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Abstract

Existing recommendation systems lack to address the need of several problems in education domain due to the availability of limited information. Study of recommendation systems to facilitate students’ education in their relevant services is being carried out in different perspective. Hence, the objective of this work is to apply existing knowledge for decision making to recommend students during their course registration process in an institution. The situation where more number of experts involves in decision making was influenced by Fuzzy Soft Expert Set (FSES). This work emphasizes the application of FSES for facilitating students’ course registration process assuming that the same course is offered by more number of faculty members during a semester. As more number of experts involved in this process and the uncertainty of the available data, using the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) along with FSES was considered. A new methodology FSES-TOPSIS was also applied in this work. The proposed method was accurate enough to enable students to rank the faculty members based on their previous performance and to enroll themselves with the faculty members of their interest. The method proposed here can be extended for various problems of similar nature.

Keywords

Recommendation system decision making fuzzy soft expert set TOPSIS course registration ideal solution

Introduction

There are certain mathematical problems in medical diagnosis, business, engineering, etc., that is difficult to solve using conventional techniques. The reason behind this fact is that the statistics that are used to solve these problems are uncertain and in diverse formats. Therefore, the need arises for mathematical techniques to address these issues. Zadeh introduced fuzzy sets in 1965 as one solution to solve the aforementioned problems. In a conventional set, the membership of each element is defined in binary as either a member or not. However, in any fuzzy sets, all the elements possess some degree of membership. Even a non-member element gets different degree of membership. Due to this ability to easily handle uncertainty, fuzzy sets have been studied by many researchers and adopted for a vast range of research domains. Fuzzy set has been combined with various soft set theories to produce more effective models for handling uncertainty. This work aims to use multiple criteria for assessing performance of faculty members who are offering same courses to facilitate students for identifying a suitable faculty member from the available list. A sample case study on the same scenario is presented in this paper to demonstrate the application of FSES.

After fuzzy set theory was introduced by Zadeh,¹ some other theories were proposed, such as rough sets,² intuitionistic fuzzy sets³ and vague sets.⁴ The inadequacy of the parameterization tool is the limitation that is identified among these theories, even though these theories have been accepted as fruitful mathematical models for handling uncertainties. In 1999, Molodtsov⁵ introduced the soft set as a tool to handle uncertainties, which is free from the previously mentioned difficulty. A pair $(F, S)$ is termed as a soft set over the universe $U$ , where $F$ is the function that maps $S$ into the subsets of $U$ . Beneficial results were produced by investigators by applying soft sets and its variations (fuzzy soft sets, bijective soft sets, exclusive disjunctive soft sets and many more) in many decision-making problems.^6–26 Since soft sets involve a single expert, there is the need for additional operations to combine the opinions from more than one expert for more effective decision making. Alkhazaleh et al. introduced soft expert sets with opinions from multiple experts in a single model without any additional operations.²⁷ Here, along with the universe $U$ and the set of evaluation parameters $P$ , the set of experts $X$ and opinions $O$ are also considered. The opinion set $O$ may contain two-valued opinions or multi-valued opinions. Here, $Z = P \times X \times O$ and S $\subseteq Z$ . A pair $(F, S)$ is termed as a soft expert set over the common universe $U$ , where $F$ maps $S$ to $P (U)$ , which is the power set of $U$ . Some works based on soft expert sets can be found in the literature.^28,29 The introduction of fuzziness in soft expert sets added more value to the stated theory. Fuzzy theory alone permits an element to be a member and nonmember of a set with different degrees of membership. Therefore, the fuzzy soft expert set became a more effective tool for decision making under uncertainty.³⁰ One can find works related to fuzzy soft expert sets in the literature.^31–35 The fuzzy soft expert set helps decision making by aggregating the agree fuzzy values and disagree fuzzy values that are given by the experts and by finding the differences to form a conclusion. However, the relation between agree and disagree values has not been considered. Buckley proposed fuzzy ranking by partitioning the ranked alternatives that are expressed by experts.³⁶ The fuzzy TOPSIS approach has been proposed by Sodhi and Prabhakar by implementing TOPSIS with fuzzy inputs.³⁷ Fuzzy TOPSIS and its variations have been analyzed in multicriteria decision making by many researchers, and a few are referred in this paper.^38–49 Several case studies were conducted to assess students’ performance using various criteria were also analyzed.^50,51 However, fuzzy TOPSIS involves only fuzzified input values. Fuzzy soft expert sets involve agree fuzzy values and disagree fuzzy values. If those values are considered in the TOPSIS technique, the decision-making accuracy might be improved.

