Modelling and prediction of complex non-linear processes by using Pareto multi-objective genetic programming

In this paper, a new multi-objective genetic programming (GP) with a diversity preserving mechanism and a real number alteration operator is presented and successfully used for Pareto optimal modelling of some complex non-linear systems using some input–output data. In this study, two different input–output data-sets of a non-linear mathematical model and of an explosive cutting process are considered separately in three-objective optimisation processes. The pertinent conflicting objective functions that have been considered for such Pareto optimisations are namely, training error (TE), prediction error (PE), and the length of tree (complexity of the network) (TL) of the GP models. Such three-objective optimisation implementations leads to some non-dominated choices of GP-type models for both cases representing the trade-offs among those objective functions. Therefore, optimal Pareto fronts of such GP models exhibit the trade-off among the corresponding conflicting objectives and, thus, provide different non-dominated optimal choices of GP-type models. Moreover, the results show that no significant optimality in TE and PE may occur when the TL of the corresponding GP model exceeds some values.     

A new approach for online multiobjective optimization of mechatronic systems

We present a new concept for online multiobjective optimization and its application to the optimization of the operating point assignment for a doubly-fed linear motor. This problem leads to a time-dependent multiobjective optimization problem. In contrast to classical optimization where the aim is to find the (global) minimum of a single function, we want to simultaneously minimize k objective functions. The solution to this problem is given by the set of optimal compromises, the so-called Pareto set. In the case of the linear motor, there are two conflicting aims which both have to be maximized: the degree of efficiency and the inverter utilization factor. The objective functions depend on velocity, force and power, which can be modeled as time-dependent parameters. For a fixed point of time, the entire corresponding Pareto set can be computed by means of a recently developed set-oriented numerical method. An online computation of the time-dependent Pareto sets is not possible, because the computation itself is too complex. Therefore, we combine the computation of the Pareto set with numerical path following techniques. Under certain smoothness assumptions the set of Pareto points can be characterized as the set of zeros of a certain function. Here, path following allows to track the evolution of a given solution point through time.     

Multi-objective optimization: A method for selecting the optimal solution from Pareto non-inferior solutions

Based on the concept of performance-price ratio, we propose a quantitative method to solve multi-objective optimization problems. A new hypothesis is established in this paper: market rules that seek a higher-performing product with a lower price are used to compare and select Pareto non-inferior solutions. After carefully observing the distribution of the Pareto front, we find that the distribution is monotonically increasing or decreasing. This means that different variability exists in the Pareto front and that new inherent disciplines can be found. Based on this discovery, we use the performance-price ratio as a reference to construct the average variability that adjacent non-inferior solutions correspond to the objective function values. Then, the sensitivity ratio that is similar to the performance-price ratio is obtained, and a quantitative method is developed to evaluate Pareto non-inferior solutions. Two important achievements are derived: (1) based on the sensitivity ratio, a new subset of the Pareto non-inferior solution set is formed in accordance with the dominance relationship. The number of Pareto non-inferior solutions is reduced, and the bias degree corresponding to every Pareto non-inferior solution is obtained for different objectives. Thus, it is convenient for decision makers to select Pareto non-inferior solutions based on their preferences. (2) In the new subset of Pareto non-inferior solutions, the solution that corresponds to the minimal absolute value difference of the sensitivity ratio for different optimization objectives is defined as an unbiased and good solution. Accordingly, we obtain the optimal solution that is acceptable for every objective. Finally, the method is illustrated with a numerical example.     

Guidance in evolutionary multi-objective optimization

Many real world design problems involve multiple, usually conflicting optimization criteria. Often, it is very difficult to weight the criteria exactly before alternatives are known. Multi-Objective Evolutionary Algorithms based on the principle of Pareto optimality are designed to explore the complete set of non-dominated solutions, which then allows the user to choose among many alternatives. However, although it is very difficult to exactly define the weighting of different optimization criteria, usually the user has some notion as to what range of weightings might be reasonable. In this paper, we present a novel, simple, and intuitive way to integrate the user's preference into the evolutionary algorithm by allowing to define linear maximum and minimum trade-off functions. On a number of test problems we show that the proposed algorithm efficiently guides the population towards the interesting region, allowing a faster convergence and a better coverage of this area of the Pareto optimal front.     

