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We address the problem of reducing the edge lengths of a network within a given budget so that the sum of weighted distances from each vertex to others is minimized. We call this problem the reverse total weighted distance problem on networks. We first show that the problem is NP-hard by reducing the set cover problem to it in polynomial time. Particularly, we develop a linear time algorithm to solve the problem on a tree. For the problem on cycles, we devise an iterative approach without mentioning the exact complexity. Additionally, if the cycle has uniform edge lengths, we can prove that the specified approach runs in O(n3) time as each edge of the cycle can be reduced at most once, where n is the number of vertices in the underlying cycle.

For a graph $G=(V, E)$ and $i, j\in V$, denote the distance between $i$ and $j$ in $G$ by $D(i, j)$ and the degrees of $i$, $j$ by $d_i$, $d_j$, respectively. Let $f(D(i, j), d_{i}, d_{j})$ be a function symmetric in $i$ and $j$. Define a matrix $W_f(G)$, called the weighted distance matrix, of $G$, with the $ij$-entry $W_f(G)(i, j)=f(D(i, j), d_{i}, d_{j})$ if $i\neq j$ and $W_f(G)(i, j)=0$ if $i=j$. In this paper, we prove that if the symmetric function $f$ satisfies that $f(D(i, j), (1+o(1))np, (1+o(1))np)=(1+o(1))f(D(i, j), np, np)$, then for almost all graphs $G_p$ in the $Erd\ddot{o}s$-$R\acute{e}nyi$ random graph model $\mathcal{G}_{n, p}$, the energy of $W_f(G_p)$ is $\{(\frac{8}{3\pi}\sqrt{p(1-p)}+o(1))\cdot|f(1, np, np)-f(2, np, np)|+o(|f(2, np, np)|)\}\cdot n^{3/2}$. As a consequence, we give the asymptotic values of energies of a variety of weighted distance matrices with function $f$ from distance-based only and mixed with degree-distance-based topological indices of chemical use. This generalizes our former result with only degree-based weights., Comment: 15 pages

Simplified neutrosophic set (SNS) is a popular tool in modelling potential, imprecise, and uncertain information within complex environments. In this paper, a method based on the integrated weighted distance measure and entropy weight is proposed for handling SNS multiple attribute group decision-making (MAGDM) problems. To this end, the simplified neutrosophic (SN) integrated weighted distance (SVNIWD) measure is first developed for overcoming the limitations of the existing methods. Afterward, the proposed SNIWD’s several properties and particular status are studied. Moreover, a flexible and useful MAGDM approach that combines the strengths of the SNIWD and the SNS is proposed, wherein the SN entropy measure is applied to calculate the unknown weight information regarding attributes. Finally, a numerical case of investment evaluation and subsequent comparative analysis are conducted to prove the superiority of the proposed framework.

In this paper, a weighted variant of the normalized pairwise angular distance metric is proposed. The inclusion of position weights aims at penalizing inversions in the top of the ranking more than inversions in the tail of the ranking. The performance of the proposed weighted distance metric for assessing ranking dissimilarity and its impact on a procedure for testing inter-group heterogeneity have been investigated via a Monte Carlo simulation study under several scenarios—differing for group size, number of ranked alternatives and system of hypotheses—and compared against those obtained for the unweighted variant.

International audience; Context To analyze the scales at which landscape structure influences ecological processes, two approaches with different underlying ecological assumptions exist; the usual threshold method and the weighted-distance method.Objectives We used abundance of species to test if the combination of weighted-distance and threshold approaches improves the explained variance of landscape metrics. Methods We developed a workflow using the two approaches to calculate metrics computed at multiple scales. The latter was developed using weighted metrics based on different weighted-distance functions, and one metric could be selected for more than one spatial scale. Then, we tested the explained variance of species distribution (the activity-density of Abax parallelepipedus) by these two approaches applied independently and then together in modeling a specific ecological response.Results The combination of metrics computed at multiple scales calculated by both weighted-distance and threshold method improved the predictive performance of the models. More precisely, adding metrics derived from the weighted-distance method to the threshold method significantly increased the explained variance when using the same environmental variables. The mean R-2 values of the selected model for the threshold method was 0.34 +/- 0.10, 0.49 +/- 0.11 with the weighted-distance method, and reached 0.71 +/- 0.07 with the two methods combined. These results demonstrate the importance of combining metrics using the weighted-distance method and the threshold method. In addition, activity-density was better explained by metrics selected at multiple scales.Conclusions This study highlights the importance of combining threshold and weighted-distance method at several scales to improve the explanation of ecological responses based on species abundance.

