A structural optimization approach for passanger car tires with consideration of uncertainties

In order to support the design process of passenger car tires, a multi-objective optimization approach based on Finite Element Analysis (FEA) can be utilized. Providing a reliable and high quality design requires the consideration of data uncertainty within the optimization process. This uncertainty is caused predominantly by instable production conditions of tire components as well as incomplete information concerning e.g. loading. In order to capture the state of information which is indeed available in such engineering applications, an uncertainty model, which enables modelling fragmentary or dubious information as well as mathematical formulation of expert specifications and evaluations has to be applied. The presented requirements are fulfilled by the uncertainty model fuzziness. Therefore, an approach of multi-objective optimization with consideration of fuzzy quantities was developed. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Synchronous and asynchronous Pareto-based multi-objective Artificial Bee Colony algorithms

Pareto-based multi-objective optimization algorithms prefer non-dominated solutions over dominated solutions and maintain as much as possible diversity in the Pareto optimal set to represent the whole Pareto-front. This paper proposes three multi-objective Artificial Bee Colony (ABC) algorithms based on synchronous and asynchronous models using Pareto-dominance and non-dominated sorting: asynchronous multi-objective ABC using only Pareto-dominance rule (A-MOABC/PD), asynchronous multi-objective ABC using non-dominated sorting procedure (A-MOABC/NS) and synchronous multi-objective ABC using non-dominated sorting procedure (S-MOABC/NS). These algorithms were investigated in terms of the inverted generational distance, hypervolume and spread performance metrics, running time, approximation to whole Pareto-front and Pareto-solutions spaces. It was shown that S-MOABC/NS is more scalable and efficient compared to its asynchronous counterpart and more efficient and robust than A-MOABC/PD. An investigation on parameter sensitivity of S-MOABC/NS was presented to relate the behavior of the algorithm to the values of the control parameters. The results of S-MOABC/NS were compared to some state-of-the art algorithms. Results show that S-MOABC/NS can provide good approximations to well distributed and high quality non-dominated fronts and can be used as a promising alternative tool to solve multi-objective problems with the advantage of being simple and employing a few control parameters.     

A hybrid method for solving multi-objective global optimization problems

Real optimization problems often involve not one, but multiple objectives, usually in conflict. In single-objective optimization there exists a global optimum, while in the multi-objective case no optimal solution is clearly defined but rather a set of optimums, which constitute the so called Pareto-optimal front. Thus, the goal of multi-objective strategies is to generate a set of non-dominated solutions as an approximation to this front. However, most problems of this kind cannot be solved exactly because they have very large and highly complex search spaces. The objective of this work is to compare the performance of a new hybrid method here proposed, with several well-known multi-objective evolutionary algorithms (MOEA). The main attraction of these methods is the integration of selection and diversity maintenance. Since it is very difficult to describe exactly what a good approximation is in terms of a number of criteria, the performance is quantified with adequate metrics that evaluate the proximity to the global Pareto-front. In addition, this work is also one of the few empirical studies that solves three-objective optimization problems using the concept of global Pareto-optimality.     

Primal and dual algorithms for optimization over the efficient set

ABSTRACT

Optimization over the efficient set of a multi-objective optimization problem is a mathematical model for the problem of selecting a most preferred solution that arises in multiple criteria decision-making to account for trade-offs between objectives within the set of efficient solutions. In this paper, we consider a particular case of this problem, namely that of optimizing a linear function over the image of the efficient set in objective space of a convex multi-objective optimization problem. We present both primal and dual algorithms for this task. The algorithms are based on recent algorithms for solving convex multi-objective optimization problems in objective space with suitable modifications to exploit specific properties of the problem of optimization over the efficient set. We first present the algorithms for the case that the underlying problem is a multi-objective linear programme. We then extend them to be able to solve problems with an underlying convex multi-objective optimization problem. We compare the new algorithms with several state of the art algorithms from the literature on a set of randomly generated instances to demonstrate that they are considerably faster than the competitors.     

