APPLICATION OF A MATHEMATICAL MODEL FOR THE MINIMIZATION OF COSTS IN A
MICRO-COMPANY OF THE GRAPHIC SECTOR
Paulo Cesar Chagas Rodrigues
Instituto Federal de Educação, Ciência e
Tecnologia de São Paulo, Brazil
E-mail:
paulo.rodrigues@ifsp.edu.br
Fernando Augusto Silva Marins
Universidade Estadual Paulista, Campus
Guaratinguetá, Brazil
E-mail:
fmarins@feg.unesp.br
Fernando Bernardi de Souza
Universidade Estadual Paulista, Campus
Bauru, Brazil
Submission: 08/04/2015
Accept: 01/05/2015
ABSTRACT
Supply chain
management, postponement and demand management are one of the operations of
strategic importance for the economic success of organizations, in times of
economic crisis or not. The objective of this article is to analyze the influence
that a mathematical model focused on the management of raw material stocks in a
microenterprise with seasonal demand. The research method adopted was of an
applied nature, with a quantitative approach and with an exploratory and
descriptive objective. The technical procedures adopted were the
bibliographical survey, documentary analysis and mathematical modeling. The
development of mathematical models for solving inventory management problems
may allow managers to observe deviations in trading methods, as well as to
support rapid decisions for possible unforeseen market or economic variability.
Keyword: Postponement; Supply Chain Management; Demand Management;
Inventory Management; Mathematical Modeling
1. INTRODUCTION
Since
ECO92, organizations have begun to worry about the conscious consumption of raw
materials and production processes, aiming to reduce waste and costs that can
be minimized.
And
mathematical modeling meets the managers' longings, as it can help in making
decisions about how much, how, where and when to buy, so that less waste is
consumed.
How to
reduce cost of production without affecting the quality and availability of the
product in the market? From the strategy of postponing inventories. This
strategy combined with management can allow a reduction in the cost of the
product, because this is talking about reducing the risk of loss of the
finished product, because it has a high value added, increased flexibility in
adapting the needs of the market.
The
research method adopted was of an applied nature, with a quantitative approach
and with an exploratory and descriptive objective. The technical procedures
adopted were the bibliographical survey, documentary analysis and mathematical
modeling.
The
purpose of this article is to analyze the influence that a mathematical model
focused on the management of raw materials inventories in a company with
make-to-stock production system and with seasonal demand.
2. THEORETICAL REFERENCE
According to Edalatkhah (2006), in
the new economy, supply chains are needed to serve varied markets around the
world, set custom product deliveries, change planning, never together with
speed and accuracy considered possible before. Managers need to work with
multiple partners to monitor the activities that are being run together in
order to solve problems and delays that may occur.
According to The Global Supply Chain
Forum, supply chain management consists of integrating key processes from
consumers to the producer of raw materials. GCS involves several areas, such as
forecasting demand, purchasing, production, distribution, inventory and
transportation, interacting in strategic, tactical and operational perspectives
(MCADAM and MCCORMACK, 2001).
According to Tan (2002), GCS
involves the integration of business processes through the supply chain,
encompassing the coordination of activities and processes not only within an
isolated organization, but among all those that make up the supply chain.
For Rodrigues and Oliveira (2009),
demand management is a practice that allows managing and coordinating the
supply chain in the opposite direction, that is, from the consumer to the
supplier, in which consumers initiate actions for the supply of products making
the Productive system.
Darú and Lacerda (2005) and Perazza
and Rodrigues (2010) describe that manufacturing for stock is a common
practice, whenever the demand can be predicted, being able to take advantage of
off-season moments to be produced, using resources better and loading them more
Balanced. But, this policy has some disadvantages, which would be the high cost
of storage and the difficulty of predicting what will be sold.
