Evaluation of Human Resources in Transportation Companies Using Multi-Criteria Model for Ranking Alternatives by Defining Relations between Ideal and Anti-Ideal Alternative (RADERIA)
Abstract
:1. Introduction
2. Literature Review
3. Methodology
3.1. The LBWA Model for Determining Weighted Coefficient of the Criteria
3.2. Ranking Alternatives by Defining the Relationship between Ideal and Anti-Ideal Alternatives (RADERIA)
- is times better than
- is times better than
- ...
- is times better than
- From the matrix we eliminate the values of all alternatives based on the weakest criteria, and from such a reduced matrix we can observe the new values and . If the condition > is satisfied, then → .
- If it is still = even after eliminating the weakest criterion, then the next weakest criterion is eliminated. We repeat this until we get ≠ . Then, if we, for example, get < , then → .
- If the equality = still exists even after eliminating all the criteria, then we analyze if any of the alternatives, and , on the criterion acquire values which are better than the value which was defined as the ideal alternative on the criterion . In the case that there is such an alternative, then that alternative is chosen as the better one.
- If the equality = still exists even after eliminating all the criteria, then we can say that the alternatives and are equal.
4. Case Study and Experimental Results
4.1. Determining Criteria Weights Using the LBWA Model
4.2. Evaluation of the Alternatives Using the RADERIA Model
- (a)
- The first group consists of the values that appear in the interval or (Figure 4a). Using Expression (1), we can define the values , , and . In this way, we get the values for .
- (b)
- The second group consists of the values which appear in the interval or (Figure 4b). Using Expression (1), we can define the values , , and . In this way, we get the values for .
5. Validity of the Results
5.1. Comparison with the Other Multi-Criteria Methods
5.2. Rank Reversal Problem
5.3. Changes of the Range in Measurement Scale
6. Discussion of Results
- The new model for data normalization: The RADERIA model has a new approach for data normalization. Most MCDM techniques that are available today use a model for data normalization in which all data are converted to values in the interval [0, 1]. However, the new model for normalization, presented in this model, allows converting data from the initial decision-making matrix into any interval that is suitable for making rational decisions. This way, it facilitates making rational decisions because the new approach enables defining intervals according to the decision-maker’s judgments. For example, in this paper, all values from the initial decision-making matrix are converted into the interval [0, 2], while in other applied models (TOPSIS, VIKOR, COPRAS, and MULTIMOORA), all values are normalized into the interval [0, 1].
- The adaptivity of the RADERIA model through the model transformation for data normalization: The model for normalization of the data which is applied in the RADERIA model represents a linear model for normalization (Figure 3a), which is based on a linear function. Adaptivity of the model can be seen in its transformation into other forms of decreasing functions, for example, quadratic functions (Figure 3b). This characteristic allows the real presentation of expert judgments and making rational decisions.
- The resistance of the RADERIA model to rank reversal problem: One of the most significant disadvantages of many MCDM methods is the rank reversal problem. The question of rank reversal was noticed by Belton and Gear [46] for the first time. To this day, this issue has received great attention. Moreover, there are many studies [48,50,51] that analyze this phenomenon and show its importance in the process of making decisions. This phenomenon especially comes to expression in dynamic conditions of making decisions where the number of alternatives changes during the process of making a decision. This phenomenon can be noticed in many traditional MCDM methods. For example, the phenomenon of rank reversal is confirmed for AHP, ELECTRE (ELimination and Choice Expressing REality), TOPSIS, and PROMETHEE (Preference Ranking Organization METhod for Enrichment of Evaluations). Besides the mentioned methods, the rank reversal problem is also confirmed in this paper in MULTIMOORA, COPRAS, and VIKOR methods. However, in the same experiment, the RADERIA model shows resistance to rank reversal. Consequently, the RADERIA method shows significant stability and reliability of the results in a dynamic environment. It is also important to mention that in numerous simulations, the RADERIA method shows stability while processing a larger number of datasets. The study case also confirmed this with 36 alternatives, as examined in this paper.
