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Journal of the Faculty of Management and Information Systems, Prefectural University of Hiroshima 2014 No.6 pp.43 56 43 The risk measure for resilience in the inventory control system Nobuyuki UENO, Yu KURISU, Koji OKUHARA and Hugang HAN 1 VaR AVaR Excel We propose risk measures for resilience in the inventory system. It is based on an expected stock-out for planning horizon. Resilience is powers of recovery against risk or business continuity. We research on risk measures in the inventory system to track the risk for resilience. Specially we investigate on risk measures concerned with multi-period. At first, we explain the representative risk measures, that is, VaR,AVaR and clarify the relationship between unfulfilled-order-rate and the representative risk measure. Next, we propose risk measures for multi-periods and clarify its validity by numerical experiences. Finally, future research is shown. VaR AVaR

44 6 1 3 1 VaR AVaR 4 5 Excel 2 3 4 5 6 6 monitoring

45 Conditional level 1 ε tail probability ε distribution functionf X (x) VaR ε ( ) valueat-riskavar ε ( ) average value-at-risk 7 8 9 1 2 AVaR ε ( )CVaR ε ( ) TVaR ε ( ) 9ε 1 VaR ε ( )1 100 1 ε 1 AVaR ε ( )VaR ε ( ) 100 1 200 1 ρ 7 8 9 monotonicity : ( ) 1 ρ( ) ρ( ) subadditivity : ρ( ) ρ( ) ρ( ) positive homogeneity : c ρ(c ) cρ( ) translation invariance : c ρ(c ) ρ( ) c AVaR ε ( )VaR ε ( ) VaR ε ( ) AVaR ε ( ) VaR ε ( ) 9

46 6 ε VaR ε ( )(1)2 f (x) 1 ε 3 VaR ε ( ) 0 ε ε VaR ε ( )Excel NORMINV ε 1 SVaR( ) Expected Stock-out value-at-risk 3 4 (1)(2) 5

47 6 7 (5) 8 ε (8) 9 10 AVaR SVaR SVaRAVaR ε 4 2 (8)0 σ 2 f ( ) 11 12 (11) 13

48 6 14 0 σ 2 (14) 0 1 15 511 Excel1 5 AVaR SVaR e AVaR SVaRAVaR 4 3 1 5 1 2 3 4 5 10 20 24 6 12 ω 3 15 1 31 1 2 11

49 1 24 19 14 9 4 2 23 18.5 14 9.5 5 3 22 18 14 10 6 4 21 17.5 14 10.5 7 5 20 17 14 11 8 6 19 16.5 14 11.5 9 7 18 16 14 12 10 8 17 15.5 14 12.5 11 9 16 15 14 13 12 10 15 14.5 14 13.5 13 11 14 14 14 14 14 3 8 1 1 28.9872992 28.004361 17.999449 21.0056793 13.0495162 2 27.9872992 26.504361 16.499449 20.0056793 13.0495162 3 26.9872992 25.004361 14.999449 19.0056793 13.0495162 4 25.9872992 23.504361 13.499449 18.0056793 13.0495162 5 24.9872992 22.004361 11.999449 17.0056793 13.0495162 6 23.9872992 20.504361 10.499449 16.0056793 13.0495162 7 22.9872992 19.004361 8.99944899 15.0056793 13.0495162 8 21.9872992 17.504361 7.49944899 14.0056793 13.0495162 9 20.9872992 16.004361 5.99944899 13.0056793 13.0495162 10 19.9872992 14.504361 4.49944899 12.0056793 13.0495162 11 18.9872992 13.004361 2.99944899 11.0056793 13.0495162 1 2.08882E-22 2.06046E-11 0.000266003 0.000232629 0.026316151 2 5.12963E-21 2.10401E-10 0.000748082 0.00042906 0.026316151 3 1.12859E-19 1.90133E-09 0.001946209 0.000770985 0.026316151 4 2.22478E-18 1.52099E-08 0.004687384 0.001349898 0.026316151 5 3.92987E-17 1.07747E-07 0.010460668 0.002303266 0.026316151 6 6.22096E-16 6.76193E-07 0.021654071 0.003830381 0.026316151 7 8.82619E-15 3.76123E-06 0.041632258 0.006209665 0.026316151 8 1.12249E-13 1.85537E-05 0.074457337 0.009815329 0.026316151 9 1.27981E-12 8.12204E-05 0.124106539 0.01513014 0.026316151 10 1.30839E-11 0.000315783 0.193238115 0.022750132 0.026316151 11 1.1996E-10 0.001091522 0.281851431 0.033376508 0.026316151

