1 Introduction

Disaster is defined as “a situation or an event which overwhelms local capacity, necessitating a request to a national or international level for external assistance; an unforeseen and often sudden event that causes great damage, destruction and human suffering” (Vos et al. 2010). In addition to earthquakes, floods and storms, landslides are also one of the most commonly experienced natural disasters in the world, representing about 10 % of the natural disasters have occurred in the first half of the year 2010 (CRED 2010). Landslide susceptibility is defined as the proneness of the terrain to produce slope failures, and it is usually expressed in a cartographic way. A landslide susceptibility map depicts areas likely to have landslides in the future by correlating some of the principal factors that contribute to landslides with the past distribution of slope failures (Brabb 1984). These maps are basic tools for land-use planning, especially in mountain areas (Yalçın 2008).

The landslide susceptibility map could be based on distribution of past landslides, slope steepness, type of soil/bedrock, structure, hydrology and other pertinent data. The unique combination of several geospatial input datasets, as stated above, creates a susceptibility map representing different classes of susceptibility: low, moderate, high and very high (Lee and Min 2001). Preparing a realistic landslide susceptibility map of an area is difficult, but several attempts have been made to minimize subjectivity and error in regional scale mapping (1:50,000–1:24,000) (Greco et al. 2007; Ruff and Czurda 2008). In other words, the reliability of such maps mostly depends on the amount and quality of available data, the working scale and the selection of the appropriate methodology of analysis and modeling.

The process of creating these maps involves several qualitative or quantitative approaches. Early attempts defined susceptibility classes by qualitative overlaying of geological and morphological slope attributes to landslide inventories (Nielsen et al. 1979). Qualitative methods depend on expert opinions. The most common types of qualitative methods simply use landslide inventories to identify sites of similar geological and geomorphological properties that are susceptible to failure. Some qualitative approaches, however, incorporate the concept of ranking and weighting and may evolve to be semiquantitative in nature (Ayalew and Yamagishi 2005). More sophisticated assessments involved AHP (Komac 2006; Feizizadeh and Blaschke 2011), bivariate, multivariate, logistic regression (Carrara 1983; Lee and Min 2001; Cheng and Wang 2007), Bayesian approach (Ozdemir 2011), fuzzy logic (Ercanoglu and Gokceoglu 2004), artificial neural network analysis (Lee and Evangelista 2006; Caniani et al. 2008), modeling approaches (Iovine et al. 2003; Iovine 2008; Perriello Zampelli et al. 2012), etc. In this study, after a brief introduction of FAHP and PROMETHEE II as well-known multi-criteria decision-making (MCDM) techniques, eight landslide-related raster-based layers have been evaluated for LSM generation with such methods.

2 General situation of the region

Minoo Dasht is located on the eastern part of Golestan province, located in northern Iran. It shares boundaries with other Golestan counties such as Kolaleh county in the north and Azadshahr County in the west (Fig. 1). In Minoo Dasht, where there are 83 recorded landslide events according to the landslide inventory database (MNR 2010), the climate is a temperate and mountainous type at heights, while in plains, temperate and semi-humid climate prevails. Similarly, mean annual precipitation within the study area varies from 138 to 335 mm.

Fig. 1
figure 1

Location map of the study area

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From a geological point of view, even though there are 25 types of rock formations in this study area, most of the study area is composed of sedimentary formations (Fig. 2h). The dominant constituents of such sedimentary rocks are as follows: well-bedded to thin-bedded, greenish-gray argillaceous limestone with intercalations of calcareous shale or marl, dark gray to black fossiliferous limestone with subordinate black shale, thick bedded gray oolitic limestone, greenish marl and shale locally including gypsum and sandstone. The study area encompasses two major fault lines in north and south which both continue toward the east where the mentioned fault lines cross each other. There are also many discontinuities, or weak planes (i.e., minor faults, bedding planes, cleavage, tension joints or shear joints) are in the center (Fig. 2e). Also, there is another important reason for the high landslide frequency in the region, regarding the presence of slopes which are nearly parallel to bedding planes or to the direction of jointing and fracturing of sedimentary parent materials.

