A Proposition for using Mathematical Models Based on a Fuzzy System with Application

Transcription

1 From the SelectedWorks of R. W. Hndoosh Winter October A Proposition for using Mathematical Models Based on a Fuzzy System with Application R. W. Hndoosh Available at:

2 International Journal of Mathematical Sciences ISSN: Vol. 33 Issue A Proposition for Using Mathematical Models Based on a Fuzzy System with Application Rana Waleed Hndoosh Department of Software Engineering College of Computers Sciences & Mathematics Mosul University Iraq. rm.mr2100@yahoo.com. ABSTRACT M. S. Saroa Department of Mathematics Maharishi Markandeshawar University Mullana India. mssaroa@yahoo.co.in. Sanjeev Kumar Department of Mathematics Dr. B. R. Ambedkar University Khandari Campus Agra India. sanjeevibs@yahoo.co.in. Some mathematical models based on fuzzy set theory fuzzy systems and neural network techniques seem very well suited for typical technical problems. This study aims to build two different models the Fuzzy Inference System (FIS) and the Adaptive Fuzzy System using neural network. We have proposed a new model of a fuzzy system that is extended from 2-dimensions to 3-dimensions using Mamdani's minimum implication the minimum inference system the Singleton fuzzifier and the Center Average Defuzzifier. Also we have extended the theorem accuracy of the fuzzy system to 3- dimensions along with changing the type of fuzzy inference system. We have applied these mathematical models on real data of Washing Machine (WM) and gotten good result of both models with better accuracy when compared between models. We have created programs for these models using MATLAB 2012b. Keywords-Fuzzy System; FIS; ANFIS; Neural Networks; Accuracy of the Fuzzy System; Fuzzy washing machine. 1. INTRODUCTION Fuzzy variables are processed using a system called a Fuzzy Inference System (FIS); it involves fuzzification fuzzy inference and defuzzification [3] [12] [15] [20]. The reason to represent a fuzzy system in terms of a neural network is to utilize the learning capability of neural networks to improve performance such as adaptation of fuzzy system [15] [17]. In the next Section we have introduced basic concepts of fuzzy rule-base with many types of fuzzy rules [4] [8]. In Section 3 we have explained the general concepts of the Fuzzy Inference System that is composition based inference and individual-rule based inference with the computational procedure of both [5] [16]. In Section 4 we have shown the detailed mathematical formulas of the Fuzzy Inference System (FIS) types [12] [13] [15] [19] [20]. In Section 5 we construct interfaces between the fuzzy inference system and the environment using fuzzifier and defuzzifier models [3] [4] [11] [14] [18]. In Section 6 we propose and construct a new model of fuzzy system that is extended from the 2-dim to 3-dim using Mamdani's minimum implication with the minimum inference system the Singleton fuzzifier and Center Average Defuzzifier where first the Mamdani fuzzy rule is used the first time and the ST fuzzy rule is used the second time. Additionally we have extended the Theorem of fuzzy system accuracy for the proposed model where this approach requires three pieces of information in order for the modeled fuzzy system to satisfy any pre-specified degree of accuracy [6] [10] [19]. In Section 7 we have adapted the structure of the fuzzy system using a neural network and we have summarized the procedures to modeling a fuzzy system that depends on three layers of network step by step [1] [5] [7] [9] [15] [16] [17]. Finally in Section 8 we have applied all the previous concepts to real application of fuzzy washing machines (WM) [2] [6] [10] and have obtained and accurate good results for both models. We have created programs for these models using MATLAB 2012b. Concluding remarks are given in Section 9. This work presents two alternative modeling approaches to find the degree of cleanness of clothes dependent on the property of inputs with concepts of the fuzzy system. The focus here is not only on how to construct the models but also on how to use these modeling systems to interpret the results and assess the uncertainty of the models. 2. FUZZY RULE-BASE A fuzzy rule-base consists of a set of fuzzy IF-THEN rules. Specifically the fuzzy rule base comprises the following fuzzy IF-THEN rules [4]: (1) where and are fuzzy set in and respectively where and are the input and output linguistic variables of the fuzzy system respectively. Let be the number of rules in the fuzzy rule base; that is 12 in Eq. (1). We call the rules in the form of (1) canonical fuzzy IF-THEN rules because they include many other types of fuzzy rules as in [8] [16]: a) Partial rules: wherem n. (2) b) Or rules: (3)

