Honors Project: A Comparison of Elite vs. Regular Ants Using the JSP

Transcription

1 Honors Project: A Comparison of Elite vs. Regular Ants Using the JSP Joshua Luchies Student # Supervising Professor: Tony White Date: March, Course: COMP

2 Table of Contents Abstract page Introduction page - Problem Description page - Solution and Methods page Results page Conclusions page Discussion page Appendices page acknowledgments: Thanking Dr. Tony White for supervising me on this Honors Project. All of the researchers which contributed indirectly to this research project through papers and texts. Joshua Luchies //

3 Abstract This paper discusses the subject of Swarm Intelligence and its applications to the Job Shop Scheduling Problem. Specifically it deals with the idea of a memory in trying to determine the shortest schedule for a set of jobs. This is modeled using an Ant Colony algorithm. Ant Colony algorithms are inspired by the ability of ants to follow a short path to a food source without using anything but local information. The research conducted here is involving an optimization method with elitist ants. Elitist ants are basically ants which can remember the best solution they have thus far encountered in their exploration. This research is very useful in many applications in which scheduling is important as well as for managers. The scope of the research performed here is to determine whether memory enhances the quality and speed of the solutions reached, or hinders these two parameters. This will be accomplished by constructing a simulation of the Scheduling Problem, and running a population of regular or no-memory ants, and a population of elitist or memory ants through it. From the results I have seen in my implementation of this problem and the simulation, I have concluded that elitist ants do better overall in a shorter period of time. This is in accordance with the literature I was directed to on the TSP implementation using elitist ants by Tony White []. Joshua Luchies //

4 Introduction Swarm Intelligence is usually based upon something I call natural algorithms, these are algorithms modeled after phenomena seen in nature. Swarm Intelligence, more specifically deals with individuals which look as if they are acting in concert to accomplish some semi-complicated task, using only local information, and having very little communication with others in the community. Some examples are a flock of birds in flight, a school of fish, army ants harvesting food, termites building their homes. In each of these examples the organism uses simple rules, and only local information to get a task accomplished, and all work independently of one another. Some examples of the rules would be to follow the leader, or follow a chemical trail, etc. The effect of the population following these simple rules results in emergent behavior when the population is seen as a single entity accomplishing a task. Each individual, can not do much, but as a whole the goal is accomplished to the benefit of all. This emergent behavior, when applied to computer science problems results in the solutions constructed by a population of agents working independently, with local information. The Job Shop Scheduling Problem is usually modeled after a factory with many optimized machines. These machines only perform one task, but do it very quickly and efficiently. The manager is given a list of jobs, each with many operations to be completed on certain machines. This implies that each job has a specific path around the factory floor, visiting the machines in a specified order. The problem for the manager is to devise a schedule so that all of the jobs are completed in the shortest amount of time. The Ant Colony algorithm is modeled after an ant called Linepithema Humile, Joshua Luchies //

5 found in Argentina, which exhibits the ability to find short paths to food using pheromones to mark the way. This is accomplished when many ants are traversing the path, a longer path receives less pheromones in the same amount of time that the shorter path does, even if the same number of ants always traverses each path equally. So an algorithm based on this behavior is applicable to finding short paths to solve problems such as the Traveling Salesman Problem and the Job Shop Scheduling Problem. Some ants also have the ability to remember the path they have taken before, and one example is the Lasius Niger, which will turn around if it finds it is going for a long time in a nonhelpful direction. This is where the inspiration for elitist ants comes from. They are able to remember the direction of the food, and tell if they are taking too long to get back to it. The elitist ants that will be used in these experiments can remember the path they have taken before. This amounts to a behavior such as the following; the ants leaving the ant hill travel randomly in search of food, and each time they come back, they leave a pheromone trail. If they take longer to get to the food than before, they will leave their pheromone on the trail they took previously on the way back to the ant hill. If they discovered a shorter path on the way out, they will follow the shorter path back, leaving their pheromone on that one instead. Joshua Luchies //

