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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-MalikyM. 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.

[2]

M. N. M. ArratiaI. F. LépezS. E. Schaeffer and L. Cruz-Reyes, Static R & D project portfolio selection in public organizations, Decision Support Systems, 84 (2016), 53-63.

[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.

[4]

W. C. Benton, A profitability evaluation of America's best hospitals, 2000-2008, Decision Sciences, 44 (2013), 1139-1153.

[5]

L. BergerJ. Emmerling and M. Tavoni, Managing catastrophic climate risks under model uncertainty aversion, Management Science, 63 (2016), 749-765.

[6]

E. BronshteinM. 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.

[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.

[8]

M. Crouhy, D. Galai and R. Mark, The Essentials of Risk Management, New Yourk: McGraw-Hill; 2005.

[9]

H. DincersU. HaciogluE. Tatoglu and D. Delen, A fuzzy-hybrid analytic model to assess investors' perceptions for industry selection, Decision Support Systems, 86 (2016), 24-34.

[10]

D. Du and P. Pardalos (eds.), Minimax and applications, Dordrecht: Kluwer; 1995. doi: 10.1007/978-1-4613-3557-3.

[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.

[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.

[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.

[14]

V. EmelichevV. Korotkov and Yu. Nikulin, Post-optimal analysis for Markowitz's multicriteria portfolio optimization problem, J. Multi-Crit. Decis. Analys., 21 (2014), 95-100.

[15]

V. EmelichevV. 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.

[16]

V. EmelichevV. 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.

[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.

[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.

[19]

V. EmelichevK. 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.

[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.

[21]

E. FernandezC. GomezG. 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.

[22]

B. Gorissenİ. Yanıkoğlu and D. den Hertog, A practical guide to robust optimization, Omega, 53 (2015), 124-137.

[23]

E. GurevskyO. 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.

[24]

M. HirschbergerR. E. SteuerS. UtzM. 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.

[25]

X. Huang and T. Zhao, Project selection and adjustment based on uncertain measure, Information Sciences, 352/353 (2016), 1-14.

[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.

[27]

K. Khalili-DamghaniS. Sadi-NezhadF. H. Lotfi and M. Tavana, A hybrid fuzzy rule-based multi-criteria framework for sustainable project portfolio selection, Information Sciences, 220 (2013), 442-462.

[28]

M. KoudstaalR. Sloof and M. van Praag, Risk, uncertainty, and entrepreneurship: Evidence from a lab-in-the-field experiment, Management Science, 62 (2015), 2897-2915.

[29]

L. KozeratskaJ. ForbesR. 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.

[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.

[31]

T. LebedevaN. 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.

[32]

M. LiburaE.S. van der PoortG. 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.

[33]

H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, New York: Willey; 1991.

[34]

G. MavrotasJ. R. Figueira and E. Siskos, Robustness analysis methodology for multi-objective combinatorial optimization problems and application to project selection, Omega, 52 (2015), 142-155.

[35]

K. Miettinen, Nonlinear Multiobjective Optimization, Boston: Kluwer; 1999.

[36]

A. MishraS. 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.

[37]

M. Note, Project Management for Information Professionals, Waltham-Kidlington: Chandos; 2016.

[38]

D. L. Olson and D. D. Wu, Enterprise Risk Management Models, Berlin-Heidelberg: Springer-Verlag; 2017.

[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.

[40]

D. PowerR. KlassenT. 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.

[41]

R. RamaswamyJ. 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.

[42]

L. Savage, The Foundations of Statistics, New York: Dover; 1972.

[43]

J. A. SefairaC. Y. MándezbO. BabatcA. 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.

[44]

P. Sironi, Modern Portfolio Management: From Markowitz to Probabilistic Scenario Optimisation, Risk books. 2015.

[45]

P. Soberanis, Risk Optimization with P-Order Conic Constraints, Ph.d. thesis University of Iowa, 2009.

[46]

Y. Sotskov, N. Sotskova, T. Lai and F. Werner, Scheduling Under Uncertainty. Theory and Algorithms, Belorusskaya nauka, Minsk, 2010.

[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.

[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.

[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.

[50]

C. Von LückenB. 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.

[51]

P. Yu, Multiple-criteria Decision Making: Concepts, Techniques, and Extensions, New York: Plenum Press; 1985. doi: 10.1007/978-1-4684-8395-6.

show all references

References:
[1]

F. Al-MalikyM. 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.

[2]

M. N. M. ArratiaI. F. LépezS. E. Schaeffer and L. Cruz-Reyes, Static R & D project portfolio selection in public organizations, Decision Support Systems, 84 (2016), 53-63.

[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.

