What would be a good heuristic to use to solve the following challenge?
Quality Blimps Inc. is looking to expand their sales to other cities
(N), so they hired you as a salesman to fly to other cities to sell
blimps. Blimps can be expensive to travel with, so you will need to
determine how many blimps to take along with you on each trip and when
to return to headquarters to get more. Quality Blimps has an unlimited
supply of blimps.
You will be able to sell only one blimp in each city you visit, but
you do not need to visit every city, since some have expensive travel
costs. Each city has an initial price that blimps sell for, but this
goes down by a certain percentage as more blimps are sold (and the
novelty wears off). Find a good route that will maximize profits.
https://www.hackerrank.com/codesprint4/challenges/tbsp
This challenge is similar to the standard Travelling Salesman Problem, but with some extra twists: The salesman needs to track both his own travel costs and the blimps'. Each city has different prices which blimps sell for, but these prices go down over his journey. What would be a fast algorithm (i.e. n log n ) to use to maximize profit?
The prices of transporting the items in a way makes the TSP simpler. If the salesman is in city A and wants to go to B, he can compare The costs of going directly to B vs. costs of going back to Headquarters first to pick up more blimps. I.e. is it cheaper to take an extra blimp to B via A or to go back in-between. This check will create a series of looped trips, which the salesman could then go through in order of highest revenue. But what would be a good way to determine these loops in the first place?
This is a search problem. Assuming the network is larger than can be solved by brute force, the best algorithm would be Monte Carlo Tree Search.
Algorithms for such problems are usually of the "run the solution lots of times, choose the best one" kind. Also, to choose what solution to try next, use results of previous iterations.
As for a specific algorithm, try backtracking with pruning, simulated annealing, tabu search, genetic algorithms, neural networks (in order of what I find relevant). Also, the monte carlo tree search idea proposed by Tyler Durden looks pretty cool.
I have read the original problem. Since the number of cities is large, it is impossible to get the exact answer. Approximation algorithm is the only choice. As #maniek mentioned, there are many choices of AA. If you have experience with AA before, that would be best, you can choose one the those which you are familar with and get a approximate answer. However, if you didn't do AA before, maybe you can start with backtracking with pruning.
As for this problem, you can easily get these pruning rules:
Bring as few blimps as possible. That means when you start from HQ and visit city A,B,C..., and then return to HQ(you can regard it as a single round), the number of blimps you brought is the same as the cities you will visit.
With the sale price gets lower, when it becomes lower than the travel expense, the city will never be visited. That gives the ending of backtracking method.
You can even apply KNN first to cluster several cities which located nearby. Then start from HQ and visit each group.
To sum up, this is indeed an open problem. There is no best answer. Maybe in case 1, using backtracking gives the best answer while in case 2, simulated annealing is the best. Just calculate the approximate answer is enough and there are so many ways to achive this goal. All in all, the really best approach is to write as many AAs as you can, and then compare these results and output the best one.
This looks like a classic optimization problem, which I know can be handled with the simulated annealing algorithm (having worked on what I think was the first commercial use of simulated annealing, the Wintek electronic CAD autoplacement program back in the 1980's). Most optimization algorithms can handle problems of many variables like yours -- the question is setting up the fitness algorithm correctly so you get the optimum solution (in your case, lowest cost).
Other optimization algorithms may be worth a look -- I just have experience in implementing the simulated annealing algorithm.
(If you go the simulated annealing route, you might want to get "Simulated Annealing: Theory and Applications (Mathematics and Its Applications" by P.J. van Laarhoven and E.H. Aarts, but you will have to hunt it down since it is out of print (it might even be the book I used back in the 1980's).)
Related
Problem: I need to drop (n) employees from office to their homes(co-ordinates available). I have (x) 7-seater & (y) 4-seater cabs available.
I have to design an algorithm to drop all the employees to their homes while travelling minimum distance.
Also, the algorithm must tell me how many 7-seater or/and 4-seater vehicles I must choose so as to travel minimum distance.
eg. If I have 15 employees then the algorithm may tell me to use 1 (7-seater) cab & 2 (4-seater) cab & have the employees in each cab as following:
[(E2, E4, E6, E8), (E1, E3, E5, E7, E9, E10, E12), (E11, E13, E14, E15)]
Approach: I'm thinking of this as a Travelling Salesman Problem with multiple salesmen with an upper limit on number of cities each can travel. Also salesmen do not need to come back to the origin. Ant's colony problem came to my mind, but I can't really choose wisely which algorithm to choose
Requirement: I really need the ALGORITHM. Either TSP or Ant's colony, doesn't matter. I'll welcome opinions, but I really need the ALGORITHM.
