a war scenario simulated by the THUNDER software. The adaptive genetic algorithm developed in this research changes the mutation and crossover rates.
Evolutionary Approach to Multi-Objective Problems Using Adaptive Genetic Algorithms Z. Bingul, A. Sekmen, and S. Zein-Sabatto Electrical and Computer Engineering Tennessee State University Nashville, TN, 37209 4.
Abstract This paper describes an adaptive genetic algorithm used to achieve multi-objectives such as minimizing the territory loses and maximizing enemy air loses by finding the optimum distribution of air-crafts fighting in a war scenario simulated by the THUNDER software. The adaptive genetic algorithm developed in this research changes the mutation and crossover rates adaptively to provide a fast convergence to the optimum possible solutions. According the population of the fitness values obtained for each generation, three distribution properties (the mean, the variance and the best fitness value) are determined and used as input to a fuzzy-logic system for modifying the mutation and crossover rates to obtain the individuals of the next generation. This enables maintaining a fast and smooth convergence to the best possible solutions.
1. Introduction Genetic algorithms (GAS) have been widely used in multi-objective optimization problems to provide a solution based on adaptive heuristic searching among a solution space [I]. In most of the cases, the objectives of optimization problems conflict each other and there may not be unique optimum solution. In other words, it may not be possible to improve some objectives without degrading some others. GAS are very powerful tools for those types of problems since they process a set of solutions in parallel and produce possible different best solutions to multi-objective optimization problems. GAP are superior to conventional optimization algorithms in multi-objective problems because of the following four features [2]: I.
GAS search with a population of points (candidate solutions), not a single point. Thus, they are less likely to be trapped in a local optimum 2. GAS use only the values of the payoff (objective function) information, and not the derivatives or other auxiliary knowledge. 3. GAS work with a coding (representation) of a parameter set not the parameters themselves. Thus the search method is naturally applicable for solving discrete and integer programming problems. 0-7803-6583-6/00/$10.00 0 2000 IEEE
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GAS use randomized parents selection and croswver from the old generation. Thus they efficiently explore the new combinations with the available knowledge to find a new generation with better fitness values.
THUNDER software is a very large campaign simulation model, which was built based on Monte-Carlo simulation. This software's internal dynamics are unknown since internal parameters of the software for every run change based on random seeds generated by an internal random number generator. This software is a stochastic, two-sided, analytical simulation of military operations developed by System Simulation Soluiions Inc. (S3I) for the Air Force Studies and Analyses Agency (AFSAA). This simulation was designed in order to examine issues involving the utility and effectiveness of air and ground power in a theater-level joint warfare context. The Thunder software can define approxiinately 25 air missions grouped under air-to-ground missions. air-to-air missions, air defense suppression missions. reconnaissance, anti-tactical ballistic missile, ancl air refueling. This software automatically plans military moves and actions in a rule-based manner. It also judges the outcome of their interactions and then it dynamically incorporates the results and uses this information into the on-going perception, planning, and execution of military operations. This software is capable of supporting campaign analysis involving the integration ol' effects over time and space. This means automatic distribution of threats and targets, the number of target kills, pround movement and deployments. This simulation also applies constraints to .the problems like defining the inventories. sortie rates, mission allocations etc. In this paper, adaptive genetic algorithms are used tor effectively allocating the optimum distribution o f aircrafts fighting in a war scenario simulated by the THUNDER software. This paper is organized as follows: Section 2 gives some background information on GA operators. Section 3 presents the statement of problem and describes the method followed. The fuzzy logic system used is explained in Section 4 and the results are presented in Section 5. Finally, some conclusions are given and future work is motivated in Section 6.
2. Background 111111
Basic genetic algorithms include three main operators selection. crossover, and mutation - as described in detail below. Figure I illustrates how basic GAS work.
2.1 Selection Selection is one of the most fundamental genetic operators. Selection operation may be modeled as follows: pop
P,, .,,.(,(ti) = . f ' ( n ) / C f ( k )
(1)
k=I
Where n is n"' string, pop is the population size andf ( n ) is the fitness function. This first population must offer a wide diversity of genetic materials. The gene pool should be as large as possible so that any solution of the search space can be engendered. Generally, the initial population is generated randomly. Some of the most commonly used selection operators are roulette wheel selection, tournament selection, Ranking selection etc.