According to Van Hoek and Dierdonck
(2000), Verol (2006) and Zang and Tan (2010), the concept of postponement is
that the risk and costs of uncertainty lie in the differentiation (of form,
place or time) of products that occurs during (a) product design: the specific
content of the delayed (delayed) operation, (b) process: the time at which the
activities are carried out, the activities of manufacturing, storage and
delivery, based on the characteristics of the product / process in the supply
chain: Are delayed in the process, and (c) place: the location where the
postponement takes place.
Ng and Chung (2008) comment that the
strategic placement of the point of decoupling of the supply chain, the
strategy of postponement can be used. The goal of postponement is to increase
the efficiency of the supply chain by moving product differentiation (at the
point of dissociation) closer to the end user. Because risk and uncertainty are
the costs linked to the differentiation of goods and the differentiation could
occur in the product itself and / or the geographic dispersion of inventories.
Form postponement consists of
manufacturing a base or standard product in sufficient quantities to achieve
economies of scale, while finishing features such as color, packaging, etc. Are
postponed until consumer applications are received and are classified into four
levels: labeling or labeling, packaging, assembly and manufacturing (FERREIRA;
BATALHA, 2007).
Mendes et al. (2008) based on Zinn (1990) describe and classify in 4 the existing subdivisions in the postponement of form that they describe a brief definition.
a) Postponement of labeling: Products are
stored without any sort of classification. The labels and labels will be
affixed when an order is made, and the customer specifies the mark that will
identify the final product;
b) Manufacturing postponement: The last
manufacturing steps are only completed after confirmation of the customer's
order. Semi-finished products or even in the form of inputs are stocked so that
the differentiation of the merchandise takes place at a time or place closer to
the demand;
c)
Product
postponement: Products can be designed following a logic of modules, or
even standardized components to facilitate further differentiation; And
d) Process postponement: Production and
distribution can be designed in a way that allows the differentiation of the
product downstream and upstream of the supply chain.
Yang, Burns and Backhouse (2003),
Engelseth (2007) argue that place postponement involves the delay of
transporting goods downstream in the chain until the orders are received, thus
keeping the goods centrally and not having them in a specific place.
According to Wallin, Rungtusanatham
and Rabinovich (2006), Bailey and Rabinovich (2006) and Drohomeretski, Cardoso
and Costa (2008), the time delay strategy assumes that the product will be
requested from the supplier only when a customer request, Which will enable the
reduction of stock levels and the obsolescence of the product.
3. MODELING
The model can be considered causal,
since it depends on present or past conditions, but also on dynamics, since the
variables vary in time, in which the solution consists of the constant and
temporary, deterministic regimes, when the result can be calculated exactly and
Because the outputs are linearly dependent on inputs and possible disturbances.
It was chosen here, to facilitate
the understanding of the modeling stage, considering only two products, being
its generalization evident.
We considered 12 months (N) of production scheduling. Decision variables, Xij,
refer to the quantities to be produced of products i in period j. The unit
production costs of each product, its storage costs, and the percentage annual
cost of annual insurance of inventories of finished products are known.
The auxiliary variables Y1 to Y7
refer to the parameters of the raw materials used for the production of the
products i. These variables will allow to measure the total quantities of raw
materials to be used to produce during the 12-month period.
It should be noted that in practice,
according to information from the managers of the companies studied, the
variables Xij are influenced by the estimated demand according to the
information of the years 2010 and 2011, in order to allow the forecast for the
years 2011 and 2012 respectively.
Indexes:
i
Is the index linked to the products, i
{1,2,...,n};
j
Is the index linked to the products,
j {1,2,...,m}.
