Topic 7 Fuzzy expert systems: Fuzzy inference

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1 Topic 7 Fuzzy expert yte: Fuzzy inference adani fuzzy inference ugeno fuzzy inference Cae tudy uary Fuzzy inference The ot coonly ued fuzzy inference technique i the o-called adani ethod. In 975, Profeor Ebrahi adani of ondon Univerity built one of the firt fuzzy yte to control a tea engine and boiler cobination. He applied a et of fuzzy rule upplied by experienced huan operator. adani fuzzy inference The adani-tyle fuzzy inference proce i perfored in four tep: fuzzification of the input variable, rule evaluation; aggregation of the rule output, and finally defuzzification. We exaine a iple two-input one-output output proble that include three rule: Rule: Rule: x i A3 project_funding i adequate OR y i B OR project_taffing i all THEN z i C THEN rik i low Rule: 2 Rule: 2 x i A2 project_funding i arginal AND y i B2 AND project_taffing i large THEN z i THEN rik i noral Rule: 3 Rule: 3 x i A project_funding i inadequate THEN z i C3 THEN rik i high tep : Fuzzification The firt tep i to take the crip input, x and y (project funding and project taffing), and deterine the degree to which thee input belong to each of the appropriate fuzzy et..5 Crip Input x A A2 A3 x µ (x = A) =.5 µ (x = A2) = X.7. B Crip Input y B2 y Y µ (y = B) =. µ (y = B2) =.7 tep 2: Rule Evaluation The econd tep i to take the fuzzified input, µ (x=a) =.5, µ (x=a2) =, µ (y=b) =. and µ (y=b2) =.7, and apply the to the antecedent of the fuzzy rule. If a given fuzzy rule ha ultiple antecedent, the fuzzy operator (AND or OR) i ued to obtain a ingle nuber that repreent the reult of the antecedent evaluation. Thi nuber (the truth value) i then applied to the conequent eberhip function.

2 To evaluate the dijunction of the rule antecedent, we ue the OR fuzzy operation.. Typically, fuzzy expert yte ake ue of the claical fuzzy operation union: µ A B (x)) = ax [µ A (x), µ B (x)] iilarly, in order to evaluate the conjunction of the rule antecedent, we apply the AND fuzzy operation interection: µ A B (x)) = in [µ A (x), µ B (x)] adani-tyle rule evaluation A3 B C.. OR. (ax) x X y Y Rule : x i A3 (.) OR y i B (.) THEN z i C (.) A2 x X A.7 B2 y Y AND (in) C Rule 2: x i A2 () AND y i B2 (.7) THEN z i ().5.5 C x X Rule 3: x i A (.5) THEN z i C3 (.5) C3 C3 C3 Now the reult of the antecedent evaluation can be applied to the eberhip function of the conequent. The ot coon ethod of correlating the rule conequent with the truth value of the rule antecedent i to cut the conequent eberhip function at the level of the antecedent truth. Thi ethod i called clipping.. ince the top of the eberhip function i liced, the clipped fuzzy et loe oe inforation. However, clipping i till often preferred becaue it involve le coplex and fater atheatic, and generate an aggregated output urface that i eaier to defuzzify. While clipping i a frequently ued ethod, caling offer a better approach for preerving the original hape of the fuzzy et. The original eberhip function of the rule conequent i adjuted by ultiplying all it eberhip degree by the truth value of the rule antecedent. Thi ethod, which generally loe le inforation, can be very ueful in fuzzy expert yte. Clipped and caled eberhip function eberhip.. eberhip.. tep 3: Aggregation of the rule output Aggregation i the proce of unification of the output of all rule. We take the eberhip function of all rule conequent previouly clipped or caled and cobine the into a ingle fuzzy et. The input of the aggregation proce i the lit of clipped or caled conequent eberhip function, and the output i one fuzzy et for each output variable.

3 . C z i C (.) Aggregation of the rule output z i ().5 z i C3 (.5) C3.5. tep 4: Defuzzification The lat tep in the fuzzy inference proce i defuzzification.. Fuzzine help u to evaluate the rule, but the final output of a fuzzy yte ha to be a crip nuber. The input for the defuzzification proce i the aggregate output fuzzy et and the output i a ingle nuber. There are everal defuzzification ethod, but probably the ot popular one i the centroid technique.. It find the point where a vertical line would lice the aggregate et into two equal ae. atheatically thi centre of gravity (COG) can be expreed a: COG b µ A = a b µ A a ( x ) ( x ) x dx dx Centroid defuzzification ethod find a point repreenting the centre of gravity of the fuzzy et, A, on the interval, ab. A reaonable etiate can be obtained by calculating it over a aple of point. µ ( x ) a A b 2 X Centre of gravity (COG): ( + + 2).+ ( ) + ( ).5 COG = = eberhip ugeno fuzzy inference adani-tyle inference, a we have jut een, require u to find the centroid of a two-dienional hape by integrating acro a continuouly varying function. In general, thi proce i not coputationally efficient. ichio ugeno uggeted to ue a ingle pike, a ingleton,, a the eberhip function of the rule conequent. A ingleton,, or ore preciely a fuzzy ingleton,, i a fuzzy et with a eberhip function that i unity at a ingle particular point on the univere of dicoure and zero everywhere ele.