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Mutation is usually used to avoid premature convergence, which is a common problem in CAS. which use fixed length, binary codlings. When proportional selection is used, all the individual chromosomes in the population become very similar before a nearly optimal solution is reached, thus preventing any further progress. In such cases mutation is essendal. Mutation acts against this, by constantly generating new chromosonies. this helps in preventing the population from getting trapped in a local maximum in a search space. Howevcr. mutation sometimes also result in loss of good individual, thus the need to prevent premature convergence has to be balanced against the loss o f efficiency due to the damage (of good genetic material. Thus there is a payoff be.tween exploitation and exploration illustrated here.
2.2 Crossover Creation aftniti,d population stnn(,s -
This is the most powerful genetic operator, and may be considered as the main engine for exploration in a GA. This operator is responsible for the shuffling and recombination of building blocks.
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Fitness functio? evaluation
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The simplest form of crossover is that, a single point is chosen on two equal length chromosomes and they are crossed at that particular point. It is possible to select two or more points for cross over, to get more genetic mixing hut sometimes while using multipoint crossover it degrades the performance. Crossover can be shown as follows.
Oenerate nshipopulation of 8Itings
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x
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'
1 1 1 1 0 0
0 0 0'0 0 0 0 0 0 0 1 1 Crossover generally consists of forming a new solution by taking some parameters from one solution and exchanging it with another at the very same point. Thus we get new offspring. Some crossover operators use complex geometric methods to generate the off springs of two parents.
2.3 Mutation This is a common genetic manipulation operator, and it involves, the random alteration of genes during the process of' copying a chromosome from one generation to the next. Mutation simply involves the incorrect copying of some parameters, which make up a solution. It may be illustrated as follows.
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1 SeleetlonZ Cruqauer 3 f$tatlon
Figure 1. The flow diagram sh'owing how GAS w o k with its three basic operators.
3.
Methodology
THUNDER software gives the results simulated for different war scenarios in which the capabilities of threat forces, conditions of the war., and the capabilities of' friendly forces are input parameters. In this research. only four missions and a 15-day war were considerecl for simulations. The missions are: Offensive Counter Air (OCA), Strategic Target Interdiction (STI), Long Ranse Air Interdiction (INT), and Lethal Direct Air Detense Suppression (DSEAD). OCA is against airbases and INT is against units moving on the network and in garrison.
logistics facilities, transportation network transshipment points, checkpoints, supply convoys, and air defense complexes. STI is against strategic targets. DSEAD is a suppression of enemy air defense missions and it is apainst air defense sites. THUNDER software can be viewed more like a two-player game in which blue represents the friendly side and red is the enemy side. Outputs from THUNDER Software (territory lost, aircraft lost, the number of strategic targets killed and the number of red armor killed) were assigned a minimum score and a maximum score. They were translated to a minimum score of 1 and a maximum score of 2 . Scores between the minimum and maximum were interpolated based on the worst case and the best case which were determined by expert knowledge. In this study, the objectives for these scenarios would be to
performance of the FLS. A set of rules that describe5 how to adjust the mutation and crossover rates was constructed to specify which actions to be taken undcr which conditions. These rules were determined using the following facts: to improve search at the beginning of' run, the mutation rate should be low and crossover rate should be high, and then the mutation rate should he increased and crossover rate should be decreased toward the end of the run. They are summarized below i n Lis1 I Adaptive Genetic Algorithms
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Mutation rate Crossover rate
Minimize the territory that blue side losses, Minimize the blue side aircraft lost, Maximize the number of red side strategic targets ki 1led, Maximize the number of red side armor killed.
Fuzzy Logic
Among many alternatives of fitness functions, the following fitness function was used after making some comparisons as explained in [ 2 ] .
Best Fitness Mean Variance
I. 2. 3.
4.
F
=iW;f,' Population of FRness values
,=I
where .f; and w, are the fitness value and weight corresponding to I"' objective, respectively. The weight show the importance of each objective.
4.
Figure 2. Schematic representation of fuzzy logic hascd GA.
Fuzzy Logic System
In order IO improve the results that would have been obtained by traditional genetic algorithms, a fuzzy logic system is combined with a GA system to adaptively change mutation and crossover rates depending on the fitness values obtained at the end of each set of runs [ 3 ] . The procedure used is exhibited in Figure 2 . First, the initial population is created randomly by the GA and then the resources are allocated and used as input in a war scenario simulated by THUNDER software. The population of fitness values is produced using the war results for each input allocation. The fuzzy logic system takes the best fitness value, the mean and variance of the fitness population as input and yields the mutation and crossover rates for each population (Figure 3 ) . Three kinds of membership (LEFT-small, TR-medium, RIGHT- large) and triangular membership functions were used to fuzzify the inputs. Figure 4 illustrates the membership functions and ranges for each fuzzy variable. The optimization of these assignments is often done through trial and error for achieving optimum
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Figure 3. The fuzzy logic system for adaptation 01' the mutation and crossover rates. List 1. Descriptions of the fuzzy system's rules 1.
