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I'm doing a research on a finite horizon decision problem with t=1,...,40 periods. In every time step t, the (only) agent has to chose an action a(t) ∈ A(t), while the agent is in state s(t) ∈ S(t). The chosen action a(t) in state s(t) affects the transition to the following state s(t+1). So there is a finite horizon markov decision problem.
In my case the following holds true: A(t)=A and S(t)=S, while the size of A is 6 000 000 and the size of S is 10^8. Further the transition function is stochastic.
Since I'm relatively new to the theory of Monte Carlo Tree Search (MCTS), i ask myself: is MCTS an appropriate method for my problem (in particular due to the large size of A and S and the stochastic transition function?)
I have already read a lot of papers about MCTS (e.g. progressiv widening and double progressiv widening, which sound quite promising), but maybe someone can tell me about his experiences applying MCTS to similar problems or about appropriate methods for this problem (with large state/action space and a stochastic transition function).
With 6 million stochastic actions per state, I don't think any kind of simulation is realistically going to differentiate between those moves without running essentially forever.
100 MM states isn't a lot however, you can store the value for all of them in less than a gigabyte of memory and something like value iteration or policy iteration would solve this optimally much faster.
This is a very difficult problem about how to maneuver a spaceship that can both translate and rotate in 3D, for a space game.
The spaceship has n jets placing in various positions and directions.
Transformation of i-th jet relative to the CM of spaceship is constant = Ti.
Transformation is a tuple of position and orientation (quaternion or matrix 3x3 or, less preferable, Euler angles).
A transformation can also be denoted by a single matrix 4x4.
In other words, all jet are glued to the ship and cannot rotate.
A jet can exert force to the spaceship only in direction of its axis (green).
As a result of glue, the axis rotated along with the spaceship.
All jets can exert force (vector,Fi) at a certain magnitude (scalar,fi) :
i-th jet can exert force (Fi= axis x fi) only within range min_i<= fi <=max_i.
Both min_i and max_i are constant with known value.
To be clear, unit of min_i,fi,max_i is Newton.
Ex. If the range doesn't cover 0, it means that the jet can't be turned off.
The spaceship's mass = m and inertia tensor = I.
The spaceship's current transformation = Tran0, velocity = V0, angularVelocity = W0.
The spaceship physic body follows well-known physic rules :-
Torque=r x F
F=ma
angularAcceleration = I^-1 x Torque
linearAcceleration = m^-1 x F
I is different for each direction, but for the sake of simplicity, it has the same value for every direction (sphere-like). Thus, I can be thought as a scalar instead of matrix 3x3.
Question
How to control all jets (all fi) to land the ship with position=0 and angle=0?
Math-like specification: Find function of fi(time) that take minimum time to reach position=(0,0,0), orient=identity with final angularVelocity and velocity = zero.
More specifically, what are names of technique or related algorithms to solve this problem?
My research (1 dimension)
If the universe is 1D (thus, no rotation), the problem will be easy to solve.
( Thank Gavin Lock, https://stackoverflow.com/a/40359322/3577745 )
First, find the value MIN_BURN=sum{min_i}/m and MAX_BURN=sum{max_i}/m.
Second, think in opposite way, assume that x=0 (position) and v=0 at t=0,
then create two parabolas with x''=MIN_BURN and x''=MAX_BURN.
(The 2nd derivative is assumed to be constant for a period of time, so it is parabola.)
The only remaining work is to join two parabolas together.
The red dash line is where them join.
In the period of time that x''=MAX_BURN, all fi=max_i.
In the period of time that x''=MIN_BURN, all fi=min_i.
It works really well for 1D, but in 3D, the problem is far more harder.
Note:
Just a rough guide pointing me to a correct direction is really appreciated.
I don't need a perfect AI, e.g. it can take a little more time than optimum.
I think about it for more than 1 week, still find no clue.
Other attempts / opinions
I don't think machine learning like neural network is appropriate for this case.
Boundary-constrained-least-square-optimisation may be useful but I don't know how to fit my two hyper-parabola to that form of problem.
This may be solved by using many iterations, but how?
I have searched NASA's website, but not find anything useful.
The feature may exist in "Space Engineer" game.
Commented by Logman: Knowledge in mechanical engineering may help.
Commented by AndyG: It is a motion planning problem with nonholonomic constraints. It could be solved by Rapidly exploring random tree (RRTs), theory around Lyapunov equation, and Linear quadratic regulator.
Commented by John Coleman: This seems more like optimal control than AI.
Edit: "Near-0 assumption" (optional)
In most case, AI (to be designed) run continuously (i.e. called every time-step).
