Baltic Journal of Economic Studies  

100

Vol. 5, No. 5, 2019

Corresponding author:
1 Central Research Institute of the Armed Forces of Ukraine, Ukraine.
E-mail: ovk1923@gmail.com
2 Central Research Institute of the Armed Forces of Ukraine, Ukraine.
E-mail: vitalikgodz@ukr.net

DOI: https://doi.org/10.30525/2256-0742/2019-5-5-100-109

GENERAL APPROACHES TO THE SUBSTANTIATION  
OF COMBAT STAFF OF THE ARMED FORCES DEPENDING  
ON MILITARY-ECONOMIC CAPABILITIES OF THE STATE

Volodymyr Mozharovskyi1, Serhii Hodz2

Abstract. The article presents the essence and main points of the methodological approach to the substantiation 
of the optimal combat staff of the armed forces from the perspective of the theory of prevented damage of 
our troops in the operation and taking into account military-economic capabilities of the state, in particular, 
financial constraints on the procurement and maintenance of combat means of relevant j-types. Inconsistency 
between tasks scope, which are entrusted to the armed forces (AF) for national security, and level of their 
combat capability has been and remain one of the fundamental problems that adversely affects the process 
of their forming, developing, preparing and using. That kind of inconsistency is caused by the impact of some 
factors, primarily factors characterizing economic-military capabilities of the state. Such factors also involve 
the capacities of the defense industrial complex in terms of development, production, modernization and 
providing military units (MU) with weapons and defense equipment ( WME). The experience of the troops 
shows that current financial constraints on the procurement and maintenance of the necessary WME negatively 
affect the level of MU fighting potential. Thus, the problem and need for the substantiation of optimal combat 
staff of the AF have been and remain inherent components of the process of their forming and developing 
taking into account military-economic capabilities of the state. The above defines the relevance of the research 
under consideration and needs new scientific approaches to the justification of optimal combat staff of the AF, 
which would ensure guaranteed execution of national security tasks. Based on the analysis of recent research 
and publications related to the problems of substantiation of the AF staff as well as the assessment of the 
impact of CP of MU on the level of their combat capability, the paper establishes that available methodologies 
(methodological approaches) don’t specify some issues, including what should the optimal AF combat staff be 
for guaranteed execution of their intended targets, taking into account the military-economic potential of the 
state? One of the alternate solutions of the problem is the methodological approach, which is presented in the 
article, to the justification of the optimal combat staff of the armed forces from the perspective of the theory 
of prevented damage of our troops in the operation and taking into account military-economic capabilities of 
the state, in particular, financial constraints on the procurement and maintenance of fighting means of relevant 
j-types. To get a much better sense of the essence and content of the research subject, the authors define a 
range of the concepts: “military-economic factors”, “prevented damage”, “prevented damage theory”, “combat 
potential”, “combat capability”, “combat staff ”, “effectiveness of combat use of forces”. The authors believe that 
compared to the available methodological approaches (methods), the methodological approach set out in the 
article allows:
formalizing the interrelation between the level of combat capability of our forces and the desired value of their 
prevented damage (CP maintenance) for a defined period of operation, which is achieved by the results of 
counteraction (fire effect) of fighting means of j-types of our troops against the enemy. This enables us to determine 
the number of combat means of all j-types of our troops required at the initiation of the operation, which (taking 
into account losses for T days of combat activities) ensures the target level of their fighting efficiency;
carrying out the differentiated approach to the determination of the role (contribution) of every kind of our forces 
in the execution of tasks set in the operation that makes it possible: to calculate the contribution of every j-type of 
combat means (military branch) to the tasks of general damages on enemy and achievement of combat actions 



Baltic Journal of Economic Studies  

101

Vol. 5, No. 5, 2019
of the required correlation of parties’ CP during T days; to define the contribution of combat means of every j- types 
(military branch) to reducing the predicted value of the average-daily costs of the original CP of our troops to the 
acceptable one, the accomplishment of the necessary absolute value of our troops and maintenance of the target 
level of their combat capability during the T-days operation;
improving (based on the indicators of prevented damage) the mathematical model of the optimization problem and 
elements of the calculation process (algorithm) that allows studying the interdependence between core indicators, 
which evaluate the effectiveness of combat use of troops during the operation, more specifically. 
Prospects for future research are the detailed elaboration of mathematical models and elements of the calculation 
process (algorithm) for upgrading methodical apparatus, which makes it possible to substantiate the optimal 
combat staff of the AF required for the successful execution of state security tasks from the perspective of the 
theory of prevented damage of our troops and taking into account military-economic capabilities of the state.

Key words: combat potential, combat capability, combat staff, combat equipment, prevented damage, 
mathematical model, optimization problem, methodical apparatus.

JEL Classification: N40, N44, C65

1. Introduction
Problem statement in general terms. The capacity of 

the Armed Forces (AF) to accomplish national security 
tasks depends on many components. In addition, a 
combat potential (CP) of the AF, which is considered 
as a set of material and spiritual factors determining 
the state and competence of the AF to deliver on the 
intended outcomes, is a major figure. CP of the AF is 
the central component of the military potential of the 
state (Combat potential).

It is known that CP of the military organization (MO) 
(unit, base, formation, union and the AF as a whole) is 
an integrated index of the effectiveness that characterizes 
the maximum number of tasks for the purpose, which 
MO is able to perform in the context of typical conditions 
under the regulatory levels of the capabilities of its 
command systems, combat (operational), logistics and 
personnel training (study skills). 152 mm towed gun-
howitzer M1955 (D-20), whose fire power potential is 
the measurement unit of firepower of different weapons 
of destruction, is accepted is an accounting unit (AU) of 
CP of the AF of Ukraine (Mozharovskyi, Godz, 2016).

