A combined constraint handling framework: an empirical study

This paper presents a new combined constraint handling framework (CCHF) for solving constrained optimization problems (COPs). The framework combines promising aspects of different constraint handling techniques (CHTs) in different situations with consideration of problem characteristics. In order to realize the framework, the features of two popular used CHTs (i.e., Deb’s feasibility-based rule and multi-objective optimization technique) are firstly studied based on their relationship with penalty function method. And then, a general relationship between problem characteristics and CHTs in different situations (i.e., infeasible situation, semi-feasible situation, and feasible situation) is empirically obtained. Finally, CCHF is proposed based on the corresponding relationship. Also, for the first time, this paper demonstrates that multi-objective optimization technique essentially can be expressed in the form of penalty function method. As CCHF combines promising aspects of different CHTs, it shows good performance on the 22 well-known benchmark test functions. In general, it is comparable to the other four differential evolution-based approaches and five dynamic or ensemble state-of-the-art approaches for constrained optimization.     

On the structure of finite elation Laguerre planes

Quasi-orderings are introduced in arbitrary affine planes and ternary rings as a common generalization of half-orderings and semiorderings. Quasi-ordered affine planes are described algebraically by quasi-ordered ternary rings. A lot of proper quasi-orderings are constructed in ternary rings. These algebraic examples show that all Lenz-Barlotti-classes, as far as they are known to be non-empty, are containing affine planes with proper quasi-orderings.Dedicated to Professor Dr. H. Lenz on the occasion of his 75th birthday     

Optimizing of bullwhip effect and net stock amplification in three-echelon supply chains using evolutionary multi-objective metaheuristics

In this article, we intend to model and optimize the bullwhip effect (BWE) and net stock amplification (NSA) in a three-stage supply chain consisting of a retailer, a wholesaler, and a manufacturer under both centralized and decentralized scenarios. In this regard, firstly, the causes of BWE and NSA are mathematically formulated using response surface methodology (RSM) as a multi-objective optimization model that aims to minimize the BWE and NSA on both chains. The simultaneous analysis of the BWE and NSA is considered as the main novelty of this paper. To tackle the addressed problem, we propose a novel multi-objective hybrid evolutionary approach called MOHES; MOHES is a hybrid of two known multi-objective algorithms i.e. multi-objective electro magnetism mechanism algorithm (MOEMA) and population-based multi-objective simulated annealing (PBMOSA). We applied a co-evolutionary strategy for this purpose with eligibility of both algorithms. Proposed MOHES is compared with three common and popular algorithms (i.e. NRGA, NSGAII, and MOPSO). Since the utilized algorithms are very sensitive to parameter values, RSM with the multi-objective decision making (MODM) approach is employed to tune the parameters. Finally, the hybrid algorithm and the singular approaches are compared together in terms of some performance measures. The results indicate that the hybrid approach achieves better solutions when compared with the others, and also the results show that in a decentralized chain, the order batching factor and the demand signal processing in wholesaler are the most important factors on BWE. Conversely, in a centralized chain, factors such as rationing, shortage gaming, and lead time are the most effective at reducing the BWE.     

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.     

Solving a new stochastic multi-mode p-hub covering location problem considering risk by a novel multi-objective algorithm

In this paper, we develop a novel stochastic multi-objective multi-mode transportation model for hub covering location problem under uncertainty. The transportation time between each pair of nodes is an uncertain parameter and also is influenced by a risk factor in the network. We extend the traditional comprehensive hub location problem by considering two new objective functions. So, our multi-objective model includes (i) minimization of total current investment costs and (ii) minimization of maximum transportation time between each origin–destination pair in the network. Besides, a novel multi-objective imperialist competitive algorithm (MOICA) is proposed to obtain the Pareto-optimal solutions of the problem. The performance of the proposed solution algorithm is compared with two well-known meta-heuristics, namely, non-dominated sorting genetic algorithm (NSGA-II) and Pareto archive evolution strategy (PAES). Computational results show that MOICA outperforms the other meta-heuristics.     

