C() 1 1.1A 4. (c) 17 . 5. (a) Jim . Joe . (b) Joe . Jim Joe . 2 2.1A 1. (a) −x1 + x2 1. (c) x1 − x2 0. (e) 0.5x1 − 0.5x2 0. 3. M1 = 4 /. 2.2A 1. (a e) C.1. 2. (a d) C.2. 5. x1 = A x2 = B max z = 20x1 + 50x2 s.t. −0.2x1 + 0.8x2 0, x1 100, x1 , x2 0 (x1 , x2 ) = (80, 20), z = 2 600 . , * . 2x1 + 4x2 240 2 C() 7. C.1 C.2 x1 = A x2 = B max z = 0.05x1 + 0.08x2 s.t. 0.75x1 − 0.25x2 0, x1 − 0.5x2 0, 0.5x1 − 0.5x2 0 x1 + x2 5 000 x1 , x2 0 (x1 , x2 ) = (2 500, 2 500), z = 325 . 11. x1 = x2 = max z = 2x1 + x2 s.t. x1 + x2 10, x1 − x2 0 x1 4, x1 , x2 0 (x1 , x2 ) = (4, 6), z = 14. 14. x1 = C1 x2 = C2 max z = 12 000x1 + 9 000x2 s.t. −200x1 + 100x2 0, 2.1x1 + 0.9x2 20 x1 , x2 0 (x1 , x2 ) = (5.13, 10.26), z = 153 846 . (a) C1:C2=0.5. C() (b) , 7 692 /. 18. x1 = HiFi1 x2 = HiFi2 min z = 1 267.2 − (15x1 + 15x2 ) s.t. 6x1 + 4x2 432, 4x1 + 6x2 422.4, 5x1 + 5x2 412.8 x1 , x2 0 (x1 , x2 ) = (50.88, 31.68), z = 28.8 . 2.2B 1. (a) C.3. C.3 5. x1 = / x2 = / min z = x1 + x2 s.t. −0.6x1 + 0.4x2 0, 0.2x1 + 0.1x2 14 0.25x1 + 0.6x2 30, 0.1x1 + 0.15x2 10 0.15x1 + 0.1x2 8, x1 = 55, x2 = 30, z = 85. 7. x1 , x2 0 3 4 C() x1 = A x2 = B min z = 100x1 + 80x2 s.t. 0.03 0.06x1 + 0.03x2 0.06, 0.03 0.03x1 + 0.06x2 0.05 0.03 0.04x1 + 0.03x2 0.07, x1 + x2 = 1, x1 , x2 0 x1 = 0.33, x2 = 0.67, z = 86 667 . 2.3A 3. xij = i j max z = 0.05(4x11 + 3x12 + 2x13 ) + 0.07(3x22 + 2x23 + x24 ) +0.15(4x31 + 3x32 + 2x33 + x34 ) + 0.02(2x43 + x44 ) s.t. x11 + x12 + x13 = 1, x43 + x44 = 1 0.25 x22 + x23 + x24 + x25 1 0.25 x31 + x32 + x33 + x34 + x35 1 5x11 + 15x31 3, 5x12 + 8x22 + 15x32 6 5x13 + 8x23 + 15x33 + 1.2x43 7 8x24 + 15x34 + 1.2x44 7, 8x25 + 15x35 7 xij 0 x11 = 0.6, x12 = 0.4, x24 = 0.255, x25 = 0.025, x32 = 0.267, x33 = 0.387, x34 = 0.346, x43 = 1, z = 523 750 . 2.3B 2. , p q. xij 2.3-2 , rij i j . , p = q ; p = q, , max z=y s.t. xij ci , ∀i = j p : I − y + i = p : j=p rji xji = j=i xij 0 ampl2.3b-2.txt. rjp xjp = —— j=i j=p xij xpj C() 5 max z = y s.t. : xij ci , ∀ i = j p : I + rjp xjp = xpj j=p q : y + j=p xqj = j=q i = p q: rjq xjq j=q rji xji = j=i xij j=i xij 0 $ →$ 1.806 4%, $ → =C 1.796 6%, $ →£ 1.828 7%, $ → 2.851 5%, $ → KD 1.047 1%. , . , . [ AMPL( ampl2.3b-2.txt) Excel ( solver 2.3b-2.xls) . 