1. INTRODUCTION
In the competitive
context in which companies are inserted, the decision process does not admit
failure in solving problems or results fall short of those achieved by
competition. In this context, it is present the importance of preparing
managers assuring they know the tools and techniques used to make decisions and
that they are able to realize the analytical methods implementation such as
linear programming (LP) for solving complex problems, thus optimizing the
results of organizations (BARCELOS, et al., 2012).
Also
according to Barcelos et al. (2012), PL is a technique for decision-making,
with problems mathematical modeling, seeking optimal solutions applied to
business reality. The manager, as decision maker, may be beneficiary with the
information extracted by LP results.
The
PL can also be used in planning public and private operations, in resource
allocation systems, arrangement of jobs, mathematical programming and
humanitarian aid, with adequate grip for application in practical and real
problems (KARSU; MORTON, 2015).
Previous studies have used linear programming in investigating transportation problems in receiving and distribution operations (OLIVEIRA, 2014), in tthe decision to buy or sell used vehicles (COSTA, et al., 2014), and it was also when could be noted the role of materials logistics (RONCARI; DELMANTO JUNIOR, 2012). Despite important role in the set of activities involving a supply chain, the expedition has not been included in the models that investigate transportation problems optimization. From this gap, this research sets out the following problem question: How to model expedition in the context of optimizing a transportation problem?In this sense, the aim of this study is to propose an optimization model of transportation operations to include in his analysis the shipping activity. In order to operationalize the research, a case study was conducted on a midsize retailer located in the great ABC/SP region, aiming to verify the usefulness of the proposed model.Besides the theoretical relevance, it is appropriate to conduct this research due to the fact that transportation operations, when not properly optimized, increase significantly the way logistics costs influence the operation speed. This can cause a decrease to the customer service levels and increase the product cost. It is important to note that the existence of transportation problems also entails higher costs due to inventory and storage.
According
to the ILOS Panorama study (2014), expenditure with logistics in companies,
considering only the transportation, inventory and storage costs, they
represent 8.7 % of net revenue. Thus, the optimization of transportation
operations with the inclusion of shipping in the model becomes an important
issue in order to obtain competitive advantage in the market. The next section
presents the theoretical framework.
2. THEORETICAL FRAMEWORK
2.1. Operational research models and linear programming
Knowledge and optimization of the production process
is essential to increase efficiency and keep the company competitive in the
market. In production environments, mathematical models are used for accurate
processes analysis of and decision-making. Decisions are made at the most
strategic, tactical and operational levels, and it means that choosing a choice
between alternative solutions, implies in influencing corporate performance.
(RAYMUNDO; GONÇALVES; RIBEIRO, 2015).
Andrade (2009) explains that the current approach, in
recent decades, with the problems analysis using linear programming, considers
the quantitative focus on the operational research, which is the recognition
that the quantitative approach to problems provides a reasoning structure and
analysis which allow to develop a systemic view of the process. This approach
can also serve as a support to decision-makers.
The idea of using models to solve problems and
make decisions is not relatively new and it is not tied exclusively to the use
of computers. These models can be simply structured, similarly to mental
models, until the use of mathematical methods, commonly used when only
intuition or experience does not provide enough information for decision making
(RAGSDALE, 2007).
Mathematical models have advantages over a verbal
description of the problem. One is that the mathematical model describes the
problem much more concisely (HILLIER; LIEBERMAN, 2010). According to Arenales
et al. (2007), mathematical optimization models are strictly related to the
study of operational research, as this area of study is not only used to
define the ideas and processes of decision problems, but also to optimize numerical
systems that use data in the models.
Once the mathematical model is consisted, it is
possible to obtain an optimal solution to the problem. Several mathematical
methods are in constant development. The software that automated these models
are important tools for solving complex problems. Among the specific software
for linear programming, it is worth mentioning the following: Solver Excel®,
which is a tool with simple commands, but able to provide accurate results for
more limited models; the LINDO® - Linear Discrete Optimizer; and CPLEX®, these
two last ones used for more complex problems (MARINS, 2011).
Mathematical models of problem solving are simplified
representations of reality. The resolution of these problems is divided into
five stages: problem formulation; construction of the mathematical model;
solution obtaining; solution testing; and implementation (MARINS, 2011).
Marins (2011) also states that the use of mathematical
models in decision making allows to describe the essence of the problem and to
identify what are the relationships among the studied variables, their relevant
data, and the most important variables. Operational research can be used to
solve several areas problems. The most common problems can be:
· Linear
Programming: production mix, raw materials mix, investment base, vehicle
routing;
· Network
Models: transportation routes, goods distribution and transportation, project
monitoring;
· Queuing
Theory: traffic congestion, hospital operations, allocation of service teams.
