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5 Problem 4 One of the motivations for using a heuristic method for optimization is the size of the search space of the problem. Discuss ways in which the size of the search space for (a) combinatorial and (b) continuous problems can be calculated or estimated. Problem 3 Discuss how you might decide whether a problem is appropriate for simulated annealing. If it is appropriate, how would you decide on a move operator and an annealing schedule? If you implemented these and the results were poor, what alterations would you try next? Homework 2 For the below two problems do the following: 1. Code a simple SA to solve the problem. To do this you need to encode the problem, develop the definition of a neighborhood, define a move operator, select an annealing schedule and select a stopping criterion. 2. Run your SA. 3. Perform the following changes on your SA code (one by one) and compare the results. change the annealing schedule twice change the stopping criterion twice change the initial starting point (initial solution) five times change the random number seed ten times 5. I am including the global optimum to each problem, but don t use it in your solution methodology. Problem 1 This is the smallest of the QAP (quadratic assignment problem) test problems of Nugent et al. Five departments are to be placed in five locations with two in the top row and three in the bottom row (see below). The objective is to minimize costs between the placed departments. The cost is (flow * rectilinear distance), where both flow and distance are symmetric between any given pair of departments. Below is the flow and distance matrix where distance is the upper half. The optimal solution is 25 (or 50 if you double the flows). Problem 2 Dept This is the six hump camelback function where x lies between ±3 and y lies between ±2. The objective is to minimize z. The global minimum lies at ( , ) where z = Homework 3 For the below problems do the following: 1. Code a simple GA to solve the problem. To do this you need to encode the problem as a binary string (first problem) or permutation (second problem), use single point or uniform crossover and bit flip or 2-opt mutation, use a form of roulette wheel selection or tournament selection, set a population size and select a stopping criterion. 2. Run your GA. 3. Perform the following changes on your GA code (one by one) and compare the results. change the initial starting points (initial solutions) 10 times change the mutation probability twice change the population size twice Continuous Optimization Problem A: This problem is a well known hub location problem where the objective is to minimize the sum of the Euclidean distances between the hub site (in 2 coordinate space) and the coordinates of 25 large U.S. cities. You can ignore curvature of the earth. The decision variables are the x and y coordinates of the hub site. The 25 city coordinate matrix is listed on the attached sheet. B: Redo this problem but use two hub sites. This requires not only minimizing the two summations of distance, but also assigning each of the 25 cities to one or the other hub. Problem 2 Use a permutation based GA to solve the QAP problem. Since the search space is small, this GA should use a relatively small population size. This is the fourth of the QAP (quadratic assignment problem) test problems of Nugent et al. Eight departments are to be placed in eight locations with four in the top row and four in the bottom row. The objective is to minimize flow costs between the placed departments. The flow cost is (flow * distance), where both flow and distance are symmetric between any given pair of departments. Below is the flow and distance matrix where rectilinear distance is the upper half. The optimal solution is 107 (or 214 if you double the flows).

6 Dept Homework 4 For the six hump camelback function below do the following: 1. Code a simple ES to solve the problem. To do this you need to encode the problem as two real variables, select values for µ and λ, select a standard deviation (the same for each variable) and change it using the 1/5 rule, randomly select an initial population and select a stopping criterion. Do not perform recombination. 2. Run your ES for ten different seeds. 3. Perform the following changes on your ES code (do this cumulatively) and compare the results. change the ratio of µ to λ add individual standard deviations for each variable to the encoding and alter them using global intermediate recombination add recombination to the two variables using dual discrete recombination Homework 5 For the 15 department QAP from Nugent (below) do the following: 1. Code a simple TS to solve the problem. To do this you need to encode the problem as a permutation, define a neighborhood and a move operator, set a tabu list size and select a stopping criterion. Use only a recency based tabu list and no aspiration criteria at this point. 2. Run your TS. 3. Perform the following changes on your TS code (one by one, being cumulative) and compare the results. change the initial starting solution 5 times change the tabu list size - smaller and larger than your original choice change the tabu list size to a dynamic one - an easy way to do this is to choose a range and generate a random uniform integer between this range every so often (i.e., only change the tabu list size infrequently) add one or more aspiration criteria such as best solution so far, or best solution in the neighborhood, or in a number of iterations use less than the whole neighborhood to select the next solution add a frequency based tabu list and/or aspiration criteria (designed to encourage the search to diversify) This is the sixth of the QAP (quadratic assignment problem) test problems of Nugent et al. Fifteen departments are to be placed in 15 locations with five departments (columns) in three rows. The objective is to minimize flow costs between the placed departments. The flow cost is (flow * distance), where both flow and distance are symmetric between any given pair of departments. Below is the flow and distance matrix where rectilinear distance is the upper half. The optimal solution is 575 (or 1150 if you double the flows). Dept

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