Multi-Objective Optimization using Genetic Algorithm or Nonlinear Goal Programming

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1 nd International Conference on Engineering Optimization September 6-9, 010, Lisbon, Portugal Multi-Objective Optimization using Genetic Algorithm or Nonlinear Goal Programming José Marcio Vasconcellos 1 1 COPPE / UFRJ, Rio de Janeiro, Brazil, jmarcio@ufrj.br Abstract Many engineering problems in its formulation has the need to address the mathematical model as a problem with multiple objectives, also called multiple criteria decision making (MCDM). In practice, this type of problem occurs due to lack of resources to meet all the project needs. One simple and illustrative example relates the objectives, cost and safety. In engineering projects we are always looking to maximize safety and minimize the project cost. These goals are often antagonistic and deserve to be treated with the application of MCDM mathematical models. To solve MCDM problems, various techniques may be used and this article discusses the application of two different solution methods: the use of non linear goal programming technique (NLGP) and the application of genetic algorithms (GA) to solve problems multiple criteria. NLGP were developed during the Second World War and require some expertise to determine the Pareto frontier. Genetic Algorithm, presented in the mid-1970 by J.Holland, was popularized by one of his students D. Goldberg in The technique presents many advantages but also some problems to be overtake. This article presents the aim of the two techniques, discusses their advantages and disadvantages and illustrates the application with two case studies where we compare the results obtained and the time needed to reach a solution to the problems presented. A discussion is made regarding the establishment of weights for the objectives and the process to decide the best design in the Pareto frontier. The two case studies were selected to illustrate the NLGP and the GA techniques application. The first one is a mathematical non linear problem and the second is selected from the naval architecture and ocean engineering field. Finally, recommendations are given for an efficient implementation of the methods in some engineering design problems. Keywords: Multi objective Optimization, Nonlinear Programming, Genetic Algorithm 1. Introduction Optimization problems are problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables within an allowed set, with the existence or not of variable restrictions. Basically, the algorithms used to solve these problems may be: deterministic or stochastic. The optimization methods based on deterministic algorithms (the majority of the classical methods) generate a deterministic sequence of possible solutions requiring in most cases the use of at least the first derivative of the objective function in relation to the design variables. In such methods the objective function and the restrictions are given as mathematical functions and functional relations. Furthermore, the objective function should be continuous and distinguishable in the search field [1]. This type of problem can be represented mathematically in the following way: Max/Min f(x 1,x,...x n ) Constraints g i (x 1,x,.x n ) or b 1... g m (x 1,x, x n ) or b m and x 1,x, x n design variables f(x 1,x, x n ) objective function g 1,g, g n constraints Many different methods can be found in the literature including excellent textbooks and papers to solve the deterministic optimization mathematical problem. Watson, D.G.M and Gilfillan, A.W., [] presented an optimization technique applied to merchant ship design. Later many others authors developed mathematical models for ship preliminary design. Different optimization 1

