Fuzzy Rule Based NetworksAlexander GegovUniversity of Portsmouth, UKalexander.gegov@port.ac.uk
Presentation OutlinePart 1 - Theory: Introduction                 Types of Fuzzy Systems                 Formal Models for...
IntroductionModelling Aspects of Systemic Complexity • non-linearity (input-output functional relationships) • uncertainty...
Types of Fuzzy SystemsFuzzification • triangular membership functions • trapezoidal membership functionsInference  • trunc...
Types of Fuzzy SystemsLogical Connections • disjunctive antecedents and conjunctive rules • conjunctive antecedents and co...
Types of Fuzzy SystemsOperation Mode • Mamdani (conventional fuzzy system) • Sugeno (modern fuzzy system) • Tsukamoto (hyb...
Types of Fuzzy SystemsFigure 1: Standard fuzzy system                                  7
Types of Fuzzy SystemsFigure 2: Hierarchical fuzzy system                                      8
Types of Fuzzy SystemsFigure 3: Fuzzy network                          9
Formal Models for Fuzzy NetworksNode Modelling by If-then RulesRule 1: If x is low, then y is smallRule 2: If x is average...
Formal Models for Fuzzy NetworksNode Modelling by Boolean Matricesx/y         1 (small) 2 (medium) 3 (big)1 (low)         ...
Formal Models for Fuzzy NetworksNetwork Modelling by Grid Structures           Layer 1Level 1    N11(x, y)Network Modellin...
Formal Models for Fuzzy NetworksNetwork Modelling by Block Schemesx N11 yNetwork Modelling by Topological Expressions[N11]...
Basic Operations in Fuzzy NetworksHorizontal Merging of Nodes[N11] (x11 | z11,12) * [N12] (z11,12 | y12) = [N11*12] (x11 |...
Basic Operations in Fuzzy NetworksVertical Merging of Nodes[N11] (x11 | y11) + [N21] (x21 | y21) = [N11+21] (x11, x21 | y1...
Basic Operations in Fuzzy NetworksOutput Merging of Nodes[N11] (x11,21 | y11) ; [N21] (x11,21 | y21) = [N11;21] (x11,21 | ...
Structural Properties of Basic OperationsAssociativity of Horizontal Merging[N11] (x11 | z11,12) * [N12] (z11,12 | z12,13)...
Structural Properties of Basic OperationsVariability of Horizontal Splitting[N11/12/13] (x11 | y13) =[N11] (x11 | z11,12) ...
Structural Properties of Basic OperationsAssociativity of Vertical Merging[N11] (x11 | y11) + [N21] (x21 | y21) + [N31] (x...
Structural Properties of Basic OperationsVariability of Vertical Splitting[N11–21–31] (x11, x21, x31 | y11, y21, y31) =[N1...
Structural Properties of Basic OperationsAssociativity of Output Merging[N11] (x11,21,31 | y11) ; [N21] (x11,21,31 | y21) ...
Structural Properties of Basic OperationsVariability of Output Splitting[N11:21:31] (x11,21,31 | y11, y21, y31) =[N11] (x1...
Advanced Operations in Fuzzy NetworksNode Transformation in Input Augmentation[N] (x | y) ⇒ [NAI] (x , xAI | y)Node Transf...
Advanced Operations in Fuzzy NetworksNode Identification in Horizontal MergingA*U =CA, C – known nodes, U – non-unique unk...
Advanced Operations in Fuzzy NetworksA*U*B=CA, B, C – known nodes, U – non-unique unknown nodeA*U=DD*B=CD – non-unique unk...
Advanced Operations in Fuzzy NetworksNode Identification in Vertical MergingA+U=CA, C – known nodes, U – unique unknown no...
Advanced Operations in Fuzzy NetworksA+U+B=CA, B, C – known nodes, U – unique unknown nodeA+U=DD+B=CD – unique unknown nod...
Advanced Operations in Fuzzy NetworksNode Identification in Output MergingA;U =CA, C – known nodes, U – unique unknown nod...
Advanced Operations in Fuzzy NetworksA;U;B=CA, B, C – known nodes, U – unique unknown nodeA;U=DD;B=CD – unique unknown nod...
Feedforward Fuzzy NetworksNetwork with Single Level and Single Layer • fuzzy system1,1[N11] (x11 | y11)                   ...
