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June 2019, 9(2): 187-209. doi: 10.3934/naco.2019014

Application of robust optimization for a product portfolio problem using an invasive weed optimization algorithm

1. 

Department of Industrial Engineering, Yazd University, Saffayieh, Yazd, Iran

2. 

School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran

 

Received  February 2018 Revised  August 2018 Published  January 2019

Product portfolio optimization (PPO) is a strategic decision for many organizations. There are several technical methods for facilitating this decision. According to the reviewed studies, the implementation of the robust optimization approach and the invasive weed optimization (IWO) algorithm is the research gap in this field. The contribution of this paper is the development of the PPO problem with the help of the robust optimization approach and the multi-objective IWO algorithm. Considering the profit margin uncertainty in real-world investment decisions, the robust optimization approach is used to address this issue. To illustrate the real-world applicability of the model, it is implemented for dairy products of Pegah Golpayegan Company in Iran. The numerical results obtained from the IWO algorithm demonstrate the effectiveness of the proposed algorithm in tracing out the efficiency frontier of the product portfolio. The average risk of efficient frontier solutions in the deterministic model is about 0.4 and for the robust counterpart formulation is at least 0.5 per product. The efficient frontier solutions obtained from robust counterpart formulation demonstrate a more realistic risk level than the deterministic model. The comparisons between CPLEX, IWO and genetic algorithm (GA) shows that the performance of the IWO algorithm is much better than the older algorithms and can be considered as an alternative to algorithms, such as GA in product portfolio optimization problems.

Citation: Alireza Goli, Hasan Khademi Zare, Reza Tavakkoli-Moghaddam, Ahmad Sadeghieh. Application of robust optimization for a product portfolio problem using an invasive weed optimization algorithm. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 187-209. doi: 10.3934/naco.2019014
References:
[1]

A. Ben-Tal and A. Nemirovski, Selected topics in robust convex optimization, Mathematical Programming, 112 (2008), 125-158. doi: 10.1007/s10107-006-0092-2.

[2] A. Ben-TalL. El Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, 2009. doi: 10.1515/9781400831050.
[3]

D. Bertsimas and M. Sim, The price of robustness, Operations Research, 52 (2004), 35-53. doi: 10.1287/opre.1030.0065.

[4]

D. Bertsimas and D. Pachamanova, Robust multiperiod portfolio management in the presence of transaction costs, Computers & Operations Research, 35 (2008), 3–17. doi: 10.1016/j.cor.2006.02.011.

[5]

R. N. Cardozo and D. K. Smith Jr, Applying financial portfolio theory to product portfolio decisions: An empirical study, The Journal of Marketing, (1983), 110–119.

[6]

L. Cruz, E. Fernandez, C. Gomez, G. Rivera and F. Perez, Many-objective portfolio optimization of interdependent projects with'a priori'incorporation of decision-maker preferences, Applied Mathematics & Information Sciences, 8 (2014), 1517–1526.

[7]

T. Cura, Particle swarm optimization approach to portfolio optimization, Nonlinear Analysis: Real World Applications, 10 (2009), 2396-2406. doi: 10.1016/j.nonrwa.2008.04.023.

[8]

L. CzaplewskiR. BaxM. ClokieM. DawsonH. FairheadV.A. FischettiS. FosterB.F. GilmoreR.E. Hancock and D. Harper, Alternatives to antibioticsa pipeline portfolio review, The Lancet Infectious Diseases, 16 (2016), 239-251.

[9]

S. El-Bizri and N. Mansour, Metaheuristics for Portfolio Optimization, in: International Conference in Swarm Intelligence, Springer, (2017), 77–84.

[10]

H. N. EsfahaniM. h. Sobhiyah and V. R. Yousefi, Project portfolio selection via harmony search algorithm and modern portfolio theory, Procedia - Social and Behavioral Sciences, 226 (2016), 51-58.

[11]

A. Fernández and S. Gómez, Portfolio selection using neural networks, Computers & Operations Research, 34 (2007), 1177–1191.

[12]

R. FernandesJ. B. Gouveia and C. Pinho, Product mix strategy and manufacturing flexibility, Journal of Manufacturing Systems, 31 (2012), 301-311.

[13]

A. Goli and S. M. R. Davoodi, Coordination policy for production and delivery scheduling in the closed loop supply chain, Production Engineering, 1 (2018), 1-11.

