Keywords

Introduction

Decision-making is a situation that every individual frequently encounters in both daily and business life. A typical decision-making process involves three stages: the definition of the decision-making problem, the development and use of a decision-making model, and the creation of action plans [24]. Although the decision-making process is completed with the creation of action plans, the adverse effects of the inefficient decision-making process are inevitable to continue [8].

In the literature, several methods have been developed by using different algorithms [12, 30, 31]. As part of these methods, Multi-Criteria Decision-Making (MCDM) methods have been developed to make decisions under conflicting criteria [1, 32]. MCDM methods combine many disciplines, predominantly mathematics, and they provide a systematic way of making decisions [14, 18, 24].

Each MCDM method was developed on different algorithms; therefore, they might reach different conclusions. In other words, different methods can suggest the selection of different alternatives. It is, at precisely this point, that the reliability of MCDM methods has been criticized by several researchers [4, 6].

However, despite all criticisms, the MCDM methods have been used in classification, selection, and ranking problems concerning various processes in many different industries. For example, Antmen and Mic [2] used fuzzy TOPSIS and Analytical Hierarchy Process (AHP) methods to select a ventilator in the pediatric intensive care unit. Ozturk and Kaya [23] used fuzzy VIKOR to select personnel in the automotive industry. Bedir and Yalcin et al. [5] used Analytical Network Process (ANP) and PROMETHEE methods to select subcontractors. Soba and Simsek et al. [29] used AHP based VIKOR to select doctoral students. Brauers and Edmundas et al. [8] used the MOORA method to select a contractor. In addition, several other MCDM methods have been used in decision-making. However, among all MCDM methods, TOPSIS, AHP and VIKOR methods were frequently used, and the MOORA method was promoted due to its ease of use and low-time requirement [13, 16, 17, 21, 34].

This study aims to use AHP-based TOPSIS, VIKOR, and MOORA methods for the selection of two researchers to an engineering faculty and to compare the findings of these MCDM methods.

Methodology

Study Design

The design of this study consists of three stages: determining criteria, estimating the criteria weights, and ranking alternatives (Fig. 1).

Fig. 1
figure 1

The research design for the selection of two researchers

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The determination of the criteria was made by reviewing similar studies in the decision-making literature [15, 17, 19, 22, 26, 28, 29] and conducting meetings with four academic members in the related faculty.

The AHP method was used to calculate the weights for each criterion. Initially, a questionnaire was designed to assess the relative importance of each criterion. In this questionnaire, a scale of 1 to 9 was used to make pairwise comparison [27]. Four faculty members completed the questionnaire, and they reached a consensus on the conflicted responses.

The VIKOR, TOPSIS, and MOORA (MOORA-rate system and MOORA-reference point theory) methods were used to rank the twenty-six candidates based on seven criteria. The application of MCDM methods was carried out using MS Office Excel.

The AHP Method

AHP is based on the general measurement theory, and it aims to solve problems identified for a specific purpose. It is possible to describe the AHP method in four stages [27].

Step 1 Creating the hierarchical structure of the decision problem: It starts from the top level. Level 1 represents the goal; level 2 represents the criteria; level 3 shows the sub-criteria, and the lowest level shows the alternatives.

Step 2 Creating the binary comparison matrix: Binary comparison matrices for each level of the hierarchical structure are created by Eq. (1). Here, n criteria \((a_{1} ,a_{2} , \ldots a_{n} )\) are compared by using the 1–9 scale of Saaty.

$$\left[ {\begin{array}{*{20}c} 1 & {a_{12} } & \ldots & {a_{1n} } \\ {a_{21} } & 1 & \ldots & {a_{2n} } \\ \vdots & \vdots & \ldots & \vdots \\ {a_{n1} } & {a_{n2} } & \cdots & 1 \\ \end{array} } \right]$$
(1)

Step 3 Determination of criterion weights: The weight values of each criterion are calculated. For this, the matrix is normalized using Eq. (2) and, then, the weights are calculated by Eq. (3).

$$a_{ij}^{*} = \frac{{a_{ij} }}{{\mathop \sum \nolimits_{i = 1}^{n} a_{ij} }}$$
(2)
$$w_{i} = \frac{{\mathop \sum \nolimits_{j = 1}^{n} a_{ij}^{*} }}{n}$$
(3)

Step 4 Making consistency calculations: Consistency is calculated to obtain reliable results. The Consistency Rate (CR) is expected to be less than 0.10. For this, λmax is calculated in Eq. (4), the Consistency Index (CI) by Eq. (5), and CR value by Eq. (6). Random Value Index (RI) in Eq. (6) is the value corresponding to n from the RI table.

