ii

i 2013 5 143 5.1...................................... 143 5.2.................................. 144 5.3....................................... 148 5.4.................................. 153 5.5................................... 155 5.6..................................... 156 5.7.................................. 159 5.8....................................... 161 5.9............................. 164 6 165 6.1....................................... 165 6.2............................... 165 6.3 P ERT......................... 169 6.4................................. 169 6.5 P ERT..................................... 175 6.6 P ERT................................ 184 6.7............................ 186 6.8....................................... 188 7 191 7.1................................. 191 7.2...................................... 192 7.3............................. 194 7.4.......................... 199 7.5 (Economic Order Quantity, EOQ)................. 200 7.6............................ 207 7.7................................. 211

ii 7.8............................... 216 7.9............................ 217 8 221 8.1....................................... 221 8.2............................... 222 8.3............................... 227 8.4..................... 227 8.5........................... 233 8.6................................. 236 9 244 9.1....................................... 244 9.2............................... 246 9.3....................................... 252 9.4................................... 254 9.5.................................. 258 9.6................... 262 10 270 10.1...................................... 270 10.2................................... 271 10.3................................... 272 10.4............................. 282 10.5............................... 284 10.6.................................. 290

5 143 5 5.1 A180 A Pareto

144 5 5.1 5.2 5.2.1 0.6 0.3 0.1 1 0.5 0.4 0.1 1 1 2

5 145 Excel sumproduct() B4 : D4 B5 : B7 C5 : C7 D5 : D7 E5 =sumproduct($b$4:$d$4,b5:d5) E6, E7 E5 M N w 1, w 2,..., w M k q k1, q k2,..., q kn i V i V i = w 1 q 1i + w 2 q 2i +... + w M q Mi, i = 1, 2,..., N (1) V 1, V 2,..., V N 5.2.2 0.6, 0.3, 0.1 0.5, 0.4, 0.1 60% 5 : 4 : 1 0.6 0.5 = 0.3 0.6 0.4 = 0.24 0.6 0.1 = 0.06

146 5 5.2.3 Analytic Hierarchy Process AHP Thomas Saaty

5 147 5.2 5.1 (1) (A) (B) (C) (2) w 1 : w 2 : w 3 (3) (A) : (B) : (C) = q 11 : q 12 : q 13 (A) : (B) : (C) = q 21 : q 22 : q 23 (A) : (B) : (C) = q 31 : q 32 : q 33 (4) 5.1 (A) = (B) = (C) =

148 5 5.3 5.3.1

5 149 1 2 3 5 7 9 5 5 1 1, 2, 3 i k i a ik k a ki a ii = 1(i = 1, 2, 3) A = (a ik ) 3 3 (1) (2) (3) (1) 1 5 (2) 1/5 1 (3) 1 5.3 5.2 1 1 1 5.3.2

150 5 (1) (2) (3) (1) 1 3 9 (2) 1/3 1 5 (3) 1/9 1/5 1 3 3 1 1 3 9 : 3 5 : 3 9 1 = 3 : 1.186 : 0.281 = 0.672 : 0.265 : 0.063 5 M M M EXCEL ˆ(1/M) 1 1 1 1 + 1 3 + 1 : 9 3 + 1 + 1 : 9 + 5 + 1 = 9 13 : 5 21 : 1 = 0.694 : 0.239 : 0.067 15 5 a A = ( 1 a 1/a 1 ) a : 1 a = a : 1 1 1 + 1 a : 1 a + 1 = a : 1

5 151 5.4 5.3 5 5.2 1 1 1 1

152 5 5.3.3 5.5 A C 6 b AC = 6, b CA = 1/6 (A) (B) (C) (D) (A) 1 (B) 1 (C) 1 (D) 1...

5 153 5.4 5.2

154 5 5.6 5.2 5.4 (1) 1 (A), (B), (C) (A) (B) (C) (A) 1 (B) 1 (C) 1 q 11 = q 12 = q 13 = (A), (B), (C) (A) (B) (C) (A) 1 (B) 1 (C) 1 q 21 = q 22 = q 23 = (A), (B), (C) (A) (B) (C) (A) 1 (B) 1 (C) 1 q 31 = q 32 = q 33 = (2) (A) (B) (C)

5 155 5.5 A = (a ik ) {w i } i k w i : w k i k w i /w k w 1 /w 1 w 1 /w 2 w 1 /w 3 D = w 2 /w 1 w 2 /w 2 w 2 /w 3 (2) w 3 /w 1 w 3 /w 2 w 3 /w 3 w 1 + w 2 + w 3 = 1 w 1 : w 2 : w 3 5.7 w 1 : w 2 : w 3 5.5.1 D D D w 1 w 2 = w 1/w 1 w 1 /w 2 w 1 /w 3 w 2 /w 1 w 2 /w 2 w 2 /w 3 w 1 w 2 w 3 w 3 /w 1 w 3 /w 2 w 3 /w 3 w 3 = 3 w 1 w 2 w 3

156 5 D 3 D 3 M D M A D D A D 5.6 5.6.1 A = {a ik } {v i } a ik v i v k

5 157 {v i } D = {d ik } d ik = v i /v k a ik /d ik 1 β = 1 9 3 i,k=1 a ik d ik = 1 9 3 i,k=1 a ik v k v i C.I. = 3 3β 3 (β 1) = 3 1 2 (3) 0 C.I. β d ik = d 1 ki A D 9 β = 1 9 3 i,k=1 a ik d ik = β = 1 9 3 a ik d ki = 1 9 i,k=1 ( 3 3 ) a ik d ki M A D A D A D M b 3β C.I. = b M M 1 C.I. A λ max i=1 k=1 C.I. = λ max M M 1 (4) b = M 0 A 0.1 0.15 5.8 5.3 C.I. b= C.I. =

