A PSO-GBR Solution for Association Rule Optimization on Supermarket Sales

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A PSO-GBR Solution for Association Rule
Optimization on Supermarket Sales
Syafrial Fachri Pane
Advanced and Creative Networks
Research Center
Telkom University
Bandung, Indonesia
fachrie@student.telkomuniversity.ac.id
Aji Gautama Putrada
Advanced and Creative Networks
Research Center
Telkom University
Bandung, Indonesia
ajigps@telkomuniversity.ac.id
Nur Alamsyah
Advanced and Creative Networks
Research Center
Telkom University
Bandung, Indonesia
nuralamsyah@student.telkomuniversity.ac.id
Mohamad Nurkamal Fauzan
Advanced and Creative Networks
Research Center
Telkom University
Bandung, Indonesia
mnurkamalfauzan@student.telkomuniversity.ac.id
Abstract—In the era of big data and cloud computing, digital
records of supermarket sales data and other accompanying fac-
tors are ubiquitous. However, less than optimal WeeklyS ale s can
occur due to several factors influencing it. This study proposes
particle swarm optimization with gradient boosting regression
(PSO-GBR) as a solution for optimizing the association rule in
supermarket sales based on a regression model that can predict
WeeklyS ales. As a benchmark for this research, we compare
our proposed GBR with two legacy prediction methods: linear
regression (LR) and AdaBoost Regression (ABR). The first step
is data preparation. Then we develop a model that can predict
sales from the dataset using GBR. The next step is to evaluate
the model with benchmark methods. Then with an optimum
regression method, we optimize sales using PSO. The last step is
to show that the method provides optimum WeeklyS ale s results.
The results show that our proposed GBR has a higher R2score
than LR and ABR, namely 0.95 and 0.94 for train data and test
data, respectively. Then PSO is proven to optimize sales using the
GBR model as a cost function. PSO can increase the ten lowest
WeeklyS ales of the actual dataset from a total of US $ 45 to a
total of US $ 2,296.9by selecting a more optimized De partment
according to the GBR prediction. This research proves that we
can use a regression model with good performance for a cost
function in PSO optimization for department sales.
Index Terms—particle swarm optimization, gradient boosting
regression, supermarket, association rule, optimization, sales
I. Introduction
Grocery shopping at the supermarket is one of the most
basic human needs [1]. In the era of big data and cloud
computing, digital records of supermarket sales data and other
accompanying factors are ubiquitous [2]. However, less than
optimal WeeklyS ales can occur due to several factors influ-
encing it. Several optimization method options can improve
The finance of this work was supported by the Penelitian dan Pengabdian
Masyarakat (PPM) Directory of Telkom University.
supermarket sales by correcting under-performing factors. In
optimizing, a regression model that can map sales with its
determinants becomes an important key [3].
Previous studies have used several regression methods to
predict sales based on association rules. For example, Von
Kirby et al. [4] has proven that adaptive boosting regression
(ABR) can predict the behavior of a product based on its
product type, season, and the type of applied discount. Kohli
et al. [5] proved that linear regression (LR) is better than k-
nearest neighbor (KNN) in predicting sales based on promos
and customers.
Several other studies, for example Srivastava et al. [6]
proved that in association rules, gradient boosting regression
(GBR) can have better performance than other methods. How-
ever, this research is on sports, not market sales prediction.
Other research can optimize predictions made using particle
swarm optimization (PSO), such as that done by Gosh et al.
[7], who applied this method to the optimization of business
capacity based on new business strength. There is an oppor-
tunity to apply this method to optimize sales.
This study proposes PSO-GBR as a solution for optimizing
the association rule in supermarket sales based on a regres-
sion model that can predict S ales. As a benchmark for this
research, we compare our proposed GBR with two legacy
prediction methods: LR and ABR.
To the best of our knowledge, there has never been a
study that applies PSO-GBR to optimize the association rule
in supermarket sales. Here are some contributions from our
research:
1) A method for optimizing sales by department type, date,
and holiday status

