Control System Synthesis as a Multiple Criteria Decision ... - Wseas.us

**Control** **System** **Synthesis** **as** a **Multiple** **Criteria****Decision** Making ProblemALEXANDRA GRANCHAROVA, JORDAN ZAPRIANOVInstitute of **Control** and **System** ResearchBulgarian Academy of SciencesAcad. G. Bonchev str., Bl.2, P.O.Box 79, Sofia 1113BULGARIAAbstract: The process of control system synthesis requires evaluation of a number of alternative controlstructures with respect to the resulting performance of the closed-loop system. At present, there is not a generalmethod for selection of best alternative for control system.In this paper, a general problem of control system synthesis is formulated and represented **as** a multiplecriteria decision making problem. Then, an approach is proposed to solve this problem through comparisonbetween alternative control structures and selection of the best one. The developed approach includes threesteps. First, the set of all possible alternatives for control structures is defined. Second, for each alternativecontrol system the optimal values of controller parameters are determined. This optimization problem can besolved by applying standard optimization techniques. Third, the best alternative control structure is selectedfrom the set of available alternatives. In order to do it, three types of fuzzy preference relations are introducedthat show how “good” the alternative A i is compared to the alternative A j with respect to optimization of themultiple criteria, w.r.t. satisfaction of the inequality constraints and w.r.t. satisfaction of the equalityconstraints. Also, the total fuzzy preference relation is defined. Then, the fuzzy subset of nondominatedalternatives is determined and the best alternative is selected **as** the one whose value of the membershipfunction is maximal. Finally, an approach is described for selection of the weight coefficients needed for thesolution of controller synthesis problem.Key Words: **Decision** making, **Control** system synthesis, **Multiple** criteria, Constraints.CSCC'99 Proceedings: - Pages 3231-32371. Introduction.The process of control system synthesis requiresevaluation of a number of alternative controlstructures with respect to the resulting performanceof the closed-loop system. At present, there are onlyfew methods which deal with the problem ofcontroller structure selection. Th**us**, in [1] afrequency domain method h**as** been proposed forcontroller design for linear plants. According to thismethod, the optimal control configuration is thesimplest one (with the lowest possible order) whichgives a satisfactory approximation of the desiredclosed-loop system. It should be noted, however, thatthis method can be applied for linear plants only andit is not shown how the tradeoff between controllercomplexity and controller performance is made.Also, sometimes it is more appropriate to choosecontrol structure that optimizes closed-loopperformance (for example, one that minimizes theintegral squared error) rather than to select controllerwhich approximates a preliminary specified closedloopresponse. At present, there is not a generalmethod for selection of best alternative for controlsystem.In this paper, a general problem of controlsystem synthesis is formulated and represented **as** amultiple criteria decision making problem. Then, anapproach is proposed to solve this problem throughcomparison between alternative control structuresand selection of the best one.2. Problem formulation.The control system under consideration is shown inFig.1, where y is an ny-dimensional vector of thevariables to be controlled, y is the set point vector,spu is an nu-dimensional vector of control variablesand ν is an nν - dimensional vector of disturbances.It is supposed