Plane Thermoelastic Waves in Infinite Half-Space Caused


Decision Making: Applications in Management and Engineering  
Vol. 3, Issue 1, 2020, pp. 1-21. 
ISSN: 2560-6018 
eISSN: 2620-0104  

 DOI: https://doi.org/10.31181/dmame2003001s 

* Corresponding author. 
 E-mail addresses: hareshshrm@gmail.com (H.K. Sharma), kriti.kri89@gmail.com (K. 
Kumari), samarjit.kar@maths.nitdgp.ac.in (S. Kar) 

A ROUGH SET THEORY APPLICATION IN FORECASTING 
MODELS 

 Haresh Kumar Sharma 1, Kriti Kumari 2 and Samarjit Kar 1* 
 

1 Department of Mathematics, National Institute of Technology, Durgapur, India 
2 Department of Mathematics, Banasthali University, Jaipur, Rajasthan, India 

 

 
Received: 5 September 2019;  
Accepted: 11 November 2019;  
Available online: 15 November 2019. 

 
Original scientific paper 

Abstract. This paper introduces the performance of different forecasting 
methods for tourism demand, which can be employed as one of the statistical 
tools for time series forecasting. The Holt-Winters (HW), Seasonal 
Autoregressive Integrated Moving Average (SARIMA) and Grey model (GM (1, 
1)) are three important statistical models in time-series forecasting. This 
paper analyzes and compare the performance of forecasting models using 
rough set methods, Total Roughness (TR), Min-Min Roughness (MMR) and 
Maximum Dependency of attributes (MDA). Current research identifies the 
best time series forecasting model among the three studied time series 
forecasting models. Comparative study shows that HW and SARIMA are 
superior models than GM (1, 1) for forecasting seasonal time series under TR, 
MMR and MDA criteria. In addition, the authors of this study showed that GM 
(1, 1) grey model is unqualified for seasonal time series data. 

Key words: Forecasting, Mean Absolute Percent Error (MAPE), Rough Set, 
Total Roughness, Maximum Dependency Degree 

1. Introduction  

The future planning has emerged as a key component for the success of a large 
number of entrepreneurs round the corners of the world. It certainly makes it easy for 
all the countries to formulate economic policies as well as act upon them efficiently. 
The perfection and accuracy are quite important in the process of highly accurate and 
reliable prediction. It helps the government in formulating development policies 
concerned with economics, infrastructure and many other sectors of the global as well 
as the domestic economy. It even improves the decision-making process. Time series 
modeling and forecasting play a key role in accurate prediction. The current trend is 

mailto:kriti.kri89@gmail.com
mailto:samarjit.kar@maths.nitdgp.ac.in


 Sharma et al./Decis. Mak. Appl. Manag. Eng. 3 (1) (2020) 1-21  

2 

incorporated for the future prediction; it thus becomes necessary to use highly 
consistent and precise forecasting tools.  

Since the last couple of decades, a wide variety of forecasting models is available 
for the study of tourist arrivals and demand forecasting (Chu, 1998, Lim and McAleer, 
2002; Wang, 2004). Suggesting Autoregressive Integrated Moving Average (ARIMA) 
model as a most suitable model for tourism demand forecast. ARIMA was first brought 
out by Box-Jenkins (Box and Jenkins, 1976) and presently it is the most accepted 
model for forecasting univariate time series data. ARIMA model is the combined result 
of autoregressive (AR) and Moving Average (MA) model. ARIMA model develops an 
optimal univariate future prediction. Moreover, the ARIMA model has received 
worldwide confidence due to its ability to handle stationary and non-stationary series 
with seasonal and non-seasonal elements (Pankratz, 1983). But, SARIMA is 
particularly designed for the time series data with trends and seasonal patterns. Holt-
Winters has also gained more popularity to capture trend and seasonality (Winters, 
1960). The HW seasonal method consists of the forecast equation and smoothing 
equations for level, trend, and seasonality. Later on, grey system theory developed by 
Deng states that a system whose internal sources such as system characteristics 
operation mechanisms and architecture are completely clear, called a white system 
(Deng, 1982). The added advantage of this system is that the theory cannot only 
estimate an uncertain system but sometimes it produces ideal results. For example, 
Tseng et al. (2001) was reported the application of the grey model to forecast Taiwan 
Machinery Industry and soft drinks time series data. However, Nguyen et al. (2013) 
studied the forecasting of tourist arrivals in Vietnam using GM (1, 1) grey model. 

From the last many years variety of forecasting criteria has been used to select the 
best time series models (Lim and McAleer, 2002; Wang, 2004). Modeling and 
forecasting consist of a large number of criteria. For instance, (Chu, 1998) employed 
MAPE and U-statistics criteria to compare the Holt winters and SARIMA models. 
However, (Chen et al, 2009) applied MAPE criterion to evaluate the forecasting 
accuracy of Holt-Winters, SARIMA, and Grey models. In earlier research accuracy of 
forecasting models have been evaluated using error based criteria (Goh and Law, 
2002; Law and Au, 1999; Law, 2000). Sometimes it may be possible that one model 
may become a good one due to some set of criteria but at the same time some other 
model may turn out to be the best one due to some other set of criteria. Moreover, 
these indicators have very much been exploited and only marginal improvements 
might be expected from their continued use. This research proposed a new approach 
that applies rough set theory to select the highly accurate models in time series 
forecasting. The rough set theory has been introduced to deal with vagueness, 
imprecision, and uncertainty. The original rough set theory depends on the 
equivalence relation (indiscernibility relation). This approach is taken into 
consideration the attributes in accordance with the normalized values (Goh and Law, 
2003). Rough set theory has been able to overcome one of its advantages in association 
with statistical analysis during the process of attribute selection using rough set 
indicators (Hassanein and Elmelegy, 2013; Herawan, 2010).  

Current research is an extended effort of Chen et al. (2009) work where they have 
used the Holt-Winters (HW) model, Seasonal Autoregressive Integrated Moving 
Average (SARIMA) model and Grey Model (GM (1, 1)). GM (1, 1) has been used to 
define monthly inbound air travel arrivals to Taiwan and to distinguish the models 
based on their respective performance. Mean Absolute Percent Error (MAPE) has been 
used as an indicator to measure forecast accuracy. Based on the results derived, they  
concluded that the HW and SARIMA models are better reliable models than GM (1, 1). 



A rough set theory application in forecasting models 

3 

       The objective of this paper is to obtain the best forecasting models using TR, MMR, 
and MDA rough set indicators. Based on rough set information table, those techniques 
are used to calculate roughness of models. Then, compare these three models in 
accordance with the roughness. The authors of this study showed that the GM (1, 1) is 
an inadequate model for forecasting with seasonality as compared to HW and SARIMA 
models. 

The rest of the research paper is organized as follows; Section 2 contains literature 
review, Section 3 briefly introduces the basic concepts rough set theory and some 
related properties. Section 4, presents an algorithmic approach for the evaluation of 
rough data using MAPE indicator. In Section 5, the experimental design and 
experimental results have been discussed. Finally, the paper is concluded in section 6.  

