The business competitiveness of SMEs
through mathematical modeling*
La competitividad empresarial de las PyMES
a través del modelado matemático
Carlos Ramiro Cepeda Godoy **
Andrés Joao Noguera Cundar***
Mónica Alexandra Moreno Barriga****
Nelson Santiago Chuquin Vasco*****
Patricio Villagomez******
Julio Mauricio Oleas*******
Received: June 2019,
accepted: December 2019
ABSTRACT
The objective of this research is to propose a
functional application model for small and mediumsized enterprises (SMEs),
with the purpose of determining the optimal production preferences of your
specific production system. The problematic that originates this research is
the methodology of projection of the costs that practice this type of companies
in the market ―which departs from a process of determination of the optimum
batch of production―, which are based on models of mathematical determination. The
applied methodology is of an analytical nature of a transversal type, the
factorial study allows the application of descriptive statistics and the
development of equations that determine variables of study of the ex ante model. The result of the research consists in the
application to a scenario that represents the optimal level of transport costs
of the SMEs products which interact in the market. It is concluded that
mathematically it is possible use variables for determining the productivity
and level of competitiveness.
Key Words: Small and medium enterprises, production,
productive system.
RESUMEN
El objetivo de esta investigación es proponer un
modelo funcional de aplicación para las pequeñas y medianas empresas (mipymes) con el propósito de determinar las preferencias de
producción óptima de su sistema productivo en concreto. La problemática que
origina esta investigación es la metodología de proyección de los costos que
practican este tipo de empresas en el mercado ―lo cual se aparta de un proceso
de determinación del lote óptimo de producción―, que estén basados en modelos
de determinación matemática. La metodología aplicada es de carácter analítico
de tipo transversal, el estudio factorial permite la aplicación de estadística
descriptiva y el desarrollo de ecuaciones que determinan variables de estudio
del modelo ex ante. El resultado de la investigación consiste en la aplicación
a un escenario que representa el nivel óptimo de costos de transporte de los
productos de las mipymes que interactúan en el
mercado. Se concluye que matemáticamente es posible usar variables para
determinar la productividad y el nivel de competitividad.
Palabras clave: pequeñas y medianas
empresas, producción, sistema productivo.
Introduction
Competition in the market has caused a constant
price discrimination, which has made the market more competitive, in this sense
companies around the world need alternatives that allow them to reduce their
operating costs in order to remain competitive and profitable (Valencia, Lambán & Royo, 2014) at the
same time, it is evident then that the optimization of resources is a key
factor that generates profits and allows it to increase the corporate value of
the organization.
According to Baykasoglu
and Kapanoglu (2006) the productive performance of
the production units depends on the management skills; in this sense, several
investigations (Sanchez, Osorio & Baena, 2007; Vélez et al., 2008; VeraColina
& MoraRiapira, 2011) have determined several
entropies that reduce the productivity levels of SMEs; which represents a
constant problem for the countries of the region, in Latin America it is
estimated that from the process of globalization of the economy, it has
resulted in a loss of competitiveness in international markets, especially for
those countries that face types of variable change in its economic system
(Bada, Rivas & Littlewood, 2017).
In this same sense, the average performance
standard of SMEs in Latin America is equivalent to less than 40 % of the productivity
of the big company (Valencia, Lambán & Royo, 2014), which leaves behind the SMEs in front of
several countries, such as those of the European Union and the United States,
where the sector reaches an average of 60 % (Mora, Vera & Melgarejo, 2015).
Several researchers (Rubio & Aragón, 2006; De
la Cruz Morales & Carrasco, 2006; Solleiro & Castañón, 2005; Quiroga, 2003) have developed new
indicators that allow a more technical measurement of the competitiveness
levels of SMEs; while other Martínez & Álvarez, 2006; Deniz, Livas & López, 2008; Santillán,
2010; Gómez, 2001; Membrillo, 2006 & Herrera,
2007). have made empirical applications to determine the competitiveness of
this sector.
In this order of ideas, models of perfect
competition can be cited in response to the processes in central assumptions,
improving on average the productivity provided by the assumption of perfect
information, namely, that all agents (companies, consumers) know how the prices
set by all companies thus provide their participation in the production chain
process and improve their activity and ability to compete.
