Document Type : Research Paper


Department of Mathematics, Semnan Branch, Islamic Azad University, Semnan, Iran.


Linear Assignment (LAM) is one of the Multi-Attribute Decision Making (MADM) methods that uses integer programming models in the solution process. In this method, only the final priority of the alternatives is determined and the distance between the alternatives is not clear. The purpose of this paper is to modify this method so that instead of the final priority of the alternatives, the final weight of the alternatives is presented. This is done using a linear programming model of Data Envelopment Analysis (DEA). In this paper, we propose a hybrid MADM-DEA method called Linear Assignment Voting (VLAM). The new method is explained with a numerical example. The method will then be implemented on a problem in the real world to demonstrate the application of the method. In this case study, VLAM demonstrates the prioritization of models proposed by experts for the purchase of excavators in a road construction company. Also, based on the results of this method, the weight of the first, second and third priorities are 0.39, 0.35 and 0.26, respectively. These results increase the decision maker's power in making the final decision and choice.


Main Subjects

  1. Aguarón, J., Escobar, M. T., & Moreno-Jiménez, J. M. (2021). Reducing inconsistency measured by the geometric consistency index in the analytic hierarchy process. European journal of operational research288(2), 576-583.
  2. Alinezhad, A., & Khalili, J. (2019). New methods and applications in multiple attribute decision making (MADM)(Vol. 277). Cham: Springer.
  3. Amiri, M., Zandieh, M., Soltani, R., & Vahdani, B. (2009). A hybrid multi-criteria decision-making model for firms competence evaluation. Expert systems with applications36(10), 12314-12322.
  4. Andersen, P., & Petersen, N. C. (1993). A procedure for ranking efficient units in data envelopment analysis. Management science39(10), 1261-1264.
  5. Babaee Tirkolaee, E., Mahdavi, I., Seyyed Esfahani, M. M., & Weber, G. W. (2020). A hybrid augmented ant colony optimization for the multi-trip capacitated arc routing problem under fuzzy demands for urban solid waste management. Waste management & research38(2), 156-172.
  6. Babaee Tirkolaee, E., Mardani, A., Dashtian, Z., Soltani, M., & Weber, G. W. (2020). A novel hybrid method using fuzzy decision making and multi-objective programming for sustainable-reliable supplier selection in two-echelon supply chain design. Journal of cleaner production250, 119517.
  7. Babaee Tirkolaee, E. B., Hosseinabadi, A. A. R., Soltani, M., Sangaiah, A. K., & Wang, J. (2018). A hybrid genetic algorithm for multi-trip green capacitated arc routing problem in the scope of urban services. Sustainability10(5), 1366.
  8. Bashiri, M., & Badri, H. (2011). A group decision making procedure for fuzzy interactive linear assignment programming. Expert systems with applications38(5), 5561-5568.
  9. Bashiri, M., Badri, H., & Hejazi, T. H. (2011). Selecting optimum maintenance strategy by fuzzy interactive linear assignment method. Applied mathematical modelling35(1), 152-164.
  10. Baykasoğlu, A., Subulan, K., & Karaslan, F. S. (2016). A new fuzzy linear assignment method for multi-attribute decision making with an application to spare parts inventory classification. Applied soft computing42, 1-17.
  11. Bernardo, J. J., & Blin, J. M. (1977). A programming model of consumer choice among multi-attributed brands. Journal of consumer research4(2), 111-118.
  12. Chen, T. Y. (2014). The extended linear assignment method for multiple criteria decision analysis based on interval-valued intuitionistic fuzzy sets. Applied mathematical modelling38(7-8), 2101-2117.
  13. Chen, T. Y. (2013). A linear assignment method for multiple-criteria decision analysis with interval type-2 fuzzy sets. Applied soft computing13(5), 2735-2748.
  14. Cook, W. D., & Kress, M. (1990). A data envelopment model for aggregating preference rankings. Management science36(11), 1302-1310.
  15. Danchick, R. (2005). A ranked linear assignment approach to Bayesian classification. Applied mathematics and computation162(1), 265-281.
  16. Academy of Sciences. (1781). History of the royal academy of sciences with the memoirs of mathematics and physics drawn from the registers of this academy. (In French). Retrieved from
  17. Dessouky, M. M., & Kijowski, B. A. (1997). Production scheduling of single-stage multi-product batch chemical processes with fixed batch sizes. IIE transactions29(5), 399-408.
  18. Doumpos, M., & Zopounidis, C. (2002). Multicriteria decision aid classification methods(Vol. 73). Springer Science & Business Media.
  19. Doumpos, M., & Zopounidis, C. (2011). Preference disaggregation and statistical learning for multicriteria decision support: a review. European journal of operational research209(3), 203-214.
  20. Janani, M. H., Ehsanifar, M., & Bakhtiarnezhad, S. (2012). Selection of portfolio by using multi attributed decision making (Tehran stock exchange). American journal of scientific research44(2), 15-29.
