Fuzzy set induced co-evolutionary approach to many objective optimisation

An optimization strategy for die design in the polymer extrusion process is proposed in the study based on the finite element simulation, the back-propagation neural network, and the non-dominated sorting genetic algorithm II (NSGA-II). The three-dimensional simulation of polymer melts flow in the extrusion process is conducted using the penalty finite element method. The model for predicting the flow patterns in the extrusion process is established with the artificial neural network based on the simulated results. The non-dominated sorting genetic algorithm II is performed for the search of globally optimal design variables with its objective functions evaluated by the established neural network model. The proposed optimization strategy is successfully applied to the die design in low-density polyethylene (LDPE) annular extrusion process. A constrained multi-objective optimization model is established according to the characteristics of annular extrusion process. The minimum of velocity relative difference, δu, and the minimum of swell ratio, Sw, that, respectively, ensure the extrinsic feature, mechanical property, and dimensional precision of the final products are taken as optimization objectives with a constrained condition on the maximum shear stress. Three important die structure parameters, including the die contraction angle α, the ratio of parallel length to inner radius L/Ri, and the ratio of outer to inner radius Ro/Ri, are taken as design variables. The Phan-Thien–Tanner constitutive model is adopted to describe the viscoelastic rheological characteristics of LDPE whose parameters are fitted by the distributions of material functions detected on the strain-controlled rheometer. The penalty finite element model of polymer melts flowing through out of the extrusion die is derived. A decoupled method is employed to solve the viscoelastic flow problem with the discrete elastic-viscous split-stress algorithm. The simulated results are selected and extracted to constitute the learning samples according to the orthogonal experimental design method. The back propagation algorithm is adopted for the training and the establishment of the predicting model for the optimization objective. A Pareto-optimal set for the constrained multi-objective optimization is obtained using the constrained NSGA-II, and the optimal solution is extracted based on the fuzzy set theory. The optimization for die parameters in the annular extrusion process of low-density polyethylene is performed and the optimization objective is successfully achieved.

fuzzy factors in the helical gear reducer design are analyzed, and a multi-objective optimization model with minimum volume and maximum bearing capacity is established in this paper. Optimal level cut-set method is introduced to conduct comprehensive evaluation on constraining fuzziness and convert the fuzzy constraint into general constraint according to the actual conditions and requirements; for multi-objectives, single-objective optimal solutions are fuzzified to form fuzzy sets to establish the membership functions of the fuzzy sets by taking the fuzziness between the single objectives and between the single objectives and multiple objectives into consideration, and multi-objective optimal solutions are obtained by maximizing the membership functions of the fuzzy set intersection. For an actual helical gear reducer, the fuzzy theory and method are applied to obtain results.

Scheduling for the flexible job shop is very important in fields of production management. To solve the multi–objective optimization in flexible job shop scheduling problem (FJSP), the FJSP multi-objective optimization model is constructed. The cost, quality and time are taken as the optimization objectives. An improved strength Pareto evolutionary algorithm (SPEA2+) is put forward to optimize the multi-objective optimization model parallelly. The algorithm uses a new model of a Multi-objective genetic algorithm that includes more effective crossover and could obtain diverse solutions in the objective and variable spaces to archive the Pareto optimal sets for FJSP multi-objective optimization. Then an approach based on fuzzy set theory was developed to extract one of the Pareto-optimal solutions as the best compromise one. The optimization results were compared with those obtained by NSGA-II and POS. At last, an instance of flexible job shop scheduling problem in automotive industry is given to illustrate that the proposed method can solve the multi-objective FJSP effectively.

To realize multi-objective optimization of the parallel-connected double-effect mechanical vapor recompression (MVR) system, this article established an optimization model based on the Strength Pareto Evolution Algorithm 2 (SPEA2), where the total power consumption and the heat exchange area were taken as the optimization objectives. The optimal combination of evaporation temperature, compression temperature rise, and emission concentration was obtained by employing the SPEA2-based multi-objective evolutionary algorithm together with the fuzzy set theory. The emission concentration was added as a variable on the basis of the original optimization, and the optimization results were compared with the original operation conditions. The results showed that the total power consumption of the system lowered by 22.9 kW, and the heat exchange area was reduced by 110.5m2; the coefficient of performance (COP) and exergy efficiency heightened by 8.4% and 24.0%, respectively, and the exergy destruction decreased by 84.6 kW. These results indicate that the established model for system optimization can make up for the deficiency of evaluating and optimizing system performance by manipulating a single-decision variable and improve the energy utilization and thermodynamic perfection of the target system.

