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- ゆめじ まつかた
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1 i P ERT P ERT P ERT (Economic Order Quantity, EOQ)
2 ii
3 A180 A Pareto
4
5 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 , 0.3, , 0.4, % 5 : 4 : = = = 0.06
6 Analytic Hierarchy Process AHP Thomas Saaty
7 (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) =
8
9 , 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)
10 150 5 (1) (2) (3) (1) (2) 1/3 1 5 (3) 1/9 1/ : 3 5 : = 3 : : = : : M M M EXCEL ˆ(1/M) : : = 9 13 : 5 21 : 1 = : : a A = ( 1 a 1/a 1 ) a : 1 a = a : a : 1 a + 1 = a : 1
11
12 A C 6 b AC = 6, b CA = 1/6 (A) (B) (C) (D) (A) 1 (B) 1 (C) 1 (D) 1...
13
14 (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)
15 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 w 1 : w 2 : w 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
16 156 5 D 3 D 3 M D M A D D A D A = {a ik } {v i } a ik v i v k
17 5 157 {v i } D = {d ik } d ik = v i /v k a ik /d ik 1 β = i,k=1 a ik d ik = i,k=1 a ik v k v i C.I. = 3 3β 3 (β 1) = (3) 0 C.I. β d ik = d 1 ki A D 9 β = i,k=1 a ik d ik = β = 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 C.I. b= C.I. =
18 (C.R.) (4) R.I. C.I. C.R. = C.I. R.I. C.R. 0.2 M R.I R.I. C.R. C.R. = 5.10 C.I. C.R
19 : 1 2 : 1 4 : 1
20
21 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 % 30%
22 162 5 (sensitivity analysis)
23
24 AHP AHP AHP 5.13 [1] (1986) [2] AHP (2005) [3] 1990 [4] 2007
25 P ERT/CP M Program Evaluation and Review Technique / Critical Path Method P ERT P ERT
26 166 6 A J 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
27 A, B, E P ERT
28 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
29 P ERT P ERT 2.1 P ERT P ERT P ERT EXCEL 6.4 P ERT B, C, D E E B, C, D 2.1
30 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
31 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
32 172 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)
33 D C B F C 6.4 F G H G H J 6.5 D A E A B F A, B, C P ERT
34 A, B, C, D, E A, B C, D A E B, D P ERT P ERT
35 P ERT P ERT T ik (i, k) In(k) k Out(k) k In(3) = {1, 2}, Out(3) = {4, 5, 6} 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))
36 176 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 = (2, 3) ES 2 + T 23 = 17 (3, 4) (2, 3) max {ES 1 + T 13, ES 2 + T 23 } = In(k) k ES k = max i In(k) {ES i + T ik } 3. PERT i
37 6 177 (i, k)( k Out(i)) i ES i , 3, 4 10, 14, 18 (2, 5), (3, 5), (4, 5) 12, 9, (5, 6), (5, 7) 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
38 178 6 LF 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 , 17, 18 25, 24, 28 (15, 16), (15, 17), (15, 18) 8, 4, (13, 15), (14, 15) ES k LF k
39 6 179 P ERT ES i LF i ES i /LF i 0/ ES k = max i In(k) {ES i + T ik } (7) LF i = min {LF k T ik } (8) k Out(i)
40 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 (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
41 / 10 A C 10 B D 5, 4 B D D B Free Float time (i, k) F F ik k i F F ik = ES k ES i T ik
42 182 6 A(4) C(6) B(3) D(2) B D 2.4 D B 5 D B 1 1 B 5 D d B D / / 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
43 critical path
44 P ERT P ERT A, B 2.5
45 6 185
46 P ERT A C E F 15 C C A, B {(1, 2), (1, 3)} {(4, 5)}
47
48 P ERT P ERT 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
49 H D G E, F, G 2, 1, 1 H 11 H K H J 10 P ERT
50 C E 6.18 J (2003)
51 revenue managemnet
52 Just In Time Kanban System
53 7 193 Supply Chain Management
54
55 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
56 (1) ( ) ( ) ( ) ( ) 0 ( ) ( )
57 7 197 (2) ( ) ( ) ( ) 2 ( ) 7.2
58 198 7 (3) 7.3
59
60 (Economic Order Quantity, EOQ) 1 T D D 1
61 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)
62 d = 10, B = 1, K = 2000 ( ) Q 200 Q OR (d = 10, B = 1, K = 2000) (d = 12, B = 1, K = 2000) 3 2 (d = 10, B = 1.2, K = 2000)
63 % ( robust)
64 ( 7.5 )d = 20, B = 1, K = 1000 (0) (Q 1 ) Q 1 = (1) d 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 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 ) =
65 f3(q1) f3(q3) f 3(Q 3) = ( ) 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 )
66 206 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 =
67 ( ) L L L
68 208 7 ( ) % 95% X m σ n nm nσ n L X X L 1.65 Lσ P ( X X L Lm ) Lσ = P ( ) X1 + + X L Lm 1.65 Lσ π e x2 /2 dx = Lσ 95% % % 10% Excel Norminv 4 95% 90% %
69 7 209 ( ) 95% 1 σ N L 1.65 N + Lσ
70 % 95%
71 ( ) ,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, ( )