2.4 .] 2.3C 2. xi = i , i = 1, 2, 3, 4 yj = j , j = 1, 2, 3, 4, 5 max z = y5 s.t. x1 + x2 + x4 + y1 10 000 0.5x1 + 0.6x2 − x3 + 0.4x4 + 1.065y1 − y2 = 0 0.3x1 + 0.2x2 + 0.8x3 + 0.6x4 + 1.065y2 − y3 = 0 1.8x1 + 1.5x2 + 1.9x3 + 1.8x4 + 1.065y3 − y4 = 0 1.2x1 + 1.3x2 + 0.8x3 + 0.95x4 + 1.065y4 − y5 = 0 x1 , x2 , x3 , x4 , y1 , y2 , y3 , y4 , y5 0 x1 = 0, x2 = $10 000, x3 = $6 000, x4 = 0, y1 = 0, y2 = 0, y3 = $6 800, y4 = $33 642, 5 z = $53 628.73. 5. xiA = i A , i = 1, 2, 3 xiB = i B , i = 1, 2, 3 max z = 3x2B + 1.7x3A s.t. x1A + x1B 100( 1 ) −1.7x1A + x2A + x2B = 0( 2 ) 6 C() −3x1B − 1.7x2A + x3A = 0( 3 ) xiA , xiB 0, i = 1, 2, 3 1 A $100 000, 2 B 170 00 . . 2.3D 3. xj j , j = 1, 2, 3. max z = 30x1 + 20x2 + 50x3 s.t. 2x1 + 3x2 + 5x3 4 000 4x1 + 2x2 + 7x3 6 000 x1 + 0.5x2 + 0.33x3 1 500 2x1 − 3x2 = 0 5x2 − 2x3 = 0 x1 200, x2 200, x3 150 x1 , x2 , x3 0 x1 = 324.32, x2 = 216.22, x3 = 540.54, z = 41 081.08 . 7. xij = j i , i = 1, 2, j = 1, 2, 3 Iij = j i , i = 1, 2, j = 1, 2, 3 3 min z = (c1j x1j + c2j x2j + 0.2I1j + 0.4I2j ) j=1 s.t. 0.6x11 800, 0.6x12 700, 0.6x13 550 0.8x21 1 000, 0.8x22 850, 0.8x23 700 x1j + I1,j−1 = x2j + I1j , x2j + I2,j−1 = dj + I2j , j = 1, 2, 3 I1,0 = I2,0 = 0, 0 dj = 500, 450, 600, j = 1, 2, 3 c1j = 10, 12, 11, j = 1, 2, 3 c2j = 15, 18, 16, j = 1, 2, 3 x11 = 1 333.33 , x13 = 216.67 , x21 = 1 250 , x23 = 300 , z = 39 720 . 2.3E 2. xs = /, xb = /, xn = /, xw = /. C() min z = 1.1xs + 1.5xb + s.t. 70 80 xn + 20 30 7 xw y = xs + xb + xn + xw y 1, xs 0.1y, xb 0.25y, xn 0.15y, xw 0.1y 1 1 10 xb xn , 50 xb xw z = 1.12 , y = 1, xs = 0.5, xb = 0.25, xn = 0.15, xw = 0.1. 