that plant dynamics is described by thefollowing general equations:d x= f ( x,u)(1)dty = g( x,u), (2)

where x is an nx-dimensional vector of statevariables, f and g are vector functions.ysp + e u + y+**Control**ler Plant-Fig.1. **Control** system.The problem to be solved is to find control law:u = u( y)(3)that will maximize the multiple optimality criteriarepresented in the following general form:t fI k ( u)= Gk[x(t f )] + ∫ f0k( x,y,u)⋅ dt → max(4)0k = 1,2,..., msubject to constraints of inequality type:ψ k ( x(t),y(t),u(t))≤ 0 , t ∈[0;t f ](5)k = 1,2,..., l1and constraints of equality type:ϕk( x(t),y(t),u(t))= 0 , t ∈[0;t f ](6)k = 1,2,..., l2Constraints (5) and (6) can include both pathconstraints that have to be satisfied in the timeinterval t ∈ 0; t ] and final time constraints that[ fhave to be kept at the end time t f . In the furtherconsiderations it is more convenient to have onlyfinal time constraints in the control problemformulation. For this purpose, the approach proposedin [2] can be applied to represent path constraints **as**end time constraints. According to [2], pathconstraints of the form:akmin≤ ak( x(t),u(t))≤ akmax(7)k = 1,2,..., lcan be converted to final time constraints throughdefinition of a new state variable **as**:dxnx+1T= ( amin− a) ⋅W1⋅ ( amin− a)+dtxnx+1(0) = 0 ,T( a − a ) ⋅W⋅( a − a )where a [ a] T1 ( x(t),u(t))... al( x(t),u(t))a = [ a] T , a [ a] Tmax2νmax(8)= ,min 1min... a l min max=1max... a l max. In(8), W 1and W2are l × l diagonal weightingmatrices whose elements reflect the satisfaction ofthe path constraints. Then, a new final time constraintis added:x ( t ) 0(9)nx+ 1 f =It is clear that when constraint (9) is satisfied, thenconstraints (7) are satisfied too.In the control problem formulated above,path constraints of types (5) and (6) are nowrepresented **as** final time constraints throughdefinition of the new state variable:dxnx+1 T= ψ ( x,y,u)⋅W1⋅ψ( x,y,u)+dtϕT( x,y,u)⋅W2⋅ϕ(x,y,u)(10)xnx+1(0)= 0 ,where:Tψ ( x , y,u)= [ ψ 1 ( x,y,u)... ψ l 1( x,y,u)](11)ϕ ( x , y,u)= [( x,y,u)](12)ϕ 1 ( x,y,u)... ϕ l 2and the elements of the weighting matrices W1andW2are:w = 0,if ψ ( x,y,u)≤ 0 ; k 1,2,... l (13)w1 kk k= , 11 kk 1,if k ( x,y,u)> 0 ; k = 1,2,... , l1w2kk= ψ (14)2kkk = 1,2,..., l2w = 1, if ϕ= 0, if −δ≤ϕ( x,y,u)≤ δkkk( x(t),y(t),u(t))< −δ∨ ϕk( x(t),y(t),u(t))> δkT(15)k = 1,2,..., l2Here, the equality constraints (6) are relaxed byintroducing the small positive value δ k. Now, newfinal time constraint is added:x ( t ) 0(17)nx+ 1 f =kk(16)When constraint (17) is satisfied, the state andcontrol trajectories are fe**as**ible with respect to theinequality and the equality path constraints (5) and(6). Th**us**, the problem of controller synthesis is tofind control law u = u( y)that will maximize themultiple optimality criteria (4) subject to thefollowing final time constraints:- constraints of inequality type:* *k x ( t f ), y(t f ), u(t f )) ≤ 0 , k =ψ ( 1,2,..., l (18)- constraints of equality type:* *k x ( t f ), y(t f ), u(t f )) = 0 , k =ϕ ( 1,2,..., l , (19)*Twhere x = [ x 1x 2... x nx x nx+1] is the augmentedstate vector.The problem of control system synthesisconsists of choosing an optimal structure and optimalparameters of the controller, i.e. it consists ofstructural and parametric synthesis. Here, the set of*1*23232

