Glushkov, V.M.

International Institute for Applied Systems Analysis (IIASA)

On Systems OptimizationGlushkov, V.M. IIASA Collaborative PaperMarch 1981Glushkov, V.M. (1981) On Systems Optimization. IIASA Collaborative Paper. Copyright © March 1981 by the author(s). http://pure.iiasa.ac.at/1793/ All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage. All copies must bear this notice and the full citation on the first page. For other purposes, to republish, to post on servers or to redistribute to lists, permission must be sought by contacting repository@iiasa.ac.at NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR ON SYSTEMS OPTIMIZATION V.M. Glushkov March 1981 CP-81- 10 Collaborative Papers r e p o r t  work which has  n o t  been performed s o l e l y  a t  t h e  I n t e r n a t i o n a l  I n s t i t u t e  f o r  Applied Systems Ana lys i s  and which has  r e c e i v e d  o n l y  l i m i t e d  review. V i e w s  or  op in ions  exp re s sed  h e r e i n  do n o t  n e c e s s a r i l y  r e p r e s e n t  t hose  o f  t h e  I n s t i t u t e ,  i t s  Nat iona l  Member Organ iza t i ons ,  o r . o t h e r  o rgan i -  z a t i o n s  s u p p o r t i n g  t h e  work. INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, A u s t r i a  This  paper was o r i g i n a l l y  prepared under t h e  t i t l e  "Modelling f o r  Management" f o r  p r e s e n t a t i o n  a t  a  Nate r  Research Cent re  (U.K. ) Conference on "River  P o l l u t i o n  Con t ro l " ,  Oxford, 9 - 1 1  A s r i l ,  1979. O n  systems optimization Institute of cybernetics Kiew USSR The modern theory and practice of optimization relies on the classical statement of optimization problems. It it known that the essence of such a statement consists of fhding the point ( or set of points ) p in the preassigned invariable feasible set P ,  where the given scalar objective function f (p) assumes its extreme value. l h s  statement is unsatisfactory for a vast number of economic planning problems and design-engineering problems in at least in two respects. For one thug, the objective function f (2) in such problems is the vector function rather than the scalar one. Furthermore, it proves to be practically irreducible to the scalar function by a particular a @mi procedure ( for instance weighting of various components of an initial vector function ). For another, the feasible set P can vary during the optimization process. What is more, its pu7posew change is the very meaning of the optimization process for a given class of problems. As the laws of admissible changes in the feasible set P are usually specified by a system of models, the described approach to the optimization problems is naturally referred to as the systems approach. It should be noted that under the systems approach the constraints, determining the feasible set in the space of speciflc parameters, generally change as a result of a sequence of solutions chosen from a discrete set of possible solutions. Consider,also, that the set itself is usually not comprehensively specifled when optimization begins and is completed during interaction between persons (planners and designers) not fully versed in formalized methods of making new decisions and optimization. Examine one of the formalized statements whch is characteristic of the systems optimization problem. To provide greater impact, let us employ its graphical representation and discuss the two-criteria case. Further assume, that the choice of values for these criteria determines uniquely the correspond- ing solution. In other words, the desired solution is sought for in the space K of optimiza- tion criteria designated by z , and z2 (see Figure). The solution starts with choosing some point A.  with coordinates a0 and b o  in the given space K , that is , the desired solution. Then the initial constraints F[') (zl,z2)20.. .   FJO)(Z l ,z 2~~~ specifying the initial teasibie set Po are con- structed. An immediate test shows whether the point A. belongs to the set Po or not. In principle, the normal (classical) procedure of optimization either by one of the c riteria zl or z2 or by their particular combination is applicable in the first cas ;e. However, the radically different method is usually employed in the systems approach. Namely, in accordance with the highest level model M ,  which controls: the choice of criteria, the point A.  is removed beyond the boundaries  of the  fern ib le  se t  Po  as is shown in the Figure. Ner :t, we isolate constraints not satisfied at  the point Ao. In the case under considel -ation, such are the constraints FA0) and ~ 1 ' ) .  Then models M a  and M4 that delfine these constraints are employed and particular solut ions  changing the con-espondent constraints in the w a n t e d  d irec t ion (if such a change hap- pens to be feasible) are tested in the dialog mode. Here the wanted direction is that which decreases the absolute value of negative discrepancies  F:(ao,b0) in the give n case FA0) ( a o , b o )  and FdO) ( a o , b o ) .  