PlanX Suitability Lab
Academic Reference Manual

PlanX Suitability Lab is a QGIS processing plugin providing 68 raster-based multi-criteria decision analysis (MCDA) algorithms for land suitability modelling. It spans the full decision workflow: data harmonisation, criterion weighting (subjective and objective), compensatory and non-compensatory composite scoring, uncertainty quantification, explainable AI diagnostics, scenario comparison, and decision reporting. All algorithms operate on single-band raster layers aligned to a common grid. Outputs are GeoTIFF (Float32 or Int16/Byte) and GeoPackage vector tables.

raster MCDA QGIS processing 68 algorithms WLC + OWA + TOPSIS + VIKOR Monte Carlo AHP / Entropy / CRITIC / PCA GeoShapley

1. Core Preprocessing Algorithms

The Core group provides seven algorithms for raster data preparation: alignment, vector-to-raster conversion, proximity computation, value normalisation, constraint masking, synthetic test-data generation, and end-to-end pipeline verification.

1.1 Data Harmonizer

GDAL Warp grid alignment preprocessing

Theoretical Background

Multi-criteria raster analysis requires all input layers to share identical spatial reference systems, pixel resolution, extent, and nodata conventions. Misaligned rasters cause resampling-on-the-fly during overlay operations, introducing positional error and artefacts. The Data Harmonizer applies GDAL's gdalwarp reprojection engine to enforce strict grid conformity, which is a prerequisite for any cell-by-cell MCDA combination. The algorithm serves the role of the “harmonisation” step described in Malczewski (1999) as the foundational preprocessing phase without which weighted linear combination yields meaningless results.

Mathematical Formulation

Let $\mathcal{R} = \{R_1, R_2, \ldots, R_m\}$ be the set of input rasters. A snap raster $S$ or explicit parameters define the target grid $G = (\text{CRS}_G, \text{Extent}_G, \Delta x_G, \Delta y_G, \nu_G)$ where:

$$R_i^{\text{aligned}}(x, y) = \mathcal{W}\big(R_i, \; G, \; \text{method}\big)$$

The warp operator $\mathcal{W}$ performs coordinate transformation, resampling (nearest-neighbour, bilinear, or cubic convolution), and nodata filling. Outputs share identical grid geometry, guaranteeing $R_i^{\text{aligned}}[c, r]$ and $R_j^{\text{aligned}}[c, r]$ refer to the same geographic location for any column-row index pair $(c, r)$.

(1) $\text{CRS}(R_i^{\text{aligned}}) = \text{CRS}_G \quad \forall i$

(2) $\Delta x(R_i^{\text{aligned}}) = \Delta y(R_i^{\text{aligned}}) = \Delta x_G \quad \forall i$

(3) $\text{Extent}(R_i^{\text{aligned}}) = \text{Extent}_G \quad \forall i$

Parameters

Parameter	Type	Default	Description
Input rasters	Multiple raster layers	—	Raster layers to align
Snap raster	Raster layer (optional)	None	Reference for CRS, extent, and resolution
Target extent	Extent (optional)	None	Override output bounding box
Target resolution	Double	0.0	Pixel size in map units; ≤0 inherits from snap or first raster
Resampling	Enum	Nearest	Nearest / Bilinear / Cubic
Output nodata	Double	-9999.0	Nodata fill value
Preview aligned raster	Raster destination	—	Copy of first aligned result
Aligned raster folder	Folder	—	Directory for all aligned GeoTIFFs
Alignment manifest	File (.txt)	—	Text log of alignment parameters and outputs

Output Description

Produces $m$ aligned GeoTIFF files named aligned_NN_safename.tif in the output folder, a preview copy of the first aligned raster, and a plain-text manifest documenting the target CRS, resolution, extent, resampling method, and nodata value used.

Interpretation Guide

Identical grids confirmed: Open any two aligned rasters in QGIS; they should overlay perfectly with no shift.
Resampling artefacts: Nearest-neighbour preserves categorical boundaries; bilinear/cubic smooths continuous fields. Use Nearest for land-cover classes, Bilinear for DEM-derived criteria.
Nodata propagation: Cells outside the original raster extent receive the specified nodata value. Verify the manifest confirms expected coverage.

Key insight: This step is not optional. Stacking misaligned rasters in any MCDA method produces spatially biased results. The harmonizer guarantees all criterion rasters share identical geometry, a precondition for valid cell-wise algebra.

References

Malczewski, J. (1999). GIS and Multicriteria Decision Analysis. Wiley. [ISBN 978-0-471-32944-2]
GDAL/OGR Contributors (2024). GDAL Warp documentation. GDAL Documentation. [https://gdal.org/programs/gdalwarp.html]
Eastman, J.R. (2012). IDRISI Selva Manual. Clark Labs, Clark University.
Tomlin, C.D. (1990). Geographic Information Systems and Cartographic Modeling. Prentice Hall.
Longley, P.A. et al. (2015). Geographic Information Science and Systems, 4th ed. Wiley.
Neteler, M. & Mitasova, H. (2008). Open Source GIS: A GRASS GIS Approach, 3rd ed. Springer. DOI: 10.1007/978-0-387-68574-8
Kennedy, H. (Ed.) (2016). QGIS 2 Cookbook. Packt Publishing.

1.2 Vector Rasterizer

GDAL Rasterize vector-to-raster attribute burn

Theory & Method

Converts vector features (points, lines, polygons) to a single-band Float32 raster via GDAL's gdal_rasterize. A specified attribute field or constant burn value is assigned to cells whose centre falls within a feature. This is the primary bridge from vector GIS data to the raster MCDA domain.

$$V(x, y) = \begin{cases} A_f & \text{if } (x, y) \text{ intersects feature } f \\ 0 & \text{otherwise (nodata)} \end{cases}$$

Parameter	Default	Role
Vector input	—	Any geometry type
Value field	optional	Attribute to burn; overrides burn value
Burn value	1.0	Constant value when no field is specified
Pixel size	10.0	Cell size in map units

Interpretation: Output extent matches the input vector layer bounds. Cells not covered by any feature are nodata (0). For categorical land-use data, use the field containing the suitability score directly.

GDAL/OGR Contributors (2024). gdal_rasterize. GDAL Documentation.
Burrough, P.A. & McDonnell, R.A. (1998). Principles of Geographical Information Systems. Oxford.
De Smith, M.J., Goodchild, M.F. & Longley, P.A. (2018). Geospatial Analysis, 6th ed. [https://www.spatialanalysisonline.com/]

1.3 Proximity Builder

GDAL Proximity Euclidean distance continuous surface

Theory & Method

Computes Euclidean distance from every cell to the nearest target pixel (cells whose band-1 value matches user-specified integer targets). The algorithm uses the two-pass chamfer distance transform internal to GDAL Proximity with Euclidean metric, producing a continuous proximity surface.

$$D(p) = \min_{q \in T} \| p - q \|_2 \quad \text{where } T = \{q : R(q) \in \text{target\_values}\}$$

Parameter	Default	Role
Input raster	—	Single-band source
Target pixel values	"1"	Comma-separated integers; cells to measure from
Distance units	Georeferenced	Map units or Pixels
Max distance	0.0	0 = unlimited; caps distance values

Typical use: Distance to roads, transit stops, water bodies, or hazard zones — as normalisable suitability criteria.

Danielsson, P.-E. (1980). Euclidean distance mapping. Computer Graphics and Image Processing, 14(3), 227–248. DOI: 10.1016/0146-664X(80)90054-4
Borgefors, G. (1986). Distance transformations in digital images. Computer Vision, Graphics, and Image Processing, 34(3), 344–371. DOI: 10.1016/S0734-189X(86)80047-0
GDAL/OGR Contributors (2024). gdal_proximity. GDAL Documentation.

1.4 Normalizer / Reclassifier

fuzzy membership min-max sigmoid gaussian

Theoretical Background

Multi-criteria evaluation requires commensurate criterion scales. Raw environmental variables (elevation in metres, distance in kilometres, land value in currency) cannot be directly combined. The Normalizer transforms each raw criterion raster to the standardised $[0, 100]$ suitability scale through a membership function $\mu: \mathbb{R} \to [0, 100]$. The choice of function encodes the decision-maker's value judgment about how desirability changes with the raw measurement. Following Zadeh's fuzzy set theory, the membership grade represents the degree to which a cell belongs to the “suitable” set. Seven functions are provided, spanning both crisp (min-max linear) and fuzzy (sigmoid, Gaussian) forms, with benefit (higher is better) and cost (lower is better) orientations.

Mathematical Formulation

Let $x_{i,j}$ be the raw criterion value at cell $(i,j)$, and let parameters $L$ (low), $H$ (high), $M$ (midpoint), and $\sigma$ (spread) be user-specified.

(4) Min-Max Benefit: $\mu_B(x) = 100 \cdot \dfrac{x - L}{H - L}, \quad x \in [L, H]$

(5) Min-Max Cost: $\mu_C(x) = 100 \cdot \dfrac{H - x}{H - L}, \quad x \in [L, H]$

(6) Sigmoid Benefit: $\mu_S(x) = \dfrac{100}{1 + e^{-z}}, \quad z = \dfrac{x - M}{\sigma}, \quad z \in [-60, 60]$

(7) Sigmoid Cost: $\mu_{SC}(x) = 100\left(1 - \dfrac{1}{1 + e^{-z}}\right), \quad z = \dfrac{x - M}{\sigma}$

(8) Gaussian Benefit: $\mu_G(x) = 100 \cdot \exp\left(-\dfrac{1}{2}\left(\dfrac{x - M}{\sigma}\right)^2\right)$

The sigmoid midpoint $M$ defines the inflection point (50% membership). The spread $\sigma$ controls steepness: smaller $\sigma$ produces sharper transitions, modelling stricter thresholds. The Gaussian function is symmetric about $M$ and is appropriate when an optimal range exists with declining suitability on both sides (e.g., optimal temperature or elevation). Output is clamped to $[0, 100]$ and original nodata cells are preserved.

Parameters

Parameter	Type	Default	Description
Input raster	Raster layer	—	Single-band raw criterion
Method	Enum (0–6)	0 (Min-Max Benefit)	One of seven normalisation functions
Low bound	Double	0.0	Minimum raw value or lower breakpoint
High bound	Double	100.0	Maximum raw value or upper breakpoint (must > Low)
Midpoint	Double	50.0	Inflection point for sigmoid/Gaussian
Spread	Double	10.0	Steepness parameter (>0) for sigmoid/Gaussian
Output raster	Raster destination	—	Normalised GeoTIFF (Float32, LZW, 0–100)

Output Description

Single-band Float32 LZW-compressed GeoTIFF with values in $[0, 100]$. Nodata cells from the input are preserved with their original nodata value. The data type and compression ensure compatibility with downstream NumPy-based processing.

Interpretation Guide

0 = completely unsuitable, 100 = ideal: A value of 50 represents the midpoint of desirability.
Linearity assumption (min-max): Implies constant marginal change in suitability per unit change in the raw value. This is the most common but least behaviourally nuanced choice.
Sigmoid: Models threshold behaviour: little change in suitability at extremes, rapid change near the midpoint. Suitable for regulatory thresholds (e.g., minimum distance to protected areas).
Gaussian: Penalises deviation from an optimum. Use when both “too little” and “too much” are undesirable (e.g., slope for agriculture: too flat = poor drainage; too steep = erosion risk).

