Function-transformation methods for multi-objective optimization

Engineering Optimization Vol. 37, No. 6, September 2005, 551–570

Function-transformation methods for multi-objective optimization R. TIMOTHY MARLER and JASBIR S. ARORA* Optimal Design Laboratory/CCAD, College of Engineering, University of Iowa, Iowa City, IA 52242, USA (Received 14 July 2004; revised 30 November 2004) It is useful with multi-objective optimization (MOO) to transform the objective functions such that they all have similar units and orders of magnitude. This article evaluates various transformation methods using simple example problems. Viewing these methods as different means to restrict function values sheds light on how the methods perform. The weighted sum approach for MOO is used to study how well different methods aid in depicting the Pareto optimal set. Whereas using unrestricted weights is well suited for providing a single solution that reflects preferences, it is found that using a convex combination of functions is desirable when generating the Pareto set. In addition, it is shown that some transformation methods are detrimental to the process of generating a diverse spread of points, and criteria are proposed for determining when the methods fail to generate an accurate representation of the Pareto set. Advantages of using a simple normalization–modification are demonstrated. Keywords: Multi-objective optimization; Transformation; Normalization; Weighted-sum

1.

Introduction

A general multi-objective optimization (MOO) problem is stated as follows: Minimize: x

F(x) = [F1 (x) F2 (x) · · · Fk (x)]T

subject to: gj (x) ≤ 0; hl (x) = 0;

(1)

j = 1, 2, . . . , m l = 1, 2, . . . , e

where k is the number of objective functions, m the number of inequality constraints, e the number of equality constraints, and x ∈ E n is a vector of design variables. The feasible design space is defined as X = {x|gj (x) ≤ 0, j = 1, 2, . . . , m; and hi (x) = 0, i = 1, 2, . . . , e}, and the feasible criterion space is defined as Z = {F(x)|x ∈ X}. The idea of a solution point for problem (1) can be unclear, because a single point that minimizes all objectives simultaneously usually does not exist. Consequently, the idea of Pareto *Corresponding author. Email: [email protected]

Engineering Optimization ISSN 0305-215X print/ISSN 1029-0273 online © 2005 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/03052150500114289

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optimality is used to describe solutions for MOO problems. A solution point for problem (1) is Pareto optimal if and only if it is not possible to move from that point and improve at least one objective function without detriment to any other objective function. Alternatively, a point is weakly Pareto optimal if it is not possible to move from that point and improve all objective functions simultaneously. Typically, problem (1) has infinitely many Pareto optimal solution points. Thus, one must determine or select a single suitable solution. Two basic approaches can be used. In the first approach, one declares preferences for various objectives, incorporates the preferences into a modified formulation of problem (1) and solves the modified problem to obtain a single Pareto optimal point. This is called a priori articulation of preferences. Preferences are often incorporated using a scalarization method in which multiple objectives are aggregated into a single function. If the resulting solution is acceptable, then the solution process is terminated. However, most scalarization methods do not transfer preferences from the user to the final solution with complete accuracy. Thus, if the solution is not acceptable, the preferences are altered, and the problem is re-solved to obtain another Pareto optimal point. Alternatively, the second approach entails generating a representation of the entire Pareto optimal set and then choosing from that set a suitable solution point that satisfies preferences. This can be achieved by solving a series of MOO problems and consistently varying parameters (such as weights) in order to yield a series of Pareto optimal points. This approach is called a posteriori articulation of preferences. With respect to a posteriori articulation of preferences, it can be difficult to select parameters (especially when the objective functions in problem (1) have significantly different orders of magnitude) that generate an accurate representation of the Pareto optimal set and allow for the selection of an appropriate final solution. Therefore, this paper studies different functiontransformation (and normalization) methods that can help in generating an approximation of the Pareto optimal set. To this end, the weighted sum method for MOO is used as representative of the many scalarization approaches, which are common, relatively fast, and relatively easy to implement. 1.1 Motivation and objectives Although the literature on MOO methods is vast (Marler and Arora 2004), not much has been written concerning function-transformation or normalization methods. In some instances, MOO methods are presented without transformed objective functions, and normalization is simply recommended for objectives of different units. In other instances, MOO methods are used with no mention of function transformation. This trend is especially prevalent with theoretical studies and with evaluations of specific MOO methods. In addition, some work concerning practical applications has also been conducted without function transformations. Although some transformation methods are mentioned in the literature, there is no systematic study or evaluation of their advantages and disadvantages. There is no indication as to why one approach or another should be used. In addition, when function transformation is used, often the intent is simply to produce functions with similar units; the complete range of potential function values resulting from a transformation is not considered. Accordingly, different transformation approaches are evaluated in this article by comparing their effect on the depiction of the Pareto optimal set. To this end, the weighted sum approach for MOO is used, which is a simple and common scalarazation approach that always provides a Pareto optimal point. The following specific objectives are pursued: 1. describe and discuss various function-transformation methods in terms of potential numerical difficulties and in terms of imposed limits on function values;

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2. evaluate these methods with respect to their ability to generate an accurate representation of the Pareto optimal set using the weighted sum approach for solving MOO problems.

