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Backjumping   W Backjumping   X Backjumping   5/COPS YCOPS ZCOPS [COPS \COPS ]COPS ^COPS _COPS `COPS aCOPS b ILOG SOLVER  c ILOG SOLVER  d ILOG SOLVER  e ILOG Schedule 1.0 f ILOG Schedule 1.0 g ILOG Schedule 1.0 h Scheduling Problem$ Constraints  i Constraints  jAlgorithm Program  kAlgorithm Program  lAlgorithm Program  mAlgorithm Program  n.Optimal Solution to the Scheduling Problem $/, 0` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> (     60* P  LUSQdkYkHrh7h_    0,   8USQdkYkHre,g7h_ ,{N~ ,{ N~ ,{V~ ,{N~    0<3 ``  X*  0@: `   Z*  0> `   Z*H  0޽h ? ̙33  ؞!jg0 ~v(    0@m P    P*    0Hp     \* d  c $ ?    0u  @  8USQdkYkHre,g7h_ ,{N~ ,{ N~ ,{V~ ,{N~    6{ `P   Z*   68 `   \* H  0޽h ? ̙3380___PPT10.p@ 0x( c8 x x 0ܑ P    X*  x 0     Z*  x 6 `P   X*  x 6̡ `   Z* H x 0޽h ? ̙3380___PPT10.@V 0 0(  x  c $P   ^B  6D>PpPx  0 p`  b ,{ Nz ~_gct S_ i -NVyf[b{b/g@b 2 ',,', C ( C   C  C    2H  0޽h ? ̙33 0 ^V (   l  C ^P     <Tb? p3.1 i 3.2 Vnl 3.3 ~_g Od 3.4 Vl 3.5 ~_gct|~COPS 3.6 ILOG SOLVER FK=C  C Cr^B  6D>PpPH  0޽h ? ̙33 0 jb( $ r  S pP     <eI  v gOS ~Nmf[@bc]v^/}XbgO: Q*NNb@w4lvh(WN*N4l4YMRbcSb4l 4l vh g'Y g\0NN`7hc MbO_;`vc egw0ُ/fN*N[Bl gOS vv vh /fw;`vce 0RgO0Dv  S $!R^B  6D>P@PH  0޽h ? ̙33: 0 z(  r  S ΡP   b  0hҡ@   OS Џy{f[ W O{l ^y~Q~ ~_gctQ  00 $ #$$#$$#$$#$Z    ^B  6D>P@PH  0޽h ? ̙33< 0N0 (    s *P<$D 0     s *<$ 0  r  N??"0@^ ^B  6D>P@PH  0޽h ? ̙339w9___PPT10W9.+tD8' = @B Dz8' = @BA?%,( < +O%,( < +Dk' =%(D' =%(D' =A@BBB B0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D%' =+4 8?pCBB+#ppt_w*sin(2.5*pi*$)CB?B*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*DY' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<* D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<* DY' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<* D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<* DY' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*DY' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(#/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*!D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*!D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*!D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*!DY' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(+/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*!)%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*!)D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*!)D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*!)D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*!)DY' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(3/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*)D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*)D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*)D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*)+p+0+0 ++0+0 + 0N0 >6(  ~  s * P     s *<$0  ^B  6D>P@PH  0޽h ? ̙33___PPT10.