Ranking function-based solutions of fully fuzzified minimal cost flow problem

The aim of minimal cost flow problem (MCFP) is to find the least transportation cost of a single commodity through a capacitated network. This paper presents a model to deal with one particular group of such problems in which the supply and demand of nodes and the capacity and cost of edges are represented as fuzzy numbers. For easier reference, hereafter, we refer to this group of problems as fully fuzzified MCFP. To represent our model, Hukuhara’s difference and approximated multiplication are used. Thereafter, we sort fuzzy numbers by an order using a ranking function and show that it is a total order, i.e., a reflexive, anti-symmetric, transitive and complete binary relation. Utilizing the proposed ranking function, we transform the fully fuzzified MCFP into three crisp problems solvable in polynomial time. From this standpoint, combinatorial algorithms are provided to solve the above-mentioned problem and find the fuzzy optimal flow. Furthermore, the proposed order is related to the importance weights of the center, the left spread and the right spread of each fuzzy number. Thus, this method is capable of handling the decision maker’s risk taking. By comparing some previous ranking function-based works with our method, the efficiency of the latter is revealed. Finally, an application of our proposed method to petroleum industry is presented.     

Decomposition's Dantzig–Wolfe applied to fuzzy multicommodity flow problems

We present, in this paper, a method for solving linear programming problems with fuzzy costs based on the classical method of decomposition's Dantzig–Wolfe. Methods using decomposition techniques address problems that have a special structure in the set of constraints. An example of such a problem that has this structure is the fuzzy multicommodity flow problem. This problem can be modeled by a graph whose nodes represent points of supply, demand and passage of commodities, which travel on the arcs of the network. The objective is to determine the flow of each commodity on the arcs, in order to meet demand at minimal cost while respecting the capacity constraints of the arcs and the flow conservation constraints of the nodes. Using the theory of fuzzy sets, the proposed method aims to find the optimal solution, working with the problem in the fuzzy form during the resolution procedure.     

Iterative inversion of fuzzified neural networks

The inversion of a neural network is a process of computing inputs that produce a given target when fed into the neural network. The inversion algorithm of crisp neural networks is based on the gradient descent search in which a candidate inverse is iteratively refined to decrease the error between its output and the target. In this paper. we derive an inversion algorithm of fuzzified neural networks from that of crisp neural networks. First, we present a framework of learning algorithms of fuzzified neural networks and introduce the idea of adjusting schemes for fuzzy variables. Next, we derive the inversion algorithm of fuzzified neural networks by applying the adjusting scheme for fuzzy variables to total inputs in the input layer. Finally, we make three experiments on the parity-three problem, examine the effect of the size of training sets on the inversion, and investigate how the fuzziness of inputs and targets of training sets affects the inversion     

Solving fully fuzzy polynomials using feed-back neural networks

Recently, there has been a considerable amount of interest and practice in solving many problems of several applied fields by fuzzy polynomials. In this paper, we have designed an artificial fuzzified feed-back neural network. With this design, we are able to find a solution of fully fuzzy polynomial with degree n. This neural network can get a fuzzy vector as an input, and calculates its corresponding fuzzy output. It is clear that the input–output relation for each unit of fuzzy neural network is defined by the extension principle of Zadeh. In this work, a cost function is also defined for the level sets of fuzzy output and fuzzy target. Next a learning algorithm based on the gradient descent method will be defined that can adjust the fuzzy connection weights. Finally, our approach is illustrated by computer simulations on numerical examples. It is worthwhile to mention that application of this method in fluid mechanics has been shown by an example.     