Thus, with the AI's tuning, Tran0 is usually near-identity, V0 and W0 are usually not so different from 0, e.g. |Seta0|<30 degree,|W0|<5 degree per time-step .
I think that AI based on this assumption would work OK in most case. Although not perfect, it can be considered as a correct solution (I started to think that without this assumption, this question might be too hard).
I faintly feel that this assumption may enable some tricks that use some "linear"-approximation.
The 2nd Alternative Question - "Tune 12 Variables" (easier)
The above question might also be viewed as followed :-
I want to tune all six values and six values' (1st-derivative) to be 0, using lowest amount of time-steps.
Here is a table show a possible situation that AI can face:-
The Multiplier table stores inertia^-1 * r and mass^-1 from the original question.
The Multiplier and Range are constant.
Each timestep, the AI will be asked to pick a tuple of values fi that must be in the range [min_i,max_i] for every i+1-th jet.
Ex. From the table, AI can pick (f0=1,f1=0.1,f2=-1).
Then, the caller will use fi to multiply with the Multiplier table to get values''.
Px'' = f0*0.2+f1*0.0+f2*0.7
Py'' = f0*0.3-f1*0.9-f2*0.6
Pz'' = ....................
SetaX''= ....................
SetaY''= ....................
SetaZ''= f0*0.0+f1*0.0+f2*5.0
After that, the caller will update all values' with formula values' += values''.
Px' += Px''
.................
SetaZ' += SetaZ''
Finally, the caller will update all values with formula values += values'.
Px += Px'
.................
SetaZ += SetaZ'
AI will be asked only once for each time-step.
The objective of AI is to return tuples of fi (can be different for different time-step), to make Px,Py,Pz,SetaX,SetaY,SetaZ,Px',Py',Pz',SetaX',SetaY',SetaZ' = 0 (or very near),
by using least amount of time-steps as possible.
I hope providing another view of the problem will make it easier.
It is not the exact same problem, but I feel that a solution that can solve this version can bring me very close to the answer of the original question.
An answer for this alternate question can be very useful.
The 3rd Alternative Question - "Tune 6 Variables" (easiest)
This is a lossy simplified version of the previous alternative.
The only difference is that the world is now 2D, Fi is also 2D (x,y).
Thus I have to tune only Px,Py,SetaZ,Px',Py',SetaZ'=0, by using least amount of time-steps as possible.
An answer to this easiest alternative question can be considered useful.
I'll try to keep this short and sweet.
One approach that is often used to solve these problems in simulation is a Rapidly-Exploring Random Tree. To give at least a little credibility to my post, I'll admit I studied these, and motion planning was my research lab's area of expertise (probabilistic motion planning).
The canonical paper to read on these is Steven LaValle's Rapidly-exploring random trees: A new tool for path planning, and there have been a million papers published since that all improve on it in some way.
First I'll cover the most basic description of an RRT, and then I'll describe how it changes when you have dynamical constraints. I'll leave fiddling with it afterwards up to you:
Terminology
"Spaces"
The state of your spaceship can be described by its 3-dimension position (x, y, z) and its 3-dimensional rotation (alpha, beta, gamma) (I use those greek names because those are the Euler angles).
state space is all possible positions and rotations your spaceship can inhabit. Of course this is infinite.
collision space are all of the "invalid" states. i.e. realistically impossible positions. These are states where your spaceship is in collision with some obstacle (With other bodies this would also include collision with itself, for example planning for a length of chain). Abbreviated as C-Space.
free space is anything that is not collision space.
General Approach (no dynamics constraints)
For a body without dynamical constraints the approach is fairly straightforward:
Sample a state
Find nearest neighbors to that state
Attempt to plan a route between the neighbors and the state
I'll briefly discuss each step
Sampling a state
Sampling a state in the most basic case means choosing at random values for each entry in your state space. If we did this with your space ship, we'd randomly sample for x, y, z, alpha, beta, gamma across all of their possible values (uniform random sampling).
Of course way more of your space is obstacle space than free space typically (because you usually confine your object in question to some "environment" you want to move about inside of). So what is very common to do is to take the bounding cube of your environment and sample positions within it (x, y, z), and now we have a lot higher chance to sample in the free space.
In an RRT, you'll sample randomly most of the time. But with some probability you will actually choose your next sample to be your goal state (play with it, start with 0.05). This is because you need to periodically test to see if a path from start to goal is available.
