50. (km) A B C C 7 B A 0

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49... 5 A B C. (. )?.. A A B C. A 4 0

50. (km) A B C..9 7. 4.5.9. 5. 7.5.0 4..4 7. 5.5 5.0 4. 4.. 8. 7 8.8 9.8. 8 5. 5.7.7 9.4 4. 4.7 0 4. 7. 8.0 4.. 5.8.4.8 8.5. 8 9 5 C 7 B 5 8 7 4 4 A 0 0 0 4 5 7 8 9 0

.. 5 B 5 9 C 7 8 45.km.7km A B C 5. A B C A B C 4..4 7...9 7..4.8 8.5 0 4. 7. 8.0 4.. 5.8 4.5.9. A:.5 4.. 8. 9.4 4. 4.7 5.5 5.0 4. B: 0.8 8 5. 5.7.7 5. 7.5.0 7 8.8 9.8. C:.8 B.km B B.9km B 4 A B.5. 4.7km(.5km).4. A C A... (vertex) (edge) (arc).7 9

5.4 A 8 9 5 C 7 B 5 8 7 4 4 A 0 0 0 4 5 7 8 9 0.5 B. B.9 4 A. B.9 7 C. A.4 8 C.7 0 A 4. 5 C 4. A 4. A 5. 9 A.4 4.7

.. 5. 8 9 5 C 7 B 5 8 7 4 4 A 0 0 0 4 5 7 8 9 0 a, b, c,... i j ij.7 9, 5,,,, 4,, 4, 5 i j ij i j ij ji n m V = {,,..., n} E = {e, e,..., e m } V E G = (V, E) (graph) ( ij ji )

54.7 4 5. V E V = f; ; ; 4; 5; g E = f; 4; ; ; 4; 45; 5g. (transshipment problem) (source) (sink)...8.8 A B C P Q 7 9 5 4 5 8.9 ( ) [ ] ( ),

.. 55.9 - [] 5 [5] [8] [8] -7 [] [5] [4] [0] 7 [4] [5] 5 4 [] [5] [9] [8] [] 8-9.. i j ij i j ( ji ij j i ) ij x ij x ij 0 E E = {, 4, 4, 5,,, 7, 4, 47, 5, 58,, 4, 78, 8} ij c ij c = 8 c 58 = 9 z z = ij E c ij x ij = 8x + 8x 4 + 5x 4 + 4x 5 + x + 5x + 5x 7 + x 4 + 0x 47 + 5x 5 + 9x 58 + 4x + x 4 + x 78 + 8x 8 x 4 x 4 5 x 5 x = x 4 + x 5

5 x x x 4 x + = x + x 4 x x x 4 = ( 8) x 58 + x 78 = x 8 + x x x 4 = x x 4 x 5 = 0 x x x 7 + x 4 + x = 5 x 4 + x 4 x 4 x 47 + x 4 = 7 x 5 x 5 x 58 = 9 x + x 5 x x 4 + x 8 = 0 x 7 + x 47 x 78 = 0 x 58 + x 78 x 8 = (.) min 8x + 8x 4 + 5x 4 + 4x 5 + x + 5x + 5x 7 + x 4 +0x 47 + 5x 5 + 9x 58 + 4x + x 4 + x 78 + 8x 8 s.t. x x x 4 = x x 4 x 5 = 0 x x x 7 + x 4 + x = 5 x 4 + x 4 x 4 x 47 + x 4 = 7 x 5 x 5 x 58 = 9 x + x 5 x x 4 + x 8 = 0 x 7 + x 47 x 78 = 0 x 58 + x 78 x 8 = x ij 0, ij E (.) (.) x = 4 x 4 = 8 x 4 = 0 x 5 = 4 x = 0 x = 0 x 7 = 0 x 4 = 5 x 47 = 0 x 5 = 0 x 58 = x = 0 x 4 = 0 x 78 = 0 x 8 = 0.. (.)