Feedforward Fuzzy NetworksNetwork with Single Level and Multiple Layers • queue of ‘n’ fuzzy systems1,1 … 1,n             ...
Feedforward Fuzzy NetworksNetwork with Multiple Levels and Single Layer • stack of ‘m’ fuzzy systems1,1…m,1               ...
Feedforward Fuzzy NetworksNetwork with Multiple Levels and Multiple Layers • grid of ‘m by n’ fuzzy systems1,1 … 1,n… … …m...
Feedback Fuzzy NetworksNetwork with Single Local Feedback • one node embraced by feedbackN(F) N     N N(F)N    N     N NN ...
Feedback Fuzzy NetworksNetwork with Multiple Local Feedback • at least two nodes embraced by separate feedbackN(F1) N(F2) ...
Feedback Fuzzy NetworksNetwork with Single Global Feedback • one set of at least two adjacent nodes embraced by feedbackN1...
Feedback Fuzzy NetworksNetwork with Multiple Global Feedback • at least two sets of nodes embraced by separate feedback wi...
Evaluation of Fuzzy NetworksLinguistic Evaluation Metrics • composition of hierarchical into standard fuzzy systems • deco...
Evaluation of Fuzzy NetworksDecomposition Algorithm1. Find N1,1 from the first two inputs and the output.2. If x=2, go to ...
Evaluation of Fuzzy NetworksFunctional Evaluation Metrics • model performance indicators • applications to case studiesFea...
Evaluation of Fuzzy NetworksAccuracy Index (AI)AI = sum i=1nl sum j=1qil sum k=1vji (|yjik – djik| / vij)nl – number of no...
Evaluation of Fuzzy NetworksEfficiency Index (EI)EI = sum i=1n (qiFID . riFID)n – number of non-identity nodesFID – fuzzif...
Evaluation of Fuzzy NetworksTransparency Index (TI)TI = (p + q) / (n + m)p – overall number of inputsq – overall number of...
Evaluation of Fuzzy NetworksCase Study 1 (Ore Flotation)Inputs:x1 – copper concentration in ore pulpx2 – iron concentratio...
Evaluation of Fuzzy Networks Figure 4: Standard fuzzy system (SFS) for case study 1                                       ...
Evaluation of Fuzzy Networks Figure 5: Hierarchical fuzzy system (HFS) for case study 1                                   ...
Evaluation of Fuzzy Networks Figure 6: Fuzzy network (FN) for case study 1                                                ...
Evaluation of Fuzzy Networks Table 1: Models performance for case study 1 Index / Model Standard       Hierarchical   Fuzz...
Evaluation of Fuzzy NetworksCase Study 2 (Retail Pricing)Inputs:x1 – expected selling price for retail productx2 – differe...
Evaluation of Fuzzy Networks             100             90             80             70             60    output        ...
Evaluation of Fuzzy Networks             100             80             60    output             40             20        ...
Evaluation of Fuzzy Networks             100             80             60    output             40             20        ...
Evaluation of Fuzzy Networks Table 2: Models performance for case study 2 Index / Model Standard       Hierarchical   Fuzz...
Evaluation of Fuzzy NetworksComposition of Hierarchical into Standard Fuzzy System • full preservation of model feasibilit...
Fuzzy Network ToolboxToolbox Details • developed by the PhD student Nedyalko Petrov • implemented in the Matlab environmen...
ConclusionTheoretical Significance of Fuzzy Networks • novel application of discrete mathematics and control theory • deta...
ConclusionOther Applications of Fuzzy Networks • finance (mortgage evaluation) • healthcare (radiotherapy margins) • robot...
ReferencesA.Gegov, Complexity Management in Fuzzy Systems,Springer, Berlin, 2007A.Gegov, Fuzzy Networks for Complex System...