Lachtermacher (2007) clarifies that the Operational
Research (OR) is an area of study that integrates computers, statistics and
mathematics to solve real problems applied to businesses under three main
objectives: to convert data into meaningful information; to support the
decision-making process; and to create useful computer systems to non-technical
users.
The OR is closely related to the decision-making
process within companies that use it, especially when it comes to raising a
problem or identifying an opportunity with the appropriate action lines
(LACHTERMACHER, 2007). Basically, modeling a problem occurs through: physical
models, analogous models and mathematical or symbolic models.
Mathematical models are the most commonly used in
management analyses involving the issue of distribution of limited resources
among the company’s activities (VANDERBEI, 2008).
Within the mathematical programming, there is a
subdivision that will depend on the type of functions used in
functions-objective and constraints. In linear programming, all of the
functions-objective and constraints are represented by linear functions. On the
other hand, regarding the nonlinear programming, at least one of the
functions-objective and/or constraints are represented by nonlinear functions
(LACHTERMACHER, 2007).
As mentioned earlier, the linear programming is a
mathematical technique that aims to find the best solution to problems that
have their models represented by linear expressions. In many ways, linear
programming is one of the most used Operational Research techniques when
involving optimization problems. In this sense, its modeling seeks the
efficient distribution of limited resources, targeting to meet a specific goal,
in general, to maximize profits or minimize costs (ANDERSON; SWEENEY; WILLIAMS,
2007). In the context of linear programming, the next section specifies the
transportation problems.
2.2. Transportation problems
The transportation problem is considered the most
representative of linear programming problems. It is of wide practical application
and has been studied by many researchers, although it was Danzig (1953; 1963)
the first to establish its formulation in LP and to propose a systematic
resolution method, as well as being the creator of the simplex method
(CANAVARRO, 2005).
Charges urban transportation is a fundamental activity
for cities economic development. However, adverse effects on the environment
are observed, such as the commitment of vehicular traffic conditions, high
energy consumption and pollutants emission (CASTRO, 2013).
In recent years, many studies have been conducted in
regards to traffic levels, and its impact in large cities. These focus mainly
on the analysis of public transportation and on private vehicles with
relatively little concern about urbanely transporting charges (BROWNE et al.,
2005).
The Vehicle Routing Problem (VRP) is also an issue of
strategic importance and difficult computational solution (LAPORTE, 1992). So,
it has been studied for several decades by several researchers around the
world. A route is a sequence of delivery and/or collecting points that the
vehicle must cover neatly, starting and ending in a warehouse. A Vehicle
Routing Problem (VRP) basically consists in establishing and organizing
efficient routes for vehicles carrying out the distribution of goods, then
minimizing costs. By featuring a fleet of vehicles, identical or not, it is
desired to meet a set of n customers,
each of them with a specific demand (PINHEIRO, 2015).
Based on the concepts reviewed in section 2, and on
the problem outlined in the introduction, this study sets out the following
research hypothesis:
H1: The
expedition model making use of linear programming allows to increase a
company’s operations efficiency.
With the hypothesis highlighted, the next section
deals with the methodology.
3. METHODOLOGY
This study was conducted in a
distribution center - CD of a midsize retailer located in ABC Paulista region,
which supplies 20 stores nearby. For the execution of this study, interviews
were carried out with professionals involved in the expedition operation, as
well as field visits to observe the operations at the distribution center and,
in addition, documents that contributed to the understanding of the
characteristics of Company operation were assessed. The data obtained in the
research have qualitative origin (interviews) and quantitative origin (data
operation) too.
In total, there were six field
visits. The first visit aimed an overview of the company, where it was possible
to identify its market strategy. The second visit, aimed to recognize the chain
distribution center operations deeply. In the third, fourth, fifth and sixth
visits, the expedition operations were studied, once it is this study object,
stage in which there was a comprehension about the way the operation is
performed, what its characteristics are, its strengths and weaknesses. Then,
individual interviews with important people in expedition operation were held,
which they are, the Logistics Operator, the charge and expedition Leader and
the Supply Chain Manager. The opinion of each of them was asked in order to
confront the information obtained in interviews with what was seen on field
visits.
Next, operation documents were analyzed targeting to
measure important indexes to sustain the information obtained from the
interviews with professionals and the conclusions taken from field visits. At
last, the data obtained through field visits, interviews and specific documents
were triangulated with the intension to validate data obtained throughout the
case study stages (YIN, 2010).The next section presents results analysis.
