Bike Trail Difficulty Rating in the South Moravian Region Modelled Using Fuzzy Sets Pavel Kolisko Regional Authority of the South Moravian Region Žerotínovo nám. 3/5, Brno Department of Geography, Faculty of Science, Masaryk University Kotlářská 2, Brno Abstract The fuzzy sets are more suitable for modelling of the vagueness than the clas- sical crisp sets. They present vague phenomenon and relations which are not exactly bounded but they are associated with their verbal expression. Inaccuracies of characteristics of the bike trail difficulty are connected to the area changes and it is necessary to evaluate and update them regularly. The analysis is solved by the compositional rule of inference methods especially by Mamdani’s and Larsen’s method. The difficulty is the result of rules processing with verbal variables for the type of road and slope. The suitability of methods is tested by certified and categorized parts of the bike trails. The modelling has been performed by rasters using software ArcGIS 10.1 and its geoprocessing tools. Keywords: GIS, fuzzy set, fuzzy logic, fuzzy inference, modus ponens, compositional rule of inference, defuzzification, centroid, center of gravity, center of sums 1. Introduction The term “fuzzy“ is used in meaning of wispy, unclear, misty, vague, uncertain [1]. Although we can describe the phenomenon exactly and clearly, we often utilize unclear, unconfined terms in usual life. We apply terms as moderate slope, near the road. We speak about “linguistic variables” (slope, road) which assume linguistic values (moderate, near) [2]. We sometimes modify linguistic terms by adding expressions called hedges, for example very moderate slope, slightly near the road. We are able to model real situations better using fuzzy sets, sets with unclear boundary. Each element is in the set more or less. It is indicated by a degree of membership to a fuzzy set expressed by value between zero and one. Fuzzy sets are perceived as generalization of classical crisp sets which are their special case. Quality “to be fuzzy” is often expressed as ambiguity, not as inaccuracy or uncertainty, it is relative and subjective. Look at the definition of fuzzy set using the characteristic function. Let X be a universe set (crisp set). A fuzzy set A of the universe X is defined by a character- istic function called membership function µA such that µA : X→ 〈0, 1〉 where µA (x) is the Geoinformatics FCE CTU 11, 2013 5 Kolisko, P.: Bike Trail Difficulty Rating in the South Moravian Region . . . membership value of x in A. The membership value assigns a degree of membership to a fuzzy set to any element. µA (x) = 1 element x belongs to a fuzzy set for sure µA (x) = 0 element x doesn´t belong to a fuzzy set for sure 0 < µA (x) < 1 we aren´t sure if element x belongs to a fuzzy set. Each function X→ 〈0, 1〉 determines any fuzzy set definitely. We can understand the fuzzy set as the complete universe, but only some elements are not definitely in it. The membership degree to the fuzzy set is specified by mathematical function [3]. We usually compose the membership functions of elementary linear functions. These are trapezoidal, triangular, S-shaped and L-shaped membership functions. We often use more complicated rounded functions, too – Gaussian function, bell-shaped function, sinusoidal function etc. 2. Operations on fuzzy sets and fuzzy logic We can define operations complement, union and intersection on fuzzy sets in similar way as on crisp sets. The standard intersection of two fuzzy sets A and B is a fuzzy set with the membership function defined by µA∩B (x) = min (µA (x) , µB (x)). Zadeh´s intersection The standard union of two fuzzy sets A and B is a fuzzy set with the membership function defined by µA∪B (x) = max (µA (x) , µB (x)) Zadeh´s union The standard