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doi: 10.3934/jimo.2019003

## Multicriteria investment problem with Savage's risk criteria: Theoretical aspects of stability and case study

 1 Economics and Management School, University of Chinese Academy of Sciences, 100190 Beijing, China 2 Faculty of Mechanics and Mathematics, Belarusian State University, 220030 Minsk, Belarus 3 Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland

* Corresponding author: vladimir.korotkov08@gmail.com

Received  June 2017 Revised  November 2018 Published  March 2019

A discrete variant of a multicriteria investment portfolio optimization problem with Savage's risk criteria is considered. One of the three problem parameter spaces is endowed with Hölder's norm, and the other two are endowed with Chebyshev's norm. The lower and upper attainable bounds on the stability radius of one Pareto optimal portfolio are obtained. We illustrate the application of our theoretical results by modeling a relevant case study.

Citation: Vladimir Korotkov, Vladimir Emelichev, Yury Nikulin. Multicriteria investment problem with Savage's risk criteria: Theoretical aspects of stability and case study. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019003
##### References:
 [1] F. Al-Maliky, M. Hifi and H. Mhalla, Sensitivity analysis of the setup knapsack problem to perturbation of arbitrary profits or weights, International Transactions in Operational Research, 25 (2018), 637-666. doi: 10.1111/itor.12373. Google Scholar [2] M. N. M. Arratia, I. F. Lépez, S. E. Schaeffer and L. Cruz-Reyes, Static R & D project portfolio selection in public organizations, Decision Support Systems, 84 (2016), 53-63. Google Scholar [3] T. Belgacem and M. Hifi, Sensitivity analysis of the optimum to perturbation of the profit of a subset of items in the binary knapsack problem, Discrete Optimization, 5 (2008), 755-761. doi: 10.1016/j.disopt.2008.05.001. Google Scholar [4] W. C. Benton, A profitability evaluation of America's best hospitals, 2000-2008, Decision Sciences, 44 (2013), 1139-1153. Google Scholar [5] L. Berger, J. Emmerling and M. Tavoni, Managing catastrophic climate risks under model uncertainty aversion, Management Science, 63 (2016), 749-765. Google Scholar [6] E. Bronshtein, M. Kachkaeva and E. Tulupova, Control of investment portfolio based on complex quantile risk measures, J. of Comput. and Syst. Sci. Int., 50 (2011), 174-180. doi: 10.1134/S1064230711010084. Google Scholar [7] N. Chakravarti and A. Wagelmans, Calculation of stability radii for combinatorial optimization problem, Oper. Res. Lett., 23 (1998), 1-7. doi: 10.1016/S0167-6377(98)00031-5. Google Scholar [8] M. Crouhy, D. Galai and R. Mark, The Essentials of Risk Management, New Yourk: McGraw-Hill; 2005.Google Scholar [9] H. Dincers, U. Hacioglu, E. Tatoglu and D. Delen, A fuzzy-hybrid analytic model to assess investors' perceptions for industry selection, Decision Support Systems, 86 (2016), 24-34. Google Scholar [10] D. Du and P. Pardalos (eds.), Minimax and applications, Dordrecht: Kluwer; 1995. doi: 10.1007/978-1-4613-3557-3. Google Scholar [11] V. Emelichev and D. Podkopaev, Quantitative stability analysis for vector problems of 0-1 programming, Discret. Optim., 7 (2010), 48-63. doi: 10.1016/j.disopt.2010.02.001. Google Scholar [12] V. Emelichev and K. Kuzmin, Stability criteria in vector combinatorial bottleneck problems in terms of binary relations, Cybernetics and Syst. Analys., 44 (2008), 397-404. doi: 10.1007/s10559-008-9001-4. Google Scholar [13] V. Emelichev and O. Karelkina, Postoptimal analysis of the multicriteria combinatorial median location