With respect to multi-criteria GSES problems with interval 2-tuple linguistic information, a new approach that uses distance measures is proposed in this chapter. Motivated by the ordered weighted distance (OWD) measures, we develop some interval 2-tuple linguistic distance operators such as the interval 2-tuple weighted distance (ITWD), the interval 2-tuple ordered weighted distance (ITOWD), and the interval 2-tuple hybrid weighted distance (ITHWD) operators. These aggregation operators are very useful for the treatment of input data in the form of interval 2-tuple linguistic variables. We study some desirable properties of the ITOWD operator and further generalize it by using the generalized and the quasi-arithmetic means. Finally, the new approach is utilized to complete a green supplier selection study for an actual hospital from the healthcare industry.

In this paper, we consider a maximum weighted k distance-d independent set problem, which is a variant of maximum weighted distance-d independent set problems, on interval graph. We propose a pseudo-polynomial time algorithm for maximum weighted k distance-d independent set problems on interval graphs, based on a solution to an exact weighted independent set problem. In addition, the performance of the proposed algorithm was evaluated by computational experiments. According to the experiments, it is found that the parameter k has a significant influence on the calculation time compared with parameters d, k

The purpose of this study is to propose an integrated distance-based methodology for multiple attribute group decision making (MAGDM) within single-valued neutrosophic linguistic (SVNL) environments. A new SVNL distance measure, namely the SVNL integrated weighted distance (SVNLIWD) measure, is first developed for achieving the aim. The remarkable feature of the SVNLIWD is that it integrates both merits of ordered weighting and average weighting into aggregating SVNL distances; therefore, it can account for both the importance of aggregated deviations as well as ordered positions. Thus, it can highlight the decision makers’ subjective risk attitudes and combine the importance of objective decision information. Some distinctive characteristics and special forms of the presented distance framework are then specifically studied. Moreover, a MAGDM model on the basis of the proposed SVNLIWD form is formulated. Finally, an illustrative numerical case regarding selecting low-carbon supplier is used to test the performance of the designed method.

The chapter considers linear combinations of digital metrics for approximating Euclidean metrics. It discusses the linear and convex combination of underestimated and overestimated norms. In particular, it presents an analysis on approximation of Euclidean metrics by a linear combination of weighted t-cost and chamfering weighted distance functions. The same theory is applied to get new results and insights in the approximation of Euclidean metrics by other sub-classes such as m-neighbor, t-cost, weighted t-cost, and hyperoctagonal distances.

In this paper, we consider an interval coverage problem. We are given [Formula: see text] intervals of the same length on a line [Formula: see text] and a line segment [Formula: see text] on [Formula: see text]. Each interval has a nonnegative weight. The goal is to move the intervals along [Formula: see text] such that every point of [Formula: see text] is covered by at least one interval and the maximum moving cost of all intervals is minimized, where the moving cost of each interval is its moving distance times its weight. Algorithms for the “unweighted” version of this problem have been given before. In this paper, we present a first-known algorithm for this weighted version and our algorithm runs in [Formula: see text] time. The problem has applications in mobile sensor barrier coverage, where [Formula: see text] is the barrier and each interval is the covering interval of a mobile sensor.

Distance and similarity measures are very important topics in fuzzy set theory. In this paper, we propose some new distance measures for generalized hesitant fuzzy sets. We investigate the connections of the aforementioned distance measures and further develop a number of generalized hesitant ordered weighted distance measures.