CP of MO determines its combat potential – a state 
which permits MO to realize its combat capabilities 
(fighting, fire, maneuver), conduct operations under any 
conditions and exercise tasks for the purpose intended. 
Combat potential and combat capabilities of MO are 
evaluated by several indicators where the provision 
of MOs with fighting equipment plays an important 
role – key samples of weapons and military equipment 
(WME). This indicator is one of the decisive when 
substantiating the combat staff – the manning strength 
of units, formations, unions, which involves organic 
forces as well as reinforcements that are intended to 
carry out combat missions. Despite that combat staff 
of the formations and units is determined by their 
major (combat) units, combat staff of military branches 
(forces), uniformed services and the AF as a whole is 
defined by the available number and combat staff of 

relevant formations and units and thus, the total number 
of fighting equipment which is in service of their major 
units. Consistency of the combat staff with the assigned 
stare security tasks mainly determines the effectiveness of 
combat use – a degree of the implementation of combat 
capabilities of forces, which are engaged in conducting 
operations (military actions). At the same time, the 
effectiveness of combat use of forces in the operation 
is considered as expected target or achievable result 
upon completion of the operation, which is evaluated 
by the following core indicators: the extent of damage 
inflicted on the enemy; the level of survivability (loses 
of human and material-technical resources shouldn’t 
exceed admissible ones); the volume of spent material 
and technical resources in the course of operation; 
completion time (terms) of missions defined by the 
operation (Mozharovskyi, Hodz, 2018; Mozharovskyi, 
Hodz, 2017). 

The analysis of the principal impact factors on 
the formation of the combat staff, operational and 
mobilization readiness of the AF as well as the 
conclusions on the assessment of the influence of the 
level of combat capability of the troops (forces) on 
the degree of realization of their fighting efficiency 
shows that there is an inconsistency between the 
scope of state security tasks entrusted on the AF and 
the level of their combat capability in the practice of 
forming, preparing and using the AF. That kind of 
inconsistency is a consequence of the impact of some 
factors, in particular, those that characterize the 
country’s military and economic capabilities, namely: 
macroeconomic indicators of the state (development) 
of the state economy in a particular period; trends 
in the military budget of the state; possibilities for 
the development of military manufacturing in a 
special period; the cost indicators of maintaining 
and providing formations (units) with WME and 
expandable material and technical resources (MTR); 
industrial capacity on the autonomous production 



Baltic Journal of Economic Studies  

102

Vol. 5, No. 5, 2019
of WME and other types of MTR ; the development 
level of science and technology; state of infrastructure 
and transport capacities of the country; the degree of 
dispersion, security and vulnerability of important 
economic entities (Review of force structures in 
implementation of partnership goal 0028…; The 
Norwegian Navy under the 2017–2020 Long Term 
Defence Plan; French and German defense…; 
Le Roux L.; The Voluntary Sector Knowledge 
Network…). There are important factors determining 
the capabilities of the defense industrial complex 
(DIC) of the state on the development and launching 
of production of samples (groups) of WME, 
modernization and term extension for the operation 
of the available WME; the existence of WME reserve 
and WME delivery terms of procurement (additional 
supply) for military organizations; the conformity of 
WME samples with current requirements (Poland’s 
Defense Modernization Plan; The Danish Defense 
Agreement 2013–2017; Bassford, Matt, Kristin 
Weed, Samir Puri, Gregory Falconer and Anaіs 
Reading, 2010).

When conducting operations (combat activities), MO 
is considered to be combat-effective if its procurement 
with basic key WME samples is not less than 80%, with 
others – not less than 60%, provided that the technical 
condition of these samples meets the established 
standards. MO is considered to be combat-ineffective 
if it has lost more than 50% of its original CP (combat 
equipment) (Mozharovskyi, Hodz, 2018).

However, the experience of development, preparation 
and use of the AF shows that the current financial 
constraints on the procurement and maintenance of the 
necessary WME negatively affect the level of MO combat 
capability. Therefore, the problem and the need to justify 
a rational (optimal) combat staff of the AF, taking into 
account the military and economic potential of the state 
(in particular concerning the elaboration, production, 
modernization and providing MO with WME) have 
always been and remain integral components of the 
process of their forming and developing. The above 
defines the relevance of the authors’ research and needs 
new scientific approaches to the substantiation of the 
optimal combat staff of the state AF, which taking into 
account military-economic factors of the influence 
on the level of their combat capability, provides a 
guaranteed execution of the entrusted tasks by the AF. 

The purpose of the article is to present the essence and 
main points of one of such scientific (methodological) 
approaches.

2. Analysis of recent research and publications.
According to the results of recent research and 

publications related to the problems of substantiation 
of the rational (optimal) MO combat staff as well as 
the assessment of the influence of CP of MO on the 

level of their combat capability, one establishes that 
available methods (methodological approaches) don’t 
come up with responses to the following important 
questions: first, what is an acceptable level of loses 
for the AF to execute the tasks on purpose during 
conducting operations (combat actions)? second, what 
number of combat means (WME samples), including 
MOs of relevant military branches (forces), should AF 
staff have to keep the target level of fighting efficiency 
of our troops in the T-days operation and their capacity 
to exercises tasks by designation with the necessary 
effectiveness? third, considering military-economic 
capabilities of the state, what should the optimal 
AF combat staff be for the guaranteed performance 
of the tasks by designation? (Mozharovskyi, Hodz, 
2017; Romanchenko, Shuenkin, Mozharovskyi, 2017; 
Mozharovskyi, Godz, 2018).

To get a well-argued answer to the above questions, 
the authors propose to use the methodological 
approach to the substantiation of optimal combat staff 
of the armed forces from the perspective of the theory of 
prevented damage of our troops in the operation and taking 
into account military-economic capabilities of the state 
( financial constraints on the procurement and maintenance 
of some j- types).

3. On the theory of prevented damage  
of troops in the operation

The Armed Forces of the state are a complex 
organizational and technical system, which is 
considered as a set of interdependent and organically 
operating elements (sub-systems) according to a unified 
plan for solving a range of challenges in the context of 
predetermined factor space (Mozharovskyi, Hodz, 
2017). The study of substantiation of the AF combat 
staff is proposed to be carried out from the perspective 
of the theory of prevented damage of our troops in the 
operation, which is comprehended as an integral part of 
the systems analysis. The authors believe that the above 
approach makes it possible to conduct morphological, 
functional and parametric representation of the complex 
organizational and technical system, to examine all its 
elements concerning unified positions of the general 
and basic property of this system (appropriate criterion) 
holistically – a given coefficient of the combat capability 
of our troops in the T-days operation. 

