The 7th International Conference on INFOrmatics and Systems (INFOS 2010) – 28-30 March Applied Optimization and Metaheuristics Track Fuzzy Cognitive Map with Dynamic Fuzzification and Causality Behaviors Ahmed M. Gadallah #1, Hesham A. Hefny#2 # Department of Computer and Information Science, Institute of Statistical Studies and Research (ISSR), Cairo University. Institute of Statistical Studies and Research (ISSR), 5 Ahamed Zewal Street, Orman, Giza, Egypt 1 2 ahmgad10@yahoo.com hehefny@hotmail.com Abstract— This paper introduces a proposed enhanced model of fuzzy cognitive map (FCM) called dynamic fuzzy cognitive map (DFCM). The aim is to make both of fuzzification mapping and causal relationships between concepts are environment sensitive. Accordingly, knowledge representation and inference process will be more flexible, reliable, and dynamic especially when dealing with complex and dynamic environments. A special set of environment dominant variables (V) is proposed that represents the environment prevailing conditions. The behavior of DFCM depends on V. Hence, the same fuzzy cognitive map knowledge and concept values will infer deferent outcomes in response to changes in V. Accordingly, it will be very appropriate to apply DFCM when modeling a wider spectrum of real world problems in various fields like decision support systems, control systems, medical diagnoses systems, effect-based planning, and prediction systems. Keywords— Fuzzy cognitive map, Dynamic causal networks, Knowledge representation, Dynamic inference, Fuzzy systems, Complex systems modeling. I. INTRODUCTION Fuzzy cognitive map is a fuzzy signed directed graph that incorporates a set of concepts and a set of causal relationships between such concepts [1, 2 and 6]. FCM represents a modeling technique for complex systems originating from a combination of fuzzy logic and neural networks. Modeling dynamic systems using fuzzy cognitive maps (FCM) has several advantages [2, 5, 7 and 10]: Firstly, it is very simple and intuitive to understand in terms of the underlying formal model and its execution. Secondly, FCM is characterized by flexibility in system analysis and design, ability to deal with fuzzy information, and adaptability to any given domain. Finally, inferences are carried out by numerical calculations instead of symbolic deductions. The potential of this work relies on enhancing the classical fuzzy cognitive map approach to make it able to model and reason under dynamism that represents the natural metaphor of many real world problems. The main contribution of this work is to make FCM approach more flexible, reliable and dynamic. This paper aims to MAKE FCM approach able to take into considerations the prevailing conditions in the environment under which the FCM will: firstly, represent the knowledge that may be sensitive to changes in its surrounding environment and secondly, make the inferences required to take decisions under recent prevailing conditions. Also, we note that in a wide spectrum of real world problems, there may be a set of special type of concepts which: firstly, does not affect directly any FCM base concepts but affects one or more of the defined FCM causal relationships; secondly, may affect some concepts and its effect should be passed in just some subsequent paths of the effected concepts. Such prevailing environment conditions and such special type of concepts will be called the environment dominant variables set (V). Accordingly, the aim is to make DFCM approach more generic by taking into consideration the effect of V on both of knowledge representation and inference process. This paper is organized as follows: section 2 gives an overview of FCM. Previous and related works were introduced in section 3. Section 4 presents the extended DFCM. An illustrative example was introduced in section 5. Section 6 addresses the conclusion. II. FUZZY COGNITIVE MAPS Fuzzy cognitive maps are fuzzy signed directed graphs with feed back [9]. FCM represents a modeling technique for complex systems originating from a combination of fuzzy logic and neural networks [6]. FCM describes the structure of a system in a set of concepts and a set of causal relationship between these concepts. Inference or reason in FCM is achieved via