6 Problem Description Job Shop Problem The general shop problem can be defined as follows. There are n jobs i=,...,n and k machines M,...,M k. Each job i consists of a set of operations O ij (j=,...,n i ) with processing times p ij. The processing times are deterministic and known in advance. Each operation has to be processed on a dedicated machine m ij ε {M,...,M k } for an uninterrupted processing period. There may be precedence relations between the operations of all jobs. Each job can only be processed only by one machine at a time and each machine can only process one job at a time. The objective is to find a feasible schedule that minimizes some objective function of the finishing times C i of the jobs i=,...,n. The objective functions are assumed to be regular. The job shop problem is a special case of the general shop in which each job i consists of m operations O ij with processing times p ij (j=,...,m) where O ij has to be processed on machine M ij, and the operations have to be scheduled in predetermined given order. Each job has its own individual flow pattern through the machines which is independent of the other jobs. The problem is to find a feasible schedule, i.e. to determine starting times for each operation, in order to minimize the objective function while satisfying all precedence and capacity constraints[]. Joshua Luchies //

7 The objective of this research is to determine whether the Ant Colony Optimization algorithms can be improved for the purpose of solving this problem. It has already been observed that the method of improvement explored in this research is practical for the TSP problem. The question is whether this can be extended to other problems of the same type, shortest path optimization. The area of exploration is whether an ant that remembers the best schedule it has come across, will speed up and enhance the new schedules, or whether remembering will deter it from exploring new areas and possibly be able to come up with better schedules in the future. For the purpose of this implementation, I am working strictly within the guidelines of the Job Shop Problem, the list of jobs has a set of operations, each with a processing time, and machine to be processed on. The Ant algorithm schedules each operation in order, and takes into account the fact that different machines cannot work on the same job at the same time, but may work on different jobs, provided all operations before the one currently being processed are finished. Joshua Luchies //

8 Solutions and Methods The solution presented here to the overall problem of the benefits of having an ant with memory as over against an ant without, is to run two populations of ants through the same problem instance, and compare them. The first population will be of regular ants without memory, and the second will be solely of elitist ants with memory. In this research the mixing of the populations is not explored, but that could be done fairly easily, and is an option for further exploration. The solution for the problem of representing the Job Shop instance is as follows. A graph representation, as outlined in a text on swarm intelligence [] was used. The graph is an almost complete graph of the operations organized like so: each job is a row, and connected by directed edges from operations (... n). Each operation is then connected to all operations of all other jobs. Finally, there is one extra node, at the start, to signify the start position for the schedule (see Appendix B for a diagram). In my implementation for the simulation, I represented the graph vertices as a list of operations, and the edges were in an adjacency matrix format. The matrix stored the amount of pheromone on the directed edge from operation i to operation j. Each of the vertices in the graph contains a job operation, with the machine number and time needed to complete the job. I used the matrix representation because it provides instant access to the pheromone count on the edges needed for the selection of a new operation and updating of the schedules is much faster. Also there is not much wasted memory usage because the graph is nearly complete as well, so nearly all of the spaces are filled. This greatly increases the speed of my program compared to using an adjacency list representation of the graph. These two attributes, the pheromone on the edges, and the operation times are necessary Joshua Luchies //

