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An adaptive optimization system applied to machine synthesis
Mechanism and Machine Theory, 1973. Vol. 8. pp. 419-436. Pergamon Press. Printed in Great Britain
An Adaptive Optimization System Applied to Machine Synthesis* J. Golinski~
Received 23 April 1973 Abstract The paper presents the foundations of an optimization system for medium size computers. The system may be used to solve technical or economic problems with nonlinear objective functions and constraints. The system contains five algorithms of the deterministic and stochastic category. The master algorithm, based on works of Bush and Mosteller, randomly seeks one of these five. It performs computations for a certain time and examines its effectiveness. This experience is then taken as the basis for modification of the probability vector used in the choice of algorithm. The technical examples solved by different algorithms are presented. The results needed for definition of the initial vector illustrate difficulties met by an engineer in the proper choice of algorithm. These difficulties, which occur often in practical cases, prove the demand for further studies on optimization systems. 1. Introduction IT IS is often considered that computer tasks are properly fulfilled if computations are performed quicker and more precisely than in the case of traditional computation method application. However, the opinion that this is all a designer can achieve using new techniques seems to be wrong. It happens that a computer, radically changes the designer's traditional scheme and allows him to reach new qualitative results. This has forced researchers to look more attentively at the process of designing, and various tests have been carried out in process formulations. The repetition of various stages which can be differentiated in the course of designing is characteristic of this process. For instance, the analysis of the solution concept can influence the task specification which, therefore, may be extended or diminished. The construction process is characterized by some short, undoubtedly most important, creative moments, and relatively long, arduous periods of computing, drawing and documentation. It is evident that work mentioned in the second category is of a mechanical, routine character. This also weakens the designer's creative ability. The hierarchic structure of the design process is characteristic. It consists of a set of operations of various values. Various levels can be differentiated in it, and every level can be connected with a creative and a work element. On each level there appears an *Submission of this paper was invited by the Editor for inclusion in this special issue. -~Dr., Stoleczny Osrodek Elektronicznej Techniki Obliczeniowej, Przedsiebiorstwo Panstwowe, Warszawa, ul. Krolewska 27, Poland.
419
420
iteration process connecting the solution concept, the solution itself and the results. Changes on one level often influence radically other already sol~ed levels. The following features describe the design process: • division into stages, • iterational character, • characteristic, relatively short, creative moments, separated by long, time consuming analyses, • the sharing of the designer's intuition, the experience of whom is used in the case of a lack of knowledge of physics and mathematics, • multilevel process character. • knowledge of proper use of formulae, • graphic presentation of partial and final results. The question arises whether designs can be improved and systematized, their methodology be given, and both these elements be packed into a determined discipline on a scientific basis, using imprecise human intuition so as to obtain a mechanism design. An engineer who wants to use a c o m p u t e r is not interested in any design scheme. Only the very one is useful for him which allows the process algorithmization. Relatively numerous difficulties are connected with this. One of the most important and essential ones is the construction of a mathematical model for a real physical phenomenon or technical device. The description of a technical p h e n o m e n o n by mathematical formulae is often very complex. Moreover, a p h e n o m e n o n described by means of complicated differential or integral equations might be unsolvable by methods known so far. Thus, the model must be built so as not to stray a w a y f r o m reality. Simultaneously it should be simple enough to be solved by accessible methods and means. There must be a c o m p r o m i s e between mathematical exactness and practical possibilities. This is a very extensive theme and it has been treated in numerous m o n o g r a p h s [ I - 3 ] . The constructor's task is to elaborate the best construction on a certain theme. The extent to which this aim has been reached is the engineer's main concern, Generally, after having concluded the first stage of work, i.e. after the construction features are finally determined and fixed in executive documentation, e v e r y good constructor finds that many features of the constructed object can be improved. H o w e v e r , often before the work is finished, at the m o m e n t when there arise perspectives of essential possibilities, the constructor experiences m a n y doubts and difficulties connected with the choice of the final solution. A quite different procedure is possible consisting in computer application, usually described as programming. In a most general way it means seeking an optimal solution satisfying the required conditions. In such a case the programming is as follows: 1. There are n variables chosen, to which the constructor can give any values. These might be axle base in a linkage mechanism, transmission ratio, angles and so on. In such a way every machine or m e c h a n i s m can be imagined as a point in an n dimensional space. 2. All demands are formulated as m inequalities including the previously chosen variables. N u m b e r m can be arbitrarily large, but the equation number must be smaller than n. The set of a b o v e mentioned m conditions cuts out from the n dimensional space a
421
certain area every point of which satisfies all demands; thus, it corresponds to a certain admissible arrangement named the "admissible region". 3. A function of n variables is formed, called the "objective function", describing the property of the device which is essential for its evaluation. It can be, for example, efficiency. Then, out of all admissible machines the most efficient one is recognized as the best. If, for example, the objective function describes a gabarit, we generally tend to make it as small as possible. A designer who, due to modern technique development, is forced to look for optimal solutions, reaches still more frequently for optimization methods. But this provides numerous difficulties too. One of them is to choose an appropriate strategy for seeking an optimal solution. A great number of existing and used algorithms, the extensive literature in this field, and the access to various strategies in the form of ready made programs, place the constructor in a difficult situation as far as the strategy of seeking optimal solutions is concerned. These difficulties have called for a series of comparative investigations aiming at a general formula for an appropriate choice of algorithm.
2. Realization of Optimization Systems on Computers Programs for linear programming are to be found in every library[4]. Nonlinear programming is also becoming ever more important, and is to be found in a standard form in various program libraries [5, 6]. In the course of time, first of all, because it was not known how to appropriately classify various nonlinear programming systems which were developed, these comprised sets of various optimization algorithms. One such system is AID (Automated Improvement of Design), developed by General Electric [7], embracing quite a few nonlinear programming methods which can be applied to solve technical problems. In 1968 the SOETO system was developed for small digital computers. It consists of many independent programs, each serving to seek the maximum of the objective function. Each program was supplied with instructions on how to use the system. The system was foreseen for ZAM-2 computers with two drums and a SAKO translator recorded in a standard way. The system use consists in translating a sequence of "chapters" written by the user in SAKO together with a standard "chapter" adjoined at the end, an underlayed tape with data, and computation start. An example of a programmed solution with an automatic choice of algorithm is presented in the system described in [8]. During the computation process, this system chooses, from a group of optimization algorithms one which will allow finding the extremum, up to a determined probability, as fast as possible. This system has four operating algorithms: two gradient and two random. It has been realized on the IBM 7090/7094 computer. 3. Adaptation System of Nonlinear Optimization for Solving Problems in the Field of Machine Synthesis This is a system that allows one to solve optimization problems by means of a series of algorithms. The problem solution consists in finding the extremum of a function of many variables, being a mathematical formulation of a technical problem; a sequence of conditions evidently restricts the area of admissible decisions and the sequence of conditions, expressed by independent variable functions, are taken into account. The objective function does not need to satisfy classical conditions in such problems, such
422
as differentiability, continuity, convexity, unimodality and so on. The system chooses an algorithm at random, operates with it for a determined time, then evaluates its operation effect on an accepted scale of evaluation. The criterion of evaluation is uniform for all algorithms. The evaluation of an algorithm operation serves, in the course of a certain mathematical procedure, to change the preliminarily accepted probability vector of the algorithm choice during the system operation. The system "learns" dynamically. This, in the course of time, gives the chance to choose an algorithm that will ensure the best evaluation with least expenditure of work. The operation of a system solving technical problems finishes when termination conditions are satisfied. Generally, they are work expenditure evaluated in machine time operation units. The system achieves still greater "knowledge" as a consequence of solving various problems contained in the entire set as its disposal. The optimization system is an open system, i.e. newly developed strategies can be easily adjoined to it. A short description of the characteristics of each system algorithm is given below.
3.1 Stray process The stray process [9] is based on the Monte Carlo method and enables: (a) any exact checking of the whole region, depending of the number of samples performed, where points which the process goes through are placed compactly in the neighbourhood of the boundary and scattered inside the region. In this way the neighbourhood of the boundary is exactly checked and optimal solutions are often found there, (b) concentration (after a certain number of samples) on searching in the vicinity of the best point obtained in the stray process.
3.2 Algorithm searching for the extremum along the ridge of the objective function [ 10] Searching for an extremum is based on the assumption that the trend of variables improving the value of the objective function determines the direction along which a further improvement of results may be expected. This strategy is purposeful, in particular when the objective function traces out an approximately straight line. The procedure starts with a small step from an arbitrarily accepted start point. It increases the step size when successive steps produce improvement in the value of the objective function; in the opposite case, the step size is diminished. 3.3 Random-gradient algorithm* The optimization process proceeds step by step. Each of the steps consists in: (a) sampling points from a cube until a point within the permissible region is sampled, (b) searching by the gradient method with an accepted step on the basis of a gradient counted by finite differences with an increment to the edge of the region or to the maximum determined by another condition, (c) procedure repetition from 1 until the accepted number of results is reached (the best ones from the previous stage being kept), (d) ordering these results by a histogram method according to objective function value, *The conception of this algorithmwas proposed by Professor Dr. Jan Oderfeld.
