DETC2001/DAC EFFECTIVE GENERATION OF PARETO SETS USING GENETIC PROGRAMMING

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1 Proceedings of DETC 0 ASME 00 Design Engineering Technical Conferences and Computers and Information in Engineering Conference Pittsburgh, PA, September 9-, 00 DETC00/DAC-094 EFFECTIVE GENERATION OF PARETO SETS USING GENETIC PROGRAMMING John Eddy Graduate Research Assistant Department of Mechanical and Aerospace Engineering University at Buffalo johneddy@eng.buffalo.edu Kemper Lewis Assistant Professor Department of Mechanical and Aerospace Engineering Corresponding Author University at Buffalo kelewis@eng.buffalo.edu ABSTRACT Many designers concede that there is typically more than one measure of performance for an artifact. Often, a large system is decomposed into smaller subsystems each having its own set of objectives, constraints, and parameters. The performance of the final design is a function of the performances of the individual subsystems. It then becomes necessary to consider the tradeoffs that occur in a multi-objective design problem. The complete solution to a multi-objective optimization problem is the entire set of non-dominated configurations commonly referred to as the Pareto set. Common methods of generating points along a Pareto frontier involve repeated conversion of multi-objective problems into single objective problems using weights. These methods have been shown to perform poorly when attempting to populate a Pareto frontier. This work presents an efficient means of generating a thorough spread of points along a Pareto frontier using genetic programming. KEYWORDS Genetic Algorithms, Heuristic Optimization, Multi-Objective Optimization. MOGA, Pareto Frontiers INTRODUCTION Engineering design problems commonly require consideration of more than one measure of performance or objectives. The objectives in a multi-objective design problem may relate differently to one another. Two objectives may be in competition, meaning that improvement of one typically comes at the expense of the other. Tradeoffs must be considered when exploring a problem of this nature. For example, it may be desirable in a design scenario to reduce two quantities simultaneously (ex. cost and weight). However, a unit reduction in weight may necessitate an increase in cost and the designer must then decide if he/she is willing to accept the tradeoff of increased cost for a reduction in weight. Objectives may also be in cooperation with each other, meaning that improvement of one typically accompanies improvement of the other. In this case, there is a single superior design and tradeoffs do not occur. The third possibility is that there is no relationship, as is the case if the objectives have no variables in common. The later two cases are less interesting than the first and are therefore not considered further in this paper. When multiple competing objectives exist, the optimum is no longer a design point but an entire set of non-dominated design points. This set is commonly referred to as the Pareto set []. The Pareto set is composed of Pareto optimal solutions. A feasible design variable vector, x, is Pareto optimal if and only if there is no feasible design variable vector, x, with the characteristics, f i ( x) f i ( x ) for all i, i =, n () fi ( x ) < fi( x ) for at least one i, n where n is the number of objectives. Copyright 00 by ASME

2 The primary issue addressed in this paper is how to improve design concepts in a multiobjective framework. Specifically, how can a Pareto frontier be efficiently and thoroughly populated? Common methods of generating points along a Pareto frontier involve repeated conversion of multi-objective problems into single objective problems. These methods have been shown to perform poorly when attempting to populate a Pareto frontier for many types of problems. Common pitfalls of weighted sum methods include: inability to generate a uniform sampling of a frontier [, 3]; inability to generate points in non-convex portions of a frontier [4, 5]; a non-intuitive relationship between combinatorial parameters (weights, etc.) and performances; and poor efficiency (can require an excessive number of function evaluations). A great deal of effort has been put into overcoming these pitfalls. The approach taken in this paper for solving multiobjective problems involves the use of genetic programming. We present a multiobjective genetic algorithm (MOGA) tailored such that the entire Pareto frontier is sought in a single optimization run without repeated conversion from a multiobjective to single objective problem. Having presented a brief discussion of the nature of multiobjective optimization problems, Section. presents some background on means of solving these problems, Section. provides some detailed information regarding the specific genetic algorithm used for this work, and Section includes two case studies to demonstrate the effectiveness of the multiobjective genetic algorithm developed in this