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Criteria | E1 | E2 | E3 | E4 |
---|---|---|---|---|
C1 | Level 1 | Level 1 | Level 1 | Level 1 |
C2 | Level 2 | Level 1 | Level 2 | Level 1 |
C3 | Level 3 | Level 3 | Level 3 | Level 3 |
C4 | Level 4 | Level 4 | Level 4 | Level 4 |
C5 | Level 5 | Level 5 | Level 5 | Level 5 |
Experts | C1 | C2 | C3 | C4 | C5 | λ | |||||
---|---|---|---|---|---|---|---|---|---|---|---|
I1 | Level | I2 | Level | I3 | Level | I4 | Level | I5 | Level | ||
E1 | 0 | S1 | 0.1 | S2 | 1 | S3 | 1 | S4 | 0.35 | S5 | 1 |
E2 | 0 | S1 | 1.5 | S1 | 0.8 | S3 | 1 | S4 | 0.35 | S5 | 2 |
E3 | 0 | S1 | 0.25 | S2 | 1 | S3 | 1 | S4 | 0.5 | S5 | 1 |
E4 | 0 | S1 | 2 | S1 | 1.2 | S3 | 1 | S4 | 0.8 | S5 | 2 |
Criteria | E1 | E2 | E3 | E4 |
---|---|---|---|---|
C1 | ||||
C2 | ||||
C3 | ||||
C4 | ||||
C5 |
Criteria Weights | E1 | E2 | E3 | E4 | Average |
---|---|---|---|---|---|
w1 | 0.457 | 0.417 | 0.461 | 0.432 | 0.442 |
w2 | 0.223 | 0.278 | 0.217 | 0.259 | 0.244 |
w3 | 0.131 | 0.128 | 0.132 | 0.127 | 0.129 |
w4 | 0.102 | 0.096 | 0.102 | 0.100 | 0.100 |
w5 | 0.088 | 0.081 | 0.088 | 0.082 | 0.085 |
Alternative | C1 (Min) | C2 (Min) | C3 (Max) | C4 (Max) | C5 (Max) |
---|---|---|---|---|---|
A1 | 28.0 | 0.000 | 7 | 7 | 5 |
A2 | 30.0 | 0.000 | 3 | 3 | 3 |
A3 | 29.5 | 0.000 | 5 | 5 | 7 |
A4 | 32.0 | 0.000 | 5 | 5 | 9 |
A5 | 28.5 | 0.000 | 5 | 5 | 9 |
A6 | 33.5 | 0.000 | 5 | 5 | 7 |
A7 | 31.5 | 0.000 | 5 | 5 | 9 |
A8 | 32.0 | 0.000 | 7 | 7 | 9 |
A9 | 33.5 | 0.030 | 3 | 3 | 9 |
A10 | 34.0 | 0.000 | 3 | 3 | 7 |
A11 | 28.5 | 0.000 | 5 | 5 | 7 |
A12 | 29.5 | 0.000 | 9 | 9 | 9 |
A13 | 33.5 | 0.000 | 9 | 9 | 7 |
A14 | 35.0 | 0.025 | 9 | 9 | 9 |
A15 | 36.5 | 0.000 | 5 | 5 | 5 |
A16 | 33.5 | 0.000 | 5 | 5 | 5 |
A17 | 31.5 | 0.000 | 3 | 3 | 3 |
A18 | 33.5 | 0.017 | 1 | 5 | 3 |
A19 | 36.0 | 0.000 | 9 | 9 | 5 |
A20 | 32.5 | 0.000 | 9 | 9 | 7 |
A21 | 33.5 | 0.000 | 9 | 9 | 3 |
A22 | 34.0 | 0.009 | 7 | 7 | 3 |
A23 | 32.5 | 0.000 | 7 | 7 | 3 |
A24 | 35.0 | 0.000 | 7 | 7 | 3 |
A25 | 35.0 | 0.000 | 7 | 7 | 5 |
A26 | 31.0 | 0.000 | 5 | 5 | 7 |
A27 | 30.5 | 0.000 | 3 | 3 | 9 |
A28 | 28.5 | 0.000 | 3 | 3 | 9 |
A29 | 31.5 | 0.000 | 9 | 9 | 7 |
A30 | 32.5 | 0.000 | 9 | 9 | 9 |
A31 | 29.5 | 0.000 | 7 | 7 | 7 |
A32 | 34.0 | 0.000 | 1 | 5 | 9 |
A33 | 31.0 | 0.000 | 3 | 3 | 5 |
A34 | 32.5 | 0.000 | 1 | 5 | 5 |
A35 | 31.0 | 0.000 | 3 | 3 | 7 |
A36 | 34.0 | 0.000 | 5 | 5 | 7 |
Ideal/Anti-Ideal Alt. | C1 | C2 | C3 | C4 | C5 |
---|---|---|---|---|---|
Ij | 22 | 0 | 10 | 10 | 10 |
Wj | 40 | 2 | 1 | 1 | 1 |
Criteria | Ij | Wj | ||||
---|---|---|---|---|---|---|
C1 | 22 | 0 | 40 | 2 | 30 | 1 |