Robustness of mechanical systems against uncertainties

In this paper, one can propose a method which takes into account the propagation of uncertainties in the finite element models in a multi-objective optimization procedure. This method is based on the coupling of stochastic response surface method (SRSM) and a genetic algorithm provided with a new robustness criterion. The SRSM is based on the use of stochastic finite element method (SFEM) via the use of the polynomial chaos expansion (PC). Thus, one can avoid the use of Monte Carlo simulation (MCS) whose costs become prohibitive in the optimization problems, especially when the finite element models are large and have a considerable number of design parameters.The objective of this study is on one hand to quantify efficiently the effects of these uncertainties on the responses variability or the cost functions which one wishes to optimize and on the other hand, to calculate solutions which are both optimal and robust with respect to the uncertainties of design parameters.In order to study the propagation of input uncertainties on the mechanical structure responses and the robust multi-objective optimization with respect to these uncertainty, two numerical examples were simulated. The results which relate to the quantification of the uncertainty effects on the responses variability were compared with those obtained by the reference method (REF) using MCS and with those of the deterministic response surfaces methodology (RSM).In the same way, the robust multi-objective optimization results resulting from the SRSM method were compared with those obtained by the direct optimization considered as reference (REF) and with RSM methodology.The SRSM method application to the response variability study and the robust multi-objective optimization gave convincing results.     

A Post-Optimality Analysis Algorithm for Multi-Objective Optimization

Algorithms for multi-objective optimization problems are designed to generate a single Pareto optimum (non-dominated solution) or a set of Pareto optima that reflect the preferences of the decision-maker. If a set of Pareto optima are generated, then it is useful for the decision-maker to be able to obtain a small set of preferred Pareto optima using an unbiased technique of filtering solutions. This suggests the need for an efficient selection procedure to identify such a preferred subset that reflects the preferences of the decision-maker with respect to the objective functions. Selection procedures typically use a value function or a scalarizing function to express preferences among objective functions. This paper introduces and analyzes the Greedy Reduction (GR) algorithm for obtaining subsets of Pareto optima from large solution sets in multi-objective optimization. Selection of these subsets is based on maximizing a scalarizing function of the vector of percentile ordinal rankings of the Pareto optima within the larger set. A proof of optimality of the GR algorithm that relies on the non-dominated property of the vector of percentile ordinal rankings is provided. The GR algorithm executes in linear time in the worst case. The GR algorithm is illustrated on sets of Pareto optima obtained from five interactive methods for multi-objective optimization and three non-linear multi-objective test problems. These results suggest that the GR algorithm provides an efficient way to identify subsets of preferred Pareto optima from larger sets.     

A robust data envelopment analysis model with different scenarios

Input and output data, under uncertainty, must be taken into account as an essential part of data envelopment analysis (DEA) models in practice. Many researchers have dealt with this kind of problem using fuzzy approaches, DEA models with interval data or probabilistic models. This paper presents an approach to scenario-based robust optimization for conventional DEA models. To consider the uncertainty in DEA models, different scenarios are formulated with a specified probability for input and output data instead of using point estimates. The robust DEA model proposed is aimed at ranking decision-making units (DMUs) based on their sensitivity analysis within the given set of scenarios, considering both feasibility and optimality factors in the objective function. The model is based on the technique proposed by Mulvey et al. (1995) for solving stochastic optimization problems. The effect of DMUs on the product possibility set is calculated using the Monte Carlo method in order to extract weights for feasibility and optimality factors in the goal programming model. The approach proposed is illustrated and verified by a case study of an engineering company.     

