Appendix A Notations in Part I

Appendix A Notations in Part I Table A.1 List of notations used in Part I x = (x 1,,x n ) Decision variable, investment ratios X f ξ R(x, ξ) R 1,R n Z P( ) p( ) μ r i σ ij r E,r 0 R 0 t e,t t s T E T S Min Max M V VaR PVaR CVaR W-CVaR WVaR Feasible set of x Random variable(s), Random process(es), risk factors like stock prices Profit rate of portfolio x Real(1-dimension real space), n-dimension real space Set of integers Probability Price Mean of profit rate of a portfolio Mean of profit rate of asset i Covariance of R i and R j Required return value Acceptable risk value Exit time, set of possible exit times Start time Set of the exit time of an investmemt Set of the start time of an investment Minimization Maximization Mean Variance Value at Risk Period Value at Risk Conditional VaR Conditional VaR in the worst case VaR in the worst case Springer Nature Singapore Pte Ltd. 2018 C. Xu, T. Shiina, Risk Management in Finance and Logistics, Translational Systems Sciences 14, https://doi.org/10.1007/978-981-13-0317-3 173

Appendix B Historical Prices and Monthly Profit Rates of IBM and INTC: Data Used in Chap. 2 The historical prices included in this appendix were the adjusted daily close price, adjusted for both dividends and splits, of IBM (International Business Machines Corporation) & INTC (Intel Corporation) in the following period: August 1, 2016 July 31, 2017 The profit rate data of each stock is the monthly profit rate(profit rate in 22 business days), they are calculated using the historical price data. For example, the monthly profit rate of IBM stock on August 1st, 2016 is calculated as follows. R(IBM) 2016.8.1 = p(2016.8.31) p(2016.8.1) 153.09 154.23 = = 0.74% p(2016.8.1) 154.23 The price data were downloaded from Yahoo!Finance in January 8, 2018. The price data were used in Examples 2.1 2.7 of Chap. 2. Springer Nature Singapore Pte Ltd. 2018 C. Xu, T. Shiina, Risk Management in Finance and Logistics, Translational Systems Sciences 14, https://doi.org/10.1007/978-981-13-0317-3 175

176 B Historical Prices and Monthly Profit Rates of IBM and INTC: Data Used in Chap. 2 Table B.1 Prices of IBM and INTC stocks(us $) and the monthly profit rate(%) generated Date IBM INTC R(IBM) R(INTC) Date IBM INTC R(IBM) R(INTC) Date IBM INTC R(IBM) R(INTC) 2016/8/1 154.23 33.58 0.74 3.76 2016/9/29 152.35 36.23 2.80 6.56 2016/11/29 159.00 34.54 1.50 2.72 2016/8/2 153.40 33.30 0.21 5.01 2016/9/30 153.06 36.65 3.81 8.56 2016/11/30 157.72 33.94 3.06 5.48 2016/8/3 153.49 33.25 0.16 5.34 2016/10/3 151.86 36.56 3.59 8.13 2016/12/1 155.39 33.03 