Audit Of Mathematical Models For Software Specification Of The Workplace Decision Support System At The Logistics Management Point Roman Litvinchuk 1, Andrii Levchenko 2 1 Military Academy (Odesa), Fontanskaya Road 10, Odesa, 65009 2 Odesa Mechnikov National University, Dvorianska, 2, Odesa, 65026, Ukraine Abstract The article analyzes the process of functioning of the system of technical support of combat operations in order to determine it`s capabilities and areas for improvement through for solving problems in modern local wars with the restriction of the use of heavy armored vehicles through application of the models of states and transitions. In addition, the possibility of creating a mathematical basis for the management of maintenance and restoration of lightly armored vehicles for software implementation of the workplace of a logistics officer on evacuation management and lightly armored vehicles recovery, which will not only explore real support systems, but also solve complex problems of technical support of combat operations in real time - on the battlefield. Keywords 1 technical support, armament and military equipment, lightly armored vehicles, intensity, probability, graph, Kolmogorov's equation 1. Introduction artillery on the line of contact. Therefore, the main armored vehicles are light armored vehicles such as (BMP, armored personnel carrier, BBM). Formulation of the problem. It is known that many factors that affect the Analysis of the models of the main states of the management of evacuation and recovery of technical support system, which were used before arming and AMM hostilities are accompanied by beginning of hostilities in the anti-terrorist uncertainties of random, natural and antagonistic operation area, shows that the task of managing nature. the evacuation and recovery of arming and An appropriate way out of this situation is to military machinery (AMM) during hostilities is reduce the dimensionality of the analysis problem more difficult in terms of preconditions and initial by comparing it in order to rank the types of data for planning than known it`s solutions. [1-4]. technical support tasks according to some general Based on the combat experience of servicemen indicator. As such an indicator can be used the of the Ukrainian Armed Forces and other military probability stay of the evacuation management formations that directly participated in repelling system and the restoration of AMM in each state the Russian armed aggression, it is well known of solving the tasks of combat operations. The that Ukraine clearly complies with the choice of the solution of the problem is possible requirements of the Minsk agreements and does by averaging the optimal partial solutions over not use heavy armored vehicles such as tanks and III International Scientific And Practical Conference “Information Security And Information Technologies”, September 13–19, 2021, Odesa, Ukraine EMAIL: Litvinsanr@gmail.com (A. 1); katyaandreylev@gmail.com (A. 2). ORCID: 0000-0002-5681-691 (A. 1); 0000-0003-4423-8267 (A. 2) ©️ 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings (CEUR-WS.org) time on the highest level of probability. It is the explains the choice of the state model for the correspondence of the probability models to the software specification of the decision support realities that requires further research. [5]. system of the logistics officer's workplace for The purpose of the article. evacuation management and light armored Currently, Light Armored Vehicles (LAV) of vehicles recovery at the logistics management all-military units are the most common type of point. This is the system of technical support of military equipment in the armed forces. Also, warfare. A variant of the graph of states and such equipment is most often affected and fails, transitions of this system to different states is and therefore requires constant correction of presented in Figure 1. planned activities of managing the evacuation and recovery of AMM in real time. Systematic shelling of the positions of the Ukrainian Armed Forces in the area of anti- terrorist operation (ATO) leads to the decommissioning of those samples of armaments and military equipment that are located directly at the bases on the line of the collision of the parties. In the context of integration of logistics management as a mechanism for providing and managing evacuation and recovery and subsequent repair of lost samples of armaments and military equipment, it turned out that mathematical models of logistics management and operation, as well as software based on them Figure 1. Graph of transitions of the do not meet the requirements of real-time decision technical system of combat operations in the support. states: S п – preparation of LAV for use; S з – The purpose of the article is to provide a mathematical justification for the management of combat use of LAV; S в – restoration of LAV after evacuation and recovery of LAV for software its damage; S о – maintenance of LAV before or implementation of the workplace logistic officer for evacuation management and recovery of LAV after combat actions. at the logistics management point, which will not only explore real support systems, but also solve The list of transition intensities and the complex problems of technical support of combat corresponding probabilities of these transitions is operations in real time, including in the as follows: battlefield. a, A – intensity and probability of transitions of the technical support system from the state of preparation of LAV for its maintenance; 2. Analysis of the model of the main b, B – intensity and