2.

Overview of the weighted sum method

As a simple approach to solving problem (1), the weighted sum method provides a convenient foundation for this study. It entails minimizing the following scalarized objective function: U=

k


wi Fi (x)

(2)

i=1

function. In this study, a convex where wi > 0 is the weighting factor for the ith objective  combination of functions is used, which implies that ki=1 wi = 1 and wi > 0. In some sense, the weights wi serve as scale factors for the objective functions. That is, they can be selected such that each term wi Fi has a different contribution to the composite function in equation (2). This way, preferences are expressed for various functions. The minimum of the weighted sum in equation (2) is always Pareto optimal. In addition, minimizing the weighted sum can yield all of the Pareto optimal points (with systematic variation in the weights) if the MOO problem is convex (Miettinen 1999). However, the method cannot yield points on non-convex portions of a Pareto optimal hypersurface. The minimum of the weighted sum clearly depends on the relative magnitude of the terms wi Fi . However, which Pareto optimal point results from a particular set of weights can depend on other factors as well, as explained subsequently. The weights represent the gradient of the expression in equation (2) with respect to F in the criterion space. Thus, changing the weights’ relative values changes the orientation of the contours for the weighted sum. The subsequent affect on the solution depends on the shape of the Pareto optimal hypersurface in the criterion space. In addition, writing the Kuhn–Tucker necessary conditions for the minimum of equation (2) reveals that the Pareto optimal solution point resulting from the use of a specific set of weights also depends on active constraints that form part of the Pareto optimal set and on the relationship between the gradients of the different objective functions.

3. Transformation methods Although the weighted sum method is eventually used in this study to depict the Pareto optimal set, the present analysis is applicable to any MOO approach. A common approach to function transformation is incorporated in the following three schemes, the most basic of which is given as (Chen et al. 1999, Koski and Silvennoinen 1987, Zhang and Yang 2002): Fitrans (x) =

Fi (x) |Fio |

(3)

where Fio = minimum{Fi (x)|x ∈ X}. This is referred to as the lower-bound approach, and x

it provides a non-dimensional objective function. The lower limit of Fitrans (x) is restricted to negative one (it is positive one if Fio > 0), whereas the upper value is unbounded. With this approach, division by zero (or a very small number) can lead to numerical difficulties.

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Equation (3) may be modified as follows (Adali et al. 1995, Eschenauer et al. 1990, Gupta and Sivakumar 2002, Hwang et al. 1979, Osyczka 1978, Salukvadze 1979, Saravanos and Chamis 1992): Fi (x) − Fio Fitrans (x) = (4) |Fio | Equation (4) is the alternate lower-bound approach and it yields non-dimensional objective function values with a lower limit of zero. As with equation (3), computational difficulties can arise if the denominator is close to zero. A second variation on equation (3) is given in equation (5), often in the context of the global criterion method (Makaraci 2000, Osyczka 1984, Rao 1996): Fitrans (x) =

Fio − Fi (x) Fio

(5)

However, even when absolute value signs are incorporated (Osyczka 1984, Surdacki and Montusiewicz 1996), the motivation for this approach is unclear. Consequently, equation (5) is not evaluated in this study. As an alternative to the methods discussed earlier, one may use the maximum value of the function in the denominator rather than Fio . The consequent upper-bound approach is shown as follows (Proos et al. 2001): Fi (x) Fitrans = max (6) Fi where Fimax represents the maximum value for objective-i. Equation (6) provides a nondimensional function value such that Fitrans ≤ 1 with no restriction on the lower value. As with equations (3), (4), and (5), division by zero is possible though less common. The following transformation approach is called the upper-lower-bound approach (Koski 1984, Koski and Silvennoinen 1987, Rao and Freiheit 1991, Yang et al. 1994): Fitrans =

Fi (x) − Fio Fimax − Fio

(7)

In this case, Fitrans (x) generally has a value between zero and one, depending on the accuracy and the method with which Fimax (x) and Fio (x) are determined. Unlike previous approaches, the denominator is guaranteed to be positive. In addition, this is the only approach that constrains the upper and lower limits of Fitrans (x). Consequently, equation (7) provides a relatively robust approach. Rao (1987) and Rao et al. (1988) use the following transformation scheme, which is often called scaling: Fitrans = mi Fi (x)

such that m1 F1 (xs ) = m2 F2 (xs ) = · · · = mk Fk (xs ) = constant

(8)

mi are scalar coefficients and xs is a feasible starting point. This approach ensures that the objective functions have similar orders of magnitude, but only at the point at which the coefficients are determined. That is, this approach constrains the function values at the starting point rather than at the function’s upper and/or lower limits. However, the significance of the starting point may be arbitrary; its relevance to the feasible space may be unknown. In addition, the appropriate value for the constant is not always clear and may be arbitrary as well, although using a value of one can constitute a form of normalization. Technically, equation (8) is similar to equations (3) and (6). Conceptually, however, the coefficients in equations (3) and (6) have significance in terms of their range of potential function values. The coefficients are not as difficult to select as those in equation (8). For these reasons, equation (8) is not considered further.