[14]

A. GoliA. Aazami and A. Jabbarzadeh, Accelerated cuckoo optimization algorithm for capacitated vehicle routing problem in competitive conditions, International Journal of Artificial Intelligence, 16 (2018), 88-112.

[15]

N. Gülpnıar and E. Çanakoğlu, Robust portfolio selection problem under temperature uncertainty, European Journal of Operational Research, 256 (2017), 500-523. doi: 10.1016/j.ejor.2016.05.046.

[16]

M. A. Lejeune and S. Shen, Multi-objective probabilistically constrained programs with variable risk: Models for multi-portfolio financial optimization, European Journal of Operational Research, 252 (2016), 522-539. doi: 10.1016/j.ejor.2016.01.039.

[17]

C.-C. Lin and Y.-T. Liu, Genetic algorithms for portfolio selection problems with minimum transaction lots, European Journal of Operational Research, 185 (2008), 393-404.

[18]

L. L. MacedoP. Godinho and M. J. Alves, Mean-semivariance portfolio optimization with multiobjective evolutionary algorithms and technical analysis rules, Expert Systems with Applications, 79 (2017), 33-43.

[19]

H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91.

[20]

H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, Cowles Foundation Monograph, No. 16, John Wiley & Sons, 1959.

[21]

H. M. Markowitz and G. P. Todd, Mean-variance Analysis in Portfolio Choice and Capital Markets, John Wiley & Sons, 2000.

[22]

D. Maringer and P. Parpas, Global optimization of higher order moments in portfolio selection, Journal of Global Optimization, 43 (2009), 219-230. doi: 10.1007/s10898-007-9224-3.

[23]

A. R. Mehrabian and C. Lucas, A novel numerical optimization algorithm inspired from weed colonization, Ecological Informatics, 1 (2006), 355-366. doi: 10.1016/B978-0-12-416743-8.00001-4.

[24]

M. MontajabihaA. Arshadi Khamseh and B. Afshar-Nadjafi, A robust algorithm for project portfolio selection problem using real options valuation, International Journal of Managing Projects in Business, 10 (2017), 386-403.

[25]

J. M. MulveyR. J. Vanderbei and S. A. Zenios, Robust optimization of large-scale systems, Operations Research, 43 (1995), 264-281. doi: 10.1287/opre.43.2.264.

[26]

G. V. Pai and T. Michel, Metaheuristic optimization of constrained large portfolios using hybrid particle swarm optimization, International Journal of Applied Metaheuristic Computing, 8 (2017), 1-23.

[27]

A. PonsichA. L. Jaimes and C. A. C. Coello, A survey on multiobjective evolutionary algorithms for the solution of the portfolio optimization problem and other finance and economics applications, IEEE Transactions on Evolutionary Computation, 17 (2013), 321-344.

[28]

B. QuQ. ZhouJ. XiaoJ. Liang and P. Suganthan, Large-scale portfolio optimization using multiobjective evmolutionary algorithms and preselection methods, Mathematical Problems in Engineering, 17 (2017), 1-14.

[29]

S. M. SeyedhosseiniM. J. Esfahani and M. Ghaffari, A novel hybrid algorithm based on a harmony search and artificial bee colony for solving a portfolio optimization problem using a mean-semi variance approach, Journal of Central South University, 23 (2016), 181-188.

[30]

F. SolatikiaE. Kiliç and G. W. Weber, Fuzzy optimization for portfolio selection based on embedding theorem in fuzzy normed linear spaces, Organizacija, 47 (2014), 90-97.

[31]

A. L. Soyster, Convex programming with set-inclusive constraints and applications to inexact linear programming, Operations Research, 21 (1973), 1154-1157. doi: 10.1287/opre.22.4.892.

[32]

M. A. Takami, R. Sheikh and S. S. Sana, Product portfolio optimisation using teaching learning-based optimisation algorithm: a new approach in supply chain management, International Journal of Systems Science: Operations & Logistics, 3 (2015), 236–246.

[33]

E. B. Tirkolaee, A. Goli, M. Bakhsi and I. Mahdavi, A robust multi-trip vehicle routing problem of perishable products with intermediate depots and time windows, Numerical Algebra, Control & Optimization, 7 (2017), 417–433. doi: 10.3934/naco.2017026.

