Two-stage prioritization procedure for multiplicative AHP-group decision making
Abstract
In this paper, we propose two-stage prioritization procedure (TSPP) for multiplicative Analytic Hierarchy Process-group decision making (AHP-GDM), which involves determining the group priority vector based on the individual pair-wise comparison matrices (PCMs), simultaneously considering the consensus and consistency of the individual PCMs. The first stage of the TSPP involves checking and revising the individual PCMs for reaching the acceptable consensus and consistency. The second stage of the TSPP involves estimating the group priority vector using Bayesian approach. The main characteristics of the proposed TSPP are as follows: 1) It makes full use of the prior information as well as the sample information during the Bayesian revision of the individual PCMs and the Bayesian estimation of the group priority vector; 2) It ensures that the revised individual PCMs reach the acceptable consensus and consistency; 3) It enriches the aggregation methods for the collective preference in multiplicative AHP-GDM. Finally, two numerical examples are used to evaluate the applicability and effectiveness of the proposed TSPP by the comparisons with several other methods.
Keyword : group decision making, pair-wise comparison matrix, consensus, consistency, group priority vector
How to Cite
Lin, C., Kou, G., Peng, Y., & Alsaadi, F. E. (2020). Two-stage prioritization procedure for multiplicative AHP-group decision making. Technological and Economic Development of Economy, 26(2), 525-545. https://doi.org/10.3846/tede.2020.12037
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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Barzilai, J., Cook, W. D., & Golany, B. (1987). Consistent weights for judgments matrices of the relative importance of alternatives. Operations Research Letters, 6(3), 131–134. https://doi.org/10.1016/0167-6377(87)90026-5
Barzilai, J., & Golany, B. (1994). AHP rank reversal normalization and aggregation rules. INFOR, 32(2), 57–64. https://doi.org/10.1080/03155986.1994.11732238
Barzilai, J. (1997). Deriving weights from pairwise comparison matrices. Journal of the Operational Research Society, 48(12), 1226–1232. https://doi.org/10.1057/palgrave.jors.2600474
Barfod, M. B., Honert, R. V. D., & Salling, K. B. (2016). Modeling group perceptions using stochastic simulation, scaling issues in the multiplicative AHP. International Journal of Information Technology & Decision Making, 15(2), 453–474. https://doi.org/10.1142/S0219622016500103
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Basak, I. (1998). Probabilistic judgments specified partially in the analytic hierarchy process. European Journal of Operational Research, 108(1), 153–164. https://doi.org/10.1016/S0377-2217(97)00140-9
Basak, I. (2001). The categorical data analysis approach for ratio model of pairwise comparisons. European Journal of Operation Research, 128(3), 532–544. https://doi.org/10.1016/S0377-2217(99)00372-0
Bernardo, J. M. & Smith, A. F. M. (1994). Bayesian theory. Wiley. https://doi.org/10.1002/9780470316870
Ben-Arieh, D., & Easton, T. (2007). Multi-criteria group consensus under linear cost opinion elasticity. Decision Support Systems, 43(3), 713–721. https://doi.org/10.1016/j.dss.2006.11.009
Crawford, G., & Williams, C. (1985). A note on the analysis of subjective judgment matrices. Journal of Mathematical Psychology, 29(4), 387–405. https://doi.org/10.1016/0022-2496(85)90002-1
Dong, Y., Xu, Y., Li, H., & Dai, M. (2008). A comparative study of the numerical scales and the prioritization methods in AHP. European Journal of Operational Research, 186(1), 229–242. https://doi.org/10.1016/j.ejor.2007.01.044
Dong, Y., Zhang, G., Hong, W. C., & Xu, Y. (2010). Consensus models for AHP group decision making under row geometric mean prioritization method. Decision Support Systems, 49(3), 281–289. https://doi.org/10.1016/j.dss.2010.03.003
Dong, Q., & Saaty, T. L. (2014). An analytic hierarchy process model of group consensus. Journal of Systems Science and Systems Engineering, 23(3), 362–374. https://doi.org/10.1007/s11518-014-5247-8
Dong, Y., Li, C., Chiclana, F., & Herrera-Viedma, E. (2016). Average-case consistency measurement and analysis of interval-valued reciprocal preference relations. Knowledge-Based Systems, 114, 108–117. https://doi.org/10.1016/j.knosys.2016.10.005
Escobar, M. T, & Moreno-Jiménez, J. M. (2000). Reciprocal distributions in the analytic hierarchy process. European Journal of Operational Research, 123(1), 154–174. https://doi.org/10.1016/S0377-2217(99)00086-7
Fichtner, J. (1983). Some thoughts about the mathematics of the analytic hierarchy process. Hochschule der Bundeswehr.