the possible alternatives for controller structures isdenoted by:A = { A1 , A2, ... An}(20)Then, controller parameters for alternative A i aredenoted by vector p A ) and their optimal values are( idenoted by vector p A ) . Th**us**, control law canbe expressed **as**:(opt iu = u y,A , p(A ))(21)( i iIn this way, the problem of controller synthesis for agiven plant is to find the best alternative A bestforcontroller structure and the optimal controllerparameters p A ) for this structure so that the(opt bestcriteria (4) are optimized and constraints (18) and(19) are satisfied.3. Approach for control systemsynthesis.Here, an approach is presented to solve the controllersynthesis problem formulated above. It includes thefollowing steps:1. Define the set of all possible alternatives ofcontroller structure:A = { A1 , A2, ... An}(22)An example set of possible alternatives is shown inFig.2.Alternativesfor controller structuresPIDcontrollerInternal modelcontrollerFuzzycontrollerNeural networkcontrollerModel predictivecontrollerFig.2. Alternatives for controller structures.2. Determine the optimal controllerparameters p A ) for alternative Ai.(opt iThese are parameters which maximize the multipleoptimality criteria (4) and satisfy constraints (18) and(19) given that controller structure is Ai. Here, thismultiple criteria optimization problem is solved byapplying fuzzy sets theory. Three types ofmembership functions are introduced:- membership function µ ( p)that showshow “good” are the values p A ) of controllerI( iparameters with respect to optimization of themultiple criteria I k( u) , k = 1,2,... , m ;- membership function µ ( ) that showsψ * phow “good” are the parameters p with regard tosatisfying inequality constraints (18);- membership function µ ( ) that showsϕ * phow “good” are the parameters p with respect tosatisfying equality constraints (19).The following steps should be carried out tofind the optimal values p of controller parameters:opt2.1. For the current values p calculate themembership function µ ( p).IIn order to do this, m individual membershipfunctions µ ( p)correcponding to the m criteria toI kbe maximized, are considered:I k max − I k ( p)µI ( p)= 1 −,kIk max− Ik min(23)where:I = min I ( p), I = max I ( p)(24)k minp∈P0kk maxp∈P0Function (23) shows to which extent the value of thecriteria is close to its “best” (maximal) value and itbelongs to the interval [0;1]. In (24) P 0denotes theadmissible range for searching the optimal values ofp .Having determined the m individualmembership functions µ ( p), then the functionµ I ( p) is computed **as** follows:whereI kI kmµ ( p)= λ ⋅ µ ( p), (25)I∑k=1Ikλ is the weight of the k-th criterion. Theseweight coefficients are chosen in a such a way sothey will satisfy:m∑k=1Ikλ = 1 , λ > 0 (26)Ik I kk3233

In section 3, an approach for selection of theweight coefficients is described.2.2. For the current values p calculate themembership function µ ( ) .ψ * p*Similarly to the previo**us** step, l1individualmembership functions µ * ( p)correcponding to theψ k*l1constraints of type (18), are considered:⎧⎪*⎪0, if ψ k ( p)> 0µ ψ* ( p)= ⎨(27)k⎪ *⎪δ k −ψk ( p)*, if ψ ( p)≤ 0⎪*k⎩ δk−ψk minwhere:**ψ k = min ψ ( p)(28)minp∈P0and δ kis a small positive value that prevents thedenominator in (27) from being equal to zero.Function (27) shows how far is the value of * ( p )kψ kfrom its maximal allowed value (zero) and it belongsto the interval [0;1]. Function µ ( ) is nowcomputed:*l1= ∑ µψ ψ k ψ kk=1ψ * pµ * ( p)λ * ⋅ * ( p)(29)where λ is the weight of the k-th constraint.*ψ k2.3. For the current values p calculate themembership function µ ( ) .ϕ * pAnalogo**us**ly, l * 2individual membership functionsµ * ( p)correcponding to the l * 2constraints of typeϕ k(19), are considered:⎧⎪**⎪0,if ϕk( p)< −δk∨ ϕk( p)> δk⎪µ ϕ* ( p)= ⎨(30)k⎪ *⎪ ϕk( p)*⎪1 − , if − δ k ≤ ϕk( p)≤ δ k⎩ δkwhere δ k is a small positive value which is **us**ed torelax the equality constraints (19). Function (30)shows how “good” is the value of ϕ * ( p ) comparedto its “best” (zero) value and it belongs to the interval[0;1]. Function µ ( ) is computed **as** follows:ϕ * pk*l2= ∑ µϕ ϕkϕkk=1µ * ( p)λ * ⋅ * ( p)(31)where λ is the weight of the k-th constraint.