5.91 7.85 2016/8/4 154.33 33.57 0.11 5.75 2016/10/4 150.76 36.45 2.61 8.93 2016/12/2 155.58 33.42 5.42 6.41 2016/8/5 156.19 33.96 0.28 4.23 2016/10/5 151.35 36.88 2.96 10.86 2016/12/5 155.41 33.64 6.06 6.08 2016/8/8 156.13 34.02 1.88 4.00 2016/10/6 151.16 36.96 0.74 8.19 2016/12/6 155.91 33.96 4.55 5.44 2016/8/9 155.87 33.90 3.76 1.49 2016/10/7 149.99 36.99 0.58 8.13 2016/12/7 160.22 34.73 0.44 2.93 2016/8/10 156.17 33.52 2.34 4.49 2016/10/10 151.30 36.91 0.51 7.91 2016/12/8 160.78 34.92 1.45 3.50 2016/8/11 157.57 33.67 4.72 2.68 2016/10/11 149.15 36.18 4.45 6.73 2016/12/9 161.90 34.98 0.86 2.66 2016/8/12 156.05 33.56 4.88 3.04 2016/10/12 148.66 36.05 5.47 6.08 2016/12/12 160.91 35.19 1.11 2.28 2016/8/15 155.98 33.89 3.84 4.73 2016/10/13 148.12 35.89 3.85 6.03 2016/12/13 163.63 36.00 0.24 0.00 2016/8/16 154.84 34.18 4.27 6.99 2016/10/14 148.82 36.36 3.66 6.08 2016/12/14 163.84 35.75 1.01 0.57 2016/8/17 154.59 34.00 3.47 6.11 2016/10/17 149.13 36.20 3.85 5.86 2016/12/15 163.36 35.99 0.72 0.60 2016/8/18 155.48 33.95 4.28 6.21 2016/10/18 145.23 36.65 6.99 6.53 2016/12/16 162.11 35.52 2.29 1.74 2016/8/19 154.21 34.21 2.82 6.27 2016/10/19 145.75 34.48 7.00 0.83 2016/12/19 162.06 36.09 2.61 0.33 2016/8/22 154.17 34.33 2.43 6.19 2016/10/20 146.00 34.40 8.40 0.52 2016/12/20 162.95 36.40 4.95 1.10 2016/8/23 154.42 34.37 3.29 5.06 2016/10/21 144.17 34.13 9.70 1.70 2016/12/21 162.69 36.18 6.55 2.22 2016/8/24 153.25 34.13 3.19 4.27 2016/10/24 145.08 34.23 8.55 0.59 2016/12/22 162.43 36.13 6.94 1.71 2016/8/25 152.85 34.07 1.17 5.96 2016/10/25 145.38 34.08 9.11 1.73 2016/12/23 162.09 36.17 6.35 2.73 2016/8/26 152.55 34.23 0.02 6.18 2016/10/26 146.28 33.90 9.36 2.46 2016/12/27 162.51 36.26 5.18 0.94 2016/8/29 153.90 34.51 1.01 4.98 2016/10/27 147.76 33.80 7.61 2.20 2016/12/28 161.58 35.83 5.01 0.52 2016/8/30 153.59 34.66 0.35 5.74 2016/10/28 147.05 33.73 7.26 0.64 2016/12/29 161.98 35.86 4.62 0.38 2016/8/31 153.09 34.85 0.80 4.93 2016/10/31 148.09 33.85 4.93 2.45 2016/12/30 161.39 35.48 5.18 1.13

B Historical Prices and Monthly Profit Rates of IBM and INTC: Data Used... 177 2016/9/1 153.72 34.97 1.93 4.22 2016/11/1 147.22 33.51 5.68 0.29 2017/1/3 162.56 35.80 5.16 0.49 2016/9/2 153.73 35.03 1.55 5.29 2016/11/2 146.41 33.59 6.15 0.15 2017/1/4 164.57 35.62 3.90 0.33 2016/9/6 154.50 35.51 2.16 4.10 2016/11/3 146.81 33.19 6.19 2.33 2017/1/5 164.02 35.56 5.79 0.71 2016/9/7 155.75 35.40 3.69 4.50 2016/11/4 146.87 32.88 9.09 5.62 2017/1/6 164.83 35.69 4.74 0.44 2016/9/8 153.20 35.38 1.25 4.34 2016/11/7 150.04 33.94 7.15 2.91 2017/1/9 163.00 35.81 6.54 2.45 2016/9/9 150.01 