Forman, E., & Peniwati, K. (1998). Aggregating individual judgments and priorities with the analytic hierarchy process. European Journal of Operational Research, 108(1), 165–169. https://doi.org/10.1016/S0377-2217(97)00244-0
Gargallo, P., Moreno-Jiménez, J. M., & Salvador, M. (2007). AHP-group decision making, a Bayesian approach based on mixtures for group pattern identification. Group Decision & Negotiation, 16(6), 485–506. https://doi.org/10.1007/s10726-006-9068-0
Golany, B., & Kress, M. A. (1993). Multicriteria evaluation of methods for obtaining weights from ratioscale matrices. European Journal of Operational Research, 69(2), 210–220. https://doi.org/10.1016/0377-2217(93)90165-J
Hahn, E. D. (2003). Decision making with uncertain judgments, a stochastic formulation of the analytic hierarchy process. Decision Sciences, 34(3), 443–466. https://doi.org/10.1111/j.1540-5414.2003.02274.x
Hauser, D., & Tadikamalla, P. (1996). The analytic hierarchy process in an uncertain environment, a simulation approach. European Journal of Operational Research, 91(1), 27–37. https://doi.org/10.1016/0377-2217(95)00002-X
Herrera-Viedma, E., Herrera, F., & Chiclana, F. (2002). A consensus model for multiperson decision making with different preference structures. IEEE Transactions on Systems Man & Cybernetics Part A Systems & Humans, 32(3), 394–402. https://doi.org/10.1109/TSMCA.2002.802821
Honert, R. C. V. D. (1998). Stochastic group preference modelling in the multiplicative AHP, a model of group consensus. European Journal of Operational Research, 110(1), 99–111. https://doi.org/10.1016/S0377-2217(97)00243-9
Hughes, W. R. (2009). A statistical framework for strategic decision making with AHP, probability assessment and Bayesian revision. Omega, 37(2), 463–470. https://doi.org/10.1016/j.omega.2007.07.002
Jalao, E. R., Wu, T., & Dan, S. (2014). A stochastic AHP decision making methodology for imprecise preferences. Information Sciences, 270, 192–203. https://doi.org/10.1016/j.ins.2014.02.077
Jin, F., Ni, Z., Chen, H., & Li, Y. (2016). Approaches to group decision making with intuitionistic fuzzy preference relations based on multiplicative consistency. Knowledge-Based Systems, 97, 48–59. https://doi.org/10.1016/j.knosys.2016.01.017
Jin, F., Ni, Z., Pei, L., et al. (2019). A decision support model for group decision making with intuitionistic fuzzy linguistic preferences relations. Neural Computing and Applications, 31, 1103–1024. https://doi.org/10.1007/s00521-017-3071-z
Jong, P. D. (1984). A statistical approach to Saaty’s scaling method for priorities. Journal of Mathematical Psychology, 28(4), 467–478. https://doi.org/10.1016/0022-2496(84)90013-0
Kou, G., Lu, Y., Peng, Y., & Shi, Y. (2012). Evaluation of classification algorithms using MCDM and rank correlation. International Journal of Information Technology & Decision Making, 11(1), 197–225. https://doi.org/10.1142/S0219622012500095
Kou, G., Peng, Y., & Wang, G. (2014). Evaluation of clustering algorithms for financial risk analysis using MCDM methods. Information Sciences, 275(11), 1–12. https://doi.org/10.1016/j.ins.2014.02.137
Kou, G., Ergu, D., Chen, Y., & Lin, C. (2016). Pairwise comparison matrix in multiple criteria decision making. Technological and Economic Development of Economy, 22(5), 738–765. https://doi.org/10.3846/20294913.2016.1210694
Kou, G., Yang, P., Peng, Y., Xiao, F., Chen, Y., & Alsaadi, F. E. (2019). Evaluation of feature selection methods for text classification with small datasets using multiple criteria decision-making methods. Applied Soft Computing Journal. https://doi.org/10.1016/j.asoc.2019.105836
Li, Y., Zhang, H., & Dong, Y. (2017). The interactive consensus reaching process with the minimum and uncertain cost in group decision making. Applied Soft Computing, 60, 202–212. https://doi.org/10.1016/j.asoc.2017.06.056
Li, G., Kou, G., & Peng, Y. (2018). A group decision making model for integrating heterogeneous information. IEEE Transactions on Systems, Man, and Cybernetics, Systems, 48(6), 982–992. https://doi.org/10.1109/TSMC.2016.2627050
