MULTIPLE CRITERIA FUZZY COST TRANSPORTATION MODEL OF BOTTLENECK TYPE

Transcription

1 Assoiate Professor Alexadra TKACENKO PhD Departet of Applied Matheatis Moldova State Uiversity Moldova E-ail: MULTIPLE CRITERIA FUZZY COST TRANSPORTATION MODEL OF BOTTLENECK TYPE Abstrat. The ultiple riteria optiizatio probles with fuzzy obetive futios oeffiiets are the ost iportat beause of their ofte appliatios i various aagerial deisio proesses. I this paper i s preseted a iterative solvig approah for the ulti-obetive trasportatio proble with fuzzy ost oeffiiets ad tie iiizig riterio. The approah is based o iterval presetatio of eah ost futios oeffiiets. By fidig of the probabilisti paraeter of belogig of oeffiiets of obetive futios to their variatio itervals for every riterio we a fid iteratively the orrespodig set of effiiet solutios for the ultiple riteria trasportatio odel for every value of paraeter strutured by the tie iiizig riterio. Thus we ould solve the ulti-riteria trasportatio proble of fuzzy type for ay value of the probabilisti paraeter of belogig that is i fat oe of stohasti bottlee type of proble. I other words we have obtaied oe sigifiat result that ay deisio-aig situatio desribed by the proposed odel a be predited by its tie ad ost harateristis. The proposed algorith has proved to be quite effiiet beig tested o several exaples. Keywords: Fuzzy prograig ulti-riteria trasportatio odel bottlee riterio effiiet solutio optial oproise solutio. JEL Classifiatio: 90C9 90C70. Itrodutio The topi oeted with trasportatio proble a be osidered as old as the world is. Nevertheless the sietifi researh iterest of this type of probles reais ad otiues to be edless. The reaso is ot due

2 Alexadra Taeo oly to the attrative ad lear for of these ids of probles but it is also due to uerous pratial appliaes of the. Now it's worth poitig that probles of ulti-riteria optiizatio iludig the ulti-riteria trasportatio probles are of great iterest i various researh areas. This happes beause is well ow the ireasig of riteria uber leads oly to ireasig of solutio auray for ulti-riteria optiizatio probles. The effiiet solutios of these ids of probles a be ahieved usig various algoriths developed i [8] [] [3] [0] ad ay others. Fro pratial appliability poit of view iposig of iial tie to realise the solutio of odel appears as a logial oditio whih would surely iprove its quality. I the speiality literature the riterio of iiizig the axiu tie is alled a bottlee riterio. A large variety of algoriths have bee proposed for differet ids of ulti-riteria trasportatio probles of bottlee type. Thus for solvig the threeriteria trasportatio proble iludig the bottlee Aea ad Nair i [] developed a effiiet algorith but Wild ad Karwa i [9] proposed a effiiet algorith for solvig the geeralized r-riteria trasportatio proble of the sae type. It's iportat to etio that ay of eooial deisio probles lead to the fratioal optiizatio odels beause that a lot of iportat harateristis of these ay be evaluated really usig oly soe ratio relatios. The ratios lie the suary ost of total trasportatio expeditures ito axiu eessary tie to satisfy deads the total beefit or produtio value related to the eessary tie the total depreiatio ito axial usig tie ad ay other siilar riteria ofte are essetial i evaluatio of several eooi effiiey idexes that lead fially to fidig of soe orret aagerial deisios. The tie-ostraiig riterio is obviously oe of oditios so uh iportat for aor optiizatio probles. A partiular ase but quite ofte eted is of idetial deoiators lie the bottlee tie futio. Moreover we studied various ases whe "bottlee" deoiator futio is iluded as a separate riterio i the optiizatio odel. The effiiet algoriths for solvig these types of odels are proposed by Shara ad Swarup i [4] for oe-riterio fratioal trasportatio odel of bottlee type ad by Taeo i [7] for ulti-riteria fratioal trasportatio odel of the sae type. I wat to ephasize that all of the above etioed algoriths were tested o various exaples ad proved to be quite effiiet for the deteriisti type of data. Ufortuately i real life ot always be so ofte soe paraeters ad oeffiiets of the optiizatio odels are of ideteriate type [7]. That is why i the proposed wor is studied the ase whe soe of paraeters of ulti-riteria bottlee odel are of fuzzy type.

