Precedence graphs generation using assembly sequences

Transcription

1 Preedene grahs generaton usng assebly sequenes VIOREL MÎNZU* and ANTONETA BRATCU ** * Deartent of Autoat Control and Eletrons Dunãrea de Jos Unversty of Galat Str. Doneasã, Galat ROMANIA ** Laboratore d Autoatque de Besançon ENSMM - UFC - UMR 6596 Unversté de Franhe-Coté 5, Rue Alan Savary Besançon FRANCE Abstrat: - Alost all ethods dealng wth assebly systes desgn - for exale, assebly lne balanng (ALB), tasks-to-workstatons assgnent algorths, resoure lannng - are based on the artton of reedene grahs. These are dreted grahs whose nodes reresent the assebly tasks and arrows reresent the reedene relaton between tasks. In the lterature there are very effent and well-known ethods to generate assebly sequenes and a reedene grah an be obtaned by "ergng" assebly sequenes. Nevertheless, no systeat obtanng ethod for these grahs was roosed. Ths aer deals wth a roerty of a gven set of assebly sequenes, that guarantees the exstene of an "equvalent" reedene grah, after sutably defnng suh an equvalene. Ths result an be used for the general ase, when ths roerty s not et, to fnd an equvalent set of reedene grahs. Key-Words: - assebly systes, assebly sequenes, reedene grahs Introduton Preedene grahs are the ost used tools n assebly systes desgn. Assebly lne balanng ethods (see []), that gve a frst rough layout of an assebly lne, have as nut data the reedene grah. Other desgn ethods use also reedene grahs. For exale, the aroah roosed at Draer Laboratory ([8], [9]) s based on assebly task flow grah whh are essentally reedene grah wth a lnear struture. Preedene grahs n dfferent fors are also used n tasks-toworkstatons assgnent ethods ([6]) or resoure lannng ([7]). In soe aers t s roosed that reedene grahs be obtaned by "ergng" assebly sequenes ([8], [5], []). In [5], the authors roose a ethod for the generaton of the reedene grahs fro a gven set of assebly sequenes, whh s essentally based on heurst searh. The reedene grahs have two weak onts. The frst one s the fat the assebly tasks are generally not well defned. An assebly task s well defned when the base art and the seondary art are sefed. The seond weak ont s the nonexstene of a systeat obtanng ethod. On the other hand, there are very effent ethods to generate assebly sequenes (see [], [4]), whose assebly tasks are well defned. Ths aer resents a theoretal analyss, n order to answer to the queston whether there exsts a reedene grah equvalent to a gven set of assebly sequenes. Proble stateent In ths seton, we shall defne our roble, that s to fnd a reedene grah equvalent (n a sense that wll be stated below) to a gven set of assebly sequenes for a ertan rodut. Notaton: S = set of sybols, N = ard (S). We onsder that a sybol reresents a art that wll be assebled wth other arts to ake the rodut. S s the set of sybols orresondng to all arts and N s the nuber of arts. An assebly sequene s a total order over the set of arts. Hene, an assebly sequene ay be reresented by a sequene of sybols: = aa...an,, a S, () and a a f. Fro now on, we shall all olete sequene a sequene havng the for () and artal sequene a sequene of dfferent sybols whose nuber of sybols s less than N. So, an assebly sequene s reresented by a olete sequene. Let Ω be a set of olete sequenes reresentng a gven set of assebly sequenes and n = ard( Ω ): Ω = {, =,...,n}. Any olete sequene Ω, N = a a... a, ndues an order relaton over S, noted O, O S S :

