The Normalized Normal Constraint Method for Generating the Pareto Frontier

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1 The Normalized Normal Constraint Method for Generating the Pareto Frontier Achille Messac Amir Ismail-Yahaya Christopher A. Mattson Corresponding Author Prof. Achille Messac Mechanical, Aerospace, and Nuclear Engineering Department 0 8 th Street, JEC-2049 Troy, NY, USA messac@rpi.edu Tel: (58) Fa: (58) Bibliographical Information Messac, A., Ismail-Yahaya, A., and Mattson, C. A., The Normalized Normal Constraint Method for Generating the Pareto Frontier, Structural and Multidisciplinary Optimization (In Press).

2 StrucOpt manuscript No. (will be inserted by the editor) The Normalized Normal Constraint Method for Generating the Pareto Frontier Achille Messac, Amir Ismail-Yahaya, and Christopher A. Mattson Abstract The authors recently proposed the normal constraint (NC) method for generating a set of evenly spaced solutions on a Pareto frontier for multiobjective optimization problems. Since few methods offer this desirable characteristic, the new method can be of significant practical use in the choice of an optimal solution in a multiobjective setting. This paper s specific contribution is two-fold. First, it presents a new formulation of the NC method that incorporates a critical linear mapping of the design objectives. This mapping has the desirable property that the resulting performance of the method is entirely independent of the design objectives scales. We address here the fact that scaling issues can pose formidable difficulties. Secondly, the notion of Pareto Filter is presented and an algorithm thereof is developed. As its name suggests, a Pareto filter is an algorithm that retains only the global Pareto points, given a set of points in objective space. As is eplained in the paper, the Pareto filter is useful in the application of the NC and other methods. Numerical eamples are provided. Key words Design optimization, Multiobjective optimization, Normal Constraint, Pareto generation, Pareto filter Introduction Multiobjective optimization (MO) plays an important role in engineering design, management, and decision making in general. Ultimately, a designer or decision Revised by Authors: 8 Sep 2002 Achille Messac, Amir Ismail-Yahaya, and Christopher A. Mattson Multidisciplinary Design and Optimization Laboratory; Department of Mechanical, Aerospace, and Nuclear Engineering; Rensselaer Polytechnic Institute, Troy, New York, 280, USA messac@rpi.edu maker needs to make tradeoffs between disparate and conflicting design objectives (i.e., design metrics). The field of multiobjective optimization defines the art and science of making such decisions. The prevailing approach for addressing this decision making task is to solve an optimization problem, which yields a candidate solution. This optimization problem generally requires imizing an objective function constraints, some of which are part of the original problem, while others may be new and part of the MO problem formulation. The generally accepted solution of an MO problem is said to be Pareto optimal, or, a Pareto solution. A Pareto solution is one where any improvement in one objective can only take place if at least one other objective worsens. This class of solutions is central to multiobjective optimization (Pareto (964), Pareto (97), Steuer (986), Belegundu and Chandrupatla (999), Grandhi and Bharatram (993)). There are several approaches to obtaining such solutions. These approaches can be considered as belonging to two classes. The first class involves forg an objective function constraints. This objective function when properly formulated represents the designer preference in its full compleity (Osyczka and Kundu (995), Cheng and Li (998), Srinivasan and Tettamanzi (996), Miettinen (999), Messac (996)). When this objective function is correct, the resulting design can be presumed optimal, and the design process could end. Unfortunately, formulating the objective function is typically an iterative process that involves physically meaningless numerical weights, which complicate the process. Such methods as goal programg (Steuer (986)) and physical programg (Messac (996)) offer important advantages in this regard, although goal programg requires that the decision maker choose weights. The second class of methods for obtaining a Pareto optimal design involves first generating a set of solutions, from which one will be selected. This paper is concerned with this generation of the set of Pareto solutions. One of the typical approaches to generating these solutions is to systematically vary the numerical scalar weights in an aggregate objective function (AOF), where each set of weights results in a corresponding Pareto solution. It is hoped in this strategy that a set of evenly distributed