9 for determining the heuristic probability for the Ant Colony algorithm. Much of the literature that I had access to did not mention the representation used in practice of the graphs, and other data structures necessary to make the programmed simulations run smoothly and efficiently. The simulation which I have constructed for this research can solve a fairly decent complexity of problems in a good time, but I am sure there is much room for improvement in both the speed and accuracy of its evaluation. The algorithm used to solve the problem is the classical Ant Colony Optimization algorithm, with a few modifications to make it applicable to the Job Shop Problem. There is a selection equation which provides the probability that a certain operation will be selected out of all available or legal operations, as outlined in the job shop problem description. This is the algorithm I used to select the next operation to schedule. It produces a probability of choosing operation j when you are currently processing operation i : Σ [pheromone(operation i, allowed operation...n ) alpha * time(allowed operation...n ) beta] ] / (pheromone(operation i, operation j ) alpha * time(operation j ) beta ) This equation favors the shorter operations first, and then the longer ones. There are benefits and detriments to using either a longest job first, or shortest job first algorithm, I didn't see a great difference between the two different algorithms in this simulation for the problems I encountered, but that may be just because the problems had a spread of quick and long operations, or my fitness function. The fitness function would actually have much to do with this as it relies on scheduling groups of jobs at once. This would hide small jobs if one long one had to be scheduled at the same time. Conceptually with the Job Shop Problem, I don't think it matters whether you use a shortest job first, or Joshua Luchies //

10 longest job first algorithm, but I also learned in my Operating Systems design class that shortest job first algorithms generally give better results. The probability is inverted to give precedence to the better choices for operations to complete. The other important equation for the Ant Colony Optimization algorithm is the pheromone dropping equation which adds to the amount of pheromone on the edges. Pheromone dropping equation: local best tour length / current ant tour length; This algorithm is adapted from the usual algorithm found in much of the literature which uses a constant instead of the local best tour length. The constant is supposed to be near to the optimal solution to work well. Using the local best tour length provided a way to keep in step with the spirit of the algorithm in that we only want to work with local information, and this provides a workable way of doing so. This method also avoids the tedious work of estimating schedule length for each problem. Although this can be done using a greedy algorithm, then you may suffer from having the value too large to achieve the optimal tour length. Joshua Luchies //

11 Here is the pseudocode for the algorithm which was implemented for the tests: Initialize a graph of the jobs and operations; Initialize an ant population; (the ant populations were either regular ants, or elite) for each tour of the graph do: for each ant in the population do: place all ants on the start operation; while all operations are not in the schedule do: rate all the available operations; randomly select an operation according to the rating given; add it to the schedule; update the available operations; end while; end for; for each ant do: calculate the length of their tour; if the ant is elite update the length of the schedule if it is better than its previous solution as well as its best schedule found; else update the length of the schedule; end for; for each edge in the graph do: for each ant in the population do: for each edge in the schedule do: drop pheromones as per the pheromone dropping equation; end for; end for; decay the pheromone as per the pheromone decay rate; end for; keep track of any statistics or information you need here; (some examples are local and global best tours, averages, standard deviations) end for; The elitist ants have the ability to store the best tour they have formulated so far. This gives them the ability to change its tour only if it is better than the best one they have made. So after the length of the schedule is calculated, the best schedule that the ant has Joshua Luchies //

12 made is then set as the schedule to update the edges in the graph with pheromone. This is how the effect of the memory is being modeled. It is an accurate model in that it depicts the aspect that if the ant would have known the inferior schedule to be that way, then it would have chosen the other. It is like the path to the food, if it took the ant longer to get to the food, than it took to get back to the nest the previous time, then the ant will take the previous path to get back, knowing it to be quicker than the path it took to get from the nest to the food (in this analogy, the ants only release pheromone on the way back to the nest). The way I implemented this in code is by providing a subclass which had the added ability to store an extra tour representing the best one seen. Another important part of the algorithm is the fitness evaluation used to rate the schedules. I used my own version of a worst time to completion algorithm. This seemed appropriate and sufficient for my purposes. The algorithm specifics are shown here in pseudocode: while there are operations left in the schedule do get the next operation; if the operation time is greater than the max time for this set of operations, set the max as this operation's time; if the machine is unused and the job is not being processed on another machine, store the machine and job number, if either the machine is being used, or the job is currently being processed, add the maximum operation time to the fitness; reset the stored machine and job numbers; end while; This results in a worst completion time algorithm for the jobs because it relies on scheduling sets of jobs, and using the maximum time to completion for each set scheduled. This is good enough for my research as I was not looking to have exact Joshua Luchies //

