Multi-objective reliability-redundancy allocation problem using particle swarm optimization

This paper considers the multi-objective reliability redundancy allocation problem of a series system where the reliability of the system and the corresponding designing cost are considered as two different objectives. Due to non-stochastic uncertain and conflicting factors it is difficult to reduce the cost of the system and improve the reliability of the system simultaneously. In such situations, the decision making is difficult, and the presence of multi-objectives gives rise to multi-objective optimization problem (MOOP), which leads to Pareto optimal solutions instead of a single optimal solution. However in order to make the model more flexible and adaptable to human decision process, the optimization model can be expressed as fuzzy nonlinear programming problems with fuzzy numbers. Thus in a fuzzy environment, a fuzzy multi-objective optimization problem (FMOOP) is formulated from the original crisp optimization problem. In order to solve the resultant problem, a crisp optimization problem is reformulated from FMOOP by taking into account the preference of decision maker regarding cost and reliability goals and then particle swarm optimization is applied to solve the resulting fuzzified MOOP under a number of constraints. The approach has been demonstrated through the case study of a pharmaceutical plant situated in the northern part of India.     

A constrained non-linear optimization model for fuzzy pairwise comparison matrices using teaching learning based optimization

Non-linear optimization models have been recently proposed to derive crisp weights from fuzzy pairwise comparison matrices. In this paper, a TLBO (Teaching Learning Based Optimization) based solution is presented for solving an optimization model as a system of non-linear equations to derive crisp weights from fuzzy pairwise comparison matrices in AHP (Analytic Hierarchy Process). This fuzzy-AHP method is named as TLBO-1. It has been found that TLBO-1 can lead to inconsistent or less consistent weights. To solve the problem of inconsistent weights, a new constrained non-linear optimization model is proposed in this paper. This model is based on the min-max approach for fuzzy pairwise comparison ratios of weights. TLBO is again used to solve this optimization model, and crisp weights are derived. This fuzzy AHP method is named as TLBO-2. The effectiveness of the proposed model is illustrated by three examples. For each example, the consistency of the derived crisp weights is compared with other optimization models. The results show that the TLBO-2 method can derive more consistent weights for the fuzzy AHP based Multi-Criteria Decision Making (MCDM) systems as compared to the other optimization models.     

A credibility-based fuzzy location model with Hurwicz criteria for the design of distribution systems in B2C e-commerce

Facility location problem is one of the most critical elements in the design of distribution systems, and numerous studies have focused on this issue. However, facility location theory and guidelines for B2C firms are sparse. In this paper, with regard to the customer characteristics peculiar to B2C e-commerce and the turbulence of the competitive market, a new fuzzy location model is proposed to optimize the distribution system design in B2C e-commerce. The model adopts a hierarchical agglomerative clustering method to classify customers and estimate the fuzzy delivery cost. At the same time, due to the turbulence of competitive market, both market supply and customer demand are treated as fuzzy variables in the model. Afterward, the credibility measure and Hurwicz criterion are introduced to convert the model into a crisp one which has NP-hard complexity. In order to solve the crisp model, an improved genetic algorithm with particle swarm optimization is developed. Finally, the computational results of some numerical examples are used to illustrate the application and performance of the proposed model and algorithm.     

A credibility-based fuzzy location model with Hurwicz criteria for the design of distribution systems in B2C e-commerce

Facility location problem is one of the most critical elements in the design of distribution systems, and numerous studies have focused on this issue. However, facility location theory and guidelines for B2C firms are sparse. In this paper, with regard to the customer characteristics peculiar to B2C e-commerce and the turbulence of the competitive market, a new fuzzy location model is proposed to optimize the distribution system design in B2C e-commerce. The model adopts a hierarchical agglomerative clustering method to classify customers and estimate the fuzzy delivery cost. At the same time, due to the turbulence of competitive market, both market supply and customer demand are treated as fuzzy variables in the model. Afterward, the credibility measure and Hurwicz criterion are introduced to convert the model into a crisp one which has NP-hard complexity. In order to solve the crisp model, an improved genetic algorithm with particle swarm optimization is developed. Finally, the computational results of some numerical examples are used to illustrate the application and performance of the proposed model and algorithm.     

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.     