In order to handle the multi-attribute group decision making(MAGDM) problems with linguistic intuitionistic fuzzy (LF) information, a modified TOPSIS method is proposed with LF entropy and LF distance measure. Firstly, the linguistic metric function(LMF) is defined as a tool to quantity LF information, and the new LF entropy is further introduced to derive the attribute weights. Secondly, based on the classical cosine similarity, the new distance measure between linguistic intuitionistic fuzzy sets(LIFSs) is developed, which is called the LF induced ordered weighted distance(LIFIOWD) measure. Subsequently, a modified TOPSIS method based on LF entropy and the LFIOWD measure is presented for linguistic decision making(LDM). Finally, the example of a car company choosing a wheel manufacturer is used to demonstrate this approach.

Let S be a set of n points and let w be a function that assigns non-negative weights to points in S. The additive weighted distance \(d_w(p, q)\) between two points \(p,q \in S\) is defined as \(w(p) + d(p, q) + w(q)\) if \(p \ne q\) and it is zero if \(p = q\). Here, d(p, q) is the geodesic Euclidean distance between p and q. For a real number \(t > 1\), a graph G(S, E) is called a t-spanner for the weighted set S of points if for any two points p and q in S the distance between p and q in graph G is at most t.\(d_w(p, q)\) for a real number \(t > 1\). For some integer \(k \ge 1\), a t-spanner G for the set S is a (k, t)-vertex fault-tolerant additive weighted spanner, denoted with (k, t)-VFTAWS, if for any set \(S' \subset S\) with cardinality at most k, the graph \(G \setminus S'\) is a t-spanner for the points in \(S \setminus S'\). For any given real number \(\epsilon > 0\), we present algorithms to compute a \((k, 4+\epsilon )\)-VFTAWS for the metric space \((S, d_w)\) resulting from the following: (i) points in S are in the free space of the polygonal domain, (ii) points in S lying on a terrain.

Within the framework of preference rankings, the interest can lie in finding which predictors and which interactions are able to explain the observed preference structures, because preference decisions will usually depend on the characteristics of both the judges and the objects being judged. This work proposes the use of a univariate decision tree for ranking data based on the weighted distances for complete and incomplete rankings, and considers the area under the ROC curve both for pruning and model assessment. Two real and well-known datasets, the SUSHI preference data and the University ranking data, are used to display the performance of the methodology.

Let S be a set of n points and let w be a function that assigns non-negative weights to points in S. The additive weighted distance \(d_w(p, q)\) between two points \(p,q \in S\) is defined as \(w(p) + d(p, q) + w(q)\) if \(p \ne q\) and it is zero if \(p = q\). Here, d(p, q) is the (geodesic) Euclidean distance between p and q. For a real number \(t > 1\), a graph G(S, E) is called a t-spanner for the weighted set S of points if for any two points p and q in S the distance between p and q in graph G is at most t.\(d_w(p, q)\) for a real number \(t > 1\). For some integer \(k \ge 1\), a t-spanner G for the set S is a (k, t)-vertex fault-tolerant additive weighted spanner, denoted with (k, t)-VFTAWS, if for any set \(S' \subset S\) with cardinality at most k, the graph \(G \setminus S'\) is a t-spanner for the points in \(S \setminus S'\). For any given real number \(\epsilon > 0\), we present algorithms to compute a \((k, 4+\epsilon )\)-VFTAWS for the metric space \((S, d_w)\) resulting from the points in S belonging to either \(\mathbb {R}^d\) or located in the given simple polygon. Note that d(p, q) is the geodesic Euclidean distance between p and q in the case of simple polygons whereas in the case of \(\mathbb {R}^d\) it is the Euclidean distance along the line segment joining p and q.

rankdist is a recently developed R package which implements various distance-based ranking models. These models capture the occurring probability of rankings based on the distances between them. The package provides a framework for fitting and evaluating finite mixture of distance-based models. This paper also presents a new probability model for ranking data based on a new notion of weighted Kendall distance. The new model is flexible and more interpretable than the existing models. We show that the new model has an analytic form of the probability mass function and the maximum likelihood estimates of the model parameters can be obtained efficiently even for ranking involving a large number of objects.