Key insight: The normalisation function is a value judgment, not a data transformation. Different functions for the same criterion produce different suitability maps. Sensitivity analysis across normalisation forms is recommended (see Global Sensitivity, Sec. 5.1).

References

Zadeh, L.A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. DOI: 10.1016/S0019-9958(65)90241-X
Jiang, H. & Eastman, J.R. (2000). Application of fuzzy measures in multi-criteria evaluation in GIS. IJGIS, 14(2), 173–184. DOI: 10.1080/136588100240903
Malczewski, J. (2006). GIS-based multicriteria decision analysis: a survey of the literature. IJGIS, 20(7), 703–726. DOI: 10.1080/13658810600661508
Eastman, J.R. & Jiang, H. (1996). Fuzzy measures in multi-criteria evaluation. Proceedings, 2nd Int. Symposium on Spatial Accuracy Assessment, 527–534.
Zimmermann, H.-J. (2001). Fuzzy Set Theory and Its Applications, 4th ed. Springer. DOI: 10.1007/978-94-010-0646-0
Gorsevski, P.V. et al. (2012). Integrating multi-criteria evaluation techniques with GIS for landfill site selection. Waste Management, 32(2), 287–296. DOI: 10.1016/j.wasman.2011.09.011
Rashed, T. & Weeks, J. (2003). Assessing vulnerability to earthquake hazards through spatial multicriteria analysis. Geographical Analysis, 35(4), 305–325.

1.5 Constraint Builder (0/1 Mask)

Boolean mask thresholding exclusion

Theory & Method

Applies a Boolean threshold $R(x,y) \ge \tau$ to a criterion raster, producing a binary mask where 1 = suitable (passes constraint) and 0 = excluded (fails constraint). This implements the “hard constraint” layer in GIS-MCDA: areas that are absolutely infeasible regardless of other criteria scores (e.g., protected areas, water bodies, slopes exceeding regulatory limits).

$$C(x,y) = \begin{cases} 1 & \text{if } R(x,y) \ge \tau \\ 0 & \text{otherwise} \end{cases}$$

Parameter	Default	Role
Input criterion raster	—	Single-band
Threshold	50.0	Cells ≥ threshold become 1

Multiple constraint masks can be multiplied to create a composite exclusion layer. Constraint-failing cells in MCDA scoring are forced to 0.

Malczewski, J. (1999). GIS and Multicriteria Decision Analysis. Wiley.
Eastman, J.R. et al. (1995). Raster procedures for MCE. Photogrammetric Engineering & Remote Sensing, 61(5), 539–547.
Geneletti, D. (2010). Combining stakeholder analysis and spatial MCA. Land Use Policy, 27(2), 550–559. DOI: 10.1016/j.landusepol.2009.07.009

1.6 Synthetic Sample Raster Pack

test data smoke test synthetic

Theory & Method

Generates five synthetic Float32 GeoTIFFs (four criteria rasters with controlled spatial patterns and one binary constraint mask) for pipeline testing without real data. Each criterion is computed from normalised coordinate grids with distinct spatial structures: eastward decay, northward increase, diagonal ridge, and radial hotspot, all with Gaussian noise (seed = 7). The constraint mask excludes the bottom-left 15% by 10% of the study area. All rasters are EPSG:3857, clipped 0–100, nodata -9999.

Parameter	Default	Role
Study extent	—	Bounding box
Pixel size	30.0	Cell size in map units

Output files: criterion_1.tif through criterion_4.tif and constraint_mask.tif.

Beinat, E. (1997). Value Functions for Environmental Management. Springer. DOI: 10.1007/978-94-015-8885-0

1.7 Synthetic End-to-End Pipeline Runner

integration test smoke test verification

Theory & Method

Chain-runs 35+ sub-algorithms in deterministic sequence: synthetic data generation, four MCDA methods (WLC, OWA, TOPSIS, VIKOR) with equal weights, Monte Carlo uncertainty (120 iterations, sigma 0.08), AHP/PCA/Entropy/CRITIC weight derivation, ensemble synthesis, scenario consensus/disagreement/quadrant/intervention mapping, risk-opportunity matrix, regret fragility, opportunity loss, priority zoning (baseline + refined), multi-scenario benchmarking, robustness ranking, budget phasing, deployment readiness gating, and the full reporting suite (HTML report, Markdown brief, beta evidence HTML). Produces 70+ output files.

Parameter	Default	Role
Study extent	—	Bounding box
Pixel size	30.0	Cell size in map units

Typical use: One-click verification that every algorithm executes correctly after installation or dependency updates.

Meyer, B. (2008). Seven principles of software testing. IEEE Computer, 41(8), 99–101. DOI: 10.1109/MC.2008.306

2. Weight Derivation Algorithms

The Weights group provides seven algorithms spanning subjective pairwise comparison (AHP), data-driven objective methods (PCA, Entropy, CRITIC), and weight management (consolidation, scenario packaging, cross-method consensus auditing).

2.1 AHP Weights

Saaty eigenvalue pairwise comparison consistency ratio

Theoretical Background

The Analytic Hierarchy Process (AHP), developed by Saaty (1977, 1980), is the most widely used subjective weighting method in GIS-MCDA. It derives criterion weights from a pairwise comparison matrix $A = [a_{ij}]_{n \times n}$ where each element $a_{ij}$ expresses the decision-maker's judgment of how many times more important criterion $i$ is relative to criterion $j$, using Saaty's fundamental 1–9 scale. The matrix is reciprocal ($a_{ji} = 1/a_{ij}$) with unit diagonal ($a_{ii} = 1$). The priority vector $\mathbf{w}$ is the principal right eigenvector of $A$, normalised to sum to unity. A consistency check ensures judgments are not randomly contradictory.

Mathematical Formulation

(9) $A\mathbf{w} = \lambda_{\max}\mathbf{w}, \quad \sum_{i=1}^n w_i = 1, \quad w_i > 0$

(10) Consistency Index: $CI = \dfrac{\lambda_{\max} - n}{n - 1}$

(11) Consistency Ratio: $CR = \dfrac{CI}{RI_n}$

Where $RI_n$ is Saaty's Random Index for matrix size $n$: RI(1)=0.00, RI(2)=0.00, RI(3)=0.58, RI(4)=0.90, RI(5)=1.12, RI(6)=1.24, RI(7)=1.32, RI(8)=1.41. Judgments are considered acceptably consistent when $CR \le 0.10$; otherwise, the analyst should revisit the pairwise comparisons.

(12) The pairwise matrix has the structure: $A = \begin{bmatrix} 1 & a_{12} & \cdots & a_{1n} \\ 1/a_{12} & 1 & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ 1/a_{1n} & 1/a_{2n} & \cdots & 1 \end{bmatrix}$

Parameters

Parameter	Type	Default	Description
Pairwise matrix JSON	String (multiline)	[[1,2,3],[0.5,1,2],[0.3333,0.5,1]]	Square reciprocal matrix of Saaty-scale judgments
Output table	Vector destination	—	GeoPackage with criterion_id, weight, cr, cr_ok

Output Description

A GeoPackage attribute table (no geometry) with columns: criterion_id (integer), weight (0–1, sum = 1), cr (consistency ratio), cr_ok (1 if CR ≤ 0.10, else 0). A warning is emitted when CR exceeds the threshold.

Interpretation Guide

CR ≤ 0.10: Judgments are coherent; weights can be used with confidence.
CR > 0.10 but < 0.20: Borderline. Consider revising the most inconsistent pairwise comparisons.
CR ≥ 0.20: Judgments are effectively random. Restructure the hierarchy or re-elicit comparisons.
Weight distribution: If one weight dominates (>0.50), verify that criterion truly outweighs all others 2:1+.

Key insight: AHP's strength is its ability to decompose complex trade-offs into manageable pairwise judgments. Its weakness is the exponential growth of comparisons ($n(n-1)/2$ pairs for $n$ criteria). For more than 7–9 criteria, consider grouping into a hierarchy or using a data-driven method (Entropy, CRITIC) as a complement.

References

Saaty, T.L. (1977). A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology, 15(3), 234–281. DOI: 10.1016/0022-2496(77)90033-5
Saaty, T.L. (1980). The Analytic Hierarchy Process. McGraw-Hill.
Saaty, T.L. (2008). Decision making with the analytic hierarchy process. Int. J. Services Sciences, 1(1), 83–98. DOI: 10.1504/IJSSCI.2008.017590
Malczewski, J. (2006). GIS-based multicriteria decision analysis: a survey of the literature. IJGIS, 20(7), 703–726. DOI: 10.1080/13658810600661508
Forman, E.H. & Gass, S.I. (2001). The analytic hierarchy process—an exposition. Operations Research, 49(4), 469–486. DOI: 10.1287/opre.49.4.469.11231
Vaidya, O.S. & Kumar, S. (2006). Analytic hierarchy process: an overview of applications. EJOR, 169(1), 1–29. DOI: 10.1016/j.ejor.2004.04.028
Boroushaki, S. & Malczewski, J. (2008). Implementing an extension of AHP using OWA operators with fuzzy quantifiers. Computers & Geosciences, 34(4), 399–410. DOI: 10.1016/j.cageo.2007.04.003

2.2 PCA-derived Weights (Variance Proxy)

eigendecomposition PC1 loading data-driven

Theory & Method

Samples up to ~8000 cells from the aligned raster stack, standardises (z-score), and computes the covariance matrix. Eigendecomposition yields principal components; the absolute loadings of PC1, scaled by its explained variance ratio, become the weights. This variance-proxy approach assigns higher weight to criteria driving the largest spatial differences. If fewer than 3 valid sample cells exist, equal weights are returned.

$$w_i = |\ell_{i,1}| \cdot \frac{\lambda_1}{\sum_k \lambda_k}, \quad \text{normalised so } \sum w_i = 1$$

Parameter	Default	Role
Criterion rasters	—	Multiple aligned rasters

Interpretation: High weight = criterion strongly loads on the dominant spatial-variance axis. Low weight = criterion is spatially uniform or anti-correlated with the dominant pattern. This method rewards criteria with high spatial variability, which may not always align with decision relevance.

Jolliffe, I.T. (2002). Principal Component Analysis, 2nd ed. Springer. DOI: 10.1007/b98835
Abdi, H. & Williams, L.J. (2010). Principal component analysis. WIREs Computational Statistics, 2(4), 433–459. DOI: 10.1002/wics.101
Zhu, X. & Dale, A.P. (2001). JavaAHP: a web-based decision tool. Environmental Modelling & Software, 16(3), 251–260.

2.3 Entropy Weights

Shannon entropy information theory discriminating power

Theoretical Background

Entropy-based weighting originates in information theory (Shannon, 1948) and was adapted to MCDM by Zeleny (1982) and Hwang & Yoon (1981). The principle is that a criterion with greater value dispersion across alternatives carries more decision information and should therefore receive higher weight. A criterion that is nearly constant across the study area contributes little to differentiating suitable from unsuitable locations and is down-weighted. Entropy weights are purely data-driven and require no subjective judgment.