Pruned Pareto-optimal sets for the system redundancy allocation problem based on multiple prioritized objectives

In this paper, a new methodology is presented to solve different versions of multi-objective system redundancy allocation problems with prioritized objectives. Multi-objective problems are often solved by modifying them into equivalent single objective problems using pre-defined weights or utility functions. Then, a multi-objective problem is solved similar to a single objective problem returning a single solution. These methods can be problematic because assigning appropriate numerical values (i.e., weights) to an objective function can be challenging for many practitioners. On the other hand, methods such as genetic algorithms and tabu search often yield numerous non-dominated Pareto optimal solutions, which makes the selection of one single best solution very difficult. In this research, a tabu search meta-heuristic approach is used to initially find the entire Pareto-optimal front, and then, Monte-Carlo simulation provides a decision maker with a pruned and prioritized set of Pareto-optimal solutions based on user-defined objective function preferences. The purpose of this study is to create a bridge between Pareto optimality and single solution approaches.     

Multi-strategy ensemble evolutionary algorithm for dynamic multi-objective optimization

Dynamic optimization and multi-objective optimization have separately gained increasing attention from the research community during the last decade. However, few studies have been reported on dynamic multi-objective optimization (dMO) and scarce effective dMO methods have been proposed. In this paper, we fulfill these gabs by developing new dMO test problems and new effective dMO algorithm. In the newly designed dMO problems, Pareto-optimal decision values (i.e., Pareto-optimal solutions: POS) or both POS and Pareto-optimal objective values (i.e., Pareto-optimal front: POF) change with time. A new multi-strategy ensemble multi-objective evolutionary algorithm (MS-MOEA) is proposed to tackle the challenges of dMO. In MS-MOEA, the convergence speed is accelerated by the new offspring creating mechanism powered by adaptive genetic and differential operators (GDM); a Gaussian mutation operator is employed to cope with premature convergence; a memory like strategy is proposed to achieve better starting population when a change takes place. In order to show the advantages of the proposed algorithm, we experimentally compare MS-MOEA with several algorithms equipped with traditional restart strategy. It is suggested that such a multi-strategy ensemble approach is promising for dealing with dMO problems.     

A multi-objective lead time control problem in multi-stage assembly systems using genetic algorithms

In this paper, we develop a multi-objective model to optimally control the lead time of a multi-stage assembly system, using genetic algorithms. The multi-stage assembly system is modelled as an open queueing network. It is assumed that the product order arrives according to a Poisson process. In each service station, there is either one or infinite number of servers (machines) with exponentially distributed processing time, in which the service rate (capacity) is controllable. The optimal service control is decided at the beginning of the time horizon. The transport times between the service stations are independent random variables with generalized Erlang distributions. The problem is formulated as a multi-objective optimal control problem that involves four conflicting objective functions. The objective functions are the total operating costs of the system per period (to be minimized), the average lead time (min), the variance of the lead time (min) and the probability that the manufacturing lead time does not exceed a certain threshold (max). Finally, we apply a genetic algorithm with double strings using continuous relaxation based on reference solution updating (GADSCRRSU) to solve this multi-objective problem, using goal attainment formulation. The results are also compared against the results of a discrete-time approximation technique to show the efficiency of the proposed genetic algorithm approach.     

Minmax robustness for multi-objective optimization problems

In real-world applications of optimization, optimal solutions are often of limited value, because disturbances of or changes to input data may diminish the quality of an optimal solution or even render it infeasible. One way to deal with uncertain input data is robust optimization, the aim of which is to find solutions which remain feasible and of good quality for all possible scenarios, i.e., realizations of the uncertain data. For single objective optimization, several definitions of robustness have been thoroughly analyzed and robust optimization methods have been developed. In this paper, we extend the concept of minmax robustness (Ben-Tal, Ghaoui, & Nemirovski, 2009) to multi-objective optimization and call this extension robust efficiency for uncertain multi-objective optimization problems. We use ingredients from robust (single objective) and (deterministic) multi-objective optimization to gain insight into the new area of robust multi-objective optimization. We analyze the new concept and discuss how robust solutions of multi-objective optimization problems may be computed. To this end, we use techniques from both robust (single objective) and (deterministic) multi-objective optimization. The new concepts are illustrated with some linear and quadratic programming instances.     