complement of fuzzy set A is a fuzzy set with the membership function defined by µA (x) = 1− µA (x) Zadeh´s complement Functions for modelling fuzzy conjunction are called triangular norms (t-norms), for fuzzy disjunction triangular conorms (t-conorms). They are assumed as functions of two variables defined on a unit square [4]. Fundamental t-norms TM (x, y) = min (x, y) minimum t-norm TP (x, y) = xy product t-norm TL (x, y) = max (0, x+ y − 1) Łukasiewicz t-norm TD (x, y) = { min (x, y) if max (x, y) = 1 0 else drastic t-norm The drastic t-norm is the smallest t-norm and the minimum t-norm is the largest t-norm, because we have TD (x, y) ≤ TL (x, y) ≤ TP (x, y) ≤ TM (x, y). Geoinformatics FCE CTU 11, 2013 6 Kolisko, P.: Bike Trail Difficulty Rating in the South Moravian Region . . . Fundamental t-conorms SM (x, y) = max (x, y) maximum t-conorm SP (x, y) = x+ y − xy probabilistic t-conorm SL (x, y) = min (1, x+ y) Łukasiewicz t-conorm SD (x, y) = { max (x, y) if min (x, y) = 0 1 else drastic t-conorm The maximum t-conorm SM is the smallest t-conorm, drastic t-conorm is the largest t-conorm, because we have SD (x, y) ≥ SL (x, y) ≥ SP (x, y) ≥ SM (x, y). Now we can generalize expression of fuzzy sets union and intersection. The intersection of fuzzy sets based on t-norm T is the fuzzy set with the membership function defined by µA∩TB (x) = T (µA (x) , µB (x)). The union of fuzzy sets based on t-conorm T is the fuzzy set with the membership function defined by µA∪SB (x) = S (µA (x) , µB (x)). Therefore, the standard intersection and union are special casesA ∩B = A ∩TM B andA ∪B = A ∪SM B. Similarly, the fuzzy negation, the complement of the fuzzy set and various implications are defined. [5]. Fuzzy relations Let X, Y be crisp sets. A binary fuzzy relation R from X to Y is any fuzzy subset R of the set X×Y. Fuzzy relation R is described by the membership function µR: X×Y→ 〈0, 1〉. We can define intersection on t-norm T and union on t-conorm S. µA∩TB (x, y) = T (µA (x, y) , µB (x, y)) µA∪SB (x, y) = S (µA (x, y) , µB (x, y)) Definition of composition of fuzzy relations Let X, Y, Z be crisp sets, A, B binary fuzzy relations and T t-norm. Then sup-T composi- tion of fuzzy relations A and B is fuzzy relation C = A ◦T B with the membership function µC (x, z) = sup y∈Y T (µA (x, y) , µB (y, z)). 3. Fuzzy inference and generalized modus ponens The fuzzy inference is a process which is applied to reasoning based on vague concept. The inductive method modus tollens and the deductive method modus ponens are the basic rules of inference in binary logic. In modus ponens we infer validity of a propositional formula q from validity of implication p⇒ q and validity of premise of a propositional formula p. Geoinformatics FCE CTU 11, 2013 7 Kolisko, P.: Bike Trail Difficulty Rating in the South Moravian Region . . . 3.1. Generalized modus ponens In fuzzy reasoning we use a generalized modus ponens (Tab. 1) according to following state- ment, where A, B, A′ , B′ are fuzzy sets, X, Y linguistic variables. The scheme consists of a rule or a premise (prerequisite), an observing and a conclusion (consequence). The table (Tab. 1) compare generalized modus ponens to the basic deduction modus ponens. Rule if X is A, then Y is B Observing X is A′ Conclusion Y is B′ Table 1: Comparison of modus ponens method against generalized modus ponens p q p⇒ q 1 1 1 1 0 0 0 1 1 0 0 1 modus ponens generalized modus ponens p⇒ q p⇒ q p p′ q q′ The observing does not have to correspond to the premise in the rule. According to finding degree of comparison between premise X is A in the rule and current observing X is A′ it happens modification conclusion Y is B in the rule and getting value B′ of variable Y . If it is A′ = A in observing, it have to be valid B′ = B. The fact is, we operate more