423
(e) keeping a determined number of best teSU!tS Ufltii the next stage, and changing the sampling cube dimensions so as to make it embrace these points, (f) writing the results, (g) in the following stage t h e s a m e procedure is realized, point 2 being omitted. 3.4 Modified " S i m p l e x " algorithm Point X which starts the procedure and satisfies all restrictions is assumed to be known. The procedure uses k points, one of them is the starting point. The remaining k - l points form a set of points produced with the use of pseudorandom number generator. The point chosen at random must satisfy the restrictions. The objective function is evaluated at each vertex of the geometrical figure described by the points chosen. The point at which the value of the objective function is the least is replaced by another one, located on the straight line passing through this worst point and the centre of gravity of the figure formed by the remaining points. The so-found new point is a mirror reflection of the former one to the centre of gravity. If the latter point does not satisfy the restrictions the section is divided into halves between this point and the centre of gravity and conditions are checked anew until an admissible point is obtained. If the successively obtained values of the objective function are close to one another with a determined exactness, computations are stopped. So as to avoid finding the local extremum the procedure is repeated, starting from various points. 3.5 R o s e n b r o c k ' s method Rosenbrock's method [12] is a modification of Gauss-Seidl's method. The modification consists in a revolution of coordinate system after each iteration cycle in order to speed up the process of optimal point searching. Relative simplicity and good convergence are characteristic of Rosenbrock's method. Its most characteristic feature is the choice of an orthogonal set of vectors along which occur the changes of the searched parameters. The set of natural coordinates is accepted as the output set. The first iteration consists in searching for a minimum by changing in turn separate parameters. After some iteration the vector shows the approximate direction of the revolution. 4. The Model of Learning Many authors have dealt with mathematical aspects of adaptation processes. They have generally tried to adjust the mathematical description to the experiment they were carrying out [13]. Only a few papers treat this process in a general way, as for instance Bush and Mosteller's linear model [14]. They have introduced such notions as: choice, evaluation and a pair composed of choice and evaluation which they defined as an event. According to them the basic difference appearing in comparison with the notions of the statistic theory of decision consists in the event dependence on alternative choices. The feature that distinguishes Bush and Mosteller's model from others consists in its sequential character. Trial tendencies are characterized by the probability vector p (l) (where l is the successive experiment) describing these alternative choices. The authors describe the way in which the vector changes from trial to trial. They based on the following assumptions: 1. The change is given in operator form Tjkp(l) = p ( l - l), where the stochastic operator TiE depends on the event occuring in t e s t I. 2. Every operator T~k is linear.
424
The system is assumed to comprise algorithms A,, A_.... A,. Each algorithm solves the optimization problem. While solving the problem the system choses at random, according to a determined distribution, algorithms which in turn solve the problem and (a) each algorithm uses the results obtained by algorithms chosen at random previously, (b) the system chooses the algorithm on the basis of probability vector
p,(1)
p..(l)
P(l)=lp,(l)
• ~ pi(l)=l, j=t
where l is the number of randomization counted from the beginning of the system operation. Elements pi(l) of vector p(l) are the probabilities of the appropriate algorithm choice in the j-th sampling. Thus, it can be accepted, for example, that the elements of vector p(0) are equal, which means that pi(0)- 1
(J - 1, 2 . . . . .
r)
r
(c)the sampled algorithm solves the given problems within the time t. Value t determines the system depending on the number of decisive variables n, (d) after the end of the algorithm operation its effect is evaluated by the system, (e) on the basis of the evaluation which the algorithm has, the system points out a new probability vector p(l - 1). Generally speaking, a positive evaluation of a given algorithm increases the probability of its choice in the next sampling, but a negative evaluation diminishes it, simultaneously increasing the probability of the remaining algorithm choices. An algorithm with the biggest number of good evaluations will be successively used with greatest probability. Thus, the system learns to solve problems while solving them, in the sense of choosing the most effective out of the algorithms it possesses. After having solved a series of problems the system will give the priority for the successive problem to the algorithm that appeared most effective so far. A solution by means of only one algorithm, determined by its user, is admitted by the system as well. The system flow-diagram is given in Fig. 1.
Examples
4.1 The speed reducer The assumptions for the reducer shown in Fig. 2 are: transmitted power N = 100 km pinion speed n = 1500 rpm transmission ratio i = 1/3. The technical definition of the problem is given in [15]. The searched unknown parameters:
425 x~=b x _, =
m
x3=z x ~ = Ii
Xs = 12 X 6 = d~
X-/
=
dz
face width (cm) teeth module (cm) number of pinion teeth shaft length 1 (between bearings) (cm) shaft length 2 (between bearings) (cm) shaft diameter 1 (cm) shaft diameter 2 (cm)
IComblned [ olgorlthm
Was the time for I running given by the user ?
I Before calling the olgarifl~m Com~u?e ,*mem@Qn / of 7 firsfvolueS of the ObleCttvefunction ~TJ of 14 f,rst val~es cf the ODjectlvefunct+on(h~l of 21 f=rst v,31ues of the obJeCtiVefunchon(~21)~,}fter
Has the algorithm given more than 14 values ofF(x} ? Yes I NO' Operation of the algorithm
Iolgor=thm Igiven more
1
1 Sorting I Of results
Y?" N
I
IN?~N~ l
coining the ql~jorithm
Yes I
Yes
For trle pcir"algor/thrn, I IThe vector l es~imation" choose /-"lmoclif,cation the respective Pl = P l --TIR I operator T,, J I JIs the end criterion fulfilled
Yes I ?No
t F i g u r e 1.
]
426 ---l--
-!-
i/,
Z2
t_x I s
-
!q
--_
I
.)
i I i
Figure 2. The objective function which is subjected to the search for o p t i m u m is the total volume of gear wheels and transmission shafts and is to be minimized. The bending and clamp constraints, strength conditions of gear shafts, permissible magnitude of deflection etc. have been considered. The problem consists in finding the minimal value of function f ( x ) in a 7 dimensional space bounded by 13 described constraints and by 14 constraints imposed by ranges of dimensions. Therefore, we have to consider 7 variables and 28 constraints. Finally the problem has been defined in the standard f o r m of an optimizing equation. Objective function f ( x ) = 0.785x._"x~(3.33x32 + t4.93x3 - 49-09)
+ (x6" + x72)(9-52 - 1.5 Ix,) + 0.785(x6"-x, + xT:x~) with the constraints as listed below x,xz:x~ >! 27 x,x.'-x/- >i 397.5
x3> ~ 17 x,/x.. >! 5 x , / x : <~ 12 x,.x3 <~40 4
x,_x3x,, >- 1.925 x,.x3xT" >i 1.925
A'<~ 1100
B,
where A , = ~ \ x._x, / + 16"9x 106
B, = 0'lx~ 3 A_.
B._
<~ 850
,/(745xfl~'-157.5×106 +
A.. = ~ \ ~ / B._ = 0. I x73 x7~>5
427
1"5x6 + 1"9 ~ x~ l'IxT+ 1"9 ~ xs. According to the instructions for system use, the constraints listedabove have to be written in F O R T R A N ; this is the only work to be done by the system user. The described example became a very interesting subject of comprehensive experiments. From the mathematical point of view, the problem is characterized by flatness of the objective function in the neighbourhood of the extremum point. Similar results have been received for different values of variables. A significant number of local extremes has been noted; moreover the relations between the area of permissible solutions and the entire area searched have not been very convenient. For instance, in a trial of 1241sec duration performed by the crude Monte Carlo method, executed on the IBM 360/67 computer, the total number of random searchings reached 825,000; however, only 67 of them were contained in the permissible area. The ranges of particular parameters for the mentioned area have been settled as follows:
2.6~< x, = b ~<3.6 0.7 ~< x., = m ~<0.8 17<~x3 = z ~<28 7.3 ~< x, = It ~< 8.3 7.3 ~
0-6
26
8-1
8-2
3-4
5.3)
refers to a value of the objective function f(x) equal to 3643.63 cm 3. This point is situated near the boundary of the region limited by constraints of shaft bending and the constraint relation between b and m values; it has been recognized as the local extremum. Among other experiments, one example performed on the same IBM computer by the stray process has brought the solution (3.0
0.6
26
7.8
8.0
3.3
5.3)
situated almost on the edge of the area described by constraints of bending of gear teeth, constraints of bending of both shafts and constraint of b/m ratio as well. For the last solution: permissible bending stress of gear teeth, 900 kg/cm-" actual bending stress value of gear teeth, 892-73 kg/cm" permissible bending stress of shaft, 1.100 kg/cm-" actual bending stress for shaft I, 1-095 kg/cm" permissible bending stress for shaft 2, 850 kg/cm 2 actual bending stress value for shaft 2, 849-96 kg/cm'constraint of b/m ratio, 5 real value of b/m ratio, 5-16.