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Determining function maxima

With reference to equations (6) and (7), Fimax may be determined as the absolute maximum (if it exists) of Fi (x) or as an approximation of the maximum. Technically, such an approach limits the maximum of the transformed function to one. However, primary concern is with the objective function values at points within the Pareto optimal set, not just within the feasible design space. Yet, there is no guarantee that the individual maxima of the objective functions are even within the vicinity of the Pareto optimal set. Therefore, when determining Pareto optimal points, the absolute maximum of a function may be irrelevant. Alternatively, an approach more conducive to MOO is to define Fimax such that Fimax = max1≤j ≤k Fi (xj∗ ), where xj∗ is the point that minimizes the j th objective function. xj∗ is a vertex of the Pareto optimal set in the design space and F(xj∗ ) is a vertex of the Pareto optimal set in the criterion space. Such a maximum is called the Pareto maximum. It approximates the nadir vector, which has components defined by the upper bounds of the Pareto optimal set in the criterion space, but is difficult to determine exactly (Korhonen et al. 1997). Using a Pareto maximum does not necessarily eliminate the possibility of a maximum transformed function value other than one. This is because non-Pareto optimal points may be encountered during the running of the optimization algorithm. In addition, the vertices of the Pareto optimal set only provide approximate bounds on function values within the set (Weistroffer 1985). Nonetheless, using the Pareto maxima rather than the absolute maxima can improve performance, as will be demonstrated. This approach for determining Fimax mentioned briefly by Miettinen (1999) has been used with membership functions for fuzzy MOO (Dhingra et al. 1992, Rao et al. 1992) and is included as a component of some methods (Chen et al. 1999, Rao 1987, Tabucanon 1988). However, it has not been adopted consistently for function transformation in the context of MOO. Consequently, its potential benefits in terms of normalization have not been realized.

4. 4.1

Evaluation of methods Problem statement

Whereas the previous section provides a basic understanding and analytical evaluation of the transformation methods, the following example problem, illustrated in figure 1, is used to evaluate the various methods computationally:  2 2  F1 = 25(x1 − 0.5) + (2x2 − 2) + 0.1 Minimize : (9) F2 = [(x1 − 2.5)2 + 4(x2 − 1.8)2 ]2 x   F3 = (x1 − 2.0)4 + 1.5(x2 − 2.8)2 + 0.3x1 x2 + 10 subject to :

g1 = (x1 − 2.1) − 0.08(2.2 − x2 )2 ≤ 0 g2 = −x1 ≤ 0 g3 = −x2 ≤ 0 g4 = x2 − 3.0 ≤ 0

The complete Pareto optimal set for this problem is shown in the design space in figure 1 and in the criterion space in figure 2. This illustrative, constrained problem is designed to yield clearly distinguishable sets of points in both two-dimensional design space and three-dimensional criterion space, with some points resulting in active constraints. For each function, the difference between the maximum

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Figure 1.

Pareto optimal set in the design space.

and the Pareto maximum is significant, as shown in table 1. Each row shows the values of the objective functions when one of the functions is minimized independently. For example, the first row contains values of F1 , F2 , and F3 calculated at the point x1∗ , which minimizes F1 . Each column contains the values of a function at different points. For instance, the column of F1 values contains values of F1 calculated at the point that minimizes F1 , the point that minimizes F2 , and the point that minimizes F3 . The lightly shaded boxes indicate the maximum value of a function, when that function is evaluated at each of the three points; this is the Pareto maximum. The darker boxes indicate the minimum of each function, i.e. the utopia point. The last row indicates the absolute maximum of each function. Clearly, the different functions compete with one another; what reduces one function may increase the other two. No function has a minimum value of zero that could result in an

Figure 2.

Pareto optimal set in the criterion space.