probability of transitions states of the technical system of from the state of LAV maintenance to the state of combat operations (Conceptual its combat use; c, C – intensity and probability of transitions Modeling) from the state of combat use of the LAV to the state of its maintenance; To study the process of technical support, d, D – intensity and probability of transitions various types of technical support models are from the state of maintenance of LAV to the state currently used. If the models adequately reflect all of recovery of LAV after damage; the states of the system, it is better to use model of e, E – intensity and probability of transitions states and transitions [5]. from the state of preparation of LAV to the state The adequacy of the model for processes of recovery of LAV after damage; without aftereffect is explained by the fact that it f, F - intensity and probability of transitions most accurately reflects the system, in the case from the state of preparation of LAV to the state when any of its current state does not depend on of its employment; the state in which the system was before. It is the g, G – intensity and probability of transitions identity of the model to the real processes that from the state of combat use of LAV to the state of preparation of LAV for the purpose of their the conditional probability of transition over time employment; t into the state S п : Pз ( t ) 31t . h, H – intensity and probability of transitions Applying the rule of adding probabilities, we from the state of recovery of LAV after damage to obtain: the state of its combat use; i, I – intensity and probability of transitions P1 ( t + t ) = P1 ( t )(1 − 12 t ) + P3 ( t )  31t (1) from the state of combat use of LAV to the state Open the brackets on the right side, move of recovery of LAV after its damage. P1( t ) to the left and divide both parts of the It is also easy to imagine a situation where it is equation by t ; we will get: necessary to perform maintenance of LAV after P1 ( t + t ) − P1 ( t ) its preparation for use, or after its combat = 12 P1 ( t ) +  31 P3 ( t ) (2) application, as well as a situation when combat t use of LAV has shown the need for new training Now direct t to zero and go to the limit: for deployment, for example, taking into account P ( t + t ) − P1 ( t ) unsatisfactory combat results due to insufficiently lim 1 = 12 P1 ( t ) +  31 P3 ( t ) (3) t → 0 t careful preliminary preparation. The left part is nothing but a derivative of the In the process of functioning of the system of function P1( t ) : technical support of combat operations in time, it P1 ( t ) is in any state with probabilities: = −12 P1 ( t ) +  31 P3 ( t ) (4) P1( t ) - probability that the system is in a state t of preparation of weapons and ammunition for Thus, the differential equation obtained by their use; the function P1( t ) . Similar differential equations P2 ( t ) - the probability that the system is in a can be derived for other probabilities of states state of use of weapons for their intended purpose; P2 ( t ) , P3 ( t ) , P4 ( t ) , which provides initial data P3 ( t ) - the probability that the system is in a for the search of computational methods for state of recovery after damage; solving problems that replace theoretical models P4 ( t ) - the probability that the system is in a in the form of differential equations. state of maintenance. Consider the second state S з . Find the Find the probability P1( t ) . We provide t probability that at the moment small increase t and find the probability that at t + t the system will be in a state S з . This event the moment t + t the system will be in a state can occur in two ways: S п . This event can happen in two ways: - at the moment t the system was already in condition S з , by the time t did not come out - at the moment t the system was already in of this state; condition S п , but by the time t did not come or out of this state, either - at the moment t system was in condition S п ; - at the moment t the system was in the state S в , by the time t moved from it to the state S п by the time t moved from it to the state S з ; The probability of the first variant is shown as or the product of the probability P1( t ) that at the - at the moment t system was in condition S о , moment t the system was in the state S п , on the by the time t moved from it to the state S з . conditional probability that, being in a state S п , The probability of the first option is calculated system by the time t will not pass from it into a as follows: P2 ( t ) multiplied by the conditional state S з . probability that the system over time t will not This conditional probability (up to pass either S в , nor in S о . Since the events that infinitesimal higher orders of magnitude) is equal are the transition over time t into S в and from to: 1 − 12 t S з into S о , are incompatible, the probability that Similarly, the probability of the second option one of these transitions will occur is equal to the is equal to the probability of that at the moment t sum of their probabilities, to wit  23 t +  24 t system was at the state S в , which is multiplied by (up to infinitesimal higher orders). The probability that none of these transitions will occur is equal 1 − (  23t +  24 t ) . Hence the requirements for productivity and speed of the probability of the first option: hardware components of the decision support system. P2 ( t ) (  23 t +  24 t ) . Let's pay attention to the structure of equations Adding here the probabilities of the second and (7). They are all built on a general rule that can be third options, we obtain: formulated