Parameters:
Cost [R$]
inventory in the period j;
Ej Cost of purchasing the paper in
period j;
Cost [R$]
of wire purchase in period j;
Cost [R$]
of ink in period j;
Oij Cost [R$] of sale loss
associated with product i in period j;
hij Cost rate [%] of annual
insurance of product i in period j;
CPij Cost [R$] of production of
product i in period j;
Dij Demand of product i in period j;
DPj Demand of the raw material paper
in period j;
DAj Wire raw material demand in
period j;
DTj Ink raw material demand in
period j;
Variables:
Xij Estimated
quantity [unit] of product i in period j;
Quantity
[kg] of paper to be purchased in period j;
Quantity
[l] of ink to be purchased in period j;
Quantity
[kg] of wire to be acquired in period j;
ρij Quantity
ceased to be produced from product i in period j;
PAj Level of inventory of the raw material paper in period j;
AAj Leveling of inventory of the raw material wire in period
j;
TAj Leveling of the stock of raw ink in period j;
Wij Level and
stock of product i in period j;
The objective function (1) of the
proposed model for n planning periods, associated with the reduction of raw
material, product in process and finished product stocks, seeks to minimize the
respective costs of production and storage of raw material stocks And finished
product:
The objective function (1) presents
the minimization of the total cost, which is represented by Z.
|
(1) |
The Restriction (2)
leveling of paper stock in period j..
|
(2) |
A Restriction (3) wire stock leveling in period j.
|
(3) |
The Restriction (4) ink leveling in period j.
|
(4) |
The Restriction (5)
product leveling of product i in period j.
|
(5) |
The Restriction (6)
loss of sale of product i in period j.
|
(6) |
The Restriction (7)
presents the domain of the variables.
|
(7) |
4. COMPANY
The
production of the company meets part of the demand of the southwestern region
of São Paulo, the focus of its production are carbon reels for POSs aimed at
commercial institutions.
Currently
the company has approximately five hundred direct customers, who buy their
product for use in shops, bars and restaurants.
Table 1
shows the composition of the objects studied and their units of measurement.
These two products served to study the behavior of the model in relation to the
strategies adopted by the managers during the years of 2012.
Data
collection was done through interviews with the manager of the company and through
the analysis of documents referring to the production of products 1 and 2 in
the year 2012.
Table 1: Raw
Materials and Company Products C
Customized two-color POS coil X1 |
Customized one-color POS coil X2 |
Y1: Paper 55g (m2) |
Y1: Paper 55g (m2) |
Y2: Tubete 80x12x15 cm (Un) |
Y2: Tubete 80x12x15 cm (Un) |
Y3: Label (un) |
Y3: Label (un) |
Y4: Packing box
nº 4 (Un) |
Y4: Packing box
nº 4 (Un) |
Y5: Adhesive tape 48 mm (m) |
Y5: Adhesive tape 48 mm (m) |
Y6: Box label
(Un) |
Y6: Box label
(Un) |
The supply chain for the production
of the customized two-color POS coil and the one-color POS coil can involve
approximately 18 suppliers, as each raw material can have up to three
suppliers. The data presented in Tables 2 to 10 and Figures 1 to 7 refer to the
production for fiscal year 2012.
Table 2 shows the two products
analyzed and their composition in terms of the raw materials, the time and cost
of production and the sale price.
In Table 2, Y7 production time is
expressed in seconds worked to produce a unit, Y8 which is the cost of
production of one unit and the sales price Y9 are in Reais.
Table 2: Use of raw
material/product, production/product time, cost of production/product and
selling price/product 2012
Items
to produce |
|||||||||
Product |
Y1 |
Y2 |
Y3 |
Y4 |
Y5 |
Y6 |
Time Y7 |
Cost Y8 |
Sale
Price Y9 |
Customized two-color POS
coil X1 |
0,210 |
1 |
1 |
1 |
1 |
1 |
22 |
2,43 |
3,50 |
Customized one-color POS
coil X2 |
0,190 |
1 |
1 |
1 |
1 |
1 |
19 |
2,15 |
3,15 |
Table 3 deals with the levels of
safety stocks of the two-color custom POS and single-color POS coil products,
expressed as percentage values during the 12 months, since the volumes of
security stocks may vary, depending on the volume to be produced In the month.