3 International Journal of Mathematical Sciences ISSN: Vol. 33 Issue c) Statement of single fuzzy: (4) d) Gradual rules as: The colder thex the hotter they (5) e) Non-fuzzy rules: If the membership functions of and can only take values 1 or 0 then the rules (1) become non-fuzzy rules. The fuzzy rule-base consists of a set of rules. We introduce the following concepts: 1. A set of fuzzy IF-THEN rules is complete if for any there exists at least one rule in the fuzzy rule base say rule such that: (6) That means at any point in the input space there is at least one rule that "fires"; that is the membership value of the IF part of the rule at this point is non-zero. 2. A set of fuzzy IF-THEN rules is consistent if there are no rules with the same IF parts but different THEN parts [12]. 3. A set of fuzzy IF-THEN rules is continuous if there do not exist such neighboring rules whose THEN part fuzzy sets have empty intersection. That means the input-output behavior of the fuzzy system should be smooth. 3. GENERAL CONCEPTS OF FUZZY INFERENCE SYSTEM In a fuzzy inference system fuzzy logic principles are used to combine the fuzzy IF-THEN rules in the fuzzy rule base into a mapping as: : (7) There are two ways to infer with a set of rules: composition based inference and individual-rule based inference. We now show the details of these two ways; In composition based inference all rules in the fuzzy rule base are combined into a single fuzzy relation in which is then viewed as a single fuzzy IF-THEN rule. There are two opposite arguments for what a set of rules should mean. The first one views the rules as independent conditional statements [3] [4]. If we accept this view then a reasonable operator for combining the rules is union. The second one views the rules as strongly coupled conditional statements. If we adapt this view then we should use the operator intersection to combine the rules. We now show the details of these two schemes: Let be a fuzzy relation in which represents the fuzzy IF-THEN rule (1); that is We know that is a fuzzy relation in defined by: (8) where represents any t-norm operator. The implication is defined according to various implications. If we accept the first view of a set of rules then the rules in the form of (1) are interpreted as a single fuzzy relation defined by: (9) This combination is called the Mamdani combination. If we use the symbol to represent the s-norm (s-norm is used to represent a union of two fuzzy sets) then (9) can be rewritten as: µ x y µ x y µ x y (10) For the second view of a set of rules the fuzzy IF-THEN rules of (1) are explained as a fuzzy relation which is defined as: (11) or equivalently (12) where denotes t-norm (t-norm is used to represent an intersection of two fuzzy sets). This combination is called the Godel combination. Let be an arbitrary fuzzy set in X and be the input to the fuzzy inference system FIS. Then by viewing or as a single fuzzy IF-THEN rule and using the generalized rule that is defined as: (13) where is a fuzzy set which represents the antecedent can be any operator and fuzzy relation

4 International Journal of Mathematical Sciences ISSN: Vol. 33 Issue which represents the antecedent a fuzzy set which represents the consequence. Then we get the output of the FIS as: sup when we use the Mamdani combination (14) t-norm operator or as: (15) when we use the Godel combination s-norm operator. In summary the computational procedure of the composition based inference is given as follows [5] [8] [20]: Step1: For the I fuzzy IF-THEN rules in the form of (1) determine the membership functions 1 according to (8). Step2: View as the in the implications and determine 1 2 according to any one of implications. Step3: Determine according to (10) or (11). Step4: For given input the FIS gives output according to (14) or (15). In individual-rule based inference each rule in the fuzzy rule base determines an output fuzzy set and the output of the whole fuzzy inference system is the combination of the individual fuzzy sets. The combination can be taken either by union or by intersection. The computational procedure of the individual-rule based inference is summarized as follows: Steps1 and2: Same as the Steps 1 and 2 for the composition based inference. Step3: For given input fuzzy set compute the output fuzzy set for each individual according to the generalized rule (13) that is 1 2. (16) Step4: The output of the FIS is the combination of the fuzzy sets either by union that is (17) or by intersection that is (18) where and denote s-norm and t-norm operators respectively. In fuzzy inference the decision is based on the input information and the knowledge in the form of rules. The inference model is classified based on the form of the consequence. There are two types of fuzzy rules: 1. Mamdani fuzzy rules: If the rule has fuzzy suggestions combined by fuzzy connectives both in the antecedent and in the consequence then it is called a Mamdani fuzzy rule and denoted by (1). The fuzzy inference of the Mamdani fuzzy rule is done as follows. First the truth value of the rule is evaluated depending on the fuzzy connections (and/or) in the IF part (antecedent). Since our model has and only then the truth value evaluation is done by [4]: (19) Then the fuzzy set of the output is obtained depending on the truth value and the fuzzy set of the consequence by the t-norm (min). The fuzzy inference produces the fuzzy set of output by:. (20) The consequences of all the rules are aggregated in the final consequence by the s-norm (max) as: (21) whereiis the number of the rules and are functions of the inputs. 2. Sugeno fuzzy rules: When the consequence of the rule is a linear function of the inputs then it is called a Sugeno fuzzy rule. The consequent of (1) becomes: (22)

