FORECASTING DEMAND AT SUPERMARKETS USING QUANTILE REGRESSION · Square Analysis, Quantile...

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International Journal of Mechanical Engineering and Technology (IJMET)

Volume 10, Issue 09, September 2019, pp. 59-71, Article ID: IJMET_10_09_006

Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=9

ISSN Print: 0976-6340 and ISSN Online: 0976-6359

© IAEME Publication

FORECASTING DEMAND AT SUPERMARKETS

USING QUANTILE REGRESSION

Swati Jha

Student of MBA, Faculty of Management and Commerce,

Ramaiah University of Applied Sciences, Bengaluru, India

Dr. N Suresh

Head of Department of Finance Management of Faculty of Management and Commerce,

Ramaiah University of Applied Sciences, Bengaluru, India

ABSTRACT

Managing the inventory at supermarket is of utmost importance as it is the majorly

responsible for maximizing revenue and maintaining high customer service levels.

There are numerous reasons why keeping a stock is fundamental. With the cutting-

edge retail industry blasting and ecommerce business infiltrating into the market, it is

very important for a supermarket to maintain its product availability. The aim of this

project was to forecast daily demand for selected supermarkets in Bengaluru using

quantile regression analysis. The basic objective of the study was to develop a

quantile regression model to identify the relationship between daily average sales and

average number of customers per day. Secondary data was collected for a time period

of 3 months from 4 selected supermarkets in Bengaluru. Descriptive Statistics Analysis

was performed using Microsoft excel to analyse the collected secondary data. Least

Square Analysis, Quantile Regression Analysis for quantiles 0.25, 0.50, 0.80 and 0.90,

Residual Analysis, and Demand Forecasting was carried out using E-views. Based on

the results of the data analysis, it is seen that the best fit for the quantile regression is

at quantile 0.80, making it a tight fit for the regression line.

Key words: Supermarkets, Inventory Management, Reorder Point, Quantile

Regression, Demand Forecasting

Cite this Article: Swati Jha, Dr. N Suresh, Forecasting Demand at Supermarkets

using Quantile Regression. International Journal of Mechanical Engineering and

Technology 10(9), 2019, pp. 59-71.

http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&IType=9

1. INTRODUCTION

A retailer manages his Inventory to keep an eye on the available products and the sold

products so that the retailer is never out of stock. However, inventory management happens to

be a time-consuming and a complex process. It is usually a top priority for retail decision

makers and it comprises of many units of the supply chain. Inventory management is vast and

covers many aspects of the retail business. Demand forecasting, cost of storage, replenishing

Swati Jha, Dr. N Suresh


exhausted goods, managing space to store products, shipping and return/exchange processes

and policies, are a few components that are a part of inventory management. The two

foremost considerations for an efficient inventory management are to avoid remnants and to

estimate the future sales. In practice, the sales of regular products can be predicted by

application of time series analysis or just by experience of the store managers. The demand

forecast provides the estimated level of claim for the products. This is the simple input for

business planning and controlling. Hence, the decisions for all the functions of any business

firm are affected by the demand forecast [3] [4]. Demand forecasting has been pointed as a

significant and a stimulating problem for supply chain management [5]. Maximum

supermarkets like to guesstimate their upcoming trades. Forecasting can help them avoid

under-estimating or overestimating their upcoming trades which could lead to a great damage

to the companies. Using reliable sales forecasting, supermarkets may be able to possibly

allocate the products more level-headedly and make enhanced revenues [6]. The ability to

accurately forecast the demand for each item sold in each retail store is critical to the survival

and growth of a retail chain because many operational decisions such as pricing, space

allocation, availability, ordering and inventory management for an item are directly related to

its demand forecast. Order decisions need to ensure that the inventory level is not too high, to

avoid high inventory costs, and not too low to avoid stock out and lost sales [7] [8].

Undoubtedly, there has been an increase in the number of supermarkets in Bengaluru over the

last decades. Consequently, a need to maintain the stock levels so as to meet the consumer

demands without piling up the inventory in the form of excessive stock has been of utmost

concern for the supermarkets. Therefore, it is essential to figure out how the inventory

management practices employed by the supermarkets in Bengaluru impact the service

delivery process and ultimately the consumer satisfaction. By doing so, the supermarkets may

gain important information which might help them make better informed decisions in the

upcoming future and hence enhance their overall performance [9].

