# Novel Methods in Multiple Criteria Decision-Making Process (MCRAT and RAPS)—Application in the Mining Industry

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Multiple Criteria Decision Making Methods

#### 1.2. Drilling and Blasting Problem

## 2. Materials and Methods

_{i}with respect to a set of criteria

_{max}—a set of criteria that should be maximized

_{min}—a set of criteria that should be minimized

_{ij}do the weighted normalization as follows:

_{i}composed of alternative components:

_{i}is a matrix obtained by the product of matrix F and G

_{i}:

_{i}is as follows:

_{i}).

_{k}and Q

_{h}represent the base and perpendicular side of this triangle, respectively.

_{i}.

## 3. Mining Problem

## 4. Validity Test of Novel Methods

#### 4.1. The First Example for Testing

#### 4.2. The Second Example for Testing

## 5. Numerical Example for the Mining Problem

_{1}composed of A1 alternative components is as follows:

_{1}defined by the product of matrix F and G

_{i}is:

_{1}is as follows:

_{1}, T

_{2},..., T

_{12}, in the previous table, we can demonstrate the final step: ranking by descending order as presented in Table 18.

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Glossary

Acronyms | Full Names |

MCDM | Multiple Criteria Decision Making |

MCRAT | Multiple Criteria Ranking by Alternative Trace |

RAPS | Ranking Alternatives by Perimeter Similarity |

ELECTRE | Elimination Et Choice Translating Reality |

AHP | Analytic Hierarchy Process |

TOPSIS | Technique for Order of Preference by Similarity to Ideal Solution |

PROMETHEE | Preference Ranking Organization Method for Enrichment of Evaluations |

MOORA | Multi-Objective Optimization by Ratio Analysis |

WASPAS | Weighted Aggregated Sum Product Assessment |

TAOV | Total Area Based on Orthogonal Vectors |

ARAS | Additive Ratio Assessment |

SAW | Simple Additive Weighting |

COPRAS | Complex Proportional Assessment |

VIKOR | Visekriterijumska Optimizacija i Kompromisno Resenje |

GRA | Grey Relational Analysis |

CP | Compromise Programming |

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Method | Advantages | Disadvantages | Year of Introduction |
---|---|---|---|

ELECTRE | - Takes uncertainty and vagueness into account.
- No need for independence of attributes.
- Very poor performance on a single criterion may eliminate an alternative from consideration.
| - Relatively complex algorithm.
- A complete ranking of the alternatives may not be achieved.
| 1966 |

SAW | - Intuitive method with simple algorithm.
- Able to compensate between variables.
- Suitable for the evaluation of a single alternative.
| - Converting minimizing criteria to maximizing is necessary.
- Holds potential for unfounded results.
- Values should be positive.
| 1968 |

AHP | - Easy to use.
- Not data intensive.
- Hierarchy structure can easily adjust to fit many sized problems.
- Improved focus on each criterion used in the calculations.
| - Possibility for intransitive preferences.
- High number of pairwise comparisons required for large scale problems.
- Potential for inconsistencies.
- Rank reversal.
| 1980 |

VIKOR | - Usable for problems with difficulties in expressing preferences.
- Stability analysis included.
| - Quantitative information is necessary.
- Needs initial weights.
- The ranking needs can be performed with different values of variables’ weights.
| 1980 |

TOPSIS | - Works with a fundamental ranking.
- Complete use of allocated information.
- Simple computation process.
| - Not considering correlation of attributes.
- A strong deviation of an indicator from the ideal solution strongly influences the results.
- Suitable when the indicators of alternatives do not vary very strongly.
| 1981 |

PROMETHEE | - Useful with alternatives that are difficult to harmonize.
- Using both qualitative and quantitative information.
- Potential for inclusion of uncertain and fuzzy data.
| - No clear method for assigning criteria weights.
- Moderately complex computation process.
| 1986 |

GRA | - Providing more distinction in alternatives ranking.
- Suitable for solving problems with complicated interrelationships between multiple factors and variables.
| - Relatively complex procedure.
- Relatively high sensitivity to criteria weights.
| 1994 |

