a robust multi criteria optimization approach

Pergamon Mech. Mach. Theory Vol. 32, No. 7, pp. 797-810, 1997

© 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain

PII : 80094-114X(97)00007-4 o094-114x/97 $17.oo + 0.o0

A R O B U S T M U L T I - C R I T E R I A O P T I M I Z A T I O N A P P R O A C H

ARUN KUNJUR and SUNDAR KRISHNAMURTYt Department of Mechanical Engineering, University of Massachusetts, Amherst, MA 01003, U.S.A.

(Received 26 June 1996)

Abstract--The overall value of a product is typically determined by its performance with respect to multiple measures. As such, the product design task will be simplified if all these performance measures were optimized simultaneously. Another significant factor that determines product quality is its sensitivity to external or uncontrollable variations, To effectively address these issues in product design, this paper presents a novel robust multiple criteria optimization (RMCO) approach that integrates multi-objective optimization concepts with statistical robust design techniques. In this approach, Pareto-optimal robust design solution sets are obtained with the aid of design of experiment set-ups, that utilize ANOVA results to quantify relative dominance and significance of design factors. Application of this method to engineering design problems is illustrated with the aid of two case studies including a mechanism dimensional synthesis problem. © 1997 Elsevier Science Ltd

I N T R O D U C T I O N

Robust design techniques based on the concept of building quality into a design, are increasingly finding new applications in the industry mainly because of their simplicity and practicality. Taguchi's Methods [1-4] have been extensively applied in process design [5], wherein mathematical models for performance do not exist and experiments are typically conducted to determine optimum settings for design and process variables. However, Taguchi's methods have not been fully exploited in the parametric design of products [6, 7]. Conventional optimization techniques employed in product design such as direct search and gradient based methods generally rely on a large number of function and gradient evaluations to determine the optimum point. This is particularly true if multiple objectives are involved. Taguchi based methods have been used in product design primarily to reduce the number of function evaluations or to obtain a good starting point for other optimization techniques. Here, special performance measures called signal-to-noise ratios, that will lead to a reduction in the sensitivity of the product to uncontrollable variations, are employed in the orthogonal array experiments towards achieving a robust design. However, the main drawback with the application of the basic form of Taguchi's method, to product design, is its inability to address two important issues--(1) systematic treatment of constraints and (2) formulation for multiple objectives.

Multiple criteria optimization techniques can be broadly classified into preference based methods and generating methods [8-10]. Preference based methods attempt to quantify the decision makers preferences or relative importance for each criterion and use this information to obtain a single super criterion. Standard optimization techniques are then used to isolate a single optimum design. Goal programming [11] and utility analysis [12-14] are some examples of preference based methods. Goal programming requires the allocation of a goal for each criterion and a weight that reflects the relative importance of deviations from that goal. The problem then is to find a solution that is as close as possible to the set goals. Utility analysis requires quantification of attribute levels, which may not be known a priori , to reduce the objective in a probabilistic sense, to a single overall

tTo whom all correspondence should be addressed.

797

798 A. Kunjur and S. Krishnamurty

utility function. A salient feature of utility analysis is that it can be used to evaluate alternatives under uncertainty. Generating methods [8], on the other hand, have been developed to enumerate the exact non-inferior set or an approximation of it. Here a feasible solution to a multi-criteria optimization problem is considered non-inferior (Pareto-optimal) if there exists no other feasible solution that will yield an improvement in one criterion without causing a degradation in at least one other criterion. This information is then presented to the decision maker, who is required to select a design that is most suitable. A major drawback of this method is that most real world problems are too large to allow the exact non-inferior set to be found and, even if it were generated, the set would include too many alternatives for the decision maker's consideration. The weighting and constraint methods [8] are the most widely used generating techniques. They operate by converting the multiple objectives into a single-criterion using certain parameters, which are then varied to obtain different Pareto-optimal solutions.

In this paper, Taguchi's method has been further developed to handle multiple objective product design problems using generative techniques and a rational approach to treat constraints in such problems is presented. Application of Taguchi Method for constrained problems has primarily been dealt with by introducing a penalty function in the performance parameter [7]. This is not possible for most problems as the inclusion of a penalty function may make the performance parameter non-additive or even discontinuous. Furthermore, the optimization of multiple performance parameters by using weighting functions to arrive at a single measure of performance can lead to erroneous results because of the difficulty involved in making the trade-off decisions to arrive at this single expression.

This paper addresses these problems and proposes a methodology that further develops Taguchi's method to incorporate multiple objectives and constraints in product design. In this methodology, statistical analysis (ANOVA) concepts are used to obtain a well diversified, though not necessarily exhaustive, set of Pareto-optimal solutions. It is expected that this integrated procedure will provide a rational platform for the systematic identification of robust designs in a multiple objective domain.

ROBUST DESIGN FOR A SINGLE RESPONSE

The Taguchi concept of robust design is based on maximizing performance measures called signal-to-noise ratios by running a partial factorial set of experiments using orthogonal arrays. The signal-to-noise ratio is typically given by

S/N = - 10 Iog[MSD] (1)

where MSD refers to Mean Square Deviation of objective function. The S/N ratio aims at achieving the separability of design factors into control factors and signal

factors. A robust optimum design is identified by finding the optimum setting of the control factors to reduce variation and then adjusting the signal factors to shift the mean.

Additivity of factor effects is an important consideration in statistical design of experiments. It ensures that the performance measure is not adversely affected by the non-linearities of the objective function. Note that this form of the S/N ratio does not guarantee separability and additivity for all types of objective functions, and thus the use of transformations such as 2-transformations may become necessary [15, 16].

