Genetic Algorithms for Multi-Objective Optimization in
Dynamic Systems
Ceyhun Eksin
Bo aziçi University
Department of Industrial Engineering
Bo aziçi University, Bebek 34342, stanbul, Turkey
ceyhun.eksin@
Abstract:
This study us parametric arch to meet multiple goals in the behavior of dynamic systems. Parameters are arched using genetic algorithm. Main aim of this study is to discuss how multi-objective parameter arch gives esntial information about the system. A nonlinear electric circuit is one of the two dynamic models in this paper ud for parameter optimization. The electric circuit model
shows oscillatory behavior. A fitness function which evaluates period and amplitude and compares it with the desired oscillatory pattern is propod. Genetic algorithm with the propod fitness function gives satisfactory results. It is shown that time horizon for a simulation bad optimization can be crucial. The cond model is a generic System Dynamics model, the stock management problem with cond order supply line. The policy parameters are weight of stock adjustment and supply line adjustment. A fitness function that evaluates the ttling time, overshoot, and steady state error is propod. The arch results provide some insight on both the fitness function and the system. The obtained results are satisfactory and they show that the respon time of the system can be decread by small overshoot. The paper is a step towards simulation bad parameter arch becoming an esntial support toolbox for model building and policy design in System Dynamics.
1) Introduction
Policy design or improvement of the system behavior is the utmost role of a System Dynamics (SD) model. Despite the growing interest in SD, the field has not succeeded to provide strong and reliable support toolbox to aid the modeler in policy design process. The need for the integration of new optimization tools to SD in model identification, behavior analysis, nsitivity analysis and policy design, has been pointed by veral authors (Richardson 1999; Coyle 2000; Yucel and Barlas 2007).
One of the most widely ud approaches in policy design is the so-called traditional approach. Traditional approach to policy design is usually an informal process where the model builder or an expert who has an understanding of the system’s behavior applies parametric or structural changes to the system on a trial-and-error basis (Coyle 1977). The traditional approach is bounded by the model builder’s or expert’s intuitive ability to come up with acceptable and good enough policy alternatives. Though the traditional approach could not be abandoned, veral studies have been propod by SD rearchers, which emphasize optimization methods to aid the model builder in generating acceptable policies efficiently. Some of the methods applied modal control theory methods (Mohapatra and Sharma 1985; Özveren and Sterman 1989) and others
made u of optimal control theory (Burns and Malone 1974; Keloharju 1982; Coyle 1985). Optimal control theory methods often u heuristics to optimize policy parameters according to a certain objective function. In this n, they can also be regarded as simulation bad optimization methods. Other applications of similar methods to SD models, which make u of simulation bad algorithmic arch, have followed the early applications. Wolstenholme and Al-Alusi (1987) make u of DYSMOD package to optimize the strategy of an army in a defen model. Dangerfield and Roberts (1999) u hill-climbing arch algorithm to fit the model to AIDS data to obtain the distributi
on of incubation period. Graham and Ariza (2003) prent a real world consulting application of the policy design by parameter optimization using simulate annealing. One such algorithm, which has prospect in obtaining desired results efficiently and effectively, is Genetic Algorithm (GA). There are few studies in the SD literature that have also applied GA in policy design. (Grossman 2002; McSharry 2004; Yücel and Barlas 2007; Duggan 2007)
One of the main reasons why such optimization methods are not well established in the literature is the difficulty to define a good objective function. Generally, the simulation bad optimization applications to SD have been done via maximizing or minimizing the end value of a single parameter or a t of parameters in the system. (Wolstenholme and Al-Alusi 1987; Miller 1998; Grossmann 2002; McSharry 2004; Duggan 2007) Our study us parameter arch to obtain multiple goals in the behavior of dynamic systems. Parameters are arched using GA. This study shows that parameter optimization can cast doubt on the objective function which could be ud as fruitful source of information to either obtain a better objective function or to gain insight about the system. One of the main aims of this study is to show that multi-objective parameter arch according to a performance index can yield esntial information about the system.
In the next ction the GA applied will be explained. Section 3 will explain the application of the para
meter arch to nonlinear electric circuit where the definition of the objective function and results will be given. After the ctions, the arch will be applied to the generic Stock Management Model with cond-order supply line. Parameter arch will be applied with different versions of performance index to show that the parameter arch can provide valuable knowledge about the system during the policy design process. In this final ction before conclusion, there will be further analysis on the Stock Management Model.