MOORA | - Relatively simple procedure.
- Attributes are independent.
- Robust method.
| - The qualitative attributes are converted into the quantitative attributes.
- Relatively complex calculations process.
| 2006 |

COPRAS | - Suitable for the evaluation of a single alternative.
- Robust method.
- Not requiring minimization of criteria.
| - Less stable in data variation case in comparison to some other methods.
- Sensitive to slight variations in data.
| 2007 |

ARAS | - The utility degree is considered as the ranking of alternatives.
- Attributes are independent.
| - The qualitative attributes should be converted into the quantitative attributes.
| 2010 |

WASPAS | - Relatively simple calculation.
- The method weighs the beneficial and non-beneficial criteria in the problem separately.
- The method is useful for the complete ranking of alternatives.
| - Taking into consideration only minimum (for non-beneficial attributes) and maximum (for beneficial attributes) values.
- Does not consider all the performance values.
| 2012 |

TAOV | - Relatively simple procedure.
- Proposes a procedure to guarantee the independence of the criteria.
- No limitations regarding the scale of criteria.
| - Low applicability to conflict resolution.
| 2018 |

Alternative | Hole Diameter (mm) | Hole Length (m) | Stemming (m) | Burden (m) | Spacing (m) | Subdrill (m) |
---|---|---|---|---|---|---|

A1 | 76 | 16.3 | 2.6 | 2.6 | 2.6 | 0.76 |

A2 | 76 | 16.7 | 2.6 | 2.6 | 2.6 | 1.14 |

A3 | 76 | 16.3 | 3.0 | 3.0 | 3.0 | 0.76 |

A4 | 76 | 16.7 | 3.0 | 3.0 | 3.0 | 1.14 |

A5 | 89 | 16.4 | 2.9 | 2.9 | 2.9 | 0.89 |

A6 | 89 | 16.4 | 3.5 | 3.5 | 3.5 | 0.89 |

A7 | 89 | 16.9 | 2.9 | 2.9 | 2.9 | 1.34 |

A8 | 89 | 16.9 | 3.5 | 3.5 | 3.5 | 1.34 |

A9 | 106 | 16.6 | 3.4 | 3.4 | 3.4 | 1.06 |

A10 | 106 | 17.1 | 3.4 | 3.4 | 3.4 | 1.56 |

A11 | 106 | 16.6 | 4.0 | 4.0 | 4.0 | 1.06 |

A12 | 106 | 17.1 | 4.0 | 4.0 | 4.0 | 1.56 |

Alternative | Powder Factor, kg/m^{3} | Fragmentation, 1/mm | Fly Rock, m | Air Shock, m | Cost, €/Pattern |
---|---|---|---|---|---|

A1 | 0.4 | 0.002353 | 347 | 101.5 | 1500 |

A2 | 0.41 | 0.001208 | 348 | 105 | 1500 |

A3 | 0.29 | 0.002353 | 364 | 97.5 | 1500 |

A4 | 0.3 | 0.001208 | 385 | 100 | 1500 |

A5 | 0.43 | 0.002353 | 335 | 125 | 1600 |

A6 | 0.28 | 0.002353 | 365 | 130 | 1600 |

A7 | 0.45 | 0.001208 | 310 | 140 | 1700 |

A8 | 0.29 | 0.002353 | 320 | 135 | 1600 |

A9 | 0.4 | 0.000619 | 315 | 175 | 1700 |

A10 | 0.42 | 0.000619 | 325 | 180 | 1700 |

A11 | 0.28 | 0.001208 | 280 | 165 | 1700 |

A12 | 0.29 | 0.001208 | 220 | 170 | 1700 |

Min | Max | Min | Min | Min |

Room No. | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} |
---|---|---|---|---|---|---|

