Entropy maximization model for the trip distribution problem with fuzzy and random parameters

Many trip distribution problems can be modeled as entropy maximization models with quadratic cost constraints. In this paper, the travel costs per unit flow between different zones are assumed to be given fuzzy variables and the trip productions at origins and trip attractions at destinations are assumed to be given random variables. For this case, an entropy maximization model with chance constraint is proposed, and is proved to be convex. In order to solve this model, fuzzy simulation, stochastic simulation and a genetic algorithm are integrated to produce a hybrid intelligent algorithm. Finally, a numerical example is presented to demonstrate the application of the model and the algorithm.     

Extended rough set model based on known same probability dominant valued tolerance relation

Classical rough set theory is based on the conventional indiscernibility relation. It is not suitable for analyzing incomplete information. Some successful extended rough set models based on different non-equivalence relations have been proposed. The data-driven valued tolerance relation is such a non-equivalence relation. However, the calculation method of tolerance degree has some limitations. In this paper, known same probability dominant valued tolerance relation is proposed to solve this problem. On this basis, an extended rough set model based on known same probability dominant valued tolerance relation is presented. Some properties of the new model are analyzed. In order to compare the classification performance of different generalized indiscernibility relations, based on the category utility function in cluster analysis, an incomplete category utility function is proposed, which can measure the classification performance of different generalized indiscernibility relations effectively. Experimental results show that the known same probability dominant valued tolerance relation can get better classification results than other generalized indiscernibility relations.     

A note on “Generalized fuzzy rough approximation operators based on fuzzy coverings”

In this note, we show by examples that Theorem 5.3, partial proof of Theorem 5.3′, Lemma 5.4 and Remark 5.2 in [1] contain slight flaws and then provide the correct versions.     

A maximization problem and its application to canonical correlation

Let Σ be an n × n positive definite matrix with eigenvalues λ₁ ≥ λ₂ ≥ … ≥ λ_n > 0 and let M = {x, y | x?Rⁿ, y?Rⁿ, x ≠ 0, y ≠ 0, x′y = 0}. Then for x, y in M, we have that $\text{x′Σy}\text{(x′Σxy′Σy)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{≤}\text{(λ}{}_1\text{? λ}{}_{\mathrm{n}}\text{)}\text{(λ}{}_1\text{+ λ}{}_{\mathrm{n}}\text{)}$ and the inequality is sharp. If

$\text{∑=}\text{∑}{}_{11}\text{∑}{}_{12}\text{∑}{}_{21}\text{∑}{}_{22}$

is a partitioning of Σ, let θ₁ be the largest canonical correlation coefficient. The above result yields $\text{θ}{}_1\text{≤}\text{(λ}{}_1\text{? λ}{}_{\mathrm{n}}\text{)}\text{(λ}{}_1\text{+ λ}{}_{\mathrm{n}}\text{)}$.     

A modified rough c-means clustering algorithm based on hybrid imbalanced measure of distance and density

Traditional c-means clustering partitions a group of objects into a number of non-overlapping sets. Rough sets provide more flexible and objective representation than classical sets with hard partition and fuzzy sets with subjective membership function for a given dataset. Rough c-means clustering and its extensions were introduced and successfully applied in many real life applications in recent years. Each cluster is represented by a reasonable pair of lower and upper approximations. However, the most available algorithms pay no attention to the influence of the imbalanced spatial distribution within a cluster. The limitation of the mean iterative calculation function, with the same weight for all the data objects in a lower or upper approximation, is analyzed. A hybrid imbalanced measure of distance and density for the rough c-means clustering is defined, and a modified rough c-means clustering algorithm is presented in this paper. To evaluate the proposed algorithm, it has been applied to several real world data sets from UCI. The validity of this algorithm is demonstrated by the results of comparative experiments.     

A rough set approach to the characterization of transversal matroids

Rough sets are efficient for data pre-processing during data mining. However, some important problems such as attribute reduction in rough sets are NP-hard and the algorithms required to solve them are mostly greedy ones. The transversal matroid is an important part of matroid theory, which provides well-established platforms for greedy algorithms. In this study, we investigate transversal matroids using the rough set approach. First, we construct a covering induced by a family of subsets and we propose the approximation operators and upper approximation number based on this covering. We present a sufficient condition under which a subset is a partial transversal, and also a necessary condition. Furthermore, we characterize the transversal matroid with the covering-based approximation operator and construct some types of circuits. Second, we explore the relationships between closure operators in transversal matroids and upper approximation operators based on the covering induced by a family of subsets. Finally, we study two types of axiomatic characterizations of the covering approximation operators based on the set theory and matroid theory, respectively. These results provide more methods for investigating the combination of transversal matroids with rough sets.     

A priori estimates for rough PDEs with application to rough conservation laws

We introduce a general weak formulation for PDEs driven by rough paths, as well as a new strategy to prove well-posedness. Our procedure is based on a combination of fundamental a priori estimates with (rough) Gronwall-type arguments. In particular this approach does not rely on any sort of transformation formula (flow transformation, Feynman–Kac representation formula etc.) and is therefore rather flexible. As an application, we study conservation laws driven by rough paths establishing well–posedness for the corresponding kinetic formulation.     