numerical calculations through a product operation between the concept vector and the causalrelationship matrix. Fig. 1 shows a simple FCM which consists of a set of concepts represented by nodes and a set of weights represented by directed arrows. Each concept Ci has a state value that measures its strength. The state value can be a crisp value in {0, 1} or a fuzzy value within [0,1]. The weight Wij represents the causal relationship or the influence degree between two concepts: a causal concept Cj, and an effect concept Ci. The direction of a causal relationship directed arrow specifies which concept affects the other one. While, the value of the weight Wij specifies how strongly the causal Faculty of Computers and Information - Cairo University AOM -99 The 7th International Conference on INFOrmatics and Systems (INFOS 2010) – 28-30 March Applied Optimization and Metaheuristics Track concept Cj affects the effect concept Ci. While a positive value of Wij represents a proportional effect, a negative value represents an inversely proportional effect and zero Wij represents the absence of an effect. [1-5]. Fig. 1 A simple fuzzy cognitive map In a simple FCM that has n concepts, the knowledge representation model has the following n_dimension concepts vector (C) and n*n weight matrix (W). C = [c1, c2, c3,.…….cn] W = (1) w 11 w 21 w 12 w 22 w 1n w 2n w n1 wn2 w nn (2) The inference process in a FCM can be described as follows: Multiply the weights matrix (W) by the concept vector (C). The result of the multiplication operation is a ndimensional vector Y as in equation (3) and (4). Apply the augmented effect yi as an input of concept ci decision making function to compute its new value as depicted in equation (5). Accordingly, each step of the inference becomes a matrix multiplication followed by applying the accumulated results to the concept decision making function to deduce the new value of the concept as a result of changes in its causal concepts. Y = W×C = w11 w21 w12 w22 wn1 wn 2 = [y1,y2,y3………….yn] w1n w2 n c2 wnn cn c1 (3) n yi = wij c j , i = 1,2, n (4) j =1, j ≠ i where ci represents the value of concept ci, yi represents the sum of the products of each causal concept value cj and the weight value between the two concept nodes wij. cinew = f i ( y i ) + ciold (5) III. PREVIOUS AND RELATED WORK Many studies of FCM have proved that it has many advantages when modeling complex systems. Applications of FCM approach have been growing rapidly in many real world domains which almost are complex. The authors in [5] used FCM approach to design intelligent software agents. The developed intelligent agent became able to represent the domain knowledge and carry out the inferences depending on numerical representation and calculations instead of symbolic representation and inference. The used FCM model is a 2element tuple: {C, W} as depicted in equation (1) and (2). A weight may be dynamic when it depends on its causal concept. Reasoning about a new state of concept ci is computed as in equation (5). According to this model, there are a set of fuzzification mapping functions that map the interface concepts real values into fuzzy values within the range [-1, 1]. Also, the model used the dynamic weight idea in which a weight may be a function in its causal concept value. Yet, the used FCM model ignores the effects of the surrounding atmosphere on the causal relationships representation between concepts. In process control problems [9], FCM approach was applied to represent the accumulated knowledge from domain experts in order to simulate and control the system. In each step of cycling FCM, the concepts will freely interacted with each other in respect to the predefined causal relationship between them. The new value for each concept is calculated by applying the augmented effect from the causal concepts to that concept decision making function as depicted in equation (5). Yet in real world, the surrounding atmosphere may affect the causal relationship between the interacted concepts in the designed FCM for process control problems. The used FCM model ignores a natural fact that, the causal relationship may be a function of the environment prevailing conditions. For example, naturally, the higher the surrounding atmosphere