Finding nearest neighbors to a sampled state
You chose some fixed integer > 0. Let's call that integer k. Your k nearest neighbors are nearby in state space. That means you have some distance metric that can tell you how far away states are from each other. The most basic distance metric is Euclidean distance, which only accounts for physical distance and doesn't care about rotational angles (because in the simplest case you can rotate 360 degrees in a single timestep).
Initially you'll only have your starting position, so it will be the only candidate in the nearest neighbor list.
Planning a route between states
This is called local planning. In a real-world scenario you know where you're going, and along the way you need to dodge other people and moving objects. We won't worry about those things here. In our planning world we assume the universe is static but for us.
What's most common is to assume some linear interpolation between the sampled state and its nearest neighbor. The neighbor (i.e. a node already in the tree) is moved along this linear interpolation bit by bit until it either reaches the sampled configuration, or it travels some maximum distance (recall your distance metric).
What's going on here is that your tree is growing towards the sample. When I say that you step "bit by bit" I mean you define some "delta" (a really small value) and move along the linear interpolation that much each timestep. At each point you check to see if you the new state is in collision with some obstacle. If you hit an obstacle, you keep the last valid configuration as part of the tree (don't forget to store the edge somehow!) So what you'll need for a local planner is:
Collision checking
how to "interpolate" between two states (for your problem you don't need to worry about this because we'll do something different).
A physics simulation for timestepping (Euler integration is quite common, but less stable than something like Runge-Kutta. Fortunately you already have a physics model!
Modification for dynamical constraints
Of course if we assume you can linearly interpolate between states, we'll violate the physics you've defined for your spaceship. So we modify the RRT as follows:
Instead of sampling random states, we sample random controls and apply said controls for a fixed time period (or until collision).
Before, when we sampled random states, what we were really doing was choosing a direction (in state space) to move. Now that we have constraints, we randomly sample our controls, which is effectively the same thing, except we're guaranteed not to violate our constraints.
After you apply your control for a fixed time interval (or until collision), you add a node to the tree, with the control stored on the edge. Your tree will grow very fast to explore the space. This control application replaces linear interpolation between tree states and sampled states.
Sampling the controls
You have n jets that individually have some min and max force they can apply. Sample within that min and max force for each jet.
Which node(s) do I apply my controls to?
Well you can choose at random, or your can bias the selection to choose nodes that are nearest to your goal state (need the distance metric). This biasing will try to grow nodes closer to the goal over time.
Now, with this approach, you're unlikely to exactly reach your goal, so you need to define some definition of "close enough". That is, you will use your distance metric to find nearest neighbors to your goal state, and then test them for "close enough". This "close enough" metric can be different than your distance metric, or not. If you're using Euclidean distance, but it's very important that you goal configuration is also rotated properly, then you may want to modify the "close enough" metric to look at angle differences.
What is "close enough" is entirely up to you. Also something for you to tune, and there are a million papers that try to get you a lot closer in the first place.
Conclusion
This random sampling may sound ridiculous, but your tree will grow to explore your free space very quickly. See some youtube videos on RRT for path planning. We can't guarantee something called "probabilistic completeness" with dynamical constraints, but it's usually "good enough". Sometimes it'll be possible that a solution does not exist, so you'll need to put some logic in there to stop growing the tree after a while (20,000 samples for example)
More Resources:
Start with these, and then start looking into their citations, and then start looking into who is citing them.
Kinodynamic RRT*
RRT-Connect
This is not an answer, but it's too long to place as a comment.
First of all, a real solution will involve both linear programming (for multivariate optimization with constraints that will be used in many of the substeps) as well as techniques used in trajectory optimization and/or control theory. This is a very complex problem and if you can solve it, you could have a job at any company of your choosing. The only thing that could make this problem worse would be friction (drag) effects or external body gravitation effects. A real solution would also ideally use Verlet integration or 4th order Runge Kutta, which offer improvements over the Euler integration you've implemented here.
Secondly, I believe your "2nd Alternative Version" of your question above has omitted the rotational influence on the positional displacement vector you add into the position at each timestep. While the jet axes all remain fixed relative to the frame of reference of the ship, they do not remain fixed relative to the global coordinate system you are using to land the ship (at global coordinate [0, 0, 0]). Therefore the [Px', Py', Pz'] vector (calculated from the ship's frame of reference) must undergo appropriate rotation in all 3 dimensions prior to being applied to the global position coordinates.