5. xA = A /, xB = B /, xr = /, xp = /, xj = /. + + max z = 50(xr − s+ r ) + 70(xp − sp ) + 120(xj − sj ) − + − + + −(10s− r + 15sp + 20sj + 2sr + 3sp + 4sj ) − (30xA + 40xB ) s.t. xA 2 500, xB 3 000, xr = 0.2xA + 0.25xB , xp = 0.1xA + 0.3xB , xj = 0.25xA + 0.1xB − + + − + xr + s− r − sr = 500, xp + sp − sp = 700, xj + sj − sj = 400 0 z = $21 852.94, xA = 1 176.47/, xB = 1 058.82/, xr = 500/, xp = 435.29/, xj = 400/, s− p = 264.71/. 2.3F 1. xi (yi ) = i min z = 2 6 8 / (12 /) xi + 3.5 i=1 6 . yi i=1 s.t. x1 + x6 + y1 + y5 + y6 4, x1 + x2 + y1 + y2 + y6 8, x2 + x3 + y1 + y2 + y3 10, x3 + x4 + y2 + y3 + y4 7, x4 + x5 + y3 + y4 + y5 12, x5 + x6 + y4 + y5 + y6 4 x1 = 4, x2 = 4, x4 = 2, x5 = 4, y3 = 6, = 0, z = 49. = 20. 8 , = 26, z = 2 × 26 = 52, ,(8 +12 ) . 5. xi = i (i = 1 8:01, i = 9 4:01.) min z = x1 + x2 + x3 + x4 + x6 + x7 + x8 + x9 s.t. x1 2, x1 + x2 2, x1 + x2 + x3 3 C() 8 x2 + x3 + x4 4, x3 + x4 4, x4 + x6 3 x6 + x7 3, x6 + x7 + x8 3, x7 + x8 + x9 3 x5 = 0, 2 8:01, 1 10:01, 3 11:01, 9 . 3 2:01. 2.3G 2 . (b) (3,0,0), (1,1,0), (1,0,1) (0,2,0), 0, 3, 1 1 (c) 20 30 . (d) 20 50 . 1. (a) 1 150L . 6. gi , yi ri i . . . max z = 3(500/3 600)g1 + 4(600/3 600)g2 + 5(400/3 600)g3 s.t. (500/3 600)g1 + (600/3 600)g2 + (400/3 600)g3 (510/3 600)(2.2 × 60 − 3 × 10) g1 + g2 + g3 + 3 × 10 2.2 × 60, g1 25, g2 25, g3 25 g1 = 25 , g2 = 43.6 , g3 = 33.4 , = 58.04 / . 2.4A 2. (d) AppenCFile solver2.4a-2(d).xls. 2.4B 2. (c) AppenCFile ampl2.4b-2(c).txt. (f) AppenCFile ampl2.4b-2(f).txt. 3.1A 1. M1 M2, 2 / 1 /. C() 4. xij = 9 j i . max z = 10(x11 + x12 ) + 15(x21 + x22 ) x11 + x21 − x12 − x22 + s1 = 5 s.t. −x11 − x21 + x12 + x22 + s2 = 5 x11 + x21 + s3 = 200 x12 + x22 + s4 = 250 si , xij 0, ∀ i, j 3.1B 3. xj = j , j = 1, 2, 3. + max z = 2x1 + 5x2 + 3x3 − 15x+ 4 − 10x5 + 2x1 + x2 + 2x3 + x− 4 − x4 = 80 s.t. + x1 + x2 + 2x3 + x− 5 − x5 = 65 + − + x1 , x2 , x3 , x− 4 , x4 , x5 , x5 0 x2 = 65 , x− 4 = 15 , = 0, z = 325 . 3.2A 1. (c) x1 = 67 , x2 = (e) 12 7 , z= 48 7 . (x1 = 0, x2 = 3) (x1 = 6, x2 = 0) . 3. (x1 , x2 ) = 26 3 , − 34 , (x1 , x3 ) = (8, −2) , (x1 , x4 ) = (6, −4) , (x2 , x3 ) = (16, −26) , (x2 , x4 ) = (3, −13) , (x3 , x4 ) = (6, −16) . 