max | Max | Max | max | min | min | |

1. | 7.6 | 46 | 18 | 390 | 0.10 | 11 |

2. | 5.5 | 32 | 21 | 360 | 0.05 | 11 |

3. | 5.3 | 32 | 21 | 290 | 0.05 | 11 |

4. | 5.7 | 37 | 19 | 270 | 0.05 | 9 |

5. | 4.2 | 38 | 19 | 240 | 0.10 | 8 |

6. | 4.4 | 38 | 19 | 260 | 0.10 | 8 |

7. | 3.9 | 42 | 16 | 270 | 0.10 | 5 |

8. | 7.9 | 44 | 20 | 400 | 0.05 | 6 |

9. | 8.1 | 44 | 20 | 380 | 0.05 | 6 |

10. | 4.5 | 46 | 18 | 320 | 0.10 | 7 |

11. | 5.7 | 48 | 20 | 320 | 0.05 | 11 |

12. | 5.2 | 48 | 20 | 310 | 0.05 | 11 |

13. | 7.1 | 49 | 19 | 280 | 0.10 | 12 |

14. | 6.9 | 50 | 16 | 250 | 0.05 | 10 |

Weight | 0.21 | 0.16 | 0.26 | 0.17 | 0.12 | 0.08 |

Room No. | MCRAT | RAPS | TAOV | ARAS | SAW | TOPSIS | COPRAS | VIKOR | WASPAS | ELECTREE |
---|---|---|---|---|---|---|---|---|---|---|

1. | 3 | 4 | 3 | 4 | 3 | 3 | 3 | 6 | 4 | 3 |

2. | 8 | 7 | 6 | 6 | 5 | 7 | 7 | 5 | 6 | 4 |

3. | 11 | 10 | 10 | 10 | 8 | 10 | 10 | 8 | 10 | 6 |

4. | 9 | 9 | 11 | 9 | 10 | 9 | 9 | 9 | 9 | 9 |

5. | 14 | 14 | 14 | 14 | 13 | 13 | 13 | 13 | 14 | 12 |

6. | 13 | 13 | 13 | 13 | 12 | 12 | 12 | 12 | 12 | 13 |

7. | 12 | 12 | 12 | 12 | 14 | 14 | 14 | 14 | 13 | 14 |

8. | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 |

9. | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |

10. | 10 | 11 | 9 | 11 | 11 | 11 | 11 | 10 | 11 | 10 |

11. | 5 | 5 | 5 | 3 | 4 | 4 | 4 | 3 | 3 | 5 |

12. | 7 | 6 | 7 | 5 | 6 | 8 | 5 | 4 | 5 | 7 |

13. | 6 | 8 | 4 | 8 | 7 | 6 | 8 | 7 | 8 | 8 |

14. | 4 | 3 | 8 | 7 | 9 | 5 | 6 | 11 | 7 | 11 |

Correlation | MCRAT | RAPS | TAOV | ARAS | SAW | TOPSIS | COPRAS | VIKOR | WASPAS | ELECTRE |
---|---|---|---|---|---|---|---|---|---|---|

MCRAT | - | 0.98 | 0.93 | 0.93 | 0.88 | 0.97 | 0.95 | 0.79 | 0.93 | 0.93 |

RAPS | - | 0.89 | 0.95 | 0.88 | 0.96 | 0.96 | 0.80 | 0.95 | 0.78 | |

TAOV | - | 0.92 | 0.94 | 0.93 | 0.91 | 0.88 | 0.92 | 0.87 | ||

ARAS | - | 0.95 | 0.94 | 0.97 | 0.92 | 0.99 | 0.88 | |||

SAW | - | 0.93 | 0.96 | 0.96 | 0.96 | 0.96 | ||||

TOPSIS | - | 0.97 | 0.84 | 0.95 | 0.85 | |||||

COPRAS | - | 0.90 | 0.99 | 0.87 | ||||||

VIKOR | - | 0.93 | 0.93 | |||||||

WASPAS | - | 0.89 | ||||||||

ELECTRE | - |

Max | Max | Max | Min | Min | Max | Max | Max | Max | Min | Max | Min | Max | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

