TOPSIS Approach Using Interval Neutrosophic Sets for Personnel Selection

INTRODUCTION

---

Personnel selection plays a decisive role for finding the sufficient input quality for an organization. In the organization, the personnel defined as capability, knowledge, skill and other abilities¹. Personnel selection is the process of choosing individuals who match the qualifications required to perform a defined job in the best possible way². To select the most suitable personnel, many individual attributes, i.e., organizing ability, creativity, personality and leadership, must be considered in selection process³. Therefore, personnel selection can be viewed as a multi-criteria decision making problem (MCDM). The MCDM approaches for personnel selection is able to incorporate qualitative as well as quantitative data².

Personnel selection is a highly complex and messy problem in real life. It is complex problem because there is uncertainty regarding the outcomes of any choice. It is messy problem because it requires systematic integration of multiple evaluation criteria of the various decision makers involved in the personnel selection process. In addition, several issues need to identify in the personnel selection process including: (1) Identify the importance weights of criteria, (2) Evaluate the applicants under multiple criteria using linguistic values and (3) Aggregate the evaluation results and then rank the applicants⁴.

In many situations, individuals mostly prefer to express their feelings with verbal expression and they may make accurate guesses in qualitative forecasting⁵. The fuzzy set theory appears as an essential tool to provide a decision framework that incorporates imprecise judgments inherent in the personnel selection process. Because of the imprecise expressions, a fuzzy multi-criteria approach is commonly used in decision problems. However, the disadvantage of fuzzy set theory is that it only has a membership and is unable to express non-membership. On the basis of fuzzy set, Atanassov^6,7 proposed the intuitionistic fuzzy set (IFS) by adding a non-membership function. However, in practice, the decision information is often incomplete, indeterminate and inconsistent information. In order to process this kind of information, Smarandache⁸ further proposed the neutrosophic set (NS) by adding an independent indeterminacy-membership on the basis of IFS, which is a generalization of fuzzy set, interval valued fuzzy set, intuitionistic fuzzy set and so on.

Recently, NSs have become an interesting research topic and attracted widely attentions^9-19. Since the NS is difficult to be directly used in real-life applications, Wang et al.¹⁸ proposed a single valued neutrosophic set (SVNS) from scientific or engineering point of view, which is an instance of the neutrosophic set. Wang et al.¹⁹ proposed INSs in which the truth-membership, indeterminacy-membership and false-membership were extended to interval numbers and discussed some properties and comparing method of INSs.

Numerous studies in the literature have applied MCDM techniques for personal selection. Kundakci²⁰ proposed grey relational analysis for employee selection to overcome the drawbacks of the traditional methods in the case of a technology firm. The Hamming distance method was presented by Saad et al.²¹ to solve personnel selection problem. Interval valued fuzzy numbers and triangular fuzzy numbers were used to express the performance rating values as well as the weight of the criteria. Khorami and Ehsani²² reviewed recent advances on the application of MCDM methods for personnel selection problem. The authors have indicated that the literature on the application of MCDM techniques for personnel selection problems has been growing increasingly and it also seems that usage of fuzzy decision making and hybrid approaches would increase within next future years. Kabak et al.²³ adapted a fuzzy hybrid MCDM including fuzzy analytic network process (FANP), a TOPSIS and ELECTRE, to personnel selection, i.e., sniper selection problem. An intuitionistic fuzzy multi-criteria group decision making method with grey relational analysis (GRA) is proposed for personnel selection²⁴. Dursun and Karsak² developed a fuzzy MCDM algorithm for personnel selection using the principles of fusion of fuzzy information, 2-tuple linguistic representation model and TOPSIS is developed. Kelemenis and Askounis²⁵ proposed a new TOPSIS-based multi-criteria approach to personnel selection. Gungor et al.⁵ employed fuzzy Analytic Hierarchy Process (AHP) to evaluate the best adequate personnel dealing with the rating of both qualitative and quantitative criteria. Although several fuzzy MCDM methods have been developed for personnel selection, most of them used fuzzy numbers in their calculation²⁶.