2) Genetic Algorithm (GA)
The GA, as the name implies, mimic the biological evolution process to find the ‘fittest’ t of parameters. This application of biological process, named as GA was first introduced by John Holland in 1975 (Holland, 1975). GA arches the solution space in multiple and random directions. GA is obrved to be an effective algorithm for highly nonlinear solution spaces since it is not typically trapped in local optima. Below is a pudo-code for a general GA algorithm.
choo initial population
evaluate each individual's fitness
Scale every individual according to their fitness
repeat
lect individuals bad on their scaled fitness values
mate pairs at random
apply crossover operator with certain probability
apply mutation operator with certain probability
evaluate each individual's fitness
until terminating condition (e.g. until at least one individual has
the desired fitness or enough generations have pasd)
As the pudo-code implies each of the steps could be done with various different operators, functions, or values. GAs have many options which provide high flexibility in coming up with an appropriate algorithm for the arch space. The reader may refer to (Goldberg 1989) for a detailed description on GAs. For our study, we utilized the “Genetic Algorithm and Direct Search Toolbox” of
MATLAB ®. The specifications of the GA The initial population is chon randomly. The population number is t to 50. Each individual has p genes where p is the number of policy parameters. The fitness values of individuals are scaled proportionally. The individuals for candidate parents are chon by roulette wheel. In roulette wheel lection, each of the individuals is assigned a share at the roulette wheel proportional with its fitness value. Then the individuals are lected randomly with the individuals with higher share having a greater chance to carry their genetic information to the next generation. For the minimization of the fitness function, the lower the fitness value of the individual, the greater is its share on the wheel. The best individual is always carried to the next generation without any alteration. After the lection, the 70% of the next generation are created using single point crossover and the rest of the next generation is created using mutation. Each gene of a parent has a 0.05 probability of going through mutation. The single point crossover choos a random integer n between 1 and the number of policy parameters. It takes the values of 1 to n from the first parent and n+1 to the number of policy parameters from the cond parent to form the child. The mutation operator lects each gene of an individual and replaces it by a uniform random number with probability 0.05. The terminating condition is a fixed number of generations. It is chon as 200. The fitness function for each model is different and will be explained with the corresponding models. 3) Application to Nonlinear Electric Circuit The Model The equations of the
model are as shown below where I is the current and V is the voltage of the RLC circuit connected ries.
Where This system is nonlinear as the resistance ‘R’ of the circuit is modeled nonlinear, and changes with the cube of the current.
C I V L I I L V I −=+−−=••33I I R +−=
Figure 1: Stock-Flow diagram for the electric circuit model Figure 2: Behavior of the voltage with C = 2, L = 2. Initially I=0, V=1. The arch parameters are capacity (C) and inductance (L) values. The system shows constant oscillatory behavior. Fitness Function for the Electric Circuit Model The desired behavior is chon as a cosine wave with certain period and amplitude for the voltage. The fi
tness function calculates the average period, and the amplitude. To obtain average period, first the times where oscillation reaches maximum values throughout the simulation time are marked (t 1, t 2, …, t n ). Then the spotted times are subtracted concutively starting from t n to obtain periods. The initial period where the system starts from one at time zero to t 1 is not calculated to omit transitory behavior.
1
2121211t t period t t period t t period n n n n n n −=−=−=−−−−−
Finally the average period is calculated from the n-1 periods. To calculate the amplitude, max and min values of the voltage is subtracted. The calculated period and amplitude values are compared with the target period and amplitude of the cosine. Penalty for the period and amplitude difference is equal which means both goals are given equal importance. The desired cosine function is cos( /10) which has a period of 20 and oscillates between of -1 and 1. Note that this fitness function could easily be applied to any model which has potential to show constant oscillation where the desired behavior is oscillatory.
Results
The parameter boundaries are given to be between 0.001 and 5 for both capacity and inductance. Initially the simulation time is t to be 100. The best L and C values and their fitness values obtained after 200 generations by the algorithm for three runs of the algorithm show that the algorithm generally finds near optimum solutions. Since the ideal optimum value can be zero. Additionally, the behaviors of the best ts of parameter values show that the fitness function is well defined.
Table 1: Best policy parameters and their fitness values, time horizon 100
Best three individuals 1 2 3 Fitness Value 0.0114 0.0044 0.2449
L 2.187381755 2.182040847 2.580582881
C 3.860666912 3.875191806 3.423580426
Figure 2: Runs of three best parameter ts, time horizon 100
In the three runs the simulation time was t to be 100 (e Figure 2). When the simulation time is t to 110, the fitness values of the best individuals increa (e Figure 3 and Table 2). This increa is due to the fact that the period is 20 and when t1, t2, …, t n are marked t n is generally eq
ual to 100 where the time the real maximum value is