3 2 scalar weights will yield a corresponding set of evenly distributed Pareto solutions. This even distribution of Pareto solutions is an indication that the design space is well represented in the Pareto set (i.e., Pareto frontier), which is important since a final solution will be chosen from this set. Unfortunately, most methods do not yield a well-distributed set of Pareto solutions, with notable eceptions (Das and Dennis (998), Messac and Sundararaj (2000), Ismail-Yahaya and Messac (2002)). It is important to note a critical distinction between the phase of generating Pareto solutions, which is objective; and the phase of choosing a Pareto solution from among the set, which is subjective. This latter phase depends entirely on designer or decision maker preference, while the former objectively seeks to generate Pareto points in the design space regardless of their relative desirability. As we choose a method to generate the Pareto frontier, it is desirable that this method harbor practical attributes: () The method should generate an even set of Pareto points in the design space and not neglect any region, (2) The method should have the ability to generate all available Pareto solutions, (3) The method should generate only Pareto solutions, and finally (4) The method should be relatively easy to apply. To the best knowledge of these authors, only the physical programg method satisfies all the above attributes. In fact, most methods fail with regard to the first attribute. Table below depicts the relative properties of notable methods for generating Pareto solutions: (i) the physical programg (PP) method, (ii) the normal boundary intersection (NBI) method (Das and Dennis (998)), (iii) the normal constraint (NC) method, (iv) the weighted sum (WS) method and, (v) the compromise programg (CP) method (Chen et al. (999)). Table Effectiveness of methods to generate Pareto solutions Attribute () (2) (3) (4) Generates Generates Generates Relatively even all available only easy to spread Pareto apply Pareto points points PP Y Y Y Y NBI Y Y N Y NC Y Y N Y WS N N Y Y CP N Y Y Y Y : yes; N: no Above, two classes of methods were discussed for obtaining an optimal solution. We note that the PP method can be successfully used for both (Messac (996), Messac and Sundararaj (2000)). The NBI and NC methods are primarily used for generating sets of Pareto solutions. The WS and CP methods are suitable for obtaining Pareto solutions; but ill-suited for the generation of Pareto sets. Attribute provides the assurance that all regions of the design space are adequately represented in the generated sampling. Interestingly, it is important to note that optimization methods that have Attribute are likely to be effective in allowing the designer to eplore the design space when the aggregate objective function parameters (or weights) are altered between optimization runs in the search for an optimal solution. Attribute 2 is also of critical importance. In cases where the desired Pareto optimal solution cannot be generated because of structural deficiencies of the method, the process fails in a fundamental way. Therefore due caution should be eercised in the use of the weighted sum method. Das and Dennis (997), Koski (985) and Messac and Ismail-Yahaya (200) offer important insights and findings regarding the deficiencies of the weighted sum in particular, and of objective functions in general (Messac and Ismail-Yahaya (200)). Attribute 3 is important since one should use non- Pareto solutions with etreme caution. A non-pareto solution implies that it is possible to find a better solution that entails no tradeoff. In general one does not wish to use such a design as the final choice. Accordingly, the two N s in Column (3) of Table are cause for concern. It is for this reason that this paper presents the novel notion of a Pareto filter. A Pareto filter is an algorithm that, given a set of design points in objective space, returns a set of design points that are all Pareto solutions (at least with respect to the original set provided). Finally, we note that the NBI method is more prone to generating doated solutions than is the NC method, as will be shown in an eample. This behavior originates from the different structures of these formulations. The former uses equality constraints where the latter uses inequality constraints. In addition, these methods use different approaches to bring insensitivity to design metrics scaling. Finally, it is important that when one uses a gradientbased local optimization algorithm, the resulting optima in all the cases of Table are potentially only locally Pareto optimal (see definitions in Section 4). Attribute 4 has important practical consequences. We note that if an algorithmic implementation of PP is available, it is almost as easy to apply as WS and CP. If not, then the application of PP is not simple, as the development of the PP algorithm is not trivial. Only the application of PP is simple, not the development of its algorithm. With regards to the compromise programg method, it is proven to overcome some of the drawbacks of the weighted sum method. The CP method is able to produce solutions on the non-conve regions of the Pareto frontier (Chen et al. (999)). A recent study conducted by Messac and Ismail-Yahaya has shown that there is a relationship between the order of the AOF and