13 schedules, just to compare two different types of populations, and working with the same fitness function is all that is really necessary for this purpose. The constants and optimal parameters used in the experiments conducted with this algorithm are the results of previous research done by Tony White []. The constants alpha, set at, and beta set at were observed to lead to the finding of good solutions consistently to the Traveling Salesman Problem, and were also able to be generalized to other shortest path problems such as JSP and QAP by research done by Dorigo []. He has a set of optimal values for these constants. I found that the values I used provided good enough solutions, and refining those constants aren't at the core of this investigation. That is the reason for those constants being set at those values. I did attempt a range of values to try and improve my results, but they all were approximately the same. That could be an area for further exploration to improve this simulation. The reasoning behind the size of the population of ants being set at was to accommodate for the random job selection, I wanted to have a good probability of exploring all combinations of the solution. Having more ants than jobs almost guarantees that each job will have an ant which starts on it. In this way, the algorithm is capable of exploring all possibilities, as it depends on the operations which have been completed as well as the available ones in the selection algorithm. Also it was not observed to be helpful to have more ants because they just reinforce redundantly and will saturate the paths through the graph with extra pheromone. The number of iterations was initially taken from some previous research of the professor in charge, but changed when it was observed that after several iterations the values of the schedule lengths varied very little, and didn't seem to change. After about Joshua Luchies //

14 iterations of the algorithm, the solutions arrived at were close to the optimal and stagnation started to occur. The pheromone decay rate was set at. so that there was some buildup, but not enough to overwhelm the other parameters. This value signifies that at each iteration, % of the pheromone was left on the path, and % was evaporated off. This aspect of the research also was not fully explored, but only lightly experimented with until I could achieve acceptable results. There were three sets of experiments conducted. One was with the abz problem (see appendix C). This is a x Job Shop Problem, one with jobs and machines. Both populations of ants, the regular and the elite, were sent through the Job Shop times to create schedules. This was done a total of times, and the results were collected and put into charts. The purpose of this experiment was to determine whether the number of iterations greatly increased the quality of the solutions of the problems. This is to determine the long-term behavior of the ant algorithm being tested. This was the first part of the problem area explored. The second set of experiments again used the abz problem. It, however explored the problem of whether the elite ants produced better solutions more quickly than the regular ants. The populations of ants, one regular and one elite were sent through the Job Shop times. This was repeated times to produce an accurate picture of how well each population performed. The third set, used a different problem, the orb problem (see appendix D). This problem has the same format and setup as the abz problem and the purpose of this is to see whether the values in the problem affected the speed and quality of the solutions. This experiment was conducted with the same parameters and algorithms as the second experiment. Joshua Luchies //

15 Results Experiment One Results Experiment one was to determine the length of time needed for the ants to generate the best schedules they were able. In some of the literature, for these small type of problems, many of the researchers felt that iterations was generally sufficient for many algorithms to converge to the optimal solutions. If the algorithm in question was not even near to the found optimal, within iterations, then it wasn't a very good algorithm in the first place. That is the reason I decided to force the iterations. One other reason was that I was curious about whether my algorithm would converge on a single solution, or tend to maintain diversity over a long period of time. The results were that the elite ants performed much better over all. The graphs presented here clearly show this emergent behavior. Looking at the global bests of each population over the trials shows that the elite ants produced much better solutions overall. The quality of the solutions is the only practical conclusion you can make from looking at this graph, the significance here is not really debatable, the optimal solution, when run through my fitness function of the schedules, comes to. All of the elite ants trials did better than this, but the regular ants didn't get within of this solution. The elite ants hovered around the mark, and the regular ants jumped around the - level. I felt that these results were acceptable in comparison to the optimal solution to proceed with the other experiments. Joshua Luchies //

