FUZZY OPTIMIZATION

FUZZY OPTIMIZATION
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Dr.NaveenBansal,India,Teacher
Published Date:25-10-2017
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CHAPTER 14 MISCELLANEOUS TOPICS Knowing ignorance is strength, and ignoring knowledge is sickness. LaoTsu Chinese philosopher, in Tao Te Ching, circa 600 BC This chapter exposes the reader to a few of the additional application areas that have been extended with fuzzy logic. These few areas cannot cover the wealth of other applications, buttheygivetothereaderanappreciationofthepotentialinfluenceoffuzzylogicinalmost any technology area. Addressed in this chapter are just four additional application areas: optimization, fuzzy cognitive mapping, system identification, and linear regression. FUZZY OPTIMIZATION Mosttechnicalfields,includingallthoseinengineering,involvesomeformofoptimization that is required in the process of design. Since design is an open-endedproblem with many solutions, the quest is to find the ‘‘best’’ solution according to some criterion. In fact, almost any optimization process involves trade-offs between costs and benefits because finding optimum solutions is analogous to creating designs – there can be many solutions, butonlyafewmightbeoptimum,oruseful,particularlywherethereisagenerallynonlinear relationshipbetweenperformanceandcost.Optimization,initsmostgeneralform,involves finding the most optimum solution from a family of reasonable solutions according to an optimization criterion. For all but a few trivial problems, finding the global optimum (the best optimum solution) can never be guaranteed. Hence, optimization in the last three decades has focused on methods to achieve the best solution per unit computational cost. In caseswhereresourcesareunlimited and the problem can be describedanalytically and there are no constraints, solutions found by exhaustive search Akai, 1994 can Fuzzy Logic with Engineering Applications, Second Edition T. J. Ross  2004 John Wiley & Sons, Ltd ISBNs: 0-470-86074-X (HB); 0-470-86075-8 (PB) www.MatlabSite.com538 MISCELLANEOUS TOPICS guarantee global optimality. In effect, this global optimum is found by setting all the derivatives of the criterion function to zero, and the coordinates of the stationary point that satisfy the resulting simultaneous equations represent the solution. Unfortunately, even if a problem can be described analytically there are seldom situations with unlimited search resources.Iftheoptimizationproblemalsorequiresthesimultaneoussatisfactionofseveral constraints and the solution is known to exist on a boundary, then constraint boundary searchmethods suchasLagrangianmultipliersareusefuldeNeufville,1990.Insituations wherethe optimum is not known to belocatedon aboundary, methods such asthe steepest gradient, Newton–Raphson, and penalty function have been used Akai, 1994, and some very promising methods have used genetic algorithms Goldberg, 1989. For functions with a single variable, search methods such as Golden section and Fibonacci are quite fast and accurate. For multivariate situations, search strategies such as parallel tangents and steepest gradients have been useful in some situations. But most of these classical methods of optimization Vanderplaats, 1984 suffer from one or more disadvantages: the problem of finding higher order derivatives of a process, the issue of describing the problem as an analytic function, the problem of combinatorial explosion when dealing with many variables, the problem of slow convergence for small spatial or temporalstepsizes,andtheproblemofovershootforstepsizestoolarge.Inmanysituations, the precision of the optimization approach is greater than the original data describing the problem, so there is an impedance mismatch in terms of resolution between the required precision and the inherent precision of the problem itself. In the typical scenario of an optimization problem, fast methods with poorer conver- gence behavior are used first to get the process near a solution point, such as a Newton method, then slower but more accurate methods, such as gradient schemes, are used to convergeto a solution. Some currentsuccessful optimization approachesarenow based on this hybrid idea: fast, approximate methods first, slower and more precise methods second. Fuzzy optimization methods have been proposed as the first steps in hybrid optimization schemes. One of these methods will be introduced here. More methods can be found in Sakawa 1993. One-dimensional Optimization Classicaloptimizationforaone-dimensional(oneindependentvariable)relationshipcanbe ∗ formulatedasfollows.Supposewewishtofindtheoptimumsolution, x ,whichmaximizes the objective function y = f(x), subject to the constraints g (x) ≤ 0,i = 1,m (14.1) i Each of the constraint functions g (x) can be aggregated as the intersection of all the i constraints. If we let C =x g (x) ≤ 0,then i i C = C ∩C ∩···∩C =x g (x) ≤ 0,g (x) ≤ 0,...,g (x) ≤ 0 (14.2) 1 2 m 1 2 m which is the feasible domain described by the constraints C . Thus, the solution is i ∗ f(x ) = maxf(x) (14.3) x∈C www.MatlabSite.comFUZZY OPTIMIZATION 539 µ µ (x), y C f(x) C 1.0 0.5 1.0 0.2 µ µ(y) 1.0 0.5 0.2 x x x x 1 0.5 0.2 FIGURE 14.1 Function to be optimized,f(x), and fuzzy constraint, C. In a real environment, the constraints might not be so crisp, and we could have fuzzy feasible domains (see Fig. 14.1) such as ‘‘x could exceed x a little bit.’’ If we use λ-cuts 0 on the fuzzy constraints C, fuzzy optimization is reduced to the classical case. Obviously, ∼ ∗ the optimum solution x is a function of the threshold level λ, as given in Eq. (14.4): ∗ f(x ) = maxf(x) (14.4) λ x∈C λ Sometimes,thegoalandtheconstraintaremoreorlesscontradictory,andsometrade- off between them is appropriate. This can be done by converting the objective function y = f(x) into a pseudogoal G Zadeh, 1972 with membership function ∼ f(x) − m µ (x) = (14.5) G ∼ M − m where m = inf f(x) x∈X M = supf(x) x∈X Then the fuzzy solution set D is defined by the intersection ∼ D = C ∩G (14.6) ∼ ∼ ∼ membership is described by µ (x) = minµ (x), µ (x) (14.7) D C G ∼ ∼ ∼ ∗ and the optimum solution will be x with the condition ∗ µ (x ) ≥ µ (x) for all x ∈ X (14.8) D D ∼ ∼ where µ (x),C,x should be substantially greaterthan x . Figure 14.2 shows this situation. C 0 ∼ ∼ www.MatlabSite.com540 MISCELLANEOUS TOPICS µ µ( ) µ µ C 1.0 µ µ G µ µ D 0 x x x 0 FIGURE 14.2 Membership functions for goal and constraint Zadeh, 1972. Example 14.1. Suppose we have a deterministic function given by (1−x/5) f(x) = xe for the region 0 ≤ x ≤ 5, and a fuzzy constraint given by  1, 0 ≤ x ≤ 1  µ (x) = 1 C ∼  ,x 1 2 1 + (x −1) Both of these functions are illustrated in Fig. 14.3. We want to determine the solution set D ∼ ∗ ∗ ∗ and the optimum solution x , i.e., find f(x ) = y . In this case we have M = supf(x) = 5 and m = inff(x) = 0; hence Eq. (14.5) becomes f(x) −0 x (1−x/5) µ (x) = = e G ∼ 5 −0 5 whichisalsoshowninFig. 14.3.The solutionset membership function,usingEq. (14.7), then becomes  x (1−x/5) ∗   e , 0 ≤ x ≤ x 5 µ (x) = D 1 ∼  ∗  ,xx 2 1 + (x −1) µ µ(x), y f(x) y µ µ (x) C µ µ (x) G 015 x x FIGURE 14.3 Problem domain for Example 14.1. www.MatlabSite.comFUZZY OPTIMIZATION 541 ∗ and the optimum solution x , using Eq. (14.8), is obtained by finding the intersection ∗ x 1 ∗ (1−x /5) e = 2 5 1 + (x −1) and is shown in Fig. 14.3. When the goal and the constraint have unequal importance the solution set D can be ∼ obtained by the convex combination, i.e., µ (x) = αµ (x) + (1 − α)µ (x) (14.9) D C G ∼ ∼ ∼ The single-goal formulation expressed in Eq. (14.9) can be extended to the multiple-goal case as follows. Suppose we want to consider n possible goals and m possible constraints. Then the solution set, D, is obtained by the aggregate intersection, i.e., by ∼     D = ∩ C ∩ ∩ G (14.10) ∼ ∼i ∼j i=1,m j=1,n Example 14.2. A beam structure is supported at one end by a hinge and at the other end by a roller. A transverse concentrated load P is applied at the middle of the beam, as in Fig. 14.4. The maximum bending stress caused by P can be expressed by the equation σ = Pl/w , b z where w is a coefficient decided by the shape and size of a beam and l is the beam’s length. z 3 The deflection at the centerline of the beam is δ = Pl /(48EI) where E and I are the beam’s modulus of elasticity and cross-sectional moment of inertia, respectively. If 0 ≤ δ ≤ 2 mm, and 0 ≤ σ ≤ 60 MPa, the constraint conditionsare these: the span length of the beam, b  l , 0 ≤ l ≤ 100 m 1 1 l = 200 − l , 100l ≤ 200 m 1 1 and the deflection,  2 − δ , 0 ≤ δ ≤2mm 1 1 δ = 0,δ 2mm 1 To find the minimum P for this two-constraint and two-goal problem (the goals are the stress, σ , and the deflection, δ), we first find the membership function for the two goals and two b constraints. 1. The µ for bending stress σ is given as follows: G b 1 w 60 w σ σ z z b b P(0) = 0,P(60 MPa) = ,P(σ ) = ; thus,µ = b G 1 l l 60 P δ l FIGURE 14.4 Simply supported beam with a transverse concentrated load. www.MatlabSite.com542 MISCELLANEOUS TOPICS To change the argument in µ into a unitless form, let x = σ /60, where 0 ≤ x ≤ 1. G b 1 Therefore, µ (x) = x when 0 ≤ x ≤ 1. G 1 2. The µ for deflection δ is as follows: G 2 48EIδ 48EI ×2 δ P(δ) = ,P(0) = 0,P(2) = ; thus,µ = G 2 3 3 l l 2 Let x = δ/2, so that the argument of µ is unitless. Therefore, µ = x,0 ≤ x ≤ 1. G G 2 2 3. Using Eq. (14.10), we combine µ (x) and µ (x) to find µ (x): G G G 1 2 µ (x) = min(µ (x), µ (x)) = x 0 ≤ x ≤ 1 G G G 1 2 4. The fuzzy constraint function µ for the span is C 1  2x, 0 ≤ x ≤ 0.5 µ (x) = C 1 2 −2x, 0.5x ≤ 1 where x = l /200. Therefore, the constraint function will vary according to a unitless 1 argument x. 5. The fuzzy constraint function µ for the deflection δ can be obtained in the same way as C 2 in point 4:  1 − x, 0 ≤ x ≤ 1 µ (x) = C 2 0,x 1 where x = δ/2. 