A fuzzy clustering-based hybrid method for a multi-facility location problem

A fuzzy clustering-based hybrid method for a multi-facility location problem is presented in this study. It is assumed that capacity of each facility is unlimited. The method uses different approaches sequentially. Initially, customers are grouped by spherical and elliptical fuzzy cluster analysis methods in respect to their geographical locations. Different numbers of clusters are experimented. Then facilities are located at the proposed cluster centers. Finally each cluster is solved as a single facility location problem. The center of gravity method, which optimizes transportation costs is employed to fine tune the facility location. In order to compare logistical performance of the method, a real world data is gathered. Results of existing and proposed locations are reported.     

Bounds for the single source modular capacitated plant location problem

In this paper, we propose a discrete location problem, which we call the Single Source Modular Capacitated Location Problem (SS-MCLP). The problem consists of finding the location and capacity of the facilities, to serve a set of customers at a minimum total cost. The demand of each customer must be satisfied by one facility only and the capacities of the open facilities must be chosen from a finite and discrete set of allowable capacities. Because the SS-MCLP is a difficult problem, a lagrangean heuristic, enhanced by tabu search or local search was developed in order to obtain good feasible solutions. When needed, the lower bounds are used in order to evaluate the quality of the feasible solutions. Our method was tested computationally on randomly generated test problems some of which are with large dimensions considering the literature related to this type of problem. The computational results obtained were compared with those provided by the commercial software Cplex.     

Region-rejection based heuristics for the capacitated multi-source Weber problem

A new type of constructive and adaptive heuristics is put forward to generate initial solutions for the capacitated multi-source Weber problem. This technique is based on guiding the search by constructing restricted regions that forbid new locations to be sited too close to the previously found locations. In this work, a restricted region is represented by a circle whose radius is initially set to a fixed value, based on the sparsity of the customers and the number of facilities, and then a scheme that dynamically adjusts the radius at each facility is proposed. A discretisation technique that divides a continuous space into a discrete number of cells while embedding the use of restricted regions within the search is also presented. The experiments show that the proposed region-rejection methods, though simple and easy to understand, provide encouraging results with regard to both solution quality and computational effort. Some future research avenues are also briefly highlighted.     

Analytically tractable case of fuzzy c-means clustering

In this paper, we offer a simple and accurate clustering algorithm which was derived as a closed-form analytical solution to a cluster fit function minimization problem. As a result, the algorithm finds the global minimum of the fit function, and combines exceptional efficiency with optimal clustering results.     

On solutions of fuzzy random multiobjective quadratic programming with applications in portfolio problem

In this paper, a multiobjective quadratic programming problem fuzzy random coefficients matrix in the objectives and constraints and the decision vector are fuzzy variables is considered. First, we show that the efficient solutions fuzzy quadratic multiobjective programming problems series-optimal-solutions of relative scalar fuzzy quadratic programming. Some theorems are to find an optimal solution of the relative scalar quadratic multiobjective programming with fuzzy coefficients, having decision vectors as fuzzy variables. An application fuzzy portfolio optimization problem as a convex quadratic programming approach is discussed and an acceptable solution to such problem is given. At the end, numerical examples are illustrated in the support of the obtained results.     

An improved heuristic for the capacitated arc routing problem

The capacitated arc routing problem (CARP) is an important and practical problem in the OR literature. In short, the problem is to identify routes to service (e.g., pickup or deliver) demand located along the edges of a network such that the total cost of the routes is minimized. In general, a single route cannot satisfy the entire demand due to capacity constraints on the vehicles. CARP belongs to the set of NP-hard problems; consequently numerous heuristic and metaheuristic solution approaches have been developed to solve it. In this paper an “ellipse rule” based heuristic is proposed for the CARP. This approach is based on the path-scanning heuristic, one of the mostly used greedy-add heuristics for this problem. The innovation consists basically of selecting edges only inside ellipses when the vehicle is near the end of each route. This new approach was implemented and tested on three standard datasets and the solutions are compared against: (i) the original path-scanning heuristic; (ii) two other path-scanning heuristics and (iii) the three best known metaheuristics. The results indicate that the “ellipse rule” approach lead to improvements over the three path-scanning heuristics, reducing the average distance to the lower bound in the test problems by about 44%.     