2 techniques were used since. Vasconcellos J.M, [3] developed a semisubmersible platform preliminary design and applied the nonlinear goal programming (Ignizio J.P., [4]) to solve the multi-objective problem. Moraes H., Vasconcellos J.M and Almeida M.P, [5], used the same optimization technique to solve a high speed catamaran preliminary design problem. Many technical papers and books were published showing optimization techniques applied to ship and offshore design for discrete problems with single or multi objective functions. In the mid-1970s, John H. Holland presented an optimization search technique based on the principles of genetics and natural selection. This technique was popularized by one of his students, David Goldberg, who used the algorithm in his dissertation to solve a difficult optimization problem (Goldberg, [6]). A genetic algorithm allows a population of possible solutions composed of many individuals to develop, under rules of specified selection, to a state which minimizes the cost function (Haupt and Haupt, [7]). In relation to the traditional search and optimization functions, Genetic Algorithms (GA) differ principally in four aspects: GAs work with a codification of the set of parameters and not with the parameters themselves; they work with a population and not with a single point with each interaction of the algorithm; they use cost or fitness value information and not derivatives or other auxiliary knowledge, and they use probabilistic and not deterministic rules of transition.. Non Linear Goal Programming The multi-objective goal programming method is based on the Simplex Linear Programming that was developed during World War II. The method was developed to solve military strategic problems. The Simplex method provides a procedure to optimize linear mathematics problems with one objective function. Ignizio J.P. [4] presents a linear and non-linear goal programming as an extension of the Simplex method. In the multi-objective goal programming approach it is necessary to follow three steps: 1 o step - Identify the decision variables (x j ) o step - Formulate mathematical model objectives (G i ) 3 o step - Formulate achievement function (a k ) All the mathematical model constraints are converted into goals in the goal programming procedure with multiple objectives. The following criteria define the objectives: 1) Designer Criteria Example: Minimize Construction Costs, Maximize Tank Volume, Minimize Forces and Tensions, Minimize Motion, etc. ) Resource Limitation Example: Material, Cost, etc. 3) All remaining constraints that could affect the decision variables. Example: Physic Constraints (Decision variable non-negative, size constraints in the shipyard, etc.) The mathematical formulas of the goals (G i ) are function of the decision variables (f i (x)): G i = f i (x) (01) All objectives are associated to a value (b i ) in the right hand side of the equation: f i (x) = (b i ) (0) where, b is the value the objective needs to fulfill. Finally, we can write the goals as: G i => f i (x j ) + n i - p i = b i i = 1,,3,...,m (m objectives) (03) j = 1,,3,...,k (k variables) where, n i and p i are the negative and positive deviation variables, respectively, from the objective. Table 1 shows formulation procedure for the achievement function.

3 Table 1: Formulation procedure for the achievement function. Objective Procedure G i b i minimize n i G i b i minimize p i G i =b i minimize (n i +p i ) For the achievement function is necessary to assign the priority level (P1, P,...) for each objective. We can write the mathematical model as: Minimize a = {P1[g 1 (n,p)],p[g (n,p)],...,pk[g k (n,p)]} (04) Where: g k (n,p) = linear function of the deviation variables Pk is the function g k (n,p) priority k m (number of objectives) Finally, the mathematical model can be written in a short form, as follows: Where: for, and Find x o = (x 1,x,...,xj) to minimize a = (a 1,a,...,a k ) (05) f i (x j ) + n i - p i = b i a 1 = g 1 (n,p) a = g (n,p) a k = g k (n,p) x o,n i,p i 0 i=1,,...m (objectives) j=1,,...k (variables) (06).1 Non-Linear Goal Programming Griffth R.R. and Stewart R.A. [8] presented a procedure for non-linear models using Taylor series expansion. The goal programming they used takes the two first terms of the Taylor expansion to approximate the goal functions near the test point. We can write the nonlinear goal function using the mathematical model presented in equation 6 and in linearization procedure (08): G i => f i (x j ) + n i - p i = b i i = 1,,3,...,m (m objectives) (07) j = 1,,3,...,k (k variables) Considering the function f i (x j ) continuously differentiable and assuming x s one solution for the objectives, the function approximation is giving by: J f ( xs) fi( x) = bi ni + pi = fi( xs) + xj xs j ( xj) (, ) i = 1,, 3,..., m j= 1 (08) The Non-Linear Goal Programming optimization technique was developed and implemented in a FORTRAN code. 3