428 This a n d o t h e r e x a m p l e s p r o v e the v e r y s u b s t a n t i a l s t a t e m e n t , a n n o u n c e d at the b e g i n n i n g of this p a p e r , that an e n g i n e e r l o o k i n g f o r the o p t i m a l s o l u t i o n o f his d e s i g n c a n n o t define the m o s t s u i t a b l e m e t h o d for finding the s o l u t i o n of a p a r t i c u l a r p r o b l e m . T h e s e difficulties g r o w with the n u m b e r of p a r a m e t e r s a n d c o n s t r a i n t s . T h e y a l s o dep e n d on the c a p a c i t y of t h e c o m p u t e r at his d i s p o s a l . T h e r e are m a n y o t h e r difficulties of c o u r s e . U s e d a l g o r i t h m s h a v e s o m e d e g r e e of f r e e d o m a n d c h o i c e of t h e m has i m p l i c a t i o n s on the s e a r c h p r o c e s s . P a r t i c u l a r a l g o r i t h m s m a y b e c o m e m o r e or less efficient a n d this d e p e n d s u p o n the c h o i c e o f p a r a m e t e r s as s h o w n b y the d e s c r i p t i o n of the s t r a y p r o c e s s a l g o r i t h m [ 9 ] in the last e x a m p l e . T h e c o m p a r i s o n of r e s u l t s r e c e i v e d b y m e a n s of d i f f e r e n t a l g o r i t h m s ( r a n d o m - g r a d i e n t , ridge a n d c o m p l e x ) is g i v e n in T a b l e 1. T h e b e s t r e s u l t , r e a c h e d a f t e r
Table 1. Best results for the speed reducer Computer time 5 min f(x)
x~
x~
x~
Computer ZAM-41 x~
x,
x~
x=
3.451 3.422 3-522 3-407 3-568 3-484 3.660
5.750 5.649 5.491 5.438 5.562 5.450 5.579
Combined gradientstochastic searching technique 2261.429 2328.785 2348.393 2352.666 2353.491 2353.646 2353.898
3.135 3.242 3-213 3.488 3.325 2.247 3.087
0.568 0.552 0.574 0-536 0.546 0.555 0-581
26.651 27.523 26.693 27.645 27.677 27.577 26-873
8-288 8.617 7.356 8.229 8-125 7-255 8.509
8.745 8.513 8-682 8-306 8-727 8.568 8.216
The method of seeking optimum along the ridge of the objective function 2283.949 3.362 7-969 3-364 5.304 0.539 27.591 8.240 2287.237 3.361 0.539 27-571 7.743 8-162 3.462 5-592 2298.674 3.446 0.551 26.643 7.056 8.443 3.347 5.389 2330.104 3.169 8.592 3.575 5-440 0.553 27.784 8.410 2386.536 3.030 7.789 3.416 5.326 0.555 28.914 7.226 2395.835 3.559 0.527 28.184 7.748 8.146 3.348 5.414 2421.614 3.149 0.629 24.824 8.158 8.435 3.585 5.759 2250.891 2319.330 2386.706 2457.073 2795.307
3-101 3.328 3.392 3-625 3.543
0.570 0.543 0.543 0-529 0.625
The complex method 26.777 7-199 27.652 8-042 27.933 7-470 28.162 7.785 25.628 7-896
8.583 8-215 8.157 8.400 8.134
3.482 3.482 3.503 3.500 3.754
5.529 5.724 5.503 5'521 5.630
5 min of c o m p u t a t i o n on a Z A M - 4 1 P o l i s h c o m p u t e r has an o p t i m u m v a l u e e q u a l to 2250-891. T h i s c o m p u t a t i o n has b e e n p e r f o r m e d b y a c o m p l e x a l g o r i t h m . T h e d e t e r m i n e d p o i n t is d e f i n e d b y the v e c t o r : 3.101 0.57 6.777 7.199 8.583 3.482 5.529
429
The results of computation with the Systeiaa are given in Table 2. The reaching of an objective function value not greater than 2-200 cm 3 has been taken as the criterion permitting the end of computation. First the stray algorithm has been employed, leading to the point 4.02 I 0.45 I 33.19 1 8.051 8.141 3.691 5.621 The value of the objective function for this point is 2674.25. As the number of points situated in permissible area had reached 18 in 1 min, the algorithm has been classified as good. The vector of probability serving for the choice of the foregoing algorithm has been transformed as follows: 0-,02 [0.5571 0-306 [0"153 0"268 --*/0-134/
I
0.324
10.162]
0.000
L0.000J
The next choice led to the gradient algorithm, but the most satisfactory result was not improved. The average value based on the 21 most promising results reached 3362.1, i.e. more than the mean value based on the seven best results of the previous example (3318.9), but less than the mean value of the fourteen most favourable results received previously. Then the estimation for the considered algorithm is formulated as satisfactory. Therefore, the vector of algorithm choice changes to:
0.153 |0.139 0-134 --*|0.137 0.162 |0-144 0-000 U0.063 Subsequently the ridge algorithm was chosen. The best one of the results reached was improved to 2102-18 which corresponded to the point described by the vector: 3-03 0.61 24-63 6.93 7-72 3-35 6.29 As the criterion imposed by the user had been satisfied (2102.18 less than 2200), the action of the system was stopped. It has to be emphasized that the ridge algorithm, besides producing the most satisfactory result, was not recognized as sufficient, since the
430
Table 2. Processing time for individual algorithm--I rain
The best result after operation of the algorithm
,Algorithm Random process Combined gradient stochastic Ridge
fix)
x~
x._
x~
x~
x~
x~
Operatzon After operation
_6.4._~
4.02
0-45
33.18
8"05
8.14
3"69
I Operation After operation
-~ -)¢, _674.._
4.02
0.45
33"18
8.05
8.14
Operation fter operation
_ 10." 18
3.03
0-61
24.63
6-93
7.72
The 2Ist result after x,
fix)
.,,
x:
5"62 4721.36
5.34
0.57
3-69
5.62
3871.24
3-65
0-56
3'35
_ -_9
3829.85
3"70
0.65
mean value of 21 best results was equal to 3270.7 cm; and worse than the mean one of 14 best results of the preceding computation (3100.9cm3). The vector of probability used in the choice of the algorithm has been transformed to 0-139 |
0.317 | 0-144 0.063 j
F° 94 |0.195
| 0.284 /°°72 L°156
The described system has proven its superiority over independent algorithms. The total duration of computation has been shorter than the computation of particular algorithms and more favourable results have been received (Table 3). T a b l e 3. Optimization system Algorithm
Combined
Ridge
Complex
SOPT
The best result The 7th result The 21st result Computer time
2261.429 2353.898
2283.949 2421.614
2250-891 2795.307
2102.18
5 rain
5 min
5 rain
3829.85
5 rain
4.2 The planetary gear reducer The application of the optimization system is illustrated by the example of an aero-engine reducer (Fig. 3). The parameters of the considered reducer are: power of free turbine, 1200 km requested rotational speed of propeller, 1500 rpm rotational speed of disc rotor driving propeller, 18,000 rpm. The design of the reducer embraces calculations for all gear wheels, i.e. control computation for mostly loaded teeth, and considers: bending stress, tangent stress on
431
The mean of results
operation of the algorithm x3
x4
xs
x6
x~
30-59
8.30
8.32
4.06
5-42
z7
z,~
Estima- The probability of choosing the algorithm tion Rand. Comp. Comb. Ridge Ros.
z.-,
9.24
8-62
3-77
5-50
3362" I
8-75
8.51
3.90
0-551
0.153
0-338
0.294
0.268
0.324
0-000
0-134
0.162
0.000
0-139
0.317
0.144
0-063
0"195
0.284
0.072
0"156
Satis.
2822"2 3100.9
28'22
0.306
Good 3318.9 3687.1
33-57
0.102
5.70
3270.7
Unsat.
the surface c l a m p f o r c o n t a c t points, and t e m p e r a t u r e conditions at the initial point of gear t o o t h interaction. T h e s e a r c h e d p a r a m e t e r s f o r the planetary gear r e d u c e r are: b=x~ m = x2 z = x3 k = x,
face width normal module the n u m b e r o f teeth in the sun gear the n u m b e r of planet gears.
T h e constraints i m p o s e d on the c o n s i d e r e d c o n s t r u c t i o n h a v e been described in [16]. T h e description o f t h e m can be given as follows:
1. Bending stress condition 2. C l a m p surface stress condition 3. H e a t condition 4. E x t r e m a l n u m b e r o f teeth 5. R e l e v a n t width o f gear wheel rim 6. R e l e v a n t width o f gear w h e e l rim 7. Minimal n u m b e r of satelites
q~7(x)= k - 2 t> 0
8. A d j a c e n c y
~os(x) =
9. 10. 11. 12.
Assembling Overall d i m e n s i o n s Width of rim wheel M o d u l e o f single t o o t h
The o b j e c t i v e f u n c t i o n c h o s e n was f(x)=-~
bm
+1
.
qh(x) = 1.16m'-zbk - ~ -
2-2 >~ 0
q~,.(x) = O.Olbkrn'-z" - ~v/--~mz- 2.2 I> 0 q~3(x) = 0.035b. m 2 z 2 - 1 >10 q~,(x) = z - 28 ~> 0 ~s(x) = b - 3m ~> 0 q~6(x) = 15m - b / > 0
~og(x) = ~O,o(X) = qh,(x) = ~0,2(x) =
3 sin ~- - 2 - 2/> 0 MOD(z, 2) = 0 3 7 - 2 z m >10 b 1>0 m i>0
432 Z 3 7-
t
-7 i
i Zz
z~ --
,
1}--f
7_ I f ..~4
.- 5
Figure 3. The function as given above expresses the total volume of the reducer assuming its solid construction. Such a criterion seems logical, as the total weight is a crucial factor in aircraft construction elements. However, in practice, the wheels of a gear reducer are not massive. An experienced engineer may eliminate smartly all needless material. Moreover, a simple proportion could be nearly assumed between the overall weight of solid elements and framed ones (hub thickness, ribbing, gauging volume may be approximately referred as functions of gear wheel width or of the module pitch). In the considered problem the variable k (describing the number of satelites) and the number of teeth in the planetary gear may only be integer. The values of module pitch are taken according to Polish standards as 2, 2.5, 3, 3.5 . . . . . mm. The problem defined above had to be translated to FORTRAN. Subsequently, the subroutine prepared by the system user was introduced to the computer and processed by the optimizing system. The results received by system action are given in Tables 4 and 5. Table 4 illustrates computations performed with the use of 3 different algorithms. Each of them has worked 1 min. The initial probability for algorithm choice has been accepted as equal to 0.2, i.e. giving an equal chance to each of 5 considered algorithms. As the first, the stray process algorithm was employed, producing 21 points in the permissible area and achieving a good estimation. Therefore the considered probability was changed significantly. As the second, the gradient algorithm was picked and, according to the method of estimation described above, it achieved a good classification. The same estimation was given to the last acting Rosenbrock's algorithm. The final result received after 3 rain is equal to 562-30 and the parameters of the associated extremum point are: (1.55, 0.4, 34, 3). The twenty-first of achieved results has the objective function value 702-72 cm 3 and parameters (1.73, 0.4, 36, 3). The above described example has been repeated with the same value of initial probability vector: p(0) = (0.2, 0.2, 0.2, 0.2, 0-2). Subsequently working algorithms (Table 5) are: stray, Rosenbrock, gradient, ridge, stray, Rosenbrock and gradient again. The whole trial has taken 7 min, and the best one of the results refers to the point (2.15, 0.3, 36, 3), and the objective function value 561.06. The twenty-first of results appears very near to the optimum, giving the point (1.25, 0-4, 38, 3), and the objective function value of 565.33 cm s. The point I 0"3 /
36 )
._3
<
Operation '~ Afler I operation
Operalion ~After Loperation
Operation ~After Loperidion
Random
Combined
Rosenbrock
Algorithm
Processing time for individual algorithm--I nlin
Table 4.
o0 Z 9
562.30
571.54
702.50
I(x)
1-55
2.41
1.72
x,
0.4
0.25
0.4
x~
34
44
36
x,
The best resuh after operation of the algorilhm
3
3
3
x~
1.73
1.73
705-34
702.12
1.73
x,
1102.00
l(x)
0-4
0.4
0.5
x,.
