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Timed continuous Petri nets and optimization via linear programming

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(1)Timed continuous Petri nets and optimization via linear programming Bruno Gaujal, Alessandro Giua. To cite this version: Bruno Gaujal, Alessandro Giua. Timed continuous Petri nets and optimization via linear programming. [Research Report] RR-4583, LIP RR-2002-35, INRIA, LIP. 2002. �inria-00072002�. HAL Id: inria-00072002 https://hal.inria.fr/inria-00072002 Submitted on 23 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Timed continuous Petri nets and optimization via linear programming Bruno Gaujal — Alessandr0 Giua. N° 4583 October, 2002. ISSN 0249-6399. ISRN INRIA/RR--4583--FR+ENG. THÈME 1. apport de recherche.

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(291) (// ,. \ci[_KÀ[_G9U‡[\UR—[_Ui[TfgcWn9UWaT]’z7KG9U³•rfgcWaHfReUŽ€WKn9KamUi“gf‘£0U‡[TfgcWn#ciŸUh7Ka_fgcrkjfgl&z7KG9U³•rfgcWa0™ ¤'fgn9Ui“g“g]˜%K&J’U('uKN[TGjK&Ÿ cu“‘“gc‡f‘n9€Žcuzr•@fgcWVR%cuz9RTKa_•³Ui[TfgcWn¯™  ,

(292)  « b« ƒ „  «       ,  

(293) 

(294) ; ,  ,  (N, δ) . lim.

(295) (. /, . !

(296). C. . ,   ,   3,  , ;

(297)  >   .   ,

(298)    . . !. . !.     ,  $, 

(299)  . λ. 

(300)  . . > !:

(301) . ,. τ →∞. . ,

(302) > 

(303) 

(304) .  , .

(305)   .  . Z(τ ) = λ. τ. N. , 1      .  

(306) . Cλ ≥ 0 .  ,

(307) .   ,    ,  ,  

(308)    , 

(309) / >=

(310) ((

(311)   ,   ,     Cλ 0    > : !(/  >

(312) 

(313)    ,  

(314)  

(315) >

(316)     ,  λ    Cλ = 0 . Š Ÿ

(317) [TG9K’njK[a_KSUulmGjKSR&U–R^[mU‡[_f‘cun9Uia_]DzKSG9U³•rf‘cuaS{¯[TGjKSnvŸ a_cWJ [TGjKZR—[_Ui[TKK @V9Ui[TfgcWn Â5 t 7N%KG9U³•uK [_G9U‡[Ÿ cuaeUi“g“¡hj“UWlK0R p >:" . i. lim. τ →∞. Ê jê yî. BB C 1D>E. Mi (τ ) Z(τ ) = C(pi , ·) lim = C(pi , ·)λ. τ →∞ τ τ.

(318) LSp.

(319) 

(320) /. U 7;4\R_RTVjJKW{zr]4lcWn@[TamUWkjfgl[TfgcWn¯{

(321) [TGU‡[Z[_GjKa_K6KXrfR—[_R˜h9“gUulK {r[TG9Ui[efgRl“‘K0Uia_“‘]Znjci[h7cuR_R—fgzj“gKW™ lim M (τ ) < 0 z 7¯K[ p zKUŽhj“UWlK"R—V9lmG#[TGU‡[ C(p , ·)λ > 0 ™¢FHGjKn τ →∞. pi. R—VlmG [TG9Ui[. C(pi , ·)λ < 0. 

(322). ™FHGjKn. i. i. i. Mi (τ ) >0 τ →∞ τ lim. =⇒. lim Mi (τ ) = +∞. =⇒. τ →∞. UWn9k#V9R—fgnj€Ž[TG9K"aTVj“gK"ciŸ'“ ecuhjf‘[_Ui“. lim. τ →∞. Z. τ. Mi (%)d% = +∞ 0. . lim. H& .  ) . 0. τ →∞. l 7H¤jcu“‘“gc‡eR%Ÿ aTcuJz 7y™ */B6 /21CE. Rτ. Mi (%)d% = lim Mi (τ ) = +∞. τ →∞ τ. ,V4",(? 63,(8963,(E. /5*.  # @ . YF ÀO fReUŽŸ Vjnly[TfgcWn ρ : P × T −→ [0, 1] R—VlmG–[TG9Ui[ P re(p, t) = 0 =⇒ Âf>™ KW™g{ ρ(p, t) > 0 cWnj“g]#f‘Ÿ t ∈ p 7UWn9kDRTV9lmG–[TG9Ui[eŸ cWaNUi“g“ p ∈ P 9 P ρ(p, t) = 1 ™ Š Ÿ ρ(p, t) = 0 {W[_GjKnU"Ÿ amUWl[TfgcWn ρ ciŸ¡[TG9KÀJ’Uia2'ŽKn@[TKSaTfgnj€,h9“gUulK p fR