Multi-objective optimization Table 1. Function values

557

Function-comparison matrix. F1 -values

F2 -values

F3 -values

At x1∗

0.1000

43.0336

20.0725

x2∗ x3∗

67.6807

0.0224

12.6562

32.0687 84.2615

16.9994 144.2401

11.2757 37.7600

At

At Maximum

undefined value for a transformed function. This problem is used to demonstrate how solution points may cluster in certain regions, and new criteria are presented for anticipating this condition with certain transformation methods. 4.2

Unrestricted weights

As suggested in reference to the weighted sum, a convex combination of functions is used in this study, because using weights with unrestricted values can result in an arbitrary selection process when determining weights for depicting the Pareto optimal set. In fact, to highlight this potential difficulty, the consequences of not using a convex combination are first demonstrated. Two of the three weights (one for each objective function) are fixed at 1.0, whereas the other is varied from 0 to 1000.0 in increments of 20.0. Then, while one weight is fixed at 1.0, the other two are increased from 0 to 1000.0 in increments of 20.0. This results in 306 solution points. Using a weight of 1.0 essentially models the case when a function is relatively insignificant, but is still considered in the problem (i.e. does not have a weight of 0). The resulting Pareto optimal points are shown in figure 3. When these results are compared with figure 2, it is clear that the Pareto optimal set is not well represented. Another example of this approach to selecting weights is shown in figure 4. In this case, it is assumed that a weighting vector of (50,50,50) indicates equal significance between the objectives. The weights are varied around this point in the weight space. Two of the three weights are fixed at 50.0, whereas the other is varied from 0 to 100.0 in increments of 2.0. Then, while just one weight is fixed at 50.0, the other two are increased from 0 to 100.0 in

F3

Utopia Point

20

0 10

F 1-minimum 20

20 40

30

40

F3

F2

20

F 3-minimum

60 F 2-minimum

F1

20 10 0 60 40

20

30

40 F 2

F1

Figure 3.

Pareto optima without transformation and with unrestricted weights – criterion space.

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R. T. Marler and J. S. Arora F3

Utopia Point

20 F 1-minimum 0 10

20

20

30

40

F3

F2

20

F 3-minimum 40 60 F 2-minimum

F1

20 10 0 60 40

20

30

40 F 2

F1

Figure 4.

Pareto optima without transformation and with unrestricted weights (second version) – criterion space.

increments of 2.0. This results in 306 solution points. Again, the Pareto optimal set is not well represented, as seen in figure 4. Although the weight-selection processes outlined earlier are somewhat arbitrary, it is reasonable to use such approaches if a convex combination is not used. Thus, the potential for poor results is demonstrated. As minimizing the weighted sum always results in a Pareto optimal point, the solution points for the two preceding problems collectively represent the Pareto optimal sets. Clearly, the results do not provide complete representations of the Pareto optimal set. Of course, it is theoretically possible to select weights with unrestricted values such that the complete Pareto optimal set is represented. However, to do so would require a difficult trial-and-error process that one can improve upon by using a convex combination of functions, as will now be demonstrated.

4.3 Convex combination of functions In this section, the results of different transformation methods are illustrated and discussed. A series of weighted sum problems is solved using a convex combination of objective functions and each problem provides one point. The weights are incremented as follows. The first of the three weights is incrementally increased from 0.0 to 0.33 and to 0.66. With the first weight fixed at a particular value, the other two weights are varied simultaneously in increments of 0.025 (one is increased whereas the other is decreased) such that a convex combination of functions is used. This process is repeated for each weight. Then, one of the three weights is increased from 0 to 1 in increments of 0.05, whereas the other two weights, assigned the same two values, are reduced simultaneously such that a convex combination of functions is used. This results in 309 solution points, which is approximately the same number of points used in the two previous cases. Technically, using a weight of 0 can yield weakly Pareto optimal points. However, such weighting values are used with the example problems in order to illustrate the independent minima of the objective functions. As a point of reference, the Pareto optimal solution points that are obtained when no function transformation is used are shown in figures 5 and 6.Although there is a significant improvement compared with figures 3 and 4, the solution points tend to cluster in one area rather than spread out evenly over the entire Pareto optimal set. The points appear to shift away from

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F 3-minimum

x2 2.5 2.0

F 2-minimum

1.5 1.0

F 1-minimum

0.5

x1

0.0 0.0

0.5

Figure 5.

1.0

1.5

2.0

2.5

Pareto optima without transformation – design space.

the minimum of F3 , toward the minima of F1 and F2 . This clustering of Pareto points is, in part, a consequence of the values for F1 and F2 shown in table 1. Although F1 has a low minimum value and F2 has the highest ratio of Pareto maximum to minimum, at x2∗ and at x3∗ the value of F1 is relatively high, so F1 dominates the weighted sum at these points. In addition, F1 has a relatively high average value, which is a general indication of how much a function dominates the weighted sum when no transformation method is used. At x2∗ , F2 has the highest value of the three functions, so it dominates the weighted sum at this point. F3 has the lowest average value, which suggests that even slight differences in the ranges of values for objective functions (within the Pareto optimal set) can have a detrimental effect when the functions are not transformed. Theoretically, unrestricted weights can be used to compensate to some extent for differences in function values. However, as suggested earlier, using such weights in a systematic fashion can be difficult. Thus, having resolved to use weights that yield a convex combination of functions, the next issue is restricting the values and ranges of the functions themselves.

