Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol. 31 No. 5s (2024) 

 

475 

 

An Analytical Estimation on The Factors Affected Students’ Education 

After COVID-19 Pandemic Via Dempster-Shafer Theory of Evidence 

 

Devi. 𝐌𝟏, Irene Hepzibah. 𝐑𝟐, Geethalakshmi. 𝐆𝟑 
1 & 2 PG & Research Department of Mathematics, T.B.M.L College (Affiliated to Annamalai university), 

Porayar- 609 307, Tamil Nadu, India. E-mail: devipg2020@gmail.com, ireneraj74@gmail.com 

3 Department of Mathematics, A.V.C. College of Engineering, Mannampandal, Mayiladuthurai, Tamil Nadu, 

India. E-mail: geethavish294@gmail.com. 

 

Article History: 

Received: 09-05-2024 

Revised: 28-06-2024 

Accepted: 11-07-2024 

Abstract:  

Introduction: Dempster-Shafer Theory (simply D-S Theory) is developed by Arthur P. 

Dempster and Glenn Shafer. This theory is widely used in many fields. One advantageous 

aspect of this is the ability to incorporate additional data into D-S Theory. However, it is 

important to note that the computational process necessitates a significant amount of 

effort due to the complexities involved with 2𝑛 of sets. 

Objectives: This study focuses on combination rule for multiple basic probability 

assignments in uncertain environment. 

Methods: An algorithm has been developed and a python program has also been 

generated to validate combination rule for multiple basic probability assignments. 

Tableau software is utilized for illustrating data visualization. 

Results: This concept has been implemented by the factors contributing to the decline in 

students’ studies after the COVID-19 pandemic by the way of adopting fuzzy numbers.  

Conclusions: Final output is tabulated which are obtained by generated python program. 

That clearly shows the highest impact factor affected students’ studies after COVID-19 

pandemic. In future, it is decided to study more number of real-life applications based on 

fuzzy evidence theory in our Research. 

Keywords: D-S Theory; Fuzzy Sets; Fuzzy Numbers; Basic probability assignments, 

Python program. 

 

1. Introduction 

These days, numerous comparisons are required for every circumstance that could arise. At least two 

data are needed for a comparison, but occasionally we need to compare more than two, in which case 

some computations are needed to determine which is preferable. The optimum method for computing 

large amounts of data in this calculation is the dempster combination rule. In 1967, Dempster presented 

evidence theory to the world, and in 1976, Glenn Shafer [10], one of his students, reinterpreted it. The 

combination rule was calculated by George J Klir and Bo Yuan [5] using two basic probability 

assignments and its numerical problem. The book by Kari Senz and Scott Ferson [7] explains a variety 

of possible combination rules for dempster-shafer frameworks [6 & 8] and includes illustrations of 

how to execute these rules for discrete and interval-valued data. The belief function is a mathematical 

framework for describing assessed beliefs in the context of provided data. Uncertainty caused by a 



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lack of evidence can be made explicit using belief functions. The belief function is discussed 

comprehensively in Bibliographies’ [1-4], [11-13].  

Dempster's rule of combination is presented in this study for more than two basic probability 

assignments. We proposed a novel algorithm for calculating efficiency. A python program [9] was 

developed to solve the proposed method. The success of the suggested approach was validated using 

numerical examples. 

The structure of this research paper is as follows: Gives a brief review of Dempster-Shafer theory in 

Section II. We presented a new algorithm in Section III for assessing the stimulus evidence. In section 

IV, we generate a Python program for our proposed method and we utilized a numerical example to 

demonstrate the applicability of our proposed approach in section V. At last, Section VI provides some 

concluding remarks. 

2. Preliminaries 

The two primary topics wrapped by the background knowledge presented in this section are as follows: 

(a) a brief overview of Dempster-Shafer Theory (D-S Theory) definitions. (b) The definition of 

Dempster’s rule of combination. 

The D-S Theory of evidence is based on a finite set of mutually exclusive elements known as the frame 

of discernment, symbolized by Θ. 