In this project, we focus on forecasting quarterly demand at supermarkets in Bengaluru by

fitting quantile regression line. Regression analysis was developed in statistical modelling to

identify the relationship between the dependent variable i.e., the outcome and the independent

variables i.e., the covariates over some 200 years ago. The most widely used statistical

method to comprehend the effects at the mean is the ordinary least square regression. It is a

conventional method and assumes that the effects of the regression covariates/coefficients

remain constant throughout the population. Common assumptions for regression include no

missing data, a linear relationship amongst the covariates and a larger sample size than the

numeral of covariates. Therefore, it is a parametric model and depends on assumptions that

are frequently not met. Quantile regression on the other hand makes no such assumptions

about the distribution of the sample and gives a more comprehensive depiction of the effect of

the forecasters on the response variable. In addition, it lets you explore the various aspects of

the relationship between the outcome (dependent variable) and the covariates (independent

variables). Quantile regression seeks the median and other quantiles (aka percentiles) of the

variable to be forecasted instead of seeking only the mean of the variable [10] [11].

Percentiles are predominantly beneficial for inventory optimization by means of a direct

method to calculate the reorder point. Reorder point is basically the level of inventory of a

product which indicates the requirement of the replenishment order. In classical terms, the

reorder point is regarded as the sum of the lead demand and the safety stock. On further

central level, the reorder point is depicted as the quantile forecast of the significant future

demand. The typical calculation of optimized reorder point involves the demand forecast,

the lead time and the service level. Depending on an innate quantile forecast massively

advances the quality of reorder point for most of the manufacturing businesses along with the

Forecasting Demand at Supermarkets using Quantile Regression


retail business. Reorder point is a vital concept for inventory optimization as well as inventory

automation [12] [13].

2. LITERATURE REVIEW

Amid the utmost promising approaches of quantile forecasting in operational planning and

inventory management are the procedures constructed on the median, the quantile estimation

and the bootstrap techniques. Likewise, Quantile forecasting can be cast-off to enhance the

worth of the smoothing constant in the MAD equation. It is also useful in computing quantile

forecasts based on autoregressive specifications with trend functions [14]. Quantile

predictions and statistical tests of the forecasting results measures lead to improved quantile

estimates over both standard econometric approaches and widespread methods from the act of

logistics forecasting on account of microeconomic monthly data. For further time series, the

finest results are typically attained by a simple modelling of the conditional mean or median,

even though quantile smoothing motionlessly remains the greatest forecasting procedures

[15]. Tiffany Hui-Kuang proposes a quantile information measure in order to assist in the

determination of the predictability of the variables [16]. While the forecasts without quantile

information criteria correctly forecast more than 50 percent of the variables, the forecasts with

quantile information criteria outperform that lacking quantile information. Based on the

empirical results, the proposed approach advances the application quantile regression model

to forecast. The applicability of quantile regression model is greatly enhanced. A proposed

approach can benefit existing forecasting systems directly. A further advantage of the

proposed approach that is relevant to practice is that it is fully automatic and data driven, and

therefore, implementable in the context of supply chain forecasting. This work has presented

the combination scheme on the basis of two quantile forecasts, KDE and CGARCH, however,

the inclusion of more techniques in the combination is straightforward [17] .

3. AIM AND OBJECTIVE

The aim of the project was to forecast daily supermarket demand using Quantile Regression

method by analysing the secondary data collected from 4 selected supermarkets in Bengaluru.

The objectives performed to achieve the aim of the project were:

To analyze the collected secondary data: Descriptive analysis and unit root test of sales data at

selected supermarkets

To fit a regression equation for collected sales data

To Develop and validate a quantile regression model for income pattern at selected

supermarkets (residual analysis done for testing)

To forecast demand for upcoming 3 months at supermarkets based on collected sample data

4. DATABASE AND METHODOLOGY

In the progression of forecasting inventory control, numerous predictions are frequently

essential for various products. Hence, it stimulates the automation of the forecasting

processes. The case considered here is one of forecasting daily sales at 4 selected

supermarkets in Bengaluru city. Secondary data was collected for daily average sales and

average number of customers per day for a period of 3 months. Data was collected from 4

selected outlets of different supermarket chains in Bengaluru, namely, More Megastore

(Yeshwantpur), Royal Mart (Dasarahalli), Big Bazar (Heganahalli) and Nilgiris Supermarket

(Malleshwaram). Further, descriptive analysis of the data was conducted. Following which,

least square analysis, quantile regression analysis, residual analysis and demand forecasting

were carried out using E-views.