problem, Optim., 61 (2012), 1151-1167. doi: 10.1080/02331934.2010.542813. Google Scholar [14] V. Emelichev, V. Korotkov and Yu. Nikulin, Post-optimal analysis for Markowitz's multicriteria portfolio optimization problem, J. Multi-Crit. Decis. Analys., 21 (2014), 95-100. Google Scholar [15] V. Emelichev, V. Korotkov and K. Kuzmin, Multicriterial investment problem in conditions of uncertainty and risk, J. of Comput. and Syst. Sci. Int., 50 (2011), 1011-1018. doi: 10.1134/S1064230711040071. Google Scholar [16] V. Emelichev, V. Korotkov and K. Kuzmin, On stability of a Pareto-optimal solution of a portfolio optimization problem with Savage's minimax risk criteria, Bull. of the Acad. of Sci. of Moldova. Math., 3 (2010), 35-44. Google Scholar [17] V. Emelichev and V. Korotkov, Stability radius of a vector investment problem with Savage's minimax risk criteria, Cybernetics and Syst. Analys., 48 (2012), 378-386. doi: 10.1007/s10559-012-9417-8. Google Scholar [18] V. Emelichev and K. Kuzmin, A general approach to studying the stability of a Pareto optimal solution of a vector integer linear programming problem, Discret. Math. Appl., 17 (2007), 349-354. doi: 10.1515/dma.2007.029. Google Scholar [19] V. Emelichev, K. Kuzmin and Y. Nikulin, Stability analysis of the Pareto optimal solution for some vector Boolean optimization problem, Optim., 54 (2005), 545-561. doi: 10.1080/02331930500342708. Google Scholar [20] J. Frank, C. F. A. Fabozzi and H. Markowitz (editors), The Theory and Practice of Investment Management: Asset Allocation, Valuation, Portfolio Construction, and Strategies. Wiley; 2011.Google Scholar [21] E. Fernandez, C. Gomez, G. Rivera and L. Cruz-Reyes, Hybrid metaheuristic approach for handling many objectives and decisions on partial support in project portfolio optimisation, Information Sciences, 315 (2015), 102-122. doi: 10.1016/j.ins.2015.03.064. Google Scholar [22] B. Gorissen, İ. Yanıkoğlu and D. den Hertog, A practical guide to robust optimization, Omega, 53 (2015), 124-137. Google Scholar [23] E. Gurevsky, O. Battaïa and A. Dolgui, Stability measure for a generalized assembly line balancing problem, Discrete Applied Mathematics, 161 (2013), 377-394. doi: 10.1016/j.dam.2012.08.037. Google Scholar [24] M. Hirschberger, R. E. Steuer, S. Utz, M. Wimmer and Y. Qi, Computing the nondominated surface in tri-criterion portfolio selection, Operations Research, 61 (2013), 169-183. doi: 10.1287/opre.1120.1140. Google Scholar [25] X. Huang and T. Zhao, Project selection and adjustment based on uncertain measure, Information Sciences, 352/353 (2016), 1-14. Google Scholar [26] K. Khalili-Damghani and M. Tavana, A comprehensive framework for sustainable project portfolio selection based on structural equation modeling, Project Management Journal, 45 (2014), 83-97. Google Scholar [27] K. Khalili-Damghani, S. Sadi-Nezhad, F. H. Lotfi and M. Tavana, A hybrid fuzzy rule-based multi-criteria framework for sustainable project portfolio selection, Information Sciences, 220 (2013), 442-462. Google Scholar [28] M. Koudstaal, R. Sloof and M. van Praag, Risk, uncertainty, and entrepreneurship: Evidence from a lab-in-the-field experiment, Management Science, 62 (2015), 2897-2915. Google Scholar [29] L. Kozeratska, J. Forbes, R. Goebel and J. Kresta, Perturbed cones for analysis of uncertain multi-criteria optimization problems, Linear Algebra and its Appl., 378 (2004), 203-229. doi: 10.1016/j.laa.2003.09.013. Google Scholar [30] T. Lebedeva and T. Sergienko, Different types of stability of vector integer optimization problem: general approach, Cybernetics and Syst. Analys., 44 (2008), 429-433. doi: 10.1007/s10559-008-9017-9. Google