The basic introduction and the related works were examined in the section above. Foundations on Fuzzy Soft Expert Sets and the way they facilitate decision making are illustrated in the next section. Followed by this is the discussion on multi-criteria decision making technique TOPSIS. The subsequent section deals with the proposed methodology FSES-TOPSIS with illustration. Finally, the penultimate section discusses about the results followed by conclusions in the last section.

FSES and decision making

In this section, an application of the FSES in a decision-making scenario has been illustrated. In some educational institution, most of the courses might be offered by more than one faculty member. Let us assume that a student wishes to choose one among many faculty members who are offering the same course based on the opinions of other students who have already completed and earned credits for the same course. Assuming that a course is offered by five faculty members the universe is defined as

U = {f_{1}, f_{2}, f_{3}, f_{4}, f_{5}}

To illustrate, the following faculty performance evaluation criteria are considered in this case.

P = {p_{1}, p_{2}, p_{3}, p_{4}, p_{5}}

where

p_{1} : subject knowledge

p_{2} : communication skill

p_{3} : encourage interaction

p_{4} : slow learners attention

, and

p_{5} : challenging assignments

Let $X = {x, y, z}$ be the set of students offering feedback. Each student evaluates the individual faculty based on the above said criteria in the form of agree-fuzzy values $A (f_{i})$ and disagree-fuzzy values $DA (f_{i})$ in the range from 0 to 1. Table 1 provides the sample feedback data.

Table 1.

Sample feedback data.

Given by	Criteria	$A (f_{1})$	$DA (f_{1})$	$A (f_{2})$	$DA (f_{2})$	$A (f_{3})$	$DA (f_{3})$	$A (f_{4})$	$DA (f_{4})$	$A (f_{5})$	$DA (f_{5})$
x	$p_{1}$	0.4	0.6	0.5	0.6	1	0.2	0.2	0.6	0.3	0.7
	$p_{2}$	0.4	0.7	0.6	0.7	0.8	0.4	0.5	0.8	0.4	0.7
	$p_{3}$	0.6	0.5	0.4	0.5	1	0.4	0.5	0.8	0.6	0.2
	$p_{4}$	0.5	0.5	0.6	0.7	0.7	0.2	0.7	0.6	0.6	0.3
	$p_{5}$	0.7	0.6	0.5	0.6	0.7	0.2	0.6	0.7	0.5	0.2
y	$p_{1}$	0.3	0.4	0.5	0.6	0.6	0.3	0.4	0.5	0.4	0.7
	$p_{2}$	0.4	0.2	0.4	0.8	0.7	0.6	0.4	0.5	0.3	0.8
	$p_{3}$	0.4	0.6	0.6	0.7	0.8	0.5	0.7	0.5	0.6	0.9
	$p_{4}$	0.5	0.7	0.6	0.9	0.6	0.2	0.7	0.5	0.5	0.7
	$p_{5}$	0.8	0.6	0.7	0.8	0.9	0.6	0.6	0.6	0.6	0.5
z	$p_{1}$	0.2	0.4	0.6	0.2	1	0.2	0.6	0.8	0.5	0.8
	$p_{2}$	0.6	0.7	0.7	0.6	0.9	0.3	0.2	0.6	0.4	0.5
	$p_{3}$	0.5	0.7	0.6	0.5	1	0.2	0.5	0.4	0.5	0.7
	$p_{4}$	0.7	0.6	0.8	0.6	0.8	0.6	0.6	0.4	0.6	0.8
	$p_{5}$	0.6	0.7	0.8	0.7	0.8	0.6	0.7	0.6	0.7	0.8

