Short Communication
System Optimal Design Approach using Knowledge-Based Genetic Neural Networks
School of Mathematics and Computer Sciences, Xinyu University, 338003, Xinyu, People`s Republic of China, China
Recently, various systems become more and more complex and the corresponding influence factors also enjoy a great increase. Therefore, it becomes a very difficult job to make an overall system optimal design. For example, the design of weapon system is the feedback and iteration process of technical information between the whole system and equipments. Usually many conflicts exist among different performances of the weapon system and in the performance between the whole system and equipments. They should be coordinated constantly to improve the performance of the whole system (Camara et al., 2009; Yuan et al., 2008; Mukhopadhyay et al., 2009; Zhou, 2009). Therefore, most of the countries in the world have shifted their focus to the integrity of combat ability, survival ability, rapidity, maneuverability and compatibility when designing weapon systems (Wang et al., 2009; Marano et al., 2009; Zhou et al., 2009) during recent years.
Since the system optimal design is a difficult job, this study first employs structure-based neural networks to fit the input-output relationship of the weapon system, then adopts genetic algorithms to train parameters of the network and optimize the neural network model and finally achieves very satisfactory experimental results. The whole design approach can be concluded as below: Extracting some knowledge from the optimization process of genetic algorithms and then employing it to guide the subsequent optimization process. Research thinking of this study is shown in Fig. 1.
PROBLEM DESCRIPTION
The problem to be studied in this study can be concluded as: employing simulation optimization approaches to study system optimal design problems and attempting to make the system to be studied output satisfactory results through fewer times of system simulation. The corresponding target function is described as:
(1) |
where, P1, P2,..., P8 denote some indicators that need to be maximized in the system to be studied, Q1, Q2, , Qr denote ones that need to be minimized and T denotes the total time cost by the system simulation during the optimization process. Constraint conditions of the problem are described as:
(2) |
In the first constraint condition and the second one, take simulation outputs of all the subsystems as the input of the whole system and then make simulations to obtain the output value of all the indicators of the system to be studied.
Fig. 1: | Research thinking of this study |
Si(1≤i≤n) denotes the simulation output of the ith subsystem, either a single value or a vector. The third constraint condition denotes the total time cost by simulations during the optimization process. ti(1≤i≤n) denotes the average time spent by the ith subsystem on a single simulation. tn+1 denotes the average time spent by the system on the overall simulation. ci(1≤i≤n) denotes the number of simulations implemented by the ith subsystem during the optimization process and cn+1 denotes the total number of overall simulations. The fourth constraint condition denotes that simulation outputs of subsystems can be obtained solely through subsystem simulations. ki(1≤i≤n) denotes the number of input variables of the ith subsystem. The fifth constraint condition defines the feasible region of input variables of the system to be studied.
KNOWLEDGE-BASED GENETIC NEURAL NETWORKS
The knowledge-based genetic neural network approach proposed in this study employs the structure-based neural networks with highly-nonlinear mapping capability to fit the input-output relationship of the weapon system, trains parameters of the structure-based network through genetic algorithms with global searching ability. Genetic algorithms can also optimize the neural network model at the same time. This approach extracts some knowledge from the optimization process of genetic algorithms and then uses it to guide the subsequent optimization process.
Structure-based neural networks: Since traditional neural network models have black boxes, this study intends to employ structure-based neural networks to fit the input-output relationship of the system optimal design problem. The structure-based neural network is established based on the causality theory and connections between its nodes are all based on the causal relationships in real systems, so it is fairly powerful in interpreting models. The structure-based neural network modeling has settled many defects confronting traditional neural network modeling, such as the unstructured models, uncertainty of the number of neurons, slow convergence and local minimization, etc. It has acted as a new powerful processing tool for non-linear systems, especially for the system modeling and structure parameter optimization of those large-scale non-linear systems (Whittaker et al., 2009; Chung et al., 2009; Martin, 2009; Kimbrough et al., 2008; Vilcot and Billaut, 2008).
Compared with standard neural networks, the neural network used in this study owns the following advantages.
Fig. 2: | Structure-based neural network model |
(1) The structure of neural networks is corresponding to that of practical systems one to one and all connections between neurons of the neural network are based on the structure and components of practical systems, (2) All the connection weights of the neural network model own specific physical meanings, (3) The number of neurons in each layer of the neural network depends on that of subsystems obtained from the division of the whole system, (4) The structure model of subsystems is very simple since a single neuron is used to simulate a practical subsystem.
Figure 2 shows the input-output relationship model for system optimal design problems based on the structure-based neural networks. x11, x12, , x1n denote the input of the first subsystem, x21, x22, , x1n denote the input of the second one and xm1, xm2, , xmn denote that of the mth one. P1, P2, , P5 denote some indicators of the system to be studied that need to be maximized and Q1, Q2, , Qr denote those that need to be minimized.
Orthogonal genetic algorithm with quantification Initialization of the population:
• | Divide the feasible region [L, U] of the problem to be optimized into B subspaces according to the following equation: | |
(3) |
where, L = [l1, l2, , ln]T and U = [u1, u2, , un]T, respectively denote the lower boundaries and upper boundaries of n independent variables of the problem. B denotes the design parameter. 1k is the n-dimensional vector of which thekth bit is 1 and other bits are 0. Li and Ui denote n-dimensional vectors respectively similar to L and U. In this way, the feasible region of the problem can be divided into B subspaces, namely [L1, U1], [L2, U2] , , [LB, UB] | |
• | Discretize each independent variable in each subspace according to the following equation. Suppose the domain of the independent variable xi is [li, ui], then xi can be quantized into Q1 (the design parameter) levels ai1, ai2, , aiQ1 and the detailed computation method for aij is: |
• | Select M1 chromosomes from each subspace |
First construct the orthogonal Table in which N denotes the dimensionality of the problem, and J1 denote the positive integers that satisfy the condition:
Then select M1 combinations from the QN1 ones to form M1 chromosomes.