L1 | L2 | L3 | E1 | E2 | E3 | P1 | P2 | P3 | C1 | C2 | C3 | C4 | |

A1 | 2.5 | 0.76 | 0.26 | 0.49 | 0.18 | 5.01 | 0.012 | 0.11 | 0.179 | 5.36 | 0.26 | 4.97 | 18.7 |

A2 | 1.47 | 1.02 | 0.29 | 0.28 | 0.22 | 6.51 | 0.016 | 0.101 | 0.225 | 2.69 | 0.48 | 2.02 | 29 |

A3 | 1.31 | 1.36 | 0.18 | 0.43 | 0.1 | 12.65 | 0.026 | 0.23 | 0.338 | 4.2 | 0.28 | 2.57 | 17.1 |

A4 | 1.34 | 1.06 | 0.18 | 0.43 | 0.16 | 24.58 | 0.043 | 0.22 | 0.527 | 2.2 | 0.45 | 2.06 | 32 |

A5 | 1.57 | 1.5 | 0.24 | 0.44 | 0.13 | 12.9 | 0.02 | 0.1 | 0.38 | 3.57 | 0.34 | 1.91 | 25.9 |

Weight | 0.161 | 0.054 | 0.161 | 0.08 | 0.032 | 0.013 | 0.039 | 0.073 | 0.014 | 0.081 | 0.047 | 0.062 | 0.185 |

MCRAT | RAPS | SAWmax | ARAS | COPRAS | MOORA (RS) | CP (p = 1) | GRA (MinMax) | MOORA (RP) | CP (p = 2) | GRA (T) | VIKOR | TOPSIS | SAW (MinMax) | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

MCRAT | 1.00 | 0.80 | 0.60 | 0.60 | 0.60 | 1.00 | 0.90 | 0.60 | 0.70 | −0.90 | 0.90 | 0.60 | 1.00 | |

RAPS | 0.80 | 0.60 | 0.60 | 0.60 | 1.00 | 0.90 | 0.60 | 0.70 | −0.90 | 0.90 | 0.60 | 1.00 | ||

SAWmax | 0.90 | 0.90 | 0.90 | 0.80 | 0.60 | 0.00 | 0.20 | −0.90 | 0.60 | 0.60 | 0.80 | |||

ARAS | 1.00 | 1.00 | 0.60 | 0.30 | −0.20 | −0.10 | −0.70 | 0.30 | 0.70 | 0.60 | ||||

COPRAS | 1.00 | 0.60 | 0.30 | −0.20 | −0.10 | −0.70 | 0.30 | 0.70 | 0.60 | |||||

MOORA (RS) | 0.60 | 0.30 | −0.20 | −0.10 | −0.70 | 0.30 | 0.70 | 0.60 | ||||||

CP (p = 1) | 0.90 | 0.60 | 0.70 | −0.90 | 0.90 | 0.60 | 1.00 | |||||||

GRA (MinMax) | 0.70 | 0.90 | −0.70 | 1.00 | 0.20 | 0.90 | ||||||||

MOORA (RP) | 0.90 | −0.30 | 0.70 | 0.20 | 0.60 | |||||||||

CP (p = 2) | −0.40 | 0.90 | 0.00 | 0.70 | ||||||||||

GRA (T) | −0.70 | −0.70 | −0.90 | |||||||||||

VIKOR | 0.20 | 0.90 | ||||||||||||

TOPSIS | 0.60 | |||||||||||||

SAW (MinMax) |

Alternative/Criterion | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} |
---|---|---|---|---|---|

Min | Max | Min | Min | Min | |

A1 | 0.40 | 0.00235 | 347 | 101.5 | 1500 |

A2 | 0.41 | 0.00121 | 348 | 105 | 1500 |

A3 | 0.29 | 0.00235 | 364 | 97.5 | 1500 |

A4 | 0.30 | 0.00121 | 385 | 100 | 1500 |

A5 | 0.43 | 0.00235 | 335 | 125 | 1600 |

A6 | 0.28 | 0.00235 | 365 | 130 | 1600 |

A7 | 0.45 | 0.00121 | 310 | 140 | 1700 |

A8 | 0.29 | 0.00235 | 320 | 135 | 1600 |

A9 | 0.40 | 0.00062 | 315 | 175 | 1700 |

A10 | 0.42 | 0.00062 | 325 | 180 | 1700 |

A11 | 0.28 | 0.00121 | 280 | 165 | 1700 |

A12 | 0.29 | 0.00121 | 220 | 170 | 1700 |

Alternative/Criterion | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} |
---|---|---|---|---|---|