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Aleksander gegov

  1. 1. Fuzzy Rule Based NetworksAlexander GegovUniversity of Portsmouth, UKalexander.gegov@port.ac.uk
  2. 2. Presentation OutlinePart 1 - Theory: Introduction Types of Fuzzy Systems Formal Models for Fuzzy Networks Basic Operations in Fuzzy Networks Structural Properties of Basic Operations Advanced Operations in Fuzzy NetworksPart 2 - Applications: Feedforward Fuzzy Networks Feedback Fuzzy Networks Evaluation of Fuzzy Networks Fuzzy Network Toolbox Conclusion References 2
  3. 3. IntroductionModelling Aspects of Systemic Complexity • non-linearity (input-output functional relationships) • uncertainty (incomplete and imprecise data) • dimensionality (large number of inputs and outputs) • structure (interacting subsystems)Complexity Management by Fuzzy Systems • model feasibility (achievable in the case of non-linearity) • model accuracy (achievable in the case of uncertainty) • model efficiency (problematic in the case of dimensionality) • model transparency (problematic in the case of structure) 3
  4. 4. Types of Fuzzy SystemsFuzzification • triangular membership functions • trapezoidal membership functionsInference • truncation of rule membership functions by firing strengths • scaling of rule membership functions by firing strengthsDefuzzification • centre of area of rule base membership function • weighted average of rule base membership function 4
  5. 5. Types of Fuzzy SystemsLogical Connections • disjunctive antecedents and conjunctive rules • conjunctive antecedents and conjunctive rules • disjunctive antecedents and disjunctive rules • conjunctive antecedents and disjunctive rulesInputs and Outputs • single-input-single-output • single-input-multiple-output • multiple-input-single-output • multiple-input-multiple-output 5
  6. 6. Types of Fuzzy SystemsOperation Mode • Mamdani (conventional fuzzy system) • Sugeno (modern fuzzy system) • Tsukamoto (hybrid fuzzy system)Rule Base • single rule base (standard fuzzy system) • multiple rule bases (hierarchical fuzzy system) • networked rule bases (fuzzy network) 6
  7. 7. Types of Fuzzy SystemsFigure 1: Standard fuzzy system 7
  8. 8. Types of Fuzzy SystemsFigure 2: Hierarchical fuzzy system 8
  9. 9. Types of Fuzzy SystemsFigure 3: Fuzzy network 9
  10. 10. Formal Models for Fuzzy NetworksNode Modelling by If-then RulesRule 1: If x is low, then y is smallRule 2: If x is average, then y is mediumRule 3: If x is high, then y is bigNode Modelling by Integer TablesLinguistic terms for x Linguistic terms for y1 (low) 1 (small)2 (average) 2 (medium)3 (high) 3 (big) 10
  11. 11. Formal Models for Fuzzy NetworksNode Modelling by Boolean Matricesx/y 1 (small) 2 (medium) 3 (big)1 (low) 1 0 02 (average) 0 1 03 (high) 0 0 1Node Modelling by Binary Relations{(1, 1), (2, 2), (3, 3)} 11
  12. 12. Formal Models for Fuzzy NetworksNetwork Modelling by Grid Structures Layer 1Level 1 N11(x, y)Network Modelling by Interconnection Structures Layer 1Level 1 y 12
  13. 13. Formal Models for Fuzzy NetworksNetwork Modelling by Block Schemesx N11 yNetwork Modelling by Topological Expressions[N11] (x | y) 13
  14. 14. Basic Operations in Fuzzy NetworksHorizontal Merging of Nodes[N11] (x11 | z11,12) * [N12] (z11,12 | y12) = [N11*12] (x11 | y12)* symbol for horizontal mergingHorizontal Splitting of Nodes[N11/12] (x11 | y12) = [N11] (x11 | z11,12) / [N12] (z11,12 | y12)/ symbol for horizontal splitting 14
  15. 15. Basic Operations in Fuzzy NetworksVertical Merging of Nodes[N11] (x11 | y11) + [N21] (x21 | y21) = [N11+21] (x11, x21 | y11, y21)+ symbol for vertical mergingVertical Splitting of Nodes[N11–21] (x11, x21 | y11, y21) = [N11] (x11 | y11) – [N21] (x21 | y21)– symbol for vertical splitting 15