The application of the Taguchi concept to product design, which is usually characterized by continuous design variables, requires that the design space be discretized by splitting each variable range into a desired number of intervals. The non-linear effects of the response (i.e. objective function) can be captured by using at least three levels (two intervals) for each factor. Standard orthogonal arrays, developed by Taguchi, are used to design the experiments (inner array) and to simulate the noise effects (outer array), if any. The objective function is evaluated, and the S/N ratio or a suitably transformed performance measure (henceforth generalized to S/N) is determined for all the experiments. ANOVA is performed on this SIN ratio to determine the effects of different factors on performance. Specifically, the sum of squares, mean squares, F-values, and percentage contribution is computed to determine which factors contribute significantly to the variance and

A robust multi-criteria optimization approach 799

mean response. This information is then used to predict the optimum settings for the various design factors.

R O B U S T D E S I G N FOR M U L T I P L E R E S P O N S E S W I T H C O N S T R A I N T S

Taguchi's method deals with a single objective function and the only constraints it can handle are the variable bounds. To handle multiple criteria problems, an alternative is to formulate the multiple objective optimization problem as a weighted combination of single objective functions. However, such an approach will fail to adequately represent the actual design problem and will have several shortcomings as described earlier. It is then apparent that Taguchi's method, in its as is form, does not handle multiple objective formulations or constraints. This work presents a methodical approach based on Taguchi's concept to address multi-objective problems where each objective is treated independently and a rational procedure is employed to identify a design space that encompasses a diverse set of Pareto-optimal design. In this procedure ANOVA for each of the objectives and constraints (responses) are performed simultaneously and then the design space is systematically pruned and refined based on the effect of the various design variables on the multiple responses. The purpose of the ANOVA tables is to help distinguish the robust designs from the non-robust ones. The underlying principle in this scheme is the identification of the optimum of each of the objectives using its corresponding ANOVA results and subsequent isolation of the range indicated by the individual optima for later evaluation of Pareto-optimal designs.

RMCO P R O C E D U R E

A flow-chart describing this technique of robust multiple criterion optimization (RMCO) is shown in Fig. 1. The basic steps involved in the procedure are the identification of the feasible domain (based on constraint satisfaction), the development of optimality condition for each objective, and finally the generation of the designs in the Pareto-optimal solution set. Towards this end, a two-step iterative procedure is used to identify the non-inferior set. Each iteration involves:

(1) Elimination of all factor levels that cause any of the constraints to be violated. (2) Identification of those factor levels that significantly effect one or more objectives.

The results of these steps are then used towards determining the Pareto-optimal set. The development and implementation of the methodical procedure involving these two steps and the rationale behind it are described below.

In the first step, the ANOVA of the constraint function is used to identify the factor levels that cause a constraint violation. It should be noted here that the ANOVA of any experimental data is instrumental in identifying the extent to which each factor effects the response. The percentage contribution (or the F-statistic) of each of the factors is a good indicator of how much any particular factor contributes to the violation of the constraint function. All factors that have a high percentage contribution, as compared to that of the other factors, are identified. For this purpose, the designer may decide to use the factors with the highest percentage contributions or, as an alternative, a cut off value may be set. The levels of these factors that cause the constraint violation are identified. If the mean effect of this level on the constraint function is beyond the constraint limit, then this level is eliminated from further consideration. In this manner, the non-feasible designs are filtered out from the initial design space.

The ANOVA of the objective functions is then considered, noting that the design space is reduced due to the elimination of levels in the previous step. In this step the best level of a factor is identified for each objective function. The range indicated by the various optimum levels of any factor for the different objective functions will form the new bounds in the next iteration. In case the same level is the best for all objective functions, then a small neighborhood around that level is bracketed to obtain the levels for the next iteration. In some cases the objectives may be mutually conflicting, i.e. the best design for one objective is the worst design for another one and vice versa. In such situations, there is no refinement of the design space as all the considered levels will be bracketed and thus a more elaborate procedure is required to identify the better levels. This topic will be covered in the next section.


The procedure outlined above does not guarantee an exhaustive non-inferior set, but it should be noted that enumeration of such a set may not be preferred in most practical design situations as it will only compound the complexity of the final selection process. On the other hand, providing a representative set in a systematic manner as shown here will greatly facilitate the identification of appropriate designs as either the final design or for further refinement.

T R E A T M E N T OF C O N F L I C T I N G O B J E C T I V E S

To handle design problems with conflicting objective functions, a novel procedure has been developed for pruning the design space iteratively that is based on identification of relative significance of design factors on different objectives. The purpose here is to retain all such levels of a factor that significantly effect an objective and eliminate all levels that have a negligible influence. For this purpose, factors are classified into three categories---dominant, significant and insignificant factors. A factor is assumed to have a dominant effect on an objective function if its percentage contribution (or F-value) is considerably higher than that of all other factors for that

Fmmul~a~ Problem

Lay-o~:tbe exlx:ie~ts , H ]vl~lify the variable levels I

I

AblOVA for objectives and comnints

[ Id~mtify fusible desip ~ ,l

I r~-vct~ optim~ty Coaditimt

Obtm n o n ~ designs

Fig. 1. Flowchart showing the RMCO procedure.

A robust multi-criteria optimization approach

L ~ffi 200cm I p X2 4 ID r

Fig. 2. Beam design problem.

x,:i Fo:[ 801

XI

objective. A simple procedure to determine dominant factors would be to find the factor that has a percentage contribution greater than the sum of all other factors for the concerned objective function. A factor has a significant effect if the percentage contribution is greater than a designer specified cut-off value (or if the F-value is greater than the F-value identified from a designer specified confidence from the F-tables).