Multi-objective optimization of water-using systems

Industrial water systems often allow efficient water uses via water reuse and/or recirculation. The design of the network layout connecting water-using processes is a complex problem which involves several criteria to optimize. Most of the time, this design is achieved using Water Pinch technology, optimizing the freshwater flow rate entering the system. This paper describes an approach that considers two criteria: (i) the minimization of freshwater consumption and (ii) the minimization of the infrastructure cost required to build the network. The optimization model considers water reuse between operations and wastewater treatment as the main mechanisms to reduce freshwater consumption. The model is solved using multi-objective distributed Q-learning (MDQL), a heuristic approach based on the exploitation of knowledge acquired during the search process. MDQL has been previously tested on several multi-objective optimization benchmark problems with promising results [C. Mariano, Reinforcement learning in multi-objective optimization, Ph.D. thesis in Computer Science, Instituto Tecnológico y de Estudios Superiores de Monterrey, Campus Cuernavaca, March, 2002, Cuernavaca, Mor., México, 2001]. In order to compare the quality of the results obtained with MDQL, the reduced gradient method was applied to solve a weighted combination of the two objective functions used in the model. The proposed approach was tested on three cases: (i) a single contaminant four unitary operations problem where freshwater consumption is reduced via water reuse, (ii) a four contaminants real-world case with ten unitary operations, also with water reuse, and (iii) the water distribution network operation of Cuernavaca, Mexico, considering reduction of water leaks, operation of existing treatment plants at their design capacity, and design and construction of new treatment infrastructure to treat 100% of the wastewater produced. It is shown that the proposed approach can solved highly constrained real-world multi-objective optimization problems.     

A hybrid multi-objective artificial bee colony algorithm for burdening optimization of copper strip production

To achieve burdening process optimization of copper strips effectively, a nonlinear constrained multi-objective model is established on the principle of the actual burdening. The problem is formulated with two objectives of minimizing the total cost of raw materials and maximizing the amount of waste material thrown into melting furnace. In this paper, a novel approach called “hybrid multi-objective artificial bee colony” (HMOABC) to solve this model is proposed. The HMOABC algorithm is new swarm intelligence based multi-objective optimization technique inspired by the intelligent foraging behavior of honey bees, summation of normalized objective values and diversified selection (SNOV-DS) and nondominated sorting approach. Two test examples were studied and the performance of HMOABC is evaluated in comparison with other nature inspired techniques which includes nondominated sorting genetic algorithm II (NSGAII) and multi-objective particle swarm optimization (MOPSO). The numerical results demonstrate HMOABC approach is a powerful search and optimization technique for burdening optimization of copper strips.     

Generation of efficient frontiers in multi-objective optimization problems by generalized data envelopment analysis

In many practical problems such as engineering design problems, criteria functions cannot be given explicitly in terms of design variables. Under this circumstance, values of criteria functions for given values of design variables are usually obtained by some analyses such as structural analysis, thermodynamical analysis or fluid mechanical analysis. These analyses require considerably much computation time. Therefore, it is not unrealistic to apply existing interactive optimization methods to those problems. On the other hand, there have been many trials using genetic algorithms (GA) for generating efficient frontiers in multi-objective optimization problems. This approach is effective in problems with two or three objective functions. However, these methods cannot usually provide a good approximation to the exact efficient frontiers within a small number of generations in spite of our time limitation. The present paper proposes a method combining generalized data envelopment analysis (GDEA) and GA for generating efficient frontiers in multi-objective optimization problems. GDEA removes dominated design alternatives faster than methods based on only GA. The proposed method can yield desirable efficient frontiers even in non-convex problems as well as convex problems. The effectiveness of the proposed method will be shown through several numerical examples.     

An approach for solving nonlinear multi-objective separable discrete optimization problem with one constraint

We propose an exact solution approach for solving nonlinear multi-objective optimization problems with separable discrete variables and a single constraint. The approach converts the multi-objective problem into a single objective problem by using surrogate multipliers from which we find all the solutions with objective values within a given range. We call this the surrogate target problem which is solved by using an algorithm based on the modular approach. Computational experiments demonstrate the effectiveness of this approach in solving large-scale problems. A simple example is presented to illustrate an interactive decision making process.     