Plant leaf identification is a significant challenge in the fields of computer vision and pattern recognition. This article presents a new approach to plant leaf identification, one that integrates shape and texture characteristics. First, we introduce the shape and texture features used by the proposed plant leaf recognition method. The proposed multiscale triangle descriptor (MTD) is employed to characterize the shape information of a plant leaf, and the local binary pattern histogram Fourier (LBP-HF) is used as the texture feature. Then, the shape and texture features of a leaf image are combined by weighted distance measurement, where L 1 distance and chi-square distance are used for shape and texture features, respectively. The proposed approach provides a robust descriptor for the task of plant leaf recognition by combining the complementary MTD and LBP-HF features. The proposed approach has been thoroughly evaluated on three benchmark leaf datasets, including the Flavia, Swedish and MEW2012 leaf datasets. Our method achieves 77.6%, 85.7%, and 67.5% retrieval accuracy on the Flavia, Swedish and MEW2012 leaf datasets, respectively, while the corresponding classification accuracy is 99.1%, 98.4%, 95.6%. The recognition performance of our method is better or comparable to prior state-of-the-art plant leaf recognition method.

Existing distance-based outlier mining methods do not consider the impact of each attribute's importance degree, thereby resulting in poor mining accuracies. To address this problem, we propose a new outlier mining algorithm – Miner* – that makes use of information entropy and Weighted Distance Sum to substantially improve mining accuracies. Miner* employs information entropy to determine weight values indicating the importance degrees of data attributes. An input dataset is reduced by Miner* through the neighbour-radius-based pruning technologies. Thus, Miner* obtains a candidate outlier set by removing any data objects that are unlikely to be outliers. Miner* calculates the weighted distance sum value Wkof each object in the candidate outlier set; Wkvalue ranks the top n to be regarded as outliers. Due to the sum of distance, which takes full advantage of the clustering characteristics of the dataset, edge distribution data objects and local outliers can be effectively mined out. To demonstrate the effectiveness of the Miner* algorithm, we implement Miner* in a prototype system to detect star spectrum data objects with abnormal characteristic lines. Our experimental results show that the algorithm in Miner* achieves high accuracy, high scalability, and low man-made influence by utilizing UCI and star spectrum dataset. Our results also confirm that Miner* is feasible and effective in mining spectrum data with abnormal characteristic lines from massive star spectrum dataset.

A single-valued neutrosophic linguistic set (SVNLS) is a popular fuzzy tool for describing deviation information in uncertain complex situations. The aim of this paper is to study some logarithmic distance measures and study their usefulness in multiple attribute group decision making (MAGDM) problems within single-valued neutrosophic linguistic (SVNL) environments. For achieving the purpose, SVNL weighted logarithmic averaging distance (SVNLWLAD) and SVNL ordered weighted logarithmic averaging distance (SVNLOWLAD) measures are firstly developed based on the logarithmic aggregation method. Then, the SVNL combined weighted logarithmic averaging distance (SVNLCWLAD) measure is presented by unifying the advantages of the previous SVNLWLAD and SVNLOWLAD measures. Moreover, a new MAGDM model by utilizing the SVNLCWLAD measure is presented under SVNL environments. Finally, a supplier selection for fresh aquatic products is taken as a case to illustrate the performance of the proposed framework.

Distance and similarity measures are very important topics in fuzzy set theory. In this paper, we propose a variety of distance measures for generalized hesitant fuzzy sets, based on which the corresponding similarity measures can be obtained. We investigate the connections of the mentioned distance measures and further develop a number of generalized hesitant ordered weighted distance measures and generalized hesitant ordered weighted similarity measures.

