On an algebraic solution of the Rawls location problem in the plane with rectilinear metric

Krivulin, Nikolai; Plotnikov, P. V.

doi:10.3103/s1063454115020065

Cited by 6 publications

(7 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…An algebraic approach, which uses results of tropical optimization, is applied in [20,24] to the problem under consideration when all constraints are removed. The approach offers a direct, explicit solution based on a straightforward algebraic technique, rather than on geometric considerations in the classical works [36,35].…”

Section: Constrained Minimax Rectilinear Single-facility Location Promentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Furthermore, several constrained minimax location problems are examined by [14,15,16,17,18] in the context of the theory of max-separable functions, which is closely related to the tropical mathematics approach. Finally, methods of tropical optimization are applied to solve unconstrained and constrained minimax single-facility location problems with Chebyshev and rectilinear distances [19,20,21,22,23,24].The aim of this paper is twofold: first, to develop new applications of tropical optimization by solving new location problems, and second, to offer new closed-form solutions to rather general problems that are of interest to location analysis. We consider a constrained minimax single-facility location problem with addends on the plane with rectilinear distance, which can be referred to as a constrained Rawls location problem or a constrained messenger boy problem.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Using tropical optimization to solve constrained minimax single-facility location problems with rectilinear distance

Krivulin

2017

Comput Manag Sci

View full text Add to dashboard Cite

The aim of this paper is twofold: first, to extend the area of applications of tropical optimization by solving new constrained location problems, and second, to offer new closed-form solutions to general problems that are of interest to location analysis. We consider a constrained minimax single-facility location problem with addends on the plane with rectilinear distance. The solution commences with the representation of the problem in a standard form, and then in terms of tropical mathematics, as a constrained optimization problem. We use a transformation technique, which can act as a template to handle optimization problems in other application areas, and hence is of independent interest. To solve the constrained optimization problem, we apply methods and results of tropical optimization, which provide direct, explicit solutions. The results obtained serve to derive new solutions of the location problem, and of its special cases with reduced sets of constraints, in a closed form, ready for practical implementation and immediate computation. As illustrations, numerical solutions of example problems and their graphical representation are given. We conclude with an application of the results to optimal location of the central monitoring facility in an indoor video surveillance system in a multi-floor building environment.

show abstract

Section: Constrained Minimax Rectilinear Single-facility Location Promentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Using tropical optimization to solve constrained minimax single-facility location problems with rectilinear distance

Krivulin

2017

Comput Manag Sci

View full text Add to dashboard Cite

show abstract

“…A solution for the weighted problem with rectilinear distance is given in [6], which involves decomposition into independent one-dimensional subproblems solved by reducing to equivalent network flow problems. In [23,18,24,25,21], an approach based on idempotent algebra is applied to solve unweighted unconstrained and constrained location problems. Further results on both unweighted and weighted location problems can be found in the survey papers [11,1,28,2,3], as well as in the books [29,17,9,7,26].…”

Section: Introductionmentioning

confidence: 99%

Algebraic Solution of Weighted Minimax Single-Facility Constrained Location Problems

Krivulin

2018

Relational and Algebraic Methods in Computer Science

View full text Add to dashboard Cite

We consider location problems to find the optimal sites of placement of a new facility, which minimize the maximum weighted Chebyshev or rectilinear distance to existing facilities under constraints on a feasible location domain. We examine Chebyshev location problems in multidimensional space to represent and solve the problems in the framework of tropical (idempotent) algebra, which deals with the theory and applications of semirings and semifields with idempotent addition. The solution approach involves formulating the problem as a tropical optimization problem, introducing a parameter that represents the minimum value of the objective function in the problem, and reducing the problem to a system of parametrized inequalities. The necessary and sufficient conditions for the existence of a solution to the system serve to evaluate the minimum, whereas all corresponding solutions of the system present a complete solution of the optimization problem. With this approach, we obtain direct, exact solutions represented in a compact closed form, which is appropriate for further analysis and straightforward computations with polynomial time complexity. The solutions of the Chebyshev problems are then used to solve location problems with rectilinear distance in the two-dimensional plane. The obtained solutions extend previous results on the Chebyshev and rectilinear location problems without weights and with less general constraints.Key-Words: tropical mathematics, idempotent semifield, constrained optimization problem, single-facility location problem, Chebyshev and rectilinear distances MSC (2010): 65K10, 15A80, 90B85, 90C47, 90C48

show abstract

“…Подход на основе методов тропической математики находит свое применение при решении задач в различных областях науки и техники, включая задачи принятия решений при анализе результатов оценки альтернатив на основе парных сравнений [20], и задачи планирования сроков выполнения проектов [21]. На основе этого подхода в работах [22][23][24][25][26] получены решения минимаксных задач размещения точечного объекта на плоскости с прямоугольной и чебышевской метрикой без ограничений и с ограничениями на допустимую область размещения.…”

Section: Introductionunclassified

Solution of minimax location problem in three-dimensional space with rectilinear metric

Krivulin¹,

Plotnikov²

2018

CTE

View full text Add to dashboard Cite

Рассматривается минимаксная задача размещения точечного объекта в трехмерном пространстве с прямоугольной метрикой (l 1-метрикой) и предлагается ее прямое аналитическое решение при помощи методов тропической (идемпотентной) математики. Сначала задача записывается в терминах тропической математики как задача тропической оптимизации, вводится параметр для обозначения минимума целевой функции, и задача сводится к решению параметризованной системы неравенств. Эта система решается относительно одной из переменных, а условия существования решений используются для нахождения оптимальных значений второй переменной с помощью вспомогательной задачи оптимизации. Затем вспомогательная задача решается аналогичным образом, и находится значение третьей переменной. Полученное общее решение преобразуется в набор прямых решений, записанных в компактной форме для различных случаев соотношений между исходными параметрами задачи. Ключевые слова: задача 1-центра, трехмерное пространство, прямоугольная метрика, идемпотентное полуполе, тропическая оптимизация, полное решение. Цитирование: Плотников П. В., Кривулин Н. К. Решение минимаксной задачи размещения в трехмерном пространстве с прямоугольной метрикой // Компьютерные инструменты в образовании. 2018. № 1. С. 31-50. * Работа выполнена при финансовой поддержке Российского фонда фундаментальных исследований (грант № 18-010-00723).

show abstract

On an algebraic solution of the Rawls location problem in the plane with rectilinear metric

Cited by 6 publications

References 11 publications

Using tropical optimization to solve constrained minimax single-facility location problems with rectilinear distance

Using tropical optimization to solve constrained minimax single-facility location problems with rectilinear distance

Algebraic Solution of Weighted Minimax Single-Facility Constrained Location Problems

Solution of minimax location problem in three-dimensional space with rectilinear metric

Contact Info

Product

Resources

About