The factor levels violating the constraint are first eliminated, as before, and the design space is then further reduced according to the following rules:

(1) If a factor has a significant effect on all objective functions, then all the levels that optimize at least one objective are selected.

(2) If a factor has a dominant effect on a single objective, the factor level that optimizes this objective is selected regardless of its significance on other objectives.

(3) If a factor has an insignificant effect on all the objectives, then the designers discretion is used to determine the objective most effected by this factor and the best level for the factor is identified.

This procedure thus attempts to identify non-dominated designs, but note that it does not ensure the elimination of all dominated designs from the final set. The proportion of dominated designs in the final set depends to a large extent on the designer determined cut-off value for identifying significant factors. A high value for this cut-off indicates that the designer does not want any dominated designs in the final set. In this case, the designer is willing to eliminate some non-dominated designs to reduce the cost of processing the final set. Alternatively, a low cut off value will result in a large final design set, and consequently a sizable number of dominated designs, that have to be filtered out by further processing. Depending on the choice of the designer, further processing would mean either another iteration on the reduced design space or filtering out the dominated designs using a standard procedure such as the technique of dominated approximations [17]. Thus the cut-off value is a trade-off between the number of non-inferior designs that the designer requires and the amount of effort he/she is willing to put in to identify these.

The application of robust design concepts to multi-criteria optimization is illustrated with two mechanical design examples. The first one is a beam design example with conflicting objectives, i.e. weight and beam deflection. The second example is a mechanism dimensional synthesis problem with path deviation and mechanical advantage as the primary objectives.

BEAM DESIGN EXAMPLE

The application of the robust design methodology to multi-criteria optimization problems with conflicting objectives is demonstrated by a beam design problem [10] in~,olving two objectives and a single constraint. The objective of this problem is to determine the dimensions of a beam (Fig. 2), that minimizes the cross section area for a given beam length and the deflection of the beam under given loads. The various parameter values for the problem are:

Permissible bending stress of the beam material is 16 kN/cm 2. Young's Modulus of Elasticity (E) is 2 • 104 kN/cm 2. Maximal bending forces P = 600 kN and Q = 50 kN. Length of the beam (L) is 200 cm.


The problem can be mathematically expressed as follows:

Minimize Cross Section Area = fl(x) = 2x2xa + x3(xl - - 2x4)

Minimize Vertical Deflection = f2(x) = PL3/48EI

Subject to the constraint

Maximum Stress = f3(x)

180,000x~ 15,000x~ -x3(x~ - 2x4) 3 + 2x2x4(4x 2 + 3x~(xt - 2x4)) + (x~ - 2x4)x] d- 2x4x~ <~ 16

where I is the Moment of Inertia of the beam cross section.

The geometric constraints are:

1 0 ~ < X l < ~ 8 0 , 1 0 ~ < X 2 ~ < 5 0 , 0 . 9 ~ < X 3 < ~ 5 , 0.9<~x4~<5.

To capture non-linear effects of the various responses, four levels were initially chosen for each of the four design variables, within the specified range. Thus, neglecting interaction effects, the total number of degrees of freedom of the system is 12. The L~6 orthogonal array was used to design the experiments for this problem. It was found from the mean effects table, that both the area and the deflection equations satisfy the additivity condition and thus do not require any transformation or study of interactions. The constraint equation was found to be non-additive, but this does not affect the result as the analysis of constraint equations is done only to eliminate invalid designs and not to predict constraint values for the optimum design. Dimensional variations in the design variables were treated as the appropriate noise factors, and a n L9 array was used to simulate these variations. The experiments are set up as shown in Table 1.

The objective functions are computed for each combination of design variables and for each combination of noise values. The S / N ratios for the various experiments are shown in Table 1. The mean response of each factor level is then computed for each objective function and constraint equation and ANOVA is performed. These are listed in Table 2.

It is seen from the ANOVA of the constraint equation [Table 2(c)] that the factors x~, x2, and x4 have a significant effect on the constraint, assuming a 1% cut off for significance. The levels of these factors that maximize the constraint value are found from Table 2(c) to be the first level for all three factors. It is further observed that the mean effects corresponding to these levels of x~ and x4 are above the transformed constraint limit given by 10 , log(162) = 24.08. Consequently, these two levels are eliminated from further consideration. The ANOVA of the objective functions [Tables 2(a) and 2(b)] show that all the four factors (x~-x4) have a significant effect on the cross section area (objective 1), whereas only the factors x~, x2 and x4 significantly influence the deflection of the beam (objective 2). Additionally, x~ is a dominant factor for beam deflection as its percentage contribution for deflection (92.7%) is much greater than that of any other factor. The level of x~

Table 1. Experiment set-up and S / N ratios

Ex. no. x l x2 x3 x4 SNI SN2 SN3

1 10 10 0.9 0.9 -28.09 -21.62 48.89 2 10 23.3 2.3 2.3 -41.56 -9 .55 38.49 3 10 36.7 3.6 3.6 -48.77 -4 .45 33.72 4 10 50 5 5 - 53.98 - 1.59 30.98 5 33.3 10 2.3 3.6 - 42.41 11.75 15.36 6 33.3 23.3 0.9 5 - 48.10 19.66 17.65 7 33.3 36.7 5 0.9 - 46.99 15.66 20.45 8 33.3 50 3.6 2.3 - 50.46 21.93 16.52 9 56.7 10 3.6 5 -48.57 25.80 11.15