  16. 16. Basic Operations in Fuzzy NetworksOutput Merging of Nodes[N11] (x11,21 | y11) ; [N21] (x11,21 | y21) = [N11;21] (x11,21 | y11, y21); symbol for output mergingOutput Splitting of Nodes[N11:21] (x11, 21 | y11, y21) = [N11] (x11,21 | y11) : [N21] (x11,21 | y21): symbol for output splitting 16
  17. 17. Structural Properties of Basic OperationsAssociativity of Horizontal Merging[N11] (x11 | z11,12) * [N12] (z11,12 | z12,13) * [N13] (z12,13 | y13) =[N11*12] (x11 | z12,13) * [N13] (z12,13 | y13) =[N11] (x11 | z11,12) * [N12*13] (z11,12 | y13) =[N11*12*13] (x11 | y13) 17
  18. 18. Structural Properties of Basic OperationsVariability of Horizontal Splitting[N11/12/13] (x11 | y13) =[N11] (x11 | z11,12) / [N12/13] (z11,12 | y13) =[N11/12] (x11 | z12,13) / [N13] (z12,13 | y13) =[N11] (x11 | z11,12) / [N12] (z11,12 | z12,13) / [N13] (z12,13 | y13) 18
  19. 19. Structural Properties of Basic OperationsAssociativity of Vertical Merging[N11] (x11 | y11) + [N21] (x21 | y21) + [N31] (x31 | y31) =[N11+21] (x11, x21 | y11, y21) + [N31] (x31 | y31) =[N11] (x11 | y11) + [N21+31] (x21, x31 | y21, y31) =[N11+21+31] (x11, x21, x31 | y11, y21, y31) 19
  20. 20. Structural Properties of Basic OperationsVariability of Vertical Splitting[N11–21–31] (x11, x21, x31 | y11, y21, y31) =[N11] (x11 | y11) – [N21–31] (x21, x31 | y21, y31) =[N11–21] (x11, x21 | y11, y21) – [N31] (x31 | y31) =[N11] (x11 | y11) – [N21] (x21 | y21) – [N31] (x31 | y31) 20
  21. 21. Structural Properties of Basic OperationsAssociativity of Output Merging[N11] (x11,21,31 | y11) ; [N21] (x11,21,31 | y21) ; [N31] (x11,21,31 | y31) =[N11;21] (x11,21,31 | y11, y21) ; [N31] (x11,21,31 | y31) =[N11] (x11,21,31 | y11) ; [N21;31] (x11,21,31 | y21, y31) =[N11;21;31] (x11,21,31 | y11, y21, y31) 21
  22. 22. Structural Properties of Basic OperationsVariability of Output Splitting[N11:21:31] (x11,21,31 | y11, y21, y31) =[N11] (x11,21,31 | y11) : [N21:31] (x11,21,31 | y21, y31) =[N11:21] (x11,21,31 | y11, y21) : [N31] (x11,21,31 | y31) =[N11] (x11,21,31 | y11) : [N21] (x11,21,31 | y21) : [N31] (x11,21,31 | y31) 22
  23. 23. Advanced Operations in Fuzzy NetworksNode Transformation in Input Augmentation[N] (x | y) ⇒ [NAI] (x , xAI | y)Node Transformation in Output Permutation PO[N] (x | y1, y2) ⇒ [N ] (x | y2, y1)Node Transformation in Feedback Equivalence[N] (x, z | y, z) ⇒ [NEF] (x, xEF | y, yEF) 23
  24. 24. Advanced Operations in Fuzzy NetworksNode Identification in Horizontal MergingA*U =CA, C – known nodes, U – non-unique unknown nodeU*B=CB, C – known nodes, U – non-unique unknown node 24
  25. 25. Advanced Operations in Fuzzy NetworksA*U*B=CA, B, C – known nodes, U – non-unique unknown nodeA*U=DD*B=CD – non-unique unknown nodeU*B=EA*E=CE – non-unique unknown node 25
  26. 26. Advanced Operations in Fuzzy NetworksNode Identification in Vertical MergingA+U=CA, C – known nodes, U – unique unknown nodeU+B=CB, C – known nodes, U – unique unknown node 26
  27. 27. Advanced Operations in Fuzzy NetworksA+U+B=CA, B, C – known nodes, U – unique unknown nodeA+U=DD+B=CD – unique unknown nodeU+B=EA+E=CE – unique unknown node 27
  28. 28. Advanced Operations in Fuzzy NetworksNode Identification in Output MergingA;U =CA, C – known nodes, U – unique unknown nodeU;B=CB, C – known nodes, U – unique unknown node 28
  29. 29. Advanced Operations in Fuzzy NetworksA;U;B=CA, B, C – known nodes, U – unique unknown nodeA;U=DD;B=CD – unique unknown nodeU;B=EA;E=CE – unique unknown node 29
  30. 30. Feedforward Fuzzy NetworksNetwork with Single Level and Single Layer • fuzzy system1,1[N11] (x11 | y11) 30
  31. 31. Feedforward Fuzzy NetworksNetwork with Single Level and Multiple Layers • queue of ‘n’ fuzzy systems1,1 … 1,n 31