+} <DO'  = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+0 +  0N0 h`()(  ~  s *@     s *$<$0  x  ) #"*UP *4`8  <? P15 h@`  < ?  P25 h@`  <(?   P80 h@`  <2?  ^d5uS/(1R)  $ h@`   <;?\ P40 h@`   <D? \ O4 h@`   <M? \  Q300 h@`   <tX?\  fgeb/(JSHr)  $ h@`   <`? \ P30 h@`  <i?  \ P24 h@`  <Hr?  \ Q100 h@`  <8}? \ kehfb/(JSHr)  $ h@`  <|?   P80 h@`  <D?   P10 h@`  <?   R1000 h@`  <?   5uƉSb(Zf N,30)  6 h@`  <?p  P50 h@`  <̲? p  P16 h@`  <? p  Q500 h@`  <?p  5uƉSa(}v)Y,1 R)6 h@`  <?p ggv[ OHeg/USMO  $ h@`  <4? p S(ugؚ!kpe/g  $ h@`  <?  p ^JT9(u(CQ/!k)  6 h@`  <T? p n^JTe_ h@``B  0o ?ZB  s *1 ?ppZB  s *1 ?  ZB   s *1 ?  ZB ! s *1 ?\\ZB " s *1 ?`B # 0o ?`B $ 0o ?ZB % s *1 ?  ZB & s *1 ?  ZB ' s *1 ?`B ( 0o ?H  0޽h ? ̙33___PPT10.+} <Dy' = @B D4' = @BA?%,( < +O%,( < +Dk' =%(D' =%(D' =K@BBBBPB0B%(/%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+0 + 0N0 :2@(   0 HA ?@@PLH $D 0 r   ` ??"P ,$D 0H  0޽h ? ̙33B : ___PPT10 .+: yD ' = @B D ' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+_0N0 P(    s *ܘ P   9 0 0 HA ?CPpH $ 0 H  0޽h ? ̙33___PPT10.+a>D' = @B D?' = @BA?%,( < +O%,( < +Dv' =%(D' =%(D' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*+:  0N0 n(  ~  s *!     0 HA ? H $ 0 r   ` ??"0 Pp   # lGH@W??"   : ^B  6D>M7MH  0޽h ? ̙33___PPT10.+a>D' = @B D?' = @BA?%,( < +O%,( < +Dv' =%(D' =%(D' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*+5 0N0  (      s *P<$D 0      s *( <$D 0  &x h_7B   Z??"p@ ,$D 0   Z??"pp',$D 0^B   6D>P@PH   0޽h ? ̙3300___PPT10g0.+{LqD/' = @B D/' = @BA?%,( < +O%,( < +Dk' =%(D' =%(D' =A@BBB B0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D%' =+4 8?pCBB+#ppt_w*sin(2.5*pi*$)CB?B*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D_' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*  %(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*  D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*  D_' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*  %(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*  D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*  D_' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* B%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* BD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* BD_' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* BZ%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* BZD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* BZD_' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(#/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* Zf%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* ZfD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* ZfD_' =%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%()/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* fu%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* fuD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* fuDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* +p+0+ 0 ++0+ 0 + 0 z$(  $r $ S &P    $ <(, N*N~_gn(Constraint Satisfaction Problem, {yCSP) S+TN~SϑNN~Sϑv~_g0 %Sϑh:yWSpe k*NSϑ gN*NV[vP@PH $ 0޽h ? ̙33 0 Z(  r  S 7P   B  08,  %~_gS(uNcW[av'`(0vNsQ|0NR Bl0vhI{0~_g n vvh1\/f~b0R@b g SϑvN*NbY*N K

