=Paper=
{{Paper
|id=Vol-2732/20200446
|storemode=property
|title=Mathematical-Logistic Model of Integrated Production Structure of
Food Production
|pdfUrl=https://ceur-ws.org/Vol-2732/20200446.pdf
|volume=Vol-2732
|authors=Andrey Mokhnenko,Vitalina Babenko,Oleksandr Naumov,Iryna Perevozova,Oleksandr Fedorchuk
|dblpUrl=https://dblp.org/rec/conf/icteri/MokhnenkoBNPF20
}}
==Mathematical-Logistic Model of Integrated Production Structure of
Food Production==
https://ceur-ws.org/Vol-2732/20200446.pdf
Mathematical-Logistic Model of Integrated Production
Structure of Food Production
Andrey Mokhnenko1, Vitalina Babenko2, Oleksandr Naumov3, Iryna Perevozova4,
Oleksandr Fedorchuk1
1
Kherson State University, 27 Universitetska st., Kherson, 73000, Ukraine
mohnenkoas@gmail.com, 15961980@ukr.net
2
V.N. Karazin Kharkiv National University 4 Svobody Sq., Kharkiv, 61022, Ukraine
vitalinababenko@karazin.ua
3
University of State Fiscal Service of Ukraine, 31, Universytetska str., Irpin city in Kiev
Region, 08201, Ukraine,
abnaumov75@gmail.com
4
Ivano-Frankivsk National Technical University of Oil and Gas 15 Karpatska Street, Ivano-
Frankivsk, 76019, Ukraine
perevozova@ukr.net
Abstract. The article is devoted to the formation of a technological-logistic
model of the integrated structure of food production. The main goal of
corporate structure management is the integration of all its constituent units for
the fulfillment of the mission, which ensures achievement of the set goals. The
main purpose of modeling is to show how the intermediate links-enterprises are
logically formed the target object. A mathematical formulation of the problem
of choosing optimal capacities and rational location of enterprises, as well as
minimum costs for transportation of raw materials, is proposed. A complex
mathematical model for planning the production of agricultural raw materials
and processing it into ready-made food products in the system "agricultural
sector - provision / primary processing - food industry enterprises" was formed.
Model of the logistic organization of integrated food production are based on
the principles of rational organization of the technological chain and are
characterized by: complexity, universality, differentiation of the approach;
specialization. The developed mathematical model allow planning and
programming of the development processes of the integrated food production
system, assessing the impact of changes in the parameters of the system, and
adjusting plans. With the help of Statgraphics, Statistica, Excel software and
having as a basis an array of enterprise data, it is possible to plan and program
the development processes of an integrated food production system, assess the
impact of changes in system parameters, make adjustments to plans. The model
make it possible to specify the technological complex of work and the need for
raw materials, provide an opportunity to establish boundaries between
complexes of works of individual companies and, in general, the responsibility
of the entire corporate structure.
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
Keywords: logistic model, food industry, agrarian sector, integrated produc-
tion, economic-mathematical modeling
1. Introduction
The food industry is one of the few branches of the Ukrainian economy, which is in
the stage of steady development. However, the raw material base of food production –
the agricultural sector of Ukraine today is in a crisis condition, characterized by a
decline in the material and technical base, high costs of production, shortage of work-
ing capital.
Measures to implement integrated development strategy should be scientifically
sound and rely on mathematical models of processes that will take place in a new
production system.
Technological peculiarities of the production of agrarian products, as well as the
complexity of the processes of its harvesting, accumulation and, if necessary, primary
processing, processing into finished products, make a relevant mathematical modeling
of the planning of production of raw materials, taking into account possible changes
in production volumes. The purpose of the article is to study the technological and
logistical side of the rational organization of the integrated complex of enterprises
"agriculture - procurement organizations - enterprises of the food industry". The ob-
jectives of the paper are: the identification of production, economic and transport
factors of the interaction of production in the technological chain of food production
and the mathematical formulation of the task of choosing the optimal capacities and
rational allocation of the integrated production complex enterprises, as well as the
minimum costs for the transportation of raw materials.
2. Related Works
It should be noted that the issue of the placement of specialized agricultural enterpris-
es and the transportation of products produced by agrarian enterprises has been given
attention in the literature since the 60-s of the last century [1].
The issue of solving problems of optimization of processes in agrarian and pro-
cessing sectors of the economy of Ukraine and today are at the center of attention.
So in the works of Y. Brodsky [2] and S. Nakonechny [3] shows the economic and
mathematical functioning of agrarian enterprises, in particular models of production
structure, innovation processes, technical and economic processes. These models have
a high level of reliability, however, they cover only the primary link in the
technological and logistic chain of food production.