The agree and disagree fuzzy soft expert sets computed using the above table are presented in Tables 2 and 3.

\begin{matrix} (F, S) = & {[(p_{1}, x, 1), {\frac{f_{1}}{0.4}, \frac{f_{2}}{0.5}, \frac{1}{f_{3}}, \frac{f_{4}}{0.2}, \frac{f_{5}}{0.3}}], & [(p_{1}, y, 1), {\frac{f_{1}}{0.3}, \frac{f_{2}}{0.5}, \frac{f_{3}}{0.6}, \frac{f_{4}}{0.4}, \frac{f_{5}}{0.4}}], \\ [(p_{1}, z, 1), {\frac{f_{1}}{0.2}, \frac{f_{2}}{0.6}, \frac{1}{f_{3}}, \frac{f_{4}}{0.6}, \frac{f_{5}}{0.5}}], & [(p_{2}, x, 1), {\frac{f_{1}}{0.4}, \frac{f_{2}}{0.6}, \frac{f_{3}}{0.8}, \frac{f_{4}}{0.5}, \frac{f_{5}}{0.4}}], \\ [(p_{2}, y, 1), {\frac{f_{1}}{0.4}, \frac{f_{2}}{0.4}, \frac{f_{3}}{0.7}, \frac{f_{4}}{0.4}, \frac{f_{5}}{0.3}}], & [(p_{2}, z, 1), {\frac{f_{1}}{0.6}, \frac{f_{2}}{0.7}, \frac{f_{3}}{0.9}, \frac{f_{4}}{0.2}, \frac{f_{5}}{0.4}}], \\ [(p_{3}, x, 1), {\frac{f_{1}}{0.6}, \frac{f_{2}}{0.4}, \frac{1}{f_{3}}, \frac{f_{4}}{0.5}, \frac{f_{5}}{0.6}}], & [(p_{3}, y, 1), {\frac{f_{1}}{0.4}, \frac{f_{2}}{0.6}, \frac{f_{3}}{0.8}, \frac{f_{4}}{0.7}, \frac{f_{5}}{0.6}}], \\ [(p_{3}, z, 1), {\frac{f_{1}}{0.5}, \frac{f_{2}}{0.6}, \frac{1}{f_{3}}, \frac{f_{4}}{0.5}, \frac{f_{5}}{0.5}}], & [(p_{4}, x, 1), {\frac{f_{1}}{0.5}, \frac{f_{2}}{0.6}, \frac{f_{3}}{0.7}, \frac{f_{4}}{0.7}, \frac{f_{5}}{0.6}}], \\ [(p_{4}, y, 1), {\frac{f_{1}}{0.5}, \frac{f_{2}}{0.6}, \frac{f_{3}}{0.6}, \frac{f_{4}}{0.7}, \frac{f_{5}}{0.5}}], & [(p_{4}, z, 1), {\frac{f_{1}}{0.7}, \frac{f_{2}}{0.8}, \frac{f_{3}}{0.8}, \frac{f_{4}}{0.6}, \frac{f_{5}}{0.6}}], \\ [(p_{5}, x, 1), {\frac{f_{1}}{0.7}, \frac{f_{2}}{0.5}, \frac{f_{3}}{0.7}, \frac{f_{4}}{0.6}, \frac{f_{5}}{0.5}}], & [(p_{5}, y, 1), {\frac{f_{1}}{0.8}, \frac{f_{2}}{0.7}, \frac{f_{3}}{0.9}, \frac{f_{4}}{0.6}, \frac{f_{5}}{0.6}}], \end{matrix}