2) A GBR method to predict the association rule in super-
market sales
3) A PSO method to optimize supermarket sales based on
the GBR predictions
The systematics for the remainder of this paper are as
follows: Section II discusses the existing related papers.
Section III describes the method used. Section IV shows
the results of the research. Finally, Section V emphasizes
important results.
II. Related Works
There are several aspects of association rules, including
where they are applied, the types of prediction methods, and
optimization methods. Several studies have applied machine
learning in predicting supermarket sales. For example, Yusuf
et al. [8] applied a decision tree to predict discounts based
on total products, product types, and total purchases. This
research has not used an optimization method to optimize
the discount in the store so that it can become a research
opportunity.
Furthermore, several studies have applied GBR in the field
of association rule. Sasirekha et al. [9] used gradient boosting
classification for disease diagnosis. The method can classify
patient datasets as normal or abnormal. However, this research
has not utilized gradient boosting in the association rule for
supermarket sales, so it can also become a research opportu-
nity.
Several studies have applied PSO in the field of sales.
Like He et al. [10] which applied PSO in the long short-
term memory (LSTM) model to forecast the sales volume of
several products. Further, some have combined PSO with the
gradient boosting method. Li et al. [11] used PSO to find
the best parameters to train his gradient boost model for an
intrusion detection system (IDS). However, in that study, PSO
is not intended to optimize the output of GBR based on its
feature values. Applying PSO for optimizing prediction results
becomes a research opportunity. Table I contains a comparison
of the latest papers and also shows a comparison of these
papers with our proposal.
III. Proposed Method
A. Methodology
Fig. 1 describes the methodology of this research. The
first step is data preparation. Then we develop a model that
can predict sales from the first step dataset using GBR. The
next step is to evaluate the model with benchmark methods.
TABLE I: Related Works Comparison
Reference Association Rule Aspect
Supermarket
Sales
GBR
Prediction
PSO Optimizi-
sation
[8] ✓ ✗ ✗
[9] ✗ ✓ ✗
[10] ✓ ✗ ✓
[11] ✗ ✓ ✓
Proposed
Method
✓✓✓
Fig. 1: The research methodology for optimization on super-
market sales with PSO-GBR.
Then with an optimum regression method, we optimize sales
using PSO. The last step is to show that the method provides
optimum S ales results.
We obtained the supermarket dataset from ”Retail Data
Analytics” from Kaggle. The dataset contains text data, which
must be converted into numerical data so that regression is
implementable. The label encoder method converts categor-
ical data into numeric data. The dataset has three features,
including Department,Date, and I sHoliday. The I sHoliday
feature is a Boolean type data whose value is T rue when it
is on holiday, then False when it is not on holiday. Table II
shows the statistics of the dataset after undergoing the label
encoder process.
B. Gradient Boosting
Gradient boosting, the basis of GBR in this research, is
an ensemble-type machine learning method that promotes
boosting [12]. In general, the booster is iterative, where in
every next iteration, incorrectly categorized data from weak
learners gets a better treatment. The final model aggregates
all previous iterations of the weak learner. The specificity
of gradient boosting is the application of gradient descent
in each iteration by considering the boosting process as an
optimization process with a cost function parameter [13]. The
cost function parameter can use a mean squared error (MSE)
parameter with the following formula:
F(x)=1
N
N
X
i=0
(ˆyi−yi)2(1)
where F(x) is the MSE value, Nis the number of datasets, iis
the index data, ˆyiis the predicted value of the ith data index,
and yiis the actual value of the ith data index [14].
TABLE II: Dataset Statistics
Statistic Feature
Department Date I sH oliday
S ize 10244
S mallest Value 0 0 0
Largest V alue 76 142 1
Average36.9 70.9 0.1
S tandard Dev. 22.3 41.3 0.3
Median 36 71 0

Furthermore, for example there are Mboosting iterations,
to provide improvement in each iteration, each iteration uses
an additional estimator, namely hm(x). So the MSE formula
for each subsequent iteration, or Fm+1(x), becomes as follows:
Fm+1(xi)=Fm(xi)+hm(xi) (2)
where:
hm(xi)=yi−Fm(xi) (3)
Taking into account F(xi), the calculation of the loss func-
tion (LMS E) from this method is as follows:
LMS E =1
N
N
X
i=0
(yi−F(xi))2(4)
then the searched gradient descent is the negative derivative
of LMS E with F(xi), where the following equation is obtained:
−
δLMS E
δF(xi)=2
N(yi−F(xi)) =2
Nhm(xi) (5)
C. Particle Swarm Optimization
PSO is one of the swarm intelligence methods used in
evolutionary algorithms to overcome optimization problems.
The PSO used in this study uses the PySwarms library used for
Python. PSO works by generating several particles in a space
whose dimensions are pre-determined. Each particle moves in
that dimension in several iterations based on a certain direction
and velocity. Each position has a different value based on a
cost function in that space. The algorithm knows the best
position of a particle and the best position of all particles.
In each next iteration, that knowledge is used for the next
movement of each particle [15].
Algorithm 1 shows the PSO algorithm. In the algorithm,
SwarmS ize is the number of moving particles, Limits deter-
mines the limit of particle movement, and Iterations deter-
mines how many times the algorithm loops. Then Options
consists of c1, c2, and w, where c1 is the coefficient of
local movement, c2 is the coefficient of global movement,
and wis the inertia weight. Then Cost Function is the op-
timized function. This research uses the GBR method as the
CostF unction. Finally, Argsis useful when some of the inputs
for the Co stFunction are external variables.
D. Benchmark Methods and Performance Measures
This study uses LR and ABR as benchmarks. LR models
a prediction between two or more independent variables and
the dependent variable, which has a linear relationship [16].
A metric named r-squared (R2) proves that the relationship
between these variables is linear. The formula for calculating
it is as follows:
R2=