Min | Max | Min | Min | Min | |

A1 | 0.7000 | 1.0000 | 0.6340 | 0.9606 | 1.0000 |

A2 | 0.6829 | 0.5133 | 0.6322 | 0.9286 | 1.0000 |

A3 | 0.9655 | 1.0000 | 0.6044 | 1.0000 | 1.0000 |

A4 | 0.9333 | 0.5133 | 0.5714 | 0.9750 | 1.0000 |

A5 | 0.6512 | 1.0000 | 0.6567 | 0.7800 | 0.9375 |

A6 | 1.0000 | 1.0000 | 0.6027 | 0.7500 | 0.9375 |

A7 | 0.6222 | 0.5133 | 0.7097 | 0.6964 | 0.8824 |

A8 | 0.9655 | 1.0000 | 0.6875 | 0.7222 | 0.9375 |

A9 | 0.7000 | 0.2630 | 0.6984 | 0.5571 | 0.8824 |

A10 | 0.6667 | 0.2630 | 0.6769 | 0.5417 | 0.8824 |

A11 | 1.0000 | 0.5133 | 0.7857 | 0.5909 | 0.8824 |

A12 | 0.9655 | 0.5133 | 1.0000 | 0.5735 | 0.8824 |

Weight/Criterion | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} |
---|---|---|---|---|---|

w1 | w2 | w3 | w4 | w5 | |

W | 0.1197 | 0.6482 | 0.0597 | 0.1627 | 0.0098 |

Alternative/Criterion | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} |
---|---|---|---|---|---|

Min | Max | Min | Min | Min | |

A1 | 0.0838 | 0.6482 | 0.0378 | 0.1562 | 0.0098 |

A2 | 0.0817 | 0.3327 | 0.0377 | 0.1510 | 0.0098 |

A3 | 0.1155 | 0.6482 | 0.0361 | 0.1627 | 0.0098 |

A4 | 0.1117 | 0.3327 | 0.0341 | 0.1586 | 0.0098 |

A5 | 0.0779 | 0.6482 | 0.0392 | 0.1269 | 0.0092 |

A6 | 0.1197 | 0.6482 | 0.0360 | 0.1220 | 0.0092 |

A7 | 0.0745 | 0.3327 | 0.0423 | 0.1133 | 0.0086 |

A8 | 0.1155 | 0.6482 | 0.0410 | 0.1175 | 0.0092 |

A9 | 0.0838 | 0.1705 | 0.0417 | 0.0906 | 0.0086 |

A10 | 0.0798 | 0.1705 | 0.0404 | 0.0881 | 0.0086 |

A11 | 0.1197 | 0.3327 | 0.0469 | 0.0961 | 0.0086 |

A12 | 0.1155 | 0.3327 | 0.0597 | 0.0933 | 0.0086 |

Optimal Alternative /Criterion | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} |
---|---|---|---|---|---|

Min | Max | Min | Min | Min | |

q_{1} | q_{2} | q_{3} | q_{4} | q_{5} | |

Q | 0.1197 | 0.6482 | 0.0597 | 0.1627 | 0.0098 |

Optimal Alternative/Criterion | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} |
---|---|---|---|---|---|

min | max | min | min | min | |

q_{1} | q_{2} | q_{3} | q_{4} | q_{5} | |

Q^{max} | - | 0.6482 | - | - | - |

Q^{min} | 0.1197 | - | 0.0597 | 0.1627 | 0.0098 |

Alternative/Criterion | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} |
---|---|---|---|---|---|