4. RESULTS ANALYSIS
4.1. Description of the company
The case study was conducted in a
food distributor that has 20 stores and a Distribution Center (DC) in the ABC
Paulista region. The DC has 12 vehicles to meet the shops demand, 10 trucks and
2 van trucks, with capabilities to 14 and 12 pallets respectively. The DC
demand is composed by pallets of household appliances and electrical portable,
cross-docking and picking. Cross-docking goods have slow-moving turnover,
unlike the picking products, which have constant output and high inventory
turnover. Finally, household appliances and electro portable that comes from a
sector called CD2.
It was found that, in the company,
there was no method to determine which stores would be met first. Empirically,
the employees stipulated intuitively what stores would be served for first,
because the separation of goods was performed 24 hours a day, in three shifts,
while the expedition only operated from 8 a.m. to 4:20 p.m. from Monday to
Saturday.
In this scenario, a bottleneck was
identified due to the fact that the expedition could not send all the pallets
that were separated, so every day in the morning, before expedition operation
started, there was an average of 415 pallets to be sent to the shops, and there
was no plan of how these pallets would be sent. In the operational system the
company worked, as the charges came, the expedition sent them. From this
context, the next section deals with the problem modeling.
4.2. Problem modeling
It is proposed in this section, the
development of a model for expedition planning, where the pallets to be sent at
the beginning of the operation would already have defined destination, turning
the process more systematic, considering that each vehicle would have a
specific demand to serve on.
The total time of daily operation
was defined through Equation 1:
(1)
Where:
TTO :
Total time of operation
Qv :
Quantity of vehicles
Jt : Working
hours
Ha :
Lunch time
The cycle time for each store was
obtained through Equation 2::
(2)
Onde:
TCl:
Store cycle time
Tc:
Charging time
Ti:
Coming time
Td:
Discharging time
Tr:
Returning time
The store cycle time measures how
many minutes each vehicle takes to leave since goods charging, from the DC, get
to the store, discharge and return.
Thus, it has as calculation the sum of times: loading; coming; discharging and
returning.
After obtaining the cycle time and
the total time of operation it was carried the expedition problem modeling
using SOLVER to obtain a expedition programming which could maximize the amount
of pallets sent to stores. Therefore, it was modeled to function objective
shown in Equation 3:
(3)
Where:
X: is
the number of trips
C: vehicle capacity
i:
store 1, 2, ..., 20
j= 12
(van truck)
l= 14
(truck)
The vehicle capacity restriction
was modeled according to Equation 4:
(4)
Where:
Xjl: is
the number of trips to store i using the vehicle j or l;
to: van
truck
tr:
Truck
Serving restrictions demand were
also included, as shown in Equation 5:
(5)
Where:
DP: is
pallets demand
It was also necessary to add an
integer numbers restriction to variables X, because,fractional numbers would
mean an incomplete trip with the vehicle. From the definition of the problem
model under analysis, the next section presents data used as input for further
analysis.
4.3. Data collected in the company
Initially, cycle times for each
store were obtained, as shown in Table 1.
Table 1: Cycle time of the stores
|
Time (Min) |
|||||
|
Store |
Charging |
Route |
Discharging |
Returning |
Cycle |
|
1 |
55 |
24 |
26 |
20 |
125 |
|
2 |
55 |
25 |
99 |
18 |
197 |
|
3 |
55 |
42 |
61 |
38 |
196 |
|
4 |
55 |
22 |
74 |
18 |
169 |
|
5 |
55 |
15 |
34 |
14 |
118 |
|
6 |
55 |
49 |
56 |
39 |
199 |
|
7 |
55 |
30 |
56 |
25 |
166 |
|
8 |
55 |
29 |
67 |
25 |
176 |
|
9 |
55 |
40 |
65 |
37 |
197 |
|
10 |
55 |
34 |
75 |
32 |
196 |
|
11 |
55 |
43 |
69 |
35 |
202 |
|
12 |
55 |
30 |
45 |
27 |
157 |
|
13 |
55 |
33 |
36 |
29 |
153 |
|
14 |
55 |
49 |
58 |
46 |
208 |
|
15 |
55 |
53 |
27 |
44 |
179 |
|
16 |
55 |
43 |
52 |
39 |
189 |
|
17 |
55 |
33 |
61 |
36 |
185 |
|
18 |
55 |
24 |
114 |
23 |
216 |
|
19 |
55 |
46 |
85 |
42 |
228 |
|
20 |
55 |
35 |
72 |
26 |
188 |
Source: Data collected in the company
The values in Table 1 were obtained
by measuring charging average times, route, discharging and returning. These
times were measured for three months, and summarized in the average shown in
Table 1.