158 5 5.6.2 (C.R.) (4) R.I. C.I. C.R. = C.I. R.I. C.R. 0.2 M 3 4 5 6 7 8 9 10 R.I. 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49 5.9 R.I. C.R. C.R. = 5.10 C.I. C.R. 9 1 9 0.15

5 159 5.7 2 : 1 2 : 1 4 : 1

160 5 5.11

5 161 5.8 5.8.1 w 1, w 2, w 3 t 1, n 1, b 1 t 2, n 2, b 2 t 3, n 3, b 3 w 1 t 1 + w 2 t 2 + w 3 t 3 w 1 : w 2 ( w1 w 1 t 1 + w 2 t 2 + w 3 t 3 = (w 1 + w 2 ) t 1 + w ) 2 t 2 + w 3 t 3 (5) w 1 + w 2 w 1 + w 2 w 3 w 3 w 1 : w 2 w 1 /(w 1 + w 2 ), w 2 /(w 1 + w 2 ) t 1, t 2 w 1 w 1 + w 2 t 1 + w 2 w 1 + w 2 t 2 (= y) w 3 w 3 w 1 t 1 + w 2 t 2 + w 3 t 3 = (1 w 3 )y + w 3 t 3 (6) w 3 w 3 0.1 50% 30%

162 5 (sensitivity analysis) 1 5.8.2 5.8.3

5 163 5.12

164 5 5.9 AHP AHP AHP 5.13 [1] (1986) [2] AHP (2005) [3] 1990 [4] 2007

6 165 6 6.1 P ERT/CP M Program Evaluation and Review Technique / Critical Path Method P ERT 1950 6.2 P ERT

166 6 A J 10 2.1 A 3 B 2 A C 3 D 4 E 6 B, C, D F 4 E G 3 E H 5 D I 3 F, G J 4 H, I E A B E F G J 2.1

6 167 2.2 22 A, B, E P ERT

168 6 6.1 2.2 2.2 A 3 B 4 C 5 D 5 A E 2 A F 3 B G 6 B, C H 6 E, F I 2 D J 3 I, H K 4 G, H, I 6.2 2.2

6 169 6.3 P ERT P ERT 2.1 P ERT 1. 2. P ERT 3. 4. P ERT EXCEL 6.4 P ERT B, C, D E E B, C, D 2.1

170 6 2.3 i, k (i, k) i (i, k) k (i, k) A B A B A A B C, D, E A, B C, D, E

6 171 B A C A D A, B C A D A B A C A D A B P ERT D B (3, 4) (3, 4) A A (3, 4) 4 A, B A, B D C A 0

172 6 B, C A D B C A B C D B C A D (2, 3) C (2, 4) B 3 (2, 4) C (2, 3 ) (3, 4)

6 173 6.3 D C B F C 6.4 F G H G H J 6.5 D A E A B F A, B, C 170 2.3 1 9 P ERT

174 6 6.6 A, B, C, D, E A, B C, D A E B, D P ERT 6.7 168 6.1 P ERT

6 175 6.5 P ERT P ERT T ik (i, k) In(k) k Out(k) k In(3) = {1, 2}, Out(3) = {4, 5, 6} 6.5.1 Earliest Start time... k ES k ES k k ES k k (i, k) (i, k) ES i (i, k) ES i T ik k ES k ES i + T ik ( i In(k))

176 6 ES i + T ik (i, k) k ES k = max i In(k) {ES i + T ik } In(k) k 6.1 In(3) = {1, 2} ES 1 = 11, ES 2 = 10 ES 3 (1, 3) (2, 3) (1, 3) 11 ES 1 + T 13 = 16 10 (2, 3) ES 2 + T 23 = 17 (3, 4) (2, 3) max {ES 1 + T 13, ES 2 + T 23 } = 17 0 0 1. 0 2. In(k) k ES k = max i In(k) {ES i + T ik } 3. PERT i

6 177 (i, k)( k Out(i)) i ES i 6.8 5 2, 3, 4 10, 14, 18 (2, 5), (3, 5), (4, 5) 12, 9, 3 5 6 7 5 (5, 6), (5, 7) 6.5.2 Latest Finish time i LF i LF i i (i, k) (i, k) LF k T ik i i LF i LF k T ik LF i LF k T ik LF k T ik (i, k) i LF i = min {LF k T ik } k Out(i) Out(i) i 6.2 LF 3 = 21, LF 4 = 19 (2, 3) 21 LF 3 T 23 = 13 (2, 4) 19 LF 4 T 24 = 14 (1, 2) min {LF 3 T 23, LF 4 T 24 } = 13

178 6 LF 2 = 13 0 1. 2. Out(i) i LF i = min {LF k T ik } k Out(i) 3. PERT i (k, i)( k In(i)) i LF i 6.9 15 16, 17, 18 25, 24, 28 (15, 16), (15, 17), (15, 18) 8, 4, 10 15 13 14 15 (13, 15), (14, 15) ES k LF k

6 179 P ERT ES i LF i ES i /LF i 0/0 6.10 174 6.7 6.5.3 ES k = max i In(k) {ES i + T ik } (7) LF i = min {LF k T ik } (8) k Out(i) 1. 2. 3. 0 4. 5. 6.