.. 57. (.). 00 400 400 00 500 00 00 4 5 4 4 4 4.....0 9.0 8 0 9 49 59 0 x = 0 x 4 = 8 x 9 = 4 x 4 = 0 x 5 = 0 x = 0 x = 0 x 7 = 0 x 4 = 5 x 47 = 0 x 49 = 0 x 5 = 0 x 58 = x 59 = x = 0 x 4 = 0 x 78 = 0 x 8 = 0

58.0 - [] 5 [5] [0] [8] [8] -7 [] [5] [4] [0] 7 [0] 4 [] 9 0 [0] [5] [4] 5 [] [5] [9] [8] 8-9.4 V E V = f; ; ; 4g E = f; ; ; ; 4; 4; 4g. (V; E). a 4 b c 4 4 4 5 4 4 7 () a = b + c () a > b + c () a < b + c.4.4.

.4. 59 4 90 40 85 0 9 50 4 90 0 4 0 4 (40 ) (0 + 50 + 0 + 0 = 40 ) 40 40 0 4 0 (.) 90 40 + 85 40 = 47540. 7 8 9 0 0( ) 4 0 0 0 04 4 45 min 90x 0 + 85x 0 + 9x 0 + 90x 04 s.t. x 0 x 0 x 0 x 04 = 550 x 0 x x = 0 x + x 0 x x 7 = 0 x + x 0 x 4 x 8 = 0 x 4 + x 04 x 45 x 49 = 0 x 45 = 0, x = 40, x 7 = 0, x 8 = 50, x 49 = 0 x ij 0, ij (.4)

0. -550 0 [90] [85] [9] [90] 4 5 0 7 8 9 40 0 50 0 (.4) x 0 = 40, x 0 = 40, x 0 = 0, x 04 = 0, x = 0, x = 40, x = 90, x 7 = 0, x 4 = 40, x 8 = 50, x 45 = 0, x 49 = 0, (.) LP. min 90x 0 + 85x 0 + 9x 0 + 90x 04 + x + x + x 4 + x 45 s.t. x 0 x 0 x 0 x 04 = 550 x 0 x x = 0 x + x 0 x x 7 = 0 x + x 0 x 4 x 8 = 0 x 4 + x 04 x 45 x 49 = 0 x 45 = 0, x = 40, x 7 = 0, x 8 = 50, x 49 = 0 x ij 0, ij (.5) x 0 = 40, x 0 = 70, x 0 = 0, x 04 = 40, x = 0, x = 40, x = 50, x 7 = 0, x 4 = 0, x 8 = 50, x 45 = 0, x 49 = 0

.4.. -550 0 [90] [85] [9] [90] 4 5 [] [] [] [] 0 7 8 9 40 0 50 0.5. 00... [ ] [ ] [ ] [ ].4. 500 80 0 50 00 4 4 85 0 90 05. ij x ij x = 80 x 0 = 5 x = 5 x 40 = 5 x 4 = 75 x 50 = 0 x 9 = 5 x 79 = 80 x 80 = 0 x 9, = 5 x 0, = 80 x, = 0

. 0 [0] [0] [0] [0] [0] [0] [0] [0] -80 4-80 5 7-80 8 0-80 -0-0 -0-0 [50] [50] [50] [50] [500] [500] [500] [500] 9 [00] [00] [00] 85 0 90 05 5 4 0 5 5.. LP.7 ( ) 0 0 p q(> p) r j d j (7 d 7 = 0 ) 5.5.4 0 5 7 4

.5..4-0 4 5 7 4 c ij min +c x + c x + c 4 x 4 + c x + c 5 x 5 + c x +c 4 x 4 + c 5 x 5 + c x + c 7 x 7 + c 4 x 4 + c 47 x 47 + c 5 x 5 + c 7 x 7 s.t. x x x 4 = 0 +x x x 5 x = 0 +x + x x 4 x 5 x x 7 = 0 +x 4 + x 4 x 4 x 47 = 0 +x 5 + x 5 + x 5 = +x + x + x 4 x 5 + x 7 = 0 +x 7 + x 47 x 7 = 4 x ij 0, ij (.) (.).5.8.5 ( ) u v u v u v ( ) (cycle)