Given a set P of n points in the plane and a multiset W of k weights with k ≤ n , we assign each weight in W to a distinct point in P to minimize the maximum weighted distance from the weighted center of P to any point in P. In this paper, we present an algorithm which takes O ( k 2 n 2 log 3 ⁡ n ) time for the problem. We also consider the case that all weights in W are at most 1, and present an O ( k 5 n log 3 ⁡ k + k n log 3 ⁡ n ) -time algorithm. For a constant k, it takes only O ( n log 3 ⁡ n ) time, which is near linear.

We propose a computational efficient yet simple numerical algorithm to solve the surface eikonal equation on general implicit surfaces. The method is developed based on the embedding idea and the fast sweeping methods. We first approximate the solution to the surface eikonal equation by the Euclidean weighted distance function defined in a tubular neighbourhood of the implicit surface, and then apply the efficient fast sweeping method to numerically compute the corresponding viscosity solution. Unlike some other embedding methods which require the radius of the computational tube satisfies $$h=O(\varDelta x^{\gamma })$$h=O(Δx?) for some $$\gamma

Article history: Received July 11 2014 Received in Revised Format October 23 2014 Accepted October 26 2014 Available online October 26 2014 Vertex and p-center problems are two well-known types of the center problem. In this paper, a pcenter problem with uncertain demand-weighted distance will be introduced in which the demands are considered as fuzzy random variables (FRVs) and the objective of the problem is to minimize the maximum distance between a node and its nearest facility. Then, by introducing new methods, the proposed problem is converted to deterministic integer programming (IP) problems where these methods will be obtained through the implementation of the possibility theory and fuzzy random chance-constrained programming (FRCCP). Finally, the proposed methods are applied for locating bicycle stations in the city of Tabriz in Iran as a real case study. The computational results of our study show that these methods can be implemented for the center problem with uncertain frameworks. © 2015 Growing Science Ltd. All rights reserved

Person re-identification remains a challenging problem due to large variations of poses, occlusions, illumination and camera views. To learn both feature representation and similarity metric simultaneously, deep metric learning methods using triplet convolutional neural network have been applied in person re-identification. In this paper, we propose a body structure based triplet convolutional neural network (BSTCNN) for person re-identification. Specifically, a four-branch CNN architecture is built to learn features from different body parts. Body-part features are then fused in score level with a novel weighted distance layer which learns weights for each body part. We further design an improved triplet loss function to speed up convergence and boost the performance. Experimental results on two challenging datasets (CUHK01 and PRID2011) demonstrate that our approach significantly outperforms the state-of-the-art methods.

Let T be an edge weighted tree and let \(d_{min}\), \(d_{max}\) be two non-negative real numbers where \(d_{min}\le d_{max}\). A pairwise compatibility graph (PCG) of T for \(d_{min}\), \(d_{max}\) is a graph G such that each vertex of G corresponds to a distinct leaf of T and two vertices are adjacent in G if and only if the weighted distance between their corresponding leaves lies within the interval \([d_{min},d_{max}]\). A graph G is a PCG if there exist an edge weighted tree T and suitable \(d_{min}\), \(d_{max}\) such that G is a PCG of T. Knowing that all graphs are not PCGs, in this paper we introduce a variant of pairwise compatibility graphs which we call multi-interval PCGs. A graph G is a multi-interval PCG if there exist an edge weighted tree T and some mutually exclusive intervals of nonnegative real numbers such that there is an edge between two vertices in G if and only if the distance between their corresponding leaves in T lies within any such intervals. If the number of intervals is k, then we call the graph a k-interval PCG. We show that every graph is a k-interval PCG for some k. We also prove that wheel graphs and a restricted subclass of series-parallel graphs are 2-interval PCGs.

