A Spatial Decision Support System (SDSS) for Electoral Districting

Components of the SDSS

The following components were used to compute the constituency boundaries in the Kenya and South African examples:

• A location-allocation model to represent the problem.
  
  
• Data items to implement the model.
  
  
• An objective function to define the user’s objectives.
  
  
• An algorithm to solve the problem.
  
  
• Special features to find non-inferior solutions.
  
  
• A GIS to map solutions.

The p-median location-allocation model

• The SDSS uses a location-allocation model to make these maps
  
  
• The classic location-allocation model finds regions for which the sum of the populations of the enumeration areas times the distance to their respective centers is least.
  
  
• These electoral districts are maximally compact regions adjusted for the different population weights.  In the p-median model, the sum of person distances to their centers is least and the function Z is minimized.

where:

• w_i = population of the ith enumeration area
• d_ij = distance from ith enumeration area to jth regional center
• x_ij  =  { 1 for the closest regional center (j) and 0 for all others}.  This definition ensures that each enumeration area belongs to one, and only one, electoral district and that every enumeration area is assigned to an electoral district. It also ensures that electoral districts are comprised of contiguous territories.
• the total number of electoral districts to be generated by the model is denoted by p.

The model is solved by selecting p districts, defined as nodal centers and their assigned enumeration areas.  The model is solved by an algorithm that minimizes the function Z.

Data items to implement location-allocation models

Four data items for each enumeration area :

• area (km²)
• shape
• population
• distances (km) to the centers of all other enumeration areas

The Objective Function

• The objective to be optimized in generating regions
  
• Here, compactness and equality in the number of voters. 
  
• This is a multi-objective function.
  
• The two objectives are “traded-off” in multiple runs of the SDSS.
  
• Spatial constraints can be introduced:
  • Electoral districts can be prevented from crossing provincial boundaries.
  • Opposite sides of a lake can be prevented from being in the same electoral districts.