as follows. In the left part of each P2 ( t + t ) = P2 ( t ) (1 −  23t +  24 t ) + equation there is a derivative of the probability of (5) + P(1 t )12 t + P4 ( t ) 42 t the state, and the right part contains as many terms Moving P2 ( t ) to the left side, dividing by t as there are gaps connected with the given state.. and crossing to the limit, we obtain a differential If the gap leaves the state, the corresponding member has a sign "minus", and if the gap enters equation for P2 ( t ) : the state - the sign "plus". Each term is equal to P2 ( t ) the product of the intensity of the transition = − 23 P2 ( t ) −  24 P2 ( t ) + (6) t corresponding to a given gap and the probability +12 P(1 t ) +  P 42 4 ( t ) of the state from which the arc emerges. Reasoning similarly for states S в and S о , we If the matrix of transition intensities or the obtain as a result a system of differential equations state graph is known, the state probability vector composed by type (5), (6). Rejecting them for the can be determined P9 ( t ) = ( P1( t ),...,Pn ( t )) , sake of convenience argument t in functions P1 , through the matrix equation P( t ) = P( t )  . P2 , P3 , P4 rewrite the system in the form: From a practical point of view, to ensure the P1 combat effectiveness of the unit is important to = −12 P1 +  31 P3 , reduce the intensity and probability (g, G) its t P2 transition to the state ( S п ) preparation of LAV for = − 23 P2 −  24 P2 + 12 P1 −  42 P4 , t (7) the purpose of their application, and also increase P3 in intensity and probability (f, F) transition of the = − 31 P3 −  34 P3 +  23 P2 , t system to the state ( S з ) use of LAV for its P4 = − 42 P4 +  24 P2 +  34 P3 . intended purpose. This requires keeping the LAV t at a high level of its readiness factor, accelerated These equations for the probabilities of states and sufficient level of preparation of the LAV for are Kolmogorov's equations. the start of combat actions. The integration of this system of equations will It is necessary to significantly reduce the give the desired probabilities of states as a intensity and probability (i, I) transition of the function of time. The initial conditions are taken technical support system to the state ( S в ) depending on what was the initial state of the system. For example, if at the initial time recovery of LAV after damages, reduce the intensity and probability (e, E) transition of the (at t = 0 ) the system was in a state S п , then the system from the state ( S п ) preparation of LAV for initial conditions must be accepted: t = 0 , P1 = 1 , the purpose of their application in a condition P2 = P3 = P4 = 0 , which gives an understanding of ( S в ) recovery from damage, ie before the start of the universality of the model under study, in terms the use of LAV for its intended purpose. of its further use as an element of the software It is necessary to increase the intensity and specification of the workplace logistic officer for probability (h, H) transition of the system from evacuation management and recovery of light the state ( S в ) recovery of LAV after damage to armored vehicles decision support system at the logistics management point. the condition ( S з ) application for intended use. Note that all four equations for P1 ,P2 ,P3 ,P4 one The greatest attention is paid to the study of the could not write because P2 + P2 + P3 + P4 = 1 for condition ( S з ) use of LAV by purpose and all t , and any of the probabilities P1 ,P2 ,P3 ,P4 can condition ( S в ) recovery of LBT after damages is be expressed through the other three. For not accidental. This is due to the fact that these states of the technical system of combat example, P4 = 1 − P1 ,P2 ,P3 . operations are the most important in terms of the Then a special equation for P4 not necessary importance of the functions of the technical to write. In the future, this fact will reduce the support system, and the structure of unconditional are in the range from 0 to 1, respectively in the relations in this system. binary calculation system the number of It is safe to say in advance that, given the characters for the mantissa is 8 bits with the uncertainties of a random nature, namely, equally corresponding mantissa. intense and equally probable transitions of the According to the formulas (1-7) we will get: technical system of combat operations from any Pп ( t = 6...48 ) = 0,13...0,09 ; state to any other state, the total probability Pо ( t = 6...48 ) = 0,17...0,18 ; Pп + Pо = 0,30...0, 27 ( Pзв = Pз + Pв ) stay of this system in a condition . ( S з ) use of LAV on purpose and in condition ( S в Pз ( t = 6...48 ) = 0,59...0, 26 ; ) recovery of LAV after damage is always the Pв ( t = 6...48 ) = 0,11...0, 47 ; highest in comparison with other general probability, equal to the sum of the probability of Pз + Pв = 0 ,70...0 ,73 . the system in the state of preparation of LAV for Graphs of general probabilities in the form of their application and the probability of the system time functions during the process of technical in a state of maintenance, that is, with the total support of hostilities, obtained according to the probability Pоп = Pо + Pп . initial data example 1 and emphasize the validity Indeed, it is easy to see this in some arbitrary of the statement which was made earlier. but concrete example. Thus, in the system of technical support of combat actions there is a pattern, namely: under conditions of equally probable transitions of the 2.1. Verification and specification of system from state to state, it is in a state of the obtained models for their application or recovery more often further implementation