Research on quality performance conceptual design based on SPEA2+

This paper investigates an integrated production scheduling and vehicle routing problem with soft time windows and fuzzy travel times, where orders are grouped into batches for production and delivered by a limited number of multi-trip heterogeneous vehicles. A bi-objective mixed integer nonlinear programming (MINLP) model is established, which takes total cost and total early and tardy weighted penalty time as optimization objectives. First, a fuzzy superposition operation is proposed to obtain the fuzzy weighted penalty time, and it is extended to a generalized fuzzy operation law in fuzzy sets and systems. Then, we propose a knowledge-driven co-evolutionary algorithm (KDCEA) to solve this problem. The algorithm fuses a dual-subpopulation co-evolution based on different update strategies and a knowledge-driven strategy based on dynamic knowledge sets and problem-specific knowledge. Finally, the correctness of the MINLP model is verified by the CPLEX solver using the ϵ-constraint method. A computational experiment is conducted based on different scale instances and a real-world case, and the results show the superiority of KDCEA in solving this problem.

A material–structure–process–performance integration multi-objective optimization design of a magnesium/aluminum alloy punch-forming solid rivet joint based on Taguchi, grey relational analysis (GRA), and fuzzy rough set theory (FRST) weighting method was proposed. First, the material constitutive models of the rivet and riveted sheets were established through experiments. Second, simulation models of the rivet forming process and the pull-out and shear tests of the riveted joint were established, and corresponding experiments verified accuracy. Finally, the FRST weighting method was proposed. The integration multi-objective optimization of a rivet joint was conducted based on Taguchi, GRA, and FRST methods with optimization objectives (the rivet forming head’s dimensions and the riveted joint’s maximum pull-out and shear forces) and design variables (the material, structure, and riveting process parameters of the rivet joint). The optimal combination of design variable levels was obtained. Optimization results show that the rivet forming head’s dimensions were quite close to the recommended values after optimization, and the pull-out and shear mechanical strengths of the riveted joint were increased by 10.7% and 12.8%, respectively. Furthermore, a comparison was presented among FRST, principal component analysis, and entropy weighting methods. The FRST weighting method was concluded to be highly effective.

Inderscience is a global company, a dynamic leading independent journal publisher disseminates the latest research across the broad fields of science, engineering and technology; management, public and business administration; environment, ecological economics and sustainable development; computing, ICT and internet/web services, and related areas.

In solving certain optimization problems, the corresponding Lagrangian dual problem is often solved simply because in these problems the dual problem is easier to solve than the original primal problem. Another reason for their solution is the implication of the weak duality theorem which suggests that under certain conditions the optimal dual function value is smaller than or equal to the optimal primal objective value. The dual problem is a special case of a bilevel programming problem involving Lagrange multipliers as upper-level variables and decision variables as lower-level variables. Another interesting aspect of dual problems is that both lower and upper-level optimization problems involve only box constraints and no other equality of inequality constraints. In this paper, we propose a coevolutionary dual optimization (CEDO) algorithm for co-evolving two populations—one involving Lagrange multipliers and other involving decision variables—to find the dual solution. On 11 test problems taken from the optimization literature, we demonstrate the efficacy of CEDO algorithm by comparing it with a couple of nested smooth and nonsmooth algorithms and a couple of previously suggested coevolutionary algorithms. The performance of CEDO algorithm is also compared with two classical methods involving nonsmooth (bundle) optimization methods. As a by-product, we analyze the test problems to find their associated duality gap and classify them into three categories having zero, finite or infinite duality gaps. The development of a coevolutionary approach, revealing the presence or absence of duality gap in a number of commonly-used test problems, and efficacy of the proposed coevolutionary algorithm compared to usual nested smooth and nonsmooth algorithms and other existing coevolutionary approaches remain as the hallmark of the current study.

