Solving nonconvex planar location problems by nite dominating sets

It is well-known that some of the classical location problems with polyhedral gauges can be solved in polynomial time by nding a fi nite dominating set, i.e. a finite set of candidates guaranteed to contain at least one optimal location. In this paper it is fi rst established that this result holds for a much larger class of problems than currently considered in the literature. The model for which this result can be proven includes, for instance, location problems with attraction and repulsion, and location-allocation problems. Next, it is shown that the approximation of general gauges by polyhedral ones in the objective function of our general model can be analyzed with regard to the subsequent error in the optimal ob jective value. For the approximation problem two di erent approaches are described, the sandwich procedure and the greedy algorithm. Both of these approaches lead - for fixed e - to polynomial approximation algorithms with accuracy for solving the general model considered in this paper.Dirección General de Enseñanza Superio

The problem of placing a two-stage production with restrictions on the capacity of the first stage enterprises

Задачи оптимального размещения предприятий - благодатная почва для разработки новых методов моделирования, инновационных алгоритмов решения и интересных применений. В статье описывается задача размещения двухэтапного производства с ограничениями на мощности предприятий первого этапа. Такие задачи возникают, например, при стратегическом планировании развития региона, решении задач оптимального размещении предприятий и определении зон их влияния, и представляют практический интерес для коммерческих (размещение складов, магазинов, точек обслуживания и пр.) и государственных (школы, больницы, пожарные станции и пр.) компаний. Целью работы является построение математической модели двухэтапной задачи оптимального размещения-распределения при наличии ограничений на мощность предприятий первого этапа, краткое описание метода ее решения и формулирование алгоритма решения. В качестве критерия оптимальности решения задачи размещения была выбрана совокупная стоимость доставки продукта. Методы решения основаны на принципах бесконечномерной оптимизации и теории двойственности. Подход к решению такой задачи основан на решении задачи оптимального разбиения множеств и дискретной многоэтапной задачи размещения. Единый подход к решению задач оптимального разбиения множеств заключается в преобразовании исходных задач в задачи бесконечномерного математического программирования с помощью характеристических функций, а затем в конечномерную задачу оптимизации с использованием функционала Лагранжа. Разработан итерационный алгоритм решения задачи. Он объединяет метод потенциалов, применяемый для классической задачи линейного программирования транспортного типа и алгоритм Н.З. Шора, позволяющий решить задачу оптимизации негладкой функции. Был разработан программный продукт для решения двухэтапных задач оптимального размещения предприятий с непрерывно распределенным ресурсом. Результаты, полученные авторами, позволяют решать ряд практических задач, связанных со стратегическим планированием в сфере производственной и социально-экономической деятельности.Facility location problems are a fertile ground for the development of new modelling techniques, innovative solution algorithms and exciting applications. The article describes a problem of placing a two-stage production with restrictions on the capacity of the first stage enterprises. Such problems arise, for example, in strategic planning of development of the region. The problems of optimal placement companies and identifying areas of their influence are interesting for business (allocation of warehouses, shops, service outlets, etc.) and public companies (schools, hospitals, fire stations and so forth.) The tasks of research were to construct the mathematical model for a two-stage of optimal location-allocation problem considering the restrictions on the capacity of enterprises of the first stage; to describe the solution method and to formulate an algorithm. In this paper, authors gave the mathematical model of the problem, a brief description of the solution method and algorithm. Aggregate cost of product de livery was chosen as a criterion for optimal location problem. Solving methods are based on principles of infinite-dimensional optimization and duality theory. The approach to solution of this type of problem is based on the solution of the problem of optimal partitioning set and discrete multi-stage location problem. A unified approach to solving optimal partitioning set problems lies in the conversion of the initial problems into infinite-dimensional mathematical programming problems by means of the characteristic functions, and then into the finite optimization problem using Lagrangian functional. Iterative algorithm to solve the problem has been elab orated. The algorithm combines a method of potentials being applied for classical problem of linear programming of transportation type and N.Z. Shor’s algorithm making it possible to solve optimization problem of nonsmooth function. Software product has been developed to solve two-stage problems of optimal location of enterprises with continuously resources. The results obtained by authors make it possible to solve a range of practical problems connected with strategic planning in the sphere of production, social and economic activities