  32. 32. Feedforward Fuzzy NetworksNetwork with Multiple Levels and Single Layer • stack of ‘m’ fuzzy systems1,1…m,1 32
  33. 33. Feedforward Fuzzy NetworksNetwork with Multiple Levels and Multiple Layers • grid of ‘m by n’ fuzzy systems1,1 … 1,n… … …m,1 … m,n 33
  34. 34. Feedback Fuzzy NetworksNetwork with Single Local Feedback • one node embraced by feedbackN(F) N N N(F)N N N NN N N NN(F) N N N(F)N(F) – node embraced by feedbackN – nodes with no feedback 34
  35. 35. Feedback Fuzzy NetworksNetwork with Multiple Local Feedback • at least two nodes embraced by separate feedbackN(F1) N(F2) N N N(F1) NN N N(F1) N(F2) N N(F2)N(F1) N N N(F1) N N(F1)N(F2) N N N(F2) N(F2) NN(F1), N(F2) – nodes embraced by separate feedbackN – nodes with no feedback 35
  36. 36. Feedback Fuzzy NetworksNetwork with Single Global Feedback • one set of at least two adjacent nodes embraced by feedbackN1(F) N2(F) N NN N N1(F) N2(F)N1(F) N N N1(F)N2(F) N N N2(F){N1(F), N2(F)} – set of nodes embraced by feedbackN – nodes with no feedback 36
  37. 37. Feedback Fuzzy NetworksNetwork with Multiple Global Feedback • at least two sets of nodes embraced by separate feedback with at least two adjacent nodes in each setN1(F1) N2(F1) N1(F1) N1(F2)N1(F2) N2(F2) N2(F1) N2(F2){N1(F1), N2(F1)}, {N1(F2), N2(F2)} – sets of nodes embracedby separate feedback 37
  38. 38. Evaluation of Fuzzy NetworksLinguistic Evaluation Metrics • composition of hierarchical into standard fuzzy systems • decomposition of standard into hierarchical fuzzy systemsComposition Formula x–1 x–1NE,x = *p=1 (N1,p + +q=p+1 Iq,p)NE,x – equivalent node for a fuzzy network with x inputs 38
  39. 39. Evaluation of Fuzzy NetworksDecomposition Algorithm1. Find N1,1 from the first two inputs and the output.2. If x=2, go to end.3. Set k=3.4. While k≤x, do steps 5-7.5. Find NE,k from the first k inputs and the output.6. Derive N1,k–1 from the formula for NE,k, if possible.7. Set k=k+1.8. Endwhile.NE,k – equivalent node for a fuzzy subnetwork with first k inputs 39
  40. 40. Evaluation of Fuzzy NetworksFunctional Evaluation Metrics • model performance indicators • applications to case studiesFeasibility Index (FI) nFI = sum i=1 (pi / n)n – number of non-identity nodesp i – number of inputs to the i-th non-identity nodelower FI ⇒ better feasibility 40
  41. 41. Evaluation of Fuzzy NetworksAccuracy Index (AI)AI = sum i=1nl sum j=1qil sum k=1vji (|yjik – djik| / vij)nl – number of nodes in the last layerqil – number of outputs from the i-th node in the last layervji – number of discrete values for the j-th output from the i-thnode in the last layeryjik , djik – simulated and measured k-th discrete value for the j-thoutput from the i-th node in the last layerlower AI ⇒ better accuracy 41
  42. 42. Evaluation of Fuzzy NetworksEfficiency Index (EI)EI = sum i=1n (qiFID . riFID)n – number of non-identity nodesFID – fuzzification-inference-defuzzification FIDqi – number of outputs from the i-th non-identity node with anassociated FID sequenceriFID – number of rules for the i-th non-identity node with anassociated FID sequencelower EI ⇒ better efficiency 42
  43. 43. Evaluation of Fuzzy NetworksTransparency Index (TI)TI = (p + q) / (n + m)p – overall number of inputsq – overall number of outputsn – number of non-identity nodesm – number of non-identity connectionslower TI ⇒ better transparency 43
  44. 44. Evaluation of Fuzzy NetworksCase Study 1 (Ore Flotation)Inputs:x1 – copper concentration in ore pulpx2 – iron concentration in ore pulpx3 – debit of ore pulpOutput:y – new copper concentration in ore pulp 44
  45. 45. Evaluation of Fuzzy Networks Figure 4: Standard fuzzy system (SFS) for case study 1 45
