A rectilinear distance location–relocation problem with a probabilistic restriction: mathematical modelling and solution approaches

In this study, we have considered a multi-period centre facility location–relocation problem in the presence of a probabilistic polyhedral barrier uniformly distributed on a horizontal barrier route in rectilinear plane. The objective function of this location–relocation problem is the minimisation of the cost of maximum expected rectilinear barrier distance from demand points to the new facility plus the relocation cost (i.e. a changeover cost at the beginning of each period) in the form of a mixed integer quadratic-constrained mathematical programming. The computational results show that the non-linear solver of commercial software LINGO is only effective in solving small-sized problems. A linear approximation for the system constraints is proposed so that a new mixed integer linear programming model is generated which is solvable via CPLEX optimisation software. Moreover, we proposed a problem decomposition procedure that reduces the multi-period problem into a number of single-period problems with some modifications. To show the efficiency of the model and solution methodologies, a broad range of numerical examples are performed. Results indicate that the developed problem decomposition procedure obtains the near-optimal solution comparatively with the results obtained from the non-linear solver of LINGO, and that the lower bound problem can be useful for large-sized problems in a reasonable time. Moreover, a practical case example to show the model validity in real world is solved and to reality check from practice, results are compared with the problem without barrier.     

On Solving Multifacility Location Problems using a Hyperboloid Approximation Procedure

An iterative solution method is presented for solving multifacility location problems involving rectilinear and/or Euclidean distances. The iterative procedure is based on the use of an approximating function involving hyperboloids, which in the limit approach the cones in the original objective function. Given that the hyperboloid approximation procedure converges, it is shown to converge to the optimum solution. Computational experience with the procedure is described.     

Pressure with friction of a perfectly rigid die upon an elastic half space with cracks

We study the interaction of a rigid die with a base of any shape and the surface of an elastic half space containing cracks in the presence of friction in the contact zone. The solution of the plane contact problem of the theory of elasticity is obtained by the method of singular integral equations. The detailed analysis of the problem is performed for the case where the base of the die is parabolic and a crack is rectilinear and appears on the surface of half space. We also investigate the effects of the friction coefficient, crack length, its orientation, and location on stress intensity factors KI and KII at the crack tip and the distribution of contact stresses under the die.     

Fast Algorithms for the Round Trip Location Problem

This paper presents several fast algorithms for the round trip location problem. I show how to simplify the method given in an article by Chan and Hearn in which only the rectilinear case is addressed and get an algorithm with improved complexity for a given accuracy. An exact solution algorithm (the complexity of which is unknown) that solved test data problems even faster is presented and generalized for Euclidean and ℓ_p distances. Computational results are also presented.     

A Geometrical Solution Procedure for a Rectilinear Distance Minimax Location Problem1

A minimax location problem is considered which involves finding the location of a facility in the plane so that the maximum rectilinear distance between the facility and an arbitrary, but fixed, nonempty and bounded set in the plane will be minimized. A geometrical procedure is developed for finding all solutions to the problem. The derivation of the solution procedure, which is based on the fact that the minimax problem is equivalent to the problem of enclosing the representation of the set after a 45 degree rotation within a smallest square, is believed to be a particularly intuitive one.     

Multi-facility location problems in the presence of a probabilistic line barrier: a mixed integer quadratic programming model

We consider a multi-facility location problem in the presence of a line barrier with the starting point of the barrier uniformly distributed. The objective is to locate n new facilities among m existing facilities minimising the summation of the weighted expected rectilinear barrier distances of the locations of new facilities and new and existing facilities. The proposed problem is designed as a mixed-integer nonlinear programming model, conveniently transformed into a mixed-integer quadratic programming model. The computational results show that the LINGO 9.0 software package is effective in solving problems with small sizes. For large problems, we propose two meta-heuristic algorithms, namely the genetic algorithm and the imperialist competitive algorithm for optimisation. The numerical investigations illustrate the effectiveness of the proposed algorithms.     

Genetically assisted optimization of cell layout and material flow path skeleton

A continuous plane manufacturing cell layout and intercell flow path skeleton problem formulation involving rectilinear distances between cell input/output stations is mapped to a genetic search space. Certain properties of such a search space are exploited to design a very efficient method for reduction of a mixed-integer programming problem formulation to an iterative sequence of linear programming problems. This paper reports theoretical and computational insights for efficiently finding good solutions for the above problem formulation, taking advantage of the solution structure and the search stage. The scores of the objective function on a set of test cases indicate better solutions than those previously reported in the literature. The empirical results based on multiple runs also suggest that the method generates final results that are not dependent on the quality of the initial solution; hence the solution search seems to be more global than many of the previous approaches.     