5 International Journal of Mathematical Sciences ISSN: Vol. 33 Issue The output of the Sugeno rule is a crisp value of the function. The final output of the fuzzy inference is obtained by a weighted mean as: where is the number of the rules and is the truth value of the antecedent got by (19). 4. TYPES OF FUZZY INFERENCE SYSTEMS We now show the detailed formulas of a number of FIS that are commonly used in fuzzy systems [1] [11] [13]. 1. Product Inference System: In product inference system we use: (i) Individual rule-based inference with union combination (17) (ii) Mamdani's product implication. (24) (iii) Algebraic product for all the t-norm operators and max for all the s-norm operators. Specifically from (16) (17) (24) and (8) we obtain the product inference system as (25) That is given fuzzy set the product inference system gives the fuzzy set. 2. Minimum Inference System: In minimum inference system we use: (i) Individual rule-based inference with union combination (17) (ii) Mamdani's minimum implication (26) (iii) min for all the t-norm operators and max for all the s-norm operators. Specifically from (16) (17) (26) and (8) we have (27) That is given fuzzy set the minimum inference system gives the fuzzy set according to (27). 3. Lukasiewicz Inference System: In Lukasiewicz inference system we use: (i) Individual rule-based inference with intersection combination (18) (ii) Lukasiewicz implication: 1 1 (28) (iii) min for all the t-norm operators. Specifically from (18) (16) (28) and (8) we get: (23) 11 1 (29) That is for given fuzzy set the Lukasiewicz inference system gives the fuzzy set. 4. Zadeh Inference System: In Zadeh inference system we use: (i) individual rule-based inference with intersection combination (18). (ii) Zadeh implication: 1 (30) (iii) min for all the t-norm operators. Specifically from (18) (16) (30) and (8) we obtain: 1 (31) 5. Dienes-Rescher Inference System: In Dienes-Rescher inference system we use: (i) The same operations as in the Zadeh inference system. (ii) Dienes-Rescher implication. 1 (32) (iii) Min for all the t-norm operators. Specifically we obtain from (18) (16) (32) and (8) that: 1 (33)

6 International Journal of Mathematical Sciences ISSN: Vol. 33 Issue FUZZIFIER AND DEFUZZIFIER MODELS In most applications the input and output of the fuzzy system are real valued numbers. Then we must construct interfaces between the fuzzy inference system and the environment [3] [11] [14] [18]. The interfaces are the fuzzifier and defuzzifier The Fuzzifier The fuzzifier is defined as: : (34) There are some criteria in designing the fuzzifier such as the following: a. The fuzzifier should consider the fact that the input is at the crisp point that is the fuzzy set should have large membership value at. b. If the input to the fuzzy system is corrupted by noise then it is desirable that the fuzzifier should help to suppress the noise. c. The fuzzifier should help to simplify the computations involved in the fuzzy inference system. We now propose three types of fuzzifiers: (i) Singleton fuzzifier: The Singleton fuzzifier mapping is defined as: : where x is a real-valued point and A is a fuzzy Singleton that is: 1 0 (35) (ii) Gaussian fuzzifier: The Gaussian fuzzifier mapping is defined as: : is a fuzzy set which has the following Gaussian membership function: (36) whereα are positive parameters and the t-norm is usually chosen as an algebraic product or min. (iii) Triangular fuzzifier: The triangular fuzzifier mapping is defined as: : is a fuzzy set which has the following triangular membership function: (37) 0 where are positive parameters and the t-norm is usually chosen as an algebraic product or min TheDefuzzifier The defuzzifier is defined as: where is the output of the fuzzy inference system the crisp point. The task of the defuzzifier is to specify a point in Y that best represents the fuzzy set. For all the defuzzifiers we assume that the fuzzy set is obtained from one of the five fuzzy inference systems in the last section that is is given by (25) (27) (29) (31) or (33) [3] [8]. From these equations we see that is the union or intersection of individual fuzzy sets. We now propose three types of defuzzifiers: (i) The Center of Gravity Defuzzifier: The Center of Gravity Defuzzifier specifies the as the center of the area covered by the membership function of that is:. (38).. where the conventional integral. In fact the membership function is usually irregular and therefore the integrations in (38) are difficult to compute. The next defuzzifier tries to solve this problem with a simpler formula: (ii) The Center Average Defuzzifier: Because the fuzzy set is the intersection or union of fuzzy sets a good approximation of (38) is the weighted average of the centers of the fuzzy sets where the weights equal the heights of the corresponding fuzzy sets.