Thirdly, there are some implicit assumptions you failed to specify. For example, one dimension should be defined as the "landing depth" dimension and negative coordinate values should be prohibited (unless you accept a fiery crash). I developed a mockup model for this in which I assumed z dimension to be the landing dimension. This problem is very sensitive to initial state and the constraints placed on the jets. All of my attempts using your example initial conditions above failed to land. For example, in my mockup (without the 3d displacement vector rotation noted above), the jet constraints only allow for rotation in one direction on the z-axis. So if aZ becomes negative at any time (which is often the case) the ship is actually forced to complete another full rotation on that axis before it can even try to approach zero degrees again. Also, without the 3d displacement vector rotation, you will find that Px will only go negative using your example initial conditions and constraints, and the ship is forced to either crash or diverge farther and farther onto the negative x-axis as it attempts to maneuver. The only way to solve this is to truly incorporate rotation or allow for sufficient positive and negative jet forces.
However, even when I relaxed your min/max force constraints, I was unable to get my mockup to land successfully, demonstrating how complex planning will probably be required here. Unless it is possible to completely formulate this problem in linear programming space, I believe you will need to incorporate advanced planning or stochastic decision trees that are "smart" enough to continually use rotational methods to reorient the most flexible jets onto the currently most necessary axes.
Lastly, as I noted in the comments section, "On May 14, 2015, the source code for Space Engineers was made freely available on GitHub to the public." If you believe that game already contains this logic, that should be your starting place. However, I suspect you are bound to be disappointed. Most space game landing sequences simply take control of the ship and do not simulate "real" force vectors. Once you take control of a 3-d model, it is very easy to predetermine a 3d spline with rotation that will allow the ship to land softly and with perfect bearing at the predetermined time. Why would any game programmer go through this level of work for a landing sequence? This sort of logic could control ICBM missiles or planetary rover re-entry vehicles and it is simply overkill IMHO for a game (unless the very purpose of the game is to see if you can land a damaged spaceship with arbitrary jets and constraints without crashing).
I can introduce another technique into the mix of (awesome) answers proposed.
It lies more in AI, and provides close-to-optimal solutions. It's called Machine Learning, more specifically Q-Learning. It's surprisingly easy to implement but hard to get right.
The advantage is that the learning can be done offline, so the algorithm can then be super fast when used.
You could do the learning when the ship is built or when something happens to it (thruster destruction, large chunks torn away...).
Optimality
I observed you're looking for near-optimal solutions. Your method with parabolas is good for optimal control. What you did is this:
Observe the state of the system.
For every state (coming in too fast, too slow, heading away, closing in etc.) you devised an action (apply a strategy) that will bring the system into a state closer to the goal.
Repeat
This is pretty much intractable for a human in 3D (too many cases, will drive you nuts) however a machine may learn where to split the parabolas in every dimensions, and devise an optimal strategy by itself.
THe Q-learning works very similarly to us:
Observe the (secretized) state of the system
Select an action based on a strategy
If this action brought the system into a desirable state (closer to the goal), mark the action/initial state as more desirable
Repeat
Discretize your system's state.
For each state, have a map intialized quasi-randomly, which maps every state to an Action (this is the strategy). Also assign a desirability to each state (initially, zero everywhere and 1000000 to the target state (X=0, V=0).
Your state would be your 3 positions, 3 angles, 3translation speed, and three rotation speed.
Your actions can be any combination of thrusters
Training
Train the AI (offline phase):
Generate many diverse situations
Apply the strategy
Evaluate the new state
Let the algo (see links above) reinforce the selected strategies' desirability value.
Live usage in the game
After some time, a global strategy for navigation emerges. You then store it, and during your game loop you simply sample your strategy and apply it to each situation as they come up.
The strategy may still learn during this phase, but probably more slowly (because it happens real-time). (Btw, I dream of a game where the AI would learn from every user's feedback so we could collectively train it ^^)
Try this in a simple 1D problem, it devises a strategy remarkably quickly (a few seconds).
In 2D I believe excellent results could be obtained in an hour.
For 3D... You're looking at overnight computations. There's a few thing to try and accelerate the process:
Try to never 'forget' previous computations, and feed them as an initial 'best guess' strategy. Save it to a file!
You might drop some states (like ship roll maybe?) without losing much navigation optimality but increasing computation speed greatly. Maybe change referentials so the ship is always on the X-axis, this way you'll drop x&y dimensions!
States more frequently encountered will have a reliable and very optimal strategy. Maybe normalize the state to make your ship state always close to a 'standard' state?
Typically rotation speeds intervals may be bounded safely (you don't want a ship tumbling wildely, so the strategy will always be to "un-wind" that speed). Of course rotation angles are additionally bounded.
You can also probably discretize non-linearly the positions because farther away from the objective, precision won't affect the strategy much.