The symbolic algorithms for maximum flow in networks

The maximum flow problem is one of the classic combinatorial optimization problems with many applications, such as electrical powers, traffics, communications, computer networks and logistics. The problem is to find a flow of maximum value on a network from a source to a sink. Ordered binary decision diagram (OBDD) is a canonical form to represent and manipulate the Boolean functions efficiently. OBDD-based symbolic algorithms appear to give improved results for large-scale combinatorial optimization problems by searching the nodes and edges implicitly. For the maximum flow problem in networks, we present the symbolic algebraic decision diagram (ADD) formulation and symbolic algorithms. The augmenting-path-based symbolic algorithm is the combination of the Gabow's scaling algorithm with the binary decision diagram (BDD)-based symbolic algorithm for the maximum flow in 0–1 networks. The Karzanov's algorithm is implemented implicitly, resulting in the preflow-based symbolic algorithm (PSA), in which the vertices on each layer of the layered networks are partitioned by the vertex's input preflow and the total capacity of its outgoing edges, and the edges to push or pull the preflow are selected in terms of the priority function. The improved PSA is developed by integrating the heuristic of Träff's algorithm and the assistant layered networks into the PSA. The symbolic algorithms do not require explicit enumeration of the nodes and edges, and therefore can handle large-scale networks.     

Using fuzzy decision making system to improve quality-based investment

In this paper, fuzzy set theory is used to select the quality-based investment in small firm. Here a new algorithm, which will consider both exogenous and endogenous variables as factors, is proposed to formulate the problem. The structure of the algorithm is based on fuzzy decision-making system (FDMS), which uses fuzzy control rules. Hence, one exogenous factor and five endogenous factors mentioned above are determined as input variables and fuzzified using membership function concept. Then, the weights of these factors are fuzzified to ensure the consistency of the decision maker when assigning the importance of one factor over another. Applying IF-THEN decision rules, quality-based investments are scored. Also the comparison with Analytical Hierarchy Process (AHP) and Fuzzy Linguistic Approach (FLA) in respect to these scores is presented.     

基于模糊逻辑的多跳WSNs分簇算法

针对无线传感器网络节点能量消耗不均衡和网络寿命过短的问题,提出一种基于模糊逻辑的多跳WSNs分簇算法(FLCMN).该算法综合考虑节点剩余能量、节点邻居个数、邻居节点的平均剩余能量.根据预先设定模糊规则库,利用模糊系统评估出当选簇头的满意度.额外考虑邻居节点平均剩余能量,改善了簇内热点问题,均衡了簇内能量的消耗;同时,为了改善簇间热点问题,提出一种基于斐波那契序列的多跳传输方式,延长了网络的生存时间.通过仿真验证,FLCAMN算法在网络生存时间和能量消耗方面的性能都优于LEACH、EAMMH和DFLC算法.     

一种模糊CMAC神经网络

提出了一种模糊CMAC(小脑模型关节控制器)神经网络,它由输入层、模糊化层、模糊相 联层、模糊后相联层与输出层等5层节点组成,具有与CMAC相似的单层连接权,可通过BP 算法学习推论参数或模糊规则.给出了网络的连接结构与学习算法,并将其应用于函数逼近 问题中仿真结果验证了该方法较之CMAC的优越性.     

A fuzzy neural network with fuzzy impact grades

Fuzzy rule derivation is often difficult and time-consuming, and requires expert knowledge. This creates a common bottleneck in fuzzy system design. In order to solve this problem, many fuzzy systems that automatically generate fuzzy rules from numerical data have been proposed. In this paper, we propose a fuzzy neural network based on mutual subsethood (MSBFNN) and its fuzzy rule identification algorithms. In our approach, fuzzy rules are described by different fuzzy sets. For each fuzzy set representing a fuzzy rule, the universe of discourse is defined as the summation of weighted membership grades of input linguistic terms that associate with the given fuzzy rule. In this manner, MSBFNN fully considers the contribution of input variables to the joint firing strength of fuzzy rules. Afterwards, the proposed fuzzy neural network quantifies the impacts of fuzzy rules on the consequent parts by fuzzy connections based on mutual subsethood. Furthermore, to enhance the knowledge representation and interpretation of the rules, a linear transformation from consequent parts to output is incorporated into MSBFNN so that higher accuracy can be achieved. In the parameter identification phase, the backpropagation algorithm is employed, and proper linear transformation is also determined dynamically. To demonstrate the capability of the MSBFNN, simulations in different areas including classification, regression and time series prediction are conducted. The proposed MSBFNN shows encouraging performance when benchmarked against other models.     