c ve z0 USMO|pev~'`e zN NI{_ sS@b gv|pe:N -1,0,10 Na|pev~'`e zN NI{_0 ~_gv^\~T0 NpeN N҉e z0M#  6x $/^B  6D>P@PH  0޽h ? ̙33 0 @8 ((  (r ( S DNP    ( <OPP<8 LF~_gh:yfNt0xS gHe[s [wQ gN NOp: ~_gh:yAQNf'`ve_eghWwƋ h R:_ ^(u z^Sc[vvhagNSpenc vvNsQ|0V wQ g;h:yv{|P@PH ( 0޽h ? ̙33R 0 @@(  @r @ S \P   z @ <],r   ~_gd"} ~_gd"};Nxvz gPW Nv~_gn0[ gPW ~_gnN,`Q N /f N*N NP 0 ~_gJC93 ^B @ 6D>P@PH @ 0޽h ? ̙33 0 ~PD(  Dr D S XrP    D <s:$X  | Vnl0 ~_g Od0 zfVnNwP@PH D 0޽h ? ̙33 0 RJ`( $ r  S P     <̅04  ^CONSTRAINTS CHIP COPS ILOG(,$^B  6D>P@PH  0޽h ? ̙33 0 6.pH(  Hr H S P    H < B CONSTRAINTS/fN*NbT5ucv~_gh:y0 \O:NN*N~_gh:y [O(uN&{SYtb/gegBlpef[e z0 (WCONSTRAITS-N irtNvRShVNv~g(u~_gh:y0 ُN~_gN,/f~'`e zN NI{_, _NSbagNh_0~_gSϑ N,/fh:yirtϑv[Sϑ0_N gNNSyceP@PH H 0޽h ? ̙33 0 L (  Lr L S xP    L <t   CONSTRAINTS vN*NOp/f(W{|WB\!k-Nh:y~_g (u~_g egh:yirt[avRN~g0vQ:p/f:ON{|P@PH L 0޽h ? ̙33P 0 P(  Pr P S PP   x P <  & CHIP(Constraint handling in Prolog) 1\/fُ7h gq_T N*N~_g; z^ vQvv/f{O0up;m gHe0W㉳Q N'Y{|~T0[ǏcOQyev{W X:_; z^ vR gPW0^\yS gty [Nk*N{W cO gHev~_gBl㉀b/g sS gPW NvN'`b/g ^\Wv^\ TNb/gS gtpeW NvUS~Wl0ddkNY CHIP؏S+TN*N N,v^ߏ{:g6R0 CHIP ;N^(uN$N*NW: Џy{f[NlxN0 CHIP :ON{|W:g6R ُy:g6R[NhWi_/fgvQ͑ v0 >px A $$  ^B P 6D>P@PH P 0޽h ? ̙33 0 T (  Tr T S lP    T 0  6 COPS|~)R(ubT[ab/g \f'`~_ghN{|WB\!k ~Tweg0(Wb__ N8T6eN8^ĉ ;N/fbT[av z^ vW,gb__0QBleǑ(u~_gct:g6R Of'`~ _gh_N{|WB\!kv~T [swƋv~gS\ň EQRS%c $NvOp RV[sN*NwQ g:_hRTؚBlHesv ~_gn|~0,px A"^B T 6D>P@PH T 0޽h ? ̙33 0 NFX(  Xr X S P    X < t~ ZCOPSvQNoN] zv^(uBl =\ϑ\N*N Nnx[ nx[S[AQagNSN_sS N/fUS ~NR_vb__eg[sN{ Ǐ{|elv͑=h[s TN ~_gv N T[s cؚN z^vgbLHes0COPS|~ Te/fN*N nX_v_>e|~ (u7bǏ{|WB\!k[IN [sevpenc{| WTev~_gsQ|0~_gCOPSwQ gYN]zf z^ vyrp Y~_g Od0bTvhTpencqRvBl0 gP ekvVn0[aRB\-Nv~bI{0 Bpx $A$>a2;^B X 6D>P@PH X 0޽h ? ̙330 `>(    H?@0  (W[E^(u-N {lvhsb__CSSNS FO/f{lv`Q_NTpenc~g{|P@PH  0޽h ? ̙33 0 qi\(  \r \ S x0    \ 0  Exhaustive-Search(f,P) 1. if f is a total assignment of the variables in P 2. if f satisfies the constraints in P 3. answer := f 4. else 5. answer := Unsat 6. else 7. v := some variable in P that is not yet assigned a value by f 8. answer := Unsat 9. for each value while answer = Unsat 10. f(v) := 11. answer := Exhaustive-Search(f,P) 12. return answer&a`H-g^B \ 6D>z<zH \ 0޽h ? ̙33v 0 &(  x  c $P     c $D0  ,h$,4^B  6D>P@PH  0޽h ? ̙33v 0 & (   x   c $|       c $7W   ,h$,4^B   6D>z7zH   0޽h ? ̙33x 0 ( P,(  ,~ , s *k t  k   , s *'   k  "P@`Pp^B , 6D>z7zH , 0޽h ? ffD3f 0 `0^(  0~ 0 s *,k P    0 `AFig06-11B=  H 0 0޽h ? ̙33 0 8:(  8~ 8 s *P P    8 lA  Fig06-13"`A?    8 lA Fig06-14"` A4U  H 8 0޽h ? ̙33y___PPT10Y+D='  = @B + 0 <^(  <~ < s *8 t    < `A Fig06-15dg  H < 0޽h ? ̙33y___PPT10Y+D='  = @B + 0 )(  r  S 8P     0P|  S Backtracking-Top(P) 1 f := the null assignment 2 return Backtracking(f,P)(TaP`P^B  6D>P@PH  0޽h ? ̙33p 0   `(  `r ` S t+P    ` 0, 8 Backtracking(f,P) 1 if f is a total