3 Multiple Criteria Fuzzy Cost Trasportatio Model of Bottlee Type. Proble forulatio Sie for ay type of atheatial optiizatio odel the obetive futio oeffiiets have greatest ifluee o both the optial solutio ad the value of obetive futio we propose to ivestigate the ultiriteria trasportatio odel i that these oeffiiets are of fuzzy type. Beause of large pratial appliae of the trasportatio odels it should be oted the ertai real eaigs of these oeffiiets. We propose to ilude i the odel the bottlee riterio separate whih is quite iportat for ay deisioal situatio espeially fro pratial poit of view. The atheatial odel of ulti-riteria trasportatio proble of bottlee type with fuzzy osts oeffiiets is the followig: i i Z i Z i r ~ x.. i ~ x i Z ~ x () r i Z r ax ti xi i i x ai i x b a i b i 0 x 0 for all i ad where : ~ = r i= = are osts or other aouts orrespodig to orete iterpretatios of those riteria beig of fuzzy type t - eessary uit trasportatio tie fro soure i to destiatio a i - disposal at soure i b - requireet of destiatio x - aout trasported fro soure i to destiatio that is oly positive. We a otie that i odel () the firsts -r riteria are of fuzzy liear type eah of the beig of iial type. However is ot exluded to have i the iitial odel soe riteria of axiu type suh as for exaple axiizig of beefit profit or ore other. I fat this does't ot ae the optiizatio odel

4 Alexadra Taeo ore opliated as usig soe eleetary trasforatios the axiu types of riteria a be odified ito iial types as they appear i the odel (). Obviously the odel () beoes ore opliate espeially fro solvig poit of view beause of the (r+) riterio whih is of o-liear type. 3. Soe Reasos ad Stateets Sie the odel () is of ulti-riteria type as we ow these rarely adit optial solutios. For solvig these usually it builds a set of effiiet solutios ow also as Pareto-optial or o-doiated solutios solutios of the best oproise. I order to ivestigate the odel of ultiple riteria () we should propose firstly the defiitio of effiiet solutio for the deteriisti type of odel. We will osider the ext ultiple-riteria trasportatio odel of bottlee type with deteriisti data where without tyig geerality we assue that all of the firsts r riteria are of iiu type. i Z x i i Z.... i x r i Z r x i Z r ax ti xi 0 () x a i i i i i i i i oditios: x b a b x 0 i with the sae sigifiae of the odel paraeters as i odel () speifyig that all of these are of deteriisti type. Let suppose that: is oe basi solutio for the odel () where: Defiitio: if ad oly if The basi solutio... r aother at least oe idex both relatios Z X = ax t x 0. i for ay other basi solutio for whih the relatio Z X T / X T... r where T is true. If verified siultaeously with the equal sig it Z of the odel X or T X Z is a basi effiiet oe X is true there iediately exists for whih exists at least oe idex for whih at least oe of the all of these three iequalities eas that the solutiois ot are uique.

5 Multiple Criteria Fuzzy Cost Trasportatio Model of Bottlee Type Defiitio: The basi solutio ( best) oproise solutio fora ertai tie T solutiosof eah riterio. of the odel is oe of if the solutio the optial is loated losest to the optial So for eah tie level allowig plaeet of the basi solutio for the odel () we a deterie its orrespodig optial oproise solutio. Beause of the first oes r riteria i odel () are of fuzzy type we develop priarily a fuzzy tehique for solvig of the fuzzy ulti-riteria odel. 4. Fuzzy tehiques By fuzzy liear prograig we ea the appliae of the fuzzy set theory to liear ulti-riteria deisio aig probles. Defiitio A eleet x has a degree of ebership i a set A deoted by a ebership futio A x. The rag of the ebership futio is [0]. I ulti-riteria deisio aig probles the obetive futios are represeted by fuzzy sets but the deisio set is defied as the itersetio of all fuzzy sets ad ostraits. The deisio rule is to selet the solutio havig the highest ebership of the deisio set. Zadeh [] itrodued the basi oepts of fuzzy set theory. Ziera i [] ade a iovatio i the field of ulti-riteria deisio aig. He first applied fuzzy set theory oept with suitable hoies of ebership futios ad derived a fuzzy liear prograig. He shows that obtaied solutio usig the fuzzy liear prograig is always effiiet oe further it a fid a optial oproise solutio. I order to solve the odel () firstly we propose to solve the odel (3) that is obtaied by exludig of the (r+) bottlee riterio. It is the ext: i Z x i i Z.... i x r i Z x (3) x a i i i i r i i i oditios: x b a b x 0 i with siilar sese of paraeters as i the odel ().