2. 3.
If (BF is LEFT) then (MR is LEFT)(CR RIGHT) If (BF is TR) and (UN is LEFT) then ( M R LEFT)(CR is RIGHT) If (BF is TR) and (UN is TR) then ( M R TR)(CR is TR)
is
is
IS
If (UN is RIGHT) and (VF is TR) then (MR is RIGHT)(CR is LEFT) 5. 11' (BF is RIGHT) and (UN is LEFT) then (MR is LEFT)(CR is RIGHT) 6. If (BF is RIGHT) and (UN is TR) then (MR is TR)(CR is TR) 7. If' (UN is RIGHT) and (VF is LEFT) then (MR is RIGHT)(CR is LEFT) 8. If (UN is RIGHT) and (VF is RIGHT) then (MR is LEFT)(CR is LEFT)
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settings of these rates, the search is not stuck at a local area during especially start up.
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Figure 5. Variations of the mutation and crossover rates over 50 generations.
Output variables
Input variables
Figure 4. Fuzzy sets for adaptive genetic algorithm system. 5.
Results and Discussion
It is very important to understand the dynamics of the population of a CA over different generations when applied to multi-objective optimization problems. It is hard to know how the population is behaving and how to vary the GA parameters to find nondominated solutions over wide range. It is generally accepted that high rates of' learning if the mutation and crossover rates are chosen right. The relationship between population features and CA parameters are very complex and nonlinear. Thus, it is useful to describe the relationship linguistically to maintain diversity in GAS. Rule based adaptation system using population features was developed to adjust mutation and crossover rates. Figure 5 illustrates variations of the mutation and crossover rates over 50 generations. It is seen from the figure that both these rates are changed between 0.018 and 0.047, and 0.65 and 0.75. These changes allow to explore new solution points and to optimize run time of GAS. By using automatic
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Figure 6 displays a comparison of the performances 01' the static and dynamic C A systems. The stochastic nature of a CA optimization requires multiple runs t o ensure reproducibility. Thus, [:he CA was run four times and each time similar results were obtained. It is observed that there are two main advantages of the adaptive CA system on the non-adaptive system. First. the adaptive CA converges to a higher fitness value compared to the non-adaptive system. Second. the rate of convergence of the adaptive system is higher than that of' the non-adaptive system over the early generations. Comparison of adaptive and non-adaphe GA systems I
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Figure 6. Comparison of adaptive and non-adaptive GA systems.
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6.
Conclusion
In this paper, a rule based genetic algorithm is applied to a multi-objective optimization problem to find the allocations of forces that gives the optimum results in a war scenario simulated by THUNDER software. Based o n the results, the adaptive genetic algorithm introduced in this paper improves the convergence rates considerably and maintains smoother convergence to the best possible solutions than that of the conventional genetic algorithms.
References
[ I ] D.E Goldberg., Genetic Algorithms in search, optimization, und iliachine learning, AddisonWesley, 1989. [2] 2. Bingiil, A. $. Sekmen, S. Palaniappan and S. Zein-Sabatto, “Genetic Algorithms Applied To Real Time Multiobjective Optimization Problems,” IEEE SouthEusf Conference 2000, April 2000, Nashville, TN, USA. 131 Y. Shi, R. Eberhart and Y. Chen, “Implementation of Evolutionary Fuzzy Systems”, IEEE Trans. on F U ~ ZSyst., J Vol. 7, NO. 2, 1999, pp. 109-1 18. [4) 2. Bingul, A. $. Sekmen, S. Palaniappan and S. Zein-Sabatto, “An Application of MultiDimensional Optimization Problems Using Genetic Algorithms,” Proceedings of the IASTED International Conference on Intelligent Systems and Control, October 26-28, 1999, Santa Barbara, CA, USA. 151 Schaffer, J. D., “Some Experiments In Machine Learning Using Vector Evaluated Genetic Algorithms,” Ph.D. Disertation, Vanderbilt University, 1984. [6] E. Zitzler and L. Thiele, “Multiobjective Evolutionary Algorithms: A comparative Case Study and the Strength Pareto Approach”, IEEE Truns. on Evolutionury Computation., Vol. 3, No. 4, 1999, pp. 257-27 I , 1999.
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