A new Pareto set generating method for multi-criteria optimization problems

This paper proposes a new classical method to capture the complete Pareto set of a multi-criteria optimization problem (MOP) even without having any prior information about the location of Pareto surface. The solutions obtained through the proposed method are globally Pareto optimal. Moreover, each and every global Pareto optimal point is within the attainable range. This paper also suggests a procedure to ensure the proper Pareto optimality of the outcomes if slight modifications are allowed in the constraint set of the MOP under consideration. Among the set of all outcomes, the proposed method can effectively detect the regions of unbounded trade-offs between the criteria, if they exist.     

A new method for evaluating decision making units in DEA

This paper provides a new structure in data envelopment analysis (DEA) for assessing the performance of decision making units (DMUs). It proposes a technique to estimate the DEA efficient frontier based on the Arash Method in a way different from the statistical inferences. The technique allows decisions in the target regions instead of points to benchmark DMUs without requiring any more information in the case of interval/fuzzy DEA methods. It suggests three efficiency indexes, called the lowest, technical and highest efficiency scores, for each DMU where small errors occur in both input and output components of the Farrell frontier, even if the data are accurate. These efficiency indexes provide a sensitivity index for each DMU and arrange both inefficient and technically efficient DMUs together while simultaneously detecting and benchmarking outliers. Two numerical examples depicted the validity of the proposed method.     

A new bounding method for single facility location models

This paper develops a new lower bound for single facility location problems withl _p distances. We prove that the method produces superior results to other known procedures. The new bound is also computationally efficient. Numerical results are given for a range of examples with varying numbers of existing facilities andp values.     

A cutting-plane approach for large-scale capacitated multi-period facility location using a specialized interior-point method

We propose a cutting-plane approach (namely, Benders decomposition) for a class of capacitated multi-period facility location problems. The novelty of this approach lies on the use of a specialized interior-point method for solving the Benders subproblems. The primal block-angular structure of the resulting linear optimization problems is exploited by the interior-point method, allowing the (either exact or inexact) efficient solution of large instances. The consequences of different modeling conditions and problem specifications on the computational performance are also investigated both theoretically and empirically, providing a deeper understanding of the significant factors influencing the overall efficiency of the cutting-plane method. The methodology proposed allowed the solution of instances of up to 200 potential locations, one million customers and three periods, resulting in mixed integer linear optimization problems of up to 600 binary and 600 millions of continuous variables. Those problems were solved by the specialized approach in less than one hour and a half, outperforming other state-of-the-art methods, which exhausted the (144 GB of) available memory in the largest instances.     

A new design of the elimination and choice translating reality method for multi-criteria group decision-making in an intuitionistic fuzzy environment

The purpose of this paper is to design a new extension of the ELECTRE, known as the elimination and choice translating reality method, for multi-criteria group decision-making problems based on intuitionistic fuzzy sets. This method is widely utilized when a set of alternatives should be identified and evaluated with respect to a set of conflicting criteria by reflecting decision makers’ (DMs’) preferences. However, handling the exact data and numerical measure is difficult to be precisely focused because the DMs’ judgments are often vague in real-life decision problems and applications. A more realistic and practical approach can be to use linguistic variables expressed in intuitionistic fuzzy numbers instead of numerical data to model DMs’ judgments and to describe the inputs in the ELECTRE method. The proposed intuitionsitic fuzzy ELECTRE utilizes the truth-membership function and non-truth-membership function to indicate the degrees of satisfiability and non-satisfiability of each alternative with respect to each criterion and the relative importance of each criterion, respectively. Then, a new discordance intuitionistic index is introduced, which is extended from the concept of the fuzzy distance measure. Outranking relations are defined by pairwise comparisons and a decision graph is depicted to determine which alternative is preferable, incomparable or indifferent in the intuitionistic fuzzy environment. Finally, a comprehensive sensitivity analysis is employed to further study regarding the impact of threshold values on the final evaluation, and a comparative analysis is demonstrated with an application example in flexible manufacturing systems between the proposed ELECTRE method and the existing intuitionistic fuzzy technique for order preference by similarity to ideal solution (IF-TOPSIS) method.     

