Fuzzy Analytic Hierarchy Process (FAHP)

A multi-criteria decision making (MCDM) framework for evaluating soil health using 8 key indicators with expert-derived fuzzy weights.

1. Introduction

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 Saaty (1980) by incorporating fuzzy set theory to handle the inherent uncertainty and imprecision in expert judgments.

The system evaluates soil health through $n = 8$ 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.

2. Soil Health Indicators

Table 1. Soil health indicators with measurement ranges and FAHP weights.

Indicator	Unit	Min	Max	Normal Range	FAHP Weight ( $w_i$ )
Soil pH	-	0	8.5	6.0 - 7.5	0.2154
Nitrogen (N)	mg/kg	10	500	50 - 250	0.1854
Phosphorus (P)	mg/kg	10	200	20 - 100	0.1597
Potassium (K)	mg/kg	10	400	50 - 200	0.1298
EC	dS/m	0	4.00	0 - 2	0.0888
Temperature	°C	0	50	10 - 30	0.0910
Moisture	%	0	100	20 - 80	0.0622
Humidity	%	0	100	30 - 70	0.0676

3. FAHP Algorithm

3.1 Triangular Fuzzy Numbers

Definition 1.

A Triangular Fuzzy Number (TFN) is defined as a triplet $\tilde{a} = (l, m, u)$ where $l$ is the lower bound, $m$ is the modal (most likely) value, and $u$ is the upper bound, with $l \leq m \leq u$ .

Each pairwise comparison between indicators $i$ and $j$ is expressed as a TFN:

\tilde{a}_{ij} = (l_{ij}, m_{ij}, u_{ij})

The reciprocal of a TFN is defined as:

\tilde{a}_{ij}^{-1} = \left(\frac{1}{u_{ij}}, \frac{1}{m_{ij}}, \frac{1}{l_{ij}}\right)

3.2 Fuzzy Pairwise Comparison Matrix

Expert judgments compare each pair of indicators using TFNs. The resulting $8 \times 8$ fuzzy comparison matrix $\tilde{A}$ captures the relative importance of each indicator pair:

\tilde{A} = \begin{pmatrix} (1,1,1) & \tilde{a}_{12} & \cdots & \tilde{a}_{1n} \\ \tilde{a}_{21} & (1,1,1) & \cdots & \tilde{a}_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ \tilde{a}_{n1} & \tilde{a}_{n2} & \cdots & (1,1,1) \end{pmatrix}

where $\tilde{a}_{ji} = \tilde{a}_{ij}^{-1}$ and diagonal elements are $(1, 1, 1)$ .

Table 2. The $8 \times 8$ fuzzy pairwise comparison matrix used in SHDS.

	pH	N	P	K	EC	Temp	Moist	Humid
pH	(1, 1, 1)	(1, 2, 3)	(1, 2, 3)	(1, 2, 3)	(2, 3, 4)	(1, 2, 3)	(2, 3, 4)	(1, 2, 3)
N	(0.33, 0.50, 1)	(1, 1, 1)	(1, 2, 3)	(1, 2, 3)	(2, 3, 4)	(1, 2, 3)	(2, 3, 4)	(1, 2, 3)
P	(0.33, 0.50, 1)	(0.33, 0.50, 1)	(1, 1, 1)	(1, 2, 3)	(2, 3, 4)	(1, 2, 3)	(2, 3, 4)	(1, 2, 3)
K	(0.33, 0.50, 1)	(0.33, 0.50, 1)	(0.33, 0.50, 1)	(1, 1, 1)	(1, 2, 3)	(1, 2, 3)	(2, 3, 4)	(1, 2, 3)
EC	(0.25, 0.33, 0.50)	(0.25, 0.33, 0.50)	(0.25, 0.33, 0.50)	(0.33, 0.50, 1)	(1, 1, 1)	(1, 2, 3)	(1, 2, 3)	(1, 2, 3)
Temp	(0.33, 0.50, 1)	(0.33, 0.50, 1)	(0.33, 0.50, 1)	(0.33, 0.50, 1)	(0.33, 0.50, 1)	(1, 1, 1)	(1, 2, 3)	(1, 2, 3)
Moist	(0.25, 0.33, 0.50)	(0.25, 0.33, 0.50)	(0.25, 0.33, 0.50)	(0.25, 0.33, 0.50)	(0.33, 0.50, 1)	(0.33, 0.50, 1)	(1, 1, 1)	(1, 2, 3)
Humid	(0.33, 0.50, 1)	(0.33, 0.50, 1)	(0.33, 0.50, 1)	(0.33, 0.50, 1)	(0.33, 0.50, 1)	(0.33, 0.50, 1)	(0.33, 0.50, 1)	(1, 1, 1)

3.3 Fuzzy Geometric Mean

For each row $i$ of the fuzzy comparison matrix, the geometric mean of all TFNs is computed. This produces a single fuzzy number per indicator that represents its aggregate comparison against all other indicators.

\tilde{r}_i = \left( \prod_{j=1}^{n} \tilde{a}_{ij} \right)^{1/n}

For each component of the TFN, this expands to:

l_i = \left( \prod_{j=1}^{n} l_{ij} \right)^{1/n}, \quad m_i = \left( \prod_{j=1}^{n} m_{ij} \right)^{1/n}, \quad u_i = \left( \prod_{j=1}^{n} u_{ij} \right)^{1/n}

3.4 Fuzzy Weight Calculation

Fuzzy weights are derived by normalizing the fuzzy geometric means. Each indicator receives a triangular fuzzy weight $\tilde{w}_i = (l_i^w, m_i^w, u_i^w)$ representing its relative importance:

\tilde{w}_i = \frac{\tilde{r}_i}{\sum_{j=1}^{n} \tilde{r}_j}

3.5 Defuzzification (Center of Area)

The triangular fuzzy weights are converted to crisp (single-value) weights using the Center of Area (CoA) method:

M_i = \frac{l_i^w + m_i^w + u_i^w}{3}

where $l_i^w, m_i^w, u_i^w$ are the lower, modal and upper values of the fuzzy weight for indicator $i$ .