in the (approximately three times) than in a state of maintenance or training. software of decision support It is clear that this result is not a new systems (Modeling verification & discovery. The problem is solved by a known method for the new initial conditions and the new validation) content of the problem of evacuation and recovery of armaments and military equipment. It only The first test example. confirms the peculiarity of the structure and the The initial prerequisites for modeling are essence of the functioning of a complex system of equally intense and equally likely transitions of technical support of combat actions. This is what the system of technical support of hostilities from is needed carefully and always consider. any state to any state, namely (check. Figure 1): a = b = c = d = e = f = g = i = h = 1/2 hours; A = B = C = D = E = F = G = I = H = 1/9; t = (6…48) hours. 1.0 Identify the general probabilities that need to 0,8 be quantified, namely: Pзв ( t ) = Pз ( t ) + Pв ( t ) ; 0,6 Pоп ( t ) = Pо ( t ) + Pп ( t ) , 0,4 where Pп ( t ) - the probability that the system 0,2 is in a state of preparation of weapons and t, годин ammunition for their use; 8 16 24 32 40 48 Pз ( t ) - the probability that the system is in a Figure 2. General probabilities of the technical state of use of weapons for their intended purpose; support system being in combat during states: Pв ( t ) - the probability that the system is in a Pзв ( t ) application or recovery, LAV; state of recovery after damage; Pо ( t ) - the probability that the system is in a maintenance or preparing, Pоп ( t ) LAV for combat actions. state of maintenance of weapons. The solution is carried out on machines for data packaging, provided that for the Next, it is necessary to investigate (for representation of numerical values that the conditions similar to the data according to decimal system of systematization of calculations Example 1) the dependence of the time of technical support of combat operations of each of known experimental results of real events of the probabilities, namely: Pп ( t ) - the probability typical support, according to local fight. Third, the of the system being in a state of preparation of weakest point of a typical unit's technical support LAV for the purpose of their application; Pз ( t ) - system is its ability to recover weapons and military equipment (AMM) damaged during the probability that the system is in a state of use combat. of LAV for its intended purpose; Pв ( t ) - the This situation necessitates further research on probability that the system is in a state of recovery the technical system of combat operations, in LAV after its damage; Pо ( t ) - the probability that order to identify measures to increase the system is in a state of maintenance LAV. opportunities for the restoration of AAM The second test example. damaged during combat. Output data. We have equally intense and The solution of the problem of evacuation and equally probable transitions of the system of recovery using the model of states and transitions technical support of combat actions from any state confirms the adequacy of the model with real to any of its states, namely (check. Ошибка! measures of evacuation and recovery of Источник ссылки не найден.): armaments and military equipment. a = b = c = d = e = f = g = i = h= 1/2 hours; Therefore, it seems appropriate measures A = B = C = D = E = F = G = I = H = 1/9; aimed at increasing the survivability of AAM. t = (6…48) hours. According to the classical definition, the Identify and plot graph of probabilities: survivability of AAM is its ability to maintain its Pп ( t ) , Pз ( t ) , Pв ( t ) , Pо ( t ) , t = (6…48) hours. functions during the action of the enemy's means of destruction and the ability to quickly recover Regarding the representation of numerical from damage and return to service. values in the binary calculation system, the It is clear that to increase the survivability of assumption introduced in the first test example. weapons part is necessary and sufficient: first, to The results obtained from the simulation organize and implement a set of measures to results of determining and comparing the reduce its radio and optical visibility by air and probabilities of the technical support system in ground reconnaissance by the enemy and before each of the main states are typical for combat and during its intended use; secondly, to organize operations. These results characterize the full and carry out measures and means for artillery and group of phenomena, under conditions of technical reconnaissance: thirdly, to organize and commensurate intensities and commensurate carry out the use of a set of repair forces, the use probabilities of transitions of this system to of replacement units, blocks, devices and different states. They show the following. materials, to organize the evacuation and rapid First, with the start of combat actions, the recovery of damaged AAM. technical support system is: in a state of According to the graph of states and transitions preparation of weapons and ammunition for the of the technical system of combat operations, the fight with a probability 13%; in the state of use of above measures and means should clearly: first, weapons for their intended purpose - with reduce the intensity and probability of transition probability 60%; in a state of restoration of of the system from the state of use of LAV for its armament after damage - with probability 10%; intended purpose to recovery after damage; in a state of service - with probability 17%. secondly, these measures and means will increase Secondly, after