No abstract

No abstract

An optimal placement problem for special-form objects on a multiply connected domain is considered. The objects must be disjoint and belong to a convex region that contains forbidden areas. The performance criterion is described by a function of maximum of differentiable functions. The method of feasible directions and the method of directed transition were adapted to solve this problem.Keywords: optimal placement, method of feasible directions, method of directed transition, minimax problem.Optimal placement problems arise in designing various devices, freight activity, structural engineering, etc. To solve such problems, the mathematics of the theory of geometrical design is used. Mathematical models for practical placement problems are characterized by the following features that complicate their solution: large dimension, multiconnectivity, and disconnectedness of the set of feasible solutions.Problem Formulation. We consider here a minimax optimal placement problem in space R n , n = 2 3 , , on a multiply connected domain. Each object can be partitioned into components: mutually oriented rectangles (or parallelepipeds in R 3 ) whose sides (edges) are parallel to the coordinate axes. The domain is specified as a convex domain that has forbidden areas in the form of rectangles (parallelepipeds). The placement performance criterion can be described by a function of maximum of differentiable functions. A layout is feasible if all objects are pairwise disjoint and within the given domain. It is necessary to find a layout, among feasible ones, that would minimize the objective function.Literature Survey. In [1], various optimal placement problems for geometrical objects are considered. Mathematical models of packing problems, including problems of packing parallelepipeds in a parallelepiped with forbidden areas are presented. To solve the problem, an approach is proposed that employs the idea of sequential single placements. In [2], the branch and bound algorithm is applied to find the exact solution of a packing problem; it is expedient to use this algorithm to place a small number of objects (about 10). In [3], packing problems for rectangles on a convex domain are considered, including a problem of arranging a maximum number of identical rectangles on a given domain. Disjointness conditions for the objects are written as a differentiable function. The paper [4] is concerned with the optimal placement problem for rectangles in a rectangular domain with a differentiable objective function. The nonconvex multiply connected set of feasible solutions of the problem is represented by the union of convex subsets. The problem is solved by solving a sequence of subproblems on the subsets obtained. To solve the subproblems, the conditional gradient method is used. In [5], the packing problem for oriented parallelepipeds in a parallelepiped is considered as a problem of inverse-convex programming. A mathematical theory is proposed to solve optimal placement problems with a convex performance criterion. In [6], a minimax o...

The location of physical field sources in fixed places is optimized. Every source should be put in one place. The sources should not overlap and go beyond the location areas. The location quality criterion is defined by maximum values of the physical field at specified points. An algorithm based on the method of potentials is proposed to solve the problem and is analyzed for efficiency.The assignment problem was analyzed in many studies. The paper [1] deals with a two-level assignment problem to be solved with the branch-and-bound algorithm. The paper [2] generalizes the minimax assignment problem and proposes a solution algorithm of pseudo-polynomial complexity. The study [3] finds nonintersecting and noncoinciding cycles on a network with two weights of arcs and describes the main stages of the polynomial algorithm to solve the problem in the general case. To increase the performance of the programs (for example, those solving assignment problems), the paper [4] presents an efficient implementation of tree structures, which considerably reduces the run time.We will consider a special minimax assignment problem, which arises in arranging sources of a physical field [5]. The authors are unaware of algorithms to solve it. We propose an algorithm for approximate solution. Numerical results for a specific problem will be presented in the Appendix.

No abstract