Table 3: Security
stock of items to be produced [%/month] 2012
Security
Inventory of Items to Produce in % |
||||||||||||
|
Jan. |
Feb. |
March |
April |
May |
June |
July |
Aug. |
Sept. |
Oct. |
Nov. |
Dec. |
Production Days |
15 |
17 |
21 |
21 |
20 |
20 |
23 |
21 |
21 |
22 |
19 |
12 |
Resupply Interval |
9 |
14 |
12 |
12 |
15 |
13 |
12 |
12 |
14 |
10 |
10 |
9 |
ES1i |
0,0013 |
0,0016 |
0,0018 |
0,0022 |
0,0025 |
0,0018 |
0,0016 |
0,0014 |
0,0016 |
0,0012 |
0,0010 |
0,0008 |
ES2i |
0,0014 |
0,0018 |
0,0016 |
0,0018 |
0,0027 |
0,0017 |
0,0018 |
0,0016 |
0,0018 |
0,0013 |
0,0011 |
0,0008 |
Table 4 shows the proportion of
units to be produced per month of each product. Two-color custom POS coil and
one color customized POS coil. This ratio may vary depending on the monthly
requirement of what should be produced for each model and is based on
historical data for the year 2012.
Table 4: Proportion
of units to be produced [%/month] 2012
Proportion of Units to Produce per month |
||||||||||||
|
Jan. |
Feb. |
March |
April |
May |
June |
July |
Aug. |
Sept. |
Oct. |
Nov. |
Dec. |
Customized two-color POS
coil X1 |
19000 |
28000 |
32000 |
28000 |
30000 |
31000 |
32000 |
35000 |
38000 |
40000 |
40000 |
25000 |
Customized one-color POS
coil X2 |
21000 |
32000 |
29000 |
23000 |
33000 |
29000 |
36000 |
40000 |
42000 |
45000 |
45000 |
27000 |
Table 5 shows the times available
for the production of the products: Two-color customized POS coil and one color
personalized POS coil, during the 12 months of the year, times may be adjusted
during the months, and may be increased or decreased, according to need.
Table 5: Total
production time per product [min/month] 2012
Estimated
total production time per product per month (minutes) |
||||||||||||
|
Jan. |
Feb. |
March |
April |
May |
June |
July |
Aug. |
Sept. |
Oct. |
Nov. |
Dec. |
Customized two-color POS coil X1 |
8708 |
11978 |
14426 |
12481 |
10127 |
13829 |
13804 |
14202 |
14804 |
15529 |
15529 |
11987 |
Customized one-color POS coil X2 |
8313 |
11822 |
11291 |
8854 |
9621 |
11173 |
13412 |
14018 |
14131 |
15088 |
15088 |
11181 |
Table 6 shows the quantities to be
produced of the two-color custom POS and two-color POS coil products within 12
months, which are calculated by the Excel Solver and the data that were
reported in Tables 2, 3 and 4 (Cj), demand (Dj), units to be produced (UPj),
and the rate of growth of the supply chain. Cost of annual insurance (hj).
Based on this and on Tables 2, 3 and 4, the information regarding the
expectation of purchases of raw materials will be generated, already providing
for a safety margin in the stock, besides presenting in the objective function
the minimization of the costs of production, storage and Acquisition of raw
materials, among other factors that can be analyzed.
