We then identified the target groups concerned by the problem of flooding in the study area, i.e. the members of the Emergency and Organization Service (ORSEC) in charge of flood management and the people affected by flooding. This approach made it possible to collect data on the perception of flood factors by the target groups. To do this, we collected data on soil permeability using the Lefranc test. Darcy’s law was used to calculate the soil permeability coefficient ( $k$) according to the following relationship:

$kSi=Q$ (1)

$Q$: flow rate in m/s

$i$: hydraulic gradient in m/m

$S$: Infiltration surface, corresponding to all surfaces in contact with water (in mm)²

$k$: Permeability m/s

To produce the soil permeability map, we classified the soil according to Table 1 :

This classification was used to draw up the soil permeability map, which will be used as input data for mapping susceptibility to flood risk and for determining the risk susceptibility index.

Figure 3 shows that the soils with low permeability are located in the outflow zones occupied by anarchic populations. Figure 3 will be used as input data for the hierarchical analysis of fuzzy processes.

We also collected “SRTM Plus” data (ASTER, 2013) to produce the digital terrain model.

Figure 4 shows that the study area lies in a basin at the top of which, further to the east, is a ridge covering an area of 140 hectares and with a perimeter of 6,976 meters. This ridge is 124 meters high.

Below this ridge, between 46 and 89 meters above sea level, lies the administrative centre, the primary, tertiary and quaternary sectors, and the defence and security forces.

Figure 4 will be used as an input to the hierarchical analysis of fuzzy processes. To determine the Normalized Difference Vegetation Index (NDVI), we used remote sensing techniques to calculate plant biomass and identify flooded areas and soil typology. We also used Landsat 8 Oli images to determine the NDVI. The ArcGIS Raster Calculator tool was used to apply the formula for Equation (2) of the Normalized Difference Vegetation Index (NDVI), which was used to produce the forest cover map.

$\text{NDVI}=\frac{\text{NIR}-\text{R}}{\text{NIR}+\text{R}}$ (2)

We also classified the vegetation cover data into five categories with an average amplitude of 7476 pixels. These classes are noted from 1 to 5 and are presented as follows:

Figure 5 will be used as input to the hierarchical fuzzy process analysis. We also used a handheld GPS receiver to survey refuse dumps in order to them.

We have classified the household waste dumps ( Figure 6(a) ) according to surface area into small, medium and large dumps, rated 1, 3 and 5 respectively. These scores will be used in the modelling.

Figure 6(b) shows the miscellaneous network made up of a set of linear features relating to hydrography, railways, roads and stormwater drainage channels. Data on railways and roads was collected from the land registry. Data on stormwater drainage channels was collected using GPS. Data on the hydrographic network was obtained using remote sensing. After correcting and processing the Aster imagery, we produced maps of the drainage network. The steps involved in determining the hydrographic network.

By superimposing the land parcel data on the soil permeability data layer, we were able to obtain a composite layer covering the potentially flood-prone parts of the study area.

Potentially flood-prone areas ( Figure 7 ) have been classified into five categories as follows:

These data layers are rasterised, classified, declassified and weighted from 1 to 5 according to their influence on the occurrence of floods.

We used three types of software for processing, interpreting and analysing the data: Sphinx Primo version 4.5 for editing the questionnaires, IBM SPSS for processing and analysing the perception data from the pair-wise comparison of factors, and ArcGIS version 10.5 for producing the flood hazard map and determining the physical damage.

To collect data on the perception of the factors, we administered a questionnaire to fifty (50) members of the ORSEC plan (78% from the public service, 8% from the private sector, 6% from local authorities, 6% from civil society and 2% from the Regional Development Agency (RDA)) and to people affected by the floods. We also administered a questionnaire to the people to be surveyed, and applied a sampling rate of 11%. To do this, we made a selection by superimposing the data from the two soil types with the 6404 plots affected by the effect of run-off. Using the intersection tool, we found 2580 plots, i.e. 40% of the total number, to which we applied a sampling rate of 11%, giving us a representative sample of 288 people to survey. This sample was made up of notables (55%), members of women’s groups (11%), religious guides (11%), neighborhood delegates (7%), members of community-based organizations (6%), grassroots community organizations (6%) and other categories (4%).