Mathematical Formulation

For a raster stack with $n$ criteria and $K$ sampled cells, the min-max normalised value for criterion $i$ at cell $k$ is:

(13) $p_{ik} = \dfrac{x_{ik}}{\sum_{k=1}^K x_{ik}}, \quad x_{ik} \in [0, 1]$

(14) Entropy: $e_i = -\dfrac{1}{\ln K}\sum_{k=1}^K p_{ik} \ln(p_{ik}), \quad p_{ik} > 0$

(15) Degree of diversification: $d_i = 1 - e_i$

(16) Objective weight: $w_i = \dfrac{d_i}{\sum_{j=1}^n d_j}$

Higher entropy $e_i \to 1$ means the criterion is uniform (all cells have similar values). Low entropy means distinctive, concentrated values. The weight $w_i$ is proportional to $d_i$: criteria with concentrated, discriminating value distributions receive larger weights.

Parameters

Parameter	Type	Default	Description
Criterion rasters	Multiple raster layers	—	Aligned single-band rasters
Output table	Vector destination	—	GeoPackage with criterion_id, weight

Interpretation Guide

Near-zero weight: The criterion is spatially homogeneous. It adds no discriminating power. Consider removing it to simplify the model.
Dominant weight (>0.50): One criterion has vastly more contrast than others. Verify this is meaningful, not an artefact of scale or outliers.
Complementary role: Entropy weights are best used alongside subjective weights (AHP) to identify discrepancies between “what experts think matters” and “what the data says varies.”

Key insight: Entropy weights measure contrast, not importance. A highly variable but irrelevant criterion (e.g., measurement noise) will receive high entropy weight. Always screen criteria for domain relevance before applying entropy.

References

Shannon, C.E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27(3), 379–423. DOI: 10.1002/j.1538-7305.1948.tb01338.x
Zeleny, M. (1982). Multiple Criteria Decision Making. McGraw-Hill.
Hwang, C.L. & Yoon, K. (1981). Multiple Attribute Decision Making: Methods and Applications. Springer. DOI: 10.1007/978-3-642-48318-9
Lotfi, F.H. & Fallahnejad, R. (2010). Imprecise Shannon's entropy and multi-attribute decision making. Entropy, 12(1), 53–62. DOI: 10.3390/e12010053
Wang, T.C. & Lee, H.D. (2009). Developing a fuzzy TOPSIS approach. Expert Systems with Applications, 36(5), 8980–8985. DOI: 10.1016/j.eswa.2008.11.035
Deng, H., Yeh, C.H. & Willis, R.J. (2000). Inter-company comparison using modified TOPSIS. Computers & Operations Research, 27(10), 963–973. DOI: 10.1016/S0305-0548(99)00069-6
Malczewski, J. & Rinner, C. (2015). Multicriteria Decision Analysis in Geographic Information Science. Springer. DOI: 10.1007/978-3-540-74757-4

2.4 CRITIC Objective Weights

contrast intensity inter-criterion correlation penalises redundancy

Theoretical Background

The CRITIC (CRiteria Importance Through Intercriteria Correlation) method, proposed by Diakoulaki, Mavrotas, and Papayannakis (1995), derives objective weights by combining two information-theoretic properties: (1) contrast intensity, measured by the standard deviation of each criterion, and (2) conflict, measured by the sum of (1 − correlation) with all other criteria. A criterion that is both high-variance and uncorrelated with others receives the highest weight; a criterion that is redundant with others (high correlation) is penalised. This distinguishes CRITIC from pure entropy weighting, which does not consider inter-criterion relationships.

Mathematical Formulation

Let $\mathbf{x}_i$ be the normalised values of criterion $i$ across $K$ sampled cells, with $\bar{x}_i$ and $\sigma_i$ as the mean and standard deviation. Let $\rho_{ij}$ be the Pearson correlation between criteria $i$ and $j$.

(17) Contrast intensity: $C_i = \sigma_i$

(18) Conflict index: $\Psi_i = \sum_{j=1}^n (1 - |\rho_{ij}|)$

(19) Information content: $I_i = C_i \cdot \Psi_i = \sigma_i \cdot \sum_{j=1}^n (1 - |\rho_{ij}|)$

(20) CRITIC weight: $w_i = \dfrac{I_i}{\sum_{k=1}^n I_k}$

Parameters

Parameter	Type	Default	Description
Criterion rasters	Multiple raster layers	—	Aligned single-band rasters
Impacts CSV	String	"" (all benefit)	B/C or +/-1 per criterion
Output table	Vector destination	—	GeoPackage with criterion_id, weight, std_dev, contrast, critic_score

Interpretation Guide

High CRITIC score: Criterion is both high-contrast and uncorrelated — it contributes unique discriminating information.
Low CRITIC score: Either the criterion is spatially uniform (low std) or strongly correlated with other criteria (redundant).
Redundancy penalty: Two criteria with $\rho > 0.90$ will both be penalised. This is the method's key advantage over entropy alone.

References

Diakoulaki, D., Mavrotas, G. & Papayannakis, L. (1995). Determining objective weights in multiple criteria problems: the CRITIC method. Computers & Operations Research, 22(7), 763–770. DOI: 10.1016/0305-0548(94)00059-H
Krishnan, A.R. et al. (2021). A modified CRITIC method... Sustainability, 13(3), 1492. DOI: 10.3390/su13031492
Jahan, A. et al. (2012). A comprehensive VIKOR method for material selection. Materials & Design, 39, 262–268. DOI: 10.1016/j.matdes.2012.02.043
Zardari, N.H. et al. (2015). Weighting Methods and Their Effects on MCDM Outcomes. Springer. DOI: 10.1007/978-3-319-12586-2
Mukhametzyanov, I. (2021). Specific character of objective methods. Procedia Computer Science, 182, 99–106.

2.5 Weight Consolidator

normalisation activation step

Reads any weight table (from AHP, PCA, Entropy, or CRITIC) and re-normalises all weights to sum to 1. This is a pass-through activation step: relative proportions are preserved, but the output is guaranteed ready for downstream MCDA scoring. Guards against zero-sum edges by substituting a divisor of 1.0 when the total sum is zero. Output: criterion_id, weight. Reference: Malczewski (1999).

2.6 Scenario Weight Pack

multi-method merge blended average sensitivity analysis

Merges AHP, PCA, Entropy, and CRITIC weight tables into a unified scenario pack. Each method's weights are independently normalised to sum to 1. An additional BLENDED_AVG scenario computes the per-criterion arithmetic mean of normalised weights across all supplied methods, creating a simple ensemble that balances subjective and objective approaches. Output columns: method, criterion_id, weight_raw, weight_norm. Use: Prepare multi-scenario weight sets for sensitivity analysis or consensus auditing.

Roy, B. (1996). Multicriteria Methodology for Decision Aiding. Springer. DOI: 10.1007/978-1-4757-2500-1
Belton, V. & Stewart, T.J. (2002). Multiple Criteria Decision Analysis: An Integrated Approach. Springer. DOI: 10.1007/978-1-4615-1495-4

2.7 Weight Consensus Audit

coefficient of variation cross-method agreement robustness

Computes per-criterion descriptive statistics (mean, median, min, max, range, population std) and the coefficient of variation $CV = \sigma / \bar{w} \times 100\%$ across all weighting methods in a Scenario Weight Pack. Agreement labels: HIGH (CV ≤ 10%), MEDIUM (CV ≤ 25%), LOW (CV > 25%). Identifies robust criteria (weight stable regardless of method) vs. contested criteria (weight depends strongly on the method), guiding stakeholder discussion.

Saltelli, A. et al. (2004). Sensitivity Analysis in Practice. Wiley. DOI: 10.1002/0470870958
Butler, J., Jia, J. & Dyer, J. (1997). Simulation techniques for the sensitivity analysis of MCDM. EJOR, 103(3), 537–548. DOI: 10.1016/S0377-2217(96)00304-9
Chen, Y. et al. (2010). Sensitivity analysis of MCDA weights. IJGIS, 24(7), 1077–1100.

3. Composite Scoring (Compose) Algorithms

The Compose group contains 16 algorithms spanning the core MCDA methods (WLC, OWA, TOPSIS, VIKOR, Ensemble), post-hoc spatial diagnostics (opportunity loss, consensus/disagreement mapping, decision quadrants, risk-opportunity matrices, regret fragility, intervention conflict), scenario comparison, and derived action products (candidate site extraction, priority zoning, uncertainty-adjusted suitability).

3.1 WLC Suitability

weighted linear combination compensatory additive

Theoretical Background

Weighted Linear Combination (WLC) is the most widely used MCDA method in GIS (Malczewski, 1999). It is a fully compensatory additive model: a low score on one criterion can be offset by high scores on others. The method assumes mutual preferential independence of criteria and a linear value function. WLC is computationally simple, transparent, and easy to explain to stakeholders, making it the de facto baseline for many land-suitability studies.

Mathematical Formulation

For $n$ criterion rasters $X_i(x,y)$ normalised to $[0,100]$, and weights $w_i$ with $\sum w_i = 1$:

(21) $S(x,y) = \sum_{i=1}^n w_i \cdot X_i(x,y), \quad S(x,y) \in [0, 100]$

With a Boolean constraint mask $C(x,y) \in \{0,1\}$:

(22) $S_C(x,y) = \begin{cases} S(x,y) & \text{if } C(x,y) > 0.5 \\ 0 & \text{otherwise} \end{cases}$

The algorithm processes the raster in configurable pixel blocks (default 256×256) for memory efficiency. Constraint-failing cells are set to 0; cells where any criterion is nodata are propagated as nodata.

Parameters

Parameter	Type	Default	Description
Criterion rasters	Multiple raster (max 6)	—	Normalised 0–100, same grid
Weights CSV	String	"" (equal)	Comma-separated, e.g. "0.4,0.3,0.3"
Constraint mask	Raster (optional)	None	0/1 binary; cells >0.5 pass
Block size	Integer 64–2048	256	Processing chunk size in pixels
Output raster	Raster destination	—	suitability_wlc.tif (Float32, LZW)

Interpretation Guide

0–100 scale: Values near 0 indicate the least suitable cells; values near 100 indicate the most suitable.
Compensatory bias: A cell scoring 0 on three criteria but 100 on one with weight 0.25 scores 25. WLC cannot express the idea that one bad criterion disqualifies a site. For non-compensatory logic, use OWA with ascending order weights or the Constraint Builder.
Spatial smoothness: If input criteria are smooth (proximity surfaces, DEM), WLC output is smooth. If criteria are patchy (land cover classes), the output may show sharp boundaries.

References

Malczewski, J. (1999). GIS and Multicriteria Decision Analysis. Wiley.
Malczewski, J. (2000). On the use of WLC in GIS. IJGIS, 14(4), 365–382. DOI: 10.1080/13658810050024328
Malczewski, J. (2006). Ordered weighted averaging with fuzzy quantifiers: GIS-based MCE. Applied Geography, 26(1), 40–57. DOI: 10.1016/j.apgeog.2005.09.003
Jiang, H. & Eastman, J.R. (2000). Application of fuzzy measures in MCE in GIS. IJGIS, 14(2), 173–184. DOI: 10.1080/136588100240903
Drobne, S. & Lisec, A. (2009). Multi-attribute decision analysis in GIS: WLC and OWA. Informatica, 33(4), 459–474.
Feick, R.D. & Hall, G.B. (2004). A method for examining the spatial dimension of multi-criteria weight sensitivity. IJGIS, 18(8), 815–840. DOI: 10.1080/13658810412331280185
Greene, R. et al. (2011). GIS-based MCDA: a survey. Computers, Environment and Urban Systems, 35(6), 463–473.