(323) aTK0R—KSaT•uKSkŽcWnj“g]ŽŸ cWa%[TGjK ρ(p , t ) = ρ > 0 aTfgnj€˜cWŸfõ[mR\cuVY[ThjVj[\[TamUinR—f‘[TfgcWn t UWn9kDlUWnjnjci[ÀzK yV9RTKSQk À[TcZKSn9Uiz9“‘K,UWn9k–9a_K,UWn@]ci[_GjKaÀcWVY[_hjVY[ [_a_UWn9RTfõ[_f‘cun¯™"FHG@VR{¡U @YF 4ÀO RTcW“g•WK0RÀUW“‘“lcunY”9fly[_RÀ[_G9U‡[&J’U³]–UWaTfRTK,fgn6[TG9KŽn9K[&fõ[_GvU#hjaTK0UWR_R—fg€Wn9KSk 9XrK0kZh7cW“gfl]W™ Š nD;U @YF 4\O 0 ­ `  5ÂVjn@[TfgJŽK0k 7

(324) [_GjK&9a_f‘nj€K uVU‡[TfgcWn 5^L 7%fgRJcYkYf‘9KSkfgn@[Tc 4. 

(325) 

(326)  2   ! 5@ 4 7 5. •. t∈T. i. j. i,j. i,j. i. j. z ≤ min ρi,j Mi /P re(pi , tj ). • pi ∈ t j. ÀR—fgnj€v[TGjK[_a_UWn9R—f‘[TfgcWnR9a_fgnj€›lcWVjn@[_KamR. Yk KSR_la_fgzK0k fgn [_GjK–hjaTKS•rf‘cuV9RRTVjz9RTKSl[TfgcWn9RUWn9k UuR 0 RTfg€Wnjfgnj€ [Tc KSUWlmG hj“UWlK p U lcuVjn@[TKa H (τ ) [TG9Ui[ZaTKShja_KSRTKn@[’[TGjK uVUin@[Tf‘[^] cWŸ&J’UWa:'YR[_G9U‡[G9UWR KSn@[TKa_KSkZ[TG9K"hj“gUulKNŸ a_cWJ [_f‘JK 0 [Tc τ 5 fgn9l“gV9kYfgnj€[TG9K"f‘njf‘[TfUi“JUWa:'rfgnj€ 7{r[TGjK"KS•Wcu“‘VY[_f‘cun#cWŸ'U[TfgJKSk @jF4ÀO 0

(327) F ­ ` lSUinzK"a_f‘[—[TKSnDUW"R 9   5 p 7 ρ H (τ − δ ) Z (τ ) ≤ min ∀t ∈ T P re(p , t ) X 5>3q 7 H (τ ) = M (0) + P ost(p , t )Z (τ ) ∀p ∈ P. i. Zj (τ ) i. i,j. j. i. i. i. j. pi ∈ • t j. i. i. j. i. j. j. i. t j ∈ • pi. É æ B¯ÉËì.

(328) L³q.  

(329) !"

(330) $#. ÀRf‘n‹[_GjK €WKSnjKamUi“NlUWRTKvkjfgR_lV9R_RTKSk‹fgn hja_K•rfgcWV9R#R—V9z9R—K0ly[_f‘cun¯{e[_GjKvJ’UWa:'rfgnj€ U‡[[TfgJK τ lUin zK kjK[TKSaTJfgnjKSk#zr]˜[_GjKR^[mU‡[_KNK @V9U‡[_f‘cun5>t?7y™ FHG9K,K•ucW“gVY[TfgcWn–[_G9U‡[&lcuaTa_KSRThcun9kjRH[_c#lmGjcrc@R—fgnj€’[_GjK,K uVUi“gfõ[^]–fgnD[TGjK  @V9U‡[_f‘cun65 p<7efg“g“z7KlSUi“g“‘K0k

(331) 