7 International Journal of Mathematical Sciences ISSN: Vol. 33 Issue Specifically let be the center of the fuzzy set and be its height. Then the Center Average Defuzzifier determines as: (39) The center average defuzzifier is the most commonly used defuzzifier in fuzzy systems. It is computationally simple and small changes in result in small changes in. (iii) The Maximum defuzzifier: The maximum defuzzifier chooses the as the point in Y at which satisfies its maximum value. Define the set as: (40) that is is the set of all points in at which satisfies its maximum value. The maximum defuzzifier defines as an arbitrary element in that is: (41) If contains a single point then is uniquely defined. If contains more than one point then we may still use (40) or use the smallest of maxima the largest of maxima or the mean of maxima defuzzifiers. Specifically the smallest of maxima defuzzifier gives: (42) the largest of maxima defizzifier gives: (43) and the mean of maxima defuzzifier is defined as: where is the usual integration for the continuous part of and is the summation for the discrete part of. The maximum defuzzifier is computationally simple But small changes in may result in large changes in. 6. PROPOSAL OF A NEW MODEL OF A FUZZY SYSTEM ON 3-DIMENSIONS In this section we have proposed a new model of a fuzzy system that is extended from 2-dim to 3-dim [20]. Let. The general membership function of fuzzy set is a continuous function in given by: 0 ; 1 (45) 0 where 0 1is a non-decreasing function 0 1is a non-increasing function. If the universe of discourse is bounded then are finite numbers. General membership functions (MFs) include many commonly used MF as specia1 cases as: then the general MF become the trapezoid MFs. If we get the triangular MF; a triangular MF is denoted as ;. If fuzzy sets then they are called: 1. Complete on if for any there exists such that Consistent on 1 for some implies that 0 for all. 3. Normal consistent and complete with general MFs. ;. If then In the next section we model a particular type for washing machine of fuzzy systems that have some nice properties. We consider three-input fuzzy systems; however the approach and results are all valid for general N-inputs fuzzy systems. We first specify the model. 6.1 The Proposed Model Let be defined as : that is a function on the compact set and the analytic formula of be unknown. Suppose that for any we can obtain. Our task is to model a fuzzy system that approximates. We now model such a fuzzy system in a step-by-step manner [4] [11]. Step 1: Define 123 fuzzy sets which are normal consistent and complete with triangular (44) (46)

8 International Journal of Mathematical Sciences ISSN: Vol. 33 Issue membership functions and with. Define: Fig.1 shows an application model with Step 2: Construct fuzzy IF-THEN rules in the following form:. (47) where is chosen as: and the center of the fuzzy set denoted by (48) This is the case when (47) depends on the Mamdani fuzzy rule and the antecedent of our model is connected by and. Only then from (19) the truth value evaluation is given by:. From (20) the fuzzy inference produces the fuzzy set of output by: The consequences of all the rules are aggregated in the consequences by the max function as: Fig. 1: An application of the fuzzy sets defined in step1 of the model procedure. But when the consequence of the rule is a linear function (that is means the fuzzy rule is Sugeno fuzzy rule) then the output of the Sugeno rule depends on where is linear function base on thatt is defined as: and parameters can be computed by the least square model that is explained in the next section. Step3: Constructing the fuzzy system from the (49) rules of (48) using Mamdani's minimum implication (26) with

9 International Journal of Mathematical Sciences ISSN: Vol. 33 Issue the minimum inference system (27) the Singleton fuzzifier (35) and the Center Average Defuzzifier (39) we get: Since the fuzzy sets are complete at every there exist such that: 0. Hence the fuzzy system (50) is well defined that is its denominator is always nonzero. From step2 we see that the IF parts (antecedent) of the rules (48) constitute all the possible combinations of the fuzzy sets defined for each input variable. So if we generalize the model procedure to N-input fuzzy systems and define N fuzzy sets for each input variable then the total number of rules is that is by using this design method the number of rules increases exponentially with the dimension of the input space. The final observation of the model procedure is that we must know the values of since can be arbitrary points in ; this is equivalent to say that we need to know the values of. In the second part of this section we explain the accuracy of the modeled above on the unknown function. We have extended accuracy of the fuzzy system from 2-dim to 3-dim as well as changed the type of fuzzy inference system to a minimum inference system to suit our application [4] [19] [20]. 6.2 Theorem1(The Fuzzy System Accuracy for the Proposed Model) Letbe the fuzzy system in (50) and be the unknown function in (48). If is continuously differentiable on then: (51) where the infinite norm. is defined as: Proof: Let where Since then we get: (52) which implies that for any there exists such that. Now suppose that is and. Since the fuzzy sets are normal consistent and complete at least one and at most three are nonzero for 1. From the definition of 1 2 these three possible nonzero are and the two possible nonzero 1 are. Similarly the two possible nonzero are 1 2 are and. Hence the fuzzy system of (52) is simplified to: From Eq. (48) we get:... (53) (54).. (50)

