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

Alireza Goli; Hasan Khademi Zare; Reza Tavakkoli-Moghaddam; Ahmad Sadeghieh

doi:10.3934/naco.2019014

Article Contents

2019, Volume 9, Issue 2: 187-209. Doi: 10.3934/naco.2019014

This issue Previous Article Iterative methods for solving large sparse Lyapunov equations and application to model reduction of index 1 differential-algebraic-equations Next Article Homotopy method for matrix rank minimization based on the matrix hard thresholding method

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: January 31, 2018

Revised: July 31, 2018

Published: January 2019

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Pseudocode of IWO algorithm

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Figure 2. An example of solution representation

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Figure 3. Efficiency frontier of portfolio selection problem with 10 products

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Figure 4. Efficiency frontier of portfolio selection problem with 50 products

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Figure 5. Comparison of objective function values obtained from the deterministic model and the robust models with different uncertainty levels for K = 10

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Figure 6. Comparison of objective function values obtained from the deterministic model and the robust models with different uncertainty levels for K = 50

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Figure 7. The execution time of CPLEX, IWO and Ga

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Figure 8. GA and IWO portfolio efficient frontier

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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

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Table 2. Computational results of CPLEX, GA and IWO in the small-scale problems

Problem	K	CPLEX		GA			IWO
Problem	K	Z	T	Z	T	GAP(%)	Z	T	GAP(%)
P1	2	-0.678	1.6	-0.678	0.03	0	-0.678	0.347	0
P2	4	-1.112	5.6	-1.106	0.154	0.54	-1.112	0.516	0
P3	6	-1.487	21.7	-1.471	0.303	1.08	-1.468	0.682	1.28
P4	8	-1.992	79.5	-1.971	0.591	1.05	-1.971	0.792	1.05
P5	10	-2.786	162.1	-2.661	0.724	4.49	-2.6914	0.998	3.40
P6	12	-3.785	729.3	-3.662	1.371	3.25	-3.699	1.947	2.27
P7	14	-5.211	1680.9	-4.997	4.274	4.11	-5.106	3.042	2.01
P8	16	-8.633	3600	-8.012	5.291	7.19	-8.475	5.708	1.83
P9	18	-	-	-9.547	7.264	-	-9.827	8.066	-
P10	20	-	-	-11.287	11.919	-	-11.531	13.281	-
Average		-	785.0875	-	3.1921	2.171		3.5379	1.184

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Table 3. Computational results of GA and IWO in the large-scale problems

0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1	Risk aversion factor
8.24	7.39	5.99	4.89	4.98	1.53	1.24	0.83	0.82	0.33	0.21	Return	GA
38.09	15.3	7.9	5.39	5.18	2.16	2.07	1.59	1.6	1.59	1.58	Risk	GA
10.02	8.52	7.36	6.62	6.45	3.1	2.94	2.37	2.35	1.87	1.78	Return	IWO
37.1	14.58	7.75	5.5	5.01	2.03	1.95	1.6	1.53	1.65	1.54	Risk	IWO

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Table 4. Summarized results of IWO algorithm for the deterministic portfolio selection problem with 10 products

Risk aversion factor	Total return	Total risk	Objective function Value	Execution time
0	1.29630	0.69760	-56.99680	0.95800
0.1	1.19926	0.60310	-50.19030	1.39730
0.2	1.17887	0.49650	-40.04700	1.05580
0.3	1.08299	0.46870	-32.99290	0.95450
0.4	1.06606	0.44740	-26.26950	1.00600
0.5	1.02268	0.44230	-18.88370	0.79310
0.6	0.96123	0.43990	-11.05970	0.96800
0.7	0.87897	0.43960	-4.22660	0.85980
0.8	0.87608	0.43850	2.91500	1.02450
0.9	0.80853	0.42070	8.71360	0.92530
1	0.80043	0.39650	14.14370	1.03380
average	1.01558	0.48098	-19.53584	0.99783

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Table 5. Summarized results of IWO algorithm for the robust portfolio selection problem with 10 products ($\rho = 0.2$)

Risk aversion factor	Total return	Total risk	Objective function Value	Execution time
0	1.21507	0.66178	-51.84080	0.97237
0.1	1.20126	0.62716	-45.59330	1.44621
0.2	1.09569	0.51388	-35.16300	1.07660
0.3	1.09358	0.49401	-26.80290	1.07190
0.4	1.05358	0.49130	-19.26790	1.10600
0.5	1.01167	0.49102	-15.62370	0.88859
0.6	0.94073	0.48330	-8.05990	1.06238
0.7	0.90638	0.47573	0.99940	0.94625
0.8	0.86803	0.46300	9.80500	1.15400
0.9	0.80883	0.45101	15.94460	1.22232
1	0.79322	0.40662	19.17160	1.06134
Average	0.99891	0.50535	-14.22099	1.09163

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Table 6. Summarized results of IWO algorithm for the robust portfolio selection problem with 10 products ($\rho = 0.4$)