work. TECHNICAL BACKGROUND. Multi-Objective Solution Techniques Because weighted sum methods have difficulty in finding and generating Pareto frontiers, many researchers have turned to other methods to generate Pareto frontiers. A straightforward approach is to apply a grid search algorithm, as employed to develop Pareto frontiers in vehicle dynamics simulation problems [6]. The Normal Boundary Intersection (NBI) method [7] also overcomes many difficulties associated with the weighted sums. The NBI generates evenly spaced Pareto points for an even spread of weights, and the spacing of the points is independent of the relative scaling of the objectives. NBI is not limited to bi-objective problems, and is primarily used for tracing the Pareto frontier. Many other researchers have employed other heuristic techniques that do not require conversion to single objectives. Messac and Sundararaj [8] employ Physical Programming to generate a distribution of points along the Pareto frontier. Physical Programming is an optimization method that does not rely on weights, but uses designer preferences in the form of metric classes in the optimization process. The Interactive Sequential Hybrid Optimization Technique (I-SHOT) developed by Narayanan and Azarm [9] is another technique for finding Pareto points. It allows the user to interact at each iteration to help specialize the solution process. It is carried out by repeated application of a simple genetic algorithm. The advantage of this method is that each iteration generates a Pareto solution that is not overly close to a previous solution. The disadvantage of this type of method is that it typically requires a huge number of function evaluations to populate the Pareto Frontier. Some of the tradeoffs in multiobjective optimization using operators with and within genetic algorithms are discussed in Azarm, Reynolds, & Narayanan [0]. The Niched Pareto genetic algorithm presented in Horn, Nafpliotis, & Goldberg [] is a MOGA that uses tournament selection of a design versus a random sample of the current population. Two individuals are selected for reproduction and each is compared to the population sample. The design that outperforms the other in the tournament is selected for reproduction. In addition to using the objective functions as a measure of fitness, a niche count is kept in each area of the performance (criterion) space. If the niche count is high in a region, a design in that region is at a disadvantage to a design in a region of low niche count. This is done to encourage a wide spread of the Pareto frontier. The metric used to quantize the niche pressure as a function of niche count can be set by the user and has a profound effect on the optimization. Some empirical evidence generated by Horn et. al. [] suggests that too much pressure results in convergence on a small portion of the entire frontier while too little pressure results in a large number of dominated solutions in the final population (recall from above that at selection time, designs are compared to only a random sample of the population, not the entire thing). Other measures of fitness are used within GA s to facilitate the generation of Pareto points. Balling [] utilizes a maxmin formulation that takes a minimum value of normalized objectives and maximizes it over all the possible designs in the current generation. This fitness function will generate nondominated designs. Other ranking schemes have also been proposed [3, 4], but ranking does not provide a way to quantitatively measure the difference in Pareto-optimality. Because of the success of genetic algorithms as effective approaches to generating Pareto frontiers, in this paper we present a MOGA catered specifically for multiobjective Copyright 00 by ASME

3 problems by taking advantage of a novel fitness function, clone testing, and introductions of epidemics. Section. presents a discussion of genetic algorithms and some details regarding the specific implementation used here.. Genetic Algorithms in Optimization Genetic Algorithms were originally developed to imitate the processes by which living beings evolve [5]. Nearly all design methodologies incorporate the concept of evolving designs. Even designers who do not subscribe to a methodology typically evolve their designs by trial and error. Genetic Algorithms are most useful for problems involving multi-modal design spaces. The fact that they do not consider any gradient information makes it possible for the algorithm to move between the peaks of a multi-modal space. A gradientbased optimizer would typically remain within a single mode throughout the solution process likely resulting in a highly suboptimal solution. A suboptimal solution is also possible when using a GA in a multi-modal space but it is less likely. It is also likely to get closer to the true optimal than the solution given by the gradient-based optimizer. One major focus of our work is on computational efficiency both in memory usage and computational time. We have