Fig. 2
figure 2

Input geodata layers (derived from 30 m SRTM DEM, 1:100,000 topography and 1:100,000 geology maps) involving: a slope (degree), b aspect, c distance to river (m), d drainage density, e distance to faults (m), f mean annual rainfall (mm), g distance to roads (m) and h lithology

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Finally, it should be mentioned that although little is known about the predisposing and triggering factors of old landslides, but by examining the recent landslides occurrence by such high frequency in the region, it must first be recognized that the rugged topography, susceptible stratigraphy and lithology along with the temperate climate causes many new landslides appear from time to time, as a result of triggering rivers erosive power, road construction activities and heavy rainfall.

3 Materials and methods

3.1 Landslide influencing data layers

The LSM in this study started with the selection and preparation of eight geodata layers in ArcGIS environment. These eight geodata layers of the study area involving slope, aspect, distance to river, drainage density, distance to fault, mean annual rainfall, distance to road and lithology (Fig. 2) were selected on the basis of similar earlier studies (Yalçın 2008; Nandi and Shakoor 2009; Feizizadeh and Blaschke 2011; Ozdemir 2011; Moradi et al. 2012; Kayastha et al. 2012). Then, a questionnaire was designed to collect necessary information required for Minoo Dasht LSM, including local experts’ opinions to ensure the practicality and the integrity of the selected geodata layers and also the importance (weight) of the approved ones. In other words, weights of the approved geodata layers were subsequently calculated using fuzzy pairwise comparisons, based on the local expert responses to the questionnaires. Eventually, to verify the actual practicality of producing LSM posterior to the map elaboration, 83 historical landslides points (including: rotational and translational slides) were used as test landslides throughout the accuracy assessment process.

3.1.1 Slope

Slope angle is the most substantial cause of landsliding (Lee and Min 2001; Lee et al. 2004) and because it directly affects the landslide process, it is frequently used in LSM (Ercanoglu and Gokceoglu 2004; Ayalew and Yamagishi 2005; Komac 2006; Guzzetti et al. 2006; Conoscenti et al. 2007; Thiery et al. 2007; Yalçin 2008). For this reason, the slope map of the study area (Fig. 2a) was prepared from a 30 m shuttle radar topography mission (SRTM) digital elevation model (DEM) and then fitted by proper preference function (Fig. 4). It also should be mentioned that according to the present landslide inventory map, and most landslides had occurred in the 25°–35° of slope ranges.

3.1.2 Aspect

Aspect is also an important influencing factor and frequently used in landslide susceptibility mapping (Ercanoglu and Gokceoglu 2004; Ayalew and Yamagishi 2005; Komac 2006; Guzzetti et al. 2006; Conoscenti et al. 2007; Thiery et al. 2007; Yalçin 2008). Aspect identifies the steepest downslope direction from each cell to its neighbors. It also can be thought of as the horizontal direction to which a mountain slope faces. Aspect-related parameters such as exposure to sunlight, drying winds, rainfall (wetness or degree of saturation) and discontinuities may control the occurrence of landslides. The aspect map of the study area and related preference function are shown in Figs. 2b and 4, respectively. Analysis shows that more than 70 % of landslides occur in northeast, east, south and southwest aspect classes.

3.1.3 Distance to river

An important parameter that controls the stability of a slope is the saturation degree of the material on the slope (Yalçin 2008). The closeness of the slope to river structures is another important factor in terms of stability. In this region, proximity to rivers is one of the most important factors affecting the occurrence of landslides. Generally, potential of landslides increases by decrease in distance to rivers, because streams may adversely affect stability by eroding the slopes or by saturating the lower part of material, resulting in water level increases (Dai et al. 2001; Yalçin 2008). The distance to stream geodata layer and its proper preference function are presented in Figs. 2c and 4. Analysis shows that about 50 % of landslides occur within 1 km of the rivers.