We consider quantization optimized for distributed estimation, where a set of sensors at different sites collect measurements on the parameter of interest, quantize them, and transmit the quantized data to a fusion node, which then estimates the parameter. Here, we propose an iterative quantizer design algorithm with a weighted distance rule that allows us to reduce a system-wide metric such as the estimation error by constructing quantization partitions with their optimal weights. We show that the search for the weights, the most expensive computational step in the algorithm, can be conducted in a sequential manner without deviating from convergence, leading to a significant reduction in design complexity. Our experments demonstrate that the proposed algorithm achieves improved performance over traditional quantizer designs. The benefit of the proposed technique is further illustrated by the experiments providing similar estimation performance with much lower complexity as compared to the recently published novel algorithms.

Linguistic continuous ordered weighted distance (LCOWD) measure is developed.Some properties of the LCOWD measure are discussed.Different families and extensions of the LCOWD measure are investigated.The new approach is applied to group decision making. This paper develops a new method for group decision making and introduces a linguistic continuous ordered weighted distance (LCOWD) measure. It is a new distance measure that combines the linguistic continuous ordered weighted averaging (LCOWA) operator with the ordered weighted distance (OWD) measure considering the risk attitude of decision maker. Moreover, it also can relieve the influence of extremely large or extremely small deviations on the aggregation results by assigning them smaller weights. These advantages make it suitable to deal with the situations where the input arguments are represented with uncertain linguistic information. Some of the main properties of the LCOWD measure and different particular cases are studied. The applicability of the new approach is also analyzed focusing on a group decision making problem.

As a nonparametric classification algorithm, K-Nearest Neighbor (KNN) is very efficient and can be easily realized. However, for large dataset, the computational demands for classifying instances using KNN can be expensive. A way to solve this problem is through the condensing approach. It means we remove instances that will bring computational burden but do not contribute to better classification accuracy. This paper proposes a novel weighted distance KNN algorithm based on instances condensing algorithm. The proposed idea is to extract some representative instances and take the processed result as a new training sample set. Meanwhile, use the distance-weighted WDKNN algorithm to improve the prediction accuracy, our experiments show that the proposed strategy can dramatically shorten the time consumption compared with the traditional KNN. On average, the speedup ratios improve 90% while classification accuracy only has 2% decreases.

Given a triangle A1A2A3 and weights m1, m2, m3, a geometric proof is given of (a) the existence of a point F whose weighted distance sum $$m_{1}|F-A_{1}|+m_{2}|F-A_{2}|+m_{3}|F-A_{3}|$$ has the smallest possible value, (b) the value of this minimum, and (c) the construction of the point. A characterization of the weights that guarantees the point is inside the triangle is given. This is then used to supply solutions to three extremal problems, as well as to generate sharp inequalities connecting the elements of triangle A1A2A3.

In this paper, we present a new approach to obtain trapezoidal approximation of fuzzy numbers with respect to weighted distance proposed by Nasibov [5] which the main property of this metric is flexibility in the decision maker's choice. Also, we prove some properties of the proposed method such as translation invariance, scale invariance and identity. Finally, we illustrate the efficiency of proposed method by solving some numerical examples.

This article investigates the problem of conditional p-maxian on the networks. Assume a network like G = (V,E) that the weight in each of its vertices vi is positive wi . In a pmaxian problem, the goal is to find a set contains p as the vertice of the set and as a facility; in a way that the sum of maximum of weighted distance of other point is the highest one from the set. In the problem of conditional p-maxian, we assume that some facilities are already existed and there is need to add other p ones. In this article, other new algorithms are presented to solve the problem of conditional p-maxian and their results (e.g. their solution time in finding the optimum points) are compared. One of the applications of this article is to determine an optimum place for landfill wastes. The wastes must be in the farthest distance of the demand (areas that people live) and it must be in the appropriate places to serve the requests.

We discuss how to rank fuzzy numbers using weighted distance in this paper. Basic properties of this ranking method are investigated. Further, we prove that the proposed order can be generated by a ranking index. At last, in the case of a pair of specific weights, we compute the ranking index in some common cases of practical applications.