36
36
36
x~
3
3
3
x,
The 21st result after operation of the algorithm
z,,
657-34 680.02
702.67 702-74
z,
643.88
687.74
z2,
The mean t)f results
Good
Good
Good
Estimarion
0-150
0-300
0-600
0-200
R;mdonl
0.1125
0-050
0-I00
0.200
i1.275
0.550
0. I00
0.200
Comp. Comh.
0.t125
0.050
O-I(X)
0.200
Ridge
(1-525
0-O.Y~t)
0.100'
0.200
l,~o,lellbrock
The probability of choosing algorithm
CO CO
'Operation '~ After [operation
Operalion '~ After . (operation
Operation ~Afler . [ operalmn
Operation ~After . [ operalU)l)
Rosenbrock
Combined
Ridge
Randonl
Rosenbruck
Combined
Operation '[After Loperation
Random
'~ After I operation
Operalion
Operation '~ After . I operatJon
Algorithm
Processing time for individual algorithm-- I rain
Table 5.
561 "(16
561.06
561-06
561.06
561.27
561.27
716"33
f(X)
2"45
2-45
2.45
2-45
I "32
1.32
2-81
X~
0.30
0-30
0.30
0-30
0.35
0"35
0"3
Xz
36
36
36
36
42
42
38
X~
The best result after operation of the algorithm
3
3
3
3
3
3
3
X4
565.33
565.93
642.50
642,50
716.33
970-99
1942-36
]'(x)
1.25
1.65
2-81
2.81
2.81
2-80
2.74
xj
0.40
0.30
0-30
0-30
0.30
0"35
0.5
x2
38
44
36
36
38
38
38
x3
The 2 Is( result after operation of the algorithm
3
3
3
3
3
3
3
x4
~61-43 562.36
561.87 563.23
561-87 563,23
577.74 618.%
624.45 670.39
716.33 752.74
Z14
563.07
563-32
581-86
581.86
651-42
720-30
Z.~ I
The mean of results
Unsal.
Unsat.
Unsat.
Saris.
Sa(is.
Saris.
Good
Estima(ion
I).211
0.171
0-092
0.184
0.244
0.363
0-600
0,200
0.233
0.216
0.308
0"I13
O'ItR)
0280
0.309
0.119
0.113
O-I(X)
0.234
11.121 11.257
11.21~ 11"242 0.3t, S
0.186
I).122
0-119
0.113
0"100
0.21X1 11-200 0.2011
I).177
O. It)5
0.209
0"169
0.213
0.31Rl
0-1011
0.2())
The probabilily of choosing alg. Random Ct)lllp. Comb. Ridge Ros,
4~
435 referring to the value of objective the f u n c t i o n equal to 561-06 is the sought e x t r e m u m , situated on the b o u n d a r y of the region due to the heating constraint.
Summary T h e target o f the given paper is the illustration of a n e w c o n c e p t in the solution of optimization problems. T h e present stage w h e r e users have the a c c e s s to m a n y different c o m p u t i n g strategies causes a lot of trouble. T h e a b s e n c e of classification of algorithms and p r o b l e m s m a k e s m a n y difficulties f o r an engineer, w h o is not able to define an algorithm m o s t suitable to the particular design. This is the reason f o r formulation of self-acting model, dispensing the s y s t e m user f r o m the i n c o n v e n i e n t choice o f an algorithm. T h e p r e p a r e d s y s t e m itself includes different algorithms acting in turn a c c o r d i n g to the d e m a n d s of the p r o b l e m solution. T h e p e r f o r m e d e x p e r i m e n t s p r o v e the rightness of such b e h a v i o u r and argue its superiority in c o m p a r i s o n with the use o f i n d e p e n d e n t algorithms, even the same ones included in the d e s c r i b e d system, but treated separately. The s y s t e m f o r m u l a t e d by the a u t h o r as d e s c r i b e d a b o v e , has been realized on a ZAM-41 Polish c o m p u t e r and applied to practical use. T h e succesful use o f this s y s t e m has interested users of o t h e r c o m p u t e r s . T h e r e f o r e t w o similar s y s t e m s for O D R A 1304 and K-202 Polish c o m p u t e r s are in the final stages of the preparation. Simultaneously w o r k s have been u n d e r t a k e n to prepare the optimization s y s t e m for c o m p u t e r s of the u n i f o r m R I A D system.
References [I] PEYNTER H. L., Analysis and Design o[ Engineering Systems. The MIT Press (1960-1961). [2] SLAYMAKER R. R., Mechanical Design and Analysis. Wiley, New York (1959). [3] VOLMER J., Design of mechanism: synthesis by iterative analysis, Transactions ASME. Paper No. 68MECH-69 (1968). [4] GASS S. I., Programowamie Liniowe. Methdy i Zastosowanie (Linear Programming--Methods and Applications.) PWN Warszawa (1963). [5] GARTH P., MCCORMICK W. Ch., MYLANDER III and FIACCO A., IBM Catalogue of Programs: Program No 70-40-H2-3189 Sumtrac (Sumpt). [6] MUGELE R. A., A nonlinear digital optimizing program for process control systems. Proceedings for the Spring Joint Computer Conference, San Francisco, I May (1%2). [7] CASEY J. K. and RUSTAY R. C., AID--a general purpose computer program for optimization, Recent Advances in Optimization Techniques. Wiley, New York (1966). [8] FLOOD M. M. and LEON A., Adaptive code for optimization GROPE, Recent Advances in Optimization Tc.chniques. Wiley, New York (1966). [9] GOLINSKI J., O badaniu pewnego procesu bl~dzenia zastosowanego do optymalnej syntezy maszyn (Stray process research applied to optimal synthesis of mechanisms.), Arch. Bud. Maszyn XV, 2 (1968). [10] WILDE D. J., Optimum Seeking Methods. Prentice-Hall, Englewood Cliffs (1964). [ II] BOX M. J., A new method of constrained optimization and a comparison with other methods, Computer Jl 8, 42-52 0965). [12] ROSENBROCK H. H,, An automatic method for finding the greatest or the least value of a function, Computer Jl 3,(2) 0960). [13] BUSH R. R. and MOSTELLER F., Stochastic Models for Learning. Wiley, New York (1955). [14] THRALL R. M., COOMBS Ch. and DAVIS R. L., Decision Processes. Wiley, New York (1954). [15] GOLIlqSKI J., Optimal synthesis problems solved by means of nonlinear programming and random methods, J. Mechanisms S, 287-309 (1970). [16] GOLIIqSKI J., O optymalnej syntezie maszyn metodami Monte Carlo. (Optimal synthesis of mechanisms by Monte Carlo methods.), Arch. Bud. Maszyn 11, 3 (1965).
Anpassungsfahiges Lernsystern angewendet for Optimierungsaufgaben J. Golinski
Kurzfassung--Das Ziel des Adaptationssystems besteht darin, aus den vorhandenen Optimierungsmethoden for eine technische Optimierungsaufgabe
436 die effektivste Methode vom Rechner automatisch ausw~hlen zu lassen. Da for die Vielzahl technischer Optimierungsaufgaben die Form der Zielfunktionen und die form der Restrikstionen nicht bekannt sind, berOcksichtigt das Adaptationssystem diese Einfl0sse nicht. Es werden die aus der Warhscheinlichkeitstheorie bekannten Methoden zur Bestimmung eines Wahrscheinlichkeitsvektors benutzt, der das Alternativauswahlverfahren beschreibt. Die Theorie wurde durch Bush und Mosteller beschreibt. Das Adaptationssystem enth< f/3nf Optiemierungs-methoden die zur L6sung von technischen Optiemierungsaufgaben mit maximal N=20 Ver~nderlichen benutzt werden. Es kann bewiesen werden das der Wahrscheinlichkeitsvektor sich durch den Operator T,k transformieren I~sst. Es gilt: p(/+1)= Tjk. P(0. Das Adaptationssystem beginnt seine Arbeit mit einem Wahrscheinlichkeitsvektor p(0). Diese Verteilung p(0) wurde aus der Bewertung von 10 Testaufgaben die durch jeden der fenf Optiemierungsalgorithmen gel6st wurden. In der laufenden Arbeit mit dem Adaptationssystem wird deshalb die Bewertung der Algorithmen fQr jede berechnete Aufgabe durchgefLihrt und auf Grund dieser Statistik wird die Art der Wahrscheinlichkeiten ge&ndert. Der somit erhaltene neue Wahrscheinlichkeits vektor spiegelt die Effektivit&t der Algorithmen besser wider. Wie erw&hnt befinden sich im System f6nf verschiedene Algorithmen: --die Wellenberg-Methode nach Hooke und Jeves --die Raffinierte Monte Carlo Methode --die Kombination der Monte Carlo Methode mit einer Gradientenmethode/nach J. Oderfeld/ --die complex methode nach Box w d i e Rosenbrock'sche Methode. Mit diesem Adaptationssystem hat man zahlreiche technische Beispiele gel6st. In der Arbeit sind 2 Beispiele-Zahnradgetriebe und Planetengetriebe beschreibt und gel6st.
