Multi-objective optimization problem under fuzzy rule constraints using particle swarm optimization

In this paper, a fuzzy multi-objective programming problem is considered where functional relationships between decision variables and objective functions are not completely known to us. Due to uncertainty in real decision situations sometimes it is difficult to find the exact functional relationship between objectives and decision variables. It is assumed that information source from where some knowledge may be obtained about the objective functions consists of a block of fuzzy if-then rules. In such situations, the decision making is difficult and the presence of multiple objectives gives rise to multi-objective optimization problem under fuzzy rule constraints. In order to tackle the problem, appropriate fuzzy reasoning schemes are used to determine crisp functional relationship between the objective functions and the decision variables. Thus a multi-objective optimization problem is formulated from the original fuzzy rule-based multi-objective optimization model. In order to solve the resultant problem, a deterministic single-objective non-linear optimization problem is reformulated with the help of fuzzy optimization technique. Finally, PSO (Particle Swarm Optimization) algorithm is employed to solve the resultant single-objective non-linear optimization model and the computation procedure is illustrated by means of numerical examples.     

Bi-objective optimization of the reliability-redundancy allocation problem for series-parallel system

The main objective of this paper is to solve the bi-objective reliability redundancy allocation problem for series-parallel system where reliability of the system and the corresponding designing cost are considered as two different objectives. In their formulation, reliability of each component is considered as a triangular fuzzy number. In order to solve the problem, developed fuzzy model is converted to a crisp model by using expected values of fuzzy numbers and taking into account the preference of decision maker regarding cost and reliability goals. Finally the obtained crisp optimization problem has been solved with particle swarm optimization (PSO) and compared their results with genetic algorithm (GA). Examples are shown to illustrate the method. Finally statistical simulation has been performed for supremacy the approach.     

基于群智能的模糊多目标软件可靠性冗余分配

针对软件可靠性冗余分配问题,建立了一种模糊多目标分配模型,并提出了基于分布估计的细菌觅食优化算法求解该模型。将软件可靠性和成本作为模糊目标函数,通过三角形隶属函数对模糊多目标进行处理,用高斯分布对细菌觅食算法进行优化,并将该优化算法用来求解多目标软件可靠性冗余分配问题,设置不同的隶属函数参数可以得到不同的Pareto最优解,实验数据验证了该群智能算法对解决多目标软件可靠性分配的有效性和正确性,Pareto最优解可为在可靠性和成本之间决策提供依据。     

基于模糊物元模型的高维多目标FJSP研究*

为解决高维多目标柔性作业车间调度问题,提出了一种基于模糊物元模型与粒子群算法的模糊粒子群算法（Fuzzy Particle Swarm Optimization,FPSO）。该算法以模糊物元分析理论为依据,采用复合模糊物元与基准模糊物元之间的欧式贴近度作为适应度值引导粒子群算法的进化,并引入具有容量限制的外部存储器保留较优的Pareto非支配解以供决策者选择。此外,构建了优化目标为最大完工时间、设备总负荷、加工成本、最大设备负荷与加工质量的高维多目标优化模型,并以Kacem基准问题与实际生产数据为例进行仿真模拟与对比分析。结果表明,该算法具有良好的收敛性且搜索到的非支配解分布性较好,能够有效地应用于求解高维多目标柔性作业车间调度问题。     

基于模糊决策与进化算法的生产设备布局优化

为解决复杂情况下制造系统的生产设备布局优化问题,提出了一种将模糊决策与进化算法相结合的设备布局优化方法。进一步完善了优化模型,优化目标包括总成本最小、设备相邻要求最大化和面积利用率最大化等优化目标;其中总成本最小目标考虑了物料搬运成本,设备重置导致的设备拆装、移动成本,生产停工造成的产能损失成本。该方法考虑了用户对于成本、利用率及相邻性要求等存在的满意度、优先度等模糊情况,基于模糊决策理论,对多目标优化模型进行了模糊化处理,设计了模糊适应度函数,用以根据用户的优先关系评价pareto解集。基于求解模型的特点,对多目标进化算法的染色体编码方式与交叉、变异等遗传操作方式进行改进,以提高求解该模型的实用性与效率。最后以实际案例的优化结果证明了该方法的有效性。     

Genetic algorithm based multi-objective reliability optimization in interval environment

In most of the real world design or decision making problems involving reliability optimization, there are simultaneous optimization of multiple objectives such as the maximization of system reliability and the minimization of system cost, weight and volume. In this paper, our goal is to solve the constrained multi-objective reliability optimization problem of a system with interval valued reliability of each component by maximizing the system reliability and minimizing the system cost under several constraints. For this purpose, four different multi-objective optimization problems have been formulated with the help of interval mathematics and our newly proposed order relations of interval valued numbers. Then these optimization problems have been solved by advanced genetic algorithm and the concept of Pareto optimality. Finally, to illustrate and also to compare the results, a numerical example has been solved.     

