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A Comparison of Parent Selection Strategies for Evolutionary Algorithms Modeled After Human Social Interaction Michael Ames CS448 FS2005 – Semester Project November 15, 2005 Abstract Based on preliminary findings of previous research, certain strategies for parental selection improve the efficiency of Evolutionary Algorithms. The purpose of the study is to examine how EAs implementing parental selection strategies modeled after human behavior effect the functionality of an EA. By implementing four basic strategies and testing on two different EAs this study will prove that implementing EA modeled after human social interaction can perform nearly as well as the standard tournament selection available to EA designers. 1 Index 1. Introduction……………………………………………………………3 2. Problem Statement…………………………………………………..3 3. Previous Research…………………………………………………...3 4. Approach………………………………………………………………4 5. N Queens Results…………………………………..……………….10 6. Binary Knapsack Results…………………………………………12 7. Conclusion……………………………………..…………………….14 8. References………………………………….………………………..14 Appendix A Running the Program…………..……………………...15 Appendix B F and T Test Results……………………………………17 2 1. Introduction Based on preliminary findings of previous research, certain strategies for parental selection improve the efficiency of Evolutionary Algorithms, (EAs). A major majority of the research in this area has been in disallowing individuals who are related to be selected as parents. [Craighurst] [Eshelman] [Ting] This current research will determine if EAs with parental selection strategies based on human social interaction will increase the efficiency in finding optimal solutions. A minor motivation behind this research is due in part to the above apparently low amount of research on the subject of deriving new parent selection strategies for EAs. The main reason for this study is to increase the overall level of knowledge of the effect of custom parent selection strategies on EA performance. In short, will the added complexity increase convergence rate of the EA to the global optimum therefore finding the solution in fewer generations. A separate EA will be written to solve two known problems, N-Queens and Binary Knapsack. Both of the EAs will be fairly standard in regards to individual representation, population, recombination, mutation, and survivor selection. However, parent selection will be handled separately. Both EAs will be able to utilize any one of the defined parent selection strategies from a separate parent selection object. The performance of each EA utilizing a standard tournament selection strategy to select the parents will be measured and averaged over several runs. The information derived from the tournament selection strategy will be used as a standard of comparison as to an EAs performance utilizing all of the defined parent selection strategies. Additionally this will allow the comparison of performance of each selection strategy to each other as well as different EAs utilizing the same parent selection strategy. 2. Problem Statement The purpose of the study is to examine how EAs implementing parental selection strategies modeled after human behavior effect the functionality of an EA. Evolutionary algorithms implementing these strategies will hopefully provide significant improvement over standard parental selection strategies for either all or at least a specific class of problems. Currently all typical EAs resort to either some standard form of Fitness Proportional Selection (FPS) or Ranking Selection (RS) to select individuals for mating [Eiben]. Analysis provided by this research of EAs implementing parental selection strategies modeled after human behavior will help determine if higher level more complex modeling will increase the efficiency of EAs. 3. Previous Research Previous research that has implemented custom parent selection strategies to improve a subset of EAs called Genetic Algorithms (GAs). That research has been limited to disallowing incest by implementing a Tabu Multi-Parent Genetic 3 Algorithm (TMPGA) [Ting]. The TMPGA is considered the closest model of a parental selection strategy to human mating behavior. The TMPGA integrated a tabu search into the parent selection of GAs to increased genetic diversity in the population individuals. All of the individuals are divided into “clans”, and as the population matures individuals belonging to clan A may not be allowed to mate with a member of the same clan or a member of clan B if clan B is on clan A’s tabu list. The tabu list simply contains the clan designation of those clans that contain individuals that have become too genetically similar through recombination. One study compared the effect of different levels of mutation rates on standard, Assortative, and Disassortative Mating Genetic Algorithms (AMGA or DMGA). [Ochoa] A