An Adaptive Optimization System Applied to Machine Synthesis* J. Golinski~
Received 23 April 1973 Abstract The paper presents the foundations of an optimization system for medium size computers. The system may be used to solve technical or economic problems with nonlinear objective functions and constraints. The system contains five algorithms of the deterministic and stochastic category. The master algorithm, based on works of Bush and Mosteller, randomly seeks one of these five. It performs computations for a certain time and examines its effectiveness. This experience is then taken as the basis for modification of the probability vector used in the choice of algorithm. The technical examples solved by different algorithms are presented. The results needed for definition of the initial vector illustrate difficulties met by an engineer in the proper choice of algorithm. These difficulties, which occur often in practical cases, prove the demand for further studies on optimization systems. 1. Introduction IT IS is often considered that computer tasks are properly fulfilled if computations are performed quicker and more precisely than in the case of traditional computation method application. However, the opinion that this is all a designer can achieve using new techniques seems to be wrong. It happens that a computer, radically changes the designer's traditional scheme and allows him to reach new qualitative results. This has forced researchers to look more attentively at the process of designing, and various tests have been carried out in process formulations. The repetition of various stages which can be differentiated in the course of designing is characteristic of this process. For instance, the analysis of the solution concept can influence the task specification which, therefore, may be extended or diminished. The construction process is characterized by some short, undoubtedly most important, creative moments, and relatively long, arduous periods of computing, drawing and documentation. It is evident that work mentioned in the second category is of a mechanical, routine character. This also weakens the designer's creative ability. The hierarchic structure of the design process is characteristic. It consists of a set of operations of various values. Various levels can be differentiated in it, and every level can be connected with a creative and a work element. On each level there appears an *Submission of this paper was invited by the Editor for inclusion in this special issue. -~Dr., Stoleczny Osrodek Elektronicznej Techniki Obliczeniowej, Przedsiebiorstwo Panstwowe, Warszawa, ul. Krolewska 27, Poland.
419
420
iteration process connecting the solution concept, the solution itself and the results. Changes on one level often influence radically other already sol~ed levels. The following features describe the design process: • division into stages, • iterational character, • characteristic, relatively short, creative moments, separated by long, time consuming analyses, • the sharing of the designer's intuition, the experience of whom is used in the case of a lack of knowledge of physics and mathematics, • multilevel process character. • knowledge of proper use of formulae, • graphic presentation of partial and final results. The question arises whether designs can be improved and systematized, their methodology be given, and both these elements be packed into a determined discipline on a scientific basis, using imprecise human intuition so as to obtain a mechanism design. An engineer who wants to use a c o m p u t e r is not interested in any design scheme. Only the very one is useful for him which allows the process algorithmization. Relatively numerous difficulties are connected with this. One of the most important and essential ones is the construction of a mathematical model for a real physical phenomenon or technical device. The description of a technical p h e n o m e n o n by mathematical formulae is often very complex. Moreover, a p h e n o m e n o n described by means of complicated differential or integral equations might be unsolvable by methods known so far. Thus, the model must be built so as not to stray a w a y f r o m reality. Simultaneously it should be simple enough to be solved by accessible methods and means. There must be a c o m p r o m i s e between mathematical exactness and practical possibilities. This is a very extensive theme and it has been treated in numerous m o n o g r a p h s [ I - 3 ] . The constructor's task is to elaborate the best construction on a certain theme. The extent to which this aim has been reached is the engineer's main concern, Generally, after having concluded the first stage of work, i.e. after the construction features are finally determined and fixed in executive documentation, e v e r y good constructor finds that many features of the constructed object can be improved. H o w e v e r , often before the work is finished, at the m o m e n t when there arise perspectives of essential possibilities, the constructor experiences m a n y doubts and difficulties connected with the choice of the final solution. A quite different procedure is possible consisting in computer application, usually described as programming. In a most general way it means seeking an optimal solution satisfying the required conditions. In such a case the programming is as follows: 1. There are n variables chosen, to which the constructor can give any values. These might be axle base in a linkage mechanism, transmission ratio, angles and so on. In such a way every machine or m e c h a n i s m can be imagined as a point in an n dimensional space. 2. All demands are formulated as m inequalities including the previously chosen variables. N u m b e r m can be arbitrarily large, but the equation number must be smaller than n. The set of a b o v e mentioned m conditions cuts out from the n dimensional space a
421
certain area every point of which satisfies all demands; thus, it corresponds to a certain admissible arrangement named the "admissible region". 3. A function of n variables is formed, called the "objective function", describing the property of the device which is essential for its evaluation. It can be, for example, efficiency. Then, out of all admissible machines the most efficient one is recognized as the best. If, for example, the objective function describes a gabarit, we generally tend to make it as small as possible. A designer who, due to modern technique development, is forced to look for optimal solutions, reaches still more frequently for optimization methods. But this provides numerous difficulties too. One of them is to choose an appropriate strategy for seeking an optimal solution. A great number of existing and used algorithms, the extensive literature in this field, and the access to various strategies in the form of ready made programs, place the constructor in a difficult situation as far as the strategy of seeking optimal solutions is concerned. These difficulties have called for a series of comparative investigations aiming at a general formula for an appropriate choice of algorithm.
2. Realization of Optimization Systems on Computers Programs for linear programming are to be found in every library[4]. Nonlinear programming is also becoming ever more important, and is to be found in a standard form in various program libraries [5, 6]. In the course of time, first of all, because it was not known how to appropriately classify various nonlinear programming systems which were developed, these comprised sets of various optimization algorithms. One such system is AID (Automated Improvement of Design), developed by General Electric [7], embracing quite a few nonlinear programming methods which can be applied to solve technical problems. In 1968 the SOETO system was developed for small digital computers. It consists of many independent programs, each serving to seek the maximum of the objective function. Each program was supplied with instructions on how to use the system. The system was foreseen for ZAM-2 computers with two drums and a SAKO translator recorded in a standard way. The system use consists in translating a sequence of "chapters" written by the user in SAKO together with a standard "chapter" adjoined at the end, an underlayed tape with data, and computation start. An example of a programmed solution with an automatic choice of algorithm is presented in the system described in [8]. During the computation process, this system chooses, from a group of optimization algorithms one which will allow finding the extremum, up to a determined probability, as fast as possible. This system has four operating algorithms: two gradient and two random. It has been realized on the IBM 7090/7094 computer. 3. Adaptation System of Nonlinear Optimization for Solving Problems in the Field of Machine Synthesis This is a system that allows one to solve optimization problems by means of a series of algorithms. The problem solution consists in finding the extremum of a function of many variables, being a mathematical formulation of a technical problem; a sequence of conditions evidently restricts the area of admissible decisions and the sequence of conditions, expressed by independent variable functions, are taken into account. The objective function does not need to satisfy classical conditions in such problems, such
422
as differentiability, continuity, convexity, unimodality and so on. The system chooses an algorithm at random, operates with it for a determined time, then evaluates its operation effect on an accepted scale of evaluation. The criterion of evaluation is uniform for all algorithms. The evaluation of an algorithm operation serves, in the course of a certain mathematical procedure, to change the preliminarily accepted probability vector of the algorithm choice during the system operation. The system "learns" dynamically. This, in the course of time, gives the chance to choose an algorithm that will ensure the best evaluation with least expenditure of work. The operation of a system solving technical problems finishes when termination conditions are satisfied. Generally, they are work expenditure evaluated in machine time operation units. The system achieves still greater "knowledge" as a consequence of solving various problems contained in the entire set as its disposal. The optimization system is an open system, i.e. newly developed strategies can be easily adjoined to it. A short description of the characteristics of each system algorithm is given below.
3.1 Stray process The stray process [9] is based on the Monte Carlo method and enables: (a) any exact checking of the whole region, depending of the number of samples performed, where points which the process goes through are placed compactly in the neighbourhood of the boundary and scattered inside the region. In this way the neighbourhood of the boundary is exactly checked and optimal solutions are often found there, (b) concentration (after a certain number of samples) on searching in the vicinity of the best point obtained in the stray process.
3.2 Algorithm searching for the extremum along the ridge of the objective function [ 10] Searching for an extremum is based on the assumption that the trend of variables improving the value of the objective function determines the direction along which a further improvement of results may be expected. This strategy is purposeful, in particular when the objective function traces out an approximately straight line. The procedure starts with a small step from an arbitrarily accepted start point. It increases the step size when successive steps produce improvement in the value of the objective function; in the opposite case, the step size is diminished. 3.3 Random-gradient algorithm* The optimization process proceeds step by step. Each of the steps consists in: (a) sampling points from a cube until a point within the permissible region is sampled, (b) searching by the gradient method with an accepted step on the basis of a gradient counted by finite differences with an increment to the edge of the region or to the maximum determined by another condition, (c) procedure repetition from 1 until the accepted number of results is reached (the best ones from the previous stage being kept), (d) ordering these results by a histogram method according to objective function value, *The conception of this algorithmwas proposed by Professor Dr. Jan Oderfeld.