72 z b z > b ( ) = z b = Excel = = = , Excel = 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)
73 ( ) 3 ( 50 10) , 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
74 214 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 (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)) = F (z) F (z ) = F (z 1) z 65 (55 ) (10 ) 50 ( ) ( ) 7.11 g(z)
75 (1) (2) (3) ( ) (4) :
76 A B B A B B A B (A B)/A
77
78
79 7 219 ABC 7.15 Excel ABC A B C A B
80 k F k 4. (k/30, F k )(k = 0, 1,..., 30) 5. F k 0.7 k k k A
81 Yahoo 8.2 F Excel rand() PERT PERT simulation
82 Excel
83
84 Wii X(t) t 2 p X(t) ( q) X(t) d dt X(t) 1 X(t) = p + qx(t)
85 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 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
86 226 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 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
87
88 % s S (s, S) s S s S , 11 8, 11 8, 11
89 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 Excel B7:B106 =ROUND(NORM.INV(RAND(),10,2),0) E3 =average(b7:b106) E4 =stdev(b7:b106) D8:D E8:E28 E8 =FREQUENCY(B7:B106,D8:D28) Ctrl Shift Enter
90 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 n A(n) B(n) C(n) E(n) n
91 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) 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 Excel s S z s S z = f(s, S) s, S
92 232 8 s, S s, S (s, S) s, S Excel S s s
93 random numbers table 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()
94 a b a = 1389, b = 8567 a b = a = a = , 9563, 3007, 6723, 8247, 5083, 287, 8643, 5127, 1403, 8767, 7363,... 8, 9, 3, 6, 8, 5, 2, 8, 5, 1, 8, 7, , 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
95 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 B 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
96 Excel... Excel s C4 J3 1. L10:L19 10 M9 =J3 2. L9:M19 What-if 3. C4 J10:L19 C4 M10:M19 J10:L J10:J19 M10:M z 1, z 2, , 6.4
97 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 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 X n n % 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
98 238 8 P ( X µ < h ) ( ) h n = 2Φ 1 σ Φ ( z) = 1 Φ (z) Φ (2) 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 = % [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 z α % 90% 8.11 Excel 10 95%
99 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 t (10) α t (20) α t (50) α z α % 90% % 100% 0% 100%
100 % 95% % S/ n t (n 1) α/2 α 95% Φ (1) % 3 1 α t (n 1) α/2 95% 95% n 1% 1 µ t(n 1) α/2 ( S < 0.01 n > n t (n 1) α/2 µ t (n 1) α/2 z α/ , α = 0.05 ( n > ) n 39 ε n > ( ) 2 ( 1 ε t (n 1) α/2 S µ ) 2 S µ ) 2
101 % 2% % % 8.13 (2004) (2007) (1989) (2000)
102 242 8 (2012) R
103 EXCEL N N 0 (1) 0 1 (2) k 1 k k 1 k k = 2, 3,..., N (3) k m m 3.5 k Excel RAND Excel (2001)
104 ATM 9.1 AT M AT M
105 9 245 AT M (1) (2) (3) (4)
106 t A(t) Arrival A D(t) Departure D A(t) D(t) [0, T ] T [0, T ] L
107 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 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
108 248 9 N/T λ L q = λw q =
109 ATM 9.3 ATM A B A B 5
110 250 9 B A B 9.4 A(t) D(t) (1) A(t) D(t)
111 9 251 (2)
112 [0, T ] N m N Nm T T N T λ Nm < T λm < 1 λm
113 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 n
114 (1) (2) (3)
115 t W (t) W (t) x 20 x
116 = 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 m) σ/m σ m σ = 0 0 m/2 σ = m
117 [a, b](b > a > 0) 2 (b a) 2 /3/(a+b) 2 [5, 15] e 0.1t
118
119 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) 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 p M = 1
120 260 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) 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 p M M = 4 95% 20 5% a = 1 a = 20 a 5% 1% 0.01% Erlang a λ m
121 p M 9.12 a = M Excel = 1 + a + a 2 2! + a3 3! + + am 1 (M 1)! + 1 = 1 + M p M a p M = a M M! M ap M 1 1 p M 1
122 A B , 1.6, 1.2, 5.3, 4.9, 2.9, 10.2, 5.6, 6.7, 11, 0.2, , 1.9, 4.9, 1.6, 6.7, 1.2, 5.3, 2.9, 5.6, 11, 0.2,... C while (1) { ; ; ; }
123 9 263 Excel Excel RAND 10.2, 1.9, 4.9, 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() = RAND() = 12.1 RAND n [ 0, 1 n), [ 1 n, 2 n), [ 2 n, 3 n), , 1.9, 4.9, 1.6, 6.7 RAND F (x) F (x) x [F (x), F (x + t)) RAND
124 264 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
125 (1)Excel (2) A7:A B7:B106 D7 =SMALL($B$7:$B$106,A7) B7:B106 D8:D106 B7:B106 D7:D106 x 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
126 266 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, 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,...