N(Pxy)−(Px)(Py)
p(NPx2−(Px)2)(NPy2−(Py)2)







2
(6)
where Nis the number of test data, xis the actual test data,
and yis the predicted test data [17].
Algorithm 1: Particle Swarm Optimization Algorithm
Data: SwarmS ize,Limits,Iterations,Options,
CostF unction,Args
Result: BestPos,BestCost
for i in range(Iterations)do
for j in range(SwarmS ize)do
Cost j=CalculateCost(CostF unction,Args);
end
if Cost j<Cost j−1then
BestCost j←C ost j;
BestPos j←Po s j;/* Pos is Position */
end
Pos j=C alculatePos(SwarmS ize,Limits,O ptions);
BestCosti←max(BestCost j);
BestPosi←max(BestPos j);
end
BestCost ←max(BestCosti);
BestPos ←max(BestPosi);
The general concept of ADB is adaptive boosting (Ad-
aBoost). The concept of boosting in AdaBoost is the same as
gradient boosting, which is to iterate over the learning process
by managing misclassified data on every next iteration [18].
The difference between AdaBoost and gradient boosting is
that, instead of Equation 5, the algorithm performs a weight
increase of αton the problematic data, where the error of the
increase in an iteration t(Et) is as follows:
Et=
N
X
i=0
E[Ft−1(xi)+αth(xi)] (7)
where Nis the number of datasets, E[p] is an error function,
h(xi) is the prediction on the problematic data xi, and FT(x)
is the final result of the AdaBoost prediction which has the
equation as follows:
FT(x)=
T
X
t=1
ft(x) (8)
where ft(x) is the prediction of the weak learner in the tth
iteration [19].
The performance comparison of GBR, LR, and ABR as a
regression method in predicting supermarket sales uses the R2
metric, whose formula has an explanation in Equation 6.
IV. Results and Discussion
A. Results
We use Python’s scikit-learn library to train the GBR, LR,
and ABR models. The first step in this research is to form
a cost function based on the three regression methods. This
study uses 80% dataset for training and 20% dataset for
testing. Before splitting the dataset, this study conducted a
random shuffle first. After doing the split, this research applies
standard scaling, changing the range of various datasets into
the standard range [20].

Fig. 2: The bar chart that shows the R2score comparison of
GBR, LR, and ABR in S ales prediction.
After the training process, we used the R2score between the
actual data and the predicted data to compare the performance
of the three regression methods in predicting supermarket
sales. Fig. 2 shows the comparison of the R2scores. GBR
is the regression method with superior R2, which is 0.95 for
train data and 0.94 for test data. LR has the lowest R2, 0.01
for train data, then 0.02 for test data. The R2result, which is
approximate to 0, indicates no correlation between the model
and the data variation around the mean. ABR has R2=0.59
for train and test data.
A fitted linear line between the actual data and the predicted
data can explain more about the performance of each regres-
sion method. Here we use the Polyfit function from the Numpy
library in Python. Fig. 3 shows the results. The x-axis of the
figure is the actual data, while the y-axis is the prediction
counterpart. The scattered data explains each response of the
predicted data to the actual data. The closer the scattered
data are to the fitted line, the better the performance [21].
The gradient value of the line is an additional performance
indicator. A gradient value that approximates zero indicates
bad performance.
We show the predictive ability of each regression method
by visualizing the prediction results against several test data.
Fig. 4 shows the graphic. The graph contains 51 test data
or 0.5% of the total data. In the graphic, the dots represent
the actual data. Based on the data variance alone, qualitative
observation can show that GBR is the most approximate
prediction compared to LR and ABR. The actual data has
a larger data variance than LR and ABR. LR has the smallest
data variance.
Here we test PSO as an optimizer in increasing super-
market S ales. PSO optimizes S ale s by finding the optimum
Department type in a Date. The I sH Olidayvariable is addi-
tional knowledge about whether the day is a holiday or not.
PSO is in the form of a collection of particles that move
in a plane with several dimensions. The movement of each
particle is determined by several parameters, namely c1, c2,
(a)
(b)
(c)
Fig. 3: The S ales prediction scatter plot, the best fit line, and
the R2value of: (a) GBR (b) LR (c) ABR.
and w. The values of these parameters determine the optimal
results of PSO. This discussion concludes that PSO parameters
also need to go through optimization [22]. This study uses the
random search function provided by PySwarms to optimize
the proposed PSO parameters. Table III shows PSO parameters
which are partly random search results.
Fig. 5 shows the optimization history of PSO. Cost,
the y-axis on the graph, is the regression output, namely
WeeklyS ales. The unit is U S $. The visualization shows that
Fig. 4: Qualitative performance comparison of regression
methods in predicting sales.

