assigned 0) then f>,o to tlic next row As a rosult wo obtain ail initial frasiblc soluition r' = } of (P) It may also Ijc startíHỈ with any fi'asihlo solution of (P), S('t Ả- = ,U1(1 111 whit(> c e l l s o f t h o r o w f r o m left t o liftht u n t i l h a v i n g go to step S t e p 2: Ti'sl for optimality For the ohtaiiuHl frasibl(‘ solution r^' wo adopt th(‘ coiiwnTioii th at a^lls w i t h a i o callod hliic cells, and cells (not black) w i t h arc called wi i i t c cclls Uvtoiuiiur ni = - "■ = f ''* = max f = max f^: C o l u i n i i J is s a i d t o h r H f ul l c o Ui n i n if íỊ' = c a llo d a d e f i c i e n t c o h i u m if ^ < t ^ - If no (k'ficicMit cohinui exists then by virtue of Proposition 1, is an optim al solnti.oii ul ( P ) O t h e r w i s o p e r f o r m r o w s a n d C ' o l m u n s i m i i i t ) c r i n g a s ( 1(‘S( r i h o d i l l s r c t i o n I f t h e r c i> no dpficiont roliimn th at is inuiiben'cl then r^' is also optim al (by virtiic of P ro p o sitio n 3) I n t h e o p p o s i t e C'as(', w o m u s t h a v e a s o q u o n c o c o f fo rm ( 10) t h a t c o n s is ts o f a l t n n.atin^ w h i t P a n d b lu o c e l l s a n d j o i n s a f u l l C ' o l u i n n it,, a i u l a c l p f i c i o n t c o l u m n jo- G o t o s t e ' p In fho cou rs e o f nuniho ring wli cii a deficiont c o l m u n is nmnl>oiP(l, W(' g o iiniiK't.l latch to stop to im p ro v e th e so lu tio n S t e p 3: Solution impiovpiiipnt Apply Transform ation A oil the soquoncp c obtau.K'd ill s t o p A s a r e s u l t W(' got a n e w f e a s ib l e s o l u t i o n x ' w h i c h e i t h e r is (f'' < -I'M 01 A Fir.ite A lg o rith m f o r a Class of has fevoi Iiuiiilx’r of full colunins than th en Ktuii to st('p 13 (Proposition 2) Set ’ = /•' and Ả- ^ A- + P r o p c s i t i o i i T he ahovc Hlgorithm tcniiiiìíìtcs at'tci a finite numl)cr of steps Proof: If 'h(> algoiitlim is not tcnniiiatod at step then aftc'r f'acli iniprovement in step 3, eith-n- a Dcttcr fcasihl(> solution or a solution witli f('WPi immbf'r of full roluniiis than previoLS (lie is obtained As the ohioctive function of the problem can tako only a finitp nu m bo ; o: pos i tive iiit(‘g('r values and a.s t h e n u m b e r o f co lm ni is ill the problrrn is also finite (-'qi.a; to rlie ab o v e s te p s r an not l)p infinitely oxtondpd □ I l l u s t r a t i v e e x a m p l e Solve problem (P) whose d a t a are a« follows: w = 4,ĩ> = 5, Pi = 2,P2 = , ; i = 3./),4 = and / = 0\ 1 1/ \1 Sum lip (’U'incnts of A Ơ1 = 3, (hj = Caivviiif; out (Black each row and column: ill (I:i = o.