\begin{matrix} [(p_{5}, z, 1), {\frac{f_{1}}{0.6}, \frac{f_{2}}{0.8}, \frac{f_{3}}{0.8}, \frac{f_{4}}{0.7}, \frac{f_{5}}{0.7}}], & [(p_{1}, x, 0), {\frac{f_{1}}{0.6}, \frac{f_{2}}{0.6}, \frac{f_{3}}{0.2}, \frac{f_{4}}{0.6}, \frac{f_{5}}{0.7}}], \\ [(p_{1}, y, 0), {\frac{f_{1}}{0.4}, \frac{f_{2}}{0.6}, \frac{f_{3}}{0.3}, \frac{f_{4}}{0.5}, \frac{f_{5}}{0.7}}], & [(p_{1}, z, 0), {\frac{f_{1}}{0.4}, \frac{f_{2}}{0.2}, \frac{f_{3}}{0.2}, \frac{f_{4}}{0.8}, \frac{f_{5}}{0.8}}], \\ [(p_{2}, x, 0), {\frac{f_{1}}{0.7}, \frac{f_{2}}{0.7}, \frac{f_{3}}{0.4}, \frac{f_{4}}{0.8}, \frac{f_{5}}{0.7}}], & [(p_{2}, y, 0), {\frac{f_{1}}{0.2}, \frac{f_{2}}{0.8}, \frac{f_{3}}{0.6}, \frac{f_{4}}{0.5}, \frac{f_{5}}{0.8}}], \\ [(p_{2}, z, 0), {\frac{f_{1}}{0.7}, \frac{f_{2}}{0.6}, \frac{f_{3}}{0.3}, \frac{f_{4}}{0.6}, \frac{f_{5}}{0.5}}], & [(p_{3}, x, 0), {\frac{f_{1}}{0.5}, \frac{f_{2}}{0.5}, \frac{f_{3}}{0.4}, \frac{f_{4}}{0.8}, \frac{f_{5}}{0.2}}], \\ [(p_{3}, y, 0), {\frac{f_{1}}{0.6}, \frac{f_{2}}{0.7}, \frac{f_{3}}{0.5}, \frac{f_{4}}{0.5}, \frac{f_{5}}{0.9}}], & [(p_{3}, z, 0), {\frac{f_{1}}{0.7}, \frac{f_{2}}{0.5}, \frac{f_{3}}{0.2}, \frac{f_{4}}{0.4}, \frac{f_{5}}{0.7}}], \\ [(p_{4}, x, 0), {\frac{f_{1}}{0.5}, \frac{f_{2}}{0.7}, \frac{f_{3}}{0.2}, \frac{f_{4}}{0.6}, \frac{f_{5}}{0.3}}], & [(p_{4}, y, 0), {\frac{f_{1}}{0.7}, \frac{f_{2}}{0.9}, \frac{f_{3}}{0.2}, \frac{f_{4}}{0.5}, \frac{f_{5}}{0.7}}], \\ [(p_{4}, z, 0), {\frac{f_{1}}{0.6}, \frac{f_{2}}{0.6}, \frac{f_{3}}{0.6}, \frac{f_{4}}{0.4}, \frac{f_{5}}{0.8}}], & [(p_{5}, x, 0), {\frac{f_{1}}{0.6}, \frac{f_{2}}{0.6}, \frac{f_{3}}{0.2}, \frac{f_{4}}{0.7}, \frac{f_{5}}{0.2}}], \\ [(p_{5}, y, 0), {\frac{f_{1}}{0.6}, \frac{f_{2}}{0.8}, \frac{f_{3}}{0.6}, \frac{f_{4}}{0.6}, \frac{f_{5}}{0.5}}], & [(p_{5}, z, 0), {\frac{f_{1}}{0.7}, \frac{f_{2}}{0.7}, \frac{f_{3}}{0.6}, \frac{f_{4}}{0.6}, \frac{f_{5}}{0.8}}]} \end{matrix}

Table 2.

Agree-fuzzy values.

