Multi-stage distributionally robust optimization with risk aversion

Ripeng Huang; Shaojian Qu; Xiaoguang Yang; Zhimin Liu

doi:10.3934/jimo.2019109

Article Contents

2021, Volume 17, Issue 1: 233-259. Doi: 10.3934/jimo.2019109

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Multi-stage distributionally robust optimization with risk aversion

1.
Business School, University of Shanghai for Science and Technology, Shanghai, China
2.
School of Mathematics and Finance, Chuzhou University, Anhui, China
3.
Academy of Mathematics and Systems Science CAS, Beijing, China

^* Corresponding author: Shaojian Qu
^* Corresponding author: Shaojian Qu

Received: October 31, 2018

Revised: March 31, 2019

Early access: September 2019

Published: January 2021

The first author is supported by National Natural Science Foundation of China (71571055)

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Abstract

Two-stage risk-neutral stochastic optimization problem has been widely studied recently. The goals of our research are to construct a two-stage distributionally robust optimization model with risk aversion and to extend it to multi-stage case. We use a coherent risk measure, Conditional Value-at-Risk, to describe risk. Due to the computational complexity of the nonlinear objective function of the proposed model, two decomposition methods based on cutting planes algorithm are proposed to solve the two-stage and multi-stage distributional robust optimization problems, respectively. To verify the validity of the two models, we give two applications on multi-product assembly problem and portfolio selection problem, respectively. Compared with the risk-neutral stochastic optimization models, the proposed models are more robust.

Keywords:

Mathematics Subject Classification: Primary: 90C15; Secondary: 90C90.

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Figure 1. The revenue of multi-product assembly problem under uncertain demanding

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Figure 2. The revenue of multi-product assembly problem with different parameters

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Figure 3. Three-stage scenario tree with 10 different scenarios at each ancestor node

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Figure 4. Comparison of different methods

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Figure 5. Comparison of cumulative wealth with different parameters

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Table 1. The Probability of Different Demand in 10 Scenarios

Scenario	s1	s2	s3	s4	s5	s6	s7	s8	s9	s10
Product1	1253	1069	1407	1125	1293	1377	1265	1235	1155	1327
Product2	1350	1074	1122	1308	1275	1190	1390	1005	1264	1345
Product3	1446	1129	1465	1237	1459	1284	1467	1168	1082	1374
Product4	1480	1421	1175	1176	1143	1038	1065	1081	1301	1225
Product5	1274	1127	1098	1416	1379	1027	1284	1397	1131	1041
Probability	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1

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Table 2. The Optimal Solution of Single-stage Multi-product Assembly Problem

NO.	1	2	3	4	5	6	7
Pre-order quantity	8671.6	9907.3	9975.8	8664.7	11265	11212	7439.3
Unused quantity	0	0	0	0	0	0	0
Units produced	1250.6	1232.3	1311.1	1210.5	1217.4	-	-

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Table 3. The Optimal Solution of Two-stage Stochastic Multi-product Assembly Problem

NO.	1	2	3	4	5	6	7
Pre-order quantity	8719.4	9976.1	10008	8691.5	11304	11265	7487.1
Unused quantity	0	0	0	0	0	0	0
Units produced	1250.6	1232.3	1316.9	1210.5	1238.4	-	-

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Table 4. The Optimal Solution of Two-stage Distributionally Robust Optimization with Risk Aversion Multi-product Assembly Problem

NO.	1	2	3	4	5	6	7
Case 1 $ \lambda $=0.5	$ \rho $=0.5 $ \beta $=0.95
Pre-order quantity	8279	9568	9496	8281	10744	10785	7137
Unused quantity	0	0	0	0	0	0	0
Units produced	1258	1142.3	1214.5	1175.6	1173.1	-	-
Case 2 $ \lambda $=0.2	$ \rho $=0.5 $ \beta $=0.95
Pre-order quantity	8556	9912	9831	8546	11184	11187	7402
Unused quantity	0	0	0	0	0	0	0
Units produced	1278.6	1154	1285.3	1221.5	1231.2	-	-
Case 3 $ \lambda $=0.8	$ \rho $=0.5 $ \beta $=0.95
Pre-order quantity	8215	9487	9452	8224	10714	10723	7084
Unused quantity	0	0	0	0	0	0	0
Units produced	1245.5	1131.2	1227.3	1166.1	1157	-	-
Case 4 $ \rho $=0.1	$ \lambda $=0.5 $ \beta $=0.95
Pre-order quantity	8266	9550	9482	8268	10729	10766	7125
Unused quantity	0	0	0	0	0	0	0
Units produced	1252.5	1141	1213.7	1174.1	1172	-	-
Case 5 $ \rho $=1	$ \lambda $=0.5 $ \beta $=0.95
Pre-order quantity	8340	9625	9561	8336	10808	10846	7184
Unused quantity	0	0	0	0	0	0	0
Units produced	1258.3	1156.6	1225.5	1178	1182.7	-	-
Case 6 $ \rho $=0.05	$ \lambda $=0.5 $ \beta $=0.95
Pre-order quantity	8262	9543	9481	8263	10731	10762	7122
Unused quantity	0	0	0	0	0	0	0
Units produced	1249.2	1140.2	1218.1	1172.1	1171.3	-	-
Case 7 $ \beta $=0.90	$ \lambda $=0.5 $ \rho $=0.5
Pre-order quantity	8375	9678	9637	8417	10952	10940	7234
Unused quantity	0	0	0	0	0	0	0
Units produced	1250.5	1141	1220.6	1235.1	1193.5	-	-
Case 8 $ \beta $=0.99	$ \lambda $=0.5 $ \rho $=0.5
Pre-order quantity	8300	9593	9513	8299	10763	10806	7156
Unused quantity	0	0	0	0	0	0	0
Units produced	1255.6	1144	1213.7	1180.5	1181.1	-	-

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