1 Introduction

The uncertainty theories like soft expert sets find their applications in domains like business, economics, medical diagnosis, and engineering which deal with uncertainties. Innovative approaches using the aforementioned techniques were successfully employed for many real-world scenarios. Despite their applications and scope, several researchers are attempting to introduce innovative ideas over the existing findings so as to improve one or the other parameters involved in decision-making process. Nowadays, all the institutions rely on automated processes that rely on one or many idealistic approaches in their decision-making stages. Most of the problems were found to be solved using several soft computing approaches. In this work, an attempt is made to idealize the impact of soft expert set utilized in decision-making process in place of traditional soft sets. In this paper, the concepts behind soft expert sets and some important operations on them are illustrated below. To explore the behavior of this soft expert set, a case study on student course registration system was considered. In any institution, more than one faculty member may offer the same course where in such scenarios, students may find it difficult to identify and enroll themselves with one among the faculty members offering that course based on their performance criteria. An attempt is made with soft expert set to assist students for identifying the faculty with desired performance characteristics and to enroll them. The same process is illustrated later in this work.

2 Literature Survey

To deal with uncertainties, Molodtsov formulated soft set theory [1]. Based on his work, some variations were presented by researchers as in Table 1.

Table 1 Soft set theories
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Such theories were studied and applied to decision-making problems. Table 2 lists a brief summary of sample works in the domain.

Table 2 Few applications of soft set theory
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The above contributions presented the idea behind the next level of soft sets and how it could be applied to other real-time problems like medical diagnosis and many more. But these models proposed exactly one expert to derive an opinion and forcing the users to perform union and other operations in case of multi-experts opinion needed. Alkhazaleh et al. proposed a model based on the idea of soft expert set, which seeks the opinion from more than one expert without any further operations [10]. This model could be more useful in almost all the decision-making problems, and the further extension principles are presented in Table 3.

Table 3 Multi-expert theories
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3 Foundations

The fundamental ideas behind soft expert set is presented in this section. Some important operations on soft expert sets are also recalled here.

Let us consider the universe U, the set of parameters P, the set of experts E, and the set of opinions O. Here Z = P × E × O and X ⊆ Z.

Definition 3.1

Soft Expert Set

A soft expert set is a pair (F, X) over the universe U, and F is a mapping of X to P(U), which is the power set of U.

$$\displaystyle \begin{aligned} F : X \rightarrow P(U) \end{aligned}$$

Definition 3.2

Soft Expert Subset and Soft Expert Superset

A soft expert set (F, X) is termed as a soft expert subset of (G, Y ) over the common universe U, if:

  • X ⊆ Y ,

  • ∀𝜖 ∈ X, F(𝜖) ⊆ G(𝜖)

which is denoted by \((F, X) \widetilde {\subseteq } (G, Y)\).

Here, (G, Y ) is known as the soft expert superset of (F, X) and is denoted by \((G, Y) \widetilde {\supseteq } (F, Y)\).

Definition 3.3

Soft Expert Equal Sets

The soft expert sets (F, X) and (G, Y ) over the universe U are termed as soft expert equal sets if (F, X) is the soft expert subset of (G, Y ) and (G, Y ) is the soft expert set of (F, X).

Definition 3.4

NOT set of Z

Z = P × E × O where P is the set of parameters, E is the set of experts, and O is the set of opinions. The NOT of Z is denoted by ¬Z and is defined as

$$\displaystyle \begin{aligned} \neg Z = \left\{(\neg p_i,e_j,o_k), \forall i,j,k \right\}\ \mathit{where}\ \neg p_i = not p_i \end{aligned}$$

Definition 3.5

Complement Soft Expert Set

(F, X)c is termed as the complement of the soft expert set (F, X) over the universe U and defined by

$$\displaystyle \begin{aligned} (F, X)^c = (F^c, \neg X), \end{aligned}$$

where F c is a mapping of ¬X to P(U), and ∀x ∈¬X, F c(X) = U − F(¬X)

Definition 3.6

Agree-Soft Expert Set

Let (F, X) be the soft expert set over the universe U. The agree-soft expert set of (F, X) denoted as (F, X)1 is defined by (F, X)1 = {F 1(𝜖) : 𝜖 ∈ E × X ×{1}}.

Definition 3.7

Disagree-Soft Expert Set

Let (F, X) be the soft expert set over the universe U. The disagree-soft expert set of (F, X) denoted as (F, X)0 is defined by (F, X)0 = {F 0(𝜖) : 𝜖 ∈ E × X ×{0}}.