assignment of the variables in P 2 answer := f 3 else 4 v := some variable in P that is not yet assigned a value by f 5 answer := Unsat 6 for each value while answer = Umsat 7 f(v) := x 8 if f satisfies the constraints in P 9 answer := Backtracking(f,P) 10 return answer(a`6+^B ` 6D>P@PH ` 0޽h ? ̙33n 0 0h(  hr h S ;P    h 0 = 6p =\{Vnl}YNubKmՋl FO[N^s^QN6q/fNOHev0vQSV(WNd"}zz-N N T_vd"}͑ Yv Tv1Y%P[_0NNxvz:N bُyS YvSV/f@bv@\ NN'`0g{USv`b_/f@bv~p NN'`0[N*NSϑvivN*NNCQ~_g0X[(WW-NN*NP@PH h 0޽h ? ̙33> 0 @d~(  dr d S L`   ^B d 6D>P@Pl d 0A T??p(  T d 6P  *'_N'` Arc consistency ,( H d 0޽h ? ̙33  0 Pl`(  lr l S T   ^B l 6D>P@PH l 6_  \ Arc consistency Yg[vi vS_MRW-Nv@b gP@P t 6HV  All of the Mackworth algorithms make use of a Revise procedure. Let Dv be the current domain of v, Let Dw be the current domain of w, Let P be the constraint predicate that holds between v and w, then Revise updates Dv as follows:0 ( P 4%t ` t c $A W??, d  Wl t 0A X??  XH t 0޽h ? ̙33A 0 px(  xx x c $P|P   ^B x 6D>P@P x 64  +Mackworth 1977 AC-1 AC-2 AC-34,$($ #` x c $A Y??, d  YH x 0޽h ? ̙33 0 6.p(  pr p S P   ^B p 6D>P@P p 6   H:REVISE(Vi,Vj) 1 DELETE false; 2 for each x Di do 3 if there is no such yj Dj 4 such that(x,yj) is consistent, 5 then 6 delete x from Di; 7 DELETE true; 8 endif 9 endfor 10 return DELETE; 11 end REVISE v,_Yt T \0H p 0޽h ? ̙33\ 0  (  r  S P     0  1 Q ; 2 repeat 3 CHANGE false; 4 for each (Vi, Vj) Q do 5 CHANGE REVISE(Vi, Vj) CHANGE; 6 endfor; 7 until not(CHANGE); 8 end AC-1a`F` ```9`>i%&^B  6D>P@Pl  0A [??,  [H  0޽h ? ̙33 0 |t| (  |r | S P    | 0x ld\___PPT9>6  ,1 Q ; 2 While Q not empty 3 Select and delete any arc(Vk,Vm) from Q; 4 If (REVISE(Vk,Vm)) Then Q {(Vi,Vk) such that (Vi,Vk)arcs(G), ik, im}; 6 endfor; 7 endwhile; 8 end AC-3U+a`` `` ``+ ` ti   . @`^B | 6D>P@Pl | 0A \??,  \H | 0޽h ? ̙33 0 _(  r  S P   G  0H@  wBackjumping-Top(P) 1 f := the null assignment 2 := Backjumping(f,P) 3 return answer (xt`a$?^B  6D>P@PH  0޽h ? ̙33` 0 (  r  S P     04 (Backjumping(f,P) 1 if f is a total assignment of the variables in P 2 answer := <f,> 3 else 4 v := some variable in P that is not yet assigned a value by f 5 answer := Unsat 6 conflict-set := 7 for each value 8 f(v) := x 9 if f satisfies the constraints in P 10 <answer, new-conflicts> := Backjumping(f,P) N`@@@a6^B  6D>P@PH  0޽h ? ̙33r 0 "(  r  S P     0 :r11 else 12 new-conflicts := the set of variables in a violated constraint 13 if answer Unsat 14 return <answer, > 15 else if v new-conflicts 16 return <Unsat, new-conflicts> 17 else 18 conflict-set := conflict-set (new-conflicts {v}) 19 return <Unsat, conflict-set>f:@1@ @@@>~^B  6D>P@PH  0޽h ? ̙33 0 G?