423
(e) keeping a determined number of best teSU!tS Ufltii the next stage, and changing the sampling cube dimensions so as to make it embrace these points, (f) writing the results, (g) in the following stage t h e s a m e procedure is realized, point 2 being omitted. 3.4 Modified " S i m p l e x " algorithm Point X which starts the procedure and satisfies all restrictions is assumed to be known. The procedure uses k points, one of them is the starting point. The remaining k - l points form a set of points produced with the use of pseudorandom number generator. The point chosen at random must satisfy the restrictions. The objective function is evaluated at each vertex of the geometrical figure described by the points chosen. The point at which the value of the objective function is the least is replaced by another one, located on the straight line passing through this worst point and the centre of gravity of the figure formed by the remaining points. The so-found new point is a mirror reflection of the former one to the centre of gravity. If the latter point does not satisfy the restrictions the section is divided into halves between this point and the centre of gravity and conditions are checked anew until an admissible point is obtained. If the successively obtained values of the objective function are close to one another with a determined exactness, computations are stopped. So as to avoid finding the local extremum the procedure is repeated, starting from various points. 3.5 R o s e n b r o c k ' s method Rosenbrock's method [12] is a modification of Gauss-Seidl's method. The modification consists in a revolution of coordinate system after each iteration cycle in order to speed up the process of optimal point searching. Relative simplicity and good convergence are characteristic of Rosenbrock's method. Its most characteristic feature is the choice of an orthogonal set of vectors along which occur the changes of the searched parameters. The set of natural coordinates is accepted as the output set. The first iteration consists in searching for a minimum by changing in turn separate parameters. After some iteration the vector shows the approximate direction of the revolution. 4. The Model of Learning Many authors have dealt with mathematical aspects of adaptation processes. They have generally tried to adjust the mathematical description to the experiment they were carrying out [13]. Only a few papers treat this process in a general way, as for instance Bush and Mosteller's linear model [14]. They have introduced such notions as: choice, evaluation and a pair composed of choice and evaluation which they defined as an event. According to them the basic difference appearing in comparison with the notions of the statistic theory of decision consists in the event dependence on alternative choices. The feature that distinguishes Bush and Mosteller's model from others consists in its sequential character. Trial tendencies are characterized by the probability vector p (l) (where l is the successive experiment) describing these alternative choices. The authors describe the way in which the vector changes from trial to trial. They based on the following assumptions: 1. The change is given in operator form Tjkp(l) = p ( l - l), where the stochastic operator TiE depends on the event occuring in t e s t I. 2. Every operator T~k is linear.
424
The system is assumed to comprise algorithms A,, A_.... A,. Each algorithm solves the optimization problem. While solving the problem the system choses at random, according to a determined distribution, algorithms which in turn solve the problem and (a) each algorithm uses the results obtained by algorithms chosen at random previously, (b) the system chooses the algorithm on the basis of probability vector
p,(1)
p..(l)
P(l)=lp,(l)
• ~ pi(l)=l, j=t
where l is the number of randomization counted from the beginning of the system operation. Elements pi(l) of vector p(l) are the probabilities of the appropriate algorithm choice in the j-th sampling. Thus, it can be accepted, for example, that the elements of vector p(0) are equal, which means that pi(0)- 1
(J - 1, 2 . . . . .
r)
r
(c)the sampled algorithm solves the given problems within the time t. Value t determines the system depending on the number of decisive variables n, (d) after the end of the algorithm operation its effect is evaluated by the system, (e) on the basis of the evaluation which the algorithm has, the system points out a new probability vector p(l - 1). Generally speaking, a positive evaluation of a given algorithm increases the probability of its choice in the next sampling, but a negative evaluation diminishes it, simultaneously increasing the probability of the remaining algorithm choices. An algorithm with the biggest number of good evaluations will be successively used with greatest probability. Thus, the system learns to solve problems while solving them, in the sense of choosing the most effective out of the algorithms it possesses. After having solved a series of problems the system will give the priority for the successive problem to the algorithm that appeared most effective so far. A solution by means of only one algorithm, determined by its user, is admitted by the system as well. The system flow-diagram is given in Fig. 1.
Examples
4.1 The speed reducer The assumptions for the reducer shown in Fig. 2 are: transmitted power N = 100 km pinion speed n = 1500 rpm transmission ratio i = 1/3. The technical definition of the problem is given in [15]. The searched unknown parameters:
425 x~=b x _, =
m
x3=z x ~ = Ii
Xs = 12 X 6 = d~
X-/
=
dz
face width (cm) teeth module (cm) number of pinion teeth shaft length 1 (between bearings) (cm) shaft length 2 (between bearings) (cm) shaft diameter 1 (cm) shaft diameter 2 (cm)
IComblned [ olgorlthm
Was the time for I running given by the user ?
I Before calling the olgarifl~m Com~u?e ,*mem@Qn / of 7 firsfvolueS of the ObleCttvefunction ~TJ of 14 f,rst val~es cf the ODjectlvefunct+on(h~l of 21 f=rst v,31ues of the obJeCtiVefunchon(~21)~,}fter
Has the algorithm given more than 14 values ofF(x} ? Yes I NO' Operation of the algorithm
Iolgor=thm Igiven more
1
1 Sorting I Of results
Y?" N
I
IN?~N~ l
coining the ql~jorithm
Yes I
Yes
For trle pcir"algor/thrn, I IThe vector l es~imation" choose /-"lmoclif,cation the respective Pl = P l --TIR I operator T,, J I JIs the end criterion fulfilled
Yes I ?No
t F i g u r e 1.
]
426 ---l--
-!-
i/,
Z2
t_x I s
-
!q
--_
I
.)
i I i
Figure 2. The objective function which is subjected to the search for o p t i m u m is the total volume of gear wheels and transmission shafts and is to be minimized. The bending and clamp constraints, strength conditions of gear shafts, permissible magnitude of deflection etc. have been considered. The problem consists in finding the minimal value of function f ( x ) in a 7 dimensional space bounded by 13 described constraints and by 14 constraints imposed by ranges of dimensions. Therefore, we have to consider 7 variables and 28 constraints. Finally the problem has been defined in the standard f o r m of an optimizing equation. Objective function f ( x ) = 0.785x._"x~(3.33x32 + t4.93x3 - 49-09)
+ (x6" + x72)(9-52 - 1.5 Ix,) + 0.785(x6"-x, + xT:x~) with the constraints as listed below x,xz:x~ >! 27 x,x.'-x/- >i 397.5
x3> ~ 17 x,/x.. >! 5 x , / x : <~ 12 x,.x3 <~40 4
x,_x3x,, >- 1.925 x,.x3xT" >i 1.925
A'<~ 1100
B,
where A , = ~ \ x._x, / + 16"9x 106
B, = 0'lx~ 3 A_.
B._
<~ 850
,/(745xfl~'-157.5×106 +
A.. = ~ \ ~ / B._ = 0. I x73 x7~>5
427
1"5x6 + 1"9 ~ x~ l'IxT+ 1"9 ~ xs. According to the instructions for system use, the constraints listedabove have to be written in F O R T R A N ; this is the only work to be done by the system user. The described example became a very interesting subject of comprehensive experiments. From the mathematical point of view, the problem is characterized by flatness of the objective function in the neighbourhood of the extremum point. Similar results have been received for different values of variables. A significant number of local extremes has been noted; moreover the relations between the area of permissible solutions and the entire area searched have not been very convenient. For instance, in a trial of 1241sec duration performed by the crude Monte Carlo method, executed on the IBM 360/67 computer, the total number of random searchings reached 825,000; however, only 67 of them were contained in the permissible area. The ranges of particular parameters for the mentioned area have been settled as follows:
2.6~< x, = b ~<3.6 0.7 ~< x., = m ~<0.8 17<~x3 = z ~<28 7.3 ~< x, = It ~< 8.3 7.3 ~
0-6
26
8-1
8-2
3-4
5.3)
refers to a value of the objective function f(x) equal to 3643.63 cm 3. This point is situated near the boundary of the region limited by constraints of shaft bending and the constraint relation between b and m values; it has been recognized as the local extremum. Among other experiments, one example performed on the same IBM computer by the stray process has brought the solution (3.0
0.6
26
7.8
8.0
3.3
5.3)
situated almost on the edge of the area described by constraints of bending of gear teeth, constraints of bending of both shafts and constraint of b/m ratio as well. For the last solution: permissible bending stress of gear teeth, 900 kg/cm-" actual bending stress value of gear teeth, 892-73 kg/cm" permissible bending stress of shaft, 1.100 kg/cm-" actual bending stress for shaft I, 1-095 kg/cm" permissible bending stress for shaft 2, 850 kg/cm 2 actual bending stress value for shaft 2, 849-96 kg/cm'constraint of b/m ratio, 5 real value of b/m ratio, 5-16.