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。     

Intuitionistic fuzzy optimization technique for solving multi-objective reliability optimization problems in interval environment

In designing phase of systems, design parameters such as component reliabilities and cost are normally under uncertainties. This paper presents a methodology for solving the multi-objective reliability optimization model in which parameters are considered as imprecise in terms of triangular interval data. The uncertain multi-objective optimization model is converted into deterministic multi-objective model including left, center and right interval functions. A conflicting nature between the objectives is resolved with the help of intuitionistic fuzzy programming technique by considering linear as well as the nonlinear degree of membership and non-membership functions. The resultants max–min problem has been solved with particle swarm optimization (PSO) and compared their results with genetic algorithm (GA). Finally, a numerical instance is presented to show the performance of the proposed approach.     

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.     

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     

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.     

New optimization method based on energy management in microgrids based on energy storage systems and combined heat and power

Microgrids can be assumed as a solution model for green energy sources, energy storage systems, and combined heat and power (CHP) systems. In this work, the cost and emission minimization based on a demand response (DR) program is considered an optimization problem. To solve the mentioned problem a new multiobjective optimization algorithm (improved particle swarm optimization) is proposed based on a fuzzy mechanism to select the optimal value. The microgrid system includes two CHP units, fuel cell and battery systems, and the heat buffer tank. In this problem, two different feasible operating regions have been assumed in CHPs. Accordingly, to decrease the operational cost, time-of-use, and real-time pricing DR programs have been simulated, and the impacts of the mentioned models are evaluated overload profiles. The effectiveness of proposed models has been applied on different cases studies by different scenarios. The proposed model solved the DR program, time of use-DR and real-time pricing-DR problems. The proposed model could reduce the cost about 10%.     

Obtaining Crisp Priorities for Triangular and Trapezoidal Fuzzy Judgments

This paper proposes anoptimal fuzzy-based model for obtaining crisp priorities for Fuzzy-AHP comparison matrices. Crisp judgments cannot be given for real-life situations, as most of these include some level of fuzziness and complexity. In these situations, judgments are represented by the set of fuzzy numbers. Most of the fuzzy optimization models derive crisp priorities for judgments represented with Triangular Fuzzy Numbers (TFNs) only. They do not work for other types of Triangular Shaped Fuzzy Numbers (TSFNs) and Trapezoidal Fuzzy Numbers (TrFNs). To overcome this problem, a sum of squared error (SSE) based optimization model is proposed. Unlike some other methods, the proposed model derives crisp weights from all of the above-mentioned fuzzy judgments. A fuzzy number is simulated using the Monte Carlo method. A threshold-based constraint is also applied to minimize the deviation from the initial judgments. Genetic Algorithm (GA) is used to solve the optimization model. We have also conducted casestudiesto show the proposed approach’s advantages over the existingmethods. Results show that the proposed model outperforms other models to minimize SSE and deviation from initial judgments. Thus, the proposed model can be applied in various real time scenarios as it can reduce the SSE value upto 29% compared to the existing studies.     

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。     

An improved fruit fly optimization algorithm to solve the homogeneous fuzzy series–parallel redundancy allocation problem under discount strategies

In this paper, a mathematical formulation is first derived for a homogenous fuzzy series–parallel redundancy allocation problem, where both the system and its subsystems can only take two states of complete perfect and complete failure. Identical redundant components are included in order to achieve desirable system reliability. The components of each subsystem characterized by their cost, weight, and reliability, are purchased from the market under all-unit discount and incremental quantity discount strategies. The goal is to find the optimum combination of the number of components for each subsystem that maximizes the system reliability under total fuzzy cost and weight constraints. An improved fruit fly optimization algorithm (IFOA) is proposed to solve the problem, where a particle swarm optimization, a genetic algorithm, and a Tabu search algorithm are utilized to validate the results obtained. These algorithms are the most common ones in the literature to solve series–parallel redundancy allocation problems. Moreover, design of experiments using the Taguchi approach is employed to calibrate the parameters of the algorithms. At the end, some numerical examples are solved to demonstrate the applicability of the proposed methodology. The results are generally in favor IFOA.     