Prevented damage of our troops in the operation 
means the difference between the loss of our troops, 
which is potential without taking into account measures 
on its hindrance or reduction, and the loss taking into 
account such measures. The theory of prevented damage 
of our troops in the operation is an integrated system of 
general knowledge, provisions, mathematical and logical 
correlations that allows formalizing the relationship 
between the current and targeted levels of combat capability 
of the troops, the required volume of their prevented damage 



Baltic Journal of Economic Studies  

103

Vol. 5, No. 5, 2019
which is achieved by counteraction – the impact of the 
troops on the enemy by combat equipment, the damages 
which have to be inflicted on the enemy to reduce its 
combat potential, the weakening of the impact on our 
troops and descent of the average-daily casualties of the 
original CP of our troops to an acceptable level, as well 
as the number of combat weapons of all types required at 
operation’s initiation, which provides (taking into account 
the losses for T days of combat activities) the level of 
fighting efficiency of troops not lower than targeted one 
and fulfillment of the tasks defined in the operation 
with the necessary efficiency (Romanchenko, Shuenkin, 
Mozharovskyi, 2017; Mozharovskyi, Godz, 2018).

4. General scope of the methodological 
approach (calculation process stages)

Methodological approach to the substantiation of 
the optimal combat staff of the armed forces of the state 
envisages the use of indices of output data, which are 
presented in Table 1, and involves six consecutive and 
interrelated stages from the perspective of the theory 
of prevented damage of our troops in the operation and 
taking into account the state’s capabilities (financial 
restrictions on procurement and maintenance of 
combat means of some j-types) (Romanchenko, 
Shuenkin, Mozharovskyi, 2017; Mozharovskyi, Godz, 
2018; Mozharovskyi, Hodz, 2018).

The first stage. Evaluation of δ Ï – a necessary 
absolute value of prevented damage (CP conservation) 
of our troops (combat means of all j-types) (in AU 
of CP) for the guarantee of the targeted value of the 
coefficient of their combat capability θâèì in the T-days 
operation θâèì.  

It is generally thought that one of the possible ways 
for achieving the value δ Ï  in the operation, without 
considering the measures of operational (combat) 
security, is to reduce the enemy’s CP by causing damage, 
that is, the cut in the original ration of CP parties (α0) to 
a level that would meet the targeted θâèì,  and therefore, 
the current number N âèì  of combat means of all j-types 
of our troops whose combat capability (taking into 
account βï ) has to be kept. Increasing the level of fire 
influence on the enemy will lead to a decline in its CP 
(combat capabilities) and weaken the impact on our 
troops. As a result, predicted daily-average relative loses 
β0

,  of our troops will decrease to the level of acceptable 
daily-average relative losses ïβ . This will make it 
possible to hold combat capability ( )N Nâèì > of units 
of combat means of all j-types of our troops. 

The physical content of the value δ Ï  is a difference 
between required numbers of combat means N âèì  
whose combat capability has to be kept in the result of 

increasing fire effect on the enemy to provide (under the 
conditions βï Ò⋅ ) the targeted value θâèì and actual 
number of combat-effective means N of the available 
combat staff of our troops (under the conditions β0

, ⋅Ò ):

δ β β

β β

Ï N N N N Ò T N

T Ò

n
âèì

n

n

= − = − = ⋅ − ⋅ ⋅ =

= − ⋅ − − ⋅  ⋅

∆ ∆0 0 0

01 1

( )

( ) ( )

,

, NN N Ïâèì j
j

r

0 0
1

= − ⋅ =
=
∑( ) .θ θ δ

 (1)

The second stage. Determination of the number 
of fighting means necessary for the initiation of the 
operation, taking into account calculated value δ Ï , 
which will ensure the target level of combat capability  
(θâèì,)of our troops in the T-days operation. 

It is believed that achieving the desired level of 
firepower on the enemy requires an increase in the 
ammunition expenditure (firing amount), as well as 
an expansion in the number of combat means of the 
relevant j-type of our troops. In this context, the essential 
additional number of j-type combat means (combat 
means of all j-types) of our troops at the beginning of 
the operation is calculated by the need for the boost 
of fire influence on the enemy, the destruction of a 
certain number of its combat means, thus, by calculated 
means δÏj  and δ Ï which will provide targeted θ j

âèì  
and θâèì respectively. Consequently, the number N âèì0
of combat means of all j-types of our troops which is 
necessary at the operation’s outset including δ Ï  (1) 
for the achievement of the targeted θâèì is as follows:

 N N N Ï N N Nâèì j
âèì

j
ï0 0 0 0 0= ≥ + = + −∑ δ ( );∆ ∆      (2) 

and the number N j
âèì
0 of combat means of j-type of 

our troops necessary at the operation’s outset including 
δÏj  for the achievement θ j

âèì  is as follows:
 N N Ï N N Nj

âèì
j j j j ï j0 0 0 0≥ + = + −δ ( ).∆ ∆                 (3)

The third stage. Calculation of the number of combat 
means of the enemy ∆N f Ïïð

âèì = ( )δ, which must be destroyed 
for the achievement of the necessary absolute value of 
prevented damage ( )δ Ï of our troops. 

The realization of the value δ Ï is conditioned by 
the destruction of the required number ∆N f Ïïð

âèì = ( )δ  
of the enemy’s combat means by our troops. At the 
same time, the formal interrelation between the 
indices ∆N f Ïïð

âèì, ( )δ  can be established through 
β
β
n ïð

ïð

ïð

ïð

N

N

N

N0 0 0
1

,
.= = −

∆

An additional number of combat 
means of j-type and combat means of all  
j-types of our troops, which is necessary at the outset 
of the operation, will be defined by the demand for 
increasing fire influence on the enemy, that is the need for 
the destruction of a targeted number of its combat means 
( ∆N f Ïïð

âèì = ( )δ), and calculated through an obtained value δ Ï :

 
∆ ∆N N N

Ò
N

Ò
Ï
N

ïð
âèì

ïð і
âèì

і
ïð

âèì

ïð= = ⋅ −
−
⋅

= ⋅ −
⋅ −

∑ 0
0

0

0
01

1
1( ) (

,

,

θ
β

β
δ

ββ
δ
β0

0
0 0

, ,
)

⋅
= ⋅

⋅ ⋅Ò
N

Ï
N Ò

ïð                        (4)



Baltic Journal of Economic Studies  

104

Vol. 5, No. 5, 2019
Table 1
Output data indices 

Sl.No. Index Index essence

θâèì
Composite index (criterion) is a coefficient of combat capability of our 
troops (combat means of all j-types) for T days of the operation, which is 
set by the commander’s decision 