Min | Max | Min | Min | Min | |

u_{1} | u_{2} | u_{3} | u_{4} | u_{5} | |

A1 U^{max} | - | 0.6482 | - | - | - |

A1 U^{min} | 0.0838 | 0.0378 | 0.1562 | 0.0098 | |

A2 U^{max} | - | 0.3327 | - | - | - |

A2 U^{min} | 0.0817 | 0.0377 | 0.1510 | 0.0098 | |

A3 U^{max} | - | 0.6482 | - | - | - |

A3 U^{min} | 0.1155 | 0.0361 | 0.1627 | 0.0098 | |

A4 U^{max} | - | 0.3327 | - | - | - |

A4 U^{min} | 0.1117 | 0.0341 | 0.1586 | 0.0098 | |

A5 U^{max} | - | 0.6482 | - | - | - |

A5 U^{min} | 0.0779 | 0.0392 | 0.1269 | 0.0092 | |

A6 U^{max} | - | 0.6482 | - | - | - |

A6 U^{min} | 0.1197 | 0.0360 | 0.1220 | 0.0092 | |

A7 U^{max} | - | 0.3327 | - | - | - |

A7 U^{min} | 0.0745 | 0.0423 | 0.1133 | 0.0086 | |

A8 U^{max} | - | 0.6482 | - | - | - |

A8 U^{min} | 0.1155 | 0.0410 | 0.1175 | 0.0092 | |

A9 U^{max} | - | 0.1705 | - | - | - |

A9 U^{min} | 0.0838 | 0.0417 | 0.0906 | 0.0086 | |

A10 U^{max} | - | 0.1705 | - | - | - |

A10 U^{min} | 0.0798 | 0.0404 | 0.0881 | 0.0086 | |

A11 U^{max} | - | 0.3327 | - | - | - |

A11 U^{min} | 0.1197 | 0.0469 | 0.0961 | 0.0086 | |

A12 U^{max} | - | 0.3327 | - | - | - |

A12 U^{min} | 0.1155 | 0.0597 | 0.0933 | 0.0086 |

Alternative | Max | Min |
---|---|---|

Q_{k}U _{ik} | Q_{h}U _{ih} | |

Q | 0.6482 | 0.2108 |

A1 | 0.6482 | 0.1815 |

A2 | 0.3327 | 0.1761 |

A3 | 0.6482 | 0.2030 |

A4 | 0.3327 | 0.1972 |

A5 | 0.6482 | 0.1542 |

A6 | 0.6482 | 0.1749 |

A7 | 0.3327 | 0.1423 |

A8 | 0.6482 | 0.1701 |

A9 | 0.1705 | 0.1305 |

A10 | 0.1705 | 0.1258 |

A11 | 0.3327 | 0.1607 |

A12 | 0.3327 | 0.1603 |

Alternative | Trace | Value |
---|---|---|

A1 | tr(T_{1}) | 0.45847 |

A2 | tr(T_{2}) | 0.25280 |

A3 | tr(T_{3}) | 0.46299 |

A4 | tr(T_{4}) | 0.25725 |

A5 | tr(T_{5}) | 0.45271 |

A6 | tr(T_{6}) | 0.45707 |

A7 | tr(T_{7}) | 0.24568 |

A8 | tr(T_{8}) | 0.45605 |

A9 | tr(T_{9}) | 0.13803 |

A10 | tr(T_{10}) | 0.13704 |

A11 | tr(T_{11}) | 0.24956 |

A12 | tr(T_{12}) | 0.24947 |

Alternative | Rank |
---|---|

A1 | 2 |

A2 | 7 |

A3 | 1 |

A4 | 6 |

A5 | 5 |

A6 | 3 |

A7 | 10 |

A8 | 4 |

A9 | 11 |

A10 | 12 |

A11 | 8 |

A12 | 9 |

Max | Min | Perimeter | Perimeter Similarity | |
---|---|---|---|---|

Q_{k}U _{ik} | Q_{h}U _{ih} | $\mathit{P}={\mathit{Q}}_{\mathit{k}}+{\mathit{Q}}_{\mathit{h}}+\sqrt{{\mathit{Q}}_{\mathit{k}}^{2}+{\mathit{Q}}_{\mathit{k}}^{2}}$ ${\mathit{P}}_{\mathit{i}}={\mathit{U}}_{\mathit{i}\mathit{k}}+{\mathit{U}}_{\mathit{i}\mathit{h}}+\sqrt{{\mathit{U}}_{\mathit{i}\mathit{k}}^{2}+{\mathit{U}}_{\mathit{i}\mathit{h}}^{2}}$ | $\mathit{P}{\mathit{S}}_{\mathit{i}}=\frac{{\mathit{P}}_{\mathit{i}}}{\mathit{P}},\forall \mathit{i}\in \left[1,2,\dots ,\mathit{m}\right]$ | |