Separately allocated works related to the simulation and optimization of logistics
processes in the industry, such as scientific work Y. Borbot [4]. Problems of find
optimal solutions in the industrial production and logistics system have been analyzed
in articles of modern scientists, like A. Gola [5]. But we can't say about existing of
universal optimization model. Individual model of optimization of food production
and logistics system of particular kind of food product must consider all set of
specific features of the sector.
It should be emphasized that market requirements today require the formation of a
complex integrated system of agro-industrial food production, which will enable to
realize the reserves of competitiveness of the industry. Consequently, it is necessary
to approach the modeling of technological and logistic processes in a complex way.
The necessity of mathematical modeling of the planning of the enterprises of the food
industry and its raw material base is caused first of all by the possibility in this case of
more rational use of available resources and optimization of commodity-cash flows
[6, 7].
Taking into account that the question of modeling the optimal development of the
food production complex remains open, we will try to propose an own view on its
solution in this paper.
3. Research Methods
The tasks of logistic management of commodity flows in order to minimize costs and
maximize profits in the food production system is a complex methodological task, as
the technological chain of food production involves a large number of actors that
enter into interaction. Consequently, it is necessary to take into account the whole set
of participants in production and to coordinate their numerous interactions, taking into
account their functional features of their organization and the technology of the work.
To solve this problem we used the instrumental apparatus of mathematical
modeling, namely - setting up an optimization problem Z-type with a system of
constraints, which allows the most adequately describe the investigated technological
and logistic system. During the study was developed the model of optimization of
capacities of production units and minimization of expenses for transportation of raw
materials. Limitations of the model are the volumes of raw material production, the
quantities of raw materials delivered between the stages of the technological process
and the volume of raw materials transported between all the links of the production-
technological chain. Method of formation and structural composition of the model are
shown on the fig. 1.
The system of integrated agro-food production
Raw element – set of Element procurement Processing element -
Transport and distribution 1 - many deliv-
Transport and distribution 2 - many deliv-
agricultural compa- and primary processing set of food processing
nies - suppliers of raw – set of procurement / enterprises N
materials L processing enterprises
R
Enterprise 1 Enterprise 1 Enterprise 1
ery options
ery options
Enterprise 2 Enterprise 2 Enterprise 2 Sales
… … …
Enterprise l Enterprise r Enterprise n
The parameter of the The parameter of the The parameter of the The parame-
model: physical vol- model: model: ter of the
ume of production of - physical volume of raw - the physical volume of model: sales
raw materials. materials processing; processing the semi- price of
- physical output of the finished product; finished
semi-finished product. - the physical volume of products.
the finished product.
Key model Key model
parameter – parameter –
transportation costs. transportation costs.
The objective function - to minimize the total cost.
Fig. 1. Component of the model for optimizing production and logistics chains of an integrated
agrarian-food production structure
The system should strive to get close to the ideal state of operation, which will
ensure the economy of resources and maximize returns.
For the practical use of this model, the formation of an appropriate information
environment is required by monitoring and accumulating the statistical base of
parameters that characterize the use of resources, costs, production and logistics and
transport and logistics flows.
4. The Proposed Optimization Model
In the complex of integrated production, a number of small agricultural enterprises,
several harvesting organizations and / or primary processing enterprises, one or
several enterprises of the food industry are connected. All of them are, as a rule,
geographically located in one region or neighboring regions of Ukraine - in the zone
of growing of raw materials. Since each enterprise of the next stage is a consumer of
raw materials or semi-finished products produced by enterprises of the previous
stages, the desired quantities can be represented in homogeneous units of
measurement by means of conversion into a single conditional product.