127 Excel /6 4 Excel Excel Excel Excel
128 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
129 ρ = 0.5, 0.6, 0.7, 0.8 ρ [1] 2001 [2] (2006) [3] 1985
130
131 = 9 = = 2.25 = %
132 A, B A B
133 x utility U(x) x 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
134 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) = a = 10 U(a) > 0 (a, U(a)) 0 < x < a y = l(x) U(a)/a x l(x) = U(a) x ( a 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)/ U(0.1) U(1) U(10)/10 (1, l(1)) 1000
135
136 (1) (2) (3) (4) (5)
137 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) = (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
138 U(X) a = 10 r = 2 r = 2
139 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 (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)))
140 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
141 ρ (u, 1 u) = (u (ρ), 1 u (ρ)) ρ = 1 ρ = 1 0 u 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
142 derivative derived
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149 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 = 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 a = 10, b = 5 P a
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3 3. (stochastic differential equations) { dx(t) =f(t, X)dt + G(t, X)dW (t), t [,T], (3.) X( )=X X(t) : [,T] R d (d ) f(t, X) : [,T] R d R d (drift term) G(t, X) : [,T] R d R d m (diffusion term) W (t)
i I II I II II IC IIC I II ii 5 8 5 3 7 8 iii I 3........................... 5......................... 7........................... 4........................ 8.3......................... 33.4...................
5 n P j j (P i,, P k, j 1) 1 n n ) φ(n) = n (1 1Pj [ ] φ φ P j j P j j = = = = = n = φ(p j j ) (P j j P j 1 j ) P j j ( 1 1 P j ) P j j ) (1 1Pj (1 1P
![5 n P j j (P i,, P k, j 1) 1 n n ) φ(n) = n (1 1Pj [ ] φ φ P j j P j j = = = = = n = φ(p j j ) (P j j P j 1 j ) P j j ( 1 1 P j ) P j j ) (1 1Pj (1 1P](/thumbs/90/103736623.jpg "5 n P j j (P i,, P k, j 1) 1 n n ) φ(n) = n (1 1Pj [ ] φ φ P j j P j j = = = = = n = φ(p j j ) (P j j P j 1 j ) P j j ( 1 1 P j ) P j j ) (1 1Pj (1 1P")
p P 1 n n n 1 φ(n) φ φ(1) = 1 1 n φ(n), n φ(n) = φ()φ(n) [ ] n 1 n 1 1 n 1 φ(n) φ() φ(n) 1 3 4 5 6 7 8 9 1 3 4 5 6 7 8 9 1 4 5 7 8 1 4 5 7 8 10 11 1 13 14 15 16 17 18 19 0 1 3 4 5 6 7 19 0 1 3 4 5 6 7
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IA 2013 : :10722 : 2 : :2 :761 :1 (23-27) : : ( / ) (1 /, ) / e.g. (Taylar ) e x = 1 + x + x xn n! +... sin x = x x3 6 + x5 x2n+1 + (
IA 2013 : :10722 : 2 : :2 :761 :1 23-27) : : 1 1.1 / ) 1 /, ) / e.g. Taylar ) e x = 1 + x + x2 2 +... + xn n! +... sin x = x x3 6 + x5 x2n+1 + 1)n 5! 2n + 1)! 2 2.1 = 1 e.g. 0 = 0.00..., π = 3.14..., 1
f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f
22 A 3,4 No.3 () (2) (3) (4), (5) (6) (7) (8) () n x = (x,, x n ), = (,, n ), x = ( (x i i ) 2 ) /2 f(x) R n f(x) = f() + i α i (x ) i + o( x ) α,, α n g(x) = o( x )) lim x g(x) x = y = f() + i α i(x )