This is a cost minimization problem, not a travelling salesman problem. It is related to TSP in the sense that TSP is a very specific cost minimization problem.
The solution consists of three steps:
Generate a list of employee drop-off points (nodes)
Create distinct paths that do not intersect, nor branch. These will be your routes and help prevent wasteful route overlaps. Use cost(path) = distance(furthest node and origin) + taxi_cost(nodes) + sum(distance between nodes) to compare paths and/or brute-force all potential networks. Networks are layouts of paths. DO NOT BRANCH THE PATHS!!
Total distance is a line of defense against waste ensuring that routes are not too long.
Sum of distances helps the algorithm converge on neighbourhoods where many employees live (when possible).
Because this variation of the coin problem allows imperfect solutions, it reduces to a variant of the Knapsack Problem. The utility of each taxi is capacity. If you also wish to choose the cheapest way to transport your employees, utility(taxi) = capacity/cost. From this our simplest solution is to be greedy; who cares about empty space? If you really care about filling up taxis perfectly (as opposed to cost efficiently), you'll need a much more complex solution. You only specify the least distance as your metric (with each additional taxi multiplying cost). I assume this is a proxy to say 'I don't want to pay too much'.
Therefore: taxi_cost(nodes) = math.floor(amount(nodes)/max(utility(taxis)+1). This equation selects the cheapest, roomiest taxi, and figures out how many of them are required to fully service the route.
Be sure to calculate the cost of each network you examine as sum(cost(path))
Once you've found the cheapest network to service, for each path in the chosen network:
make a list of employees travelling to the furthest node
fill the preferred taxi with those employees
repeat with the next furthest node until you have a full taxi, then add the filled taxi to the list. If you run out of employees, you've finished assigning taxis to the route. (The benefit of furthest-first selection is that you can ask employees in unfilled taxis to walk if that part of the route is within blocks of the office).
The algorithm above is not perfect, but it will have many desirable tendencies.
routes will be as short as possible and cover the greatest possible area (by not looping or branching)
routes will tend to service neighbourhoods, rather than trying to overlap responsibilities. This part of the algorithm isn't optimal, but is effective. This makes it really easy to remove service routes without needing to recalculate the transportation network.
the taxis chosen will be cost-efficient, helping to avoid paying more than necessary.
routes will use as few taxis as possible, taking into account the relative cost of upgrading to roomier ones with higher capacity
because the taxis travelling furthest will be full, it has less of an impact on your employee's ability to get to work if you decide to cancel service to emptier taxis.
Every step closer to perfection costs you many times more than the previous step, so diminished returns are acceptable if the solution provide desirable features. Although the algorithm makes some potentially sub-optimal tradeoffs, they come with huge value; your network of taxi routes becomes much easier to modify.
If you'd like to make an optimal solution, the Knapsack Problem, Coin Problem, and Change-making Problem help determine the cost of taxis and routes.
Spanning Trees are the most effective way to determine routes. Center the spanning tree at the office and calculate the cost of each branch as the maximum distance from the office. Try to keep each branch servicing areas with high density to make it easier to add and remove taxi routes.
Studying pathfinding can help you learn how to determine good cost functions so that you can numerically compare different potential paths. Remember that your network consists of a set of paths, but will require its own cost function so that you can compare different layouts.
I've written an in-depth guide to pathfinding for this answer. Pathfinding articles are few and just don't go into enough depth for a lot of problem spaces. A good cost function can get you a nearly perfect solution if you have multiple priorities. Unfortunately, good cost functions are domain specific so you will need to identify them yourself. Feel free to message me if you aren't sure how to make a path with certain traits and I'll help you figure out a good cost function.
It's a constraint satisfaction problem, not really a TSP. If you're looking for a project that may be able to help you, you could look into cspdb, which is what I wrote some time ago:
https://github.com/gtoonstra/cspdb
You'd be using a database in the backend that maintains the state and write a couple of scripts in it's own grammar that manipulates that state. A couple of examples are included to solve nqueens and classroom scheduling with multiple constraints.