428 This a n d o t h e r e x a m p l e s p r o v e the v e r y s u b s t a n t i a l s t a t e m e n t , a n n o u n c e d at the b e g i n n i n g of this p a p e r , that an e n g i n e e r l o o k i n g f o r the o p t i m a l s o l u t i o n o f his d e s i g n c a n n o t define the m o s t s u i t a b l e m e t h o d for finding the s o l u t i o n of a p a r t i c u l a r p r o b l e m . T h e s e difficulties g r o w with the n u m b e r of p a r a m e t e r s a n d c o n s t r a i n t s . T h e y a l s o dep e n d on the c a p a c i t y of t h e c o m p u t e r at his d i s p o s a l . T h e r e are m a n y o t h e r difficulties of c o u r s e . U s e d a l g o r i t h m s h a v e s o m e d e g r e e of f r e e d o m a n d c h o i c e of t h e m has i m p l i c a t i o n s on the s e a r c h p r o c e s s . P a r t i c u l a r a l g o r i t h m s m a y b e c o m e m o r e or less efficient a n d this d e p e n d s u p o n the c h o i c e o f p a r a m e t e r s as s h o w n b y the d e s c r i p t i o n of the s t r a y p r o c e s s a l g o r i t h m [ 9 ] in the last e x a m p l e . T h e c o m p a r i s o n of r e s u l t s r e c e i v e d b y m e a n s of d i f f e r e n t a l g o r i t h m s ( r a n d o m - g r a d i e n t , ridge a n d c o m p l e x ) is g i v e n in T a b l e 1. T h e b e s t r e s u l t , r e a c h e d a f t e r
Table 1. Best results for the speed reducer Computer time 5 min f(x)
x~
x~
x~
Computer ZAM-41 x~
x,
x~
x=
3.451 3.422 3-522 3-407 3-568 3-484 3.660
5.750 5.649 5.491 5.438 5.562 5.450 5.579
Combined gradientstochastic searching technique 2261.429 2328.785 2348.393 2352.666 2353.491 2353.646 2353.898
3.135 3.242 3-213 3.488 3.325 2.247 3.087
0.568 0.552 0.574 0-536 0.546 0.555 0-581
26.651 27.523 26.693 27.645 27.677 27.577 26-873
8-288 8.617 7.356 8.229 8-125 7-255 8.509
8.745 8.513 8-682 8-306 8-727 8.568 8.216
The method of seeking optimum along the ridge of the objective function 2283.949 3.362 7-969 3-364 5.304 0.539 27.591 8.240 2287.237 3.361 0.539 27-571 7.743 8-162 3.462 5-592 2298.674 3.446 0.551 26.643 7.056 8.443 3.347 5.389 2330.104 3.169 8.592 3.575 5-440 0.553 27.784 8.410 2386.536 3.030 7.789 3.416 5.326 0.555 28.914 7.226 2395.835 3.559 0.527 28.184 7.748 8.146 3.348 5.414 2421.614 3.149 0.629 24.824 8.158 8.435 3.585 5.759 2250.891 2319.330 2386.706 2457.073 2795.307
3-101 3.328 3.392 3-625 3.543
0.570 0.543 0.543 0-529 0.625
The complex method 26.777 7-199 27.652 8-042 27.933 7-470 28.162 7.785 25.628 7-896
8.583 8-215 8.157 8.400 8.134
3.482 3.482 3.503 3.500 3.754
5.529 5.724 5.503 5'521 5.630
5 min of c o m p u t a t i o n on a Z A M - 4 1 P o l i s h c o m p u t e r has an o p t i m u m v a l u e e q u a l to 2250-891. T h i s c o m p u t a t i o n has b e e n p e r f o r m e d b y a c o m p l e x a l g o r i t h m . T h e d e t e r m i n e d p o i n t is d e f i n e d b y the v e c t o r : 3.101 0.57 6.777 7.199 8.583 3.482 5.529
429
The results of computation with the Systeiaa are given in Table 2. The reaching of an objective function value not greater than 2-200 cm 3 has been taken as the criterion permitting the end of computation. First the stray algorithm has been employed, leading to the point 4.02 I 0.45 I 33.19 1 8.051 8.141 3.691 5.621 The value of the objective function for this point is 2674.25. As the number of points situated in permissible area had reached 18 in 1 min, the algorithm has been classified as good. The vector of probability serving for the choice of the foregoing algorithm has been transformed as follows: 0-,02 [0.5571 0-306 [0"153 0"268 --*/0-134/
I
0.324
10.162]
0.000
L0.000J
The next choice led to the gradient algorithm, but the most satisfactory result was not improved. The average value based on the 21 most promising results reached 3362.1, i.e. more than the mean value based on the seven best results of the previous example (3318.9), but less than the mean value of the fourteen most favourable results received previously. Then the estimation for the considered algorithm is formulated as satisfactory. Therefore, the vector of algorithm choice changes to:
0.153 |0.139 0-134 --*|0.137 0.162 |0-144 0-000 U0.063 Subsequently the ridge algorithm was chosen. The best one of the results reached was improved to 2102-18 which corresponded to the point described by the vector: 3-03 0.61 24-63 6.93 7-72 3-35 6.29 As the criterion imposed by the user had been satisfied (2102.18 less than 2200), the action of the system was stopped. It has to be emphasized that the ridge algorithm, besides producing the most satisfactory result, was not recognized as sufficient, since the
430
Table 2. Processing time for individual algorithm--I rain
The best result after operation of the algorithm
,Algorithm Random process Combined gradient stochastic Ridge
fix)
x~
x._
x~
x~
x~
x~
Operatzon After operation
_6.4._~
4.02
0-45
33.18
8"05
8.14
3"69
I Operation After operation
-~ -)¢, _674.._
4.02
0.45
33"18
8.05
8.14
Operation fter operation
_ 10." 18
3.03
0-61
24.63
6-93
7.72
The 2Ist result after x,
fix)
.,,
x:
5"62 4721.36
5.34
0.57
3-69
5.62
3871.24
3-65
0-56
3'35
_ -_9
3829.85
3"70
0.65
mean value of 21 best results was equal to 3270.7 cm; and worse than the mean one of 14 best results of the preceding computation (3100.9cm3). The vector of probability used in the choice of the algorithm has been transformed to 0-139 |
0.317 | 0-144 0.063 j
F° 94 |0.195
| 0.284 /°°72 L°156
The described system has proven its superiority over independent algorithms. The total duration of computation has been shorter than the computation of particular algorithms and more favourable results have been received (Table 3). T a b l e 3. Optimization system Algorithm
Combined
Ridge
Complex
SOPT
The best result The 7th result The 21st result Computer time
2261.429 2353.898
2283.949 2421.614
2250-891 2795.307
2102.18
5 rain
5 min
5 rain
3829.85
5 rain
4.2 The planetary gear reducer The application of the optimization system is illustrated by the example of an aero-engine reducer (Fig. 3). The parameters of the considered reducer are: power of free turbine, 1200 km requested rotational speed of propeller, 1500 rpm rotational speed of disc rotor driving propeller, 18,000 rpm. The design of the reducer embraces calculations for all gear wheels, i.e. control computation for mostly loaded teeth, and considers: bending stress, tangent stress on
431
The mean of results
operation of the algorithm x3
x4
xs
x6
x~
30-59
8.30
8.32
4.06
5-42
z7
z,~
Estima- The probability of choosing the algorithm tion Rand. Comp. Comb. Ridge Ros.
z.-,
9.24
8-62
3-77
5-50
3362" I
8-75
8.51
3.90
0-551
0.153
0-338
0.294
0.268
0.324
0-000
0-134
0.162
0.000
0-139
0.317
0.144
0-063
0"195
0.284
0.072
0"156
Satis.
2822"2 3100.9
28'22
0.306
Good 3318.9 3687.1
33-57
0.102
5.70
3270.7
Unsat.
the surface c l a m p f o r c o n t a c t points, and t e m p e r a t u r e conditions at the initial point of gear t o o t h interaction. T h e s e a r c h e d p a r a m e t e r s f o r the planetary gear r e d u c e r are: b=x~ m = x2 z = x3 k = x,
face width normal module the n u m b e r o f teeth in the sun gear the n u m b e r of planet gears.
T h e constraints i m p o s e d on the c o n s i d e r e d c o n s t r u c t i o n h a v e been described in [16]. T h e description o f t h e m can be given as follows:
1. Bending stress condition 2. C l a m p surface stress condition 3. H e a t condition 4. E x t r e m a l n u m b e r o f teeth 5. R e l e v a n t width o f gear wheel rim 6. R e l e v a n t width o f gear w h e e l rim 7. Minimal n u m b e r of satelites
q~7(x)= k - 2 t> 0
8. A d j a c e n c y
~os(x) =
9. 10. 11. 12.
Assembling Overall d i m e n s i o n s Width of rim wheel M o d u l e o f single t o o t h
The o b j e c t i v e f u n c t i o n c h o s e n was f(x)=-~
bm
+1
.
qh(x) = 1.16m'-zbk - ~ -
2-2 >~ 0
q~,.(x) = O.Olbkrn'-z" - ~v/--~mz- 2.2 I> 0 q~3(x) = 0.035b. m 2 z 2 - 1 >10 q~,(x) = z - 28 ~> 0 ~s(x) = b - 3m ~> 0 q~6(x) = 15m - b / > 0
~og(x) = ~O,o(X) = qh,(x) = ~0,2(x) =
3 sin ~- - 2 - 2/> 0 MOD(z, 2) = 0 3 7 - 2 z m >10 b 1>0 m i>0
432 Z 3 7-
t
-7 i
i Zz
z~ --
,
1}--f
7_ I f ..~4
.- 5
Figure 3. The function as given above expresses the total volume of the reducer assuming its solid construction. Such a criterion seems logical, as the total weight is a crucial factor in aircraft construction elements. However, in practice, the wheels of a gear reducer are not massive. An experienced engineer may eliminate smartly all needless material. Moreover, a simple proportion could be nearly assumed between the overall weight of solid elements and framed ones (hub thickness, ribbing, gauging volume may be approximately referred as functions of gear wheel width or of the module pitch). In the considered problem the variable k (describing the number of satelites) and the number of teeth in the planetary gear may only be integer. The values of module pitch are taken according to Polish standards as 2, 2.5, 3, 3.5 . . . . . mm. The problem defined above had to be translated to FORTRAN. Subsequently, the subroutine prepared by the system user was introduced to the computer and processed by the optimizing system. The results received by system action are given in Tables 4 and 5. Table 4 illustrates computations performed with the use of 3 different algorithms. Each of them has worked 1 min. The initial probability for algorithm choice has been accepted as equal to 0.2, i.e. giving an equal chance to each of 5 considered algorithms. As the first, the stray process algorithm was employed, producing 21 points in the permissible area and achieving a good estimation. Therefore the considered probability was changed significantly. As the second, the gradient algorithm was picked and, according to the method of estimation described above, it achieved a good classification. The same estimation was given to the last acting Rosenbrock's algorithm. The final result received after 3 rain is equal to 562-30 and the parameters of the associated extremum point are: (1.55, 0.4, 34, 3). The twenty-first of achieved results has the objective function value 702-72 cm 3 and parameters (1.73, 0.4, 36, 3). The above described example has been repeated with the same value of initial probability vector: p(0) = (0.2, 0.2, 0.2, 0.2, 0-2). Subsequently working algorithms (Table 5) are: stray, Rosenbrock, gradient, ridge, stray, Rosenbrock and gradient again. The whole trial has taken 7 min, and the best one of the results refers to the point (2.15, 0.3, 36, 3), and the objective function value 561.06. The twenty-first of results appears very near to the optimum, giving the point (1.25, 0-4, 38, 3), and the objective function value of 565.33 cm s. The point I 0"3 /
36 )
._3
<
Operation '~ Afler I operation
Operalion ~After Loperation
Operation ~After Loperidion
Random
Combined
Rosenbrock
Algorithm
Processing time for individual algorithm--I nlin
Table 4.
o0 Z 9
562.30
571.54
702.50
I(x)
1-55
2.41
1.72
x,
0.4
0.25
0.4
x~
34
44
36
x,
The best resuh after operation of the algorilhm
3
3
3
x~
1.73
1.73
705-34
702.12
1.73
x,
1102.00
l(x)
0-4
0.4
0.5
x,.