[4]

W. C. Benton, A profitability evaluation of America's best hospitals, 2000-2008, Decision Sciences, 44 (2013), 1139-1153.

[5]

L. BergerJ. Emmerling and M. Tavoni, Managing catastrophic climate risks under model uncertainty aversion, Management Science, 63 (2016), 749-765.

[6]

E. BronshteinM. 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.

[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.

[8]

M. Crouhy, D. Galai and R. Mark, The Essentials of Risk Management, New Yourk: McGraw-Hill; 2005.

[9]

H. DincersU. HaciogluE. Tatoglu and D. Delen, A fuzzy-hybrid analytic model to assess investors' perceptions for industry selection, Decision Support Systems, 86 (2016), 24-34.

[10]

D. Du and P. Pardalos (eds.), Minimax and applications, Dordrecht: Kluwer; 1995. doi: 10.1007/978-1-4613-3557-3.

[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.

[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.

[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.

[14]

V. EmelichevV. Korotkov and Yu. Nikulin, Post-optimal analysis for Markowitz's multicriteria portfolio optimization problem, J. Multi-Crit. Decis. Analys., 21 (2014), 95-100.

[15]

V. EmelichevV. 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.

[16]

V. EmelichevV. 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.

[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.

[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.

[19]

V. EmelichevK. 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.

[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.

[21]

E. FernandezC. GomezG. 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.

[22]

B. Gorissenİ. Yanıkoğlu and D. den Hertog, A practical guide to robust optimization, Omega, 53 (2015), 124-137.

[23]

E. GurevskyO. 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.

[24]

M. HirschbergerR. E. SteuerS. UtzM. 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.

[25]

X. Huang and T. Zhao, Project selection and adjustment based on uncertain measure, Information Sciences, 352/353 (2016), 1-14.

[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.

[27]

K. Khalili-DamghaniS. Sadi-NezhadF. H. Lotfi and M. Tavana, A hybrid fuzzy rule-based multi-criteria framework for sustainable project portfolio selection, Information Sciences, 220 (2013), 442-462.

[28]

M. KoudstaalR. Sloof and M. van Praag, Risk, uncertainty, and entrepreneurship: Evidence from a lab-in-the-field experiment, Management Science, 62 (2015), 2897-2915.

[29]

L. KozeratskaJ. ForbesR. 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.

[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.

[31]

T. LebedevaN. 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.

[32]

M. LiburaE.S. van der PoortG. 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.

[33]

H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, New York: Willey; 1991.

[34]

G. MavrotasJ. R. Figueira and E. Siskos, Robustness analysis methodology for multi-objective combinatorial optimization problems and application to project selection, Omega, 52 (2015), 142-155.

[35]

K. Miettinen, Nonlinear Multiobjective Optimization, Boston: Kluwer; 1999.

[36]

A. MishraS. 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.

[37]

M. Note, Project Management for Information Professionals, Waltham-Kidlington: Chandos; 2016.

[38]

D. L. Olson and D. D. Wu, Enterprise Risk Management Models, Berlin-Heidelberg: Springer-Verlag; 2017.

[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.

[40]

D. PowerR. KlassenT. 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.

[41]

R. RamaswamyJ. 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.

[42]

L. Savage, The Foundations of Statistics, New York: Dover; 1972.

[43]

J. A. SefairaC. Y. MándezbO. BabatcA. 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.

[44]

P. Sironi, Modern Portfolio Management: From Markowitz to Probabilistic Scenario Optimisation, Risk books. 2015.

[45]

P. Soberanis, Risk Optimization with P-Order Conic Constraints, Ph.d. thesis University of Iowa, 2009.

[46]

Y. Sotskov, N. Sotskova, T. Lai and F. Werner, Scheduling Under Uncertainty. Theory and Algorithms, Belorusskaya nauka, Minsk, 2010.

[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.

[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.

[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.

[50]

C. Von LückenB. 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.

[51]

P. Yu, Multiple-criteria Decision Making: Concepts, Techniques, and Extensions, New York: Plenum Press; 1985. doi: 10.1007/978-1-4684-8395-6.

Figure 1.  Values for $\varphi^s_1(x^0,m,p,\infty,\infty)$
Figure 2.  Values for $\psi^s_1(x^0,m,p,\infty,\infty)$
Figure 3.  Values for $\varphi^s_2(x^0,m,\infty,p,\infty)$
Figure 4.  Values for $\psi^s_2(x^0,m,\infty,p,\infty)$
Figure 5.  Values for $\varphi^s_3(x^0,m,\infty,\infty,p)$
Figure 6.  Values for $\psi^s_3(x^0,m,\infty,\infty,p)$
Table 1.  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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