For these kind of problems there are two techniques available: bruteforce search and heuristics. Bruteforce means to recognize the problem as a blackbox with input and output parameters and the aim is to get the right input parameters for winning the game. To program such a bruteforce search, the gamephysics runs in a simulation loop (physics simulation) and via stochastic search (minimax, alpha-beta-prunning) every possibility is tried out. The disadvantage of bruteforce search is the high cpu consumption.
The other techniques utilizes knowledge about the game. Knowledge about motion primitives and about evaluation. This knowledge is programmed with normal computerlanguages like C++ or Java. The disadvantage of this idea is, that it is often difficult to grasp the knowledge.
The best practice for solving spaceship navigation is to combine both ideas into a hybrid system. For programming sourcecode for this concrete problem I estimate that nearly 2000 lines of code are necessary. These kind of problems are normaly done within huge projects with many programmers and takes about 6 months.
I understood how belief states are updated in POMDP. But in Policy and Value function section, in http://en.wikipedia.org/wiki/Partially_observable_Markov_decision_process I could not figure out how to calculate value of V*(T(b,a,o)) for finding optimal value function V*(b). I have read a lot of resources on the internet but none explain how to calculate this clearly. Can some one provide me with a mathematically solved example with all the calculations or provide me with a mathematically clear explanation.
You should check out this tutorial on POMDPs:
http://cs.brown.edu/research/ai/pomdp/tutorial/index.html
It includes a section about Value Iteration, which can be used to find an optimal policy/value function.
I try to use the same notation in this answer as Wikipedia.
First I repeat the Value Function as stated on Wikipedia:
V*(b) is the value function with the belief b as parameter. b contains the probability of all states s, which sum up to 1:
r(b,a) is the reward for belief b and action a which has to be calculated using the belief over each state given the original reward function R(s,a): the reward for being in state s and having done action a.
We can also write the function O in terms of states instead of belief b:
this is the probability of having observation o given a belief b and action a. Note that O and T are probability functions.
Finally the function τ(b,a,o) gives the new belief state b'=τ(b,a,o) given the previous belief b, action a and observation o. Per state we can calculate the new probability:
Now the new belief b' can be used to calculate iteratively: V(τ(b,a,o)).
The optimal value function can be approached by using for example Value Iteration which applies dynamic programming. Then the function is iteratively updated until the difference is smaller then a small value ε.
There is a lot more information on POMDPs, for example:
Sebastian Thrun, Wolfram Burgard, and Dieter Fox. 2005. Probabilistic Robotics (Intelligent Robotics and Autonomous Agents). The MIT Press.
A brief introduction to reinforcement learning
A POMDP Tutorial
Reinforcement Learning and Markov Decision Processes
I am writing a Time table generator in java, using AI approaches to satisfy the hard constraints and help find an optimal solution. So far I have implemented and Iterative construction (a most-constrained first heuristic) and Simulated Annealing, and I'm in the process of implementing a genetic algorithm.
Some info on the problem, and how I represent it then :
I have a set of events, rooms , features (that events require and rooms satisfy), students and slots
The problem consists in assigning to each event a slot and a room, such that no student is required to attend two events in one slot, all the rooms assigned fulfill the necessary requirements.
I have a grading function that for each set if assignments grades the soft constraint violations, thus the point is to minimize this.
The way I am implementing the GA is I start with a population generated by the iterative construction (which can leave events unassigned) and then do the normal steps: evaluate, select, cross, mutate and keep the best. Rinse and repeat.
My problem is that my solution appears to improve too little. No matter what I do, the populations tends to a random fitness and is stuck there. Note that this fitness always differ, but nevertheless a lower limit will appear.
I suspect that the problem is in my crossover function, and here is the logic behind it:
Two assignments are randomly chosen to be crossed. Lets call them assignments A and B. For all of B's events do the following procedure (the order B's events are selected is random):
Get the corresponding event in A and compare the assignment. 3 different situations might happen.
If only one of them is unassigned and if it is possible to replicate
the other assignment on the child, this assignment is chosen.
If both of them are assigned, but only one of them creates no
conflicts when assigning to the child, that one is chosen.
If both of them are assigned and none create conflict, on of
them is randomly chosen.
In any other case, the event is left unassigned.
This creates a child with some of the parent's assignments, some of the mother's, so it seems to me it is a valid function. Moreover, it does not break any hard constraints.
As for mutation, I am using the neighboring function of my SA to give me another assignment based on on of the children, and then replacing that child.
So again. With this setup, initial population of 100, the GA runs and always tends to stabilize at some random (high) fitness value. Can someone give me a pointer as to what could I possibly be doing wrong?