GA known as an AMGA or DMGA models mate selection after the mating habits of certain animal species. Parent selection and eventual mating of individuals in an AMGA, or DMGA, are based on phenotypic similarities, AMGA, or dissimilarities, DMGA, between parental candidates. In Ochoa these similarities/dissimilarities were based on the Hamming Distance (HD) of individuals. Ochoa used the expanded definition of the HD as the number of unlike values in integer strings in order for the algorithms to select a mate. The smaller the HD the greater the likelihood the candidates would mate in an AMGA. The inverse is true for DMGAs. Convergence to optimal solutions with low mutation rates were found to be favorable for the DMGA, whereas a medium level of mutation was favorable for the standard GA, and a high mutation rate was favorable for the AMGA. A couple of other research papers detail other parental selection strategies each with positive results. Both papers addressed the topic of preventing incest in parent selection. One such strategy is preventing premature convergence by preventing incest. [Eshelman] Another includes enhancing GA performance through crossover prohibitions based on ancestry. [Craighurst] 4. Approach The coding for this assignment was completed on an Intel Pentium III 600Mhz desktop PC running Fedora Core 3 Linux. The PC has a 100Mhz front side bus with 256MB SDRAM and an additional 512MB of the hard drive partitioned as a swap, technically giving the computer 768MB of RAM. However, due to the significant number of runs required to form a solid basis for the conclusion of this report, actual running of the code on the PC was inadequate. In order to complete the requirements of the project on time the program was run on UMR’s Numerically Intensive Computing (NIC) Cluster that is a batch computing system consisting of 40 Dell PowerEdge 1850 computing nodes and 2 Dell PowerEdge 1850 service nodes. All nodes have dual 3.2GHz Xeon EM64T processors and 2GB of RAM. All of the nodes of the NIC are interconnected via switched gigabit ethernet. 4 The basic characteristics of each EA utilized to test the different parent selection strategies detailed in this report are described in detail below. Table 1 the basic characteristics of the N-Queens EA as it was implemented for the experiment. Likewise Table 2 details the Binary Knapsack characteristics. As a default, all of the EAs will terminate after a maximum number of generations have been reached or the termination condition listed is true. It is because of these characteristics that several different EAs will be used to measure the performance of the different parent selection strategies. Table 3 lists the parameters that will remain constant throughout the experiment. By leaving the parameters in Table 3 constant this will form a steady basis for the results and conclusions later in Sections 5 and 6. Table 1 N-Queens Individual Representation Initialization Parent Selection Recombination Mutation Competition Termination Fixed Length Permutation of Integers Uniform Random Custom see Section 4.4 Simple One Point Crossover Swap Gene Location Elitist Valid Solution Found Table 2 Binary Knapsack Individual Representation Initialization Parent Selection Recombination Mutation Competition Termination Fixed Length Bit String Uniform Random Custom see Section 4.4 Simple One Point Crossover Bit Flipping Elitist No Improvement in 250 Generations Table 3 Constant Parameters Recombination Percentage Individual Mutation Percentage Gene Mutation N-Queens Board size (n) Items For Knapsack (n) Population Size Parent Pool Size Total Children per Generation Max Children per Parent Pair Number of Iterations 100% 100% 1/n 30 100 2000 20 15 3 30 5 4.1 Parameters Sixteen parameters that must be read in from a formatted text file are required for the proper execution of the algorithm. Details as to the format of the text file, and the valid ranges of the parameters are outlined in Appendix A “Running the Program.” The first parameter is the Recombination Probability (PR) which is used to determine if new individuals will be generated from the individuals selected as parents. The higher the probability the greater the likelihood the children will contain genetic material from both parents as a result of the crossover operation explained in the Recombination section below. The Individual Mutation Probability (PM), which is the second parameter, determines the chance that a newly generated individual has been selected as a candidate for mutation. The Number of Genes (n) is the number of genes an individual will have. For the N-Queens EA this is where the size of the board is set. For the Knapsack EA the number of genes is set by the data file read in when the EA is initialized. The default Gene Mutation Probability (P G) is 1/n. The fourth parameter determines the Population Size ()for the EA which dictates the number of individuals that will exist initially, and survive from one generation to the next. The fifth parameter determines the Parent Pool Size (which is the number of individuals chosen from each generation with the possibility of becoming parents. In order to model human social interaction as close as possible there is a Cheating Probability (PC) for individuals who are married. The number of children generated from each generation will be set by the next parameter which is the Total Number of Children per Generation ( and is exactly what it means. Parameter eight is the Maximum Number of Children per set of parents. One is added to the modulus of a random number by the Maximum Number of Children to determine how many children will be generated from each set of parents. Parent Selection and Recombination will continue until the Total Number of Children has been reached. The ninth parameter is the default stopping case the Maximum Number of Generations. If the Maximum Number of Generations is reached the EA will terminate automatically and output the best individual found at that time. The tenth parameter is the Number of Runs/Iterations each parameter set will be executed for that particular EA. If a specific Random Seed is desired it may be entered as the eleventh parameter. If the Random Seed is left at 0 then a new random seed will be generated from the CPU clock. Parameter twelve is the name of the file that will mirror all of the preformatted screen output. Only the filename is required, no filename extensions are necessary. The output will be a text file available for viewing in the executable directory after the program terminates. The next parameter also asks for an output filename to store all of the statistical data in a file that is in .csv format. The .csv format is for ease of importation into various spreadsheet programs, and can also be viewed once the program terminates. The next parameter in the 6 parameter file allows the user to select either EA to run. The next to last parameter allows the user to enter the Parent Selection Strategy, described further in Section 4.4 below. Finally the last parameter allows the user to select an input data file for the EA if it requires one. 4.2 Individual Representation Individual representation will be unique for each EA implemented in the study. Each individual for the N-Queens EA represents the placement of N number of chess queens on a chess board with N rows and N columns. The N-Queen individuals will be an array of N integers arranged in a permutation. Implementing the individuals as permutations will make the fitness calculation easier, see Section 4.7. The individuals of the Binary Knapsack will be represented as bit strings equal in length to the total number of items available for placement in the knapsack. A one at any location in the bit string will indicate that particular item is included in the knapsack, and a zero will indicate the item is not included. 4.3 Population Initialization Each population will be implemented as either a two dimensional array of bit strings or a two dimensional vector of integers. The size of the initial population is equal to the population size parameter read in from the parameter file. Each array or vector in the population represents an individual. All other vectors associated with the population like the vector of floating point numbers that maintains the fitness scores for each individual in the population are cleared and reset prior to calculating the fitness scores. Once all of the fitness scores have been calculated (Section 4.7) for the initial population the statistics are calculated. (Section 4.9) 4.4 Parent Selection This is the heart of the study. Both EAs will be run using a standard tournament selection for parent selection to be used as a standard to be measured against. Four custom parent selection strategies will be implemented for the assignment. The first custom selection strategy will simply be to implement an AMEA with an added marriage component. As with the implementation of the AMEA found in the previous research, this one will also be based on the HD of the individuals. The best fit individual of the parent pool is selected as the first parent. The first parent will be mated with the individual with the lowest HD in the parent pool, as long as it is not 0, and the individual is not already married. Once two individuals have been selected for mating they in effect have been married, and the location of the opposite spouse is recorded in a hash that keeps track of 7 all the marriages of the population. Once two individuals have been married any time either one of them is selected out of the parent pool for recombination they will only mate with their designated spouse. If an individual is selected for mating, but all of the other individuals in the parent pool are