[34]

M. Tuba and N. Bacanin, Upgraded firefly algorithm for portfolio optimization problem, in: Computer Modelling and Simulation, 2014 UKSim-AMSS 16th International Conference on, IEEE, (2014), 113–118.

show all references

References:
[1]

A. Ben-Tal and A. Nemirovski, Selected topics in robust convex optimization, Mathematical Programming, 112 (2008), 125-158. doi: 10.1007/s10107-006-0092-2.

[2] A. Ben-TalL. El Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, 2009. doi: 10.1515/9781400831050.
[3]

D. Bertsimas and M. Sim, The price of robustness, Operations Research, 52 (2004), 35-53. doi: 10.1287/opre.1030.0065.

[4]

D. Bertsimas and D. Pachamanova, Robust multiperiod portfolio management in the presence of transaction costs, Computers & Operations Research, 35 (2008), 3–17. doi: 10.1016/j.cor.2006.02.011.

[5]

R. N. Cardozo and D. K. Smith Jr, Applying financial portfolio theory to product portfolio decisions: An empirical study, The Journal of Marketing, (1983), 110–119.

[6]

L. Cruz, E. Fernandez, C. Gomez, G. Rivera and F. Perez, Many-objective portfolio optimization of interdependent projects with'a priori'incorporation of decision-maker preferences, Applied Mathematics & Information Sciences, 8 (2014), 1517–1526.

[7]

T. Cura, Particle swarm optimization approach to portfolio optimization, Nonlinear Analysis: Real World Applications, 10 (2009), 2396-2406. doi: 10.1016/j.nonrwa.2008.04.023.

[8]

L. CzaplewskiR. BaxM. ClokieM. DawsonH. FairheadV.A. FischettiS. FosterB.F. GilmoreR.E. Hancock and D. Harper, Alternatives to antibioticsa pipeline portfolio review, The Lancet Infectious Diseases, 16 (2016), 239-251.

[9]

S. El-Bizri and N. Mansour, Metaheuristics for Portfolio Optimization, in: International Conference in Swarm Intelligence, Springer, (2017), 77–84.

[10]

H. N. EsfahaniM. h. Sobhiyah and V. R. Yousefi, Project portfolio selection via harmony search algorithm and modern portfolio theory, Procedia - Social and Behavioral Sciences, 226 (2016), 51-58.

[11]

A. Fernández and S. Gómez, Portfolio selection using neural networks, Computers & Operations Research, 34 (2007), 1177–1191.

[12]

R. FernandesJ. B. Gouveia and C. Pinho, Product mix strategy and manufacturing flexibility, Journal of Manufacturing Systems, 31 (2012), 301-311.

[13]

A. Goli and S. M. R. Davoodi, Coordination policy for production and delivery scheduling in the closed loop supply chain, Production Engineering, 1 (2018), 1-11.

[14]

A. GoliA. Aazami and A. Jabbarzadeh, Accelerated cuckoo optimization algorithm for capacitated vehicle routing problem in competitive conditions, International Journal of Artificial Intelligence, 16 (2018), 88-112.

[15]

N. Gülpnıar and E. Çanakoğlu, Robust portfolio selection problem under temperature uncertainty, European Journal of Operational Research, 256 (2017), 500-523. doi: 10.1016/j.ejor.2016.05.046.

[16]

M. A. Lejeune and S. Shen, Multi-objective probabilistically constrained programs with variable risk: Models for multi-portfolio financial optimization, European Journal of Operational Research, 252 (2016), 522-539. doi: 10.1016/j.ejor.2016.01.039.

[17]

C.-C. Lin and Y.-T. Liu, Genetic algorithms for portfolio selection problems with minimum transaction lots, European Journal of Operational Research, 185 (2008), 393-404.

[18]

L. L. MacedoP. Godinho and M. J. Alves, Mean-semivariance portfolio optimization with multiobjective evolutionary algorithms and technical analysis rules, Expert Systems with Applications, 79 (2017), 33-43.

[19]

H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91.

[20]

H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, Cowles Foundation Monograph, No. 16, John Wiley & Sons, 1959.

[21]

H. M. Markowitz and G. P. Todd, Mean-variance Analysis in Portfolio Choice and Capital Markets, John Wiley & Sons, 2000.

[22]

D. Maringer and P. Parpas, Global optimization of higher order moments in portfolio selection, Journal of Global Optimization, 43 (2009), 219-230. doi: 10.1007/s10898-007-9224-3.