6. The fuzzy constraint function µ (x) for the problem can be found by the combination of C µ (x) and µ (x), using Eq. (14.10): C C 1 2 µ (x) = min(µ (x), µ (x)) C C C 1 2 and µ (x) is shown as the bold line in Fig. 14.5. C Now, the optimum solutions P can be found by using Eq. (14.10): D = (G ∩C) µ (x) = µ (x) ∧ µ (x) D C G µ µ( ) µ µ (x) C 2 1.0 µ µ (x) C 1 µ µ (x) C 0 0.5 1.0 x FIGURE 14.5 Minimum of two constraint functions. www.MatlabSite.comFUZZY OPTIMIZATION 543 µ µ( ) 1.0 x µ µ (x) G µ µ (x) C 0 0.5 1.0 x FIGURE 14.6 Graphical solutionto minimization problem of Example 14.2. ∗ The optimum value can be determined graphically, as seen in Fig. 14.6, to be x = 0.5. From this, we can obtain the optimum span length, l = 100 m, optimum deflection, δ = 1 mm, and optimum bending stress, σ = 30 MPa. The minimum load P is b   σ w 48EIδ b z P = min , 3 l l Suppose that the importance factor for the goal function µ (x) is 0.4. Then the solution G for this same optimization problem can be determined using Eq. (14.9) as µ = 0.4µ +0.6µ D G C where µ can be expressed by the function (see Fig. 14.5) C  1 2x, 0 ≤ x ≤ 3 µ (x) = C 1 1 − x, x ≤ 1 3 Therefore,  1 1.2x, 0 ≤ x ≤ 3 0.6µ (x) = C 1 0.6 −0.6x, x ≤ 1 3 and 0.4µ (x) = 0.4x. The membership function for the solution set, from Eq. (14.9), then is G  1 1.6x, 0 ≤ x ≤ 3 µ (x) = D 1 0.6 −0.2x, x ≤ 1 3 ∗ The optimum solution for this is x = 0.33, which is shown in Fig. 14.7. µ µ (x) D x 0.5 0 0.5 1.0 x FIGURE 14.7 Solution of Example 14.2 considering an importance factor. www.MatlabSite.com544 MISCELLANEOUS TOPICS FUZZY COGNITIVE MAPPING Cognitive maps (CMs) were introduced by Robert Axelrod 1976 as a formal means of modeling decisionmakinginsocialandpolitical systems.CMsareatypeofdirectedgraph that offers a means to model interrelationships or causalities among concepts; there are variousformsofCMs,suchassigneddigraphs,weightedgraphs,andfunctionalgraphs.The differences amongst these various forms can be found in Kardaras and Karakostas 1999. CMs can also be used for strategic planning, prediction, explanation, and for engineering concept development. The use of simple binary relationships (i.e., increase and decrease) is done in a conventional (crisp) CM. All CMs offer a number of advantages that make them attractive as models for engineering planning and concept development. CMs have a clear way to visually represent causal relationships, they expand the range of complexity that can be managed, they allow users to rapidly compare their mental models with reality, they make evaluations easier, and they promote new ways of thinking about the issue being evaluated. Concept variables and causal relations CMs graphically describe a system in terms of two basic types of elements: concept variablesandcausalrelations.Nodesrepresentconceptvariables, C ,where x = 1,...,N. x A concept variable at the origin of an arrowis a cause variable, whereas a concept variable atthe endpoint of anarrowis aneffectvariable.Forexample,for C → C ,C is the cause h i h variable that impacts C , which is the effect variable. Figure 14.8 represents a simple CM, i in which therearefour concept variables(C representsutilization of waste steam for heat, h C represents the amount of natural gas required to generate heat, C represents economic i k gain for the local economy, and C represents the market value of waste steam (which is j dependent on the price of natural gas)). Arrows represent the causal relations between concept variables, which can be − positive ornegative. Forexample, for C →C ,C has anegative causalrelationship on C . h i h i Therefore,an increase in C results in a decrease in C . h i Amount of natural gas required to generate heat C i −+ Utilization Economic of waste C h C gain k steam for heat ++ C j Market value of waste steam FIGURE 14.8 A conventional cognitive map for the utilization of waste steam. www.MatlabSite.comFUZZY COGNITIVE MAPPING 545 Paths and cycles A path between two concept variables, C and C , denoted by P(h,k), is a sequence of h k all the nodes which are connected by arrows from the first node (C ) to the last node (C ) h k (Fig. 14.8)Kosko,1986.Acycleisapaththathasanarrowfromthelastpointofthepath to the first point. Indirect effect The indirect effect of a path from the cause variable C to the effect variable C,which h k is denoted by I(h,k), is the product of the causal relationships that form the path from the cause variable to the effect variable Axelrod, 1976. If a path has an even number of negative arrows, then the indirect effect is positive. If the path has an odd number of negative arrows,then the indirect effectis negative.In Fig. 14.8 theindirect effectofcause variable C on the effect variable C through path P(h,i,k) is negative; the indirect effect h k of the cause variable C on the effect variable C through path P(h,j,k) is positive. h k Total effect The total effect of the cause variable C on the effect variable C , which is denoted by h k T(h,k), is the union of all the indirect effects of all the paths from the cause variable to the effect variable Axelrod, 1976. If all the indirect effects are positive, the total effect is positive. If all the indirect effects are negative, so is the total effect. If some indirect effectsarepositive andsome arenegative,the sumis indeterminate Kosko, 1986.Alarge CM, that is one with a large number of concepts and paths, will therefore be dominated by the characteristic of being indeterminate. In Fig. 14.8 the total effect of cause variable C h to effect variable