双种群模糊粒子动态优化多能互益微网配置

为增强风光柴储多能互益微网运行稳定性、降低运行成本,并解决传统微网优化多目标决策随机性大、收敛慢及精度低等问题,提出一种双种群模糊粒子动态优化多能互益微网方法.首先从微网系统可靠性、经济运行性和环保性角度建立多能互益优化模型.然后依据模糊理论对多目标决策函数进行动态优化,避免人为设置权重因素,其中采用双种群惯性权重线性...     

新零售下生鲜产品闭环物流网络模糊规划

针对我国新零售模式的快速发展，消费者对生鲜产品需求与退货的模糊不确定性问题，考虑最低物流总成本、最佳设施选址以及最优配送车辆运输路径的决策，构建了新零售下生鲜产品闭环物流网络模糊规划模型。为求解该模型，将需求量与退货量看成三角模糊参数，利用模糊机会约束方法将模糊约束转化为等价的清晰条件。以上海市某生鲜电商企业为实例，通过置信水平的敏感性分析以及遗传算法与粒子群算法的双求解，验证了模型的有效性与可行性，进而为相关决策者提供了借鉴。     

Fuzzy global optimization of complex system reliability

The problem of optimizing the reliability of complex systems has been modeled as a fuzzy multi-objective optimization problem. Three different kinds of optimization problems: reliability optimization of a complex system with constraints on cost and weight; optimal redundancy allocation in a multistage mixed system with constraints on cost and weight; and optimal reliability allocation in a multistage mixed system with constraints on cost, weight, and volume have been solved. Four numerical examples have been solved to demonstrate the effectiveness of the present methodology. The influence of various kinds of aggregators is also studied. The inefficiency of the noncompensatory min operator has been demonstrated. One of the well-known global optimization meta-heuristics, the threshold accepting, has been invoked to take care of the optimization part of the model. A software has been developed to implement the above model. The results obtained are encouraging     

逆向物流供应链最佳供应商选择及订单量分配*

为解决逆向物流供应链中,供应商选择、订单量分配和提货点位置等不确定问题,建立了一个新的模糊多目标数学模型来确定最佳供应商选择、供应量及提货点位置,为避免在解决多目标模型时人为主观赋权,运用基于模糊目标规划的蒙特卡罗仿真模型来求解帕累托(pareto)理想解,采用遗传算法进行求解,并给出了相应优化方案,在此基础上研究讨论了不同权重分配下结果的优劣性及供应商选择风险,最后,针对不同权重分配,比较了遗传算法和Gurobi求解,实验表明,对于该问题模型遗传算法在解的优劣性上优于Gurobi。     

Multi-objective decision support system in numerical reliability optimization of modern electronic packaging

Nowadays, numerical prototyping methods in electronic packaging are widely used. This is mainly due to cost and time reduction and improved functionality and reliability of final products. Recently, there has been a lot of interest and work conducted on advanced numerical optimization, which can be directly applied to prototyping. So far, the optimization is focused on one criteria while neglecting problem of multi-objectivity, which is not the best approach from practical point of view. Nevertheless, such an approach is jusitified from the point of view of complex analysis, interdisciplinary issues and reduced accuracy of numerical models. In reality, there are usually many criteria which, in order to solve the problem, have to be taken into consideration. There are many multi-objective methods, of which the Pareto set approach is mostly cited in the literature. The “problem” of multi-objective optimization is that not a single optimal solution has resulted but the set of equivalent optimal solutions. This set of equivalent optimal solutions is referenced as “the Pareto set”. From the mathematical point of view, every value from this set can be treated as optimal for certain assumed constraints. However, there could be some additional conditions which cannot be applied to optimization process and some of the results from the Pareto set are more likely (i.e., the fabrication process will be more repeatable) then the others. So, the question is: which value from the Pareto set should be taken to further processing? There are two possibilities: asking an expert for the advice or use the decision making system. Decision making methods based on multi-objective optimization could be referenced as “Multiple criteria decision making” (MCDM) or “Multiple criterial decision aid” (MCDA) systems. There are several groups of these methods: (a) mathematical multi-objective programming, (b) artificial intelligence methods, (c) simple arithmetic methods, and (d) advanced mathematical methods. The current paper will focus on designing and application of the decision support system for multi-objective numerical reliability optimization of electronic packaging. The work will be based on the self developed numerical tool based on Python Scrippting language and will present its application to selected microelectronic packages based on its numerical model elaborated in ABAQUS.     