From a list d destinations you can make the array of pairwise travel costs c. where c[a,b] is the travel cost from a to b...
Now you have a start point p. add to the array c2 values for p to each point in d.
So you now have the concept of groups.
You can look at this as a greedy algorithm. Given the list c2 you can take the cheapest option given your state.
Your state is the vector all you cab vectors (the costs of getting from where ever they are to where ever they could go next) .* the assignment vector where k == 0 for k in the. You find the minimum option given your state (considering adding another onerous to a cab of 4 costs the difference between the 4 person cab and the 7 person cab and adding a person to a zero person cab or adding a new cab also has a cost. Once all your people have been assigned to their cabs you have an answer.
The Idea of a greedy algorithm is most often characterized by the backpack problem but it can also be implemented for statistical methods such as feature selection.
Like #Aaron3468 this approach is not perfect and does not guarantee the best solution.
If you want the best possible solutions you can iterate through all the combinations but this becomes impractical quickly.
From my point of view your algorithm should solve 2 problems: the number of cars of each type and the shortest distance (how you number your employees depends on you or you should give more details). Sorry I'm a using a phone and I don't have have all features of the site.
For the number of cars you can use below algorithm. To solve issues related to distances, you should give more info about the paths and their lengths. A graph algorithm may then be combined with this to do the trick. Here 11=7+4.
Begin
Integer temp:= n/11
Integer rem:= n mod 11
If rem=0
x:=temp
y:=temp
Else if rem<=4
x:=temp
y:=temp+1
Else if rem<=7
x:=temp+1
y:=temp
Else
x:=temp+1
y:=temp+1
Endif
End
I'd like to set up a system that crowd sources the best 10 items from a set that can vary from 20-2000 items (the ranking amoung the top ten is not important). There is an excellent stackoverflow post on algorithms for doing the actual sort at
How to rank a million images with a crowdsourced sort. I am leaning toward asking users which they like best between two items and then using the TrueSkill algorithm.
My question is given I am using something like TrueSkill, what is the best algorithm for deciding which pairs of items to show a user to rate? I will have a limited number of opportunities to ask people which items they like best so it is important that the pairs presented will give the system the most valuable information in identifying the top 10. Again, I am mostly interested in finding the top ten, less so how the rest of the items rank amongst themselves or even how the top ten rank amongst themselves.
This problem is very similar to organizing a knock-out tournament where skills of the players are not well known and number of players is very high (think school level tennis tournaments). Since round robin ( O(n^2) matches) is very expensive, but a simple knock-out tournament is too simplistic, the usual option is to go with k-elimination structure. Essentially, every player (in your context a item) is knocked out of contention after losing k games. Take a look at the double elimination structure: http://en.wikipedia.org/wiki/Double-elimination_tournament .
Perhaps you can modify it sufficiently to meet your needs.
Another well known algorithm for this was produced to calculate rankings in Go or Chess tournaments. You can have a look at the MacMahon Algorithms which calculate such pairings and the ranks at the same time. It should be possible to truncate this algorithm, so that it will only produce a set of 10 best Items.
You can find more details in Christian Gerlach's thesis, where he describes the actual optimization algorithm (unfortunately the thesis is in German).
What are concrete examples (e.g. Alpha-beta pruning, example:tic-tac-toe and how is it applicable there) of heuristics. I already saw an answered question about what heuristics is but I still don't get the thing where it uses estimation. Can you give me a concrete example of a heuristic and how it works?
Warnsdorff's rule is an heuristic, but the A* search algorithm isn't. It is, as its name implies, a search algorithm, which is not problem-dependent. The heuristic is. An example: you can use the A* (if correctly implemented) to solve the Fifteen puzzle and to find the shortest way out of a maze, but the heuristics used will be different. With the Fifteen puzzle your heuristic could be how many tiles are out of place: the number of moves needed to solve the puzzle will always be greater or equal to the heuristic.
To get out of the maze you could use the Manhattan Distance to a point you know is outside of the maze as your heuristic. Manhattan Distance is widely used in game-like problems as it is the number of "steps" in horizontal and in vertical needed to get to the goal.