C is the collection of the indirect effect of C to C through the paths k h k P(h,i,k) and P(h,j,k). Since one indirect effects is positive and the other is negative in this case, this means that the total effect is indeterminate. Indeterminacy The character of a conventional CM being indeterminate can be resolved, but it comes at a computational and conceptual price. To do so, the CM must accommodate a numerical weightingschemeKosko,1986.Ifthecausaledgesareweightedwithpositiveornegative real numbers, then the indirect effect is the product of each of the weights in a given path, and the total effect is the sum of the path products. This scheme of weighting the path relationships removes the problem of indeterminacy from the total effect calculation, but it also requires a finer causal discrimination. Such a fineness may not be available from the analysts or experts who formulate the CM. This finer discrimination between concepts in the CM would make knowledge acquisition a more onerous process–forced numbers from insufficient decision information, different numbers from different experts, or from the same expert on different days, and so on. However, causal relationships could be representedby linguistic quantities as opposed to numerical ones. Such is the context of a fuzzy CM (FCM). Fuzzy Cognitive Maps IfoneweretoemphasizethatthesimplebinaryrelationshipofaCMneededtobeextended to include various degrees of increase or decrease (small decrease, large increase, almost www.MatlabSite.com546 MISCELLANEOUS TOPICS no increase,etc.), then a fuzzy cognitive map (FCM) is more appropriate. An FCM extends the idea of conventional CMs by allowing concepts to be representedlinguistically with an associated fuzzy set, rather than requiring them to be precise. Extensions by Taber 1994 andKosko1992allowfuzzynumbersorlinguistictermstobeusedtodescribethedegree of the relationship between concepts in the FCM. FCMs are analyzed either geometrically ornumericallyPelaezandBowles,1996.Ageometricanalysisisusedprimarilyforsmall FCMs, where it simply traces the increasing and decreasing effects along all paths from one concept to another. For larger FCMs, such as those illustrated later in this section, a numerical analysis is required,wherethe conceptsarerepresentedby a state vector and the relationsbetweenconceptsarerepresentedbyafuzzyrelationalmatrix,calledanadjacency matrix.This,alongwithafewotherkeyfeaturesofFCMsthatdistinguish themfromCMs, are mentioned below. Adjacency matrix A CM can be transformed using a matrix called an adjacency matrix Kosko, 1986. An adjacency matrix is a square matrix that denotes the effect that a cause variable (row) given in the CM has on the effect variable (column). Figure 14.9 is an adjacency matrix for the CM displayed in Fig. 14.8. In other words, the adjacency matrix for a CM with n nodes uses an n × n matrix in which an entry in the (i, j) position of the matrix denotes an arrow between nodes C and C . This arrow (as shown in Fig. 14.8) simply represents the h i ‘‘strength’’oftheeffectbetweenthetwonodes(i.e.,a‘‘+1’’representsthattheeffectisto increase,whereas a ‘‘−1’’represents that the effect is to decrease). Threshold function Concept states are held within defined boundaries through the threshold function. The type of threshold function chosen determines the behavior of a CM. A bivalent threshold function requiresconcepts to have a value of 1or 0,which is equivalentto ‘‘on’’or ‘‘off’’: f(x ) = 0,x ≤ 0 i i f(x ) = 1,x 0 i i The trivalent threshold function includes negative activation. Therefore, concepts have a valueof1,0,or −1, which is equivalent to ‘‘positive effect’’, ‘‘no effect,’’ and ‘‘negative effect’’,respectively: f(x )=−1,x ≤−0.5 i i C C C C h i k j C 0 −10 +1 h C 00 +10 i E = C 0 00 0 k FIGURE 14.9 C 00 +10 j The adjacency matrix for the cognitive map in Fig. 14.8. www.MatlabSite.comFUZZY COGNITIVE MAPPING 547 f(x ) = 0, −0.5x 0.5 i i f(x ) = 1,x ≥ 0.5 i i Concepts are multiplied by their connecting causal relation weights to give the total input to theeffectconcept.Incaseswheretherearemultiple paths connectingaconcept,the sum of all the causal products is taken as the input Tsadiras and Margaritis, 1996: n x = C w (14.11) i j ji j = 1 j = i where x = input i C = concept state j w = weight of the causal relations ji Feedback For FCMs we can model dynamic systems that are cyclic, and therefore, feedback within a cycle is allowed. Each concept variable is given an initial value based on the belief of the expert(s) of the current state. The FCM is then free to interact until an equilibrium is reached Kosko, 1997. An equilibrium is defined to be the case when a new state vector is equal to a previous state vector. Min–max inference approach The min–max inference approach is a technique that can be used to evaluate the indirect and total effects of an FCM. The causal relations between concepts are often defined by linguistic variables, which are words that describe the strength of the relationship. The min–max inference approach can be utilized to evaluate these linguistic variables Pelaez and Bowles, 1995. The minimum value of the