A genetic fuzzy system based optimized zone based energy efficient routing protocol for mobile sensor networks (OZEEP)

Wireless sensor networks have become increasingly popular because of their ability to cater to multifaceted applications without much human intervention. However, because of their distributed deployment, these networks face certain challenges, namely, network coverage, continuous connectivity and bandwidth utilization. All of these correlated issues impact the network performance because they define the energy consumption model of the network and have therefore become a crucial subject of study. Well-managed energy usage of nodes can lead to an extended network lifetime. One way to achieve this is through clustering. Clustering of nodes minimizes the amount of data transmission, routing delay and redundant data in the network, thereby conserving network energy. In addition to these advantages, clustering also makes the network scalable for real world applications. However, clustering algorithms require careful planning and design so that balanced and uniformly distributed clusters are created in a way that the network lifetime is enhanced. In this work, we extend our previous algorithm, titled the zone-based energy efficient routing protocol for mobile sensor networks (ZEEP). The algorithm we propose optimizes the clustering and cluster head selection of ZEEP by using a genetic fuzzy system. The two-step clustering process of our algorithm uses a fuzzy inference system in the first step to select optimal nodes that can be a cluster head based on parameters such as energy, distance, density and mobility. In the second step, we use a genetic algorithm to make a final choice of cluster heads from the nominated candidates proposed by the fuzzy system so that the optimal solution generated is a uniformly distributed balanced set of clusters that aim at an enhanced network lifetime. We also study the impact and dominance of mobility with regard to the variables. However, before we arrived at a GFS-based solution, we also studied fuzzy-based clustering using different membership functions, and we present our understanding on the same. Simulations were carried out in MATLAB and ns2. The results obtained are compared with ZEEP.     

Novel methodologies and a comparative study for manufacturing systems performance evaluations

The purpose of this work is to establish complex fuzzy methodologies in the evaluation of a manufacturing system’s performance. Many empirical studies have been presented about the evaluation of manufacturing system’s performance. However, the performance evaluation is quite subjective, since it relies on the individual judgment of the managers who have different, various and multi-factor assessment methods of a system’s performance. In this study, two fuzzy modeling designs were developed and in the construction of the models, a hierarchy process was used. In the first method, the performance factors and the Analytic Hierarchy Process (AHP) were fuzzified and the use of fuzzy numbers and a fuzzy AHP for this problem was recommended. Also, the relative importance of these factors with respect to each other and their contribution to the overall performance was quantified with fuzzy linguistic terms. In the other method, we proposed Approximate Reasoning (AR) based on experts’ knowledge which is represented with the collection of the rules. These fuzzy rule bases are “if-then” linguistic rules that are formed with linguistic variables such as poor, below average, average, above average and superior. Additionally, the problem was structured with the normal AHP and System-With-Feedback (SWF), Finally, these methods were compared. The results showed that fuzzy AHP leads to the best result. It is expected that the recommended models would have an advantage in the competitive manufacturing including cost, flexibility, quality, speed and dependability.     

Particle swarm optimization for determining fuzzy measures from data

Fuzzy measures and fuzzy integrals have been successfully used in many real applications. How to determine fuzzy measures is the most difficult problem in these applications. Though there have existed some methodologies for solving this problem, such as genetic algorithms, gradient descent algorithms and neural networks, it is hard to say which one is more appropriate and more feasible. Each method has its advantages and limitations. Therefore it is necessary to develop new methods or techniques to learn distinct fuzzy measures. In this paper, we make the first attempt to design a special particle swarm algorithm to determine a type of general fuzzy measures from data, and demonstrate that the algorithm is effective and efficient. Furthermore we extend this algorithm to identify and revise other types of fuzzy measures. To test our algorithms, we compare them with the basic particle swarm algorithms, gradient descent algorithms and genetic algorithms in literatures. In addition, for verifying whether our algorithms are robust in noisy-situations, a number of numerical experiments are conducted. Theoretical analysis and experimental results show that, for determining fuzzy measures, the particle swarm optimization is feasible and has a better performance than the existing genetic algorithms and gradient descent algorithms.     