10 56.7 23.3 5 3.6 -52.37 30.57 3.18 11 56.7 36.7 0.9 2.3 - 46.68 28.66 11.72 12 56.7 50 2,3 0.9 - 46.70 26.17 14.04 13 80 10 5 2.3 - 52.33 33.91 15.35 14 80 23.3 3.6 0.9 - 50.20 32.42 17.50 15 80 36.7 2.3 5 - 54.45 41.32 - 0.46 16 80 50 0.9 3.6 - 52.58 40.90 2.48


Table 2. Analysis of variance for all objectives and constraints

803

(a) Objective 1

Mean effects Level 1 Level 2 Level 3 Level 4 SS DOF MS F0 % Cont

xl --43.0977 --46.98891 --48.57791 -52.439 179.5706 3 59 ,85686 11.14517 26.25771 x2 --42.89953 -48.05369 --49.22116 --50.92913 143.5511 3 47,85038 8 . 9 0 9 6 20.47161 x3 --43.86072 -46.28051 -49.49674 -51.46554 136.5584 3 45 .51947 8 . 4 7 5 5 9 19,34831 x4 -42.99387 --47.80442 -49.03123 -51.27399 146.7245 3 48.90818 9 .106559 20.98138 Residual 16.11196 Error 16.11196 3 5.370654 12.94099 Total 622.5166 15 100

(b) Objective 2

Level 1 Level 2 Level 3 Level 4 SS DOF MS F0 % Cont

xl -9.30278 17.24992 27.80081 3 7 . 1 3 6 4 4832.267 3 1610.756 2172.463 92.76362 x2 12.46102 18.27419 20.29889 21.85025 202.677 3 67 .55899 91.11835 3.849803 x3 16.89814 17.42388 18.92493 1 9 . 6 3 7 4 19.54822 3 6 .516072 8.788375 0.332715 x4 13.15836 18.73512 19.69314 21.29773 150.1122 3 50.03741 6 7 . 4 8 6 6 2.840268 Residual 2.224327 Error 2.224327 3 0.741442 0.213597 Total 5206.829 15 100

(c) Constraint

Level 1 Level 2 Level 3 Level 4 SS DOF MS F0 % Cont

xl 38.02051 17.49263 10.02144 8.718296 2196.467 3 732.8224 104.2603 80.33045 x2 22.68757 19.20507 16.35471 16.00554 115.3661 3 38.45536 5 .471134 3.478278 x~ 20.18324 16.85627 19.72158 17.49179 32.11156 3 10.70385 1.522862 0.406756 .v4 25.2187 20 .52064 13.68373 14.82981 343.4986 3 114.4995 16 ,29012 11.89481 Residual 21.08632 Error 21.08632 3 7.028774 3.889705 Total 2710.53 15 100

that optimizes deflection (level 4) is thus singled out for further consideration, regardless of its effect on the weight of the beam. The factors x: and x4 have a significant effect on both the objectives and therefore a range of levels is obtained for these factors by identifying the levels that optimize each of the objective functions. The factor x3 has a significant effect only on the weight of the beam, which is optimized by setting x3 to the first level as seen from Table 2(a). The factor levels that optimize the respective objectives, after eliminating the levels that violate the constraints, are shown highlighted in Tables 2(a) and (b). Subsequently, the design space is narrowed down the new levels (bounds) for the design factors are identified as follows:

x l = 8 0 , 10~<x2~<50, x3=0.9, 2.3<~5.

At this point, the designer has the option of either (1) performing another iteration on the reduced space to further narrow down the search or (2) identifying the non-inferior set by combining the various levels of each factor. In the case of another iteration, the reduced bounds (as for x2 and x4), or the neighborhood (as for x~ and x3) of the predicted levels can be used to formulate new design levels. Alternatively, the combination of the various levels in the reduced design space results in a design set as shown in Table 3. The graphical representation of the resulting objective functions for all the non-dominated designs is shown in Fig. 3. All the designs in this set, for this particular example, are Pareto-optimal designs. This will not be true in general and the final solution set may require further processing to identify the non-dominated designs. Note that a lower cut-off value would have resulted in fewer factor levels being eliminated and consequently the final design set would have had a larger number of designs, possibly including some dominated designs.

M E C H A N I S M D I M E N S I O N A L S Y N T H E S I S

Typically, mechanism dimensional synthesis involves the minimization of structural error subject to a set of size and geometric constraints such as Grashof and crank rocker conditions. For example, in path generation problems, the coupler point is required to trace a path with minimum


Table 3. Non-inferior design set

Ex. No. xl x2 x3 x4 Area Deft. Stress

1 80 10 0.9 2.3 113.86 0.049213 20.41247 2 80 I 0 0.9 3.2 130.24 0.0402 i 6 13.59076 3 80 10 0.9 4.1 146.62 0.034253 9.955928 4 80 10 0.9 5 163 0.03002 7.719161 5 80 23.3 0.9 2.3 175.04 0.025778 0.185818 6 80 23.3 0,9 3.2 215.36 0,020007 0.487314 7 80 23.3 0.9 4.1 255,68 0.016491 0.590065 8 80 23.3 0.9 5 296 0.01413 0.629231 9 80 36.7 0.9 2.3 236.68 0.01742 1.760328

10 80 36.7 0.9 3.2 301.12 0.013282 i.447877 11 80 36.7 0.9 4.1 365.56 0.010832 !.241703 12 80 36,7 0.9 5 430 0.009215 1.09808 13 80 50 0.9 2.3 297.86 0.013179 1.858706 14 80 50 0.9 3.2 386.24 0.009959 1.45284 15 80 50 0.9 4.1 474.62 0.00808 1.2075 i 16 80 50 0.9 5 563 0.00685 1.044104