6 Alexadra Taeo We will apply the fuzzy liear prograig tehique [3] for solvig the odel (3). It should be oted that i this ase we apply the fuzzy approah oly to the obetive futios of the odel. By applyig of fuzzy liear prograig tehique to the ulti-obetive liear trasportatio odel (3) we will fid its optial oproise solutio. At the first we assig for eah obetive futio two values upper ad lower bouds for the obetive futio Z : L - aspired level of ahieveet for obetive ; U - highest aeptable level of ahieveet for obetive ; U ad L as d U L is obviously a degradatio allowae for obetive. We build the fuzzy odel beause of aspiratio ad degradatio levels for eah obetive have bee speified. O the ext step we will trasfor the fuzzy odel ito oe of deteriisti type odel of liear prograig. The solvig fuzzy tehique is the followig: Step. Solvig of r oe-riterio trasportatio probles. Step. Buildig the table of values i whih are registered values of the all obetive futios i the optial solutios of every obetive futio. Step 3. Aordig to the table of values we ay hoose the best - L ad the worst U values fro the set of solutios. The iitial fuzzy odel is built eepig the aspiratios of eah riterio as the follows: Z L r x ~ x ai i x b (4) i a i b i The ebership futio U U Z L 0 x 0 for all i ad if if if L Z Z Z x X is defied as the ext: Taig ito aout the relatios (4) ad the above defiitio of the x x L U U

7 Multiple Criteria Fuzzy Cost Trasportatio Model of Bottlee Type ebership futio X the equivalet liear prograig proble for the ulti-obetive trasportatio proble (3) is the followig: i Max i oditios: U Z U L r x ai i x b (5) a i b x i 0 for all i ad By siplifyig the odel (5) we will obtai the ext liear prograig optiizatio odel: Max i oditios: i i x U L U r (6) x ai i x b a i b x i 0. 0 for all i ad Thus we a say that usig the fuzzy tehique for solvig the odel (6) we easily fid a oproise solutio for the ulti-obetive trasportatio odel (3). Rear. The above desribed algorith is appliable to all types of ultiobetive trasportatio probles as well to the vetor iiu as to the vetor axiu probles. Rear. The optial oproise solutio of the odel (3) does t eessarily to be of iteger type. I order to solve the odel () by applyig the fuzzy tehique we propose the ext algorith: Step. Orderig the tie atrix - T aordig to ell values i asedig order ad assigig for eah ell a serial uber thus we will get all ordered ells. 0.

8 Alexadra Taeo Step. Seletig the firsts at least ( + -) ells aordig to the arrageet order util we a plae the iitial basi solutio for the odel () supposig that the other ells are bloed. Step 3. By applyig the algorith of fuzzy tehique for the proble with uloed ells we get the optial oproise solutio for the odel () usig oly the uloed ells whih orrespods to the followig * tie: t i ax ti xi 0. i Step 4. Uloig iteratively i ireasig order of tie the ext atrix ell (or ells with the sae tie ad ost values) we will retur to the step 3 of the algorith ad we will fid the ext optial oproise solutio of * odel with tie of its realizatio obviously higher tha the previous t tie. The step 4 is repeated util all of ells i the atrix of tie will be ubloed. Thus the proposed algorith will highlight a fiite set of optial oproise solutios for the odel () eah of the orrespodig to the sallest tie possible of its realisatio. Beause the proble has fiite diesios the algorith is realized i a fiite uber of steps. 5. Theoretial aalysis of fuzzy ost ulti-riteria odel We a state that the fuzzy tehique is very effiiet for solvig the various optiizatio probles. Applyig the fuzzy tehique for solvig the ulti-riteria odel ot oly of liear type we build a optial oproise solutio where oly the obetive futios are represeted i the fuzzy for. This ethod is applied ost ooly whe the paraeters ad oeffiiets of odel are of deteriisti type. Ufortuately i everyday life we eet ot oly with suh ases. Beause the paraeters ad oeffiiets of trasportatio ulti-riteria odels have real pratial sigifiaes suh as uit pries uit osts ad ay other as we ow they vary quite frequetly. Moreover it was foud that all of the are iteroeted i.e. they are hagig siultaeously eve with the sae paraeter of variatio. For exaple if eletriity beoes ore expesive with oe oeffiiet the i a stable eooy ertaily the values of produts of idustries diretly depedet of it will irease by the sae oeffiiet. For other brahes these ay be differet but they ay be alulated owig a priori the orrelatio oeffiiets betwee all the brahes. The last a be alulated by applyig of various statistial solvig ethods for ertai data of produtio or osuptio betwee the for seleted brahes. We propose to alulate the paraeters for the odel () usig the forula:

