The overall value of a product is typically determined by its
performance with respect to multiple measures. As such the product
design task is very much simplified if all these performance measures
were optimized simultaneously rather than sequentially. 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 methodical
approach that integrates multi-criteria optimization concepts
with statistical robust design techniques. In this approach,
which includes a systematic treatment of constraints, the results
are presented as a set of non-inferior design solutions. These
solution sets are obtained from the initial design space based
on a statistical design of virtual experiment set-up, that utilizes
ANOVA results to quantify the relative dominance and significance
of design factors.
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 (Roy, 1990; Kacker, 1985) have been extensively applied in process design, 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 (Kota and Chiou, 1993 ; Otto and Antonsson, 1991). 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) a systematic treatment of constraints and (2) a formulation for multiple objectives.
Multiple criteria optimization techniques can be broadly classified into preference based methods and generating methods (Cohon, 1985). 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 (Charnes and Cooper, 1961) and utility analysis (Keeny and Raiffa, 1976 ; vonNeuman and Morgenstern, 1947) 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 utility function. A salient feature of utility analysis is that it can be used to evaluate alternatives under uncertainty. Generating methods (Cohon, 1985), 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 (Cohon, 1985) 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 (Otto and Antonsson, 1991). 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 log [MSD] 
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 -transformations may become necessary (Box, 1988).
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 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 S/N 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 mean
response. This information is then used to predict the optimum
settings for the various design factors.
Robust Design For Multiple Responses With Constraints
Taguchi's method, in its as is form, does not handle multiple objective formulations or constraints. This paper extends Taguchi's method to address such cases. The problem setup for the multiple responses case is similar to the single response case except that each objective and constraint function will result in a separate ANOVA table. A two step procedure is then followed to identify the non-inferior set :
1) Eliminate all factor levels that cause any of the constraints to be violated.
2) Identify those factor levels, in the reduced space, that significantly effect any of the objectives.
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 objective. It has a significant effect if the percentage contribution is greater than a designer specified cut-off value. Details on this cut-off value and its selection are addressed later in this section.
The signal-to-noise ratio for the constraint functions are set up for the larger is better case if it is a 'lesser than' constraint (and vice versa), so that the factors that cause a violation of the constraint can be easily identified. Using the ANOVA table, the effect of the various factor levels on the constraint functions are first considered. If the ANOVA of a constraint function indicates that it is significantly effected by a factor, the level of the factor causing the violation is 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. Note that before comparing constraint mean effects with the constraint limit, the latter has to be transformed by the same transformation that is used to obtain the S/N ratio from the constraint equation. In this manner, the non-feasible designs are filtered out from the initial design space.
The ANOVA of each of the objective functions are then considered, and the design space is 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 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 (Majchrzak, 1989). 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 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.
Problem Formulation And Results
The application of this methodology to multi-criteria optimization problems is demonstrated by a beam design problem (Osyczka, 1985) involving two objectives and a single constraint. The objective of this problem is to determine the dimensions of a beam (Figure 1), 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/cm2.
Young's Modulus of Elasticity (E) is 2*104 kN/cm2.
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 = f1(x) = 2 x2 x4 + x3 ( x1- 2 x4 )
Minimize Vertical Deflection = f2(x) = P L3 / 48 E I
Subject to the constraint
Maximum Stress = f3(x)
where I is the Moment of Inertia of the beam cross section.
The geometric constraints are :
10 x1 80, 10 x2 50, 0.9 x3
5, 0.9 x4 5.
Figure 1 : Beam design problem
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 degrees of freedom of the system is 12. The L16 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 effect 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 variation in the design variables were treated as the appropriate noise factors and an 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. These are listed in Table 2. Table 3 lists the results of ANOVA for each objective function and constraint equation.
It is seen from the ANOVA of the constraint equation (Table 3c) that the factors x1, 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 2c to be the first level for all three factors. It is further observed that the mean
Table 1 : Experiment set-up and S/N ratios
effects corresponding to these levels of x1 and x2 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 3a and 3b) show that all the four factors (x1 - x4) have a significant effect on the cross section area (objective 1) whereas only the factors x1, x2, and x4 significantly influence the deflection of the beam (objective 2). Additionally, x1 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 x1 that optimizes deflection (level 4) is thus singled out for further consideration, regardless of its effect on the weight of the beam. The factors x2 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.
a) Objective 1
b) Objective 2
Table 2 : Mean response for all objectives and constraints
Table 3 : Analysis of variance for all objectives and constraints
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 2a. The factor levels that optimize the respective objectives, after eliminating the levels that violate the constraints, are shown highlighted in Tables 2a and 2c. Subsequently, the design space is narrowed down and the new levels (bounds) for the design factors are identified as follows:
x1 = 80, 10 x2 50, x3 = 0.9, 2.3 x4 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 x1 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 4. The
graphical representation of the resulting objective functions
for all the non-dominated designs is shown in Figure 2. 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.
Table 4 : Non-inferior design set
Figure 2 : Graph of cross section area against deflection
In this paper, an efficient framework for the identification of
a non-dominated design set for multi-criteria optimization problems
with constraints is presented. Its application to engineering
design problems is discussed with the aid of an illustrative beam
design problem. This work employs statistical robust design techniques.
In particular, it utilizes ANOVA results to uniquely identify
a Pareto-optimal design solution set based on the relative dominance
and significance of the factors on the objective functions. The
nature and size of this final solution set is a function of the
cut-off value specified by the designer for the identification
of significant factors. The determination of this value is, in
general, a much simpler task than making the trade off decisions
necessary for computing weights to reduce the multiple objectives
to a single expression. For the purposes of optimization, these
salient features make this efficient robust design method an effective
solution procedure in engineering design, especially for constrained
multiple objective problems.
The authors gratefully
acknowledge the support of the National Science Foundation under
Grant No. CMS-9402608.
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