two days of combat actions, the the intensity and probability of the transition of technical support system is in a state of the system from the state of recovery of LAV after preparation of weapons and ammunition - with a damage to the state of use for its intended probability 9%; in the state of use of weapons for purpose.. their intended purpose - with probability 26%; in We will further determine the direction of a state of restoration of armament after damage - change in the operation of the technical system of with probability 47%; in the state of service of combat operations for some specific conditions armaments - with probability 17%. that differ (from the conditions of Example 2) by This shows that the data obtained (under reducing the intensity and probability of transition conditions of equally intense and equally probable of the system from the intended use to the transitions of the system to different states) using recovery state after damage, for example, in two the model, in the presence of random and times, in addition, differ in the increase in the antagonistic uncertainties, do not contradict the intensity and probability of the transition of the system from the state of recovery of LAV after 3. Under the conditions of creating a software damage to the state of use for its intended purpose product and implementing a dialog-information also in two times. model of the operation of the technical system of The system of technical support of combat combat operations using a personal computer, it is operations is: in the state of preparation of the possible not only to explore real support systems, LAV for combat with a probability of (13… but also to solve complex problems of technical 14)%; in the state of application of LAV for the support of combat operations in real time and in purpose - with a probability of (62… 47)%; in a including on the battlefield. state of recovery after damage - with probability; in the state of service - with a probability of (18… 4. References 29)%. The implementation of measures aimed at increasing the survivability of LAV, compared [1] Military-political, strategic, operational and with measures, showed that the probability of recovery of LAV after damage and its return to tactical content of local wars and armed conflicts. НАОУ, 2001. 222 p. service increases more than four times; the [2] Varvanets Y.B. Analysis of the use and probability of the maintenance system in the state development trends of rolling stock of use for its intended purpose (as of two days) is maintenance and repair of military equipment. doubled, the probability of being in the state of maintenance is also increased by one and a half Proceedings of the International Scientific and times. Technical Conference: Prospects for the development of weapons of the Land Forces.- Lviv, 17-18 May 2018 20 p. 3. Conclusions [3] Technical support of troops (forces) in operations and combat. Textbook: НАОУ, 1. The weakest point of the standard system of 2001. – 616 p. technical support of combat operations of the unit [4] Technical support of service and combat is its ability to restore weapons damaged during operations (application) Internal troops of the combat. The solution of the problem of evacuation Ministry of Internal Affairs of Ukraine. and recovery using the model of states and Tutorial. R. О. Kaidalov, G. M. Marenki, В. transitions confirms the adequacy of the model О. Temnikov, В. І. Kuzhelovich. – Х.: with real measures of evacuation and recovery of Academy of Internal troops of the Ministry weapons. This situation necessitates further of Internal Affairs of Ukraine, 2013. – 111 p. research on the technical system of combat [5] Demianchuck B.О. Basics of automotive operations, in order to identify measures to support. Process modeling / Demianchuck increase the ability to restore weapons damaged B.О, Werpivskyi S.M., Melenchuck B.M. during combat. Textbook with a stamp Ministry of Defense 2. Analysis of the functioning of the technical of Ukraine. – Odesa: Military Academy. – system of combat operations in order to determine 2015. – 391 p. its capabilities and areas for improvement in [6] 2. Application of units and military units of conditions of random and antagonistic technical support. Part.1: Technical support uncertainties - all this necessitates the search for units; Tutorial.- К:НУОУ named after Ivan and application of effective models and Chernyakhovsky, 2017.-136p . appropriate quantitative analysis and synthesis for [7] Anolovich B.Y. Reliability of machines in adequate scientific management of technical tasks and examples В.Y. Anolovich, О.S. problems. Grinchenko, В.L. Litvinenko. – Х.: Eye, 3. The use of models of states and transitions 2001. – 320 p. allows by building an adequate model and [8] Methodological guide for comparative appropriate simple calculations, even in assessment of the quality of military conditions of random and antagonistic engineering equipment - Moscow.: uncertainties to obtain sufficiently reliable Publisher. Ministry of Defence, 1991. – 50 p. quantitative estimates of the capabilities of the [9] Technical support management documents. technical system of combat operations, and to Part 1. Tutorial. – Кyiv.: Kyiv Institute of determine appropriate directions and ways to Land Forces, 1996. – 121 p. improve it and increase important parameters. its [10] 3. Sedov S.G., Bubliy В.А., Revutskyi А.А. functioning. Analysis and forecasts of the development of protection of lightly armored combat vehicles from small arms. Proceedings of the University. National Defense University of Ukraine. 2019. №1, p. 129–137.