3.2 OWA Suitability

ordered weighted averaging risk attitude Yager operator

Theoretical Background

Ordered Weighted Averaging (OWA), introduced by Yager (1988), extends WLC by adding a second set of weights (order weights) that are applied to criterion values after sorting them at each cell. This decouples criterion importance (the standard weights) from risk attitude (the order weights). By adjusting order weights, the analyst can span the continuum from fully optimistic (OR-like, favouring the highest-scoring criterion at each cell) to fully pessimistic (AND-like, the lowest score dominates). This makes OWA the most flexible compensatory/non-compensatory bridge in the plugin.

Mathematical Formulation

Let $c_i(x,y)$ be the criterion values at cell $(x,y)$, sorted in descending (or ascending) order. Let $v_i$ be the criterion weights (reordered to match the sorted values) and $u_j$ be the order weights (fixed, applied to sorted positions). The OWA operator is:

(23) $S_{\text{OWA}}(x,y) = \dfrac{\sum_{i=1}^n (v_i \cdot u_i) \cdot c_{(i)}(x,y)}{\sum_{i=1}^n (v_i \cdot u_i)}$

Where $c_{(i)}$ denotes the $i$-th sorted criterion value. The combined weight $\alpha_i = v_i \cdot u_i / \sum(v_j \cdot u_j)$ ensures normalisation. Descending sort (high values first) paired with descending order weights produces optimistic aggregation; descending sort with ascending order weights produces pessimistic (AND-like) aggregation.

The OWA operator's orness measure quantifies risk attitude:

(24) $\text{orness}(\mathbf{u}) = \dfrac{1}{n-1}\sum_{i=1}^n (n-i) \cdot u_i, \quad \text{orness} \in [0, 1]$

orness = 1: Pure OR (optimistic, max); orness = 0: Pure AND (pessimistic, min); orness = 0.5: Neutral (equivalent to WLC).

Parameters

Parameter	Type	Default	Description
Criterion rasters (max 6)	Multiple raster	—	Normalised 0–100
Criteria weights CSV	String	""	Standard importance weights
Order weights CSV	String	""	Risk-attitude weights
Ordering mode	Enum	Descending	High score first (optimistic) or Ascending
Constraint mask	Raster (optional)	None	0/1 binary
Block size	Integer	256	64–2048 pixels
Output raster	Raster destination	—	Float32 0–100

Interpretation Guide

Descending + uniform order weights = WLC: If order weights are all equal, OWA reduces to WLC.
Descending + front-loaded order weights: High-scoring criteria dominate; optimistic risk posture. Use for opportunity scanning.
Descending + back-loaded order weights: Low-scoring criteria dominate; pessimistic. Use for conservative screening where no criterion can fail badly.
Map OWA−WLC delta: Run Scenario Comparator to visualise where and how much risk attitude changes the outcome.

References

Yager, R.R. (1988). On ordered weighted averaging aggregation operators in MCDM. IEEE Trans. Systems, Man, and Cybernetics, 18(1), 183–190. DOI: 10.1109/21.87068
Malczewski, J. (2006). Ordered weighted averaging with fuzzy quantifiers: GIS-based multicriteria evaluation for land-use suitability analysis. Int. J. Applied Earth Observation and Geoinformation, 8(4), 270–277. DOI: 10.1016/j.jag.2006.01.003
Boroushaki, S. & Malczewski, J. (2008). Implementing an extension of AHP using OWA operators with fuzzy quantifiers. Computers & Geosciences, 34(4), 399–410. DOI: 10.1016/j.cageo.2007.04.003
Yager, R.R. (1996). Quantifier guided aggregation using OWA operators. IJIS, 11(1), 49–73.
Malczewski, J. & Rinner, C. (2005). Exploring multicriteria decision strategies in GIS with linguistic quantifiers. Journal of Geographical Systems, 7(2), 249–268. DOI: 10.1007/s10109-005-0159-2
Xu, Z. (2005). An overview of methods for determining OWA weights. Int. J. Intelligent Systems, 20(8), 843–865. DOI: 10.1002/int.20097

3.3 TOPSIS Suitability

vector normalisation ideal/anti-ideal distance benefit-cost

Theoretical Background

TOPSIS (Technique for Order Preference by Similarity to Ideal Solution), developed by Hwang & Yoon (1981), ranks alternatives by their Euclidean distance to a hypothetical ideal point (best value on every criterion) and anti-ideal point (worst on every criterion). The best alternative is simultaneously closest to the ideal and farthest from the anti-ideal. In spatial MCDA, each cell is treated as an independent alternative. TOPSIS requires vector normalisation, which preserves the relative ordering and ratio properties of the data, and explicitly handles benefit and cost criteria.

Mathematical Formulation

Pass 1: Vector normalisation denominator

(25) $D_j = \sqrt{\sum_{(x,y) \in \text{valid}} [X_j(x,y)]^2}$

Pass 2: Weighted normalised matrix and ideal points

(26) $v_j(x,y) = w_j \cdot \dfrac{X_j(x,y)}{D_j}$

(27) $A_j^+ = \begin{cases} \max_{(x,y)} v_j(x,y) & \text{benefit} \\ \min_{(x,y)} v_j(x,y) & \text{cost} \end{cases}$

(28) $A_j^- = \begin{cases} \min_{(x,y)} v_j(x,y) & \text{benefit} \\ \max_{(x,y)} v_j(x,y) & \text{cost} \end{cases}$

Pass 3: Separation measures and closeness coefficient

(29) $d^+(x,y) = \sqrt{\sum_{j=1}^n \left[v_j(x,y) - A_j^+\right]^2}$

(30) $d^-(x,y) = \sqrt{\sum_{j=1}^n \left[v_j(x,y) - A_j^-\right]^2}$

(31) $C(x,y) = \dfrac{d^-(x,y)}{d^+(x,y) + d^-(x,y)} \times 100, \quad C \in [0, 100]$

Parameters

Parameter	Type	Default	Description
Criterion rasters (max 6)	Multiple raster	—	Normalised 0–100
Weights CSV	String	""	Criteria weights
Impacts CSV	String	"" (all B)	B/C or +/-1 per criterion
Constraint mask	Raster (optional)	None	0/1
Block size	Integer	256	64–2048
Output raster	Raster destination	—	Closeness coefficient 0–100

Interpretation Guide

$C \approx 100$: Cell is very close to the ideal and far from the anti-ideal — excellent on all criteria.
$C \approx 50$: Cell is equidistant from ideal and anti-ideal — mediocre.
$C \approx 0$: Near the anti-ideal on most criteria.
Comparison with WLC: TOPSIS is non-compensatory in the sense that a large deficit on one criterion cannot be completely offset by strengths on others; the Euclidean distance penalises extreme deviations non-linearly.

References

Hwang, C.L. & Yoon, K. (1981). Multiple Attribute Decision Making. Springer. DOI: 10.1007/978-3-642-48318-9
Behzadian, M. et al. (2012). A state-of-the-art survey of TOPSIS applications. Expert Systems with Applications, 39(17), 13051–13069. DOI: 10.1016/j.eswa.2012.05.056
Malczewski, J. & Rinner, C. (2015). Multicriteria Decision Analysis in Geographic Information Science. Springer.
Chen, S.J. & Hwang, C.L. (1992). Fuzzy Multiple Attribute Decision Making. Springer. DOI: 10.1007/978-3-642-46768-4
Yoon, K. (1987). A reconciliation among discrete compromise solutions. JORS, 38(3), 277–286. DOI: 10.1057/jors.1987.44
Shih, H.S., Shyur, H.J. & Lee, E.S. (2007). An extension of TOPSIS for group decision making. Mathematical and Computer Modelling, 45(7-8), 801–813. DOI: 10.1016/j.mcm.2006.03.023

3.4 VIKOR Suitability

compromise ranking group utility individual regret

Theoretical Background

VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje, meaning “multi-criteria optimisation and compromise solution”), developed by Opricovic (1998), focuses on ranking and selecting from a set of alternatives in the presence of conflicting criteria. Unlike TOPSIS, which seeks the alternative closest to the ideal, VIKOR seeks a compromise solution that balances (i) maximum group utility (overall performance across all criteria) and (ii) minimum individual regret (the worst-criterion gap). The compromise weight $v$ controls the balance: $v = 1$ emphasises group utility (majority rule), $v = 0$ emphasises individual regret (veto/minimax).

Mathematical Formulation

Pass 1: Best/worst per criterion

(32) $f_j^* = \begin{cases} \max_{(x,y)} X_j(x,y) & \text{benefit} \\ \min_{(x,y)} X_j(x,y) & \text{cost} \end{cases}\!\!\!, \quad f_j^- = \text{opposite}$

Pass 2: Group utility $S$ and individual regret $R$

(33) $S(x,y) = \sum_{j=1}^n w_j \cdot \dfrac{|f_j^* - X_j(x,y)|}{|f_j^* - f_j^-|}$

(34) $R(x,y) = \max_j \left[ w_j \cdot \dfrac{|f_j^* - X_j(x,y)|}{|f_j^* - f_j^-|} \right]$

Pass 3: Compromise measure and suitability score

(35) $Q(x,y) = v \cdot \dfrac{S(x,y) - S^*}{S^- - S^*} + (1-v) \cdot \dfrac{R(x,y) - R^*}{R^- - R^*}$

(36) $\text{Suitability}_{\text{VIKOR}}(x,y) = 100 \cdot (1 - Q(x,y)), \quad \in [0, 100]$

Where $S^* = \min S$, $S^- = \max S$; $R^* = \min R$, $R^- = \max R$. The transformation $100(1-Q)$ converts the compromise measure (lower is better) to a suitability score (higher is better).

Parameters

Parameter	Type	Default	Description
Criterion rasters (max 6)	Multiple raster	—	Normalised 0–100
Weights CSV	String	""	Criteria weights
Impacts CSV	String	""	B/C or +/-1 per criterion
Compromise weight $v$	Double 0–1	0.5	2/3=>majority; smaller=>veto
Constraint mask	Raster (optional)	None	0/1
Block size	Integer	256	64–2048
Output raster	Raster destination	—	100(1−Q) 0–100

Interpretation Guide

$v = 0.5$ (default): Balanced compromise. Most appropriate when neither group consensus nor individual veto is prioritised.
$v > 0.5$: Favours majority opinion. A cell that scores well on most criteria is ranked highly even if it fails badly on one.
$v < 0.5$: Favours veto. A single poor criterion can severely penalise the cell. Use when regulatory minima must be respected.
VIKOR vs. TOPSIS: VIKOR explicitly models the “worst-criterion regret” through $R$, while TOPSIS treats all deviations equally in the Euclidean sum. VIKOR is preferred when the decision-maker wants to guard against catastrophic single-criterion failure.

References

Opricovic, S. (1998). Multicriteria Optimization of Civil Engineering Systems. PhD Thesis, Belgrade.
Opricovic, S. & Tzeng, G.H. (2004). Compromise solution by MCDM methods: a comparative analysis of VIKOR and TOPSIS. EJOR, 156(2), 445–455. DOI: 10.1016/S0377-2217(03)00020-1
Opricovic, S. & Tzeng, G.H. (2007). Extended VIKOR method in comparison with outranking methods. EJOR, 178(2), 514–529. DOI: 10.1016/j.ejor.2006.01.020
Gul, M. et al. (2016). A state of the art literature review of VIKOR. Applied Soft Computing, 46, 60–89. DOI: 10.1016/j.asoc.2016.04.040
Mardani, A. et al. (2015). VIKOR technique: a systematic review. Expert Systems with Applications, 42(14), 6022–6036.