Definition 3.8

Union operation

Let (F, X) and (G, Y ) be the two soft expert sets over the universe U. The union of these soft expert sets (H, Z) is the soft expert set denoted by \((F, X) \widetilde {\cup } (G, Y)\), where Z = X ∪ Y ∀𝜖 ∈ Z.

$$\displaystyle \begin{aligned} H(\epsilon)= \begin{cases} F(\epsilon) & if \epsilon \in X - Y \\ G(\epsilon) & if \epsilon \in Y - X \\ F(\epsilon) \cup G(\epsilon) & if \epsilon \in X \cup Y \end{cases} \end{aligned}$$

Definition 3.9

Intersection operation

Let (F, X) and (G, Y ) be the two soft expert sets over the universe U. The intersection of these soft expert sets (H, Z) is the soft expert set denoted by \((F, X) \widetilde {\cap } (G, Y)\), where Z = X ∩ Y ∀𝜖 ∈ Z.

$$\displaystyle \begin{aligned} H(\epsilon)= \begin{cases} F(\epsilon) & if \epsilon \in X - Y \\ G(\epsilon) & if \epsilon \in Y - X \\ F(\epsilon) \cap G(\epsilon) & if \epsilon \in X \cap Y \end{cases} \end{aligned}$$

Definition 3.10

AND operation

Let (F, X) and (G, Y ) be the two soft expert sets over the universe U. Now (F, X) AND (G, Y ) is denoted as (F, X) ∧ (G, Y ) and defined by (F, X) ∧ (G, Y ) = (H, X × Y ), where \(H(\alpha ,\beta )= F(\alpha ) \widetilde {\cap } G(\beta )\ \forall (\alpha ,\beta ) \in X \times Y\).

Definition 3.11

OR operation

Let (F, X) and (G, Y ) be the two soft expert sets over the universe U. Now (F, X) OR (G, Y ) is denoted as (F, X) ∨ (G, Y ) and defined by (F, X) ∨ (G, Y ) = (I, X × Y ), where \(I(\alpha ,\beta )= F(\alpha ) \widetilde {\cup } G(\beta )\ \forall (\alpha ,\beta ) \in X \times Y\).

4 Soft Expert Set and Decision-Making

As aforementioned, the students registration case study considered here would enable them to identify one faculty member for enrolling into a course. Institutions rely on feedback from students’ community as one of the criteria to evaluate the performance of their faculty members. So, the feedback collected thus would enable students to understand and analyze the performance based on the previous year’s performance of an individual. The factors to evaluate an individual faculty member by students vary among institutions. The following factors are considered for evaluating ten faculty members by five students in this case so as to make illustration precise: subject knowledge, communication skill, encourage interaction, slow learners attention, and challenging assignments.

The universe is given by U = {f 1, f 2, f 3, f 4, f 5, f 6, f 7, f 8, f 9, f 10}. Let the evaluation parameters be represented as P = {e 1, e 2, e 3, e 4, e 5}, where,

  • e 1 : subject knowledge

  • e 2 : communication skill

  • e 3 : encourage interaction

  • e 4 : slow − learner attention

  • e 5 : challenging assignments

Let E = {p, q, r, s, t} be the set of students providing feedback on the evaluation criteria. Assume that the following soft expert set (F, X) is obtained based on the students feedback. It can be represented as agree-soft expert set and disagree-soft expert set as in Tables 4 and 5, respectively.

Table 4 Agree-soft expert set
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Table 5 Disagree-soft expert set
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To make the final decision on faculty, the following procedure is to be followed:

  1. (a)

    Input the soft expert set (F, X).

  2. (b)

    Find the corresponding agree-soft expert set and disagree-soft expert set.

  3. (c)

    For the agree-soft expert set, compute a j =∑i f ij.

  4. (d)

    For the disagree-soft expert set, compute d j =∑i f ij.

  5. (e)

    Compute x j = a j − d j.

  6. (f)

    Find max(x j) which is the optimal choice.

