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.

我们考虑位置问题，以找到新设施的最佳放置位置，从而在可行的位置域约束下将与现有设施的最大加权Chebyshev或直线距离最小化。我们研究了多维空间中的切比雪夫位置问题，以表示和解决热带（幂等）代数框架中的问题，该代数讨论了幂等加法的半环和半场的理论和应用。解决方案包括将问题表述为热带优化问题，引入代表问题中目标函数最小值的参数，并将问题简化为参数化不等式系统。存在系统解决方案的必要和充分条件有助于评估最小值，而系统的所有相应解决方案都代表了优化问题的完整解决方案。通过这种方法，我们获得了紧凑紧凑形式表示的直接，精确解，适用于进一步分析和具有多项式时间复杂度的直接计算。然后，将切比雪夫（Chebyshev）问题的解用于解决二维平面中直线距离的位置问题。所获得的解扩展了先前在Chebyshev和直线位置问题上的结果，而没有权重且约束较少。以紧凑的封闭形式表示的精确解，适用于进一步分析和具有多项式时间复杂度的直接计算。然后，将切比雪夫（Chebyshev）问题的解用于解决二维平面中直线距离的位置问题。所获得的解决方案扩展了先前在切比雪夫（Chebyshev）和直线位置问题上的结果，而没有权重且约束较少。以紧凑的封闭形式表示的精确解，适用于进一步分析和具有多项式时间复杂度的直接计算。然后，将切比雪夫（Chebyshev）问题的解用于解决二维平面中直线距离的位置问题。所获得的解扩展了先前在Chebyshev和直线位置问题上的结果，而没有权重且约束较少。