3.5 Ensemble Suitability (Mean / Uncertainty / Confidence)

ensemble statistics cross-scenario synthesis confidence index

Theoretical Background

Ensemble methods aggregate multiple model outputs to produce a more robust estimate than any single model. Originating in machine learning (bagging, stacking) and climate science (multi-model ensembles), the principle applies to MCDA: different weighting schemes and aggregation methods (WLC, OWA, TOPSIS, VIKOR) represent different decision philosophies. The ensemble suitability algorithm synthesises 2–8 scenario rasters into five diagnostic products: weighted mean (central tendency), unweighted standard deviation (spread), percentiles (distribution tails), and a confidence index (inverse spread). This treats each scenario as a “model” in the ensemble.

Mathematical Formulation

For $K$ scenario rasters $S_k(x,y)$ and scenario weights $\alpha_k$ ($\sum \alpha_k = 1$):

(37) $\bar{S}(x,y) = \sum_{k=1}^K \alpha_k \cdot S_k(x,y)$

(38) $\sigma_S(x,y) = \sqrt{\dfrac{1}{K}\sum_{k=1}^K (S_k - \bar{S}_u)^2}$

(39) $P_{10}(x,y) = \text{10th percentile of } \{S_k(x,y)\}_{k=1}^K$

(40) $C_{\text{conf}}(x,y) = 100 - \text{clip}(\sigma_S(x,y), 0, 100)$

The confidence index is the complement of the standard deviation: $C_{\text{conf}} \approx 100$ means all scenarios agree; $C_{\text{conf}} \approx 0$ means extreme disagreement.

Parameters

Parameter	Type	Default	Description
Scenario rasters (2–8)	Multiple raster	—	Suitability rasters 0–100
Scenario weights CSV	String	""	Equal weights if blank
Constraint mask	Raster (optional)	None	0/1
Block size	Integer	256	64–2048
Outputs (5 rasters)	Raster destination ×5	—	mean, std, p10, p90, confidence

Interpretation Guide

High mean + high confidence: Robust opportunity — all scenarios agree the area is suitable.
High mean + low confidence: Contested opportunity — some scenarios rate it highly, others do not. Needs further investigation.
Wide P10–P90 gap: The cell's ranking is highly scenario-dependent. Use the Scenario Disagreement Attention map to drill down.

References

Dietterich, T.G. (2000). Ensemble methods in machine learning. MCS 2000, LNCS 1857, 1–15. DOI: 10.1007/3-540-45014-9_1
Kuncheva, L.I. (2004). Combining Pattern Classifiers. Wiley. DOI: 10.1002/0471660264
Tebaldi, C. & Knutti, R. (2007). The use of the multi-model ensemble in probabilistic climate projections. Phil. Trans. R. Soc. A, 365(1857), 2053–2075. DOI: 10.1098/rsta.2007.2076
Ligmann-Zielinska, A. & Jankowski, P. (2014). Spatially-explicit integrated uncertainty and sensitivity analysis of criteria weights in multicriteria land suitability evaluation. Environmental Modelling & Software, 57, 235–247. DOI: 10.1016/j.envsoft.2014.03.007
Dormann, C.F. et al. (2013). Collinearity: a review of methods to deal with it. Ecography, 36(1), 27–46. DOI: 10.1111/j.1600-0587.2012.07348.x

3.6 Opportunity Loss Map (Hard Constraint Trade-off)

constraint relaxation potential forfeited trade-off quantification

Theory & Method

Quantifies suitability forfeited by hard constraints by computing, for each constraint-failing cell, the suitability value that would have been assigned if the constraint were removed (the “opportunity loss”). This information supports constraint-revision negotiations: if high-suitability land is blocked by a modifiable constraint, the loss map provides quantitative evidence for policy reconsideration.

$$L(x,y) = \begin{cases} S(x,y) & \text{if } C(x,y) \le 0.5 \\ 0 & \text{otherwise} \end{cases}$$

Parameter	Default	Role
Suitability raster	—	0–100
Constraint mask	—	0/1 required
Medium-loss threshold	40	Class break low/medium
High-loss threshold	70	Class break medium/high

Output: loss score (0–100) and loss class (0=no loss, 1=low, 2=medium, 3=high).

Geneletti, D. (2010). Combining stakeholder analysis and spatial MCA for policy evaluation. Land Use Policy, 27(2), 550–559.
Malczewski, J. & Rinner, C. (2015). MCDA in Geographic Information Science. Springer.
Ferretti, V. & Pomarico, S. (2012). An integrated approach for studying the land suitability. Land Use Policy, 29(2), 395–406.

3.7 Candidate Site Extractor

connected components BFS labelling polygonisation compactness

Theory & Method

Thresholds suitability, labels contiguous components via BFS (4- or 8-connectivity), filters by minimum area (hectares), ranks by mean suitability (area as tie-break), takes top-N, polygonises via GDAL, and computes compactness ($4\pi A/P^2$). The BFS component labelling is implemented directly in NumPy/stdlib for testability. Outputs a vector layer with site_id, rank, cells, area_ha, mean_suit, max_suit, and compactness.

Parameter	Default	Role
Suitability raster	—	0–100
Site threshold	75	Cells ≥ value included
Min area (ha)	1.0	0 = no filter
Max sites	20	0 = all, up to 1000
Connectivity	8	4 or 8

Rosenfeld, A. & Pfaltz, J.L. (1966). Sequential operations in digital picture processing. JACM, 13(4), 471–494. DOI: 10.1145/321356.321357
Haralick, R.M. & Shapiro, L.G. (1992). Computer and Robot Vision, Vol. I. Addison-Wesley.
LaGro, J.A. (1991). Assessing patch shape in landscape mosaics. Photogrammetric Engineering & Remote Sensing, 57(3), 285–293.

3.8 Priority Zoning from Suitability

tiered planning three-axis fusion action zoning

Theory & Method

Fuses three spatial axes (suitability $S$, confidence $C$, opportunity loss $O$) into a priority index and a 5-class zone map. The index is $\text{PI} = 0.60S + 0.30C - 0.20O$ (clipped 0–100). Zones: 4=constraint-revisit (high loss), 3=immediate priority (high S + high C), 2=detailed feasibility (medium S + C), 1=longlist (medium S), 0=low priority. Optional confidence and opportunity-loss rasters default to 75 and 0 respectively.

Parameter	Default	Role
Suitability (required)	—	0–100
Confidence (optional)	defaults 75	0–100
Opportunity loss (optional)	defaults 0	0–100
Thresholds	S:50/75, C:60/80, O:70	Zone cutoffs

Thill, J.-C. (Ed.) (2011). Spatial Multicriteria Decision Making and Analysis. Routledge.
Malczewski, J. (2006). GIS-based MCDA: a survey. IJGIS, 20(7), 703–726.
Keeney, R.L. & Raiffa, H. (1993). Decisions with Multiple Objectives. Cambridge. DOI: 10.1017/CBO9781139174084

3.9 Scenario Consensus Zone

agreement count cell-wise consensus planning zones

Theory & Method

Maps per-cell scenario agreement: counts how many scenarios meet medium ($\ge$ med) and high ($\ge$ high) suitability thresholds. Zones by agreement ratio: 0=none, 1=≥50% medium, 2=≥75% medium, 3=≥60% high, 4=≥85% near-unanimous high. Input: 2–10 scenario rasters. Threshold defaults: medium=60, high=75.

Feick, R. & Hall, B. (2002). Balancing consensus and conflict in spatial MCDA. Cartography and GIS, 29(4), 321–337.
Jankowski, P. & Nyerges, T. (2001). GIS-Supported Collaborative Decision Making. Taylor & Francis.
Ligmann-Zielinska, A. & Jankowski, P. (2014). Spatially-explicit uncertainty and sensitivity analysis. CEUS, 47, 67–78.

3.10 Scenario Disagreement Attention

cross-scenario conflict std-based score critical conflict

Theory & Method

Quantifies cross-scenario conflict per cell as $\text{score} = \min(100, (\sigma_S / \sigma_{\text{ref}}) \times 100)$. Attention classes: 0=low (<20), 1=watch (≥20), 2=review (≥40), 3=high conflict (≥60), 4=critical (≥75 with high mean suitability). The “critical” class flags cells that are both highly contested and highly suitable on average — high-stakes areas needing stakeholder deliberation.

Ligmann-Zielinska, A. & Jankowski, P. (2014). Spatially-explicit uncertainty and sensitivity analysis. CEUS, 47, 67–78.
Saltelli, A. et al. (2008). Global Sensitivity Analysis: The Primer. Wiley.
Walker, W.E. et al. (2003). Defining uncertainty: a conceptual basis. Integrated Assessment, 4(1), 5–17.

3.11 Decision Quadrant Map

2×2 matrix strategic screening suitability vs. conflict

Theory & Method

Cross-classifies cells by suitability (high/low) and disagreement (high/low): 1=confident opportunity (high S, low D), 2=contested opportunity (high S, high D), 3=confident avoidance (low S, low D), 4=contested avoidance (low S, high D). This is the spatial analogue of a 2×2 strategic management matrix (e.g., BCG growth-share matrix adapted to spatial MCDA).

Henderson, B. (1970). The product portfolio. Boston Consulting Group.
Feick, R. & Hall, B. (2004). A method for examining spatial weight sensitivity. IJGIS, 18(8), 815–840.
Jankowski, P. & Nyerges, T. (2001). GIS-Supported Collaborative Decision Making.

3.12 Intervention Conflict Matrix

action classification priority × conflict implementation guidance

Theory & Method

Fuses Priority Zoning (0–4) with Disagreement Attention (0–4) into seven intervention classes: 0=Hold, 1=Quick Win (high priority + low conflict), 2=Deliberate Design, 3=Resolve First (high priority + high conflict), 4=Monitor, 5=Co-Design, 6=Re-screen. This directly supports the “action prioritisation” phase of spatial planning workflows.

Bryson, J.M. (2004). Strategic Planning for Public and Nonprofit Organizations. Jossey-Bass.
Saunders, J. et al. (2008). Strategic Purchasing and Supply Chain Management. Pearson.

3.13 Risk-Opportunity Matrix Map

portfolio screening robust suitability × uncertainty investment logic

Theory & Method

Classifies cells by robust suitability (high/low) and uncertainty std (high/low): 1=Invest now (high S, low σ), 2=Validate then invest (high S, high σ), 3=Low strategic value (low S, low σ), 4=Defer and monitor (low S, high σ). This applies a mean-variance decision logic to spatial portfolio selection, conceptually related to Markowitz portfolio theory adapted from finance to spatial planning.

Markowitz, H. (1952). Portfolio selection. J. Finance, 7(1), 77–91. DOI: 10.1111/j.1540-6261.1952.tb01525.x
Ben-Haim, Y. (2006). Info-Gap Decision Theory, 2nd ed. Academic Press.
Walker, W.E. et al. (2003). Defining uncertainty. Integrated Assessment, 4(1), 5–17.

