Fuzzy Analytic Hierarchy Process
A multi-criteria decision making (MCDM) framework for evaluating soil health using eight key indicators with expert-derived fuzzy weights.
Overview
The Soil Health Diagnostic System (SHDS) employs the Fuzzy Analytic Hierarchy Process (FAHP) as its core decision-making model. FAHP extends the classical AHP method developed by Thomas L. Saaty by incorporating fuzzy set theory to handle the inherent uncertainty and imprecision in expert judgments. The system evaluates soil health through eight critical indicators, each assigned a weight derived from expert pairwise comparisons expressed as triangular fuzzy numbers. The final Soil Health Index (SHI) provides a single composite score that classifies soil into one of seven health categories.
Soil Health Indicators
| Indicator | Min | Max | Optimal Range | Unit | FAHP Weight |
|---|---|---|---|---|---|
| Soil pH | 0 | 8.5 | 6 - 7.5 | - | 0.215391 |
| Nitrogen (N) | 10 | 500 | 50 - 250 | mg/kg | 0.185427 |
| Phosphorus (P) | 10 | 200 | 20 - 100 | mg/kg | 0.159677 |
| Potassium (K) | 10 | 400 | 50 - 200 | mg/kg | 0.129830 |
| EC | 0 | 4 | 0 - 2 | dS/m | 0.088770 |
| Temperature | 0 | 50 | 10 - 30 | °C | 0.091019 |
| Moisture | 0 | 100 | 20 - 80 | % | 0.062241 |
| Humidity | 0 | 100 | 30 - 70 | % | 0.067645 |
Weight Distribution
FAHP Algorithm Pipeline
Define Soil Health Indicators
Eight soil health indicators are identified as evaluation criteria: pH, Nitrogen (N), Phosphorus (P), Potassium (K), Electrical Conductivity (EC), Temperature, Moisture and Humidity. Each indicator has defined acceptable ranges and optimal values based on agronomic research.
Construct Fuzzy Pairwise Comparison Matrix
Expert judgments are collected to compare each pair of indicators using triangular fuzzy numbers (l, m, u). The 8x8 fuzzy comparison matrix captures the relative importance of each indicator pair with inherent uncertainty modeled through fuzzy sets.
Compute Fuzzy Geometric Means
For each row of the fuzzy comparison matrix, the geometric mean of all fuzzy numbers is calculated. This produces a single fuzzy number per indicator that represents its aggregate comparison against all other indicators.
Calculate Fuzzy Weights
Fuzzy weights are derived by normalizing the fuzzy geometric means. Each indicator receives a triangular fuzzy weight (l, m, u) representing its relative importance in the overall soil health assessment.
Defuzzification (Center of Area)
The triangular fuzzy weights are converted to crisp (single-value) weights using the Center of Area method: W = (l + m + u) / 3. A consistency ratio check ensures the comparison matrix is logically consistent (CR < 0.10).
Soil Health Index Computation
Each indicator value is normalized to a 0-1 scale. The Soil Health Index (SHI) is computed as the weighted sum of all normalized indicator scores. The SHI is then classified into one of seven health categories.
Health Classification
The Soil Health Index (SHI) is a weighted composite score ranging from 0 to 1. It is mapped to seven health categories based on the following thresholds:
SHI >= 0.0 and < 0.2
SHI >= 0.2 and < 0.4
SHI >= 0.4 and < 0.6
SHI >= 0.6 and < 0.7
SHI >= 0.7 and < 0.8
SHI >= 0.8 and < 0.9
SHI >= 0.9
Crop Recommendations
Soil is in optimal condition for all major crops. Maintain current practices.
Suitable for most crops. Minor nutrient supplementation may benefit yields.
Balanced soil. Regular monitoring recommended for sustained productivity.
Consider organic amendments to improve soil structure and nutrient content.
Soil improvement needed. Apply compost and consider cover cropping.
Significant soil rehabilitation required. Prioritize organic matter addition.
Intensive soil restoration needed before productive cultivation. Use green manure crops.
References
- Saaty, T.L. (1980). The Analytic Hierarchy Process. McGraw-Hill, New York.
- Chang, D.Y. (1996). Applications of the extent analysis method on fuzzy AHP. European Journal of Operational Research, 95(3), 649-655.
- Buckley, J.J. (1985). Fuzzy hierarchical analysis. Fuzzy Sets and Systems, 17(3), 233-247.
- Doran, J.W. and Parkin, T.B. (1994). Defining and assessing soil quality. In Defining Soil Quality for a Sustainable Environment, SSSA Special Publication No. 35, pp. 1-21.
- Karlen, D.L. et al. (1997). Soil quality: A concept, definition and framework for evaluation. Soil Science Society of America Journal, 61(1), 4-10.
- Andrews, S.S., Karlen, D.L. and Cambardella, C.A. (2004). The Soil Management Assessment Framework: A quantitative soil quality evaluation method. Soil Science Society of America Journal, 68(6), 1945-1962.