tailored our Genetic Algorithm accordingly. In the following paragraphs, we describe the operators for our GA along with some of the methods we have employed to help ensure computational efficiency. Binary Encoding Each design variable is represented as a single 3 bit signed integer value within the computer. Therefore, 30 bits are available for genetic encoding. The desired decimal precision for each design variable is input by the user. Floating point numbers are converted to integers by multiplying by 0 prec(i) where prec(i) is the desired decimal precision of the i th design variable. An array of 30 short integers for genetic encoding would require at least 60 bytes or 40 bits compared to 4 bytes or 3 bits and accommodations would still be necessary for a sign and a decimal place. Another advantage of using the signed integer representation of design variables is that conversion from a binary representation to a decimal representation requires no code. It is done at the hardware level, which saves a great deal of computational time. Dynamic Memory Allocation All the arrays in the genetic algorithm are sized at either compile or run time using dynamic memory allocation commands according to the problem parameters (number of design variables, number of constraints, etc). The alternative is to allocate arbitrarily large arrays and perhaps use only a portion of them. We have incorporated the basic components of reproduction and selection in our Genetic Algorithm. They are presented and described in the following paragraphs. Generation of an Initial Population The initial population is simply a collection of designs with design variable values taken as random numbers between the design variable bounds. Each design is tested to be sure that it is unique to encourage diversity amongst the initial population. Clone Testing An efficient algorithm for duplicate design point detection is available with this GA. This is useful for 4 reasons.. To avoid design point re-evaluation.. To encourage design space exploration. 3. To avoid wasting memory to store duplicate design point information, and 4. To avoid premature convergence in a suboptimal mode. Each new design is tested to see that it is unique amongst every other design evaluated during the optimization run (including those that were discarded). Evaluation of Design Fitness A set of objective function values must be computed for each design. The form of the functions is not important to the algorithm. Either calls to other software packages or actual analytical functions are suitable. Constraints are computed for each design and used to penalize infeasible designs at selection and replacement time. A value is stored for each design that indicates how infeasible it is. The value is computed as shown in Equation (L norm). D = m j= l {max[ 0, g ( x)]} + h ( x) () j k = In equation, m is the number of inequality constraints and l is the number of equality constraints. Side constraints are handled separately. Reproduction Reproduction is accomplished in three stages. First, a mating pool is created based on the feasibility metric of Equation. All individuals are represented at least once in the mating pool. Individuals with a lower value from Equation are represented more times in the mating pool making it more likely that they will be given an opportunity to reproduce. All Feasible designs are given an equal chance of reproducing. Individuals are randomly chosen from the mating pool. The k 3 Copyright 00 by ASME

4 number of individuals selected is determined by the size of the population, the number of children produced per crossover, and the user input crossover rate. Crossover Two individuals selected for reproduction are chosen as a mating pair. Each mating pair can produce any even number of offspring (value input by user). The offspring will be referred to as children. Two types of crossover are available in this GA. The first is Single Point Uniform Parameterized Crossover [6] and is performed for each design variable as shown in Figure. Binary (encoded) Decimal Parent 03 Parent Child Child Figure : Single Point Uniform Parameterized Crossover The vertical line serves as a randomly selected crossover point. The actual bit switch is accomplished using bitwise logical operators. The second is Arithmetic Crossover and is carried out according to equations 3 and 4 below. Child = ( r)( Parent) + ( r)( Parent) (3) Child = ( r)( Parent) + ( r)( Parent) (4) In the above equations, r is U[0,]. In this way, the children are the result of a convex combination of the parents [7]. Mutation Children from the crossover stage are randomly selected for mutation. There are two types of mutation available in this GA. The first is referred to as random bit mutation. In random bit mutation, a design variable is randomly selected, as is a bit location for mutation. The selected bit is negated. If it was a 0 it becomes a and vice versa. The number of mutations is determined by the number of children created, the total