Genetic algorithm approach in solving minisum facility location problem with fixed line barrier

Facility location problem is a field of study in Operational Research that required in considering locating a facility or a set of new facilities on the plane to serve a finite set of existing demand points. A facility location problem usually formulated as a minimization or maximization problem with an objective function involving distances between the facility and demand points. Generally, facility location problems can be classified into several problems. However in this study, minisum facility location problem involving fixed line barrier is considered since line barrier is the most applicable one in real life problem. This is because the line barrier such as rivers, highways, borders or mountain ranges are frequently encountered in practice or real problem. The main objective of this study is to concentrate on solving the minisum facility location problem with fixed line barrier using meta-heuristic approach namely as Genetic Algorithm (GA). The basic concepts of facility location with barrier as well as formulation of the problem are also had been discussed in this study. Subsequently, the developed genetic algorithm for solving the problem is proposed in this study. The procedure is coded using C++ programming and implemented on generated data of 50 fixed points

Simulation and Optimization Of Ant Colony Optimization Algorithm For The Stochiastic Uncapacitated Location-Allocation Problem

This study proposes a novel methodology towards using ant colony optimization (ACO) with stochastic demand. In particular, an optimizationsimulation-optimization approach is used to solve the Stochastic uncapacitated location-allocation problem with an unknown number of facilities, and an objective of minimizing the fixed and transportation costs. ACO is modeled using discrete event simulation to capture the randomness of customers’ demand, and its objective is to optimize the costs. On the other hand, the simulated ACO’s parameters are also optimized to guarantee superior solutions. This approach’s performance is evaluated by comparing its solutions to the ones obtained using deterministic data. The results show that simulation was able to identify better facility allocations where the deterministic solutions would have been inadequate due to the real randomness of customers’ demands

Geometric-based Optimization Algorithms for Cable Routing and Branching in Cluttered Environments

The need for designing lighter and more compact systems often leaves limited space for planning routes for the connectors that enable interactions among the system’s components. Finding optimal routes for these connectors in a densely populated environment left behind at the detail design stage has been a challenging problem for decades. A variety of deterministic as well as heuristic methods has been developed to address different instances of this problem. While the focus of the deterministic methods is primarily on the optimality of the final solution, the heuristics offer acceptable solutions, especially for such problems, in a reasonable amount of time without guaranteeing to find optimal solutions. This study is an attempt to furthering the efforts in deterministic optimization methods to tackle the routing problem in two and three dimensions by focusing on the optimality of final solutions. The objective of this research is twofold. First, a mathematical framework is proposed for the optimization of the layout of wiring connectors in planar cluttered environments. The problem looks at finding the optimal tree network that spans multiple components to be connected with the aim of minimizing the overall length of the connectors while maximizing their common length (for maintainability and traceability of connectors). The optimization problem is formulated as a bi-objective problem and two solution methods are proposed: (1) to solve for the optimal locations of a known number of breakouts (where the connectors branch out) using mixed-binary optimization and visibility notion and (2) to find the minimum length tree that spans multiple components of the system and generates the optimal layout using the previously-developed convex hull based routing. The computational performance of these methods in solving a variety of problems is further evaluated. Second, the problem of finding the shortest route connecting two given nodes in a 3D cluttered environment is considered and addressed through deterministically generating a graphical representation of the collision-free space and searching for the shortest path on the found graph. The method is tested on sample workspaces with scattered convex polyhedra and its computational performance is evaluated. The work demonstrates the NP-hardness aspect of the problem which becomes quickly intractable as added components or increase in facets are considered