Pareto optimal generalized <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>-control and vibroprotection problems

We consider a novel multi-objective control problem where the criteria are generalized H ₂-norms of transfer matrices of individual channels from the disturbance input to various objective outputs. We obtain necessary conditions for Pareto optimality. We show that synthesis of Pareto optimal controls can be done in terms of linear matrix inequalities based on optimizing Germeier’s convolution, which also turns out to be the generalized H ₂-norm of the closed-loop system with output composed of the objective outputs multiplied by scalars. As applications we consider multi-objective problems of vibration isolation and oscillation suppression with new types of criteria.     

A preference ordering among various Pareto optimal alternatives

It is often necessary to choose a Pareto optimal point from a set of many. This paper introduces the concept of order of efficiency, which provides a notion that is stronger than Pareto optimality and allows us to set up a preference ordering amongst various alternatives that are Pareto optimal. This approach does not resort to setting up a ranking on the basis of an arbitrary criterion of merit obtained by combining the multiple decision criteria into one scalar index. Examples are cited and it is argued that using the procedure described in this paper, it is possible to rule out Pareto alternatives with extreme components and retain alternatives in the middle of the Pareto set without the help of plots or other visualization aids. This makes the approach applicable for cases where the number of criteria is very high and visualization is intractable.     

Pareto-optimal solutions for multicriteria optimization of a chemical engineering process using a diploid genetic algorithm

In many, if not most, optimization problems, industrialists are often confronted with multi‐objective decision problems. For example, in manufacturing processes, it may be necessary to optimize several criteria to take into account all the market constraints. Hence, the purpose is to choose the best trade‐offs among all the defined and conflicting objectives. This paper presents a multi‐objective optimization procedure based on a diploid genetic algorithm, which yields an optimal zone containing the solution under the concept of Pareto dominance. Pair‐wise points are compared, and non‐dominated points are collected in the Pareto region. Then a ranking is established, and the decision maker selects the first‐best solution. Finally, the procedure is applied to the chemical engineering process of cattle feed manufacture.     

TRANSMax II: Designing a flexible model for transit route optimization

Covering path problems date from the pioneering work of Current et al. (1984, 1985). Two basic forms were defined in their work: the shortest covering path problem and the maximal covering shortest path problem. These two problems differ in that one requires complete coverage by the defined path and the other involves determining path alternatives which cover as much as possible while keeping the path length as short as possible. The latter of these two problems, the maximal covering shortest path problem, embodies the two major goals in transit planning: that is, finding efficient paths which serve as many people as possible. Often transit routes are restricted to major road segments, and when that occurs, routes do not compete with one another unless they overlap along a street segment or at an intersection. In addition, coverage distances can be quite small, barely extending to other streets. Given this type of situation, Curtin and Biba (2011) developed a model called TRANSMax (Transit Route Arc-Node Service Maximization), which maximizes node and arc service, where service coverage is defined for only those street and node segments that are part of a route. They based their model on a structure first proposed by Vajda (1961) in formulating and solving the traveling salesman problem. Because of this structure, we demonstrate that it is possible that a route generated by their original TRANSMax model may not be Pareto optimal with respect to both distance and access. In this paper, we develop a flexible TRANSMax model formulation that finds Pareto Optimal solutions when the original form does not. We also present computational experience in solving this new model on the same street network of Curtin and Biba involving Richardson, Texas. This application allows us to make comparisons between this work and the original work of Curtin and Biba. Overall, we show that this new model can identify new, improved routes over the existing TRANSMax model.     

A bi-objective approach for shortest-path network interdiction

In the literature, solution approaches to the shortest-path network interdiction problem have been developed for optimizing a single figure-of-merit of the network configuration when considering limited amount of resources available to interdict network links. This paper presents a newly developed evolutionary algorithm that allows approximating the optimal Pareto set of network interdiction strategies when considering bi-objective shortest path problems. Thus, the paper considers the concurrent optimization of two objectives: (1) maximization of shortest-path length and (2) minimization of interdiction strategy cost. Also, the paper considers the transformation of the first objective into the minimization of the most reliable path reliability. To solve these multi-objective optimization problems, an evolutionary algorithm has been developed. This algorithm is based on Monte Carlo simulation, to generate potential network interdiction strategies, graph theory to analyze strategies’ shortest path or most reliable path and, an evolutionary search driven by the probability that a link will appear in the optimal Pareto set. Examples for different sizes of networks and network behavior are used throughout the paper to illustrate and validate the approach.     