34.41 0.58 5.16 2016/11/8 150.87 33.98 7.31 2.94 2017/1/10 160.93 35.74 8.80 2.59 2016/9/12 152.52 35.03 2.53 2.91 2016/11/9 150.52 33.99 6.91 3.51 2017/1/11 163.10 36.15 7.77 2.42 2016/9/13 150.13 34.57 1.34 3.82 2016/11/10 155.78 33.75 5.04 6.67 2017/1/12 163.30 35.91 8.10 1.43 2016/9/14 148.43 34.58 0.26 5.14 2016/11/11 156.80 33.86 4.49 5.61 2017/1/13 162.70 35.99 9.43 1.31 2016/9/15 149.99 35.50 0.57 2.00 2016/11/14 153.83 33.73 6.20 6.70 2017/1/17 163.24 36.00 8.92 0.35 2016/9/16 148.23 36.57 2.03 0.21 2016/11/15 154.27 34.15 5.08 4.01 2017/1/18 162.18 35.96 9.17 0.05 2016/9/19 149.22 36.08 2.33 4.44 2016/11/16 154.88 34.08 4.64 5.88 2017/1/19 162.19 35.77 8.92 0.58 2016/9/20 148.82 36.06 1.90 4.60 2016/11/17 155.37 34.26 4.88 6.25 2017/1/20 165.82 36.14 7.06 1.66 2016/9/21 149.86 36.36 3.79 6.14 2016/11/18 155.94 34.19 4.33 5.81 2017/1/23 166.29 35.97 7.05 0.90 2016/9/22 150.42 36.46 3.55 6.10 2016/11/21 158.26 34.22 2.64 5.57 2017/1/24 171.02 36.80 3.91 2.20 2016/9/23 149.33 36.11 2.65 5.62 2016/11/22 158.16 34.71 2.48 4.20 2017/1/25 173.35 36.98 1.42 2.72 2016/9/26 148.37 35.58 1.41 4.72 2016/11/23 157.49 34.43 3.19 5.31 2017/1/26 173.71 36.74 1.45 2.93 2016/9/27 151.05 36.10 2.18 6.37 2016/11/25 158.62 34.67 1.87 3.36 2017/1/27 172.39 37.15 3.43 4.72 2016/9/28 152.52 36.35 3.59 7.21 2016/11/28 159.96 34.74 1.26 3.24 2017/1/30 170.93 36.61 3.50 3.35

178 B Historical Prices and Monthly Profit Rates of IBM and INTC: Data Used in Chap. 2 Table B.2 Historical prices of IBM and INTC stocks and the monthly profit rate generated Date IBM INTC R(IBM) R(INTC) Date IBM INTC R(IBM) R(INTC) Date IBM INTC R(IBM) R(INTC) 2017/1/31 169.68 36.02 3.98 1.80 2017/3/31 170.65 35.54 8.91 3.29 2017/6/1 151.07 35.85 1.91 7.36 2017/2/1 169.46 35.73 4.36 1.91 2017/4/3 171.01 35.63 8.85 2.67 2017/6/2 150.46 36.05 1.07 5.45 2017/2/2 169.74 35.88 4.14 1.70 2017/4/4 171.02 35.74 11.16 2.24 2017/6/5 150.82 36.07 0.03 7.46 2017/2/3 170.95 35.98 2.87 2.46 2017/4/5 169.42 35.68 10.62 1.63 2017/6/6 150.78 35.86 0.37 6.23 2017/2/6 170.99 35.73 1.55 1.24 2017/4/6 169.00 35.50 10.93 1.69 2017/6/7 149.40 35.99 1.62 7.20 2017/2/7 173.51 35.81 0.43 1.21 2017/4/7 168.69 35.50 11.28 0.69 2017/6/8 150.51 36.21 0.72 7.02 2017/2/8 172.64 35.84 0.16 3.35 2017/4/10 167.77 35.27 11.14 0.43 2017/6/9 152.49 35.44 0.26 4.09 2017/2/9 173.66 34.94 0.84 0.79 2017/4/11 167.16 35.21 10.99 0.15 2017/6/12 153.56 35.46 1.00 4.17 2017/2/10 175.10 34.82 1.61 0.68 2017/4/12 167.24 35.10 10.35 0.74 2017/6/13 152.64 35.61 0.01 3.34 2017/2/13 175.77 35.27 1.18 1.84 2017/4/13 166.13 34.73 8.46 2.37 2017/6/14 152.20 