married then the individual will mate with the first unmarried individual in the population. This strategy is modeled after is the common human mate selection process of mating with candidates with similar phenotypic traits. The second strategy will be to implement a DMEA the same way. Keeping in mind the best fit individual will mate with an unmarried individual with the greatest HD. By doing so the interaction strategy closely models the human mate selection strategy commonly known as “opposites attract.” The third and fourth strategies will be the same as the first two with an added “Cheating” component. There will be a flat cheating probability for the entire population, 0.25 for the third strategy, and 0.75 for the fourth. If a randomly generated value from [0, 1] is less than the cheating probability then the individual will cheat on its spouse and mate. Note that by setting the cheating parameter to 0 will result in an EA performing exactly as parent selection strategies one and two. Likewise setting the cheating value to 1 will result in an EA that acts as a normal AMEA, in effect turning the marriage property off. 4.5 Recombination The first step of Recombination is to generate a random value over the range of [0, 1]. If the random value is less than or equal to the individual recombination probability (PR) then recombination will occur. If there is no recombination then the child created is an exact copy of one of a random parent. If recombination is to occur a random location in the range of [1, individual size-1] is chosen. This random location represents the point of crossover for selecting genetic material to create the new child. Each new child is created by copying values in corresponding locations from the first parent into corresponding locations of the child up to the crossover point. After the crossover point values are copied from corresponding locations of the second parent. Because duplicate values are discouraged in the NQueens EA the remaining positions of the child after the crossover are filled with the values from the second parent that are not already in the child. Looping through all of the locations of the second parent if a value is not already in the child it is now placed in the child’s next empty location. 4.6 Mutation Individual mutation will occur if a randomly generated value from [0, 1] is less than the Individual Mutation Probability (PM) that was read in from the 8 parameter file mentioned earlier. If mutation is to occur in the N-Queens EA, two locations are chosen at random, and the values simply switch locations. Mutation for individuals of the Binary Knapsack EA is simple enough. If an individual is selected for mutation starting at the beginning of the individual check each Boolean value to see if the bit should be “flipped.” Flip the bit to the opposite value if the random value is less than PG. 4.7 Individual Fitness Calculation Of course both EAs will each have their own way of calculating fitness. Both EAs are designed so that a higher fitness score is interpreted as a better fitness score. To calculate the fitness for the N-Queens problem each array location of an individual represents a unique column on the chess board, and each value at that array location represents a unique row of the chess board. Because the values in the array are arranged as a permutation of integers, by design only diagonal attacks have to be accounted for. Additionally only diagonal attacks to the right need to be considered as considering diagonal attacks to the left would be redundant. For the Binary Knapsack EA fitness will be based on if the knapsack is overfilled or not. If the knapsack is overfilled the individual will receive a negative fitness score based on the percentage the knapsack is overfull. A negative fitness will be over the range of [-1, 0), where -1 represents every item in the list has been placed in the knapsack. If the individual is not overfull then the fitness will be the sum of the profit scores of each item in the individual. 4.8 Survivor Selection Newly generated individuals of both EAs will be added to the population if their fitness is greater than any individual currently in the population. A starting point is selected at random and the entire population is checked for an individual with a lower fitness score. Once a new individual has been placed in the population the next child is selected for placement. As long as the fitness score of a newly generated child is greater than the fitness score of a single individual in the population it will be placed in the population, otherwise it will not. If an individual has been selected for replacement its location and the location of its mate are reset to -1 in the marriage hash, and the “spouse” of the replaced individual is once again eligible to remarry. 