Aimed at solving difficulties of managing test routing and matching resource ,this paper brings out an instrument resource matching algorithm based on minimum weighted distance .The algorithm adopts weighted distance to describe similarity between signals , while weight value is determined by experts' experience with consideration of actual condition ,which can describe similarity more reasonable .At the same time the strategy and algorithm of resource matching are given ,serial and parallel testing are optimized with simpler and rapider criterion. Application example tests the method possess better routing management capability ,strengthens system reliability and improves resource matching and routing management efficiency on the basis of testing demand models and instrument resource models.

Traditional PSO(Particle Swarm Optimization) algorithm has the problems of particle cross-border and premature convergence while solving the normalized constrained optimization problem. Our paper uses the attractor and spatial zoom method, and presented the normalized PSO algorithm. Secondly, each attribute is treated equally in the traditional classification algorithm, without considering the differences in attribute measure and contribution, which cause the problem of low classification accuracy. Our paper introduce the use of the normalized PSO algorithm to solve the optimal attribute normalized weighted distance. For example, in KNN classifier, leave-one-out experimental results with multiple UCI data sets show that the classification accuracy using normalization PSO algorithm to calculate normalized weighted distance is higher than using PSO algorithm and traditional none-attribute weighted classifiers.

© 2016 IEEE. The aim of the paper is to develop new aggregation operators using Bonferroni means, ordered weighted averaging (OWA) operators and some distance measures. We introduce the Bonferroni-Hamming weighted distance, Bonferroni OWA distance, and Bonferroni distances with OWA operators and weighted averages. The main advantages of using these operators are that they allow considering different aggregations contexts, multiple-comparison between each argument and distance measures in the same formulation.

The aim of this paper is to develop the continuous intuitionistic fuzzy ordered weighted distance (C-IFOWD) measure by using the continuous intuitionistic fuzzy ordered weighted averaging (C-IFOWA) operator in the interval distance. We investigate some desirable properties and different families of the C-IFOWD measure. We also generalize the C-IFOWD measure. The prominent characteristics of the C-IFOWD measure are that it is not only a generalization of some widely used distance measure, but also it can deal with interval deviations in aggregation on interval-valued intuitionistic fuzzy values (IVIFVs) by using a controlled parameter, which can decrease the uncertainty of argument and improve the accuracy of decision. The desirable characteristics make the C-IFOWD measure suitable to wide range situations, such as decision making, engineering and investment, etc. In the end, we introduce a new approach to group decision making with IVIFVs in human resource management. First published online:18 Sep 2015

In this paper we introduce the weighted distances on the Khalimsky grid. There are two types of natural neighborhood relations and one semi-neighborhood is also defined. Based on them two types of weighted distances are defined. We give formulae for computing the weighted distance of any point-pair on the Khalimsky grid in both cases.

A natural methodology for discriminating functional data is based on the distances from the observation or its derivatives to group representative functions (usually the mean) or their derivatives. It is proposed to use a combination of these distances for supervised classification. Simulation studies show that this procedure performs very well, resulting in smaller testing classification errors. Applications to real data show that this technique behaves as well as-and in some cases better than-existing supervised classification methods for functions.

Xu and Chen (J Syst Sci Syst Eng 17:432–445, 2008) introduced a new decision-making technique called the ordered weighted distance (OWD) measure, having been proved useful for the treatment of situation where the available information is represented in exact numerical values. In this paper, we consider the situations with intuitionistic fuzzy and interval-valued intuitionistic information, and develop some intuitionistic fuzzy weighted distance measures such as intuitionistic fuzzy ordered weighted distance (IFOWD) measure, interval-valued intuitionistic fuzzy ordered weighted distance (IVIFOWD) measure, intuitionistic fuzzy hybrid weighted distance (IFHWD) measure and interval-valued intuitionistic fuzzy hybrid weighted distance (IVIFHWD) measure. These developed weighted distance measures are very suitable to deal with the situation where the input data are represented in intuitionistic fuzzy numbers or interval-lvalued intuitionistic fuzzy numbers. Then we present a consensus reaching process for group decision making with intuitionistic fuzzy preference information based on the developed distance measures. Finally, a practical application of he developed approach to the problem of evaluating university faculty for tenure and promotion is given.

