Robustness analysis in Multi-Objective Mathematical Programming using Monte Carlo simulation

A multi-objective particle swarm optimization (MOPSO) approach for multi-objective economic load dispatch problem in power system is presented in this paper. The economic load dispatch problem is a non-linear constrained multi-objective optimization problem. The proposed MOPSO approach handles the problem as a multi-objective problem with competing and non-commensurable fuel cost, emission and system loss objectives and has a diversity-preserving mechanism using an external memory (call “repository”) and a geographically-based approach to find widely different Pareto-optimal solutions. In addition, fuzzy set theory is employed to extract the best compromise solution. Several optimization runs of the proposed MOPSO approach were carried out on the standard IEEE 30-bus test system. The results revealed the capabilities of the proposed MOPSO approach to generate well-distributed Pareto-optimal non-dominated solutions of multi-objective economic load dispatch. Comparison with Multi-objective Evolutionary Algorithm (MOEA) showed the superiority of the proposed MOPSO approach and confirmed its potential for solving multi-objective economic load dispatch.

The two primary goals of a multi-objective evolutionary algorithm (MOEA) for solving multi-objective optimization problems are to search as much non-dominated solutions as possible towards the true Pareto front and to maintain diversity of Pareto optimal solutions along tradeoff curves. This study presents a new hybrid MOEA, the niched Pareto tabu search combined with genetic algorithm (NPTSGA), to find Pareto-optimal solutions to multi-objective optimization problems. The NPTSGA is developed on the thoughts of integrating genetic algorithm (GA) with the improved tabu search (TS) based MOEA, niched Pareto tabu search (NPTS). The proposed NPTSGA is then tested through a simple test example and compared with other two techniques, NPTS and niched Pareto genetic algorithm (NPGA). Computational results indicate that the proposed NPTSGA is an efficient and effective method for solving multi-objective problems, while keeping the balance between the intensification of non-domination to the true Pareto-optimal solutions (TPS) and the diversification of the near Pareto-optimal solutions (NPS) along the tradeoff curves.

Most practical problems are formulated as multiobjective optimization problems (MOP) so as to meet the diversified demands of a decision maker. Usually, there is a trade-off relation among objective functions, and thus there does not necessarily exist the solution that optimizes all objective functions simultaneously in MOP. Therefore, Pareto optimal solution is used as a definition of solution to MOP. Recently, evolutionary algorithms have been developed remarkably in order to obtain approximate solutions to optimization problems. Particularly, multiobjective genetic algorithms (MOGA) have been developed for generating Pareto optimal solutions. However, there are two problems in MOGA: how to assign the fitness to individuals, and how to keep the diversification of individuals. Many existing MOGAs have made an effort in order to overcome these problems, and so does this paper. First, this paper suggests a fitness function in MOGA using generalized data envelopment analysis (GDEA) which was suggested for evaluating the relative efficiency of individuals under several items of assessment management science. It is shown that the GDEA method can approximate Pareto optimal solutions more effectively and faster than the ranking method which is mostly used in MOGA, and generate well-distributed Pareto optimal solutions. Furthermore, this paper suggests the aspiration-level based GDEA method to generate the most interesting part (not the whole Pareto optimal solutions) to an aspiration level of decision maker for choosing a final solution from many Pareto optimal solutions. Finally, this paper illustrates the effectiveness of the methods using GDEA through several numerical examples.