4.5 ij 4 5 4 5 7 4 47 5 7 () c ij 9.0 9.5.4 4.4 4.8.5.4 5.4 7. 4.0..9 8.7. x ij 0 4 0 0 0 0 0 0 0 4 0 0 () c ij.5 9.8.5..9.7.4 0..7. 0.7 8..9 0.4 x ij 0 0 0 0 0 0 0 0 4 0 0 0 0 () c ij 7.4 5.4.8.7 0. 8.9 8.7. 5.7..0.4. 8.7 x ij 0 4 0 0 0 0 0 0 0 4 0 0 (4) c ij 9.0. 7.4 4. 4.0 5..7 5..5 0. 8.4 8. 7.0 4.7 x ij 0 0 0 0 0 0 0 0 4 0 0 0 0 (5) c ij 0. 5.7 4. 9..9 0.7 4.8 9.9 9. 5.7. 7.8. 0. x ij 4 0 0 0 0 0 0 4 0 0 0 () c ij 5.5 0. 4..0 7.9. 0. 9.0 7.7 9. 7..9.4 5. x ij 0 0 0 0 0 0 4 0 0 0 4 0 0 (7) c ij 9..8 0..0 4.0..9 0.8 0. 7.9 0. 8.8. 7. x ij 0 0 0 0 0 0 0 0 0 0 4 0 (8) c ij.0...4 0. 4.0.5 0.9 7.7 9.7 7. 7.9.4.0 x ij 0 4 0 0 0 0 0 0 0 4 0 0 (tree) (spanning tree).. ( ). ( 0 ) 4..(A).. 0

.5. 5. - [] 5 [5] (A) (B) (C) [8] 0 - [8] 0 - [8] [5] [8] 4 4 [] [5] [4] [5] 5 [4] 0 5 [9] 4 [5] [] [8] -9-7 4 [] 4 [] [5] [4] 0 5 [9] 9 [5] [] [8] 0-9 -7 4 [] 9 5 [5] [4] [4] 5 [9] -9-7 [] 7 [] [5] 5 [5] [4] 8 [0] [5] [8] [0] [8] [5] 0 [0] [8] 8 7 [] 8 7 [] 8 7 [] 8

4 4 4 4 4 5 4 5 8 + + = 7 4 4 0 4 = ( + ) 4 = 5 4 0 4 9 58 9 8 5 0.(B) 4 7.(C) ( leaf ) ( root ) (0 ) ( ). ( ) min [ ] s.t. [ ] [ ] = [ or ], [ ]

.. 7.... (= ) (= ) i (i =,..., ) j (j = A, B, C) x ij (i =,..., ; j = A, B, C) x ia + x ib + x ic =, i =,..., x A + x A + + x A = x B + x B + + x B = x C + x C + + x C = x ij 0, i =,..., ; j = A, B, C.x A + 4.5x A + +.4x A +.9x B +.9x B + +.8x B + 7.x C +.x C + + 8.5x C (.7) min.x A + 4.5x A + +.4x A +.9x B +.9x B + +.8x B +7.x C +.x C + + 8.5x C s.t. x ia + x ib + x ic =, i =,..., x A + x A + + x A = x B + x B + + x B = x C + x C + + x C = x ij 0, i =,..., ; j = A, B, C (.8)

8 x A = x B = x A = x 4A = x 5C = x B = x 7C = x 8C = x 9B = x 0A = x A = x A = 0 0 A,,4,0,, B,,9 C 5,7,8 4.(km).4.7 C. A.7.7 8 9 5 C 7 B 5 8 7 4 4 A 0 0 0 4 5 7 8 9 0.9 (.8)

.. 9.. 8.8.8 4 5 7 8 a 8 5 4 7 b 5 4 c 7 8 7 7 8 d 4 8 5 8 4 e 7 4 8 8 f 4 4 7 5 g 8 7 5 5 h 5 5 4 7.9.9 ( ).9 4 5 7 8 a b c d e f g h