A new orthogonal evolutionary algorithm based on decomposition for multi-objective optimization

The diversity of solutions is very important for multi-objective evolutionary algorithms to deal with multi-objective optimization problems (MOPs). In order to achieve the goal, a new orthogonal evolutionary algorithm based on objective space decomposition (OEA/D) is proposed in this paper. To be specific, the objective space of an MOP is firstly decomposed into a set of sub-regions via a set of direction vectors, and OEA/D maintains the diversity of solutions by making each sub-region have a solution to the maximum extent. Also, the quantization orthogonal crossover (QOX) is used to enhance the search ability of OEA/D. Experimental studies have been conducted to compare this proposed algorithm with classic MOEA/D, NSGAII, NICA and D2MOPSO. Simulation results on six multi-objective benchmark functions show that the proposed algorithm is able to obtain better diversity and more evenly distributed Pareto fronts than other four algorithms.     

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.     

Unique NSGA-II and MOPSO algorithms for improved dynamic cellular manufacturing systems considering human factors

In this study, we present a new mathematical model of a multi-objective dynamic cellular manufacturing system (MDCMS) that considers human factors. Human factors are incorporated into the proposed model in terms of human reliability and decision-making processes. Three objective functions are considered simultaneously. The first objective minimizes the total cost of the MDCMS. The second objective function minimizes inconsistency in the decision-making style of operators in the common manufacturing cells. The third objective function balances the workload of cells with respect to the efficiency of operators, which is calculated based on human reliability analysis. Various studies have been conducted in the field of MDCMS, but human factors have not received sufficient attention as important elements. Due to the NP-hardness of the MDCMS problem, two innovative meta-heuristic algorithms are developed, i.e., a non-dominated sorting genetic algorithm (NSGA-II) and a multi-objective particle swarm optimization method. The results obtained by the algorithms were compared and analyzed using different criteria. Several test problems were considered to verify and validate the proposed model and solution methods. To the best of our knowledge, this is the first study to consider human reliability and decision-making styles in a large MDCMS in an actual production setting.     

Solving multi-objective production scheduling problems using metaheuristics

Most of research in production scheduling is concerned with the optimization of a single criterion. However the analysis of the performance of a schedule often involves more than one aspect and therefore requires a multi-objective treatment. In this paper we first present (Section 1) the general context of multi-objective production scheduling, analyze briefly the different possible approaches and define the aim of this study i.e. to design a general method able to approximate the set of all the efficient schedules for a large set of scheduling models. Then we introduce (Section 2) the models we want to treat––one machine, parallel machines and permutation flow shops––and the corresponding notations. The method used––called multi-objective simulated annealing––is described in Section 3. Section 4 is devoted to extensive numerical experiments and their analysis. Conclusions and further directions of research are discussed in the last section.     

QoS and energy consumption aware service composition and optimal-selection based on Pareto group leader algorithm in cloud manufacturing system

Service composition and optimal selection (SCOS) is one of the key issues for implementing a cloud manufacturing system. Exiting works on SCOS are primarily based on quality of service (QoS) to provide high-quality service for user. Few works have been delivered on providing both high-quality and low-energy consumption service. Therefore, this article studies the problem of SCOS based on QoS and energy consumption (QoS-EnCon). First, the model of multi-objective service composition was established; the evaluation of QoS and energy consumption (EnCon) were investigated, as well as a dimensionless QoS objective function. In order to solve the multi-objective SCOS problem effectively, then a novel globe optimization algorithm, named group leader algorithm (GLA), was introduced. In GLA, the influence of the leaders in social groups is used as an inspiration for the evolutionary technology which is design into group architecture. Then, the mapping from the solution (i.e., a composed service execute path) of SCOS problem to a GLA solution is investigated, and a new multi-objective optimization algorithm (i.e., GLA-Pareto) based on the combination of the idea of Pareto solution and GLA is proposed for addressing the SCOS problem. The key operators for implementing the Pareto-GA are designed. The results of the case study illustrated that compared with enumeration method, genetic algorithm (GA), and particle swarm optimization, the proposed GLA-Pareto has better performance for addressing the SCOS problem in cloud manufacturing system.     