links in a path is considered to be the path strength. If more than one path exists between the cause variable and the effect variable, the maximum value of all the paths is considered to be the overall effect. In other words, the indirect effect amounts to specifying the weakest linguistic variable in a path, and the total effectamounts to specifying the strongest of the weakest paths. Example 14.3. Figure 14.10 depicts an FCM with five concept variables (C represents 1 utilization of waste steam for heat, C represents amount of natural gas required to produce 2 heat, C represents the resulting carbon dioxide (CO ) emissions produced from the burning 3 2 of a methane-based gas, C represents carbon credits that would need to be purchased, and 4 C represents the economic gain). Carbon credits are credits a company would receive from 5 reducing its CO emissions below the required level stipulated by the government’s Kyoto 2 implementation plan. Those companies not meeting their required level may need to purchase credits from others. In the FCM the ‘‘effects’’ of the paths, P, are linguistic instead of simple binary quantities like a ‘‘+1’’ (increase) or a ‘‘−1’’ (decrease). However, the numerical quantities +1,0, and −1 for a trivalentFCM are stillused to convey the signs of the linguistic term.Forexample,alinguisticeffectof‘‘significant, +1’’meansthattheeffectis‘‘significantly positive.’’ A linguistic effect of ‘‘a lot, −1’’ means that the effect is ‘‘negatively a lot.’’ The valuesofthepathscanbeinmattersofdegreesuchas‘‘none,’’‘‘some,’’‘‘much,’’or‘‘alot.’’ www.MatlabSite.com548 MISCELLANEOUS TOPICS C 1 A lot (−1) C 2 A lot (−1) A lot Natural gas (+1) requirements Economic CO 2 C C 3 5 gain emissions Much (+1) Some (−1) C 4 Purchasing of carbon credits FIGURE 14.10 A fuzzy cognitive map involvingwaste steam and greenhouse gas emissions. In this example, then, P =none some much alot.These P values would be the linguisticvalues that would be contained within the adjacency matrix of the FCM:   0 −1 000 00 +10 −1   00 0 +10     0 000 −1 0 0000 To implement the FCM we start by activating C (i.e., we begin the process by assessing the 1 impact of an increase in waste steam for a facility); this results in the initial state vector 1,0,0,0,0 This state vector is activating only concept C in Fig. 14.10. Causal flow in the FCM was 1 determined with repeated vector–matrix operations and thresholding Pelaez and Bowles, 1995. The new state is the old state multiplied by the adjacency matrix Pelaez and Bowles, 1996:   C ... C 11 1n   . . . C C ...C = C C ...C ∗ . . . (14.12)   1 2 n new 1 2 n old . . . C ... C n1 nn www.MatlabSite.comFUZZY COGNITIVE MAPPING 549 The values of the state vector were thresholded to keep their values in the set −1,0,1,and the activated concept (in this case C ) was reset to 1 after each matrix multiplication. Using 1 the algorithm developed by Pelaez and Bowles 1995 we premultiply the trivalent adjacency matrix shown above by this initial state vector. At each iteration of this multiplication, the trivalent threshold function is invoked.This multiplicationis continued until the outputvector reaches a limiting state (i.e., it stabilizes). For this simple example, the resulting state vector stabilized after four iterations to the following form: 1.0,−1.0,−1.0,−1.0,1.0 This stabilized output vector can be understood in the following sense. For an increase in the waste steam (+1), the natural gas requirements will decrease (−1),theCO emissions 2 will decrease (−1), and carbon credits also decrease (−1). Finally, there is an increase in economic gain (+1). With a conventional CM we would get the following results. First, we see that path I = (1,2,5) has two negative causal relationships, and its indirect path effect would be 1 positive (two negatives yield a positive). For path I = (1,2,3,4,5) we see that it has one 2 negative causal relationship(between C and C ) and three positive relationshipsfor the other 1 2 three elements of the path; hence, the indirect effect of this path is negative (one negative and three positives yield a negative effect). Hence, in the conventional CM characterization of this simple example, the results would be indeterminate (one positive indirect effect and a negative indirect effect). For an FCM, we can accommodate linguisticcharacterizations of the elements, as discussed previously, and as seen in Fig. 14.10. Two unique paths that exist from the cause variable (C ) to the effect variable (C ) are I = (1,2,5) and I = (1,2,3,4,5). 