Optimization of agricultural enterprises based on
the methodology of optimal aggregation
Taisa Borovska, Inna Shulgan
Abstract - This article discusses a new approach to
the optimization of agricultural enterprises on
complex processing of raw materials on the basis of
the methodology of the optimal aggregation, which
allows you to replace the multidimensional problems
of nonlinear programming system of onedimensional problems. The process of solving the
problem: the mapping of the resource structure of
the enterprise in a binary tree of optimal aggregation
and development of the binary operator optimal
aggregation. An example is given for a typical dairy.
Keywords – information technology, optimization,
aggregation, operator algebra, the production
function.
I. INTRODUCTION
Modern distributed production system integrates:
production, logistics, development and innovation to
ensure the release of an effective product line.
Therefore, the problem of optimal production control
and enterprise development are relevant today. The
methodology of optimal aggregation of production
systems based on information technology to build
"working models" - mathematical models implemented
in an environment of mathematical packages. The
essence of methodologies - the replacement of a
multidimensional nonlinear programming system of onedimensional optimization problems. Computing costs
while increasing approximately linearly with the
dimensionality of the problem. In [1] presented optimal
solution to the problem of aggregation of the integrated
system "production, development". Substantially, the
problem lies in the optimal distribution of of affordable
"quantum" of the resource between the subsystems of
"production" and "the development of production". The
peculiarity of this mission is the parametric resource
relationship between subsystems. In [2] based on the
methodology optimal aggregation solved the problem
optimal aggregation integrated subsystem of the
"production, development, innovation." In [3] shows a
system model of the optimal development of the
production system taking into account the of
uncertainties of "active environment" - competitors and
consumers.
Taisa Borovska - Vinnitsa National Technical University,
Khmelnytske shosse 95, Vinnitsa, Ukraine, 21021,
E-mail: taisaborovska@gmail.com
Inna Shulgan - Vinnitsa National Technical University,
Khmelnytske shosse 95, Vinnitsa, Ukraine, 21021,
E-mail: shulganinna29@gmail.com
In this paper solves the problem of optimal
distribution of resources between different types of
agricultural products taking into account the uncertainty
in prices resources and products of production.
II. METHODOLOGY OF OPTIMAL
AGGREGATION. EXAMPLE OF
OPTIMIZATION OF DISTRIBUTION OF
RESOURCES BETWEEN PRODUCTS
The methodology of optimal aggregation occurred as
a result of the decisions and studies the classical
problem of optimal distribution of limited resources
between the production elements. The aim of the
development was to create a method of optimization that
has no restrictions of a mathematical nature: linearity,
convexity, the existence of continuous the derivatives.
The method has no direct analogs in information
technology for solving optimization problems. Consider
the procedure for solving optimization problems by the
method of optimal aggregation on the example (Fig. 1):
- structuring of the production system (selection of
elements - converters "resource, product", identification
of resource links); - the formation of base functions of
the "expenses, production" (production functions (PF))
for the elements; - record PF in a standard format
operand algebra; - forming the base of the binary
operators optimal aggregation for typical links; displaying the production system structure in binary tree
structure optimal aggregation; - meeting the challenge
(getting the optimal equivalent production functions
(OEPF) system, and vector function of optimal resource
allocation); - parameterization OEFP (getting the system
of functions influence).
Method of optimum aggregating of parallel structures
– algebraic; algebra-matrix structure, in the first column
(Fig. 1b) - the value of the discrete function output (PF),
in the following columns - the relevant dimensionless
resource values for each subsystem. A binary tree in
Fig. 1a - is a structural formula that runs in the
environment of simulation; f1, f2, f3 – discretized PF of
subsystems - matrix corresponding dimension; f2o –
operator which takes two production function and
returns the optimal equivalent production function
(OEFP). EOFP belongs to a plurality of carriers algebra.
Strictly proven associativity and operator f2o
commutativity [1, 2]. Analogues: algebra numbers addition, algebra of transfer function. Unlike
counterparts - optimization, built-in operator.
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OEFP values and functions of the optimal allocation of
resources (Fig. 2b). Is the text of the module that
implements operator F2opr, optimal aggregation
structure with nonlinear parametric relationship (fig. 2c).