(  r  S P     0$  _o Constraint : predicate expression P(t1, ..., tn) where P is built in function, such as sum times eq(equal) neq(not equal) ge(great than or equal to) gt(great than) also can be defined by users`pa ``+```l0*2^B  6D>P@PH  0޽h ? ̙33d 0  (  r  S pP     0  ,D Conditional constraint condition1: constraint1; . . conditionn: constraintn where condition1, ..., conditionn are boolean expressions. constraint1,... constraintn are constraints or contraints table. FEa@`a ` ~~  5 /   ^B  6D>P@PH  0޽h ? ̙33d 0  (  r  S +P     0D- ,2 RULE Rule is used to define new function, method, predicate, or add new constraint into object. RULE [class::] predicate(varibles) (boolean expression) { constraint_1; constraint_n; CASE boolean expression_1: constraint_1; boolean expression_m: constraint_m; } |3a```b````Z~<($^B  6D>P@PH  0޽h ? ̙33; 0 {(  r  S >P   c  0@   For example: RULE multiple(INTEGER: *x, INTEGER: y, INTEGER: z) (neq(y, 0)) { equal(x, divide(z, y)); } z = x * y`" ES^B  6D>P@PH  0޽h ? ̙33v 0 & (  r  S IP     0 Kt > CLASS [class_name][:superclass_name] { // attributes definition date type: attribute_name; ... // rule definition rule_name;P ... //function definition function_name; ... // method definition method_name; ... }:``@@";^B  6D>pH  0޽h ? ̙33% 0 0e(  r  S 4XP   M  0Y  Implementation Program written by COPS consists of classes and rules. COPS constraint programming language is a declarative language, providing classes, methods which are exist in object oriented language. It is similar with C++ . COPS has the features: constraint object oriented logic programming production system 0`@@^B  6D>pH  0޽h ? ̙33 0 kc@(  r  S $fP     0g4   COPS_Compiler 1 { 2 Call yacc to parse the program and 3 to generate internal structures. 4 Initializatiion 5 Create Cops Constant trueNode; 6 Allocate memories for global variables. >`@` @ F L4^B  6D>pH  0޽h ? ̙33 0 ogP(  r  S  tP     0lu4  !7 Interprte the program with the internal structures. 8 Constraint networks are built up for Unsolved 9 constraints and variables. 10 while some constraints in the constraint networks are triggered, 11 inteprete the triggered constraints. 12 }""@,  $^B  6D>pH  0޽h ? ̙33 0 `Q(  r  S P   9  0L aInterpreter: 1 { 2 switch (constraint type) 3 case Constant: 4 return Constant: 5 case global variable: 6 interprete global variable: 7 case local variable or argument: 8 interprete local variable or argument: 9 case object-attribute pair; 10 interprete object-attribute pair: 11 case function call:bb@> ; K 0^B  6D>pH  0޽h ? ̙33 0 MEp(  r  S P     0H  e12 interprete function call: 13 case method call: 14 interprete method call: 15 case CASE expression: 16 interprete CASE expression: 17 ... 18 default: 20 report error 21 }@> 3 6 Z^B  6D>pH  0޽h ? ̙33 0 ME(  r  S <P     0t eCombines object oriented programming with constraint logic programming, containing logic variables, incremental constraint satisfaction and backtracking. variables : C++ object integer variable CtIntVar floating variable CtFloatVar boolean variable CtBoolVar Memory Management new: delete:8@@@@P1  Q^B  6D>pH  0޽h ? ̙33 0 X(  r  S P   @  0@ Constraints CtTell(x == (y + z)); Basic constraints: =, , , <, >, +, -, *, /, subset, superset, union, intersection, member, boolean or, boolean and, boolean not, boolean xor, CtTell((x==0) || (y==0)); CtIfThen (x < 100, x = x+1); Search Hom@@@&! .