Filter allocation and replacement strategies in fluid power system under uncertainty: a fuzzy robust nonlinear programming approach

A fuzzy robust nonlinear programming model is developed for the assessment of filter allocation and replacement strategies in hydraulic systems under uncertainty. It integrates the methods of fuzzy mathematic programming (FMP) and robust programming (RP) within the mixed integer nonlinear programming framework, and can facilitate dynamic analysis and optimization of filters allocation and replacement planning where the uncertainties are expressed as fuzzy membership functions. In modeling formulation, theory of contamination wear of typical hydraulic components is introduced to strengthen the presentation of relationship between system contamination and work performance. The fuzzy decision space is delimited into a more robust one by specifying the uncertainties through dimensional enlargement of the original fuzzy constraints. The piecewise linearization approach is employed to handle the nonlinearities of problem. The developed method has been applied to a case of planning filter allocation and replacement strategies under uncertainty and the generated optimal solution will help to reduce the total system cost and failure-risk level of the FPS.     

区间数多产品多计划期生产计划问题的目标规划求解方法

针对多目标、多产品、多计划期和需求、生产费用、生产能力等参数不确定的综合生产计划问题进行了研究。引入区间数描述生产计划问题中存在的不确定性,建立了以生产成本最小和设备利用率最大为优化目标的目标规划模型。为求解模型,运用区间规划理论和基于区间序关系的可能度定义,实现了区间目标规划模型的清晰等价转换,并采用Lingo软件完成模型求解。该方法解决了传统不确定优化方法在获取概率分布和模糊隶属度函数较为困难的不足,能根据决策者的偏好以交互方式分析出不同置信水平对目标的影响,为决策者在不确定环境下进行生产计划决策提供理论依据。最后,通过算例说明方法的有效性和灵活性。     

An uncertain multidisciplinary design optimization method using interval convex models

This article proposes an uncertain multi-objective multidisciplinary design optimization methodology, which employs the interval model to represent the uncertainties of uncertain-but-bounded parameters. The interval number programming method is applied to transform each uncertain objective function into two deterministic objective functions, and a satisfaction degree of intervals is used to convert both the uncertain inequality and equality constraints to deterministic inequality constraints. In doing so, an unconstrained deterministic optimization problem will be constructed in association with the penalty function method. The design will be finally formulated as a nested three-loop optimization, a class of highly challenging problems in the area of engineering design optimization. An advanced hierarchical optimization scheme is developed to solve the proposed optimization problem based on the multidisciplinary feasible strategy, which is a well-studied method able to reduce the dimensions of multidisciplinary design optimization problems by using the design variables as independent optimization variables. In the hierarchical optimization system, the non-dominated sorting genetic algorithm II, sequential quadratic programming method and Gauss–Seidel iterative approach are applied to the outer, middle and inner loops of the optimization problem, respectively. Typical numerical examples are used to demonstrate the effectiveness of the proposed methodology.     

Inexact rough-interval type-2 fuzzy stochastic optimization model supporting municipal solid waste management under uncertainty

In this study, an inexact rough-interval type-2 fuzzy stochastic linear programming (IRIT2FSLP) approach is developed for addressing uncertainties presented as rough-interval, type-2 fuzzy and random variables. The proposed method is applied to the case of a long-term municipal solid waste management system. The IRIT2FSLP approach is an extension of the inexact interval linear programming for handling nonlinear stochastic optimization problems where rough-interval and type-2 fuzzy parameters are integrated into a general framework. The results indicate that IRIT2FSLP normally leads to rough-interval solutions. Comparisons of the proposed model with scenarios without rough-interval and type-2 fuzzy parameters are also conducted. The results indicate the significant impact of dual-uncertain information on the system, which implies the reliability of IRIT2FSLP in handling waste flow allocation.     

Interval-Based Uncertain Multi-Objective Optimization Design of Vehicle Crashworthiness

In this paper, an uncertain multi-objective optimization method is suggested to deal with crashworthiness design problem of vehicle, in which the uncertainties of the parameters are described by intervals. Considering both lightweight and safety performance, structural weight and peak acceleration are selected as objectives. The occupant distance is treated as constraint. Based on interval number programming method, the uncertain optimization problem is transformed into a deterministic optimization problem. The approximation models are constructed for objective functions and constraint based on Latin Hypercube Design (LHD). Thus, the interval number programming method is combined with the approximation model to solve the uncertain optimization problem of vehicle crashworthiness efficiently. The present method is applied to two practical full frontal impact (FFI) problems.     