Lin, C., Kou, G., & Ergu, D. (2013). An improved statistical approach for consistency test in AHP. Annals of Operations Research, 211(1), 289–299. https://doi.org/10.1007/s10479-013-1413-5
Lin, C., & Kou, G. (2015). Bayesian revision of the individual pair-wise comparison matrices under consensus in AHP-GDM. Applied Soft Computing, 35(C), 802–811. https://doi.org/10.1016/j.asoc.2015.02.041
Lipovetsky, S., & Tishler, A. (1999) Interval estimation of priorities in the AHP. European Journal of Operational Research, 114(1), 153–164. https://doi.org/10.1016/S0377-2217(98)00012-5
Liu, J., Song, J., Xu, Q., Tao, Z., & Chen, H. (2019). Group decision making based on DEA crossefficiency with intuitionistic fuzzy preference relations. Fuzzy Optimization and Decision Making, 18(3), 345–370. https://doi.org/10.1007/s10700-018-9297-0
Moreno-Jiménez, J. M. (2011). An AHP/ANP multicriteria methodology to estimate the value and transfers fees of professional football players. In Proceedings ISAHP. Sorrento, Italia.
Press, S. J. (1989). Bayesian statist, principle, models and applications. John Wiley & Sons.
Ramanathan, R., & Ganesh, L. S. (1994). Group preference aggregation methods employed in AHP, An evaluation and an intrinsic process for deriving members’ weightages. European Journal of Operational Research, 79(2), 249–265. https://doi.org/10.1016/0377-2217(94)90356-5
Rosenbloom, E. S. (1996). A probabilistic interpretation of the final rankings in AHP. European Journal of Operational Research, 96(2), 371–378. https://doi.org/10.1016/S0377-2217(96)00049-5
Saaty, T. L. (1977). A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology, 15(3), 234–281. https://doi.org/10.1016/0022-2496(77)90033-5
Saaty, T. L. (1980). The analytic hierarchy process. McGraw-Hill. https://doi.org/10.21236/ADA214804
Schotten, P. C., & Morais, D. C. (2019). A group decision model for credit granting in the financial market. Financial Innovation, 5, 6. https://doi.org/10.1186/s40854-019-0126-4
Srdjevic, B., & Srdjevic, Z. (2013). Synthesis of individual best local priority vectors in AHP-group decision making. Applied Soft Computing Journal, 13(4), 2045–2056. https://doi.org/10.1016/j.asoc.2012.11.010
Vargas, L. G. (1982). Reciprocal matrices with random coefficients. Mathematical Modelling, 3(1), 69–81. https://doi.org/10.1016/0270-0255(82)90013-6
Wu, W., & Kou, G. (2016). A group consensus model for evaluating real estate nvestment alternatives. Financial Innovation, 2, 8. https://doi.org/10.1186/s40854-016-0027-8
Wu, Z., & Xu, J. (2012). A consensus and consistency based decision support model for group decision making with multiplicative preference relations. Decision Support Systems, 52(3), 757–767. https://doi.org/10.1016/j.dss.2011.11.022
Wu, P., Zhou, L., Chen, H., & Tao, Z. (2019). Additive consistency of hesitant fuzzy linguistic preference relation with a new expansion principle for hesitant fuzzy linguistic term sets. IEEE Transactions on Fuzzy Systems, 27(4), 716–730. https://doi.org/10.1109/TFUZZ.2018.2868492
Xu, Z., & Wei, C. (1999). A consistency improving method in the analytic hierarchy process. European Journal of Operational Research, 116(2), 443–449. https://doi.org/10.1016/S0377-2217(98)00109-X
Zhang, H., Kou, G., & Peng, Y. (2019). Soft consensus cost models for group decision making and economic interpretations. European Journal of Operational Research, 277(3), 964–980. https://doi.org/10.1016/j.ejor.2019.03.009
Altuzarra, A., Moreno-Jiménez, J. M., & Salvador, M. (2007). A Bayesian priorization procedure for AHP-group decision making. European Journal of Operational Research, 182(1), 367–382. https://doi.org/10.1016/j.ejor.2006.07.025
Altuzarra, A., Moreno-Jiménez, J. M., & Salvador, M. (2010). Consensus building in AHP-group decision making, a Bayesian approach. Operations Research, 58(6), 1755–1773. https://doi.org/10.1287/opre.1100.0856
Arrow, K. J. (1963). Social choice and individual values (2 ed.). Wiley.