2. Literature review 

In recent years, rough set theory have been employed in various literature to select 
the clustering attributes. For example, Mazlack et al. (2000) proposed Bi-Clustering 
(BC) technique depend on balanced/unbalanced bi-valued attributes and Total 
Roughness (TR) technique based on the average accuracy of roughness (Pawlak, 1982; 
Pawlak & Skowron, 2007). The TR technique is useful for selecting the clustering 
attributes in the data set, where the maximum TR is the maximum accuracy for 
selecting clustering attributes. Three indicators i.e. TR, MMR, and MDA of the rough 
set theory have been successfully used. For instance, Parmar et al. (2007) developed a 
new method called Min-Min Roughness (MMR) to develop BC technique for the 
information system with many valued attributes. In this technique, attributes for 
approximation are calculated using well-known corporate to the lower and upper 
approximations of a subset of the universe in the information system. Herawan et al. 
(2010) developed a new technique known as Maximum Dependency of Attributes 
(MDA) to select clustering attributes. MDA technique is based on the dependency of 
attributes using rough set theory in an information system. These three techniques 
TR, MMR and MDA provide the same outcome in selecting the attributes. This makes 
the rough set criteria a very useful to select the different attributes. However, in 
previous literature, there is no any link of rough set theory with relationship time 
series modeling to select the best forecasting models.  In time series analysis and 
forecasting, the selection of highly accurate model is very important to evaluate the 
best time series model. Hence, this research proposed a rough set criterion for strong 
evidence in the selection of best suitable time series models that is different another 
traditional statistical indicator. 

Rough set theory has been consistently employed in a variety of research areas for 
the extraction of decision rules (Law & Au 1998, 2000, Goh & Law, 2003; Liou et al. 
2016).  Celotto et al. (2012) applied rough set theory based forecasting model in data 
of tourist service demand.. Moreover, Li et al. (2011) predicted tourism in Tangshan 
city of China using rough set model. Golmohammadi and Ghareneh (2011) analyze the 
importance of travel attributes by rough set approach. Celotto et al. (2015) applied 
rough set theory to summarize tourist evaluations of a destination. , Faustino et al. 
(2011) present a rough set analysis of electrical charge demand in the United States 
and the level of the Sapucal river in Brazil. Liou (2016) used the rough set theory to 
study the airline service quality to Taiwan. Sharma et al. (2019) proposed hybrid 
rough set based forecasting model and applied on tourism demand of air 
transportation passenger data set in Australia tourism demand. 



 Sharma et al./Decis. Mak. Appl. Manag. Eng. 3 (1) (2020) 1-21  

4 

Rough set theory use to alter the roughness of a data, which has been successfully 
applied to various real life decision making problems ( Karavidić & Projović, 2018; Roy 
et al., 2018; Vasiljević et al., 2018). Moreover, the rough set concept can definitely be 
implemented to sets categorized by means of immaterial facts wherein statistical tools 
fail to provide fruitful outcomes (Pawlak, 1991).  Pamučar et al. (2018) proposed 
interval rough number enabled AHP-MABAC model for web pages evaluation. Sharma 
et al. (2018) applied Modified Rough AHP-Mabac Method for Prioritizing Indian 
Railway Stations.  

3. Rough Set Theory 

The rough set theory was first introduced by Pawlak (1982). The rough set concept 
is a new mathematical technique to tackle vagueness, imprecision, and uncertainty 
(Pawlak, 1982; Pawlak & Skowron, 2007). It is a vital tool to examine the degree of 
dependencies and minimize the number of attributes within the dataset. Its success is 
partly owed to the following properties: (1) Analysis is performed on the hidden fact 
of the data; (2) Supplementary information on data is not required like specialist 
awareness or thresholds; (3) Equivalent relation is a basic idea of classical rough set 
theory. Whereas, the attribute might be assign with both the values symbolic or real. 

Pawlak proposed that the rough set theory is established on the assumption that 
with every member of the universe of discourse we relate some information. For 
example, symptoms of the disease develop a crucial part of information where objects 
are the patients suffering from the certain disease. The objects become indiscernible 
(similar) when characterized by the same information in view of the available 
information about them. The indiscernibility relation created in this way is the 
mathematical foundation of the rough set theory.  

The original concept of the rough set theory is the induction of approximation. The 
main aim of the rough set theory is the approximation of a set by a pair of two crisp 
sets called the lower and upper approximations of the sets. 

3.1 Indiscernibility Relation 

Let U be the non-empty finite set of all objects known as the universe and 𝐴 is the 
finite set of all attributes, then the couple 𝑆 = (𝑈, 𝐴) is known as an information 
system. For any non-empty subset 𝐵 of 𝐴 is associated with an equivalence relation 
INDS(B) relation,  

𝐼𝑁𝐷𝑆(𝐵) = { (𝑦𝑖 , 𝑦𝑗 ) ∈ 𝑈 × 𝑈 ∣∣ ∀ 𝑏 ∈ 𝐵, 𝑏(yi) = b(yj) }    (1) 

where 𝑏(𝑦𝑖 ) represents the value of attribute 𝑏 for the element 𝑦𝑖 . 𝐼𝑁𝐷𝑆(𝐵) is 
called the Indiscernibility relation on 𝑈. The notion [𝑦𝑖 ]𝐼𝑁𝐷𝑆(𝐵) represent the 

equivalence class of the indiscernibility relation. [𝑦𝑖 ]𝐼𝑁𝐷𝑆(𝐵) is also called as 

elementary set with respect to the attribute 𝐵. 

3.2. Lower and Upper Approximation 

Lower approximation and upper approximation (Pawlak, 1982; Pawlak, 1991) of 
any set can be defined as follows: 

For an information system S = (U, A) Given the set of attribute B ⊆ A, Y ⊆U, the 
lower and upper approximation of Y are defined as follows respectively, 
𝑌𝐵 = ∪ {𝑦𝑖 |[𝑦𝑖 ]𝐼𝑁𝐷𝑆(𝐵) ⊆ 𝑌}                 (2) 

𝑌𝐵 = ∪ {𝑦𝑖 |[𝑦𝑖 ]𝐼𝑁𝐷𝑆(𝐵) ∩ 𝑌 ≠ ∅}  (3) 



A rough set theory application in forecasting models 

5 

 Clearly, lower approximation contains all members which certain objects of Y and 
upper approximation consists all members which possible objects of Y. 
The boundary region is the set of members that can possible member, but not surely, 
defined as follow: 

𝐵𝑁𝐷𝐵 (𝑌) = 𝑌𝐵 − 𝑌𝐵   (4) 

The boundary region of an exact (crisp) set is an empty set like the lower 
approximation and upper approximation of exact set are similar. If the boundary 
region of a set is non-empty i.e. 𝐵𝑁𝐷𝐵 (𝑌) ≠ ∅, then the set Y has been referred to as 
rough (vague). 