An uncertain optimization method for overall ballistics based on stochastic programming and a neural network surrogate model

A nonlinear stochastic programming method is proposed in this article to deal with the uncertain optimization problems of overall ballistics. First, a general overall ballistic dynamics model is achieved based on classical interior ballistics, projectile initial disturbance calculation model, exterior ballistics and firing dispersion calculation model. Secondly, the random characteristics of uncertainties are simulated using a hybrid probabilistic and interval model. Then, a nonlinear stochastic programming method is put forward by integrating a back-propagation neural network with the Monte Carlo method. Thus, the uncertain optimization problem is transformed into a deterministic multi-objective optimization problem by employing the mean value, the standard deviation, the probability and the expected loss function, and then the sorting and optimizing of design vectors are realized by the non-dominated sorting genetic algorithm-II. Finally, two numerical examples in practical engineering are presented to demonstrate the effectiveness and robustness of the proposed method.     

汽车车身耐撞性设计参数的区间稳健性优化

基于区间分析,提出了一种考虑公差的汽车车身耐撞性稳健优化设计模型,可在有效降低耐撞性能对设计参数波动敏感性的同时实现公差范围的最大化。该模型首先利用对称公差来描述汽车碰撞模型中车身关键耐撞部件的主要尺寸、位置和形状等设计参数本身的不确定性,然后将参数设计和公差设计相结合,建立了以稳健性评价指标和公差评价指标为优化目标,设计变量名义值和公差同步优化的多目标优化模型。再次,利用区间可能度处理不确定约束,将该优化模型转换为确定性多目标优化模型。最后,将该模型应用于两个汽车耐撞性优化设计问题,并通过序列二次规划法和改进的非支配排序遗传算法进行求解,结果表明该方法及稳健优化设计模型可行且实用。     

基于区间法的发动机曲轴不确定性优化研究

该文基于非线性区间数规划方法和区间分析方法,针对某型发动机曲轴的不确定性优化问题进行了研究。载荷中的不确定参数采用区间描述,极限工况下的最大等效应力作为目标函数且通过有限元方法求解。非线性区间数规划方法用以处理不确定目标函数,区间分析方法用以快速求解目标函数在每一个设计矢量下的区间,隔代映射遗传算法作为优化求解器。应用算例说明了该文算法的有效性。     

A hybrid interval-parameter fuzzy robust programming method and its application to filter management strategy in fluid power systems

An interval-parameter fuzzy robust programming (IFRP) method is developed for the assessment of filter allocation and replacement strategies in a fluid power system (FPS) under uncertainty. The developed IFRP can effectively handle the uncertainties expressed as fuzzy sets, interval values, and their combinations, which exist in contaminant ingression/generation of the system and contaminant-holding capacity of filter without making assumptions on their probabilistic distributions. The fuzzy decision space can be delimited into a more robust one with the uncertainties being specified through dimensional enlargement of the original fuzzy constraints, leading to enhanced robustness for the optimization process. Results indicate that the developed IFRP can not only help decision-maker to identify optimal filter allocation and replacement strategies to control the contamination level of FPS with a minimized system-cost and system-failure risk under multiple uncertainties, but also mitigate uncertainties through abating interval widths of the replacement periods and service life under different contamination ingression/generation rates.     

Robust interval linear programming for environmental decision making under uncertainty

In this study, a robust interval linear programming (RILP) method is developed for dealing with uncertainties expressed as intervals with deterministic boundaries. An enhanced two-step method (ETSM) is also advanced to solve the RILP model. The developed RILP improves upon the conventional interval linear programming (ILP) method since it can generate solution intervals within a larger feasible zone. The decision space based on ETSM contains all feasible solutions, such that no useful information is neglected. Moreover, the RILP can guarantee the stability of the optimization model due to no violation for the best-case constraints. The results also suggest that the RILP is effective for practical environmental and engineering problems that involve uncertainties.     