5. RESULTS AND DISCUSSIONS

5.1 Descriptive Analysis

First, the descriptive statistics for the collected secondary data was calculated using Microsoft

excel and then the unit root test was carried out for the collected dataset using E-views.

Descriptive statistics are divided into two parts. The first is the measure of the central

tendency and the second is the measure of the variability i.e., the spread. Central tendency

comprises of the mean, and the median, and the variability comprises of the standard

deviation, the kurtosis and the skewness. Descriptive analysis of the raw data gives us

knowledge about the distribution of the collected dataset [18]. The following figures depict

the graph of the calculated descriptive statistics for the collected secondary data used in this

project:

Figure 1 Descriptive statistics (Independent Variable)

0

50

100

150

200

250

300

Mean Median SD

Descriptive Analysis (No. of Customers)

Mon Tue Wed Thu Fri Sat Sun

-2

-1

0

1

2

3

4

Skewness Kurtosis

Descriptive Analysis (No. of Customers)




Figure 2 Descriptive statistics (Dependent Variable)

After the descriptive statistics, unit root test was carried out for the raw data using E-

views. The following tables depict the unit test results for the collected secondary data:

Table 1 Unit Test Results for Independent Variable

Null Hypothesis: Number of Customers has a Unit Root

Lag Length: 0

T-statistics Probability

Augmented Dickey-

Fuller Test Statistics

-1.538626 0.505

Test critical values:

1% level

5% level

10% level

-3.577723

-2.925169

-2.600658

0

50000

100000

150000

200000

Mean Median SD

Descriptive Statistics (Daily Sales in INR)


-2

-1

0

1

2

Skewness Kurtosis

Descriptive Analysis (Daily Sales in INR)




Table 2 Unit Test Results for Dependent Variable

Null Hypothesis: Daily Sales has a Unit Root

Lag Length: 0


Augmented Dickey-Fuller Test

Statistics

-1.049663 0.727


1% level

5% level

10% level

-3.577723

-2.925169

-2.600658

As the null hypothesis is accepted for both dependent and independent variable, it means

that the collected secondary data for the study has a unit root, i.e., the raw data is not

stationary. Hence, we cannot fit a regression line for the collected sample data. Therefore, we

convert the data into 1st level difference data and using that we fit a regression line. The

results for 1st difference level are as follows:

Table 3 Unit Root Test for Independent Variable (1st Difference Level)

Null Hypothesis: Number of Customers has a Unit Root

Lag Length: 0



Statistics

-6.657768 0.000


1% level

5% level

10% level

-3.581152

-2.926622

-2.601424

Table 4 Unit Root Test for Dependent Variable (1st Difference Level)

Null Hypothesis: Daily Sales has a Unit Root

Lag Length: 0



Statistics

-6.776104 0.000


1% level

5% level

10% level

-3.581152

-2.926622

-2.601424

As we can see from the results above, the 1st difference level data rejects the null

hypothesis as probability value is less than 5% and accepts the alternate hypothesis which

means that the data does not have a unit root. Hence, it gives us a stationary data.



5.2 Ordinary least square method

To analyse the relationship between daily sales and average number of customers, we start the

analysis with least square test, Here the table 5.2.1 shows the result of least square where

dependent variable is average daily income and independent variable is average number of

customers per day. Here we check if the variables are statistically viable or not. To become

significant, probability should be less than 5%. If the probability is more than 5%, then it is

not significant.