Q | 0.6482 | 0.2108 | 1.54067 | |

A1 | 0.6482 | 0.1815 | 1.50294 | 0.9755 |

A2 | 0.3327 | 0.1761 | 0.88528 | 0.5746 |

A3 | 0.6482 | 0.2030 | 1.53049 | 0.9934 |

A4 | 0.3327 | 0.1972 | 0.91669 | 0.5950 |

A5 | 0.6482 | 0.1542 | 1.46879 | 0.9533 |

A6 | 0.6482 | 0.1749 | 1.49451 | 0.9700 |

A7 | 0.3327 | 0.1423 | 0.83688 | 0.5432 |

A8 | 0.6482 | 0.1701 | 1.48845 | 0.9661 |

A9 | 0.1705 | 0.1305 | 0.51574 | 0.3348 |

A10 | 0.1705 | 0.1258 | 0.50820 | 0.3299 |

A11 | 0.3327 | 0.1607 | 0.86296 | 0.5601 |

A12 | 0.3327 | 0.1603 | 0.86232 | 0.5597 |

Alternative | Rank |
---|---|

A1 | 2 |

A2 | 7 |

A3 | 1 |

A4 | 6 |

A5 | 5 |

A6 | 3 |

A7 | 10 |

A8 | 4 |

A9 | 11 |

A10 | 12 |

A11 | 8 |

A12 | 9 |

Criterion | MCRAT | RAPS |
---|---|---|

Simplicity | High | High |

Comprehensive structure: breadth and depth | Medium | Medium |

Comprehensive structure consisting of merit substructures | High | High |

Logical procedure | High | High |

Justification | High | High |

Measurement scale | High | High |

Synthesis of judgements | Medium | Medium |

Ranking of tangibles | High | High |

Generalization of ranking | High | High |

Rank preservation | High | High |

Sensitivity analysis | Medium | Medium |

Validation of decision problems | High | High |

Generalizability to dependence | High | High |

Applicability to conflict resolution | Medium | Medium |

Trustworthiness and validity of the approach | Medium | Medium |

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## Share and Cite

**MDPI and ACS Style**

Urošević, K.; Gligorić, Z.; Miljanović, I.; Beljić, Č.; Gligorić, M.
Novel Methods in Multiple Criteria Decision-Making Process (MCRAT and RAPS)—Application in the Mining Industry. *Mathematics* **2021**, *9*, 1980.
https://doi.org/10.3390/math9161980

**AMA Style**

Urošević K, Gligorić Z, Miljanović I, Beljić Č, Gligorić M.
Novel Methods in Multiple Criteria Decision-Making Process (MCRAT and RAPS)—Application in the Mining Industry. *Mathematics*. 2021; 9(16):1980.
https://doi.org/10.3390/math9161980

**Chicago/Turabian Style**

Urošević, Katarina, Zoran Gligorić, Igor Miljanović, Čedomir Beljić, and Miloš Gligorić.
2021. "Novel Methods in Multiple Criteria Decision-Making Process (MCRAT and RAPS)—Application in the Mining Industry" *Mathematics* 9, no. 16: 1980.
https://doi.org/10.3390/math9161980