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📗 [4 points] When using the Genetic Algorithm, suppose the states are \(\begin{bmatrix} x_{1} & x_{2} & ... & x_{T} \end{bmatrix}\) = , , , . Let \(T\) = , the fitness function (not the cost) is \(\mathop{\mathrm{argmax}}_{t \in \left\{0, ..., T\right\}} x_{t} = 1\) with \(x_{0} = 1\) (i.e. the index of the last feature that is 1). What is the reproduction probability of the state with the highest reproduction probability?
📗 Answer: .
📗 [4 points] When using the Genetic Algorithm, suppose the states are \(\begin{bmatrix} x_{1} & x_{2} & ... & x_{T} \end{bmatrix}\) = , , , . Let \(T\) = , the fitness function (not the cost) is \(\mathop{\mathrm{argmin}}_{t \in \left\{1, ..., T + 1\right\}} x_{t} = 1\) with \(x_{T + 1} = 1\) (i.e. the index of the first feature that is 1). What is the reproduction probability of the state with the highest reproduction probability?
📗 Answer: .
📗 [4 points] When using the Genetic Algorithm, suppose the states are \(\begin{bmatrix} x_{1} & x_{2} & ... & x_{T} \end{bmatrix}\) = , , , . Let \(T\) = , the fitness function (not the cost) is \(\mathop{\mathrm{argmax}}_{t \in \left\{0, ..., T\right\}} x_{t} = 1\) with \(x_{0} = 1\) (i.e. the index of the last feature that is 1). What is the reproduction probability of the first state: ?
📗 Answer: .
📗 [4 points] When using the Genetic Algorithm, suppose the states are \(\begin{bmatrix} x_{1} & x_{2} & ... & x_{T} \end{bmatrix}\) = , , , . Let \(T\) = , the fitness function (not the cost) is \(\mathop{\mathrm{argmin}}_{t \in \left\{1, ..., T + 1\right\}} x_{t} = 1\) with \(x_{T + 1} = 1\) (i.e. the index of the first feature that is 1). What is the reproduction probability of the first state: ?
📗 Answer: .
📗 [4 points] When using the Genetic Algorithm, suppose the states are \(\begin{bmatrix} x_{1} & x_{2} & ... & x_{T} \end{bmatrix}\) = , , , . Let \(T\) = , the fitness function (not the cost) is \(\mathop{\mathrm{argmax}}_{t \in \left\{0, ..., T\right\}} x_{t} = 1\) with \(x_{0} = 1\) (i.e. the index of the last feature that is 1). What is the reproduction probability of the first state: ?
📗 Answer: .
📗 [3 points] Four individuals (i.e. candidate solutions) in the current generation are given by -digit ( dimensional) sequences: . Individual 1: ; Individual 2: ; Individual 3: ; Individual 4: . The fitness function is . What is the result of performing 1-point crossover for the sequences with the highest fitness (break ties by preferring the sequence that appears earlier in the list) with a cross-point between digit and digit .
📗 Note: the first line representing the first child should start with the sequence with the highest fitness, and the second line representing the second child should start with the sequence with the second highest fitness.
📗 Calculator: .
📗 Answer (matrix with 2 lines, each line is a comma separated vector): .
📗 [4 points] Suppose the score (fitness) of a state \(\left(d_{1}, d_{2}, d_{3}, d_{4}\right)\) is \(d_{1} + d_{2} + d_{3} + d_{4}\), and only 1-point crossover with the cross-over point between \(d_{2}\) and \(d_{3}\) is used in a genetic algorithm (i.e. mutation probabilities are 0). Two states are chosen as parents at random according to the reproduction probabilities, what is the probability that one of their children is the optimal state (i.e. \(\left(1, 1, 1, 1\right)\)? Enter a number between 0 and 1.
📗 Note: the two parents are sampled with replacement, meaning the probability that two states are chosen as parents is the product of their reproduction probabilities.
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