F3

Utopia Point

20 F 1-minimum 0

10

20

30

20

40

F3

F2

20

F 3-minimum 40 60 F 2-minimum

F1

20 10 0 60 40

20

F1

Figure 6.

Pareto optima without transformation – criterion space.

30

40 F 2

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F 3-minimum

x2 2.5 2.0

F 2-minimum

1.5 1.0

F 1-minimum

0.5

x1

0.0 0.0 Figure 7.

0.5

1.0

1.5

2.0

2.5

Pareto optima with upper-bound approach and absolute maximum – design space.

The upper-bound approach for transformation, given in equation (6), acts to equalize the upper bounds of the functions by dividing by the maximum of each function. This improves distribution of the Pareto optimal points, as shown in figure 7. However, only Pareto optimal points are of interest, and there is no guarantee that the point that maximizes a function is close (in the design space or in the criterion space) to the Pareto optimal set. For example, the absolute maximum of F2 is 144.2401, whereas the Pareto maximum is 43.0336. Thus, it makes more sense to use the Pareto maximum. The results from such an approach, shown in figure 8, reflect slight improvements. A similar increase in the dispersion of solution points occurs in the criterion space. Intuitively and practically, using the Pareto maximum provides better results than using the absolute maximum. In addition, there is no guarantee that an absolute maximum exists for every objective function. Although the denominator is the function minimum rather than the maximum, the lowerbound approach given in equation (3) is conceptually similar to the upper-bound approach. Both approaches essentially scale each objective function with a constant. However, the results can be significantly different as demonstrated in figures 9 and 10. The transformation actually F 3-minimum

x2 2.5 2.0

F 2-minimum 1.5 1.0

F 1-minimum

0.5

x1

0.0 0.0

Figure 8.

0.5

1.0

1.5

2.0

2.5

Pareto optima with upper-bound approach and Pareto-maximum – design space.

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F 3-minimum

x2 2.5 2.0

F 2-minimum 1.5 1.0

F 1-minimum

0.5

x1

0.0 0.0

Figure 9.

0.5

1.0

1.5

2.0

2.5

Pareto optima with lower-bound approach – design space.

worsens the results, and in this case, using the alternate lower-bound approach in equation (4) yields similar consequences. F3 plays an insignificant role in governing which Pareto optimal solution points are provided. That is, the results shown in figures 9 and 10 are the same as they would be if F3 was not considered in the problem. The results in figures 5–8 can be improved upon and the degradation shown in figures 9 and 10 can be avoided by using the upper-lower-bound approach given in equation (7), as shown in figures 11 and 12. In conjunction with a Pareto maximum, equation (7) provides a reliable approach that consistently improves the distribution of Pareto optimal solution points. 4.4 Discussion Despite the success of the upper-lower-bound approach, the question arises as to why the degradation occurs with the lower-bound approach and when it can be anticipated. In this F3

Utopia Point

20 F 1-minimum 0 10

20

20

30

40

F3 F2

20

F 3-minimum 40 60

F1

F 2-minimum

20 10 0 60 40

20

F1

Figure 10.

Pareto optima with lower-bound approach – criterion space.

30

40 F 2

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x2 2.5 2.0

F 2-minimum 1.5 1.0

F 1-minimum

0.5

x1

0.0 0.0

Figure 11.

0.5

1.0

1.5

2.0

2.5

Pareto optima with upper-lower-bound approach and Pareto-maximum – design space.

section, simple criteria are developed that can be used to anticipate the performance of the lower-bound and upper-bound transformation methods. The afore-mentioned difficulties may seem to surface because the minima of F1 and F2 are relatively small when compared with their respective Pareto maxima. Thus, when the functions are transformed using the lower-bound approach, the transformed versions of F1 and F2 dominate the weighted sum, and F3 essentially becomes irrelevant. This does, in fact occur, but it is not the only cause for difficulty. If this was the only cause for difficulty, then the following ratio would indicate whether one transformed function would dominate other transformed functions: Pareto-maximum value ρ= (10) minimum value The values of equation (10) for each function would indicate whether using the lower-bound approach would be effective. However, if the ratio in equation (10) was always applicable, then one would expect similar difficulties if the upper-bound approach was used with functions that have a relatively large Pareto maximum. 1/ρ would be used in place of ρ, and division F3

Utopia Point

20 F 1-minimum 0 10

20

20

30

40

F2

F 3-minimum 40

F3 20

60 F 2-minimum F1

F1 Figure 12.

20 10 60 40 0

20

30

40 F 2

Pareto optima with upper-lower-bound approach and Pareto-maximum – criterion space.