TABLE III: PSO Parameters
Optimized Parameter
Name Value
Yes c1 4.7
Yes c2 9.9
Yes w4.2
No Iterations 400
No N particles 300
the initial Iterations =400 is sufficient for the optimization
to converge. GBR has the highest Best Cost, which is 4,213.
LR has the lowest Best Cost, which is 478. The Best C ost of
ABR is 3,093.
Based on the actual dataset, the highest value of
WeeklyS ales can reach U S $ 693,099.36. So the previously
mentioned GBR optimization values may not always be able to
give better results than they should. However, this optimization
can be a solution in situations where sales are small. Here
we collect ten dataset items with the lowest WeeklyS ale s
value. The idea is to test the PSO optimization on the ten
items. Fig. 6 shows the result of the accumulation. The
image shows that PSO optimization can increase the value
of the actual WeeklyS ales by searching for a more opti-
mum Department. Of the three compared regression methods,
our proposed method gives the highest S ales Accumulation,
namely US $ 43,557. The LR regression function gives the
lowest S ales Accumulation, which is U S $ 10,073. The ABR
function returns S ales Accumulat ion =U S $ 30,136.
PSO can optimize WeeklyS ales by choosing one of the
99 Departments that can provide the highest W eeklyS ale s
on a certain date based on holidays or not. Table IV shows
the optimized Department from the original 10 data items
with the worst WeeklyS ales in the dataset. For example on
the first row, Department 5 has US$0 sales. If on the same
day we replace Department 5 with Department 78, we get
an increase of sales for up to US$263.7. Some PSO results
have WeeklyS ale s =N/A, which means that the optimization
results suggest a Department enumeration that is out of range.
If it occurs, we replace the value with the actual WeeklyS ale s
on that data item when added to the final total.
B. Discussion
Several studies have shown that LR performs better than
other methods predicting S ales [23]. However, in our study,
Fig. 5: Comparison of PSO optimization using the three
regression methods as cost functions through 400 iterations.
Fig. 6: Comparison of PSO optimization using the three re-
gression methods as cost functions on optimizing supermarket
sales.
linear regression has an approximate zero value of R2. The
possible cause is the correlation between the features and
the output value. In our dataset, no feature has a Pearson
correlation coefficient (PCC) more than 0.5 with the output
value [24]. Another solution is to choose another non-linear
regression method.
GBR and ABR have the same boosting concept but with
different approaches. Then several studies have indeed proven
that GBR has a better performance than ABR [25]. This study
strengthens the existing research by showing that the ability
of gradient descent to boost gives more effective results in
predictive model performance.
We provide an algorithm with much potential. For future
work, we can test and compare it with several studies with
approximate approaches, for example, for feature selection
and prediction of the ability of teachers to teach during the
coronavirus disease (COVID-19) pandemic [26]. In addition,
PSO can also be used to optimize learning rates in artificial
neural network (ANN) training models, usually useful for
wireless sensor network (WSN) localization [27]. Another
potential application of PSO-GBR is in dynamic pricing,
where many researchers have applied it to optimize sales [28].
V. Conclusion
Here we create a regression model using gradient boosting
regression (GBR) to predict WeeklyS ales based on department
type, date, and holiday status. This regression method is useful
for optimizing sales based on Department, which uses particle
TABLE IV: Optimized Departments and S ales with PSO-
GBR Optimization
Actual Optimized
Department SalesaDepartment Salesa
5 0 78 263.7
42 1 104 N/A
42 2 91 365.9
42 3 75 232.0
56 4 71 315.3
42 5 77 301.2
17 6 101 N/A
50 7 75 261.1
50 8 83 216.0
50 9 82 282.7
T otal 45 Tot al 2296.9
aSales are in US $.

swarm optimization (PSO). The results show that our proposed
GBR has a higher R2score than linear regression (LR) and
adaptive boosting regression (ABR), namely 0.95 and 0.94 for
train data and test data, respectively. Then PSO is proven to
optimize sales using the GBR model as a cost function. PSO
can increase the ten lowest WeeklyS ale s of the actual dataset
from a total of US $ 45 to a total of U S $2,296.9 by selecting a
more optimized Department according to the GBR prediction.
Acknowledgment
The authors thank Telkom University’s Informatics Doctoral
Program for continuously encouraging students to publish
papers. We also thank our colleagues from The Inspiration
Room, who always maintain a conducive environment for
research. We hope that our cooperation will increase and
become more intertwined in the future.
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