How can I best visualize the results of my combinatorial optimization problem?
Not a solution in the least but more of suggestion so you change your approach: I find it strange that you are using a scatter plot as a way to solve this. Normally for optimization problems, you use linear programming or non-linear programming optimization. http://en.wikipedia.org/wiki/Nonlinear_programming Random sampling will never prove an optimal solution after all. There are some free programs to download to do LP and NLP. If you are convinced you want to do this random sampling, you could always look for solutions that _dominate_ other solutions. By this I mean, solutions that are better in all evaluation criteria than others, the others are discarded and you continue until you are left with. If there is a boundary of solutions that are dominated/not dominated, I guess a really, really extensive random sampling might reveal this. On the other hand, this is just a real shot in the dark. I think that Nonlinear programming is your best bet.
Not a solution in the least but more of suggestion so you change your approach: I find it strange that you are using a scatter plot as a way to solve this. Normally for optimization problems, you use linear programming or non-linear programming optimization. http://en.wikipedia.org/wiki/Nonlinear_programming Random sampling will never prove an optimal solution after all. There are some free programs to download to do LP and NLP. If you are convinced you want to do this random sampling, you could always look for solutions that _dominate_ other solutions. By this I mean, solutions that are better in all evaluation criteria than others, the others are discarded and you continue until you are left with only the non-dominated. If there is a boundary of solutions that are dominated/not dominated, I guess a really, really extensive random sampling might reveal this. On the other hand, this is just a real shot in the dark. I think that Nonlinear programming is your best bet.
Oh, I think I’ve confused some people. I agree with you all about how to solve the optimization problem. LP and NLP are definitely tasks. And it’s actually a genetic algorithm-based approach that I’m using as a metaheuristic. The random sampling is just a way to compare this approach to other approaches. What I’m really interested in is visualizing the results of these generate and test algorithms. I’d like to see what sort of results have been generated in the decision space. The scatterplot gives me an idea of what it looks like in objective space.
On second thought, since your data is categorical, I’m not so sure that a straight-up PCA (in the formulation I proposed) would be the way to go here. Any assignment of these discrete variables (i.e.: resources) to an axis would be largely arbitrary. There appears to be some kind of version of PCA for categorical variables, but I don’t know anything at all about it. There’s also something called “joint correspondence analysis”, but I know even less about that. However, we might be able to do something slightly different. Instead of trying to directly visualize the solution space, perhaps we can try visualizing properties of the solution space. For example, for each solution, we can form a vector (w1, w2, …, wm) where wk is the number of tasks assigned to resource k (or maybe wk is the total costs or total value associated with tasks assigned to resource k). By formulating the solution space this way, we get vectors in Rm instead of some discrete space and thus you can apply the usual
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