35.27 0.52 2.98 2017/2/14 176.52 35.40 2.49 1.84 2017/4/17 167.67 34.96 10.93 0.51 2017/6/15 152.61 35.05 0.14 2.21 2017/2/15 178.04 35.52 3.29 1.72 2017/4/18 166.64 35.24 10.47 0.81 2017/6/16 153.76 34.95 5.05 1.85 2017/2/16 177.80 35.87 4.16 3.76 2017/4/19 158.45 35.38 5.09 0.69 2017/6/19 153.22 35.25 4.64 2.14 2017/2/17 177.05 35.94 3.26 3.04 2017/4/20 159.05 35.65 5.03 0.40 2017/6/20 153.33 34.60 5.08 0.37 2017/2/21 176.65 35.98 3.02 3.42 2017/4/21 157.17 35.78 4.28 0.53 2017/6/21 152.18 34.32 5.07 0.23 2017/2/22 177.52 35.54 4.04 2.52 2017/4/24 157.53 36.21 4.20 0.98 2017/6/22 152.79 34.10 5.32 0.90 2017/2/23 178.01 35.65 4.34 2.18 2017/4/25 157.18 36.33 3.55 0.92 2017/6/23 152.50 33.94 5.68 1.64 2017/2/24 177.72 35.99 3.77 2.55 2017/4/26 156.85 36.38 3.80 1.08 2017/6/26 153.61 33.82 6.55 2.64 2017/2/27 175.81 35.97 3.04 2.57 2017/4/27 157.11 36.88 4.43 2.62 2017/6/27 153.13 33.40 6.76 4.93

B Historical Prices and Monthly Profit Rates of IBM and INTC: Data Used... 179 2017/2/28 176.22 35.66 3.31 1.24 2017/4/28 157.08 35.62 3.85 0.63 2017/6/28 153.70 33.95 6.86 3.71 2017/3/1 178.31 35.40 4.29 0.39 2017/5/1 155.66 35.77 2.95 0.22 2017/6/29 152.52 33.29 2017/3/2 176.91 35.38 3.34 0.70 2017/5/2 155.91 36.42 3.50 1.03 2017/6/30 152.22 33.49 2017/3/3 176.44 35.37 3.07 1.06 2017/5/3 155.45 36.70 2.98 1.73 2017/7/3 153.95 33.21 2017/3/6 176.86 35.04 4.21 1.83 2017/5/4 155.86 36.58 3.26 1.95 2017/7/5 152.06 34.08 2017/3/7 176.77 35.27 4.40 0.64 2017/5/5 151.94 36.55 1.67 1.52 2017/7/6 150.77 33.38 2017/3/8 175.86 35.09 4.07 1.15 2017/5/8 151.43 36.27 0.61 0.16 2017/7/7 151.34 33.63 2017/3/9 173.63 35.29 3.38 0.06 2017/5/9 150.52 36.10 1.31 1.81 2017/7/10 151.82 33.40 2017/3/10 174.27 35.38 4.08 0.47 2017/5/10 149.67 35.74 2.60 0.78 2017/7/11 151.59 33.67 2017/3/13 172.93 34.64 3.29 1.34 2017/5/11 149.07 35.42 2.39 0.53 2017/7/12 152.09 33.99 2017/3/14 172.20 34.66 3.52 0.20 2017/5/12 148.80 35.27 2.29 0.00 2017/7/13 152.02 33.98 2017/3/15 172.29 34.58 2.68 1.08 2017/5/15 149.93 35.36 1.79 0.90 2017/7/14 152.63 34.42 2017/3/16 173.69 34.62 4.06 1.79 2017/5/16 152.07 35.55 1.11 1.70 2017/7/17 151.41 34.21 2017/3/17 172.13 34.75 7.95 1.81 2017/5/17 149.35 34.78 2.59 1.34 2017/7/18 152.39 34.27 2017/3/20 172.18 34.91 7.63 2.12 2017/5/18 149.20 34.96 2.77 1.02 2017/7/19 145.99 34.30 2017/3/21 170.40 34.52 7.76 3.65 2017/5/19 150.39 35.14 1.19 2.32 2017/7/20 146.12 34.49 2017/3/22 171.28 34.85 8.03 3.90 2017/5/22 151.04 35.50 1.15 3.94 2017/7/21 145.54 34.47 2017/3/23 171.32 34.75 8.25 4.54 2017/5/23 150.44 35.59 1.37 4.66 2017/7/24 144.46 34.24 2017/3/24 170.35 