MATHEMATICAL MODELLING OF COMBAT OPERATIONS WITH THE POSSIBILITY OF REDISTRIBUTING COMBAT RESOURCES BETWEEN THE AREAS OF CONTACT AND DISTRIBUTING RESERVES

Abstract

Context. Mathematical and computer models of the dynamics of combat operations are an important tool for predicting their outcome. The known Lanchester-type models were simulation models and did not take into account the ultimate goal and redistribution of resources during combat operations. This paper proposes an optimisation model of the dynamics of combat operations between parties A and B in two areas of collision, based on the method of dynamic programming with maximisation of the objective function as a function of enemy losses. The article develops a mathematical and computer model of a typical situation in modern warfare of combat operations between parties A and B in two areas of collision with the aim of inflicting maximum losses of combat resources on the enemy. This goal is achieved by redistributing resources between the areas of collision and introducing appropriate reserves to these areas.
Objective. To build a mathematical and computer model of the dynamics of combat operations between parties A and B in two areas of collision, in which the goal of party A is to maximise the losses of party B by using three resources (the first is the number of combat units that party A can distribute across the areas of collision at the initial moment of time; the second is the number of combat units that party A must transfer from one area to another at some subsequent moment of time; the third is the number of combat units that party A must distribute using the reserve) and by modelling the
Method. The mathematical model is based on the method of dynamic programming with the objective function as a function of enemy losses, and the parameters are units of combat resources in different areas of the clash. Their number is changed by redistributing them between these areas and introducing reserve combat units. The enemy’s losses are determined using Lanchester’s systems of differential equations. Given the complexity of the objective function, the Python programming language is used to find its maximum.
Results. A mathematical model of the problem has been constructed and implemented, based on a combination of the dynamic programming method with the solution of Lanchester’s systems of differential equations of battle dynamics with certain initial conditions at each of the three stages of the battle. With the help of a numerical experiment, the admissibility of the parameters of the optimisation problem (the number of combat units of side A, which are appropriately distributed, transferred from area to area or from the reserve at each stage of the battle) is analysed. The developed Python program allows, for any initial data, to give an answer to the optimal allocation of resources of party A, including from the reserve, at three stages of the battle and to calculate the corresponding largest enemy losses at a given time or to give an answer that there are no valid values of the problem parameters, i.e. the problem has no solution for certain initial data.
Conclusions. The scientific novelty lies in the development of mathematical and computer models of the dynamics of combat in two areas of collision, which takes into account the redistribution of combat resources and reserves in order to inflict maximum losses on the enemy. Numerical modelling made it possible to analyse the admissibility of redistribution and reserve parameters. Based on the examples considered, it is concluded that if the problem is unsolvable with certain data, it means that it is necessary to reduce the time of redeployment of combat units at one or more stages of the battle, i.e. to reduce the duration of the battle at a certain stage, thus allowing to predict the time of redeployment of combat resources.