U	$f_{1}$	$f_{2}$	$f_{3}$	$f_{4}$	$f_{5}$
${(p}_{1}, x)$	0.4	0.5	1	0.2	0.3
${(p}_{2,} x)$	0.4	0.6	0.8	0.5	0.4
${(p}_{3}, x)$	0.6	0.4	1	0.5	0.6
${(p}_{4}, x)$	0.5	0.6	0.7	0.7	0.6
${(p}_{5}, x)$	0.7	0.5	0.7	0.6	0.5
${(p}_{1}, y)$	0.3	0.5	0.6	0.4	0.4
${(p}_{2,} y)$	0.4	0.4	0.7	0.4	0.3
${(p}_{3}, y)$	0.4	0.6	0.8	0.7	0.6
${(p}_{4}, y)$	0.5	0.6	0.6	0.7	0.5
${(p}_{5}, y)$	0.8	0.7	0.9	0.6	0.6
${(p}_{1}, z)$	0.2	0.6	1	0.6	0.5
${(p}_{2,} z)$	0.6	0.7	0.9	0.2	0.4
${(p}_{3}, z)$	0.5	0.6	1	0.5	0.5
${(p}_{4}, z)$	0.7	0.8	0.8	0.6	0.6
${(p}_{5}, z)$	0.6	0.8	0.8	0.7	0.7
$a_{j} = \sum_{i} u_{ij}$	$a_{1} = 7.6$	$a_{2} = 8.9$	$a_{3} = 12.3$	$a_{4} = 7.9$	$a_{5} = 7.5$

The above FSES $(F, S)$ is obtained from the collected feedback data. To rank the performance of the chosen faculty members, the difference between the aggregates of Tables 2 and 3 are considered. Table 4 refers to the result of this above process.

Table 3.

Disagree-fuzzy values.

U	$f_{1}$	$f_{2}$	$f_{3}$	$f_{4}$	$f_{5}$
${(p}_{1}, x)$	0.6	0.6	0.2	0.6	0.7
${(p}_{2,} x)$	0.7	0.7	0.4	0.8	0.7
${(p}_{3}, x)$	0.5	0.5	0.4	0.8	0.2
${(p}_{4}, x)$	0.5	0.7	0.2	0.6	0.3
${(p}_{5}, x)$	0.6	0.6	0.2	0.7	0.2
${(p}_{1}, y)$	0.4	0.6	0.3	0.5	0.7
${(p}_{2,} y)$	0.2	0.8	0.6	0.5	0.8
${(p}_{3}, y)$	0.6	0.7	0.5	0.5	0.9
${(p}_{4}, y)$	0.7	0.9	0.2	0.5	0.7
${(p}_{5}, y)$	0.6	0.8	0.6	0.6	0.5
${(p}_{1}, z)$	0.4	0.2	0.2	0.8	0.8
${(p}_{2,} z)$	0.7	0.6	0.3	0.6	0.5
${(p}_{3}, z)$	0.7	0.5	0.2	0.4	0.7
${(p}_{4}, z)$	0.6	0.6	0.6	0.4	0.8
${(p}_{5}, z)$	0.7	0.7	0.6	0.6	0.8
$d_{j} = \sum_{i} u_{ij}$	$d_{1} = 8.5$	$d_{2} = 9.5$	$d_{3} = 5.5$	$d_{4} = 8.9$	$d_{5} = 9.3$

Table 4.

FSES decision making.

Faculty	Agree value $a_{j} = \sum_{i} u_{ij}$	Disagree value $d_{j} = \sum_{i} u_{ij}$	Final value $s_{j} = a_{j} - d_{j}$	Rank
f ₁	$a_{1} = 7.6$	$d_{1} = 8.5$	$s_{1} = - 0.9$	3
f ₂	$a_{2} = 8.9$	$d_{2} = 9.5$	$s_{2} = - 0.6$	2
f ₃	$a_{3} = 12.3$	$d_{3} = 5.5$	$s_{3} = + 6.8$	1
f ₄	$a_{4} = 7.9$	$d_{4} = 8.9$	$s_{4} = - 1.0$	4
f ₅	$a_{5} = 7.5$	$d_{5} = 9.3$	$s_{5} = - 1.8$	5

According to Table 4, $\max (s_{j})$ is $s_{3}$ ; hence, the student may opt for $facult y_{3}$ based on the opinions that were given by other students.

TOPSIS fundamentals

Hwang and Yoon developed TOPSIS as a method for multiattribute decision making.⁵² The steps involved in TOPSIS are as follows:

Obtain the decision matrix D for ‘n’ alternatives over ‘m’ evaluation criteria.