ị = 1; hi = 6;i = b \ = 65 = and p = 10 of the al};oiitlnn, \V(' obtain an initial f(-asiỉ)l(' Kolurioii of (P): / 1 X X \ 1 X {) X 1 \ X 0 0 / arc luaikcd 1)\- x) f - t c] S u m m in g I1Ị1 a l l o l c i i K ' n t s i n e a c h c o l u m n o f = (oh-iiiiiH 1, , an- f\ = full, = :]j\ coluiiiiis number'd A’ltli \\'(' search column ill so h i = t h( ' S( = ì t ị ^ ị ) 4, an' aiul t' (Icficii'ut wo obtain: = C olum ns 1, aif' first for a (l)lu(' (-('11) and find it in rows 1, 2, 4, lo.v.s arc iiuinlx'K'd with ( subscript o f coluiiiii 1) Tht'ii, in coluniii there is a ill rov not vet immlx'K'd) St) that this row is iminhorod with (subscript of coluinn 2) All li( lovvH liavf' IxM'ii iminbi'K’d, coliunns and arc not V(‘t nnniherf'd \V(’ now search lu n Jx ’icd row for a (\vhit(' ('('11) and find it in coluinn (not yet n u m b e m l), so coluiiii ị is niinilx'K'd with (subscript of row 1) At this point, (lefirient coluimi is munbei'd vith (row 1), row is imiiilx'ml with (coluinn 1) Cohiniii 1is full Thus we o b tan 'I k' soqiK'iK'o of ci'lls: (1, 1) - (1.4) joiiiiufj, full colunui anti deficient column S ('p3 ChaiiRÌní-, ,r‘ oil jiust foiind scqiionce of colls, wo obtain new feasible solution Vo Van Tuan Dunq, Tran Vu Thĩen 14 X 1 V1 X ( 1 X X \ X 1 0 Ì ;5 Rf'tuni to step Summ ing up elements in each colunui of s ’, we obtain t'ị = 2, f'ị = /3 = 3, f'ị = 2, iị = and t'~ = Columns 2, are full, column is (lefirimit Columns are first iiuinbcK'd with We search C'olmnii in J-‘ for a (blue coll) and find it in rows 2, so those rows are m in ib e r p c l with (subscript of coUinin 2) In full column thoro is a in row ypt num beied) so th a t this row is nunib(np(l with (subscript of column 3) (nut Wc seairh iiuinbeiPtl row for a (white cell) and tiiul it in cohiinn (not yet nuinbeiecl), so coliniin is numbered w ith (subscript of row 1) Then, we search luinilx'red row for a (I (white cell) and find it in column (not yof num beipd), so column is nuniborod w ith (subscript of row 2) At this point, deficipnt column is mimbered with (row 2), Ỉ o\v is num bered with (colmnn 2) Colum n is full Thus, we obtain the soqueiico of colls: (2,2) - (2,5) joining full column and doficiont colmnn S t o p C h a n g i n g r'^ oil j u s t f o u i u l s o q u f i i c e o f rol ls , WP o b t a i n n o w f pas i bk ' s o l u t i o n Í ,t = X \ 0/ X R eturn to step Suimnine, up ploinents in pach column of r^, we obtain /■] = t ị - 3, t'i = 2, f'ị = and C o h u n n is full, r o l u m n is cloficient = C o h u i i n is first Iiuniborod w i t h w v search column in T'* for a (blur coll) and find it ill rows 2, 3, 4, so these row s aiirí l ỉ ỉiỉtg ĩỉhii luụt KÍHìỉì l ì oc T i r Ỉiỉiỉèii \v'ỉ c \ j t i g It gl i ệ Q u ố c J í l i i u / I i ^ p li,íị> l i i í u nlur s - Transfonnatioii A on c Argniiig as in tli(' proof of Loiniiia 1, wr obtain tlir lolatioiis (11) -(13) As J„ is a drfirieiit column, from (11) - (13) it follows th at if JA- is a uniqiuc... li'ninia sliows that this transforination does not change t h r objective functoi value of r L e m m i Assiiine tìiiìt r ' is uhtHÌned from r Ị)y Tiansfoiinatioii A oil sonic seqtience of íììteriaiug... improve' feasible solutioiis coninioiily used ill solving probiMn.’ of transportatioii type II FOUNDATION OF T H E SO LU TIO N M E T P O D Ai usually for the convniience \v a g n v th a t a m atrix