  46. 46. Evaluation of Fuzzy Networks Figure 5: Hierarchical fuzzy system (HFS) for case study 1 46
  47. 47. Evaluation of Fuzzy Networks Figure 6: Fuzzy network (FN) for case study 1 47
  48. 48. Evaluation of Fuzzy Networks Table 1: Models performance for case study 1 Index / Model Standard Hierarchical Fuzzy Fuzzy System Fuzzy System Network Feasibility 3 2 2 Accuracy 4.35 4.76 4.60 Efficiency 343 126 343 Transparency 4 1.33 1.33 FI – FN better than SFS and equal to HFS AI – FN worse than SFS and better than HFS EI – FN equal to SFS and worse than HFS TI – FN better than SFS and equal to HFS 48
  49. 49. Evaluation of Fuzzy NetworksCase Study 2 (Retail Pricing)Inputs:x1 – expected selling price for retail productx2 – difference between selling price and cost for retail productx3 – expected percentage to be sold from retail productOutput:y – maximum cost for retail product 49
  50. 50. Evaluation of Fuzzy Networks 100 90 80 70 60 output 50 40 30 20 10 0 0 20 40 60 80 100 120 140 input permutations Figure 7: Standard fuzzy system (SFS) for case study 2 50
  51. 51. Evaluation of Fuzzy Networks 100 80 60 output 40 20 0 0 20 40 60 80 100 120 140 input permutations Figure 8: Hierarchical fuzzy system (HFS) for case study 2 51
  52. 52. Evaluation of Fuzzy Networks 100 80 60 output 40 20 0 0 20 40 60 80 100 120 140 input permutations Figure 9: Fuzzy network (FN) for case study 2 52
  53. 53. Evaluation of Fuzzy Networks Table 2: Models performance for case study 2 Index / Model Standard Hierarchical Fuzzy Fuzzy System Fuzzy System Network Feasibility 3 2 2 Accuracy 2.86 5.57 3.64 Efficiency 125 80 125 Transparency 4 1.33 1.33 FI – FN better than SFS and equal to HFS AI – FN worse than SFS and better than HFS EI – FN equal to SFS and worse than HFS TI – FN better than SFS and equal to HFS 53
  54. 54. Evaluation of Fuzzy NetworksComposition of Hierarchical into Standard Fuzzy System • full preservation of model feasibility • maximal improvement of model accuracy • fixed loss of model efficiency • full preservation of model transparencyDecomposition of Standard into Hierarchical Fuzzy System • no change of model feasibility • minimal loss of model accuracy • fixed improvement of model efficiency • no change of model transparency 54
  55. 55. Fuzzy Network ToolboxToolbox Details • developed by the PhD student Nedyalko Petrov • implemented in the Matlab environment • to be uploaded on the Mathworks web site • to be uploaded on the Springer web siteToolbox Structure • Matlab files for basic operations on nodes • Matlab files for advanced operations on nodes • Matlab files for auxiliary operations • Word files with illustration examples 55
  56. 56. ConclusionTheoretical Significance of Fuzzy Networks • novel application of discrete mathematics and control theory • detailed validation by test examples and case studiesMethodological Impact of Fuzzy Networks • extension of standard and hierarchical fuzzy systems • bridge between standard and hierarchical fuzzy systems • novel framework for non-fuzzy rule based systemsApplication Areas of Fuzzy Networks • modelling and simulation in the mining industry • modelling and simulation in the retail industry 56
  57. 57. ConclusionOther Applications of Fuzzy Networks • finance (mortgage evaluation) • healthcare (radiotherapy margins) • robotics (path planning) • games (obstacle avoidance) • sports (boxer training) • security (terrorist threat) • environment (natural preservation) 57
  58. 58. ReferencesA.Gegov, Complexity Management in Fuzzy Systems,Springer, Berlin, 2007A.Gegov, Fuzzy Networks for Complex Systems,Springer, Berlin, 2010 58

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