After having the measured cycle
times, it was necessary to identify the operation times available. The drivers,
who were outsourced, worked from Monday to Saturday, and each working week was
equivalent to 44 hours. Thus, it has been 440 minutes of daily operation per
vehicle, once considered the lunch time. Therefore, the total daily operating
time of Trucks and Van trucks was obtained as shown in Equations 6 and 7.
(6)
(7)
That is, the daily operation of the
trucks is 4400 minutes while the van trucks use 880 minutes. Finally, it was
stipulated the average demand for pallets to be sent per store, as shown in
Table 2.
Table
2: Pallets demand to send to the store
|
Store |
Picking |
Cross |
Electro |
Total |
Demand |
|
1 |
9,71 |
2,76 |
2,81 |
15,29 |
16,00 |
|
2 |
18,24 |
2,76 |
2,00 |
23,00 |
23,00 |
|
3 |
31,35 |
4,35 |
2,88 |
38,58 |
39,00 |
|
4 |
20,65 |
3,29 |
2,06 |
26,00 |
26,00 |
|
5 |
19,53 |
4,82 |
2,94 |
27,29 |
28,00 |
|
6 |
41,41 |
3,94 |
3,12 |
48,47 |
49,00 |
|
7 |
12,82 |
3,00 |
2,41 |
18,24 |
19,00 |
|
8 |
21,50 |
3,56 |
3,36 |
28,42 |
29,00 |
|
9 |
15,29 |
3,00 |
2,06 |
20,35 |
21,00 |
|
10 |
17,35 |
2,94 |
2,50 |
22,79 |
23,00 |
|
11 |
22,44 |
2,88 |
4,75 |
30,07 |
31,00 |
|
12 |
6,14 |
2,24 |
1,92 |
10,30 |
11,00 |
|
13 |
10,31 |
2,47 |
2,00 |
14,78 |
15,00 |
|
14 |
12,73 |
2,53 |
2,29 |
17,55 |
18,00 |
|
15 |
9,35 |
3,24 |
2,40 |
14,99 |
15,00 |
|
16 |
9,15 |
2,82 |
2,20 |
14,18 |
15,00 |
|
17 |
4,65 |
1,71 |
1,08 |
7,44 |
8 |
|
18 |
7,69 |
2,24 |
2,09 |
12,01 |
13,00 |
|
19 |
6,06 |
1,88 |
9,06 |
17,01 |
18,00 |
|
20 |
6,82 |
1,76 |
1,20 |
9,79 |
10,00 |
Source: Designed by the study authors
Table 2 values, similarly to those
in Table 1 were measured for 3 months, being adopted pallets expedition mean of
each sector for each store chain. Once these values had been added, it was
finally obtained the average demand of pallets per store, after rounding up the
total, because often there were incomplete pallets or that were not fully
utilized, and these were also considered in the calculation.
The next section presents the
results of the study.
4.4. Results
After problem modelling and data
preparation shown in the previous sections, SOLVER was conducted on a computer
with Intel Core i5 processor with 4 Giga RAM. The results obtained are
presented in Table 3.
The SOLVER showed an optimal
solution where 342 pallets were sent from a total of 427 pallets, remaining 85
pallets to be sent. However, this amount would be impractical to deliver
because they are settings that would not meet the vehicle restriction in
traveling at full charge capacity, as shown in Table 4.
Table
3: Pallets to send at the end of the day
|
Store |
Pallets to send |
|
1 |
2 |
|
2 |
9 |
|
3 |
1 |
|
4 |
0 |
|
5 |
0 |
|
6 |
7 |
|
7 |
5 |
|
8 |
1 |
|
9 |
7 |
|
10 |
9 |
|
11 |
3 |
|
12 |
11 |
|
13 |
1 |
|
14 |
4 |
|
15 |
1 |
|
16 |
1 |
|
17 |
8 |
|
18 |
1 |
|
19 |
4 |
|
20 |
10 |
|
Total |
85 |
Source: Designed by the study authors.
Concerning the cycle times, it was
quantified the use 3792 minutes of the trucks and 777 minutes of the van
trucks. This represents 86.18% and 88.30% of the vehicles use in operation,
respectively. While the trucks traveled 21 times, the van trucks traveled 4.
The stores which most used this system were the units 6, 3:11, with a use of
13.07%, 12.87% and 8.84% respectively. Table 4 shows the resolution obtained.