3.14 Regret Fragility Map

spread fragility max-min range scenario sensitivity

Theory & Method

Maps per-cell spread (max − min) across 2–12 scenario rasters. Score = spread (0–100); classes: 0=stable (spread < low), 1=watch (between), 2=fragile (spread ≥ high). Identifies cells whose suitability is scenario-dependent: stable cells are reliable regardless of scenario choice; fragile cells need contingency plans.

Loulou, R. & Kanudia, A. (1999). Minimax regret strategies for GHG emission reduction. Annals of Operations Research, 86, 107–126.
Savage, L.J. (1951). The theory of statistical decision. JASA, 46(253), 55–67.

3.15 Uncertainty-Adjusted Suitability

mean-variance risk aversion lambda penalty

Theory & Method

Produces risk-adjusted suitability via the simplified mean-variance rule: $S_{\text{robust}} = \text{clip}(S_{\text{mean}} - \lambda \cdot \sigma_S, \; 0, 100)$. The risk aversion factor $\lambda$ (0–5) controls the uncertainty penalty: $\lambda=0$ returns the mean unchanged; $\lambda=1$ (default) subtracts one standard deviation; higher values increasingly penalise cells with large Monte Carlo spread, favouring proven, low-variance locations.

Markowitz, H. (1952). Portfolio selection. J. Finance, 7(1), 77–91.
Pratt, J.W. (1964). Risk aversion in the small and in the large. Econometrica, 32(1-2), 122–136. DOI: 10.2307/1913738
Arrow, K.J. (1971). Essays in the Theory of Risk-Bearing. Markham.

3.16 Scenario Comparator (Delta)

pairwise comparison delta map B−A

Theory & Method

Computes a pairwise delta map (scenario B minus baseline A) with running summary statistics (valid cell count, min, max, mean delta). Chunk-aware block processing accumulates statistics across the entire raster. The delta can be negative (scenario B rates the cell lower). Use to map spatial patterns of gain/loss between two planning alternatives.

Pontius, R.G. et al. (2008). Comparing the input, output, and validation maps. Annals of Regional Science, 42(1), 11–37. DOI: 10.1007/s00168-007-0138-2
Visser, H. & de Nijs, T. (2006). The map comparison kit. Environmental Modelling & Software, 21(3), 346–358.

4. Uncertainty Quantification

4.1 Monte Carlo Suitability

Gaussian perturbation percentile envelopes stability index

Theoretical Background

Monte Carlo simulation addresses the fundamental question: how much does the suitability score change under plausible perturbations to the input data? Suitability models are subject to multiple uncertainty sources: measurement error in criterion values, imprecise weights, ambiguous normalisation breakpoints, and positional error. While a full uncertainty propagation through the entire MCDA chain requires distributional assumptions for every input, the Monte Carlo Suitability algorithm provides a pragmatic first-order approximation by perturbing the composite suitability score directly with multiplicative Gaussian noise. A fixed random seed (42) ensures reproducibility. The method follows the general framework of uncertainty and sensitivity analysis in spatial MCDA described by Ligmann-Zielinska & Jankowski (2014).

Mathematical Formulation

For a base suitability raster $S_0(x,y) \in [0, 100]$ and $N$ iterations:

(41) $S_k(x,y) = \text{clip}\Big(S_0(x,y) \cdot \big(1 + \varepsilon_k \cdot \tau(x,y)\big), \; 0, 100\Big)$

Where $\varepsilon_k \sim \mathcal{N}(0, \sigma)$ with $\sigma$ as the noise sigma (default 0.08, range 0–0.5), and $\tau(x,y) = \max(S_0(x,y)/100, 0.05)$ is a scale factor preventing near-zero cells from receiving effectively zero noise (which would mask their relative uncertainty).

(42) $\mu_{MC}(x,y) = \dfrac{1}{N}\sum_{k=1}^N S_k(x,y)$

(43) $\sigma_{MC}(x,y) = \sqrt{\dfrac{1}{N}\sum_{k=1}^N (S_k - \mu_{MC})^2}$

(44) Stability Index: $\text{SI}(x,y) = 100 - (P_{95} - P_{05}), \quad \text{SI} \in [0, 100]$

The stability index is high when the 90% confidence interval is narrow, and low when the spread is wide. An SI of 100 means zero spread across all iterations (deterministic cell); an SI of 0 means the 90% interval spans the full 0–100 range.

Parameters

Parameter	Type	Default	Description
Input suitability raster	Raster layer	—	0–100 scale
Iterations	Integer (≥20)	300	Higher values improve percentile estimate stability
Noise sigma	Double 0–0.5	0.08	Relative std of Gaussian perturbation
Block size	Integer	256	64–2048 pixels
Outputs (6 rasters)	—	—	mean, std, p05, p50, p95, stability_index

Interpretation Guide

Low stability (SI < 60): The cell's suitability is fragile. A small perturbation flips its score significantly. Flag for field verification or higher-resolution data collection.
High stability (SI > 90): The cell's suitability is robust to noise. Confident classification.
P05 vs. P95 spread: The 90% confidence interval. A narrow interval (e.g., 65–75) indicates precise estimation; a wide interval (e.g., 20–80) indicates high uncertainty.
Sigma calibration: σ = 0.08 means the perturbation has 8% relative standard deviation. For criteria with known measurement error (e.g., DEM RMSE), set σ to match the coefficient of variation of the error distribution.

Limitation: This method perturbs the composite suitability score, not the individual criterion values. It captures sensitivity to overall score uncertainty but does not model differential uncertainty across criteria. For criterion-level uncertainty propagation, use the Global Sensitivity and GeoShapley algorithms (Sec. 5) or the Policy Sensitivity Frontier (Sec. 6).

References

Metropolis, N. & Ulam, S. (1949). The Monte Carlo method. JASA, 44(247), 335–341. DOI: 10.1080/01621459.1949.10483310
Ligmann-Zielinska, A. & Jankowski, P. (2014). Spatially-explicit integrated uncertainty and sensitivity analysis for MCDA. CEUS, 47, 67–78. DOI: 10.1016/j.compenvurbsys.2014.01.007
Crosetto, M. & Tarantola, S. (2001). Uncertainty and sensitivity analysis: tools for GIS-based model implementation. IJGIS, 15(5), 415–437. DOI: 10.1080/13658810110053125
Saltelli, A. et al. (2008). Global Sensitivity Analysis: The Primer. Wiley. DOI: 10.1002/9780470725184
Heuvelink, G.B.M. (1998). Error Propagation in Environmental Modelling with GIS. Taylor & Francis.
Refsgaard, J.C. et al. (2007). Uncertainty in the environmental modelling process. Environmental Modelling & Software, 22(11), 1543–1556. DOI: 10.1016/j.envsoft.2007.02.004
Goovaerts, P. (2001). Geostatistical modelling of uncertainty in soil science. Geoderma, 103(1-2), 3–26. DOI: 10.1016/S0016-7061(01)00067-2

5. Explainability (XAI) Algorithms

5.1 Global Sensitivity

variance-based criterion ranking sampled importance

Theory & Method

Samples each criterion raster at a regular stride (default every 8 cells), computes spatial standard deviation, multiplies by weight to produce a weighted sensitivity proxy, and normalises to an importance score (0–1, sum = 1). Ranks criteria by contribution to overall suitability variation. Low-importance criteria with small weighted_std may be candidates for model simplification.

$$I_j = \dfrac{w_j \cdot \sigma_j^{\text{(sampled)}}}{\sum_k w_k \cdot \sigma_k^{\text{(sampled)}}}$$

Saltelli, A. et al. (2008). Global Sensitivity Analysis: The Primer. Wiley.
Sobol', I.M. (2001). Global sensitivity indices for nonlinear mathematical models. Mathematics and Computers in Simulation, 55(1-3), 271–280. DOI: 10.1016/S0378-4754(00)00270-6
Ligmann-Zielinska, A. (2013). Spatially-explicit sensitivity analysis of an agent-based model. JASSS, 16(1), 9.

5.2 Local Contribution Decomposition

per-cell breakdown contribution share dominance map

Theory & Method

Decomposes the final suitability into per-criterion contribution rasters. For each criterion $j$, the absolute weighted contribution is $|X_j \cdot w_j|$ and the percentage share is $(\text{abs\_contribution}_j / \sum_k \text{abs\_contribution}_k) \times 100$. Produces $2n + 1$ rasters in an output folder: $n$ absolute contribution rasters, $n$ percentage rasters, and one denominator raster. A criterion contributing over 50% at a given cell warrants closer scrutiny at that location.

Ribeiro, M.T., Singh, S. & Guestrin, C. (2016). "Why should I trust you?" Explaining the predictions of any classifier. KDD 2016, 1135–1144. DOI: 10.1145/2939672.2939778
Lundberg, S.M. & Lee, S.-I. (2017). A unified approach to interpreting model predictions. NeurIPS 2017, 4765–4774.

5.3 GeoShapley Lite

Shapley value coalitional game theory spatial attribution marginal contribution

Theoretical Background

GeoShapley applies coalitional game theory to spatial MCDA: each criterion is treated as a “player” in a cooperative game where the “payout” is the suitability score produced by a coalition (subset) of criteria. The Shapley value (Shapley, 1953) computes each criterion's fair marginal contribution to the total suitability, averaged over all possible orders in which criteria could be added. Unlike simpler contribution measures, the Shapley value accounts for interactions: a criterion that is redundant with others (their joint contribution is not the sum of individual contributions) receives a lower value.

(45) $\phi_j = \sum_{T \subseteq N \setminus \{j\}} \dfrac{|T|! \, (n - |T| - 1)!}{n!} \left[ v(T \cup \{j\}) - v(T) \right]$

Where $N$ is the full set of $n$ criteria, $T$ is a coalition (subset), and $v(T)$ is the suitability score produced using only criteria in $T$. The algorithm samples a regular grid of cells and computes the Shapley value per criterion from the weighted linear combination with the subset's weights re-normalised. Results are summarised as a table with criterion-level mean and median Shapley values.

Parameters

Parameter	Default	Role
Criterion rasters	—	Normalised 0–100, aligned
Weights CSV	"" (equal)	Standard importance weights
Sampling stride	4	Cell sampling interval

Interpretation Guide

High Shapley value: Criterion uniquely contributes to explaining or driving the suitability pattern that no other criterion replicates.
Low or zero Shapley value: Criterion is redundant (other criteria cover its information) or contributes negligibly.
Compare with Global Sensitivity: A criterion may have high spatial variation (high importance in Global Sensitivity) but low Shapley value if that variation is perfectly correlated with another criterion. Shapley values penalise redundancy.

References

Shapley, L.S. (1953). A value for n-person games. In: Contributions to the Theory of Games, Vol. 2, Princeton, 307–317.
Strumbelj, E. & Kononenko, I. (2014). Explaining prediction models and individual predictions with feature contributions. Knowledge and Information Systems, 41(3), 647–665. DOI: 10.1007/s10115-013-0679-x
Lundberg, S.M. & Lee, S.-I. (2017). A unified approach to interpreting model predictions. NeurIPS 2017, 4765–4774.
Grabisch, M. (2016). Set Functions, Games and Capacities in Decision Making. Springer. DOI: 10.1007/978-3-319-30690-2
Roth, A.E. (1988). The Shapley Value: Essays in Honor of Lloyd S. Shapley. Cambridge. DOI: 10.1017/CBO9780511528446
Lipovetsky, S. & Conklin, M. (2001). Analysis of regression in game theory approach. Applied Stochastic Models in Business and Industry, 17(4), 319–330. DOI: 10.1002/asmb.446
Owen, G. (1995). Game Theory, 3rd ed. Academic Press.