Fig.1. Example of optimal aggregating parallel structure
The methodology of optimal aggregation in practical
terms - the solution of problems of optimization of
production systems when the production system there
are new classes of inter-element connection. Consider
the procedure for solving optimization problems on the
example (Fig. 2): - analysis of the resource structure of
the chosen segment of production systems; - analysis of
resource links between elements; - the formalization of
new classes of resource links, structure development of
the operands and binary operators optimal aggregation; solving optimization problem by the method of optimal
aggregation.
In Fig. 2 decision represented the optimal aggregation
of integrated structures “production, development". The
relevance of this problem is due to the fact that in
today's
high-tech
industries
development
(modernization, replacement of production capacity,
product models) actually occurs continuously.
Informative statement of the problem: the enterprise
solves the problem of optimum distribution of
"quantum" resource between existing production and
increasing production efficiency. In mathematical terms,
the relationship between the subsystems is parametric:
"development" modifies parameters of "production".
Optimization of object-system of two subsystems, each
of which consists of subsystems "production" and
"development". Shows the structural formula of optimal
aggregation (Fig. 2a) and optimization results-matrix
Fig.2. Example of optimum aggregating of “production,
development”
Solution of the problem of optimal aggregation
structures "production development" allowed us to
posing the problem and solve the problem of the class
"localization of production in the region", "optimization
of assembly production management structure
components". Without a solution to such problems does
not make sense to do formulation of the problem of
sustainable of optimal development of regional
structures (settlements, districts, regions).
Consider the features of solution of optimization tasks
of production systems based on the methodology at the
detailed example model dairy. Dairy provides a specific
region of dairy products, creates jobs and provides a
reliable product sales dairy farms and small suppliers, as
well as supplies under contracts remote users (including
for export) products such as butter, powdered and
condensed milk. This is potentially an effective
sustainable socio-techno-economic system.
Perform the formulation and solution of the problem
of optimizing the distribution of the primary resource milk from suppliers of production between products.
"Problem Statement" in this case, the methods for
Applied Systems Analysis. We consider the system as a
“COMPUTER SCIENCE & INFORMATION TECHNOLOGIES” (CSIT’2015), 14-17 SEPTEMBER 2015, LVIV, UKRAINE
whole (Fig. 3) as a technological converter "resources product" or "input - output".
Fig.4. Optimal aggregation product line "milk"
Fig.3. Displaying the resource structure of the enterprise in a
binary tree of optimal aggregating
We perform display resource structure of
technological system (Fig. 3a) in the binary tree
structure optimal aggregating (Fig. 3b). We are doing it
for a part of the circuit - from the delivery of milk to the
total output. On the scheme presents the elements of
production system, which will be included in the model
the following of approximations - subsystem
development, service, and patterns of consumption and
of consumers, competition models, advertising. The
company produces a large quantity of products, but they
can be grouped by purpose and production technologies.
In Fig. 3 shows the product line "butter" (a fat content of
60%, 70%, 80%), "milk" (a fat content of 2%, 4%, 6%),
"powdered
milk",
"condensed
milk".
The
multidimensional problem of optimal aggregating is
divided into tasks aggregating lines of products, and
then - aggregating types of products.
In Fig. 4 for lines milk are shown: the formula
aggregating (Fig. 4a), price charts of production
functions milk of various fat (Fig. 4b) and the result of
optimization - Matrix Fml (Fig. 4c), according to which
the graphs in Fig. 5.
Fig.5. Results of optimal aggregation product line "milk"
In Figs. 6 and 7 shows the results of optimal
aggregating product line "butter".
Fig.6. Optimal aggregation product line "butter"
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1500
1350
Butr1kc  kc   c1200
1050
Butr2kc  kc   c 900
750
Butr3kc  kc   c
600
Butr4kc  kc   c 450
300
150
0
0
Темп випуску
Доля выпуска продукции
1500

Functions
optimal
distribution of
resources
0
2400 0
2160
1920
1680
1 1440
1200
Fmb
kc 960
720
480
240
0
0
400
800
1200
1600
kc  c
Затраты ресурса суммарные
2000
2000

Optimal
equivalent
öåíà
ðåñóðñà
production
Cm0
3
function
араспределения
400
800 1200
1600 2000
kcproduct
 c ресурсов
Fig.5. Results of optimal aggregation
line " butter
Темп витрат
The last step in the optimization of resource allocation
between the products of production - the optimal
aggregation of OEFP lines "butter", "milk", "condensed
milk", "powdered milk" - aggregation of second level
(Fig. 8).