• | Select the best G (the size of the initial population) chromosomes from the M1B potential ones to form the initial population according to their fitness value |
Crossover operation: Select two parent chromosomes for crossover operation according to the crossover probability. Suppose that the two selected parents are:
and the defined solution space [lparent, uparent] is:
(4) |
• | Discetize the solution space [lparent, uparent] of two parent individuals for crossover operation into Q2 parts |
• | Select some independent variables that will suffer crossover operations. The number of potential child individuals generated by each pair of parent individuals should be controlled to avoid a huge evaluation on the populations during the crossover operation. In this study, the crossover only operates on F genes of the parent chromosome. Discretize these F independent variables in each subspace |
• | Select potential child points from the solution space of parents according to the orthogonal table. First generate the orthogonal table in which Q2 is an odd number, and J2 is the smallest positive integer that satisfies the condition: |
Then select M2 combinations from these QF2 ones to form M2 potential child individuals | |
• | Select two with the best fitness value from the M2 potential child individuals and two parents as the result of this crossover operation |
• | If the number of the implemented crossover operations has reached the preset value, stop crossover immediately, or turn to Eq. 1 |
Mutation operation:
• | Randomly select a parent chromosome for mutation operation according to the mutation probability |
• | Obtain the mutated child chromosomes according to the fine perturbation method. Here, the perturbation means generating four mutated child chromosomes through tuning the value of the selected parent gene respectively to its original 1-2σ, 1-σ, 1+σ times and 1+2σ times |
• | Select the best one of the parent and child chromosomes as the result of this mutation operation |
• | If the number of the implemented mutation operations has reached the preset value, stop mutation immediately, or turn to Eq. 1 |
Knowledge extraction and learning
Summary of expertise: The summary form of expertise is shown in Fig. 3. The system to be studied becomes more and more complex and its influence factors are also on the increase, so it is necessary to make an analysis on the influence of input variables on target functions according to expertise. Figure 3a shows that the value of target function f(x) increases as the variable x does and Fig. 3b is just the opposite. Fig. 3c shows that f(x) decreases first and then increases as x increases and Fig. 3d is just the opposite. Figure 3e shows that the influence of x on f(x) is slight and finally 3(f) shows that f(x) decreases, increases and then again decreases as x increases.
Extraction of optimal knowledge: In this study, the knowledge-based genetic neural network first selects several elitists from the current population, then extracts the sensitivity of variables to the output from these elitists through the sensitivity analysis method and finally uses the sensitivity to guide the subsequent optimization process. The form of optimal knowledge extracted in this study is shown Fig. 4 which indicates that f(x) increases first and then decreases as x increases.
Fig. 3(a-f): | Summary form of expertise |
Application of expertise and optimal knowledge: As to optimization problems involving multiple variables, the sensitivity of each variable to the target value is different. Some are very sensitive to the target value, namely that even a slight adjustment to the variable will cause a great change on the output, whereas, some are not so sensitive and any fine adjustments to the variable will not cause big changes on the output.
Sensitivity knowledge of variables can be employed to assist intelligent optimization approaches in improving their optimization performance in solving these complex optimization problems. In a certain region of the feasible space of the optimization problem, if some variable is relatively sensitive to the target value, then try to adjust its input value to improve the target value to the largest extent and if it is not sensitive, then any fine adjustments will be in vain. Therefore, in order to improve the efficiency of intelligent optimization approaches, it should try to make fine adjustments on variables with high sensitivity to better the target value to the largest extent and at the same time avoid adjustments on variables with low sensitivity to save the calculation to the largest extent.
Fig. 4: | Representation form of optimal knowledge |
Table 1: | Comparative analysis on experimental results |
/: Means that no researches have been made |
EXPERIMENTAL RESULTS
XX system exerts its damage on the target through target search, target identification and target attack. The system owns multiple factors and its exertion process is complicated, so we can make full use of its high cost-effectiveness only by optimizing structure parameters and performance parameters and coordinating their relationships. Performance factors selected in this study for the system mainly include: The fall velocity Vy, rotating speed T, scanning angle 2S, operating distance H, position error of sensors and dispersion error of warhead E, ambient wind velocity F. The hit probability is selected as the target function.
According to the practical simulation model of XX system, the obtained simulation hit probability corresponding to the above-mentioned optimization parameters is 0.812. Compare the optimal hit probability with the simulation hit probability and we can find that the simulation model based on the performance factors of neural network is feasible and accurate. The author has made a comparative analysis between the result obtained in this study and that in literature (Rongzhong, 1996; Kun and Rongzhong, 1999, 2001; Kun et al., 2004), shown in Table 1. It shows that the two are in close proximity to each other which further indicates that it is totally feasible to employ genetic neural networks for optimal design.
Main contributions of this study include: Employing the structure-based neural networks with highly-nonlinear mapping capability to fit the input-output relationship of the weapon system, using genetic algorithms with global searching ability to train parameters of the structure-based network and optimize neural network models. This approach extracts some knowledge from the optimization process of genetic algorithms and then uses it to guide the subsequent optimization process. The experimental result indicates that the approach outperforms several other existing methods.