MIN 113.86 0.00685

e r ror relat ive to a given curve, specified by a set o f points . The object ive funct ion is a measure o f the e r ror between the pa th ob ta ined and the desired pa th . This e r ror is usual ly expressed as the sum o f the squares o f the e r ror at each po in t in the pa th . The object ive funct ion is eva lua ted by de te rmin ing the coord ina tes o f the precis ion po in t a t var ious pos i t ions o f the mechan i sm for a pa r t i cu la r design and this is accompl i shed using basic geometr ic and t r igonomet r ic relat ions. The m i n i m u m mechanica l advan tage for one comple te cycle is also an i m p o r t a n t pe r fo rmance cr i te r ion as it de te rmines the a m o u n t o f power tha t can be t r ansmi t t ed by the mechanism. This is general ly c o m p u t e d by pe r fo rming a veloci ty analysis o f the mechan i sm and is expressed as the ra t io o f input to ou tpu t velocity. The op t imiza t ion p rob l em for a fou r -ba r mechan i sm is thus s ta ted as fol lows (Fig. 4):

P =~ ( p , _ p~)2 + ( p , _ p~.,)2

Minimize F 2p (2)

Plot of non4nfedor set

i

0.05

0.045

0.04

0.035

0.03

0.025

0.02

0.015

0.01

0.005

0 I I I I I

100 200 300 400 503 Cross Section

Fig. 3. Graph of cross section area against deflection.

e00

I O ~ 1 1

A robust multi-criteria optimization approach 805

x4

x3

xl xO (x

x7

Fig. 4. Typical four bar mechanism.

Maximize MA = Min[xl COo/V~] (3)

where i = position

p = number of points COo = angular velocity of input link V, = velocity of coupler point at position i

Subject to

Grashof Condition--xL + Xs <~ xe + xo

Crank Rocker--x~ < x0, x2, x3 Variable Bounds--x~nj ~< xj ~< Xmaxj

where P~, P,* are the x and y coordinates of the obtained points, P]~, P~, are the x and y coordinates of the desired points, xjs are the link lengths, xL is the longest link length, Xs is the shortest link length, xp and xe are the lengths of the other two links of the four bar. The Grashof condition checks assemblability at all positions. The crank rocker constraint ensures that the crank is the smallest link in the four-bar and hence can be rotated by external means. The desired curve is specified as a set of coupler coordinates that the coupler point traces at discrete angular rotations of the crank, i.e. it is a path synthesis problem with prescribed timing.

The variable bounds constraint is easily taken care of in Taguchi's method by ensuring that the chosen levels are within the specified range. The treatment of the other two constraints requires special consideration since the violation of either of them results in non-assemblability of the mechanism or in a non-drivable crank. The crank of an assemblable four-bar mechanism has to be the smallest link to satisfy the crank rocker constraint. This can be ensured by normalizing all the design variables with respect to the crank and selecting variable bounds so that the length of any link is greater than that of the crank. This procedure essentially reduces the crank length to a constant (say unity) and each design represents a family of coupler curves obtained by varying the crank length and proportionally modifying the design variables. The coupler curves generated by the various designs of any family are similar in shape but vary in the magnitude of the coupler points obtained. This is analogous to similar triangles (or any geometry) wherein two similar triangles may be of different sizes. Moreover, two similar curves will have the same mechanical advantage (objective function # 2) at analogous points since the ratio of input to output velocity does not change for different crank lengths. Thus we now need a procedure to parametrically compare two curves so that the amount of deviation (objective function # 1) can be quantified irrespective of the size (crank size in particular) of the mechanisms generating them. This procedure is briefly described in the following paragraph. It should be noted that we have conveniently taken

MMT 32/7--B


care of two of the constraints and the third constraint, namely the Grashoff criterion, needs to be accounted for during the optimization process.

Kota [6] presents a methodology to parametrically compare two discretized symmetric curves by computing the parametric length and angle at each of the coupler points for both the curves. He then determines the deviation in the parametric angle between the two curves at specified parametric lengths. The square of these deviations summed over all the points gives the total deviation between the two curves. For the purposes of the problem considered in our example, this procedure is modified to include non-symmetric curves and prescribed timing and is detailed below.

The first step is to obtain the coupler points of the generated curve and arbitrarily select a starting point. The parametric angle at any other point is computed as the difference between the tangential angle of the curve at this point and that at the starting point. The deviation, F, is then obtained as follows:

•/,• (0" - 0~)2

F = (4) n

where 0* and 0~ are the parametric angles of the generated and desired curves respectively at position i of the coupler. In order to match the two curves irrespective of their position or orientation, F is measured with every point on one of the curves as the starting point and the minimum of these values gives the deviation between the two curves.

Thus the prescribed timing mechanism path synthesis problem has been reduced to a 5 variable problem with two objectives and one constraint as shown below (see Fig. 5):

Minimize F (as in 4)

Maximize MA (as in 3)

such that

XL + 1 < xe-t- xQ.

The orientation of the curve has been accounted for in the objective function computation. As a result, the coordinates of the crank-ground joint (x5, x6), the orientation of the ground link (x7), and the crank angle (x8) are no longer variables (note that variable x9 has been changed to x5 in Fig. 5).