[23]

A. R. Mehrabian and C. Lucas, A novel numerical optimization algorithm inspired from weed colonization, Ecological Informatics, 1 (2006), 355-366. doi: 10.1016/B978-0-12-416743-8.00001-4.

[24]

M. MontajabihaA. Arshadi Khamseh and B. Afshar-Nadjafi, A robust algorithm for project portfolio selection problem using real options valuation, International Journal of Managing Projects in Business, 10 (2017), 386-403.

[25]

J. M. MulveyR. J. Vanderbei and S. A. Zenios, Robust optimization of large-scale systems, Operations Research, 43 (1995), 264-281. doi: 10.1287/opre.43.2.264.

[26]

G. V. Pai and T. Michel, Metaheuristic optimization of constrained large portfolios using hybrid particle swarm optimization, International Journal of Applied Metaheuristic Computing, 8 (2017), 1-23.

[27]

A. PonsichA. L. Jaimes and C. A. C. Coello, A survey on multiobjective evolutionary algorithms for the solution of the portfolio optimization problem and other finance and economics applications, IEEE Transactions on Evolutionary Computation, 17 (2013), 321-344.

[28]

B. QuQ. ZhouJ. XiaoJ. Liang and P. Suganthan, Large-scale portfolio optimization using multiobjective evmolutionary algorithms and preselection methods, Mathematical Problems in Engineering, 17 (2017), 1-14.

[29]

S. M. SeyedhosseiniM. J. Esfahani and M. Ghaffari, A novel hybrid algorithm based on a harmony search and artificial bee colony for solving a portfolio optimization problem using a mean-semi variance approach, Journal of Central South University, 23 (2016), 181-188.

[30]

F. SolatikiaE. Kiliç and G. W. Weber, Fuzzy optimization for portfolio selection based on embedding theorem in fuzzy normed linear spaces, Organizacija, 47 (2014), 90-97.

[31]

A. L. Soyster, Convex programming with set-inclusive constraints and applications to inexact linear programming, Operations Research, 21 (1973), 1154-1157. doi: 10.1287/opre.22.4.892.

[32]

M. A. Takami, R. Sheikh and S. S. Sana, Product portfolio optimisation using teaching learning-based optimisation algorithm: a new approach in supply chain management, International Journal of Systems Science: Operations & Logistics, 3 (2015), 236–246.

[33]

E. B. Tirkolaee, A. Goli, M. Bakhsi and I. Mahdavi, A robust multi-trip vehicle routing problem of perishable products with intermediate depots and time windows, Numerical Algebra, Control & Optimization, 7 (2017), 417–433. doi: 10.3934/naco.2017026.

[34]

M. Tuba and N. Bacanin, Upgraded firefly algorithm for portfolio optimization problem, in: Computer Modelling and Simulation, 2014 UKSim-AMSS 16th International Conference on, IEEE, (2014), 113–118.

