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White SITE University of Ottawa white@site.uottawa.ca"@**#Examples of very difficult problemsmedical personnel in hospitals contact centre personnel judicial staff assignments examination scheduling portfolio management     Examples3These are all examples of NP-hard assignment/scheduling problems. They are characterized by having a series of non-linear constraints We wish to find solutions such that all constraints are satisfied If this is not possible, we wish to find solutions such that a maximum number of constraints are satisfied.04Z/  *!The class of NP-complete problems""  The complexity class of decision problems for which answers can be checked for correctness, given a certificate, by an algorithm whose run time is polynomial in the size of the input and no other NP problem is more than a polynomial factor harder. +The class of NP-Hard problemsF The complexity class of decision problems that are intrinsically harder than those that can be solved by a nondeterministic Turing machine in polynomial time. When a decision version of a combinatorial optimization problem is proved to belong to the class of NP-complete problems, then the optimization version is NP-hard. GZG OptimizationThere is often more than one possible solution. In this case we want the one that is best (i.e. we want to optimize some property of the schedule) total wages paid overall satisfaction personnel coverage separation bZDZZU3D  OptimizationThis implies that we must optimize some cost function(to the best value permitted by the constraints and the time available). unidimensional optimization multidimensional optimization8~:5G:,- F,  OptimizationThis also means that we will have to use an approximation algorithm to find good solutions. Exact solutions require too much time for real life problems. tabu search particle swarm optimization simulated annealing great deluge partialcol IDWalk etc :ZbZ\>b>E   yor-s-83      yor-s-83    The shape of the curvenat some time in the future it seems reasonable to assume that the best penalty values will reach a limit, i.e.$oj  kthe form of dP/dt is unknown but it is reasonable to assume that it is some function of the current penalty.l Y [ ,expanding this as a Maclauren series yields    Qwe want to simplify this equation as much as possible (but no further) so we try .R6 ! try" the next simplest form is # at the limiting value $ =this often appears in the literature with symbol substitution% !The solution for this equation is& The limiting value of P(t) is $' hTo estimate the limiting penalty of a data set Collect the data representing the  current champion over time. Fit a curve to this data. Calculate the limiting value of this curve.,//(ProblemsLack of data: The largest number of points for any of the data sets is 6. Number of parameters: 3 parameters Uncertain and irregular spacing in data: Curious data points: The first (1996) data points: J ; ^)Problems}Therefore, the numerical results must be regarded as preliminary estimates, subject to review as more data becomes available.  yor-s-83     .  Similar behaviour has been observed for other data sets of the same type. Work continues on other sets of data from other real-world problems.- 1 Thank you George White white@site.uottawa.ca ,0  0/  0` ___ff3ff` ___33f̙ffff` CCC` d___a%qw` *(Z___fFQt` 6CCCff3` CCCff` CCCf3f` f333ff` f333f3̙>?" dd@$?Fd@l A@lF`< n?" dd@   @@``PP   @ ` `(p>>   @  (  XT ``   "`` B  C H<@AB}C}EvFGxGH`I{QUNVWX}} }}}} `$ }} }}  G H I }} }} }}           `  `` "(`  B0 B  C H@AB}C}EvFGFH`I}QUNVWX}} }}}} `$ }} }}  G H I }} }} }}           `  `` "  hB  s *D1"``  <H "_^  T Click to edit Master title style! !