The standard L25 orthogonal array allows five levels for up to six factors and is considered suitable for this problem. The levels of the five design factors along with the experiment layout for the initial iteration are shown in Table 4. Uncertainty is deliberately introduced into the design

x4

xl

c x 3

////////!

Fig. 5. Modified dimensional synthesis problem.


Table 4. Initial experiment set-up

807

xl x2 x3 x4 x5

1 1. 5000 2. 5000 2. 5000 O. 5000 0.0000 2 1.5000 3.2500 3.2500 1.5000 1.2571 3 1. 5000 4,0000 4.0000 2. 5000 2.5143 4 1.5000 4.7500 4.7500 3.5000 3.5968 5 1. 5000 5.0000 5. 5000 4. 5000 4.8540 6 2.2500 2.5000 3.2500 2.5000 3.5968 7 2.2500 3.2500 4.0000 3. 5000 4.8540 8 2.2500 4.0000 4.7500 4.5000 0.0000 9 2.2500 4.7500 5.5000 0.5000 1.2571

10 2.25 O0 5. 5000 2. 5000 1. 5000 2.5143 I1 3.0000 2.5000 4.0000 4.5000 1.2571 12 3.0000 3. 2500 4. 7500 O. 5000 2.5143 13 3.0000 4.0000 5.5000 1.5000 3.5968 14 3.0000 4.7500 2. 5000 2. 5000 4.8540 15 3.0000 5.5000 3.2500 3.5000 0.0000 16 3.7500 2. 5000 4.7500 1. 5000 4.8540 17 3. 7500 3.2500 5. 5000 2. 5000 0.0000 18 3.7500 4.0000 2.5000 3.5000 1.2571 19 3.7500 4.7500 3.2500 4. 5000 2.5143 20 3.7500 5.5000 4.0000 0.5000 3.5968 21 4. 5000 2. 5000 5. 5000 3. 5000 2.5143 22 4. 5000 3.2500 2. 5000 4. 5000 3. 5968 23 4.5000 4.0000 3.2500 0.5000 4.8540 24 4.5000 4.7500 4.0000 1.5000 0.0000 25 4. 5000 5. 5000 4. 7500 2. 5000 1.2571

by the L8 o r thogona l a r r a y (noise a r ray) which al lows two levels o f noise for each o f the factors. The levels in the noise a r r a y are set by assuming a + 1% var ia t ion in the link lengths and angles. Each i te ra t ion will thus involve 25 experiments . Here, each exper iment is being repl icated eight t imes with differing values o f noise factors resul t ing in a to ta l o f 200 funct ion evalua t ions o f each object ive and cons t ra in t funct ion. The mechan i sm op t imiza t ion p rob lem is different f rom typical mechanica l design op t imiza t ion p rob lems as the object ive funct ions will have an inde te rmina te value at non-assemblab le posi t ions, i f the G r a s h o f condi t ion is not satisfied. To a c c o m m o d a t e this s i tuat ion, A N O V A is pe r fo rmed only on the cons t ra in t funct ion until all fac tor levels causing a cons t ra in t v io la t ion are e l iminated. Once this is accompl ished , the object ive funct ions can be eva lua ted for any design in the exper imenta l design set-up. No te tha t this will general ly not be necessary in mos t design op t imiza t ion p rob lems and the A N O V A of the cons t ra in t and object ive funct ions can be pe r fo rmed in paral lel .

The A N O V A tables o f the cons t ra in t funct ion for the first two i tera t ions are shown in Tables 5 and 6. In Table 5 it is seen tha t design factors 2 and 3 have a p r e d o m i n a n t effect (i.e. with the highest percentage con t r ibu t ions ) on the cons t ra in t funct ion and levels 5 and 1 respectively o f these factors are found to have large mean effect values. Thus these levels are e l iminated and the second i te ra t ion is pe r fo rmed on the reduced space. Similarly, the A N O V A o f the second i te ra t ion indicates tha t the first level o f the first fac tor causes a v io la t ion and hence it is e l iminated. Al l the designs cons idered in the th i rd i te ra t ion are found to satisfy the cons t ra in t and thus A N O V A can be pe r fo rmed on the object ive functions. No te tha t as the object ives for this p rob lem are not conflicting, only the mean effect values o f the var ious factors are required to de te rmine their o p t i m u m range. The design factors are b racke ted between the o p t i m u m levels o f the indiv idual

Table 5. ANOVA of constraint function for first iteration

Mean effects x Level l Level 2 Level 3 Level 4 Level 5 SOS DOF MS F % Cont

0 0.000 0.350 0.I00 0.000 0.000 0.46 4 0.12 4.230 0.84 2 0.000 0.000 0.000 0.050 0.400 0.61 4 0.15 5.609 5.86 3 0.400 0.050 0.000 0.000 0.000 0.61 4 0.15 5.609 5.86 4 0.000 0.350 0.050 0.050 0.000 0.44 4 0. l I 4.000 0.00 5 0.050 0.000 0.350 0.000 0.050 0.44 4 0.11 4.000 0.00 Error 0.43 4 0. l I 87.44 Total 2.98 24 100.00


Table 6. A N O V A of constraint function for second iteration

x Level 1 Level 2 Level 3 Level 4 Level 5 SOS D O F MS F % Cont

1 0.2500 0.1000 0.0000 0.0000 0.0000 0.24 4 0.061 70.668 87.41 2 0.0625 0.0625 0.0625 0.0625 0.1000 0.01 4 0.00 4.000 0.00 3 O.lO00 0.0625 0.0625 0.0625 0.0625 0.01 4 0.00 4.000 0.00 4 0.0625 0.1000 0.0625 0.0625 0.01 4 0.00 4.000 0.00 5 0.0625 0.0625 0.1000 0.0625 0.0625 0.01 4 0.00 4.000 0.00 Error 0.01 4 0.00 12.59 Total 0.27 24 100.00

objectives, which are shown highlighted in Table 7. For the subsequent iteration, the bracketed levels are used as the new range, and the A N O V A (Table 8) is again performed on both objectives. The optimum levels of each factor for each objective for this iteration are shown highlighted in Table 8 and the reduced design space is:

x0 = 3.9375, 4.6094 < x2 < 4.75 (2 levels), X3 = 3.5312,

0.5 < x4 < 1.5 (5 levels), 3.9548 < x5 < 4.854 (2 levels).

