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Attempts to find an optimum solution penalty value for certain classes of NP-Hard problems Attempts to find an optimum solution penalty value for certain classes of NP-Hard problems George M. White SITE University of Ottawa white@site. uottawa. ca CORS - Ottawa

Examples of very difficult problems medical personnel in hospitals ¡ contact centre personnel ¡ Examples of very difficult problems medical personnel in hospitals ¡ contact centre personnel ¡ judicial staff assignments ¡ examination scheduling ¡ portfolio management ¡ CORS - Ottawa

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Examples These are all examples of NP-hard assignment/scheduling problems. They are characterized by having 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. ¡ CORS - Ottawa

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Optimization ¡ There is often more than one possible solution. In this case we 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) l l total wages paid overall satisfaction personnel coverage separation CORS - Ottawa

Optimization ¡ This implies that we must optimize some cost function(to the best value Optimization ¡ This implies that we must optimize some cost function(to the best value permitted by the constraints and the time available). l l unidimensional optimization multidimensional optimization CORS - Ottawa

Optimization ¡ This also means that we will have to use an approximation algorithm 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. l l l l tabu search particle swarm optimization simulated annealing great deluge partialcol IDWalk etc CORS - Ottawa

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The shape of the curve ¡ at some time in the future it seems 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. CORS - Ottawa

¡ the form of d. P/dt is unknown but it is reasonable to assume ¡ the form of d. P/dt is unknown but it is reasonable to assume that it is some function of the current penalty CORS - Ottawa

¡ expanding this as a Maclauren series yields or CORS - Ottawa ¡ expanding this as a Maclauren series yields or CORS - Ottawa

¡ we want to simplify this equation as much as possible (but no further) ¡ we want to simplify this equation as much as possible (but no further) so we try d. P/dt = a 0 ¡ this doesn’t work CORS - Ottawa

d. P/dt = ao + a 1 P this doesn’t work ¡ try CORS d. P/dt = ao + a 1 P this doesn’t work ¡ try CORS - Ottawa

¡ the next simplest form is it turns out that this is a plausible ¡ the next simplest form is it turns out that this is a plausible form CORS - Ottawa

¡ at the limiting value CORS - Ottawa ¡ at the limiting value CORS - Ottawa

and the equation is written ¡ this often appears in the literature with symbol and the equation is written ¡ this often appears in the literature with symbol substitution CORS - Ottawa

¡ The solution for this equation is where P 0 = P(0) CORS - ¡ The solution for this equation is where P 0 = P(0) CORS - Ottawa

¡ The limiting value of P(t) is CORS - Ottawa ¡ The limiting value of P(t) is CORS - Ottawa

¡ To estimate the limiting penalty of a data set 1. 2. 3. Collect ¡ To estimate the limiting penalty of a data set 1. 2. 3. Collect the data representing the “current champion” over time. Fit a curve to this data. Calculate the limiting value of this curve. CORS - Ottawa

Problems ¡ Lack of data: The largest number of points for any of the 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: ¡ CORS - Ottawa

Problems ¡ Therefore, the numerical results must be regarded as preliminary estimates, subject to Problems ¡ Therefore, the numerical results must be regarded as preliminary estimates, subject to review as more data becomes available. CORS - Ottawa

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Similar behaviour has been observed for other data sets of the same type. Work 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. CORS - Ottawa

Thank you George White white@site. uottawa. ca CORS - Ottawa Thank you George White white@site. uottawa. ca CORS - Ottawa