Blog GIS & Territories

Introduction

In spatial analysis, criteria are often applied strictly: yes/no, inside/outside, above/below a threshold.

But reality is rarely so clear-cut: is a municipality with 4,900 inhabitants really so different from another with 5,100?

Fuzzy logic allows these nuances to be taken into account and produces more flexible results that are better suited to territorial decisions.

Data set and criteria

To illustrate the difference, we cross-reference three layers of information:

• Municipalities: population threshold set at 5,000 inhabitants.
• Housing by IRIS: threshold set at 1,500 dwellings.
• PLU zoning: agricultural areas coded A.

Boolean cross-referencing (traditional approach)

• Municipalities with more than 5,000 inhabitants → 1 (green), otherwise 0 (red).

• IRIS with more than 1,500 dwellings → 1 (green), otherwise 0 (red).

• PLU Zone A → 1 (green), otherwise 0 (red).

The final cross-reference (intersection of the three criteria) is a binary selection: entities that meet all criteria are valued at 1 (green), all others are valued at 0 (red).

Fuzzy transition: definitions of membership functions

Rather than a clear threshold, each criterion is transformed into a fuzzy function (degrees of membership between 0 and 1).

• Municipal population:

  • < 2,000 inhabitants → membership 0
  • between 2,000 and 5,000 → gradual transition
  • 5,000 inhabitants → membership 1

• Housing by IRIS:

  • < 800 → membership 0
  • between 800 and 1,500 → gradual transition
  • 1,500 → membership 1

• PLU zoning:

  • Zone A → 1
  • Zone AU / As → intermediate values (e.g., 0.5)
  • Other zones → 0

Fuzzy aggregations: three strategies

Once each criterion has been converted into a fuzzy value, they can be combined in different ways:

• Average aggregation (balanced compromise)

  • Result = average of the 3 fuzzy criteria.
  • Reflects an overall balance between conditions.

1. Optimistic aggregation (opportunity logic)

   • Result = maximum value of the criteria.
   • If a criterion is favorable, the site is considered potentially suitable.

1. Pessimistic aggregation (precautionary logic)

   • Result = minimum value of the criteria.
   • If a criterion is unfavorable, the site is rejected.

Comparison of results

• The Boolean approach gives a very strict selection, which is often limited.
• The fuzzy approach highlights:

  • Intermediate areas (neither totally suitable nor totally excluded).
  • Sensitivity to the choice of aggregation (optimistic vs pessimistic).
  • A useful scale for decision-making (prioritization, trade-offs, consultation).

With the FuzzyAttributes plugin, this process is carried out directly in QGIS, without the need for manual calculations or external scripts. You can transform your criteria into fuzzy values using the built-in membership functions (linear, trapezoidal, sigmoid, etc.), then test different aggregation modes (average, optimistic, pessimistic). The advantage is that you can quickly compare a strict (Boolean) intersection with several fuzzy scenarios, in order to better reflect the reality on the ground and explore the sensitivity of the results to the chosen thresholds.

Conclusion

Fuzzy logic in a GIS does not replace thresholds but allows for a better representation of the complexity of the terrain.

It offers a more flexible view, capable of distinguishing potential, significant constraints, and areas for discussion.

It is a valuable tool for the environment, land use planning, and risk management.

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Atilio Francois