3.1.4 Drainage density

Here, drainage density is the length of streams per unit area (defined by search radius) within which to calculate density. In mountainous regions such as Minoo Dasht, drainage density provides an indirect measure of groundwater conditions, which have an important role to play in landslide activity. Generally, the higher the drainage density, the lower the infiltration and the faster the movement of the surface flow (Yalçin 2008). The drainage density map and its related preference function are presented in Figs. 2d and 4. It was found that about 50 % of landslides occur in the basins with drainage density varying between 1.3 and 1.7 km/km2.

3.1.5 Distance to faults

Faults are the structural features which describe a zone of weakness with relative movement, along which landslide susceptibility is higher. It has generally been observed that the probability of landslide occurrence increases as the mentioned distance decreases, which not only affect the surface material structures but also make contribution to terrain permeability causing slope instability (Kanungo et al. 2006). As it can be seen in Fig. 2e, not only landslides are apparently concentrated near faults but also occur in the blocks between faults in some parts of the map area. Resultantly, a Gaussian preference fuzzy function was applied (Fig. 4), in order to produce the conditioning geodata layer. Primary analysis results show that about 40 % of landslides occur within 2 km of the faults.

3.1.6 Mean annual rainfall

Rainfall is a recognized trigger of landslides, and investigators have long attempted to determine the amount of precipitation needed to trigger slope failures (Hong et al. 2005; Guzzetti et al. 2007). Most of the landslides occurs after heavy rainfall; thus the rainfall is one of the main parameters in LSM. Water infiltrates rapidly upon heavy rainfall and increases the degree of saturation and potential of landslide occurrence. Regarding physical characteristics of Minoo Dasht which is outlined with the rugged topography, temperate weather and high mean annual rainfall, mentioned geodata layer (Fig. 2f) fitted by proper preference function (Fig. 4), are used in LSM. It also should be noted that most of the recorded landslide events of Minoo Dasht had occurred where mean annual rainfall ranges between 220 and 300 mm.

3.1.7 Distance to roads

Proximity to roads is an important parameter in analyzing instability along roadsides in the study area. In other words, landslides may occur on the slopes intersected by roads (Ayalew and Yamagishi 2005; Yalçin 2008; Perriello Zampelli 2009). Road construction along slopes often causes local worsening of equilibrium conditions, generally due to inadequate support of cuts and/or poor compaction and drainage of fills. As a result, although a slope is balanced before the road construction, some instability may be observed because of negative effects of excavation. The proximity to road geodata layer and its related preference function are shown in Figs. 2g and 4. Analysis shows that about 30 % of landslides occur within 300 m of the roads.

3.1.8 Lithology

Landslides are greatly controlled by the lithological properties of the land surface, because lithological and structural variations often lead to a difference in the strength and permeability of rocks and soils (Ayalew and Yamagishi 2005). Lithology is one of the important factors for landslide susceptibility mapping (LSM) (Ercanoglu and Gokceoglu 2004; Ayalew and Yamagishi 2005; Conoscenti et al. 2007; Thiery et al. 2007; Yalçin 2008). Therefore, a lithology map of the study area is digitized from the existing geology map at the scale of 1:100,000 from the geological survey of Iran (GSI). As we discussed earlier, the study area consists of 25 types of rock formations which are mainly composed of typical sedimentary rock. Lithology classes and related preference function are presented in Figs. 2h and 4, respectively.

3.2 Proposed methodology

Proposed methodology has two steps: in step 1, AHP is improved by fuzzy set theory. By using fuzzy set theory in AHP method, the qualitative judgment can be qualified to make comparison more intuitionistic and reduce or eliminate assessment bias in pairwise comparison process. In step 2, obtained results have been used as input weights in PROMETHEE II algorithm.

3.2.1 Fuzzy sets theory and FAHP

To deal with vagueness of human thought, Zadeh (1965) first introduced the fuzzy set theory, which was oriented to the rationality of uncertainty due to imprecision or vagueness. A major contribution of fuzzy set theory is its capability of representing vague data. The theory also allows mathematical operators and programming to apply to the fuzzy domain. A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function, which assigns to each object a grade of membership ranging between zero and one.