Duality theory in fuzzy mathematical programming problems with fuzzy coefficients

In this paper, the notions of subgradient, subdifferential, and differential with respect to convex fuzzy mappings are investigated, which provides the basis for the fuzzy extremum problem theory. We consider the problems of minimizing or maximizing a convex fuzzy mapping over a convex set and develop the necessary and/or sufficient optimality conditions. Furthermore, the concept of saddle-points and minimax theorems under fuzzy environment is discussed. The results obtained are used to formulate the Lagrangian dual of fuzzy programming. Under certain fuzzy convexity assumptions, KKT conditions for fuzzy programming are derived, and the “perturbed” convex fuzzy programming is considered. Finally, these results are applied to fuzzy linear programming and fuzzy quadratic programming.     

Vector quantization in DCT domain using fuzzy possibilistic c-means based on penalized and compensated constraints

In this paper, fuzzy possibilistic c-means (FPCM) approach based on penalized and compensated constraints are proposed to vector quantization (VQ) in discrete cosine transform (DCT) for image compression. These approaches are named penalized fuzzy possibilistic c-means (PFPCM) and compensated fuzzy possibilistic c-means (CFPCM). The main purpose is to modify the FPCM strategy with penalized or compensated constraints so that the cluster centroids can be updated with penalized or compensated terms iteratively in order to find near-global solution in optimal problem. The information transformed by DCT was separated into DC and AC coefficients. Then, the AC coefficients are trained by using the proposed methods to generate better codebook based on VQ. The compression performances using the proposed approaches are compared with FPCM and conventional VQ method. From the experimental results, the promising performances can be obtained using the proposed approaches.     

A guided reactive GRASP for the capacitated multi-source Weber problem

The capacitated multi-source Weber problem entails finding both the locations of capacitated facilities on a plane and their customer allocations. A framework that uses adaptive learning and functional representation to construct the restricted candidate list (RCL) within a greedy randomized adaptive search procedure (GRASP) is put forward. An implementation of restricted regions that forbids new facilities to be located too close to the previously found facilities is also embedded into the search to build up the RCL more efficiently. The performance of this GRASP based approach is tested on three classes of instances with constant and variable capacities. Very competitive results are obtained when compared to the best known results from the literature.     

A convex programming with nested constraints on the amount of resources

This paper investigates the resource allocation problem of maximizing a strictly concave objective function under nested constraints on a weighted amount of resources. The resources are distributed in a multi-dimensional space and the distribution gives a planner some reward. The constraints on the resources are set in such a way that a weighted amount of resources is limited in each level of subspace as well as in the whole space. We call such constraints nested. The purpose of this paper is to devise efficient methods for an optimal distribution of resources under the constraints. For the concave maximization problem, first we derive necessary and sufficient conditions for an optimal solution and, secondly, propose two methods to solve the problem. Both methods manipulate the so-called Lagrangian multipliers and make a sequence of feasible solutions so as to satisfy all the necessary and sufficient conditions. Numerical examination confirms that the proposed methods perform better than some well-known methods of nonlinear programming in terms of computational time.     

Efficient approximate solution methods for the multi-commodity capacitated multi-facility Weber problem

The capacitated multi-facility Weber problem is concerned with locating I capacitated facilities in the plane to satisfy the demand of J customers with the minimum total transportation cost of a single commodity. This is a nonconvex optimization problem and difficult to solve. In this work, we focus on a multi-commodity extension and consider the situation where K distinct commodities are shipped subject to capacity constraints between each customer and facility pair. Customer locations, demands and capacities for each commodity, and bundle restrictions are known a priori. The transportation costs, which are proportional to the distance between customers and facilities, depend on the commodity type. We address several location-allocation and discrete approximation heuristics using different strategies. Based on the obtained computational results we can say that the alternate solution of location and allocation problems is a very efficient strategy; but the discrete approximation has excellent accuracy.     

A size-insensitive integrity-based fuzzy c-means method for data clustering

Fuzzy c-means (FCM) is one of the most popular techniques for data clustering. Since FCM tends to balance the number of data points in each cluster, centers of smaller clusters are forced to drift to larger adjacent clusters. For datasets with unbalanced clusters, the partition results of FCM are usually unsatisfactory. Cluster size insensitive FCM (csiFCM) dealt with “cluster-size sensitivity” problem by dynamically adjusting the condition value for the membership of each data point based on cluster size after the defuzzification step in each iterative cycle. However, the performance of csiFCM is sensitive to both the initial positions of cluster centers and the “distance” between adjacent clusters. In this paper, we present a cluster size insensitive integrity-based FCM method called siibFCM to improve the deficiency of csiFCM. The siibFCM method can determine the membership contribution of every data point to each individual cluster by considering cluster's integrity, which is a combination of compactness and purity. “Compactness” represents the distribution of data points within a cluster while “purity” represents how far a cluster is away from its adjacent cluster. We tested our siibFCM method and compared with the traditional FCM and csiFCM methods extensively by using artificially generated datasets with different shapes and data distributions, synthetic images, real images, and Escherichia coli dataset. Experimental results showed that the performance of siibFCM is superior to both traditional FCM and csiFCM in terms of the tolerance for “distance” between adjacent clusters and the flexibility of selecting initial cluster centers when dealing with datasets with unbalanced clusters.     