temperature the higher the steaming rate and the lower the temperature lose rate and vice versa. Also, the wetter the atmosphere the lower the steaming rate and vice versa. The used model assumes that the environment has no effect on the designed FCM causal relationships. In the work of [6], a fuzzy cognitive agent has been developed. This agent provides personalized recommendations to on-line customers in e-commerce sites. Fuzzy cognitive agents are designed to give personalized suggestions based on the on-line customer preferences and experts' domain knowledge. The agent knowledge model of FCM is defined as a 2-element tuple: FCM = {C, W} as depicted in equations (1) and (2). The inference will take place as depicted in equations (3) and (5). An algorithm based on the self organized map learning technique is used to learn the membership function for transferring the base case into fuzzy case base. The learned membership function for each concept represents its fuzzification mapping function. In this model of FCM, the fuzzification mapping function were learned to map where fi is the decision making function of the concept ci. Faculty of Computers and Information - Cairo University AOM -100 The 7th International Conference on INFOrmatics and Systems (INFOS 2010) – 28-30 March Applied Optimization and Metaheuristics Track the real concept value into a fuzzy value within [-1, 1]. Also, the weights of a FCM can be learned from a fuzzy training set for each concept. The main problem here concerns with the natural dependency that may exist of both of the fuzzification mapping functions and the causal relationships between concepts on the environment prevailing conditions. Also, the learned weights became fixed values and don't have the flexibility to respond to changes in the environment prevailing conditions. Dynamic cognitive networks (DCN) proposed in [3] have their roots in the domain of cognitive mapping. DCN aimed to overcome some shortages of FCM when dealing with many real-world problems: firstly, FCM is based on static causeeffect schemes and their inference mechanism is able to provide just the potential of causal relationship between their concepts, secondly, FCM does not provide a robust and dynamic mechanism to represent the strength of the cause and the degree of the effect and finally, FCM lacks the temporal concept that is crucial in many real world applications. The authors proposed equation (6) for compute concepts new values trying to overcome the above shortages. Such equation provides non-linear causal functions, decaying mechanism, and temporal effect mechanism. Yet, such causal functions are mainly function of only its causal concept value. c t +1 i t c + s (1 − c ) c + s (1 + c ) c +s t = F(ci , si ) = t t t i i i t t t i i i t t i i t t i i t t i i (7) c ≥ 0, s ≥ 0 c < 0, s < 0 otherwise where n t wij c j , j ≠ i and c i is the value of concept Ci at time t. Sit = j =1 This mechanism has the advantage of keeping the same relative strength among weights because of dividing each of them by same saturating value s t max using equations (8) and (9) [Yaman et al., 2009], [Tsadiras et al., 1999]. Yet, as the number of concepts increases, the absolute values of the weights decrease as a result of dividing it by a bigger value. Accordingly, concepts can't effectively affect each other [Tsadiras et al., 1999]. n n Sit,max = | wij | | c j | = | wij | i = 1,2,....n (8) (s1t,max, s2t ,max,.....sit,max) (9) j =1 j =1 t S max = max S t i On the other hand, it lacks keeping the relative effect of over concept ci. That is: the first two cases in equation (7); that apply the whole value of the external influence c t +1 j = ( ( t min 1, max − 1, (1 − d j ). c j + R )) (6) where R= c t +1 j + w. ij i =1, t i ≠ j , i >0 c t i to the concept not-activated degree; have different relative effect of third case. The average effect of S t over ci in first i n and S t max −t t . + t max ci i =1, n − w. ij t max −t t . t max ci t c i ≠ j , i <0 two cases in equation (7) can be computed respectively as follows: AVG case 1 c represents the new value of concept cj at time t+1, dj is a decaying factor, wij+:[0, 1] to [-1, 1] represents the dynamic weights of the positive domain of the causal concept, wij-:[0, -1] to [-1, 1] represents the dynamic weights of the negative domain of the causal concept, tmax is the time horizon until the deactivation of the existing causal relationship. FCM model used in [10] was developed to model and handle effect-based operations aiming to determine or predict alternative courses of actions to realize the aims of a military effect-based operation. In this model of FCM a new set of attributes were added to the concept value calculation algorithm. These attributes include influence possibility, influence duration, influence permanence and dynamic influence. At each step of cycling the FCM, the value of concept ci will be obtained using equation (7). The dynamic unit influence degree through the influence duration is modeled by reducing the links influence duration by one unit and recalculating the unit influence degree for every step. ( s it ) = AVG t ≥ 0 , s it ≥ 0 i t case 2 s it (1 − 0 ) + s it ( 1 − 1 ) 1 t = si 2 2 ( S i) = S t i and, (1 + 0 ) + S t i (10) ( 1 + ( − 1 )) 2 c t i ≤ 0, s t i < 0 = 1 2 S t i (11) Reasonably, in order to keep the relative effect of s it over ci in the third case, the decision making function of concepts may be replaced by c it + 1 S it . 2 As a result of studying previous models of FCM, a need has been appeared for a more flexible, reliable and dynamic FCM for more effective knowledge representation and inference models to cope efficiently with a wider spectrum of real world problems. IV. THE PROPOSED EXTENDED DYNAMIC FUZZY COGNITIVE MAP DFCM The definition of FCM in previous works [1-13] deal with FCM as a 2-element tuple that includes a set of nodes and a set of directed arrows. A node represents a concept that indicates an entity, a state or a characteristic of the system. On the other hand, a directed arrow represents the weighted causal relationship between the causal and the effect concepts. Faculty of Computers and Information - Cairo University AOM -101 The 7th International Conference on INFOrmatics and Systems (INFOS 2010) – 28-30 March Applied Optimization and Metaheuristics Track Such weight may be static crisp value or fuzzy value and may depend on the value of its causal concept. Yet, none of these previous works take into considerations the prevailing conditions in the environment under which the FCM will: firstly, represent the knowledge that may be sensitive to changes in its surrounding environment and secondly, make the inferences required to take decisions under recent prevailing conditions. Also, it ignores the existence of special type of concepts which: firstly, does not affect directly any FCM base concepts but affects one or more of the defined FCM causal relationships; secondly, may affect some concepts and its effect should be pass in just some subsequent paths. Such prevailing environment conditions and such special type of concepts we will call the set of environment dominant variables V. Accordingly, we propose to identify a set of environment dominant variables V which is comprehensive to represent the current environment state. Actually, this assertion aims to make the FCM approach more flexible and reliable in representing a wider range of real world problems through satisfying the following needs: As a natural metaphor, there may be some special set of concepts that have no direct causal relationship with the base concepts of the targeted FCM although they may affect the strength of some of its causal relationships. If such special concepts were added to the designed FCM it would be isolated-nodes forever which may duplicate the complexity and complicate both of knowledge representation and inference. There may be some concepts that affect other ones but its effect shouldn't be passed forward in subsequent paths. Hence, they should be represented in the designed FCM in a flexible and reliable manner that insures their natural effect behavior. The dynamic weights proposed in [6] make the weight value depends on its causal concepts value. But in real world, the strength of the causal relationship between two concepts may depend not only on the strength of its causal concept but on the environment prevailing conditions which almost are dynamic. The proposed DFCM approach aims to fulfill the above cases through a set of enhancements on both of knowledge representation and inference phases. Assuming a DFCM that has n concepts and m environment dominant variables, the model has a concepts vector, weight matrix and variable vector as depicted in equation (1), (2) and (13) respectively. V = [v1, v2, v3,.