4 ugeno-tyle fuzzy inference i very iilar to the adani ethod. ugeno changed only a rule conequent. Intead of a fuzzy et, he ued a atheatical function of the input variable. The forat of the ugeno-tyle fuzzy rule i x i A AND y i B THEN z i f (x, y) y where x, y and z are linguitic variable; A and B are fuzzy et on univere of dicoure X and Y, repectively; and f (x, y) i a atheatical function. The ot coonly ued zero-order order ugeno fuzzy odel applie fuzzy rule in the following for: x i A AND y i B THEN z i k where k i a contant. In thi cae, the output of each fuzzy rule i contant. All conequent eberhip function are repreented by ingleton pike. A3 B.. OR. x X y Y (ax) Rule : x i A3 (.) A2 x X A ugeno-tyle rule evaluation OR y i B (.) THEN z i k (.).7 B2 y Y AND (in) Rule 2: x i A2 () AND y i B2 (.7) THEN z i k2 () x X Rule 3: x i A (.5) THEN z i k3 (.5).5.5 k k2 k3 ugeno-tyle aggregation of the rule output. z i k (.) z i k2 () z i k3 (.5).5.5. k k2 k3 k k2 k3 Weighted average (WA): µ ( k) k+ µ ( k2) k2 + µ ( k3) k WA = = = 65 µ ( k) + µ ( k2) + µ ( k3) ugeno-tyle defuzzification z Crip Output z How to ake a deciion on which ethod to apply adani or ugeno? adani ethod i widely accepted for capturing expert knowledge. It allow u to decribe the expertie in ore intuitive, ore huan-like anner. However, adani-type fuzzy inference entail a ubtantial coputational burden. On the other hand, ugeno ethod i coputationally effective and work well with optiiation and adaptive technique, which ake it very attractive in control proble, particularly for dynaic nonlinear yte.

5 Building a fuzzy expert yte: cae tudy A ervice centre keep pare part and repair failed one. A cutoer bring a failed ite and receive a pare of the ae type. Failed part are repaired, placed on the helf, and thu becoe pare. The objective here i to advie a anager of the ervice centre on certain deciion policie to keep the cutoer atified. Proce of developing a fuzzy expert yte. pecify the proble and define linguitic variable. 2. Deterine fuzzy et. 3. Elicit and contruct fuzzy rule. 4. Encode the fuzzy et, fuzzy rule and procedure to perfor fuzzy inference into the expert yte. 5. Evaluate and tune the yte. tep : : pecify the proble and define linguitic variable There are four ain linguitic variable: average waiting tie (ean delay),, repair utiliation factor of the ervice centre ρ,, nuber of erver, and initial nuber of pare part n. inguitic variable and their range inguitic Variable: ean Delay, inguitic Value Notation Nuerical Range (noralied) Very hort hort ediu V [,.3] [.,.5] [.4,.7] inguitic Variable: Nuber of erver, inguitic Value Notation Nuerical Range (noralied) all ediu arge [,.35] [.3,.7] [.6, ] inguitic Variable: Repair Utiliation Factor, ρ inguitic Value Notation Nuerical Range ow ediu High H [,.6] [.4,.8] [.6, ] inguitic Variable: Nuber of pare, n inguitic Value Notation Nuerical Range (noralied) Very all all Rather all ediu Rather arge arge Very arge V R R V [,.3] [,.4] [5,.45] [.3,.7] [.55,.75] [.6, ] [.7, ] tep 2: : Deterine fuzzy et Fuzzy et can have a variety of hape. However, a triangle or a trapezoid can often provide an adequate repreentation of the expert knowledge, and at the ae tie, ignificantly iplifie the proce of coputation. eberhip V Fuzzy et of ean Delay ean Delay (noralied)