4 3. Multi Criteria Decision Making with Genetic Algorithm Genetic Algorithms (GAs) are adaptive heuristic search algorithms premised on the evolutionary ideas of natural selection and genetics. The basic concept of GAs is designed to simulate processes in natural systems necessary for evolution, specifically those that follow the principles of survival of the fittest first laid down by Charles Darwin. As such they represent an intelligent exploitation of a random search within a defined search space to solve a problem [6], [7] and [9]. The selective mechanisms carry out the changes that determine the evolution of a population over generations. Such changes can occur due to the interactions between the individuals or due to the influences of the environment on the individuals. Three basic mechanisms derive from this: crossing or crossover, reproduction and mutation. They are called genetic operators and are responsible for carrying out the evolution of the algorithm. The application of these operators is preceded by a selection process of the best adapted individuals, which uses a function called the fitness function, also known as the adaptation function. An implementation of a genetic algorithm begins with a random population of chromosomes, that is to say, the initial population can be obtained by choosing a value for the parameters or variables of each chromosome randomly between its minimum and maximum value. In general, the size of the initial population is directly related to the number of generations or iterations in order to obtain convergence. In general the GA converges more rapidly, in terms of the number of iterations for the solution, when the initial population is larger. And the larger the population, the greater the number of evaluations of the cost function. At this stage of definitions, there follows an evaluation of each individual of the population through the objective function which, in this context, is called the fitness function. The fittest individuals (with the best adaptation values) have the greatest probability of reproducing (selection). The genetic crossover and mutation operators are at work on those selected. The new individuals replace totally or partially the previous population, thus concluding a generation. The selection operates from the current population allowing the transmission to the new population of the individuals of the present population, with greater probability for the individuals with a better performance (fitness value), and with less probability for individuals with a worse performance. To the survivors the crossover and mutation operators are applied. A way of carrying out the selection operation would be to order the chromosomes from the best to the worst, and normalize the cost value of each chromosome. Then, a value would be chosen at random valuing the hierarchical position of the chromosome and its normalized cost. The crossover is the operator responsible for interchanging and combining characteristics of the parents during the reproduction process, allowing the next generations to inherit these characteristics. The idea is that the new descendent individuals can be better than their parents if they inherit the best characteristics of each parent. There are various crossover operators. The simplest consists of choosing one or various random cut-off scores in the chromosome, and permute from these points the variables of the parent chromosomes to generate the offspring. Figure 1 illustrates an example of the action of the operator in binary chromosomes: Figure 1: Example of crossover in a binary chromosome Other possibilities are: to make a weighted average between the values of the parameters (blending); or even a linear combination between the values of the parameters (linear crossover). The mutation operator is designed to introduce diversity into the chromosomes of the population of the GA, in order to ensure that the latter is not trapped in minimum areas. However, if the rate of mutation is very high the GA runs the risk of losing former optimum parameters and executing a purely random search, which is not desirable, as the speed of convergence for solving the problem will tend to diminish. The scheme below (Table ) exemplifies how this operator functions in binary chromosomes: Table : Example of mutation in binary chromosomes Original descent Original descent Mutated descent Mutated descent