Multi-objective optimization Table 2.

F1 F2 F3

563

Ratio values. ξ

ς

675.807 1920.143 0.780

0.999 0.999 0.439

by the large number would result in relatively small, insignificant values. However, figures 7 and 8 show that this is not necessarily the case. Is there a unified process for explaining the results of both the lower-bound approach and the upper-bound approach? Additional examples demonstrate that what dictates the success of the lower-bound approach is the following ratio: ξ=

(Pareto-maximum value) − (minimum value) minimum value

(11)

Generally, if this ratio is similar for different objective functions, then using the lower-bound approach will not be detrimental. Thus it is found that both the range of function values and the ratio between the Pareto maximum and minimum values have an effect on transformation results. A similar metric is provided for the upper-bound approach (with the Pareto maximum), as follows: (Pareto-maximum value) − (minimum value) ς= (12) Pareto-maximum value If the ratio in equation (12) is similar for different functions, then using the upper-bound approach will not be detrimental. The ratios given in equations (11) and (12) are calculated for this problem and shown in table 2. Note that the values for ξ , which relate to the lower-bound approach, are significantly different, thus explaining the poor results depicted in figures 9 and 10. Although the values for ς are similar, the value corresponding to F3 is slightly lower than the other two, explaining the acceptable, but less than ideal results shown in figure 8.

5. 5.1

Example problems Problem 1

With this first example, which concerns the optimal design of the three-bar truss shown in figure 13, the advantages of the upper-lower-bound approach and the Pareto maximum are again demonstrated. In addition, whereas previous results illustrate the clustering of solution points, this example demonstrates more severe potential consequences when functions are not transformed. In figure 13, L = 0.1 m, the modulus of elasticity E = 200 × 109 N/m2 , and the load F = 20 × 103 N. F is applied simultaneously in both the negative x- and y-directions. This structural design problem is a variation of the problem presented in Koski (1985) and Koski and Silvennoinen (1987). The areas of the members (in units of m2 ) are the design variables. The total volume of the truss and the stress in member 1 are the objectives, both of which are minimized. Member 1 is considered because it is most likely in compression and thus subject

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y

L 3

L

2

1

L

3

F

x F

Figure 13. Three-bar truss.

to buckling. Limits are placed on the design variables. The MOO problem is formulated as follows:  F1 = stress in member 1 = σ12 √ (13) Minimize : A1 ,A2 ,A3 F2 = total volume = A1 L 2 + A2 L + A3 2L subject to : 0.0004 ≤ Ai ≤ 0.002;

i = 1, 2, 3

Although one is typically concerned with the absolute value of stress, σ 2 is minimized to avoid computational difficulties. Table 3, which is similar to table 1, shows the values of the objective functions when each function is minimized independently. The vector A∗s is the point in the design space that minimizes stress and A∗v is the point that minimizes volume. In this case, the two different functions have substantially different orders of magnitude and ranges. However, the absolute maxima are not significantly different from the Pareto maxima, as was the case previously. A series of weighted sum problems is solved using a convex combination of objective functions. The weight for stress is increased from 0 to 1 in increments of 0.02, while the weight for volume is decreased in the same increment. When no function transformation is used, the only two resulting solution points are the individual minima for each objective function, because stress dominates the weighted sum. Consequently, the minimum stress is found for all sets of weights until w1 is zero, in which case the minimum volume is found. Note that the minimum value for the stress in member 1 is ∼0, which occurs when the areas of the three members are such that the displacement of the node where the loads are applied is perpendicular to element 1. The minimum volume is achieved at the point where all areas are at their lower limit, i.e., A1 = A2 = A3 = 0.0004. Table 3.

Function-comparison matrix for problem 1.

Function values At A∗s A∗v

At Maximum

(Stress)2 -values

Volume-values

2.9504E−05

6.57550E−04

3.13822E+14 8.93531E+14

1.76569E−04 8.82843E−04

Multi-objective optimization Table 4.

565

Ratio values for Problem 1.