An efficient dynamic programming algorithm for a special case of the capacitated lot-sizing problem

In this paper we consider the capacitated lot-sizing problem (CLSP) with linear costs. It is known that this problem is NP-hard, but there exist special cases that can be solved in polynomial time. We derive a new O(T²) algorithm for the CLSP with non-increasing setup costs, general holding costs, non-increasing production costs and non-decreasing capacities over time, where T is the length of the model horizon. We show that in every iteration we do not consider more candidate solutions than the O(T²) algorithm proposed by [Chung and Lin, 1988. Management Science 34, 420–6]. We also develop a variant of our algorithm that is more efficient in the case of relatively large capacities. Numerical tests show the superior performance of our algorithms compared to the algorithm of [Chung and Lin, 1988. Management Science 34, 420–6].     

Savings based ant colony optimization for the capacitated minimum spanning tree problem

The problem of connecting a set of client nodes with known demands to a root node through a minimum cost tree network, subject to capacity constraints on all links is known as the capacitated minimum spanning tree (CMST) problem. As the problem is NP-hard, we propose a hybrid ant colony optimization (ACO) algorithm to tackle it heuristically. The algorithm exploits two important problem characteristics: (i) the CMST problem is closely related to the capacitated vehicle routing problem (CVRP), and (ii) given a clustering of client nodes that satisfies capacity constraints, the solution is to find a MST for each cluster, which can be done exactly in polynomial time. Our ACO exploits these two characteristics of the CMST by a solution construction originally developed for the CVRP. Given the CVRP solution, we then apply an implementation of Prim's algorithm to each cluster to obtain a feasible CMST solution. Results from a comprehensive computational study indicate the efficiency and effectiveness of the proposed approach.     

Robust color image segmentation using fuzzy c-means with weighted hue and intensity

In this paper we propose a novel method for color image segmentation. This method uses only hue and intensity components (which are chosen rationally) of image and combines those by adaptive tuned weights in a specially defined fuzzy c-means cost function. The tuned weights indicate how informative every color component (hue and intensity) is. Obtaining tuned weights begins with finding peaks of hue and intensity's histograms and continues by obtaining the table of the frequencies of hue and intensity values and computing entropy and contrast of every color component. Also this method specifies proper initial values for cluster centers with the aim of reducing the overall number of iterations and avoiding converging of FCM to wrong centroids. Experimental results demonstrate that our algorithm achieves better segmentation performance and also runs faster than similar methods.     

A fuzzy multi-objective covering-based vehicle location model for emergency services

Timeliness is one of the most important objectives that reflect the quality of emergency services such as ambulance and firefighting systems. To provide timeliness, system administrators may increase the number of service vehicles available. Unfortunately, increasing the number of vehicles is generally impossible due to capital constraints. In such a case, the efficient deployment of emergency service vehicles becomes a crucial issue. In this paper, a multi-objective covering-based emergency vehicle location model is proposed. The objectives considered in the model are maximization of the population covered by one vehicle, maximization of the population with backup coverage and increasing the service level by minimizing the total travel distance from locations at a distance bigger than a prespecified distance standard for all zones. Model applications with different solution approaches such as lexicographic linear programming and fuzzy goal programming (FGP) are provided through numerical illustrations to demonstrate the applicability of the model. Numerical results indicate that the model generates satisfactory solutions at an acceptable achievement level of desired goals.     

Robust capacitated facility location model for acquisitions under uncertainty

We study a new robust formulation for strategic location and capacity planning considering potential company acquisitions under uncertainty. Long-term logistics network planning is among the most difficult decisions for supply-chain managers. While costs, demands, etc. may be known or estimated well for the short-term, their future development is uncertain and difficult to predict.A new model formulation for the robust capacitated facility location problem is presented to cope with uncertainty in planning. Minimizing the expectation of the relative regrets across scenarios over multiple periods is the objective. It is achieved by dynamically assigning multi-level production allocations, locations and capacity adjustments for uncertain parameter development over time. Considering acquisitions for profit maximization and its supply-chain impact is new as well as the simultaneous decision of capacity adjustment and facility location over time. The solution of the novel robust formulation provides a single setup where good results can be achieved for any realized scenario. Hence, the solution may not be optimal for one particular scenario but may be good, i.e. the highest expected profit to gain, for any highly probable future realization. We show that robust mixed-integer linear programming model achieves superior results to the deterministic configurations in exhaustive computational tests. This dynamic robust formulation allows the supply-chain to favorably adapt to acquisitions and uncertain developments of revenue, demand and costs and hence reduces the potential negative impacts of uncertainty on supply-chain operations.     