A multi-objective selection procedure of determining a Pareto set

Ranking and selection (R&S) procedures have been widely studied and applied in determining the required sample size (i.e., the number of replications or batches) for selecting the best system or a subset containing the best system from a set of k alternatives. Most of the studies in the R&S have focused on a single measure of system performance. In many practical situations, however, we need to select systems based on multiple criteria. A solution is called Pareto optimal if there exists no other solution which is better in all criteria. This paper discusses extending a R&S procedure to select a Pareto set containing non-dominated systems. Computational results show that the proposed procedures are effective in obtaining non-dominated systems.     

Designing pareto optimal nonlinear filters for image processing

We consider a method of designing multidimensional polynomial filters characterized by an interval of a discrete functional Volterra series. This method allows designing Pareto optimal nonlinear filters by several criteria. We prove that finding a set of Pareto optimal alternatives is equivalent to minimizing a weighted target function. The designed nonlinear filter is characterized by the plane tangent to a convex optimal solving function, whose curvature is determined by how contradictory the given criteria are. We show examples of polynomial filter design for image processing and compare them with known filters of the same class.     

Complementary metamaterial transmission line for monoband and dual‐band bandpass filters application

A novel composite right‐/left‐handed transmission line (CRLH TL) and its equivalent circuit model are proposed based on cascaded complementary single split ring resonator (CCSSRR). It features an intrinsically balanced wider band and an additional transmission zero above the right‐handed band relative to CRLH TL using complementary single split ring resonator and complementary split ring resonators. Moreover, two single negative (SN) metamaterial (MTM) TLs constructed by using complementary electric inductive‐capacitive resonator on the conductor strip and on the ground, respectively, are researched. Both SN MTM TLs exhibit electric resonance above the fundamental magnetic resonance. For application, a monoband (MB) bandpass filter (BPF) covered WLAN band, and a dual‐band (DB) BPF covered satellite DMB band and WiMAX band are designed, fabricated, and measured. The SN MTM TLs are adopted for the sake of deep and wide out‐of‐band suppression while CRLH MTM TLs using square‐shaped and Sierpinski‐shaped CCSSRR are critical factors of the MB and DB behavior. © 2011 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2012.     

一种有效的数据共享环境多数据源选择算法

针对数据共享环境多数据源选择MDSS(multiple data sources selection)问题,基于Pareto最优理论提出了MDSSA(MDSS algorithm)算法.该算法借助崭新的基于法线测量的非线性路径代价方程计算出到每个数据源的最优路径集合,进而通过代价对比确定实施数据访问的最佳数据源及路径,极大地缩小了搜索空间,在搜索到有效路径的同时,确保了算法的响应时间.大量仿真实验表明,MDSSA算法是有效的.     

Image thresholding based on Pareto multiobjective optimization

A new image thresholding method based on multiobjective optimization following the Pareto approach is presented. This method allows to optimize several segmentation criteria simultaneously, in order to improve the quality of the segmentation. To obtain the Pareto front and then the optimal Pareto solution, we adapted the evolutionary algorithm NSGA-II (Deb et al., 2002). The final solution or Pareto solution corresponds to that allowing a compromise between the different segmentation criteria, without favouring any one. The proposed method was evaluated on various types of images. The obtained results show the robustness of the method, and its non dependence towards the kind of the image to be segmented.     

异构多目标差分-动态窗口算法 及其在移动机器人中的应用

为了实现在多移动机器人和多窄通道的复杂动态环境中机器人的节能运动规划,提出异构多目标差分-动态窗口法(heterogeneous multi-objective differential evolution-dynamic window algorithm,HMODE-DWA).首先,建立行驶时间、执行器作用力和平滑度的3目标优化模型,设计具有碰撞约束的异构多目标差分进化算法来获得3个目标函数的最优解,进而在已知的静态环境中获得帕累托前沿,利用平均隶属度函数获得起点与终点间最优的全局路径;其次,定义基于环境缓冲区域的模糊动态窗口法使机器人完成动态复杂环境中避障,利用所提出的HMODE-DWA算法动态避障的同时实现节能规划.仿真和实验结果表明,所提出的混合路径规划控制策略能够有效降低移动机器人动态避障过程中的能耗.     