A triangular fuzzy number (TFN), \( \tilde{M} \), is shown in Fig. 3. A TFN is denoted simply (m 1, m 2, m 3). The parameters m 1, m 2 and m 3, respectively, denote the smallest possible value, the most promising value and the largest possible value that describe a fuzzy event (Kahraman et al. 2003).

Fig. 3
figure 3

A triangular fuzzy number

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The analytic hierarchy process (AHP) is one of the extensively used multi-criteria decision-making methods. One of the main advantages of this method is the relative ease with which it handles multiple criteria. In addition to this, AHP is easier to understand, and it can effectively handle both qualitative and quantitative data. The use of AHP does not involve cumbersome mathematics. AHP involves the principles of decomposition, pairwise comparisons and priority vector generation and synthesis. Though the purpose of AHP is to capture the expert’s knowledge, the conventional AHP still cannot reflect the human thinking style. Therefore, fuzzy analytic hierarchy process (FAHP), a fuzzy extension of AHP, was developed to solve the hierarchical fuzzy problems. In the FAHP procedure, the pairwise comparisons in the judgment matrix are fuzzy numbers that are modified by the designer’s emphasis (Kahraman et al. 2003).

3.2.2 Extent analysis method on FAHP

In the following, first, the outlines of the extent analysis method on FAHP are given, and then, the method is applied to a supplier selection problem. Let:

$$ X = \left\{ {x_{1} ,x_{2} ,x_{3} , \ldots ,x_{n} } \right\} $$
(1)

be an object set, and

$$ U = \left\{ {u_{1} ,u_{2} ,u_{3} , \ldots ,u_{n} } \right\} $$
(2)

be a goal set.

According to Chang’s (1992) method, for extent analysis, each object is taken, and extent analysis for each goal is performed, respectively. Therefore, m extent analysis values for each object can be obtained, with the following signs:

$$ M_{{g_{i} }}^{1} ,M_{{g_{i} }}^{2} ,M_{{g_{i} }}^{3} , \ldots ,M_{{g_{i} }}^{m} \quad i = 1,2, \ldots ,n $$
(3)

where all the \( M_{{g_{i} }}^{m} \) (i = 1, 2, …, n) are TFNs. The value of fuzzy synthetic extent with respect to the ith object is defined as:

$$ \mathop S\nolimits_{i} = \sum\limits_{j = 1}^{m} {\mathop M\nolimits_{{g_{i} }}^{j} \otimes \mathop {\left[ {\sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{m} {\mathop M\nolimits_{{g_{i} }}^{j} } } } \right]}\nolimits^{ - 1} } $$
(4)

The degree of possibility of \( M_{1} \ge M_{2} \) is defined as:

$$ V(M_{1} \ge M_{2} ) = \mathop {\text{SUP}}\limits_{x \ge y} \left[ {\hbox{min} \left( {\mu M_{1} (x)} \right),\hbox{min} \left( {\mu M_{2} (y)} \right)} \right] $$
(5)

When a pair (x, y) exists such that \( x \ge y \) and \( \mu M_{1} (x) = \mu M_{2} (y) \), then, we have \( V(M_{1} \ge M_{2} ) \). Since M 1 and M 2 are convex fuzzy numbers, we have that:

$$ \left\{ {\begin{array}{*{20}c} {V(M_{1} \ge M_{2} ) = 1} \hfill & {\mathop \to \limits^{\text{if}} m_{1} \ge m_{2} } \hfill \\ {V(M_{1} \ge M_{2} ) = {\text{hgt}}\left( {M_{1} \cap M_{2} } \right) = \mu M_{1} (d)} \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right. $$
(6)

where d is the ordinate of the highest intersection point D between \( \mu M_{ 1} \) and \( \mu M_{2} \). The ordinate of D is given by Eq. (7):

$$ V(M_{1} \ge M_{2} ) = {\text{hgt}}(M_{1} \cap M_{2} ) = \frac{{m_{1} - m_{3} }}{{\left( {m_{2}^{'} - m_{3}^{'} } \right) + \left( {m_{2} - m_{3} } \right)}} $$
(7)