1 5 1 2 The indirect effects of C on C , expressed in terms of the linguistic values of P, are Kosko, 1 5 1992 I (C ,C ) = mine ,e = minalot,alot 1 1 5 12 25 =alot I (C ,C ) = mine ,e ,e ,e = mina lot, a lot, much, some 2 1 5 12 23 34 45 = some Therefore, the linguistictotal effect is expressed as T(C ,C ) = maxI (C ,C ), I (C ,C ) 1 5 1 1 5 2 1 5 = maxa lot, some=alot Applying these linguistic results to the stabilized vector above, i.e., 1.0, −1.0,−1.0,−1.0, 1.0, we come to the conclusion that an increase in waste steam for this facility results in ‘‘a lot of increase’’ in economic gain. Toconcludetheexample,fuzzycognitivemappingdoessufferincomparisonwithother methodsinthatthereisalargedegreeofsubjectivity.Butfuzzycognitivemappingdoesallow forvaryingdegreesofmagnitudeorsignificanceofrelationships,whichisalimitationofother crisp or standard methods. Therefore, much of the grayness in subjectivity is captured and accounted for, resulting in a more balanced assessment. Thus, with appropriate expert-based professional judgment (likely by a panel of experts in the field of the subject matter), FCM can bean effective assessment tool.Problems 14.12–14.14at theend ofthischapter are given as exercises to illustrate how various changes to the paths in this example result in different results and conclusionsof the FCM approach. www.MatlabSite.com550 MISCELLANEOUS TOPICS SYSTEM IDENTIFICATION ◦ Suppose we have a standard fuzzy relational equation of the form B = A R. In the normal ∼ ∼ ∼ situationwehaveafuzzyrelationRfromeitherrulesordata,andtheinformationcontained ∼ inAisalsoknownfromdataorisassumed.ThedeterminationofBusuallyisaccomplished ∼ ∼ through some form of composition. Suppose, however, that we know B and R,andwe ∼ ∼ are interested in finding A. There are many physical problems where this situation arises. ∼ Foremostamongtheseisthefieldofsystemidentification.Wemighthaveasystem(modeled by R) which is subjected to an input which is unknown (representedby A), but we are able ∼ ∼ to observe or measure the output of the system (given by B). So, we want to find out what ∼ possible sets of the input could generate the observed output. If the relational equation were linear, we would find the inverse of the equation, i.e., −1 ◦ A = B R . ∼ ∼ ∼ Unfortunately,afuzzyrelationalequationisnotlinear,andtheinversecannotprovide a unique solution, in general, or any solution in some situations. In fact, the inverse is difficult to find for most situations Terano et al., 1992. A fuzzy relational equation can be expressed in expanded form by (a ∧ r ) ∨ (a ∧ r )∨···∨ (a ∧ r ) = b 1 11 2 12 n 1n 1 (a ∧ r ) ∨ (a ∧ r )∨···∨ (a ∧ r ) = b 1 21 2 22 n 2n 2 (14.13) . . . (a ∧ r ) ∨ (a ∧ r )∨···∨ (a ∧ r ) = b 1 m1 2 m2 n mn m where a ,r ,and b are membership values for A,R,andB, respectively. i ij i ∼ ∼ ∼ To solve A=a ,a ,...,a given r and b (i = 1,2,...,n and j = 1,2,...,m), 1 2 n ij j ∼ we can use a method reported by Tsukamoto and Terano 1977 that produces interval values for the solution. In this approachthere aretwo standard definitions that first must be presented: a fuzzy equality and a fuzzy inequality. An equality is expressed as Equality a ∧ r =b(14.14) The inverse solution for a in the equality in Eq. (14.14), given r and b are known, is generally an interval number and is denoted by an operator b r,  b, r b b r = b,1,r = b (14.15) ∅,rb An inequality is defined as Inequality a ∧ r ≤b(14.16) For the inequality (14.16) the inverse solution for a is also an interval number, denoted by ˆ the operator b r, and is given by  0,b,rb ˆ b r = (14.17) 0,1,r ≤ b www.MatlabSite.comSYSTEM IDENTIFICATION 551 Atypical rowofthestandardfuzzyrelationalequationsystem (seeEqs. (14.13)),say the first row, can be represented in a simpler form, (a ∧ r ) ∨ (a ∧ r )∨···∨ (a ∧ r ) =b(14.18) 1 1 2 2 n n The expression in Eq. (14.18) can be subdivided into n equalities of the type (a ∧ r ) = b, (a ∧ r ) = b, ..., (a ∧ r ) =b(14.19) 1 1 2 2 n n and n inequalities of the type (a ∧ r ) ≤ b, (a ∧ r ) ≤ b, ..., (a ∧ r ) ≤b(14.20) 1 1 2 2 n n Asolution,representedasanintervalvector,tothefuzzyrelationalEq. (14.18)existsifand only if there is at least one equality and no more than (n −1) inequalities in the solution. That is, the inverse solution for a in Eq. (14.18), denoted W,is i i a = W or W or ... or W (14.21) i 1 2 n where W = (b ˆ r ,...,b ˆ r ,b r ,b ˆ r ,...,b ˆ r)(14.22) i 1 i−1 i i+1 n Note that the ith term in Eq. (14.22) is an equality and the other terms are inequalities. Example 14.4. Suppose we want to solve for a (i = 1,2,3,4) in the single inverse fuzzy i equation (a ∧0.7) ∨ (a ∧0.8) ∨ (a ∧0.6) ∨ (a ∧0.3) = 0.6 1 2 3 4 Makinguseofexpressions(14.19)–(14.20),wesubdividethesinglefuzzyequationinto n = 4 equalities: Y =b r ,b r ,...,b r eq 1 2 n =0.6 0.7,0.6 0.8,0.6 0.6,0.6 0.3 =0.6,0.6,0.6,1,∅ (four equality values) and into n = 4 inequalities: Y =b ˆ r ,b ˆ r ,...,b ˆ r ineq 1 2 n ˆ ˆ ˆ ˆ =0.6 0.7,0.6 0.8,0.6 0.6,0.6 0.3 =0,0.6,0,0.6,0,1,0,1 (four inequality values) Then Eq. (14.22) provides for the n = 4 potential solutions, W,where i = 1,2,3,4: i W =0.6,0,0.6,0,1,0,1 (position1 is the first equality value) 1 W =0,0.6,0.6,0,1,0,1 (position2 is the second equality value) 2 W =0,0.6,0,0.6,0.6,1,0,1 (position 3 is the third equality value) 3 W =0,0.6,0,0.6,0,1,∅ = ∅ (position 4 is the fourth equality value) 4 www.MatlabSite.com552 MISCELLANEOUS TOPICS whereallvaluesaretheinequalities,exceptthoseequalitiesnotedspecifically.Equation (14.21) provides for the aggregated solution in interval form; note that W does not contribute to the 4 aggregated solution because it has a value of null. Hence, a = W or W or W i 1 2 3 Its maximum solutionis a =0.6,0.6,1,1. Its minimum solutionsare max a =0.6,0,0,0 from W min 1 =0,0.6,0,0 from W 2 =0,0,0.6,0 from W 3 Nowsupposeinsteadofasingleequationwewanttofindthesolutionforanequation set; that is, a collection of m simultaneous equations of the form given in Eqs. (14.13). Then the solution set will consist of an m set of n equalities, expressed in an m × n matrix ˆ (denoted Y) and an m set of n inequalities, also expressed in an m × n matrix (denoted Y):   b r b r ... b r 1 11 1 12 1 1n   b r b r ... b r 2 21 2 22 2 2n   Y = (14.23)   . .   . b r b r ... b r m m1 m m2 m mn   b ˆ r b ˆ r ... b ˆ r 1 11 1 12 1 1n   b ˆ r b ˆ r ... b ˆ r 2 21 2 22 2 2n   ˆ Y =  (14.24) . .   . b ˆ r b ˆ r ... b ˆ r m m1 m m2 m mn ˆ Taking an element for each row from Y and replacing the corresponding element in Y,we get an array solution for each element, ij,inthe m × n equation matrix:   b ˆ r ... b r ... b ˆ r 1 11 1 1i 1 1n   b ˆ r ... b r ... b ˆ r 2 21 2 2i 2 2n   ∗ ∗ W = = (w)(14.25)   . (i ,i ,...,i ) ij 1 2 m .   . ˆ ˆ b r ... b r ... b r m m1 m mi m mn where indices i = (1,2,...,n),i = (1,2,...,n),and i = (1,2,...,n) represent m 1 2 m arrays that are all of length n. Each array is a solution to one row of the original equations (i.e., of Eqs. (14.13)), and we need at most m arrays for the complete solution. m Hence, we can have (n ) solutions, including null solutions, of the type expressed by ∗ Eq. (14.25). To establish a sign convention we denote a particular solution, (w ),ofthe ij m × narraybyidentifyingitslocation (ij)inthearray.Thecompletesolutionincorporating all m × n possible solutions will be denoted W =w ,w ,...,w (14.26) (i ,i ,...,i ) 1 2 m 1 2 m where ∗ w =∩ w (14.27) j ij i and where i = 1,2,...,n and j = 1,2,...,m. www.MatlabSite.comSYSTEM IDENTIFICATION 553 In the foregoing development, each possible solution w ,w ,...,w in Eq. (14.26) 1 2 m is found by taking the intersection of all the solutions in the jth column, i.e.,Eq. (14.27), of the solution arrays described by Eq. (14.25). Example 14.5. Suppose we have a system of three simultaneous equations as given here. In 3 this example, m =3and n = 3. Hence, there is a potential for 3 = 27 distinct solutions that need to be explored in order to develop the full solution. Our goal is to find interval values for the three unknown quantities, a ,a ,a , in the following inverse equations: 1 2 3   0.30.50.2 ◦ a ,a ,a 0.200.4 = 0.20.40.2 1 2 3 00.60.1 To begin the solution process, we need to find the individual equality sets, Y, and ˆ inequality sets, Y, using Eqs. (14.23) and (14.24), respectively. So for the first row of Y we operate b on the first column of the r matrix, i.e., using Eq. (14.15), 1 b r ,b r ,b r =0.2 0.3,0.2 0.2,0.2 0=0.2,0.2,1, ∅ 1 11 1 21 1 31 The second row of Y is found, by operating b on the second column of the r matrix, to be 2 b r ,b r ,b r =0.4 0.5,0.4 0,0.4 0.6=0.4, ∅,0.4 2 12 2 22 2 32 The third row of Y is found, by operating b on the third column of the r matrix, to be 3 b r ,b r ,b r =0.2 0.2,0.2 0.4,0.2 0.1=0.2,1,0.2,∅ 3 13 3 23 3 33 Therefore, we have   0.20.2,1 ∅ Y = 0.4 ∅ 0.4 0.2,1 0.2 ∅ ˆ To calculate the inequality matrix, Y, we have the same sequence of operations of ˆ elements in the b vector on columns in the r matrix, but we use the operator .Forthefirst row of Y we operate b on the first column of the r matrix, i.e., using Eq. (14.17), 1 ˆ ˆ ˆ ˆ ˆ ˆ b r ,b r ,b r =0.2 0.3,0.2 0.2,0.2 0=0,0.2,0,1,0,1 1 11 1 21 1 31 The second row of Y is found, by operating b on the second column of the r matrix, to be 2 b ˆ r ,b ˆ r ,b ˆ r =0.4 ˆ 0.5,0.4 ˆ 0,0.4 ˆ 0.6=0,0.4,0,1,0,0.4 2 12 2 22 2 32 The third row of Y is found, by operating b on the third column of the r matrix, to be 3 ˆ ˆ ˆ ˆ ˆ ˆ b r ,b r ,b r =0.2 0.2,0.2 0.4,0.2 0.1=0,1,0,0.2,0,1 3 13 3 23 3 33 Therefore, we have   0,0.2 0,1 0,1 ˆ 0,0.4 0,1 0,0.4 Y = 0,1 0,0.2 0,1 www.MatlabSite.com554 MISCELLANEOUS TOPICS ∗ ∗ Now, using Eq. (14.25), we can construct the W matrices. The first matrix is W ,the ij 111 ∗ subscripts denoting that the equality element for the first row of W comes from the first 111 position in Y (the first subscript 1) in the first row; the equality element for the second row of ∗ W comesfromthefirst positioninY(thesecondsubscript1)inthesecondrow;theequality 111 ∗ element for the third row of W comes from the first position in Y (the third subscript 1) in 111 ∗ the third row. All other elements for W come from the same positions they are in for the 111 ∗ ˆ matrix Y. Hence, W looks like 111   0.20,1 0,1 ∗ W = 0.40,1 0,0.4 111 0.2,1 0,0.2 0,1 Finally, using Eqs. (14.26)–(14.27), we take the intersection of the elements in each ∗ column of W to get 111 W = (∅,0,0.2,0,0.4)=∅ 111 Since there is a null element in W , we set the entire value equal to null. 111 ∗ Continuing in a similar fashion, the second matrix is W , with the subscripts denoting 112 ∗ that the equality element for the first row of W comes from the first position in Y (the first 112 ∗ subscript 1) in the first row; the equality element for the second row of W comes from the 112 first positioninY(the secondsubscript1)in thesecondrow;theequalityelement forthethird ∗ row of W comes from the second position in Y (the third subscript 2) in the third row. All 112 ∗ ˆ other elements for W come from the same positions they are in for the matrix Y. Hence, 112 ∗ W looks like 112   0.20,1 0,1 ∗ W = 0.40,1 0,0.4 112 0,1 0.20,1 and using Eqs. (14.26)–(14.27) again, we get W = (∅,0.2,0,0.4)=∅ 112 ∗ because there is at least one null element in W . 