3.6 Normalization

The defuzzified weights are normalized so they sum to 1, producing the final crisp FAHP weights:

w_i = \frac{M_i}{\sum_{j=1}^{n} M_j}

Table 3. Final normalized FAHP weights.

pH	N	P	K	EC	Temp	Moist	Humid
0.2154	0.1854	0.1597	0.1298	0.0888	0.0910	0.0622	0.0676

Figure 1. Weight distribution across the 8 soil health indicators.

4. Soil Health Index Computation

The Soil Health Index (SHI) is computed as the weighted sum of normalized indicator values. Each raw indicator value is first normalized to a $[0, 1]$ scale using min-max normalization:

S_i = \frac{x_i - x_{\min,i}}{x_{\max,i} - x_{\min,i}}

where $x_i$ is the measured value, $x_{\min,i}$ and $x_{\max,i}$ are the defined minimum and maximum bounds for indicator $i$ .

The Soil Health Index is then computed as:

SHI = \sum_{i=1}^{n} w_i \cdot S_i

where $w_i$ is the FAHP weight for indicator $i$ and $S_i$ is the normalized score. Since $\sum w_i = 1$ , the SHI falls in the range $[0, 1]$ .

5. Health Classification

The SHI value is mapped to one of seven health categories. Each category has a defined score range and associated color for visualization.

Table 4. Soil health classification based on SHI score.

Rating	SHI Range	Color
Very Poor	< 0.2	#ef4444
Poor	0.2 to < 0.4	#f97316
Below Average	0.4 to < 0.6	#eab308
Average	0.6 to < 0.7	#84cc16
Above Average	0.7 to < 0.8	#22c55e
Good	0.8 to < 0.9	#14b8a6
Excellent	>= 0.9	#06b6d4

Figure 2. Visual scale of soil health classification.

Very Poor

Poor

Below Average

Average

Above Average

Good

Excellent

0.00.20.40.60.70.80.91.0

6. Crop Recommendations

SHI: 0.0 - 0.2

Focus on soil remediation. Grow leguminous cover crops like Cowpea, Horse gram, or Sunn hemp.

SHI: 0.2 - 0.4

Grow Millets (Sorghum, Pearl millet), Pulses (Pigeon pea, Chickpea), and Oilseeds (Safflower, Castor).

SHI: 0.4 - 0.6

Grow Maize, Soybean, Groundnut, Cotton, and incorporate legumes into the cropping system.

SHI: 0.6 - 0.8

Grow Vegetables (Tomato, Brinjal, Chili), Fruits (Mango, Banana, Citrus), and Cash crops (Sugarcane, Tobacco).

SHI: 0.8 - 1.0

Grow high-yielding crops like Rice, Wheat, Potato, and Vegetables. Explore organic farming and cultivation of medicinal and aromatic plants.

7. Fertilizer Recommendations

SHI: 0.0 - 0.2

Apply organic amendments like Compost (5-7.5 tonnes/ha), Vermicompost (2.5-3.5 tonnes/ha), or Well-decomposed Farmyard manure (10-12.5 tonnes/ha). Incorporate green manure crops like Dhaincha (Sesbania aculeata) (5-6 tonnes/ha), Sunhemp (Crotalaria juncea) (4-5 tonnes/ha), or Cowpea (Vigna unguiculata) (3-4 tonnes/ha). Avoid applying chemical fertilizers until soil health improves.

SHI: 0.2 - 0.4

Apply organic amendments like Compost (3-5 tonnes/ha), Vermicompost (1.5-2.5 tonnes/ha), or Well-decomposed Farmyard manure (7.5-10 tonnes/ha). Use biofertilizers like Rhizobium (200-300 g/ha), Azotobacter (200-300 g/ha), and Phosphate Solubilizing Bacteria (PSB) (500-750 g/ha). Apply chemical fertilizers at 50% of the recommended dose based on soil test results.

SHI: 0.4 - 0.6

Apply organic amendments like Compost (2-3 tonnes/ha), Vermicompost (1-1.5 tonnes/ha), or Well-decomposed Farmyard manure (5-7.5 tonnes/ha). Use biofertilizers like Rhizobium (200-300 g/ha), Azotobacter (200-300 g/ha), and PSB (500-750 g/ha). Apply chemical fertilizers at 75% of the recommended dose based on soil test results and crop requirements.

SHI: 0.6 - 0.8

Apply organic amendments like Compost (1-2 tonnes/ha) or Vermicompost (0.5-1 tonne/ha). Follow integrated nutrient management practices. Apply chemical fertilizers like Urea (0.08-0.12 tonnes/ha), Single Superphosphate (0.06-0.09 tonnes/ha), and Muriate of Potash (0.04-0.06 tonnes/ha) as per soil test recommendations and crop requirements.

SHI: 0.8 - 1.0

Maintain soil health by incorporating Crop residues (2-3 tonnes/ha) and applying organic amendments like Compost (0.5-1 tonne/ha) or Vermicompost (0.25-0.5 tonne/ha). Apply chemical fertilizers judiciously based on soil test results and crop needs. Consider using slow-release fertilizers like Neem-coated Urea (0.04-0.06 tonnes/ha) or Sulfur-coated Urea (0.03-0.05 tonnes/ha) to improve nutrient use efficiency.

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