$$\lambda_{{{\text{max}}}} = \frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {\frac{{d_{i} }}{{w_{i} }}} \right)}}{n}, \quad \left[ {d_{i} } \right]_{n \times 1} = \left[ {a_{ij} } \right]_{n \times n} \times \left[ {w_{i} } \right]_{n \times 1}$$
(4)
$$CI = \frac{{\lambda_{{{\text{max}}}} - n}}{n - 1}$$
(5)
$$CR = \frac{CI}{{RI}}$$
(6)

The VIKOR Method

The VIKOR method is developed to calculate the closeness of the alternatives to the ideal solution, and, thus, it provides a compromise solution to the problem [20]. It is possible to describe the VIKOR method in five stages:

Step 1 The best \(f_{i}^{ \divideontimes }\) and the worst \(f_{i}^{ - }\) values are determined: The decision matrix is created with the scores of the alternatives for each criterion (i = 1, 2,.. n), and the values of \(f_{i}^{ \divideontimes }\) and \(f_{i}^{ - }\) are calculated based on the criterion features. Here, Eq. (7a) is for the criterion with the benefit feature and Eq. (7b) for cost.

$$\begin{array}{*{20}c} {f_{i}^{ \divideontimes } = max_{j} x_{ij} } \\ {f_{i}^{ - } = min_{j} x_{ij} } \\ \end{array}$$
(7a)
$$\begin{array}{*{20}c} {f_{i}^{ \divideontimes } = min_{j} x_{ij} } \\ {f_{i}^{ - } = max_{j} x_{ij} } \\ \end{array}$$
(7b)

Step 2 Calculation of \(S_{j }\) and \(R_{j }\) values: \(S_{j }\) (average group score) the score is calculated in Eq. (8) and \(R_{j }\) (worst group score) score in Eq. (9) for each alternative \(( j = 1, 2,.. J)\).

$$S_{j} = \mathop \sum \limits_{i = 1}^{n} w_{i} \frac{{f_{i}^{ \divideontimes } - x_{ij} }}{{f_{i}^{ \divideontimes } - f_{i}^{ - } }}$$
(8)
$$R_{j} = max_{i} \left[ {w_{i} \frac{{f_{i}^{ \divideontimes } - x_{ij} }}{{f_{i}^{ \divideontimes } - f_{i}^{ - } }}} \right]$$
(9)

Step 3 Calculation of \(Q_{j }\) value: For alternatives \(\left( {j = 1, 2, \ldots J} \right)\), the maximum group benefit \((Q_{j } )\) is calculated by Eq. (10). The parameters \(S^{ \divideontimes } , S^{ - } ,R^{ \divideontimes } , R^{ - }\) required to calculate \(Q_{j }\) are shown by Eq. (11). The v value in Eq. (11) represents the maximum group utility.

$$Q_{j} = \frac{{v\left( {S_{j} - S^{ \divideontimes } } \right)}}{{\left( {S^{ - } - S^{ \divideontimes } } \right)}} + \frac{{\left( {1 - v} \right)\left( {R_{j} - R^{ \divideontimes } } \right)}}{{\left( {R^{ - } - R^{ \divideontimes } } \right)}}$$
(10)
$$\begin{gathered} S^{ \divideontimes } = min_{j} S_{j} ; R^{ \divideontimes } = min_{j} R_{j} \hfill \\ S^{ - } = max_{j} S_{j} ;R^{ - } = max_{j} R_{j} \hfill \\ \end{gathered}$$
(11)

Step 4 Sorting \(S_{j }\), \(R_{j } , Q_{j }\) values: These three values obtained by each alternative are sorted from lowest to highest.

Step 5 Checking the conditions: The reliability of the ranking ordering of alternatives is controlled by two conditions: acceptable advantage condition and acceptable stability condition.

Under the condition of acceptable advantage in Eq. (12), \(A^{1}\) is the first (the lowest value) alternative \(\left( {j = 1, 2, \ldots J} \right)\) that ranks from lowest to highest and \(A^{2}\) is the second.

$$Q_{{A^{2} }} - Q_{{A^{1} }} \ge \frac{1}{j - 1}$$
(12)

Under the acceptable stability condition, \(A^{1}\) is ranked the best by \(S_{j }\) and/or \(R_{j }\).

When the first (acceptable advantage) of these conditions is met, but the second condition (acceptable stability) is not met, \(A^{1}\) and \(A^{2}\) are considered together as a compromised solution.

If the first condition is not met: all of the alternatives from \(A^{1}\), \(A^{2}\) \(A^{3}\)… \(A^{m}\) are considered as compromised solutions. The value of m is determined according to Eq. (13).

$$Q_{{A^{m} }} - Q_{{A^{1} }} < \frac{1}{j - 1}{\text{ for}}\;{\text{maximum}}\;m$$
(13)

The TOPSIS Method

The TOPSIS method chooses the alternative that is closest to the ideal solution, but the farthest to the negative ideal solution [11]. It is possible to apply the TOPSIS method in six steps.