Manhattan distance = abs(x2-x1) + abs(y2-y1)
It's easy to see that in the best case (there are no walls) that will be the exact distance to the goal, in the rest you will need more. This is important: your heuristic must be optimistic (admissible heuristic) so that your search algorithm is optimal. It must also be consistent. However, in some applications (such as games with very big maps) you use non-admissible heuristics because a suboptimal solution suffices.
A heuristic is just an approximation to the real cost, (always lower than the real cost if admissible). The better the approximation, the fewer states the search algorithm will have to explore. But better approximations usually mean more computing time, so you have to find a compromise solution.
Most demonstrative is the usage of heuristics in informed search algorithms, such as A-Star. For realistic problems you usually have large search space, making it infeasible to check every single part of it. To avoid this, i.e. to try the most promising parts of the search space first, you use a heuristic. A heuristic gives you an estimate of how good the available subsequent search steps are. You will choose the most promising next step, i.e. best-first. For example if you'd like to search the path between two cities (i.e. vertices, connected by a set of roads, i.e. edges, that form a graph) you may want to choose the straight-line distance to the goal as a heuristic to determine which city to visit first (and see if it's the target city).
Heuristics should have similar properties as metrics for the search space and they usually should be optimistic, but that's another story. The problem of providing a heuristic that works out to be effective and that is side-effect free is yet another problem...
For an application of different heuristics being used to find the path through a given maze also have a look at this answer.
Your question interests me as I've heard about heuristics too during my studies but never saw an application for it, I googled a bit and found this : http://www.predictia.es/blog/aco-search
This code simulate an "ant colony optimization" algorithm to search trough a website.
The "ants" are workers which will search through the site, some will search randomly, some others will follow the "best path" determined by the previous ones.
A concrete example: I've been doing a solver for the game JT's Block, which is roughly equivalent to the Same Game. The algorithm performs a breadth-first search on all possible hits, store the values, and performs to the next ply. Problem is the number of possible hits quickly grows out of control (10e30 estimated positions per game), so I need to prune the list of positions at each turn and only take the "best" of them.
Now, the definition of the "best" positions is quite fuzzy: they are the positions that are expected to lead to the best final scores, but nothing is sure. And here comes the heuristics. I've tried a few of them:
sort positions by score obtained so far
increase score by best score obtained with a x-depth search
increase score based on a complex formula using the number of tiles, their color and their proximity
improve the last heuristic by tweaking its parameters and seeing how they perform
etc...
The last of these heuristic could have lead to an ant-march optimization: there's half a dozen parameters that can be tweaked from 0 to 1, and an optimizer could find the optimal combination of these. For the moment I've just manually improved some of them.
The second of this heuristics is interesting: it could lead to the optimal score through a full depth-first search, but such a goal is impossible of course because it would take too much time. In general, increasing X leads to a better heuristic, but increases the computing time a lot.
So here it is, some examples of heuristics. Anything can be an heuristic as long as it helps your algorithm perform better, and it's what makes them so hard to grasp: they're not deterministic. Another point with heuristics: they're supposed to lead to quick and dirty results of the real stuff, so there's a trade-of between their execution time and their accuracy.
A couple of concrete examples: for solving the Knight's Tour problem, one can use Warnsdorff's rule - an heuristic. Or for solving the Fifteen puzzle, a possible heuristic is the A* search algorithm.
The original question asked for concrete examples for heuristics.
Some of these concrete examples were already given. Another one would be the number of misplaced tiles in the 15-puzzle or its improvement, the Manhattan distance, based on the misplaced tiles.
One of the previous answers also claimed that heuristics are always problem-dependent, whereas algorithms are problem-independent. While there are, of course, also problem-dependent algorithms (for instance, for every problem you can just give an algorithm that immediately solves that very problem, e.g. the optimal strategy for any tower-of-hanoi problem is known) there are also problem-independent heuristics!
Consequently, there are also different kinds of problem-independent heuristics. Thus, in a certain way, every such heuristic can be regarded a concrete heuristic example while not being tailored to a specific problem like 15-puzzle. (Examples for problem-independent heuristics taken from planning are the FF heuristic or the Add heuristic.)