2 x, y S, (x, y) O x = a, y = a k, < k We say that x reedes y and we shall note ths relaton by x < y. Obvously, O s a total order relaton over S. One an defne : I n O o = O = whh s, obvously, a artal order over S. Note that ths order s deterned by the ntal set of sequenes. Ω. The dgrah of ths relaton s G = (S, O0) and we all t reedene grah. For a gven set of sybols S and for a gven set of olete sequene Ω, the artal order relaton O0 and ts reedene grah G are unquely defned. We onsder another relaton over S: I = S S - {(a, b) S S ((a, b) O0) ((b, a) O0 )} We all I ndfferene relaton and we note (x, y) I by x?y. Defnton : A olete sequene = a eets a...a N the reedene grah G = (S, O0) f for any two values < N ether t exsts a ath n G fro a to a or a? a. Fro now on, we note by Θ the set of olete sequenes eetng the reedene grah G=(S, O0). One an say that the grah G s equvalent to the set Θ, beause G an be obtaned fro Θ and ve versa. To slfy the reresentaton of G, any arrow (a, a) s erased, f there s a ath n G fro a to a fored by at least two arrows. Exale : Let us onsder S={a, b,, d, e, f} and Ω={abdef, abdef, abdef}. The grah G orresondng to the sets S and Ω s resented here-after: a b f d e Fg. Preedene grah G=(S, O0) One an verfy that the set of sequenes eetng the reedene grah s Θ ={abdef, abdef, abdef}=ω. Hene, the reedene grah G s equvalent to the set Ω. Exale : Let s onsder S={a, b,, d, e, f, g, h} and Ω={abdefgh, abfdegh}. The grah G orresondng to the set S and Ω s resented here-after: b d e a g h f Fg. Preedene grah G= (S, O0) One an verfy that the set of sequenes eetng the reedene grah G s: Θ = abdefgh, abdefgh abdfegh, abdfegh. abfdegh, abfdegh It s easy to verfy that ths te the grah G s not equvalent to the set Ω Θ. Obvously, fro the way we have onstruted G, t holds: Ω Θ () The favorable stuaton s when Θ =Ω, beause one an onlude that the grah G= (S, O0) s equvalent to the set Ω. One an regard the onstruton Ω G Θ lke a a that defnes the set Θ for a gven Ω. One an ask the queston: does the set Ω have a roerty that guarantees the equalty Θ =Ω? In the next setons we shall show that suh a roerty exsts. The general for of a olete sequene belongng to Θ We shall note by GI = (S, I) the undreted grah of the ndfferene relaton, sly alled below as ndfferene grah. a b f d e Fg. Indfferene grah for exale. Generally, GI s not a onneted grah. Let C, =,,, be the onneted oonents of GI that ontan ore than one sybol. For the exale above, t exsts only one suh onneted oonent: C={, d, e}, whle for exale there are C={b, } and C ={d, e, f}. The varablty of olete sequenes s gven only by the sybols belongng to the onneted oonents. The other sybols have fxed ostons nsde of a olete sequene (for nstane, the sybols a, b and f n exale ). We shall all segent a gener artal sequene ade u of all sybols belongng to a onneted oonent C, =,,,. We shall note by S the segent orresondng to C. In a gven olete sequene the segent S, =,, s nstaned by the artal sequene S, =,,. In exale, there are two segents, S and S, orresondng to the two onneted oonents. A olete sequene has the for assgh, where S s ether b or b and S s ether def, dfe or fde. Theore (the for of a olete sequene eetng a reedene grah G): Θ = a b...s ( ).d e...f.s ( ).g h... ( ). k... n. Hene, Θ wll aear as the set of all nstanes of the exresson: a b...s.d e...f.s.g h.... k... n.

3 Lea : Let Θ. (ù = σ.a. σ.b. σ, a?b, σ φ) x ó :(x?a) (x?b). The roof s obvous, beause a, b and all the sybols of σ belong to the sae onneted oonent of GI. 4 The ase Ω=Θ. Neessary and suffent ondton Let onsder X a set of olete sequenes fored wth sybols of S. Fro X t an be dedued ts assoated reedene grah G X, usng the sae onstruton as above. Let s(x)={(a, b) a, b S, a? b} be the set of ars of ndfferent sybols, dedued fro G X. Note that s(ω)=s(θ). Defnton (roerty Π): Let X be a set of olete sequenes fored wth the sybols of S and E S S. If ( X, (a, b) E S S, = α.ab. β) ' = α.ba. β X we say that X has the roerty Π n relaton to the set E. In the ase E=s(X), t s sly sad that X has the roerty Π. Reark: Θ has the roerty Π. Notaton: ()=a - the sybol a has the oston n the olete sequene (a, b) - the erutaton of the sybols a and b "." - the uxtaoston oerator Lea : If there exst two olete sequenes, Θ, suh that ()=a, ()=a and <, then t exsts a olete sequene eetng G so that (+)=a and, oreover, an be obtaned fro by alyng a sequene of erutatons. Beause ()=a, one an wrte: =σ.a.σ, where σ s a artal sequene ontanng - sybols. We shall show that b σ, a?b. Suosng that x σ: a<x, t results that ( Θ, (k)=a) k. But Θ wth ()=a and >. Therefore, the suoston s false. So, we an note by b the frst sybol of σ whh s ndfferent wth a. Hene, we an wrte: =σ.axx...xb.σ and x, =... : a < x. () But a? b, so, alyng lea, we obtan: (x? a) or (x? b), x, =... (4) Fro () and (4), t follows that: x? b, x, =... Usng the roerty Π of Θ, one an aly the sequene of erutatons: (x,b) (x,b) σ.ax x (x,b)...bx x. σ σ.abx x...x. σ ' Θ (a,b) σ.bax x...x. σ ' = ' Θ... Θ; (a,b) (x,b) σ.ax x...bx. σ ' Θ We note: :(x, b), (x, b),..., (x, b), (a,b). (x,b) It s known that σ ontans - sybols, so (+)=a; besdes,, q.e.d. Consequene: If there are olete sequenes, belongng to Θ, so that ()=a, ()=a and <, then, for eah nuber k, <k<, t exsts another olete sequene belongng to Θ so that (k)=a and, oreover, an be obtaned fro by alyng a sequene of erutatons. Usng the result of theore, we ntrodue the followng Notaton: Θ : = () ()... () ( + )... ( + l) ( + l + ) l... (... ( >, =... ) ( + )... ( + l ) S ) ( + )... ( + l )... (N), S S Theore : Let, Θ be two olete sequenes: = () = ()... S (N); (N) For any,, t exsts a sequene of erutatons suh that: ' = ()... S... (N). We shall fx. To slfy the notaton, we shall wrte: =... aa...a l... ; =...bb... bl S S Obvously: { a } = { b }. If l l l k k=,,... lk k =,,... (5) a b, then frstly we shall try to obtan fro a sequene havng the sybol Beause b l ( q) = b, l (r) = a, l (r) = b l on the oston l of S. and q<r, one an aly the onsequene of lea ; hene, t exsts a sequene of erutatons,, suh that = K K l b l In the sae way, fro we shall obtan a sequene havng the sybol bl on the oston l- of S. Therefore K