rules, input and output variables. Example: Rule if the slope is moderate, the bike trail difficulty is easy Observing slope is steeper Conclusion bike trail difficulty is harder 3.2. Compositional Rule of Inference Practically we need to interpret verbal values of sets A, B mathematically and define the rule of fuzzy relation R between variables X, Y . We use the compositional rule of inference for assignment value B′ of variable Y , which corresponds with value A′ of variable X. We can get term, where the set B′ is the sup-min composition of the fuzzy set A′ and the fuzzy relation R, written as B′ = A ′ ◦R with the membership [6] µB′ (y) = sup x∈X min ( µA′ (x) , µR (x, y) ) standard intersection or generally µB′ (y) = sup x∈X T ( µA′ (x) , µR (x, y) ) union based on t-norm T (X, Y ) is R (A, B) X is A ′ Y is B ′ , B ′ = A ′ ◦T R (A, B) compositional rule of inference on t-norm T Geoinformatics FCE CTU 11, 2013 8 Kolisko, P.: Bike Trail Difficulty Rating in the South Moravian Region . . . We have to keep generalized modus ponens during relational reasoning, too, i.e. A ◦T R (A, B) = B. The fuzzy relations can be modelled by a logical implication or by a cartesian product T ∗ based on t-norm. We confine to the second possibility and we get µR(A,B) (x, y) = T ∗ (µA (x) , µB (y)) µB′ (y) = sup x∈X min ( µA′ (x) , T ∗ (µA (x) , µB (y)) ) We can generalize the properties to t-norm T . µB′ (y) = sup x∈X T ( µA′ (x) , T ∗ (µA (x) , µB (y)) ) If we choose T = T ∗ = TM we get Mamdani’s method. µB′ (y) = sup x∈X min ( µA′ (x) , min (µA (x) , µB (y)) ) For T = TM and T ∗ = TP , it is Larsen’s method [7]. µB′ (y) = sup x∈X min ( µA′ (x) , µA (x) · µB (y) ) 4. Mamdani’s method Let's have a look at Mamdani’s method in detail [8]. LetB = {P1, P2, . . . , Pk} be a knowledge base with k rules for n input variablesX1, X2, . . . , Xn and one output variable Y . Each of the variables Xi have the verbal value Ai,j in j-th rule, variable Y has the verbal value Bj , where i = 1, 2, . . . , n, j = 1, 2, . . . , k. For Mamdani’s regulator are defined: Rules P1 : if X1 is A11 and X2 is A21 and . . . and Xn is An1 , then Y is B1 P2 : if X1 is A12 and X2 is A22 and . . . and Xn is An2 , then Y is B2 . . . Pk : if X1 is A1k and X2 is A2k and . . . and Xn is Ank, then Y is Bk Observing X1 is A′1 and X2 is A′2 and . . . and Xn is A′n Conclusion Y is B′ Because the effort with the whole of the relation is numerically arduous, it is preferable to use the approach FITA (first inference then aggregation), which means reasoning of conclusion rule-by-rule, where the final aggregate conclusion is B′ = ∪kj=1B ′ j . Therefore µB′ (y) can be presented as µB′ (y) = kmax j=1 µ B ′ j (y) = kmax j=1 min ( wj , µBj (y) ) , where wj = min (w1j , w2j , . . . , wnj) is the total weight of j-th rule, numbers w1j , w2j , . . . , wnj are particular degrees of fulfilment of the premises in j-th rule X1 is A1j , X2 is A2j , . . . , Xn is Anj . We can generalize the properties to t-norm T . Geoinformatics FCE CTU 11, 2013 9 Kolisko, P.: Bike Trail Difficulty Rating in the South Moravian Region . . . Consider the generalisation of t-norm T for an intersection and t-norm T ∗ for an assignment of the relation (Fig. 1). The membership function for degrees wj = T (w1j , w2j , . . . , wnj) is defined as µB′ (y) = kmax j=1 µ B ′ j (y) = kmax j=1 T ∗ ( wj , µBj (y) ) . For Larsen’s method is written T = TM and T ∗ = TP . w1 = T (w11, w21) w2 = T (w12, w22) µ B ′ 1 (y) = T ∗ (w1, µB1 (y)) µ B ′ 2 (y) = T ∗ (w2, µB2 (y)) µB′ (y) = max ( µ B ′ 1 (y) , µ B ′ 2 (y) ) µB′ (y) = 2max j=1 T ∗ ( T (w1j , w2j) , µBj (y) ) Figure 1: Illustrative scheme of the universal regulator with two rules, two input variables and one output variable 5. Defuzzification If we apply crisp inputs, the results of inference are fuzzy outputs. We