To compare M 1 and M 1, we need both the values of \( V(M_{1} \ge M_{2} ) \) and \( V(M_{1} \ge M_{2} ) \). The degree possibility for a convex fuzzy number to be >k convex fuzzy numbers \( M_{i} (i = 1,2, \ldots ,n) \) can be defined by:

$$ V\left( {M \ge M_{1} ,M_{2} , \ldots ,M_{k} } \right) = \hbox{min} \left( {V\left( {M \ge M_{i} } \right)} \right)\quad i = 1,2, \ldots ,k $$
(8)

Assume that:

$$ W^{'} (A_{i} ) = \hbox{min} \left\{ {V(S_{i} \ge S_{k} )} \right\}\quad k = 1, 2, \ldots ,n;\quad k \ne i $$
(9)

Then, the weight vector is given by:

$$ W^{'} (A_{i} ) = \left[ {W^{'} (A_{1} ),W^{'} (A_{2} ), \ldots ,W^{'} (A_{n} )} \right]^{T} $$
(10)

where \( A_{i} (i = 1,2, \ldots ,n) \) are n elements. Via normalization, the normalized weight vectors are:

$$ W(A_{i} ) = \left[ {W(A_{1} ),W(A_{2} ), \ldots ,W(A_{n} )} \right]^{T} $$
(11)

where W is a nonfuzzy number.

3.2.3 PROMETHEE II

The family of PROMETHEE methods was developed by Brans (1982) and further extended by Vincke and Brans (1985) to help a decision-maker (DM) rank partially (PROMETHEE I) or completely (PROMETHEE II) a finite number of options using the outranking principle. There are a considerable number of PROMETHEE applications currently available for various fields because despite its comprehensive features, and it is easily implementable (Brans et al. 1986; Goumas and Lygerou 2000; Macharis et al. 2004; Dagdeviren 2008). PROMETHEE is a superior method for ranking and selecting among a finite set of alternatives while considering a number of conflicting criteria. In addition, it is also a rather simple MCDM ranking method, both in concept and in practice, in comparison with other ones (Brans et al. 1986). Therefore, the number of practitioners who are applying the PROMETHEE method to practical MCDM problems and of researchers who are interested in the sensitivity aspects of the PROMETHEE method, has increased in recent years, as evidenced by increased number of scholarly papers and conference presentations that have used PROMETHEE (Behzadian et al. 2010). A considerable number of successful applications have been treated by the PROMETHEE methodology in various fields such as banking, industrial location, manpower planning, water resources, investments, health care, tourism, dynamic management, etc. The success of the methodology is basically due to its mathematical properties and to its particular friendliness of use (Brans and Mareschal 2005). So far, different versions of the PROMETHEE have been developed including PROMETHEE II, which is the most frequently applied version because it enables a DM to find a full-ranked vector of alternatives.

The basic principle of PROMETHEE II is based on the pairwise comparison of alternatives along each selected criterion. It requires two additional types of information: (1) information on the weights of the criteria and (2) a DM’s preference functions, which were used for comparing the alternatives (Dagdeviren 2008). PROMETHEE II stepwise procedure is presented below: (Al-Shiekh Khalil et al. 2004; Figueira et al. 2005; Behzadian et al. 2010):

Step 1

Construction of an evaluation matrix: the basic data must be prepared in the evaluation matrix in which the performance of each alternative with respect to each criterion is provided.

Step 2

Transformation of the raw data matrix to a difference matrix: for each criterion, the column entries, y, of the raw data matrix are subtracted from each other in all possible ways to create a difference, d, matrix.

$$ d_{j} (a,b) = g_{i} (a) - g_{j} (b) $$
(12)

where \( g_{i} (a) \) and \( g_{j} (b) \) show the performance of alternatives a and b, respectively, with regard to criterion j, and \( d_{j} (a,b) \) denotes the difference between these performances.

Step 3

Application of the preference function: for each criterion, the selected preference function P(a, b) is applied to decide how much the outcome a is preferred to b. Typically, there are six available choices for preference functions (Brans and Mareschal 2005), including: (1) usual criterion, (2) U shape criterion, (3) V shape criterion, (4) level criterion, (5) V shape with indifference criterion and (6) Gaussian criterion. Furthermore, for each preference function, at most two out of three parameters namely indifference (q k ), preference (p k ) and Gaussian threshold \( (\sigma_{k} ) \) must be determined by the DM (Fig. 4).