Barzilai, J., Cook, W. D., & Golany, B. (1987). Consistent weights for judgments matrices of the relative importance of alternatives. Operations Research Letters, 6(3), 131–134. https://doi.org/10.1016/0167-6377(87)90026-5
Barzilai, J., & Golany, B. (1994). AHP rank reversal normalization and aggregation rules. INFOR, 32(2), 57–64. https://doi.org/10.1080/03155986.1994.11732238
Barzilai, J. (1997). Deriving weights from pairwise comparison matrices. Journal of the Operational Research Society, 48(12), 1226–1232. https://doi.org/10.1057/palgrave.jors.2600474
Barfod, M. B., Honert, R. V. D., & Salling, K. B. (2016). Modeling group perceptions using stochastic simulation, scaling issues in the multiplicative AHP. International Journal of Information Technology & Decision Making, 15(2), 453–474. https://doi.org/10.1142/S0219622016500103
Basak, I. (1990). Testing for the rank ordering of the priorities of the alternatives in Saaty’s ratio-scale method. European Journal of Operational Research, 48(1), 148–152. https://doi.org/10.1016/0377-2217(90)90071-I
Basak, I. (1998). Probabilistic judgments specified partially in the analytic hierarchy process. European Journal of Operational Research, 108(1), 153–164. https://doi.org/10.1016/S0377-2217(97)00140-9
Basak, I. (2001). The categorical data analysis approach for ratio model of pairwise comparisons. European Journal of Operation Research, 128(3), 532–544. https://doi.org/10.1016/S0377-2217(99)00372-0
Bernardo, J. M. & Smith, A. F. M. (1994). Bayesian theory. Wiley. https://doi.org/10.1002/9780470316870
Ben-Arieh, D., & Easton, T. (2007). Multi-criteria group consensus under linear cost opinion elasticity. Decision Support Systems, 43(3), 713–721. https://doi.org/10.1016/j.dss.2006.11.009
Crawford, G., & Williams, C. (1985). A note on the analysis of subjective judgment matrices. Journal of Mathematical Psychology, 29(4), 387–405. https://doi.org/10.1016/0022-2496(85)90002-1
Dong, Y., Xu, Y., Li, H., & Dai, M. (2008). A comparative study of the numerical scales and the prioritization methods in AHP. European Journal of Operational Research, 186(1), 229–242. https://doi.org/10.1016/j.ejor.2007.01.044
Dong, Y., Zhang, G., Hong, W. C., & Xu, Y. (2010). Consensus models for AHP group decision making under row geometric mean prioritization method. Decision Support Systems, 49(3), 281–289. https://doi.org/10.1016/j.dss.2010.03.003
Dong, Q., & Saaty, T. L. (2014). An analytic hierarchy process model of group consensus. Journal of Systems Science and Systems Engineering, 23(3), 362–374. https://doi.org/10.1007/s11518-014-5247-8
Dong, Y., Li, C., Chiclana, F., & Herrera-Viedma, E. (2016). Average-case consistency measurement and analysis of interval-valued reciprocal preference relations. Knowledge-Based Systems, 114, 108–117. https://doi.org/10.1016/j.knosys.2016.10.005
Escobar, M. T, & Moreno-Jiménez, J. M. (2000). Reciprocal distributions in the analytic hierarchy process. European Journal of Operational Research, 123(1), 154–174. https://doi.org/10.1016/S0377-2217(99)00086-7
Fichtner, J. (1983). Some thoughts about the mathematics of the analytic hierarchy process. Hochschule der Bundeswehr.