Solution of fuzzy integrated production and marketing planning based on extension principle

The integration of production and marketing planning is crucial in practice for increasing a firm’s profit. However, the conventional inventory models determine the selling price and demand quantity for a retailer’s maximal profit with exactly known parameters. When the demand quantity, unit cost, and production rate are represented as fuzzy numbers, the profit calculated from the model should be fuzzy as well. Unlike previous studies, this paper develops a solution method to find the fuzzy profit of the integrated production and marketing planning problem when the demand quantity, unit cost, and production rate are represented as fuzzy numbers. Based on Zadeh’s extension principle, we transform the problem into a pair of two-level mathematical programming models to calculate the lower bound and upper bound of the fuzzy profit. According to the duality theorem of geometric programming technique, the two-level mathematical program is transformed into the one-level conventional geometric program to solve. At a specific α-level, we can derive the global optimum solutions for the lower and upper bounds of the fuzzy profit by applying well-developed theories of geometric programming. Examples are given to illustrate the whole idea proposed in this paper.     

Multicriteria distribution network planning using simulated annealing

The classical long-range distribution network planning problem involves deciding network investments to meet future demand at a minimum cost while meeting technical restrictions (thermal limits and maximum voltage drop). The decision whether to construct facilities and branches leads to a mixed integer programming problem with a large number of decision variables. The great deal of uncertainty associated with data that cannot be modeled using probabilistic methods leads to the use of fuzzy models to capture the uncertainty. In addition, several criteria must be taken into account that is resulting in the problem being fuzzy multiobjective. The combinatorial nature of the problem limits the use of traditional mathematical tools to limited size problems. This contribution presents a methodology that generates a sample of efficient solutions for the fuzzy multiobjective problem, based on a meta-heuristic, simulated annealing (SA). The results obtained with this approach are shown to be satisfactory compared to other methods under similar conditions.     

A genetic algorithm application using fuzzy processing times in non-identical parallel machine scheduling problem

There are many scheduling problems which are NP-hard in the literature. Several heuristics and dispatching rules are proposed to solve such hard combinatorial optimization problems. Genetic algorithms (GA) have shown great advantages in solving the combinatorial optimization problems in view of its characteristic that has high efficiency and that is fit for practical application [1]. Two different scale numerical examples demonstrate the genetic algorithm proposed is efficient and fit for larger scale identical parallel machine scheduling problem for minimizing the makespan. But, even though it is a common problem in the industry, only a small number of studies deal with non-identical parallel machines. In this article, a kind of genetic algorithm based on machine code for minimizing the processing times in non-identical machine scheduling problem is presented. Also triangular fuzzy processing times are used in order to adapt the GA to non-identical parallel machine scheduling problem in the paper. Fuzzy systems are excellent tools for representing heuristic, commonsense rules. That is why we try to use fuzzy systems in this study.     

An efficient time series forecasting model based on fuzzy time series

In this paper, we present a new model to handle four major issues of fuzzy time series forecasting, viz., determination of effective length of intervals, handling of fuzzy logical relationships (FLRs), determination of weight for each FLR, and defuzzification of fuzzified time series values. To resolve the problem associated with the determination of length of intervals, this study suggests a new time series data discretization technique. After generating the intervals, the historical time series data set is fuzzified based on fuzzy time series theory. Each fuzzified time series values are then used to create the FLRs. Most of the existing fuzzy time series models simply ignore the repeated FLRs without any proper justification. Since FLRs represent the patterns of historical events as well as reflect the possibility of appearances of these types of patterns in the future. If we simply discard the repeated FLRs, then there may be a chance of information lost. Therefore, in this model, it is recommended to consider the repeated FLRs during forecasting. It is also suggested to assign weights on the FLRs based on their severity rather than their patterns of occurrences. For this purpose, a new technique is incorporated in the model. This technique determines the weight for each FLR based on the index of the fuzzy set associated with the current state of the FLR. To handle these weighted FLRs and to obtain the forecasted results, this study proposes a new defuzzification technique. The proposed model is verified and validated with three different time series data sets. Empirical analyses signify that the proposed model have the robustness to handle one-factor time series data set very efficiently than the conventional fuzzy time series models. Experimental results show that the proposed model also outperforms over the conventional statistical models.     