3.3A 3. (a) (A, B) , AB . (b) (i) . (ii) , C I . (iii) , 5. (a) x3 1 , D z = 3. 3.3B 3. x1 x2 x3 x4 1.5 1 0 0.8 x7 x7 x8 x5 . A. 10 C() 6. (b) x2 , x5 x6 z . x2 , x8 , ∆z = 5 × 4 = 20. x5 , x1 , ∆z = 0, , x5 0. x6 , , x6 . ∆z = ∞, x6 . 9. s2 , z = 20. 3.4A 3. (a) min z = (8M − 4)x1 + (6M − 1)x2 − M s2 − M s3 = 10M . (b) min z = (3M − 4)x1 + (M − 1)x2 = 3M . 6. x1 x2 x3 x4 z −1 −12 0 0 −8 x3 1 1 1 0 4 x4 1 4 0 1 8 3.4B 1. , . II 7. I , , I ; , I . 3.5A 1. (a) A → B → C → D. (b) A 1, B 1, C C42 = 6, 3.5B 1. 1 0, 0, 10 3 , (0, 5, 0), 1, 4, 3 . D 1. 1 α3 , 5α2 + 4α3 , 10 3 α1 + 3 α3 , α1 + α2 + α3 = 1, 0 αi 1, i = 1, 2, 3. 3.5C 2. (a) x2 . (b) . x2 , z 10 . 3.5D 1. 275 . C() 11 3.6A 2. x1 = 1 x2 = 2 max z = 8x1 + 5x2 s.t. 2x1 + x2 400 x1 150, x2 200, x1 , x2 0 (a) C.4x1 = 100, x2 = 200, z = 1 800 (b) (200, 500) 2 4 . , (c) (d) B. (100, ∞) 0 . 1 . 2 = 200. (100, 400) C.4 3.6B c1 c2 2. cc12 = 1. 3. (a) 0 (b) . 3.6C 2. (a) , = 1 ( 10 ), 0.83 /. (b) 2 / ( 400 )= 240 . 12 C() = 110 . = 130 . , 0, . (d) D1 = 10 . D1 10, = 1 /. x1 = 0, x2 = 105, x3 = 230, = ($1 350 + $1 × 10) − $40 60 × 10 = $1 353.33. (e) D2 = −15. D2 −20, = 2 /. = 30 . = 7.5 . . 6. x1 = , x2 = , x3 = (c) max z = x1 + 50x2 + 5x3 s.t. 15x1 + 300x2 + 50x3 + s1 = 10 000, x3 − S2 = 5, x1 + s3 = 400, −x1 + 2x2 + s4 = 0, x1 , x2 , x3 0, s1 , S 2 , s3 , s4 0 (a) x1 = 59.09 , x2 = 29.55 , x3 = 5 , z = 1 561.36. (b) TORA, z + 0.158s1 + 2.879S2 + 0s3 + 1.364s4 = 156.364. (c) 11. (a) (b) (c) 13. (b) 0.158, −2.879, 0, 1.364. (= −2.879). , , 0( ). TORA, x1 = 59.909 1 + 0.006 06D1 0, x3 = 5, s3 = 340.909 09 + 0.006 06D1 0, x2 = 29.545 45 + 0.003 03D1 0. , = 0.158, −9 750 D1 56 250. , 50% (D1 = 5 000 ), . ; . 1.25 , 0.25 0 . D3 = 350 − 800 = −450 D3 −400 . . ∆ > 0, x1 = x2 = 2 + ∆3 . 0 < ∆ 3, ∆ r1 + r2 = ∆ 3 1 ⇒ . 3 ∆ < 6, r1 + r2 = 3 > 1 ⇒ . ∆ > 6, D1 D2 . 3.6D 2. (a) x1 =A1 , x2 =A2 , x3 =BK . max z = 80x1 + 70x2 + 60x3 s.t. x1 + x2 + x3 500, x1 100, 4x1 − 2x2 − 2x3 0 C() 13 x1 , x2 , x3 0 x1 = 166.67, x2 = 333.33, x3 = 0, z = 36 666.67. (b) TORA , BK = 10. 