Scholar [31] T. Lebedeva, N. Semenona and T. Sergienko, Stability of vector problems of integer optimization: relationship with the stability of sets of optimal and nonoptimal solutions, Cybernetics and Syst. Analys., 41 (2005), 551-558. doi: 10.1007/s10559-005-0090-z. Google Scholar [32] M. Libura, E.S. van der Poort, G. Sierksma and J. A. A. van der Veen, Stability aspects of the traveling salesman problem based on k-best solutions, Discrete Applied Mathematics, 87 (1998), 159-185. doi: 10.1016/S0166-218X(98)00055-9. Google Scholar [33] H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, New York: Willey; 1991.Google Scholar [34] G. Mavrotas, J. R. Figueira and E. Siskos, Robustness analysis methodology for multi-objective combinatorial optimization problems and application to project selection, Omega, 52 (2015), 142-155. Google Scholar [35] K. Miettinen, Nonlinear Multiobjective Optimization, Boston: Kluwer; 1999. Google Scholar [36] A. Mishra, S. R. Das and J. J. Murray, Risk, process maturity, and project performance: An empirical analysis of US federal government technology projects, Production and Operations Management, 25 (2016), 210-232. Google Scholar [37] M. Note, Project Management for Information Professionals, Waltham-Kidlington: Chandos; 2016.Google Scholar [38] D. L. Olson and D. D. Wu, Enterprise Risk Management Models, Berlin-Heidelberg: Springer-Verlag; 2017.Google Scholar [39] A. Özkış and A. Babalık, A novel metaheuristic for multi-objective optimization problems: The multi-objective vortex search algorithm, Information Sciences, 402 (2017), 124-148. Google Scholar [40] D. Power, R. Klassen, T. J. Kull and D. Simpson, Competitive goals and plant investment in environment and safety practices: Moderating effect of national culture, Decision Sciences, 46 (2015), 63-100. Google Scholar [41] R. Ramaswamy, J. B. Orlin and N. Chakravarti, Sensitivity analysis for shortest path problems and maximum capacity path problems in undirected graphs, Mathematical Programming, 102 (2005), 355-369. doi: 10.1007/s10107-004-0517-8. Google Scholar [42] L. Savage, The Foundations of Statistics, New York: Dover; 1972. Google Scholar [43] J. A. Sefaira, C. Y. Mándezb, O. Babatc, A. L. Medagliab and L. F. Zuluaga, Linear solution schemes for mean-semivariance project portfolio selection problems: An application in the oil and gas industry, Omega, 68 (2017), 39-48. Google Scholar [44] P. Sironi, Modern Portfolio Management: From Markowitz to Probabilistic Scenario Optimisation, Risk books. 2015.Google Scholar [45] P. Soberanis, Risk Optimization with P-Order Conic Constraints, Ph.d. thesis University of Iowa, 2009. Google Scholar [46] Y. Sotskov, N. Sotskova, T. Lai and F. Werner, Scheduling Under Uncertainty. Theory and Algorithms, Belorusskaya nauka, Minsk, 2010.Google Scholar [47] Y. Sotskov and T. Lai, Minimizing total weighted flow under uncertainty using dominance and a stability box, Computers & Operations Res., 39 (2012), 1271-1289. doi: 10.1016/j.cor.2011.02.001. Google Scholar [48] Y. Sotskov and F. Werner, A stability approach in sequencing and scheduling, Chapter in the book sequencing and Scheduling with Inaccurate Data? Y. Sotskov, F. Werner (Editors). Nova Science Publishers, Inc., New York, USA, (2014), 283-344.Google Scholar [49] S. Van Hoesel and A. Wagelmans, On the complexity of postoptimality analysis of 0-1 programs, Discret. Appl. Math., 91 (1999), 251-263. doi: 10.1016/S0166-218X(98)00151-6. Google Scholar [50] C. Von Lücken, B. Barán and C. Brizuela, A survey on multi-objective evolutionary algorithms for many-objective problems, Computational Optimization and Applications, 58 (2014), 707-756. doi: 10.1007/s10589-014-9644-1. Google Scholar [51] P. Yu, Multiple-criteria Decision Making: Concepts, Techniques, and Extensions, New York: Plenum Press; 1985. doi: 10.1007/978-1-4684-8395-6. Google Scholar