Table 6: Quantities
to be produced [product/month] 2012
Period |
Qtd X1 |
Qtd X2 |
Cj |
Dj |
CPj |
UPj |
UCj |
UPi |
PCj |
TCi |
ACj |
ECj |
ADj |
ESj |
January |
23750 |
26250 |
0,128300 |
50000 |
87000 |
50000 |
50000 |
50000 |
9975 |
50000 |
50000 |
50000 |
50000 |
0,0014 |
February |
32667 |
37333 |
0,099200 |
70000 |
87000 |
70000 |
70000 |
70000 |
13953 |
70000 |
70000 |
70000 |
70000 |
0,0017 |
March |
39344 |
35656 |
0,122500 |
75000 |
87000 |
75000 |
75000 |
75000 |
15037 |
75000 |
75000 |
75000 |
75000 |
0,0017 |
April |
34039 |
27961 |
0,122500 |
62000 |
87000 |
62000 |
62000 |
62000 |
12461 |
62000 |
62000 |
62000 |
62000 |
0,0021 |
May |
27619 |
30381 |
0,116700 |
58000 |
87000 |
58000 |
58000 |
58000 |
11572 |
58000 |
58000 |
58000 |
58000 |
0,0026 |
June |
37717 |
35283 |
0,116700 |
73000 |
87000 |
73000 |
73000 |
73000 |
14624 |
73000 |
73000 |
73000 |
73000 |
0,0018 |
July |
37647 |
42353 |
0,134700 |
80000 |
87000 |
80000 |
80000 |
80000 |
15953 |
80000 |
80000 |
80000 |
80000 |
0,0017 |
August |
38733 |
44267 |
0,122500 |
83000 |
87000 |
83000 |
83000 |
83000 |
16545 |
83000 |
83000 |
83000 |
83000 |
0,0015 |
September |
40375 |
44625 |
0,122500 |
85000 |
87000 |
85000 |
85000 |
85000 |
16958 |
85000 |
85000 |
85000 |
85000 |
0,0017 |
October |
42353 |
47647 |
0,128300 |
90000 |
95000 |
90000 |
90000 |
90000 |
17947 |
90000 |
90000 |
90000 |
90000 |
0,0012 |
November |
42353 |
47647 |
0,110800 |
90000 |
95000 |
90000 |
90000 |
90000 |
17947 |
90000 |
90000 |
90000 |
90000 |
0,0011 |
December |
32692 |
35308 |
0,110500 |
68000 |
87000 |
68000 |
68000 |
68000 |
13574 |
68000 |
68000 |
68000 |
68000 |
0,0008 |
Annual insurance cost rate % |
0,0155 |
0,0167 |
||||||||||||
Function Purpose |
34201,05 |
Table 7 presents the production time
data (in minutes) that should be reported by the PCP manager. These data will
influence the results of Tables 8, 9 and 10.
Table 7: Total
production time estimated by managers per product [min/month] 2012
Estimated total production time per product per month (minutes) |
||||||||||||
|
Jan. |
Feb. |
March |
April |
May |
June |
July |
Aug. |
Sept. |
Oct. |
Nov. |
Dec. |
Customized two-color POS
coil X1 |
6000 |
12000 |
15000 |
12000 |
11000 |
13500 |
14000 |
14000 |
14900 |
15500 |
15530 |
14000 |
Customized one-color POS
coil X2 |
7000 |
13000 |
11000 |
9000 |
9700 |
11100 |
13400 |
14000 |
14100 |
15100 |
15100 |
14000 |
Table 8 presents the results
regarding the quantity of product that was no longer produced or could be
produced, according to the times reported in Table 7 and those generated during
the execution of the Solver (Table 5).
Table 8: Production
losses [product/month] 2012
Table 9 presents the result
regarding possible production losses (in minutes), which refer to what could
not be produced or could be produced during the year.
Table 9: Production
losses [min/month] 2012
Table 10 presents the costs of sales
losses and refers to what could not be produced or could be produced during the
year.
Table 10: Cost of
sales losses [R $ / month] 2012
In an attempt to explain the values
in Table 2, Figures 1 to 4 were elaborated. Figure 1 shows the volume to be
produced in 2011.
Table 6 presents the monthly demand
(DJ) that the company has of the two products, in which it can be observed that
it has a significant growth between the months of June and October. Figure 1
shows that the quantities to be produced seek to keep pace with this demand.
Figure 1:
Graphic referring to the volume to be produced 2012
Figure 2 shows the overall volume of
units to be produced during the 12 months of this year 2011.
Figure 2:
Graph of units to be produced in the 12-month period 2012
Figure 3 shows the evolution of the
global security stock in the 12-month period, the percentage values show the
period in which it may be smaller or larger, in order to meet emergency
requests that may exist.