In recent years, TOPSIS (technique for order performance by similarity to ideal solution)²⁷ has been one of the well-known methods for solving MCDM problems. The fundamental idea of TOPSIS is that the chosen alternative should have the shortest distance from the positive-ideal solution and the farthest distance from the negative-ideal solution. Some recent applications can be found in Nadaban et al.²⁸, Alizadeh et al.²⁹, Kutlu and Ekmekcioglu³⁰, Roszkowska and Wachowicz³¹, Mohammadi et al.³² and Sengul et al.³³. To the best of our knowledge, till now no one has used INSs for solving the personnel selection problems. Therefore, this study proposed a TOPSIS approach using INS was developed to support for personnel selection process.

PRELIMINARIES

Definition 1
Neutrosophic set (NS)⁸: Let X be a space of points and let x∈X. A neutrosophic set in X is characterized by a truth membership function an indeterminacy membership function and a falsehood membership function and are real standard or non-standard subsets of ]0‾ , 1⁺[. To use neutrosophic set in some real-life applications, such as engineering and scientific problems, it is necessary to consider the interval [0, 1] instead of ]0‾ , 1⁺[, for technical applications. The neutrosophic set can be represented as:

where, one has that and and are subsets of the unit interval [0, 1].

Definition 2
Single valued neutrosophic set¹⁹: Let X be a universe of discourse, with a generic element in X denoted by x. A single valued neutrosophic set A in X is A = [x{T_A(x), I_A(x), F_A(x)}|x∈X], where, T_A(x), I_A(x) and F_A(x) are the truth-membership function, indeterminacy-membership function and the falsity-membership function, respectively. For each point x in X, we have T_A(x), I_A(x), F_A(x)∈[0, 1] and 0<T_A(x)+I_A(x)+F_A(x)<3.

Definition 3
Interval neutrosophic set¹⁹: Let X be a universe of discourse, with a generic element in X denoted by x. A interval neutrosophic set A in X is A = [x {T_A(x), I_A(x), F_A(x)}|x∈X], where, T_A(x), I_A(x) and F_A(x) are the truth-membership function, indeterminacy-membership function and the falsity-membership function, respectively. For each point x in X, we have T_A(x), I_A(x), F_A (x)⊆[0,1] and 0<sup T_A(x)+sup I_A(x)+sup F_A(x)<3. For convenience, we can use x = ([T^L, T^U], [I^L, I^U], [F^L, F^U]) to represent a value in INS and call interval neutrosophic value (INV).

Definition 4
Operational rules of the INV¹⁹: Let:

and:

be two INVs. The operational rules are then defined as follows:

The complement of x is:

(1)

(2)

(3)

(4)

(5)

Definition 5
Distance between two neutrosophic values¹⁹: Let:

and:

be two INVs.

•	The Hamming distance between x and y is defined as follows:

(6)

•	The Euclidian distance between x and y is defined as follows:

(7)

PROPOSED TOPSIS APPROACH USING INTERVAL NEUTROSOPHIC SET

In this section, the TOPSIS method using INS is developed for personnel selection. Let us assume that a committee of k decision-makers (D_t, t = 1,…, k) is responsible for evaluating n alternatives (A_i, i = 1,…, n) under m selection criteria (C_h, h = 1,…, m), where the suitability ratings of alternatives under each criterion, as well as the weights of all criteria are assessed in INS. The steps of the proposed TOPSIS method are as follows.

Aggregate ratings of alternatives versus criteria: Let be the suitability rating assigned to alternative A_i by decision-maker D_t for criterion C_h i = 1,…, n; t = 1,…, k; h = 1,…, m. Using the operational rules of the INS, the averaged suitability rating can be evaluated as:

(8)

Where:

Aggregate the importance weights: Let be the weight assigned by decision-maker D_t to criterion C_h, t = 1,…, k; h = 1,…, m. Using the operational rules of the INS, the average weight can be evaluated as:

(9)

Where:

Aggregate the weighted ratings of alternatives versus criteria: The weighted ratings of alternatives can be developed via the operations of INS as follows:

(10)

Calculation of A⁺, A‾, and : The positive-ideal solution (FPIS, A⁺) and fuzzy negative ideal solution (FNIS, A‾) are obtained as:

A+ = ([1, 1], [0, 0], [0, 0])

(11)

A‾ = ([0, 0], [1, 1], [1, 1])

(12)

The distances of each alternative A_i, i = 1,…, n from A⁺ and A‾ are calculated as:

(13)

(14)

where, represent the shortest distances of alternative A_i and represents the farthest distance of alternative A_i.