70.5 8 min s.t. n c ij x ij i= j J x ij =, i =,..., n j J n x ij =, i= x ij 0, j J i =,..., n; j J c ij i j J = {a, b,,..., h}. 8 5 7 4.0.0 ( ) 8 5 7 4 h a c f b e d g 4 8. 4 5 7 8 a b c d e f g h.0 5 5 B D E 4 5

.. 7. ( ) 4 5 A 5 4 8 7 B 4 9 5 C 5 8 5 4 D 7 9 E 8 0.. 4 j i u ji. ( ).... ( 8 8 ) 0.. a, e, j, c, f, a, b, e, 4 d, f, 5 d, g, a, h, i, k, 7 d, f, 8 g, h, 9 f, g, 0 d, k,

7.4.4 s t,,..., 0 s t a, b,..., k t s,,..., 0 a, b,..., k s t 0 ts 0 (.5).5 (cover) M (jmj) C (jcj) jcj jmj M C jc j = jm j

.. 7 x j = x c = x e = x 4s = x 5g = x i = x 7d = x 8h = x 9f = x 0k = x ta = x tb = x ts = 9. 9 a b. 7 8.7.7 4 5 7 8 7/ 7/ 7/ 7/9 7/ 7/ 7/5 7/0 7/ 7/9 7/ 7/ 7/9 7/4 7/9 7/ 8 7 8 5 4 i j i j i j i j

74.8 ( ) ( ) - 4 5 4 5 7-4 - 8-8 9 - - - - 4 5 7 5 7 8 0 7 8 8-8 [-] 8 9 8 0 i(i=,..., 8) i (i =,..., 8) 09 min x 09 s.t. x 8 + x 9 =, x 5 + x + x 7 + x 8 + x 9 = x 8 + x 9 =, x 45 + x 4 + x 47 + x 49 + x 49 = x 59 =, x 7 + x 8 + x 9 =, x 78 + x 79 = x 89 =, x 0 =, x 0 =, x 0 =, x 04 = x 5 + x 45 + x 05 =, x + x 4 + x 0 = x 47 + x 7 + x 07 =, x 8 + x 8 + x 8 + x 8 + x 78 + x 08 = 8 x i9 + x 09 = 8 i= 8 x 0j + x 09 = 8 j= x ij 0 ij

.7. 75 x 8 = 0 x 9 = x 5 = x = 0 x 7 = 0 x 8 = 0 x 9 = 0 x 8 = 0 x 9 = x 45 = 0 x 4 = x 47 = 0 x 49 = 0 x 49 = 0 x 59 = x 7 = x 8 = 0 x 9 = 0 x 78 = x 79 = 0 x 89 = x 0 = x 0 = x 0 = x 04 = x 05 = 0 x 0 = 0 x 07 = 0 x 08 = 0 x 09 = 4 x ij = 5 4 7 7 8 4 4 7 8 ( ) 4 7 n M M n M.7.7..9 <> <> <5> <> <4> 5 <0> s <8> 4 <7> <> <9> t.9 s t t s ts (.0 ) x ts

7 ( ).0 <> <> <5> <> <4> 5 <0> s <8> 4 <7> <> <9> t [] max x ts s.t. x ts x s x s = 0 x s x x 4 = 0 x s x = 0 x x 5 = 0 x 4 x 45 x 4 = 0 x 5 + x 45 x 5t = 0 x 4 + x x t = 0 x 5t + x t x ts = 0 x s, x s 8, x 5, x 4, x 7 x 5, x 45 4, x 4, x 5t 0, x t 9 x ij 0, ij E (.9) x s = 5 x s = 7 x = x 4 = x = 7 x 5 = x 45 = x 4 = 0 x 5t = 5 x t = 7 x ts =.7. A B C.. ( S )

.7. 77 C S 4 C S 4 C. a b c d 4 e 4 f 5 g h 4 i 4 C j 5 A k B l C. (.) A B C. a b c d e f g h i j k l 5 8 7 5 8 9 7 9 7 7 9. T A B C T T S.0 (.9) A B 9 C 4..7. (.9)