number of bits making up all the children, and the user input mutation rate. The second mutation type is referred to as random design variable mutation. In this form of mutation, a design variable is chosen at random and reassigned to a random value within the upper and lower bounds for that variable. The number of mutations is determined by the number of children, the number of design variables, and the user input mutation rate. Insertion of Children into Population Once the children have been created and mutated, the process of inserting them into the population begins. This is where the Pareto dominance of the designs is considered. If there are infeasible members in the population, then the one with the largest D (from Equation ) is selected for comparison and possible replacement. If the child that it is being compared to is feasible or has a lesser D, then the child replaces it. Otherwise, the child is not allowed entrance into the population. If all members of the population are feasible and the child is feasible, then the population is searched for a member that is dominated by the child. The function that performs the search returns the design that is most dominated by the child. This is done by keeping track of the current worst design as it searches and comparing each successive design to the current worst. If it is found that the child does not dominate any members of the population, then a check is run to see if any members of the population dominate the child. If the child is dominated by a member of the population, then it is not permitted entrance into the population. If the child is not dominated by any member of the population, then the population size is increased and the child is granted entrance. Because of the way new designs are admitted into the population, a sub-optimal design has a chance of remaining within the population and reproducing for a significant number of generations. This can cause the population size to become unnecessarily large which significantly affects the speed and efficiency of the algorithm. Therefore, an additional capability is developed for this work whereby the user can periodically eliminate all dominated designs from the population. This is termed an introduction of an epidemic, as it eliminates all the unfit members of a population. The frequency at which epidemics occur is set by the user. The two parameters that define the frequency of the epidemics are the Cut-off Point and the Cut-off Point Adder. The Cut-off Point is the size that the population must reach before an epidemic is introduced. After an epidemic, the Cut-off Point is incremented by the Cut-off Point Adder and the population is allowed to expand to the new Cut-off Point before another epidemic is introduced. Pareto Set Quality Metrics Upon completion of the optimization, two quality metrics are computed for the resulting Pareto set. These quality metrics were developed by Wu & Azarm [8] because of the lack of standard metrics to evaluate the effectiveness of any method to 4 Copyright 00 by ASME

5 generate points on a Pareto frontier. The two metrics are described in the following sections. Distinct Point Metric The first metric is the number of distinct choices in the set. To compute this metric, the design space is divided into a hypergrid. A distinct choice is one that exists alone in a grid location as in Figure. third constraint limits the stress in bar BC to no more than 00,000 kpa. The three design variables are the cross sectional areas of bars AC and BC labeled x and x respectively and the overall height of the truss labeled y. Side constraints are placed on all three design variables. 4m m Pareto Distinct A B x x y Figure : Distinct Point Quality Metric The grid spacing along each objective in this work is determined by a user input Cluster Percentage and the range of objective function values in the population. Cluster Metric The second metric is the Cluster Metric. This metric gives an indication of how uniformly spaced the final population is. It is computed as the number of Pareto points divided by the number of distinct Pareto points. The target value for the cluster metric is, which corresponds to a Pareto set in which every member is distinct. The worst possible value is equal to the number of Pareto points which is the case when all the points are tightly clustered such that only a single design is considered distinct. The next section contains two case studies used to verify the approach presented above. 3 CASE STUDIES All Trials for the following case studies were carried out on a PC with a 600 MHz Pentium III processor and 56 MB of RAM. 3. Two Bar Truss This case study involves a multi-objective -bar truss problem adapted from Azarm, Reynolds, & Narayanan [0]. A diagram of the truss is shown in Figure 3. The two objectives in this problem are to minimize the overall volume of material used and to minimize the stress in bar AC (see Figure 3). The system is subject to three inequality constraints. The first two are constraints on the objective function values. They are intended to limit the size of the Pareto set. The Pareto set is unbounded if these constraints are not in place. The side constraints will limit the Pareto set but in a less concise manner than the inequality constraints. The The problem formulation is given below. Minimize: Subject to: f volume = x 6 + y f stress, AC + y + x 0 = 6 + y yx g ( x, ) f 0. y 00kN Figure 3: Truss for Case Study 3 volume g ( x, y) f stress, AC 00, y g 3( x, y) 00, 000 yx C y 3 (meters) > x > 0. (meters ) > x > 0. (meters ) 5 Copyright 00 by ASME