1. Ò Duration (number of days) of the operation

2. N Nïð і ïð
i

v

0 0
1

=
=
∑

The original CP of the enemy taking into account its combat means of 
all і-types that is defined by the number of tangible combat means or the 
number of combat means expressed in the accounting units (AU) of CP 

3. N N j
j

r

0 0
1

=
=
∑

The original CP of our troops taking into account combat means of all 
j-types (military branches) that is defined by the number of tangible 
combat means or the number of combat means expressed in AU of CP 

4. a
N

N

N

N
ïð

і ïð
i

j
j

0
0

0

0

0

= =
∑
∑

General original correlation of parties’ CP 

5. β0 j
Average standard relative average daily loses of the combat means of 
j-type of our troops due to the enemy’s influence 

6. β β0 0 0j ô ÎÒÂ ó j ja ê ê ê
, = ⋅ ⋅ ⋅ ⋅

The predicted value of the relative average daily expenditures of the 
original CP of j-type of our troops (combat means of j-type) for T days 
of operational activities caused by the combat means of the enemy in 
view of the specified value β0 j  and coefficients: a0 ; êô  – forms of the 
engagement of own troops; êÎÒ  – an operational-tactical significance of 
the troops; êó j  – original staffing level of j-type of troops in accordance 
with the staff-basic needs 

7.
β

β

0

0 0
1

0

,

,

=
⋅

=
∑N

N

j j
j

r
The predicted value of average daily loses, which is unified for all branches 
of our troops (combat means of all j-types), of their original CP for 
T-days of combat activities

8.
β

β

0

0 0
1

0

1,
,

⋅ =
⋅

⋅ ≤=
∑

Ò
N

N
Ò

j j
j

r
The predicted value of average daily loses, which is average for all 
branches of our troops (combat means of all j-x types), of their original 
CP for T days combat activities 

9.
∆ ∆N N Ò N j

j

r

0 0 0 0
1

= ⋅ ⋅ =
=
∑β , ;

∆N N Òj j j0 0 0= ⋅ ⋅β
,

The predicted number of lost ∆N0 from N0 combat means of all j-types 
of our troops (tangible and in AU in CP) under the conditions β0

, for Т 
days of combat activities; the number of lost ∆N0 j  combat means of j-type 
of our troops under the conditions β0 j

,
 for – days of combat activities

10.
N N N N N Tj

j

= = − = ⋅ − ⋅∑ 0 0 0 01∆ ( );,β

N N N N Tj j j j j= − = ⋅ − ⋅0 0 0 01∆ ( )
,β

The predicted current number of N combat-effective means of all 
j-types of our troops (tangible ones and in AU of CP) on the results of 
the influence of enemy’s combat means of all types on them under the 
conditions β0

,  for Т days of combat activities; the current number N j  
of combat-effective means of j-type of our troops under the conditions 
β0

,  for Т days of combat activities

11.
θ = =

−
= −

⋅
⋅ =

N
N

N N
N

N
T N

T
0

0 0

0

0

0

1
∆ ∆

= − ⋅ = −
− ⋅

=
∑

1 1
1

0

0
1

0

( )
( )

]

,β
θ

T
N

N

j j
j

r

The predicted current coefficient value of combat capability (relative 
value of CP) of our troops of the available combat staff according to the 
actual number N  of combat-effective means of all j-types of our troops 
under the conditions β0

,  for Т days of combat activities

12.
θ j

j

j

j j

j

N

N

N N

N
= =

−
=

0

0 0

0

∆

= −
⋅

⋅ = − ⋅1 10

0
0

∆N

T N
T Tj

j
jβ

,

The predicted current coefficient value of combat capability of j-types 
of troops according to the current number N j  of combat-effective 
means of relevant j-type under the conditions β0 j

, for Т days of combat 
activities

 13. β
θ

βï
âèì

Ò
=

−
≤

1
0
,

Acceptable value of daily average relative loses of the original CP of 
combat means of all j-х types of our troops for Т days of combat activities 
for the achievement of the targeted value θâèì



Baltic Journal of Economic Studies  

105

Vol. 5, No. 5, 2019

Sl.No. Index Index essence

14. β
β
β

βï j
ï

j= ⋅
0

0,
,

Acceptable value of daily average relative loses of the original CP of 
combat means of all j-х types of our troops for Т days of combat activities 
for the achievement of the targeted value θ j

âèì

15. θ βj
âèì

ï j Ò≥ − ⋅1
Targeted value of the coefficient of combat capability of j military branch 
(combat means of j-type)

16. ∆ ∆N N Ò N N Tï ï n j
j

r

j
j

r

n j= ⋅ ⋅ = = ⋅ ⋅
= =
∑ ∑0

1
0

1

β β
Acceptable number of the lost ∆Nn for the achievement θ

âèì from N0 
combat means of all j-types of our troops under the conditions β βï ≤ 0

,  
for Т days of combat activities

17.
N N N N

N Ò N

âèì
j
âèì

j
ï

ï
âèì

= = − =

= ⋅ − ⋅ = ⋅

∑ 0

0 01

∆

( )β θ

The current number   of combat means of all j-types of our troops whose 
combat capability has to be kept for the achievement of the given one θâèì  

18. θ j
âèì j

âèì

j
âèì

j
âèì

n j

j
âèì

N

N

N N

N
≥ =

−

0

0

0

∆ The targeted value of the coefficient of the combat capability of j-type of 
troops in view of the necessary number θ j

âèì  of combat means of j-type 
of our troops whose combat capability has to be kept 

19. θâèì
âèì

âèì

âèì
n

âèì

N
N

N N

N
≥ =

−

0

0

0

∆ The targeted value of the coefficient of the combat capability ( )θ
âèì  of 

our troops in view of the necessary number ( )N âèì  of combat means of 
j-x type of our troops whose combat readiness has to be kept 

(End of Table 1)

However, the outcome of such calculations is the 
determination of not only indices N Nâèì j

âèì

j
0 0= ∑  (2)–(3)  

and ∆ ∆N N f Ïïð
âèì

ïð і
і

= =∑ ( )δ  (4) but also the 
number ∆ ∆N N f Ïïð
âèì

ïð і
і

= =∑ ( )δ of the destroyed combat means of і-type 
of the enemy by combat means of j-type of our troops. 
Wherefore, it is essential to calculate the number of 
combat means of every j-type of our troops necessary at 
the outset of the operation – N j