(332) 2 5>UWR¯RTcrcWn"UWR¯hc@RTRTf‘z9“‘K 7©zK0lUWV9R—K¢f‘[lcWa_aTK0R—h7cWn9k9R7[TcH9a_f‘n9€eUH[TamUin9RTf‘[TfgcWn"UWR¯RTcrcWn&UWR©[_GjK”VjfgkN[_G9U‡[ GUWRNKn@[TKSaTK0kD[_GjK’f‘n9hjVY["hj“UWlK0RNzK0lcWJK0R"U³•³UWf‘“Uiz9“‘K,Ÿ cWaN[TGjKaTfgnj€9™ŽFHGjKlcWa_a_KSRThcun9kYfgnj€#lcWVjn@[TKSa_R UWaTK&kYKSnjci[_KSk Z = [Z · · · Z ] ™ FHG9KÀ[_f‘JK0k–•WKamRTf‘cunZciŸ'hja_cWh7cuRTfõ[_f‘cun–|fRH€Wfg•WKnz7K“gc‡"™    > N Q   

(333)  ;

(334)   > ,   

(335) >

(336) (   3 > ?, 

(337)  « b« ƒ „  «   C > 3, 

(338)  ;

(339) V( ! 

(340) (C  (/  !; , 2(3 > 3,    ,

(341)   

(342)  ,     *  M = M (0) + Cz  δ   ,  , ,  >

(343) $ ,      /;    , !

(344) 

(345)  , (/  .  , 2   , . , = (N, δ)  τ  Z(τ )  =  ,    

(346) (

(347)  4. asap 1. asap. asap T n. . τ. . Z(τ ) = z . M (τ ) = M. ]UuRTRTVjJhY[TfgcWn$4L 5ÂRTKKHa_KJ’Uia2'q37'Ui“g“@[TamUin9RTf‘[TfgcWn9RlUWnz7K

(348) 9a_KSk¡™ \RTf‘nj€&hja_cWh7cuRTf‘[TfgcWnŽ|j{‡%K 'rnjc‡œ[_G9U‡[N[TG9Ka_KKXrfR—[_R&UZRTK @VjKSn9lK RTV9lmG6[_G9U‡[ M (0) → M → (j , z ), · · · (j , z ), · · · ™ M ··· → M ]#

(349) UWfõ[_f‘nj€’“gcWn9€’KnjcuVj€WG 5ÂkYKShKSn9kYfgnj€Zcun[TGjK•‡UW“‘VjK"cWŸ δ 7y{UW“‘“JUW:a 'YRh9aTK0R—KSnu[eUi[e[TfgJK 0 zK0lcWJK U³•‡UWf‘“Uizj“gKW{UW“‘“gc‡fgnj€ t [Tcv9a_K#U @V9Uin@[Tf‘[^] z ™ŒwÀn9lK#J’Ui2a 'rf‘n9€ M fRŽaTK0UWlmGjK0k¡{¢cWn9KZnjKSKSkjR[Tc HUif‘[Tfgnj€ RTcWJK6JcuaTKž[TfgJŽKvVjn@[Tfg“NUW“‘“NJ’Ui2a 'YR˜fgn M z7KSlcWJKvU³•‡Uifg“Uizj“gKW™FHGjKSn t J’U³] aTKžU @V9UWn@[Tf‘[^] z UWn9k–R—cŽŸ cua—[_G¯™ eci[_K&[TG9Ui[[TGjfRKS•WcW“gVY[_f‘cunZfRnjci[njK0lK0RTR_Uia_f‘“g]

(350)

(351)  ™ >:" . 1. (j1 ,z1 ). 1. k. k. 1. (j2 ,z2 ). (jk ,zk )···. 2. 1. j1. 1. 1. j2. 2. š›K–Uia_KZfgnu[_Ka_KSR—[TK0kf‘n 9n9kYfgnj€›cWVY[f‘ŸÀU

(352) F ­ `  aTK0UWlmGjK0RUž%K0U('R—[_U‡[_f‘cun9Uia_]›zKSG9U³•rf‘cuaS{fE™ Ku™‘{bU lcWn9kjfõ[_f‘cunf‘n G9fglmG[_GjKZJK0UinnrVjJ,z7KaŽciŸH9a_fgnj€6ciŸeK0UWlmG [_a_UWn9R—f‘[TfgcWn t h7Ka,[TfgJKZVjnjf‘[aTK0UWlmGjK0R UuR—]rJhY[_ci[TflUW“‘“g]˜Ulcun9R^[mUin@[•‡Ui“gVjK39 j. λj = lim. τ →∞. Z(τ ) . τ. HF G9K\cWn9“‘]$'rnjc‡n’a_KSRTVj“õ[Hlcun9lKSaTnjfgnj€"[_GjKNKXYfgR—[TKSn9lKÀciŸUR^[mU‡[TfgcWnUia_],z7KG9U³•rfgcWaŸ cua @jF4ÀO 0