Fig. 4
figure 4

All types of used preference functions including: (criterion I) V shape with indifference criterion (including: slope and aspect) (criterion II) Gaussian criterion (including: distance to river, drainage density, distance to faults, mean annual rainfall and distance to road) and (criterion III) level criterion (lithology)

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Step 4

Calculation of an overall or global preference index:

$$ \pi (a,b) = \sum\limits_{j = 1}^{k} {w_{j} } \otimes P_{j} (a,b) $$
(13)

where \( \pi (a,b) \) denotes the overall preference of a over b, and w j is the weight associated with the jth criterion.

Step 5

Calculation of outranking flows:

$$ \varphi^{ + } (a) = \sum\limits_{x \in A} {\pi (a,x)} \quad {\text{for each}}\,a $$
(14)
$$ \varphi^{ - } (a) = \sum\limits_{x \in A} {\pi (x,a)} \quad {\text{for}}\,{\text{each}}\,a $$
(15)

The positive outranking flow, \( \varphi^{ + } (a) \), indicates how an object outranks all others, while the negative, \( \varphi^{ - } (a) \), shows how all others outrank each object. The higher the \( \varphi^{ + } (a) \)and the lower the \( \varphi^{ - } (a) \) the higher the preference for an object.

Step 6

Calculation of net outranking flows: the net outranking flow of alternative a can be calculated as follows:

$$ \varphi (a) = \varphi^{ + } (a) - \varphi^{ - } (a)\quad {\text{for}}\,{\text{each}}\,a $$
(16)

Using these net outranking flows, PROMETHEE II can provide a complete ranking of the alternatives from best to worst (Macharis et al. 2004).

4 Numerical example

Assuming that a LSM could be based on the proposed methodology, a 2-step procedure has been applied. In step 1, using MATLAB programming, hierarchy structure for assessment of geodata layer weights was assessed with the help of AHP improved by fuzzy set theory; the adopted tree is shown in Table 1. In Step 2, according to PROMETHEE II algorithm’s activities which need further programming, resultant outputs are obtained and then integrated and further analyzed using ArcGIS 9.3. The results are displayed in Figs. 5, 6, 7 and 8.

Table 1 Fuzzy evaluation matrix
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Fig. 5
figure 5

Resultant output after applying related preference functions on each parameter. a Slope, b aspect, c distance to river, d drainage density, e distance to faults, f mean annual rainfall, g distance to roads and h lithology

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Fig. 6
figure 6

PROMETHEE II landslide susceptibility map (LSM)

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Fig. 7
figure 7

ROC curve for the PROMETHEE II landslide susceptibility map (LSM)

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Fig. 8
figure 8

Histogram of calculated landslide susceptibility map showing the relative areas for each susceptibility class (susceptibility classes are labeled with the numbers of the observed landslide points accordingly)

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4.1 Step 1: assessment of weights with FAHP

After forming fuzzy pairwise comparison matrix, weights of all criteria are determined by the help of FAHP.

According to the FAHP method, firstly, synthesis values must be calculated. From Table 1, the following values are obtained:

$$ \mathop {\left[ {\sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{m} {M_{{g_{i} }}^{j} } } } \right]}\nolimits^{ - 1} = \left( {\begin{array}{*{20}c} {61.58} \hfill & {82.41} \hfill & {124.34} \hfill \\ \end{array} } \right)^{ - 1} = \left( {\begin{array}{*{20}c} {0.008} & {0.012} & {0.016} \\ \end{array} } \right) $$

S Aspect = (3.70 5.00 6.67) × (0.008 0.012 0.016) = (0.03 0.06 0.10), S River = (6.90 10.53 19.17) × (0.008 0.012 0.016) = (0.05 0.12 0.31), S Road = (5.92 9.05 13.82) × (0.008 0.012 0.016) = (0.04 0.11 0.22), S Drainage = (5.31 7.60 10.78) × (0.008 0.012 0.016) = (0.04 0.09 0.17), S Fault = (6.02 8.53 12.75) × (0.008 0.012 0.016) = ((0.05 0.10 0.21), S Lithology = (10.65 15.70 25.61) × (0.008 0.012 0.016) = (0.08 0.19 0.41), S Rain = (3.20 4.43 7.55) × (0.008 0.012 0.016) = (0.02 0.05 0.12), S Slope = (12.15 21.55 28) × (0.008 0.012 0.016) = (0.09 0.26 0.45).