隐形矫治方案中的牙齿运动路径规划方法研究

针对隐形矫治方案制定过程中传统牙齿运动路径规划方法准确度及效率低下问题， 根据牙颌评价参数提出新的目标函数，再以传统的人工蜂群算法(ABC)为基础，通过外部存储 存放Pareto 解集，然后以改进的Harmonic 距离对Pareto 解集进行更新，从而提高种群的多样 性。随后通过Slerp 球面线性插值以及线性插值获取牙齿运动路径初始值，与人工蜂群算法中 的初始食物源生成方式相结合，生成更好的食物源。通过改进后的人工蜂群算法采用优先级方 案对新目标函数进行优化，得到牙齿的无碰撞运动路径。通过验证本文方法的矫治方案效果， 并与传统目标函数进行比较，结果表明目标函数可以生成更符合临床治疗要求的矫治方案，改 进ABC 算法相比基本ABC 能够获得更优的路径，缩短了矫治阶段数，具有实用价值。     

Multi-objective design and tolerance allocation for single- and multi-level systems

In this work we develop a method to perform simultaneous design and tolerance allocation for engineering problems with multiple objectives. Most studies in existing literature focus on either optimal design with constant tolerances or the optimal tolerance allocation for a given design setup. Simultaneously performing both design and tolerance allocation with multiple objectives for hierarchical systems increases problem dimensions and raises additional computational challenges. A design framework is proposed to obtain optimal design alternatives and to rank their performances when variations are present. An optimality influence range is developed to aid design alternatives selections with an influence signal-to-noise ratio that indicates the accordance of objective variations to the Pareto set and an influence area that quantifies the variations of a design . An additional tolerance design scheme is implemented to ensure that design alternatives meet the target tolerance regions. The proposed method is also extended to decomposed multi-level systems by integrating traditional sensitivity analysis for uncertainty propagation with analytical target cascading. This work enables decision-makers to select their best design alternatives on the Pareto set using three measures with different purposes. Examples demonstrate the effectiveness of the method on both single- and multi-level systems.     

An improved TOPSIS approach for the competitiveness analysis of provincial information resource industries in China

The development of information resources industries is becoming increasingly important for improving the competitiveness of individual economies. As a result, effectively evaluating the performance of individual information resources industries becomes a critical problem that needs to be adequately addressed. This paper presents an improved Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) approach for evaluating the performance of provincial information resources industries in an objective manner in China. The concept of information entropy is appropriately used for determining the objective weighting of the evaluation criteria in the evaluation process. The Mahalanobis distance is adequately adopted for better reflecting the relative distance between individual alternatives and the idea solution in determining the overall performance of individual alternatives across all the evaluation criteria. This leads to the determination of consistent performance evaluation outcomes, which are better accepted by all the stakeholders in the evaluation process. An example of evaluating the competitiveness of provincial information resources industries in China is presented. The result shows that the improved TOPSIS approach is effective for evaluating the competitiveness of information resources industries in the real‐world setting.     

A new compromise solution method for fuzzy group decision-making problems with an application to the contractor selection

This paper presents a novel compromise solution method for solving fuzzy group decision-making problems by a group of experts, which can determine the best alternative by considering both conflicting quantitative and qualitative evaluation criteria in real-life applications. The compromise solution method is developed based on the concept that the chosen alternative should be as close as possible to the positive ideal solution and as far away from the negative ideal solution as possible concurrently. The performance rating values of alternatives versus conflicting criteria as well as the weights of criteria are described by linguistic variables with multi-judges and are converted to triangular fuzzy numbers. Then, a new collective index is introduced to distinguish among potential alternatives in the assessment process with respect to subjective judgment and objective information. Finally, a real case study and an application example for a contractor selection problem are provided in construction industry to demonstrate the implementation process of the proposed method.     

An ant colony optimization algorithm for the bi-objective shortest path problem

Multi-objective shortest path problem (MOSP) is an extension of a traditional single objective shortest path problem that seeks for the efficient paths satisfying several conflicting objectives between two nodes of a network. MOSP is one of the most important problems in network optimization with wide applications in telecommunication industries, transportation and project management. This research presents an algorithm based on multi-objective ant colony optimization (ACO) to solve the bi-objective shortest path problem. To analyze the efficiency of the algorithm and check for the quality of solutions, experimental analyses are conducted. Two sets of small and large sized problems that generated randomly are solved. Results on the set problems are compared with those of label correcting solutions that is the most known efficient algorithm for solving MOSP. To compare the Pareto optimal frontiers produced by the suggested ACO algorithm and the label correcting algorithm, some performance measures are employed that consider and compare the distance, uniformity distribution and extension of the Pareto frontiers. The results on the set of instance problems show that the suggested algorithm produces good quality non-dominated solutions and time saving in computation of large-scale bi-objective shortest path problems.