3.3. Roughness (R) 

Inexactness of a category (set) is one of the reasons behind the existence of 
boundary line region. As the boundary line region of a category increases, the accuracy 
of the category decreases. To model such kind of imprecision the concept of accuracy 
of approximation (Pawlak, 1991) is very much required. Accuracy measure 
represented as follow: 

        𝛼𝐵(𝑌) =
𝑐𝑎𝑟𝑑 𝑌𝐵

𝑐𝑎𝑟𝑑 𝑌𝐵
 

The accuracy is intended to compute the degree of satisfaction of our knowledge 
about the category (set). Obviously 0 ≤ 𝛼𝐵 (𝑌) ≤ 1. If  𝛼𝐵 (𝑌) =1, Y is exact with 
respect to B, if 𝛼𝐵 (𝑌) < 1, Y is rough with respect to B.  

Assume that an attribute 𝑎𝑖 ∈ 𝐴 having k-distinct values, say 𝛼𝑘 , 𝑘 = 1,2, … . , 𝑚. 
Suppose 𝑌(𝑎𝑖 = 𝛼𝑘 ),𝑤ℎ𝑒𝑟𝑒 𝑘 = 1,2, . . … , 𝑚 𝑖 a subset of the objects consists k-distinct 
values of attribute 𝑎𝑖 . The roughness of TR (Mazlack, 2000) of the set(𝑎𝑖 = 𝛼𝑘 ), 𝑘 =
 1, 2, … . , 𝑚, with respect to aj, where 𝑖 ≠ 𝑗, represented by 𝑅𝑎𝑗 (𝑌 ∣ 𝑎𝑖 = 𝛼𝑘 ) as is 

defined by 

𝑅𝑎𝑗 ( 𝑌 ∣∣ 𝑎𝑖 = 𝛼𝑘 ) =
|Yaj

(𝑎𝑖=𝛼𝑘)|

|Yaj
(𝑎𝑖=𝛼𝑘)|

,𝑘 =  1, 2, … . , 𝑚                                                                     (5) 

3.3.1. Mean roughness (MR) 

The values of Mean roughness of an attribute 𝑎𝑖  ∈ 𝐴 with respect to another 
attribute 𝑎𝑗 ∈ 𝐴, where, 𝑖 ≠  𝑗, represented by the following formula 

𝑅𝑜𝑢𝑔ℎ𝑎𝑗 (𝑎𝑖 )=
∑ 𝑅𝑎𝑗

(𝑌∣𝑎𝑖=𝛼𝑘)
|𝑉(𝑎𝑖)|

𝑘=1

|𝑉(𝑎𝑖)|
                             (6) 

where 𝑉(𝑎𝑖 ) is the set of all values of attribute 𝑎𝑖 ∈ A. 

3.3.2. Total Roughness (TR) 

The total roughness of the attribute 𝑎𝑖 ∈ A with respect to the attribute 𝑎𝑗 ∈  𝐴, where, 

𝑖 ≠ 𝑗, represented by 𝑇𝑅 (𝑎𝑖 ), is defined by  

𝑇𝑅(𝑎𝑖 ) =
∑ 𝑅𝑜𝑢𝑔ℎ𝑎𝑗

(𝑎𝑖)
|𝐴|
𝑗=1

|𝐴|−1
  (7) 

 The maximum value of TR, the finest selection choice of clustering attributes. 

3.3.3. Minimum – Minimum Roughness (MMR) 

From the TR system, the mean roughness of attribute ai with respect to attribute  
aj, where, 𝑖 ≠ 𝑗 is define by  



 Sharma et al./Decis. Mak. Appl. Manag. Eng. 3 (1) (2020) 1-21  

6 

𝑀𝑀𝑅𝑜𝑢𝑔ℎ𝑎𝑗 (𝑎𝑖 ) =1 −
∑ 𝑅𝑎𝑗

(𝑌∣𝑎𝑖=𝛼𝑘)
|𝑉(𝑎𝑖)|

𝑘=1

|𝑉(𝑎𝑖)|
 

𝑀𝑀𝑅𝑜𝑢𝑔ℎ𝑎𝑗 (𝑎𝑖 ) = 1 – 𝑅𝑜𝑢𝑔ℎ𝑎𝑗 (𝑎𝑖 )     (8) 

3.4. Maximum Dependency Attribute (MDA) [Herawan et al. (2010)] 

Suppose S = (U, A) is information system and let ai and aj be any subsets of A. 
Dependency attribute ai on aj in a degree  k (0<k<1), is denoted by 𝑎𝑗 ⟹ 𝑘𝑎𝑖 . The 

degree k is derived:  

𝑘 =
∑ |𝑎𝑗(𝑌)|𝑌∈𝑈/𝑎𝑖

|𝑈|
  (9) 

If degree k =1, ai  is fully depends on aj. Otherwise ai partially depends on aj. 
      For an assortment of clustering attributes a maximum degree of dependency of 
attributes is most appropriate. Because as greater the degree of dependence of 
attribute as much accurate for the assortment of clustering attribute. 

3.5. Accuracy of TR, MMR and MDA Techniques 

Section 4 involves the applications of TR, MMR and MDA techniques, used for the 
selection of better forecasting models. The calculation of the accuracy consists of (i) 
TR which makes the use of the total average of mean roughness (ii) MMR which relies 
upon the minimum of mean roughness and (iii) MDA depends on the maximum degree 
of dependency to select the forecasting attribute model. After determining the 
roughness and mean roughness of the attributes, we select the best model based on 
maximum TR, maximum MDA values and lower values of minimum roughness.   

Example 1. To demonstrate the degree of dependency of attributes, we consider the 
information system as shown in Table 1.  

Table 1. Representation of Food Junction database with ten objects and five 

attributes 

Customer 
 

Food 
quality 

(𝐹1) 

Menu 
variety 

(𝐹2) 

Service 
(𝐹3) 

Atmosphere 
(𝐹4) 

Cleanliness 
(𝐹5) 

𝑦1 Good Good Poor Good Average 
𝑦2 Poor Poor Poor Average Good 
𝑦3 Poor Good Average Good Average 
𝑦4 Good Average Poor Average Average 
𝑦5 Good Average Poor Good Poor 
𝑦6 Average Good Average Good Poor 
𝑦7 Average Good Good Good Good 
𝑦8 Poor Poor Poor Average Good 
𝑦9  Good Average Poor Average Average 
𝑦10 Good Good Average Average Good 

Table 1 shows the food junction data set in which 𝑈 =  {𝑦1, 𝑦2 , . . , 𝑦10} (where 𝑦𝑗  

represents a customer) and 𝐴 =  (𝐹1, 𝐹2, 𝐹3, 𝐹4, 𝐹5), being 𝐹1, 𝐹2, 𝐹3, 𝐹4, and  𝐹5 
attribute set. 