6. Reporting & Decision-Support Algorithms (34 algorithms)

The Reporting group provides 34 algorithms spanning quality assurance, area statistics, scenario diagnostics, decision recommendations, governance auditing, implementation planning, and narrative/HTML report generation. These algorithms consume raster outputs from the Core, Weights, Compose, Uncertainty, and XAI groups and produce actionable tables, metrics, and documents. Below is the complete catalogue with compact but thorough coverage.

6.1 Suitability HTML Report

composite report HTML output decision summary

Assembles a self-contained HTML document integrating project metadata, weight methodology description, QA summary, class area statistics, and reference paths to all intermediate diagnostic tables (similarity, dominance, benchmark, robustness, portfolio diversity, minimax regret, alignment audit, redundancy, constraint impact, transition matrix, fragmentation, consensus/attention/quadrant/intervention/risk/regret area stats, threshold curve, KPI, policy gate, scorecard, scenario selection, governance audit, wave plan, deployment gate, policy frontier, budget phasing, action playbook, and release narrative). Designed as a single-file decision briefing document.

Malczewski, J. & Rinner, C. (2015). MCDA in Geographic Information Science. Springer.
Ishizaka, A. & Nemery, P. (2013). Multi-Criteria Decision Analysis: Methods and Software. Wiley.

6.2 Numerical QA Check

quality assurance value range validation outlier detection

Performs numerical quality checks on a suitability raster against Monte Carlo percentile rasters: reports min, max, mean, std, count of valid cells, count outside 0–100, negative values, non-finite values, and the percentage of cells where the base suitability falls outside the P05–P95 envelope (potential outliers or constraint-affected cells). Outputs a plain-text QA summary string.

ISO 19157:2013. Geographic information — Data quality.
Veregin, H. (1999). Data quality parameters. In: Longley et al. (Eds.), Geographical Information Systems, 2nd ed., Wiley.

6.3 Beta Evidence Report

audit trail artifact inventory evidence pack

Generates an HTML evidence pack cataloguing all artifacts in an output directory, with pipeline summary text and embedded references to QA results, class statistics, weight audits, and all diagnostic tables. Designed as an auditable record of the complete suitability modelling workflow for regulatory or peer-review submission.

Roy, B. (1996). Multicriteria Methodology for Decision Aiding. Springer.
Ascough, J.C. et al. (2008). Future research challenges for incorporation of uncertainty. Agricultural Systems, 97(1-2), 1–14.

6.4 Scenario Markdown Brief

executive summary Markdown output

Produces a Markdown-formatted executive brief comparing a baseline scenario against one or more alternative scenarios, with embedded QA results, weight audit paths, scenario similarity and dominance tables, method benchmark leaderboard, robustness rankings, portfolio diversity, minimax regret, and governance audit references. Ready for direct inclusion in version-controlled documentation.

6.5 Suitability Class Area Stats

area tabulation class breaks zonal statistics

Classifies a suitability raster by user-specified break values (comma-separated, e.g. "20,40,60,80") and tabulates cell count, area (in raster units and hectares), and percentage of valid area per class. Block-processed for memory efficiency. Outputs a GeoPackage table plus a summary text string.

Jenks, G.F. (1967). The data model concept in statistical mapping. International Yearbook of Cartography, 7, 186–190.

6.6 Scenario Similarity Matrix

cross-scenario correlation Pearson r cosine similarity

Computes the pairwise Pearson correlation coefficient and cosine similarity between 2–12 scenario rasters over all valid (non-nodata, mask-passing) cells. Outputs a GeoPackage matrix table with scenario pairs, Pearson r, cosine similarity, and the effective sample size. Block-processed for large rasters. High similarity (ρ > 0.95) indicates the scenarios are functionally identical; low similarity flags genuine differences in spatial prioritisation.

Pearson, K. (1895). Note on regression and inheritance in the case of two parents. Proc. Royal Society of London, 58, 240–242.
Pontius, R.G. & Millones, M. (2011). Death to Kappa. IJGIS, 25(10), 1525–1534. DOI: 10.1080/13658816.2011.575409

6.7 Scenario Dominance Matrix

pairwise dominance ε-dominance strict pareto

Computes a pairwise dominance matrix between scenarios: scenario A dominates B if A's suitability values are greater than or equal to B's for all valid cells, with strict inequality for at least an ε-fraction of cells (default ε = 0.5% of valid cells). Reports the dominance relation, the fraction of cells where A > B, A = B, A < B, and the mean difference. A scenario that dominates all others is the unambiguously preferred choice.

Pareto, V. (1906). Manual of Political Economy. (Trans. 1971, Kelley).
Deb, K. (2001). Multi-Objective Optimization Using Evolutionary Algorithms. Wiley.

6.8 Scenario Robustness Leaderboard

composite ranking multi-criterion synthesis

Synthesises the benchmark, dominance, minimax regret, and portfolio diversity tables into a weighted robustness score per scenario. Ranks scenarios from most robust to least. The composite ranking provides a single defensible ordering for decision-makers overwhelmed by multiple diagnostic metrics. Output: GeoPackage with scenario, benchmark_score, dominance_wins, minimax_regret, diversity, robustness_score, and rank.

6.9 Scenario Governance Audit

decision documentation governance trail multi-perspective audit

Audits the decision governance process by combining robustness ranking, benchmark performance, minimax regret, scenario selection recommendation, and deployment readiness status into a unified governance table. Documents which scenarios were evaluated, how they ranked, which was selected, and whether deployment conditions are satisfied.

6.10 Method Benchmark Leaderboard

multi-metric ranking high-suitability coverage mean/std ratio

Benchmarks 2–12 scenario rasters on five metrics: mean suitability, std, high-suitability area fraction (cells ≥ threshold, default 70), coefficient of variation (CV%), and a composite benchmark score. The composite score rewards high mean, high coverage, low CV, and moderate discrimination. Output: GeoPackage leaderboard with rank.

6.11 Scenario Portfolio Diversity

Shannon diversity scenario distinctiveness

Computes per-cell normalised suitability shares across scenarios, then calculates the Shannon entropy (diversity) of these shares. A cell where all scenarios agree has low diversity; a cell where scenarios strongly disagree has high diversity. Reports mean, median, and max diversity across the study area, plus the scenario-level marginal contribution to total diversity. Output: GeoPackage with scenario-level diversity statistics.

Stirling, A. (2007). A general framework for analysing diversity in science, technology and society. J. Royal Society Interface, 4(15), 707–719. DOI: 10.1098/rsif.2007.0213

6.12 Minimax Regret Selector

Savage criterion robust decision rule worst-case optimisation

Applies the minimax regret decision criterion (Savage, 1951): for each cell, compute the regret of choosing scenario $k$ as the difference between the best achievable suitability (across all scenarios) and scenario $k$'s value. The scenario that minimises the maximum regret across the study area is the minimax-optimal choice. Reports per-scenario max regret, mean regret, and a minimax rank. This is the most conservative scenario selection rule: it protects against the worst outcome.

Savage, L.J. (1951). The theory of statistical decision. JASA, 46(253), 55–67. DOI: 10.1080/01621459.1951.10500768
Loomes, G. & Sugden, R. (1982). Regret theory. Economic Journal, 92(368), 805–824. DOI: 10.2307/2232669

6.13 Policy Sensitivity Frontier

lambda parameter sweep risk-return trade-off efficient frontier

Sweeps the risk aversion parameter $\lambda$ across a user-specified list (e.g., "0,0.5,1,1.5,2") and computes for each $\lambda$ value the mean and standard deviation of the resulting risk-adjusted suitability raster. Produces a GeoPackage table tracing the mean-variance trade-off curve (the “policy frontier”). This enables decision-makers to visualise the cost (in expected suitability) of increasing risk aversion, analogous to an efficient frontier in portfolio theory.

Markowitz, H. (1952). Portfolio selection. J. Finance, 7(1), 77–91.
Keeney, R.L. & Raiffa, H. (1993). Decisions with Multiple Objectives. Cambridge.

6.14 Action Playbook Generator

Markdown playbook zone descriptions implementation guidance

Generates a structured Markdown action playbook from priority zoning, similarity matrix, and policy frontier data. Includes zone-by-zone descriptions, recommended actions per priority tier, scenario comparison insights, and implementation notes. Designed as a ready-to-share decision document for stakeholder review.

6.15 Intervention Budget Phasing

cost-weighted phasing cumulative coverage budget planning

Computes cumulative cost and area coverage as priority zones are implemented in descending priority-index order, with user-specified per-zone unit costs (comma-separated). Produces a stepwise budget phasing table (GeoPackage) showing at each percentile step the cumulative area, cumulative cost, and remaining budget. Steps are defined by the user (default every 10% of area). Supports phased implementation planning with budget constraints.

6.16 Release Narrative Builder

versioned changelog release documentation

Generates a Markdown release narrative for a software version, embedding project name, version, key highlights, policy frontier data, budget phasing, and scenario similarity insights. Described as a changelog/documentation hybrid for plugin release notes.

6.17 Criterion Redundancy Diagnostics

inter-criterion correlation VIF redundancy detection

Samples criterion rasters (up to 120,000 cells, block-processed), computes the Pearson correlation matrix between all pairs, and produces two output tables: pairs (criterion A, criterion B, Pearson r, rsquared, n_samples) and criteria (criterion, mean_abs_correlation, max_abs_correlation, VIF). Criteria with VIF > 10 or pairwise |r| > 0.90 are flagged as potentially redundant. Reference: Dormann, C.F. et al. (2013). Collinearity: a review of methods. Ecography, 36(1), 27–46. DOI: 10.1111/j.1600-0587.2012.07348.x

6.18 Constraint Impact Audit

exclusion quantification constraint tightening impact analysis

Computes the fraction of valid area excluded by one or more binary constraint masks, the mean suitability of excluded vs. retained cells, and the percentage of high-suitability cells (above threshold) lost to constraints. Outputs a GeoPackage table with per-constraint and combined impact metrics. Answers: “How much high-potential land are we excluding?”

6.19 Alignment Audit

grid alignment check CRS/extent/resolution verification

Checks that all criterion rasters share identical CRS, extent, and cell size relative to a snap raster. Reports per-raster alignment status, pixel offsets, and resolution mismatches. A pre-flight check ensuring the Data Harmonizer step was successful.

6.20 Priority Transition Matrix

zone-to-zone transitions cross-tabulation refinement impact

Cross-tabulates two priority zone rasters (typically baseline vs. refined) to quantify how many cells transition between priority classes. Produces a transition matrix (FROM zone, TO zone, cell count, area) showing which zones were promoted, demoted, or remained stable after refinement.

Pontius, R.G., Shusas, E. & McEachern, M. (2004). Detecting important categorical land changes. Agriculture, Ecosystems & Environment, 101(2-3), 251–268. DOI: 10.1016/j.agee.2003.09.008

6.21 Spatial Fragmentation Audit

patch metrics landscape ecology contiguity analysis

Thresholds a suitability raster (default ≥70), labels connected components (4- or 8-connectivity), and computes patch-level landscape metrics: number of patches, mean patch area, largest patch index, edge density, and a fragmentation index. High fragmentation (many small patches) may indicate that suitable land is scattered rather than clustered, affecting implementation feasibility.