Figure 1.  Pseudocode of IWO algorithm
Figure 2.  An example of solution representation
Figure 3.  Efficiency frontier of portfolio selection problem with 10 products
Figure 4.  Efficiency frontier of portfolio selection problem with 50 products
Figure 5.  Comparison of objective function values obtained from the deterministic model and the robust models with different uncertainty levels for K = 10
Figure 6.  Comparison of objective function values obtained from the deterministic model and the robust models with different uncertainty levels for K = 50
Figure 7.  The execution time of CPLEX, IWO and Ga
Figure 8.  GA and IWO portfolio efficient frontier
Table 1.  Optimal setting of IWO Algorithm parameters
Parameter Symbol Optimal Value
Max iteration Maxit 100
Number of first population Npop0 50
Maximum population size Pmax 100
Minimum seed Smin 20
Maximum seed Smax 50
Reducing power of standard deviation (pu) N 0.01
Initial standard deviation Sigma_final 0.5
Final standard deviation Sigma_final 0.1
Parameter Symbol Optimal Value
Max iteration Maxit 100
Number of first population Npop0 50
Maximum population size Pmax 100
Minimum seed Smin 20
Maximum seed Smax 50
Reducing power of standard deviation (pu) N 0.01
Initial standard deviation Sigma_final 0.5
Final standard deviation Sigma_final 0.1
Table 2.  Computational results of CPLEX, GA and IWO in the small-scale problems
ProblemKCPLEXGAIWO
ZTZTGAP(%)ZTGAP(%)
P12-0.6781.6-0.6780.030-0.6780.3470
P24-1.1125.6-1.1060.1540.54-1.1120.5160
P36-1.48721.7-1.4710.3031.08-1.4680.6821.28
P48-1.99279.5-1.9710.5911.05-1.9710.7921.05
P510-2.786162.1-2.6610.7244.49-2.69140.9983.40
P612-3.785729.3-3.6621.3713.25-3.6991.9472.27
P714-5.2111680.9-4.9974.2744.11-5.1063.0422.01
P816-8.6333600-8.0125.2917.19-8.4755.7081.83
P918---9.5477.264--9.8278.066-
P1020---11.28711.919--11.53113.281-
Average-785.0875-3.19212.1713.53791.184
ProblemKCPLEXGAIWO
ZTZTGAP(%)ZTGAP(%)
P12-0.6781.6-0.6780.030-0.6780.3470
P24-1.1125.6-1.1060.1540.54-1.1120.5160
P36-1.48721.7-1.4710.3031.08-1.4680.6821.28
P48-1.99279.5-1.9710.5911.05-1.9710.7921.05
P510-2.786162.1-2.6610.7244.49-2.69140.9983.40
P612-3.785729.3-3.6621.3713.25-3.6991.9472.27
P714-5.2111680.9-4.9974.2744.11-5.1063.0422.01
P816-8.6333600-8.0125.2917.19-8.4755.7081.83
P918---9.5477.264--9.8278.066-
P1020---11.28711.919--11.53113.281-
Average-785.0875-3.19212.1713.53791.184
Table 3.  Computational results of GA and IWO in the large-scale problems
00.10.20.30.40.50.60.70.80.91Risk aversion factor
8.247.395.994.894.981.531.240.830.820.330.21ReturnGA
38.0915.37.95.395.182.162.071.591.61.591.58Risk
10.028.527.366.626.453.12.942.372.351.871.78ReturnIWO
37.114.587.755.55.012.031.951.61.531.651.54Risk
00.10.20.30.40.50.60.70.80.91Risk aversion factor
8.247.395.994.894.981.531.240.830.820.330.21ReturnGA
38.0915.37.95.395.182.162.071.591.61.591.58Risk
10.028.527.366.626.453.12.942.372.351.871.78ReturnIWO
37.114.587.755.55.012.031.951.61.531.651.54Risk
Table 4.  Summarized results of IWO algorithm for the deterministic portfolio selection problem with 10 products
Risk aversion factorTotal returnTotal riskObjective function ValueExecution time
01.296300.69760-56.996800.95800
0.11.199260.60310-50.190301.39730
0.21.178870.49650-40.047001.05580
0.31.082990.46870-32.992900.95450
0.41.066060.44740-26.269501.00600
0.51.022680.44230-18.883700.79310
0.60.961230.43990-11.059700.96800
0.70.878970.43960-4.226600.85980
0.80.876080.438502.915001.02450
0.90.808530.420708.713600.92530
10.800430.3965014.143701.03380
average1.015580.48098-19.535840.99783
Risk aversion factorTotal returnTotal riskObjective function ValueExecution time
01.296300.69760-56.996800.95800
0.11.199260.60310-50.190301.39730
0.21.178870.49650-40.047001.05580
0.31.082990.46870-32.992900.95450