4 : b = K dd K 444 K d l b l l 4444 Reark: To eet exatly the ondtons of lea, we ust have the guarantee that does not hange the oston of b l n. Ths eans we ust rove that b s not the frst sybol ndfferent wth bl. In the l S ooste stuaton, we should have < =...b l σ' b l...and x σ': b l 44 4 S So, n every sequene of Θ, all the sybols of σ should sueed bl. But s not suh a sequene. Hene, our suoston s false. We roeed n the sae way for eah sybol of S. Fnally we obtan l... x =...bb...b l and, so, the sequene of erutatons :,,..., l realses the transfer S, q.e.d. Theore : Let, Θ be two olete sequenes. Then t exsts a sequene of erutatons suh that. We wrte the sequenes and n the for (5) and we use tes the result of theore as follows: () () () : = ()... S... (N); () () () () : = ()... (N); () () ( ) () : = ()... (N); Therefore, the sequene of erutatons = (). (). (). aheves the transfer: () = ()... (N) = = ()... (N) =. In the rustanes of the onstruton Ω G Θ, resented before, we gve the an result. Theore 4 (neessary and suffent ondton for the ase Ω=Θ): Ω=Θ Ω has the roerty Π. Neessty: We know that Ω=Θ. Beause Θ has the roerty Π and s ( Θ) = s ( Ω), t follows that Ω has the roerty Π. Suffeny: We know that Ω has the roerty Π and we shall rove that Θ = Ω. Suosng that Ω Θ and onsderng the relaton (), t holds Ω Θ. Hene suh that Θ and Ω. We shall fx suh a olete sequene. Let onsder 0 Ω. Aordng to theore, there s a sequene of erutaton o, suh that : (a,b ) (ak,bk ) = (a, b ).(a, b )...(a, b ), where (a, b ) s ( Ω) = s... k k ( Θ), =...k. (a,b ) (ak,bk ) k (a,b)... Fgure 4 llustrates ths stuaton. The last erutaton nvolves by reflexvty the followng one: (b k,a k ) 0 k. Suosng that κ Ω, t follows that Ω has not the roerty Π, fat that ontradts our suoston. Hene, κ Ω. κ 0 κ Fg. 4. Sequene of erutatons By alyng the sae reasonng, t results that Ω, whh s a ontradton to our suoston. Therefore, Ω=Θ, q.e.d. 5 Use of the roerty Π One an ake an algorth that dedes whether a gven set Ω of olete sequenes has the roerty Π or not. If the answer s ostve, the reedene grah equvalent to the set of sequenes s G. If Ω has not the roerty Π, the algorth should generate the artton of Ω nto subsets havng the roerty Π, eah of the beng reresented by a reedene grah. The arttonng anner ay be subet to two rtera: (a) the nal nuber of subsets, and (b) at eah teraton hoosng the subset of axal ardnal. The base dea of the algorth s to aly sequenes of erutatons to eah olete sequene of Ω, n order to obtan ts syetr sequenes. The algorth an start fro any sequene of Ω. When obtanng a syetr sequene that overasses the set Ω, the algorth wll sto and dede that Ω has not the roerty Π. In ths ase, usng the sae dea, t ust be found the axal subset of Ω havng the roerty Π n relaton Ω Θ

