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The classical TODIM method solves problems by expressing the attribute values as crisp numbers, thus incurring information loss. This paper, noting the ambiguity and fuzziness of information especially in a probabilistic hesitant fuzzy environment, develops a TODIM dynamic emergency decision-making method based on hybrid weighted distance. This method is suitable for emergency decision-making as it supports the rapidity and evolving nature of emergency responses, and accommodates the uncertainty of the external environment. A case study of a huge fire in China is used to validate the proposed method. The proposed TODIM method outperforms the methods proposed by Gao et al. in Int. J. Fuzzy. Syst. 19(5), 1261–1278 (2017) and Zhang et al. in Syst. 61, 48–58 (2014), suggesting its appeal in dealing with fuzzy information during emergency decision-making.

In this paper, we propose a novel approach for outlier detection, called local projections, which is based on concepts of Local Outlier Factor (LOF) (Breunig et al., 2000) and RobPCA (Hubert et al., 2005). By using aspects of both methods, our algorithm is robust towards noise variables and is capable of performing outlier detection in multi-group situations. We are further not reliant on a specific underlying data distribution. For each observation of a dataset, we identify a local group of dense nearby observations, which we call a core, based on a modification of the k-nearest neighbours algorithm. By projecting the dataset onto the space spanned by those observations, two aspects are revealed. First, we can analyze the distance from an observation to the center of the core within the projection space in order to provide a measure of quality of description of the observation by the projection. Second, we consider the distance of the observation to the projection space in order to assess the suitability of the core for describing the outlyingness of the observation. These novel interpretations lead to a univariate measure of outlyingness based on aggregations over all local projections, which outperforms LOF and RobPCA as well as other popular methods like PCOut (Filzmoser et al., 2008) and subspace-based outlier detection (Kriegel et al., 2009) in our simulation setups. Experiments in the context of real-word applications employing datasets of various dimensionality demonstrate the advantages of local projections.

Image segmentation is the basis of image analysis, object tracking, and other fields. However, image segmentation is still a bottleneck due to the complexity of images. In recent years, fuzzy clustering is one of the most important selections for image segmentation, which can retain information as much as possible. However, fuzzy clustering algorithms are sensitive to image artifacts. In this study, an improved image segmentation algorithm based on patch-weighted distance and fuzzy clustering is proposed, which can be divided into two steps. First, the pixel correlation between adjacent pixels is retrieved based on patch-weighted distance, and then the pixel correlation is used to replace the influence of neighboring information in fuzzy algorithms, thereby enhancing the robustness. Experiments on simulated, natural and medical images illustrate that the proposed schema outperforms other fuzzy clustering algorithms.

The hesitant fuzzy linguistic term set (HFLTS) and its extensions have been intensively investigated as a useful tool for the qualitative information problems in recent years. In the present paper, the concept of hesitant fuzzy linguistic term soft sets (HFLTSSs), a combination of HFLTSs and soft sets, are developed. First of all, relationships between any two HFLTSSs are provided, including inclusion, equivalence, and complementation. Then, several basic set operations for HFLTSSs are studied, such as AND, OR, union, and intersection. Meanwhile, corresponding properties of these operations are also discussed. In view of decision-making (DM) problem, we propose two algorithms for HFLTSSs, using the level soft sets and the hesitant fuzzy linguistic weighted distance (HFLWD) operator. Finally, a numerical “third-party evaluation”-related example in green development is used to present the utility and effectiveness of our algorithms. We also make comparisons between proposed method and some existing ones to confirm its feasibility and rationality. The main contribution of this paper possesses three points: (1) Enriching soft set theory by proposing the HFLTSSs, a soft set that can reflect the evaluation on objects more reasonably. (2) Redefining the HFLWD operator based on a novel distance of any two hesitant fuzzy linguistic term elements (HFLTEs), which can make the operations among HFLTEs much easier. (3) Applying the level soft set and new distance operator to develop two algorithms for decision-making problem with the information of HFLTSs.