Thanks
Edit: Formatting and clear some things
I think GA only makes sense if part of the solution (part of the vector) has a significance as a stand alone part of the solution, so that the crossover function integrates valid parts of a solution between two solution vectors. Much like a certain part of a DNA sequence controls or affects a specific aspect of the individual - eye color is one gene for example. In this problem however the different parts of the solution vector affect each other making the crossover almost meaningless. This results (my guess) in the algorithm converging on a single solution rather quickly with the different crossovers and mutations having only a negative affect on the fitness.
I dont believe GA is the right tool for this problem.
If you could please provide the original problem statement, I will be able to give you a better solution. Here is my answer for the present moment.
A genetic algorithm is not the best tool to satisfy hard constraints. This is an assigment problem that can be solved using integer program, a special case of a linear program.
Linear programs allow users to minimize or maximize some goal modeled by an objective function (grading function). The objective function is defined by the sum of individual decisions (or decision variables) and the value or contribution to the objective function. Linear programs allow for your decision variables to be decimal values, but integer programs force the decision variables to be integer values.
So, what are your decisions? Your decisions are to assign students to slots. And these slots have features which events require and rooms satisfy.
In your case, you want to maximize the number of students that are assigned to a slot.
You also have constraints. In your case, a student may only attend at most one event.
The website below provides a good tutorial on how to model integer programs.
http://people.brunel.ac.uk/~mastjjb/jeb/or/moreip.html
For a java specific implementation, use the link below.
http://javailp.sourceforge.net/
SolverFactory factory = new SolverFactoryLpSolve(); // use lp_solve
factory.setParameter(Solver.VERBOSE, 0);
factory.setParameter(Solver.TIMEOUT, 100); // set timeout to 100 seconds
/**
* Constructing a Problem:
* Maximize: 143x+60y
* Subject to:
* 120x+210y <= 15000
* 110x+30y <= 4000
* x+y <= 75
*
* With x,y being integers
*
*/
Problem problem = new Problem();
Linear linear = new Linear();
linear.add(143, "x");
linear.add(60, "y");
problem.setObjective(linear, OptType.MAX);
linear = new Linear();
linear.add(120, "x");
linear.add(210, "y");
problem.add(linear, "<=", 15000);
linear = new Linear();
linear.add(110, "x");
linear.add(30, "y");
problem.add(linear, "<=", 4000);
linear = new Linear();
linear.add(1, "x");
linear.add(1, "y");
problem.add(linear, "<=", 75);
problem.setVarType("x", Integer.class);
problem.setVarType("y", Integer.class);
Solver solver = factory.get(); // you should use this solver only once for one problem
Result result = solver.solve(problem);
System.out.println(result);
/**
* Extend the problem with x <= 16 and solve it again
*/
problem.setVarUpperBound("x", 16);
solver = factory.get();
result = solver.solve(problem);
System.out.println(result);
// Results in the following output:
// Objective: 6266.0 {y=52, x=22}
// Objective: 5828.0 {y=59, x=16}
I would start by measuring what's going on directly. For example, what fraction of the assignments are falling under your "any other case" catch-all and therefore doing nothing?
Also, while we can't really tell from the information given, it doesn't seem any of your moves can do a "swap", which may be a problem. If a schedule is tightly constrained, then once you find something feasible, it's likely that you won't be able to just move a class from room A to room B, as room B will be in use. You'd need to consider ways of moving a class from A to B along with moving a class from B to A.
You can also sometimes improve things by allowing constraints to be violated. Instead of forbidding crossover from ever violating a constraint, you can allow it, but penalize the fitness in proportion to the "badness" of the violation.
Finally, it's possible that your other operators are the problem as well. If your selection and replacement operators are too aggressive, you can converge very quickly to something that's only slightly better than where you started. Once you converge, it's very difficult for mutations alone to kick you back out into a productive search.
I think there is nothing wrong with GA for this problem, some people just hate Genetic Algorithms no matter what.
Here is what I would check:
First you mention that your GA stabilizes at a random "High" fitness value, but isn't this a good thing? Does "high" fitness correspond to good or bad in your case? It is possible you are favoring "High" fitness in one part of your code and "Low" fitness in another thus causing the seemingly random result.
I think you want to be a bit more careful about the logic behind your crossover operation. Basically there are many situations for all 3 cases where making any of those choices would not cause an increase in fitness at all of the crossed-over individual, but you are still using a "resource" (an assignment that could potentially be used for another class/student/etc.) I realize that a GA traditionally will make assignments via crossover that cause worse behavior, but you are already performing a bit of computation in the crossover phase anyway, why not choose one that actually will improve fitness or maybe don't cross at all?