âèì
0 (combat staff of every 

j-type of our troops). 
The fourth stage. Calculation of the current number 

of i-type combat means of the enemy (∆ ∆N N f Ïïð
âèì

ïð і
і

= =∑ ( )δ) which 
have to be destroyed by combat means of j-type of 
our troops relying on their fighting potential and 
without considering the value δ Ï.  For this reason, 
the calculated value N j

âèì
0  is converted in the tangible 

combat means N N Nj ð
âèì

j
âèì

j ð
âèì

0 0 0...( ).↔  Every tangible 
combat means has its inherent fighting capabilities 
(official parameters). Combat means of all j-types of 
our troops can participate in destructing i-type combat 
means of the enemy with some probability Ð j і  of 
elimination of i-type combat means of the enemy (or at 
least one combat means) from the whole number N і ïð0  
of units by the tangible combat means of j-type of our 
troops composed of N N N Ïj

âèì
j ð

âèì
j j0 0 0↔ = + δ  etc. 

Taking into account the amount n j і  of conducted fire 
activity of every combat means during T days of military 
operation, it is calculated the number of destroyed  ∆ ∆N N f Ïïð

âèì
ïð і

і

= =∑ ( )δ 
from N і ïð0  of combat means of i-type of the enemy 
during T days of military actions. The number ∆ ∆N N f Ïïð

âèì
ïð і

і

= =∑ ( )δ of 
destroyed i-type combat means of the enemy for T days 
of military operation from the available 0 ïð iN  units in 
terms of the influence on them by combat means of all 
types in the composition 0N  of units of our troops is 
calculated:

 ∆N N Ò N Ð N Pïð і ïð і ïð і ïð і ïð і ïð і
j

r

j i= ⋅ ⋅ = ⋅ = ⋅ − −










=
∏0 0 0

1

1 1β ( ) . (5)

Then, it is calculated the total number of destroyed 
(without δ Ï ) combat means of all types of the enemy 
from N ïð0  for T days of military operations:

 ∆N N Ð Nïð ïð і ïð і
і

ïð і
і

= ⋅ = ⋅ ⋅∑∑ 0 .                          (6)
To carry out calculations at the fourth stage, it is also 

used individual mathematical models [15; [16].
The fifth stage. The comparison of the current 

number ∆N N Ð Nïð ïð і ïð і
і

ïð і
і

= ⋅ = ⋅ ⋅∑∑ 0 .  of combat 
means of all types of the enemy and particularly 

combat means of every i-type of the enemy, which 
were destroyed by combat means of every j-type 
of our troops, proceeding from their inherent 
fighting potential (official parameters 1 jiR , jin ) and 
without considering δ Ï  with required number of 
the enemy ’s destroyed combat means of all types 

∆ ∆N N N
Ï

N Ò
ïð
âèì

ïð і
âèì

і
ïð= = ⋅ ⋅ ⋅

∑ 0
0 0

δ
β ,

,  which has 

been calculated at the third stage according to the 
necessary δ Ï  of our troops obtained at the first 
stage. If the condition ∆ ∆N Nïð ïð

âèì≥  is not performed , 
there is a need to advance combat capabilities of our 
troops either by improving military characteristics 
(modernization) of WME samples or by increasing 
the number of tangible 0 j ð îáìN  combat means of 
j-types of our troops or scheduling required fire 
activity jin  to blow up the value j iP , ïð iP   given 

1 jiR , ïjβ , 0 j ðN   under unvaried value 0 ïðN . At 
the same time, the amount jin  of conducted fire 
activity during the T-days operation can be realized 
by relevant tactical and technical characteristics 



Baltic Journal of Economic Studies  

106

Vol. 5, No. 5, 2019
of combat means as well as due to the available 
reserve of missiles and ammunition of every combat 
means, which is allocated for disbursements for 
every day of combat actions. Thus, one gets back to 
the fourth stage of calculations. Using the method of 
successive approximations (iteration), it is achieved 
the condition where ∆ ∆N Nïð ïð

âèì≥ .  In this context, it 
is considered that the troops, according to the level of 
their fighting efficiency, can complete the tasks defined 
in the operation with the desired result. Then, the 
number of combat means of every j-type and combat 
means of all types of our troops (in AU of CP and 
tangible combat means) required at the operation’s 
initiation is calculated as a total value of their original 
initial number (N0 j), (N0) and an absolute value 
of prevented damage of every j-type of our troops 
( )δÏj  based on the results of calculations and an 
absolute value of prevented damage of combat means 
of all types of our troops ( )δ Ï . In prospect, it will 
allow determining what number of different types 
of combat means should be in the Armed Forces of 
the state at the beginning of the operation (rational 
combat staff of the AF) to ensure (taking into account 
the losses for T days of combat operations) the 
current coefficient of combat capability ( )θï of our 
troops not lower than the given one ( ).θâèì

The sixth stage. Statement of the optimization 
problem and its solution based on the use of the 
mathematical model for the substantiation of the 
desirable combat staff of the national AF from the 
perspective of prevented damage theory of our troops 
in the operation and taking into account military and 
economic capabilities of the state (Mozharovskyi, 
Hodz, 2018; Mozharovskyi, Godz, 2018). 

It is generally believed that the coefficient of combat 
capability âèìθ of our troops, which is fixed by the 
commander’s decision, in the T-days operation is 
achieved through varying the number of combat means 
of j-type, which are necessary at the operation’s onset 
and similar in terms of their purpose and potential. 
Combat means of j-type can be expressed both in WME 

hardware 0 0
1

ð
âèì

jð
âèì

j

r

N N=
=
∑  and AU of CP 0 0âèì jâèì

j
N N= ∑ .  

It is important to note that combat means of j-types 
(military branches) can significantly differ not only 
in terms of the purpose and capabilities (tactical and 
technical characteristics) as well as cost indicators jñ . 
Thus, if it is specified the cost jñ  of one combat means 
of j-type, the cost of all combat means (weapons 
pieces) of this type (military branch) jC  is interpreted 
as follows:

 j j j ðâèìC c N= ⋅ 0 ;                           (7)
and the total cost of combat means of all types of our 

troops is defined:

 C C Nj
j

j j ð
âèì

j
c= = ⋅∑ ∑ 0 .                         (8)

In the light of the existing trends of the state budget 
expenditures on defense needs, the option of imposing 
financial restrictions îáìC  on the procurement 
(purchase, production, modernization) and 
maintenance (operation, storage, overhaul and repair) 
of combat means of certain j-types (military branches) is 
not excluded. Therefore, there may be the need to select 
such combat means of j-types numerical strength of 
which will make it possible to achieve the targeted value 
of the combat capability âèìθ of our troops and maximize 
the damage to be inflicted on the enemy max ∆N пр.