Compute the normalized matrix $R$ in which each component is computed as follows

r_{ij} = \frac{d_{ij}}{\sqrt{\sum_{k = 1}^{m} d_{kj}^{2}}}

Determine the weight $w$ for each evaluation criteria and produce a $1 \times m$ matrix $W$ . Compute the normalized weighted normalized matrix as $V = W \times R$ by multiplying each $w_{i}$ by $r_{ij}$ to produce $v_{ij}$ .

Compute the respective ideal solutions viz., positive and negative (PIS and NIS).

Find separation measurement over each criterion for the PIS and NIS.

Determine relative closeness of alternatives.

Order or prioritize the alternatives based on the values that are obtained in step (f).

Proposed methodology FSES-TOPSIS

In this section, a novel approach called FSES-TOPSIS has been proposed. Both agree and disagree fuzzy values from the fuzzy soft expert set are considered for TOPSIS to solve any multicriteria decision-making problem. The proposed idea is to derive the positive ideal solution using agree FSES and negative ideal solution using disagree-FSES. The steps for the process of arriving at the solution are:

Construct the agree-decision matrix $D^{+}$ and disagree-decision matrix $D^{-}$ for ‘n’ alternatives over ‘m’ evaluation criteria.

Compute the agree-normalized matrix $R^{+}$ and disagree-normalized matrix $R^{-}$ .

Determine the weighted agree-normalized decision matrix $V^{+}$ and weighted disagree-normalized decision matrix $V^{-}$ .

Compute PIS and NIS using $V^{+}$ and $V^{-}$ matrices.

Find separation measurements $S^{+}$ and $S^{-}$ using $V^{+}$ and $V^{-}$ , respectively.

Compute relative closeness of alternatives.

Prioritize or rank alternatives.

FSES-TOPSIS and decision making

Assume same universal set U of faculty members and set of evaluation criteria P as

U = {f_{1}, f_{2}, f_{3}, f_{4}, f_{5}}

P = {p_{1}, p_{2}, p_{3}, p_{4}, p_{5}}

Let $X = {x, y, z}$ represents students involved in decision making.

Step 1: Construct agree-decision matrix $D^{+}$ and disagree-decision matrix $D^{-}$ using the given FSES.

The agree-fuzzy values of experts x, y and z are extracted as

x = (\begin{matrix} 0.4 & 0.4 & 0.6 & 0.5 & 0.7 \\ 0.5 & 0.6 & 0.4 & 0.6 & 0.5 \\ 1 & 0.8 & 1 & 0.7 & 0.7 \\ 0.2 & 0.5 & 0.5 & 0.7 & 0.6 \\ 0.3 & 0.4 & 0.6 & 0.6 & 0.5 \end{matrix})

y = (\begin{matrix} 0.3 & 0.4 & 0.4 & 0.5 & 0.8 \\ 0.5 & 0.4 & 0.6 & 0.6 & 0.7 \\ 0.6 & 0.7 & 0.8 & 0.6 & 0.9 \\ 0.4 & 0.4 & 0.7 & 0.7 & 0.6 \\ 0.4 & 0.3 & 0.6 & 0.5 & 0.6 \end{matrix})

z = (\begin{matrix} 0.2 & 0.6 & 0.5 & 0.7 & 0.6 \\ 0.6 & 0.7 & 0.6 & 0.8 & 0.8 \\ 1 & 0.9 & 1 & 0.8 & 0.8 \\ 0.6 & 0.2 & 0.5 & 0.6 & 0.7 \\ 0.5 & 0.4 & 0.5 & 0.6 & 0.7 \end{matrix})

The agree-decision matrix $D^{+}$ is obtained as $D^{+} = \cup_{i, j}$ .