4.9 Statistics Calculations 9 The statistics of each EA population includes the average fitness, the variance, and standard deviation. The fitness score and individual representation of the best fit individual for all three EAs is constantly maintained. Additionally the fitness score of the worst individual is also maintained to show an additional level of improvement. 4.10 Stopping Criteria As noted earlier in Tables 1 and 2 both of the EAs has its own unique stopping criteria. In addition to those criteria previously mentioned the following default criteria will be used. Each parameter set is kept until the EA has iterated equal to the number of runs declared in the parameter file. Every iteration of the EA will terminate by default when it reaches the maximum number of generations also specified in the parameter file. Upon reaching the end of a iteration the program will display on the screen the best fit individual and the fitness score of that individual. In addition the N-Queens EA will display the total time it took the EA to run after initialization. The Knapsack EA will also display the time as well as the capacity of the knapsack used for the solution. 5. N Queens Results The results section will be broken up into two sections; one for the N-Queens implementation and another for the Binary Knapsack implementation. Starting with the performance of the N-Queens EA, Chart 1 show the average rate of convergence for all seven parental selection strategies implemented for the study. You can see by Chart 1 it is not very descriptive and extremely hard to read, as all of the data is overlapping. However Chart 2 is an exploded view of the last 4000 generations of Chart 1. It is clear from Char 2 that the standard tournament selection outperformed the six custom strategies. As you can also see from Chart 2 that the standard tournament selection strategy not only maintained an overall better average fitness it also converged to a solution on average 1000 to 1500 generations sooner, around 4000. All six of the custom strategies converged on a solution in the 5000 to 5500 generation range. Table 4 shows a brief general statistical breakdown of the performance of each strategy as compared to each other. One interesting point is Minimum convergence rate for the Assortative EA with a low cheating probability and the Disassortative EA Table 4 Minimum Maximum Average Solutions Found Tournament 1450 8700 3904.86 30 AMEA M 1700 8200 4905.20 30 AMEA L C 800 8450 4822.98 30 10 AMEA H C 2400 8400 4646.59 30 DMEA M 2300 8900 4732.15 30 DMEA L C 2150 7700 4886.52 30 DMEA H C 500 9150 4542.84 30 Chart 1 N-Queens Convergence 0 1 0 2 5 4 3 -1 -2 Tournament AMEA Marriage AMEA L C AMEA H C DMEA Marriage DMEA L C DMEA H C Fitness -3 -4 -5 -6 -7 -8 Generations X 1000 Chart 2 N Queen Convergence Exploded View -0.8 2 3 4 5 -1 Tournamet AMEA Marriage AMEA L C AMEA H C DMEA Marriage DMEA L C DMEA H C Fitness -1.2 -1.4 -1.6 -1.8 -2 Generations X 1000 11 with the high cheating probability. Further study will be required to prove if a low cheating probability was the cause of the extremely fast convergence of the AMEA, and if the high cheating probability caused the DMEA to converge so rapidly, or if it was mere coincidence. Higher level statistical analysis of the N Queen problem was as follows: The two sample f test was performed on all six custom strategies comparing them to the tournament selection. Out of the six comparisons it was found that the variances of all of the strategies were not equal to the variance of the Tournament selection strategy, with the exception of the AMEA with a high cheating probability. However, upon further analysis utilizing the appropriate two tailed t tests, it was determined that the means of all six custom strategies were in fact equal to that of the tournament selection strategy. The results of the f and t tests can be seen in Appendix B. 6. Knapsack Results The Binary Knapsack implementation also did not show any improvement over the tournament strategy. As you can see by Chart 3, like Chart 1, it is also hard to read due to the fact all of the data is overlapping. However, Chart 4 is an exploded view of the last 4000 generations which clearly shows the tournament selection strategy outperforming the six custom strategies ever so slightly. The DMEA with a high cheating probability performed the best out of the six, but was not enough to beat out the tournament selection. Table 5 simply shows some low level statistical analysis of all seven strategies as compared to each other. The Minimum and Maximum in Table 5 is the worst and best solutions found by each EA. Marriage is implied for the strategies with the cheating component. Not much can be discerned from Table 5 so further analysis is warranted. Higher level statistical analysis of the Knapsack problem was as follows: The two sample f test was performed on all six custom strategies comparing them to the tournament