Having made the previous observation, the
assumption of atomicity and homogeneity of the production process, therefore,
considers how the effective administration of business resources involves the
way of managing both expenses and income in an appropriate manner, from which
the effectiveness in business activity and its correct execution in the proper
determination of the optimal production lot, improving productivity of SMEs.
However, in a perfectly competitive market,
marginal revenues are equal to the price where; if the company sets a price
above that of the other companies, then it sells nothing; if on the contrary
the company sets a price below the other companies, then it receives all the
market demand that; compared to its capacity, it is a large amount of
production (Tarziján, 2006).
It means then that the assumption of equal access
in the market that all companies have allow the access to all production
technologies (Tarziján, 2006); finally, we conclude
that the assumption of free entry namely allows any company to enter or exit
the market as desired.
As a final summary, the work begins with the
bibliographic and textual revision of perfect competition models that are
related to the thesis of business competitiveness, continues with the
delineation of the state on SMEs in the country, as well as the correspondence
analysis between the performance of the commerce sector as an innovative aspect
of proactive outputs to the problems that affect SMEs. The methodology part
estimates the process of characterization of the relationship of variables that
describe the levels of competitiveness of SMEs. Consecutively, in the final
phase a discussion of the results is made and finalized with the conclusions.
Bases for the determination of the mathematical
model
In this sense several models were reviewed, one
of them is the perfect competition model; model that shows that competition is
a good thing; specifically, because the balance under perfect competition is
efficient. Made this consideration, the model is directed in two ways: first,
where each company sets the level of efficient production, that is, the level
of production in such a way that the price is equal to cost; a lower level of
production would be less efficient, since it increases more than the cost; on
the contrary, a higher level of production would also be inefficient. Second,
the group of longterm active companies demonstrates a more efficient degree,
due to their way of entering the market, where their price is equal to the
minimum average cost; this gives a greater or lesser number of companies
involved in a higher total cost for the same level of production.
In that same sense, a linear demand function can
function as a useful approximation for local analysis, but not very likely to
work in a global dynamic; except that a linear demand function results in
negative values of all its variables; which almost always make no sense in the
economy, since it contradicts basic economic theory, because it cannot be the
solution to the maximization of any limited budgetary utility.
In this sense, we analyze the discriminatory
monopoly markets where it is stated that q1 and q2 are amounts of prices determined
by brand p1 and p2.
In the theory of market firms
the quantities sold in the Cournot market are described as q1 and q2 which
means that (Q1 + q1) (Q2 + q2) are the total amount of product sold by the two
companies.
If we substitute in the budget constraint
we have:
*
Budget actions will not be added to a constant
budget.
Suppose we assume:
As suggested by replacing the budget constraint
we will obtain:
To make sense of the model we match for all
eligible p1 and p2 so that it does not produce a logical error in the model. In
the Cournot market we only need the “inelastic” demand model as suggested by
several authors (Puu and Norin,
2003). It is derived from a simple utility function CobbDouglas, therefore,
using the suggested notation, it is expressed:
and maximizing under a usual budget constraint is
represented:
where the budget normalizes the unit.
The solutions to maximize the usefulness of
CobbDouglas subject to linear budgetary restrictions such as the optimal
budget quotas are constant, in this case symmetrical, half of each one, that
is, it is up to consumers to decide which one can be chosen: the price or
quantity.
The problem is quite logical, because a product
cannot be homogeneous in the “Cournot” sense and not homogeneous in the
“Bertrand” sense at the same time, as it is nothing that sellers can choose
when devising their actions (Puu and Norin, 2003).
Then, suppose a slight modification of the
utility function for Bertrand markets
(
It has a CobbDouglas type factor q1 and q2
although we multiply by a factor dependent on the sum (q1 + q2). There is a
budget restriction again
The economic purpose of this proposed model is to
make a combination of Cournot and Bertrand, in this direction have been a bit
confusing the ideas that duopolists could choose one
or the other, quantity of offer or price, as an action parameter. However, it
can only be consumers who decide whether they consider the product supplied to
be homogeneous or not. What suppliers can do is just conclude and market the
product to convince consumers that there are different brands with their
defined advantages.