10 International Journal of Mathematical Sciences ISSN: Vol. 33 Issue We have: : : : From the Mean Value model we can write (55) as: (55) (56) Since which means that we have that for 1 1 and 1 Then (56) becomes: (57) Since then we get: (58) From (57) we can conclude that fuzzy systems in the form of (50) are Specifically since are finite numbers for any given 0 we can choose and small enough such that. Hence from (51) we have: From (58) we see that in order to model a fuzzy system with a pre-specified accuracy we must know the bounds of the derivatives of with respect to that is. In the model process we need to know the value of Therefore this approach requires these three pieces of information in order for the modeled fuzzy system to satisfy any pre-specified degree of accuracy. 7. MODEL OF AN ADAPTIVE FUZZY SYSTEM USING A NEURAL NETWORK In this section we adapt the structure of the fuzzy system that is specified first with the structure of some parameters that are free to change. Then these free parameters are determined according to the input-output pairs [1] [17]. First we specify the structure of the fuzzy system to be modeled [15]. Here we choose the fuzzy system with a minimum inference system a Singleton fuzzifier a Center Average Defuzzifier and a Triangular MF given by: (59) where I is fixed and are free parameters. The fuzzy system (59) has not been modeled because the parameters are not specified. To determine these parameters in some optimal manner it is helpful to represent the fuzzy system of (59) as a feed-forward network. Specifically the mapping from the input

11 International Journal of Mathematical Sciences ISSN: Vol. 33 Issue to the output can be performed according to the following operations: first the input minimum triangular operator to become [20]: is passed through a ; then the are passed through a summation operator and a weighted summation operator to get: and ; finally the output of the fuzzy system is computed as. This three-stage operation is shown in Fig.2 as a three layer feed forward network. We now summarize the procedures to model a fuzzy system that depends on three layers of network. Step1: Structure specification and initial parameters Select the fuzzy system (59) and determine the. The larger results more parameters and in more computation but gives better accuracy. Specify the initial parameters. Then the initial fuzzy system becomes as in (60). These initial parameters may be determined according to the linguistic rules from human experts as in our application. (60) Fig.2: Network representation of the fuzzy system. Step2: Calculating the outputs of the fuzzy system. For a given inputs-output pair and at the training stage present to the input layer of the fuzzy system in Fig.2 and compute the outputs of layers. That is compute [1] [5] [6] [7] [16] [17] [19]: First output: Every node produces MF of an input parameter. The node output is explained by:. (61) where are the inputs; are linguistic fuzzy sets associated with nodes. is the degree of MFs of a fuzzy set. Second output: Every node is a fixed node whose outpu is the minimum of all MF: where is declared by triangular MF. Then we get: (62) Third output: Depending on (62) the node calculates all rules as: (63)

12 International Journal of Mathematical Sciences ISSN: Vol. 33 Issue Fourth output: Every node here is an adaptive node with a node MF of output. (64) where and from (49) we get: where 1 4. is the parameter set of the node. Fifth output: The single node is a fixed node labeled which computes the final output as the summation of all result (65) Suppose: (66) Then the final output obtained is: / (67) Now that we have noted a are free parameters to be modeled. We can collect these parameters into the -dim vector as: (68) When select the initial parameters 1 if there are linguistic rules from human experts then choose 1to be the centers of the THEN part fuzzy sets in these linguistic rules; otherwise choose 1 arbitrarily in the output space. In this way we can say that the initial fuzzy system is constructed from human experts. Step3: Update the parameters Use the training algorithm to compute the updated parameters where and equal those computed in step2. That means compute the new parameters using the least squares model as: 1 1 (69) 1 1 where 1 is chosen in step2 and 1 is a large constant. The modeled fuzzy system in (59) with the parameters is equal to the corresponding elements in. (70) (71) Step4: Repeat by going to step2 with 1 until the error or until theequals a pre-specified number. Step5: Repeat by going to step2 with 1 12 ; that is update the parameters using the next input-output pair ; Keep in mind that the parameters are the centers of the fuzzy sets in the THEN parts of the rules and the parameters are the left and right base points the centers (triangle peak) of the Triangular fuzzy sets in the IF parts of the rules [3] [9] [15]. Therefore given a modeled fuzzy system (59) we can improve the fuzzy IF- THEN rules that modeled the fuzzy system. These improved fuzzy IF-THEN rules may help to explain the model fuzzy system in a user friendly manner. 8. APPLICATION (THE FUZZY SYSTEM OF WASHING MACHINES) Fuzzy washing machines were the first major consumer products to use fuzzy systems [10]. They use a fuzzy system to automatically set the cleanness of clothes depending on the amount of the dirty in water. There are multi-input: (washing time) (temperature of water) and (washing powder). More specifically the fuzzy system used is a three input one output system where the three inputs are measurements of dirtiness: washing time temperature of water and the washing powder and the output is the cleanness of clothes. Clearly the more cleanness of the clothes the more washing time temperature of water and washing powder is needed. This work aims at presenting the idea of controlling the cleanness of clothes using a fuzzy system where it describes the procedure that can be used to get a suitable cleanness of clothes. When one uses a washing machine the person generally selects the length of wash time based on the dirtiness