50 6 1 0 0 0.9427806 0.6649073 648.16203 649.76972 2 0 0 4.8991956 2.1066624 648.16203 655.16789 3 0 0 20.636295 7.2807268 648.16203 676.07905 4 0 0 69.815893 18.504658 648.16203 736.48258 5 0 0 193.39814 33.509882 648.16203 875.07005 6 0 0 434.32945 63.611303 648.16203 1146.1028 7 0 0 884.36845 115.7605 648.16203 1648.291 8 0 0 1721.984 197.08877 648.16203 2567.2348 9 0 0.7065355 3167.4197 322.99781 648.16203 4139.286 10 0 3.4427881 5497.7285 515.06055 648.16203 6664.3939 11 0 15.180264 9025.5091 800.47131 648.16203 10489.323 1 0.304032322 0.616251492 1.316966462 1.508347589 2.551970409 2 0.31443992 0.648201776 1.409136467 1.567302456 2.551970409 3 0.325569315 0.683492233 1.513617766 1.63057021 2.551970409 4 0.337496522 0.722649565 1.632738137 1.69859193 2.551970409 5 0.350308349 0.766311346 1.769365384 1.771864003 2.551970409 6 0.364104337 0.815255017 1.927043431 1.850945862 2.551970409 7 0.378999106 0.870435822 2.110158515 1.936468786 2.551970409 8 0.395125243 0.93303678 2.324135991 2.029145892 2.551970409 9 0.41263684 1.0045349 2.57566144 2.129783392 2.551970409 10 0.431713881 1.086789364 2.872906458 2.239293197 2.551970409 11 0.452567698 1.182159337 3.225715784 2.358706876 2.551970409 1 0 0 9.4278E-05 6.6491E-05 0.0648162 2 0 0 0.00048992 0.00021067 0.0648162 3 0 0 0.00206363 0.00072807 0.0648162 4 0 0 0.00698159 0.00185047 0.0648162 5 0 0 0.01933981 0.00335099 0.0648162 6 0 0 0.04343295 0.00636113 0.0648162 7 0 0 0.08843684 0.01157605 0.0648162 8 0 0 0.1721984 0.01970888 0.0648162 9 0 7.0654E-05 0.31674197 0.03229978 0.0648162 10 0 0.00034428 0.54977285 0.05150605 0.0648162 11 0 0.00151803 0.90255091 0.08004713 0.0648162

51 ε 1 6.35068E-23 1.26976E-11 0.000256039 0.000284395 0.002341837 2 1.61296E-21 1.36382E-10 0.00056423 0.000461801 0.002341837 3 3.67434E-20 1.29954E-09 0.000882186 0.000529072 0.002341837 4 7.50854E-19 1.09914E-08 0.000671682 0.00044246 0.002341837 5 1.37667E-17 8.25678E-08 0.00083107 0.000730086 0.002341837 6 2.26508E-16 5.5127E-07 0.001704609 0.000728697 0.002341837 7 3.34512E-15 3.27391E-06 0.00058618 0.000448773 0.002341837 8 4.43523E-14 1.73113E-05 0.000850572 0.000207857 0.002341837 9 5.28098E-13 1.09352E-05 0.002914462 7.58603E-05 0.002341837 10 5.64851E-12 1.08882E-06 0.005382178 0.000561839 0.002341837 11 5.42901E-11 0.000227673 0.0066217 0.001321733 0.002341837 SVaR e AVaR 0.006611 3 AVaR 2.5 8 0.0017 6 3 0.0013 11 4 10 1 2 5 1 2 i 14 i 4 3 i 15

52 6 AVaR O n 0 i 16 AVaR 2 1 AVaR AVaR Unfulfilled-order-rate for aggregation 17 2 AVaRUnfulfilled-order-rate for multi-period 18 19 20 AVaR (agr) (multi) (16)(17) O n O n O n 1 (20) (17)(20) 2 AVaR AVaR