These problem-independent heuristics base on a general description language and then they perform a problem relaxation. That is, the problem relaxation only bases on the syntax (and, of course, its underlying semantics) of the problem description without "knowing" what it represents. If you are interested in this, you should get familiar with "planning" and, more specifically, with "planning as heuristic search". I also want to mention that these heuristics, while being problem-independent, are dependent on the problem description language, of course. (E.g., my before-mentioned heuristics are specific to "planning problems" and even for planning there are various different sub problem classes with differing kinds of heuristics.)
I have a problem that has been effectively reduced to a Travelling Salesman Problem with multiple salesmen. I have a list of cities to visit from an initial location, and have to visit all cities with a limited number of salesmen.
I am trying to come up with a heuristic and was wondering if anyone could give a hand. For example, if I have 20 cities with 2 salesmen, the approach that I thought of taking is a 2 step approach. First, divide the 20 cities up randomly into 10 cities for 2 salesman each, and I'd find the tour for each as if it were independent for a few iterations. Afterwards, I'd like to either swap or assign a city to another salesman and find the tour. Effectively, it'd be a TSP and then minimum makespan problem. The problem with this is that it'd be too slow and good neighborhood generation of swapping or assigning a city is hard.
Can anyone give an advise on how I could improve the above?
EDIT:
The geo-location for each city are known, and the salesmen start and end at the same places. The goal is to minimize the max travelling time, making this sort of a minimum makespan problem. So for example, if salesman1 takes 10 hours and salesman2 takes 20 hours, the maximum travelling time would be 20 hours.
TSP is a difficult problem. Multi-TSP is probably much worse. I'm not sure you can find good solutions with ad-hoc methods like this. Have you tried meta-heuristic methods ? I'd try using the Cross Entropy method first : it shouldn't be too hard to use it for your problem. Otherwise look for Generic Algorithms, Ant Colony Optimization, Simulated Annealing ...
See "A Tutorial on the Cross-Entropy Method" from Boer et al. They explain how to use the CE method on the TSP. A simple adaptation for your problem might be to define a different matrix for each salesman.
You might want to assume that you only want to find the optimal partition of cities between the salesmen (and delegate the shortest tour for each salesman to a classic TSP implementation). In this case, in the Cross Entropy setting, you consider a probability for each city Xi to be in the tour of salesman A : P(Xi in A) = pi. And you work, on the space of p = (p1, ... pn). (I'm not sure it will work very well, because you will have to solve many TSP problems.)
When you start talking about multiple salesmen I start thinking about particle swarm optimization. I've found a lot of success with this using a gravitational search algorithm. Here's a (lengthy) paper I found covering the topic. http://eprints.utm.my/11060/1/AmirAtapourAbarghoueiMFSKSM2010.pdf
Why don't you convert your multiple TSP into the traditional TSP?
This is a well-known problem (transforming multiple salesman TSP into TSP) and you can find several articles on it.
For most transformations, you basically copy your depot (where the salesmen start and finish) into several depots (in your case 2), make the edge weights in a way to force a TSP to come back to the depot twice, and then remove the two depots and turn them into one.
Voila! got two min cost tours that cover the vertices exactly once.
I wouldn't start writing an algorythm for such complicated issue (unless that's my day job - to write optimization algorythms). Why don't you turn to a generic solution like http://www.optaplanner.org/ ? You have to define your problem and the program uses algorithms that top developers took years to create and optimize.
My first thought on reading the problem description would be to use a standard approach for the salesman problem (googling for an appropriate one as I've never actually had to write code for it); Then take the result and split it in half. Your algorithm then could be to decide where "half" is -- maybe it is half of the cities, or maybe it is based on distance, or maybe some combination. Or search the result for the largest distance between two cities and choose that as the separation between salesman #1's last city and salesman #2's first city. Of course it does not limit to two salesman, you would break into x pieces; But overall the idea is that your standard 1 salesman TSP solution should have already gotten the "nearby" cities next to each other in the travel graph, so you don't have to come up with a separate grouping algorithm...
Anyway, I'm sure there are better solutions, but this seems like a good first approach to me.
Have a look at this question (562904) - while not identical to yours there should be some good food for thought and references for further study.
As mentioned in the answer above the hierarchical clustering solution will work very well for your problem. Instead of continuing to dissolve clusters until you have a single path, however, stop when you have n, where n is the number of salesmen you have. You can probably improve it some by adding in some "fake" stops to improve the likelihood that your clusters end up evenly spaced from the initial destination if the initial clusters are too disparate. It's not optimal - but you're not going to get an optimal solution for a problem like this. I'd create an app that visualizes the problem and then test many variants of the solution to get a feel for whether your heuristic is optimal enough.