5 In addition to these, there are other factors that influence the performance of a GA, adapted to the particularities of certain classes of problems. Among the most important are: the size of the population, the number of generations, the crossover rate and the mutation rate. A very small population offers a small search space coverage, which can cause a drop in performance. A population which is sufficiently large provides a better coverage of the field of action of the problem and tends to prevent premature convergence for local solutions. With a very large population, however, greater computational resources become necessary, or a longer processing time of the problem, bearing in mind that a larger number of evaluations of the objective function are carried out. One should seek accordingly an equilibrium point for the value attributed to this parameter. The number of generations varies according to the complexity of the problem in question and should be determined experimentally. As the GA resolves optimization problems, the ideal thing would be for the algorithm to stop as soon as the optimum point was found. In most applications, one cannot affirm that a given optimum point found is a global optimum. In this case another criterion is adopted for the termination of the processing of the algorithm. In general, the maximum number of generations criterion is used or a time limit is established to terminate the process. The generation interval controls the percentage of the population that will be substituted during the next generation (total substitution, substitution with elitism, substitution of the worst individuals of the present population, partial substitution of the population without duplicates). The crossing rate determines if the crossing between two chromosomes will be made. For this, a random number is generated in the interval [0.1] and is compared to the rate. If the number is less than the rate, the crossing is affected. The higher the rate, the more rapidly new structures will be introduced into the population. But if this is very high, structures with good aptitudes can be removed more rapidly giving a high value, the majority of the population will be substituted, but with very high values a loss of high aptitude structures can occur. With a low value, the algorithm can become very slow. The mutation rate determines if the genes of the chromosomes of the selected population will undergo mutation or not. A random number is generated in the interval [0.1] and compared to the mutation rate. If the number is less than the rate, the gene will be modified according to the operator chosen. A low mutation rate prevents a given position remaining stagnant in one value, besides making it possible to reach any point in the search space. With a very high rate the search becomes essentially random. 4. Case Study Two case studies were selected to highlight the nonlinear goal programming (NLGP) application and genetic algorithm (GA) in multi objective problems. For NLGP a Fortran programming was created and for GA the commercial software ModeFrontier TM was used. The first problem is a chemical manufacturing company [9] producing two primary chemical processing agents. The second case is a semisubmersible platform preliminary design where the optimization is used to indicate the main characteristics. 4.1 Chemical Manufacturing Problem This problem is established in [9] and can be mathematically written as: Two agents (x 1 and x ) should be mix under the limitations of 9 hours of weekly aging time and limit mixing process time of 6 hours a week. The company wants to maximum profit that is approximated by the product of the amounts of agents. The mathematical problem written as a NLGP is: Minimize a = {(p 3 ), (n 1 +p ) G 1 = x 1.x + n 1 p 1 = C G = (x 1-3) + x + n - p = 9 G 3 = x 1 + x + n 3 p 3 = 6 Where C is the total profit the company set up as a goal. The Fortran NLGP found the solution (x 1 = 3,0 and x = 3,0) in less than sec. The GA application was done using Mode Frontier TM commercial software. The problem workflow is presented in figure. The G 1 objective was set up to maximum and G and G 3 to minimum but with constrains. Figure 3 presents the graph showing the feasible solutions and the optimum point (x 1 =3, x =3). To achieve a widely feasible solutions the GA used 30 minutes. 5

6 Figure : Workflow Chemical Manufacturing Problem Figure 3: Results from Genetic Algorithm 4.. Semisubmersible Design Problem This case study is presented to indicate the optimization in the offshore platform design. The example presented herein is for a semisubmersible platform preliminary design based on the platform requirements for Campos field in Brazil. The case is a production system where no load oil tank is required. The plant capacity is for 100 million barrels/day. The environmental conditions are water depth of 800 meters and waves with 8.0 meters of significant height and 1 seconds of period. The platform geometry is presented in Figure 4 and table 3 describes the main variables. Figure 4: Semisubmersible Geometry 6

7 Table 3: Main Variables (see Figure 4) Variables X1 X X3 X4 X5 X6 Width Length Pontoon height Pontoon width Diameter Column height The mathematical model is presented in the following equations: Design Parameters Equation Water Line Area Equivalent Diameter Transversal Inertia Longitudinal Inertia Columns Volume Columns Submerse Volume Pontoons Volume ncol π D Awl = i i= equi aib i π ncol I yy = I yy ( i) + dx ( i) A ( i) wl i= 1 ncol I yy = I yy ( i) + dx ( i) A ( i) wl i= 1 ncol V c = A wl. h tcol i=1 ncol V mc = A wl. h mcol i=1 npont D V p π = i. l i i= 1 4 D = (10) Displacement Δ = ( V +V ). γ mc p (16) Columns Steel Weight 1,61 Pcol = 0.56 D h col (17) Pontoons Steel Weight P pont ( ) R op (18) S p = L ( b + h ) p p p (19) Deck Steel Weight Pdeck = 0. 18dA (0) Heave Motion RAO ( Z / H ) = f f f 1 3 (1) [ 4 π / T d / g ] f 1 = e () f 4π b = cos (3) T The first model result was obtained by an optimization using a non linear goal programming technique approach. The solver indicates the following solution for the mathematical model. The process is fast (less than 30 sec) and consistent (the solution was repeated for different starting point). The objective functions are the area under the RAO curve, the steel weight and maximum KG. These objectives indicate a good design for platform behavior in waves. g (4π d / g ) 3 TN T f = (9) (11) (1) (13) (14) (15) (4) 7