In this thesis we investigate the existence of solutions to boundary value problems (BVPs) for nonlinear systems of first-order ordinary differential equations (ODEs) and difference equations. The thesis is divided into two parts. Part one is devoted to the study of difference equations where two types of problems are studied. First, we derive the existence of solutions to first-order systems of difference equations that arise when one applies the trapezoidal rule to approximate solutions of second-order scalar ODEs. Strict discrete lower solutions are used with maximum principle arguments in the discrete problem to obtain a priori bounds on solutions. The a priori bounds on difference quotients for solutions of the discrete problem are obtained from the Bernstein-Nagumo condition. We use homototopy to find solutions of the discrete amroximations. Second, motivated by Ma [60] we consider existence and uniqueness of solutions to difference equations which arise as discrete approximations to three-point BVPs for systems of first-order ODES after employing the Euler method. Several existence and uniqueness results for three-point BVPs are established, using the contraction mapping theorem and the Brouwer fixed point theorem in Rn. In part two, two types of problems for systems of first-order ODES are analyzed. First, we study the existence of a unique solution to a system of first-order differentia1 equations under multi-point BVPs with nonlinear boundary conditions. We prove the existence and uniqueness of solutions for a system of first-order three-point BVP under which the properties of linear three-point BVP are preserved under small nonlinear perturbations of both the differentia1 equation and the boundary conditions. The method of proof of the existence of solutions using the Banach contraction mapping principle is similar to that used by Rodriguez [75], who studied the discrete analogue of this BVP. Second, motivated by Cronin 120,211 we investigate the existence of solutions to three-point BVPs in perturbed systems of first-order ODEs at resonance where the associated homogeneous linear problem has nontrivial solutions. Based on the second part of the investigation of three-point BVPs in perturbed systems of first-order ODEs at resonance, four different results were obtained: (i) The existence of solutions via a version of the Brouwer Fixed Point Theorem which is due to Miranda. (ii) The existence of solutions to the problem through the application of the Implicit Function Theorem. These two results extend the work of Feng and Webb [34], and Gupta [39] for the problems at resonance in Euclidean 2-space. (iii) Existence results for three-point BVPs at resonance for general BVPs through the application of Brouwer degree theory; we show the degree is non zero through the application of Borsuk's Theorem for one problem and through a result of Cronin involving pairs of polynomials in two variables whose terms of highest order have no common factors. (iv) The existence of solutions to the entrainment of frequency problem in Euclidean 2-space. We will investigate conditions under which entrainment of frequency for three-point as well as two-point BVPs by adapting the work of Cronin in periodic case. We will describe such conditions and then show the degree is non zero through a result of Cronin involving pairs of polynomials in two variables whose terms of lowest order have no common factors.

<p><b>The real world is full of problems with multiple conflicting objectives. However, Reinforcement Learning (RL) traditionally deals with only a single learning objective. Recently, several Multi-Objective Reinforcement Learning (MORL) algorithms have been proposed. Nevertheless, many of these algorithms rely on tabular representations of the value function, which are only suitable for solving small-scale problems. In addition, although some existing MORL techniques can learn the Pareto optimal solutions, they can only be applied to simple multi-step problems in a discrete Markov environment. However, many real-world problems involve partially-observable continuous environments, where continuous environments are often discretise before the systems learn the best policies from a starting to a goal state. Therefore, effective MORL techniques are needed to address these problems. </b></p> <p>Learning Classifier Systems (LCSs) have been widely used to tackle RL problems as they have a good generalization ability and provide a simple, understandable rule-based solution. The rule-based solution has good scalability to solve large-scale problems. In addition, the rule-based solution is explainable. Thus the user is not only able to know what the solution is but understand why the solution is suited to the task and when to trust the solution. Moreover, the accuracy-based LCS, XCS, has been adopted successfully for single-objective RL problems. Thus, LCSs, especially XCS, have huge potential to solve MORL problems.</p> <p>In this thesis, LCSs are enabled to learn the Pareto optimal policies for discrete, partially-observable, discrete and continuous MORL problems. The objectives are to develop XCS-based algorithms to learn the Pareto optimal policies for MORL problems, improve the generalization ability of the multi-objective XCS-based algorithm where possible, evaluate the generalization ability of developed multi-objective XCS-based algorithms for solving MORL problems, and apply multi-objective XCS-based algorithms to solve large-scale partially-observed MORL problems. The following tasks have been completed in this thesis.</p> <p>A new multi-objective LCS-based algorithm MO-XCS has been developed based on XCS for multi-objective learning. This algorithm is designed to learn a group of Pareto optimal solutions through a single learning process. For this purpose, four technical issues in XCS have been identified and addressed. Experimental studies on three bi-objective maze problems further demonstrate the effectiveness of MO-XCS.</p> <p>A new multi-objective LCS-based algorithm MOXCS, different from MO-XCS, has been developed based on XCS and MOEA/D. It employs a decomposition strategy based on MOEA/D in XCS to approximate complex Pareto Fronts. The experimental results show that on complex bi-objective maze problems, MOXCS can learn a group of Pareto optimal solutions for MORL problems without requiring huge storage. Analysis of the learned policies shows successful trade-offs between the distance to the reward and the amount of reward itself. With integer inputs, MOXCS can address the partially observable Markov problem Deep Sea Treasure. Lastly, the generalization ability of MOXCS is tested in the Multi-Maze and Multi-Maze Connection environments. However, it is shown that the Multi-Maze Connection domain is not a proper benchmark for testing the generalization ability of the MORL algorithm.</p> <p>CoinRun is a continuous domain, unlike the previous Maze and Deep Sea Treasure, which are discrete. Two bi-objective CoinRun environments are developed in this thesis for testing the generalization ability of the MORL algorithm. Several changes in the developed bi-objective CoinRun environments have been made. First, they have been changed from continuous environments to the discretized environments by the technique of discretizing continuous inputs. Second, those environments have been changed from the non-Markov environments to Markov environments by adding extra characters. Third, in order to enable the agent to sense the environment a sub-actions technique has been developed; when the agent takes action several following actions will be taken automatically. With these changes and techniques, the developed bi-objective CoinRun problems are solved by MOXCS. The generalization ability of MOXCS will be evaluated by exchange those two bi-objective CoinRun environments as the training and testing environment. The evidence shows MOXCS has the generalization ability for solving the MORL problem in an unseen environment with proper training in a similar environment. </p> <p>The single-objective Mountain Car is extended as a bi-objective MORL problem by adding another objective for the mountain car to take more action, `zero-throttles'. MOXCS is implemented to resolve the multi-objective mountain car problem. The result shows that MOXCS has good potential to resolve this large-scale Markov problem. </p> <p>This research work has shown that the LCS-based algorithms can learn Pareto optimal policies for the discrete, partially-observed, and adapted continuous MORL problems and have the potential to scale to complex, large-scale MORL problems.</p>