0.41.066060.44740-26.269501.00600
0.51.022680.44230-18.883700.79310
0.60.961230.43990-11.059700.96800
0.70.878970.43960-4.226600.85980
0.80.876080.438502.915001.02450
0.90.808530.420708.713600.92530
10.800430.3965014.143701.03380
average1.015580.48098-19.535840.99783
Table 5.  Summarized results of IWO algorithm for the robust portfolio selection problem with 10 products ($\rho = 0.2$)
Risk aversion factorTotal returnTotal riskObjective function ValueExecution time
01.215070.66178-51.840800.97237
0.11.201260.62716-45.593301.44621
0.21.095690.51388-35.163001.07660
0.31.093580.49401-26.802901.07190
0.41.053580.49130-19.267901.10600
0.51.011670.49102-15.623700.88859
0.60.940730.48330-8.059901.06238
0.70.906380.475730.999400.94625
0.80.868030.463009.805001.15400
0.90.808830.4510115.944601.22232
10.793220.4066219.171601.06134
Average0.998910.50535-14.220991.09163
Risk aversion factorTotal returnTotal riskObjective function ValueExecution time
01.215070.66178-51.840800.97237
0.11.201260.62716-45.593301.44621
0.21.095690.51388-35.163001.07660
0.31.093580.49401-26.802901.07190
0.41.053580.49130-19.267901.10600
0.51.011670.49102-15.623700.88859
0.60.940730.48330-8.059901.06238
0.70.906380.475730.999400.94625
0.80.868030.463009.805001.15400
0.90.808830.4510115.944601.22232
10.793220.4066219.171601.06134
Average0.998910.50535-14.220991.09163
Table 6.  Summarized results of IWO algorithm for the robust portfolio selection problem with 10 products ($\rho = 0.4$)
Risk aversion factorTotal returnTotal riskObjective function ValueExecution time
01.265270.69400-50.321801.05357
0.11.204620.64976-45.066901.68514
0.21.138010.59878-36.048601.13963
0.31.070960.55574-24.828901.01938
0.40.995640.53676-18.000801.27179
0.50.934290.53060-9.884301.01640
0.60.849200.51546-5.038101.06400
0.70.839770.48991-0.629601.10240
0.80.755940.4483410.179001.23596
0.90.728470.3974216.839601.04254
10.724230.3917020.170701.20169
Average0.955130.52804-12.966341.16659
Risk aversion factorTotal returnTotal riskObjective function ValueExecution time
01.265270.69400-50.321801.05357
0.11.204620.64976-45.066901.68514
0.21.138010.59878-36.048601.13963
0.31.070960.55574-24.828901.01938
0.40.995640.53676-18.000801.27179
0.50.934290.53060-9.884301.01640
0.60.849200.51546-5.038101.06400
0.70.839770.48991-0.629601.10240
0.80.755940.4483410.179001.23596
0.90.728470.3974216.839601.04254
10.724230.3917020.170701.20169
Average0.955130.52804-12.966341.16659
Table 7.  Summarized results of IWO algorithm for the robust portfolio selection problem with 10 products ($\rho = 0.6$)
Risk aversion factorTotal returnTotal riskObjective function ValueExecution time
01.222550.76641-45.061801.24743
0.11.182290.72025-37.850501.72660
0.21.093490.72213-32.922601.47810
0.30.998920.64398-19.874301.36637
0.40.960530.64205-10.739801.26848
0.50.925690.57386-1.758301.01640
0.60.904230.570493.983541.06400
0.70.853090.566881.540401.10240
0.80.786590.4553316.305501.21371
0.90.756710.4317325.855601.27669
10.752420.4195828.116821.45525
Average0.948770.59206-6.582311.29231
Risk aversion factorTotal returnTotal riskObjective function ValueExecution time
01.222550.76641-45.061801.24743
0.11.182290.72025-37.850501.72660
0.21.093490.72213-32.922601.47810
0.30.998920.64398-19.874301.36637
0.40.960530.64205-10.739801.26848
0.50.925690.57386-1.758301.01640
0.60.904230.570493.983541.06400
0.70.853090.566881.540401.10240
0.80.786590.4553316.305501.21371
0.90.756710.4317325.855601.27669
10.752420.4195828.116821.45525
Average0.948770.59206-6.582311.29231
Table 8.  Summarized results of IWO algorithm for the deterministic portfolio selection problem with 50 products
Risk aversion factorTotal returnTotal riskObjective function ValueExecution time
01.43270.6733-32.30220.9262