8 Table 4: Results Non linear Goal Programming Variables meters X1 Width 90 X Length 100 X3 Pontoon height 4.5 X4 Pontoon width X5 Diameter X6 Column height 5.65 Deck height 8.0 Air gap 6.90 Top (draft) 3.0 Displacement ton GM.0 Max KG 45.0 x4/x x4-x5>1.0 OK Tn Min. area under RAO 1.4 The GA method was tested and the ModeFrontier TM workflow created as shown in figure 5. Two constraints (displacement and b/h relation) and three objectives (KG maximum, minimum area under the heave motion curve and minimum weight) Figure 5: Workflow Semisubmersible Platform Design Figures 6, 7 and 8 present the graphs results. The graphs indicate the feasible designs and the Pareto frontier. Area under the heave motion curve and total weight are objectives in conflict (Figure 6). The GA method allowed a very broad field problem overview. Although the necessary computer time is much high, the method indicates many feasible designs and give to the designer the possibility to choose the best solution to the specific design problem. Figure 6: Area under the heave motion curve versus weight 8

9 Figure 7: KG maximum versus area under the heave motion curve Figure 8: KG maximum versus weight Table 5 presents 9 designs selected from graphs in figures 6, 7 and 8. The points were chosen by minimum weight, maximum KG and minimum area in each graph. The NLGP solution was also included in the last column. The minimum and maximum objectives were underlined to indicate the best in each goal. Designer can selected the best design from the options using some different criteria as preference any technical characteristic difficult to be established in the mathematical the model. Designers can also use weight to each objective to select the best choice. Table 5: Semisubmersible Platform Design GA method - Some feasible designs From Graphic Fig. 6 From Graphic Fig. 7 From Graphic Fig. 8 Design ID NLGP Top X X X X X X b/h Displacement KG Weight Area

10 5. Conclusion The paper compares NLGP and GA methods to solve multi objective decision problems. Although faster, the non linear goal programming do not indicates a broad overview of the feasible possibilities. To investigate the feasible space, users should change systematically the weight attributed to each objective. To improve the method the weight credited of each objective could be selected from a random generation. The genetic algorithms allow a complete investigation of the designs possibilities but in the other hand takes higher computer time. The two case studies presented a very close solution using both method and in the case of semisubmersible platform design the software used is a powerful tool to solve complex optimization problems. 6. References [1] Aror, Jabir S. Introduction to optimum design. McGraw-Hill Inc., [] Watson, D.G.M and Gilfillan, A.W., 1977, "Some Ship Design Methods". Transaction of the Royal Institution of Naval Architects, Vol [3] Vasconcellos, J.M., 1994, Semisubmersible Platform Preliminary Design, D.Sc Thesis, Rio de Janeiro, Brazil. [4] Ignizio, J.P., 1976, Goal programming and extensions, Lexington Books. Lexington, MA. [5] Moraes, H.B., Vasconcellos, J.M., Almeida M.P., 007, Multiple criteria optimization applied to high speed catamaran preliminary design, Ocean Engineering Journal, Elsevier, 34, [6] Goldberg D. E., Genetic Algorithms in Search Optimization and Machine Learning. Reading, Mass: Addison-Wesley, [7] R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms, ed., Wiley, 004. [8] Griffith, R. E., and R. A. Stewart, A Nonlinear Programming Technique for the Optimization of Continuous Processing Systems, Management Science, Vol 7, pp , [9] Deb K. (001), Multi-Objective Optimization using Evolutionary Algorithms, Wiley-Iterscience Series in Systems and Optimization. 10