6 The MOGA parameters used for the optimization are given in Table. Mutation Type Random Design Variable Crossover Type Sing. Pt. Uniform Par. Clone Test Enabled Initial Population Size 00 # of Children (per crossover) Mutation Rate 7% Crossover Rate 0% Initial Cutoff Point 000 Cutoff Point Adder 500 Max # of Generations 000 Decimal Prec. (all variables) 4 Cluster % (all objectives) % Table : Optimization Parameters for -bar Truss The results of applying the multi-objective genetic algorithm presented in this paper to the truss example are presented in Table. This problem was selected because it is used by Azarm, et al [0] to compare MOGA s with repeated applications of single objective genetic algorithms. The results in Table above are compared to the results given by Azarm, et al [0]. Time of execution 5 Seconds # of non-dom. Pts # of distinct non-dom. Pts. 6 Cluster Metric # of obj. fctn. evaluations 9090 # of Generations 0 Table : Results for Optimization of -bar Truss Before presenting the comparison, it is extremely important to note that implementation in this work is not the same as that of Azarm, et. al. [0]. First, the genetic parameters used here are not the same nor are the genetic operators. For instance, Azarm uses -point uniform crossover and random bit mutation where single point uniform parameterized crossover and random design variable mutation are used in this work. Second, Azarm computes distinct solutions by computing the distance between points on the Pareto frontier and comparing the result to some limiting value. This is similar to but not the same as the grid implementation presented by Wu & Azarm [8] and also used in this work. No indication was given by Azarm regarding the limiting distance for distinct points and therefore it cannot be compared to the Cluster Percentage used here. Third, the MOGA used by Azarm incorporates niche pressure and this work does not in the approach presented in this paper. Fourth, Azarm did not supply upper bounds on the cross sectional areas ( x ). The algorithm used in this work requires them. Therefore, the upper bounds were set to very large values. Finally, the decimal precision used by Azarm was not reported and certainly has an effect on the efficiency of the optimization. With this in mind, the results are now presented for general comparison purposes in Table 3. Method Total Pts. Pareto Pts. Distinct Pts. I-SHOT 0, MOGA 3, This MOGA 9,090 3,049 6 Table 3: Comparison of Previous work to this work The I-SHOT method visited,974 points for every Pareto point found, the MOGA of Azarm visited 97 points for every Pareto point found, and the MOGA implementation of this work visited 3 points for every Pareto point found. The number of distinct points is strongly related to the limiting distance in Azarm, et. al. [0] and the Cluster percentage used in this work. Since Azarm did not report the limiting distance, the two cannot be compared. Therefore the number of distinct points should not be considered as a strong indicator of the relative efficiencies of the Azarm methods and the method presented here. Azarm also mentions that some duplication occurred and was part of the reason that there were fewer distinct Pareto points than total Pareto points. Duplication was not allowed in this work and this should be considered as a point for comparison. Based on this information, the algorithm used in this work compares favorably to the ones used by Azarm, et al [0]. There are significantly fewer evaluations per non-dominated point in both cases. It is important to note however that many of the points generated by this MOGA are non-distinct. Such a dense sampling of the Pareto set is likely unnecessary. Figure 4 contains the points generated along the Pareto frontier for this problem using the MOGA presented in this paper. 6 Copyright 00 by ASME