78 (.9) min ξ s + 8ξ s + 5ξ + ξ 4 + 7ξ + ξ 5 +4ξ 45 + ξ 4 + 0ξ 5 + 0ξ 5t + 9ξ t s.t. λ s + λ t λ s λ + ξ s 0 λ s λ + ξ s 0 λ λ + ξ 0 λ λ 4 + ξ 4 0 λ λ + ξ 0 λ λ 5 + ξ 5 0 λ 4 λ 5 + ξ 45 0 λ 4 λ + ξ 4 0 λ 5 λ t + ξ 5t 0 λ λ t + ξ t 0 ξ ij 0, ij E (.0) (.0) λ i (i = s,,,...,, t) λ i (i = s,,,...,, t) ξ ij (ij E) δ λ i = λ i + δ, i = s,,,...,, t λ i (i = s,,,...,, t) ξ ij (ij E) λ s = 0 (.0) (λ s, λ,..., λ t, ξ s,..., ξ t ) λ λ 4 + ξ 4 > 0 (ξ ij ) (.0) min ξ s + 8ξ s + 5ξ + ξ 4 + 7ξ + ξ 5 +4ξ 45 + ξ 4 + 0ξ 5 + 0ξ 5t + 9ξ t s.t. λ s + λ t = λ s λ + ξ s = 0 λ s λ + ξ s = 0 λ λ + ξ = 0 λ λ 4 + ξ 4 = 0 λ λ + ξ = 0 λ λ 5 + ξ 5 = 0 λ 4 λ 5 + ξ 45 = 0 λ 4 λ + ξ 4 = 0 λ 5 λ t + ξ 5t = 0 λ λ t + ξ t = 0 λ s = 0 ξ ij 0, ij E (.)

.7. 79 s t : s 5 t : s 4 5 t : s 4 t 4 : s t P = {s,, 5, 5t} P = {s, 4, 45, 5t} P = {s, 4, 4, t} P 4 = {s,, t} λ s λ + ξ s = 0 λ λ + ξ = 0 λ λ 5 + ξ 5 = 0 λ 5 λ t + ξ 5t = 0 λ s + λ s = 0 λ t = ij P ξ ij λ t = 0 ij P ξ ij = min ξ s + 8ξ s + 5ξ + ξ 4 + 7ξ + ξ 5 s.t. +4ξ 45 + ξ 4 + 0ξ 5 + 0ξ 5t + 9ξ t ξ ij =, k =,,, 4 ij P k ξ ij 0, ij E (.) ( ) (.0) : λ s = λ = λ = λ = λ 4 = 0 λ 5 = 0 λ = 0 λ t = 0 ξ = 0 ξ 4 = ξ = ξ 5 = ξ 45 = 0 ξ 4 = 0 ξ 5 = 0 ξ 5t = 0 ξ t = 0 ξ s = 0 ξ s = 0

80. 7 5 4 5 4 ( ).8 (Shortest Path Problem) A B.4?.4 A B ¾ ½ ½ ½¼ ¾ ¾ ¾ A B.5 ( )

.8. (Shortest Path Problem) 8 LP (Dijkstra) V = {s,,,..., n} E ij E d ij > 0 s j [ ] : v s = 0 v j =, M = {s} N = j V, j s M i v i = min j M {v j} M N N N {i} M M/{i} 4 v j j V/M ij E v i + d ij < v j 5 v j v i + d ij M M {j} M = d j s j.4.5, (dynamic programming, DP) V =,,..., n, t t (.4 ) i t n k J k (i) k = 0,,..., n J k (i) { J k (i) = min j=,...,n [d ij + J k+ (j)], k = 0,,..., n ; i =,..., n J n (i) = a it, i =,..., n DP (.) (.)