 In dataset 1, indiscernibility relations can described as follows: 𝑈/𝐹1 =
{(𝑦2, 𝑦3, 𝑦8), (𝑦6, 𝑦7), (𝑦1, 𝑦4, 𝑦5, 𝑦9, 𝑦10)} 

             Since 𝐹1(𝑦1)  =  𝐹1(𝑦4)  =  𝐹1(𝑦5)  =  𝐹1(𝑦9)  =  𝐹1(𝑦10)  =  𝑔𝑜𝑜𝑑 
             𝐹1(𝑦2) = 𝐹1(𝑦3) = 𝐹1(𝑦8) = 𝑝𝑜𝑜𝑟  



A rough set theory application in forecasting models 

7 

             𝐹1(𝑦6) = 𝐹1(𝑦7) = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 

 𝑈/𝐹2 = {(𝑦2, 𝑦8), (𝑦4, 𝑦5, 𝑦9), (𝑦1, 𝑦3, 𝑦6, 𝑦7, 𝑦10),  
 𝑈/𝐹3 = {(𝑦1, 𝑦2, 𝑦4, 𝑦5, 𝑦8, 𝑦9), (𝑦3, 𝑦6, 𝑦10), (𝑦7),  
 𝑈/𝐹4 = {(𝑦2, 𝑦4, 𝑦8, 𝑦9, 𝑦10), (𝑦1, 𝑦3, 𝑦5, 𝑦6, 𝑦7)},  
 𝑈/𝐹5 = {(𝑦5, 𝑦6), (𝑦1, 𝑦3, 𝑦4, 𝑦9, ), (𝑦2, 𝑦7, 𝑦8, 𝑦10)},  

Lower and upper approximation of subsets of 𝑈 of attribute 𝐹1 with respect to 
attribute 𝐹2,  𝐹3,  𝐹4  and 𝐹5 are given below: 

(i) 𝐹1 with respect to 𝐹2 

𝑋(𝐹1 = 𝑝𝑜𝑜𝑟) = {𝑦2, 𝑦8}, 𝑋(𝐹1 = 𝑝𝑜𝑜𝑟) =

{𝑦1, 𝑦2, 𝑦3, 𝑦6, 𝑦7, 𝑦8, 𝑦10} 

𝑋(𝐹1 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒) = {∅}, 𝑋(𝐹1 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒) = {𝑦1, 𝑦3, 𝑦6, 𝑦7, 𝑦10} 

𝑋(𝐹1 = 𝑔𝑜𝑜𝑑) = {𝑦4, 𝑦5, 𝑦9},  𝑋(𝐹1 = 𝑔𝑜𝑜𝑑) =

{𝑦1, 𝑦3, 𝑦4, 𝑦5, 𝑦6, 𝑦7, 𝑦9, 𝑦10}. 
(ii) 𝐹1 with respect to 𝐹3 

𝑋(𝐹1 = 𝑝𝑜𝑜𝑟) = {∅}, 𝑋(𝐹1 = 𝑝𝑜𝑜𝑟) =

{𝑦1, 𝑦2, 𝑦3, 𝑦4, 𝑦5, 𝑦6, 𝑦8, 𝑦9, 𝑦10} 

𝑋(𝐹1 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒) = {𝑦7}, 𝑋(𝐹1 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒) = {𝑦3, 𝑦6, 𝑦7, 𝑦10} 

𝑋(𝐹1 = 𝑔𝑜𝑜𝑑) = {∅},  𝑋(𝐹1 = 𝑔𝑜𝑜𝑑) =

{𝑦1, 𝑦2, 𝑦3, 𝑦4, 𝑦5, 𝑦6, 𝑦8, 𝑦9, 𝑦10}. 
(iii) 𝐹1  with respect to 𝐹4 

𝑋(𝐹1 = 𝑝𝑜𝑜𝑟) = {∅}, 𝑋(𝐹1 = 𝑝𝑜𝑜𝑟) =

{𝑦1, 𝑦2, 𝑦3, 𝑦4, 𝑦5, 𝑦6, 𝑦7, 𝑦8, 𝑦9, 𝑦10} 

𝑋(𝐹1 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒) = {∅}, 𝑋(𝐹1 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒) = {𝑦1, 𝑦3, 𝑦5, 𝑦6, 𝑦7} 

𝑋(𝐹1 = 𝑔𝑜𝑜𝑑) = {∅},  𝑋(𝐹1 = 𝑔𝑜𝑜𝑑) =

{𝑦1, 𝑦2, 𝑦3, 𝑦4, 𝑦5, 𝑦6, 𝑦7, 𝑦8, 𝑦9, 𝑦10}. 
(iv) 𝐹1 with respect to 𝐹5 

𝑋(𝐹1 = 𝑝𝑜𝑜𝑟) = {∅}, 𝑋(𝐹1 = 𝑝𝑜𝑜𝑟) = {𝑦1, 𝑦2, 𝑦3, 𝑦4, 𝑦7, 𝑦8, 𝑦9, 𝑦10} 

𝑋(𝐹1 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒) = {∅}, 𝑋(𝐹1 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒) =

{𝑦2, 𝑦5, 𝑦6, 𝑦7, 𝑦8, 𝑦10} 

𝑋(𝐹1 = 𝑔𝑜𝑜𝑑) = {∅},  𝑋(𝐹1 = 𝑔𝑜𝑜𝑑)

= {𝑦1, 𝑦2, 𝑦3, 𝑦4, 𝑦5, 𝑦6, 𝑦7, 𝑦8, 𝑦9, 𝑦10} 

To explain in finding the degree of dependency of the attributes, we examine the 
information system as shown in Table1. From Table1, depend on each attribute there 
are five classes of 𝑈 induced by indiscernibility relation on each attribute. Using (9), 
the degree of dependency of attribute 𝐹2  on attribute 𝐹1, denoted by, 𝐹1 ⟹𝑘 𝐹2can be 
computed as follows: 
 

𝑘 =
∑ |𝐹1(𝑌)|𝑌∈𝑈/𝐹2

|𝑈|
=

|{𝑦6, 𝑦7}|

|{𝑦1, 𝑦2,   𝑦3, 𝑦4, 𝑦5,   𝑦6, 𝑦7, 𝑦8, 𝑦9,   𝑦10}|
 = 0.2 



 Sharma et al./Decis. Mak. Appl. Manag. Eng. 3 (1) (2020) 1-21  

8 

Similarly, we can compute 𝐹2 ⟹𝑘 𝐹3 as 

𝑘 =
∑ |𝐹2(𝑌)|𝑌∈𝑈/𝐹3

|𝑈|
=

|{𝑦2,   𝑦4,   𝑦5,   𝑦8,   𝑦9}|

|{𝑦1, 𝑦2,   𝑦3,  𝑦4,  𝑦5,   𝑦6, 𝑦7, 𝑦8, 𝑦9, 𝑦10}|
 = 0.5.  

The degree of dependency of all the attributes in Table 1 can be outlined as in Table 
2. In MDA performance, if the maximum degree of an attribute is equal with other 
attributes, then we check the next maximum value of the attribute. From the Table 2 
the first highest degree of the attribute, i.e., 0.5 comes out in attributes 𝐹1 and 𝐹3. Then 
the second maximum degree of attribute 𝐹1 is 0.1 even as in attribute 𝐹3  is 0. Hence, 
attribute 𝐹1 has been elected as a good attribute as compared with 𝐹2,  𝐹3,  𝐹4  and 𝐹5 
attributes. Therefore, Food quality (𝐹1) is the most appropriate attribute for Food 
Junction data among rest of the attributes. 