Consequently, it is considered the optimization 
problem, which involves determining the possible 
(limited) number 0 0  îáì j îáì

j
N N= ∑  of combat means 

of some j-x types (military branches) which would give 
an option of inflicting maximum damage max ∆N пр on 
the enemy in the T-days operation as well as achieving 
the required degreed of prevented damage δ Ï  of our 
troops and thus, the current coefficient of combat 
capability value of the troops ( )θï  not lower than given 
one ( ).âèìθ

It should be pointed out that the problem statement 
envisages the condition when a certain number of combat 
means of j-type can be purchased and maintained in the 
context of financial constraints (allocated funds) îáìC .  
There may be also a need to justify the favorable ratio 
between the number 0 j ð îáìN  of actual combat means 
and the amount βï ≤ 0 04, , of firing activity, which is 
conducted by every combat means of j-type of our 
troops during T days of operational activities to achieve 
a particular degree (probability j iP ) of destruction of 
і-type combat means of the enemy.

To set the optimization problem, one must also 
bear in mind that the achievement of the value ∆Nпрвим  
is provided by the actions of a possible (limited) 
number 0 0îáì j îáì

j
N N= ∑  of combat means of all 

j-types of our troops, which are characterized by the 
inherent parameters 1 jiR , jin , before the operation. In 
this regard, it is assumed that for maximum damage to 
the enemy ïð ïð i

i
N Nmax ∆ ∆ =  ∑ , actual combat means 

ïð ïð i
i

N Nmax ∆ ∆ =  ∑ of our troops can act together 
with some probability ïð i j i

j

r

P P = − −
=
∏1 1

1

( )  against any 
combat means of i-type of the enemy. Consequently, a 
value ïð i ïð i ïð іN P N∆ = ⋅ 0  can be carried. Calculations 
results of the actions of tangible combat means of our 
troops depend both on the values 0 j ð îáìN  and 1 jiR ,  

jin . There may also be a need to justify the optimum 
ratio between the number 0 j ð îáìN  of actual combat 
means and the amount βï ≤ 0 04, ,  of firing conducted 
by each combat means of j-type of our troops during T 
days of military operations to achieve varying degrees 
(probability j iP ) of destruction of i-type combat 
means of the enemy. 

Considering the above, the target function of the 
optimization problem is as follows: 



Baltic Journal of Economic Studies  

107

Vol. 5, No. 5, 2019

In addition, there are some restrictions. 
The first restriction. Financial resources (funds) Ñ,

which are necessary for the procurement and maintenance 
of the possible (limited) number 0 0ð îáì j ð îáì

j
N N  = ∑  

of actual combat means of all types of our troops, 
shouldn’t exceed allocated financial resources (funds) 
 îáìÑ . In other words, it is essential to adhere to the 
condition where Ñ c N Ñj j ð îáì

j
îáì= ⋅ ≤∑ 0   .  Moreover, 

the initial (approximate) value of the allocated funds 
îáìÑ  is decided in the context of overrun of the total 

cost 0 0 0Ñ Ñ c Nj
j

j j ð
j

= = ⋅∑ ∑  of combat means 
(WME samples) of all types of the original combat staff 

0 0N N j
j

= ∑  of our troops as well as considering the cost 
δ δ δ β βÑ Ñ c Ï c N Tj

j
j j

j
j ï j ð

j

= = ⋅ = ⋅ − ⋅ ⋅ ∑ ∑ ∑ ( ),0 0
 

of combat means of all j-types of our troops, which 
should be added to their original combat staff for 
achieving expected absolute value of prevented damage 
(CP maintenance) δ δÏ Ïj

j

r

=
=
∑

1

 of our troops in the 
T-days operation: 

 îáì j
j

j
j

j jð
j

j j
j

Ñ Ñ Ñ C C c N c Ï≥ + = + = ⋅ + ⋅∑ ∑ ∑ ∑0 0 0δ δ δ . (10)

The second restriction. A calculated possible 
(limited) number N ð îáì0  of combat means of all 
j-types of our troops before starting the T-days 
operation should provide the current value of the 
coefficient of combat capability θï of the troops 
not lower than given one âèìθ .  Moreover, the actual 
absolute value of prevented damage of our troops 
has to be not lower than calculated required absolute 
value of the prevented damage ( )δ Ï  of the troops. 
Thus, the condition should be met: 

 N N N Ï N Ïð îáì j ð îáì
j

ð j ð
j

j
j

r

0 0 0 0
1

= ≥ + = +∑ ∑ ∑
=

δ δ  (11)

The third restriction. Taking into account the damages 
which our troops will cause to the enemy during 
the T-days operation with the probability ïð iP  , the 
current number ïð ïð i

i
N N= ∑   of the enemy’s combat 

means of all i-types, whose combat capability will 
be kept, shouldn’t exceed the acceptable number 
N N Nïð

âèì
ïð ïð

âèì
0 = − ∆  of combat-effective means 

of all i-types of the enemy, which can inflict on our 
troops average-daily relative losses not more than the 
acceptable losses ïβ  in the T-days operation (table 1). 
Thus, one should adhere to the condition:

ïð ïð i
i

ïð i
i

ïð і ïð і
і

R N

N N P N N å

j i j ð îáì

max ∆ ∆ =    ∑ ∑ ∑= ⋅ = ⋅ −
−

⋅ ⋅

0 0 1

1 0 ïï

N

j i ï j
T

j

r

ïð і

⋅ −
=
∑

















( )

.