Several models have allowed the improvement of
the relationship of intrinsic variables in the optimal batch process, among
which we mention the Hall and Mendoza model that incorporate the cost of
distribution in the calculation of optimal lots (Valencia, Lambán
& Royo, 2014). Exploring in this direction the
optimal production batch model is based on the “recognized EOQ model” (Economic
Order Quantity) which optimizes the volume of the purchase or production lot by
reducing the result of combining two important management costs on “Total
Annual Cost of Inventory” that defines the sum of acquisition or purchase costs
(D*C), order issuance costs (D/Q)*S and storage costs (Q/2)*H.
In the EOQ/EPQ model, it seeks to determine the
optimum production lot (represented as Q*) from only the following data:
H = Annual unit cost of maintaining inventory.
D = Demand.
S = Fixed purchase or production cost.
The EOQ Order Economic Size model formula
represents the total cost of the inventory is as follows:
(1)
Equation (1) allows to identify that the total of
the sum of all costs is detailed by the absence of the logistic index, but, as
the name implies, it should be considered only in the process of transport or
storage of the merchandise.
The optimal lot is found by deriving and
equalizing the management cost illustrated in the following equation to zero:
(2)
The logistic index was calculated with:
Several investigations identify that the economic
order quantity can be established with the EOQ model, a model that establishes
the optimal process for the quantity to be produced, or purchased; they affirm
that in an organization it is known, representing a fixed quantity model which
seeks to determine through quantitative equality the costs of ordering and
maintenance costs at the lowest possible total cost (Hall, 1996; Jamal, Sarker and Mondal, 2004; Yuan et al., 2011).
As a result of this, different researchers have
recently proposed modifications to the model to obtain optimal ones closer to
the real ones; among the models developed in recent years are the Hall’s model
(Hall, 1996) and Mendoza’s model (Mendoza and Ventura, 2008) who incorporate
the cost of distribution in the calculation of optimal lots.
EOQ model
= Variable cost of raw material + Fixed cost + Variable cost of production +
Variable cost of rework + Variable cost of scrap disposal + Logistic index of transport
(Fixed cost of transportation to the customer + Variable cost of
transportation to the customer + Variable cost of internal transport) + Logistic index of storage (Variable cost of
storage during production + Variable cost of storage during rework + Variable
cost of storage during deliveries) + Variable cost of inspection after
production + Variable cost of inspection after rework + Variable cost of
maintenance by production + Variable cost of maintenance by rework.
Other outstanding works (Sarker
and Khan, 1999; Jamal, Sarker and Mondal, 2004),
include raw material costs, and identify the need to incorporate costs known as
“Reprocessing”; defining the importance of these in the process of calculating
the logistic index (Yuan et al., 2011).
(3)
In equation (3) it is observed that not all costs
are described so that they can influence the calculation of the volume of the
logistic index, but, as the name implies, this should only be considered in
processes such as transport or storage, which is represented below:
Figure 1. EOQ transport and storage processes. Source:
Bada, Rivas and Littlewood (2017).
However, this model is still widely applied in
several companies, several authors (Jaber, Bonney and Moualek,
2009) consider that the reduced calculation of the number of costs taken into
account is too simplistic and therefore it makes the model inaccurate.
These contributions allow production times to be
counted as nonconstant, but to follow a normal distribution, as well as to
integrate the logistic index, a cost inducer that allows logistic costs to be
adjusted to a specific reference.
Material and Methods
The research determined as the subject of
application and measurement instrument to the managers and owners of SMEs, the
same that according to Ecuadorian legislation can be found observations on SMEs
in articles 53 and 56 of the Organic Code of Production, Commerce and
Investments, that speak of the definitions and the unique register of SMEs.