selection. Out of the six comparisons it was found that the variances of all of the AMEA strategies were not equal to the variance of the Tournament selection strategy, and all of the DMEA variances were equal. However, upon further analysis utilizing the appropriate two tailed t tests, it was determined that the means of all six custom strategies were in fact equal to that of the tournament selection strategy. The results of the f and t tests can be seen in Appendix B. Table 5 Minimum Maximum Average Tournament 26.7901 27.9844 27.42215 AMEA M 26.7743 27.9302 27.38704 AMEA L C 26.7323 27.8754 27.38704 12 AMEA H C 26.6994 27.9586 27.28795 DMEA M 26.7344 27.7919 27.3204 DMEA L C 26.6205 27.7609 27.28731 DMEA H C 26.6767 27.8572 27.3967 Chart 3 Knapsack Convergence 30 25 Fitness 20 Tournament AMEA Marriage AMEA L C AMEA H C DMEA Marriage DMEA L C DMEA H C 15 10 5 0 0 1 2 3 4 5 6 7 8 9 10 -5 Generations X 1000 Chart 4 Knapsack Convergence Exploded View 27.45 27.4 27.35 Tournament AMEA Marriage AMEA L C AMEA H C DMEA Marriage DMEA L C DMEA H C Fitness 27.3 27.25 27.2 27.15 27.1 27.05 27 6 7 8 9 Generations X 1000 13 10 7. Conclusions Though there was not an improvement in the convergence rates of the EA utilizing any of the six custom strategies there is substantial proof that implementing an EA with one of the above selection strategies would perform nearly as well as the standard tournament selection. With some minor adjustments the strategies may perform as well as the tournament selection. Due to the results above, and the relative ease in implementation, it is possible that implementing EA with a marriage and cheating component could become an alternative to the tournament selection strategy. 8. References (1) Craighurst, Rob and Martin, Worthy N. Enhancing GA Performance through Crossover Prohibitions Based on Ancestry Proc. Of the 6th International Conference on Genetic Algorithms. San Francisco, 1995. (2) Eiben, A. E. and Smith, J. E. Natural Computing Series: Introduction to Evolutionary Computing Springer 2003 (3) Eshelman, Larry j. and Schaffer, J. David Preventing Premature Convergence in GeneticAlgorithms by Preventing Incest. Proc. Of the 4th International Conference on Genetic Algorithms. San Diego, 1991. (4) Ochoa, Gabriel; Madler-Kron, Christian, Rodriguez, Ricardo; and Jaffe, Klaus Assortive Mating in Genetic Algorithms for Dynamic Problems EvoWorkshops 2005, LNCS 3449, pp. 617-622, 2005. (5) Ting, Chuan-Kang and Buning, Hans Kleine A Mating Strategy for Multiparent Genetic Algorithms by Integrating Tabu Search IEEE 2003 14 Appendix A Running the Program Part 1 The Parameter File The parameter file is simply a tab delimited text file that contains eleven values with the following restrictions on individual parameters: 1. Recombination Probability (float) [0, 1] 2. Individual Mutation Probability (float) [0, 1] 3. Gene Mutation Probability (float) [0, 1] 4. Population Size (uint) [1, 10000] 5. Parent Pool Size (uint) [1, Population Size] Note (a): Recombination will occur if Parent Pool Size > 1, otherwise strictly Mutation will occur. Note (b): This parameter will default to 1 if Population Size is 1. 6. Cheating Probability (float) [0, 1] 7. Total Children per Generation (uint) [1, + infinity) 8. Maximum Number of Children/Parent(s) (uint) [1, Total Children] 9. Maximum Number of Generations (uint) [10, + infinity) 10. Number of Runs (uint) [1, + infinity) 11.Random Seed (long uint) [0, + infinity) Note : If 0 a new seed will be generated from the cpu clock. 12.This is the name of the file that the screen output will be directed to. (string) [filename extensions are not required] 13.This is the name of the file that the statistical output will be directed to. (string) [filename extensions are not required] 14.Selects the EA you wish to run. (int) [1=N-Queens, 2=Binary Knapsack] 15.Selects the Parent Selection Strategy you wish to use. (int) [0=Standard Tournament, 1=AMEA w/marriage and cheating, 2=DMEA w/marriage and cheating] 16.Selects the name of the data file if one is required by the EA (string) Part 2 Executing the Program 1. 2. 3. 4. Copy the entire directory to a directory of your choosing. cd to the new directory and compile all the cpp files. Create your parameter file in the new directory according to Part 1 above. Once the make file has completed type “./” followed by the executable name of the binary on the command line. 5. The program will then execute and prompt you to enter the name of the parameter file you wish to use. Enter it at this time. 6. If there are any errors in your parameter file, values out of range for example, the program will notify you of the error. 15 7. If there is an error you can edit your parameter file without terminating the program then reload it when prompted again. 