Definition 2.1. Basic Probability Assignment [5] 

If Θ is a frame of discernment, then a function m: 2Θ  → [0,1]  is called a Basic Probability Assignment 

whenever  

 (1) m(∅) = 0 

And  (2) ∑ 𝑚(𝐴)𝐴⊂Θ = 1. 

Definition 2.2. Belief Measure [5] 

A function Bel: 2Θ  → [0,1] is called a Belief function over Θ,  

If   Bel(A) = ∑ 𝑚(𝐵)𝐵⊂A  

For all proper subsets of B of A. 

Furthermore,  Bel(Θ) = 1 but Bel (A) = 0 for all A≠Θ. 

Definition 2.3. Plausibility Measure [5] 

Belief functions and Plausibility functions are mutually dual. 

A function Pl: 2Θ  → [0,1] is called a Plausibility function over Θ, 

If  Pl(A) = ∑ 𝑚(𝐵)𝐴∩B≠∅  

For all proper subsets of B of A. 

Pl (A) = 1 – Bel (�̅�) 

Since Bel (A) + Bel (�̅�) ≤ 1. 



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Definition 2.4. Dempster’s Rule of combination [5] 

The general way of combining evidence is denoted by the formula 

𝑚1⨁𝑚2(𝐴) =  
∑ 𝑚1(𝐴𝑖). 𝑚2(𝐵𝑗)𝐴𝑖∩𝐵𝑗=A

1 − 𝐾
 

For all A≠∅ and 𝑚1,2(∅) =  0 , where 

              𝐾 = ∑ 𝑚1(𝐴𝑖). 𝑚2(𝐵𝑗)

𝐴𝑖∩𝐵𝑗=∅

 

3. Dempster’s Rule of combination for Multiple BPA [8]: 

In this section, we propose a method to estimate more than two basic probability assignment (BPA). 

Through this we can calculate various data at the same time.  

Let 𝐵𝑒𝑙1 ⋯ 𝐵𝑒𝑙𝑛are belief functions over the same frame Θ, with basic probability assignments 

𝑚1  ⋯ 𝑚𝑛 and focal elements 𝐴1  ⋯ 𝐴𝑘 and 𝐵1  ⋯ 𝐵𝑙 respectively. 

The combination rule for the multiple BPA is 

      𝑚(𝐴) =  
∑ (∏ 𝑚𝑗(𝐴𝑗)𝑛

𝑗=1 )𝐴1∩𝐴2∩⋯∩𝐴𝑛=𝐴

1−𝑘
               ⋯(3.1)  

Where            𝐾 = ∑ (∏ 𝑚𝑗
𝑛
𝑗=1 (𝐴𝑗))   𝐴1∩𝐴2∩⋯∩𝐴𝑑=∅                ⋯(3.2) 

3.1 Algorithm: 

An algorithm has been developed to solve the problem with simplicity. 

step 1 – START the python program. 

step 2 – Create a new file and save your file using Ctrl + s. 

step 3 − Define the values of 𝒎𝒋(𝑨𝒋)for j = 1,2, ⋯, n. 

step 4 – Using 1 −  ∑ ∏ 𝑚𝑗
𝑛
𝑗=1 (𝐴𝑗)𝐴1∩𝐴2∩⋯∩𝐴𝑗=∅  for finding K value. 

Step 5 – Finding combined evidence, put the values in 𝑚(𝐴) =  ∑ ∏ 𝑚𝑗
𝑛
𝑗=1 (𝐴𝑖)𝐴1∩𝐴2∩⋯∩𝐴𝑖=A  . 

step 6 – Run the coding using function key F5. 

step 7 – Print the output. 

step 8 – STOP the program. 