A Genco self-scheduling problem with correlated prices using a new robust optimization approach

In this research, a self-scheduling problem for a power generation company (Genco), participating in a day-ahead power market is studied. A robust optimisation approach is followed to tackle uncertainty on the market prices. Due to the existing correlations among hourly market prices and in order to enhance the value of the objective function in an uncertain environment, a new robust optimisation approach is developed and presented to prevent over-conservative solutions. A couple of polyhedral uncertainty sets are applied to protect the optimal solution solely against any correlated perturbation. In addition two robust self-scheduling models are formulated under these uncertainty sets. The results of this study justify the performance of the proposed models compared to those of the existing robust self-scheduling model applied for conventional polyhedral uncertainty set.     

An improved facility layout construction method

This study presents a new distance-based facility layout construction technique. Given a two-dimensional (i.e. single-floor) facility layout construction problem in which the order of placement of individual departments is known (a challenging problem in itself), the technique presented herein proposes the use of sub-departments and expected distance functions instead of centroid-to-centroid distances for the placement of departments. In this paper an expected distance function is defined as the probabilistic expectation of the particular distance metric of interest (rectilinear, Euclidean, etc.) in which the parameters involved are defined by random vectors in 2-dimensional Euclidean space. This study presents an enhanced facility layout construction technique that incorporates several enhancements over the well-known systematic layout procedure (SLP). The goal herein is to minimise the error induced by the use of the centroid-to-centroid distances between the departments inherent to the SLP.     

Dominant graphs for rectilinear network design with barriers

Given a set of points on a Cartesian plane and the coordinate axes, the rectilinear network design problem is to find a network, with arcs parallel to either one of the axes, that minimizes the fixed and the variable costs of interactions between a specified set of pairs of points. We show that, even in the presence of arbitrary barriers, an optimal solution to the problem (when feasible) is contained in a grid graph defined by the set of given points and the barriers. This converts the spatial problem to a combinatorial problem. Finally we show connections between the rectilinear network design problem and a number of well-known problems. Thus this paper unifies the known dominating set results for these problems and extends the results to the case with barriers.     

A Network Flow Solution to a Rectilinear Distance Facility Location Problem

The problem of locating new facilities is considered with respect to existing facilities so as to minimize a sum of costs which consists of costs proportional to the rectilinear distances between new and existing facilities, and costs proportional to the rectilinear distances among new facilities. The location problem decomposes into two independent sub-problems, each of which is equivalent to a linear programming problem which is essentially the dual of a minimal cost network flow problem. Fulkerson's out-of-kilter algorithm provides an efficient means of solving each of the network flow problems as well as the location problem. The dual variables in each of the optimum tableaus to the two flow problems give the x and y coordinates respectively of the optimum locations of the new facilities. Several alternative approaches to solving the equivalent linear programming problems are also discussed, and some research questions are identified.     

Fourier series solution for a rectangular thick plate with free edges on an elastic foundation

A Fourier series solution is presented for a system of first-order partial differential equations which describe the linear elastic behaviour of a thick rectangular plate resting on an elastic foundation and carrying an arbitrary transverse load. The lateral edges of the plate are unstressed. A central step in the method for solving the system of equations is to combine a complementary function with a particular solution of the system in order to satisfy the boundary conditions. The complementary function is the sum of two series. The terms of the first series are products of a Fourier term in one space variable with the solution of an eigenvalue problem in the other space variable. The second series is similar and comes from reversing the roles of the space variables.     

Sliding contact of a punch with elastic wedge

The problem of contact interaction between a punch with rectilinear base and an elastic wedge with one fixed face is studied with regard for the friction in the contact zone. By using the Wiener–Hopf method, we obtain the analytic solution of the problem. The results of evaluation of the contact stresses and displacements of points of the free face of the wedge are presented.     

Computing the survival probability density function in jump-diffusion models: A new approach based on radial basis functions

We propose a numerical method to compute the survival (first-passage) probability density function in jump-diffusion models. This function is obtained by numerical approximation of the associated Fokker–Planck partial integro-differential equation, with suitable boundary conditions and delta initial condition. In order to obtain an accurate numerical solution, the singularity of the Dirac delta function is removed using a change of variables based on the fundamental solution of the pure diffusion model. This approach allows to transform the original problem to a regular problem, which is solved using a radial basis functions (RBFs) meshless collocation method. In particular the RBFs approximation is carried out in conjunction with a suitable change of variables, which allows to use radial basis functions with equally spaced centers and at the same time to obtain a sharp resolution of the gradients of the survival probability density function near the barrier. Numerical experiments are presented in which several different kinds of radial basis functions are employed. The results obtained reveal that the numerical method proposed is extremely accurate and fast, and performs significantly better than a conventional finite difference approach.     