FPAD, a CAD tool for the design of analog circuits, is presented. For a set of input specifications, FPAD deduces the device sizes that optimize the performance objectives while satisfying the constraint specifications. Fuzzy set theory is used to measure the degree of fulfilment of the objectives and constraints, and hence provides a measure of the design quality. Trade-offs are handled by manipulating the shape of the membership functions that reflect the fulfilment or violation of the performance specifications. FPAD allows the optimization of several objectives simultaneously since multiple objective optimization is used. The mathematical formulation of the design problem is fuzzified to support real-world terms like high, good, small, acceptable, etc. The optimization is based on analytic models to avoid a simulator in the inner optimization loop. Using this approach, a two-stage unbuffered CMOS op amp and a simple emitter follower were designed, with promising results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Ranking solutions of the population in an evolutionary algorithm that solves a many objective optimization problem is a challenging task which has been vastly studied in recent years. Loss in the hypervolume of the population when a solution is omitted could be a good measure for ranking solutions but calculating this value for high dimensional problems is not tractable. In the selection operator of evolutionary algorithms, the actual hypervolume values are not important, with only a relative knowledge of these values we can select best solutions. In this paper we have proposed a novel method for approximating the ranking induced by the hypervolume indicator. This method is compared to the similar methods and its efficiency and performance is proved via proper experiments. The method is tested using benchmark test problems from the Walking Fish Group problem family which are scalable both in number of variables and objectives.

Demand uncertainty is an important issue that influences the strategic, tactical, and operational-level decision making in the transportation/logistics/supply chain planning. In this study, we explore the effect of demand uncertainty on the operational-level freight routing problem in the capacitated multimodal transportation network that consists of schedule-based rail transportation and time-flexible road transportation. Considering the imprecise characteristic of the demand, we adopt fuzzy set theory to model its uncertainty and use trapezoidal fuzzy numbers to represent the fuzzy demands. We set multiple transportation orders as the optimization object and employ soft time windows to reflect the customer requirement on on-time transportation. Under the above situation, we establish a fuzzy mixed integer nonlinear programming (FMINLP) model to formulate the capacitated road–rail multimodal routing problem with demand uncertainty and time windows. We first use the fuzzy expected value model and credibility measure based fuzzy chance-constrained programming to realize the defuzziness of the model and then adopt linearization technique to reformulate the crisp model to finally generate an equivalent mixed integer linear programming (MILP) model that can be solved by standard mathematical programming software. Finally, a numerical case is presented to demonstrate the feasibility of the proposed method. Sensitivity analysis and fuzzy simulation are combined to quantify the effect of demand uncertainty on the routing problem and also reveal some helpful insights and managerial implications.

Multi-objective optimisation handles the optimisation of multiple objectives on a multi-dimensional space (Lootsma in Fuzzy Multi-Objective Optimization. Springer, Boston, 1997 [1]). There are various classical methods and a wide variety of genetic algorithms for determining the Pareto-optimal front in MOOP. Most of the MOOP algorithms dealing with fuzzy systems treat fuzzy parameters (Young-Jou and Ching-Lai in Fuzzy multiple objective decision making: Methods and applications, Springer, Berlin, 1994 [2]), fuzzy inequalities (Chuntian in Hydrological Sciences Journal 44(4): 573–582, 1999 [3]) and fuzzy objective function (Young Jou and Ching-Lai in Fuzzy Sets and Systems 54(2): 135–146, 1993 [4]). In this article, an algorithm for multi-objective optimisation using neural network is presented where the variables are fuzzy. The paper deals with the core of the issue that is the fuzzy variables in multi-objective optimisation. Here, the variables are treated as triangular fuzzy variables. The arithmetic on these fuzzy variables is defined, according to the existing available work. As a numerical illustration, the new algorithm has been tested on two fractional functions. The results obtained after implementing the new algorithm using MATLAB code is presented. The algorithm uses neural network to approximate the Pareto front. This proposed algorithm is an illustration of possible optimisation technique in the fuzzy domain using Neural Network.

Feature similarity learning based on fuzziness minimization for semi-supervised medical image segmentation

Fuzzy resource constrained project scheduling problem (FRCPSP) is an extended problem of RCPSP considering uncertainty. It is a very important research issue, as a NP-hard combinatorial optimization problem and actual application of project scheduling. This paper proposes a hybrid genetic algorithm that combines a non-random initialization, a neighborhood search-based mutation, and two local search strategies. Fuzzy RCPSP uses fuzzy set method to describe uncertainty. It assumes that the activities with random duration changed in an interval, which is composed of optimistic time, pessimistic time and possible time. This paper innovatively converts the interval into 3 optimization objectives, reformulates FRCPSP into a multiobjective optimization model, and designs a hybrid multiobjective genetic algorithm based on NSGA-II for solving this FRCPSP. Finally, benchmarks of RCPSP and extended datasets with fuzzy processing time are adopted to test our approach. Computational results show that our approach performs better than the existing state-of-the-art methods.