4. Numerical Illustration 

The quality of education has completely lagged behind during the COVID-19 pandemic. We collecting 

the data from students. The below factors play an important role in affecting students’ studies. That 

factors are listed below: 

• Comprehension of the subject is decreased during Online classes – denoted as C 

• Increase in mobile phone use leads to decline in interest in studies – denoted as P 



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• Difficulty in memorizing concepts – denoted as M 

• Decreased writing skills – denoted as W 

Example 4.1: 

Factors M1 M2 M3 M4 

C 0.038 0.037 0.039 0.038 

P 0.040 0.039 0.039 0.040 

M 0.037 0.036 0.037 0.038 

W 0.038 0.037 0.038 0.039 

C, P 0.058 0.057 0.058 0.057 

C, M 0.056 0.056 0.057 0.057 

C, W 0.057 0.057 0.056 0.056 

P, M 0.058 0.058 0.057 0.057 

P, W 0.057 0.056 0.058 0.056 

M, W 0.056 0.056 0.055 0.055 

C, P, M 0.087 0.086 0.088 0.087 

C, P, W 0.088 0.087 0.087 0.087 

C, M, W 0.086 0.085 0.087 0.086 

P, M, W 0.087 0.087 0.086 0.087 

C, P, M, W 0.157 0.166 0.158 0.160 

Table 4.1 

Table 4.1 gives the input getting from experts to find the combination rule for Multiple Basic 

Probability Assignment, by using the relation (3.1) and the proposed algorithm to solve the 

computation. 

To the calculation, we first compute K value, by using the following  

𝐾 =  ∑ ∏ 𝑀𝑗(𝑍𝑗)

4

𝑗=1𝑍1∩𝑍2∩𝑍3∩𝑍4=∅

                                                                 ⋯ (4.1) 

Where 𝑧1 = {𝐶}, {𝑃}, {𝑀}, {𝑊} 

𝑍2 = {𝐶, 𝑃}, {𝐶, 𝑀}, {𝐶, 𝑊}, {𝑃, 𝑀}, {𝑃, 𝑊}, {𝑀, 𝑊} 

𝑍3 = {𝐶, 𝑃, 𝑀}, {𝐶, 𝑃, 𝑊}, {𝐶, 𝑀, 𝑊}, {𝑃, 𝑀, 𝑊} 

𝑍4 = {𝐶, 𝑃, 𝑀, 𝑊} 

Now, find combined evidence for 𝑀1,2 

𝐶 =  ∑ ∏ 𝑀𝑗(𝑍𝑗)

4

𝑗=1𝑍1∩𝑍2∩𝑍3∩𝑍4=𝐶

                                                            ⋯ (4.2) 

𝐶𝑃 =  ∑ ∏ 𝑀𝑗(𝑍𝑗)           

4

𝑗=1

                                               ⋯ (4.3)

𝑍1∩𝑍2∩𝑍3∩𝑍4=𝐶𝑃

 



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𝐶𝑃𝑀 =  ∑ ∏ 𝑀𝑗(𝑍𝑗)

4

𝑗=1𝑍1∩𝑍2∩𝑍3∩𝑍4=𝐶𝑃𝑀

                                                        ⋯ (4.4) 

𝐶𝑃𝑀𝑊 =  ∑ ∏ 𝑀𝑗(𝑍𝑗)

4

𝑗=1𝑍1∩𝑍2∩𝑍3∩𝑍4=𝐶𝑃𝑀𝑊

                                                     ⋯ (4.5) 

Where 𝑧1 = {𝐶}, {𝑃}, {𝑀}, {𝑊} 

𝑍2 = {𝐶, 𝑃}, {𝐶, 𝑀}, {𝐶, 𝑊}, {𝑃, 𝑀}, {𝑃, 𝑊}, {𝑀, 𝑊} 

𝑍3 = {𝐶, 𝑃, 𝑀}, {𝐶, 𝑃, 𝑊}, {𝐶, 𝑀, 𝑊}, {𝑃, 𝑀, 𝑊} 

𝑍4 = {𝐶, 𝑃, 𝑀, 𝑊} 

Similarly, we can calculate P, M, W, CM, CW, PM, PW, MW, CPW, CMW, PMW and 𝑀2,3, 𝑀3,4. 

4.2. Python program 

The equations (4.1), (4.2), (4.3), (4.4) & (4.5) clearly shows that, solving the given problem requires 

a significant amount of calculating efforts. So, we generated new python code for Dempster-shafer 

theory, to compute the combined evidence. Using this code, the computations are done without any 

difficulty. 

4.2.1 Sample code 

Here, we give a sample python code for 

Dempster-Shafer theory. 