Hierarchical Fuzzy Process Analysis (HFPA) is the method used in this study. It is based on hierarchical process analysis (HPA), which is a multi-criteria analytical approach to decision support developed by Saaty and before its implementation [14] [15] . The AHP method is used in many fields, including multi-criteria decision making [16] [17] . It is based on complex calculations using matrix algebra [16] [18] . The AHP method reduces the complexity of a decision problem to a sequence of pairwise comparisons, which are synthesised into a ratio matrix that provides a clear rationale for ordering the more and less desirable decision alternatives [19] .

Consequently, the AHP is a fundamental step in the implementation of the AHFP method. This is why we go through the conceptualisation of the AHP, the matrix comparison by pair of factors (CMPF) which concerns the linguistic variables, the calculation of the coherences and the validation of the approach. The linguistic variables are qualitative data derived from the lessons learned by flood management stakeholders and relate to: slopes (relief), soil type (permeability), vegetation cover, the various networks (roads, railways, waterways, drainage channels), household waste and buildings ( Table 2 and Table 3 ).

P: Slope; TSP: Soil type (permeability); CV: Plant cover; RD: Miscellaneous networks; OM: Household waste; B: Buildings.

The values of these matrices are obtained by transforming the judgements into numerical values according to the Saaty scale, while respecting the principle of reciprocity [18] : each occurrence of a factor (F_ij) in Table 4 is interpreted in both rows and columns.

Using the ${ƛ}_{\max }$ value, the AHP method is used to determine the coherence index.

$\text{IC}=\frac{{ƛ}_{\max }-n}{n-1}$ (3)

IC of the members of the ORSEC Plan is 0.27 and is equal to 0.28 for people by flooding.

The consistency ratio is the ratio of the consistency index (CI) to the sum of the factors [16] [21] . The consistency ratio (CR) is used to indicate the probability that the judgements in the matrix were randomly generated [22] [23] . Its comparison with Saaty’s random Table 5 (TAS) is used to assess the degree of acceptability of the matrix [16] .

$\text{RC}=\frac{\text{IC}}{n}=5\%$ (4)

We then carry out a comparative analysis of the coherence ratio and the Saaty table. The consistency ratio should be compared with the values shown in Table 5 , which relates to the number of factors and the value of the random indices (AI) developed by [10] . It is used to assess the consistency of the approach.

Indeed, according to authors [9] [24] [25] , if the consistency ratio RC is greater than or equal to (≥) 10%, this shows that the approach is acceptable. Consequently, we found a CI of 0.27 and an OR of 5%. We assume that the data collected from the players followed an acceptable and consistent pairwise comparison approach.

After validating the consistency approach of the hierarchical process analysis, we proceed with the design and implementation of the Hierarchical Fuzzy Process Analysis (HFPA). Developing an AHPF model means first validating the AHP model, hence the existence of a hyphen between the models. This means that it first integrates the AHP conceptual data model, then validates the consistency of the comparison by pairs of factors and finally approves the approach.

This model ( Figure 8 ) shows that the AHPF is based on fuzzy logic, a discipline used in the exact sciences to take account of uncertainty and imprecision [26] . The important aspects of the fuzzy conceptual data model are the membership function, the transformation of linguistic variables into qualitative and quantitative data, and their framing by close values through the processes of fuzzification, inference engines and defuzzification.

In implementing AHPF, we follow four (4) steps with seven (7) techniques. In step 1, we design the fuzzy logic conceptual data model, which is an extension of the AHP model ( Figure 8 ).

In step 2, we carry out fuzzification, which aims to determine the degree of membership of a fuzzy input via membership functions (e.g. triangular, Gaussian, trapezoidal) towards a clear and precise output for decision support [27] . To this end, we fuzzify stakeholder perceptions by pairs of factors. Step 2 consists of four (4) techniques. Technique 1 consists of framing the arithmetic means by the close values as defined by [20] .