C2 | 0 | 0 | 2 | 2 | 0.02 | 1 |
C3 | 1 | 2 | 10 | 0 | 5 | 1 |
C4 | 1 | 2 | 10 | 0 | 5 | 1 |
C5 | 1 | 2 | 10 | 0 | 5 | 1 |
Alternative | C1 | C2 | C3 | C4 | C5 |
---|---|---|---|---|---|
A1 | 0.75 | 0.00 | 0.60 | 0.60 | 1.00 |
A2 | 1.00 | 0.00 | 1.50 | 1.50 | 1.50 |
A3 | 0.94 | 0.00 | 1.00 | 1.00 | 0.60 |
A4 | 1.20 | 0.00 | 1.00 | 1.00 | 0.20 |
A5 | 0.81 | 0.00 | 1.00 | 1.00 | 0.20 |
A6 | 1.35 | 0.00 | 1.00 | 1.00 | 0.60 |
A7 | 1.15 | 0.00 | 1.00 | 1.00 | 0.20 |
A8 | 1.20 | 0.00 | 0.60 | 0.60 | 0.20 |
A9 | 1.35 | 1.01 | 1.50 | 1.50 | 0.20 |
A10 | 1.40 | 0.00 | 1.50 | 1.50 | 0.60 |
A11 | 0.81 | 0.00 | 1.00 | 1.00 | 0.60 |
A12 | 0.94 | 0.00 | 0.20 | 0.20 | 0.20 |
A13 | 1.35 | 0.00 | 0.20 | 0.20 | 0.60 |
A14 | 1.50 | 1.00 | 0.20 | 0.20 | 0.20 |
A15 | 1.65 | 0.00 | 1.00 | 1.00 | 1.00 |
A16 | 1.35 | 0.00 | 1.00 | 1.00 | 1.00 |
A17 | 1.15 | 0.00 | 1.50 | 1.50 | 1.50 |
A18 | 1.35 | 0.85 | 2.00 | 1.00 | 1.50 |
A19 | 1.60 | 0.00 | 0.20 | 0.20 | 1.00 |
A20 | 1.25 | 0.00 | 0.20 | 0.20 | 0.60 |
A21 | 1.35 | 0.00 | 0.20 | 0.20 | 1.50 |
A22 | 1.40 | 0.45 | 0.60 | 0.60 | 1.50 |
A23 | 1.25 | 0.00 | 0.60 | 0.60 | 1.50 |
A24 | 1.50 | 0.00 | 0.60 | 0.60 | 1.50 |
A25 | 1.50 | 0.00 | 0.60 | 0.60 | 1.00 |
A26 | 1.10 | 0.00 | 1.00 | 1.00 | 0.60 |
A27 | 1.05 | 0.00 | 1.50 | 1.50 | 0.20 |
A28 | 0.81 | 0.00 | 1.50 | 1.50 | 0.20 |
A29 | 1.15 | 0.00 | 0.20 | 0.20 | 0.60 |
A30 | 1.25 | 0.00 | 0.20 | 0.20 | 0.20 |
A31 | 0.94 | 0.00 | 0.60 | 0.60 | 0.60 |
A32 | 1.40 | 0.00 | 2.00 | 1.00 | 0.20 |
A33 | 1.10 | 0.00 | 1.50 | 1.50 | 1.00 |
A34 | 1.25 | 0.00 | 2.00 | 1.00 | 1.00 |
A35 | 1.10 | 0.00 | 1.50 | 1.50 | 0.60 |
A36 | 1.40 | 0.00 | 1.00 | 1.00 | 0.60 |
Alternative | C1 | C2 | C3 | C4 | C5 | Rank | |
---|---|---|---|---|---|---|---|
A1 | 0.325 | 0.000 | 0.077 | 0.058 | 0.087 | 0.547 | 2 |
A2 | 0.434 | 0.000 | 0.191 | 0.145 | 0.130 | 0.900 | 25 |
A3 | 0.407 | 0.000 | 0.128 | 0.096 | 0.052 | 0.683 | 11 |
A4 | 0.521 | 0.000 | 0.128 | 0.096 | 0.017 | 0.762 | 16 |
A5 | 0.353 | 0.000 | 0.128 | 0.096 | 0.017 | 0.594 | 4 |
A6 | 0.586 | 0.000 | 0.128 | 0.096 | 0.052 | 0.862 | 20 |
A7 | 0.499 | 0.000 | 0.128 | 0.096 | 0.017 | 0.740 | 13 |
A8 | 0.521 | 0.000 | 0.077 | 0.058 | 0.017 | 0.672 | 9 |
A9 | 0.586 | 0.257 | 0.191 | 0.145 | 0.017 | 1.196 | 35 |
A10 | 0.607 | 0.000 | 0.191 | 0.145 | 0.052 | 0.996 | 33 |
A11 | 0.353 | 0.000 | 0.128 | 0.096 | 0.052 | 0.629 | 7 |
A12 | 0.407 | 0.000 | 0.026 | 0.019 | 0.017 | 0.469 | 1 |
A13 | 0.586 | 0.000 | 0.026 | 0.019 | 0.052 | 0.683 | 10 |
A14 | 0.651 | 0.256 | 0.026 | 0.019 | 0.017 | 0.969 | 29 |
A15 | 0.716 | 0.000 | 0.128 | 0.096 | 0.087 | 1.027 | 34 |