180 6 7. 0 6.11 166 2.1 2.1 A 3 B 2 A C 3 D 4 E 6 B, C, D F 4 E G 3 E H 5 D I 3 F, G J 4 H, I 6.5.4 (i, k) i ES i k LT k (i, k) ES i + T ik < LF k (i, k) LF k ES i T ik ( T F ik ) (i, k) Total Float time T F ik

6 181 6.3 / 10 A C 10 B D 5, 4 B 5 5 10 D D B 6.5.5 Free Float time (i, k) F F ik k i F F ik = ES k ES i T ik

182 6 A(4) C(6) B(3) D(2) B D 2.4 D B 5 D B 1 1 B 5 D 10 2 4 d B D 6.12 168 2.2 / / 2.2 A 3 B 4 C 5 D 5 A E 2 A F 3 B G 6 B, C H 6 E, F I 2 D J 3 I, H K 4 G, H, I

6 183 6.5.6 critical path 6.13 6.12 0

184 6 6.6 P ERT P ERT A, B 2.5

6 185

186 6 6.7 P ERT A C E F 15 C 4 3 14 13 C A, B 10 14 15 11 {(1, 2), (1, 3)} {(4, 5)}

6 187 6.14 6.15 6.16 13 15 11

188 6 6.8 P ERT P ERT 18 6.4 A, B, C D, E, F E D, E, F A, B, C D, E, F 3, 7, 6 E I J D, F, G E

6 189 14 13 6.5 H D G E, F, G 2, 1, 1 H 11 H K H J 10 P ERT

190 6 6.17 168 2.2 C E 6.18 J (2003)

7 191 7 7.1 7.1 7.2 7.3 7.4 7.5 revenue managemnet

192 7 7.2 Just In Time Kanban System

7 193 Supply Chain Management

194 7 7.3 1 2

7 195 t I(t) O(t) t Z(t) Z(t) = I(t) O(t) + Z(0) Z(0) 0 ( ) Z(0) = 0 n n n n

196 7 7.3.1 (1) ( ) ( ) ( ) ( ) 0 ( ) ( ) 0 0... 7.1

7 197 (2) ( ) ( ) ( ) 2 ( ) 7.2

198 7 (3) 7.3

7 199 7.4 1. 2. 3. 4. 5.

200 7 7.5 (Economic Order Quantity, EOQ) 1 T D D 1

7 201 1 K ( 1 ) 1 B ( ) [0, T ] 1 Q [0, T ] D Q 1 1 [0, T ] Q/2 T [0, T ] f T (Q) Q f T (Q) = Q Q = D d = D/T Q (Economic Order Quantity) EOQ (Economic Lot Size ) U U = 7.4 f(q ) = 7.5 d = D/T = 20, B = 1, K = 1000 Excel f(q)/t Q U f(q )/T Q = U = f(q ) T = [0, T ] T = 1 T = 1 f 1 (Q) f(q)

202 7 7.5.1 d = 10, B = 1, K = 2000 ( ) Q 200 Q OR (d = 10, B = 1, K = 2000) 1 2 2 (d = 12, B = 1, K = 2000) 3 2 (d = 10, B = 1.2, K = 2000)

7 203 1 200 2 219.1 3 182.6 3 200 5 3% ( robust)

204 7 7.6 ( 7.5 )d = 20, B = 1, K = 1000 (0) (Q 1 ) Q 1 = (1) d 2 7.5 f 2 (x) 7.5 (Q 2 ) f 2 (Q 2 ) ( ): Q 2 = f 2 (Q 2 ) = (2)( ) d Q 1 ( f 2 (Q 1 ) f 2 (Q 2 ) f 2 (Q 1 ) f 2 (Q 2 ) = 100 f 2(Q 1 ) f 2 (Q 2 ) f 2(Q 2) = (3) B 2 7.5 f 3 (x) 7.5 (Q 3 ) f 3 (Q 3 ) ( ): Q 3 = f 3 (Q 3 ) = (4)( ) B Q 1 ( f 3 (Q 1 ) f 3 (Q 3 ) f 3 (Q 1 ) f 3 (Q 3 ) =

7 205 100 f3(q1) f3(q3) f 3(Q 3) = 7.5.2 ( ) Q V A f(q) = { B Q 2 + K d Q + Ad if Q < V B Q 2 + K d Q + 0.9Ad if Q V Q (?) Q V f(q) f(v )

206 7 V 7.7 A EOQ d = 20, B = 1, K = 1000, A = 100 Excel V = 400 V = 500 (1) EOQ = = (2)V = 400 = (3)V = 500 =

7 207 7.6 ( ) 7.6.1 L L L

208 7 ( ) 20 1 5% 95% X m σ n nm nσ n L X 1 + + X L 1.65 Lσ P ( X 1 + + X L Lm + 1.65 ) Lσ = P ( ) X1 + + X L Lm 1.65 Lσ 1.65 1 2π e x2 /2 dx = 0.05 1.65 Lσ 95% 1.65 5% 7.8 90% 10% Excel Norminv 4 95% 90% 7.6.2 1 95%

7 209 ( ) 95% 1 σ N L 1.65 N + Lσ

210 7 7.9 7 2 10 95% 95%

7 211 7.7 ( ) 7.6 50 105 1 10 50 158,146,159,151,146,151,156,153,140,128,156,140,138,148,166, 149,150,160,171,158,141,161,150,144,166,144,150,154,154,151, 155,151,151,150,144,142,162,154,159,133,147,151,146,165,142, 136,154,143,144,132 50 55 10 65 ( ) 50 50 150 9.2