34.64 7.92 5.03 2017/5/24 150.92 35.85 1.78 5.68 2017/7/25 144.66 34.41 2017/3/27 170.29 34.87 7.74 5.76 2017/5/25 151.60 35.99 1.01 7.20 2017/7/26 143.84 34.49 2017/3/28 171.01 35.07 8.15 1.54 2017/5/26 150.90 35.99 1.86 5.68 2017/7/27 143.55 34.71 2017/3/29 170.46 35.04 8.68 2.08 2017/5/30 150.14 35.91 1.58 7.30 2017/7/28 142.78 35.05 2017/3/30 170.38 35.22 8.49 3.41 2017/5/31 151.03 35.84 0.79 6.56 2017/7/31 143.16 35.21

Appendix C Historical Prices of 10 Components of the DJIA: Data Used in Chap. 5 The historical prices included in this appendix were the adjusted daily close price, adjusted for both dividends and splits, of the following 10 component stocks of DJIA(Dow Jones Industrial Average) in the following period: January 3, 2012 December 29, 2017 GE APPL XOM WMT UTX JNJ INTC GS VZ MSFT Symbol Company Name General Electric Company Apple Inc. Exxon Mobil Corporation Wal-Mart Stores, Inc. United Technologies Corporation Johnson & Johnson Intel Corporation The Goldman Sachs Group, Inc. Verizon Communications Inc. Microsoft Corporation *Only partial price data were included in this appendix for saving space The price data were downloaded from Yahoo!Finance on January 8, 2018. The price data were used in Examples 5.1 5.4 of Chap. 5, in generating scenarios for the profit rates of component stocks. Springer Nature Singapore Pte Ltd. 2018 C. Xu, T. Shiina, Risk Management in Finance and Logistics, Translational Systems Sciences 14, https://doi.org/10.1007/978-981-13-0317-3 181

182 C Historical Prices of 10 Components of the DJIA: Data Used in Chap. 5 Table C.1 Prices of the 10 component stocks of Dow Jones Industrial Average(US $) Date GE APPL XOM WMT UTX JNJ INTC GS VZ MSFT 2012/1/3 15.01 52.66 71.42 51.67 64.73 55.23 20.20 87.71 29.85 22.80 2012/1/4 15.18 52.95 71.44 51.14 65.07 54.89 20.67 87.14 29.46 23.33 2012/1/5 15.17 53.53 71.22 50.89 64.44 54.83 20.91 86.99 29.26 23.57 2012/1/6 15.25 54.09 70.69 50.53 64.07 54.35 20.79 85.93 29.18 23.94 2012/1/9 15.42 54.01 71.01 50.69 64.24 54.43 20.97 87.10 29.21 23.62 2012/1/10 15.31 54.20 71.19 50.57 65.92 54.66 21.07 90.44 29.36 23.71 2012/1/11 15.44 54.11 70.66 50.87 66.50 54.60 21.24 91.76 29.61 23.60 2012/1/12 15.48 53.96 70.38 50.96 66.96 54.69 21.20 93.09 29.63 23.84 2012/1/13 15.41 53.76 70.49 50.99 65.96 54.71 20.70 91.02 29.63 24.06 2012/1/17 15.32 54.39 71.17 51.26 66.79 54.59 20.61 89.85 29.70 24.06 2012/1/18 15.55 54.95 71.80 51.40 67.28 54.73 20.90 95.94 29.69 24.04 2012/1/19 15.66 54.78 72.28 51.91 66.92 54.65 21.10 99.04 29.69 23.94 2012/1/20 15.66 53.82 72.66 52.25 66.49 54.72 21.72 100.02 