Application of a mixed fuzzy decision making and optimization programming model to the empty container allocation

Containerization transportation has been growing fast in the past few decades. International trades have been growing fast since the globalization of world economies intensified in the early 1990s. However, these international trades are typically imbalanced in terms of the numbers of import and export containers. As a result, the relocation of empty containers has become one of the important problems faced by liner shipping companies. In this paper, we consider the empty container allocation problem where we need to determine the optimal volume of empty containers at a port and to reposition empty containers between ports to meet exporters’ demand over time. We formulate this empty container allocation problem as a two-stage model: in stage one, we propose a fuzzy backorder quantity inventory decision making model for determining the optimal quantity of empty container at a port; whereas in stage two, an optimization mathematical programming network model is proposed for determining the optimal number of empty containers to be allocated between ports. The parameters such as the cost of loading container, cost of unloading container, leasing cost of empty container, cost of storing container, supplies, demands and ship capacities for empty containers are considered in this model. By taking advantages of the fuzzy decision making and the network structure, we show how a mixed fuzzy decision making and optimization programming model can be applied to solve the empty container allocation problem. The utilization of the proposed model is demonstrated with a case of trans-Pacific liner route in the real world. Six major container ports on the trans-Pacific route are considered in the case study, including the Port of Kaohsiung, the Port of Hong Kong, the Port of Keelung, the Port of Kobe, the Port of Yokohama and the Port of Los Angles. The results show that the proposed mixed fuzzy decision making and optimization programming model can be used to solve the empty container allocation problem well.     

Supply chain modeling in uncertain environment with bi-objective approach

Supply chain is viewed as a large-scale system that consists of production and inventory units, organized in a serial structure. Uncertainty is the main attribute in managing the supply chains. Managing a supply chain (SC) is very difficult, since various sources of uncertainty and complex interrelationships among various entities exist in the SC. Uncertainty may result from customer’s demand variability or unreliability in external suppliers. In this paper we develop an inventory model for an assembly supply chain network (each unit has at most one immediate successor, but any number of immediate predecessors) which fuzzy demand for single product in one hand and fuzzy reliability of external suppliers in other hand affect on determination of inventory policy in SCM. External supplier’s reliability has determined using a fuzzy expert system. Also the performance of supply chain is assessed by two criteria including total cost and fill rate. To solve this bi-criteria model, hybridization of multi-objective particle swarm optimization and simulation optimization is considered. Results indicate the efficiency of proposed approach in performance measurement.     

模糊需求的多周期可靠性绿色选址-路径问题

选址-路径问题作为供应链管理中的重要问题已经得到大量关注。针对模糊需求下的可靠性绿色选址-路径问题,建立多周期的模糊机会约束优化模型,在满足运输线路可靠性、设施能力和车辆能力模糊机会约束条件下最小化物流及燃油消耗成本。为了对模型进行求解,设计一种混合遗传算法（HGA）。为了验证所提出算法的性能和模型的合理性,进行了不同规模的仿真实验,结果表明了算法的有效性和模型的合理性。最后通过数值实验分析了置信水平和可靠性水平对最终解的影响。     

Connectionist model for solving static structural problems with fuzzy parameters

In this article, a novel approach for solving static structural problems based on Artificial Neural Network (ANN) has been presented. Various numerical methods are available to solve static problems of structures having crisp parameters by converting the problem to algebraic systems. However, sometimes these structures involve uncertainties. In that case, these problems may not be handled by the usual/existing methods. In the present study, uncertainties are considered as fuzzy. As such, one may get fuzzy linear system of equations. In order to handle these type of fuzzy linear systems, the concept of ANN has been developed. Detail procedure with algorithm has been included for solving the titled problems. Here, various example problems have been examined and results have also been compared with the existing ones in special cases.     

Application of firefly algorithm for multi-stage transmission expansion planning with adequacy-security considerations in deregulated environments

Electric energy is the most popular form of energy because it can be transported easily at high efficiency and reasonable cost. Nowadays the real-world electric power systems are large-scale and highly complex interconnected transmission systems. The transmission expansion planning (TEP) problem is a large-scale optimization, complicated and nonlinear problem that the number of candidate solutions increases exponentially with system size. Investment cost, reliability (both adequacy and security), and congestion cost are considered in this optimization. To overcome the difficulties in solving the non-convex and mixed integer nature of this optimization problem, this paper offers a firefly algorithm (FA) to solve this problem. In this paper it is shown that FA, like other heuristic optimization algorithms, can solve the problem in a better manner compare with other methods such genetic algorithm (GA), particle swarm optimization (PSO), Simulated Annealing (SA) and Differential Evolution (DE). To show the feasibility of proposed method, applied model has been considered in IEEE 24-Bus, IEEE 118-Bus and Iran 400-KV transmission grid case studies for TEP problem in both adequacy and security modes. The obtained results show the capability of the proposed method. A comprehensive analysis of the GA, PSO, SA and DE with proposed method is also presented.     

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.