1
1

0

β

                

(9)

 
ïð ïð ïð ïð i

i
N N N N= − =∑0  = ∆ N ïð і

і
0∑ ⋅

å

R N ï

N

j i j ð îáì j i ï j
T

j

r

ïð і
−

⋅ ⋅ ⋅ −
=
∑

≤

1 0
1

0

1( )β

0
0 0

0

0

1 1 
 

ïð
ïð âèì

ïð
âèì

N
Ï

N T
N
T

N≤ ⋅ −
⋅ ⋅

=
⋅

⋅ − =( ) ( ) ,
, ,

δ
β β

θ

 

(12)

where 0 0j ð îáìN  ≥  is a value that determines the 
possible (limited) number of actual j-type combat 
means of our troops with specific fighting potential  
( 1 j iR , j in ), which can be purchased and maintained 
using special allocated funds îáìÑ . .

The above optimization problem (7)–(12) belongs 
to the nonlinear programming problems. Thus, this 
problem can be successfully solved through, for 
example, the method of Lagrange multipliers. 

5. Example of calculations
The solution of the optimization problem involves 

an advance consideration of the results of calculations 
of the rational combat staff of our troops in T-days 
operation. Consequently, according to the calculation 
results (1)–(6) (the first-fifth stage), one marks that 
the number of combat means of all types of our troops
N ÐÎ N ÐÎâèì0 0585 6 582 3

, , ,= > =  that is essential 
at the beginning of tactical activities guarantees the 
coefficient value of fighting potential of our troops not 
lower than given one: θ θ, , ,= > ≥0 805 0 80âèì . The 
number of combat means (rational combat staff of the 

AF) makes it possible to inflict damage on the enemy at 
the level ∆ ∆N ÐÎ N ÐÎïð ïð

âèì, , , ,= ≥ =713 3 697 1  which 
meets the specified requirements.

In view of the aforesaid, the solution of problem 
optimization includes: 

1) the valuation ∆C' of an additional quantity 
∆ ∆N N j

j

= ∑  of combat means of all types of 
our troops, which is necessary for the condition’s 
fulfillment θ θ, .≥ âèì  It is established that the 
valuation ∆C' of the additional number of combat 
means ∆ ∆N N j

j

= ∑  of all types of our troops 
required for the condition fulfillment θ θ, ,≥ âèì  is 
∆ ∆Ñ ñ Cj

j

, , , .= = < =∑ 173 1 177 9  million UAH , million UAHδ
In other words, ∆C' is less than the value 

δ δ δÑ Ñ c Ïj
j

j j
j

= = ⋅∑ ∑  – cost of combat means of 
all types of our troops that should be added to their 
original combat staff N 0  for the achievement of an 
essential value δ δÏ Ïj

j

r

=
=
∑

1

 of the troops in the T-days 
operation. 



Baltic Journal of Economic Studies  

108

Vol. 5, No. 5, 2019

2) the cost calculation Ñ0
,  of the total number of 

combat means of all types of our troops required at the 
operation beginning to fulfill the condition θ θ, .≥ âèì  
Thus, taking into consideration the cost of original 
number of combat means of all types of our troops 
Ñ ñ j

j
0 0 517 1= =∑ , million UAH , the cost Ñ0,  of the 

general number of combat means of all types of our 
troops needed to start the operations N ÐÎ0 585 6

, ,= is 
Ñ Ñ Ñ Ñ Ñ Ñ Ñj

j

âèì
0 0 0 0 0690 2 695 0
, , , , ,= = + = < = + =∑ δ δmillion UAH milli oon UAH.

Ñ Ñ Ñ Ñ Ñ Ñ Ñj
j

âèì
0 0 0 0 0690 2 695 0
, , , , ,= = + = < = + =∑ δ δmillion UAH milli oon UAH.

3) determining the demand for funds C to purchase and 
maintain a possible (limited) number 0 0  îáì j îáì

j
N N= ∑  of 

combat means of all types of our troops, which shouldn’t 

exceed the allocated funds Собм, that is when the condition 
Ñ c N Ñj j ð îáì

j
îáì= ⋅ ≤∑ 0    is fulfilled. Considering that Собм≥С0вим, 

from this perspective, one accepts a condition under which С 

≥ Собм. According to the results of calculation of the target 
function through the method of Lagrange multipliers, 
given the restrictions, we obtain the following data: 
Ñ Ñ Ñ Cîáì

âèì= = = > =0 0695 0 690 0, , ;
,million UAH million UAH  

N N ÐÎ N ÐÎ N ÐÎîïò îáì
âèì

0 0 0 0593 6 585 6 582 3= = > = > =, , , ,
,

 where N îïò0 – a combat staff of our troops, which is 
optimal at the outset of operational activities (an optimal 
number of combat means of all j-types) resulting from 
solving the optimization problem. At the same time, 

ïð ïð ïð
âèì

N ÐÎ N ÐÎ N ÐÎ= < = < =488 7 498 4 514 6, , , .
,  

Respectively, indicator values regarding the infliction 
of the maximum damage on the enemy are as follows: 
max , , .,∆ ∆ ∆N ÐÎ N ÐÎ N ÐÎïð ïð ïð

âèì= > = > =723 713 3 697 1  
Findings provide (including loses during Т=5 days of 
operation activities) the current value of the coefficient 
of combat capability θ ï  of our troops not lower than 
given one: θ θ θï

âèì= > = > ≥0 816 0 805 0 80, , , .,

6. Conclusions and prospects  
for future research

One can note that the problem and need to substantiate 
the optimal combat staff of the AF of the state (taking into 
account the financial constraints on the procurement and 
maintenance of combat means of particular j-types) have 
been and remain an integral component of the process of 
their formation and development. One of the options for 
problem solution is the methodological approach, which is 
presented in the article, to the justification of the optimal 
combat staff of the armed forces from the perspective of 
prevented damage theory of our troops in the operation 
and taking into account military-economic capabilities 
of the state, in particular, financial limits on the purchase 
and maintenance of fighting equipment of j-types. 