10 . (c) d1 = d2 = d3 = −5 . TROA , x3 : 10 + d2 − d3 0, s1 : 73.33 + 0.67d2 + 0.33d1 0, s3 : 1.67 − 0.17d2 + 0.17d1 0, . 5. (a) xi = i . max z = 60x1 + 40x2 + 25x3 + 30x4 s.t. 8x1 + 5x2 + 4x3 + 6x4 8 000, x1 500, x2 500, x3 800, x4 750, x1 , x2 , x3 0 x1 = 500, x2 = 500, x3 = 375, x4 = 0, z = 59 375 . (b) TORA , 8.75 + d2 0. 2 8.75 . (c) d1 = −15 , d2 = −10 , d3 = −6.25 , d4 = −7.50 . TORA , x4 : 7.5 + 1.5d3 − d4 0, s1 : 6.25 + 0.25d3 0, s2 : 10 − 2d3 + d1 0, s3 : 8.75 − 1.25d3 + d2 0, , z 25%. (d) x4 = 7.5, 7.50 . 3.6E 5. , x1A + x1B 100 . 9. (a) A 10.27 . 12 , A. (b) B 0 . , . 5.10 . 14 C() 4 4.1A 2. y1 , y2 , y3 . max w = 3y1 + 5y2 + 4y3 y1 + 2y2 + 3y3 15, 2y1 − 4y2 + y3 12 s.t. y1 0, y2 0, y3 4. (c) y1 y2 . min w = 5y1 + 6y2 s.t. 2y1 + 3y2 = 1, y1 − y2 = 1 y1 , y2 5. y2 −M . −∞, y2 . , M → ∞ ⇒ y 4.2A . (e) V 2 A = (−14 − 32). 1. (a) AV 1 4.2B 1. (a) ⎛ 1 4 − 12 ⎜ 1 ⎜ − ⎜ = ⎜ 8 ⎜ 3 ⎝ 8 ⎞ − 54 ⎟ 0 0 ⎟ ⎟ ⎟ 1 0 ⎟ ⎠ − 34 0 1 3 4 1 8 0 0 4.2C 3. y1 y2 . min w = 30y1 + 40y2 s.t. y1 + y2 5, 5y1 − 5y2 2, 2y1 − 6y2 3 y1 −M (⇒ y1 ), y2 0 y1 = 5, y2 = 0, w = 150. 6. y1 y2 . min w = 3y1 + 4y2 s.t. y1 + 2y2 1, 2y1 − y2 5, y1 3 y2 C() 15 y1 = 3, y2 = −1, w = 5. 8. (a) (x1 , x2 ) = (3, 0), z = 15, (y1 , y2 ) = (3, 1), w = 14. = (14, 15). 9. (a) , , z = w = 17. 4.2D 2. (a) (x2 , x4 ) = (3, 15) ⇒ . (x1 , x3 ) = (0, 2) ⇒ . 4. z x1 x2 x3 x4 x5 0 0 − 25 − 51 0 −1 1 1 12 5 3 5 6 5 x1 1 0 x2 0 1 x5 0 0 − 35 4 5 1 5 − 53 0 0 0 , . 7. , z = c1 x1 + c2 x2 ; , w = b1 y1 + b2 y2 + b3 y3 . b1 = 4, b2 = 6, b3 = 8, c1 = 2, c2 = 5 ⇒ z = w = 34. 4.3A 2. (a) (x1 , x2 , x3 , x4 ) =SC320, SC325, SC340 SC370 max z = 9.4x1 + 10.8x2 + 8.75x3 + 7.8x4 s.t. 10.5x1 + 9.3x2 + 11.6x3 + 8.2x4 4 800 20.4x1 + 24.6x2 + 17.7x3 + 26.5x4 9 600 3.2x1 + 2.5x2 + 3.6x3 + 5.5x4 4 700 5x1 + 5x2 + 5x3 + 5x4 4 500 x1 100, x2 100, x3 100, x4 100 (b) , (= 0.494 4). (c) 0(−0.684 7, −1.361, 0 −5.300 3), . (d) 0.494 4 /, (8 920, 10 201.72), 6.26%. 4.3B , = −2. 3. PP3 PP4 . 