These levels identify 20 designs (2 x 5 x 2) from which the Pareto-optimal set shown in Table 9 is determined. Thus we obtain six non-dominated designs that can be considered to be optimal in a robust sense.

Table 7, A N O V A of objectives 1 and 2 for third iteration

A N O V A table for objective No. 1 x Level 1 Level 2 Level 3 Level 4 Level 5 SOS DOF MS F % Cont

0 42.958 41.309 54,595 54.950 56.163 1043.48 4 260.87 13.140 15.12 2 47,245 50.664 53.322 53.795 44.946 295.08 4 73.77 3.716 - 0 . 4 7 3 59.890 54.012 44.459 45,953 45.657 899.29 4 224.82 11.324 12.12 4 47.754 59.199 49.657 46.605 46.756 559.17 4 139.79 7.041 5.03 5 44.988 56.922 56.637 55.287 36.136 1686.19 4 421.55 21,233 28.51 Error 317.65 4 79.41 39.70 Total 4800.86 24 100.00

A N O V A table for objective No. 2 x Level 1 Level 2 Level 3 Level 4 Level 5 SOS DOF MS F %Cont

0 - 19.903 - 15.171 - 11.663 -- 11.611 - 11.859 259.80 4 64.95 15.97 10.35 2 - 16.491 -14 .989 -15 .906 - 11.827 -- 10.995 122.81 4 30.70 7.553 3.07 3 -- 10.491 - 13.269 -14 .911 -15 .044 -16 .492 104.86 4 26.22 6.44 2.12 4 -- 5.023 -8 .668 -14 .787 -17 .140 -24 .590 1158.1 4 289.53 71.22 58.10 5 -11 .760 -16.581 -16 .598 -15 .083 -- 10.186 170.68 4 42.67 10.49 5.62 Error 65.04 4 16.26 20.74 Total 1881.3 24 100.00

Table 8. A N O V A of objectives 1 and 2 for fourth iteration

A N O V A table for objective No. 1

x Level 1 Level 2 Level 3 Level 4 Level 5 SOS DOF MS F % Cont

0 50.189 49.021 48.288 48.226 48.344 13.89 4 3.47 29.05 4.94 2 48.547 48.870 48.472 49.183 48.996 1.80 4 0.45 3.767 - 0 . 0 5 3 48.807 48,683 48.755 48.737 49.087 0.51 4 0.13 1.060 - 0 . 5 8 4 44.860 46.711 48.754 50.866 52.878 203,9 4 50.99 426.59 83.34 5 47.507 48.082 49.277 50.034 49.169 20.36 4 5.09 42,575 7.61 Error 1.91 4 0.48 4.73 Total 242.4 24 100.00

A N O V A table for objective No. 2

x Level 1 Level 2 Level 3 Level 4 Level 5 SOS DOF MS F % Cont

0 -- 4.596 - 4 . 7 1 2 -4 .811 -5 .037 - 5 . 1 0 6 0.93 4 0.23 4.142 0.04 2 --5.278 - 4 , 7 1 4 - 5 . 3 1 0 -4 ,699 --4.261 3.92 4 0.98 17.418 3.41 3 --4.822 --5.033 - 4 . 9 1 9 -4 .945 - -4 .543 0.71 4 0.18 3.166 --0.21 4 -- 3.233 - 3 . 9 4 0 - 4 . 9 0 6 - 5 . 5 3 8 -6 ,645 35.71 4 8.93 158.76 39,31 5 - 5 . 0 4 4 - 6 . 4 1 4 - 5 . 8 3 8 -4 ,503 - -2 .463 46.38 4 11.60 206.20 51.36 Error 0.90 4 0.22 6,10 Total 88.56 24 100.00


Table 9. Non-dominated designs

809

xO x2 x3 x4 x5 S/NI Obj. 1 S/N2 Obj. 2

I 3.937 4.75 3.5312 0.50 4.854 44.896 0.106 -- 1.631 0.922 2 3.937 4.75 3.5312 0.75 4.854 46.994 0.095 - !.891 0.910 3 3.937 4.75 3.5312 1.00 4.854 49.443 0.084 -2.171 0.897 4 3.937 4.75 3.5312 1.25 4.854 52.284 0.073 --2.470 0.884 5 3.937 4.75 3.5312 1.50 3.954 56.993 0.058 -5.606 0.755 6 3.937 4.75 3.5312 1.50 4.854 55.529 0.062 -2.784 0.870

0.95

0.9

0.85

0.8 ]E

0.75

0.7 0.O4

Plot of NonJnfedor Set

I I I I I t

0.05 0.06 0.07 0.08 0.09 0.1 0.11

Deviation

Fig. 6. Non-dominated design set.