A numerical method for identifying the location of a fluid leak in a pipeline

On the basis of mathematical models for studying gas or liquid flows in long-distance pipelines, a numerical method is developed that allows us to determine the coordinates of the fluid leak points by measuring data for the flow parameters (pressure and flow rate) both in stationary and nonstationary cases. The method is a modification of the Newton method applied to minimizing the functional of the mean-square deviation of the calculated parameters from the measured data.     

A method and a computer program for determining the thermal diffusivity in a solid slab

The authors describe a method for computing the thermal diffusivity of a solid, based on a computer assisted evaluation of the solution of the transient inverse heat conduction problem.The program computes either the unknown diffusivity or simulates the one-dimensional unsteady heat transfer problem. The user may model the boundary conditions by a choice of different functions.The program provides instruction and information at all stages of input and provides tabular output of results. It may be used by anybody wishing to solve or simulate heat transfer processes.     

An interior-point method for the single-facility location problem with mixed norms using a conic formulation

We consider the single-facility location problem with mixed norms, i.e. the problem of minimizing the sum of the distances from a point to a set of fixed points in , where each distance can be measured according to a different p-norm. We show how this problem can be expressed into a structured conic format by decomposing the nonlinear components of the objective into a series of constraints involving three-dimensional cones. Using the availability of a self-concordant barrier for these cones, we present a polynomial-time algorithm (a long-step path-following interior-point scheme) to solve the problem up to any given accuracy. Finally, we report computational results for this algorithm and compare with standard nonlinear optimization solvers applied to this problem. This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The first author acknowledges support by an FSR grant from Université Catholique de Louvain and a FRIA grant from Fonds de la Recherche Scientifique-FNRS. The scientific responsibility rests with its authors.     

A projection method for the uncapacitated facility location problem

Several algorithms already exist for solving the uncapacitated facility location problem. The most efficient are based upon the solution of the strong linear programming relaxation. The dual of this relaxation has a condensed form which consists of minimizing a certain piecewise linear convex function. This paper presents a new method for solving the uncapacitated facility location problem based upon the exact solution of the condensed dual via orthogonal projections. The amount of work per iteration is of the same order as that of a simplex iteration for a linear program inm variables and constraints, wherem is the number of clients. For comparison, the underlying linear programming dual hasmn + m + n variables andmn +n constraints, wheren is the number of potential locations for the facilities. The method is flexible as it can handle side constraints. In particular, when there is a duality gap, the linear programming formulation can be strengthened by adding cuts. Numerical results for some classical test problems are included.     

A generalized Weiszfeld method for the multi-facility location problem

An iterative method is proposed for the K facilities location problem. The problem is relaxed using probabilistic assignments, depending on the distances to the facilities. The probabilities, that decompose the problem into K single-facility location problems, are updated at each iteration together with the facility locations. The proposed method is a natural generalization of the Weiszfeld method to several facilities.     

A new bilevel formulation for the vehicle routing problem and a solution method using a genetic algorithm

The Vehicle Routing Problem (VRP) is one of the most well studied problems in operations research, both in real life problems and for scientific research purposes. During the last 50 years a number of different formulations have been proposed, together with an even greater number of algorithms for the solution of the problem. In this paper, the VRP is formulated as a problem of two decision levels. In the first level, the decision maker assigns customers to the vehicles checking the feasibility of the constructed routes (vehicle capacity constraints) and without taking into account the sequence by which the vehicles will visit the customers. In the second level, the decision maker finds the optimal routes of these assignments. The decision maker of the first level, once the cost of each routing has been calculated in the second level, estimates which assignment is the better one to choose. Based on this formulation, a bilevel genetic algorithm is proposed. In the first level of the proposed algorithm, a genetic algorithm is used for calculating the population of the most promising assignments of customers to vehicles. In the second level of the proposed algorithm, a Traveling Salesman Problem (TSP) is solved, independently for each member of the population and for each assignment to vehicles. The algorithm was tested on two sets of benchmark instances and gave very satisfactory results. In both sets of instances the average quality is less than 1%. More specifically in the set with the 14 classic instances proposed by Christofides, the quality is 0.479% and in the second set with the 20 large scale vehicle routing problems, the quality is 0.826%. The algorithm is ranked in the tenth place among the 36 most known and effective algorithms in the literature for the first set of instances and in the sixth place among the 16 algorithms for the second set of instances. The computational time of the algorithm is decreased significantly compared to other heuristic and metaheuristic algorithms due to the fact that the Expanding Neighborhood Search Strategy is used.     