often need to find the particular real value of output by defuzzification. There are several methods to defuzzify for miscellaneous usage (Fig. 2). We can distribute them to methods searching the most acceptable solution and methods of the best compromise [9]. The methods of the most acceptable solution are presented by the methods of the most important maximum with selection of the biggest value of the membership functions placed leftmost, middlemost or rightmost - Left of Maximum (LoM), Mean of Maximum (MoM), Right of Maximum (RoM). Methods of the best compromise include: Center of Gravity (CoG) – the centroid of area (the centroid of the plane figure given by union of the part areas bounded by particular membership functions). Geoinformatics FCE CTU 11, 2013 10 Kolisko, P.: Bike Trail Difficulty Rating in the South Moravian Region . . . Center of Sums (CoS) – the centroid of sums (the centroid of the plane figure given by function, which is equal to the sum of the particular membership functions in the rules) Center of Maximum (CoM) – the centroid of singletons (the centroid of the typical values, e.g. MoM, for the particular membership functions of the rules). Method Bisector of Area (BoA) divides the area of the plane figure into two sub-regions of the equal area. Figure 2: Defuzzification methods CoG It makes for finding the first coordinate of the centroid of area bounded by the membership function µB′ . The method is mathematically difficult because we need to know the member- ship function and calculate the Riemann integrals. In the reasoning of conclusion rule-by-rule B ′ = ∪kj=1B ′ j is µB′ (y) = max 1 ≤ j ≤ k µ B ′ j (y). The situation is simpler, if the universe of the output variable is discrete subset of real numbers Y = {y1, y2, . . . , yr}. yCoGB′ = ∫ Y µ B ′ (y) y dy∫ Y µ B ′ (y) dy = ∫ Y ( max 1 ≤ j ≤k µ B ′ j (y) ) y dy∫ Y ( max 1 ≤ j ≤k µ B ′ j (y) ) dy continuous membership function yCoGB′ = r∑ i=1 µ B ′ (yi)yi r∑ i=1 µ B ′ (yi) discrete membership function CoS [3] It serves to find the first coordinate of the centroid of area which is bounded by the function defined as sum of the membership functions µ B ′ j . The method is easy-to-use because it does Geoinformatics FCE CTU 11, 2013 11 Kolisko, P.: Bike Trail Difficulty Rating in the South Moravian Region . . . not need to determine the conclusion B′ . If the particular conclusions of rules do not overlap, the result of the method CoS is the same as for the method CoG. yCoSB′j = ∫ Y ( ∑ 1 ≤ j ≤ k µB′ j (y) ) y dy ∫ Y ( ∑ 1 ≤ j ≤ k µB′ j (y) ) dy = ∑ 1 ≤ j ≤ k (∫ Y µB′ j (y) y dy ) ∑ 1 ≤ j ≤ k (∫ Y µB′ j (y) dy ) continuous membership function yCoSB′j = r∑ i=1 yi k∑ j=1 µB′ j (yi) r∑ i=1 k∑ j=1 µB′ j (yi) discrete membership function CoM The first coordinate of the membership function is written for each conclusion of rule by the method of the most important maximum (Mean of Maximum) and the result is the centroid of singletons. yCoMB′j = k∑ j=1 yj ·µB′j (yj) k∑ j=1 µB′j (yj) 6. The application of fuzzy methods in solution of bike trail difficulty rating Bike trail difficulty is the basic characteristic to recognize during the cycle route planning. It gives us to qualify whether the route is suitable for families with children, for recreational sportsmen, maybe for athletes. In 2003 and 2005 projects were made with intent to collect information about cycle routes and their facilities. In 2007 the data were updated by terrain research - especially the status of surface and difficulty (demandingness) of bike trail. The data are published on the web cycling portal of the South Moravian Region http: //www.cyklo-jizni-morava.cz, including the interactive bike trail map with choosing routes and view points of interest. During actual checking well-known routes it was verified that