4 3 that of the Pareto frontier for the compromise programg method to successfully generate Pareto solutions (Messac and Ismail-Yahaya (200)). Even though the CP method is capable of generating the Pareto frontier for conve and non-conve regions, it still shares some deficiencies of the weighted sum method; namely, it fails to generate a set of well-distributed solutions for a corresponding even distribution of weights. In addition to the methods presented in Table, other notable techniques have been employed. The Genetic Algorithm (GA) method was introduced by Holland (975). This method is based on the Darwinian theory of evolution. The method typically depends on a single-valued objective function (fitness function). Based on this principle, Osyczka and Kundu (995) have developed a method to transform a multicriteria optimization problem into a single criterion optimization, which is able to generate a set of Pareto solutions. However, doated solutions are also generated using this method. Cheng and Li (998) introduced the Pareto GA method based on the characteristic of the GA method to search for nondoated solutions. This technique necessitates certain rules to be followed, which are discussed comprehensively in Cheng and Li (998). This method also claims to generate fewer doated solutions compared to other GA methods. However, the Pareto GA method lacks the ability to generate a set of well-distributed Pareto solutions. Following the development of the GA method, Srinivasan and Tettamanzi (996) introduced the Evolutionary Algorithm (EA) method, which is based on the GA method. An EA generates candidate solutions by applying a set of operators, known as mutation, recombination, reproduction and selection. The EA method tends to locate the global imum solutions. The efficiency in generating the Pareto frontier through this method is claimed to be better in comparison with the GA. We note that the method does not have the ability to generate a set of well-distributed Pareto solutions. Within the contet of the literature discussed above, this paper presents a significant etension of the normal constraint method. This paper is structured as follows. Section 2 presents requisite mathematical preliaries. Section 3 presents the analytical development of the normalized normal constraint method, which redresses some scaling deficiencies of the original development of the NC method. Section 4 develops the Pareto filter, which eliates doated solutions from the generated set of points. Section 5 provides numerical eamples; and concluding remarks are provided in Section 6. 2 Mathematical Preliaries This section provides requisite mathematical preliaries. A generic formulation of the multiobjective optimization problem is presented. 2. The Multiobjective Optimization Problem We define the mathematical representation of the multiobjective optimization problem as follows. Problem P { µ () µ 2 () µ n () } (n 2) () g j () 0 ( j r) (2) h k () =0 ( k s) (3) li i ui ( i n ) (4) The vector denotes the design variables and µ i denotes the i th generic design metric (i.e., objective). In Problem P, (2) and (3) denote the inequality and equality constraints, respectively; and (4) is the side constraint. As stated, Problem P does not yield a unique solution. To acquire a single optimum solution, a statement of user preference is required. As part of the development of the normal constraint method, we require what we call anchor points, µ i,or optimum vertices. The generic i th anchor point (or, end point of the Pareto frontier) is obtained when the generic i th objective is imized independently. We call the line joining two anchor points in bi-objective cases the Utopia line, and the plane that comprises all anchor points in the multiobjective case the Utopia hyperplane. The word Utopia is used here to indicate that the plane contains the n optimum vertices, components of which form the Utopia point. We note that since the Utopia point is generally unattainable, it is not part of the Utopia plane. The anchor points (or optimum vertices) are obtained by solving Problem PUi, defined as follows. Problem PUi {µ i ()} ( i n) (5) g j () 0 ( j r) (6) h k () =0 ( k s) (7) li i ui ( i n ) (8) We now define the following quantities, which result from solving Problem PUi.