Forman, E., & Peniwati, K. (1998). Aggregating individual judgments and priorities with the analytic hierarchy process. European Journal of Operational Research, 108(1), 165–169. https://doi.org/10.1016/S0377-2217(97)00244-0
Gargallo, P., Moreno-Jiménez, J. M., & Salvador, M. (2007). AHP-group decision making, a Bayesian approach based on mixtures for group pattern identification. Group Decision & Negotiation, 16(6), 485–506. https://doi.org/10.1007/s10726-006-9068-0
Golany, B., & Kress, M. A. (1993). Multicriteria evaluation of methods for obtaining weights from ratioscale matrices. European Journal of Operational Research, 69(2), 210–220. https://doi.org/10.1016/0377-2217(93)90165-J
Hahn, E. D. (2003). Decision making with uncertain judgments, a stochastic formulation of the analytic hierarchy process. Decision Sciences, 34(3), 443–466. https://doi.org/10.1111/j.1540-5414.2003.02274.x
Hauser, D., & Tadikamalla, P. (1996). The analytic hierarchy process in an uncertain environment, a simulation approach. European Journal of Operational Research, 91(1), 27–37. https://doi.org/10.1016/0377-2217(95)00002-X
Herrera-Viedma, E., Herrera, F., & Chiclana, F. (2002). A consensus model for multiperson decision making with different preference structures. IEEE Transactions on Systems Man & Cybernetics Part A Systems & Humans, 32(3), 394–402. https://doi.org/10.1109/TSMCA.2002.802821
Honert, R. C. V. D. (1998). Stochastic group preference modelling in the multiplicative AHP, a model of group consensus. European Journal of Operational Research, 110(1), 99–111. https://doi.org/10.1016/S0377-2217(97)00243-9
Hughes, W. R. (2009). A statistical framework for strategic decision making with AHP, probability assessment and Bayesian revision. Omega, 37(2), 463–470. https://doi.org/10.1016/j.omega.2007.07.002
Jalao, E. R., Wu, T., & Dan, S. (2014). A stochastic AHP decision making methodology for imprecise preferences. Information Sciences, 270, 192–203. https://doi.org/10.1016/j.ins.2014.02.077
Jin, F., Ni, Z., Chen, H., & Li, Y. (2016). Approaches to group decision making with intuitionistic fuzzy preference relations based on multiplicative consistency. Knowledge-Based Systems, 97, 48–59. https://doi.org/10.1016/j.knosys.2016.01.017
Jin, F., Ni, Z., Pei, L., et al. (2019). A decision support model for group decision making with intuitionistic fuzzy linguistic preferences relations. Neural Computing and Applications, 31, 1103–1024. https://doi.org/10.1007/s00521-017-3071-z
Jong, P. D. (1984). A statistical approach to Saaty’s scaling method for priorities. Journal of Mathematical Psychology, 28(4), 467–478. https://doi.org/10.1016/0022-2496(84)90013-0
Kou, G., Lu, Y., Peng, Y., & Shi, Y. (2012). Evaluation of classification algorithms using MCDM and rank correlation. International Journal of Information Technology & Decision Making, 11(1), 197–225. https://doi.org/10.1142/S0219622012500095
Kou, G., Peng, Y., & Wang, G. (2014). Evaluation of clustering algorithms for financial risk analysis using MCDM methods. Information Sciences, 275(11), 1–12. https://doi.org/10.1016/j.ins.2014.02.137
Kou, G., Ergu, D., Chen, Y., & Lin, C. (2016). Pairwise comparison matrix in multiple criteria decision making. Technological and Economic Development of Economy, 22(5), 738–765. https://doi.org/10.3846/20294913.2016.1210694
Kou, G., Yang, P., Peng, Y., Xiao, F., Chen, Y., & Alsaadi, F. E. (2019). Evaluation of feature selection methods for text classification with small datasets using multiple criteria decision-making methods. Applied Soft Computing Journal. https://doi.org/10.1016/j.asoc.2019.105836
Li, Y., Zhang, H., & Dong, Y. (2017). The interactive consensus reaching process with the minimum and uncertain cost in group decision making. Applied Soft Computing, 60, 202–212. https://doi.org/10.1016/j.asoc.2017.06.056
