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CPSC 433 Artificial Intelligence Or-Tree Modeling Example CPSC 433 T01 & T02 Andrew Kuipers

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CPSC 433 Artificial Intelligence Problem Description Let T be a set of tasks, and W a set of workers Let e:T W [0,1] be a function mapping tasks and workers onto efficiency ratings For a given sequence of tasks S = t 1, …, t n , assign tasks to workers such that : –the overall time to complete the sequence is minimized, and –no worker is assigned to a task for which he has an efficiency rating of 0

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CPSC 433 Artificial Intelligence Or-Tree Model Defining a Problem Instance pr = S = t 1, …, t n , A = (w i, t’ 1i, …, t’ mi ) | w i W, t’ ij T S : the sequence of tasks to complete (not yet assigned) A : an assignment of workers to sequences of tasks W: the set of workers T : the set of tasks

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CPSC 433 Artificial Intelligence Or-Tree Model Evaluating Solutions f(pr) = max( { i=1..mj ( e(t ij, w j ) -1 ) | (w j, t 1j, …, t mj A } ) For each worker, calculate the time to complete its assigned tasks Time taken by worker is 1 / efficiency for a particular task Workers work in parallel The total time is the max of the times taken by each worker

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CPSC 433 Artificial Intelligence Or-Tree Model When is a branch Solvable / Unsolvable? Erw ,wt ((pr, ?), (pr, no)) (w i, t 1i, …, t mi ) A, j {1…m i } | e(t ji, w i ) = 0 ie: if a worker is assigned a task for which it has 0 efficiency Erw ,wt ((pr, ?), (pr, yes)) Erw ,wt ((pr, ?), (pr, no)) and |S| = 0 * where pr = S, A ie: is not unsolvable, and no tasks left to assign

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CPSC 433 Artificial Intelligence Or-Tree Model Branching Definition Altern(pr 0, pr 1, …, pr n ) pr 0 = s 0 : S, A pr i = S, Assign(A, w i, s 0 ) w i W Assign the first task in S to each worker on a different branch Assign(A, w, s) = A’ s.t : if (w, t 1, …, t n ) A then (w, t 1, …, t n, s ) A’ else (w, s ) A’

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CPSC 433 Artificial Intelligence Or-Tree Model pr = S = t 1, …, t n , A = (w i, t’ 1i, …, t’ mi ) | w i W, t’ ij T f(pr) = max( { i=1..mj ( e(t ij, w j ) -1 ) | (w j, t 1j, …, t mj A } ) Erw ,wt ((pr, ?), (pr, no)) (w i, t 1i, …, t mi ) A, j {1…m i } | e(t ji, w i ) = 0 Erw ,wt ((pr, ?), (pr, yes)) Erw ,wt ((pr, ?), (pr, no)) and |S| = 0 * where pr = S, A Altern(pr 0, pr 1, …, pr n ) pr 0 = s 0 : S, A pr i = S, Assign(A, w i, s 0 ) w i W Assign(A, w, s) = A’ s.t : if (w, t 1, …, t n ) A then (w, t 1, …, t n, s ) A’ else (w, s ) A’

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CPSC 433 Artificial Intelligence Example Problems W = { A, B, C } T = { p, q, r } e : S = p, p, q, r, p, q W = { A, B, C } T = { p, q, r } e: S = q, r, q, p, r, q, p ABC p10.20 q0.510.2 r00.51 ABC p0.200 q0.500.2 r0.10.20.5

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