8. The program will display various information during it’s execution to the screen. Upon completing each iteration the power index of and best fit individual will be displayed on the screen as well as written to the output file. Part 3 .CSV Output File Details At the start of each run the random seed that was generated by the program or the seed that was passed into it from the parameter file is written to the output file, proceeded only by the character string “Random_seed,”. Immediately following the random seed is the rest of the parameter set used for the current run separated by parameter name and comma. After the parameters the generational statistics are included. For this study the statistics for every 50th generation was written to the output file. The statistics for each line of output includes the following comma delimited data: 1. 2. 3. 4. 5. 6. The number of the iteration The number of the generation The fitness of the best and worst individuals The average fitness of the population at that generation The variance of the fitness scores at that generation The standard deviation of the fitness scores at that generation Following the last generation the string “,end_iteration,” will denote the end of an iteration and the start of another. Part 4 .TXT Output File Details This file will mirror everything that is output to the screen during execution. The text file includes everything as it was seen on the screen to include the initial input of the parameter file. Screen output is preformatted for ease of reading as opposed to the format of the .csv file. The program will display the same data detailed in 1 thru 6 in part 3 above, but without all of the commas. At the end of each iteration the fitness score of the best individual and the best solution found at the time of termination are written to the file as they were displayed to the screen. If additional iterations are desired the statistics for the first generation of the next iteration will immediately follow the output of the best individual of the previous iteration. 16 Appendix B F and T test Results N-Queens Tournament vs. AMEA w/ marriage reject H0, the varainces are unequal F-Test Two-Sample for Variances Mean Variance Observations df F P(F<=f) one-tail F Critical one-tail Variable 1 3986.666667 2796540.23 30 29 0.989965017 0.489260419 0.537399965 Tournament vs. AMEA w/amrriage & low cheating reject H0, the variances are unequal F-Test Two-Sample for Variances Variable 2 5045 2824888 30 29 Tournament vs. AMEA w/marriage & high cheating accept H0, the variances are equal F-Test Two-Sample for Variances Mean Variance Observations df F P(F<=f) one-tail F Critical one-tail Variable 1 3986.666667 2796540.23 30 29 1.17283476 0.335289143 1.860811434 Variable 2 4888.333 2384428 30 29 Tournament vs. DMEA w/ marriage reject H0, the variances are unequal F-Test Two-Sample for Variances Mean Variance Observations df F P(F<=f) one-tail F Critical one-tail Variable 1 3986.666667 2796540.23 30 29 0.865382341 0.349823678 0.537399965 Mean Variance Observations df F P(F<=f) one-tail F Critical one-tail Variable 1 3986.666667 2796540.23 30 29 0.979187649 0.477613615 0.537399965 Variable 2 4971.667 2855980 30 29 Tournament vs. DMEA w/marriage & high cheating reject H0, the variancea are unequal F-Test Two-Sample for Variances Mean Variance Observations df F P(F<=f) one-tail F Critical one-tail Variable 1 5146.666667 2396540.23 30 29 0.759539138 0.231755018 0.537399965 Variable 2 4698.333 3155256 30 29 Tournament vs. DMEA w/amrriage & low cheating reject H0, the variances are unequal F-Test Two-Sample for Variances Variable 2 4591.667 3231566 30 29 17 Mean Variance Observations df F P(F<=f) one-tail F Critical one-tail Variable 1 5146.666667 2396540.23 30 29 0.741603347 0.212856375 0.537399965 Variable 2 4591.667 3231566 30 29 Knapsack Tournament vs. AMEA w/ marriage reject H0, the variances are unequal F-Test Two-Sample for Variances Mean Variance Observations df F P(F<=f) one-tail F Critical one-tail Variable 1 27.42215 0.078228718 30 29 0.96132138 0.458069651 0.537399965 Tournament vs. AMEA w/amrriage & low cheating reject H0, the variances are unequal F-Test Two-Sample for Variances Variable 2 27.38704 0.081376 30 29 Tournament vs. AMEA w/marriage & high cheating reject H0, the variances are unequal F-Test Two-Sample for Variances Mean Variance Observations df F P(F<=f) one-tail F Critical one-tail Variable 1 27.42215 0.078228718 30 29 0.739570232 0.210745476 0.537399965 Variable 2 27.28795 0.105776 30 29 Tournament vs. DMEA w/ marriage accept H0, the variances are equal F-Test Two-Sample for Variances Mean Variance Observations df F P(F<=f) one-tail F Critical one-tail Variable 1 27.42215 0.078228718 30 29 1.393482501 0.188416575 1.860811434 Mean Variance Observations df F P(F<=f) one-tail F Critical one-tail Variable 1 27.42215 0.078228718 30 29 0.820162214 0.298471009 0.537399965 Variable 2 27.30497 0.095382 30 29 Tournament vs. DMEA w/marriage & high cheating