Table 2. Degree of Dependency using MDA criteria 

 𝐹1 𝐹2 𝐹3 𝐹4 𝐹5 MDA 

𝐹1 ….. 0.5 0.1 0 0 
0.5 
0.1 

𝐹2 0.2 ….. 0.4 0 0 
0.4 
0.2 

𝐹3 0 0.5 ….. 0 0 
0.5 
0 

𝐹4 0.2 0.2 0.1 ….. 0.2 0.2 

𝐹5 0 0.2 0.1 0 ….. 
0.2 
0.1 

4. Data description  

4.1. Mean absolute percentage error (MAPE) 

The mean absolute percentage error (MAPE), as well renowned as mean absolute 
percentage deviation (MAPD), is used as a measure of accuracy for constructing actual 
and fitted time series data. It usually denoted by the formula: 

 

𝑀𝐴𝑃𝐸 =
∑ (|𝐴𝑐𝑡− 𝑃𝑟�̂�|)/ 𝐴𝑐𝑡

𝑛
𝑡=1

𝑛
   (10) 

     Where 𝐴𝑐𝑡  (𝑡 =  1, 2, . … , 𝑛) is the actual value, Prt  ̂ (𝑡 =  1, 2, . … , 𝑛) represents 
the predicted values and 𝑛 is the total number of observations. 

4.2. Evaluation of rough set data 

According to Lewis (1982), the range of MAPE is fixed for time series models. But 
there is no any strong evidence for others criteria. Therefore MAPE is suitable criteria 
in the rough set analysis as compared to different indicators. For the measurement of 
the accuracy of roughness, we normalize the MAPE values in the following two ways.  
     Case1. MAPE range according to Lewis (1982), the value of MAPE being less than 10% 
denotes a high degree of accuracy. Moreover, when it lies between 10-20% predictions is 

good, 20-50% is reasonable and more than 50% depicts inaccuracy in prediction. 



A rough set theory application in forecasting models 

9 

Case 2. The accuracy in the level of prediction relies on the minimum value of 
MAPE. Henceforth in the paper, we have proposed our assumptions which offer us a 
valid and highly reliable result. In this sense value of MAPE, less than 5% provides 
highly accurate results. However, when it lies in the range of 5% to 10% the prediction 
is appreciably good and more than 10% shows inaccuracy in the prediction.  

The proposed procedures of modeling are: first normalized decision table is 
constructed in the range (0 to 1). Then TR, MMR and MDA are calculated to select best 
criteria for forecasting. Also, the best criteria are selected on the basis of minimum 
MMR value and maximum TR and MDA. All the steps also described in Figure 1. 

 

                           Figure 1. The Stages of Building best forecasting models. 

4.3. Data 

The study of Chen et al. (2009) has been used to evaluate MAPE value for this 
research as shown Table 3. In this research, the author analysed total arrivals of 12 
countries objects, of Japan, Hong Kong, and the US during the period 1996: 1-2007:12 
and three attributes SARIMA, Holt-Winters and Grey time series models. The entirety 
data series are separated into two groups, Tourism, and Non-Tourism.  

5. Result analysis and discussions 

In this section, we begin with the analysis of the data and make a comparison of 
their precise results. The comparison of the performance is based on the accuracy of 
the models. We have considered two test cases to bring out the comparison and 

Start 

Normalize the MAPE of Tourism, Non-Tourism and All-
purpose objects into its equivalent rough set information 

system 

Compute TR, MMR and MDA of all the attributes of the 
data sets 

Select the best attribute based on minimum MMR and 
maximum TR and MDA 

Stop 



 Sharma et al./Decis. Mak. Appl. Manag. Eng. 3 (1) (2020) 1-21  

10 

evaluation of the accuracy of the three models using three different rough set criteria 
TR, MMR, and MDA. 

5.1. Case 1: The rough set information system 

We have incorporated three models which are Holt-Winters, SARIMA, and Grey 
model. All these three models have three normalized values, i.e. reasonable, good and 
high accuracy. These values are mention in table 4 for considered twelve objectives. 

Table 3.   MAPE of fitted models (%) (Chen et al., 2009) 

Objects Holt-Winters SARIMA Model Grey Model 

Tourism purpose    

Japan 17.90 10.24 29.80 

Hong Kong 19.27 15.97 24.61 

US 5.23 5.01 16.37 

Total 16.56 7.03 31.16 

Non-tourism 
purpose 

   

Japan 4.75 4.02 8.85 

Hong Kong 4.03 4.87 19.57 

US 3.54 3.24 7.55 

Total 2.69 2.94 4.25 

All-purposes    

Japan 10.54 8.50 23.12 

Hong Kong 11.52 13.80 22.58 

US 2.81 2.72 8.54 

Total 5.55 8.95 16.04 

 

  



A rough set theory application in forecasting models 

11 

Table 4. Rough set information system of three models (Case1) 

Objects Holt-Winters SARIMA Model Grey Model 

Tourism purpose    

Japan Reasonable Reasonable Reasonable 

Hong Kong Reasonable Reasonable Reasonable 

Us Good Good Reasonable 

Total Reasonable Good Reasonable 

Non-tourism 
purpose 

   

Japan High accuracy High accuracy Good 

Hong Kong High accuracy High accuracy Reasonable 

Us High accuracy High accuracy Good 

Total High accuracy High accuracy High accuracy 

All-purposes   Reasonable 

Japan Reasonable Good Reasonable 

Hong Kong Reasonable Reasonable Good 

US High accuracy High accuracy Reasonable 

Total Good Good Reasonable 

To serve this purpose we obtain the equivalence classes determined by 
indiscernibility relation from equation (1). 
 

(i) 𝑌( 𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 =  𝑟𝑒𝑎𝑠𝑜𝑛𝑎𝑏𝑙𝑒 ) =  { 1,2 4, 9, 10}, 𝑌(𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 =
 𝑔𝑜𝑜𝑑) =  {3,12}, 𝑌(𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 =  𝐻𝑖𝑔ℎ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦) =  {5,6,7,8,11},
𝑈 / 𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 =  {{1,2,4,9,10}, {3,12}, {5,6,7,8,11}} 
 

(ii) 𝑌 (𝑆𝐴𝑅𝐼𝑀𝐴 =  𝑟𝑒𝑎𝑠𝑜𝑛𝑎𝑏𝑙𝑒) =  {1,2,10}, 𝑌(𝑆𝐴𝑅𝐼𝑀𝐴 =  𝑔𝑜𝑜𝑑)  =
 {3,4,9,12}, (𝑆𝐴𝑅𝐼𝑀𝐴 =  𝐻𝑖𝑔ℎ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦)  =  {5,6,7,8, 11}, 𝑈/𝑆𝐴𝑅𝐼𝑀𝐴 =
 {{1,2,10}, {3,4,9,12}, {5,6,7,8,11}} 

 

(iii) 𝑌 (𝐺𝑀(1,1)  =  𝑟𝑒𝑎𝑠𝑜𝑛𝑎𝑏𝑙𝑒)  =  {1,2,3,4,6,9,10,12}, 𝑌(𝐺𝑀(1,1)  =
 𝑔𝑜𝑜𝑑)  =  {5,7,11}, 𝑌(𝐺𝑀(1,1)  =  𝐻𝑖𝑔ℎ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦)  =  {8}, 𝑈/
 𝐺𝑀(1,1)  =  {{1,2,3,4,6,9,10,12}, {5,7,11}, {8}}. 