13 International Journal of Mathematical Sciences ISSN: Vol. 33 Issue of clothes. Unfortunately there is no easy way to formulate an exact mathematical relationship between time temperature & powder with the degree of cleanness required [2] [6]. The real data we was obtained from a group of experts working at Jumbo Electronics Company Ltd in the United Arab Emirates and contains 50 cases. We have built two models the first with the Mamdani fuzzy rule and the second with the ST fuzzy rule see Fig.(3a) and (3b). Fig.3a: FIS of WM using Mamdani model. Fig.3b: FIS of WM using ST model Structure of the Applied Model Now we consider multi-inputs: (temperature of water) (washing powder) and (washing time) with single-output y (cleanness of clothes). : &. The fuzzy system is composed of a fuzzifier fuzzy rule-base fuzzy inference and defuzzifier. To apply steps of FIS step by step first we described the three inputs with output as: 1. Input2 (Temperature of water): The range of temperature is 0 to 50 and the unit is C; this data was treated and measured so that it becomes trapped between 0 and 1 see Fig.(4a). The temperature of water (TW) has five different degrees of linguistic variables: Very Cold (Vcold) Cold (cold) Good (good) Hot (hot) and Very Hot (Vhot). See Fig.(4b). 2. Input3 (Amount of washing powder): The range of washing powder is from 0 to 100 and the unit is grams in this case also the data was measured so that is converted also between 0 and 1 see Fig.(5a). Washing powder (WP) also represented by five linguistic variables: Very Little (Vlittle) Little (little) Medium (med) Much (much) and Very Much (Vmuch). See Fig.(5b). 3. Input1(Washing time): The range of time is from 0 to 15 and the unit is minutes but this data was treated and measured so that it is trapped between 0 and 1 see Fig.(6a). Washing time (WT) is represented by three linguistic variables Less Time (Ltime) Average Time (Atime) and More Time (Mtime) see Fig.(6b). 4. Output (Cleanness of clothes):the range of cleanness of clothes is from 0 to 1 and the unit is percent see Fig.(7a). Cleanness of clothes (CC) is represented by five linguistic variables Not Clean (Nclean) Less Average (Laverage) Average (Average) More Average (Maverage) and Full Clean (Fclean). See Fig.(7b) Discussion and Results for the Model of the Fuzzy System with Mamdani and ST Models Now we build the proposed fuzzy system step by step on our application: Step 1: Define 123 : 1 5 : 1 5 : 1 3 fuzzy sets and with triangular membership functions ; ; and with 0 1. Define: fuzzifier performs a mapping: : where with the Singleton fuzzifier (30). We used MATLAB to created the programs. We have loaded the inputs-output variables for our application into the workspace see table (2). Three vectors of inputs with one vector of output are loaded with 50 cases. For the first observation that consisted of (temperature=0.108 powder= and time=0.1493) see Fig (8a). We can build the first rule as: [(temperature is Vcold) AND (powder is Vmuch) AND (time is Ltime)] Step2: We note the fuzzy rule-base consists of fuzzy IF-THEN rules and the centers of are evaluated at the 75 red points shown in Fig. (1) then (42) becomes:

14 International Journal of Mathematical Sciences ISSN: Vol. 33 Issue Fig.4a: Representation RD of TW. Fig.4b: Representation TW by MFs Fig.6a: Representation RD of WT. Fig.6b: Representation WT by MFs. Fig.5a: Representation RD of WP. Fig.5b: Representation WP by MFs. Fig.7a: Representation RD of CC. Fig.7b: Representation CC by MFs. : : (72a).... : : (72b).. Form (72a) is used when we create the Mamdani fuzzy rule but form (72b) is used when we create the Sugeno fuzzy rule where A & B are fuzzy sets are input variables y is the output variable i is the number of rules and the fuzzy set A consists of fuzzy subsets called linguistic terms characterized by MF ( ) are triangular MFs with form as in (40) and (41) when and the linear function depends on inputs are defined as (49). The first linguistic term 1 5. depends on linguistic variable and consists of: = very cold ; = cold ; = good ; = hot ; and = very hot ; The second linguistic term 1 5 depends on linguistic variable and consists of: ; ; ; = much ; and = very much ; The third linguistic term 1 23 depends on linguistics variable and consists of: = less time ; = good time ; and = more time ; The center of the fuzzy set is evaluated at the 50 red points as in Fig. (1). Also it is characterized by triangular MFs ( ) denoted by We have applied the fuzzy operator and (t-norm) on all observations through 75-rules that are built in this system as forms (see Fig. (8a)): IF (tem is Vcold) and (powder is Vmore) and (time is Ltime) THEN (cleanness is Average) The fuzzy inference process determines the mapping as: : isinput fuzzy sets in the input space X the fuzzy sets in the output space Y. Each of the 75 rules specifies a fuzzy set Y given by the compositional rule of inference: and from (8) we get an MF: 123 is a number of rules. (73) From (14) fuzzy sets are characterized by MF: µ y (74)