Supervoxel segmentation has become an essential tool to medical image analysis for three-dimension MR image. However, in no consideration of the intensity inhomogeneity in 2D/3D MR image, the state-of-the-art supervoxel segmentation methods do not satisfy the further analysis, such as tissue classification according to intensity feature. In order to overcome the above-mentioned issues, we propose a modified supervoxel segmentation method for three-dimension MR image, which integrates the bias field into the weighted distance metric to determine the nearest cluster center. The supervoxel segmentation and bias correction can be simultaneously completed in our method. Especially, the bias corrected image lays the foundation for the supervoxel classification in accordance with the intensity feature. The experimental results and quantitative evaluation showed that the supervoxels obtained by our method are adherence to the MR tissue boundaries, and the bias corrected image is positive for the intensity feature extraction.

In this paper, after providing an appropriate coordinate system, we investigate the weighted distances on the Khalimsky grid. There are two types of natural neighborhood relations, and one semi-neighborhood is also defined. Weighted distances are defined for both cases, i.e., allowing or not the semi-neighborhood. We give formulae for computing the weighted distance of any point-pair on the Khalimsky grid in these cases. Digital disks based on the weighted distances are also investigated. In some cases, these disks may not be convex; moreover, they may contain holes. Sometimes, if semi-neighborhood is allowed, they are not connected, i.e., they contain islands. The conditions of concavities, holes and islands are characterized.

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

In this paper we consider the single facility collection depots location problem in the plane where n weighted demand points and p collection depots are given. The service of a demand point consists of the travel from the facility to the demand point and returning back to the facility through a collection depot which provides the shortest route. The weights are drawn from independent continuous distributions , and the objective is to find the location such that the probability that the maximum weighted distance of the round trip exceeds a threshold is minimized. The problem is formulated and analyzed. Computational experience is reported.

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.

We consider the p-median problem which is to find the location of p-facilities so as to minimize the average weighted distance or time between demand points and service centers. Many heuristic algorithms have been proposed for this problem. In this paper we present a simple new heuristic which is effective for moderately size problem. The heuristic uses a reduction and an exchange procedure. Our methodology is tested on 400 randomly generated problems with 10 to 50 customer locations as well as 6 well known literature test problems. We also compare our method with the Branch and Bound method in terms of quality and computational time using a larger problem size of 150 customer locations. For the random problems the generated solutions were on average within 0.61 % of the optimum. A similar result was achieved for the literature test problems. A comparative analysis with literature heuristics supports the superiority of our method. The computational time of our heuristic is 0.75 % of the Branch and Bound Method. We also apply our heuristic to a case study involving the location of emergency vehicles (ambulances) in Perth City (Australia).

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.

This paper calculated spatial accessibility of all counties (city, urban district) in China with cost weighted distance method. Region divisions of county accessibility were conducted, and relation of traffic accessibility and population aggregation was discussed in this paper. The results indicated that county accessibility in China had mainly low values and a distribution structure of circle layer and reverse-to-natural gradient. There was an obvious correlation between county accessibility and population density in China. With these analyses, inner mechanisms of population migration in different traffic conditions and region types were revealed, and can provide useful proposals to regional planning, traffic planning and smart distribution of people in China.

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.

In this paper, we consider the problem of finding a point on a general network using two objectives, maximizing the minimum weighted distance from the point to the vertices (Maximin) and maximizing the sum of weighted distances between the point and the vertices (Maxisum). This bicriterion model can be used to locate an obnoxious facility on a network. We will identify the model properties, develop a polynomial algorithm for generating the efficient set and provide a numerical example.

A new method is introduced which can explain individual differences in dissimilarity judgments by N subjects. The method is based on a weighted distance model which is a generalization of DeLeeuw and Pruzansky’s model to Minkowski r metric. A set of N dissimilarity matrices is analyzed nonmetrically to derive a group stimulus configuration and a subject configuration just like as INDSCAL. The method optimizes the goodness of fit measure which is based on the stress formula two by the method of steepest descent. This optimizing measure makes the computing process independent on Minkowski constant. The method was applied to schematic face data, and gave satisfactory results.

This paper describes a computational method for weighted euclidean distance scaling which combines aspects of an “analytic” solution with an approach using loss functions. We justify this new method by giving a simplified treatment of the algebraic properties of a transformed version of the weighted distance model. The new algorithm is much faster than INDSCAL yet less arbitrary than other “analytic” procedures. The procedure, which we call SUMSCAL (subjectivemetricscaling), gives essentially the same solutions as INDSCAL for two moderate-size data sets tested.