On a method of Atkinson for evaluating domain integrals in the boundary element method

We have shown how to couple a numerical method due to Atkinson for computing particular solutions to a class of elliptic differential equations with a variety of boundary element methods, which alleviates the problem of domain discretization for solving inhomogeneous equations. When the inhomogeneous term is of the form typically used in the dual reciprocity method, Atkinson's formula is such that only boundary integrals need to be approximated numerically. If this method is coupled with the method of potentials, it results in a computational technique which requires neither boundary nor domain discretization. Some numerical results are given validating our approach.     

A new approximation algorithm for the multilevel facility location problem

In this paper we propose a new integer programming formulation for the multilevel facility location problem and a novel 3-approximation algorithm based on LP-rounding. The linear program that we use has a polynomial number of variables and constraints, thus being more efficient than the one commonly used in the approximation algorithms for these types of problems.     

A multi-criteria evaluation of diseases in a study for public-health planning

This paper describes an assessment of the seriousness of (clusters of) diseases in two rounds, one with the criteria which are predominant for policy makers, and one with the criteria of the patients. The project was carried out in a Group Decision Room by 12 high-ranking officials of the Ministry of Health, Welfare, and Sport in The Netherlands. The evaluation remained manageable by a context-related scaling technique which accelerated the decision process and enhanced the consistency of the evaluators.     

A note on new exact solutions for the Kawahara equation using Exp-function method

Exact solutions of the Kawahara equation by Assas [L.M.B. Assas, New Exact solutions for the Kawahara equation using Exp-function method, J. Comput. Appl. Math. 233 (2009) 97-102] are analyzed. It is shown that all solutions do not satisfy the Kawahara equation and consequently all nontrivial solutions by Assas are wrong.     

A new method for solving fuzzy transportation problems using ranking function

In the literature, several methods are proposed for solving transportation problems in fuzzy environment but in all the proposed methods the parameters are represented by normal fuzzy numbers. [S.H. Chen, Operations on fuzzy numbers with function principal, Tamkang Journal of Management Sciences 6 (1985) 13–25] pointed out that in many cases it is not to possible to restrict the membership function to the normal form and proposed the concept of generalized fuzzy numbers. There are several papers in the literature in which generalized fuzzy numbers are used for solving real life problems but to the best of our knowledge, till now no one has used generalized fuzzy numbers for solving the transportation problems. In this paper, a new method is proposed for solving fuzzy transportation problems by assuming that a decision maker is uncertain about the precise values of the transportation cost, availability and demand of the product. In the proposed method transportation cost, availability and demand of the product are represented by generalized trapezoidal fuzzy numbers. To illustrate the proposed method a numerical example is solved and the obtained results are compared with the results of existing methods. Since the proposed method is a direct extension of classical method so the proposed method is very easy to understand and to apply on real life transportation problems for the decision makers.     

An interactive method for parametric identification of the mathematical model of a parachute system using computer graphics

A simple mathematical model of the motion of a parachute system in space is described and an interactive algorithm for parametric identification of the model is proposed. The algorithm selects the model parameters that minimize the deviation of the calculated dependences from experimental observations on the computer graphic monitor.Moscow. Translated from Dinamicheskie Sistemy, No. 10, pp. 106–111, 1992.     

A method for evaluating the stability of fixed points in perturbed symplectic maps

An efficient method for determining the linear stability of fixed points of perturbed symplectic maps is presented. The method is applied to the perturbed twist maps, the near-identity and the near π/2-rotation symplectic maps.