15 International Journal of Mathematical Sciences ISSN: Vol. 33 Issue Then (74) becomes: µ y (75) where. is an implementation. Since in this model we used Mamdani's minimum implication (26) then: (76) from that we get: µ y The aggregation operator was applied in order to get the fuzzy set that uses the s-norm or t-norm operator depending on the type of fuzzy implication. The aggregation is obtained by: (77) The MF of is computed by using s-norm as:. (78) Stepe3: The defuzzifier performs a mapping as: where fuzzy set is a crisp point. The center of the area is a final system output that is defined by the following formula: : : (79) Fig.8a: Representation rules editor. Fig.8b: The rule view of inputs with output. The fuzzy system has been built from the 75 rules of (72a) using Mamdani's minimum implication (26) with the minimum inference system (27) the Singleton fuzzifier (35) and the Center Average Defuzzifier (39).See Fig. (8b). We have created two programs the first depended on the Mamdani model and the second depended on the ST model. After applying the programs of FIS that is given by model (79) we have obtained good target results and we have applied measure accuracy for all data between the actual output and the target by: (where Rv is actual output & FISv is the output target by FIS). This value is very small where the min error=0 and the max error=0.15 when FIS is used with the Mamdani fuzzy rule model. But when the FIS is used with the ST fuzzy rule model the min error=0 and the max error=0.11. We have represented all data with results in table (1) and (2). The surfview tool is the surface viewer that helps view the input-output surface of the FIS. This conception is very helpful to understand how the system is going to behave for the entire range of values in the inputs space. In the plots below the first surface viewer shows the output surface for two inputs Time (T) & Powder (P) see Fig.(9a) the second surface shows the output surface between Time & Temperature (Temp) see Fig.(9b) and the third surface shows the output surface for two inputs Temperature & Powder (see Fig.(9c)).

16 International Journal of Mathematical Sciences ISSN: Vol. 33 Issue Fig.9a: Surface Fig.9b: Surface Fig.9c: Surface b/w T & P. b/w T & Temp. b/w Temp. & P Discussion and Results for the Model of the Adaptive Fuzzy System Using Neural Network We specify the structure of the fuzzy system to be modeled. Here we choose the fuzzy system with a minimum inference system a Singleton fuzzifier the Center Average Defuzzifier and a Triangular MF given by: (80) where I=75 is the number of rules. Model (80) hasn t been modeled because the free parameters are not specified. These parameters should be determined in order to represent the : Step1: Determine the initial parameters according to the linguistic rules from human experts as (when 1 1 then similarly for all and for each input. Step2: From a given inputs-output pair layers on Fig. (2) as: (i) The node of first output is represented by: where is the degree of MFs of a fuzzy set. ; cases then we can compute the outputs of (ii) To compute the second output we should use the minimum function of all MFs as: Since is a Triangular MF then we get: when p=1 we have ( ; 0.462) then Similarly we can find all with respect to all inputs. (iii) For all rules we have calculated the node as: 0 0

17 International Journal of Mathematical Sciences ISSN: Vol. 33 Issue (iv) The fourth output depends on a node MF of output with an adaptive node. Then: Since then we should determine the initial parameters of 1 4. such as Then the center of the first consequence fuzzy set is Similarly we determined all parameters of linear functions for each rules. (v) To compute the final output we must take the summation of all results as: Step3: In step2 we have specified initial parameters of when 0. But now we update these parameters at 1 using the least squares model and repeat all the procedure in step2 to compute and. From (69) to (71) we compute the new parameters 1 as: θ2 θ1 t o θ1 P2o t P2o o P o Similarly we can update all parameters of for all training data. Repeat all procedures in step2 with 1 until the last specific number of qis reached and with 1 to update the parameters use the next input-output pair. We have created a program and selected initial parameters. After applying the programs of AFS using a Neural Network that is given by model (80) we have obtained better results with the Neural Network than with the FIS. we have applied measure of accuracy to all data where the min error=0 and the max error=0.008 (see table (1) and (2)) with the best training error ( ) and average testing error of training data ( ) see Fig.(10a) and (10b). Fig.10a: Representation of training error with epochs 55. Fig.10b: Representation of training data and FIS output. Table1: Representation of Minimum and Maximum error with Average error. Type of model Min error Max error Average error of training data FIS (Mamdani model) FIS (ST model) ANFIS (ST model)

