Mathematical Model and Algorithm of Branch and Boundary Method for Optimizing Solutions for Package Compositions in Multi-Stage Systems
Modern methods for solving problems of planning of task packages execution in multi-stage systems are characterized by the presence of restrictions on their dimension, the impossibility of obtaining guaranteed best results in comparison with fixed packages for different values of the input parameters of tasks. The problem of optimizing the composition of task packages executed in multi-stage systems using the method of branches and borders is solved in the paper. Studies of various ways of forming the order of execution of task packages in multi-stage systems (heuristic rules for ordering task packages in the sequences of their execution on MS devices) have been carried out. The method of ordering packets in the sequence of their execution (a heuristic rule), which minimizes the total time for implementing actions with them on the devices, is defined. The method of ordering the types of tasks, according to which their packages are considered in the procedure of the method of branches and borders, is formulated on the basis of the obtained rule. A mathematical model of the process of implementing actions with packages on the system devices, which provides the calculation of its parameters, has been built. The construction of a method for forming all possible solutions for the composition of task packages for a given number of them has been completed. Decisions on the composition of task packages of different types are interpreted in the procedure of the method of branches and borders in order to build the optimal combination of them. To implement the method of branches and borders, a branching (splitting) procedure is formulated, which assumes the formation of subsets of solutions that include packages of different compositions of tasks of the same type. Expressions for calculating the lower and upper estimates of the values of the optimization criterion for the composition of packages for subsets formed in the branching procedure are constructed. The dropout procedure involves the exclusion of subsets whose lower estimate is not less than the record. To find optimal solutions, a breadth-first search strategy is applied, which provides for the study of all subsets of solutions that include various packages of tasks of the same type obtained as a result of the procedure for splitting subsets of tasks that are not excluded from consideration after the implementation of the dropout procedure. The developed algorithms are implemented programmatically, which allowed to obtain the results of planning the execution of task packages in a multi-stage system, which are on average 30 % better than fixed packages.