…….vm] The state value of any concept Ci can be within [-1, 1] that represents the strength of the concept. There is a fuzzification mapping function for each interface concept that maps its real world value into a fuzzy value within [-1, 1]. There are many types of functions [14, 15] that can be used as fuzzification mapping functions like logistic, sigmoid (logistic or hyperbolic), Gaussian and S membership functions. Occasionally, the prevailing environment conditions may affect the shape, the slops and the control points of such functions. Hence, such membership functions may be sensitive to the ongoing changes in the environment. Hence, the same actual value for a concept Ci will be mapped to different fuzzy values with different environment states. Dynamic effects of V over the DFCM causal relationships may be represented using one or more of equations (14), (15), (16) and (17). Such equations are subjective and depend on the nature of the problem and the expert point of view. fw∞− (wij , vk , α ) = where f ∞ w− w −αv (1+ w ) v ≥ 0 w +αv w v 0 k ij k ij k ij (14) k relationship strength wij in response to a positive value of an environment variable vk and vice versa, and is an effect factor. f w∞+ (wij , vk ,α ) = where f ∞ w+ w + αv (1 − w ) v ≥ 0 w + αv w v 0 ij k ij k ij k ij (15) k indicates an increase in the positive causal relationship strength wij in response to a positive value of an environment variable vk and vice versa, and is an effect factor. A. Knowledge Representation Phase where f x w− w − αv w v ≥0 w + αv (1 + w ) v 0 ij k ij k ij k ij (16) k indicates a decrease in the negative causal relationship strength wij in response to a positive value of an environment variable vk and vice versa, and is an effect factor. f wx+ (wij , vk ,α ) = where The value of any concept ci, causal relationship wij or environment dominant variable Vk can be within the bipolar interval [-1, 1]. ij indicates an increase in the negative causal f wx− ( wij , vk , α ) = The proposed model of DFCM is defined as a 3-element tuple as follows: DFCM = {C, W, V} (12) where, C = { ci | ci ∈ [-1, 1], i = 1, 2, …n} is a set of concepts. W={wij|wij ∈ [-1, 1], i, j = 1,2,…n } is a weight matrix. V = {vk | vk ∈ [-1, 1], k = 1,2,…..m } is a set of environment dominant variables. (13) f x w+ w − αv w v ≥0 w + αv (1 − w ) v 0 ij k ij k ij k ij (17) k indicates a decrease in the positive causal relationship strength wij in response to a positive value of an environment variable vk and vice versa, and is an effect factor. Faculty of Computers and Information - Cairo University AOM -102 The 7th International Conference on INFOrmatics and Systems (INFOS 2010) – 28-30 March Applied Optimization and Metaheuristics Track B. Reasoning Phase The inference process in the enhanced DFCM is carried out by multiplying the adjusted weights values matrix (Wd); as a response to the effect of its causal concept and the environment dominant variables at time t (Vt); by the concept matrix at time t (Ct). The result of the multiplication operation will be a new 1 x n vector Y as shown in equation (18). w11d w12d w1dn c1t d w d w22 w2dn c 2t (18) Y = W d × C t = 21 wnd1 wnd2 = [y1,y2,y3………….yn] wijd c tj , yi = j = 1,2, d wnn variables. Fig. 3 shows the fuzzification mapping function for car price attribute C1. It is a hyperbolic tangent function whose output is in the range [-1, 1]. c nt n (19) Fig. 2 The Knowledge model of car_evaluation problem j ≠i where yi represents the augmented influence from all causal concepts over concept xi. Price strength Accordingly, the effects between the concepts of the designed DFCM will reflect any changes in the prevailing environment conditions as depicted in equation (19). Hence, applying yi as an input of the decision making function of the concept x;, that takes into consideration the values of V using equation (20), will result in the new value of the concept ci. t +1 c i ( = f i yi , c )= t i c + y (1− c ) c + y (1+ c ) t t t i i i t t