18 International Journal of Mathematical Sciences ISSN: Vol. 33 Issue Table2: Representation of results of FIS with (Mamdani & ST model) ANFIS and errors. No. Actual FIS with FIS with ANFIS E(FIS output (Mamdani model) M ) E(FIS ( ST model) ST ) (ST model) E(ANFIS ST ) max min 1 0 min max min max CONCLUSION In this work we have suggested first a new fuzzy system at 3-dim depending on Mamdani's minimum implication with the minimum inference system the Singleton fuzzifier and the Center Average Defuzzifier and second a model of the Adaptive Fuzzy System using a Neural Network. We have applied these models on 50 real observations of fuzzy washing machine models and obtained very good results for all data. We have recorded a minimum error of 0 when we applied FIS with Mamdani & ST but recorded a maximum error of 0.15 and 0.11 with Mamdani & ST models respectively. Also we have obtained the average error of training data equal to with the Mamdani model and with the ST model. We have recorded better results with the model of the Adaptive Fuzzy System using a Neural Network that depends on the ST model. The maximum error is the equal to and the average error of training data is equal to We conclude that the best results with the lesser error are obtained with Adaptive Fuzzy System. In the future we can develop these models of fuzzy system to generate many outputs rather than one output or extend the number of input variables such as weight of clothes or types of clothes or types of dirt. Also we can change the fuzzy inference system with another types or with different type of fuzzifier or defuzzifier.

19 International Journal of Mathematical Sciences ISSN: Vol. 33 Issue ACKNOWLEDGEMENTS The author wishes to thank the reviewers for their excellent suggestions that have been incorporated into this paper. I would like to thank and acknowledge Professor. Deepak Gupta Head of Department of Mathematics M.M. University Mullana Ambala India who advised me to improve my work. REFERENCE [1] L. Aik& Y. Jayakumar (2008) A Study of Neuro-fuzzy System in Approximation-based Problems Matematika Vol. 24 No.2 pp [2] M.Agarwal (2011) Fuzzy logic control of washing machines. Department of Mechanical Engineering Indian Institute of Technology Kharagpur India pp [3] R.Belohlavek and G. Klir (2011) Concepts and Fuzzy Logic. Massachusetts Institute of Technology London England. [4] Y.Chai L. Jia and Z. Zhang (2009) Mamdani model based adaptive neural fuzzy inference system and its application. World Academy of Science Engineering and Technology Vol. 51 pp [5] E.Doğan L. Saltabaş and E. Yıldırım (2007) Adaptive neuro-fuzzy inference system application for estimating suspended sediment loads. International Earthquake Symposium Kocher Vol. 5 pp [6] R.Hndoosh M. Saroa and S. Kumar (2012) Fuzzy and adaptive neuro-fuzzy inference system of washing machine European Journal of Scientific Research Vol. 86 No 3 pp [7] C.Jose R. Neil and W. Curt (1999) Neural and Adaptive Systems John Wiley & SONS INC. New York. [8] G.Jandaghi R. Tehrani D. Hosseinpour R. Gholipour and S. Shadkam (2010) Application of fuzzy-neural networks in multi-ahead forecast of stock price. African Journal of Business Management Vol. 4 pp [9] T.Kamel and M. Hassan (2009) Adaptive neuro fuzzy inference system (ANFIS) for fault classification in the transmission lines The Online Journal on Electronics and Electrical Engineering Vol. 2 pp [10] R.Milasi M.Jamali and C. Lucas (2010) Intelligent washing machine: A bioinspired and multi-objective approach International Journal of Control Automation and Systems Vol. 5 No. 4 pp [11] V.Marza and A. Seyyedi (2008) Estimating development time of software projects using a neuro fuzzy approach. World Academy of Science Engineering and Technology Vol. 46 pp [12] P.Nayak K.Sudheer D. Rangan& K. Ramasastri (2004) A neuro-fuzzy computing technique for modeling hydrological time series. Journal of Hydrology Vol. 291 pp [13] A.Rameshkumar and S. Arumugam (2011) A neuro-fuzzy integrated system for non-linear buck and quasiresonant buck converter. European Journal of Scientific Research Vol.51 No.1 pp [14] D.Ruan and P. Wang (1997) Intelligent Hybrid System: Fuzzy Logic Neural Network and Genetic Algorithm Kluwer Academic Publishers. [15] R. Rojas (1996) Neural Networks Springer-Verlag Berlin. [16] A.Samandar (2011) A model of adaptive neural-based fuzzy inference system (ANFIS) for prediction of friction coefficient in open channel flow. Academic Journals Scientific Research and Essays Vol. 6 No. 5 pp [17] J.Shing and R. Jang (1993) ANFIS: Adaptive network-based fuzzy inference system. IEEE Transactions on Systems Man and Cybernetics Vol. 23 No. 3 pp [18] M.Singh M. Sharma and S. Kumar (2009) Conjugate descent formulation of back propagation error in feed forward neural network. ORION Vol. 25 pp [19] S.Sivanandam and S. Deepa (2008) Principal of Soft Computing Wiley India (P) Ltd. [20] L. Wang (1997) A Course in Fuzzy Systems and Control. Prentice-Hall International Inc.