36
36
36
x~
3
3
3
x,
The 21st result after operation of the algorithm
z,,
657-34 680.02
702.67 702-74
z,
643.88
687.74
z2,
The mean t)f results
Good
Good
Good
Estimarion
0-150
0-300
0-600
0-200
R;mdonl
0.1125
0-050
0-I00
0.200
i1.275
0.550
0. I00
0.200
Comp. Comh.
0.t125
0.050
O-I(X)
0.200
Ridge
(1-525
0-O.Y~t)
0.100'
0.200
l,~o,lellbrock
The probability of choosing algorithm
CO CO
'Operation '~ After [operation
Operalion '~ After . (operation
Operation ~Afler . [ operalmn
Operation ~After . [ operalU)l)
Rosenbrock
Combined
Ridge
Randonl
Rosenbruck
Combined
Operation '[After Loperation
Random
'~ After I operation
Operalion
Operation '~ After . I operatJon
Algorithm
Processing time for individual algorithm-- I rain
Table 5.
561 "(16
561.06
561-06
561.06
561.27
561.27
716"33
f(X)
2"45
2-45
2.45
2-45
I "32
1.32
2-81
X~
0.30
0-30
0.30
0-30
0.35
0"35
0"3
Xz
36
36
36
36
42
42
38
X~
The best result after operation of the algorithm
3
3
3
3
3
3
3
X4
565.33
565.93
642.50
642,50
716.33
970-99
1942-36
]'(x)
1.25
1.65
2-81
2.81
2.81
2-80
2.74
xj
0.40
0.30
0-30
0-30
0.30
0"35
0.5
x2
38
44
36
36
38
38
38
x3
The 2 Is( result after operation of the algorithm
3
3
3
3
3
3
3
x4
~61-43 562.36
561.87 563.23
561-87 563,23
577.74 618.%
624.45 670.39
716.33 752.74
Z14
563.07
563-32
581-86
581.86
651-42
720-30
Z.~ I
The mean of results
Unsal.
Unsat.
Unsat.
Saris.
Sa(is.
Saris.
Good
Estima(ion
I).211
0.171
0-092
0.184
0.244
0.363
0-600
0,200
0.233
0.216
0.308
0"I13
O'ItR)
0280
0.309
0.119
0.113
O-I(X)
0.234
11.121 11.257
11.21~ 11"242 0.3t, S
0.186
I).122
0-119
0.113
0"100
0.21X1 11-200 0.2011
I).177
O. It)5
0.209
0"169
0.213
0.31Rl
0-1011
0.2())
The probabilily of choosing alg. Random Ct)lllp. Comb. Ridge Ros,
4~
435 referring to the value of objective the f u n c t i o n equal to 561-06 is the sought e x t r e m u m , situated on the b o u n d a r y of the region due to the heating constraint.
Summary T h e target o f the given paper is the illustration of a n e w c o n c e p t in the solution of optimization problems. T h e present stage w h e r e users have the a c c e s s to m a n y different c o m p u t i n g strategies causes a lot of trouble. T h e a b s e n c e of classification of algorithms and p r o b l e m s m a k e s m a n y difficulties f o r an engineer, w h o is not able to define an algorithm m o s t suitable to the particular design. This is the reason f o r formulation of self-acting model, dispensing the s y s t e m user f r o m the i n c o n v e n i e n t choice o f an algorithm. T h e p r e p a r e d s y s t e m itself includes different algorithms acting in turn a c c o r d i n g to the d e m a n d s of the p r o b l e m solution. T h e p e r f o r m e d e x p e r i m e n t s p r o v e the rightness of such b e h a v i o u r and argue its superiority in c o m p a r i s o n with the use o f i n d e p e n d e n t algorithms, even the same ones included in the d e s c r i b e d system, but treated separately. The s y s t e m f o r m u l a t e d by the a u t h o r as d e s c r i b e d a b o v e , has been realized on a ZAM-41 Polish c o m p u t e r and applied to practical use. T h e succesful use o f this s y s t e m has interested users of o t h e r c o m p u t e r s . T h e r e f o r e t w o similar s y s t e m s for O D R A 1304 and K-202 Polish c o m p u t e r s are in the final stages of the preparation. Simultaneously w o r k s have been u n d e r t a k e n to prepare the optimization s y s t e m for c o m p u t e r s of the u n i f o r m R I A D system.
References [I] PEYNTER H. L., Analysis and Design o[ Engineering Systems. The MIT Press (1960-1961). [2] SLAYMAKER R. R., Mechanical Design and Analysis. Wiley, New York (1959). [3] VOLMER J., Design of mechanism: synthesis by iterative analysis, Transactions ASME. Paper No. 68MECH-69 (1968). [4] GASS S. I., Programowamie Liniowe. Methdy i Zastosowanie (Linear Programming--Methods and Applications.) PWN Warszawa (1963). [5] GARTH P., MCCORMICK W. Ch., MYLANDER III and FIACCO A., IBM Catalogue of Programs: Program No 70-40-H2-3189 Sumtrac (Sumpt). [6] MUGELE R. A., A nonlinear digital optimizing program for process control systems. Proceedings for the Spring Joint Computer Conference, San Francisco, I May (1%2). [7] CASEY J. K. and RUSTAY R. C., AID--a general purpose computer program for optimization, Recent Advances in Optimization Techniques. Wiley, New York (1966). [8] FLOOD M. M. and LEON A., Adaptive code for optimization GROPE, Recent Advances in Optimization Tc.chniques. Wiley, New York (1966). [9] GOLINSKI J., O badaniu pewnego procesu bl~dzenia zastosowanego do optymalnej syntezy maszyn (Stray process research applied to optimal synthesis of mechanisms.), Arch. Bud. Maszyn XV, 2 (1968). [10] WILDE D. J., Optimum Seeking Methods. Prentice-Hall, Englewood Cliffs (1964). [ II] BOX M. J., A new method of constrained optimization and a comparison with other methods, Computer Jl 8, 42-52 0965). [12] ROSENBROCK H. H,, An automatic method for finding the greatest or the least value of a function, Computer Jl 3,(2) 0960). [13] BUSH R. R. and MOSTELLER F., Stochastic Models for Learning. Wiley, New York (1955). [14] THRALL R. M., COOMBS Ch. and DAVIS R. L., Decision Processes. Wiley, New York (1954). [15] GOLIlqSKI J., Optimal synthesis problems solved by means of nonlinear programming and random methods, J. Mechanisms S, 287-309 (1970). [16] GOLIIqSKI J., O optymalnej syntezie maszyn metodami Monte Carlo. (Optimal synthesis of mechanisms by Monte Carlo methods.), Arch. Bud. Maszyn 11, 3 (1965).
Anpassungsfahiges Lernsystern angewendet for Optimierungsaufgaben J. Golinski
Kurzfassung--Das Ziel des Adaptationssystems besteht darin, aus den vorhandenen Optimierungsmethoden for eine technische Optimierungsaufgabe
436 die effektivste Methode vom Rechner automatisch ausw~hlen zu lassen. Da for die Vielzahl technischer Optimierungsaufgaben die Form der Zielfunktionen und die form der Restrikstionen nicht bekannt sind, berOcksichtigt das Adaptationssystem diese Einfl0sse nicht. Es werden die aus der Warhscheinlichkeitstheorie bekannten Methoden zur Bestimmung eines Wahrscheinlichkeitsvektors benutzt, der das Alternativauswahlverfahren beschreibt. Die Theorie wurde durch Bush und Mosteller beschreibt. Das Adaptationssystem enth< f/3nf Optiemierungs-methoden die zur L6sung von technischen Optiemierungsaufgaben mit maximal N=20 Ver~nderlichen benutzt werden. Es kann bewiesen werden das der Wahrscheinlichkeitsvektor sich durch den Operator T,k transformieren I~sst. Es gilt: p(/+1)= Tjk. P(0. Das Adaptationssystem beginnt seine Arbeit mit einem Wahrscheinlichkeitsvektor p(0). Diese Verteilung p(0) wurde aus der Bewertung von 10 Testaufgaben die durch jeden der fenf Optiemierungsalgorithmen gel6st wurden. In der laufenden Arbeit mit dem Adaptationssystem wird deshalb die Bewertung der Algorithmen fQr jede berechnete Aufgabe durchgefLihrt und auf Grund dieser Statistik wird die Art der Wahrscheinlichkeiten ge&ndert. Der somit erhaltene neue Wahrscheinlichkeits vektor spiegelt die Effektivit&t der Algorithmen besser wider. Wie erw&hnt befinden sich im System f6nf verschiedene Algorithmen: --die Wellenberg-Methode nach Hooke und Jeves --die Raffinierte Monte Carlo Methode --die Kombination der Monte Carlo Methode mit einer Gradientenmethode/nach J. Oderfeld/ --die complex methode nach Box w d i e Rosenbrock'sche Methode. Mit diesem Adaptationssystem hat man zahlreiche technische Beispiele gel6st. In der Arbeit sind 2 Beispiele-Zahnradgetriebe und Planetengetriebe beschreibt und gel6st.