5 4 i : Optimal decision vector ( i R n ) µ i : Generic i-th optimal objective specifically, µ i = µ i (i )(µ i Rn ) µ u : Utopia Point µ u = [µ,µ 2,...,µ n] T (µ u R n ) µ i : i-th Anchor Point (µ i R n ) P u : Utopia Plane. Hyperplane in n-dimension that comprises the n anchor points, µ i (i =,...,n). 3 Overview of the Normalized Normal Constraint Method (a) General Design Metric Space This section provides the analytical development of the normalized normal constraint method for multiobjective optimization problems. This section is divided into two subsections: bi-objective and multiobjective cases. This separation of the bi-objective from the multiobjective cases is intended to promote clarity and simplicity of presentation. For simplicity, from here on the word normalized may be omitted, and implied. When we intend to discuss the original non-normalized case, we will simply say so. 3. Normal Constraint for Bi-Objective Case Let us begin with a graphical perspective of the normal constraint method. Figure (a) shows the nonnormalized design space and the Pareto frontier of a generic bi-objective problem. Figure (b) represents the normalized Pareto frontier in the normalized design space. In the normalized objective space, all anchor points are one unit away from the Utopia point, and the Utopia point is at the origin by definition. A bar over a variable implies that it is normalized. We now present the normal constraint method by defining a seven-step process for its application. To understand the idea of the NC method, consider Figures 2 and 3, for the bi-objective case. In Figure 2, we observe the feasible space and the corresponding Pareto frontier. We also note the two anchor points that are obtained by successively imizing the first and second design metrics (Problem PUi). A line joining the two anchor points is drawn, and is called the Utopia line. The Utopia line is divided into m segments, resulting in m points. In Figure 3, one of the generic points intersecting the segments is used to define a normal to the Utopia line. This normal line is used to reduce the feasible space as indicated in Figure 3. As can be seen, if we imize µ 2,the resulting optimum point is µ. By translating the normal line, we can see that a corresponding set of solutions will be generated. Importantly, we note that the generation of the set of Pareto points is performed in the normalized objective (b) Normalized Design Metric Space Fig. Design metric space for a bi-objective case 0 0 Fig. 2 A set of evenly spaced points on the Utopia line for a bi-objective problem space, which results in critically beneficial scaling properties. We now proceed to define the seven-step process that formalizes the preceding description. Step- : ANCHOR POINTS. Obtain the two anchor points, denoted by µ and µ 2, resulting from solving Problem PU and PU2 respectively. The line joining these two points is the Utopia line.

6 5 Feasible Region Infeasible Region Line NU δ = m (4) Step-5 : GENERATE UTOPIA LINE POINTS. Evaluate a set of evenly distributed points on the Utopia line as (see Fig. 2) 0 0 utopia line X Pj = α j µ + α 2j µ 2 (5) where Fig. 3 Graphical representation of the normal constraint method for bi-objective problems Step-2 : OBJECTIVES MAPPING/Normalization. To avoid scaling deficiencies, the optimization takes place in the normalized design metric space (design objective space). Let µ be the normalized form of µ. We define the Utopia point, µ u,as µ u = [ µ ( ) µ 2 ( 2 ) ] T (9) and we let l and l 2 be the distances between µ 2 and µ, and the Utopia point, µ u, respectively (see Fig. (a)). We have l = µ ( 2 ) µ ( ) (0) l 2 = µ 2 ( ) µ 2 ( 2 ) () Using the above definitions, the normalized design metrics can be evaluated as µ = { µ () µ ( ) µ 2 () µ 2 ( 2 } T ) (2) l l 2 Following the normalization of the design metrics, we can proceed to generate the Pareto points, as indicated in Figures 2 and 3. Step-3 : UTOPIA LINE VECTOR. Define N as the direction from µ to µ 2, yielding N = µ 2 µ (3) Step-4 : NORMALIZED INCREMENTS. Compute a normalized