A fully polynomial time approximation scheme for the Replenishment Storage problem

The Replenishment Storage problem (RSP) is to minimize the storage capacity requirement for a deterministic demand, multi-item inventory system with specified individual reorder cycle lengths. The reorders can only take place at integer time units. This problem was shown to be weakly NP-hard for constant joint cycle length (the least common multiple of all individual cycle lengths). We devise here the first known FPTAS for the RSP with different individual cycle lengths and constant joint cycle length.     

带有惩罚和软容量约束的下界设施选址问题的双标准近似算法研究

研究带惩罚和软容量约束的下界设施选址问题. 扩展Guha等(Guha S, Meyerson A, Munagala K. Hierarchical placement and network design problems [C]//Proceedings of Foundations of Computer Science, 2000: 892328, DOI: 10.1109/SFCS.2000.892328)和Karger等(Karger D R,Minkoff M. Building steiner trees with incomplete global knowledge [C]//Proceedings of Foundations of Computer Science, 2000: 892329, DOI: 10.1109/SFCS.2000.892329)的工作到带有惩罚的下界约束设施选址问题,提出了一个新的双标准近似算法,得到了同样的近似比ρ(1+α)/(1-α). 进一步考虑带惩罚和软容量约束的下界设施选址问题,得到了近似比为2ρ(1+α)/(1-α)的双标准近似算法.     

An algorithm to solve a quantile optimization problem with loss function having a separable structure,and its application to an aerospace problem

A quantile minimization problem with loss function having separable structure is considered. The distribution of the random parameters is assumed to be normal. To solve the problem, the confidence method and the sample average approximation method are used. Thus, the problem is reduced to a combinatorial one, which is solved by using the variable neighborhood search. The suggested algorithm is applied to optimization of runway area. Parameters of the runway are selected to minimize the area taking into account random wind speed.     

一般梯形模糊逼近算子及其在模糊运输问题中的应用

研究运输成本信息为一般模糊数的模糊运输问题.首先,在保持一般模糊数的核不变的条件下,建立一般模糊数与一般梯形模糊数的距离最小优化模型,通过求解模型得到一般模糊数的一般梯形模糊逼近算子,并给出该逼近算子具有的性质如数乘不变性、平移不变性、连续性等.然后利用该逼近算子将一般模糊运输信息表转换成一般梯形模糊运输信息表,再根据已有GFLCM和GFMDM算法得到模糊运输问题的近似最优解,最后给出具体算例分析说明方法的有效性和合理性.     

A production-repairing inventory model with fuzzy rough coefficients under inflation and time value of money

In this paper, a production-repairing inventory model in fuzzy rough environment is proposed incorporating inflationary effects where a part of the produced defective units are repaired and sold as fresh units. Here, production and repairing rates are assumed as dynamic control variables. Due to complexity of environment, different costs and coefficients are considered as fuzzy rough type and these are reduced to crisp ones using fuzzy rough expectation. Here production cost is production rate dependent, repairing cost is repairing rate dependent and demand of the item is stock-dependent. Goal of the research work is to find decisions for the decision maker (DM) who likes to maximize the total profit from the above system for a finite time horizon. The model is formulated as an optimal control problem and solved using a gradient based non-linear optimization method. Some particular cases of the general model are derived. The results of the models are illustrated with some numerical examples.     

Log-det approximation based on uniformly distributed seeds and its application to Gaussian process regression

Maximum likelihood estimation (MLE) of hyperparameters in Gaussian process regression as well as other computational models usually and frequently requires the evaluation of the logarithm of the determinant of a positive-definite matrix (denoted by C hereafter). In general, the exact computation of is of O(N³) operations where N is the matrix dimension. The approximation of could be developed with O(N²) operations based on power-series expansion and randomized trace estimator. In this paper, the accuracy and effectiveness of using uniformly distributed seeds for approximation are investigated. The research shows that uniform-seed based approximation is an equally good alternative to Gaussian-seed based approximation, having slightly better approximation accuracy and smaller variance. Gaussian process regression examples also substantiate the effectiveness of such a uniform-seed based log-det approximation scheme.     

Mean-chance model for portfolio selection based on uncertain measure

This paper discusses a portfolio selection problem in which security returns are given by experts’ evaluations instead of historical data. A factor method for evaluating security returns based on experts’ judgment is proposed and a mean-chance model for optimal portfolio selection is developed taking transaction costs and investors’ preference on diversification and investment limitations on certain securities into account. The factor method of evaluation can make good use of experts’ knowledge on the effects of economic environment and the companies’ unique characteristics on security returns and incorporate the contemporary relationship of security returns in the portfolio. The use of chance of portfolio return failing to reach the threshold can help investors easily tell their tolerance toward risk and thus facilitate a decision making. To solve the proposed nonlinear programming problem, a genetic algorithm is provided. To illustrate the application of the proposed method, a numerical example is also presented.