Clustering Algorithm for Paper Boys

Question

I need help selecting or creating a clustering algorithm according to certain criteria.

Imagine you are managing newspaper delivery persons.

• You have a set

Answer 1

Have you thought about using an economic/market based solution? Divide the set up by an arbitrary (but constant to avoid randomness effects) split into even subsets (as determined by the number of delivery persons).

Assign a cost function to each point by how much it adds to the graph, and give each extra point an economic value.

Iterate allowing each person in turn to auction their worst point, and give each person a maximum budget.

This probably matches fairly well how the delivery people would think in real life, as people will find swaps, or will say "my life would be so much easier if I didn't do this one or two. It is also pretty flexible (for example, would allow one point miles away from any others to be given a premium fairly easily).

Answer 2

I would approach it differently: Considering the street network as a graph, with an edge for each side of each street, find a partitioning of the graph into n segments, each no more than a given length, such that each paperboy can ride a single continuous path from the start to the end of their route. This way, you avoid giving people routes that require them to ride the same segments repeatedly (eg, when asked to cover both sides of a street without covering all the surrounding streets).

Answer 3

Perhaps a minimum spanning tree of the customers, broken into set based on locality to the paper boy. Prims or Kruskal to get the MST with the distance between houses for the weight.

Answer 4

This is a very quick and dirty method of discovering where your "clusters" lie. This was inspired by the game "Minesweeper."

Divide your entire delivery space up into a grid of squares. Note - it will take some tweaking of the size of the grid before this will work nicely. My intuition tells me that a square size roughly the size of a physical neighbourhood block will be a good starting point.

Loop through each square and store the number of delivery locations (houses) within each block. Use a second loop (or some clever method on the first pass) to store the number of delivery points for each neighbouring block.

Now you can operate on this grid in a similar way to photo manipulation software. You can detect the edges of clusters by finding blocks where some neighbouring blocks have no delivery points in them.

Finally you need a system that combines number of deliveries made as well as total distance travelled to create and assign routes. There may be some isolated clusters with just a few deliveries to be made, and one or two super clusters with many homes very close to each other, requiring multiple delivery people in the same cluster. Every home must be visited, so that is your first constraint.

Derive a maximum allowable distance to be travelled by any one delivery person on a single run. Next do the same for the number of deliveries made per person.

The first ever run of the routing algorithm would assign a single delivery person, send them to any random cluster with not all deliveries completed, let them deliver until they hit their delivery limit or they have delivered to all the homes in the cluster. If they have hit the delivery limit, end the route by sending them back to home base. If they could safely travel to the nearest cluster and then home without hitting their max travel distance, do so and repeat as above.

Once the route is finished for the current delivery person, check if there are homes that have not yet had a delivery. If so, assign another delivery person, and repeat the above algorithm.

This will generate initial routes. I would store all the info - the location and dimensions of each square, the number of homes within a square and all of its direct neighbours, the cluster to which each square belongs, the delivery people and their routes - I would store all of these in a database.

I'll leave the recalc procedure up to you - but having all the current routes, clusters, etc in a database will enable you to keep all historic routes, and also try various scenarios to see how to best to adapt to changes creating the least possible changes to existing routes.

Answer 5

This is a classic example of a problem that deserves an optimized solution rather than trying to solve for "The OPTIMUM". It's similar in some ways to the "Travelling Salesman Problem", but you also need to segment the locations during the optimization.

I've used three different optimization algorithms to good effect on problems like this:

1. Simulated Annealing
2. Great Deluge Algorithm
3. Genetic Algoritms

Using an optimization algorithm, I think you've described the following "goals":

1. The geographic area for each paper boy should be minimized.
2. The number of subscribers served by each should be approximately equal.
3. The distance travelled by each should be about equal.
4. (And one you didn't state, but might matter) The route should end where it began.

Hope this gets you started!

* Edit *

If you don't care about the routes themselves, that eliminates goals 3 and 4 above, and perhaps allows the problem to be more tailored to your bonus requirements.

If you take demographic information into account (such as population density, subscription adoption rate and subscription cancellation rate) you could probably use the optimization techniques above to eliminate the need to rerun the algorithm at all as subscribers adopted or dropped your service. Once the clusters were optimized, they would stay in balance because the rates of each for an individual cluster matched the rates for the other clusters.

The only time you'd have to rerun the algorithm was when and external factor (such as a recession/depression) caused changes in the behavior of a demographic group.