(Recibido: 2015/01/29 - Aceptado: 2015/03/25)En la presente investigación se propone un modelo matemático que permite enfocar el proceso de creación de equipos de desarrollo de software como un problema multi objetivo discreto. Los objetivos considerados son el nivel de competencia y el nivel de utilización de los profesionales en el equipo. Dada la complejidad del espacio de búsqueda del problema, se propone la aplicación de un método de optimización aproximado. Específicamente se seleccionó el algoritmo genético NSGA-II basado en el concepto de Dominancia de Pareto. Este método fue aplicado en seis escenarios diferentes con el objetivo de analizar la calidad de soluciones que obtiene. En general se puede afirmar que el método es eficiente y obtiene soluciones (asignaciones) de alta calidad.

A multi-objective endocrine PSO algorithm and application

This paper proposed a new method for humanoid robot gait generation based on several objective functions. The proposed method is based on multiobjective evolutionary algorithm. In our work, we considered two competing objective functions: MCE and MTC. Based on simulation and experimental results, we conclude: Multiobjective evolution is efficient because optimal humanoid robot gaits with completely different characteristics can be found in one simulation run. The nondominated solutions in the obtained Pareto-optimal set are well distributed and have satisfactory diversity characteristics. The optimal gaits generated by simulation gave good performance when they were tested in the real hardware of "Bonten-Maru" humanoid robot. The optimal gait reduces the energy consumption and increases the stability during the robot motion. In the future, it will be interesting to investigate if the robot can learn in real time to switch between different gaits based on the environment conditions. In uneven terrains MTC gaits will be more

Dynamic multiobjective optimization problem (DMOP) denotes the multiobjective optimization problem, which contains objectives that may vary over time. Due to the widespread applications of DMOP existed in reality, DMOP has attracted much research attention in the last decade. In this article, we propose to solve DMOPs via an autoencoding evolutionary search. In particular, for tracking the dynamic changes of a given DMOP, an autoencoder is derived to predict the moving of the Pareto-optimal solutions based on the nondominated solutions obtained before the dynamic occurs. This autoencoder can be easily integrated into the existing multiobjective evolutionary algorithms (EAs), for example, NSGA-II, MOEA/D, etc., for solving DMOP. In contrast to the existing approaches, the proposed prediction method holds a closed-form solution, which thus will not bring much computational burden in the iterative evolutionary search process. Furthermore, the proposed prediction of dynamic change is automatically learned from the nondominated solutions found along the dynamic optimization process, which could provide more accurate Pareto-optimal solution prediction. To investigate the performance of the proposed autoencoding evolutionary search for solving DMOP, comprehensive empirical studies have been conducted by comparing three state-of-the-art prediction-based dynamic multiobjective EAs. The results obtained on the commonly used DMOP benchmarks confirmed the efficacy of the proposed method.