increment, δ along the direction N for a prescribed number of solutions, m,as 0 α j (6) 2 α kj = (7) k= We note that α ij is incremented by δ between 0 and (Fig. 2), and we use values of j where j {, 2,..., m }. Step-6 : PARETO POINTS GENERATION. Using the set of evenly distributed points on the Utopia line, generate a corresponding set of Pareto points by solving a succession of optimization runs of Problem P2. Each optimization run corresponds to a point on the Utopia line. Specifically, for each generated point on the Utopia line, solve for the j th point. Problem P2 (For j -th point) µ 2 (8) g j () 0 ( j r) (9) h k () =0 ( k s) (20) li i ui ( i n ) (2) N ( µ XPj ) T 0 (22) µ =[ µ () µ 2 ()] T (23) Step-7 : PARETO DESIGN METRICS VALUES. Evaluate the non-normalized design metrics that correspond to each Pareto point. This evaluation can be done in two ways. First, since the function µ() is known, the evaluation is direct. Alternatively, if the normalized design metrics were saved from Step-6, the non-normalized design metrics can be obtained through an inverse mapping of (2) by using the relation

7 6 µ = [ µ l + µ ( ) µ 2 l 2 + µ 2 ( 2 ) ] T (24) Up to this point we have not considered the possibility that some of the points generated in some pathological cases will be doated by other points in the set. This important situation is eaed in Section 4, where a Pareto filter is developed. 3.2 Normal Constraint for n-objective Case Here, we present the development of the normal constraint method for a general multiobjective case. This development will be terse to avoid repetitions with respect to the bi-objective case. The basic steps are similar to those of the bi-objective case. Step- : ANCHOR POINTS. Obtain the anchor points, µ i for i {, 2,..., n}, which are obtained by solving Problem PUi. We define the hyperplane, which comprises all the anchor points. This plane is called the Utopia hyperplane (or, Utopia plane). Figure 4 shows the Utopia plane for three design metrics. Recall that the optimum design variables obtained from solving Problem PUi are denoted by i. Step-2 : OBJECTIVES MAPPING/Normalization. To avoid scaling deficiencies, the optimization is performed in the normalized design metric space. In order to obtain the required mapping parameters, we need to define the two points: the Utopia point and the Nadir point, which are respectively evaluated as follows. µ i = µ i µ i ( i ) l i, i =, 2,..., n (29) Step-3 : UTOPIA PLANE VECTORS. Define the direction, Nk from µ k to µ n for k {, 2,..., n } as N k = µ n µ k (30) Step-4 : NORMALIZED INCREMENTS. Compute a normalized increment, δ k along the direction N k for a prescribed number of solutions, m k, along the associated N k direction. δ k = m k ( k n ) (3) Care must be taken in choosing the number of points, m k, for each direction N k. To ensure an even distribution of points on the n-dimensional Utopia plane, the following relationship can be used. Given a specified number of points, m, along the vector N, m k is given as m k = m Nk N (32) Step-5 : GENERATE HYPERPLANE POINTS. Evaluate a set of evenly distributed points on the Utopia hyperplane as µ u = [ µ ( ) µ 2 ( 2 ) µ n ( n ) ] T (25) n X Pj = α kj µ k (33) k= µ N = [ µ N µ N 2 µ N n where ] T (26) where 0 α kj (34) µ N i = ma [ µ i ( ),µ i ( 2 ),..., µ i ( n ) ] (27) i {, 2,.., n} We define the matri L (see Fig. (a)) l l 2... L = = µ N µ u (28) l n which leads to the normalized design metrics as n α kj = (35) k= Figure 4 describes how generic points are generated in the Utopia plane, where two planes serve as constraints (see (40)). Figure 5 shows the resulting uniformly distributed points on the Utopia plane for a threedimensional case in the normalized objective space. Step-6 : PARETO POINTS GENERATION. We generate a set of well-distributed Pareto solutions in the normalized objective space. For each value of X pj generated