M = 1 

m1c =float(input('Enter m1c: ')) 

m1p =float(input('Enter m1p: ')) 

m1m =float(input('Enter m1m: ')) 

m1w =float(input('Enter m1w: ')) 

m1cp=float(input('Enter m1cp:')) 

m1cm=float(input('Enter m1cm:')) 

m1cw=float(input('Enter m1cw:')) 

⋮ 

print(f'value of pmw is {div(v,K)}') 

print(f'value of cpmw is {div(x,K)}') 

4.2.2 Final Output 

Enter m1c: .107 

Enter m1p: .111 

Enter m1m: .105 

Enter m1w: .106 

Enter m1cp: .064 

Enter m1cm: .062 

Enter m1cw: .064 

Enter m1pm: .065 

Enter m1pw: .064 

Enter m1mw: .063 

Enter m1cpm: .04 

Enter m1cpw: .041 

Enter m1cmw: .04 

Enter m1pmw: .04 

Enter m1cpmw: .03 

Enter m2c: .109 

Enter m2p: .111 

Enter m2m: .106 



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Enter m2w: .107 

Enter m2cp: .064 

Enter m2cm: .064 

Enter m2cw: .063 

Enter m2pm: .064 

Enter m2pw: .064 

Enter m2mw: .062 

Enter m2cpm: .04 

Enter m2cpw: .04 

Enter m2cmw: .04 

Enter m2pmw: .04 

Enter m2cpmw: .029 

value of K is 0.637446 

value of c is 0.1956401012791672 

value of p is 0.20162335319383923 

value of m is 0.19143739234382207 

value of w is 0.19310027829808332 

value of cp is 0.03359657131741356 

value of cm is 0.032890629167019646 

value of cw is 0.03332203825892704 

value of pm is 0.03370481578047396 

value of pw is 0.03359657131741356 

value of mw is 0.032619233629201536 

value of cpm is 0.006212290923466458 

value of cpw is 0.0063205353865268595 

value of cmw is 0.006212290923466458 

value of pmw is 0.006212290923466458 

value of cpmw is 0.0013648214907615705 

> 

4.2.3 Output Table: The final output values are tableted below. 

factors M1,2 M3,4 M Bel 

C 0.107 0.109 0.196 0.19 

P 0.111 0.111 0.201 0.201 

M 0.105 0.106 0.191 0.191 

W 0.106 0.107 0.193 0.193 

C, P 0.064 0.064 0.034 0.431 

C, M 0.062 0.064 0.033 0.420 

C, W 0.064 0.063 0.033 0.422 

P, M 0.065 0.064 0.034 0.426 

P, W 0.064 0.064 0.034 0.428 

M, W 0.063 0.062 0.033 0.417 

C, P, M 0.040 0.040 0.006 0.661 

C, P, W 0.041 0.040 0.006 0.663 

C, M, W 0.040 0.040 0.006 0.652 

P, M, W 0.040 0.040 0.006 0.659 

C, P, M, W 0.030 0.029 0.001 1 

Table 4.2 



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4.3. Data Visualization: 

A clear diagram for the provided computation is illustrated here, by utilizing Tableau software. 

 

Fig 4.1 

5. Conclusion 

As we see in the book of “Mathematical Theory of evidence”, by Shafer the calculations for two basic 

probability assignments have been made. But here, in this research work we compared more than two 

basic probability assignments to find out the high impact factor affecting studies. This concept has 

been established with the help of an illustration by the factors contributing to the decline in students’ 

studies after the COVID-19 pandemic. An algorithm has been used to solve a proposed method based 

on given data and a python program has been generated in this research work. Data visualization is 

done by Tableau software. Finally, based on the results, mobile phone usage has been increased among 

students, which is one of the factors which affect students' education after the Covid-19 pandemic.  

The reason is that during and after the pandemic period, students learned through online mode, so that 

their writing skills and comprehension of the subject declines.  And they are distracted by unnecessary 

advertisements and social media while using mobile phones. Further, it is decided to study more 

number of real-life applications based on fuzzy evidence theory in our future Research. 

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[3] Flora. M, Jouselme. A.L., et al. (2003), “Combining belief functions and fuzzy membership functions”, The 

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