Entropy based region reducing genetic algorithm for reliability redundancy allocation in interval environment

This research paper presents a multi-objective reliability redundancy allocation problem for optimum system reliability and system cost with limitation on entropy of the system which is very essential for effective sustainability. Both crisp and interval-valued system parameters are considered for better realization of the model in more realistic sense. We propose that the system cost of the redundancy allocation problem depends on reliability of the components. A subpopulation and entropy based region reducing genetic algorithm (GA) with Laplace crossover and power mutation is proposed to determine the optimum number of redundant components at each stage of the system. The approach is demonstrated through the case study of a break lining manufacturing plant. A comprehensive study is conducted for comparing the performance of the proposed GA with the single-population based standard GA by evaluating the optimum system reliability and system cost with the optimum number of redundant components. Set of numerical examples are provided to illustrate the effectiveness of the redundancy allocation model based on the proposed optimization technique. We present a brief discussion on change of the system using graphical phenomenon due to the changes of parameters of the system. Comparative performance studies of the proposed GA with the standard GA demonstrate that the proposed GA is promising to solve the reliability redundancy optimization problem providing better optimum system reliability.     

基于改进人工蜂群算法的配电网重构方法

为提高配电网运行的经济性和供电的可靠性,本文选取系统平均停电频率和系统平均停电持续时间两个指标来表征配电网的供电可靠性,并同时考虑有功网损的因素,建立了计及供电可靠性指标的配电网多目标重构模型.本文将量子理论和Metropolis准则引入到人工蜂群算法中,并通过模糊满意度决策方法来确定多目标重构模型的最优解,提出了基于改进人工蜂群算法的配电网多目标重构模型优化方法.建立配电网重构实例仿真系统,通过与其它智能方法的重构对比分析证明了本文重构模型及求解方法的可行性和优越性.     

A multi-objective assembly line balancing problem with worker’s skill and qualification considerations in fuzzy environment

In this study a new multi-objective assembly line balancing problem is studied. Objectives like the number of stations, the equipment purchasing cost, the worker time dependent wage, and worker dependent dis-quality level of the stations is to be minimized simultaneously with worker allocation and equipment assignment possibilities. The problem also is formulated in a fuzzy environment. To solve such problem, a new hybrid fuzzy interactive approach is proposed in two stages. In the first stage, the fuzzy formulation is converted to a crisp multi-objective formulation using a credibility-based chance constrained programming approach. Then in the second stage, the obtained crisp multi-objective formulation is solved by a new hybrid fuzzy programming approach. To evaluate the proposed approach, two generated examples and a case study from garment production industries are used for computational experiments. The extensive computational study prove the superiority of the proposed approach over the well-known approaches of the literature.     

基于模糊时间窗的多目标冷链配送优化

随着生鲜冷链行业竞争逐渐白热化,成本高、时效性强、新鲜度难以保持等问题已成为制约冷链物流配送的瓶颈。为提高生鲜配送效率,考虑客户满意度,以货损成本、惩罚成本等综合配送成本最低为目标函数,构建了一个多目标配送路径优化模型。设计带精英策略的非支配排序遗传算法（Elitist Non-dominated Sorting Genetic Algorithm,NSGA-II）求解该问题,利用Solomon标准数据集进行仿真模拟实验。实验结果对比分析表明,考虑满意度时冷链物流配送所需车辆更少,总路径长度更短,设计的算法可以在较短的时间内获取到帕累托最优解集,能够有效地解决模糊时间窗下的配送路径优化问题。     

Necessity constraint in two plant optimal production problem with imprecise parameters

The purpose of this paper is to present and solve a real-life problem of two plants producing the same item under fuzzy-stochastic environment. Here, an item alongwith random defective units is produced at two different plants situated in different locations under a single management. The rates of demand, production and defectiveness at these places are different. Demands of the item are primarily met locally from the respective plants but if a stock-out situation occurs in a plant, immediately some stock, from the other plant if available, is rushed to the stock-out plant. The demands are known but production rates are unknown, functions of time are taken as control variables. The available budget for the management house is imprecise. The holding, shortage and transportation costs are assumed to be imprecise and represented by fuzzy numbers which are transformed to corresponding interval numbers. Following interval mathematics and nearest interval approximation, the objective function is changed to respective multi-objective functions and thus the single-objective fuzzy problem is reduced to a crisp multi-objective decision making (MODM) problem. The MODM problem is then again transformed to a single crisp objective function with the help of weighted sum method. Using fuzzy relations, the imprecise budget constraint expressed in the form of necessity constraint is transformed into an equivalent crisp one. Then, total cost consisting of production, holding, shortage and transportation (from one plant to another) costs is expressed as an optimal control problem and solved using weighted sum method, the Kuhn-Tucker conditions, Pontryagin’s Optimal Control principle and generalized reduced gradient (GRG) technique. The model has been illustrated by numerical data. The optimum results are presented in both tabular and graphical forms.     