29.66 25.30 2012/1/23 15.49 54.73 72.64 52.17 66.63 54.49 21.99 99.51 29.23 25.32 2012/1/24 15.41 53.84 72.40 52.58 67.43 54.49 22.15 100.14 28.77 24.98 2012/1/25 15.64 57.20 72.44 52.65 67.32 54.68 22.15 99.59 28.69 25.17 2012/1/26 15.59 56.94 72.06 52.22 67.11 55.08 22.02 99.85 28.42 25.12 2012/1/27 15.56 57.28 71.28 52.00 67.29 54.96 22.01 102.81 28.32 24.89 2017/11/30 18.16 171.85 83.29 96.72 121.45 139.33 44.84 247.64 50.32 84.17 2017/12/1 17.76 171.05 83.46 96.84 120.12 139.98 44.68 248.95 50.67 84.26 2017/12/4 17.83 169.80 83.57 96.50 120.04 139.01 44.49 250.65 51.14 81.08 2017/12/5 17.64 169.64 82.89 97.32 120.29 139.67 43.44 248.33 50.35 81.59 2017/12/6 17.54 169.01 82.28 96.77 121.20 141.06 43.45 245.95 50.11 82.78 2017/12/7 17.59 169.32 82.55 96.78 122.40 140.01 43.08 248.56 49.85 82.49 2017/12/8 17.59 169.37 82.66 96.55 122.81 140.59 43.35 250.35 50.51 84.16 2017/12/11 17.53 172.67 83.03 96.93 123.30 141.14 43.66 250.13 51.26 85.23 2017/12/12 17.79 171.70 82.76 96.70 123.48 142.60 43.33 257.68 52.59 85.58 2017/12/13 17.64 172.27 83.12 97.76 124.30 142.89 43.34 255.56 52.29 85.35 2017/12/14 17.52 172.22 82.90 97.13 123.76 141.65 43.26 255.48 51.75 84.69 2017/12/15 17.70 173.97 83.03 97.11 126.17 142.46 44.56 257.17 52.08 86.85 2017/12/18 17.64 176.42 82.94 97.90 126.71 141.80 46.26 260.02 52.65 86.38 2017/12/19 17.47 174.54 82.44 98.80 126.78 141.78 47.04 256.48 52.24 85.83 2017/12/20 17.33 174.35 82.87 98.75 127.00 141.16 47.56 255.18 52.18 85.52 2017/12/21 17.35 175.01 83.85 98.06 127.31 141.06 46.76 261.01 52.41 85.50 2017/12/22 17.38 175.01 83.97 98.21 127.23 140.12 46.70 258.97 52.59 85.51 2017/12/26 17.43 170.57 83.98 99.16 127.14 140.09 46.08 257.72 52.62 85.40 2017/12/27 17.38 170.60 83.90 99.26 127.58 140.57 46.11 255.95 52.68 85.71 2017/12/28 17.36 171.08 84.02 99.40 128.12 140.56 46.22 256.50 52.83 85.72 2017/12/29 17.45 169.23 83.64 98.75 127.57 139.72 46.16 254.76 52.33 85.54

Index A Absolute deviation, 40, 55 Aggregate level decision, 112, 115 Algorithm, 35, 38, 42, 44, 46 48, 57, 90 93, 100, 108, 109, 111, 115, 116, 121, 124, 132, 136, 138, 141 145, 149, 156 160, 162, 163, 168 A Modeling Language for Mathematical Programming (AMPL), 39, 136, 145, 162 B Benders decomposition, 109, 115, 156 Beta distribution, 126 Block separable recourse, 111 115 Branch-and-cut, 121, 124, 125 C Chance-constrained programming, 125 130 Complete recourse, 142, 160 Conditional value at risk (CVaR), 27 31, 34, 44, 53 55, 57, 61 64, 70, 71, 73, 75, 77, 78, 80, 82, 90, 91, 95, 97 99, 153, 156, 161 162, 164 168, 173 Confidence level, 18, 21, 23, 26, 27, 