Attempts to find an optimum solution penalty value for certain classes of NP-Hard problems George M....
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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 [email protected]
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Transcript of Attempts to find an optimum solution penalty value for certain classes of NP-Hard problems George M....
Attempts to find an optimum solution penalty value for certain classes of NP-Hard problems
George M. WhiteSITE
University of [email protected]
CORS - Ottawa
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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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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yor-s-83
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yor-s-83
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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
...!
)0(...
!2
)0(")0(')0()(
)(2 n
n
Pn
fP
fPffPf
or
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we want to simplify this equation as much as possible (but no further) so we try
dP/dt = a0
this doesn’t work
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try
dP/dt = ao + a1P
this doesn’t work
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the next simplest form is
it turns out that this is a plausibleform
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at the limiting value
2lim2lim1
221 0 PaPaPaPa
dt
dP
2
1lim a
aP
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this often appears in the literature with symbol substitution
21 aa
and the equation is written
2PPdt
dP
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The solution for this equation is
teP
tP
0
)(
where P0 = P(0)
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The limiting value of P(t) is
limP
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To estimate the limiting penalty of a data set
1. Collect the data representing the “current champion” over time.
2. Fit a curve to this data.3. 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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yor-s-83
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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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