Stress Volume

ξ

ς

1.06366E+19 2.72404

1.0 0.73147

The same results are obtained when the lower-bound and alternate lower-bound approaches are used. These results can be anticipated using the values for the ratios given in equations (11) and (12), as shown in table 4. ξ is 1.06366E+19 for stress and 2.72404 for volume, which clearly indicates the dominance of stress. These results can have severe consequences, as they imply that there are only two Pareto optimal designs. However, the use of proper transformation schemes can alleviate this misleading difficulty. In reference to the upper-bound approach (with a Pareto maximum), ς is 1.0 for stress and 0.73147 for volume. These values are similar, and consequently, results are improved, as shown in figures 14 and 15. Note that although the maximum value for stress is relatively large, using it as a denominator does not yield detrimental results. The Pareto optimal solution is a function of A3 primarily, as shown in figure 14. For most of the trials, A1 and A2 have a shared value of 0.0004 (their lower limit), thus reducing the volume. A3 varies between 0.0004 and 0.002 and redirects the consequent displacement, thus changing the value of stress in member 1. The gap between the minimum point for stress and the rest of the Pareto optimal set is a consequence of an extremely high slope for the Pareto optimal set in the criterion space. If additional weighted sum problems are solved with the increment in the weights reduced to the order of 1 × 10−6 , then solution points begin to fill this gap. Further improvements in the distribution of Pareto optimal points are achieved by using the upper-lower-bound approach with the Pareto maximum. 5.2 Problem 2 With this example, the significance of equations (11) and (12) are clearly illustrated. In addition, the importance of restricting both upper and lower bounds of a transformed function A3 0.002

0.001 Stressminimum

0

0.0015

0.00053 0.0015

0.00053

Volumeminimum

A2

A1

Figure 14.

Problem 1, Pareto optima with upper-bound approach and Pareto-maximum – design space.

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Volume 7.0E-04 6.0E-04 5.0E-04 4.0E-04 3.0E-04 2.0E-04 1.0E-04 0.0E+00 0.0E+00 Figure 15.

5.0E+06

1.0E+07

1.5E+07

Stress 2.0E+07

Problem 1, Pareto optima with upper-bound approach and Pareto-maximum – criterion space.

is illustrated. Although this problem is similar to Problem 1, here the stress in member 3 is minimized, rather than the stress in member 1. The stress in member 3 is objective function 1, and the volume is objective function 2. With the given loading condition, there is no chance that member 3 is subject to compression. Consequently, the stress values for this member are always positive and are not squared as they are in Problem 1. Table 5 shows the values of the objective functions when each function is minimized independently. As with Problem 1, the two different functions have substantially different orders of magnitude. However, in this case, the range of values for stress is significantly smaller than for Problem 1. In addition, the absolute maxima are identical to the Pareto maxima. When no transformation is used, results are similar to those in Problem 1; there are only two solution points and they represent the individual minima of the two functions. However, in contrast to Problem 1, results with the lower-bound approach in equation (3), the alternate lower-bound approach in equation (4), and the upper-bound approach in equation (5), all are nearly identical and are illustrated in figures 16 and 17. One can anticipate these results using the ratios from equations (11) and (12), as shown in table 6. The values in table 6 suggest that either the lower-bound or the upper-bound approach may be used to improve original results and this prediction is substantiated by the results. Using the upper-lower-bound approach in equation (7) yields results similar to those provided in figures 16 and 17. In fact, with this particular problem, every transformation method produces results in which the minimum value for the transformed version of stress is the same as the minimum value for the transformed version of volume. Concurrently, the maximum value for the transformed version of stress is the same as the maximum value for the transformed version of volume. In addition, as is always the case when there are only two objectives, the minimum of one function corresponds to the Pareto maximum of the other function and viceversa. Consequently, it is not just the upper-lower-bound approach, but every approach that Table 5.

Function-comparison matrix for problem 2.

Function values

Stress-values

Volume-values

At A∗s

8.65417E+06

8.82843E−04

At A∗v Maximum

4.32708E+07 4.32708E+07

1.76569E−04 8.82843E−04

Multi-objective optimization

Figure 16.

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Problem 2, Pareto optima with upper-bound approach – design space.

constrains the upper and lower limits of the functions. Although this can occur naturally with certain problems, using the upper-lower-bound approach with a Pareto maximum ensures that it occurs regardless of the nature of the problem. Thus, although using the proposed normalization approach is not necessary for improving results, it is sufficient. In other words, the upper-lower-bound approach is not always required for acceptable results, but it provides a safeguard against potential difficulties. Figure 16 is similar to figure 14 (for Problem 1) in that much of the Pareto optimal curve is a function of A3 . However, in this case, once A3 reaches its upper limit of 0.002, then A2 is

Figure 17.

Problem 2, Pareto optima with upper-bound approach – criterion space.

Table 6. Ratio values for Problem 2.

Stress Volume

ξ

ς

4.0 4.0

0.8 0.8

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forced to increase. Eventually, A1 increases until all areas reach their upper limits and stress is minimized.

6.