212 7 150 z b z > b ( ) = z b = 150 50 Excel 150 105 150 50 150 10 8 = 8170 105 146 50 150 = 7830 105 150 50 150 10 9 = 8160 50 392, 145 7.10 Excel 149 150 151 152 153 154 155 = Excel 128, 129,... b n b b 50 z b z b 50 g(z) g(z) = n b (105b 50z) + n b (105z 50z 10(b z)) b z b>z = n b (55b 50(z b)) + n b (55z 10(b z)) b z b>z = n b (55b 50(z b)) + n b (55b 65(b z)) b z b>z = 55 b n b b 50 b z n b (z b) 65 b>z n b (b z)

7 213 2 ( ) 3 ( 50 10) 1 151 392, 520 151 50 50 50 130 170 150 X f(x) n b f(b) (!!!) g(z) g(z) = 55 b bf(b) 50 b z (z b)f(b) 65 b>z(b z)f(b) z z

214 7 g(z 1) g(z) g(z + 1) z g(z + 1) g(z) = 50 (z + 1 b)f(b) 65 (b z 1)f(b) b z+1 b>z+1 + 50 (z b)f(b) + 65 z)f(b) b z b>z(b = 50 f(b) + 65 f(b) b z b>z F (z) = b z f(b) g(z + 1) g(z) = 50 f(b) + 65 f(b) b z b>z = 50F (z) + 65 (1 F (z)) = 65 115F (z) F (z ) 65 115 = 65 50 + 65 F (z 1) z 65 (55 ) (10 ) 50 ( ) ( ) 7.11 g(z)

7 215 7.12 1 200 400 100 50 100 91 2 96 10 101 8 106 2 92 2 97 8 102 12 107 1 93 2 98 9 103 4 108 0 94 4 99 9 104 2 109 2 95 4 100 12 105 6 110 1 (1) (2) (3) ( ) (4) 104 105 1 : 3 105 104 100

216 7 7.8 10 A B B A B B A B (A B)/A 7.13... 7.14

7 217 7.9... 80-20

218 7 80-20

7 219 ABC 7.15 Excel ABC A B C 1 150 116 11 560 76 21 40 425 2 3770 653 12 280 118 22 800 92 3 810 1603 13 2980 241 23 1470 73 4 670 82 14 50 123 24 940 338 5 1630 151 15 310 45 25 570 84 6 1810 131 16 2030 60 26 1820 412 7 1420 3483 17 800 325 27 140 49 8 40 97 18 2340 754 28 60 1503 9 30 69 19 80 138 29 940 430 10 1280 72 20 1490 98 30 420 57 A B

220 7 1. 2. 3. k F k 4. (k/30, F k )(k = 0, 1,..., 30) 5. F k 0.7 k k k A

8 221 8 8.1 8.1 Yahoo 8.2 F1 http://www.fs21.com/fs-e.htm 8.3 http://www.nilim.go.jp/lab/bbg/kasai/h23/top.htm 8.4 http://www.nagata.co.jp/news/news0611.htm 8.5 8.6 Excel rand() 8.7 3 PERT PERT simulation

222 8 8.2 8.2.1 Excel

8 223 8.2.2 http://www.bosai.go.jp/hyogo/ddtpj/soil/subsoil1/soilthema5.htm http://www.jamstec.go.jp

224 8 8.2.3 Wii X(t) t 2 p X(t) ( q) X(t) d dt X(t) 1 X(t) = p + qx(t)

8 225 X(t + t) X(t) t (p + qx(t)) (1 X(t) X(t + t) = X(t) + (p + qx(t)) (1 X(t) t x n = X(n t) x n+1 = x n + (p + qx n ) (1 x n ) t x 0 Excel 8.1 (1)x 0 = 0.01, p = 0.01, q = 0.5, t = 0.1 {x 0, x 1,..., x 100 } Excel (2)p = 0.05, q = 0.5 (3)p = 0.05, q = 0.8 (4)x 0 8.8 100 110 120 50 A B A t x(t) B t y(t) dx(t) dt = b y(t), dy(t) dt = a x(t) a, b A B dx(t) dt x(t + t) x(t) t

226 8 x(t + t) x(t) b y(t) t y(t + t) y(t) a x(t) t x(t), y(t) t x n = x(n t), y n = y(n t) x n+1 = x n by n t y n+1 = y n ax n t x 0, y 0 8.2 Excel dx(t) dt = by(t), dy(t) dt = ax(t) x(0) = x 0, y(0) = 1000 a = a 0, b = 1 (x 0, a 0 ) 8.3 B A C ABC BC a b A B L Excel

8 227 8.3 1 8.4 8.4.1 10

228 8 95% s S (s, S) s S 500 20 50 1 10 s S 100 4 15 8 12 12 7 10 11 11 7 11 10 8 9 8 8 15 9 10 12 11 9 11 10 13 12 11 8 14 11 8 12 8 8 9 11 12 13 9 9 12 8 10 10 9 12 6 4 10 10 10 11 14 13 8 11 8 8 8 14 8 8 9 13 11 11 7 8 12 5 14 11 8 10 8 7 14 6 12 11 11 11 12 7 11 9 9 13 9 9 10 13 8 11 11 10 11 10 13 10 8 12 10 6 8.4.2 100 1 8, 11 8, 11 8, 11

8 229 10 2 Excel =NORM.INV(RAND(),heikin,hensa) heikin,hensa =ROUND(MAX(NORM.INV(RAND(),heikin,hensa),0),0) ROUND(x,0) x MAX(x,0) x 0 8.4 8.5 Excel 10 2 100 B7:B106 =ROUND(NORM.INV(RAND(),10,2),0) E3 =average(b7:b106) E4 =stdev(b7:b106) D8:D28 0 20 E8:E28 E8 =FREQUENCY(B7:B106,D8:D28) Ctrl Shift Enter