Figure 3:
Chart of the volume of security stocks in% in the period 2012
Figure 4 shows the quantities that
were not produced during the year 2011, with the green line referring to the
total quantity not produced.
Figure 4: Chart of quantity that was not
produced 2012
The company manager commented that
the model was interesting, as it would complement some features that it finds
very complex in ERP5 and allows a simple tool to be used and that its employees
know, in this case Excel.
Strong points:
·
It allowed
the flexibility of decisions on what to produce and the amount of raw material.
ü The model
allowed flexibility in decisions about what to produce, the amount of raw
material needed and the time of production, since these decisions can be altered
and observe the possible consequences. The model also allows a dynamism to be
maintained regarding the alteration of the data and consequently to the
presented information.
·
It allowed
the validation of the strategies adopted and the results presented by the ERP5.
ü
The results obtained with the model for the year 2012
show that the strategies of purchase of raw material and of production were
close to what the company defined. From the graphs presented, it was possible
to observe the deviations between the model and the decisions made.
ü
With the results presented by the model and ERP5, it was
possible to observe how the two software complement each other, when analyzing
the strengths and weaknesses of the two applications.
·
It allowed
to analyze the demand, productive capacity and seasonality.
ü
The results that are based on the demand and the
productive capacity of the company were interesting, since it could be observed
that the company can study the increase of productivity in certain periods of
the year.
ü
With the application of the model, it was possible to
observe the seasonality that influences the demand and consequently the
productivity and thus review the productive strategies, in order to maintain a
production that meets the needs not only of the month in question, but also of
the next.
·
It allowed
the application of the model in a considered small company.
ü
Because it is a small company and because it does not
have so much financial resources, the model has shown to be interesting as
support for decision making. And if the model is documented in its routines and
the mode of operation can facilitate its application.
ü
Based on the model and meetings with the researcher, it
was possible to review the way and the variables that made up the cost of
production, since the company composed the production cost with the hourly man
/ machine value, which was then seen as A fixed cost, that is, that regardless
of the quantity produced it does not change, starting from this premise the
cost of production was revised.
ü
The model made it possible to observe more critically the
raw materials involved directly in the production of a given product, for
example: packaging was considered as a raw material directly involved in the
production of the product, which made the cost Of the packaging was part of the
cost of production.
Weak point:
·
Lack of costing of raw materials.
ü
The model could make it possible to predict the total
costs for the acquisition of raw materials and labor.
Recommendations for improvement:
·
Expansion of results, allowing
prediction of inventory levels.
ü
The model could be improved with regard to the
formulation of the calculation of the safety stock and until it allows the
calculation of the minimum stock, informing the maximum time between the
request and the arrival of the raw material.
ü
A table could also be presented with the result of what
should actually be produced, and the quantities of raw material needed for
production.
5. FINAL CONSIDERATIONS
After the test of the model with the
data collected in the company focus of the study, some analyzes were made
regarding the consistency of the model and consequent adherence regarding its
applicability when defining the productive strategies.
The model allowed to observe about
the ways in which some information is obtained and to present the possible
deviations that information conflicting or not validated by the management can
cause during the execution of the model and the results generated.
The difference between the
production times generated by the model and those defined by the manager of
company C is relatively low, but one must observe the productive capacity,
demand, consumer market of its products and the quantity of products that the
company.
The Company's interviewee commented
that changes in the definition of production times are being carried out by the
company aiming at a production planning that reduces the costs associated with
meeting the demand of a particular item, observing the possible seasonality.
The general objective of this work
was to "Analyze the influence that a mathematical model focused on the
management of raw material stocks in a microenterprise with seasonal
demand". It can be affirmed that this objective was successfully achieved,
because the mathematical model adhered to the reality of the company's focus of
the study.
As can be observed in the previous
graphs, this objective was reached, since it presented results that
collaborated to minimize the inventory and consequent minimization of storage
costs.
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