16 I performed a t test on the global sets, and came up with a score of.. This I believe is high enough to show that the two sets of data are significantly different (the degrees of freedom are ). By examining the local best schedules, I can draw some conclusions about the behavior of how the ants formulate their solutions, and the time needed for them to achieve acceptable schedules. Joshua Luchies // Global Bests for iterations Regular Ants Elite Ants regular ants local bests

17 These two graphs show that the regular ants have a hard time improving their schedules, and after many iterations the elite ants start to stagnate. This is shown better however in the standard deviations. The deviation of the regular ants stays very high, around to, while the elite ants drop their deviation to about by iterations. Joshua Luchies // elite ants local bests Elite Ants Standard Deviation iteration deviation Regular Ants Standard Deviations Iteration deviation

18 This behavior makes sense when you think about how each of the populations is progressing through the shop. The elite ants always keep in mind their best schedule, while the regular ants only follow the pheromones. The populations of regular ants are always updating their own scheduled path, while the elite ants update the best scheduled path they have seen so far. The elite ants therefore, are piling pheromones on the best schedules, whereas the regular ants put down pheromones more equally on all the paths explored. This more equal pheromone deposition seems to keep the paths equally good, so diversity is maintained equally throughout the iterations. I don't see this as a good sign, the algorithm should converge on a single solution, not keep roughly the same number of solutions around. The elite ants are better at converging on the good solutions but suffer from stagnation when certain paths become established, so they are unable to follow new paths after a while. The behavior exhibited as the results of this experiment are validated in the results shown in the following experiments. I used only iterations because after that the gains from either population were very minimal, they either did not improve, or improved very little. This is, I hypothesize because the ants begin to follow the same paths all the time. The pheromone and time considerations re enforcing each other along good paths, and the ants choose these paths consistently. I conducted t tests on the data of experiment one, with the iterations and the data of experiment with the iterations. The sets I used were the same type of ants, so the t tests were to show that the values belong to the same set. The scores received were. for the regular ants, and. for the elite ants. Each test had degrees of freedom. These scores show that the data could have come from the same samples, and so I could get away with only doing generations, Joshua Luchies //

19 instead of a whopping. Experiment two was executed to verify my results from observations on the behavior of experiment one. The objective was to determine if the elite ants performed better than the regular ants, and they do. This graph shows that over the runs, the ants, for the main part performed fairly constantly, there is only one major dip, where the regular ants got lucky and stumbled on a good solution, but still not comparable to the solutions found be the elitist ants. The smoothness of this graph indicates to me that the algorithm is driven with some purpose in mind as well, and isn't just evaluating random solutions. Abz Global Bests Schedule Length Regular Ants Elite Ants Trial If we consider the local behavior of the ants over these trials, we can more clearly see that the elite ants are searching through a smaller space and are therefore able to come to better solutions more often and after a shorter time. Joshua Luchies //

20 In comparing these two graphs, it is evident that the elite ants have a more focused search, resulting in their ability to find better paths more often. The regular ants have a harder time because they update all the paths they take, not just the best one they have ever taken. My results for the best solution have apparently improved on the optimal found for this problem, when I ran their solution through my algorithm, it came up with a value of, and my best solution for this problem came to. The regular ants had a best solution of, which is within about % of the optimal. This is a good indication that using the elite ants does improve the quality, speed and directness of the Ant Colony algorithm. Experiment three was performed with another instance of the problem. This was done to determine if the values for the operations had any bearing on the performance of the algorithm, and either population of ants. This proved to be untrue, the populations exhibited the same trends as in the previous experiments, and the graphs look very similar. This graph of the global best solutions is fairly smooth and indicates constant Joshua Luchies // Regular Ants Local Bests for abz Iteration Schedule Length Elite Ants Local Bests for abz Iteration Schedule Length