  21. Ehsanifar, M., Bakhtiarnezhad, S., Anvari, F., Anvari, N., Farahani, H. R., & Mohajerfar, M. (2012).
    Linear assignment and its application in financial management and portfolio. Archives Des Sciences,
    65(7), 333-58.
  22. Emami Saleh, A., Yavari, A., Anbari, Y., & Derakhshani, H. (2014). Local parking positioning by
    using the linear assignment method (case study: Qazvin, Iran). International journal of architecture
    and urban development
    , 4(4), 39-44.
  23. Foroughi, A., & Esfahani, M. (2002). An Empirical study for ranking risk factors using linear Assignment: a case study of road construction. Management science letters2(2), 615-622.
  24. Foroughi, A. A., Jones, D. F., & Tamiz, M. (2005). A selection method for a preferential election. Applied mathematics and computation163(1), 107-116.
  25. Foroughi, A. A., & Tamiz, M. (2005). An effective total ranking model for a ranked voting system. Omega33(6), 491-496.
  26. Gal, T., Stewart, T., & Hanne, T. (1999). Multi criteria decision making advances in MCDM models: algorithms, theory, and applications. Springer.
  27. Galanis, V. I., Ikonomakis, E. K., Meletiou, G. C., & Vrahatis, M. N. (2010). An e-voting-based data gathering scheme for decision support systems. International journal of decision sciences, risk and management2(1-2), 36-45.
  28. Green, R. H., Doyle, J. R., & Cook, W. D. (1996). Preference voting and project ranking using DEA and cross-evaluation. European journal of operational research90(3), 461-472.
  29. Razavi Hajiagha, S. H., Shahbazi, M., Amoozad Mahdiraji, H., & Panahian, H. (2018). A bi-objective score-variance based linear assignment method for group decision making with hesitant fuzzy linguistic term sets. Technological and economic development of economy, 24(3), 1125-1148.
  30. Hashimoto, A. (1997). A ranked voting system using a DEA/AR exclusion model: a note. European journal of operational research97(3), 600-604.
  31. Hwang, C. L., & Yoon, K. (1987). Multiple attribute decision making. Springer, Berlin, Heidelberg.
  32. Jahan, A., Ismail, M. Y., Mustapha, F., & Sapuan, S. M. (2010). Material selection based on ordinal data. Materials & design31(7), 3180-3187.
  33. Kahraman, C. (Ed.). (2008). Fuzzy multi-criteria decision making: theory and applications with recent developments(Vol. 16). Springer Science & Business Media.
  34. Keramati, M. A., Ehsanifar, M., & Rezaei, Z. (2016). Modification of classic linear assignment method including the impact of the distance between the performances of alternatives for alternatives evaluation and ranking. Advances in industrial engineering50(1), 69-81.
  35. Köksalan, M., & Zionts, S. (2001). Multiple criteria decision making in the new millennium. Springer, Berlin, Heidelberg.
  36. Komijan, A. R., & Koupaei, M. N. (2012). A multi_attribute decision_making and mathematical model for university examination timetabling. Journal of basic and applied scientific research2(10), 10258-10262.
  37. Kou, G., Ergu, D., Lin, C., & Chen, Y. (2016). Pairwise comparison matrix in multiple criteria decision making. Technological and economic development of economy22(5), 738-765.
  38. LeBlanc, L. J., & Farhangian, K. (1981). Efficient algorithms for solving elastic demand traffic assignment problems and mode split-assignment problems. Transportation science15(4), 306-317.
  39. Llamazares, B., & Pena, T. (2009). Preference aggregation and DEA: An analysis of the methods proposed to discriminate efficient candidates. European journal of operational research197(2), 714-721.
  40. Liang, D., Darko, A. P., Xu, Z., & Zhang, Y. (2020). Partitioned fuzzy measure-based linear assignment method for Pythagorean fuzzy multi-criteria decision-making with a new likelihood. Journal of the operational research society71(5), 831-845.
  41. Lin, C., Kou, G., Peng, Y., & Alsaadi, F. E. (2020). Aggregation of the nearest consistency matrices with the acceptable consensus in AHP-GDM. Annals of operations research.
  42. Ma, C. Y., Buontempo, F. V., & Wang, X. Z. (2011). Inductive data mining: automatic generation of decision trees from data for QSAR modelling and process historical data analysis. International journal of modelling, identification and control12(1-2), 101-106.
  43. Macharis, C., Springael, J., De Brucker, K., & Verbeke, A. (2004). PROMETHEE and AHP: The design of operational synergies in multicriteria analysis: strengthening PROMETHEE with ideas of AHP. European journal of operational research153(2), 307-317.
  44. Mianabadi, H., & Afshar, A. (2008). Multi-attribute decision-marking to rank urban water supply schemes. Journal of water and wastewater66, 34-45.