An efficient network flow code for finding all minimum cost s–t cutsets

Cutset algorithms have been well documented in the operations research literature. A directed graph is used to model the network, where each node and arc has an associated cost to cut or remove it from the graph. The problem considered in this paper is to determine all minimum cost sets of nodes and/or arcs to cut so that no directed paths exist from a specified source node s to a specified sink node t. By solving the dual maximum flow problem, it is possible to construct a binary relation associated with an optimal maximum flow such that all minimum cost s–t cutsets are identified through the set of closures for this relation. The key to our implementation is the use of graph theoretic techniques to rapidly enumerate this set of closures. Computational results are presented to suggest the efficiency of our approach.Scope and purposeThis paper describes the technical details of a network flow algorithm used to find all minimum cost s–t cutsets in any network topology. The motivation for this work was to provide additional automated analysis capability to a military network targeting system. Specifically, the problem is to identify a minimum cost set of nodes and/or arcs that when removed from the network, will disconnect a selected pair of origin–destination nodes. Algorithms for solving this problem are well understood, with an active research thrust in both the operations research and computer science academic communities in developing more efficient algorithms for larger networks. The main contribution of this paper is in extending these algorithms to quickly find all minimum cost cutset solutions. The implementation described in this paper outperformed conventional methods by several orders of magnitude on networks having thousands of nodes and arcs, with empirical solution times that grew linearly with the network size. The results translate to a real-time cutset analysis capability to support military targeting applications.     

Fuzzy random programming models for location-allocation problem with applications

Three types of fuzzy random programming models based on the mean chance for the capacitated location-allocation problem with fuzzy random demands are proposed according to different criteria, including the expected cost minimization model, the α-cost minimization model, and the chance maximization model. In order to solve the proposed models, some hybrid intelligent algorithms are designed by integrating the network simplex algorithm, fuzzy random simulation, and genetic algorithm. Finally, some numerical examples about a container freight station problem are given to illustrate the effectiveness of the devised algorithms.     

MOFCA: Multi-objective fuzzy clustering algorithm for wireless sensor networks

This study introduces a new clustering approach which is not only energy-efficient but also distribution-independent for wireless sensor networks (WSNs). Clustering is used as a means of efficient data gathering technique in terms of energy consumption. In clustered networks, each node transmits acquired data to a cluster-head which the nodes belong to. After a cluster-head collects all the data from all member nodes, it transmits the data to the base station (sink) either in a compressed or uncompressed manner. This data transmission occurs via other cluster-heads in a multi-hop network environment. As a result of this situation, cluster-heads close to the sink tend to die earlier because of the heavy inter-cluster relay. This problem is named as the hotspots problem. To solve this problem, some unequal clustering approaches have already been introduced in the literature. Unequal clustering techniques generate clusters in smaller sizes when approaching the sink in order to decrease intra-cluster relay. In addition to the hotspots problem, the energy hole problem may also occur because of the changes in the node deployment locations. Although a number of previous studies have focused on energy-efficiency in clustering, to the best of our knowledge, none considers both problems in uniformly and non-uniformly distributed networks. Therefore, we propose a multi-objective solution for these problems. In this study, we introduce a multi-objective fuzzy clustering algorithm (MOFCA) that addresses both hotspots and energy hole problems in stationary and evolving networks. Performance analysis and evaluations are done with popular clustering algorithms and obtained experimental results show that MOFCA outperforms the existing algorithms in the same set up in terms of efficiency metrics, which are First Node Dies (FND), Half of the Nodes Alive (HNA), and Total Remaining Energy (TRE) used for estimating the lifetime of the WSNs and efficiency of protocols.     

Bi-level programming model and hybrid genetic algorithm for flow interception problem with customer choice

This paper investigates how to optimize the facility location strategy such as to maximize the intercepted customer flow, while accounting for “flow-by” customers’ path choice behaviors and their travel cost limitation. A bi-level programming static model is constructed for this problem. An heuristic based on a greedy search is designed to solve it. Consequently, we proposed a chance constrained bi-level model with stochastic flow and fuzzy trip cost threshold level. For solving this uncertain model more efficiently, we integrate the simplex method, genetic algorithm, stochastic simulation and fuzzy simulation to design a hybrid intelligent algorithm. Some examples are generated randomly to illustrate the performance and the effectiveness of the proposed algorithms.