Annealing-tabu PAES: a multi-objective hybrid meta-heuristic

Most real-life optimization problems require taking into account not one, but multiple objectives simultaneously. In most cases these objectives are in conflict, i.e. the improvement of some objectives implies the deterioration of others. 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 solutions. In the last decade most papers dealing with multi-objective optimization use the concept of Pareto-optimality. The goal of Pareto-based multi-objective strategies is to generate a front (set) of non-dominated solutions as an approximation to the true Pareto-optimal front. However, this front is unknown for problems with large and highly complex search spaces, which is why meta-heuristic methods have become important tools for solving this kind of problem. Hybridization in the multi-objective context is nowadays an open research area. This article presents a novel extension of the well-known Pareto archived evolution strategy (PAES) which combines simulated annealing and tabu search. Experiments on several mathematical problems show that this hybridization allows an improvement in the quality of the non-dominated solutions in comparison with PAES, and also with its extension M-PAES.     

Transportation policies for single and multi-objective transportation problem using fuzzy logic

In this paper, single and multi-objective transportation models are formulated with fuzzy relations under the fuzzy logic. In the single-objective model, objective is to minimize the transportation cost. In this case, the amount of quantities transported from an origin to a destination depends on the corresponding transportation cost and this relation is verbally expressed in an imprecise sense i.e., by the words ‘low’, ‘medium’, ‘high’. For the multi-objective model, objectives are minimization of (i) total transportation cost and (ii) total time for transportation required for the system. Here, also the transported quantity from a source to a destination is determined on the basis of minimum total transportation cost as well as minimum transportation time. These relations are imprecise and stated by verbal words such as ‘very high’, ‘high’, ‘medium’, ‘low’ and ‘very low’. Both single objective and multi-objective problems using Real coded Genetic Algorithms (GA and MOGA) are developed and used to solve the single level and bi-level logical relations respectively. The models are illustrated with numerical data and optimum results are presented.     

模糊多目标半定规划的最优性条件

将模糊集理论应用到多目标半定规划中来,提出了有约束的模糊多目标半定规划模型,并首次给出了其最优有效解的定义.通过构造确定的隶属度函数,将以矩阵为决策变量的模糊多目标半定规划转化为一种目标函数的某些分量由约束函数决定的确定性多目标半定规划,并证明了前者最优有效解与后者有效解的一致性.在此基础之上,讨论了二者的最优性条件.     

A numerical method for nonconvex multi-objective optimal control problems

A numerical method is proposed for constructing an approximation of the Pareto front of nonconvex multi-objective optimal control problems. First, a suitable scalarization technique is employed for the multi-objective optimal control problem. Then by using a grid of scalarization parameter values, i.e., a grid of weights, a sequence of single-objective optimal control problems are solved to obtain points which are spread over the Pareto front. The technique is illustrated on problems involving tumor anti-angiogenesis and a fed-batch bioreactor, which exhibit bang–bang, singular and boundary types of optimal control. We illustrate that the Bolza form, the traditional scalarization in optimal control, fails to represent all the compromise, i.e., Pareto optimal, solutions.     

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.     

Quasi-ordered Desarguesian affine spaces

Starting from Euclidean spaces with a betweenness function, which is compatible with the congruence relation, we introduce quasi-orderings as a common generalization of half-orderings and semi-orderings. Quasi-ordered Desarguesian affine spaces are described algebraically by guasi-ordered skew fields.Dedicated to Professor Dr. W. Benz on the occasion of his 60 th birthday