Models of scheduling synchronized movement of many objects

Abstract

The paper deals with the problem of determining movement schedule of many objects, used in many domains such as: routing in computer networks, movement planning of mobile robots, tasks processing in parallel or distributed computing systems, arms control of independent robots, planning and synchronization of the movement of many objects in computer simulation games (e.g., in Computer Generated Forces (CGF) systems or Semi-Automated Forces (SAF) systems). A lot of movement scheduling models are discussed. Two groups of criteria which are essential from the point of view of schedule estimation are described: a group connected with movement time of all objects and a group connected with “parallelization” of their movement (in the sense of location and times of reaching specified checkpoints). A nonlinear movement scheduling problem in order to minimize the sum of delays of all objects at checkpoints with some additional constraints is defined. Two equivalent formulations of two-criteria mathematical programming problems are also presented. It is proved that constraint coefficient matrices for both problems are totally unimodular and we can use effective algorithms for solving linear programming problems to find lexicographic solution of two-criteria problems. Similarities and differences between the defined problem and classical tasks scheduling problem before critical lines on parallel processors are discussed. Some extensions of the problem are presented, one of which is the scheduling movement problem of many objects according to a group pattern. Methods of solving formulated problems are indicated.movement scheduling and synchronization, shortest paths, disjoint paths, multicriteria shortest paths problems