230 8 8.4.3 s S (s, S) 1 S s n W (n) Z(n) n D(n) Z(n) D(n) W (n) 0 W (n) Z(n 1) s, D(n) W (n) Z(n) =, D(n) > W (n), Z(n 1) > s W (n) =, Z(n 1) s 1 50 20 500 10 n A(n) B(n) C(n) E(n) n

8 231 P (n) A(n) = B(n) = C(n) = E(n) = P (n) = A(n) B(n) C(n) E(n) 10 2 Z(0) D(n) 8.4.4 Excel C, Java, V isual Basic SLAM,SIMAN,Simul8 Excel A B W (n) C D(n) D Z(n) E R(n) F, G, H, I, J A(n) B(n) C(n) E(n) P (n) What-If 8.4.5 Excel s S z s S z = f(s, S) s, S

232 8 s, S s, S 10 10 9 9 10 8.6 1 500 10 8.7 (s, S) s, S Excel S s s

8 233 8.5 20 20 0 9 2 0 9 random numbers table 54 79 01 84 56 50 50 13 10 03 70 95 19 24 84 13 36 83 69 87 34 35 78 12 43 65 92 60 99 72 09 13 86 37 06 68 66 26 24 50 21 26 81 93 88 46 69 32 72 29 31 30 29 49 89 69 91 45 55 23 84 06 13 47 81 61 88 38 21 49 06 69 41 03 18 8.5.1 pseudo-random numbers Excel RAND() RANDBETWEEN() C rand() RAND() C 0 1 RANDBETWEEN() Excel 8.8 Excel Excel 600 Excel RANDBETWEEN(m,n) m n m n FREQUENCY()

234 8 8.5.2 a b a = 1389, b = 8567 a b 11899563 9563 1389 8567 = 11899563 4 a 13283007 4 3007 1389 9563 = 13283007 4 a 4176723 4 6723 1389 3007 = 4176723 4 8567, 9563, 3007, 6723, 8247, 5083, 287, 8643, 5127, 1403, 8767, 7363,... 8, 9, 3, 6, 8, 5, 2, 8, 5, 1, 8, 7,... 1 4 3, 7 3 multiplicative congruential method a, P, x 0 a x 0 P x 1 a x 1 P x 2 mod P P x n ax n 1 mod P

8 235 a, P, x 0 Excel x 0, x 1, x 2,... 0, 1, 2,..., P 1 P n x n n + m x n+m x n+1 x n+m+1 x n+2 x n+m+2 a, P a, P, x 0 C rand() srand() 47 rand() srand() Excel rand() C rand() 8.9 Excel (1) A7 : A107 0 (2) B3 131 B4 10000 C7 (3) B8 = $B$3 C7 C8 = mod(b8, $B$4) D8 = C8/$B$4 (4) B8 : D6 B9 : D107 (5) D8 : D106 X D9 : D107 Y (6) B3 C 15

236 8 8.6 160 200 100 10 10 100 25 8.6.1 Excel... Excel s C4 J3 1. L10:L19 10 M9 =J3 2. L9:M19 What-if 3. C4 J10:L19 C4 M10:M19 J10:L19 10 10 J10:J19 M10:M19 8.6.2 z 1, z 2,... 200 200 205.7, 6.4

8 237 X Z Z = f(x) Z z 1, z 2,... Z µ σ µ n Z 1, Z 2,..., Z n Z = Z 1 + Z 2 +... + Z n n µ n Z µ E( Z) = E(Z 1) + E(Z 2 ) +... + E(Z n ) n µ Z point estimation = µ X 1, X 2,... µ n µ X = X 1 + X 2 +... + X n n 2500 2% Z 1, Z 2,..., Z n µ σ nµ nσ 2 Z µ σ/ n Z µ σ/ n P ( Z µ σ/ n z ) = z 1 2π e x2 /2 dx( Φ(z)) Z µ h P ( ( ) X µ X µ < h = P σ/ n < h ) ( ) n h n = Φ σ σ P ( ( ) ( ) X µ n X µ > h = P σ/ n > h = 1 Φ h ) n σ σ

238 8 P ( X µ < h ) ( ) h n = 2Φ 1 σ Φ ( z) = 1 Φ (z) Φ (2) 0.973 h n/σ = 2 h h = 2σ/ n 0.95 µ 2σ/ n [ X 2 σ n, X + 2 σ n ] µ 95% 95% n = 200, X = 205.7, σ = 6.4 95% [204.8, 206.6] 1 Φ(z) = α z 100α% z α z α [ ] σ X z α/2 n, X σ + z α/2 n 100(1 α)% interval estimation 1 α confidence level α z α α 0.05 0.025 0.005 z α 1.645 1.96 2.58 8.10 10 187.6 97.5 95% 90% 8.11 Excel 10 95%

8 239 8.6.3 t 95% σ σ t t X 1, X 2,..., X n µ σ 2 n X = 1 n X k, S = 1 n ( Xn n n 1 X ) 2 k=1 k=1 ( X µ ) / (S/ n) n 1 t S 2 σ 2 n n 1 100(1 α)% [ X t (n 1) α/2 ] S, X + t (n 1) S n n t n α n t 100α% α n z α α/2 α 0.05 0.025 0.005 t (10) α 1.812 2.228 3.169 t (20) α 1.725 2.086 2.845 t (50) α 1.676 2.009 2.678 z α 1.645 1.960 2.576 50 8.6.4 95% 90% 0.95 20 20 19 20 19 0 0% 100% 0% 100%