Discussion and conclusions

This article has provided an analysis and comparison of methods for transforming objective functions in a MOO problem. The idea of using transformation methods to limit the potential values of functions in different ways has been introduced and exploited. In general, it has been shown that function transformation is beneficial. In particular, the use of the upper-lowerbound approach in equation (7), along with the Pareto maximum, is robust and ensures that the transformed functions are dimensionless and have similar orders of magnitude. This can help prevent any single objective from dominating an aggregated function, such as the weighted sum. It has been shown that some transformation methods can actually worsen results, and criteria have been proposed for determining when such methods are detrimental. Although the performance of MOO methods depends on more than just relative function values, function transformation is one way to improve performance. The criteria involving the ratios in equations (11) and (12) are valid regardless of whether a convex combination of functions is used (whether the values of the weights are restricted), except for one caveat. When unrestricted weights are used, one of the reasons for clustering is that there is no systematic approach for ensuring an even, consistent sampling of the weight space in terms of the weights’ relative values. If such sampling is carefully enforced, then a convex combination of functions is inadvertently used, with which each function is multiplied by the same scalar that represents the largest common denominator among the weights. Multiplying the weighted sum by a scalar does not affect the solution point, and the results are thus the same as with a convex combination of functions. The clustering of points in a specific area can reduce the effectiveness of a method. This phenomenon can depend on four factors: the relative function values, which can be altered using transformations; the shape of the Pareto optimal set in the criterion space; the active constraints; and the relationship between function gradients. Discussing each of these facets is beyond the scope of this study. Rather, it has focused on the first issue: the significance of function values and transformation methods. The example problems presented earlier have direct relevance to a posteriori articulation of preferences. Transformed objective functions can improve a method’s ability to depict the complete Pareto optimal set in the design space and in the criterion space, with consistent variation in parameters such as weights. The examples also apply to an a priori articulation of preferences. Clustering of Pareto optimal points can be caused by, among other things, the numerical dominance of a function or functions. When one selects a single set of weights with an a priori articulation of preferences, the weights themselves should indicate preference, rather than the values of the functions. However, depending on how the weights are set, the numerical dominance of a function may work against this goal, thus reducing the effectiveness of the method. That is, if functions are not transformed, the solution may be determined in part by the relative magnitude of the functions, rather than by the relative size of the weights alone. Thus, transforming functions so that they have comparable values can help one set parameters more accurately. Note that using transformations to eliminate the potential for the numerical dominance of a function is not always sufficient for ensuring acceptable results. Rather, one must eliminate the potential for numerical dominance in any one of the terms, wi Fi , in equation (2). As these terms suggest, the value of a weight is significant relative to the values of the other weights and

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relative to the values of the objective functions. This study has demonstrated that one achieves superior results when a convex combination of normalized objective functions is used, because the weights have the same range of values as the transformed objective functions. Thus, it is best to use normalization in conjunction with a convex combination of functions. In considering the terms, wi Fi , it is apparent that there is subtle distinction between using weights to transform functions and using weights to indicate preferences (or to depict the Pareto optimal set). This idea is touched on indirectly by Hunt et al. (2004) and is elaborated here. In equation (8), weights are used as a transformation method to change the nature of fi , and as suggested, they are relatively ineffective in this capacity. Thus, throughout this study, the weights have been treated somewhat independently from the objective functions. To clarify this distinction, consider a weighted sum formulated as follows: U=

k


wi [γi Fi (x)]

(14)

i=1

where wi provide the kind of weight discussed in reference to the weighted sum and γi represent separate weights used for transformation. Obviously, these two numbers can be multiplied together to provide a single coefficient. However, these two types of weights have separate roles, and thus, they should be considered separately. The weighted terms in equation (2) appear in many different scalarization methods, not just the weighted sum method. Consequently, the findings in this study are significant with respect to other MOO methods. Conclusions based on the present work are presented as follows. (1) Although using unconstrained weights (with no limits on their values) is possible for a priori articulation of preferences, it results in an arbitrary process for a posteriori articulation of preferences (depicting the complete Pareto optimal set). Using a convex combination of functions is advantageous in the latter case. (2) Using the minimum or maximum of a function as the denominator in a transformation scheme can result in numerical difficulties and/or poor performance in terms of depicting the complete Pareto optimal set, as predicted by the ratios in equations (11) and (12). (3) When using the upper-bound approach or the upper-lower-bound approach, one should use the Pareto maximum rather than the absolute maximum. (4) Using the upper-lower-bound approach improves the performance of the weighted sum method, and using this approach with a Pareto maximum provides superior results. This is the most robust approach to function transformation, and is recommended in conjunction with a convex combination of weights. Acknowledgement Support for this research provided by the Virtual Soldier Research Program, Center for Computer Aided Design at the University of Iowa, is gratefully acknowledged. References Adali, S., Richter, A. and Verijenko, V.E., Multiobjective design of laminated cylindrical shells for maximum pressure and buckling load. Microcomput. Civil Eng., 1995, 10, 269–279. Chen, W., Wiecek, M.M. and Zhang, J., Quality utility – a compromise programming approach to robust design. J. Mech. Des., 1999, 121, 179–187. Dhingra, A.K., Rao, S.S. and Kumar, V., Nonlinear membership functions in multiobjective fuzzy optimization of mechanical and structural systems. AIAA J., 1992, 30(1), 251–260.

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