It s:hould be remembered that on numerous occasions the constraints Fi turn out; to be in t e rdependen t  ,so a change in one of them induces changes in a certain part of the other constraints. The so lu f ion  choice control exercised for changin, g the constraints is aimed at the minimization of a p e n a l t y  funct ion g O ( a o , b  ). The ~ u m  absolute  v a l u e  of nega t i ve  discrepancias &Fi(0)(ao, bo)  ( where A, are positive weighting coefficients ) is usually chosen as such a func- tion. I f .  the discrepancies do not exist, then by definition g o ( a o , b o )  = 0. In ,g ieneral, the control results in a number of solutions cRl,.. .,R, reducing the peni dty function value which after the m-th solution is denoted by g, (ao ,  bo ) .  Changin g the constraints of each of the adopted decisions is responsible for the adequat e change in the feasible set. The figure illustrates two such changes. The firs1 t one alters the constraints FAo) and FdO) and substitutes them by the constrai .nts F$') and ~ 4 ' )  respectively. The second affects only the constraint Fi0)  sub~stituting it by the constraint FA') The feasible set P2, resulting from the above t- no  changes, is bounded by the Lines F[O),Fd1) , F J 1 )  and ~ 1 ' )  whle the corresponding value of the penalty function equals gz(ao,bo)  # 0. Note, that it is impossible to choose the h t e  feasible set in advance, as the sequence of sets Po,P1,. , . .  may not be completely ordered with respect to the imbedding. More- over, the laboriousness of the construction of new constraints, when one would have to waste a great deal of effort in changing insignificant constraints, inter- feres with completion of this work in good time. If (as in our Figure) g2(ao,bo) = 0. , and there are no solutions which decrease the penalty function value turther, then we return to the hghest  model M which controls the choice of the desired problem solution A(a ,b) .  A succession of solutions D1,De, .... , will change the initial problem solution Ac(ao,bo) by step-by-step substitution of the latter by Al(a1,b I), Az(a2,bz). ~ ~ ( a s , b ~ ) ,  .... until the successive point 4 ( a k , b r )  falls within the feasible set (in Figure k=l). The solutions to change are chosen from the feasible set of solutions for the pur- pose of minimizing the penalty function. The described process is similar to the classic optimization process except that the steps are not chosen arbitrarily, but rather to match the admissible (with respect to the model M) solutions. At last, once the point 4 is in the Anal feasible set P, an  additional pro- cedure of optimization by some combinations of criteria z 1  and z 2  may be applied withn this feasible set. Such a procedure differs from the classic one by the fact that the choice of optimization steps is not arbitrary, but is controlled by the hlghest level model M. If some constraints, changeable in the desired direction, impede further improvement of the chosen criterion, then the optimi- zation process may be continued with the inclusion of successive solutions to . attain such changes. I t  should be emphasized that the unambiguous d e h t i o n  of the problem solution by choosing all values of the optimization criteria is not so infrequent as it may seem at  Arst sight. It is applied, for instance, to econometric planning problems, where the net output of diversified products is the (vector) criterion, while the gross output is the problem solution(Glushkovi'5a). In the absence of such unambiguousness the set where the solution is sought can have coordinates other that those corrseponding to the optimization criteria. Then the above optimization process becomes more complicated owing to the fact that points Ai(q,bi) are substituted by hyperplanes. The definition of the penalty function also acquires more sophistication: any distance from the chosen hyperplane to the next feasible set in space with specified compressions (extensions) along axes corresponding to the optimization criteria may serve as that. In the most general case, arbitrary punctiform sets can appear instead of the hyperplanes. Statements are possible by which criteria values are not deter- mined uniquely on those sets, and the corresponding weighting functions are .specified (by the highest level model M) on the sets to discriminate between more or less preferable solutions. However, the important systems optimization feature inherent in all the approaches is an interaction between models of different levels, aside from the multicriteria nature and changeability of the feasible set. In the case of economic planning problems, decisions are made by managers at different levels, and in the case of design-engineering problems, by designers handling various parts of the project. The described principles have found application in '?)isplan", one of the specific optimization systems, developed by the author. Glushkov75a. V.M. Glushkov, Macroeconomic models  a n d  principles undurlf ing design of d l -  Union d o m a t e d  s y s t e m ,  Statistika, Moscow (1975). 

On Systems Optimization

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