Table 5 Ordinary Least Square Results

Method: Ordinary Least Square

Dependent Variable (Y): Sales (INR)

Variable Coefficient Probability

X1 = Mon 432.473 0.370

X2 = Tue 2516.838 0.000

X3 = Wed 964.608 0.057

X4 = Fri 1719.324 0.000

X5 = Sat -630.045 0.130

X6 = Sun -1422.370 0.001

C 392585.1 0.000

Pseudo R-squared 98%

Adjusted R-squared 97.7%

Prob (F-statistics) 0.000

In table 5.2.1, F-statistics is 0.000 which is less than 5% that means it is significant. Also,

the R-square value gives a tight fit [19]. Hence, the regression equation becomes:

5.3 Fitting Regression Model:

The regression equation assumed for the study is:

Where,

Y = Average daily sales at a supermarket

m = Regression Co-efficient

X = Number of customers per day at the supermarket

C = Constant (Co-efficient of intercept)

The results were obtained as follows:



For quantile 0.25

Table 6 Quantile Regression Analysis (τ=0.25)

Method: Quantile Regression



X1 = Mon 181.927 0.809

X2 = Tue 3017.355 0.000

X3 = Wed 25.479 0.972

X4 = Fri 1959.264 0.002

X5 = Sat -387.27 0.567

X6 = Sun -1106.936 0.229

C 296672.5 0.006

Pseudo R-squared 88.1%

Adjusted R-

squared

86.4%

Prob (Quasi-LR

stat)

0.000

Table 5.3.1 shows that quantile regression method is used for estimation. The probability

value of Quasi-LR statistics is 0.000 which is less than 5% that means the variables are

significant. Also, the goodness-of-fit measure (pseudo R-squared), and adjusted version of the

statistic, is valued at 88.1%, which is a good fit. But, the probability value for only two

coefficients are below 5% so the model does not hold good. And, the equation for the quantile

becomes:

For Quantile 0.5

Table 7 Quantile Regression (τ=0.50)




X1 = Mon 600.387 0.388

X2 = Tue 2626.801 0.000

X3 = Wed 661.832 0.424

X4 = Fri 1936.194 0.007

X5 = Sat -387.27 0.129

X6 = Sun -1106.936 0.097

C 342884.1 0.001

Pseudo R-

squared

88.3%

Adjusted R-

squared

86.6%

Prob (Quasi-LR

stat)

0.000




statistic, is valued at 88.3% which is a tight fit. But, the probability values for only two



variables are less than 5%. Hence, the model does not hold good. And, the equation for the

quantile becomes:

For Quantile 0.80





X1 = Mon -313.076 0.685

X2 = Tue 2426.975 0.000

X3 = Wed 1554.834 0.037

X4 = Fri 1571.2 0.021

X5 = Sat -706.504 0.232

X6 = Sun -1670.714 0.008

C 517757.5 0.000

Pseudo R-

squared

82.8%

Adjusted R-

squared

80.2%

Prob (Quasi-LR

stat)

0.000




statistic, is valued at 82.8% which is a tight fit. The probability value for 4 variables out of the

6 variables considered is less than 5%. Hence, the model holds good and the equation for the

quantile becomes:

For Quantile 0.90





X1 = Mon 328.334 0.657

X2 = Tue 1578.133 0.014

X3 = Wed 2241.28 0.003

X4 = Fri 1634.511 0.010

X5 = Sat -745.399 0.204

X6 = Sun -2261.051 0.000

C 604346.9 0.000

Pseudo R-squared 80.8%

Adjusted R-

squared

78.05%

Prob (Quasi-LR

stat)

0.000






statistic, is valued at 80.8% which is a tight fit. The probability value for 4 variables out of the

6 variables considered is less than 5%. Hence, the model holds good and the equation for the

quantile becomes:

To summarize,

Table 9 Comparison of R2 at various quantiles

Quantile 0.10 0.5 0.80 0.90

R-squared

value

88.1% 88.3% 82.8% 80.8%

Evidently, the desired value is obtained for the 0.80 quantile i.e., 82.8% and it is the best fit.

5.4 Residual analysis

The following tests were performed for the residual analysis:

Stability test (Cusum Test)

Figure 3 Cusum Test

As we can see from the chart above, both the plots remain within critical bounds at 5%

level of significance, the model is structurally stable [20].