In this paper, we propose a variety of distance measures for hesitant fuzzy sets, based on which the corresponding similarity measures can be obtained. We investigate the connections of the aforementioned distance measures and further develop a number of hesitant ordered weighted distance measures and hesitant ordered weighted similarity measures. They can alleviate the influence of unduly large (or small) deviations on the aggregation results by assigning them low (or high) weights. Several numerical examples are provided to illustrate these distance and similarity measures.

We are interested in locations of multiple facilities in the plane with the aim of minimizing the sum of weighted distance between these facilities and regional customers, where the distance between a facility and a regional customer is evaluated by the farthest distance from this facility to the demand region. By applying the well-known location-allocation heuristic, the main task for solving such a problem turns out to solve a number of constrained Weber problems (CWPs). This paper focuses on the computational contribution in this topic by developing a variant of the classical Barzilai-Borwein (BB) gradient method to solve the reduced CWPs. Consequently, a hybrid Cooper type method is developed to solve the problem under consideration. Preliminary numerical results are reported to verify the evident effectiveness of the new method.

The accuracy and efficiency of case retrieval decline with individuality and diversity of customer needs in current product configuration design. To cope with this problem, this paper proposes the method of case retrieval based on NRS and KNN. Firstly, setting neighbor value can effectively avoid errors generated by discretization of continuous attributes in classical RS. Secondly, NRS is used for redundant reduction and weight allocation of attributes, and to calculate distance of each sample points by weighted distance. Thirdly, K similar cases are selected according to Euclidean distance with weight, then the case that meet customer needs best is acquired. Lastly, product configuration of coal mining machine is used to illustrate that accuracy and efficiency of the method are superior to the existing methods of case retrieval.

Considerable interest has been directed in the past to developing approaches for solving the p-median problem with maximum distance constraints. All current solution techniques consider potential facilities to be located only at nodes of the network. This paper deals with the solution of this problem under the condition where facility placement is not restricted to nodes. The examples given show that improvement in weighted distance can be obtained by solving the unrestricted site problem. In addition, feasible solutions can be obtained for smaller numbers of facilities than possible by all nodal facility placement.

By terms of extracting quantization values of each index making contributions to classification, this paper defines index classification weight; and also defines class representative points, weighted distance between samples and representative points; provides an iterative algorithm of searching class representative points, establishes a supervised clustering method based on representative points and it is apply into Fault diagnosis of Diesel Engine.

In 2007, Zeng and Li proposed a weighted triangular approximation of a fuzzy number. Unfortunately, this approximation may fail to be a fuzzy number. In this paper, we improve this approximation and propose a generalization by the name of weighted trapezoidal approximation. Their algorithms are also presented. Finally, some examples and relevant properties are discussed.

Nonuniform tubular neighborhoods of curves in R n are studied by using weighted distance functions and generalizing the normal exponential map. Different notions of injectivity radii are introduced to investigate singular but injective exponential maps. A generalization of the thickness formula is obtained for nonuniform thickness. All singularities within almost injectivity radius are classified by the Horizontal Collapsing Property. Examples are provided to show the distinction between the different types of injectivity radii, as well as showing that the standard differentiable injectivity radius fails to be upper semicontinuous on a singular set of weight functions.

This paper considers the problem of locating semi-obnoxious facilities assuming that ''too close'' demand nodes can be expropriated by the developer at a given price. The objective is to maximize the minimum weighted distance from the facility to the non-expropriated demand nodes given a limited budget while taking into account the fact that customers do not want to be too far away from the facility. Two models of this problem on a network are presented. One is to minimize the difference between the maximum and the minimum weighted distances. The other one is to maximize the minimum weighted distance subject to an upper bound constraint on the maximum weighted distance. The dominating sets are determined and efficient algorithms are presented.