Next, the classification of SMEs in Ecuador is
shown in the following table:
Table 1. Classification of SMEs in Ecuador
Classification
of the companies 
Annual Sales
Volumes 
Occupied
staff 
Microenterprise 
Less than or equal to 100,000 
1 to 9 
Small
Company 
From 100,001 to 1’000,000 
10 to 49 
Medium
Company “A” 
From 1’000,001 to 2’000,000 
50 to 99 
Medium
Company “B” 
From 2’000,001 to 5’000,000 
100 to 199 
Big
Company 
From 5’000,001 onwards 
200 onwards 
Source: Author’s
elaboration.
Table 2. Number of companies by company size and
national participation
Company Size 
No. of
Companies 
% Total 
Total 
843,745 
100,0 % 
Micro
Enterprise 
763,636 
90,5 % 
Small
Company 
63,400 
7,5 % 
Medium
Company “A” 
7,703 
0,9 % 
Medium
Company “B” 
5.143 
0,6 % 
Big
Company 
3.863 
0,5 % 
Source:
Author’s elaboration.
Of this number of companies, according to the
National Institute of Statistics and Census, about 200 SMEs have obtained an
ISO mark. Of these, about 100 are companies from Quito and 50 from Guayaquil;
the rest comes from other cities in the country.
We proceed to establish the formula to estimate
the sample size from companies with ISO certification and based on it we
proceed to build the table where a sample is determined in relation to a population.
The research elements to whom the measurement instrument was applied are the
managers and owners of commercial SMEs.
The type of probabilistic sampling, based on the
table by Krejcie and Morgan (Valencia, Lambán & Royo, 2014), where
the population size and the number of errors determine the size of the randomly
selected sample.
Formula:
A sample of 48 SMEs was determined. In this
sense, a sample of 48 SMEs was obtained, placing the population in the table; a
pilot test was applied to 20 SMEs.
The instrument that was designed for the
measurement was based on the variables diagram of the expost
model, which juxtaposes to the sagittal diagram of variables, where the
methodological matrix of variables is defined in a conceptual, operational way,
and by dimensions, which establish indicators, as well as the level of
measurement.
Figure 2. Ex ante Model. Source: Author’s elaboration.
Results
With respect to the determination of the level of
reliability of the measuring instrument, the statistical test of the pilot test
was performed where a KaiserMeyerOlkin alpha = 0.874 was obtained, indicating
a level of significant relationship of the variables of study.
Table 3. KMO and Bartlett’s
test
KMO
and Bartlett’s test 

KaiserMeyerOlkin sample adequacy measure 
,874 

Bartlett’s sphericity test 
Chisquare approximate 
11609,268 
Gl 
5 

Sig. 
,000 
Source: Author’s elaboration.
The results of the correlations of KMO and Bartlett’s
test^{1} and the coefficient of determination R2^{2} showed significant correspondences (Tables 4,
5); the values of the established variables are between 0.993 and 0.992,
correlation level, which indicates a moderate correlation while the independent
variables as a whole show a correlation level of 0.993.
Table 4. Independent
variables

Actors 
Logistics 
Enviroment 
Public
Policies 
Association 
Actors 
1 
0.050 
0.720 
0.513 
0.923 
Logistics 
0.050 
1 
0.213 
0.313 
0.622 
Enviroment 
0.720 
0.720 
1 
0.477 
0.923 
Public
Policies 
0.313 
0.813 
0.420 
1 
0.713 
Association 
0.923 
0.623 
0.414 
0.803 
1 
Source: Author’s elaboration.
Table 5. R2 coefficient of determination
Model 
R 
R2 
Setting of
R2 
Standard
estimation error 

1 
0997 
0.9933 
0.992 
2.015 

Source: Author’s elaboration.
Discussion
From the preliminary results it can be inferred
that the logistics dimension has an obvious impact on the appreciation of the
competitiveness of entrepreneurs, observing lower correlation coefficients in
this second analysis; in addition to identifying that several of the
relationships between the different variables lose significance statistics.
This allows us to identify that associativity, in addition to contributing
significantly to improve the competitiveness of SMEs, as it demonstrates that
it contributes to the other dimensions established in the study harmonize and
work according to the same objective.
Next, in Figure 3 the methodological congruence
is observed, with the approach of the variables^{3} and the degree of relationship of each
independent variable.