FEMOEA: a fast and efficient multi-objective evolutionary algorithm

A multi-objective evolutionary algorithm which can be applied to many nonlinear multi-objective optimization problems is proposed. Its aim is to quickly obtain a fixed size Pareto-front approximation. It adapts ideas from different multi-objective evolutionary algorithms, but also incorporates new devices. In particular, the search in the feasible region is carried out on promising areas (hyperspheres) determined by a radius value, which decreases as the optimization procedure evolves. This mechanism helps to maintain a balance between exploration and exploitation of the search space. Additionally, a new local search method which accelerates the convergence of the population towards the Pareto-front, has been incorporated. It is an extension of the local optimizer SASS and improves a given solution along a search direction (no gradient information is used). Finally, a termination criterion has also been proposed, which stops the algorithm if the distances between the Pareto-front approximations provided by the algorithm in three consecutive iterations are smaller than a given tolerance. To know how far two of those sets are from each other, a modification of the well-known Hausdorff distance is proposed. In order to analyze the algorithm performance, it has been compared to the reference algorithms NSGA-II and SPEA2 and the state-of-the-art algorithms MOEA/D and SMS-EMOA. Several quality indicators have been considered, namely, hypervolume, average distance, additive epsilon indicator, spread and spacing. According to the computational tests performed, the new algorithm, named FEMOEA, outperforms the other algorithms.     

A general approach to solving a wide class of fuzzy optimization problems

Results of research into the use of fuzzy sets for handling various forms of uncertainty in the optimal design and control of complex systems are presented. A general approach to solving a wide class of optimization problems containing fuzzy coefficients in objective functions and constraints is described. It involves a modification of traditional mathematical programming methods and is associated with formulating and solving one and the same problem within the framework of mutually conjugated models. This approach allows one to maximally cut off dominated alternatives from below as well as from above. The subsequent contraction of the decision uncertainty region is associated with reduction of the problem to multicriteria decision making in a fuzzy environment. The general approach is applied within the context of a fuzzy discrete optimization model that is based on a modification of discrete optimization algorithms. Prior to application of these algorithms there is a transition from a model with fuzzy coefficients in objective functions and constraints to an equivalent analog with fuzzy coefficients in objective functions alone. The results of the paper are of a universal character and are already being used to solve problems of power engineering.     

Lexicographical dynamic goal programming approach to a robust design optimization within the pharmaceutical environment

The primary objective of this paper is to develop a new robust design (RD) optimization procedure based on a lexicographical dynamic goal programming (LDGP) approach for implementing time-series based multi-responses, while the conventional experimental design formats and frameworks may implement static responses. First, a parameter estimation method for time-dependent pharmaceutical responses (i.e., drug release and gelation kinetics) is proposed using the dual response estimation concept that separately estimates the response functions of the mean and variance, as a part of response surface method. Second, a multi-objective RD optimization model using the estimated response functions of both the process mean and variance is proposed by incorporating a time-series components within a dynamic modeling environment. Finally, a pharmaceutical case study associated with a generic drug development process is conducted for verification purposes. Based on the case study results, we conclude that the proposed LDGP approach effectively provides the optimal drug formulations with significantly small biases and MSE values, compared to other models.     

The use of possibility theory in the definition of fuzzy Pareto-optimality

Pareto-optimality conditions are crucial when dealing with classic multi-objective optimization problems. Extensions of these conditions to the fuzzy domain have been discussed and addressed in recent literature. This work presents a novel approach based on the definition of a fuzzily ordered set with a view to generating the necessary conditions for the Pareto-optimality of candidate solutions in the fuzzy domain. Making use of the conditions generated, one can characterize fuzzy efficient solutions by means of carefully chosen mono-objective problems and Karush-Kuhn-Tucker conditions to fuzzy non-dominated solutions. The uncertainties are inserted into the formulation of the studied fuzzy multi-objective optimization problem by means of fuzzy coefficients in the objective function. Some numerical examples are analytically solved to illustrate the efficiency of the proposed approach.     

Environmental supply chain network design using multi-objective fuzzy mathematical programming

The concern about environmental impact of business activities has spurred an interest in designing environmentally conscious supply chains. This paper proposes a multi-objective fuzzy mathematical programming model for designing an environmental supply chain under inherent uncertainty of input data in such problem. The proposed model is able to consider the minimization of multiple environmental impacts beside the traditional cost minimization objective to make a fair balance between them. A life cycle assessment-based (LCA-based) method is applied to assess and quantify the environmental impact of different options for supply chain network configuration. Also, to solve the proposed multi-objective fuzzy optimization model, an interactive fuzzy solution approach is developed. A real industrial case is used to demonstrate the significance and applicability of the developed fuzzy optimization model as well as the usefulness of the proposed solution approach.     