These fuzzy values are compared by using Eq. (9), and obtained values are shown in Table 2:

Table 2 The ordinate of the highest intersection point and the degree possibility for TFNs
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The weight vector from Table 2 is calculated as:

$$ W^{'} { (}x_{i} )= \mathop {\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {0.32} & {0.62} & {0.52} \\ \end{array} } & {\begin{array}{*{20}c} {0.43} & {0.48} & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {0.78} & {0.33} \\ \end{array} } & 1 \\ \end{array} } \\ \end{array} } \\ \end{array} } \right\}}\nolimits^{T} $$
$$ W{ (}x_{i} )= \mathop {\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {0.0724} & {0.1390} & {0.1155} \\ \end{array} } & {\begin{array}{*{20}c} {0.0954} & {0.1071} & {\begin{array}{*{20}c} {0.1753} & {\begin{array}{*{20}c} {0.0731} & {0.2222} \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right\}}\nolimits^{T} $$

Finally, after a defuzzification process which converts TFNs to easily understandable definite values, we obtained the crisp weights (Table 3) to be used in geodata layer integration in the next step. It also should be noted that slope and lithology are the two most important geodata layers according to the obtained results, respectively.

Table 3 The calculated weight vector from FAHP and TFNs
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4.2 Step 2: application of PROMETHEE II in the Minoo Dasht LSM

To use the PROMETHEE II method, an evaluation matrix is first constructed. The eight layers are considered to be key criteria, as illustrated in Table 1. Cell values of raster datasets associated with these criteria are extracted and stored in eight separate columns and 1,698,300 rows in a table of database. Afterward, fuzzy scores of two layers and real values of other layers are applied to construct the respective \( G_{8 \times 1 6 9 , 8 3 0 , 0} \) matrix. The performance value of each alternative with regard to each criterion is extracted to generate the evaluation matrix in the first step of applying the PROMETHEE II technique. Differences between the performance values of alternatives regarding each criterion (geodata layer) are calculated and stored to be used as inputs for constructing the preference functions in the second step. There is no optimal method for choosing the most appropriate preference functions and their respective parameters at the third step; these are generally selected according to the preferences of the decision makers (DMs). Nevertheless, all applied functions of geodata layers and the resultant output raster files are shown in Figs. 4 and 5, respectively.

The FAHP method was previously applied by DMs to determine the weights of each criterion (Mahmoodzadeh et al. 2007). Normalized weights are indicated in Table 3, and the aggregated preference indices are calculated using Eq. (13) at step 4. At step 5, the positive and negative outranking flows of each alternative are calculated using Eqs. (14) and (15), respectively. Finally, the net outranking flows of the alternatives are determined during step six; these, in turn, are used to provide a complete ranking of the alternatives, from best to worst. Consequently, for providing the final susceptibility map, the net outranking flows of PROMETHE II are divided into five equal-sized intervals. Higher class labels are related to both higher values of net outranking flows and higher landslide susceptibility and vice versa. Accordingly, having combined eight resultant output in ArcGIS environment, final result is shown in Fig. 6.

4.3 Validation of the results

Validation is a fundamental step in the development of a susceptibility map and determination of its prediction ability. The prediction capability of LSM and its resultant output is usually estimated by using independent information that is not available in LSM process (i.e., landslide inventory map). Therefore, here, where we have used no training set, the accuracy of the proposed PROMETHE II and FAHP technique in Minoo Dasht was evaluated by calculating relative operating characteristics (ROC) (Fawcett 2006; Nandi and Shakoor 2009) and percentage of known landslides in various susceptibility classes.