Obtain the closeness coefficient: The closeness coefficient of each alternative, which is defined to determine the ranking order of all alternatives, is calculated as:

(15)

A higher value of the closeness coefficient indicates that an alternative is closer to PIS and farther from NIS simultaneously. The closeness coefficient of each alternative is used to determine the ranking order of all alternatives and identify the best one among a set of given feasible alternatives.

APPLICATION OF THE PROPOSED TOPSIS METHOD

In this section, the proposed TOPSIS approach is applied to select the academic staffs at Academy of Finance (AOF). Recently, AOF has 13 administrative divisions, 14 academic faculties and 02 research institutes.

Suppose that AOF needs to select a candidate for the teaching position. After preliminary screening, four candidates, namely A₁,…, A₃ and A₄ are chosen for further selection. A committee of four decision makers from AOF’s Board of management, office of human resources and department head, i.e., D₁, D₂, D₃ and D₄, conducts the selection of the four candidates. Based on the discussion with the committee members, six selection criteria are considered including number of publications (C₁), quality of publications (C₂), personality factors (C₃), activity in professional society (C₄), classroom teaching experience (C₅) and fluency in a foreign language (C₆). The computational procedure is summarized as follows:

Aggregation of the ratings of candidates versus the criteria: Four decision makers determine the suitability ratings of four potential candidates versus the criteria using the INS S = {VP, P, M, G, VG} where, VP = Very poor = ([0.1, 0.2], [0.6, 0.7], [0.7, 0.8]), P = Poor = ([0.2, 0.3], [0.5, 0.6], [0.6, 0.7]), M = Medium = ([0.3, 0.5], [0.4, 0.6], [0.4, 0.5]), G = Good = ([0.5, 0.6], [0.4, 0.5], [0.3, 0.4]) and VG = Very good = ([0.6, 0.7], [0.2, 0.3], [0.2, 0.3]), to evaluate the suitability of the candidates under six criteria. Using Eq. 8, the aggregated ratings of four candidates (A₁, A₂, A₃, A₄) versus six criteria (C₁,…, C₆) from four decision-makers (D₁, D₂, D₃, D₄) are shown in Table 1.

Aggregate the importance weights: After determining the lecturer selection criteria, the committee members are asked to determine the level of importance of each criterion using the INS, V = {UI, OI, I, VI and AI}, where UI = Unimportant = ([0.1, 0.2], [0.5, 0.6], [0.7, 0.8]), OI = Ordinary important = ([0.2, 0.4], [0.5, 0.6], [0.5, 0.6]), I = Important = ([0.4, 0.6], [0.4, 0.5], [0.3, 0.4]), VI = Very important = ([0.6, 0.8], [0.3, 0.4], [0.2, 0.3]) and AI = Absolutely Important = ([0.7, 0.9], [0.2, 0.3], [0.1, 0.2]). Table 2 displayed the importance weights of the six criteria from the four decision-makers. The aggregated weights of criteria obtained by Eq. 9 are shown in the last column of Table 2.

Aggregate the weighted ratings of alternatives versus criteria: Table 3 presented the weighted ratings of alternatives of each candidate using Eq. 10.

Calculation of A⁺, A‾, and : As shown in Table 4, the distance of each candidate from A⁺ and A‾ can be calculated using Eq. 11-14.

Obtain the closeness coefficient: The closeness coefficient of each candidate can be calculated by Eq. 15, as shown in Table 5. Therefore, the ranking order of the four candidates was A₂›A₃›A₄›A₁ Consequently, the best candidate is A₂.

Table 1:	Aggregated ratings of alternatives versus the criteria
G: Good, M: Medium, VG: Very good, P: Poor

Table 2:	Importance and aggregated weights of the criteria

Table 3:	Weighted ratings of each candidate

Table 4:	Distance of each candidate from A+ and A‾

Table 5:	Closeness coefficients of candidates