*ϕ k2.4. For the current values p calculate themembership function µ S ( p).This function shows to what extent p can beaccepted **as** the optimal solution of the problem, i.e.how “good” p is with respect to optimization of themultiple criteria and satisfaction of all constraints oftype (18) and (19). It is computed **as** follows:µ S p)= λI⋅ µ I ( p)+ λ * ⋅ µ * + λ * ⋅ * , (32)where( µψ ψ ϕ ϕλ , λ * and λ * are weight coefficientsIψϕreflecting the importance respectively of optimalitycriteria, inequality constraints and equalityconstraints. They should satisfy:λ I λ ψ+ λ 1(33)+ * * =ϕ2.5. Determine the optimal values p optbymaximizing µ S ( p).The optimal values p are found through solutionoptof the following optimization problem:p = arg max µ ( p)(34)optp∈P0Problem (34) can be solved by applying standardoptimization techniques. In order to execute this step,all previo**us** steps (from 2.1 to 2.4) have to berepeated several times.3. Determine the best alternative A bestfromthe set of alternatives A = { A1 , A2, ... An}.The problem of best alternative selection represents amultiple criteria decision making problem underconstraints. Here, a strategy is proposed to solve thisproblem by introducing three types of fuzzypreference relations:- preference relation Q I with respect to the mcriteria to be optimized:Q I = {( Ai, Aj) : Ai, Aj∈ A,µ I ( Ai, Aj)} (35)where*- preference relation Q ψ with respect to the l * 1inequality constraints to be satisfied:Q = {( A , A ) : A , A ∈ A,* ( A , A )} (36)* i j i j µψ ψ i j*- preference relation Q ϕ with respect to the l * 2equality constraints to be satisfied:Q = {( A , A ) : A , A ∈ A,* ( A , A )} , (37)* i j i j µϕ ϕ i jµ , µ * and µ * are membership functions.IψI ( Ai, AjϕFunction µ ) shows how “good” isS3234

alternative A icompared to alternative A j withrespect to optimization of the multiple criteria (4),function µ * ( A i , Aj) reflects how “good” is Aψiincomparison to A j with regard to satisfaction of theinequality constraints (18) and µ * ( A i , Aj) showsϕhow “good” is Aicompared to A j with respect tosatisfaction of the equality constraints (19).The following steps are to be carried out inorder to select the best alternative Abestof controllerstructure:3.1. Determine the fuzzy preference relationwith respect to the criteria to be optimized.In order to determine this preference relation, mindividual preference relations Q corresponding tothe m criteria (4) to be optimized, are considered:QI= {( Ai, Aj) : Ai, Aj∈ A,µI ( Ai, Akk j )}(38)k = 1,2,..., mwhere it is proposed for the membership functionµ A , A ) to be computed **as** follows:I (k i j⎧0,if I k ( Ai) < I k ∧ I k ( Aj) > I k⎪⎪⎪1,if I k ( Ai) > I k ∧ I k ( Aj) < I k⎪⎪⎪ ( I k ( Ai) − I k)⎪,µ = ⎨( k i − k ) + ( k j − k )I( Ai, Aj) I ( A ) I I ( A ) I (39)k⎪ if I k ( Ai) > I k ∧ I k ( Aj) > I k⎪⎪⎪ ( I k − I k ( Ai))⎪1−,⎪ ( I k − I k ( Ai)) + ( I k − I k ( Aj))⎪⎩ if I k ( Ai) < I k ∧ I k ( Aj) < I kHere, I kis the average value of the criterion I kamong all available alternatives, i.e.:n1I k = ⋅∑I k ( Ai)(40)n i=1The membership function µ A , A ) shows howI kI (k i j“good” the alternative A i is compared to thealternative A j with respect to the maximization ofcriterion I k . The average value I kof the criterionserves **as** a b**as**is for comparison.Having determined the m individual preferencerelations Q , then the membership function of theI ktotal preference relation Q I with respect to alloptimization criteria is computed **as** follows:wheremµ ( A , A ) = λ ⋅ µ ( A , A ) , (41)I kIij∑k=1Ikλ is the weight of the k-th criterion. Theseweight coefficients are chosen in a such a way sothey will satisfy:m∑k=1Ikiλ = 1 , λ > 0 (42)Ik I k3.2. Determine the fuzzy preference relationwith respect to the inequality constraints to besatisfied.In