8 7 N k ( µ Xpj ) T 0 ( k n ) (40) µ = { µ (),..., µ n ()} (4) In solving Problem P3 using a gradient-based algorithm, the initial point used contributes to the efficiency of the Pareto frontier generation. In the case of the normal constraint method, a good choice for the starting point is the point X pj. This automated scheme works well in practice. Step-7 : PARETO DESIGN METRICS VALUES. The design metrics values for the Pareto solutions obtained in Step 6 can be obtained using the equation µ i = µ i l i + µ i ( i ),i=, 2,..., n (42) Fig. 4 Utopia hyperplane for a three-objective case 4 Pareto Filter Development As indicated earlier, under contrived circumstances, the normal constraint method can generate non-pareto solutions (see Table ). This unfortunate situation can occur for eample in the case of a feasible space depicted in Figure 6, which would not be common. In such cases, we propose using a Pareto filter. A Pareto filter is an algorithm that, given a set of points in objective space, produces a subset of the given points where none will be doated by any other. That is, the filter eliates all doated points from the given set. Fig. 5 Evenly-spaced points on the Utopia plane for a threeobjective case in Step 5, we obtain the corresponding Pareto solution by solving Problem P3. Problem P3 (for j -th Point) µ n (36) g j () 0 ( j r) (37) h k () =0 ( k s) (38) li i ui ( i n ) (39) Fig. 6 Normal constraint generates a non-pareto solution under a contrived feasible space To facilitate the ensuing discussion, it is important to differentiate between local Pareto optimality and global Pareto optimality. We note that a global Pareto optimal

9 8 point is also a local Pareto optimal point, but the reverse is not typically true. Definition : A design metric vector µ is globally Pareto optimal if there does not eist another design metric vector µ such that µ i µ i for all i {, 2,..., n}, and µ j <µ j for at least one inde of j, j {, 2,..., n} in the feasible design space. Definition 2: A design metric vector µ is locally Pareto optimal if there does not eist another design metric vector µ such that µ i µ i for all i {, 2,..., n}, and µ j <µ j for at least one inde of j, j {, 2,..., n} in a neighborhood of µ. Consider Figure 6, where a highly concave feasible region is depicted. Arcs AB and FG represent the (global) Pareto frontier. Regions BC and DE are local Pareto frontiers. Arcs CD and EF are neither globally nor locally Pareto. We now make the important note that generating Pareto points using the normal constraint method with the anchor points A and G will yield non-pareto solutions. Point S for eample, a non-pareto solution, would be generated by the normal constraint method when the Line NU is used as the normal constraint. Using a different optimization starting point in a gradientbased scheme, we could obtain the point R. The NBI method would also suffer from this deleterious behavior. In light of the fact that certain methods yield non- Pareto solutions, it is important to develop a means to avoid retaining doated points in the set of solutions from which an optimum will be chosen. The Pareto filter does just that. The filter works by comparing a point on the Pareto frontier with every other generated point. If a point is not globally Pareto, it is eliated. Figure 7 shows the functional diagram of the Pareto filter. Figure 8 shows a detailed flow diagram of the Pareto filter. The algorithm is described in the following section. Fig. 7 Pareto Filter 4. Pareto Filter Algorithm This section presents the Pareto filter algorithm, both, through the presentation of a flow diagram and the description of a four-step process. Step- : Initialize Initialize the algorithm indices and variables: Fig. 8 Flow diagram of Pareto filter i =0,j =0,k =and m = number of generated solutions; m = f(m k ) Step-2 : Set i = i+; j =0 Step-3 : (enclosed in dashed bo): Eliate non-global Pareto points by doing the following j = j + If i = j go to the beginning of Step 3 Else continue If µ i µ j and ( µ i µ j) 0, s s then µ i is not a global Pareto point. Go to Step 4. Else if j = m Then µ i is a global Pareto point. p k = µ i k = k + Go to Step 4. Else go to the beginning of Step 3. Step-4 :Ifi m, go to Step 2, else end.