基于模糊QoS和偏好权重的Web服务组合方法研究

为了充分体现服务质量(QoS)的不确定性和用户偏好的模糊性,本文将模糊集理论引入基于QoS的Web服务组合中,将不适合精确表示的QoS属性和用户偏好等信息用三角模糊数表示.然后基于权重和法计算模糊总目标,通过设计新的模糊数比较方法,改写Pareto支配关系,将基于模糊数比较的单目标优化问题转化为多目标优化问题,并设计模糊多目标遗传算法(FMOGA)求得Pareto最优解集.该方法不仅能够得到更加贴近实际情况的优化解,同时也解决了多属性决策方法无法对大量候选服务进行全局优化的问题.最后通过实验验证了该算法的有效性和优越性.     

A fuzzy particle swarm optimization algorithm for computer communication network topology design

Particle swarm optimization (PSO) is a powerful optimization technique that has been applied to solve a number of complex optimization problems. One such optimization problem is topology design of distributed local area networks (DLANs). The problem is defined as a multi-objective optimization problem requiring simultaneous optimization of monetary cost, average network delay, hop count between communicating nodes, and reliability under a set of constraints. This paper presents a multi-objective particle swarm optimization algorithm to efficiently solve the DLAN topology design problem. Fuzzy logic is incorporated in the PSO algorithm to handle the multi-objective nature of the problem. Specifically, a recently proposed fuzzy aggregation operator, namely the unified And-Or operator (Khan and Engelbrecht in Inf. Sci. 177: 2692–2711, 2007), is used to aggregate the objectives. The proposed fuzzy PSO (FPSO) algorithm is empirically evaluated through a preliminary sensitivity analysis of the PSO parameters. FPSO is also compared with fuzzy simulated annealing and fuzzy ant colony optimization algorithms. Results suggest that the fuzzy PSO is a suitable algorithm for solving the DLAN topology design problem.     

A gamma type-2 defuzzification method for solving a solid transportation problem considering carbon emission

This paper intends to develop a multi-objective solid transportation problem considering carbon emission, where the parameters are of gamma type-2 fuzzy in nature. This paper proposed the defuzzification process for gamma type-2 fuzzy variable using critical value (CV ) and nearest interval approximation method. A chance constraint programming problem is generated using the CV based reduction method to convert the fuzzy problem to its equivalent crisp form. Applying the \(\alpha \)-cut based interval approximation method, a deterministic problem is developed. Some real life data are used to minimize the cost and carbon emission. LINGO standard optimization solver has been used to solve the multi-objective problem using weighted sum method and intuitionistic fuzzy programming technique. The Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) algorithm are implemented to generate efficient optimal solution by converting the multi-objective problem to a single objective problem using penalty cost for carbon emission. After solving the problem, analysis on some particular cases has been presented. The sensitivity analysis has been shown to different credibility levels of cost, emission, source, demand, conveyance to find total cost, emission and transported amount in each level. A comparison study on the performance of three algorithms (LINGO, GA and PSO) is presented. At the end, some graphs have been plotted which shows the effect of emission with different emission parameters.     

A study of distribution center location based on the rough sets and interactive multi-objective fuzzy decision theory

In a distribution network, the location of the distribution center(s) is recognized to be one of the most crucial elements in determining the success and efficiency of the product flow. Determination of optimal distribution center location is an abiding challenge for researchers and professionals in the logistics sector. Such determination usually involves both quantitative and qualitative factors. For this reason, this paper proposes a new hybrid heuristic algorithm to deal with the problem, which combines rough set methods and fuzzy logic. The algorithm first employs rough set methods to obtain the targeted weights. Upon building up the targeted values, the synthetic evaluation values are obtained for alternative distribution centers. Since the distribution is different from the special work system of the ordinary retail trade, the multi-objective programming model is then developed taking into account the cost and combined earnings as the goal. After that, an interactive multi-objective fuzzy decision theory is used to formulate the mathematical model and the opinion interaction among the policy-makers. Thus, the distribution center location problem can be solved more efficiently. Analysis using numerical examples has proven the optimization reliability of the algorithm, and the result obtained is promising.