21 behavior throughout the trials of the simulation. Orb Global Bests Regular Ants Elite Ants The fact that it has the same shape as the others of the global bests substantiates the claim that the actual values of the operation times does not matter in the solving of this algorithm. Schedule Length Regular Ants Local Bests for orb Iteration Joshua Luchies //

22 Schedule Length Elite Ants Local Bests for orb Iteration Once again, even though the values are different, the important thing to notice is that the shape of the charts is not radically different. This is to be anticipated as the same algorithm should not behave differently on two different sets of information representing the same type of problem. I have compared my solutions to those provided, and it seems as if I have improved on the best ones they have given. The method I used for comparison was to run their solutions through my fitness algorithm to determine the length of the schedule, and could then take the best solution from my data and just use the fitness of that solution to determine that mine was better. The solution given for the orb problem came to when put through my evaluation algorithm. My ants came up with a best solution of. The average global best solution for the regular ants was., which is within % of the optimal solution that was previously found for this problem. These solutions were generated within iterations of the algorithm, so there is probably most definitely room for improvement on these. As a disclaimer, I did not check the solutions carefully, or sufficiently to prove that they are a better optimal than the ones given as it was not my intent to try and best the people posting these problems, Joshua Luchies //

23 merely to evaluate my algorithm and attempt to achieve reasonable results for the purpose of comparing the two types of ants. I did not bother to conduct t tests on this data as it looks the same as that found in experiment two and I felt it unnecessary to do this step of the analysis. Joshua Luchies //

24 Conclusions The conclusions that I was able to arrive at concerning the use of elite ants to solve the Job Shop Problem coincide with some research done previously with the TSP. The ants are better when they remember the best path they have explored, and have outperformed the regular ants with only very little effort for modification of the algorithm. The time taken for the algorithm to run is also indistinguishable from when you are using elite ants, or regular ants. Therefore I propose that the gains made in the quality of the solutions far outweighs the efforts needed to implement this optimization of the algorithm. The solutions found with these algorithms are also consistently better than the solutions previously found, so this just reinforces the idea that Ant Colony algorithms are very applicable for solving more and more shortest path problems. Furthermore, the elite ants optimization is easy to implement and gives great gains to the direction of the algorithm's search. I anticipate that if you used a mixed population of both elitist and regular ants, there would be greater gains in that the elite ants update the better paths more often, but the regular ants are able to deviate and enforce finding new solutions later on in the problem. This would avoid the stagnation behavior of updating the same paths all the time. Joshua Luchies //

25 Discussion What can I say here that hasn't already been said. I have spewed out the details of my implementation of the Problem instances. I used a matrix and list representation of the graph, which is conceptually equivalent to the description of the graph in both Dorigo's paper [] and the book by Theraulaz et al []. I also used a very adapted version of the classical Ant colony algorithm. In my selection algorithm, the summation is on top, and the values for the next operation are on the bottom. This was necessary because otherwise the probabilities were wrong, and favored the longer schedules when I used a roulette wheel type random selection algorithm. I did try the algorithm at first with the summation on the bottom, but that lead the regular ants to come up with a graph that favored longer solutions to the problem. From this observed behavior I concluded that my ants were confused about which operations to choose, and the behavior also disappeared when I inverted the probability. Stagnation behavior was much talked about in the texts and journals I read, and also seemed to dominate the research I looked into. Many of the researchers seemed to want the algorithms to keep going, and not converge upon one definite solution. I see this as a problem because then you are trying to cause the algorithm to search in less profitable areas by trying to extend the search in hopes of finding more optimal solutions. This extends the time needed to get to the optimal, and potentially the program could run forever without coming to any definite solution. The flip side to this problem is that if the algorithm is too deterministic, the solutions generated are much worse because they are arrived at too early in the run. A trade-off is needed and some small number of elite Joshua Luchies //