 

In the next step, we finalize the lower and upper approximations of subset Y of U 
based on Holt-Winters with respect to SARIMA and Grey model using the formula in 
Equation (2) and (3). 

 
 



 Sharma et al./Decis. Mak. Appl. Manag. Eng. 3 (1) (2020) 1-21  

12 

(i) Holt-Winters with respect to SARIMA 
𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 =  𝑟𝑒𝑎𝑠𝑜𝑛𝑎𝑏𝑙𝑒) = {1, 2, 10}, 

                    𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 =  𝑟𝑒𝑎𝑠𝑜𝑛𝑎𝑏𝑙𝑒) = {1, 2, 3, 4, 9, 10, 12}, 
𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 =  𝑔𝑜𝑜𝑑)   =   {∅} , 

𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 =  𝑔𝑜𝑜𝑑) = {3, 4, 9, 12}, 
𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 =  𝐻𝑖𝑔ℎ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦)  = {5, 6, 7, 8, 11},  

𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 =  𝐻𝑖𝑔ℎ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦 ) =  {5, 6, 7, 8, 11}. 
 

(ii) Holt-Winters with respect to grey 
𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 =  𝑟𝑒𝑎𝑠𝑜𝑛𝑎𝑏𝑙𝑒) = {∅},  

𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 =  𝑟𝑒𝑎𝑠𝑜𝑛𝑎𝑏𝑙𝑒)  = {1, 2, 3, 4, 6, 9, 10, 12}, 
𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 =  𝑔𝑜𝑜𝑑) = {∅}, 

𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 =  𝑔𝑜𝑜𝑑)  = {1, 2, 3, 4, 6, 9, 10}, 
𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 =  𝐻𝑖𝑔ℎ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦)  =  {5, 6, 7, 8, 11}, 

𝑌 (𝐻𝑜𝑙𝑡 𝑊𝑖𝑛𝑡𝑒𝑟𝑠 =  𝐻𝑖𝑔ℎ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦) = {5, 6, 7, 8, 11}. 
 

Then, we obtain the roughness of all models, where TR makes the use of formula 
as in the Equation (5). The roughness of subsets of U consist distinct value of Holt-
Winters with respect to SARIMA and Grey models are given below. 
 

(i) Roughness(R) of Holt Winters with respect to SARIMA  model  
𝑅𝑟𝑒𝑎𝑠𝑜𝑛𝑎𝑏𝑙𝑒  =  0.42857, 𝑅𝑔𝑜𝑜𝑑 =  0 and 𝑅ℎ𝑖𝑔ℎ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦 =  1 

 
(ii) Roughness of Holt-Winters with respect to Grey  model 

𝑅𝑟𝑒𝑎𝑠𝑜𝑛𝑎𝑏𝑙𝑒 = 0, 𝑅𝑔𝑜𝑜𝑑 =  0  and 𝑅ℎ𝑖𝑔ℎ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦 =  1. 

 
Further, we obtain the mean roughness of all attributes model. Where TR uses the 

formula in Equation (6). 
 

(i) Mean roughness(MR) of Holt-Winters with respect to SARIMA  model 
𝑀𝑅𝐻𝑊  =  0.47619. 

 
(ii) Mean roughness of Holt-Winters with respect to grey model 

𝑀𝑅𝐻𝑊  =  0.11. 
 

Now finally, we obtain the total roughness of Holt-Winters with respect to the 
SARIMA and Grey model. 

 𝑇𝑅(𝑎𝑖 ) = (0.47 + 0.11) /2= 0.2931, where ai is the numbers of attributes model, 
𝑖 =  1, 2. 

Now, from Table 5, we observe that the TR value 0.2931 of Holt-Winters is much 
higher than TR value 0.1944 of Grey model. Thus, the Holt-Winters consider as a better 
model than the Grey model. Similarly, TR value of SARIMA is greater than that of the 
Grey model. Hence, the SARIMA model serves us as a better model in view of the Grey 
model.  
  



A rough set theory application in forecasting models 

13 

Table 5. The total roughness of three model using TR criteria 

Attribute Holt-Winters 
SARIMA 
Model 

Grey Model 
Total Roughness 

 

Holt Winters ……… 0.47619 0.11 0.2931 

SARIMA 
Model 

0.42857 ……….. 0.11 0.26929 

Grey Model 0.19444 0.19444 ………….. 0.19444 

 

     Fourthly, we find the mean roughness of the three attributes using the principle of 
the MMR technique as in the equation (8) which differs from that in the TR technique. 
Mean roughness of the model Holt-Winters with respect to SARIMA and Grey model is 
as below. 
 

(i) MMR of Holt Winters with respect to SARIMA model 
𝑀𝑀𝑅𝐻𝑊 = 1 − 𝑀𝑅𝐻𝑊 = 1 − 0.47619 = 0.52381 

 
(ii)  MMR of Holt Winters with respect to Grey model 

𝑀𝑀𝑅𝐻𝑊 = 1 − 𝑀𝑅𝐻𝑊 = 1 − 0.11 = 0.89 
 

Consequently, the MMR of the entire models is depicted in Table 6. We perceive 
that the MMR of the Holt-Winters and SARIMA have attained their minimum values 
respectively i.e., 0.52381 and 0.57143. These values are lower than that of the MMR of 
Grey model i.e., 0.80556. Thus, Holt-Winters and SARIMA are preferred as the good 
models. Also, the results of the experiment were summarized in Fig. 2 and Fig. 3 shows 
the accuracy of three models with TR, MMR, and MDA values. However, as far as MDA 
techniques are considered, Holt-Winters and SARIMA have the maximum degree of 
dependency in Table 7 in contrast with the Grey model. As a result, the Holt-Winters 
and SARIMA are preferred as the good models.  