In any case I would not randomize the clusters, that would cause the majority of the clusters to be sub-optimal.
just by starting to read your question using genetic algorithm came to my head. just use two genetic algorithm in the same time one can solve how to assign cities to salesmen and the other can solve the TSP for each salesman you have.
Given a list of cities and the cost to fly between each city, I am trying to find the cheapest itinerary that visits all of these cities. I am currently using a MATLAB solution to find the cheapest route, but I'd now like to modify the algorithm to allow the following:
repeat nodes - repeat nodes should be allowed, since travelling via hub cities can often result in a cheaper route
dynamic edge weights - return/round-trip flights have a different (usually lower) cost to two equivalent one-way flights
For now, I am ignoring the issue of flight dates and assuming that it is possible to travel from any city to any other city.
Does anyone have any ideas how to solve this problem? My first idea was to use an evolutionary optimisation method like GA or ACO to solve point 2, and simply adjust the edge weights when evaluating the objective function based on whether the itinerary contains return/round-trip flights, but perhaps somebody else has a better idea.
(Note: I am using MATLAB, but I am not specifically looking for coded solutions, more just high-level ideas about what algorithms can be used.)
Edit - after thinking about this some more, allowing "repeat nodes" seems to be too loose of a constraint. We could further constrain the problem so that, although nodes can be repeatedly visited, each directed edge can only be visited at most once. It seems reasonable to ignore any itineraries which include the same flight in the same direction more than once.
I haven't tested it myself; however, I have read that implementing Simulated Annealing to solve the TSP (or variants of it) can produce excellent results. The key point here is that Simulated Annealing is very easy to implement and requires minimal tweaking, while approximation algorithms can take much longer to implement and are probably more error prone. Skiena also has a page dedicated to specific TSP solvers.
If you want the cost of the solution produced by the algorithm is within 3/2 of the optimum then you want the Christofides algorithm. ACO and GA don't have a guaranteed cost.
Solving the TSP is a NP-hard problem for its subcycles elimination constraints, if you remove any of them (for your hub cities) you just make the problem easier.
But watch out: TSP has similarities with association problem in the meaning that you could obtain non-valid itineraries like:
Cities: New York, Boston, Dallas, Toronto
Path:
Boston - New York
New York - Boston
Dallas - Toronto
Toronto - Dallas
which is clearly wrong since we don't go across all cities.
The subcycle elimination constraints serve just to this purpose. Including a 'hub city' sounds like you need to add weights to the point and make an hybrid between flux problems and tsp problems. Sounds pretty hard but the first try may be: eliminate the subcycles constraints relative to your hub cities (and leave all the others). You can then link the subcycles obtained for the hub cities together.
Good luck
Firstly, what is approximate number of cities in your problem set? (Up to 100? More than 100?)
I have a fair bit of experience with GA (not ACO), and like epitaph says, it has a bit of gambling aspect. For some input, it might stop at a brutally inefficient solution. So, what I have done in the past is to use GA as the first option, compare the answer to some lower bound, and if that seems to be "way off", then run a second (usually a less efficient) algorithm.
Of course, I used plenty of terms that were not standard, so let us make sure that we agree what they would be in this context:
lower bound - of course, in this case, MST would be a lower bound.
"Way Off" - If triangle inequality holds, then an upper bound is UB = 2 * MST. A good "way off" in this context would be 2 * UB.
Second algorithm - In this case, both a linear programming based approach and Christofides would be good choices.
If you limit the problem to round-trips (i.e. the salesman can only buy round-trip tickets), then it can be represented by an undirected graph, and the problem boils down to finding the minimum spanning tree, which can be done efficiently.
In the general case I don't know of a clever way to use efficient algorithms; GA or similar might be a good way to go.
Do you want a near-optimal solution, or do you want the optimal solution?
For the optimal solution, there's still good ol' brute force. Due to requirement 1 involving repeat nodes, you'll have to make sure you search breadth-first, not dept-first. Otherwise you can end up in an infinite loop. You can slowly drop all routes that exceed your current minimum until all routes are exhausted and the minimal route is discovered.