ParEGO extensions for multi-objective optimization of expensive evaluation functions

This paper deals with multi-objective optimization in the case of expensive objective functions. Such a problem arises frequently in engineering applications where the main purpose is to find a set of optimal solutions in a limited global processing time. Several algorithms use linearly combined criteria to use directly mono-objective algorithms. Nevertheless, other algorithms, such as multi-objective evolutionary algorithm (MOEA) and model-based algorithms, propose a strategy based on Pareto dominance to optimize efficiently all criteria. A widely used model-based algorithm for multi-objective optimization is Pareto efficient global optimization (ParEGO). It combines linearly the objective functions with several random weights and maximizes the expected improvement (EI) criterion. However, this algorithm tends to favor parameter values suitable for the reduction of the surrogate model error, rather than finding non-dominated solutions. The contribution of this article is to propose an extension of the ParEGO algorithm for finding the Pareto Front by introducing a double Kriging strategy. Such an innovation allows to calculate a modified EI criterion that jointly accounts for the objective function approximation error and the probability to find Pareto Set solutions. The main feature of the resulting algorithm is to enhance the convergence speed and thus to reduce the total number of function evaluations. This new algorithm is compared against ParEGO and several MOEA algorithms on a standard benchmark problems. Finally, an automotive engineering problem allowing to illustrate the applicability of the proposed approach is given as an example of a real application: the parameter setting of an indirect tire pressure monitoring system.     

A Bayesian approach to constrained single- and multi-objective optimization

This article addresses the problem of derivative-free (single- or multi-objective) optimization subject to multiple inequality constraints. Both the objective and constraint functions are assumed to be smooth, non-linear and expensive to evaluate. As a consequence, the number of evaluations that can be used to carry out the optimization is very limited, as in complex industrial design optimization problems. The method we propose to overcome this difficulty has its roots in both the Bayesian and the multi-objective optimization literatures. More specifically, an extended domination rule is used to handle objectives and constraints in a unified way, and a corresponding expected hyper-volume improvement sampling criterion is proposed. This new criterion is naturally adapted to the search of a feasible point when none is available, and reduces to existing Bayesian sampling criteria—the classical Expected Improvement (EI) criterion and some of its constrained/multi-objective extensions—as soon as at least one feasible point is available. The calculation and optimization of the criterion are performed using Sequential Monte Carlo techniques. In particular, an algorithm similar to the subset simulation method, which is well known in the field of structural reliability, is used to estimate the criterion. The method, which we call BMOO (for Bayesian Multi-Objective Optimization), is compared to state-of-the-art algorithms for single- and multi-objective constrained optimization.     

Non-dominated sorting procedure for Pareto dominance ranking on multicore CPU and/or GPU

Evolutionary multi-objective optimization algorithms aim at finding an approximation of the Pareto set. For hard to solve problems with many conflicting objectives, the number of functions evaluations to represent the Pareto front can be large and time consuming. Parallel computing can reduce the wall-clock time of such algorithms. Previous studies tackled the parallelization of a particular evolutionary algorithm. In this research, we focus on improving one of the most time consuming procedures—the non-dominated sorting—, which is used in the state-of-the-art multi-objective genetic algorithms. Here, three parallel versions of the non-dominated sorting procedure are developed: (1) a multicore (based on Pthreads); (2) a Graphic Processing Unit (GPU) (based on CUDA interface); and (3) a hybrid (based on Pthreads and CUDA). The user can select the most suitable option to efficiently compute the non-dominated sorting procedure depending on the available hardware. Results show that the use of GPU computing provides a substantial improvement in terms of performance. The hybrid approach has the best performance when a good load balance is established among cores and GPU.     

Branch and probability bound methods in multi-objective optimization

An approach to non-convex multi-objective optimization problems is considered where only the values of objective functions are required by the algorithm. The proposed approach is a generalization of the probabilistic branch-and-bound approach well applicable to complicated problems of single-objective global optimization. In the present paper the concept of probabilistic branch-and-bound based multi-objective optimization algorithms is discussed, and some illustrations are presented.