  45. Mohammed, A. (2020). Towards a sustainable assessment of suppliers: an integrated fuzzy TOPSIS-possibilistic multi-objective approach. Annals of operations research293(2), 639–668.
  46. Mostafaeipour, A., Goli, A., Rezaei, M., Qolipour, M., Arabnia, H. R., Goudarzi, H., & Behnam, E. (2021). Performance of different hybrid algorithms for prediction of wind speed behavior. Wind engineering45(2), 245-256.
  47. Cao, N. V., Fragniere, E., Gauthier, J. A., Sapin, M., & Widmer, E. D. (2010). Optimizing the marriage market: An application of the linear assignment model. European journal of operational research202(2), 547-553.
  48. Noguchi, H., Ogawa, M., & Ishii, H. (2002). The appropriate total ranking method using DEA for multiple categorized purposes. Journal of computational and applied mathematics146(1), 155-166.
  49. Nourozi, S. A., & Shariati, A. R. (2013). Study of locating fire stations using linear assignment method: case study Maku City. Global journal pf human social science interdisciplinary, 13(3), 28-35.
  50. Obata, T., & Ishii, H. (2003). A method for discriminating efficient candidates with ranked voting data. European journal of operational research151(1), 233-237.
  51. Ortiz-Barrios, M., Cabarcas-Reyes, J., Ishizaka, A., Barbati, M., Jaramillo-Rueda, N., & de Jesús Carrascal-Zambrano, G. (2021). A hybrid fuzzy multi-criteria decision making model for selecting a sustainable supplier of forklift filters: a case study from the mining industry. Annals of operations research307(1), 443-481.
  52. J. C & Romero, S. B. (2000). Multi-criterion decision in management: principles and practices. Springer, Boston, MA.
  53. Post, G. V. (2010). Using re-voting to reduce the threat of coercion in elections. Electronic government, an international journal7(2), 168-182.
  54. Danila, N. (1986). Roy B.: Multi-criteria decision support methodology. (In French). Retrieved from
  55. Sadeghravesh, M. H., Khosravi, H., & Ghasemian, S. (2015). Application of fuzzy analytical hierarchy process for assessment of combating-desertification alternatives in central Iran. Natural hazards75(1), 653-667.
  56. Sharafi, H., Lotfi, F. H., Jahanshahloo, G., Rostamy-malkhalifeh, M., Soltanifar, M., & Razipour-GhalehJough, S. (2019). Ranking of petrochemical companies using preferential voting at unequal levels of voting power through data envelopment analysis. Mathematical sciences13(3), 287-297.
  57. Shirouyehzad, H., Tavakoli, M. M., & Badakhshian, M. (2016). The linear assignment method for ranking of organizations with service quality approach: a case study of hotels in city of Isfahan. Journal of applied research on industrial engineering3(14), 49-57.
  58. Soltanifar, M. (2011). Introducing an interval efficiency for each candidate in ranked voting data using data envelopment analysis. International journal of society systems science3(4), 346-361.
  59. Soltanifar, M. (2011). Ranking of different common set of weights models using a voting model and its application in determining efficient DMUs. International journal of advanced operations management3(3-4), 290-308.
  60. Soltanifar, M. (2020). A‎ n‎ew voting model for groups with members of unequal power and proficiency‎. International journal of industrial mathematics12(2), 121-134.
  61. Soltanifar, M., Ebrahimnejad, A., & Farrokhi, M. M. (2010). Ranking of different ranking models using a voting model and its application in determining efficient candidates. International journal of society systems science2(4), 375-389.
  62. Soltanifar, M., & Hosseinzadeh Lotfi, F. (2011). The voting analytic hierarchy process method for discriminating among efficient decision making units in data envelopment analysis. Computers & industrial engineering60(4), 585-592.
  63. Tzeng, G. H., & Huang, J. J. (2011). Multiple attribute decision making: methods and applications. CRC press.
  64. Venkatesh, V. G., Zhang, A., Deakins, E., Luthra, S., & Mangla, S. (2019). A fuzzy AHP-TOPSIS approach to supply partner selection in continuous aid humanitarian supply chains. Annals of operations research283(1), 1517-1550.
  65. Wang, Y. M., & Chin, K. S. (2007). Discriminating DEA efficient candidates by considering their least relative total scores. Journal of computational and applied mathematics206(1), 209-215.
  66. Wei, G., Alsaadi, F. E., Hayat, T., & Alsaedi, A. (2017). A linear assignment method for multiple criteria decision analysis with hesitant fuzzy sets based on fuzzy measure. International journal of fuzzy systems19(3), 607-614.
  67. Xian-Ying, M. (2012). Application of assignment model in PE human resources allocation. Energy procedia16, 1720-1723.
  68. Xu, L., & Yang, J. B. (2001). Introduction to multi-criteria decision making and the evidential reasoning approach(Vol. 106). Manchester: Manchester School of Management.