Mixed interval–fuzzy two-stage integer programming and its application to flood-diversion planning

Innovative prevention, adaptation, and mitigation approaches as well as policies for sustainable flood management continue to be challenges faced by decision-makers. In this study, a mixed interval–fuzzy two-stage integer programming (IFTIP) method is developed for flood-diversion planning under uncertainty. This method improves upon the existing interval, fuzzy, and two-stage programming approaches by allowing uncertainties expressed as probability distributions, fuzzy sets, and discrete intervals to be directly incorporated within the optimization framework. In its modelling formulation, economic penalties as corrective measures against any infeasibilities arising because of a particular realization of the uncertainties are taken into account. The method can also be used for analysing a variety of policy scenarios that are associated with different levels of economic penalties. A management problem in terms of flood control is studied to illustrate the applicability of the proposed approach. The results indicate that reasonable solutions have been generated. They can provide desired flood-diversion alternatives and capacity-expansion schemes with a minimized system cost and a maximized safety level. The developed IFTIP is also applicable to other management problems that involve uncertainties presented in multiple formats as well as complexities in policy dynamics.     

I-VFRP: An interval-valued fuzzy robust programming approach for municipal waste-management planning under uncertainty

In this study, an interval-valued fuzzy robust programming (I-VFRP) model has been developed and applied to municipal solid-waste management under uncertainty. The I-VFRP model can explicitly address system uncertainties with multiple presentations, and can directly communicate the waste manager's confidence gradients into the optimization process, facilitating the reflection of weak or strong confidence when subjectively estimating parameter values. Parameters in the I-VFRP model can be represented as either intervals or interval-valued fuzzy sets. Thus, variations of the waste manager's confidence gradients over defining parameters can be effectively handled through interval-valued membership functions, leading to enhanced robustness of the optimization efforts. The results of a theoretical case study indicate that useful solutions for planning municipal solid-waste-management practices can be generated. The waste manager's confidence gradients over various subjective judgments can be directly incorporated into the modeling formulation and solution process. The results also suggest that the proposed methodology can be applied to practical problems that are associated with complex and uncertain information.     

Fuzzy structural analysis using α-level optimization

In this paper new concepts and developments are presented for structural analysis involving uncertain parameters. Based on a classification of the uncertainties in structural analysis the uncertainty “fuzziness” is identified and its quantification is demonstrated. On the basis of fuzzy set theory a general method for fuzzy structural analysis is developed and formulated in terms of the α-level optimization with the application of a modified evolution strategy. Every known analysis algorithm for the realistic simulation of load-bearing behavior may be applied in the fuzzy structural analysis in the sense of a deterministic fundamental solution. By way of example, geometrically and physically nonlinear algorithms are adopted in the presented study as a deterministic fundamental solution for the analysis of steel and reinforced concrete structures. The paper also describes coupling between α-level optimization and the deterministic fundamental solution. Received 9 May 2000     

Robust design optimization of nonlinear energy sink under random system parameters

Generally, in designing nonlinear energy sink (NES), only uncertainties in the ground motion parameters are considered and the unconditional expected mean of the performance metric is minimized. However, such an approach has two major limitations. First, ignoring the uncertainties in the system parameters can result in an inefficient design of the NES. Second, only minimizing the unconditional mean of the performance metric may result in large variance of the response because of the uncertainties in the system parameters. To address these issues, we focus on robust design optimization (RDO) of NES under uncertain system and hazard parameters. The RDO is solved as a bi-objective optimization problem where the mean and the standard deviation of the performance metric are simultaneously minimized. This bi-objective optimization problem has been converted into a single objective problem by using the weighted sum method. However, solving an RDO problem can be computationally expensive. We thus used a novel machine learning technique, referred to as the hybrid polynomial correlated function expansion (H-PCFE), for solving the RDO problem in an efficient manner. Moreover, we adopt an adaptive framework where H-PCFE models trained at previous iterations are reused and hence, the computational cost is less. We illustrate that H-PCFE is computationally efficient and accurate as compared to other similar methods available in the literature. A numerical study showcasing the importance of incorporating the uncertain system parameters into the optimization procedure is shown. Using the same example, we also illustrate the importance of solving an RDO problem for NES design. Overall, considering the uncertainties in the parameters have resulted in a more efficient design. Determining NES parameters by solving an RDO problem results in a less sensitive design.     