53 9 10 9 AVaR i 1 2 3 4 5 6 7 8 9 10 11 SVaR(i) 0 0 9.42781E-05 6.64907E-05 0.064816203 SOn(i) 2.08882E-22 2.06046E-11 0.000266003 0.000232629 0.026316151 AVaR(i) 0.304032322 0.616251492 1.316966462 1.508347589 2.551970409 SVaR(i) 0 0 0.00048992 0.000210666 0.064816203 SOn(i) 5.12963E-21 2.10401E-10 0.000748082 0.00042906 0.026316151 AVaR(i) 0.31443992 0.648201776 1.409136467 1.567302456 2.551970409 SVaR(i) 0 0 0.00206363 0.000728073 0.064816203 SOn(i) 1.12859E-19 1.90133E-09 0.001946209 0.000770985 0.026316151 AVaR(i) 0.325569315 0.683492233 1.513617766 1.63057021 2.551970409 SVaR(i) 0 0 0.006981589 0.001850466 0.064816203 SOn(i) 2.22478E-18 1.52099E-08 0.004687384 0.001349898 0.026316151 AVaR(i) 0.337496522 0.722649565 1.632738137 1.69859193 2.551970409 SVaR(i) 0 0 0.019339814 0.003350988 0.064816203 SOn(i) 3.92987E-17 1.07747E-07 0.010460668 0.002303266 0.026316151 AVaR(i) 0.350308349 0.766311346 1.769365384 1.771864003 2.551970409 SVaR(i) 0 0 0.043432945 0.00636113 0.064816203 SOn(i) 6.22096E-16 6.76193E-07 0.021654071 0.003830381 0.026316151 AVaR(i) 0.364104337 0.815255017 1.927043431 1.850945862 2.551970409 SVaR(i) 0 0 0.088436845 0.01157605 0.064816203 SOn(i) 8.82619E-15 3.76123E-06 0.041632258 0.006209665 0.026316151 AVaR(i) 0.378999106 0.870435822 2.110158515 1.936468786 2.551970409 SVaR(i) 0 0 0.172198404 0.019708877 0.064816203 SOn(i) 1.12249E-13 1.85537E-05 0.074457337 0.009815329 0.026316151 AVaR(i) 0.395125243 0.93303678 2.324135991 2.029145892 2.551970409 SVaR(i) 0 7.06535E-05 0.316741967 0.032299781 0.064816203 SOn(i) 1.27981E-12 8.12204E-05 0.124106539 0.01513014 0.026316151 AVaR(i) 0.41263684 1.0045349 2.57566144 2.129783392 2.551970409 SVaR(i) 0 0.000344279 0.549772852 0.051506055 0.064816203 SOn(i) 1.30839E-11 0.000315783 0.193238115 0.022750132 0.026316151 AVaR(i) 0.431713881 1.086789364 2.872906458 2.239293197 2.551970409 SVaR(i) 0 0.001518026 0.902550909 0.080047131 0.064816203 SOn(i) 1.1996E-10 0.001091522 0.281851431 0.033376508 0.026316151 AVaR(i) 0.452567698 1.182159337 3.225715784 2.358706876 2.551970409

54 6 (agr) (multi) O n O n 0.0162 11 28 20 0.009210 (agr) (multi) O n O n AVaR 11

55 VaR AVaR (multi) AVaR O n 10 2 AVaR AVaR AVaR RMWG (C)-25350452. 1 2011 2 28 1 pp.27 36 2011 3 N. Ueno K.Okuhara H.Ishii H. Shibuki and T. Kuramoto Multi-item Production Planning and Management System Based on Unfulfilled Order Rate in Supply Chain Journal of the Operations Research Society of Japan 50 3 pp.201-218 2007 4 23 7 pp.147 156 2010 5 24 3

56 6 pp.43 53 2011 6 E.Hollnagel D.D.Woods N.Leveson 2012 7 S.T.Rachev, S.V.Stoyanov and F.J.Fabozzi : Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization, John Wiley & Sons,Inc 2008 8 R.T.Rockafellar and S.Uryasev : Conditional value-at-risk for general loss distributions, Journal of Banking & Finance, Vol.26, No.7, pp1443-1471 2002 92010

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