D^{+} = (\begin{matrix} 0.4 & 0.6 & 0.6 & 0.7 & 0.8 \\ 0.6 & 0.7 & 0.6 & 0.8 & 0.8 \\ 1 & 0.9 & 1 & 0.8 & 0.9 \\ 0.6 & 0.5 & 0.7 & 0.7 & 0.7 \\ 0.5 & 0.4 & 0.6 & 0.6 & 0.7 \end{matrix})

Similarly, the disagree-fuzzy values of experts $x, y$ and $z$ are extracted as

\neg x = (\begin{matrix} 0.6 & 0.7 & 0.5 & 0.5 & 0.6 \\ 0.6 & 0.7 & 0.5 & 0.7 & 0.6 \\ 0.2 & 0.4 & 0.4 & 0.2 & 0.2 \\ 0.6 & 0.8 & 0.8 & 0.8 & 0.6 \\ 0.7 & 0.7 & 0.2 & 0.3 & 0.2 \end{matrix})

\neg y = (\begin{matrix} 0.4 & 0.2 & 0.6 & 0.7 & 0.6 \\ 0.6 & 0.8 & 0.7 & 0.9 & 0.8 \\ 0.3 & 0.6 & 0.5 & 0.2 & 0.6 \\ 0.5 & 0.5 & 0.5 & 0.5 & 0.6 \\ 0.7 & 0.8 & 0.9 & 0.7 & 0.5 \end{matrix})

\neg z = (\begin{matrix} 0.4 & 0.7 & 0.7 & 0.6 & 0.7 \\ 0.2 & 0.6 & 0.5 & 0.6 & 0.7 \\ 0.2 & 0.3 & 0.2 & 0.6 & 0.6 \\ 0.8 & 0.6 & 0.4 & 0.4 & 0.6 \\ 0.8 & 0.5 & 0.7 & 0.8 & 0.8 \end{matrix})

The disagree-decision matrix $D^{-}$ is obtained as $D^{-} = \cap_{i, j}$ .

D^{-} = (\begin{matrix} 0.4 & 0.2 & 0.5 & 0.5 & 0.6 \\ 0.2 & 0.6 & 0.5 & 0.6 & 0.6 \\ 0.2 & 0.3 & 0.2 & 0.2 & 0.2 \\ 0.5 & 0.5 & 0.4 & 0.4 & 0.6 \\ 0.7 & 0.5 & 0.2 & 0.3 & 0.2 \end{matrix})

Step 2: Find agree and disagree normalized decision matrices $R^{+}$ and $R^{-}$ .

R^{+} = (\begin{matrix} 0.274 & 0.417 & 0.374 & 0.432 & 0.457 \\ 0.411 & 0.487 & 0.374 & 0.494 & 0.457 \\ 0.685 & 0.626 & 0.624 & 0.494 & 0.514 \\ 0.411 & 0.348 & 0.437 & 0.432 & 0.400 \\ 0.343 & 0.278 & 0.374 & 0.371 & 0.400 \end{matrix})

R^{-} = (\begin{matrix} 0.404 & 0.201 & 0.581 & 0.527 & 0.557 \\ 0.202 & 0.603 & 0.581 & 0.632 & 0.557 \\ 0.202 & 0.302 & 0.232 & 0.211 & 0.186 \\ 0.505 & 0.503 & 0.465 & 0.422 & 0.557 \\ 0.707 & 0.503 & 0.232 & 0.316 & 0.186 \end{matrix})

Step 3: Determine weighted agree and disagree normalized decision matrices $V^{+}$ and $V^{-} .$ The criterion weights are as follows:

$p_{1} : subject knowledge : 0.25$ , $p_{2} : communication skill : 0.2$ , $p_{3} : encourage interaction : 0.1$ , $p_{4} : slow learners attention : 0.3$ , $p_{5} : challenging assignments : 0.15$ i.e.,

W = {[w_{j}]}_{1 \times n} = [\begin{matrix} 0.25 & 0.2 & 0.1 & 0.3 & 0.15 \end{matrix}]

R^{+}

matrix elements are multiplied by

W

v_{ij}^{+} = w_{j} . r_{ij}^{+}

and

V^{+}

is obtained as follows

V^{+} = (\begin{matrix} 0.069 & 0.094 & 0.037 & 0.130 & 0.069 \\ 0.103 & 0.097 & 0.037 & 0.148 & 0.069 \\ 0.171 & 0.125 & 0.062 & 0.148 & 0.077 \\ 0.103 & 0.070 & 0.044 & 0.130 & 0.060 \\ 0.086 & 0.056 & 0.037 & 0.111 & 0.060 \end{matrix})