Interval-parameter robust quadratic programming for water quality management under uncertainty

Effective planning of water quality management is important for facilitating sustainable socio-economic development in watershed systems. An interval-parameter robust quadratic programming (IRQP) method is developed by incorporating techniques of robust programming and interval quadratic programming within a general optimization framework. The IRQP improves upon existing quadratic programming methods, and can tackle uncertainties presented as interval numbers and fuzzy sets as well as their combinations. Moreover, it can deal with nonlinearities in the objective function such that economies-of-scale effects can be reflected. The developed method is applied to a case study of a water quality management under uncertainty. A number of decision alternatives are generated based on the interval solutions as well as the projected applicable conditions. They represent multiple decision options with various environmental and economic considerations. Willingness to accept a low economic revenue will guarantee satisfying the water quality requirements. A strong desire to acquire a high benefit will run the risk of violating environmental criteria.     

An optimization model under interval and fuzzy uncertainties for a by-product gas system of an iron and steel plant

The effective utilization of by-product gas is essential for achieving the targets of energy conservation and emission reduction of iron and steel plants in China. The application of deterministic optimization methods may lead to oversimplification and inaccurate estimation of system parameters, and even to system failure. The major contributions made by this study are the development of a gas scheduling optimization model under fuzzy and interval uncertainties and it application to the gas scheduling system of the Baotai steel plant. The integration of type-1 and type-2 fuzzy sets and interval numbers was first used to describe specific model parameters, and the reduced fuzzy chance-constrained programming algorithm and interactive two-step interval algorithm were used for model solution. Compared with practical allocation patterns, it is shown that the proposed model could offer better solutions with more outstanding performance in rapid response to production fluctuations, as well as increases in system revenue.     

Incentive contract design in project management with serial tasks and uncertain completion times

This article investigates an incentive contract design problem for a project manager who operates a project consisting of multiple tasks performed sequentially by different subcontractors in which all task completion times are uncertain and described by fuzzy variables. On the basis of an expected value criterion and a critical value criterion, two classes of fuzzy bilevel programming models are developed. In the case where the uncertain task completion times are mutually independent, each model can first be decomposed into multiple equivalent sub-models by taking advantage of the structural characteristics, and then a two-step optimization method is employed to derive the optimal incentive contract in each sub-model. In a more general case where the uncertain task completion times are correlative, the approximation approach (AA) technique is adopted first in order to evaluate the objective functions involving fuzzy parameters, which are usually difficult to convert into their crisp equivalents. Then, an AA-based hybrid genetic algorithm integrated with the golden search method and variable neighbourhood search is designed to solve the proposed fuzzy bilevel programming models. Finally, a numerical example of a construction project is conducted to demonstrate the modelling idea and the effectiveness of the proposed methods.     

An Efficient Reliability-based Optimization Method for Uncertain Structures Based on Non-probability Interval Model

In this paper, an efficient interval optimization method based on a reliability-based possibility degree of interval (RPDI) is suggested for the design of uncertain structures. A general nonlinear interval optimization problem is studied in which the objective function and constraints are both nonlinear and uncertain. Through an interval order relation and a reliability-based possibility degree of interval, the uncertain optimization problem is transformed into a deterministic one. A sequence of approximate optimization problems are constructed based on the linear approximation technique. Each approximate optimization problem can be changed to a traditional linear programming problem, which can be easily solved by the simplex method. An iterative framework is also created, in which the design space is updated adaptively and a fine optimum can be well reached. Two numerical examples are investigated to demonstrate the effectiveness of the present method. Finally, it is employed to perform the optimization design of a practical automobile frame.     

A rolling horizon approach for material requirement planning under fuzzy lead times

This paper proposes a fuzzy multi-objective integer linear programming (FMOILP) approach to model a material requirement planning (MRP) problem with fuzzy lead times. The objective functions minimise the total costs, back-order quantities and idle times of productive resources. Capacity constraints are included by considering overtime resources. Into the crisp MRP multi-objective model, we incorporate the possibility of occurrence of each uncertain lead time using fuzzy numbers. Then FMOILP is transformed into an auxiliary crisp mixed-integer linear programming model by a fuzzy goal programming approach for each fuzzy lead time combination. In order to defuzzify the set of solutions associated with each fuzzy lead time combination, a solution method based on the centre of gravity concept is addressed. Model validation with a numerical example is carried out by a novel rolling horizon procedure where uncertain lead times are updated during each planning period according to the centre of gravity obtained. For illustration purposes, the proposed solution approach is satisfactorily compared to a rolling horizon approach in which lead times are allocated when the possibility of occurrence is established at one.