show all references

##### References:
 [1] F. Al-Maliky, M. Hifi and H. Mhalla, Sensitivity analysis of the setup knapsack problem to perturbation of arbitrary profits or weights, International Transactions in Operational Research, 25 (2018), 637-666. doi: 10.1111/itor.12373. Google Scholar [2] M. N. M. Arratia, I. F. Lépez, S. E. Schaeffer and L. Cruz-Reyes, Static R & D project portfolio selection in public organizations, Decision Support Systems, 84 (2016), 53-63. Google Scholar [3] T. Belgacem and M. Hifi, Sensitivity analysis of the optimum to perturbation of the profit of a subset of items in the binary knapsack problem, Discrete Optimization, 5 (2008), 755-761. doi: 10.1016/j.disopt.2008.05.001. Google Scholar [4] W. C. Benton, A profitability evaluation of America's best hospitals, 2000-2008, Decision Sciences, 44 (2013), 1139-1153. Google Scholar [5] L. Berger, J. Emmerling and M. Tavoni, Managing catastrophic climate risks under model uncertainty aversion, Management Science, 63 (2016), 749-765. Google Scholar [6] E. Bronshtein, M. Kachkaeva and E. Tulupova, Control of investment portfolio based on complex quantile risk measures, J. of Comput. and Syst. Sci. Int., 50 (2011), 174-180. doi: 10.1134/S1064230711010084. Google Scholar [7] N. Chakravarti and A. Wagelmans, Calculation of stability radii for combinatorial optimization problem, Oper. Res. Lett., 23 (1998), 1-7. doi: 10.1016/S0167-6377(98)00031-5. Google Scholar [8] M. Crouhy, D. Galai and R. Mark, The Essentials of Risk Management, New Yourk: McGraw-Hill; 2005.Google Scholar [9] H. Dincers, U. Hacioglu, E. Tatoglu and D. Delen, A fuzzy-hybrid analytic model to assess investors' perceptions for industry selection, Decision Support Systems, 86 (2016), 24-34. Google Scholar [10] D. Du and P. Pardalos (eds.), Minimax and applications, Dordrecht: Kluwer; 1995. doi: 10.1007/978-1-4613-3557-3. Google Scholar [11] V. Emelichev and D. Podkopaev, Quantitative stability analysis for vector problems of 0-1 programming, Discret. Optim., 7 (2010), 48-63. doi: 10.1016/j.disopt.2010.02.001. Google Scholar [12] V. Emelichev and K. Kuzmin, Stability criteria in vector combinatorial bottleneck problems in terms of binary relations, Cybernetics and Syst. Analys., 44 (2008), 397-404. doi: 10.1007/s10559-008-9001-4. Google Scholar [13] V. Emelichev and O. Karelkina, Postoptimal analysis of the multicriteria combinatorial median location problem, Optim., 61 (2012), 1151-1167. doi: 10.1080/02331934.2010.542813. Google Scholar [14] V. Emelichev, V. Korotkov and Yu. Nikulin, Post-optimal analysis for Markowitz's multicriteria portfolio optimization problem, J. Multi-Crit. Decis. Analys., 21 (2014), 95-100. Google Scholar [15] V. Emelichev, V. Korotkov and K. Kuzmin, Multicriterial investment problem in conditions of uncertainty and risk, J. of Comput. and Syst. Sci. Int., 50 (2011), 1011-1018. doi: 10.1134/S1064230711040071. Google Scholar [16] V. Emelichev, V. Korotkov and K. Kuzmin, On stability of a Pareto-optimal solution of a portfolio optimization problem with Savage's minimax risk criteria, Bull. of the Acad. of Sci. of Moldova. Math., 3 (2010), 35-44. Google Scholar [17] V. Emelichev and V. Korotkov, Stability radius of a vector investment problem with Savage's minimax risk criteria, Cybernetics and Syst. Analys., 48 (2012), 378-386. doi: 10.1007/s10559-012-9417-8. Google Scholar [18] V. Emelichev and K. Kuzmin, A