8.5 step v = 0, v j =, j =,, 4, 5, M = {}, N = { } step min j M {v j } = v = 0 step M = { }, N = {} step 4 : min{v 0 + d, v } = min{0 +, } = : min{v 0 + d, v } = min{0 + 7, } = 7 M = {, } step 5 M step step min j M {v j } = min{v, v } = min{, 7} = step M = {}, N = {, } step 4 : min{v + d, v } = min{ +, 7} = 4 4: min{v + d 4, v 4 } = min{ +, } = 7 5: min{v + d 5, v 5 } = min{ + 0, } = M = {, 4, 5} step 5 M step step min j M {v j } = min{v, v 4, v 5 } = min{4, 7, }=4 step M = {4, 5}, N = {,, } step 4 4: min{v + d 4, v 4 } = min{4 +, 7} = 5: min{v + d 5, v 5 } = min{4 + 8, } = M = {4, 5} step 5 M step 4 step min j M {v j } = min{v 4, v 5 } = min{, }= 4 step M = {5}, N = {,,, 4} step 4 5: min{v 4 + d 45, v 5 } = min{ +, } = 8 : min{v 4 + d 4, v } = min{ + 5, } = M = {5} step 5 M step 5 step min j M {v j } = min{v 5, v } = min{8, }=8 5 step M = {}, N = {,,, 4, 5} step 4 : min{v 5 + d 5, v } = min{8 +, } = 0 M = {} step 5 M step step min j M {v j } = min{v } = 0 step M = { }, N = {,,, 4, 5, } step 4 M = { } step 5 M = v = 0, v =, v = 4, v 4 =, v 5 = 8, v = 0 : 4 5

.8. (Shortest Path Problem) 8 d ij ij d ii = 0 k = n 0 (.) ( ) DP.4 k = 4 ( t ) J 4 (4) = 5 J 4 (5) = k = ( t ) J (5) = min {d 5j + J 4 (j)} = d 55 + J 4 (5) = 0 + = j=,...,n J (4) = min{d 44 + J 4 (4), d 45 + J 4 (5)} = {0 + 5, + } = 4 J () = min{d 4 + J 4 (4), d 5 + J 4 (5)} = { + 5, 8 + } = 7 J () = min{d 4 + J 4 (4), d 5 + J 4 (5)} = { + 5, 0 + } = k = ( t ) J (5) = min{d 55 + J (5)} = 0 + = J (4) = min{d 44 + J (4), d 45 + J (5)} = {0 + 4, + } = 4 J () = min{d + J (), d 4 + J (4), d 5 + J (5)} = {0 + 7, + 4, 8 + } = J () = min{d + J (), d + J (), d 4 + J (4), d 5 + J (5)} = {0 +, + 7, + 4, 0 + } = 0 J () = min{d + J (), d + J ()} = min{ +, 7 + 7} = k = (4 t ) J (5) = min{d 55 + J (5)} = 0 + = J (4) = min{d 44 + J (4), d 45 + J (5)} = {0 + 4, + } = 4 J () = min{d + J (), d 4 + J (4), d 5 + J (5)} = {0 +, + 4, 8 + } = J () = min{d + J (), d + J (), d 4 + J (4), d 5 + J (5)} = {0 + 0, +, + 4, 0 + } = 9 J () = min{d + J (), d + J (), d + J ()} = min{0 +, + 0, 7 + } = k = 0 (5 t ) J 0 (5) = min{d 55 + J (5)} = 0 + = J 0 (4) = min{d 44 + J (4), d 45 + J (5)} = {0 + 4, + } = 4 J 0 () = min{d + J (), d 4 + J (4), d 5 + J (5)} = {0 +, + 4, 8 + } = J 0 () = min{d + J (), d + J (), d 4 + J (4), d 5 + J (5)} = {0 + 9, +, + 4, 0 + } = 9 J 0 () = min{d + J (), d + J (), d + J ()} = min{0 +, + 9, 7 + } = 0

84. J k (i) k = 0 i = ( ) 4 5 (0). DP 5 4 i 4 9 0 4 4 4 5 9 0 7 0 0 4 k. 4 00 8 00 80 4 8 8 4 0.5 ( ) 5 00 [ ].7 0 4 = 00 + 4 0.5 00 0 8 = 00 + 8 0.5 (00 + 00) 0 = 00 + 0.5 (00 + 00 + 80) 4 8 = 00 + 8 0.5 00 4 = 00 + 0.5 (00 + 80) 8 = 00 + 0.5 80