7 Stress in Bar AC (MPa) Pareto Frontier Volume of Material (m^3) 0. R 36.0 (inches) 0.5 T 6.0 (inches) 0. L 40.0 (inches) The values for the constants in the problem are listed in Table 4. Parameter Value P 3.89 ksi S t 35.0 ksi ρ 0.83 lbs/in 3 3. Pressure Vessel The second case study is a multi-objective pressure vessel problem taken from Winer [9]. The two objectives are to maximize the volume and to minimize the weight of a pressure vessel. The objective functions are both non-linear. There are four inequality constraints the first of which is non-linear and the rest of which are linear. The first inequality constraint limits the maximum circumferential stress in the vessel while the other three are geometric constraints. There are 3 design variables, they are the radius of the tank (R), the thickness of the tank (t), and the length of the tank (L). There are side constraints on all the design variables. All dimensions are in inches and all weights are in pounds. The problem statement is given below. Maximize: Minimize: f wgt Subject to: Figure 4: Pareto Set Generated 4 3 f vol = π R + πr L 3 = ρ π ( R + T ) + π ( R + T ) L ( πr + πr L) 3 3 PR g ( R, T) S t 0 T g ( R, ) 5T R 0 T g ( R, ) R + T T g ( R, L, ) L + R + T T Table 4: Constants for Case Study The parameters used for the optimization are given in Table 5. Mutation Type Random Design Variable Crossover Type Sing. Pt. Uniform Par. Clone Test Enabled Initial Population Size 00 # of Children (per crossover) Mutation Rate 7% Crossover Rate 0% Initial Cutoff Point 000 Cutoff Point Adder 000 Max # of Generations 000 Decimal Prec. (all variables) 5 Cluster % (all objectives) 5% Table 5: Optimization Parameters for Pressure Vessel The results of the applying the multiobjective optimization genetic algorithm presented in this paper to pressure vessel example are presented in Table 6. Time of execution 8 Seconds # of non-dom. Pts. 4 # of distinct non-dom. Pts. 39 Cluster Metric # of obj. fctn. evaluations 3,588 # of Generations 50 Table 6: Results for Optimization of Pressure Vessel The most noticeable value in Table 6 is the cluster metric. Recall from Section that the target value for the cluster metric is. This high value can be caused by one of three things. The first possible cause is that the algorithm quickly moved to distinct portions of the space and remained in these 7 Copyright 00 by ASME

8 areas creating clusters. The second possible cause is that the cluster percentage is high forcing the space between distinct points to be large and thus only a small percentage of the generated points are considered distinct. This may be desired in which case a high cluster metric is not necessarily a bad thing accepting that there were far more function evaluations than necessary. The final possible cause is that the Pareto frontier is well represented but the high decimal precision allows for many closely spaced performance points and the clustering occurs on the entire frontier (recall that no explicit niche forming was used). The algorithm was able to produce non-dominated point for every 5 points visited. Figure 5 shows all the non-dominated designs generated. The plot shows the objective space, given by Volume vs. Weight, where larger volumes are preferred and smaller weights are preferred. The data is displayed using an in-house visualization package called Cloud Visualization [0]. The visualization package allows a user to visualize various sets of points, including dominated points, non-dominated points, distinct Pareto points (see Section.) as clouds of points. Figure 6 is the same data as that of Figure 5 with all the nondistinct points filtered out. Volume (in 3 ) Desired Region Weight (lbs) Figure 5: Case Study Performance Space Nondominated Points Only Volume (in 3 ) Weight (lbs) Figure 6: Case Study Performance space distinct non-dominated points only. 4 CONCLUSIONS This problem was selected because it is used by Azarm, et al [0] to compare MOGA s with repeated applications of single objective genetic algorithms. The results in Table above are compared to the results given by Azarm, et al [0]. In this paper we have presented an efficient means of populating a Pareto frontier using genetic programming. The method used here was shown to provide a far better ratio of total points to Pareto points than other methods for the first test case presented. A thorough sampling of the Pareto frontier was also obtained for the second case study with a good ratio of total points to Pareto points. The multiple objective genetic algorithm presented takes advantage of a number of conceptual and computational developments to effectively generate Pareto frontiers quite efficiently. The use of binary encoding, dynamic memory allocation, clone testing, and epidemics allow the genetic algorithm to generate many Pareto points in one optimization process, giving a designer better information about the problem at hand. The algorithm has also been applied in parallel computational environments [] to optimize and simulate computationally expensive vehicle dynamics design problems. Future work includes the addition of niche forming to this algorithm as well as real-time visualization of the evolutionary process. ACKNOWLEDGEMENTS We acknowledge the support of the National Science Foundation, grant DMII in this work. REFERENCES [] Pareto, V., 906, Manuale di Econòmica Polìttica, Società Editrice Libràia, Milan, Italy; translated into English by A. S. Schwier, as Manual of Political Economy, Macmillan, New York, Copyright 00 by ASME

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