Table 4: SOLVER resolution
|
Demand |
Variables |
Vehicle capacity |
Pallets sent |
Cycle time |
Route |
|||||||
|
Store |
Total |
Truck |
Van
Truck |
Truck |
Van
Truck |
Truck |
Van
Truck |
Total |
Truck |
Van
Truck |
Total |
|
|
1 |
16 |
1 |
0 |
14 |
12 |
14 |
0 |
14 |
125 |
125 |
0 |
125 |
|
2 |
23 |
1 |
0 |
14 |
12 |
14 |
0 |
14 |
197 |
197 |
0 |
197 |
|
3 |
39 |
1 |
2 |
14 |
12 |
14 |
24 |
38 |
196 |
196 |
392 |
588 |
|
4 |
26 |
1 |
1 |
14 |
12 |
14 |
12 |
26 |
169 |
169 |
169 |
338 |
|
5 |
28 |
2 |
0 |
14 |
12 |
28 |
0 |
28 |
118 |
236 |
0 |
236 |
|
6 |
49 |
3 |
0 |
14 |
12 |
42 |
0 |
42 |
199 |
597 |
0 |
597 |
|
7 |
19 |
1 |
0 |
14 |
12 |
14 |
0 |
14 |
166 |
166 |
0 |
166 |
|
8 |
29 |
2 |
0 |
14 |
12 |
28 |
0 |
28 |
176 |
352 |
0 |
352 |
|
9 |
21 |
1 |
0 |
14 |
12 |
14 |
0 |
14 |
197 |
197 |
0 |
197 |
|
10 |
23 |
1 |
0 |
14 |
12 |
14 |
0 |
14 |
196 |
196 |
0 |
196 |
|
11 |
31 |
2 |
0 |
14 |
12 |
28 |
0 |
28 |
202 |
404 |
0 |
404 |
|
12 |
11 |
0 |
0 |
14 |
12 |
0 |
0 |
0 |
157 |
0 |
0 |
0 |
|
13 |
15 |
1 |
0 |
14 |
12 |
14 |
0 |
14 |
153 |
153 |
0 |
153 |
|
14 |
18 |
1 |
0 |
14 |
12 |
14 |
0 |
14 |
208 |
208 |
0 |
208 |
|
15 |
15 |
1 |
0 |
14 |
12 |
14 |
0 |
14 |
179 |
179 |
0 |
179 |
|
16 |
15 |
1 |
0 |
14 |
12 |
14 |
0 |
14 |
189 |
189 |
0 |
189 |
|
17 |
8 |
0 |
0 |
14 |
12 |
0 |
0 |
0 |
185 |
0 |
0 |
0 |
|
18 |
13 |
0 |
1 |
14 |
12 |
0 |
12 |
12 |
216 |
0 |
216 |
216 |
|
19 |
18 |
1 |
0 |
14 |
12 |
14 |
0 |
14 |
228 |
228 |
0 |
228 |
|
20 |
10 |
0 |
0 |
14 |
12 |
0 |
0 |
0 |
188 |
0 |
0 |
0 |
As it can be seen in Table 4, the proposed model sent
427 pallets to the stores, 26 pallets more than the operation performed on the
DC in question. This optimization was mainly given because developers were
sending suboptimal amounts, despite already knowing the total demand for
pallets to be sent. This occurs because of the way the expedition was operated,
in view of the pallets were randomly sent, that is, the easiest charge to be
grouped together for charging the vehicle was the one chosen.
The next section presents the final
considerations.
5. FINAL CONSIDERATIONS
It is concluded that the hypothesis
was not rejected, ie the linear programming tool was suitable for the
expedition planning activities in the case analyzed. In this particular study,
the model allowed to suggest enlarging the use of vehicles, the demand absorbed
by each store and identified important points, such as the fact that the
pallets demand and DC expedition capacity are aligned, having no fleet sizing
problems nor installations.
The model designed made possible to
obtain information on how to allocate charges for expedition before
operationalizing activities, allowing to plan the operation more efficiently
based on data collected in the company. It was possible to increase the number
of pallets sent to stores, when compared with the historical decisions made by
the company.
The Excel SOLVER, which was the
tool used for the LP modeling problem in this study was an appropriate tool for
the development and operationalizing of the proposed model, proving useful for
optimizing development with simplified models.
It is important to mention that, in
order to obtain results that actually contribute to an improvement in a
company’s expedition planning, it is necessary that the data collected fairly
represent the company's reality, that perhaps it is a question involving more
difficult aspects than modeling problem itself.
Regarding the limitations this
study shows, one of them would be not considering the operation after the
expedition, which is transporting DC materials to stores. Future studies may
expand the analysis undertaken in this study and consider this operation.
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