Python 3 library for solving multi-criteria decision-making (MCDM) problems.
You can download and install pymcdm library using pip:
pip install pymcdmThe library contains:
- MCDA methods:
| Acronym | Method Name | Reference |
|---|---|---|
| TOPSIS | Technique for the Order of Prioritisation by Similarity to Ideal Solution | [1] |
| VIKOR | VIseKriterijumska Optimizacija I Kompromisno Resenje | [2] |
| COPRAS | COmplex PRoportional ASsessment | [3] |
| PROMETHEE I & II | Preference Ranking Organization METHod for Enrichment of Evaluations I & II | [4] |
| COMET | Characteristic Objects Method | [5] |
| SPOTIS | Stable Preference Ordering Towards Ideal Solution | [6] |
| ARAS | Additive Ratio ASsessment | [7,8] |
| COCOSO | COmbined COmpromise SOlution | [9] |
| CODAS | COmbinative Distance-based ASsessment | [10] |
| EDAS | Evaluation based on Distance from Average Solution | [11,12] |
| MABAC | Multi-Attributive Border Approximation area Comparison | [13] |
| MAIRCA | MultiAttributive Ideal-Real Comparative Analysis | [14,15,16] |
| MARCOS | Measurement Alternatives and Ranking according to COmpromise Solution | [17,18] |
| OCRA | Operational Competitiveness Ratings | [19,20] |
| MOORA | Multi-Objective Optimization Method by Ratio Analysis | [21,22] |
- Weighting methods:
| Acronym | Method Name | Reference |
|---|---|---|
| - | Equal/Mean weights | [23] |
| - | Entropy weights | [23,24,25] |
| STD | Standard Deviation weights | [23,26] |
| MEREC | MEthod based on the Removal Effects of Criteria | [27] |
| CRITIC | CRiteria Importance Through Intercriteria Correlation | [28, 29] |
| CILOS | Criterion Impact LOS | [30] |
| IDOCRIW | Integrated Determination of Objective CRIteria Weight | [30] |
| - | Angular/Angle weights | [31] |
| - | Gini Coeficient weights | [32] |
| - | Statistical variance weights | [33] |
- Normalization methods:
| Method Name | Reference |
|---|---|
| Weitendorf’s Linear Normalization | [34] |
| Maximum - Linear Normalization | [35] |
| Sum-Based Linear Normalization | [36] |
| Vector Normalization | [36,37] |
| Logarithmic Normalization | [36, 37] |
| Linear Normalization (Max-Min) | [34,38] |
| Non-linear Normalization (Max-Min) | [39] |
| Enhanced Accuracy Normalization | [40] |
- Correlation coefficients:
| Coefficient name | Reference |
|---|---|
| Spearman's rank correlation coefficient | [41,41] |
| Pearson correlation coefficient | [43] |
| Weighted Spearman’s rank correlation coefficient | [44] |
| Rank Similarity Coefficient | [45] |
| Kendall rank correlation coefficient | [46] |
| Goodman and Kruskal's gamma | [47] |
- Helpers
- rankdata
- rrankdata
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[3] Zavadskas, E. K., Kaklauskas, A., Peldschus, F., & Turskis, Z. (2007). Multi-attribute assessment of road design solutions by using the COPRAS method. The Baltic Journal of Road and Bridge Engineering, 2(4), 195-203.
[4] Brans, J. P., Vincke, P., & Mareschal, B. (1986). How to select and how to rank projects: The PROMETHEE method. European journal of operational research, 24(2), 228-238.
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[7] Zavadskas, E. K., & Turskis, Z. (2010). A new additive ratio assessment (ARAS) method in multicriteria decision‐making. Technological and economic development of economy, 16(2), 159-172.
[8] Stanujkic, D., Djordjevic, B., & Karabasevic, D. (2015). Selection of candidates in the process of recruitment and selection of personnel based on the SWARA and ARAS methods. Quaestus, (7), 53.
[9] Yazdani, M., Zarate, P., Zavadskas, E. K., & Turskis, Z. (2019). A Combined Compromise Solution (CoCoSo) method for multi-criteria decision-making problems. Management Decision.
[10] Badi, I., Shetwan, A. G., & Abdulshahed, A. M. (2017, September). Supplier selection using COmbinative Distance-based ASsessment (CODAS) method for multi-criteria decision-making. In Proceedings of The 1st International Conference on Management, Engineering and Environment (ICMNEE) (pp. 395-407).