50, 54, 61 64, 77, 87 Convex envelope, 116, 121, 122 Convex hull, 121 Correlative, 126, 129 Covariance, 15, 16, 18 20, 26, 34, 39, 71 74, 95, 173 Credit risk, 5, 6 Cumulative, 114, 128, 165 167 Customer, 153, 154, 159, 160, 162, 163 CVaR in the worst case(w-cvar), 63, 64, 77, 78, 82, 91, 92, 173 CVaR minimization, 44, 54, 55, 91, 92, 156, 162, 164 168 D Derivatives, 4, 6, 7 Detailed level decisions, 112, 114, 115 Deterministic equivalent, 108, 112 114, 126, 128 Dirichlet distribution, 126 Distribution center, 139 141 Distribution function, 18, 19, 74, 125 128 Down-side risk, 32, 34 E Emergency lateral transshipment, 138, 141, 143, 144, 146, 147, 149 Exit time, 9, 10, 60 64, 68 70, 72, 75, 78, 80 83, 86, 89 91, 93 97, 99, 100, 173 F Factory, 153, 154, 162, 163 Feasibility cut, 109, 110, 159, 160 Financial investment, 3 11, 13 34, 37 Financial market, 3 7, 36 Financial risk, 3 11 First stage decision, 108, 115, 116, 157 Flexible investments, 59 100 Future, 4 7, 9, 10, 15, 20, 22, 24, 25, 34, 36, 61, 65, 67, 86, 140, 149, 168 Springer Nature Singapore Pte Ltd. 2018 C. Xu, T. Shiina, Risk Management in Finance and Logistics, Translational Systems Sciences 14, https://doi.org/10.1007/978-981-13-0317-3 183

184 Index G Gamma distribution, 126 Gauss quadrature, 129 Geometric Brownian motion (GBM), 65, 66, 67 H Heuristic, 43 45, 57, 82, 153 Historical simulation, 22 24, 27, 30, 34, 42, 51, 52, 62, 64, 67, 68, 70, 75, 77, 78, 83, 84, 86, 88, 90, 97 Multistage stochastic programming with recourse, 110 Multivariate normal distribution, 128 MV(t) model, 93, 94 N Nested decomposition, 110 112 Nonlinear programming (NLP), 38, 42, 57, 90 93, 100, 125, 129 Normal distribution, 19, 20, 24, 25, 34, 47, 67, 71, 73, 74, 76 78, 90 92, 100, 128, 144 I IBM ILOG CPLEX, 136, 162, 163 Inventory, 137 149 Inventory distribution problem, 137 149 J Joint chance-constraint, 126 L Lateral transshipment problem, 139 144 Linear programming (LP), 30, 31, 40, 42, 44, 45, 53 55, 57, 71, 75, 82, 85, 86, 91, 98, 108, 111 114, 126, 140 Logarithmic concave probabilistic measure, 126 Logistics network, 151 168 Lower bound, 121 123, 125, 131 136, 143, 156, 162 Lower semicontinuous, 120, 121 L-shaped method, 109, 110, 145, 153, 156, 162 164, 168 M Market risk, 6, 7, 10, 13 57, 59 100 Master problem, 116, 142, 143, 156, 157, 159, 160, 162, 164 M-CVaR(t) model, 95, 99 Mean variance model (MV model), 37 42, 93, 96 MIP, 139, 142 145, 149 Mixed integer linear programming (MILP), 82, 84 87, 89, 100 Monte Carlo simulation, 19, 24 27, 62, 64, 65, 67, 77 M-PVaR model, 81 89 M-risk models, 55 57, 81, 89 93, 