R^{-}

matrix elements are multiplied by

W

v_{ij}^{-} = w_{j} . r_{ij}^{-}

and

V^{-}

is obtained as follows

V^{-} = (\begin{matrix} 0.101 & 0.040 & 0.058 & 0.158 & 0.084 \\ 0.051 & 0.121 & 0.058 & 0.190 & 0.084 \\ 0.051 & 0.060 & 0.023 & 0.063 & 0.028 \\ 0.126 & 0.101 & 0.047 & 0.127 & 0.084 \\ 0.177 & 0.101 & 0.023 & 0.095 & 0.028 \end{matrix})

Step 4: Determine the PIS and NIS

A^{+} = {0.171, 0.125, 0.062, 0.148, 0.077}

A^{-} = {0.051, 0.040, 0.023, 0.063, 0.028}

Step 5: Compute the separation measurements ${S_{i}}^{+}$ and ${S_{i}}^{-}$

{S_{i}}^{+} = {0.164, 0.102, 0.047, 0.045, 0.027}

{S_{i}}^{-} = {0.155, 0.120, 0.055, 0.174, 0.097}

Step 6: Compute the relative closeness of the alternatives

{C_{i}}^{+} = {0.514, 0.459, 0.461, 0.205, 0.218}

Now, the ranking appears as in Table 5, which gives a different perception on making a decision for choosing a faculty by considering multiple criteria.

Table 5.

FSES-TOPSIS decision making.

Faculty	Relative closeness	Rank
f ₁	0.514	1
f ₂	0.459	3
f ₃	0.461	2
f ₄	0.205	5
f ₅	0.218	4

Results and discussion

In our case, it was assumed that five different faculty members were handling the same course. Various performance indicators (as shown in Table 1) were used to arrive at basic values of FSES in equation (9). Then, agree and disagree FSES sets were extracted from Table 1. They were presented in Tables 2 and 3 along with the aggregates. Then the process of decision making using FSES was calculated from the aggregated values of Tables 2 and 3. Final result of this computation was presented in Table 4 which shows that the faculty member 3 was recommended to the students as their first choice. To improve the recommendation still further, as mentioned, use of FSES with TOPSIS was equated. Table 5 shows the results of the procedure employed to obtain the values of proposed FSES-TOPSIS method. It was evident that the ideal values used to rank the faculty members were different than the previous results obtained. As more number of criteria are involved in this FSES-TOPSIS method when compared to just only feedback obtained from the students in traditional FSES method, it is evident that this FSES-TOPSIS method would be more appropriate to recommend the students need.

Conclusion

This paper started with the theoretical background of fuzzy soft expert sets with the possible operations. To best understand the same, a simple application of students’ course registration for an institution was considered where it was assumed that the more than a faculty was about to offer the same course and the students need to identify suitable faculty member based on teachers previous academic performance. The traditional FSES method was applied and results were obtained. As this approach only has simple arithmetic, a new idea of combining FSES with TOPSIS was proposed. This method was found to be more reasonable as the number of criteria to rank and recommend the faculty members was more when compared to the normal method. It was evident that both the methods can be used for recommendations in educational domain. As insisted, based on the number of parameters considered to rank and recommend, it can be concluded that the ideal solution would be FSES-TOPSIS method as presented here. Rural students enrolling into higher studies after their school would find it difficult in their initial stages. But our methodology would facilitate them to choose the more appropriate faculty member for their course registrations assuming that the same course is to be offered in more number of batches by more number of faculty members in a very big institution. This would help students to increase their interest in learning by registering with the more appropriate faculty members based on the ranking mechanism proposed in this approach. The same technique can be used to any problems of similar nature for any multicriteria decision making and recommendations.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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Recommendation system for under graduate students using FSES-TOPSIS

Abstract

Abstract

Keywords

Introduction

FSES and decision making

TOPSIS fundamentals

Proposed methodology FSES-TOPSIS

FSES-TOPSIS and decision making

Results and discussion

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

References