*  6 "_^  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  6 #" `` `  `*0    6 #" ``   b*0    6 #" `` `  b*0 H  0޽h ? ___ff3ff80___PPT10. 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S  6# _P  # P*    6ؐ# _  # R*  H  0޽h ? 3380___PPT10.P@pt^3 0  0(  x  c $um] u x  c $ uo]  u H  0޽h ? 33___PPT10i.$|+D=' #= @B +}  0 p$(  r  S z_^  z r  S u_^ u H  0޽h ? ___ff3ff___PPT10i.$`dC+D=' #= @B + 0 8(  8X 8 C 0Atest_engine@pH 8 0޽h ? ___ff3ff___PPT10i.% ׁ +D=' #= @B +  0 0x(  0r 0 S <7z_^  z r 0 S 8z_^ z L 0 C $A patatp@@pH 0 0޽h ? ___ff3ff___PPT10i.$Ʉ+D=' #= @B +   0  P(   r  S az_^  z   S bz_^<$ 0 z H  0޽h ? ___ff3ffme___PPT10E.$@Ƽ+[W_D' #= @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 4%(+8+0+ 0 +  0 |(  r  S sz_^  z r  S tz_^ z P  C (ANPStuffp@@pH  0޽h ? ___ff3ff___PPT10i.A Xg?+D=' #= @B +  0L0 $(  r  S zz_^  z r  S {z_^ z H  0޽h ? ___ff3ff___PPT10i.N@I+D=' #= @B +  0L0 $(  r  S \z_^  z r  S ̆z_^ z H  0޽h ? ___ff3ff___PPT10i.O@ ^U+D=' #= @B +   0 $P(  $r $ S z_^  z  $ S dz_^<$ 0 z H $ 0޽h ? ___ff3ff  ___PPT10 .$W5+[W_D* ' #= @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(+8+0+$0 +<  0 (P(  (r ( S z_^  z  ( S z_^<$ 0 z H ( 0޽h ? ___ff3ff$___PPT10.$4+[W_D' #= @B DS' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(~%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(+8+0+(0 +}  0 ,$(  ,r , S lz_^  z r , S z_^ z H , 0޽h ? ___ff3ff___PPT10i.$ z5+D=' #= @B +  0 Lv(  Lr L S z_^  z r L S z_^ z J L C "A Yor0tgH L 0޽h ? ___ff3ff___PPT10i.C@灗+D=' #= @B +  0  Pv(  Pr P S z_^  z r P S z_^ z J P C "A Yor1/t,YH P 0޽h ? ___ff3ff___PPT10i.C@PĈ+D=' #= @B +  0 0Tr(  Tx T c $z_^  z x T c $z_^ z : T 3 A  H T 0޽h ? ___ff3ff___PPT10i.K .F+D=' #= @B +K  0 bZ@X(  Xx X c $z_^ z : X 3 A t H X 0޽h ? ___ff3ff___PPT10i.S+D=' #= @B +z   0 RJP\(  \~ \ s *`z_` z  \0 TA ? ?0   z|z  s \  s,$D  0B \ 3 A ` s \ <z Q$  4orH \ 0޽h ? ___ff3ff___PPT10.T&+eD|' = @B D7' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*\D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*\+k   0 ``(  `x ` c $Xu_^ u  ` 0uP  p dP/dt = a00   ` 6ux ` ,$  0 d$ this doesn t work$ FH ` 0޽h ? ___ff3ff ___PPT10.U~+׸D' u= @B DD' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*`D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*`+8+0+`0 +A   0 p dd(  dx d c $z_^  z ~ d s *lz_.  z  d0 TA ? ?3  z d <z@  dP/dt = ao + a1PJ   $ d <# Z  :  d <|#  k  :  d B #@  ,$  0 T"this doesn t work  d0 TA ? ? @  #`  d c $A ??, d  H d 0޽h ? ___ff3ff ___PPT10.Y!_+'JkD'  #= @B DD' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*dD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*d+8+0+d0 +   0 h(  hx h c $X#_^ #  h <# Z  : : h 3 A    h B#  ,$  0 \*it turns out that this is a plausible form++H h 0޽h ? ___ff3ff ___PPT10.l`i+QD' #= @B DD' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*hD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*h+8+0+h0 +_  0 vnl(  l~ l s *##_ #  l0 TA ? ?Y   # l0 TA ? ?P 0   #H l 0޽h ? ___ff3ff___PPT10i.= +D=' #= @B +  0 +#p(  p~ p s **#_` #  p0 TA ? ?PP   # p 0Zu` P  Mand the equation is written p0 TA ? ?@    #H p 0޽h ? ___ff3ff___PPT10i.l0F`+D=' #= @B +  0 tB(  t~ t s *0#_` #  t0 TA  ? ?)   # t 0t2#`   |where P0 = P(0) P  H t 0޽h ? ___ff3ff___PPT10i.ngq+D=' #= @B +  0 x^(  x~ x s *<#_.  #  x0 TA  ? ?@   #H x 0޽h ? ___ff3ff___PPT10i.+D=' #= @B +v  0 bZ|(  | | c $H#_^<$D  0 # "\LHDdH | 0޽h ? ___ff3ff___PPT10. 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