t i i i 1 t ci + 2 yi t t t i i t t c ≥ 0, y ≥ 0 c < 0, y < 0 i Price real value (20) i otherwise where fi is the decision making function of the concept ci that takes into consideration the current values of the environment dominant variables V. C. AN ILLUSTRATIVE EXAMPLE The following example illustrates the ability of applying the proposed DFCM to an intelligent software agent to make it able to reason in a dynamic environment. The case study assumes a car-purchasing domain. An intelligent software agent that represents a customer is in charge of taking a purchasing decision about a specific class of cars produced at same year. The agent is concerned with three attributes about the exhibited cars: price C1, warranty C2, and fuel consuming rate C3. According such attributes values the agent should decide how much a specific car satisfies the buyer in accordance to the prevailing environment conditions. The set of environment dominant variables V includes: competition degree V1, fuel price V2 and buyer financial state V3. According to the proposed model, the representation of carevaluation problem is depicted in Fig. 2. According to the proposed model any fuzzification mapping function may be uncertain, i.e., its shape may depend on the values of one or more of the environment dominant Fig. 3 Different possible shapes of price fuzzification mapping function at different times depending on competition degree and customer financial state. Intuitively, we can note that both of the competition degree and the financial state of the buyer affects sharply the shape and the steepness of the price fuzzification mapping function. Hence, the fuzzification mapping function should be flexible to respond dynamically to changes in both of them. That is, the same real price will be mapped to different fuzzy values in different competition degrees and different buyer financial states. As a result the price mapping function can be defined as depicted in equation (21). Accordingly, the user financial state and the competition degree will affect the behavior of measuring the strength of a given price i.e. how much a given price is high or strong. 2 (21) −1 1+ e Where θ = pavg + v3 ( pmax − pmin ) / 2, λ = 0.25v1, x is the price real value, controls the steepness of the function, and is the function threshold value, pmax, pmin and pavg represent highest, lowest and average prices. FC1 (x) = − λ ( x −θ ) Also, the fuzzification mapping functions for warranty FC2 can be defined as follows: Faculty of Computers and Information - Cairo University AOM -103 The 7th International Conference on INFOrmatics and Systems (INFOS 2010) – 28-30 March Applied Optimization and Metaheuristics Track 2 FC2 (x) = 1 + e − 0.225 ( x − 6 ) −1 (22) Assuming that, the initial causal relationship matrix W0 is set as shown in equation (24). where Fc2 represents the fuzzification mapping function of warranty concept. W 0 C1 = C2 Fuel consuming rate fuzzy value C1 C2 C3 C4 0 − 0 .1 0 .3 0 0 0 0 0 C3 0 0 0 0 C4 − 0 .6 0 .4 − 0 .5 0 (24) Assuming that, the following notes represent an expert's point of view indicating which variables in V affect which weights in W and how much an effect will take place. • An increase in competition degree V1 will increase the negative influences of both of price W41 and fuel consuming rate W43 and the positive influence of warranty W42 on satisfaction degree; and vise versa. Such effects can be Fuel consuming rate real value represented by f w∞− (w41, v1 ,0.5) , f w∞− (w43 , v1 ,1) and ∞ w+ f (w42, v1,0.5) respectively using equations (14) and (15) Fig. 4 Fuel consuming rate fuzzification mapping function with threshold depends on fuel price. Fig. 4 and equation (23) represent the fuzzification mapping function for fuel consuming rate C3 as a hyperbolic tangent function with a threshold depends on fuel price V3. That is, the higher the fuel price, the lower the threshold value and vice versa. FC3 (x) = 2 1+ e (23) −1 − λ ( x −θ ) where θ = f avg − v2 ( f max − f min ) / 2, λ = 0.25 , Fc3 represents the fuzzification mapping function of fuel consuming rate, fmax , fmin , favg represent highest, lowest and average fuel consuming rates for 20 Km, V3 denotes the strength of current fuel price. TABLE I • An increase in fuel price