PrelimExam2010_sols

# PrelimExam2010_sols - Prelim Exam Solutions 2010 1(25...

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Prelim Exam Solutions 2010 1 1. (25 points) Hypothesis Testing Assume you are implementing SA and GA to maximize an expensive cost function. You run 10 trials of 200 iterations and the best solutions, the means and the standard deviations are given below: Trial SA GA 1 108.71 79.08 2 95.22 88.33 3 105.15 90.89 4 108.77 77.50 5 96.97 83.53 6 103.42 82.40 7 78.22 77.64 8 95.38 80.43 9 90.24 67.04 10 96.70 84.33 Mean 97.88 81.12 Std 8.83 6.27 (a) Which hypothesis test (2-sample t-test or paired t-test) would you perform on the data and why? Perform 2-sample t-test. Paired t-test cannot be performed because the two algorithms do not have the same starting conditions, i.e. SA starts with a single solution while GA starts with a population of initial solutions Perform the hypothesis test as stated above: (b) State your null and alternative hypothesis H o : Means of the distribution are the same, μ SA = μ GA H a : Means of the distribution are not the same, μ SA μ GA Students may also choose to perform a 1-sided t-test, but it is less rigorous than a 2-tailed t-test (c) What is your test statistic? (d) Based on the table given below, what is your rejection region for α = 0.05? α 0.1 0.05 0.025 0.01 0.005 0.001 0.0005 t α 1.330 1.734 2.101 2.552 2.878 3.610 3.922 Z α 1.282 1.645 1.960 2.326 2.576 3.090 3.291

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Prelim Exam Solutions 2010 2 For 2-tailed test: Reject if |T| 2.101 For 1-tailed test: Reject if |T| ≥ 1.734 depending on lower- or upper-tailed test (e) What is your conclusion from the hypothesis test, i.e. does the hypothesis test tell you one algorithm performs better than the other or does it tell you that there is no statistical evidence that they are different? Reject H o . Can perform a 1-tailed t-test to verify that μ SA > μ GA , or just infer from observation that μ SA > μ GA . 2. (13 points) Short answer questions: (a) Four algorithms are compared using an empirical CDF shown below. (i) Which algorithm performs best if we wish to maximize the objective function? Give one reason why you think so. DDS performs the best. Furthest to the right, indicating that it obtains higher cost function values overall compared to other algorithms. Does not intersect with any other empirical CDFs (ii) For each algorithm, indicate if it is stochastically dominated by any other algorithm and indicate which algorithms stochastically dominate it. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 120 140 160 SA GA DDS GS
Prelim Exam Solutions 2010 3 SA stochastically dominated by DDS GA stochastically dominated by DDS GS stochastically domnated by DDS and SA DDS not stochastically dominated by any other algorithm 3. (25 points) Assume you want to do deterministic minimization Tabu search for a problem where the decision variable S is a binary string of length 8. Assume you want to use a neighborhood that flips two bits at once. Let (j,k) denote the two bits that at flipped at location j and k. Note the bits do not need to be adjacent so a possible way to flip of two bits is to flip the bit in location 2 and in location 7 simultaneously [denoted as a (2,7) flip or equivalently as a (7,2) flip] The tenure duration is 0 (i.e there are no ―tabu‖ solutions). . The objective function is the sum of 1’s in the string. (See Hint at bottom before calculating an answer.)

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