A16 | 0.586 | 0.000 | 0.128 | 0.096 | 0.087 | 0.897 | 24 |
A17 | 0.499 | 0.000 | 0.191 | 0.145 | 0.130 | 0.965 | 28 |
A18 | 0.586 | 0.217 | 0.255 | 0.096 | 0.130 | 1.285 | 36 |
A19 | 0.694 | 0.000 | 0.026 | 0.019 | 0.087 | 0.826 | 19 |
A20 | 0.542 | 0.000 | 0.026 | 0.019 | 0.052 | 0.639 | 8 |
A21 | 0.586 | 0.000 | 0.026 | 0.019 | 0.130 | 0.761 | 15 |
A22 | 0.607 | 0.115 | 0.077 | 0.058 | 0.130 | 0.987 | 32 |
A23 | 0.542 | 0.000 | 0.077 | 0.058 | 0.130 | 0.807 | 17 |
A24 | 0.651 | 0.000 | 0.077 | 0.058 | 0.130 | 0.915 | 27 |
A25 | 0.651 | 0.000 | 0.077 | 0.058 | 0.087 | 0.872 | 22 |
A26 | 0.477 | 0.000 | 0.128 | 0.096 | 0.052 | 0.753 | 14 |
A27 | 0.456 | 0.000 | 0.191 | 0.145 | 0.017 | 0.809 | 18 |
A28 | 0.353 | 0.000 | 0.191 | 0.145 | 0.017 | 0.706 | 12 |
A29 | 0.499 | 0.000 | 0.026 | 0.019 | 0.052 | 0.596 | 5 |
A30 | 0.542 | 0.000 | 0.026 | 0.019 | 0.017 | 0.605 | 6 |
A31 | 0.407 | 0.000 | 0.077 | 0.058 | 0.052 | 0.593 | 3 |
A32 | 0.607 | 0.000 | 0.255 | 0.096 | 0.017 | 0.977 | 30 |
A33 | 0.477 | 0.000 | 0.191 | 0.145 | 0.087 | 0.900 | 25 |
A34 | 0.542 | 0.000 | 0.255 | 0.096 | 0.087 | 0.981 | 31 |
A35 | 0.477 | 0.000 | 0.191 | 0.145 | 0.052 | 0.865 | 21 |
A36 | 0.607 | 0.000 | 0.128 | 0.096 | 0.052 | 0.884 | 23 |
MCDM Model | RADERIA | VIKOR | COPRAS | TOPSIS | MULTIMOORA |
---|---|---|---|---|---|
RADERIA | 1.000 | 0.832 | 0.968 | 0.938 | 0.930 |
VIKOR | - | 1.000 | 0.712 | 0.644 | 0.620 |
COPRAS | - | - | 1.000 | 0.968 | 0.987 |
TOPSIS | - | - | - | 1.000 | 0.982 |
MULTIMOORA | - | - | - | - | 1.000 |
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Jakovljevic, V.; Zizovic, M.; Pamucar, D.; Stević, Ž.; Albijanic, M. Evaluation of Human Resources in Transportation Companies Using Multi-Criteria Model for Ranking Alternatives by Defining Relations between Ideal and Anti-Ideal Alternative (RADERIA). Mathematics 2021, 9, 976. https://doi.org/10.3390/math9090976
Jakovljevic V, Zizovic M, Pamucar D, Stević Ž, Albijanic M. Evaluation of Human Resources in Transportation Companies Using Multi-Criteria Model for Ranking Alternatives by Defining Relations between Ideal and Anti-Ideal Alternative (RADERIA). Mathematics. 2021; 9(9):976. https://doi.org/10.3390/math9090976
Chicago/Turabian StyleJakovljevic, Vladimir, Mališa Zizovic, Dragan Pamucar, Željko Stević, and Miloljub Albijanic. 2021. "Evaluation of Human Resources in Transportation Companies Using Multi-Criteria Model for Ranking Alternatives by Defining Relations between Ideal and Anti-Ideal Alternative (RADERIA)" Mathematics 9, no. 9: 976. https://doi.org/10.3390/math9090976