5 wth a subset of s(ω). One ay verfy that the roble of arttonng s NP-hard. In the exale below t has been onsdered as otal artton that one orresondng to a nal nuber of reedene grahs. Exale : Let onsder a set of olete sequenes: Ω = = abdegfh, = abdegfh, = abdegfh 4 = abdefgh, 5 = abdegfh Ω = s( ) ( b, ), (, d), ( {, e), ( f, g) 4 abdef ;abdef; abdfe; abdfe; abedf; abedf; Θ = abefd; abefd; abfde; abfde; abfed; abfed Ω Θ Ω has not the roerty Π. The algorth has been arbtrarly started wth =abdegfh. Reark: To obtan syetr sequenes startng wth a ertan sequene eans, n fat, to buld the grah of the syetry relaton, whose onts are the sequenes of Ω and whose edges are labelled by the ars of s(ω). Fgure 5 resents ths onstruton. The otal artton that has been obtaned s: ( ) () = {,, }; = {, }. X ot 4 5 X ot * 4 * * * * * Fg. 5 The grah of the syetry relaton (sequenes overassng Ω are reresented by * ) The obtaned artton: One an verfy that the subset Xot () s equvalent to the ( ) ( ) () ( ) Ω = X ot X ot ; X X ot = ; grah ot G and the grah G s equvalent to the subset ( ) X ot has the roerty Π n relaton wth {, } Xot () (see fgure 6). ( ) X ot has the roerty Π n relaton wth {} = abdegfh G () = ab deg fh X ot a b g f h = abdegfh d e Ω = G 4 = abdefgh b () X ot a d e f g h 5 = abdefgh Fg. 6 Preedene grahs reresentng the set Ω * 6 Conluson In the feld of assebly systes, the reresentaton of a gven set of assebly sequenes by a reedene grah s a ont of nterest. In ths aer, we have analyzed the equvalene between a set of assebly sequenes and a reedene grah. When ths equvalene exsts, we have roved that a roerty of the ntal set of sequenes (roerty Π ) s et. The test of ths roerty an be leented by a sle algorth. For a gven set of assebly sequenes that does not eet the roerty Π, we have suggested a way to obtan an equvalent set of reedene grahs. Obvously, ths roble s NP-hard and, so, any algorth oneved to solve t wll be exonentally olex. Referenes [] S. Ghosh and R.J. Gagnon, A orehensve lterature revew analyss of the desgn, balanng and shedulng of assebly systes, Internatonal Journal of Produton and Researh, Vol.7, No.4, 989, [] D.W. He and A. Kusak, Desgn of Assebly Systes for Modular Produt, IEEE Transatons on Robots and Autoaton, Vol., No.5, 997, [] J.M. Henroud and A. Bourault, LEGA: a outeraded generator of assebly lans, Couter Aded Mehanal Assebly Plannng, Kluwer Aade Publshers, 99 [4] L. S. Hoe de Mello and A. C. Sanderson, A Corret and Colete Algorth for the Generaton of Mehanal Assebly Sequenes, IEEE Transatons

6 on Robots and Autoaton, Vol.7, No., Arl 99, [5] C. Ke and J.M. Henroud, Systeat generaton of assebly reedene grahs, IEEE Inernatonal. Conferene on Robots and Autoaton, Vol., San Dego, Calforna, U.S.A., May 994, [6] V. Mînzu and J.M. Henroud, Assgnent Stohast Algorth n Mult-Produt Assebly Lnes, Proeedngs of the IEEE Internatonal Syosu on Assebly and Task Plannng, Marna Del Rey, Calforna, U.S.A., August 7-9, 997. [7] B. Rekek, E. Falkenauer and A. Delhabre, Mult- Produt Resoure Plannng, Proeedngs of the 997 IEEE Internatonal Syosu On Assebly and Task Plannng, Marna Del Rey, Calforna, U.S.A., August 997,. 5. [8] D.E. Whtney, et al., Couter-Aded Desgn of Flexble Assebly Systes, Frst Reort, C.S. Draer Laboratory, In. Reort No.CSDL 947, Cabrdge, Massahusetts, August 986. [9] D.E. Whtney, et al., Couter-Aded Desgn of Flexble Assebly Systes, Fnal Reort, C.S. Draer Laboratory, In. Reort No.CSDL -R-0, Cabrdge, Massahusetts, U.S.A., January 988.