general approach to studying the stability of a Pareto optimal solution of a vector integer linear programming problem, Discret. Math. Appl., 17 (2007), 349-354. doi: 10.1515/dma.2007.029. Google Scholar [19] V. Emelichev, K. Kuzmin and Y. Nikulin, Stability analysis of the Pareto optimal solution for some vector Boolean optimization problem, Optim., 54 (2005), 545-561. doi: 10.1080/02331930500342708. Google Scholar [20] J. Frank, C. F. A. Fabozzi and H. Markowitz (editors), The Theory and Practice of Investment Management: Asset Allocation, Valuation, Portfolio Construction, and Strategies. Wiley; 2011.Google Scholar [21] E. Fernandez, C. Gomez, G. Rivera and L. Cruz-Reyes, Hybrid metaheuristic approach for handling many objectives and decisions on partial support in project portfolio optimisation, Information Sciences, 315 (2015), 102-122. doi: 10.1016/j.ins.2015.03.064. Google Scholar [22] B. Gorissen, İ. Yanıkoğlu and D. den Hertog, A practical guide to robust optimization, Omega, 53 (2015), 124-137. Google Scholar [23] E. Gurevsky, O. Battaïa and A. Dolgui, Stability measure for a generalized assembly line balancing problem, Discrete Applied Mathematics, 161 (2013), 377-394. doi: 10.1016/j.dam.2012.08.037. Google Scholar [24] M. Hirschberger, R. E. Steuer, S. Utz, M. Wimmer and Y. Qi, Computing the nondominated surface in tri-criterion portfolio selection, Operations Research, 61 (2013), 169-183. doi: 10.1287/opre.1120.1140. Google Scholar [25] X. Huang and T. Zhao, Project selection and adjustment based on uncertain measure, Information Sciences, 352/353 (2016), 1-14. Google Scholar [26] K. Khalili-Damghani and M. Tavana, A comprehensive framework for sustainable project portfolio selection based on structural equation modeling, Project Management Journal, 45 (2014), 83-97. Google Scholar [27] K. Khalili-Damghani, S. Sadi-Nezhad, F. H. Lotfi and M. Tavana, A hybrid fuzzy rule-based multi-criteria framework for sustainable project portfolio selection, Information Sciences, 220 (2013), 442-462. Google Scholar [28] M. Koudstaal, R. Sloof and M. van Praag, Risk, uncertainty, and entrepreneurship: Evidence from a lab-in-the-field experiment, Management Science, 62 (2015), 2897-2915. Google Scholar [29] L. Kozeratska, J. Forbes, R. Goebel and J. Kresta, Perturbed cones for analysis of uncertain multi-criteria optimization problems, Linear Algebra and its Appl., 378 (2004), 203-229. doi: 10.1016/j.laa.2003.09.013. Google Scholar [30] T. Lebedeva and T. Sergienko, Different types of stability of vector integer optimization problem: general approach, Cybernetics and Syst. Analys., 44 (2008), 429-433. doi: 10.1007/s10559-008-9017-9. Google Scholar [31] T. Lebedeva, N. Semenona and T. Sergienko, Stability of vector problems of integer optimization: relationship with the stability of sets of optimal and nonoptimal solutions, Cybernetics and Syst. Analys., 41 (2005), 551-558. doi: 10.1007/s10559-005-0090-z. Google Scholar [32] M. Libura, E.S. van der Poort, G. Sierksma and J. A. A. van der Veen, Stability aspects of the traveling salesman problem based on k-best solutions, Discrete Applied Mathematics, 87 (1998), 159-185. doi: 10.1016/S0166-218X(98)00055-9. Google Scholar [33] H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, New York: Willey; 1991.Google Scholar [34] G. Mavrotas, J. R. Figueira and E. Siskos, Robustness analysis methodology for multi-objective combinatorial optimization problems and application to project selection, Omega, 52 (2015), 142-155. Google Scholar [35] K. Miettinen, Nonlinear Multiobjective Optimization, Boston: Kluwer; 1999. Google Scholar [36] A. Mishra, S. R. Das