9 Multiple Criteria Fuzzy Cost Trasportatio Model of Bottlee Type where: oeffiiet p (7) - are the liit values of variatio iterval for eah where: We a observe that i r. p [ 0;] for i where i r. Moreover ay values of ost oeffiiets aordig of their variatio itervals [ ; ] for the odel () a be alulate usig the followig forula: p for i r. (8) Agreeig to the forula (7) the paraeters { ost p } a be osidered as the probabilisti paraeters of belogig for every value of oeffiiets { } fro their orrespodig variatio itervals [7]. This is suessfully applied for both iiu riteria type ad axiu type. Supposig that the variables { } for i where i r are otiuous o theirs orrespodig itervals the paraeters { p } appear as the distributio futios of these variable. Therefore the futios { p } eoys all properties of distributio futios iludig the ootoy ad otiuity property. So whe it fid the ireasig of pries for produts the lielihood of a higher prie for eah produt type aordig of its prie variatio iterval is greater tha of a lower prie. Aalogial it will our whe it is foud the heaper of produts the lielihood that the prie of oe produt fro its prie iterval deteried priori will be lower is greater tha for the higher prie. Thus are true the ext relatios: for the axiu type of riteria: if for ay two values ad the relatio: is true ad [ ; ]

10 Alexadra Taeo p p for i t... r ad sigifies the axiu type of riterio; for the iiu type of riteria: if for ay two values l l l ad the relatio: l l l is true ad l l [ ; ] ; l l p p for i t... r ad l sigifies the iiu type of riterio; Partiular ases of the odel () without of the bottlee riterio were aalysed i [4] by Chaas ad Kuhta. The authors proposed a ethod of iterval for solvig oe riterio trasportatio odel with fuzzy ost oeffiiets. The idea be applied to ulti-riteria proble [5] but it leads to osiderable ireasig of the uber of obetive futios whih really opliates the solvig proess of the proble. I the papers [6][9][0][][5][6][8] [] are proposed ertai aalyses of various poits of view about the ultiple riteria trasportatio odel with fuzzy paraeters ad are developed differet algoriths i order to its solvig. It should be oted the pratial ipossibility of solvig these types of odels usig soe paraetri ethods. The ai idea of the ethod that will be developed is the siultaeously ad iteroeted variatio of obetive futios oeffiiets fat resultig diretly fro the pratial appliatios of the studied odels ad iposig of the bottlee riterio. 6. The solvig Algorith. We will suppose that the set of all oeffiiets variatio itervals is give by their variatio liit values they are the followig: [ ; ] i r ;. Let be for soe idies: i r we ow at least oe oeffiiet value for exaple of alulate by applyig the forula (7) the probabilisti paraeter of belogig to its iterval [ ] that is the followig: p i i i i i i i i the we ay

11 Multiple Criteria Fuzzy Cost Trasportatio Model of Bottlee Type Assuig that it s the sae for all obetive futio oeffiiets it will be deoted by p ; 3. Aordig of the stateet about the siultaeously ad iteroetio of variatio of obetive futios oeffiiets we a alulate a set of all values of obetive futios oeffiiets: applyig the ext forula: i r by p for the ireasig oeffiiets values ad p for the dereasig oeffiiets values. 4. Appliatio of algorith of fuzzy tehique. So for ay value of probabilisti paraeter of belogig we have obtaied the ulti-riteria trasportatio deteriisti odel of bottlee type. For its solvig we ay apply the above proposed algorith lie as for the odel (). I this ase for every tie level we a apply the fuzzy tehique thus obtaiig the orrespodig optial oproise solutio. By odifyig the tie level we a obtai the set of all oproise solutios orrespodig to all tie levels. Thus for every tie level we will have the orrespodig optial oproise solutio. It s eessary to rear that for a ew value of probabilisti paraeter of belogig we will solve agai oe deteriisti odel of type (). We a solve the odel () also by fidig of a set of effiiet solutios for eah value of the probabilisti paraeter of belogig. This algorith is ore diffiult but for eah value of this paraeter it offers for the deider oe large set of effiiet solutios whih are very iportat i order to elaborate oe orret deisioal strategy. 7. The Solvig Algorith Proedure. Priarily we will perfor the first three steps of Algorith. Proedure. Proedure is oe iterative. Eah iteratio supposes a deeper searh level ad fidig of a ew effiiet basi solutios ad opletig of the ultitude of effiiet basi solutios with other ew aordig of the ew ubloed ells. The exploratio proedure of eah tie level is fiite i depth ad eds o every brah whe the sae solutios have bee foud at oe upper level of ay other brahes or whe all possibilities of iproveet have bee exhausted at a