112 ∗ ∗ Now, moving to other elements in w ,suchas W ,weget ij 131   0.20,1 0,1 ∗ W = 0,0.4 0,1 0.4 131 0.2,1 0,0.2 0,1 and the resulting solutionafter performing intersections on the elements in each column is W = (0.2,0,0.2,0.4) 131 This process continues for the other 24 solutions, e.g., W ,W ,W , etc., out of the total 211 232 333 3 of 27 (3 ). Of all 27 possible solutions,only four are non-null (nonempty). These four are W = (0.2,0,0.2,0.4) 131 W = (0.2,0.2,0.4) 132 W = (0.2,0.2,0.4) 231 W = (0,0.2,0.2,0.4) 232 www.MatlabSite.comFUZZY LINEAR REGRESSION 555 a 3 W 232 0.4 a min a min a = W = W max 132 231 W 131 0.2 a 2 0.2 a 1 FIGURE 14.11 Solution for Example 14.5. By inspection we can see that W ⊇ W and W ⊇ W ,andthat W = W ; 131 132 232 231 231 132 hence, thesolutioncan beexpressedbythe intervals W and W . Thissolutionspace inthe 132 232 three dimensions governed by the original coordinates a ,a ,and a is shown in Fig. 14.11. 1 2 3 In the figure we see that the minimum solution is given by two points a =a ,a ,a =0.2,0,0.4 from W min,1 1 2 3 131 a =a ,a ,a =0,0.2,0.4 from W min,2 1 2 3 232 and that the maximum solution is given by the single point a =a ,a ,a =0.2,0.2,0.4 max 1 2 3 Theentiresolutioninthisthree-dimensionalexamplecomprisesthetwoedgesthataredarkened in Fig. 14.11. It is perhaps clear from Examples 14.4 and 14.5 that when the cardinal numbers n and m are large, the number of analytical solutions becomes exponentially large; also, sometimes the final solution can be null. For both these cases other approaches, such as those in pattern recognition, might be more practical. In any case, fuzzy inverses are only approximate; they are not unique in general, even for linear equations. FUZZY LINEAR REGRESSION Regression analysis is used to model the relationship between dependent and independent variables. Inregression analysis, the dependent variable, y, is a function of the independent variables; and the degree of contribution of each variable to the output is represented by coefficients on these variables. The model is empirically developed from data collected fromobservationsandexperiments.AcrisplinearregressionmodelisshowninEq. (14.28), y = f(x,a) = a + a x + a x +···+ a x (14.28) 0 1 1 2 2 n n www.MatlabSite.com556 MISCELLANEOUS TOPICS In conventional regression techniques, the difference between the observed values and the values estimated from the model is assumed to be due to observational errors, and the difference is considered a random variable. Upper and lower bounds for the estimated value are established, and the probability that the estimated value will be within these two bounds represents the confidence of the estimate. In other words, conventional regression analysis is probabilistic. But in fuzzy regression, the difference between the observed and theestimatedvaluesisassumedtobeduetotheambiguityinherentlypresentinthesystem. The output for a specified input is assumed to be a range of possible values, i.e., the output can take on any of these possible values. Therefore, fuzzy regression is possibilistic in nature.Moreover,fuzzyregressionanalysesusefuzzyfunctionstorepresentthecoefficients as opposed to crisp coefficients used in conventional regression analysis Terano et al., 1992. Equation (14.29) shows a typical fuzzy linear regression model, Y = f(x,A) = A +A x +A x +···+A x (14.29) 1 2 n ∼ ∼ ∼ ∼ ∼ ∼ 0 1 2 n where A is the ith fuzzy coefficient (usually a fuzzy number). ∼i Fuzzy regression estimates a range of possible values that are represented by a possibility distribution (a more rigorous definition of possibilities is given in Chapter 15), termed here a membership function. Membership functions are formed by assigning a specificmembershipvalue(degreeofbelonging)toeachoftheestimatedvalues(Fig.14.12). Suchmembershipfunctionsarealsodefinedforthecoefficientsoftheindependentvariables. Triangularmembership functions forthefuzzycoefficients,like those shown in Fig. 14.12, allowforthe solution tobefoundviaalinearprogrammingformulation; othermembership functions for the coefficients require alternative approaches Kikuchi and Nanda, 1991. The membership function µ for each of the coefficients is expressed as A ∼  p − a i i 1 − ,p − c ≤ x ≤ p + c i i i i i µ (a ) = (14.30) A i c i ∼ i 0, otherwise The fuzzy function A is a function of two parameters, p and c, known as the middle value ∼ and the spread, respectively. The spread denotes the fuzziness of the function. The figure shows the membership function for a fuzzy number ‘‘approximately p.’’ A more detailed i explanation of membership functions, fuzzy numbers, and operations on fuzzy numbers µ µ A 1.0 A i 0 a c p c i i i FIGURE 14.12 Membership function for the fuzzy coefficient A. ∼ www.MatlabSite.com

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