6 Fuzzy et of Nuber of erver Fuzzy et of Repair Utiliation Factor ρ Fuzzy et of Nuber of pare n eberhip. eberhip. eberhip..8.8 H.8 V R R V Nuber of erver (noralied) Repair Utiliation Factor Nuber of pare (noralied) tep 3: : Elicit and contruct fuzzy rule To accoplih thi tak, we ight ak the expert to decribe how the proble can be olved uing the fuzzy linguitic variable defined previouly. Required knowledge alo can be collected fro other ource uch a book, coputer databae, flow diagra and oberved huan behaviour. The quare FA repreentation R V R V V The rule table Rule ρ n Rule ρ n Rule ρ n V V V 9 V H V 2 V V 2 H 3 V 2 V 2 H 4 V V 3 V R 22 V H 5 V 4 23 H 6 V 5 V 24 H 7 V 6 V 25 V H R 8 7 R 26 H 9 V 8 27 H R

7 Rule Bae. If (utiliation_factor i ) then (nuber_of_pare i ) 2. If (utiliation_factor i ) then (nuber_of_pare i ) 3. If (utiliation_factor i H) then (nuber_of_pare i ) 4. If (ean_delay i V) and (nuber_of_erver i ) then (nuber_of_pare i V) 5. If (ean_delay i ) and (nuber_of_erver i ) then (nuber_of_pare i ) 6. If (ean_delay i ) and (nuber_of_erver i ) then (nuber_of_pare i ) 7. If (ean_delay i V) and (nuber_of_erver i ) then (nuber_of_pare i R) 8. If (ean_delay i ) and (nuber_of_erver i ) then (nuber_of_pare i R) 9. If (ean_delay i ) and (nuber_of_erver i ) then (nuber_of_pare i ).If (ean_delay i V) and (nuber_of_erver i ) then (nuber_of_pare i ).If (ean_delay i ) and (nuber_of_erver i ) then (nuber_of_pare i ) 2.If (ean_delay i ) and (nuber_of_erver i ) then (nuber_of_pare i V) Cube FA of Rule Bae 2 V V H V ρ R R V V V V R R V V H ρ tep 4: : Encode the fuzzy et, fuzzy rule and procedure to perfor fuzzy inference into the expert yte To accoplih thi tak, we ay chooe one of two option: to build our yte uing a prograing language uch a C/C++ or Pacal, or to apply a fuzzy logic developent tool uch a ATAB Fuzzy ogic Toolbox or Fuzzy Knowledge Builder. V tep 5: : Evaluate and tune the yte The lat, and the ot laboriou, tak i to evaluate and tune the yte. We want to ee whether our fuzzy yte eet the requireent pecified at the beginning. everal tet ituation depend on the ean delay, nuber of erver and repair utiliation factor. The Fuzzy ogic Toolbox can generate urface to help u analye the yte perforance. Three-dienional plot for Rule Bae nuber_of_pare ean_delay nuber_of_erver Three-dienional plot for Rule Bae nuber_of_pare ean_delay utiliation_factor

8 Three-dienional plot for Rule Bae 2.35 Three-dienional plot for Rule Bae 2.5 However, even now, the expert ight not be atified with the yte perforance. nuber_of_pare nuber_of_pare.4.3 To iprove the yte perforance, we ay ue additional et Rather all and Rather arge on the univere of dicoure Nuber of erver, and then extend the rule bae..4 ean_delay nuber_of_erver.4 ean_delay utiliation_factor odified fuzzy et of Nuber of erver eberhip R R Nuber of erver (noralied) R R Cube FA of Rule Bae 3 V V V R R H V V V ρ R R R R R V V V V V V R R R R R V R R V V H ρ Three-dienional plot for Rule Bae 3 nuber_of_pare ean_delay nuber_of_erver

9 Three-dienional plot for Rule Bae 3 nuber_of_pare ean_delay utiliation_factor.8 Tuning fuzzy yte. Review odel input and output variable, and if required redefine their range. 2. Review the fuzzy et, and if required define additional et on the univere of dicoure. The ue of wide fuzzy et ay caue the fuzzy yte to perfor roughly. 3. Provide ufficient overlap between neighbouring et. It i uggeted that triangle-to to-triangle triangle and trapezoid-to to-triangle triangle fuzzy et hould overlap between 25% to 5% of their bae. 4. Review the exiting rule, and if required add new rule to the rule bae. 5. Exaine the rule bae for opportunitie to write hedge rule to capture the pathological behaviour of the yte. 6. Adjut the rule execution weight. ot fuzzy logic tool allow control of the iportance of rule by changing a weight ultiplier. 7. Revie hape of the fuzzy et. In ot cae, fuzzy yte are highly tolerant of a hape approxiation.