V2 will increase the negative influence of fuel consuming rate W43 on satisfaction degree; and vice versa. Such effect can be represented as f w∞− (w43 , v2 ,1) using equation (14) • An increase in the buyer financial state V3 will decrease the negative influences of both of price W41 and fuel consuming rate W43; and vice versa. Such effects can be represented as f w−x (w41, v3 ,0.6) and f w−x (w43, v3 ,1) using equation (16). According to defining such dynamic causal relationship functions will reflex the environment prevailing condition as shown in TABLE . TABLE II DYNAMIC EFFECTS OF V ON W Wij0 Wij THE CONCEPTS MAPPED FUZZY VALUES WITH RESPECT TO V AT TIMES T1 AND T2. Mapping the concepts real values into fuzzy values V At time C V t1 Real Value t2 V1 0.3 0.4 V2 -0.5 0.3 V3 0.6 -0.2 C1 C2 C3 90,000 $ 9 month 3 L/ 20km Fuzzy value at t2 W42 -0.149 0.889 W43 0.325 0.325 -0.245 -0.075 Accordingly for specific car, the fuzzification mapping process for interface concepts real values will be sensitive to the environment dominant variables V. Assuming that pmax=110, pmin=30, fmax =6 and fmin=2. The result of the fuzzification mapping process of the interface concepts C1, C2, and C3 at times t1 and t2 is shown in TABLE I. Wijd (t1) Wijd (t2) f w∞− (w41, v1 ,0.5) -0.66 -0.68 f wx− (w41, v3 ,1) -0.264 -0.744 fw∞+ (w42, v1,0.5) 0.49 0.52 f w∞− (w43 , v1 ,1) -0.65 -0.7 f w∞− (w43 , v2 ,1) -0.325 -0.79 f wx− (w43, v3 ,1) -0.130 -0.83 -0.6 W41 Fuzzy value at t1 The effect of V on Wij 0.4 -0.5 Accordingly, the causal relationship matrices after the influences of V at times t1 and t2 become as shown in equations (25) and (26) respectively W Faculty of Computers and Information - Cairo University t1 C1 = C2 C3 C4 C1 0 0 0 - 0.264 C2 − 0 .1 0 0 0 . 49 C3 0 .3 0 0 - 0.130 AOM -104 C4 0 0 0 0 (25) The 7th International Conference on INFOrmatics and Systems (INFOS 2010) – 28-30 March Applied Optimization and Metaheuristics Track C1 0 C2 − 0 .1 C3 0 .3 C4 0 2 0 0 0 0 3 0 0 0 0 − 0 . 744 0 . 52 − 0 . 83 0 C1 W t2 = C C C 4 (26) Accordingly, the inference is carried out at times t1 and t2 as follows: w1t1j C tj1 = w12t1 * C 2t1 + w13t1 * C 3t1 y1t1 = j = -0.1 * 0.325 + 0.3 * -0.245= - 0.106 ( C1t1 +1 = f C C1t1 , y1t1 ) =c t1 1 t + y1t1 (1− c11) = -0.149 - 0.106 * 0.851= - 0.239 t t t t t t w1t1j C tj1 = w411 * C11 + w421 * C 21 + w431 * C 31 t1 4 y = j = -0.264 * -0.239 + 0.49 * 0.325 + - 0.130 * -0.245 = 0.254 ( C4t1 +1 = f C C4t1 , y4t1 ) = in some cases, the designer may not able to add such concepts because of its partial effect in some subsequent paths of the effected concept. The proposed model enables the designer of a DFCM to make any dynamic causal relationship function in not only its causal concept but also in the environment prevailing conditions if needed. Also, the proposed model allows dynamic fuzzification mapping for interface concepts. This assertion aims to reflex the environment prevailing conditions on the behavior of fuzzification process. Accordingly, same interface concept real value will be mapped to different fuzzy values in response to changes in the environment. Such contribution makes the inference process environment sensitive. Accordingly, the result will be more reliable and robust. An illustrative example shows that: when evaluating specific car, although it may be satisfying at time t1 it may become not satisfying at time t2. Hence, the same fuzzy cognitive map knowledge and concept values infer deferent outcomes in response to changes in the environment prevailing conditions and the result became more reliable. f C (0,0.254 ) = 0.254 REFERENCES [1] In the same way, the satisfaction degree at time t2 will be computed as follows: w1t 2j C tj2 = w12t2 * C 2t2 + w13t2 * C 3t2 y1t2 = [2] [3] j = -0.1 * 0.325 + 0.3 * -0.075 = -0.0325 - 0.0225= - 0.055 ( C1t2 +1 = f C C1t 2 , y1t 2 ) =c t2 1 [4] t + y1t2 (1− c12 ) = 0.889 – 0.5 * 0.055 = 0.861 t2 y4 = w1t 2j C tj2 = [5] [6] j t2 41 t2 1 t2 t2 w * C + w42 * C 2t2 + w43 * C 3t2 = -0.744 * 0.861 + 0.52 * 0.325 + - 0.83 * -0.075 = -0.409 ( ) C 4t2 +1 = f C C 4t2 , y 4t2 = f C (0,0.254 ) =-0.409 D. CONCLUSION In this paper, we introduced an enhanced version of fuzzy cognitive map approach called "DFCM". 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