Table 6. The minimum-minimum roughness of three models using MMR 

criteria 

Models 
Holt-

Winters 
SARIMA Model Grey Model 

MMR 
 

Holt Winters …….. 0.52381 0.89 
0.52381 

 

SARIMA Model 0.57143 ……….. 0.89 
0.57143 

 

Grey Model 0.80556 0.80556 ……….. 
0.80556 

 

 

  



 Sharma et al./Decis. Mak. Appl. Manag. Eng. 3 (1) (2020) 1-21  

14 

Table 7. Degree of Dependency using MDA criteria of three models 

 

 

 

       Figure 2. The roughness of HW, SARIMA and GM (1, 1) models using 
TR criterion for case 1 

 

Models Holt-Winters SARIMA Model Grey Model 
MDA 

 

Holt 
Winters 

….. 0.6667 0.333 0.6667 

SARIMA 
Model 

0.583 ….. 0.333 
0.583 
0.333 

Grey 
Model 

0.583 0.583 ….. 0.583 

0.2931

0.26929

0.19444

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

HW SARIMA GM(1,1)

R
o

u
g
h
n
e
ss

 M
e
a
su

re
m

e
n
t

Forecasting Models



A rough set theory application in forecasting models 

15 

Figure 3. The roughness of HW, SARIMA and GM (1, 1) models using MMR 

and MDA criteria for Case 1 

5.3. Case 2: The rough set information system 

Table 8 shows in the information system of rough set data of case2. In this case, the 
analysis to evaluate TR, MMR, and MDA of each method is similar as in case1. With TR 
techniques Holt-Winters model is considered an as good model because this attribute 
has the highest total roughness in Table 9 i.e. 0.5625 as compared to SARIMA and Grey 
model i.e. 0.29165 and 0.2065. Thus, we select Holt-Winters model is good as compare 
to SARIMA and Grey model. The results of MMR and MDA are given in Table 10 and 
Table 11. According to MMR and MDA reports, Holt-Winters models are chosen to 
select good model as compared to SARIMA and Grey model. Also, Fig. 4 and Fig. 5 
illustrate the accuracy of three models with TR, MMR and MDA values which are 
involving the datasets of case 2. From the above considerations, we can see that Case 
1 gives better results of three models in terms of rough accuracy compare to Case 2 
results. Therefore, case 1 is an appropriate range of MAPE for applying the rough set 
approach in the field of time series modeling. 
  

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

R
o
u
g
h
n
e
ss

Models

MMR

MDA



 Sharma et al./Decis. Mak. Appl. Manag. Eng. 3 (1) (2020) 1-21  

16 

Table 8. Rough set information system of three models (Case 2) 

Objects Holt-Winters SARIMA Model Grey Model 

Tourism purpose    

Japan Good Good Reasonable 

Hong Kong Good Good Reasonable 

US High accuracy High accuracy Good 

Total Good High accuracy Reasonable 

Non-tourism 
purpose 

   

Japan High accuracy High accuracy High accuracy 

Hong Kong High accuracy High accuracy Good 

US High accuracy High accuracy High accuracy 

Total High accuracy High accuracy High accuracy 

All-purposes    

Japan Good High accuracy Reasonable 

Hong Kong Good Good Reasonable 

US High accuracy High accuracy High accuracy 

Total High accuracy High accuracy Good 

Table 9. The total roughness of three models using TR criteria 

Attribute Holt-Winters 
SARIMA 
Model 

Grey Model 
Total Roughness 

 

Holt Winters ……… 0.125 1 0.5625 

SARIMA 
Model 

0.29165 ……….. 0.29165 0.29165 

Grey Model 0.33 0.083 ………….. 0.2065 

 

Table 10. The minimum-minimum roughness of three models using MMR 

criteria 

Attribute Holt-
Winters 

SARIMA Model Grey Model MMR 
 

Holt Winters …….. 0.875 0 0 
0.875 

SARIMA Model 0.70835 ……….. 0.70835 0.70835 
 

Grey Model 0.67 0.917 ……….. 0.67 
0.917 

 



A rough set theory application in forecasting models 

17 

Table 11. Degree of Dependency using MDA criteria 

Attribute Holt-Winters SARIMA Model Grey Model 
MDA 

 

Holt Winters ….. 0.25 1 
1 
 

SARIMA Model 0.5833 ….. 0.5833 
0.5833 

 

Grey Model 0.4167 0.25 ….. 
0.4167 

 

 

 

Figure 4. The roughness of HW, SARIMA and GM (1, 1) models based on 

TR criterion for Case 2 

 

 

 

Figure 5. The accuracy of HW, SARIMA and GM (1,1) models based on 

MMR and MDA criteria for Case 2 

0

0.1

0.2

0.3

0.4

0.5

0.6

HW SARIMA GM(1,1)

R
o
u
g
h
n
e
ss

Models

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

HW-SARIMA SARIMA-HW GM(1,1)-HW GM(1,1)-SARIMA

R
o
u
g
h
n
e
ss

Models

MMR

MDA



 Sharma et al./Decis. Mak. Appl. Manag. Eng. 3 (1) (2020) 1-21  

18 

6. Conclusions  

Current research proposes a new technique to select the best forecasting model 
using rough set approach. This technique is based on the rough set criteria i.e., TR, 
MMR and MDA. In the present study, the problems concerning the selection of good 
forecasting model using rough set methods, such as normalized MAPE values. Two test 
cases are considered to implement rough set and three models i.e. Holt winters, 
SARIMA, and Grey models are employed. At first, we normalize MAPE values into the 
rough set information system and then we calculate the accuracy of each model. A 
close analysis of all the results furnished by the respective models, it reports that Holt-
Winters and SARIMA models are far better models when compared to Grey model. 

     According to the analysis reports, Holt-Winters and SARIMA are good models in 
consideration of seasonal time series. Also, Grey model is an unqualified model to 
forecast seasonal time series under TR, MMR, and MDA rough set criteria. In this 
research, we would recommend that the proposed research is feasible and it offers a 
powerful statistical evidence for rough set methods. . Herewith these concepts, we 
suppose that various applications through rough set will be relevant in time series 
forecasting. The projected approach could also be useful in large real-life datasets. In 
future, one can apply these techniques in other time series forecasting models. 
    Our newly proposed technique differs from the traditional statistical method in the 
sense, that it provides an alternative way to selection best forecasting model for time 
series forecasting. 

Author Contributions: Each author has participated and contributed sufficiently to 
take public responsibility for appropriate portions of the content.  

Funding: This research received no external funding. 

Conflicts of Interest: The authors declare no conflicts of interest. 

References 

Box, G. E. P., & Jenkins, G. M. (1976). Time Series Analysis: Forecasting and Control,  

Revised. San Fransisco: Holden-Day, 575. 

Celotto, E., Ellero, A. &Ferretti, P. (2012).  Short-medium term tourist services demand 
to forecast with rough ret theory, Procedia Economics, and Finance, 3, 62-67. 

Chen, C.-F., Chang,Y.-H &Chang, Y.-W. (2009). Seasonal ARIMA Forecasting of inbound 
air Travel arrivals to Taiwan, Transportmetrica, 5(2), 125-140. 

Chiao, Wang, C.H. (2002). Reliability improvement of fluorescent lamp using grey 
Forecasting model, Microelectronics Reliability, 42, 127-134. 

Chu, F. L. (1998). Forecasting tourism demand in Asian-Pacific countries, Annual of 
Tourism Research, 25(3), 597-615. 

Deng, J. L. (1982). Control problems of grey systems, Systems and Control Letters, 1(1), 
288-294. 

Faustino, P. C., Pinheiro, A. C., Carpinteiro, A. O. &Lima, I. (2011). Time series 
forecasting through rule-based models obtained via rough sets. Artif Intell Rev, 36, 
299-310.   



A rough set theory application in forecasting models 

19 

Goh, C. &Law, R. (2002). Modeling and forecasting tourism demand for arrivals with 
stochastic nonstationary seasonality and intervention, Tourism Management, 23, 499-
510. 

