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

George M. WhiteSITE

University of Ottawawhite@site.uottawa.ca

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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Thank you

George Whitewhite@site.uottawa.ca