240 8 95% 95% 20 19 95% S/ n t (n 1) α/2 α 95% Φ (1) 0.6413 68% 3 1 α t (n 1) α/2 95% 95% 8.6.5 n 1% 1 µ t(n 1) α/2 ( S < 0.01 n > 10000 n t (n 1) α/2 µ t (n 1) α/2 z α/2 205.7, 6.4 2 α = 0.05 ( n > 10000 2 6.4 ) 2 205.7 n 39 ε n > ( ) 2 ( 1 ε t (n 1) α/2 S µ ) 2 S µ ) 2

8 241 10 100 8.12 10 10 203.2 42.3 95% 2% 8.6.6 7 14.286% 1.39884 14.286% 8.13 (2004) (2007) (1989) (2000)

242 8 (2012) R

8 243 8.14 EXCEL N + 1 0 N 0 (1) 0 1 (2) k 1 k k 1 k k = 2, 3,..., N (3) k m m 3.5 k 1 0 2 3 10 50 Excel RAND Excel 20 100 (2001) 163 176

244 9 9 9.1 ATM 9.1 AT M AT M 9.2 9.3 9.4 9.5 9.1.1

9 245 AT M 9.1.2 9.1 (1) (2) (3) (4)

246 9 9.2 9.2.1 t A(t) Arrival A D(t) Departure D A(t) D(t) [0, T ] T [0, T ] L

9 247 L = 1 T T 0 (A(t) D(t)) dt [0, T ] N N N W W = 1 N T 0 (A(t) D(t)) dt L W T L = N W L = N T W N T λ l L = λw 9.2.2 D(t) D(t) D(t) A(t) D(t) L q W q L q = 1 T T 0 ( A(t) D(t) ) dt, W q = 1 T ( A(t) N D(t) ) dt 0 L q = N T W q

248 9 N/T λ L q = λw q 9.6 10 60 10 60 10 60 = 600 600 600 800 1000 500

9 249 9.2 ATM 9.3 ATM 4.5 20 9.2.3 A B A B 5

250 9 B A B 9.4 A(t) D(t) (1) A(t) D(t)

9 251 (2)

252 9 9.3 9.3.1 [0, T ] N m N Nm T T N T λ Nm < T λm < 1 λm

9 253 ρ r ρ λm < 1 λ λ m λ < 1 m 1/m [0, T ] N 4 m < T λm 4 < 1 λ < 4 m 1/m 4 4/m 9.5 5 n

254 9 9.4 10 10 (1) (2) (3) 10 5 5 15 5 2.5 15 7.5 5

9 255 9.6 5 15 5 t W (t) W (t) 9.7 10 9.8 2 18 x 20 x 9.9 5 20 25 40 40 2.5 7.5 10 2.5 30

256 9 7.5 40 10 2.5 10 + 7.5 30 40 = 6.25 2.5 7.5 0 T N t 1, t 2,..., t N x k = t k t k 1 a k ax k x k /2 N at 1 at N ( ax k x k 2 k=1 ) = 1 T N k=1 x 2 k 2 = N T 1 N N k=1 x 2 k 2 X 1, X 2,... m = E(X), σ 2 = V (X) x 1, x 2,... [0, T ] N T/N E(X) 1 N N k=1 x2 k N X2 E(X 2 ) = V (X) + E(X) 2 = σ 2 + m 2 w w = 1 E(X 2 ) = 1 ( σ 2 + m 2) = m m 2 2m 2 ( ( ) σ 2 1 + m) σ/m σ m σ = 0 0 m/2 σ = m

9 257 9.10 [a, b](b > a > 0) 2 (b a) 2 /3/(a+b) 2 [5, 15] 10 0.1e 0.1t

258 9 9.5 9.5.1 1 1 2

9 259 λ k T k λt k k k + 1 k k + 1 λt k k k 1 k 3 1 k 3 1 m 1 k/m T k k k 1 T k k/m k 1( 0) k k k 1 2 λt k 1 = k m T k T T k /T k p k M p k = λm k p k 1 (k = 1, 2,..., M) 9.5.2 a = λm p k = a k p k 1 = a k a k 1 p k 2 = = ak k! p 0 (k = 1, 2,..., M) T k p k 1 p 0 + p 1 + + p M = 1

260 9 p 0 = (1 + a + a2 2! + a3 3! + + am M! ) 1 p k = a k k! 1 + a + a2 2! + a3 3! + + am M! (k = 0, 1,..., M) 9.5.3 M p M = a M M! 1 + a + a2 2! + a3 3! + + am M! a = 1 95% a = 1 p M < 0.05 M M 1 2 3 4 p M 0.5 0.2 0.0625 0.0154 M = 4 95% 20 5% a = 1 a = 20 a 5% 1% 0.01% Erlang a λ m

9 261 9.11 p M 9.12 a = 20 0.05 M Excel 9.5.4 1 = 1 + a + a 2 2! + a3 3! + + am 1 (M 1)! + 1 = 1 + M p M a p M = a M M! 1 1 + M ap M 1 1 p M 1

262 9 9.6 A B... 9.6.1 1.9, 1.6, 1.2, 5.3, 4.9, 2.9, 10.2, 5.6, 6.7, 11, 0.2,... 10.2, 1.9, 4.9, 1.6, 6.7, 1.2, 5.3, 2.9, 5.6, 11, 0.2,... C while (1) { ; ; ; }

9 263 Excel Excel RAND 10.2, 1.9, 4.9, 1.6, 6.7 1.6 [0, 0.2) 1.9 [0.2, 0.4) 4.9 [0.4, 0.6) 6.7 [0.6, 0.8) 10.2 [0.8, 1) RAND = RAND() 0.49871 4.9 = RAND() 0.80982 10.2 1.9 + 10.2 = 12.1 RAND n [ 0, 1 n), [ 1 n, 2 n), [ 2 n, 3 n),... 10.2, 1.9, 4.9, 1.6, 6.7 RAND F (x) F (x) x [F (x), F (x + t)) RAND