Goh, C. &Law, R. (2003). Incorporating the rough sets theory into travel demand 
analysis, Tourism Management, 24, 511-517. 

Golmohammadi, A., &Ghareneh, N. S. (2011). Importance analysis of travel attributes 
using a rough set-based neural network: The case of Iranian tourism industry Journal 
of Hospitality and Tourism Technology 2(2), 155-171. 

Hassanein, W. A., &Elmelegy, A. (2013). An algorithm for selecting clustering attribute 
using significance of attributes, International Journal of Database Theory and 
Application, 5, 53-66. 

Herawan, T., Deris, M., &Abawajy, J. (2010). A rough set approach for selecting 
clustering Attribute, Knowledge-Based Systems, 23, 220-231. 

Herawan, T., Ghazali, R., & Deris, M. (2010). Rough Set approach for categorical data 
clustering, International Journal of Database Theory and Application, 3(1), 33-52. 

Jensen, R., &Shen, Q. (2009). New Approaches to Fuzzy-Rough Feature Selection, IEEE 
Transactions on Fuzzy systems, 17(4), 824-838. 

Kaya, Y., & Uyar, M. (2013). A hybrid decision support system based on rough set and 
extreme learning machine for diagnosis of hepatitis disease, Applied Soft Computing, 
13(8), 3429-3438. 

Karavidić, Z., & Projović, D. (2018). A multi-criteria decision-making (MCDM) model in 
the security forces operations based on rough sets. Decision Making: Applications in 
Management and Engineering, 1(1), 97-120.  

Law, R., &Au, N. (1998). A rough set approach to hotel expenditure decision rules 
induction, Journal of Hospitality and Tourism Research, 22(4), 359-375. 

Law, R., & Au, N. (2000). Relationship modeling in tourism shopping: a decision rules 
induction approach, Tourism Management, 21(3), 241-249. 

Law, R., & Au, N., (1999). A neural network model to forecast Japanese demand for 
travel to Hong Kong, Tourism Management, 20, 89-97. 

Law, R. (2000). Back-propagation learning in improving the accuracy of neural 
network based Tourism demand forecasting, Tourism Management, 21, 331-340. 

Lewis, C.D. (1982). International and business forecasting methods, London: 
Butterworths. 

Li, J., Li F., & Zhou, G. (2011). Travel Demand Prediction in Tangshan City of China 
Based on Rough Set. Springer-Verlag Berlin Heidelberg, 440-446. 

Lim, C., & McAleer, M. (2002). Time series forecasts of international travel demand for 
Australia,Tourism Management, 23(4), 389-396. 

Liou, J. J. H., Chuang, Y. C., & Hsu, C. C. (2016). Improving airline service quality based 
on rough set theory and flow graphs. Journal of Industrial and Production Engineering. 
33(2), 123-133. 



 Sharma et al./Decis. Mak. Appl. Manag. Eng. 3 (1) (2020) 1-21  

20 

Lu, I. J., Lewis C., & Lin, S. J., (2009). The forecast of a motor vehicle, energy demand 
and Co2 emission from Taiwans road transportation sector, Energy Policy, 37, 2952-
2961. 

Mahapatra, S., & Mahapatra S. S. (2010). Attribute selection in marketing: A rough Set 
approach, IIMB Management Review, 22(1-2), 16-24. 

Mamat, R., Herawan, T., & Deris, M. (2013). MAR: Maximum Attribute Relative of soft 
set for clustering attribute selection, Knowledge-Based Systems, 52, 11-20. 

Mazlack, L., He., A., Zhu, Y., & Coppock, S. (2000). A rough set approach in choosing 
partitioning attributes, Proceedings of the ISCA 13th, International Conference 
(CAINE-2000), 1-6. 

Nguyen, L.T., Shu, H.M., Huang, F.Y., & Hsu, M.B. (2013). Accurate forecasting models 
in predicting the inbound tourism demand in Vietnam, Journal of Statistics and 
Management Systems, 16(1), 25-43. 

Pamučar, D., Stević, Z., & Zavadskas, E. K. (2018). Integration of interval rough AHP and 
interval rough MABAC methods for evaluating university web pages, Applied Soft 
Computing, 67, 141-163. 

Pankratz, A. (1983). Forecasting with univariate Box-Jenkins models: concepts and 
cases, New York: Wiley. 

Parmar, D., Wu, T., & Blackhurst, J. (2007). MMR: an algorithm for clustering 
categorical data using rough set theory, Data and knowledge Engineering, 63, 879-893. 

Pawlak, Z. (1982). Rough sets, International Journal of Computer and Information 
Science, 11, 341-356. 

Pawlak, Z. (1991). Rough sets, A Theoretical Aspect of Reasoning about data, Kluwer 
Academic Publisher, Boston. 

Pawlak, Z., & Skowron, A. (2007). Rudiments of rough sets, Information Sciences, 
177(1), 3-27. 

Qin, H., Ma, X., Zain, J.M., & Herawan, T. (2012). A novel soft set approach in selecting 
clustering attribute, Knowledge-Based Systems, 36, 139-145. 

Roy, J., Adhikary, K., Kar, S., & Pamucar, D. (2018). A rough strength relational 
DEMATEL model for analysing the key success factors of hospital service quality. 
Decision Making: Applications in Management and Engineering, 1(1), 121-142.  

Sharma, H. K., Kumari, K., & Kar, S. (2019). Short-term Forecasting of Air Passengers 
based on Hybrid Rough Set and Double Exponential Smoothing Models, Intelligent 
Automation and Soft Computing, 25(1), 1-13. 

Sharma, H., Roy, J., Kar, S., & Prentkovskis, O. (2018). Multi-Criteria Evaluation 
Framework for Prioritizing Indian Railway Stations Using Modified Rough AHP-Mabac 
Method. Transport and Telecommunication Journal, 19(2), 113-127.  

Tseng, F.M., Yu, H.C., & Tzeng, G. H. (2001). Applied hybrid Grey model to forecast 
seasonal time series, Technological Forecasting and Social Change, 67(2), 291-302. 

Vasiljević, M., Fazlollahtabar, H., Stević, Ž., & Vesković, S. (2018). A rough multicriteria 
approach for evaluation of the supplier criteria in automotive industry. Decision 
Making: Applications in Management and Engineering, 1(1), 82-96.  



A rough set theory application in forecasting models 

21 

Wang, C.H. (2004). Predicting tourism demand using fuzzy time series and hybrid Grey 
Theory, Tourism Management, 25(3), 367-374. 

Winters, P.R. (1960). Forecasting sales by exponentially weighted moving averages, 
Management  Science, 6(3), 324-342. 

Wu, H., Wu, Y., & Luo, J. (2009). An Interval Type-2 Fuzzy Rough Set model forattribute 
Reduction, IEEE Transactions on fuzzy systems, 17(2), 301-315.  

© 2018 by the authors. Submitted for possible open access publication under the 

terms and conditions of the Creative Commons Attribution (CC BY) license 

(http://creativecommons.org/licenses/by/4.0/).