[11] Keshavarz Ghorabaee, M., Zavadskas, E. K., Olfat, L., & Turskis, Z. (2015). Multi-criteria inventory classification using a new method of evaluation based on distance from average solution (EDAS). Informatica, 26(3), 435-451.
[12] Yazdani, M., Torkayesh, A. E., Santibanez-Gonzalez, E. D., & Otaghsara, S. K. (2020). Evaluation of renewable energy resources using integrated Shannon Entropy—EDAS model. Sustainable Operations and Computers, 1, 35-42.
[13] Pamučar, D., & Ćirović, G. (2015). The selection of transport and handling resources in logistics centers using Multi-Attributive Border Approximation area Comparison (MABAC). Expert systems with applications, 42(6), 3016-3028.
[14] Gigović, L., Pamučar, D., Bajić, Z., & Milićević, M. (2016). The combination of expert judgment and GIS-MAIRCA analysis for the selection of sites for ammunition depots. Sustainability, 8(4), 372.
[15] Pamucar, D. S., Pejcic Tarle, S., & Parezanovic, T. (2018). New hybrid multi-criteria decision-making DEMATELMAIRCA model: sustainable selection of a location for the development of multimodal logistics centre. Economic research-Ekonomska istraživanja, 31(1), 1641-1665.
[16] Aksoy, E. (2021). An Analysis on Turkey's Merger and Acquisition Activities: MAIRCA Method. Gümüşhane Üniversitesi Sosyal Bilimler Enstitüsü Elektronik Dergisi, 12(1), 1-11.
[17] Stević, Ž., Pamučar, D., Puška, A., & Chatterjee, P. (2020). Sustainable supplier selection in healthcare industries using a new MCDM method: Measurement of alternatives and ranking according to COmpromise solution (MARCOS). Computers & Industrial Engineering, 140, 106231.
[18] Ulutaş, A., Karabasevic, D., Popovic, G., Stanujkic, D., Nguyen, P. T., & Karaköy, Ç. (2020). Development of a novel integrated CCSD-ITARA-MARCOS decision-making approach for stackers selection in a logistics system. Mathematics, 8(10), 1672.
[19] Parkan, C. (1994). Operational competitiveness ratings of production units. Managerial and Decision Economics, 15(3), 201-221.
[20] Işık, A. T., & Adalı, E. A. (2016). A new integrated decision making approach based on SWARA and OCRA methods for the hotel selection problem. International Journal of Advanced Operations Management, 8(2), 140-151.
[21] Brauers, W. K. (2003). Optimization methods for a stakeholder society: a revolution in economic thinking by multi-objective optimization (Vol. 73). Springer Science & Business Media.
[22] Hussain, S. A. I., & Mandal, U. K. (2016). Entropy based MCDM approach for Selection of material. In National Level Conference on Engineering Problems and Application of Mathematics (pp. 1-6).
[23] Sałabun, W., Wątróbski, J., & Shekhovtsov, A. (2020). Are mcda methods benchmarkable? a comparative study of topsis, vikor, copras, and promethee ii methods. Symmetry, 12(9), 1549.
[24] Lotfi, F. H., & Fallahnejad, R. (2010). Imprecise Shannon’s entropy and multi attribute decision making. Entropy, 12(1), 53-62.
[25] Li, X., Wang, K., Liu, L., Xin, J., Yang, H., & Gao, C. (2011). Application of the entropy weight and TOPSIS method in safety evaluation of coal mines. Procedia engineering, 26, 2085-2091.
[26] Wang, Y. M., & Luo, Y. (2010). Integration of correlations with standard deviations for determining attribute weights in multiple attribute decision making. Mathematical and Computer Modelling, 51(1-2), 1-12.
[27] Keshavarz-Ghorabaee, M., Amiri, M., Zavadskas, E. K., Turskis, Z., & Antucheviciene, J. (2021). Determination of Objective Weights Using a New Method Based on the Removal Effects of Criteria (MEREC). Symmetry, 13(4), 525.
[28] Diakoulaki, D., Mavrotas, G., & Papayannakis, L. (1995). Determining objective weights in multiple criteria problems: The critic method. Computers & Operations Research, 22(7), 763-770.