Step 1 Calculate the normalized decision matrix: The normalization of the decision matrix is calculated by finding \(r_{ij}\) (normalized values) as in Eq. (14). Here, the criteria are specified with i \(\left( {i = 1, 2,.. n} \right){\text{and}}\) alternatives with j \(\left( {j = 1, 2,.. J} \right).\)

$$r_{ij} = \frac{{x_{ij} }}{{\sqrt {\mathop \sum \nolimits_{j = 1}^{J} x_{ij}^{2} } }}$$
(14)

Step 2 Creating the weighted normalized decision matrix: The weights \((w_{1} , w_{2} , \ldots w_{n} )\) of each criterion \(\left( {i = 1, 2, \ldots n} \right)\) are determined by the decision-maker. The weighted normalized value \(v_{ij}\) is calculated, as shown by Eq. (15).

$$v_{ij} = w_{i} r_{ij}$$
(15)

Step 3 Determination of the ideal and negative ideal solutions: The ideal solution \(\left( {A^{*} } \right)\) takes the maximum value when associated with benefit criterion \((I^{\prime } )\), and the minimum value when associated with cost criterion \((I^{\prime \prime } )\) (Eq. 16a). The negative ideal solution \(\left( {A^{ - } } \right)\) applies the opposite (Eq. 16b).

$$\begin{aligned} A^{*} & = \left\{ {v_{1}^{*} ,v_{2}^{*} , \ldots v_{n}^{*} } \right\} \\ & = \left\{ {\left( {max_{j} v_{ij} \parallel {\text{i}} \epsilon I^{\prime } } \right),\left( {min_{j} v_{ij} \parallel {\text{i}} \epsilon I^{\prime \prime } } \right)} \right\} \\ \end{aligned}$$
(16a)
$$\begin{aligned} A^{ - } & = \left\{ {v_{1}^{ - } ,v_{2}^{ - } , \ldots v_{n}^{ - } } \right\} \\ & = \left\{ {\left( {min_{j} v_{ij} \parallel {\text{i}} \epsilon I^{\prime } } \right),\left( {max_{j} v_{ij} \parallel {\text{i}} \epsilon I^{\prime \prime } } \right)} \right\} \\ \end{aligned}$$
(16b)

Step 4 Calculate distance values: The distance from the ideal solution \((D_{j}^{*} )\) is calculated by Eq. (17a) by using the Euclidean distance, and the distance from the negative ideal solution \((D_{j}^{ - } )\) is calculated by Eq. (17b).

$$D_{j}^{*} = \sqrt {\sum\nolimits_{i = 1}^{n} {\left( {v_{ij} - v_{i}^{*} } \right)^{2} } } , j = 1, 2, \ldots , J$$
(17a)
$$D_{j}^{ - } = \sqrt {\sum\nolimits_{i = 1}^{n} {\left( {v_{ij} - v_{i}^{ - } } \right)^{2} } } , j = 1, 2, \ldots , J$$
(17b)

Step 5 Calculation of the relative proximity to the ideal solution: Relative proximity \((C_{j}^{*} )\) of alternative \(a_{j}\) to \(A^{*}\) is calculated by Eq. (18).

$$C_{j}^{*} = \frac{{D_{j}^{ - } }}{{\left( {D_{j}^{*} + D_{j}^{ - } } \right)}}, j = 1, 2, \ldots , J$$
(18)

Step 6 Rank the preference order: The ranking is made from the alternative having the largest \(C_{j}^{*}\) values (the best alternative) to the lowest.

The MOORA Method

The MOORA method is for multi-objective optimization with discrete alternatives. It has two approaches: the ratio system and reference point theory [7, 8].

The ratio system is carried out in two steps. In the first step, the normalization process is applied by Eq. (19). Here, i represents the objective \(\left( {i = 1, 2, \ldots ,n} \right)\) and j alternative \(\left( {{\text{j}} = 1,{ }2, \ldots ,{\text{J}}} \right)\).

$$x_{ij}^{*} = \frac{{x_{ij} }}{{\sqrt {\mathop \sum \nolimits_{j = 1}^{J} x_{ij}^{2} } }}$$
(19)

In the second step of the ratio system, the evaluation of the degree of the alternative (j) meeting the objective (i) is found by the optimization of the normalized values. Equation (20) is applied depending on the objectives of the criteria (i.e., maximization or minimization).

$$y_{j}^{*} = \mathop \sum \limits_{i = 1}^{i = g} x_{ij}^{*} - \mathop \sum \limits_{i = g + 1}^{i = n} x_{ij}^{*}$$
(20)

In Eq. (20), \(i = 1, 2, \ldots , g\) represents objectives (of the criteria) to be maximized and \(i = g + 1, g + 2, \ldots , n\) objectives to be minimized. The ranking order of each alternative is obtained by sorting the values of \(y_{j}^{*}\) from highest to lowest. The largest \(y_{j}^{*}\) values will be the best alternative.