100 M-risk(t) models, 93 100 O Operational risk, 6, 9 Optimality cut, 110, 115, 123 125, 142 144, 156, 159, 162 Option, 4, 6, 7, 156 P Period Value at Risk (PVaR), 61 69, 78, 80 89, 100, 173 Piecewise linear, 45, 109 Portfolio, 7, 14, 16, 17, 19 27, 29 33, 36 40, 42, 44, 45, 50 55, 57, 62, 63, 67 73, 75 77, 80, 83, 84, 86 91, 95, 96, 98, 173 Portfolio selection, 36 37, 40, 42 44, 50, 51, 53, 54, 57, 60, 81, 82, 86 92 Preventive lateral transshipment, 138 141, 144, 146, 147, 149 Probabilistic constraint, 108 Probability distribution, 18, 28, 125, 126, 167 Probability space, 108, 111, 112, 125 Profit rate, 7, 8, 14 16, 19, 20, 22 26, 30 34, 36, 39, 40, 50, 51, 53 55, 66 72, 74, 76 78, 80, 83, 86 92, 94, 95, 97, 98, 173, 175, 178, 181 PVaR minimization, 82 89, 100 Q Quadratic programming (QP), 38, 56, 57, 95, 136 R Recourse function, 108, 109, 110, 113, 115 122, 124, 131, 133, 134, 141 143 Reformulation, 146 Relative error, 145, 146 Relatively complete recourse, 117

Index 185 Return, 7 9, 18, 22, 27, 31, 36 38, 44, 50, 53, 54, 80, 82, 87, 94 96, 98 100, 132, 133, 143 measure, 96, 99 Risk countermeasures, 6 7 diversification, 7 9 hedging, 6 7 measure, 10, 13 34, 37, 40, 42, 44, 45, 53, 57, 59 78, 81, 82, 89 93, 95, 100, 161 S Safety-first criterion, 31, 32, 34 Scenario, 16, 22, 24, 25, 27, 29 32, 34, 39, 40, 44, 45, 50, 53, 54, 55, 57, 64, 67 70, 83, 84, 86, 88, 89, 91, 94, 95, 97, 98, 108, 109, 111 114, 117, 122, 123, 125, 141, 143, 144, 145, 147, 153, 154, 156, 157, 159, 160, 163 168, 181 Scenario simulation, 64 70, 78, 83, 94 Scenario tree, 111, 112, 114 Second stage decision, 108, 116 Semi-variance, 32, 33, 55, 120, 121 Soft optimization approach, 43, 45 46, 82, 88 90, 100 Stochastic integer programming, 115 125 Stochastic programming, 107 136, 138, 141, 149, 155, 156 Stochastic programming problem with recourse, 108, 110 Supply chain, 138 140 Support, 108, 111, 117, 124, 130, 131 T Tender, 117, 118, 132, 134 Transshipment, 140, 146, 147 Two-stage stochastic problem, 108, 114, 115 U Uncertain exit time, 60 64, 72, 81, 89 Uncorrelated, 126, 128 Uniform distribution, 118, 163 Uniformly sampling, 48, 88 V Value at risk (VaR), 17 28, 34, 42 46, 50 53, 57, 61 64, 70 78, 80, 82, 87, 88, 90, 98, 99, 154, 161, 167, 173 Variance, 14 18, 20, 21, 25, 34, 37, 39, 40, 55, 57, 66, 67, 71 73, 75, 76, 82, 95, 130 136, 144, 147, 148, 149, 173 Variance-covariance, 19 21, 26, 34, 72, 74 VaR in the worst case (WVaR), 63, 64, 75, 76, 77, 82, 91, 92, 173 VaR minimization, 42, 45, 46, 50, 57, 82, 88, 90 W Warehouse, 139, 152 154, 156, 158, 159, 162, 163 Wishart distribution, 126