12 Alexadra Taeo ertai level. I this ase it will ublo a ew value of tie i the ireasig order therefore the ost of ells i all the atries for whih it will be resued the Proedure. Ubloig proedure otiues util all ells of tie ad those that orrespod to atries of ost beoe available. So fially we get the set of all sets of effiiet solutios of the trasportatio ulti-riteria proble of bottlee type eah of the orrespodig to a ertai tie level. The solvig proedure is perforig aordig to the ext logial shee: The logial shee S 0 If i. e 0 l=0 l= S S 0 S 0 S 0r S S S S S S r S r The set of effiiet basi solutios S rr where: u v i are defied by the proble diesio is a orderig idex of ells aordig with the tie table of data The expliit presetatio of the solvig Proedure The first step. We arrage the values t fro the atrix T i ireasig order usig for this a orderig idex let it be. We outlie that i the odel () there are supplies ad deads ad ( r ) obetive-futios (iludig tie riterio). The seod step. We try to fid a iitial basi solutio usig oe of the riteria aely the atrix of this riterio fro that we will use oly the ells aordig to orderig ireasig p-idex. Obviously the iitial basi solutio will be plaed i at least ( ) ells. Thus i the atrix i whih we plaed the iitial

13 Multiple Criteria Fuzzy Cost Trasportatio Model of Bottlee Type basi solutio will be ubloed p 0 ells where p The obtaied solutio at this iteratio will ar the 0-level of the logial tree of effiiet solutios. We will osider that the followig ells with orderig idexes greater tha p 0 are bloed. For the 0-level we will alulate the orrespodig T 0 aordig to the followig forula: T 0 = ax t i xi 0. The third step (exploratio of the deep brah). We shall try to iprove the solutio fro the atual level usig for this oly the ubloed ells. For this purpose we shall alulate the values: u v. All ofiguratios of basi solutios a be reorded at the ext level l=. Thus the logial tree will otai o level o ore tha p brahes where p p0 ( ). The rd proedure fro the 3 step is iterative oe ad explores the possibility to irease the uber of logial brahes o the ext level usig every the brahes fro the previous level. If all possibilities of plaeet have bee explored as to iprove at least oe of riteria usig for this purpose ust the p 0 ells ( aordig the th desribed orderig) the oe a go to the 4 step. The fourth step. We will ublo the ext ( p 0 +) ell ad will obtai a ew ahieveet tie for a ew effiiet solutio whih will be obviously greater or equal tha the previous tie. I d lie to outlie that after eah ubloig iteratively proedure there are agai ( p0 ) bloed ells beause after every uloig we osider: p 0 p 0. If the relatio i 0 is true at least rd for oe riterio for this ell we will repeat the proedure of the 3 step otherwise we shall otiue to ublo the ext ell aordig to the orderig th idex util we will get p 0. After fiishig up the 4 step the set of all basi solutios of odel () will have bee reorded out of whih we a easily selet the oes that are basi ad effiiet. Oe a see that the logial solvig tree iteratively irease its brahes by exploratio of a ew ofiguratios of the basi solutios o every level. The ireasig of both the uerous of brahes of eah level as well as the uber of levels is ostraied by the fat that the proble is of fiite diesios o the oe had ad o the other had by the request that the ew solutio ofiguratio should ot be repeated. The orretess of the above algorith is based fro the followig theore. Theore. The set of all effiiet basi solutios for the ultiple riteria trasportatio proble with fuzzy ost oeffiiets ad bottlee riterio () is foud by applyig the above Algorith ad. i i