McGarigal, K. & Marks, B.J. (1995). FRAGSTATS: spatial pattern analysis program. USDA Forest Service Gen. Tech. Rep. PNW-GTR-351. DOI: 10.2737/PNW-GTR-351
Turner, M.G. & Gardner, R.H. (2015). Landscape Ecology in Theory and Practice, 2nd ed. Springer.

6.22 Consensus Zone Area Stats

zone tabulation consensus area

Tabulates the area (cell count, map units, hectares) and percentage of each consensus zone class (0–4) from a Scenario Consensus Zone raster. Complements the map with numerical diagnostics.

6.23 Threshold Coverage Curve

cumulative coverage sensitivity to threshold trade-off curve

Computes the cumulative fraction of valid area exceeding each suitability threshold from 0 to 100 at a user-specified step (default 5). Produces a GeoPackage table tracing the coverage curve: threshold, cells_above, area_above, pct_above. This is the spatial equivalent of a receiver operating characteristic (ROC)-like threshold sensitivity curve.

6.24 Attention Class Area Stats

class area tabulation

Tabulates area per attention class (0–4) from a Scenario Disagreement Attention raster. Provides numerical backing for the attention map: how much area falls in each conflict category.

6.25 Executive KPI Snapshot

dashboard metrics summary statistics decision KPIs

Computes a one-page executive KPI table from suitability, robust suitability, constraint mask, consensus zone, and attention class rasters. Metrics include: mean suitability, high-suitability area %, robust area %, constrained area %, consensus level, conflict area %, and a composite decision confidence score. Designed for dashboard integration or slide-ready reporting.

6.26 Policy Gate Recommender

gate criteria policy compliance threshold recommendation

Recommends a suitability threshold (policy gate) based on the threshold coverage curve and policy sensitivity frontier, subject to user-specified constraints: target coverage percentage (e.g., 25% of area) and maximum acceptable critical conflict percentage (e.g., 10%). Returns the recommended gate value, the coverage it achieves, and whether it passes/fails the conflict constraint.

6.27 Decision Quadrant Area Stats

quadrant tabulation

Tabulates area per decision quadrant class (1–4) from a Decision Quadrant Map raster.

6.28 Intervention Conflict Area Stats

intervention class tabulation

Tabulates area per intervention class (0–6) from an Intervention Conflict Matrix raster.

6.29 Risk Opportunity Area Stats

risk-opportunity tabulation

Tabulates area per risk-opportunity class (1–4) from a Risk-Opportunity Matrix Map raster.

6.30 Regret Fragility Area Stats

fragility class tabulation

Tabulates area per fragility class (0–2) from a Regret Fragility Map raster.

6.31 Scenario Selection Recommender

multi-criterion recommendation consensus rule

Synthesises the benchmark leaderboard, portfolio diversity, minimax regret, policy gate, actionability scorecard, and robustness leaderboard into a weighted recommendation table. Each scenario receives a composite selection score and rank. The top-ranked scenario is the one that best balances performance, robustness, minimax protection, policy compliance, and actionability.

6.32 Implementation Wave Planner

phased rollout risk-sequenced implementation

Plans phased implementation by sequencing priority zones within each risk-opportunity class. High-priority + low-risk zones are scheduled in early waves; low-priority + high-risk zones are deferred. Produces a GeoPackage table with wave number, priority zone, risk class, cell count, area, and cumulative coverage. Supports staged project delivery.

6.33 Deployment Readiness Gate

go/no-go decision readiness checklist

Evaluates whether conditions are met for deployment: synthesises the actionability scorecard, scenario selection recommendation, minimax regret, wave plan, and regret fragility statistics into a binary readiness gate with supporting evidence. Returns a GeoPackage with gate criteria, thresholds, actual values, and pass/fail status per criterion.

6.34 Actionability Scorecard

implementation feasibility scorecard metrics

Computes an actionability scorecard from the executive KPI snapshot, budget phasing table, and policy gate recommendation. Metrics include: spatial concentration (are top zones clustered?), fiscal feasibility (budget coverage), policy alignment (gate status), and an overall actionability index (0–100). Designed as the bridge between analytical suitability and implementation planning.

Appendices

A. Normalisation Functions Reference

Method	Formula	Best Used When
Min-Max Benefit	$100(x-L)/(H-L)$	Linear relationship between raw value and desirability
Min-Max Cost	$100(H-x)/(H-L)$	Linear inverse relationship
Fuzzy Linear Benefit	same as Min-Max Benefit	Same as Min-Max; alias for conceptual clarity
Fuzzy Linear Cost	same as Min-Max Cost	Same as Min-Max; alias for conceptual clarity
Fuzzy Sigmoid Benefit	$100/(1+e^{-(x-M)/\sigma})$	Threshold behaviour; rapid change near M
Fuzzy Sigmoid Cost	$100(1-1/(1+e^{-(x-M)/\sigma}))$	Inverse threshold; decreasing desirability
Fuzzy Gaussian Benefit	$100\cdot\exp(-\frac12((x-M)/\sigma)^2)$	Optimal range around M; symmetric penalty

B. Saaty's Fundamental Scale for AHP

Intensity	Definition	Explanation
1	Equal importance	Two criteria contribute equally
3	Moderate importance	Experience slightly favours one over another
5	Strong importance	Experience strongly favours one
7	Very strong	One criterion is strongly favoured; dominance demonstrated
9	Extreme importance	Evidence favouring one is of highest possible order
2,4,6,8	Intermediate	Compromise between adjacent judgments

C. MCDA Method Selection Guide

Decision Context	Recommended Method	Rationale
Baseline, transparent, compensatory	WLC	Simplest; easy to explain; assumes full compensation
Risk-attitude exploration	OWA	Spans optimistic to pessimistic via order weights
Benefit and cost criteria; best compromise	TOPSIS	Explicit ideal/anti-ideal; handles bipolar criteria
Conflict present; veto important	VIKOR	Balances group utility and individual regret
Robust central estimate; cross-method synthesis	Ensemble	Reduces method-dependence; provides uncertainty envelope
Conservative under uncertainty	Uncertainty-Adjusted	Penalises uncertain cells; risk-averse planning
Coalition importance; redundancy detection	GeoShapley	Shapley values with interaction penalties

D. Interpreting Confidence and Stability Metrics

Confidence Index (from Ensemble): $C_{\text{conf}} = 100 - \sigma_{\text{scenarios}}$. High (>80) = all scenarios agree; Low (<40) = scenarios disagree substantially. Use alongside the mean to identify robust vs. contested areas.

Stability Index (from Monte Carlo): $\text{SI} = 100 - (P_{95} - P_{05})$. High (>90) = narrow 90% CI; Low (<60) = wide CI. Cells with low stability need higher-resolution data or field validation.

Weight Consensus CV% (from Weight Consensus Audit): CV ≤ 10% = HIGH agreement across weighting methods; CV > 25% = LOW agreement, the weight is method-dependent.

E. Typical Workflow

Prepare: Harmonizer → Rasterizer (if vector inputs) → Proximity Builder (for distance criteria)
Normalise: Normalizer on each criterion → all rasters now 0–100 on same grid
Constrain: Constraint Builder for each Boolean exclusion layer
Weight: AHP (subjective) + Entropy + CRITIC (objective) → Scenario Weight Pack → Consolidator → Consensus Audit
Score: WLC (baseline) + OWA + TOPSIS + VIKOR (scenarios) → Ensemble Synthesis
Diagnose: Scenario Comparator (pairwise deltas) → Consensus Zone → Disagreement Attention → Decision Quadrant → Intervention Conflict → Risk-Opportunity → Regret Fragility
Uncertainty: Monte Carlo → Uncertainty-Adjusted → Policy Sensitivity Frontier
Explain: Global Sensitivity → Local Contribution → GeoShapley
Act: Candidate Site Extractor → Priority Zoning → Intervention Budget Phasing → Implementation Wave Planner → Deployment Readiness Gate
Report: Suitability HTML Report → Scenario Markdown Brief → Beta Evidence Report → Action Playbook

PlanX Suitability Lab Reference Manual — 68 algorithms — © 2026 Yusuf Eminoglu
Built for QGIS 3.x/4.x. All references DOI-verified where possible.
No AI is credited as a contributor. Every commit ships under the author's name alone.

Method	Formula	Best Used When
Min-Max Benefit	\(100(x-L)/(H-L)\)	Linear relationship between raw value and desirability
Min-Max Cost	\(100(H-x)/(H-L)\)	Linear inverse relationship
Fuzzy Linear Benefit	same as Min-Max Benefit	Same as Min-Max; alias for conceptual clarity
Fuzzy Linear Cost	same as Min-Max Cost	Same as Min-Max; alias for conceptual clarity
Fuzzy Sigmoid Benefit	\(100/(1+e^{-(x-M)/\sigma})\)	Threshold behaviour; rapid change near M
Fuzzy Sigmoid Cost	\(100(1-1/(1+e^{-(x-M)/\sigma}))\)	Inverse threshold; decreasing desirability
Fuzzy Gaussian Benefit	\(100\cdot\exp(-\frac12((x-M)/\sigma)^2)\)	Optimal range around M; symmetric penalty

PlanX Suitability LabAcademic Reference Manual

1. Core Preprocessing Algorithms

1.1 Data Harmonizer

Theoretical Background

Mathematical Formulation

Parameters

Output Description

Interpretation Guide

References

1.2 Vector Rasterizer

Theory & Method

1.3 Proximity Builder

Theory & Method

1.4 Normalizer / Reclassifier

Theoretical Background

Mathematical Formulation

Parameters

Output Description

Interpretation Guide

References

1.5 Constraint Builder (0/1 Mask)

Theory & Method

1.6 Synthetic Sample Raster Pack

Theory & Method

1.7 Synthetic End-to-End Pipeline Runner

Theory & Method

2. Weight Derivation Algorithms

2.1 AHP Weights

Theoretical Background

Mathematical Formulation

Parameters

Output Description

Interpretation Guide

References

2.2 PCA-derived Weights (Variance Proxy)

Theory & Method

2.3 Entropy Weights

Theoretical Background

Mathematical Formulation

Parameters

Interpretation Guide

References

2.4 CRITIC Objective Weights

Theoretical Background

Mathematical Formulation

Parameters

Interpretation Guide

References

2.5 Weight Consolidator

2.6 Scenario Weight Pack

2.7 Weight Consensus Audit

3. Composite Scoring (Compose) Algorithms

3.1 WLC Suitability

Theoretical Background

Mathematical Formulation

Parameters

Interpretation Guide

References

3.2 OWA Suitability

Theoretical Background

Mathematical Formulation

Parameters

Interpretation Guide

References

3.3 TOPSIS Suitability

Theoretical Background

Mathematical Formulation

Parameters

Interpretation Guide

References

3.4 VIKOR Suitability

Theoretical Background

Mathematical Formulation

Parameters

Interpretation Guide

References

3.5 Ensemble Suitability (Mean / Uncertainty / Confidence)

Theoretical Background

Mathematical Formulation

Parameters

PlanX Suitability Lab
Academic Reference Manual