Optional Comment to Consider : Although your iterative construction approach is quite interesting, this may cause you to have an overly complex Gene representation that could be causing problems with your crossover. Is it possible to model a single individual solution as an array (or 2D array) of bits or integers? Even if the array turns out to be very long, it may be worth it use a more simple crossover procedure. I recommend Googling "ga gene representation time tabling" you may find an approach that you like more and can more easily scale to many individuals (100 is a rather small population size for a GA, but I understand you are still testing, also how many generations?).
One final note, I am not sure what language you are working in but if it is Java and you don't NEED to code the GA by hand I would recommend taking a look at ECJ. Maybe even if you have to code by hand, it could help you develop your representation or breeding pipeline.
Newcomers to GA can make any of a number of standard mistakes:
In general, when doing crossover, make sure that the child has some chance of inheriting that which made the parent or parents winner(s) in the first place. In other words, choose a genome representation where the "gene" fragments of the genome have meaningful mappings to the problem statement. A common mistake is to encode everything as a bitvector and then, in crossover, to split the bitvector at random places, splitting up the good thing the bitvector represented and thereby destroying the thing that made the individual float to the top as a good candidate. A vector of (limited) integers is likely to be a better choice, where integers can be replaced by mutation but not by crossover. Not preserving something (doesn't have to be 100%, but it has to be some aspect) of what made parents winners means you are essentially doing random search, which will perform no better than linear search.
In general, use much less mutation than you might think. Mutation is there mainly to keep some diversity in the population. If your initial population doesn't contain anything with a fractional advantage, then your population is too small for the problem at hand and a high mutation rate will, in general, not help.
In this specific case, your crossover function is too complicated. Do not ever put constraints aimed at keeping all solutions valid into the crossover. Instead the crossover function should be free to generate invalid solutions and it is the job of the goal function to somewhat (not totally) penalize the invalid solutions. If your GA works, then the final answers will not contain any invalid assignments, provided 100% valid assignments are at all possible. Insisting on validity in the crossover prevents valid solutions from taking shortcuts through invalid solutions to other and better valid solutions.
I would recommend anyone who thinks they have written a poorly performing GA to conduct the following test: Run the GA a few times, and note the number of generations it took to reach an acceptable result. Then replace the winner selection step and goal function (whatever you use - tournament, ranking, etc) with a random choice, and run it again. If you still converge roughly at the same speed as with the real evaluator/goal function then you didn't actually have a functioning GA. Many people who say GAs don't work have made some mistake in their code which means the GA converges as slowly as random search which is enough to turn anyone off from the technique.
I want to program a chess engine which learns to make good moves and win against other players. I've already coded a representation of the chess board and a function which outputs all possible moves. So I only need an evaluation function which says how good a given situation of the board is. Therefore, I would like to use an artificial neural network which should then evaluate a given position. The output should be a numerical value. The higher the value is, the better is the position for the white player.
My approach is to build a network of 385 neurons: There are six unique chess pieces and 64 fields on the board. So for every field we take 6 neurons (1 for every piece). If there is a white piece, the input value is 1. If there is a black piece, the value is -1. And if there is no piece of that sort on that field, the value is 0. In addition to that there should be 1 neuron for the player to move. If it is White's turn, the input value is 1 and if it's Black's turn, the value is -1.
I think that configuration of the neural network is quite good. But the main part is missing: How can I implement this neural network into a coding language (e.g. Delphi)? I think the weights for each neuron should be the same in the beginning. Depending on the result of a match, the weights should then be adjusted. But how? I think I should let 2 computer players (both using my engine) play against each other. If White wins, Black gets the feedback that its weights aren't good.
So it would be great if you could help me implementing the neural network into a coding language (best would be Delphi, otherwise pseudo-code). Thanks in advance!
In case somebody randomly finds this page. Given what we know now, what the OP proposes is almost certainly possible. In fact we managed to do it for a game with much larger state space - Go ( https://deepmind.com/research/case-studies/alphago-the-story-so-far ).
I don't see why you can't have a neural net for a static evaluator if you also do some classic mini-max lookahead with alpha-beta pruning. Lots of Chess engines use minimax with a braindead static evaluator that just adds up the pieces or something; it doesn't matter so much if you have enough levels of minimax. I don't know how much of an improvement the net would make but there's little to lose. Training it would be tricky though. I'd suggest using an engine that looks ahead many moves (and takes loads of CPU etc) to train the evaluator for an engine that looks ahead fewer moves. That way you end up with an engine that doesn't take as much CPU (hopefully).