*^B  6D>pH  0޽h ? ̙33  0 I(  r  S P   1  0  CTGOALn: how to execute CTGOAL1(CtInstantiate, CtIntVar* x){ CtInt a = x->chooseValue(); CtOr(Constraint(x == a), CtAnd(Constraint(x != a), CtInstantiate(x))); } @~4 !.^B  6D>pH  0޽h ? ̙33I 0 (  r  S P   q  0   Schedule CtSchedule class Global object: time original ---tineMin time horizon ---timeMax @> 55^B  6D>pH  0޽h ? ̙33 0 7/(  r  S P     00h OResources CtResource CtDiscreteResource CtUnaryResource CtDiscreteEnergy CtStateResource $ @@b   ^B  6D>pH  0޽h ? ̙33 0 6.(  r  S P     0,  NActivities CtActivity class CtIntervalActivity An activity is defined by its start time, end time and duration Activities require, provide, consume and produce resources@, ^B  6D>pH  0޽h ? ̙33 0 _W(  x  c $|P   ^B  6D>ppp!  6Ppp Y Prices paid as tasks begin $1000 per day Availability: Day 0:$20000, Day 15: +$9000 F*0 ) .   ZA  ~_gct1"/dI H  0޽h ? ̙334 0 t(   r  S 4P   \  < ^ // To create a schedule with origin 0 and given horizon. CtSchedule* schedule = new CtSchedule(0, horizon); // To create an activity with the given duration. CtIntervalActivity* act = new CtIntervalActivity(schedule, duration); //To post a precedence constraint between act1 and act2. act2->startsAfterEnd(act1,0); &_C [`H9 hj^B  6D>P@PH  0޽h ? ̙33 0 SK(  r  S P     <p _//To create a total budget of limited capacity (here 29000). CtDiscreteResource* res = new CtDiscreteResource(schedule, CtRequiredResource, capacity); // To state that only cap (here 20000) is available prior to a // given date (here 15). res->setCapacityMax(0,date,cap); // To state that an activity act consumes c units of res. act->consumes(res, c); $`AP=0 ^B  6D>P@PH  0޽h ? ̙33 0 [S(  r  S  P     <"p|  gCtBoolean IsUnScheduled(CtActivity* act){ // Return true if act does not have a fixed start time. if (act->getStartVariable()->isBound()) return CtFalse; else return CtTrue; }@l M^B  6D>P@PH  0޽h ? ̙33b 0   (  r  S @,P     <0p VCtBoolean IsMoreUrgent(CtActivity* act1, CtActivity* act2){ // Returns true if act1 is more urgent than act2. // Returns true if act2 is unbound (==0) if (act2 == 0) return CtTrue; else if (act1->getStartMax() < act2->getStartMax()) return CtTrue; else return CtFalse; }WW@ 2   ^B  6D>P@PH  0޽h ? ̙33 0 ph0(  r  S L<P     <Ap |CtActivity* SelectActivity(CtSchedule* schedule){ // Returns the unscheduled activity with the smallest latest // statrt time. Returns 0 if all activities are scheduled. CtActivity* bestActivity = 0; //Creates an iterator to iterate on all activities. CtActivityIterator* iterator(schedule); CtActivity* newActivity; while(iterator.next(newactivity)) if((IsUnScheduled(newActivity)) && (IsMoreUgent(newActivity, bestActivity))) bestactivity = newActivity; return bestActivity; }@2 Q4  !       ^B  6D>P@PH  0޽h ? ̙33 0 @E(  r  S HVP   -  <WpYb  void SolveProblem(CtSchedule* schedule){ // Solve the problem assuming constraints have been posted. 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