10 9 5 Numerical Eamples In this section, we use the normalized normal constraint method to generate Pareto frontiers in three eamples. The first eample deals with scaling issues where one design metric is orders of magnitude larger than the other, and compares the NC and WS methods. The second eample demonstrates a case where non-pareto points are generated, and compares the behaviors of the NC and NBI methods for the same problem. In the third eample, a truss problem is used, where we deal with a concave Pareto frontier and compare the relative behaviors of the NC and WS methods. 5. Eample Scalingissues Consider the multiobjective optimization problem below, for which we wish to generate the Pareto frontier { µ µ 2 } (43) µ = (44) µ 2 = 2 (45) ( ) 8 ( ) (46) 20 The normalized normal constraint method leads to the following single criterion optimization problem. superiority in the performance of the normalized NC method. The same data is plotted in Figure 9(b) and Figure 9(c). Figure 9(c) provides some insight as to why the second design metric is not well represented in the Pareto points generated in the non-normalized NC case. It is important to note that because the ais are not equally scaled in the figures, the Line NU does not appear normal to the Utopia line. We also note that the discussed evenness of the distribution of the Pareto points refers to good distribution for all design metrics. In other words, Figure 9(b) is said to have poor distribution because certain ranges of the second design metric are well represented, while other ranges are not. In Figure 9(a), such is not the case, and we say that we have an even distribution. Since the WS method is popularly used, we perform a comparison. We use the following formulation, and again use 20 equal increments of the weight w. J() =wµ ()+( w)µ 2 () (52) Figure 9(d) depicts the resulting Pareto points, which displays the usual poor performance of the WS method. 5.2 Eample 2 NC and NBI Methods, and Pareto Filtering Eample 2 compares the NC and NBI methods, and uses the Pareto filter. Consider the multiobjective optimization problem, for which we wish to generate the Pareto frontier. We are concerned with the feasible space shown in Figure 0. The global Pareto frontier is from A to B, and D to E. The region from C to D denotes a local Pareto frontier. The curve BC denotes a non-pareto frontier. The optimization problem takes the form µ 2 (47) { µ µ 2 } (53) µ = (48) µ 2 = 2 (49) ( ) 8 ( ) (50) 20 ) T N ( µ Xpj 0 (5) Solving the above problem with a normalized increment (δ ) of 0.05 yields twenty-one Pareto solutions. Figure 9(a) shows the generated Pareto solutions for the normalized NC method. For comparison purposes, the problem is also solved using the original non-normalized NC method (Ismail-Yahaya and Messac (2002)), which is shown in Figure 9(b). As can be seen, there is marked µ = (54) µ 2 = 2 (55) 5e +2e 0.5( 3)2 2 (56) Using the normal constraint method, the above problem leads to the single criterion optimization problem presented below for the j th point. µ 2 (57) µ = (58)

11 0 (a) Normalized Normal Constraint Fig. 0 Boundary of feasible space for Eample 2 µ 2 = 2 (59) 5e +2e 0.5( 3)2 2 (60) N ( µ XPj ) T 0 (6) (b) Non-normalized Normal Constraint Solving the above problem with a total of 60 solutions yields the Pareto sets as shown in Figure (a) and Figure (b) for the NBI and NC methods, respectively. The global Pareto points that result from the application of the Pareto filter are shown in Figure (c). We note that the NBI method generated global, local, and non-pareto points. In this eample, the NC method generated global and local Pareto points, but no doated solutions. We then applied the Pareto filter, which eliated all but the global Pareto points (c) Non-normalized Normal Constraint 5.3 Eample 3 Three-bar truss under static loading: NC, WS and Pareto Filtering (d) Weighted Sum Fig. 9 Pareto sets for Eample In this eample, we consider a three-bar truss structure from Koski (985). The structure and the loading conditions of the problem are shown in Figure 2. For this particular eample, the design metrics are: () the volume of the structure, and (2) a linear combination of the displacements at node P,. The design metrics are to be imized. The cross sectional areas of the three-bar truss are the design variables, which are allowed to vary between 0. cm 2 and 2 cm 2. The stresses in each bar are limited to 200 MPa. The length L is fied to 00 cm. The forces, F, which are applied at node P, have the same value of 20 kn. The modulus of elasticity of the material used is 200 GPa. Koski (985) used a linear combination of the displacement design metric,, at node P to yield nonconveity. The coefficients of δ and δ 2 are 0.25 and 0.75,