the characteristic of bike trail difficulty has already completely disagreed with the reality. Each rating depends on time, it is affected by the subjective view and data collection is a hard task in terrain. Therefore, we need to utilize another approach for instance by fuzzy reasoning. The slope and the quality or type of the road surface, which were chosen as analytical inputs, impact on the difficulty. The modelling is accomplished over rasters in ArcGIS 10.1 using ModelBuilder and geopro- cessing tools, especially Spatial Analyst Tools – Fuzzy Membership, Fuzzy Overlay, Raster Calculator, Cell Statistics. Geoinformatics FCE CTU 11, 2013 12 http://www.cyklo-jizni-morava.cz http://www.cyklo-jizni-morava.cz Kolisko, P.: Bike Trail Difficulty Rating in the South Moravian Region . . . 6.1. Methods We use two input variables, X1 for the type of the road surface and X2 for the angle of the slope (both defined by crisp values) and output variable Y for the bike trail difficulty. Assume the following input and output fuzzy subsets which are given by verbal values and rules representing their relationship. Type of road surface (data StreetNet 2012) K1 - paved roads (asphalt, pavement, concrete) K2 - maintained roads (unpaved, gravel) K3 - other unpaved roads (forest and cart roads) Angle of slope (DMT, in degrees) S1 - moderate slope S2 - steep slope Bike trail difficulty D1 - small difficulty – easy difficult roads (suitable for families with children) D2 - intermediate difficulty – more difficult roads (suitable for recreational sportsmen) D3 - hard difficulty - very difficult roads (suitable for athletes) Rules P1 : if X1 is K1 and X2 is S1, then Y is D1 P2 : if X1 is K2 and X2 is S1, then Y is D1 P3 : if X1 is K3 and X2 is S1, then Y is D2 P4 : if X1 is K1 and X2 is S2, then Y is D2 P5 : if X1 is K2 and X2 is S2, then Y is D3 P6 : if X1 is K3 and X2 is S2, then Y is D3 Observing X1 is K ′ and X2 is S′ Conclusion Y is D′ The fuzzy sets K1, K2, K3 were given by the bell-shaped membership function Near (Mid- point 0, Spread 0,0001) available in the geoprocessing tools of ArcMap in the category Fuzzy Membership (Fig. 3). The function expresses the close localization of the road as a fuzzy line [10] in network of roads. The tool Kernel Density was selected at first. It more high- lighted density of roads to the chosen area. But the results were not satisfactory because they characterised roads inaccurately as fuzzy lines in regions with small density of roads. Next figures show settings that define S1, S2 and D1, D2, D3 (Fig. 4 and Fig. 5). We will use and compare several regulators and defuzzification methods. We will do the interpretation rule-by-rule. We declare wj as the total weight of the j-th rule worked from particular weights of premises (roads, slope) w1j , w2j . The membership function of conclusion of the j-th rule is written µD′j (y). This is summary and specification of applied methods. Mamdani’s method (COS-TM-TM, COM-TM-TM) µD′ (y) = kmax j=1 TM ( TM (w1j , w2j) , µDj (y) ) = kmax j=1 min ( min (w1j , w2j) , µDj (y) ) Geoinformatics FCE CTU 11, 2013 13 Kolisko, P.: Bike Trail Difficulty Rating in the South Moravian Region . . . µ (x1) = 1 1+0,0001x12 Figure 3: Membership function for road surface Figure 4: Membership function for slope Figure 5: Membership function for difficulty of road Geoinformatics FCE CTU 11, 2013 14 Kolisko, P.: Bike Trail Difficulty Rating in the South Moravian Region . . . Larsen’s method (COS-TP-TM) µD′ (y) = kmax j=1 TP ( TM (w1j , w2j) , µDj (y) ) = kmax j=1 ( min (w1j , w2j) ·µDj (y) ) Product t-norm and product t-norm (COS-TP-TP) µD′ (y) = kmax j=1 TP ( TP (w1j , w2j) , µDj (y) ) = kmax j=1 ( w1j ·w2j ·µDj (y) ) Łukasiewicz t-norm and minimum t-norm (COS-TL-TM) µD′ (y) = kmax j=1 TL ( TM (w1j , w2j) , µDj (y) ) = kmax