In the ROC method, the area under the ROC curve (AUC) values, ranging from 0.5 to 1.0, is used to evaluate the accuracy of the model. The area under curve (AUC) defines the quality of the probabilistic model by describing its ability to reliably predict the occurrence or non-occurrence of an event. The ideal model shows an AUC value close to 1.0, whereas a value close to 0.5 indicates inaccuracy in the model (Fawcett 2006; Nandi and Shakoor 2009). In order to apply the ROC method, a concise and representative dataset was prepared using 83 recent and historic landslide points and 83 randomly selected non-landslide locations of the study area. The AUC value of ROC curve for the output map was found to be 0.752, with an estimated standard error of 0.04 (Fig. 7).

The LSM results were also verified using the landslides inventory map itself. Accordingly, these 83 landslide locations were overlaid on the proposed PROMETHE II map (Fig. 8). The result shows that 43 recorded landslides events (52 % of the recorded landslides) occurred in the high and very high susceptibility zones, which they only cover 73.443 km² (13.3 %) of the study area. In addition, no recorded landslide appears in the very low susceptibility zone.

In addition to the above, it should be mentioned that about 4 landslide points (4.8 % of all recorded landslides) fall into the low susceptibility class of the PROMETHEE II map which covers approximately about 615.15 km² (40.25 %) of study area (Fig. 8). These results indicate that the PROMETHE II technique is a promising tool for integrating multiple raster-based geodata layers for LSM.

5 Short discussion and conclusion

Considering the most important factors which cause the present conditions of slope instability being in place in several areas, there was a demand to conduct LSM. Two different methodologies, fuzzy analytical hierarchy process and PROMETHEE, were combined to define spatial distribution of landslide susceptibility in the study area, at a regional scale of 1:100,000. The study was conducted by means of fuzzy pixel-based analysis in ArcGIS within which the size of each pixel is 30 m × 30 m, square grid. In other words, understanding the processes that lead to landsliding in the region and the effort for subsequent LSM, we attempted to present the results of an assessment of LSM, carried out by combinational fuzzy analytical hierarchy process and PROMETHEE evaluation, of some factors that may affect the landslide susceptibility of Minoo Dasht. Therefore, the landslide susceptibility map prepared in the present study is the result of a combination of eight geodata layers responsible for landslide susceptibility, in which each one has relative importance (expressed in terms of local experts’ opinions) with respect to probable landslide activity. According to the resultant weight of FAHP method (Table 3) and PROMETHEE II landslide susceptibility map (Fig. 6), most of the landslides that occurred over the last several years in the region has been triggered by rivers erosive power, road construction and also heavy rainfalls, respectively, where there is susceptible lithological formation and slope angle.

As a matter of fact, this study presents an integrated strategic LSM framework with emphasis on structuring the decision problem including careful selection and weighting of criteria and alternative evaluation. Accordingly, by using improved AHP with fuzzy set theory, the qualitative judgment can be qualified to make geodata layer comparison more intuitionistic and reduce or eliminate assessment bias in pairwise comparison process (Mahmoodzadeh et al. 2007). Also, a number of favorable characteristics of the AHP method could enhance PROMETHEE, namely at the level of structuring of the decision problem and of the determination of weights (Macharis et al. 2004). Accordingly, this article introduces an approach that integrates FAHP with PROMETHEE II algorithm, which could be a useful geospatial tool for integrating multiple features/attributes that affect the LSM process. Further, it also should be mentioned that along with the simple overlay technique, the receiver operating characteristics (ROC) curves analysis was used to further validate results. According to the obtain AUC of 0.752 with standard error of 0.0404, the produced map has exhibited good performance.

Finally, the obtained landslide susceptibility map with respective explanation is of great importance for authorities have responsibility for developing landslide risk management strategies of Minoo Dasht. The provided information by this research shall help citizens, planners and engineers to reduce the financial and life losses caused by existing and future landslides by means of prevention, mitigation and avoidance. The results are therefore useful for explaining the driving factors of the known existing landslides, for supporting emergency decisions and also for supporting the efforts on the mitigation of future landslide hazards of the region.