Correlogram Q statistics: The figure below displays the autocorrelation and partial

autocorrelation functions of the residuals:



Date: 07/08/19 Time: 22:45

Sample: 1 48

Included observations: 48

Autocorrelation Partial Correlation AC PAC Q-Stat Prob

1 0.447 0.447 10.188 0.001

2 0.284 0.105 14.384 0.001

3 0.157 -0.004 15.695 0.001

4 -0.054 -0.184 15.856 0.003

5 0.097 0.211 16.385 0.006

6 0.010 -0.070 16.391 0.012

7 -0.128 -0.187 17.348 0.015

8 0.000 0.115 17.348 0.027

9 -0.193 -0.179 19.651 0.020

10 -0.150 -0.036 21.077 0.021

11 -0.022 0.096 21.108 0.032

12 -0.112 -0.044 21.949 0.038

13 0.043 0.043 22.079 0.054

14 -0.018 -0.060 22.101 0.077

15 -0.093 -0.025 22.724 0.090

16 -0.047 -0.105 22.891 0.117

17 -0.113 -0.001 23.886 0.123

18 -0.005 0.093 23.888 0.159

19 0.078 0.003 24.391 0.182

20 -0.043 -0.086 24.553 0.219

21 0.015 0.016 24.573 0.266

22 0.078 0.173 25.130 0.291

23 0.085 0.016 25.826 0.309

24 0.182 0.016 29.130 0.215

25 0.075 -0.027 29.712 0.235

26 0.005 -0.078 29.715 0.280

27 -0.034 -0.085 29.849 0.321

28 -0.212 -0.157 35.238 0.163

Figure 4 Correlogram Q statistics

As we can see from the figure above that the Q-statistics probability value are less than

5% at all lags, which indicates that there is significant serial correlation in the residuals.

5.5 Demand Forecasting

Demand forecast is conducted on the basis of 3 Linear Regression and Quantile Regression

[21] [22] . We collected the sales data for 3 months at 4 different supermarket outlets in

Bengaluru and fitted the regression line. We obtained the best fit at quantile 0.80. Next, we

move on to the sales forecasting for the upcoming quarter, i.e., the next 3 months using E-

views in order to determine the demand for the upcoming months. The forecasting results

obtained for the next quarter is as follows:



Figure 5 Forecasting Results

6. CONCLUSIONS AND FUTURE DIRECTIONS

Based on the data analysis, the best fit for the proposed model was obtained at 0.80 quantile

which gave R-squared value of 82.8%. Hence, the regression equation that fits the model

becomes:

From the results we obtained from the sample data, we can conclude that our proposed

model gives the supermarket an accurate forecast about the sales and helps in refining the

supermarket‟s strategies. Also, as the forecasted results are close to the actual results, it helps

in preserving the supply chain steadiness. The model can be further improved if we work out

on the accurateness of the forecasted results by the use of hybrid model of Statistical and AI

methods. The old-fashioned mean-based linear regression was incapable of fully describing

the relationship between the variables. Hence, the use of quantile regression method

recognised a diverse association by modelling the coefficients across the distribution of the

outcome. It gave us a more complete picture of the relationship between the variables. These

particular findings from the quantile regression results can further be applied towards

developing a more operative inventory management practice. Also, instead of using only two

variables for the model, more variables could be considered in order to achieve a more robust

model.

REFERENCES

[1] D. Battini, M. Faccio, A. Persona, and F. Sgarbossa, “„supermarket warehouses‟: Stocking

policies optimization in an assembly-to-order environment,” Int. J. Adv. Manuf. Technol.,

vol. 50, no. 5–8, pp. 775–788, 2010.

[2] R. Ganeshan, “Managing supply chain inventories: A multiple retailer, one warehouse,

multiple supplier model,” Int. J. Prod. Econ., vol. 59, no. 1, pp. 341–354, 1999.

[3] C. Dubelaar, G. Chow, and P. D. Larson, “Relationships between inventory, sales and

service in a retail chain store operation,” Int. J. Phys. Distrib. Logist. Manag., vol. 31, no.

2, pp. 96–108, 2001.

[4] T. L. Urban, “The interdependence of inventory management and retail shelf

management,” Int. J. Phys. Distrib. Logist. Manag., vol. 32, no. 1, pp. 41–58, 2002.