order to determine this preference relation, l * 1individual preference relations corresponding toQ ψ*kthe l * 1inequality constraints (18) to be satisfied, areconsidered:Q * = {( Ai, Aj) : Ai, Aj∈ A,µ * ( Ai, Aj)}ψ kψ k(43)*k = 1,2,..., l1where it is proposed for the membership functionµ ( A i , A ) to be computed **as** follows:*ψ kj**⎧0,if ψk( Ai) > 0∧ψk( Aj) ≤ 0⎪⎪⎪**1, if ψ k ( Ai) ≤ 0∧ψk ( Aj) > 0⎪⎪⎪*( δ k −ψk ( Ai))⎪,**µ = ⎨( − ( )) + ( − ( ))* ( Ai, A ) δkψk Aiδkψk Ajj (44)ψ k⎪**if ψ k ( Ai) ≤ 0∧ψk ( Aj) ≤ 0⎪⎪⎪*( ψ ( ) − )⎪kAiδk1−,**⎪ ( ψ k ( Ai) − δ k) + ( ψ k ( Aj) −δk)⎪**⎪⎩if ψ k ( Ai) ≥ 0∧ψk ( Aj) ≥ 0Here, the small positive value δkguarantees that thedenominator in the expression for µ ( A i , A ) willj*ψ kjnot be equal to zero. The membership functionµ ( A i , A ) shows how “good” the alternative A i is*ψ kjcompared to the alternative A j with respect tosatisfaction of the k-th inequality constraint of type(18). The third and fourth expressions in (44) allowto compare A i with A j depending on the distance ofψ * k( A ) from the boundary (zero) of the inequalityconstraint. Th**us**, if both alternatives satisfy thisconstraint, it is accepted that the better alternative isthe one whose value ψ * k( A ) is more far fromconstraint boundary and respectively, if bothalternatives don’t satisfy this constraint, the better3235

alternative is the one whose value ψ * k( A ) is moreclose to the boundary. The membership function of*the total preference relation Q ψ with respect to allconstraints (18) is computed **as** follows:where*l1i j = ∑ µψ ψ k ψ kk=1µ * ( A , A ) λ * ⋅ * ( A , A ) , (45)λ is the weight of the k-th constraint. These*ψ kweight coefficients should satisfy:*l1∑ 1 λ *ψ kψ kk=1λ * = , > 0 (46)3.3. Determine the fuzzy preference relationwith respect to the equality constraints to be satisfied.Similarly, l * 2individual preference relations *ijQ ϕkcorrecponding to the l * 2equality constraints (19) tobe satisfied, are considered:Q * = {( Ai, Aj) : Ai, Aj∈ A,µ * ( Ai, Aj)}ϕ kϕk(47)*k = 1,2,..., l2where it is proposed for the membership functionµ ( A i , A ) to be computed **as** follows:*ϕ kj**⎧0,if { ϕ k ( Ai) > δ k ∨ ϕ k ( Ai) < −δk}⎪*⎪ ∧ − δk≤ ϕk( Aj) ≤ δk⎪*⎪1,if −δk ≤ ϕ k ( Ai) ≤ δ k ∧⎪ **{ ϕ⎪ k( Aj) > δk∨ ϕk( Aj) < −δk}⎪*ϕ ( )⎪k Ai1−,⎪**2 + ϕk( Ai) + ϕk( Aj)µ ϕ* ( Ai, Aj) = ⎨k*⎪ if − δk≤ ϕk( Ai) ≤ δk⎪*⎪∧ − δ k ≤ ϕ k ( Aj) ≤ δ k⎪*⎪ϕk( Ai)1−,⎪ **ϕ k ( Ai) + ϕ k ( Aj)⎪⎪ **if { ϕ ( ) > ∨ ( ) < − }⎪k Aiδ k ϕ k Aiδ k⎪**⎩ ∧{ϕk( Aj) > δk∨ ϕk( Aj) < −δk}(48)Here, the small positive value δkis **us**ed to relax theequality constraints (19). The third expression in (48)guarantees that µ ϕ* ( A , A ) µ * ( A , A ) = 0. 5*k*= k ji j = j ik ϕ kwhen ϕ ( A ) ϕ ( A ) = 0 . The membershipifunction µ ( A i , A ) shows how “good” the*ϕ kjalternative A i is compared to the alternative A j withrespect to satisfaction of the k-th equality constraintof type (19). The membership function of the totalpreference relation Q * with respect to all constraintsϕ(19) is computed **as** follows:where*l2i j = ∑ µϕ ϕkϕkk=1µ * ( A , A ) λ * ⋅ * ( A , A ) , (49)λ is the weight of the k-th constraint. These*ϕ kweight coefficients should satisfy:*l2∑k=1λ * = , > 0 (50)1 λ *ϕkϕ k3.4. Determine the total fuzzy preferencerelation and the fuzzy subset of nondominatedalternatives.The total fuzzy preference relation is defined **as**:Q T = {( Ai, Aj) : Ai, Aj∈ A,µ T ( Ai, Aj)} (51)and it shows how “good” the alternative A i iscompared to the alternative A j with respect tooptimization of the multiple criteria (4), thesatisfaction of the inequality constraints of type (18)and the satisfaction of the equality constraints of type(19). The membership function µ A , A ) iscomputed **as** follows:µ ( A , A ) = λ ⋅ µ ( A , A ) + λTijI+ λI*ϕi⋅ µ*ϕj*ψ( A , A )where the weight coefficientsiji⋅ µI*ψjT ( i j( A , Aij),(52)λ , λ * and *reflect the importance of the optimization criteriacompared to that of the inequality and the equalityconstraints. It is evident that:µ ( A , A ) µ ( A , A ) = 1 (53)sT ( i jTij+ T j iLet µ A , A ) be the corresponding strictly fuzzypreference realtion to µ A , A ) with theT ( i jmembership function [3]:sµ A , A ) = max{[ µ ( A , A ) − µ ( A , A )],0} (54)T ( i jT i j T j iThen the fuzzy subset of nondominated alternativesis described with a membership function **as** [4]:n.d.sµ ( A ) = 1 − max µ ( A , A ) (55)TiAj∈A3.5. Select the best alternative A best .The best alternative is the one that maximizes. d.µ n ( A ) [4], i.e.:TiTn.d.A best= arg max µ ( A )(56)Ai∈A4. Results from the solution of controllersynthesis problem.The results of performing steps 1 through 3 is that thebest alternative A best of controller structure isselected (in step 3) with the corresponding optimalvalues of controller parameters p A )(determined in step 2).Tijiψλ ϕ(opt best3236

4. Approach for selection of weightcoefficients.Here, the approach proposed in [5] is accepted toselect the weight coefficients. In [5] this method h**as**been applied to the problem of determination ofweight coefficients of the uncertain parameters whensolving optimization problems under parameteruncertainty. Here, it is shown that this method can besuccessfully **us**ed in the solution of controllersynthesis problem described in this paper. In section2 several weight coefficients were introduced:λ , k = 1,2,..., mλλIk*ψ k*ϕkλ , λI, k = 1,2,..., l, k = 1,2,..., l*ψ, λ*ϕ*1*2(57)As an example, the approach in [5] will be applied todetermine the weight coefficients λ , k = 1,2,... mI ,kof the optimality criteria (4). The following fuzzypreference relation is considered:Q = {( I j , I k ) : I j , I k ∈{I1,I2,...I m},µ ( I j , I k )} ,(58)where I k, k = 1,2,... , m are the optimality criteria(4) to be optimized. The membership functionµ I j , I )} can take only three values and is( kdetermined **as** follows:⎧1,if I j f Ik⎪µ ( Ij, Ik) = ajk= ⎨ 0, if Ijp Ik (59)⎪⎩ 0.5, if Ij↔ IkIn (59) the symbol “f” means that the optimalitycriterion I j is more important than criterion I k, thesymbol “p” means that I j is less important than Ikand the symbol ↔ is **us**ed to denote that I j and I kare equally important. Then the table with the valuesa of the membership function (59) is constructed.jkIt is accepted that:a jj = 1 , j = 1,2,... , m (60)In Table 1, the coefficients S j are computed **as**:mS j = ∑a jk −1(61)k=1Then, the weight coefficientsfollows:and they satisfy:∑k=1kλ are determined **as**I jSjλI=j m(62)Sm∑j=1λ = 1(63)I jThe other weight coefficients in (57) can bedetermined in a similar way.Table 1. Values of the membership function of the fuzzy preference relation (58).I k →I 1 I 2 … I k … I m S i λI j ↓I jI 1 a 11 a 12 … a 1k … a 1m S 1 λ I 1I 2 a 21 a 22 … a 2k … a 2m S 2 λ I 2... …. …I j a j1 a j2 … a jk … a jm S j λ I j... …. …I m a m1 a m2 … a mk … a mm S m λ I m......References[1]. S. Engell, R. Muller, “F**as**t and efficientselection of control structures”, Proceedings ofESCAPE-1, Elsinore, Denmark, 24-28 May,1992, pp.157-164.[2]. G. Sullivan, “Development of feed changeoverpolicies for refinery distillation units”, Ph.D.thesis, University of London, London, England,1977.[3]. I. Popchev, V. Peneva, “An algorithm forcomparison of fuzzy sets”, Fuzzy Sets and**System**s, vol.60, 1993, pp.59-65.[4]. S. Orlovski, “**Decision** making with a fuzzypreference relation”, Fuzzy Sets and **System**s,vol.1, 1978, pp.155-167.[5]. R. Pavlova, “Optimization of technologicalsystems under uncertainty”, Ph.D. thesis (InBulgarian), University of Chemical Technologyand Metallurgy, Sofia, Bulgaria, 1988.3237