14 Alexadra Taeo Proof. Let L T be a list of basi effiiet solutios of odel () beig foud by applyig the above algorith ad for oe value of probabilisti paraeter of belogig p. We suppose that exists oe basi effiiet solutio S for the odel () that was foud usig aother algorith differet of the above oe so it results that S LT. Let S orrespods to T. We will fix it o the brah that orrespods to the T begiig with the level 0. Wide exploratio of the fixed brah leads to the registratio of all basi solutios of the brah T. So all the basi solutios that orrespod to tie T belog to this set. We will separate i the set L the effiiet basi solutios that orrespod to tie T. It is obvious that L T l L T. As a result if S L the S is a basi effiiet T solutio foud by applyig the above algorith or if S L T the S is ot a basi solutio ad oreover it is ot oe basi effiiet. So is true the followig: either S is a basi effiiet solutio ad it belogs i the list L T or it is ot a basi effiiet solutio. We proved that for oe value of probabilisti paraeter of belogig p we obtaied the set of all orrespodig effiiet solutios for the odel (). Therefore by odifyig this paraeter we get aother lot of effetive solutios for the odel (). Buildig the set of effiiet solutios for odel () for ay value of the paraeter p i the iterval [0] i fat we fully solve the proposed odel. The theore is proved. 8. Colusios I this paper is developed a itegrate ultistage proedure to solve the ultiobetive trasportatio proble of bottlee type with fuzzy obetive futios oeffiiets. By applyig the hypothesis about the iteroetio ad siilarly variatio of the odel s obetive futios oeffiiets we redue the odel to oe of deteriisti type. After for eah of possible tie level we ostrut its orrespodig set of effiiet solutios. I would lie to ephasize that at this stage we ay apply the fuzzy tehique for fidig the optial oproise solutio orrespodig to the early established tie level. However as it s ow the set of effiiet solutios offers several optios for developig optial aageet strategies. By odifyig of the tie level we a obtai all sets of effiiet solutios eah of the orrespodig to its tie of realizatio but it should be oted for oly oe value of probabilisti paraeter of belogig. I depedee of the eooi stability the paraeter ay followig differet laws of distributio. Fially we olude that these id of odels are very atually ad utile espeially fro the deisioal ad aagerial poit of view therefore these deserve to be further ivestigated ad studied.

15 Multiple Criteria Fuzzy Cost Trasportatio Model of Bottlee Type Exaple: Let be the followig 3-riteria proble with 3 supplies ad 4 deads. Supposig that we ow the set of variatio itervals of obetive ost oeffiiets that is the follow: [ ; ] 0.5;.5 ; [ ; ] 0.8;. ; [ ; ] 6; 0 ; [ ; ] ; 3 ; [ ; ] 5; 3 ; [ ; ] 7; ; [ ; ] 4;0 ; [ ; ] ; 4 ; [ ; ] 3; 5 ; [ ; ] 4;0 ; [ ; ] ; 6 ; [ ; ] 4; 8 ; 3 3 [ ; ] ; 6 ; [ ; ] 3; 7 ; [ ; ] 4; 8 ; 3 3 [ ; ] 3; 5 ; [ ; ] 7; 9 ; [ ; ] ; 3 ; [ ; ] ; 4 ; [ ; ] 8; 0 ; [ ; ] 3; 7 ; [ ; ] ;7 ; [ ; ] 8; ; [ ; ] 0.6;. 4 ; Let's assue that we have oe value for exaple =; We wat to build the set of all effiiet solutios owig the tie atrix that is the ext: Tie= b \a i Solutio proedure: Kowig the value = we apply forula: p i order to deterie the probabilisti paraeter of belogig that is: p. Let suppose that it is the sae for all ost oeffiiets. Beause i the variatio iterval [ ; ] for the value = we foud the irease i ost to deterie the other ost oeffiiets we apply the ext forula:

16 Alexadra Taeo p oeffiiets:. We obtai the ext data for the obetive futios Cost = By usig the above proposed Algorith we have foud the followig effiiet basi solutios: X x x 3 x x 4 x x 6 S ; X x 8 x 3 x 3 x4 3 x33 4 x34 3 S 6 x4 x 5 x3 4 x3 3 x34 8 x 3 x3 4 x4 x3 3 x X x 8 x 3 x4 6 x3 3 x33 4 S ; 6 6 x3 8 x x x3 6 x3 x34 6 S 7 7 x3 6 x4 x x3 8 x3 3 x34 4 S 8 8 x x3 6 x x3 8 x3 x34 6 S 9 9 x 3 x3 5 x x3 8 x33 x34 6 S 0 X x 5 x 3 x 6 x4 3 x33 4 x ; 3 3 X x 4 S ; 4 4 X x 4 S 7368 ; X ; X ; X ; X ; 0 S ; X x x 3 x 6 x 3 x x 6 S Fially I d lie to etio that we solved the proble oly for the probabilisti paraeter of belogig: p. We observe that for this paraeter value the data of proble oiide with the data fro the exaple of Aea ad Nair artile []. We a etio that usig the proposed Algorith we obtaied with effiiet basi solutios ore opared as the authors results fro this artile.

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