Attempts to find an optimum solution penalty value for certain classes of NPHard problems

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Attempts to find an optimum solution penalty
value for certain classes of NP-Hard problems

• George M. White
• SITE
• University of Ottawa
• white_at_site.uottawa.ca

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Examples of very difficult problems

• medical personnel in hospitals
• contact centre personnel
• judicial staff assignments
• examination scheduling
• portfolio management

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Examples

• These 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.

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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.

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The class of NP-Hard problems

• 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.

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Optimization

• There 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

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Optimization

• This implies that we must optimize some cost
  function(to the best value permitted by the
  constraints and the time available).
• unidimensional optimization
• multidimensional optimization

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Optimization

• This 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

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The shape of the curve

• at some time in the future it seems reasonable to
  assume that the best penalty values will reach a
  limit, i.e.

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• the form of dP/dt is unknown but it is reasonable
  to assume that it is some function of the current
  penalty

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• expanding this as a Maclauren series yields

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• we want to simplify this equation as much as
  possible (but no further) so we try

dP/dt a0

• this doesnt work

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• try

dP/dt ao a1P
this doesnt work
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• the next simplest form is

it turns out that this is a plausible form
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• at the limiting value

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• this often appears in the literature with symbol
  substitution

and the equation is written
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• The solution for this equation is

where P0 P(0)
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• The limiting value of P(t) is

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• To 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.

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Problems

• Lack 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

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Problems

• Therefore, the numerical results must be regarded
  as preliminary estimates, subject to review as
  more data becomes available.

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• 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.

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• Thank you
• George White
• white_at_site.uottawa.ca