j=1 max ( 0, min (w1j , w2j) +µDj (y)−1 ) Łukasiewicz t-norm and product t-norm (COS-TL-TP) µD′ (y) = kmax j=1 TL ( TP (w1j , w2j) , µDj (y) ) = kmax j=1 max ( 0, w1j ·w2j+µDj (y)−1 ) Łukasiewicz t-norm and Łukasiewicz t-norm (COS-TL-TL) µD′ (y) = kmax j=1 TL ( TL (w1j , w2j) , µDj (y) ) = kmax j=1 max ( 0, max (0, w1j+w2j−1) + µDj (y)−1 ) 6.2. Mamdani’s method (COS-TM-TM) Considering evaluation of the road surface and reasoning of conclusion rule-by-rule, we will choose (COS-TM-TM) the centroid of sums which means calculation. yCoSD′j = ∫ Y µD′1 (y) y dy+ ∫ Y µD′2 (y) y dy+ ∫ Y µD′3 (y) y dy+ ∫ Y µD′4 (y) y dy+ ∫ Y µD′5 (y) y dy+ ∫ Y µD′6 (y) y dy∫ Y µD′1 (y) dy + ∫ Y µD′2 (y) dy+ ∫ Y µD′3 (y) dy+ ∫ Y µD′4 (y) dy+ ∫ Y µD′5 (y) dy+ ∫ Y µD′6 (y) dy The total weight of the j-th rule wj is the minimum of the particular weights of the premises (roads, slope) w1j , w2j in this rule (simply signed w). The membership function of the con- clusion of the j-th rule is presented as µD′j (y) = min ( wj , µDj (y) ) . The membership µDj (y) is simply denoted µ (y). The model in ArcGIS ModelBulder is shown in Fig. 9. In the first and the second rule we evaluate small difficulty D1 (Fig. 6). A = w [ y2 2 ]−2w+3 0 + [ −y 3 6 + 3y2 4 ]3 −2w+3 = 2 3w 3 − 3w2 + 9 2w Geoinformatics FCE CTU 11, 2013 15 Kolisko, P.: Bike Trail Difficulty Rating in the South Moravian Region . . . −2w+3∫ 0 w y dy + 3∫ −2w+3 ( −y2 + 3 2 ) y dy (A) and −2w+3∫ 0 w dy + 3∫ −2w+3 ( −y2 + 3 2 ) dy (B) Figure 6: Membership function for small difficulty 2w+1∫ 1 ( y 2 − 1 2 ) y dy + −2w+5∫ 2w+1 w y dy+ + 5∫ −2w+5 ( −y2 + 5 2 ) y dy (C) and 2w+1∫ 1 ( y 2 − 1 2 ) dy + −2w+5∫ 2w+1 w dy+ + 5∫ −2w+5 ( −y2 + 5 2 ) dy (D) Figure 7: Membership function for intermediate difficulty B = w [y] −2w+3 0 + [ −y 2 4 + 3y 2 ]3 −2w+3 = −w2 + 3w In the third and the fourth rule we evaluate intermediate difficulty D2 (Fig. 7). C = [ y3 6 − y2 4 ]2w+1 1 + w [ y2 2 ]−2w+5 2w+1 + [ −y 3 6 + 5y2 4 ]5 −2w+5 = −6w2 + 12w D = [ y2 4 − y 2 ]2w+1 1 + w [y] −2w+5 2w+1 + [ −y 2 4 + 5y 2 ]5 −2w+5 = −2w2 + 4w In the fifth and the sixth rule we evaluate hard difficulty D3 (Fig. 8). Geoinformatics FCE CTU 11, 2013 16 Kolisko, P.: Bike Trail Difficulty Rating in the South Moravian Region . . . 2w+3∫ 3 ( y 2 − 3 2 ) y dy + 6∫ 2w+3 w y dy (E) and 2w+3∫ 3 ( y 2 − 3 2 ) dy + 6∫ 2w+3 w dy (F ) Figure 8: Membership function for hard difficulty E = [ y3 6 − 3y2 4 ]2w+3 3 + w [ y2 2 ]6 2w+3 = −2 3w 3 − 3w2 + 27 2 F = [ y2 4 − 3y 2 ]2w+3 3 + w [y] 6 2w+3 = −w2 + 3w Figure 9: The model of Mamdani’s method (COS-TM-TM) Geoinformatics FCE CTU 11, 2013 17 Kolisko, P.: Bike Trail Difficulty Rating in the South Moravian Region . . . 6.3. Mamdani’s method (COM-TM-TM) We evaluate by the centroid of singletons Center of Maximum (COM-TM-TM) using the mean of the maximum. yCoMD′j = y1·µD′1 (y1)+y2·µD′2 (y2)+y3·µD′3 (y3)+y4·µD′4 (y4)+y5·µD′5 (y5)+y6·µD′6 (y6) µD′1 (y1) +µD′2 (y2)+µD′3 (y3)+µD′4 (y4)+µD′5 (y5)+µD′6 (y6) = By substituting values: = 0−2w1+3 2 · w1 + 0−2w2+3 2 · w2 + 2w3+1−2w3+5 2 · w3 + 2w4+1−2w4+5 2 · w4 + 2w5+3+6 2 · w5 + 2w6+3+6 2 · w6 w1 + w2 + w3 + w4 + w5 + w6 = −w2 1 + 3 2w1 − w2 2 + 3 2w2 + 3 · w3 + 3 · w4 + w2 5 + 9 2w5 + w2 6 + 9 2w6 w1 + w2 + w3 + w4 + w5 + w6 6.4. Comparison of defuzzification methods CoS and CoM The raster analysis result is in range between 1,085 and 4,916 for CoS method, between 0,501 and 5,500 for CoM method. In ArcMap we see that the results are comparable. The value difference of both processes CoM-CoS gives results from -0,564 to 0,626. Negative values of the difference are related to the flat land and the closeness to the paved and maintained roads (CoS>CoM), CoM gives the less difficulty of the roads. Positive values are related to the steep slope and the closeness to the forest and the cart roads (CoS