0.11.33330.6593-29.60121.0732
0.21.20030.5725-23.48241.0453
0.31.17570.5570-19.88221.1412
0.41.16360.5561-15.93301.1919
0.51.11840.5254-11.21730.7735
0.61.03640.4975-6.49510.9450
0.71.02680.4714-2.28120.7081
0.80.92910.45801.67481.0726
0.90.90800.42264.61301.0580
10.89630.40527.74440.9811
Average1.11100.5271-11.56020.9924
Risk aversion factorTotal returnTotal riskObjective function ValueExecution time
01.43270.6733-32.30220.9262
0.11.33330.6593-29.60121.0732
0.21.20030.5725-23.48241.0453
0.31.17570.5570-19.88221.1412
0.41.16360.5561-15.93301.1919
0.51.11840.5254-11.21730.7735
0.61.03640.4975-6.49510.9450
0.71.02680.4714-2.28120.7081
0.80.92910.45801.67481.0726
0.90.90800.42264.61301.0580
10.89630.40527.74440.9811
Average1.11100.5271-11.56020.9924
Table 9.  Summarized results of IWO algorithm for the robust portfolio selection problem with 50 products ($\rho = 0.2$)
Risk aversion factorTotal returnTotal riskObjective function ValueExecution time
01.49130.7941-29.20691.1059
0.11.42430.7584-25.60671.1246
0.21.31920.6295-19.41841.1503
0.31.19590.5930-14.71741.2152
0.41.16030.5929-12.83851.3426
0.51.10070.5813-4.15611.0164
0.61.04270.52961.64491.0640
0.70.94590.48733.84281.1024
0.80.94560.47726.32381.1423
0.90.91260.465510.67421.2695
10.90990.443510.76381.1976
Average1.13170.5775-6.60861.1573
Risk aversion factorTotal returnTotal riskObjective function ValueExecution time
01.49130.7941-29.20691.1059
0.11.42430.7584-25.60671.1246
0.21.31920.6295-19.41841.1503
0.31.19590.5930-14.71741.2152
0.41.16030.5929-12.83851.3426
0.51.10070.5813-4.15611.0164
0.61.04270.52961.64491.0640
0.70.94590.48733.84281.1024
0.80.94560.47726.32381.1423
0.90.91260.465510.67421.2695
10.90990.443510.76381.1976
Average1.13170.5775-6.60861.1573
Table 10.  Summarized results of IWO algorithm for the robust portfolio selection problem with 50 products ($\rho = 0.4$)
Risk aversion factorTotal returnTotal riskObjective function ValueExecution time
01.49540.8165-27.15321.0138
0.11.38150.7981-25.33691.1246
0.21.29730.7045-18.37601.1505
0.31.18370.6121-13.75621.2547
0.41.13880.6095-9.93201.2114
0.51.09320.5819-3.18321.0164
0.60.99350.5793-0.47911.0640
0.70.98660.57762.84181.1024
0.80.95940.513310.78081.3515
0.90.95040.510612.62531.2808
10.88170.449611.77541.0439
Average1.12380.6139-5.47211.1467
Risk aversion factorTotal returnTotal riskObjective function ValueExecution time
01.49540.8165-27.15321.0138
0.11.38150.7981-25.33691.1246
0.21.29730.7045-18.37601.1505
0.31.18370.6121-13.75621.2547
0.41.13880.6095-9.93201.2114
0.51.09320.5819-3.18321.0164
0.60.99350.5793-0.47911.0640
0.70.98660.57762.84181.1024
0.80.95940.513310.78081.3515
0.90.95040.510612.62531.2808
10.88170.449611.77541.0439
Average1.12380.6139-5.47211.1467
Table 11.  Summarized results of IWO algorithm for the robust portfolio selection problem with 50 products ($\rho = 0.6$)
Risk aversion factorTotal returnTotal riskObjective function ValueExecution time
01.41890.8471-26.13821.0433
0.11.33620.7674-23.47521.1246
0.21.24200.7352-19.25141.3819
0.31.19470.7170-12.86061.3767
0.41.17670.7040-8.86901.3882
0.51.17660.6931-2.09091.0164
0.61.14910.66681.52951.0640
0.71.04110.63382.35881.1024
0.81.00950.56406.80081.1792
0.90.94350.501511.64701.2315
10.91860.445516.96941.6090
Average1.14610.6614-4.85271.2288
Risk aversion factorTotal returnTotal riskObjective function ValueExecution time
01.41890.8471-26.13821.0433
0.11.33620.7674-23.47521.1246
0.21.24200.7352-19.25141.3819
0.31.19470.7170-12.86061.3767
0.41.17670.7040-8.86901.3882
0.51.17660.6931-2.09091.0164
0.61.14910.66681.52951.0640
0.71.04110.63382.35881.1024
0.81.00950.56406.80081.1792
0.90.94350.501511.64701.2315
10.91860.445516.96941.6090
Average1.14610.6614-4.85271.2288
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