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle (F, X) =\\ &\displaystyle &\displaystyle \hspace{-3pt}\left\{ \begin{matrix} ((e_1,p,1),\{f_1,f_2,f_4,f_6,f_8,f_{10}\}), ((e_1,q,1),\{f_1,f_3,f_7,f_9\}), ((e_1,r,1),\{f_2,f_6,f_8,f_{10}\}), \\ ((e_1,s,1),\{f_3,f_4,f_5,f_6,f_9\}), ((e_1,t,1),\{f_1,f_2,f_4,f_5,f_6,f_8\}), ((e_2,p,1),\{f_1,f_3,f_5,f_7,f_9,f_{10}\}), \\ ((e_2,q,1),\{f_2,f_3,f_6,f_7\}), ((e_2,r,1),\{f_4,f_8,f_9,f_{10}\}), ((e_2,s,1),\{f_3,f_4,f_5,f_9,f_{10}\}), ((e_2,t,1),\{f_1,f_9,f_{10}\}), \\ ((e_3,p,1),\{f_3,f_6,f_8,f_9\}), ((e_3,q,1),\{f_2,f_6,f_8\}),((e_3,r,1),\{f_9,f_{10}\}), ((e_3,s,1),\{f_2,f_3,f_4,f_5,f_{10}\}), \\ ((e_3,t,1),\{f_6,f_7,f_8,f_{10}\}), ((e_4,p,1),\{f_2,f_4,f_6,f_8,f_{10}\}), ((e_4,q,1),\{f_1,f_9,f_{10}\}), ((e_4,r,1),\{f_2,f_3,f_5,f_6\}), \\ ((e_4,s,1),\{f_3,f_7,f_8,f_9\}), ((e_4,t,1),\{f_1,f_2,f_5,f_6\}), ((e_5,p,1),\{f_2,f_3,f_6\}), ((e_5,q,1),\{f_7,f_8,f_9\}), \\ ((e_5,r,1),\{f_2,f_4,f_6\}), ((e_5,s,1),\{f_1,f_2,f_6,f_8,f_{10}\}), ((e_5,t,1),\{f_2,f_3,f_5,f_6,f_8,f_{10}\}), ((e_1,p,0),\{f_3,f_5,f_7,f_9\}),\\ ((e_1,q,0),\{f_2,f_4,f_5,f_6,f_8 f_{10}\}), ((e_1,r,0),\{f_1,f_3,f_4,f_5,f_7,f_9\}), ((e_1,s,0),\{f_1,f_2,f_7,f_8,f_{10}\}), \\ ((e_1,t,0),\{f_3,f_7,f_9,f_{10}\}), ((e_2,p,0),\{f_2,f_4,f_6,f_8\}), ((e_2,q,0),\{f_1,f_4,f_5,f_8,f_9,f_{10}\}), \\ ((e_2,r,0),\{f_1,f_2,f_3,f_5,f_6,f_7\}), ((e_2,s,0),\{f_1,f_2,f_6,f_7,f_8\}), ((e_2,t,0),\{f_2,f_3,f_4,f_5,f_6,f_7,f_8\}), \\ ((e_3,p,0),\{f_1,f_2,f_4,f_5,f_7,f_{10}\}), ((e_3,q,0),\{f_1,f_3,f_4,f_5,f_7,f_9,f_{10}\}), \\ ((e_3,r,0),\{f_1,f_2,f_3,f_4,f_5,f_6,f_7,f_8\}), ((e_3,s,0),\{f_1,f_6,f_7,f_8,f_9\}), ((e_3,t,0),\{f_1,f_2,f_3,f_4,f_5,f_9\}), \\ ((e_4,p,0),\{f_1,f_3,f_5,f_7,f_9\}), ((e_4,q,0),\{f_2,f_3,f_4,f_5,f_6,f_7,f_8\}), ((e_4,r,0),\{f_1,f_4,f_7,f_8,f_9,f_{10}\}), \\ ((e_4,s,0),\{f_1,f_2,f_4,f_5,f_6,f_{10}\}), ((e_4,t,0),\{f_3,f_4,f_7,f_8,f_9,f_{10}\}), ((e_5,p,0),\{f_1,f_4,f_5,f_7,f_8,f_9,f_{10}\}), \\ ((e_5,q,0),\{f_1,f_2,f_3,f_4,f_5,f_6,f_{10}\}), ((e_5,r,0),\{f_1,f_3,f_5,f_7,f_8,f_9,f_{10}\}), ((e_5,s,0),\{f_3,f_4,f_5,f_7,f_9\} ), \\ ((e_5,t,0),\{f_1,f_4,f_7,f_9\}) \end{matrix} \right\} \end{array} \end{aligned} $$

According to Table 6, max(x j) is x 6, hence the students may opt for faculty 4 based on the previous performance feedback.

Table 6 x j = a j − d j
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5 Conclusion

The basic idea of soft expert set was presented in this work. Fundamental operations on the same set were also discussed. Finally, the same concept was applied for the students’ course registration process of an institution and concluded with a simple idea. The same idea can also be applied to many problems involving decision-making. The number of factors and the number of students considered for faculty performance evaluation in this case were limited. This case can also be extended with fuzzy soft expert set.