264 9 u F 1 (.) F 1 (u) m F (x) = 1 e x/m (x 0) F 1 (x) = m log(1 x) Excel = LN() Excel x 1 x F 1 (x) = m log x

9 265 9.13 (1)Excel 5 100 (2) 100 10 100... A7:A106 1 100 B7:B106 D7 =SMALL($B$7:$B$106,A7) B7:B106 D8:D106 B7:B106 D7:D106 x 9.6.2 n T n W n S n D n D n = T n + W n + S n = T n+1 + W n+1 n n + 1

266 9 A n+1 = T n+1 T n W n+1 = W n + S n A n+1 n W n+1 = 0 n + 1 W n+1 = max(0, W n + S n A n+1 ) W 1 = 0 W 2, W 3,... 0 0 w 1 = 0 1. s 1 a 2 2. w 2 max(0, w 1 + s 1 a 2 ) 3. s 2 a 3 4. w 3 max(0, w 2 + s 2 a 3 ) 5. s 3 a 4... s 1, s 2, s 3,... a 2, a 3,...

9 267 9.14 Excel 500 6 1/6 4 Excel Excel 1 2 3 4 9.15 5 8 10 Excel 4 4 5 8 10 9.16 4 8 Excel 10 5 5 8 10

268 9 9.6.3 a s w (s/a, w) 500 ρ m W q W q = m 1 + c2 2 ρ 1 ρ c L = λw = λ(w q + m) = λw q + ρ

9 269 9.17 4 ρ = 0.5, 0.6, 0.7, 0.8 ρ 0.5 0.6 0.7 0.8 9.18 [1] 2001 [2] (2006) [3] 1985

270 10 10 10.1 100

10 271 10.1 10.2 0.1 10 0.9 1 0.1 10 2 1 = 9 = 3 2 3 0.2 4 0.8 2500 1 0.2 4 2 + 0.8 0.25 2 1 = 2.25 = 1.5 2 1.5 1 1000 10%

272 10 A, B A B 10.3 10 10 10 10 10 10

10 273 x utility U(x) x 0.999 x U(x) U(11) U(1) U(10010) U(10000) U(11) U(1) > U(10010) U(10000) x < y h > 0 U(x + h) U(x) > U(y + h) U(y) U(x) U(x + h) U(x) > U(y + h) U(y) U(x + h) U(x) h > U(y + h) U(y) h h 0 U (x) > U (y) (for x < y) U(x) 18 x U(x) du(x) dx = 1 x U(x) = log x + C

274 10 x 1 x 1 2 x 2 x 0 C x U(x) y U(y) 10.2 f(x) f (x) f (x) = x 1, f(1) = 0 10.3.1 a = 10 U(a) > 0 (a, U(a)) 0 < x < a y = l(x) U(a)/a x l(x) = U(a) x ( a + 0 1 x ) a l(x) p = x/a U(a) 1 p 0 l(pa) p U(x) U(a) a = 10 p = 0.1 l(1) = U(10)/10 1000 0.1 10 1000 U(0.1) 1000 1000 10000 10000 U(1) U(10)/10 (1, l(1)) 1000

10 275 1000 0.1

276 10 10.3 (1) 1000 0.1 10 (2) 2000 0.1 10 (3) 0.1 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 (4) 0.05 100 500 1000 2000 3000 4000 5000 0.2 1000 2000 3000 4000 5000 6000 10000 15000 20000 0.3 1000 2000 3000 4000 5000 6000 10000 15000 20000 30000 (5)

10 277 10.3.2 X E(X) S(X) U(X) U(X) E(U(X)) U(x) = x 1 2a x2 = a 2 1 2a (x a)2 (0 x a) a = 10 u(x) = 5 1 20 (x 10)2 µ, σ X U(X) ( E(U(X)) = E X 1 ) 2a X2 = E(X) 1 2a E(X2 ) = E(X) 1 ( V (X) + E(X) 2 ) 2a = µ 1 2a (σ2 + µ 2 ) = 1 ( a 2 (µ a) 2 σ 2) 2a µ, σ a - r (0, a) a 2 2ar a = 10, r = 3.75 r = 1

278 10 10.4 U(X) a = 10 r = 2 r = 2

10 279 10.3.3 10.5 A, B X, Y µ, ν σ, τ ρ σ X,Y ρ σ X,Y = E((X µ)(y ν)) ( ) X µ Y ν ρ = E = σ X,Y σ τ στ A u B 1 u (u, 1 u) A B 10.3.4 (u, 1 u) Z = Z(u) = ux + (1 u)y E(Z) V (Z) S(Z) E(Z) = ue(x) + (1 u)e(y ) = uµ + (1 u)ν V (Z) = E(Z 2 ) (E(Z)) 2 = u 2 V (X) + (1 u) 2 V (Y ) + 2u(1 u)σ X,Y = u 2 σ 2 + (1 u) 2 τ 2 + 2u(1 u)ρστ S(Z) = V (Z) = u 2 σ 2 + (1 u) 2 τ 2 + 2u(1 u)ρστ S(Z), E(Z) u (S(Z(u)), E(Z(u))) 0 u 1 (S(Z(u)), E(Z(u)))