Figure 6 shows a plot of the six non-dominated designs obtained after the fourth iteration. It should be noted that the number of non-dominated designs obtained and the accuracy of the solution depends on the number of iterations, each of which involves a total of 200 function evaluations of each objective. Generally, a large number of iterations will result in a larger non-dominated design set. For example, it was seen that for three iterations, three non-dominated designs were obtained and for five iterations 16 were obtained. Moreover, as each iteration refines the design space, the discretization interval reduces and therefore the quality of the solution becomes better with every iteration. Thus the termination criterion for the procedure is determined by the trade-off between the number of non-dominated designs required, the solution accuracy and the number of function evaluations.

An added advantage in obtaining a Pareto-optimal set of designs as shown in Fig. 6 is that it clearly depicts the amount of one criterion that must be sacrificed so as to obtain a particular improvement in the other criterion. Thus the designer has a clear idea of the available range of choice and is in a position to make an intelligent decision.

CONCLUSIONS

In this paper, an efficient framework for the identification of a non-dominated design set for multi-criteria optimization problems with constraints is presented. This work employs a statistical design of experiment formulation for the purposes of finding robust design solutions to problems with multiple objectives. In particular, a novel two-step procedure is developed that utilizes ANOVA results to handle constraints and to identify a Pareto-optimal solution set based on relative effects o f the various factors on the objective functions. A salient feature of this approach is its consistent treatment of conflicting objectives that enables determination of such


Pareto-optimal solutions based on a rigorous categorization of design factors. Thus, this technique eliminates the need for dealing with trade off decisions necessary for computing pre-set weights to reduce the multiple objectives to a single expression. The potential application of this technique to engineering design problems is discussed with the aid of an illustrative beam design and a mechanism dimensional synthesis problem.

Acknowledgements--The authors gratefully acknowledge the support of the National Science Foundation under Grant No. CMS-9402608.

REFERENCES

1. Roy, R. K., A Primer on the Taguchi Method. Van Nostrand Reinhold, New York, 1990. 2. Kacker, R. N., Journal of Quality Technology, 1985, 17(4), 176-188. 3. Bagchi, T. P., Taguchi Methods Explained--Practical Steps to Robust Design. Prentice-Hall of India, New Delhi, 1993. 4. Dehnad, K., Quality Control, Robust Design, and the Taguchi Method. Wadsworth & Brooks/Cole, California, 1989. 5. Song, A. A., Mathur, A. and Pattipati, K. R., IEEE Transactions on Systems, Man and Cybernetics, 1995, 25(11),

1437-1446. 6. Kota, S. and Chiou, S. J., Mechanism and Machine Theory, 1993, 2~(6), 777-794. 7. Otto, K. N. and Antonson, E. K., ASME Design Theory and Methodology, DE-Vol. 31, 1991. 8. Cohon, J. L., in Design Optimization, ed. John S. Gero. Academic Press, New York, 1985. 9. Cohon, J. L., Multiobjective Programming and Planning. Academic Press, New York, 1978.

10. Osyczka, A., in Design Optimization, ed. John S. Gero. Academic Press, New York, 1985. 11. Charnes, A. and Cooper, W. W., Management Models and Industrial Applications of Linear Programming, Vol. 1. Wiley, New York, 1961. 12. Keeney, R. L. and Raiffa, H., Decisions and Multiple Objectives: Preferences and Value Tradeoffs. Wiley and Sons, New York, 1976. 13. Von Neumann, J. and Morgenstern, O., Theory of Games and Economic Behavior, 3rd edn. Princeton University Press, Princeton, N.J., 1953. 14. Thurston, D. L., Carnahan, J. V. and Liu, T., ASME Design Theory and Methodology, DE-Vol. 31, 1991. 15. Box, G., Technometrics, 1988, 30(1), 1-17. 16. Leon, R. V., Shoemaker, A. C. and Kacker, R. N., 1987, Technometrics, 1987, 29(3), 253-265. 17. Majchrzak, J., in Aspiration Based Decision Support Systems, eds A. Lewandowski and A. P. Wierzbicki. Springer-Verlag, New York, 1989. 18. Tribus, M. and Szonyi, G., in Quality Progress, May 1989.

E I N R O B U S T E R M E H R K R I T E R I E N - A N S A T Z Z U R O P T I M I E R U N G

Zusammen[asstmg--Der gesamte Wert eines Produktes wird allgemein dutch seine Leistungsf'~ihigkeit bezi~glich vielfiiltiger Faktoren bestimmt. Somit wird die Aufgabe des Produktentwurfs vereinfacht, wenn all diese Leistungsmerkmale gleichzeitig optimiert werden k6nnen. Ein anderer bedeutsamer Faktor, der die Qualitaumlautt des Produkts bestimmt, ist dessen Empfindlichkeit gegeniiber aiisseren oder nicht kontrollierbaren Schwankungen. Fiir das Einbeziehen dieser Punkte in den Produktentwurf wird in diesem Aufsatz ein neuer robuster Mehrkriterien-Ansatz zur Optimierung vorgestellt, der Mehrzweck- Optimierungskonzepte mit statistischen robusten Entwurfstechniken erg/inzl. In diesem Ansatz werden Mengen von Pareto-optimalen robusten Entwurfsl6sungen mit Hilfe des Entwurfs yon Versuchsaufbauten gewonnen, die ANOVA Ergebnisse zur Quantifizierung von relativer Dominanz und Signifikanz von Entwuffsfaktoren ausnutzen. Die Anwendung dieser Methode auf ingenieurswissenschaftliche Entwurfsprobleme wird anhand zweier Fallbeispiele dargestellt, die ein Dimensionales Syntheseproblem eines Mechanismus beinhalten.