Figure 3. Structural equation
modeling. Source: Author’s elaboration.
Figure 3 estimates the modeling of structural
equations, where the dependent variable “Associativity” has a greater
correlation in the development of the productive chain; it is explained by the Direct
Actors, Logistics and Public Policies, which are in the parameter of 0.764,
0.7567 and 0.7608, correspondingly, as shown, through ovals of the variables of
less significance that obtained a parameter of 0.30 and 0.040, the alphas of
the variables do not contribute to the prediction of the independent variable,
that is due to their low alpha and the size of the sample.
Conclusions
The purpose of the investigation determined that
the economic aspects of the proposed models of Cournot and Bertrand, they apply
duopolist ideas that could determine that the
microentrepreneurs could choose one or the other, quantity of offer or price,
as a parameter of action in the lower index impact for the logistic volume and
optimal production lot.
However, it can only be consumers who decide
whether they consider the product supplied to be homogeneous or not. What makes
suppliers can do is just conclude and market the product to convince consumers
that there are different brands with their defined advantages.
This is treated under the title that business
competitiveness can apply mathematical procedures that allow measuring the risk
of the application of price discrimination in standard microeconomics. And this
is precisely what we propose to do, although we need three different markets;
the common duopoly market (Cournot) and two discriminatory monopoly markets
(Bertrand), one for each supplier.
However, there is a direct and significant
relationship between the associativity and the dimensions of the competitiveness
of SMEs in the commerce sector of the study area, so the logic can be rectified
how competitors can influence the sizes of consumer groups to through several
devices. Mathematically the model produces bifurcations this gives the
possibility to study more variables much more intriguing before the exposed
ones.
The Associativity variable of the productive
chain is more accepted and explained by the SMEs of the sector than the other
study variables established such as; direct actors, logistics and government
policies.
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* This article is the result of the research
project: “Business competitiveness of SMEs.”
** Master in Industrial Security. ESPOCH. Riobamba, Ecuador.
Email: ccepeda@espoch.edu.ec. ORCID: 0000000245664180. Google Scholar: https://scholar.google.es/citations?hl=es&user=DtQLXpYAAAAJ
*** Master in Mechanical Engineering. ESPOCH. Riobamba, Ecuador. Email: andres.noguera@espoch.edu.ec. ORCID: 0000
000167639288. Google
Scholar: https://scholar.google.es/citations?hl=es&user=Uskz P1EAAAAJ
**** Master in Integrated Quality, Environment and Safety Management Systems. ESPOCH. Riobamba,
Ecuador. Email: monica.moreno@espoch.edu.ec. ORCID: 0000000298816360. Google Scholar: https://scholar.google.es/citations?hl=es &user=g6osJr IAAAAJ
***** Master in Hydraulic Engineering and Environment. ESPOCH. Riobamba,
Ecuador. Email: nelson.chuquin@espoch.edu.ec. ORCID: 0000000189981156. Google Scholar: https://scholar.google.es/citations ?hl=es&user=gurKt_QAAAAJ
****** Master in Educational Management. ESPOCH. Riobamba, Ecuador. Email: pvillagomez@espoch.edu.ec. ORCID:
00000001 8490272X. Google
Scholar: https://scholar.google.es/citations?hl=es&user= Eo0qQkQAAAAJ
******* Master in Industrial Security. ESPOCH. Riobamba, Ecuador. Email: julio.oleas@espoch.edu.ec. ORCID:
000000028576 248X. Google
Scholar https://scholar.google.es/citations?hl=es&user=cE18sAAAAJ
1 The measure of the
KaiserMeyerOlkin sample adequacy contrasts whether the partial correlations
between the variables are small. Bartlett’s test of sphericity contrasts
whether the correlation matrix is an identity matrix, which would indicate that
the factorial model is inadequate.
2 R2 is the percentage
of variation of the response variable that explains its relationship with one
or more predictor variables.
3 The ovals
are the latent variables or constructs, the arrows that join the ovals are beta
coefficients and the small arrows that are next to the rectangles are
estimation errors.
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