-80,000

-60,000

-40,000

-20,000

0

20,000

40,000

400,000

500,000

600,000

700,000

800,000

5 10 15 20 25 30 35 40 45

Residual Actual Fitted



[5] Y. S. P. Chiu, V. Chiu, H. D. Lin, and H. H. Chang, “Meeting multiproduct demand with

a hybrid inventory replenishment system featuring quality reassurance,” Oper. Res.

Perspect., vol. 6, no. December 2018, p. 100112, 2019.

[6] [6] M. S. N, V. N. N, K. P. K, and R. K. H. S, “Predicting Sales In Supermarkets,” Int.

Res. J. Eng. Technol., pp. 994–997, 2018.

[7] F. Caro and J. Gallien, “Inventory management of a fast-fashion retail network,” Oper.

Res., vol. 58, no. 2, pp. 257–273, 2010.

[8] Y. Boulaksil, “Safety stock placement in supply chains with demand forecast updates,”

Oper. Res. Perspect., vol. 3, pp. 27–31, 2016.

[9] B. K. Irungu, K. L. Wanjau, L. Orange, and P. O. Box, “Effectiveness of vendor managed

inventory systems in retail supermarkets in Kenya,” J. Bus., vol. 1, no. April, pp. 85–89,

2011.

[10] Y. Keming, L. Zudi, and S. Julian, “Quantile Regression : Applications and Current

Research Areas Author ( s ): Keming Yu , Zudi Lu and Julian Stander Published

by :Wiley for the Royal Statistical Society Stable URL :

https://www.jstor.org/stable/4128208 Linked references are available on,” vol. 52, no. 3,

pp. 331–350, 2003.

[11] A. U. Haque, M. H. Nehrir, and P. Mandal, “A hybrid intelligent model for deterministic

and quantile regression approach for probabilistic wind power forecasting,” IEEE Trans.

Power Syst., vol. 29, no. 4, pp. 1663–1672, 2014.

[12] S. K. Chaharsooghi and J. Heydari, “Supply chain coordination for the joint determination

of order quantity and reorder point using credit option,” Eur. J. Oper. Res., vol. 204, no. 1,

pp. 86–95, 2010.

[13] L. Aburto and R. Weber, “Improved supply chain management based on hybrid demand

forecasts,” Appl. Soft Comput. J., vol. 7, no. 1, pp. 136–144, 2007.

[14] J. Bruzda, “Quantile Forecasting in Operational Planning and Inventory Management – an

Initial Empirical Verification,” Dyn. Econom. Model., vol. 16, no. 1, p. 5, 2016.

[15] J. Bruzda, “Multistep quantile forecasts for supply chain and logistics operations:

bootstrapping, the GARCH model and quantile regression based approaches,” Cent. Eur.

J. Oper. Res., pp. 87–100, 2018.

[16] K. H. Huarng and T. H. K. Yu, “A new quantile regression forecasting model,” J. Bus.

Res., vol. 67, no. 5, pp. 779–784, 2014.

[17] J. R. Trapero, M. Cardós, and N. Kourentzes, “Empirical safety stock estimation based on

kernel and GARCH models,” Omega (United Kingdom), vol. 84, pp. 199–211, 2019.

[18] H.~Oja, “Descriptive statistics for multivariate distributions,” Stat. Probab. Lett., vol. 1,

no. October, pp. 327–332, 1983.

[19] A. C. Cameron and F. A. G. Windmeijer, “An R-squared measure of goodness of fit for

some common nonlinear regression models,” J. Econom., vol. 77, no. 2, pp. 329–342,

1997.

[20] S. Lee, J. Ha, O. Na, and S. Na, “The Cusum Test for Parameter Change in Time Series

Models,” Scand. J. Stat., vol. 30, no. 4, pp. 781–796, 2003.

[21] K. NainSukhia, A. Ashraf Khan, and M. Bano, “Introducing Economic Order Quantity

Model for Inventory Control in Web based Point of Sale Applications and Comparative

Analysis of Techniques for Demand Forecasting in Inventory Management,” Int. J.

Comput. Appl., vol. 107, no. 19, pp. 1–8, 2014.

[22] L. Suganthi and A. A. Samuel, “Energy models for demand forecasting - A review,”

Renew. Sustain. Energy Rev., vol. 16, no. 2, pp. 1223–1240, 2012.

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