Comprehensive Operations: The fun stuff

Objectives of Lecture:

1. Location-Allocation, the most comprehensive operations
2. A warning about even harder problems to solve
3. Connection between GIS and statistical analysis

Chapter 8: in two halves

• Incremental operations, presented last time, produce a more global result by applying more local operations, tacked on to each other incrementally.
• These are the tools of 'spatial analysis' that some have said are missing from the traditional GIS toolkit. There is a lot of interest in bridging this gap.

Truly comprehensive results come from a further set of operations:

Location-Allocation:

A Family of Problems; a common approach

Some problems involve a global measure or constraint, phrased as a minimum overall cost or as a measure of equality. Many seemingly different problems can be reformulated into a small number of basic techniques.

The simple statement involves:

• A collection of fixed demand, such a consumers who might buy a product or need a certain administrative service
• A certain number of suppliers must be located to serve the demand.

For example: Area Education Authority Offices serve the School Districts of Iowa...

NOTE: This means a RELATIONSHIP, a higher-order of information than a simple overlay or indeed most of the prior operations...

The problem can be constrained in a number of ways (after (Rushton 1979, p.33)):

1. Minimize the total distance between demand and supply
2. Minimize the maximum distance to the closest facility
3. Equalize the number assigned to each facility
4. Ensure that the number assigned is greater than a threshold
5. Ensure that the number assigned does not exceed capacity
6. Minimize the total distance subject to an upper limit on distance (combines 1&2)
7. Set the minimum number of facilities such that the maximum distance is less than some upper limit
8. Route all traffic from origin to dest. along the least cost path subject to the capacity constraints and the interaction between capacity and cost

Solving a location allocation problem

• Warehousing Problem (1, 2, 6) - use iterative applications of incremental network operations, see last lecture.
• Transportation Problem - deals with the rest, using linear optiomization techniques

Additional complexity involves finding the best number of 'suppliers' (rather than assuming that a certain number must exist.

Warning: NP Complete

These comprehensive solutions seem to provide the 'best' location, but in every case, they must use a 'heuristic', not a global optimum. These problems skirt very close to the edge of the BIG problems for computation. (NP-Complete).

For example:

• The Traveling Salesperson Problem requires the shortest path to visit a set of 'cities' over some network without visiting any city twice.
• The Knapsack Problem requires the best fit of a set of 'parcels' (integers) into a set of 'knapsacks' (containers, or regions). [Participatory experiment...]

These problems belong to a group of graph problems that may not ever be solved in 'polynomial' time (hence they are Non-Polynomial or NP). The reason is that you cannot find an iterative solution that allows you to say that the salesman MUST follow this particular path or that this particular parcel fits in that particular knapsack until everything has been fit into place. Essentially, these problems require examining all the possible combinations, a set of numbers that rise very very fast. [all the possible combinations of 59 items would require a number as large as all the baryons in the universe...]

NP problems can be approximated rather closely using iterative heuristics, just without a guarantee that the solution is optimal.

Statistical Summaries:

Summarize the whole distribution

A statistical model set out a framework of relationships that are then confirmed or rejected by some measure of 'goodness of fit'. To discover these relationships, a series of axioms are implied (homogeneity of 'population', source of error, etc.)

The estimation for the model is based on a model of error, usually based on the error in sampling from a 'population'. Yet, most spatial data is an exhaustive partitioning of a region (as with census tracts, counties, etc.). The connections between these two assumptions are not as direct as some hope... The calculations are totally global: means and deviations are abstracted from their neighborhoods totally.

Data model for statistics: a 'case' as a replicate in an experiment; matrix of cases by variables (the old 'geographical matrix' of Berry) : THE SPACE PART CAN BE LOST...

Example: Gold deposits for Nova Scotia [overheads]: pixels or watersheds are not indivisible 'cases'

Resources about statistics and GIS