26 ants in a population of regular ants is what I believe to be a good compromise in the context of Ant Colony algorithms. My fitness function was very innovative on my part, it is a worst time estimate that I think is specific enough to differentiate between the viable schedules. It takes into account the ability to schedule two different jobs on different machines simultaneously. The main downfall is that it doesn't provide a very accurate estimation of the actual length of the schedule. It has to wait for all currently scheduled jobs to finish before scheduling the next set. This means that if there is only one long operation scheduled, the machines are effectively idle until that operation is finished. Even so, my algorithms came up with some very good solutions to the problems encountered. The regular ants were fairly close to the optimal given, and the elite ants were way below. On another note, I am not very satisfied with my simulation as a whole, the code is very messy, and hard to understand. The algorithms taken from the texts that I read aren't very clear and straightforward to put into code. The algorithm itself is confusing and the matrix implementation, while being much faster than much else takes up lots of memory and is very rigid in the type and amount of data it can hold. This is the first time I have coded a simulation such as this, so I anticipate being able to improve greatly on this implementation so it may be able to solve the larger job shop problems easily (the order of x or something like that). Currently I have tried solving a x problem and the computer could do it, but it took some time (half hour) for iterations. I may even have some fun and code up a gui to show the ants making their rounds and displaying the solutions as they generate them. However that is all in the future. Joshua Luchies //

27 Appendix A: References. Tony White, Simon Kaegi, Terri Oda Revisiting Elitism in Ant Colony Search. In proceedings of Genetic and Evolutionary Computation Conference (GECCO).. webpage: [ ]. Eric Bonabeau, Marc Dorigo, Guy Theraulaz. () Swarm Intelligence: from Natural to Artificial Systems. Oxford University Press.. Marco Dorigo, Vittorio Maniezzo, Alberto Colorni. () The Ant System: Optimization by a colony of cooperating agents. Published in IEEE Transactions on Systems, Man, and Cybernetics, Part B, Vol., No., pp.. Joshua Luchies //

28 Appendix B: Job Shop Graph Figure. AS graph for a jobs and machines JSP. Connections with no arrows (in bold) are intended to represent a pair of directed arcs. Joshua Luchies //

29 Appendix C: abz Problem Problem Representation: The format for the problem is the first line is the problem size, number of jobs and number of machines. The second and consecutive lines are one for each job, and consist of machines and times for each operation of that job. The machine number comes first, then the time needed for that operation. Optimal Solution Given: There is one line per machine, and the numbers are the jobs scheduled for that machine. The times and operations can be found by referencing the problem description. Joshua Luchies //

30 My Best solution: J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O The format here is just a list of the jobs and operations in their scheduled order. To check the total completion times, reference them to the representation. Joshua Luchies //

31 Appendix D: orb Problem Problem Representation: This is the same format as the abz problem. Optimal Solution Given: This also is in the same format as the abz problem. My Best Solution: J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O J O The format here is just a consecutive list of the jobs and operations scheduled. To check the time for completion, reference them with the problem representation. Joshua Luchies //

32 Appendix E: The Simulation The simulation was written in Java, and can only be run from the console as of yet. The command line is: java JobShop The problem files must be in the same format as the ones in Appendices C and D. There are options for different population sizes and configurations, you do not need to include both elite and regular ants. The generations referred to is the number of times you wish the population to traverse the graph. There is no criteria for prematurely halting the number of generations to complete, so each trial will necessarily do all of the generations. You may specify the number of trials to complete, this means that after each trial the populations as well as the problem instance are reset to the original default values. There is no option for changing the alpha, beta, pheromone decay rate as of yet. You may, if you wish, go into the code to change these. There are variables called alpha and beta for those parameters, one instance each in the code just under a comment for the selection algorithm. The pheromone decay rate is specified in a function parameter, the function is called decaypheromone(double); You must also provide an output file name when asked, or the program will come back with an error message. The output file contains the trial number, generation numbers, and local best fitness, standard deviation, and average schedule length for each generation, also for each trial it outputs the best global solution in the format for my best solutions in the appendices as well as its fitness. Joshua Luchies //