Edit: I wrote the above in 2010, and now in 2020 Stockfish NNUE has done it. "The network is optimized and trained on the [classical Stockfish] evaluations of millions of positions at moderate search depth" and then used as a static evaluator, and in their initial tests they got an 80-elo improvement when using this static evaluator instead of their previous one (or, equivalently, the same elo with a little less CPU time). So yes it does work, and you don't even have to train the network at high search depth as I originally suggested: moderate search depth is enough, but the key is to use many millions of positions.
Been there, done that. Since there is no continuity in your problem (the value of a position is not closely related to an other position with only 1 change in the value of one input), there is very little chance a NN would work. And it never did in my experiments.
I would rather see a simulated annealing system with an ad-hoc heuristic (of which there are plenty out there) to evaluate the value of the position...
However, if you are set on using a NN, is is relatively easy to represent. A general NN is simply a graph, with each node being a neuron. Each neuron has a current activation value, and a transition formula to compute the next activation value, based on input values, i.e. activation values of all the nodes that have a link to it.
A more classical NN, that is with an input layer, an output layer, identical neurons for each layer, and no time-dependency, can thus be represented by an array of input nodes, an array of output nodes, and a linked graph of nodes connecting those. Each node possesses a current activation value, and a list of nodes it forwards to. Computing the output value is simply setting the activations of the input neurons to the input values, and iterating through each subsequent layer in turn, computing the activation values from the previous layer using the transition formula. When you have reached the last (output) layer, you have your result.
It is possible, but not trivial by any means.
https://erikbern.com/2014/11/29/deep-learning-for-chess/
To train his evaluation function, he utilized a lot of computing power to do so.
To summarize generally, you could go about it as follows. Your evaluation function is a feedforward NN. Let the matrix computations lead to a scalar output valuing how good the move is. The input vector for the network is the board state represented by all the pieces on the board so say white pawn is 1, white knight is 2... and empty space is 0. An example board state input vector is simply a sequence of 0-12's. This evaluation can be trained using grandmaster games (available at a fics database for example) for many games, minimizing loss between what the current parameters say is the highest valuation and what move the grandmasters made (which should have the highest valuation). This of course assumes that the grandmaster moves are correct and optimal.
What you need to train a ANN is either something like backpropagation learning or some form of a genetic algorithm. But chess is such an complex game that it is unlikly that a simple ANN will learn to play it - even more if the learning process is unsupervised.
Further, your question does not say anything about the number of layers. You want to use 385 input neurons to encode the current situation. But how do you want to decide what to do? On neuron per field? Highest excitation wins? But there is often more than one possible move.
Further you will need several hidden layers - the functions that can be represented with an input and an output layer without hidden layer are really limited.
So I do not want to prevent you from trying it, but chances for a successful implemenation and training within say one year or so a practically zero.
I tried to build and train an ANN to play Tic-tac-toe when I was 16 years or so ... and I failed. I would suggest to try such an simple game first.
The main problem I see here is one of training. You say you want your ANN to take the current board position and evaluate how good it is for a player. (I assume you will take every possible move for a player, apply it to the current board state, evaluate via the ANN and then take the one with the highest output - ie: hill climbing)
Your options as I see them are:
Develop some heuristic function to evaluate the board state and train the network off that. But that begs the question of why use an ANN at all, when you could just use your heuristic.
Use some statistical measure such as "How many games were won by white or black from this board configuration?", which would give you a fitness value between white or black. The difficulty with that is the amount of training data required for the size of your problem space.
With the second option you could always feed it board sequences from grandmaster games and hope there is enough coverage for the ANN to develop a solution.
Due to the complexity of the problem I'd want to throw the largest network (ie: lots of internal nodes) at it as I could without slowing down the training too much.
Your input algorithm is sound - all positions, all pieces, and both players are accounted for. You may need an input layer for every past state of the gameboard, so that past events are used as input again.
The output layer should (in some form) give the piece to move, and the location to move to.
Write a genetic algorithm using a connectome which contains all neuron weights and synapse strengths, and begin multiple separated gene pools with a large number of connectomes in each.
Make them play one another, keep the best handful, crossover and mutate the best connectomes to repopulate the pool.
Read blondie24 : http://www.amazon.co.uk/Blondie24-Playing-Kaufmann-Artificial-Intelligence/dp/1558607838.
It deals with checkers instead of chess but the principles are the same.
Came here to say what Silas said. Using a minimax algorithm, you can expect to be able to look ahead N moves. Using Alpha-beta pruning, you can expand that to theoretically 2*N moves, but more realistically 3*N/4 moves. Neural networks are really appropriate here.
Perhaps though a genetic algorithm could be used.