Manufacturing sector is regularly facing the challenges caused due to high market dynamics and mass customization. Traditionally designed machine cells fail to address this issue due to lack of flexibility in capacity and functionality. However, the benefits of cell based production can be achieved by changing the machines involved and the design of cells. This paper presents a hybrid model of machine cells comprising of reconfigurable machine tools (RMTs) that are acting as part feeders for a lean assembly line of discrete products. The strategies of lean manufacturing to maintain the Takt time and synchronized one-piece-flow are considered in the model. A multi-objective optimization problem is formulated and solved to minimize the inter-cellular part movement, the error for Takt time among machine cells and the total reconfiguration time of the RMTs using NSGA-II metaheuristic. A numerical case example for the model is solved using MATLAB© and illustrated along with computational steps and Pareto optimal solutions.

This article is dedicated to unifying the framework used to derive the Wiener–Hopf equations arising from some discrete and continuous wave diffraction problems. The main tools are the discrete Green’s identity and the appropriate notion of discrete normal derivative. The resulting formal analogy between the Wiener–Hopf equations allows one to effortlessly move between the discrete and continuous formulations. The validity is illustrated through several famous two-dimensional canonical diffraction problems and extended to three-dimensional problems. It is shown that the analogy preserves kernels of the Chebotarev–Daniele–Khrapkov class, which allows the solution of certain discrete problems.

It is shown that finding the equivalence set for solving multiobjective discrete optimization problems is advantageous over finding the set of Pareto optimal decisions. An example of a set of key parameters characterizing the economic efficiency of a commercial firm is proposed, and a mathematical model of its activities is constructed. In contrast to the classical problem of finding the maximum profit for any business, this study deals with a multiobjective optimization problem. A method for solving inverse multiobjective problems in a multidimensional pseudometric space is proposed for finding the best project of firm’s activities. The solution of a particular problem of this type is presented.

Although various optimization methods for solving single-objective problems have been developed in the last few decades, these methods have lost their eligibility due to the fact that today’s engineering problems are toward multiple objective optimization problems, in real applications. For single-objective optimization problems, for example, in case of a minimization problem, this value is the decision vector giving the smallest objective that can be achieved within the specified constraints. Hence the minimum decision vector within all possible (feasible) solution vectors is the so-called optimal solution and/or optimal design. However, in multi-objective optimization problems, since a different objective value is generated against each decision vector, the superiority of the solutions over each other is determined by considering the trade-off among the objective values. Therefore, the solution of multi-objective optimization problems, unlike single-objective problems, is a set of vectors rather than a single decision vector. In multi-objective optimization problems, especially if there are intricate objectives, the computational cost of the problem increases. In other words, while synchronously trying to maximize one of the objectives and to minimize another one makes it difficult to find the global optimum design. One of the important techniques used in multi-objective optimization problems is Pareto optimality which enables to select the global optimum solution taking into account the trade-off among all objectives. In this context, using of derivative-based methods has decreased, but the use of metaheuristic methods has increased due to the rapid availability of global optimum solution. This is because the improvements in the field of optimization are progressing in proportion to technology and varying according to the needs. In this chapter, one of the recent metaheuristic optimization methods based on swarm intelligence that is so-called a Pareto-based bat algorithm inspired by the behavior of determining the direction and distance of an object using the echo of the sound called the echolocation of bats is used in order to obtain optimum solutions for multi-objective engineering design problems. In this regard, a four-bar planar truss, a real-sized welded steel beam as well as a multi-layer radar absorber are selected as multi-objective engineering design optimization problems. In case the obtained results (optimal designs) are examined, the potency and the reliability of the proposed multi-objective Pareto-based bat algorithm are demonstrated.

Optimal design of energy systems using constrained grey-box multi-objective optimization

Multi-objectives optimization problems are often solved constructing the Pareto front and applying a decision-making strategy to select the preferred solution among the Pareto-optimal solutions. With the aim to reduce the computational effort in multi-objective optimization problems, this paper presents a procedure for the direct evaluation of the preferred updated model, without the need to evaluate the whole Pareto front. For this purpose, the objective function to minimize is defined as the distance between a candidate point and the equilibrium point in the objective function space. The choice of the criterion of the minimum distance from the equilibrium point comes from a preliminary study carried out to assess the performances of different selection criteria. The robustness and the efficiency of the proposed procedure are assessed through the comparison with the results obtained from the estimation of the Pareto-optimal solutions and the subsequent selection of the preferred one for two numerical case studies. The proposed procedure is finally applied to the calibration of a complex FE model with respect to experimental modal data. Results show that the proposed procedure is effective and considerably reduces the computational effort. Moreover, the procedure is able to directly estimate the optimal weighting factor that allows to know the relative importance between the selected objectives and can be used to solve the multi-objective optimization with the weighed sum method.