Compared to the available methodological approaches 
(methods), the article’s methodological approach allows:

formalizing the interrelation between the level of 
combat capability of our forces and the desired degree 
of prevented damage (maintenance of CP) for a defined 
period of operation, which is achieved by the results of 
counteraction (fire effect) of combat means of all types 
of our troops against the enemy. This makes it possible 
to determine the number of combat means of all types 
of our troops required at the initiation of the operation, 
which (taking into account losses for T days of combat 
activities) provide the target level of their fighting 
efficiency, due to the calculated data of the needed 
absolute value of prevented damage;

carrying out the differentiated approach to the definition 
of the role (contribution) of every military branch in the 
execution of tasks set in the operation that enables us: to 
calculate the contribution of every type of combat means 
(military branch) to the tasks of general damages to enemy 
and achievement of the required correlation of parties’ CP 
for T days of tactical activity; to define the contribution of 
combat means of every type (military branch) to reducing 
the predicted number of the average-daily costs of the initial 
CP of our troops to the acceptable one, the accomplishment 
of the required absolute value of prevented damage of our 
troops (CP conservation) and maintenance (taking into 
account loses for T days of tactical activities) of the target 
level of their fighting potential;

improving (based on the indices of prevented damage) 
the mathematical model of the optimization problem and 
elements of the calculation process (algorithm) that allows 
studying the interdependence between core indicators, 
which evaluate the effectiveness of combat use of troops 
in the operation, more specifically. In the future, this will 
allow determining the number of combat means of every 
type (military branches), which is optimal at the beginning 
of the operation. The above will ensure (taking into account 
the losses for T days of combat operations) the required 
amount of prevented damage and the level of combat 
capability of our troops not lower than the specified one. 

In the authors’ view, the application of article’s 
methodological approach in the practice of forming, 
preparing and using the AF will enable one to give a 
substantiated response to the following questions: 
what the optimal number of combat means (WME 
samples), as well as MO of relevant military branches 
(forces), should the AF have in its composition to keep 
the targeted level of fighting capability of our troops and 
their potential to exercise the tasks on purpose with 
necessary effectiveness during T days operation, taking 
into account financial constraints? 

Prospects for future research are a specification of 
mathematical models and elements of the calculating 
process (algorithm) for improving methodical apparatus, 
which allows substantiating the optimal combat staff of 
the AF needed for the successful execution of national 
security tasks from the perspective of the theory of 
prevented damage of our troops in the operation and 
considering military and economic potential of the state.



Baltic Journal of Economic Studies  

109

Vol. 5, No. 5, 2019

References:
Combat potential. Military Encyclopedic Dictionary. Moscow: ONICS 21 Century Publishing House, 2002. 1036 p.
Mozharovskyi, V. M., & Godz, S. V. (2016). Combat potential as the primary factor of the combat strength of the 
Ukrainian Forces. Works of University: coll. of sciences. works of NUD of Ukraine named after Ivan Cherniakhovskyi,  
no 2(68), pp. 279–297.
Mozharovskyi, V. M., & Hodz, S. V. (2018). The effect of weapons and military equipment on the combat  
potential and level of combat readiness of troops. S. Direct 24. Safety, Society, Science. International scientific  
journal published under the auspices of NATO Defence Education Enhancement Program (ISSN 2545-0697). 
Warszawa, Republic of Poland, no. 1; 01 march.
Mozharovskyi, V. M., & Hodz, S. V. (2017). Methodological bases for substantiation of rational composition of 
organizational and technical systems of military designation: Coll. of mater. of the international science-pract. 
conferences. The Іnternational research and practicе conference “Modern methods, innovations, and experience of 
practical application in the field of technical sciences” (General issues of technical sciences). Radom, Republic of 
Poland, December 27–28, pp. 264–269. 
Review of force structures in implementation of partnership goal 0028, preliminary report for consultations 
with NATO / Ministry of Defense. Sofia, 28 May 2011, 142 p. 
The Norwegian Navy under the 2017–2020 Long Term Defence Plan. URL: http://www.navyrecognition.com/
index.php/focus-analysis/naval-technology/4153-analysis-the-nor wegian-navy-under-the-2017-2020-long-
term-defence-plan.html
French and German defense: The opportunities of transformation. URL: http://www.iris-france.org/wp-content/
uploads/2015/03/IRIS-Note-march-2015-IRIS-SWP.pdf
Poland’s Defense Modernization Plan: A Race Against Time. URL: http: //www.avascent.com/2016/09/Poland’s-
defense-modernization-plan-race-time
The Danish Defense Agreement 2013–2017. 
Le Roux L. The Military Budgeting Process: An Overview (Defense Planning, Programming and Budgeting).  
URL: ww. sipri.org/contents/milap/milex/le_roux. pdf/download
The Voluntary Sector Knowledge Network, Leadership: Strategic Planning and Strategic Management.  
URL: http://vskn.ca/lead/strategy.htm
Bassford, Matt, Kristin Weed, Samir Puri, Gregory Falconer and Anaіs Reading. 2010. Strengths and Weaknesses  
of the Netherlands Armed Forces. Cambridge: Rand Europe Technical Report.
Romanchenko, I. S., Shuenkin, V. A., & Mozharovskyi, V. M. (2017). Prevented Damage Theory: Monogram.  
Lviv: NASW, 244 p.
Mozharovskyi, V. M., & Godz, S. V. (2018). Methodical aspects of substantiation of the military composition  
of the armed forces of the state from the standpoint of the theory of prevented damage. International scientific- 
theoretical. journal of Cybernetics and Systems Analysis, vol. 54, no. 1, pp. 154–167.
Mozharovsky, V. M., & Godz, S. V. (2016). A mathematical model for determining the magnitude of the 
possible loss of combat potential of their troops by days of operation that lasts T days. Coll. of sciences. KhNUAF  
named after Ivan Kozhedub, vol. 4(49), pp. 23–27.
Romanchenko, I. S., & Mozharovskyi, V. M. (2017). A mathematical model for determining the right amount  
of destroyed legal enemy units to achieve a given value of the prevented damage to their troops in an operation  
lasting T days. Coll. of sciences. CMSR of NUD of Ukraine named after Ivan Chernyakhovskyi, no. 1(59), pp. 85–93.
Mozharovskyi, V. M., & Hodz, S. V. (2018). Substantiation of the optimal composition of organizational and 
technical systems of military designation: coll. of mater. of international science-pract. conferences. The international  
research and practical conference “The development of technical sciences: problems and solutions” (General issues  
of engineering sciences). Вrno: Baltija Publishing, the Czech Republic, April 27–28, pp. 165–169. 
Mozharovskyi, V. M., & Godz, S. V. (2018). The Mathematical Model of Substantiation of the Optimal  
Combat Staff of the Armed Forces of the State from the Standpoints of the Theory of Prevented Damage  
and Taking into account Economic Factors. International Scientific Theoretic. journal of Cybernetics and Systems 
Analysis, vol. 54, no. 5, pp. 150–159.