Dyadic Green's function of a sphere with an eccentric spherical inclusion

An exact, analytical solution to the problem of point-source radiation in the presence of a sphere with an eccentric spherical inclusion has been obtained by combined use of the dyadic Green's function formalism and the indirect mode-matching technique. The end result of the analysis is a set of linear equations for the vector wave amplitudes of the electric Green's dyad. The point source can be anywhere, even within the aforesaid nonspherical body, and there is no restriction with regard to the electrical properties in any part of space. Several checks confirm that this solution obeys the energy conservation and reciprocity principles. Numerical results are presented for an electric Hertz dipole radiating from within an acrylic sphere, which contains an eccentric spherical cavity.     

Pareto navigator for interactive nonlinear multiobjective optimization

We describe a new interactive learning-oriented method called Pareto navigator for nonlinear multiobjective optimization. In the method, first a polyhedral approximation of the Pareto optimal set is formed in the objective function space using a relatively small set of Pareto optimal solutions representing the Pareto optimal set. Then the decision maker can navigate around the polyhedral approximation and direct the search for promising regions where the most preferred solution could be located. In this way, the decision maker can learn about the interdependencies between the conflicting objectives and possibly adjust one’s preferences. Once an interesting region has been identified, the polyhedral approximation can be made more accurate in that region or the decision maker can ask for the closest counterpart in the actual Pareto optimal set. If desired, (s)he can continue with another interactive method from the solution obtained. Pareto navigator can be seen as a nonlinear extension of the linear Pareto race method. After the representative set of Pareto optimal solutions has been generated, Pareto navigator is computationally efficient because the computations are performed in the polyhedral approximation and for that reason function evaluations of the actual objective functions are not needed. Thus, the method is well suited especially for problems with computationally costly functions. Furthermore, thanks to the visualization technique used, the method is applicable also for problems with three or more objective functions, and in fact it is best suited for such problems. After introducing the method in more detail, we illustrate it and the underlying ideas with an example.     

Drag on a sphere accelerating rectilinearly in an elastico-viscous fluid

The exact solution for the drag on a sphere moving in an arbitrary manner along a rectilinear path in an otherwise-still elastico-viscous fluid of infinite extent is presented. The Fourier transform technique is used to derive the solution. This technique is based on the fact that drag on an accelerating body can be obtained by integrating the drag on an oscillating body over all possible frequencies. The solution is also expressed in terms of infinite series which is suitable for numerical evaluation of the drag. The solution for the drag on a sphere suddenly brought to uniform motion is presented as an example of this study.     

Meshless collocation method by Delta-shaped basis functions for default barrier model

In this paper we first approximate a ‘nearly singular’ function, which tends to be the Dirac-delta function, to high degree of accuracy by using a recently developed Delta-shaped basis function. The Hermite-based meshless collocation method based on radial basis functions is then applied to solve a default barrier model, which is a time-dependent boundary value problem with a singularity at the initial condition. For numerical verification on the accuracy and efficiency of the newly proposed method, we compare the results with an analytical solution of the default barrier model under an assumption on the affine boundary. Numerical results indicate that the proposed method has potential advantage to solve problems with Dirac-type singularities.     

LOCATING A NOXIOUS FACILITY WITH RESPECT TO SEVERAL POLYGONAL REGIONS USING ASYMMETRIC DISTANCES

This paper presents an interactive computer graphical method for locating a noxious facility that has nuisance costs which vary with the direction as well as with the distance from the facility. The objective is to minimize the maximum cost incurred within several polygonal, possibly non-convex, regions. The directionally dependent cost functions are based on the asymmetric distance functions of Drezner and Wesolowsky. The maximin location problem with rectilinear, Euclidean, or ℓ_p-distances can be formulated as a special case of this minimax problem.     

Drag on a sphere accelerating rectilinearly in a maxwell fluid

The exact solution for the drag on a sphere moving in an arbitrary manner along a rectilinear path in an otherwise still Maxwell fluid of infinite extent is presented. The method of solution is that used by Landau and Lifshitz. This technique is based on the fact that drag on an accelerating body can be obtained by integrating the drag on an oscillating body over all possible frequencies. The solutions for the drag for some particular type of accelerating motions are also presented. Discussion on the limits of applicability of the solution is also presented.