In implementing this technique, we apply the framing of each occurrence ( Table 6 ) by the values close to it. For example, the occurrence [PCV] which corresponds to the value of 4.24 is framed by 3 and 6, so we get the matrix [3 4,24 6]. When using technique 2, we use matrix multiplication. For example, as part of the fuzzification by pairs of factors of the data from the perception of the members of the ORSEC Plan ( Table 7 ), the PP matrix [1 1 3] is multiplied by the following matrices: PTSP [2 4.32 6]; P-CV [3 4.24 6]; PRD [3 4.92 6]; P-OM [3 4.6 6]; P-B [3 5.32 7]. We obtain the matrix [162.00 2205.38 27216.00] ( Table 8 ). We also carried out fuzzification (technique 2) by pairs of factors of the data derived from the perception of people affected by flooding for the year 2022 ( Table 8 ). The PP matrix [1 1 3] is multiplied by the following matrices: P-TSP [5 4.77 5]; P-CV [1 3.87 5]; P-RD [3 5.45 7]; P-OM [5 4.96 7]; P-B [3 5.02 7] and we obtain the matrix [225.00 2472.66 36015.00] ( Table 9 ). The fuzzification operation allows us to move from the real domain to the fuzzy domain and consists of determining the degree of membership of a value [28] .

If no information is obtained using technique 2, the fuzzification process (technique 1) must be corrected. This correction constitutes technique 3. If the application of technique 2 generates information, we proceed with technique 4, in which we calculate the normalisation of the pairwise comparison matrices using the equations below:

${\beta }_{\text{facteur}}=\text{PP}\ast \text{PPTSP}\ast \text{PPCV}\ast \text{PPRD}\ast \text{PPOM}\ast \text{PPB}$ (5)

${\alpha }_{\text{facteur}}=\sqrt[n]{\text{PP}\ast \text{PPTSP}\ast \text{PPCV}\ast \text{PPRD}\ast \text{PPOM}\ast \text{PPB}}$ (6)

Technique 4, which consists of calculating the geometric mean (Equations (5) and (6)), is also used to produce Table 8 and Table 9 .

This defuzzification stage produces precise, clear-cut information. The transformation of fuzzy information into specific information is called defuzzification [29] . In short, defuzzification is the transition from the “blurred world” to the “real world” [28] . Defuzzification can use a threshold defined subjectively or objectively by a fuzzy value [30] .

Figure 9 shows the virtual organisation of the fuzzy logic system, which starts with the input data from the universe (fuzzification or blurring), lays down rules using a set of codes for perception, control, monitoring, processing and analysis (inference rules) and ends with an understanding of the data universe (defuzzification or defuzzing). It is, therefore, the process of decomposing a data input into an output using fuzzy inference rules [31] . The literature shows the existence of triangular, trapezoidal and Gaussian membership functions. With this in mind, we have taken the triangular membership function determined by Figure 9 .

${\delta }_{\text{facteur}}={\displaystyle {\sum }_1^n{\alpha }_{\text{facteur}}}$ (7)

In step 3, we use technique 5, where each factor occurrence is normalised to unity; this involves dividing the occurrence of the factor by the sum of the occurrences of all the factors (Equation (7)).

${\theta }_{\text{facteur}}=\frac{{\alpha }_{\text{facteur}}}{{\delta }_{\text{facteur}}}$ (8)

${\theta }_{\text{facteur}}$ corresponds to the perception of a normalised factor divided by the sum of the perceptions of the normalised factors

${\lambda }_{\text{facteur}}=\frac{{{\displaystyle \sum }}_1^n\left({\theta }_{\text{facteur}}\right)}{n}$ (9)

${\lambda }_{\text{facteur}}$ the perception of a normalised factor divided by the sum of the perceptions of the normalised factors and the whole divided by the number of factors.

${\Phi }_{\text{facteur}}=\frac{{\lambda }_{\text{facteur}}}{{{\displaystyle \sum }}_1^n\left({\theta }_{\text{facteur}}\right)}$ (10)

${\Phi }_{\text{facteur}}$ corresponds to the fuzzy weighting of a factor.

In step 4, we determine the net flood risk index (NFRI), which involves the following techniques.

In technique 6, we determine the Global Factor Vulnerability Index (GFVI) according to the following relationship:

$\text{GFVI}={\displaystyle \sum {\text{CP}}_{\text{facteur}}}$ (11)

CP_facte is the factor weighting criterion.

In technique 7, we calculate the NFRI, which is the product of the rainfall hazard and the overall factor vulnerability index (IGVF). To do this, we apply Equation (12) below:

$\text{IRI}={\text{Hazard}}_{\text{rainfall}}\ast \text{GFVI}$ (12)

The rainfall hazard is uniform across the study area. The spatial uniformity of the rainfall hazard means that its value is a constant and uniformly influences the NFRI over the entire study area. Consequently, if Aléapluviometry is equal to 1, the NFRI is equal to the IGVF.

$\text{NFRI}=\text{GFVI}$ (13)