The mathematical model of the optimal development of the food production
complex consists of the objective function F (x), which expresses the general
minimum expected costs of growing the raw material, its harvesting and primary
processing and transportation of products:
n l n l n l r n l r n r p
F ( x) C jk x jk C jk x jk S jki x jki C jki x jki S jim x jim →
* *
j 1 k 1 j 1 k 1 j 1 k 1 i 1 j 1 k 1 i 1 j 1 i 1 m 1
min, (6)
where: k is one of the plurality (1,2,3, ..., k, ..., l) of agricultural enterprises
producing agrarian raw materials and supplying it to procurement organizations and /
or primary processing enterprises; j is one of the plural (1,2,3, ..., j, ..., n) type of raw
material produced by the kth agricultural enterprise; i is one of the plurality (1,2,3, ...,
i, ..., r) of procurement organizations and / or primary processing enterprises; m is one
of the plural (1,2,3, ..., m, ..., p) enterprises of the food industry, which produces the
final product; τ - one of the plural (1,2,3, ..., τ, ..., t) years of the planned period of
development of food production; Cjk - production costs of the unit of raw material of
the j-th species in the k-th agricultural enterprise; C * jk - costs related to the
expansion of the sown area / livestock to obtain an additional unit of raw material of
the j-th species in the k-th agricultural enterprise; Cjki - expenses for processing of
the unit of volume of raw material of the j type, received from the k-th agricultural
enterprise at the i-th enterprise of primary processing; Sjki - transportation costs per
unit of raw material of j-th type from k-th agricultural enterprise to i-th enterprise of
primary processing and / or procurement organization; Sjim - transportation costs per
unit of raw material j-th type from the i-th primary processing enterprise and / or
procurement organization to the m-e enterprise of the food industry; νj - the largest
volume of raw materials of the j-th type, which should be developed by all l
agricultural enterprises; Vj - the largest volume of j-th type raw material, which can
be taken for processing from all l agricultural enterprises all r of the enterprise of
primary processing; Wj - the largest volume of j-th type raw material, which can be
taken for processing by all r of procurement organizations and / or enterprises of
primary processing all p enterprises of the food industry; xjk - the desired amount of
raw material of the j-th species, which should be developed in the k-th agricultural
enterprise; x * jk - the required additional amount of j-th type raw material to be
produced in the k-th agricultural enterprise; xjki is the desired amount of j-th type raw
material delivered from the k-th agricultural enterprise to the i-th primary processing
enterprise and / or procurement organization; xjim - the required volume of j-th type
raw material, delivered from the i-th enterprise of primary processing and / or
procurement organization to the m-th enterprise of the food-processing industry; Gojk
- quantity (in units of measurement) of j-th type raw material at k-th agricultural
enterprise; Gnji - the need (in units of measurement) of j-th type raw material at the i-
th enterprise of primary processing and / or stockpiling organization; Goji - the
quantity (in units of measurement) of j-th type raw material at the i-th enterprise of
primary processing and / or procurement organization; Gnjm - demand (in units of
measurement) of raw material of the j-th type at the m-th enterprise of the food-
processing industry/
Restrictions of model:
1. The gross output of j-type raw material by all l agricultural enterprises must be
agreed in advance with all enterprises of primary processing and / or procurement
organizations:
l
( x jk x* jk ) ≤ j ; (7)
k 1
2. The total volume of deliveries of raw materials of the j-th type to all l agricultural
enterprises should not exaggerate the possibilities of its processing by all enterprises
of primary processing:
l r
x jki ≤ V j ; (8)
k 1 i 1
3. The total volume of supplies of j-type raw materials by all enterprises of primary
processing and / or procurement organizations should not exaggerate the possibilities
for its processing by all enterprises of the food industry:
r p
x jim ≤ W j ; (9)
i 1 m 1
In this case, to perform compatibility of the conditions of the problem, it is
necessary that there are fair inequalities:
j V j W j
(10)
In the set task the criterion of optimality is taken the minimum of production and
transport costs. With the closed model of the transport task to the specified
restrictions, the following is added:
4. The gross volume of j-type consignments shipped from all l agricultural
enterprises should correspond to the total demand for these cargoes at the destination
points of each of the r enterprises of primary processing and / or procurement
organizations:
n r n l l r
x jki G o jk ; (11) x jki G н ji ; (12) G o jk G н ji (13)
j 1 i 1 j 1 k 1 k 1 i 1
The same value should occur when sending materials to the food industry:
n p n r
x jim G o ji ; (14) x jim G н jm ; (15) G o ji G н jm . (16)
r p
j 1 m 1 j 1 i 1 i 1 m 1
If the problem under consideration is to be formulated in a dynamic statement, then
the mathematical model to be derived should be classified in a particular year, which
we accept for the first (τ = 1) in the planned period t. In this case, the target function Z
(x, τ) takes the form:
t
Z ( x, ) F ( x) →min, (17)
1
where Fτ (x) means that the parameter τ is present as an additional index for all
parameters and variables of the function F (x).
Since the dynamical model implies the need to increase the production plan with
each passing year, the relationship must be fulfilled:
0 x jk x jk( 1) x jki x jki( 1) x jim x jim( 1) (18)
5. Results and discussion
The offered mathematical model allows to carry out planning and programming of
processes of development of the new integrated food production system, to estimate
influence of changes in system parameters and to make adjustments of plans. The
output data for calculating the optimal amount of raw materials for production and
processing are given in the table 1.