12 Fig. 2 Three-bar truss under static loading (a) Normal Boundary Intersection respectively. The resulting non-conveity is shown in Figure 3, which shows the feasible space boundary of the problem. 900 A Volume (cm 3 ) C B (b) Normal Constraint (cm) D Fig. 3 Boundary of feasible space of three-bar truss (c) Normal Constraint with Pareto filter Fig. Pareto sets for Eample 2 Before we proceed to generate the discrete solutions, let us comment on the region of interest in Figure 3. Regions AB and CD denote the Pareto frontier, while region BC consists of the doated solutions. With the normalized NC method, we generate the whole frontier. This generation includes regions AB and CD, but unfortunately also includes arc BC, which is not Pareto optimal. The result is generated with a total of 60 points. Figure 4(a) depicts the solutions generated by the normalized NC method. Figure 4(b) depicts the resulting set of points after applying the Pareto filter. Figure 4(c) shows results using the WS method, again using 60 evenly spaced weights in (52). We have represented the two obtained Pareto points by solid points to make them more visible. The salient observation here is that the weighted sum method performed quite poorly. Because of the order of magnitude difference between the two design metrics, it was not even possible to capture parts of the Pareto frontier that are conve using a resolution of 60 increments in weights. This is the case

13 2 even though in theory they can be captured by the WS method. To capture these points, one needs to iterate to find the appropriate scale of the weight Concluding Remarks This paper presented an important etension of the normal constraint method that redresses numerical scaling deficiencies of the original NC method. Specifically, a mapping is implemented at the level of the design metrics, which results in highly favorable numerical properties and in the ability to generate a well distributed set of Pareto points even in numerically demanding (illconditioned) situations. This paper also introduced the notion of a Pareto filter, which performs the function of eliating all but the global Pareto solutions when given a set of candidate solutions. This filtering approach is significantly simpler than using analytical means. Acknowledgement This research was supported by NSF award number DMI We also acknowledge the contributions of the Multidisciplinary Optimization Laboratory members in the development of the Pareto filter. We are also especially grateful to Anoop Mullur for his invaluable assistance in the final phase of this work. References Belegundu, A.; Chandrupatla, T. 999: Optimization concepts and applications in engineering. New Jersey: Prentice Hall, pp Chen, W.; Wiecek, M. M.; Zhang, J. 999: Quality utility - a compromise programg approach to robust design. Journal of Mechanical Design 2, pp Cheng, F.; Li, D. 998: Genetic algorithm development for multiobjective optimization of structures. AIAA Journal 36, No. 6, pp Das, I.; Dennis, J. 997: A closer look at drawbacks of imizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Structural Optimization 4, pp Das, I.; Dennis, J. E. 998: Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM Journal on Optimization 8, No. 3, pp Grandhi, R. V.; Bharatram, G. 993: Multiobjective optimization of large-scale structures. AIAA Journal 3, No. 7, pp Holland, J. H. 975: Adaptation in natural and artificial systems, Ann Arbor, MI: The University of Michigan Press Ismail-Yahaya, A.; Messac, A. 2002: Effective generation of the Pareto frontier using the normal constraint method. 40th Aerospace Sciences Meeting and Ehibit, Paper No. AIAA , Reno, Nevada Volume (cm 3 ) Volume (cm 3 ) Volume (cm 3 ) (cm) (a) Normal Constraint (cm) (b) Normal Constraint with Filtering (cm) (c) Weighted Sum Fig. 4 Pareto sets for Eample 3

14 3 Koski, J. 985: Defectiveness of weighting methods in multicriterion optimization of structures. Communications in Applied Numerical Methods, No. 6, pp Messac, A. 996: Physical programg: effective optimization for computaional design. AIAA Journal 34, No., pp Messac, A.; Sundararaj, G. J. 2000: Physical programg s ability to generate a well-distributed set of Pareto points. 4st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Paper No. AIAA , Atlanta, Georgia Messac, A.; Ismail-Yahaya, A. 200: Required relationship between objective function and Pareto frontier orders: practical implications. AIAA Journal, No., pp Miettinen, K. 999: Nonlinear multiobjective optimization, Massachusetts: Kluwer Academic Publishers, pp. 2-3 Osyczka, A.; Kundu, S. 995: New method to solve generalized multicriteria optimization problems using the simple genetic algorithm. Structural Optimization 0, No. 2, pp Pareto, V. 964: Cour d economie politique, Librarie Droz- Geneve (the first edition in 896) Pareto, V. 97: Manuale di economica politica, societa editrice libraria, Milano, Italy: MacMillan Press Ltd (the first edition in 906), (translated into English by A. S. Schwier as Manual of Political Economy) Srinivasan, D.; Tettamanzi, A. 996: Heuristic-guided evolutionary approach to multiobjective generation scheduling. IEE Proceedings. Generation, Transmission and Distribution 43, pp Steuer, R. 986: Mutiple criteria optimization: theory, computation, and applications, New York: John Wiley & Sons, Chapter 3