(353) F ­ `  fRH€Wfg•WKSn#fgn[TGjK&Ÿ cu“‘“gc‡fgnj€,[_GjKcuaTKSJ–™

(354) ¦r« ¦     

(355)  > : !(/   , 1 ,       

(356) (

(357)  

(358)   > ,   

(359) >

(360) ( . .

(361) (    

(362)    ,      ,3, /

(363)   ,  ,

(364)  ,   >

(365)       , . Ê jê yî. BB C 1D>E. . . .   C //   , /   . limτ →∞ Zjasap (τ )/τ.

(366) L.

(367) 

(368) /. 

(369). t3. t1. ÂU. α1. 5 7. [ρ3 ] p1 [ρ4 ]. α2. t4. t2 t1.  z. 5 7. t3. p31. α1 ρ 3 α2 ρ 3 α1 ρ 4 α2 ρ 4. p41. t2. t4. ¤'fg€WVja_Kq 9 >5 U?7 œ4 hj“UWlK&f‘[TGDlcunY”9fly[ 5 z 7%[_a_UWn9R—Ÿ cWa_JUi[TfgcWnZ[Tc˜RTcW“g•WKÀ[TGjKlcWnY”9fly[V9RTfgnj€˜U;@YF4ÀO"™ H&. >=",

(370) I(/5476. ? 6 1C= 8 63=1 I(I. 8;:<4 ,(8 6  13/2/. 4/26. \nr]

(371) F ­ `  (N, δ) Gjc@R—K"lcWnY”9fly[mReUia_K"R—cu“‘•uKSkV9RTf‘n9€U @jF4ÀO ρ lUWn#z7K"[_a_UWn9R^Ÿ cuaTJK0kZfgn@[Tc˜Uin K @Vjf‘•‡UW“‘KSnu[ >    2

(372) // (  1  , ,,njK[5 ) t+ 7 (N , δ) {¢fE™ KW™g{U6njK[fgn GjflmG KSUulmGhj“UWlK#G9UWRU‡[ Jc@R^[bcunjKecuVY[ThjVj[b[TamUinR—f‘[TfgcWn¯™ bUulmG’hj“gUulK p fgn[TGjKÀcWa_f‘€uf‘nUi“jnjK[%fRbRThj“gfõ[

(373) f‘n@[Tc h hj“UWlKSR p , · · · , p 5GjKa_K fRH[TG9K,R—fg£K"cWŸ 7y™%FHG9KnrVjJ,z7KaecWŸ'h9“gUulKSRef‘n fRekYKn9ci[TK0k m ™ 4e“g“¯[_GjKhj“UWlKSR p h p (N , δ) GU³•WK&[TG9K,RTUWJKGjcW“kYfgnj€’[_f‘JK,UuR[TG9KcWa_f‘€uf‘nUi“¯hj“UWlK p 9 δ = δ {9Ÿ cua\Ui“g“ 1 ≤ k ≤ h ™eFHGjKfgnjfõ[_fgUW“ J’UW:a 'rfgnj€˜fgn6hj“UWlK p fR M (0) = ρ(p , t )M (0) ™ÀFHGjKnjK h9aTK0R—K[ÀUin9k6hc@R^[mR—K[NUia_K,RTV9lmGD[_G9U‡[ { Uink P re(p , t ) = 0 fõŸ P ost(p , t ) = P ost(p , t ) × ρ(p , t ) P re(p , t ) = P re(p , t ) ™ 4en#KXjUiJhj“gK\cWŸR—VlmGZU,lcWn9R—[Ta_V9ly[_f‘cun’fgR

(374) RTGjc‡n’f‘nZ9€WV9aTKÀqGjKa_Ke[_GjK\a_cWVj[Tfgnj€,lcrK

(375) ’lf‘KSn@[_R  j 6= k UWaTK&kYfR—h9“gU³]uKSk#zK[^%KSKnzja_Uul 'WK[_R%[Tcz7KkYfgR—[Tfgnj€uVjfgRTGjK0kZŸ a_cWJ[TGjK"

(376) Kfg€WG@[_RHcWŸ[_GjKUiamlRS™       hN, δ, M (0), ρi

(377)    ,

(378)   , 3,  , >    :

(379) //  « b« ƒ „  «     ,  4. .