Reference point theory measures the distances between alternatives \((x_{ij}^{*} )\) and a reference point \((r_{i} )\) by Eq. (21).

$$min_{j} max_{i} \left( {\left| {r_{i} - x_{ij}^{*} } \right|} \right)$$
(21)

In cases where criteria weights \(\left( {s_{i} } \right)\) are known, \(x_{ij}^{*}\) values in Eq. (20) and \(r_{i}\) and \(x_{ij}^{*}\) values in Eq. (21) are multiplied by the coefficient \(s_{i}\), as in Eqs. (22) and (23).

$$y_{j}^{*} = \mathop \sum \limits_{i = 1}^{i = g} s_{i} x_{ij}^{*} - \mathop \sum \limits_{i = g + 1}^{i = n} s_{i} x_{ij}^{*}$$
(22)
$$min_{j} max_{i} \left( {\left| {s_{i} r_{i} - s_{i} x_{ij}^{*} } \right|} \right)$$
(23)

Results

In this study, seven criteria were identified, as shown in Table 1. These criteria were used to evaluate twenty-six candidates for the selection of two researchers.

Table 1 Candidate selection criteria and their explanations
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Table 2 represents the evaluation findings of the candidates based on seven criteria. The relevant data for criteria C1 to C4 were directly obtained from the candidates; C5 was from the faculty; C6 was from a university ranking list; C7 were from faculty members.

Table 2 Evaluation of candidates in terms of criteria
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In this study, AHP was used to calculate the weight of each criterion. Table 3 shows the criterion comparison matrix. Four faculty members participated in a questionnaire to generate the comparison matrix.

Table 3 Criterion comparison matrix
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The comparison matrix has been normalized by the operations in Eq. (2). Then, the criterion significance weight in Table 4 has been calculated by using Eq. (3). Following that, the consistency ratio was calculated, as CR = 0.03. By reason of \(CR{ } < 0.1\), the values obtained are considered to be consistent.

Table 4 The weight of importance of each criterion
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After the calculation of the weights of the criteria, VIKOR was applied to calculate \(Q_{j }\) values; TOPSIS was used to calculate \(C_{j}^{*}\) values; the MOORA-rate system method was used to calculate \(y_{j}^{*}\) values; the MOORA-reference point theory method was used to calculate \({\text{max}}_{i} \left( {\left| {r_{i} - x_{ij}^{*} } \right|} \right)\) and \({\text{max}}_{i} \left( {\left| {s_{i} r_{i} - s_{i} x_{ij}^{*} } \right|} \right)\) values. Table 5 shows the ranking order of the twenty-six alternatives by using each method.

Table 5 The ranking order of alternatives with the use of VIKOR, TOPSIS, and MOORA methods
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Discussion and Conclusion

This study was conducted to select two researchers to an engineering faculty and to compare the findings of the different methods. The findings revealed that the VIKOR and TOPSIS methods suggest the selection of the same candidates. However, the ranking order of the rest of the candidates was considerably different. Moreover, the MOORA-reference method suggests the selection of entirely different candidates. Figure 2 illustrates the comparative ranking orders for twenty-six alternatives.

Fig. 2
figure 2

Comparative ranking results of VIKOR, TOPSIS and MOORA methods

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In this respect, although MCDM methods are useful to assist in decision-making, they might not always be reliable. It should be taken into account that MCDM methods might not always give the best results. Several researchers have criticized the use of MCDM methods from this perspective [4, 9, 32]. Here, two points should be mentioned. Firstly, MCDM methods have been built on linear mathematical algorithms. Decision-making might depend on the conditions, and thus, the use of linear methods might not fit well in real-life decisions. Individuals might use a non-linear algorithm when making decisions. Indeed, in real life, the decision-making process is rather complex. Secondly, the selection of the criteria and the evaluation of the candidates were based on expert judgments, which can be subjective. Researchers suggested using fuzzy logic to reduce the subjectivity and deal with uncertainty [3, 15, 23]. However, the use of fuzzy logic would still not solve the problem with the dynamic and non-linear features of the decision-making process. At this point, non-linear decision-making methods might provide more reliable results [10, 25, 33].

However, despite their limitations, MCDM methods have still been used to support decision-making due to their simplicity. Future studies might investigate the ideal selection of MCDM methods for specific conditions.