# American Institute of Mathematical Sciences

January  2021, 17(1): 233-259. doi: 10.3934/jimo.2019109

## 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

Received  November 2018 Revised  April 2019 Published  September 2019

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

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.

Citation: Ripeng Huang, Shaojian Qu, Xiaoguang Yang, Zhimin Liu. Multi-stage distributionally robust optimization with risk aversion. Journal of Industrial & Management Optimization, 2021, 17 (1) : 233-259. doi: 10.3934/jimo.2019109
##### References:

show all references

##### References:
The revenue of multi-product assembly problem under uncertain demanding
The revenue of multi-product assembly problem with different parameters
Three-stage scenario tree with 10 different scenarios at each ancestor node
Comparison of different methods
Comparison of cumulative wealth with different parameters
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
 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
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 - -
 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 - -
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 - -
 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 - -
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 - -
 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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