.9. PERT/CPM 85.7 <700> 0 <800> <4700> <400> <7700> 8 4 <400>.9 PERT/CPM.9. [?] PERT/CPM.9. PERT program evaluation and review technique 957 958 Booz Allen & Hammilton ( )

8 PERT PERT PERT ([4] ) PERT ( activity ).8 i. d. f. d f.8 ( ) a. b. a 4 c. b 0 d. c 7 e. c 4 f. e 5 g. c h. g 7 i. d, f 8 j. e, h 9 k. i 4 l. i 5 m. j n. k, l PERT (activity on node; AoN) (activity on arc; AoA).9..8.9

.9. PERT/CPM 87 i d f d f i a n s t 0.9 (AoN ) s (0) (7) (8) (4) d i k n () () (0) (4) a c e f (5) l (5) b (4) () g h j (7) m (9) () (0) t.9. AoA (event) /.8 AOA.40 i d f d f 7 i 7 7 f d i 0.40 (AoA ) g () a () b (4) 4 c (0) e (4) 5 h (7) 8 (0) 7 f (5) j (9) m () i (8) 0 9 k (4) l (5) n () d (7)

88.9.4 (earliest start time) (latest finish time) (critical path). ( ) j E j E j = max i P(j) {E i + d i } P(j) j a h i P(i) = {d, f} E d = d d = 7 E f = 0 d f = 5 E i = max + 7, 0 + 5 = 5.8 E t = 44 44. ( )

.9. PERT/CPM 89 j L j L j = min i S(j) {L i d i } S(j) j t k n i S(i) = {k, l} L k = 8 d l = 4 L l = 8 d l = 5 L i = min 8 4, 8 5 =. E j L j E j + d j = L j.4.4 d j, [E j, L j ] s 0 [0,0] 7 [,5] d 8 [5,] i 4 [,8] k n [8,44] [0,] 0 [,] 4 [,0] a c e f 5 [0,5] l 5 [,8] [,] b g h j m t 4 [,] 7 [,] 9 [9,4] [8,44] 0 [ 44,44] (total slack, total float) j E j j L j d j j TS j TS j = L j E j d j j

90 (free slack, free float) j S(j) j FS j FS j = min i S(j) {E i} E j d j j j (safety slack, safety float) j P(j) j SS j SS j = L j d j max i P(j) {L i} j j.4.4 s. 0 0 0 0 0 0 a. s 0 0 0 0 b. a 4 0 0 0 c. b 0 0 0 0 d. c 7 5 e. c 4 0 0 0 0 f. e 5 0 5 0 0 0 g. c 4 0 4 h. g 7 4 0 0 i. d, f 8 5 0 0 0 j. e, h 9 9 4 4 0 0 k. i 4 8 l. i 5 8 0 0 0 m. j 8 44 4 4 0 n. k, l 8 44 0 0 0 t. m, n 0 44 44 0 0 0.4.8 j. 9 5.4

.9. PERT/CPM 9.9.5 a b a b a b.9. PERT (Three-Estimated Approach).44 /

9.4 probability 5 duration.44 S T 4 0.5 7.5 4 S T ( ) 5...9.7 CPM PERT

.9. PERT/CPM 9 ( ) CPM(Critical Path Method).45 α j cost T j c n T j time CPM (time cost curve).45 (normal point) (crash point) CPM CPM V j V T n j T c j β j α j E x j (j V) y j (j V) s t T min j V (α j + β j x j ) s.t. y s = 0, x s = 0, x t = 0 y j + x j y k 0, jk E y t T Tj c x j Tj n, j V y j 0, j V (.4) T T (.4)

94 T T ( ).5.4.4 A 5 B 0 C D A E C F B G A H C,G I 8 E,F..47 CPM.47 (normal time 0 ) a 8-0.5 b 4 7-0. c 9-0.8 a d -0. a e 5 5-0.5 b, c

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