(380). 1 i. i. h i k i. • i. i. u. i. i. i. u. i. k. k. i. k i. k. i. k i. k. . . i. k. k i. k i. k i. j. . (

(381)  1   , , 

(382)  , %, $  

(383) (/ ,      , 1//    !A , C , 1  

(384) //  hN , δ, M (0)i τ  ;

(385) 

(386) /     ,  , =1  

(387)    /;  

(388) ;

(389) /     ,  , =1. Z(τ ). N. Z(τ ). ∈ R+  N. É æ B¯ÉËì.

(390) L.  

(391) !"

(392) $#. HF G9K"hjaTcrcWŸfRz9UWRTKSk#cWn#[TG9KK•ucW“gVY[TfgcWnK @V9Ui[TfgcWn9RHcWŸ Z(τ ) ™bš›KkYKnjcW[TK"zr] Z Uin9k H [_GjK lcWVjn@[_Kab•uKSl[TcWamRcWŸ¯[TG9KÀnjK[ N ™ ÀR—fgnj€,[TGjKNhja_K•rf‘cuV9Rn9ci[_Ui[TfgcWn9RS{@cWnjKNG9UWRbŸ cWaHUi“g“[TamUin9RTfõ[_f‘cun t Uink UW“‘“'hj“UWlK p { δ = δ Uin9k P ost(p , t ) = P ost(p , t ) × ρ(p , t ) f‘Jhj“g]–[TG9Ui[ H (τ − δ ) = ™%wÀn[TGjK"cW[TGjKSaG9Uin9k¯{ ρ = 1 ™¢FHGjKSaTKŸ cWa_KW{ Z(τ ) = Z(τ ) ™ ρ(p , t )H (τ − δ ) >:" . j. k. i. i. k. i. i. k i. i. k. j. i. j. i. k. i. i. k i,j. i. HF G9fgRJK0Uin9RH[_G9U‡[[_GjK"hja_cWh7KaT[TfgKSRcWŸ'“gfg•WKn9KSR_R{Yz7cWVjnkYKSkYn9KSR_ReUia_KN[TGjK,R_UiJKNŸ cWa\zcW[TGDnjK[_RS™ 8DcWa_K10 c‡•uKa0{f‘Ÿ N fgR&R—[Ta_cWnj€u“‘]DlcWnjnjK0ly[_KSkžUWn9kD[_GjK @YF 4\O lcrK

(393) ˜lfgKn@[mRNUWaTKŽUi“g“hc@R—f‘[Tfg•WKu{©[TGjKSn N fgR&Ui“RTc R—[Ta_cWn9€W“g]˜lcWn9njKSl[TKSk¯™ 4

(394) [[_GjfRhcuf‘n@[0{jcWnjK&J’U³]˜njcW[TflKN[TGU‡[e

(395) K&lcWVj“kG9U³•WK&lcun9RTfgkYKSaTK0k#UŽJcWa_K&€WKSnjKamUi“©a_cWVY[_f‘nj€{YGjflmG [m(U 'uKSRHfgnu[_c˜UWlSlcWV9nu[[_GjKcuaTfg€Wfgn#cWŸ[_GjKJ’Ui2a 'YR™ Š nDhj“gUulK p {j[_GjKa_cWVY[_f‘n9€lcrK

(396) ˜lfgKn@[[Tc˜[TamUin9RTfõ[_f‘cun lcWVj“kUi“RTc kYKh7KnkcWn [_GjK›cWa_f‘€uf‘n 5>U[TamUinR—f‘[TfgcWn t ∈ p 7˜ciŸ"[_GjKvJ’UW:a 'YRS{ekYKnjcW[TK0k t ∈ p ™ Š n [_GjKZlcunY”9fly:[ 0>Ÿ a_KK’njK[S{'[TG9fgR,JŽK0Uin9R"[TG9Ui[[TGjKZ%KSf‘€uGu[z7KSlcWJKS"R 9 P ost(p , t ) = ρ(t , p , t ) Š n[_GjK

(397) Ÿ cW“g“‘c‡fgnj€{0

(398) Kbfg“g“ 'WKKSh"[TG9K%a_cWVY[_f‘n9€\fgn9kYKh7KnkYKn@[¢ciŸj[_GjK

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