280 10 ρ = 1 E(Z) = uµ + (1 u)ν S(Z) = uσ + (1 u)τ u E(Z) = µ ν (S(Z) τ) + ν σ τ (σ, µ) (τ, ν) ρ = 1 E(Z) = uµ + (1 u)ν S(Z) = u 2 σ 2 + (1 u) 2 τ 2 2u(1 u)στ = (uσ (1 u)τ) 2 uσ (1 u)τ E(Z) = µ ν τµ + νσ S(Z) + σ + τ σ + τ µ ν τµ + νσ S(Z) + σ + τ σ + τ u τ σ + τ u < τ σ + τ τµ+νσ σ+τ 0 0 ρ = 1

10 281 ρ (u, 1 u) = (u (ρ), 1 u (ρ)) ρ = 1 ρ = 1 0 u 1 10.6 µ = 5, ν = 20, σ = 2, τ = 5, ρ = 0.5 EXCEL 0 u 1 u (S(Z(u)), E(Z(u))) 10.7 (u, 1 u) V (Z) u σ, τ, ρ

282 10 10.4 1970 1000 90 1000 900 100 1000 90 1000 1000 1000 1000 derivative derived 10.1 10.4.1 1000 90

10 283 10.4.2 Z P P Z Z P P max {Z P, 0} Z 10.8 max {P Z, 0}

284 10 10.5 S S up S down P S down < P < S up P S up S up P 0 P P 10.9 5 A 8 3 A 6

10 285 10.10 8000 10 8 12 20 A 4 S up P a(< 1) S a S (1) C (2) as up (1) S up P (2) as C as down as C + as up (S up P ) as = C + as down as a a = S up P S up S down C S a

286 10 P = 6, S up = 8, S down = 3 a = 0.4 10 4 0.4 0 C + a S down a S = 0 C = a (S S down ) = S up P S up S down (S S down ) C 10.11 2000 1000 3000 (1) P = 2000 C (2)P = 2500 (3)C P 10.5.1 a C as C B B(1 + r) r B B(1 + r) 1 B B(1 + r) 1 (1 + r) 1 as a(1+r)s a C = a (1 + r)s a S down 1 + r = S ( up P S S ) down S up S down 1 + r 1. a S C 2. C 3. S a 4.

10 287 5. (1 + r)(a S C) a S down (1 + r)(a S C) = 0 0 10.12 0 10.13 2000 1000 3000 r = 0.1 P = 2000 10.5.2 { Sup P C C 0 S S down C = S up P S up S down ( S S ) down 1 + r r, P S S down 10.14 (1) 2000 1000 3000 P (2) 1500 3000 0

288 10 10.5.3 * S up, S down C = 1 E (max {Z P, 0}) 1 + r 10.2 Z a { 1 f(z) = b (z a + b) (a b z a) 2 1 b (z a b) (a z a + b) 2 P (> a) (1 + r) E (max {Z P, 0}) = 1 b 2 a+b P (z P )(z a b)dz = 1 (a P + b)3 6b2 P = a b/6 b σ S up, S down S p S up 1 p S down p (S up P ) + (1 p) 0 1 1 + r p(s up P ) p p p S S down 1+r S up S down (S up P ) = 1 1 + r p(s up P ) p = (1 + r)s S down S up S down p

10 289 10.5.4 Z P max {Z P, 0} P Z P 0 0 Z P E (max {Z P, 0}) E(Z P ) = E(Z) P y P P E(Z) P 0 max {Z P, 0} y = E(Z) P y = 0 10.3 Z a { 1 f(z) = b (z a + b) (a b z a) 2 1 b (z a b) (a z a + b) 2 P (< a) (1 + r) E (max {Z P, 0}) = 1 a b 2 (z P )(z a + b)dz 1 a+b b 2 (z P )(z a b)dz P = 1 6b 2 (P a)2 (P a + 3b) + (a P ) + b 2 6 10.15 a = 10, b = 5 P a

290 10 10.6... 1990 2001 8, 9 25.5 0.1 5000 26.5 0.1 5000 7 10.16

10 291 10.4 (1) 12 5cm 100 1000 514, 000 (2) 7, 8 26.5 C 1 C 500 1500 220 http://www.77bank.co.jp (3) 3cm 250 1750 (4) 13 (5) 10 22 C 5 200 2000 http://www.sompo-japan.co.jp 10.17 10.6.1

292 10 10.5 100 100 1 1 90 9000 1 110 1 1000 1 1

10 293 10.6.2 25 C 0.1 C 500 5000

294 10 10.6.3 25 C 0.1 C 500 5000 30 25 C 23.6 C, 24.4 C, 24.7 C, 24.9 C 30 1 (5000 + 3000 + 1500 + 500) = 333.33 30 30 23.6 C 29

10 295 1 (3000 + 1500 + 500) = 172.41 29 10.18 5.5 C 0.1 C 100 1000 20 3.8, 5.4, 6.6, 7.5, 5.0, 6.3, 6.6, 6.7, 5.5, 5.5, 5.5, 5.8, 5.4, 6.1, 6.9, 4.7, 6.6, 5.0, 5.7, 5.6 X X = 24.9 500 X = 24 5000 X = 23 5000 X = 26 0 X 36 27.1, 1.28 24 C 24.1 24.2... 24.8 24.9 25 C 0.0082 0.0019 0.0023... 0.0061 0.007 0.9549

296 10 5000P (X < 24) + 10 k=1 500k P (X = 25 0.1k) 114.7 25 C 500 5000 (1970 2006) 0.2 C 10.19 5.5 C 0.1 C 100 1000 Excel 20 x =NORMDIST(x+0.05,m,s,TRUE)-NORMDIST(x-0.05,m,s,TRUE) m, s

10 297 10.6.4 10.20 6 0.1 100 Excel 7 8 9 [1] (2000) [2] (Against the Gods) (2001) [3] (2002)