Li, G., Kou, G., & Peng, Y. (2018). A group decision making model for integrating heterogeneous information. IEEE Transactions on Systems, Man, and Cybernetics, Systems, 48(6), 982–992. https://doi.org/10.1109/TSMC.2016.2627050
Lin, C., Kou, G., & Ergu, D. (2013). An improved statistical approach for consistency test in AHP. Annals of Operations Research, 211(1), 289–299. https://doi.org/10.1007/s10479-013-1413-5
Lin, C., & Kou, G. (2015). Bayesian revision of the individual pair-wise comparison matrices under consensus in AHP-GDM. Applied Soft Computing, 35(C), 802–811. https://doi.org/10.1016/j.asoc.2015.02.041
Lipovetsky, S., & Tishler, A. (1999) Interval estimation of priorities in the AHP. European Journal of Operational Research, 114(1), 153–164. https://doi.org/10.1016/S0377-2217(98)00012-5
Liu, J., Song, J., Xu, Q., Tao, Z., & Chen, H. (2019). Group decision making based on DEA crossefficiency with intuitionistic fuzzy preference relations. Fuzzy Optimization and Decision Making, 18(3), 345–370. https://doi.org/10.1007/s10700-018-9297-0
Moreno-Jiménez, J. M. (2011). An AHP/ANP multicriteria methodology to estimate the value and transfers fees of professional football players. In Proceedings ISAHP. Sorrento, Italia.
Press, S. J. (1989). Bayesian statist, principle, models and applications. John Wiley & Sons.
Ramanathan, R., & Ganesh, L. S. (1994). Group preference aggregation methods employed in AHP, An evaluation and an intrinsic process for deriving members’ weightages. European Journal of Operational Research, 79(2), 249–265. https://doi.org/10.1016/0377-2217(94)90356-5
Rosenbloom, E. S. (1996). A probabilistic interpretation of the final rankings in AHP. European Journal of Operational Research, 96(2), 371–378. https://doi.org/10.1016/S0377-2217(96)00049-5
Saaty, T. L. (1977). A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology, 15(3), 234–281. https://doi.org/10.1016/0022-2496(77)90033-5
Saaty, T. L. (1980). The analytic hierarchy process. McGraw-Hill. https://doi.org/10.21236/ADA214804
Schotten, P. C., & Morais, D. C. (2019). A group decision model for credit granting in the financial market. Financial Innovation, 5, 6. https://doi.org/10.1186/s40854-019-0126-4
Srdjevic, B., & Srdjevic, Z. (2013). Synthesis of individual best local priority vectors in AHP-group decision making. Applied Soft Computing Journal, 13(4), 2045–2056. https://doi.org/10.1016/j.asoc.2012.11.010
Vargas, L. G. (1982). Reciprocal matrices with random coefficients. Mathematical Modelling, 3(1), 69–81. https://doi.org/10.1016/0270-0255(82)90013-6
Wu, W., & Kou, G. (2016). A group consensus model for evaluating real estate nvestment alternatives. Financial Innovation, 2, 8. https://doi.org/10.1186/s40854-016-0027-8
Wu, Z., & Xu, J. (2012). A consensus and consistency based decision support model for group decision making with multiplicative preference relations. Decision Support Systems, 52(3), 757–767. https://doi.org/10.1016/j.dss.2011.11.022
Wu, P., Zhou, L., Chen, H., & Tao, Z. (2019). Additive consistency of hesitant fuzzy linguistic preference relation with a new expansion principle for hesitant fuzzy linguistic term sets. IEEE Transactions on Fuzzy Systems, 27(4), 716–730. https://doi.org/10.1109/TFUZZ.2018.2868492
Xu, Z., & Wei, C. (1999). A consistency improving method in the analytic hierarchy process. European Journal of Operational Research, 116(2), 443–449. https://doi.org/10.1016/S0377-2217(98)00109-X
Zhang, H., Kou, G., & Peng, Y. (2019). Soft consensus cost models for group decision making and economic interpretations. European Journal of Operational Research, 277(3), 964–980. https://doi.org/10.1016/j.ejor.2019.03.009