Risk aversion factor	Total return	Total risk	Objective function Value	Execution time
0	1.26527	0.69400	-50.32180	1.05357
0.1	1.20462	0.64976	-45.06690	1.68514
0.2	1.13801	0.59878	-36.04860	1.13963
0.3	1.07096	0.55574	-24.82890	1.01938
0.4	0.99564	0.53676	-18.00080	1.27179
0.5	0.93429	0.53060	-9.88430	1.01640
0.6	0.84920	0.51546	-5.03810	1.06400
0.7	0.83977	0.48991	-0.62960	1.10240
0.8	0.75594	0.44834	10.17900	1.23596
0.9	0.72847	0.39742	16.83960	1.04254
1	0.72423	0.39170	20.17070	1.20169
Average	0.95513	0.52804	-12.96634	1.16659

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Table 7. Summarized results of IWO algorithm for the robust portfolio selection problem with 10 products ($\rho = 0.6$)

Risk aversion factor	Total return	Total risk	Objective function Value	Execution time
0	1.22255	0.76641	-45.06180	1.24743
0.1	1.18229	0.72025	-37.85050	1.72660
0.2	1.09349	0.72213	-32.92260	1.47810
0.3	0.99892	0.64398	-19.87430	1.36637
0.4	0.96053	0.64205	-10.73980	1.26848
0.5	0.92569	0.57386	-1.75830	1.01640
0.6	0.90423	0.57049	3.98354	1.06400
0.7	0.85309	0.56688	1.54040	1.10240
0.8	0.78659	0.45533	16.30550	1.21371
0.9	0.75671	0.43173	25.85560	1.27669
1	0.75242	0.41958	28.11682	1.45525
Average	0.94877	0.59206	-6.58231	1.29231

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Table 8. Summarized results of IWO algorithm for the deterministic portfolio selection problem with 50 products

Risk aversion factor	Total return	Total risk	Objective function Value	Execution time
0	1.4327	0.6733	-32.3022	0.9262
0.1	1.3333	0.6593	-29.6012	1.0732
0.2	1.2003	0.5725	-23.4824	1.0453
0.3	1.1757	0.5570	-19.8822	1.1412
0.4	1.1636	0.5561	-15.9330	1.1919
0.5	1.1184	0.5254	-11.2173	0.7735
0.6	1.0364	0.4975	-6.4951	0.9450
0.7	1.0268	0.4714	-2.2812	0.7081
0.8	0.9291	0.4580	1.6748	1.0726
0.9	0.9080	0.4226	4.6130	1.0580
1	0.8963	0.4052	7.7444	0.9811
Average	1.1110	0.5271	-11.5602	0.9924

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Table 9. Summarized results of IWO algorithm for the robust portfolio selection problem with 50 products ($\rho = 0.2$)

Risk aversion factor	Total return	Total risk	Objective function Value	Execution time
0	1.4913	0.7941	-29.2069	1.1059
0.1	1.4243	0.7584	-25.6067	1.1246
0.2	1.3192	0.6295	-19.4184	1.1503
0.3	1.1959	0.5930	-14.7174	1.2152
0.4	1.1603	0.5929	-12.8385	1.3426
0.5	1.1007	0.5813	-4.1561	1.0164
0.6	1.0427	0.5296	1.6449	1.0640
0.7	0.9459	0.4873	3.8428	1.1024
0.8	0.9456	0.4772	6.3238	1.1423
0.9	0.9126	0.4655	10.6742	1.2695
1	0.9099	0.4435	10.7638	1.1976
Average	1.1317	0.5775	-6.6086	1.1573

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Table 10. Summarized results of IWO algorithm for the robust portfolio selection problem with 50 products ($\rho = 0.4$)

Risk aversion factor	Total return	Total risk	Objective function Value	Execution time
0	1.4954	0.8165	-27.1532	1.0138
0.1	1.3815	0.7981	-25.3369	1.1246
0.2	1.2973	0.7045	-18.3760	1.1505
0.3	1.1837	0.6121	-13.7562	1.2547
0.4	1.1388	0.6095	-9.9320	1.2114
0.5	1.0932	0.5819	-3.1832	1.0164
0.6	0.9935	0.5793	-0.4791	1.0640
0.7	0.9866	0.5776	2.8418	1.1024
0.8	0.9594	0.5133	10.7808	1.3515
0.9	0.9504	0.5106	12.6253	1.2808
1	0.8817	0.4496	11.7754	1.0439
Average	1.1238	0.6139	-5.4721	1.1467

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Table 11. Summarized results of IWO algorithm for the robust portfolio selection problem with 50 products ($\rho = 0.6$)

Risk aversion factor	Total return	Total risk	Objective function Value	Execution time
0	1.4189	0.8471	-26.1382	1.0433
0.1	1.3362	0.7674	-23.4752	1.1246
0.2	1.2420	0.7352	-19.2514	1.3819
0.3	1.1947	0.7170	-12.8606	1.3767
0.4	1.1767	0.7040	-8.8690	1.3882
0.5	1.1766	0.6931	-2.0909	1.0164
0.6	1.1491	0.6668	1.5295	1.0640
0.7	1.0411	0.6338	2.3588	1.1024
0.8	1.0095	0.5640	6.8008	1.1792
0.9	0.9435	0.5015	11.6470	1.2315
1	0.9186	0.4455	16.9694	1.6090
Average	1.1461	0.6614	-4.8527	1.2288

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