and J. J. Murray, Risk, process maturity, and project performance: An empirical analysis of US federal government technology projects, Production and Operations Management, 25 (2016), 210-232. Google Scholar [37] M. Note, Project Management for Information Professionals, Waltham-Kidlington: Chandos; 2016.Google Scholar [38] D. L. Olson and D. D. Wu, Enterprise Risk Management Models, Berlin-Heidelberg: Springer-Verlag; 2017.Google Scholar [39] A. Özkış and A. Babalık, A novel metaheuristic for multi-objective optimization problems: The multi-objective vortex search algorithm, Information Sciences, 402 (2017), 124-148. Google Scholar [40] D. Power, R. Klassen, T. J. Kull and D. Simpson, Competitive goals and plant investment in environment and safety practices: Moderating effect of national culture, Decision Sciences, 46 (2015), 63-100. Google Scholar [41] R. Ramaswamy, J. B. Orlin and N. Chakravarti, Sensitivity analysis for shortest path problems and maximum capacity path problems in undirected graphs, Mathematical Programming, 102 (2005), 355-369. doi: 10.1007/s10107-004-0517-8. Google Scholar [42] L. Savage, The Foundations of Statistics, New York: Dover; 1972. Google Scholar [43] J. A. Sefaira, C. Y. Mándezb, O. Babatc, A. L. Medagliab and L. F. Zuluaga, Linear solution schemes for mean-semivariance project portfolio selection problems: An application in the oil and gas industry, Omega, 68 (2017), 39-48. Google Scholar [44] P. Sironi, Modern Portfolio Management: From Markowitz to Probabilistic Scenario Optimisation, Risk books. 2015.Google Scholar [45] P. Soberanis, Risk Optimization with P-Order Conic Constraints, Ph.d. thesis University of Iowa, 2009. Google Scholar [46] Y. Sotskov, N. Sotskova, T. Lai and F. Werner, Scheduling Under Uncertainty. Theory and Algorithms, Belorusskaya nauka, Minsk, 2010.Google Scholar [47] Y. Sotskov and T. Lai, Minimizing total weighted flow under uncertainty using dominance and a stability box, Computers & Operations Res., 39 (2012), 1271-1289. doi: 10.1016/j.cor.2011.02.001. Google Scholar [48] Y. Sotskov and F. Werner, A stability approach in sequencing and scheduling, Chapter in the book sequencing and Scheduling with Inaccurate Data? Y. Sotskov, F. Werner (Editors). Nova Science Publishers, Inc., New York, USA, (2014), 283-344.Google Scholar [49] S. Van Hoesel and A. Wagelmans, On the complexity of postoptimality analysis of 0-1 programs, Discret. Appl. Math., 91 (1999), 251-263. doi: 10.1016/S0166-218X(98)00151-6. Google Scholar [50] C. Von Lücken, B. Barán and C. Brizuela, A survey on multi-objective evolutionary algorithms for many-objective problems, Computational Optimization and Applications, 58 (2014), 707-756. doi: 10.1007/s10589-014-9644-1. Google Scholar [51] P. Yu, Multiple-criteria Decision Making: Concepts, Techniques, and Extensions, New York: Plenum Press; 1985. doi: 10.1007/978-1-4684-8395-6. Google Scholar
Values for $\varphi^s_1(x^0,m,p,\infty,\infty)$
Values for $\psi^s_1(x^0,m,p,\infty,\infty)$
Values for $\varphi^s_2(x^0,m,\infty,p,\infty)$
Values for $\psi^s_2(x^0,m,\infty,p,\infty)$
Values for $\varphi^s_3(x^0,m,\infty,\infty,p)$
Values for $\psi^s_3(x^0,m,\infty,\infty,p)$
Value function for portfolios
 a b c d e f g h CSME 81 63 110 102 79 161 168 61 EAEU 120 68 155 92 137 149 231 90 MERCOSUR 144 50 186 100 124 152 146 119 GCC 125 58 182 192 125 136 254 116 SICA 58 66 171 94 126 139 323 106
 a b c d e f g h CSME 81 63 110 102 79 161 168 61 EAEU 120 68 155 92 137 149 231 90 MERCOSUR 144 50 186 100 124 152 146 119 GCC 125 58 182 192 125 136 254 116 SICA 58 66 171 94 126 139 323 106
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