[29] Tuş, A., & Adalı, E. A. (2019). The new combination with CRITIC and WASPAS methods for the time and attendance software selection problem. Opsearch, 56(2), 528-538.
[30] Zavadskas, E. K., & Podvezko, V. (2016). Integrated determination of objective criteria weights in MCDM. International Journal of Information Technology & Decision Making, 15(02), 267-283.
[31] Shuai, D., Zongzhun, Z., Yongji, W., & Lei, L. (2012, May). A new angular method to determine the objective weights. In 2012 24th Chinese Control and Decision Conference (CCDC) (pp. 3889-3892). IEEE.
[32] Li, G., & Chi, G. (2009, December). A new determining objective weights method-gini coefficient weight. In 2009 First International Conference on Information Science and Engineering (pp. 3726-3729). IEEE.
[33] Rao, R. V., & Patel, B. K. (2010). A subjective and objective integrated multiple attribute decision making method for material selection. Materials & Design, 31(10), 4738-4747.
[34] Brauers, W. K., & Zavadskas, E. K. (2006). The MOORA method and its application to privatization in a transition economy. Control and cybernetics, 35, 445-469.
[35] Jahan, A., & Edwards, K. L. (2015). A state-of-the-art survey on the influence of normalization techniques in ranking: Improving the materials selection process in engineering design. Materials & Design (1980-2015), 65, 335-342.
[36] Gardziejczyk, W., & Zabicki, P. (2017). Normalization and variant assessment methods in selection of road alignment variants–case study. Journal of civil engineering and management, 23(4), 510-523.
[37] Zavadskas, E. K., & Turskis, Z. (2008). A new logarithmic normalization method in games theory. Informatica, 19(2), 303-314.
[38] Jahan, A., & Edwards, K. L. (2015). A state-of-the-art survey on the influence of normalization techniques in ranking: Improving the materials selection process in engineering design. Materials & Design (1980-2015), 65, 335-342.
[39] Peldschus, F., Vaigauskas, E., & Zavadskas, E. K. (1983). Technologische entscheidungen bei der berücksichtigung mehrerer Ziehle. Bauplanung Bautechnik, 37(4), 173-175.
[40] Zeng, Q. L., Li, D. D., & Yang, Y. B. (2013). VIKOR method with enhanced accuracy for multiple criteria decision making in healthcare management. Journal of medical systems, 37(2), 1-9.
[41] Binet, A., & Henri, V. (1898). La fatigue intellectuelle (Vol. 1). Schleicher frères.
[42] Spearman, C. (1910). Correlation calculated from faulty data. British Journal of Psychology, 1904‐1920, 3(3), 271-295.
[43] Pearson, K. (1895). VII. Note on regression and inheritance in the case of two parents. proceedings of the royal society of London, 58(347-352), 240-242.
[44] Dancelli, L., Manisera, M., & Vezzoli, M. (2013). On two classes of Weighted Rank Correlation measures deriving from the Spearman’s ρ. In Statistical Models for Data Analysis (pp. 107-114). Springer, Heidelberg.
[45] Sałabun, W., & Urbaniak, K. (2020, June). A new coefficient of rankings similarity in decision-making problems. In International Conference on Computational Science (pp. 632-645). Springer, Cham.
[46] Kendall, M. G. (1938). A new measure of rank correlation. Biometrika, 30(1/2), 81-93.
[47] Goodman, L. A., & Kruskal, W. H. (1979). Measures of association for cross classifications. Measures of association for cross classifications, 2-34.
Here's a small example of how use this library to solve MCDM problem. For more examples with explanation see examples.
import numpy as np
from pymcdm.methods import TOPSIS
from pymcdm.helpers import rrankdata
# Define decision matrix (2 criteria, 4 alternative)
alts = np.array([
[4, 4],
[1, 5],
[3, 2],
[4, 2]
], dtype='float')
# Define weights and types
weights = np.array([0.5, 0.5])
types = np.array([1, -1])
# Create object of the method
topsis = TOPSIS()
# Determine preferences and ranking for alternatives
pref = topsis(alts, weights, types)
ranking = rrankdata(pref)
for r, p in zip(ranking, pref):
print(r, p)And the output of this example (numbers are rounded):
3 0.6126
4 0.0
2 0.7829
1 1.0