accept H0, the variances are equal F-Test Two-Sample for Variances Mean Variance Observations df F P(F<=f) one-tail F Critical one-tail Variable 1 27.42215 0.078228718 30 29 1.07855776 0.420015231 1.860811434 Variable 2 27.28731 0.072531 30 29 Tournament vs. DMEA w/amrriage & low cheating accept H0, the variances are equal F-Test Two-Sample for Variances Variable 2 27.3204 0.056139 30 29 18 Mean Variance Observations df F P(F<=f) one-tail F Critical one-tail Variable 1 27.42215 0.078228718 30 29 1.07855776 0.420015231 1.860811434 Variable 2 27.28731 0.072531 30 29 N-Queens Tournament vs. AMEA w/ marriage The means are equal t-Test: Two-Sample Assuming Unequal Variances Mean Variance Observations Hypothesized Mean Difference df t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail Variable 1 3986.6667 2796540.2 30 Variable 2 5045 2824888 30 0 58 2.4448925 0.0087741 1.6715528 0.0175482 2.0017175 Mean Variance Observations Pooled Variance Hypothesized Mean Difference df t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail Mean Variance Observations Hypothesized Mean Difference df t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail Tournament vs. AMEA w/marriage & high cheating The means are equal t-Test: Two-Sample Assuming Equal Variances Variable 1 3986.6667 2796540.2 30 Tournament vs. AMEA w/amrriage & low cheating The means are equal t-Test: Two-Sample Assuming Unequal Variances Variable 2 4888.333 2384428 30 2590484.2 0 58 2.1697074 0.0170695 1.6715528 0.034139 2.0017175 19 Variable 2 4971.667 2855980 30 Variable 1 3986.666667 2796540.23 30 0 58 -2.2692158 0.013495384 1.671552763 0.026990768 2.001717468 Tournament vs. DMEA w/marriage & high cheating The means are equal t-Test: Two-Sample Assuming Unequal Variances Mean Variance Observations Hypothesized Mean Difference Variable 1 4698.333333 3155255.747 30 0 df t Stat 58 -0.61062046 P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail 0.27191818 1.671552763 0.54383636 2.001717468 Variable 2 4971.667 2855980 30 Tournament vs. DMEA w/ marriage The means are equal t-Test: Two-Sample Assuming Unequal Variances Mean Variance Observations Hypothesized Mean Difference df t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail Variable 1 4591.6667 3231566.1 30 Variable 2 4971.667 2855980 30 0 58 0.8435738 0.2011867 1.6715528 0.4023734 2.0017175 Tournament vs. DMEA w/amrriage & low cheating The means are equal t-Test: Two-Sample Assuming Unequal Variances Mean Variance Observations Hypothesized Mean Difference df t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail Variable 1 5146.666667 2396540.23 30 Variable 2 4971.667 2855980 30 0 58 0.418229646 0.338661867 1.671552763 0.677323733 2.001717468 Knapsack Tournament vs. AMEA w/ marriage The means are equal t-Test: Two-Sample Assuming Unequal Variances Mean Variance Observations Hypothesized Mean Difference df t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail Variable 1 27.42215 0.0782287 30 Variable 2 27.38704 0.081376 30 0 58 0.4813581 0.3160366 1.6715528 0.6320732 2.0017175 20 Tournament vs. AMEA w/amrriage & low cheating The means are equal t-Test: Two-Sample Assuming Unequal Variances Mean Variance Observations Hypothesized Mean Difference df t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail Variable 1 27.42215 0.078228718 30 0 57 1.540328994 0.064506904 1.672028889 0.129013808 2.002465444 Variable 2 27.30497 0.095382 30 Tournament vs. AMEA w/marriage & high cheating The means are equal t-Test: Two-Sample Assuming Unequal Variances Mean Variance Observations Hypothesized Mean Difference df t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail Variable 1 27.42215 0.0782287 30 Variable 2 27.28795 0.105776 30 0 Tournament vs. DMEA w/ marriage The menas are equal t-Test: Two-Sample Assuming Equal Variances Mean Variance Observations Pooled Variance Hypothesized Mean Difference df t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail Mean Variance Observations Pooled Variance Hypothesized Mean Difference df t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail 57 1.7135153 0.0460261 1.6720289 0.0920521 2.0024654 Variable 1 27.42215 0.0782287 30 0.0671839 Tournament vs. DMEA w/marriage & high cheating The means are equal t-Test: Two-Sample Assuming Equal Variances Variable 2 27.3204 0.056139 30 0 58 1.5203142 0.0669326 1.6715528 0.1338652 2.0017175 21 Variable 1 27.39670333 0.084338039 30 Variable 2 27.3204 0.056139 30 0.070238521 0 58 1.115019566 0.134720451 1.671552763 0.269440902 2.001717468 Tournament vs. DMEA w/amrriage & low cheating The means are equal t-Test: Two-Sample Assuming Equal Variances Mean Variance Observations Pooled Variance Hypothesized Mean Difference df t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail Variable 1 27.28731 0.072530857 30 0.06433493 0 58 -0.50531562 0.307626041 1.671552763 0.615252081 2.001717468 Variable 2 27.3204 0.056139 30