Fig. 8. Binary tree optimal aggregation product lines
Also, like the previous aggregating, the last
aggregation - routine operation subject to the availability
of adequate models of production functions elements.
Production function should reflect the dependence
"costs - Issue " for some production. This is quite a
difficult task for technologists, economists and
consumers even in the case of a known product. In this
challenging and discussion task, choose the approaches
developed in the Applied System Analysis.
Possible ways of obtaining production functions (FP):
- FP-regularly forms the manufacturer based on
standard cost accounting and production in natural and
monetary units. PF in this case empirical dependence,
sophisticated fuzzy accounting procedures and data
confidentiality;
- PF receives a system analyst based on gathering
descriptive information about "generating mechanisms",
well-known mathematical models of technological
processes. Statistics available is drawn at the stage of
verification of the model. This methodology of J
Forrester;
- PF is determined based on modern data processing
ICS, PCs online.
It should be noted that classical methods of analysis
and optimization, empirical PF approximates the
restricted method used: linear, quadratic, convex,
continuous. Limitations of the method of optimal
aggregation: weakly positive and weakly monotonic
bounded PF. Inability to get PF some enterprise
accounting standard database reveals deficiencies in the
Organization and technology of production.
In many cases technological processes stochastic. For
standard calculations of uncertainties and risks of
production it is necessary to determine the probability
distribution production system as a convolution of the
probability distributions of subsystems. This topical and
complex problem is solved on the basis of methodology
of optimal aggregating [4]: we created an operator of
optimal aggregating for stochastic PF.
III. CONCLUSION
In this paper goal to develop effective work of
mathematical models and techniques to solve the
problem of optimal allocation of resources between
various types of agricultural products, taking into
account the uncertainty of the resource prices and
products. Proposed solution to the problem based on the
methodology for the optimal aggregation of production
systems. The effectiveness of the methodology due to
the use of information technology to build working
models in an environment modeling packages. We have
defined the procedure for solving optimization problems
of varying complexity. On this basis, we have performed
the task of optimal aggregation for dairy. We have
developed a method of optimal aggregating product
lines. We performed analysis of alternatives to the
formation of production functions. We have shown the
need for sharing "technology" and "price" of the
functions of production.
In a modern economy, the prices of raw materials
and product only on average reflect real costs and value.
To account for the effects of random fluctuations is
designed
and
implemented
programmatically
parameterized solution of optimization tasks when
optimal function equivalent production system depends
not only on the magnitude of a system resource, but also
on resource prices and products. This solution is unique
among the known methods. Resource allocation
optimization model developed dairy is open to changes
in the assortment of products, production technology.
The model may include additional restrictions on the
volume of production of individual products. Possible to
configure the optimal allocation of system resources to
the mating tasks: minimizing the cost of production at
the specified output volume. Theoretical solution of the
tasks set out in article is an extension of the algebra of
optimal aggregation is the basis for a holistic, algebra
theory of production systems, such as the theory of
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linear dynamic systems. Fast parametric and structural
changes of modern production systems require the
development of information technology is continuously
immediately to modify models of the production system.
REFERENCES
[1]. Borovska, T. N. (2014). Optimal aggregation of
production systems with parametric connections.
Eastern-European Journal of Enterprise Technologies, 4(11(70)), 9-19. DOI: 10.15587/17294061.2014.26306.
[2]. Borovska, T. N., Kolesnik, I. S., Severilov, V. A.,
Severilov, P. V. (2014). Optimal models of innovation development production systems. EasternEuropean Journal of Enterprise Technologies,
5(2(71)), 42-50. DOI: 10.15587/1729-4061.2014.
28030.
[3]. Borovska, T. M., Severilov, P. V., Khomyn Y. P.
(2014). Alternative models optimal development industrial systems under uncertainty. System Research
and Information Technologies, 4, 121–136.
[4]. Borovska T. M., Kolesnik I. S., Severilov P. V.,
Malinochka A. A. (2014). Optimal aggregation
systems with stochastic production functions.
Vestnik of National University "Lviv Polytechnic",
792, 41–52.
“COMPUTER SCIENCE & INFORMATION TECHNOLOGIES” (CSIT’2015), 14-17 SEPTEMBER 2015, LVIV, UKRAINE