Robust spatially constrained fuzzy c-means algorithm for brain MR image segmentation

Objective

Accurate brain tissue segmentation from magnetic resonance (MR) images is an essential step in quantitative brain image analysis, and hence has attracted extensive research attention. However, due to the existence of noise and intensity inhomogeneity in brain MR images, many segmentation algorithms suffer from limited robustness to outliers, over-smoothness for segmentations and limited segmentation accuracy for image details. To further improve the accuracy for brain MR image segmentation, a robust spatially constrained fuzzy c-means (RSCFCM) algorithm is proposed in this paper.

Method

Firstly, a novel spatial factor is proposed to overcome the impact of noise in the images. By incorporating the spatial information amongst neighborhood pixels, the proposed spatial factor is constructed based on the posterior probabilities and prior probabilities, and takes the spatial direction into account. It plays a role as linear filters for smoothing and restoring images corrupted by noise. Therefore, the proposed spatial factor is fast and easy to implement, and can preserve more details. Secondly, the negative log-posterior is utilized as dissimilarity function by taking the prior probabilities into account, which can further improve the ability to identify the class for each pixel. Finally, to overcome the impact of intensity inhomogeneity, we approximate the bias field at the pixel-by-pixel level by using a linear combination of orthogonal polynomials. The fuzzy objective function is then integrated with the bias field estimation model to overcome the intensity inhomogeneity in the image and segment the brain MR images simultaneously.

Results

To demonstrate the performances of the proposed algorithm for the images with/without skull stripping, the first group of experiments is carried out in clinical 3T-weighted brain MR images which contain quite serious intensity inhomogeneity and noise. Then we quantitatively compare our algorithm to state-of-the-art segmentation approaches by using Jaccard similarity on benchmark images obtained from IBSR and BrainWeb with different level of noise and intensity inhomogeneity. The comparison results demonstrate that the proposed algorithm can produce higher accuracy segmentation and has stronger ability of denoising, especially in the area with abundant textures and details.

Conclusion

In this paper, the RSCFCM algorithm is proposed by utilizing the negative log-posterior as the dissimilarity function, introducing a novel factor and integrating the bias field estimation model into the fuzzy objective function. This algorithm successfully overcomes the drawbacks of existing FCM-type clustering schemes and EM-type mixture models. Our statistical results (mean and standard deviation of Jaccard similarity for each tissue) on both synthetic and clinical images show that the proposed algorithm can overcome the difficulties caused by noise and bias fields, and is capable of improving over 5% segmentation accuracy comparing with several state-of-the-art algorithms.     

A genetic algorithm for a bi-objective capacitated arc routing problem

The capacitated arc routing problem (CARP) is a very hard vehicle routing problem for which the objective—in its classical form—is the minimization of the total cost of the routes. In addition, one can seek to minimize also the cost of the longest trip.In this paper, a multi-objective genetic algorithm is presented for this more realistic CARP. Inspired by the second version of the Non-dominated sorted genetic algorithm framework, the procedure is improved by using good constructive heuristics to seed the initial population and by including a local search procedure. The new framework and its different flavour is appraised on three sets of classical CARP instances comprising 81 files.Yet designed for a bi-objective problem, the best versions are competitive with state-of-the-art metaheuristics for the single objective CARP, both in terms of solution quality and computational efficiency: indeed, they retrieve a majority of proven optima and improve two best-known solutions.     

Design of capacitated minimum spanning tree with uncertain cost and demand parameters

The classical Capacitated Minimum Spanning Tree Problem (CMSTP) deals with finding a minimum-cost spanning tree so that the total demand of the vertices in each subtree does not exceed the capacity limitation. In most of the CMSTP models, the edge costs and the demands of the vertices in the network are assumed to be known with certainty. This paper considers the CMSTP model, where the edge costs and/or the demands are only approximately known. A fast approximate reasoning algorithm, which is based on the Esau-Williams savings heuristic and fuzzy logic rules, is proposed. The computational results of the study based on the proposed approach are also reported.