Table 1. The output data for calculating the optimal amount of raw materials
The raw material base
Available amount
of raw materials,
Distribution of
costs (Cjk), uah/t
Unit production
Crop capacity,
Transportation
raw materials
Area under
crops, ha
costs (Sjki), UAH/t
(G0jk), t
t/ha
Farms
t
Kherson
Kherson
Product
Product
Nash
Nash
1. Chaika 10 70 700 700 - 7000 30 35
2. Lotos 27 60 1620 800 - 6000 25 30
3. Ukraine 20 55 1100 500 - 5000 20 25
4. Druhba 50 40 2000 - 2000 5000 30 24
5. Ahrkom 35 30 1050 - 1000 4000 20 15
Primary processing
Need for raw Output Unit production Costs of transportation
Processing
materials (Gнjk), products, costs (Cji), UAH/t to the food business
enterprises
t t (Sjim), UAH/t
1. Kherson 2000 350 11000 15
2. Nash product 3000 500 10000 25
Production of finished products
Volume of Unit production
Price of the finished Total cost F(x),
Enterprises consumed raw costs (Cjim),
product, UAH/t UAH
materials (Gнjm), t UAHt
1. Pani 850 40000
5000 39431750
Kritina
A real system of production of tomato raw materials (tomatoes → tomato paste →
ketchup), localized on the territory of Belozersky district of the Kherson region, was
selected to test the model. The system consists of three production steps. The first
stage is the raw material base, which is represented by farms (F) “Chaika”, “Lotos”,
“Ukraine”, “Druzhba” and “Ahrokom”, which grow tomatoes. The second stage is the
primary processing, which is represented by processing enterprises LLC Fruit and
Vegetable Plant “Kherson” and PE “Nash Product”, which produce tomato paste. The
third stage is the enterprise of the food industry of PICF "Pani Kristina", which
produces the final product - ketchup under the trademark "Holiday".
The calculation of the model was made using the SAS Model Manager software.
Results of optimization of the model are presented in the table 2.
Table 2. Calculating the optimal amount of raw materials
The raw material base
Unit Transportatio
amount of
materials,
Distribution of raw
Available
crops, ha
capaity,
producti n costs (Sjki),
under
Сrop
Area
materials (G0jk), t
t/ha
raw
Farms on costs UAH /t
t
Nash (Cjk), Khers Nash
Kherson
Product uah/t on Product
1. Chaika 10 70 700 - - 7000 30 35
2. Lotos 27 60 1620 850 - 6000 25 30
3. Ukraine 20 55 1100 1100 - 5000 20 25
4. Druzhba 50 40 2000 50 1950 5000 30 24
5. Ahrоkom 35 30 1050 - 1050 4000 20 15
Primary processing
Costs of transportation
Processing Need for raw Output Unit production
to the food business
enterprises materials (Gнjk), t products, t costs (Cji), UAH/t
(Sjim), UAH/t
1. Kherson 2000 350 11000 15
2. Nash Product 3000 500 10000 25
Production of finished products
Volume of Unit production Price of the
Total cost
Enterprises consumed raw costs (Cjim), finished product,
F(x), UAH
materials (Gнjm), t UAH/t UAH/t
1. Pani Kristina 850 5000 40000 38025050
According to the results of the calculation, optimal volumes of production and
supply of raw materials and semi-finished products in the technological chain of
production and processing of tomatoes were determined. In the basic, actually
existing (non-optimal) version of production, the total amount of expenses is
39431750 UAH, the optimal amount of expenditures is F(x)= 38025050. The obtained
data allow to reduce expenses for production and transportation of products, increase
production efficiency. In particular, the cost saving is 1406700 UAH.
It is possible to recommend "Chaika" to refuse to produce raw materials in favor of
other types of products, due to economic impracticability. It is recommended to
reduce volumes of tomato crop area for Lotus, and it is advisable to revise programs
for the supply of raw materials to processing plants. In particular, the part of raw
materials from "Druzhba" should be sent to the processing plant LLC Fruit and
Vegetable Complex "Kherson".
6. Conclusions and Outlook
The construction of a cost management system in integrated food production should
be based on the principle of feedback, that is, on the needs of food industry enterpris-
es, which are conditioned by the market conditions. The resource management cycle,
like the whole system of control of the technological chain, should cover all stages of
product creation.
The offered model of the technological-logistic integrated structure of food produc-
tion should be used in practice in the activity of the enterprise of the food sector.
With the help of Statgraphics, Statistica, Excel software, and based on the enter-
prise data array, it is possible to plan and program the development processes of the
integrated food production system, to evaluate the impact of changes in the parame-
ters of the system, to make adjustments to the plans.
These programs are most user-friendly for beginners due to the lack of targeting a
specific subject area, a wide range of statistical techniques, and a user-friendly inter-
face. They are more accessible to practice and can be used by a wide range of special-
ists of different profiles.
Using the proposed model will significantly reduce the need for raw materials in
the enterprise. In addition, a significant reduction in the likelihood of errors when
making managerial decisions.
The presented model allow to specify the technological complex of works and the
need for raw materials, provide an opportunity to establish boundaries between the
complex of works, for which the producers-executors are responsible and, in general,
the responsibility of the entire corporate structure of food production.
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