0.142 9 1.142 9. , PP3, 0.142 9 , PP4 1.142 9 . 2. 16 C() 4.4A 1. (b) . E , . 4. (c) x1 M . . 4.5A 4. Q (= 5 200, 9 600, 15 000, 20 000, 26 000, 32 000, 38 000, 42 000, = 1, 2, · · · , 8). = 0.028Q, = 0.649Q, = 0.323Q. = 0.812 21Q. 4.5B 1. (a) . 4.5C = 12 , 0, 0, 0 . . 1 11 (c) = − 8 , 4 , 0, 0 . z − 0.125s1 + 2.75s2 = 13.5. x2 = 2, x3 = 4, z = 14. 2. (a) x1 = 2, 4.5D 1. p 100 (y1 3. (a) + 3y2 + y3 ) − 3 0. y1 = 1, y2 = 2 y3 = 0, p 42.86%. = 3y1 + 2y2 + 4y3 − 5 = 2 > 0. , . 5 5.1A 4. , M . 6. (a b) M = 10 000. , =$49 710. 1 1 2 (c) 1 600 23 320 500 3 4 2 13 700 17 25 1000 36 =$13 000. 400 25 300 350 480 450 5 1000 42 3 M 30 25 40 30 13 C() 9. (100 ). 2 A1 1 2 3 200 . =$304 000. A2 12 4 A3 18 M 10 8 2 6 30 4 1 20 5 25 12 6 6 M 4 50 50 2 2 4 8 7 5.2A 2. =$804. . 2 3 24 6 6 18 0 12 12 0 0 0 2 14 0 0 0 0 0 20 0 0 0 14 0 0 4 0 2 0 0 12 0 0 0 0 22 5. =$190 040. . 1 500 500 1 400, 2 100 2 600 600 2 200, 3 220, 3 200 200 3 200 4 300 200 4 200 4 180 5.3A 3. (a) =$42. =$37. Vogel =$37. 5.3B 5. (a) =$1 475. (b) c12 3, c13 8, c23 13, c31 7. 5.4A 5. (, ) 17 . , (D,3)–(A,7) 18 C() 63 , 6 7 =$1 180. . (A,7) (D,3) (D,10) (D,17) (D,25) 6. (A,12) 400 300 300 300 300 400 300 300 , $400. (A,21) , (A,28) 300 300 400 300 280 300 300 400 I-d,II-c,III-a,IV-b. 5.5A 4. =$1 550. , 1 1 2 2 50 50 0 200 . 3 0 50 6 6.1A 1. (i); (a)1-3-4-2; (b)1-5-4-3-1; (c d) C.5. C.5 4. , . 1 , . . C.6. 8 C.6 C() 19 6.2A 2. (a) 1-2,2-5,5-6,6-4,4-3. =14 . 5. 1-2-3-4-6. 1-5-7 5-9-8. 6.3A 1. 1 4 . =8 900 . C.7. C.7 4. (i, vi ) − (i + 1, vi+1 ), p(q) = 1 2. ( i ). 80 . C.8. C.8 6.3B 1. (c) 4,5,6,7,8, 4-6-8. 6.3C 1. (a) 5-4-2-1, =12. . 8, 4-5-6-8 20 4. C() C.9. , . Bob-Kay-Rae-Kim-Joe. 4. Bob Joe C.9 6.3D 1. (a) 1 5 1-3-5 1-3-4-5, 90. 1 −1, 0. 6.4A 1. 11-2, 1-4, 3-4, 3-5, 60. 6.4B 1. (a) (2-3)=40, (2-5)=10, (4-3)=5. (b) 220 . 330 . 420 . (c) . 1 . 4. Rif–3, Mai–1, Ben–2, Kim–5. Ken . 7. 6.5A 3. C.10. C.10 C() 6.5B 1. 1-3-4-5-6-7. 19. 6.5C 3. (a) 10; (b) 5; (c) 0. 5. (a) 1-3-6, 45 . (b) A, D, E. (c) C, D, G 5 . E (d) 2 . . 7 7.1A − + 1. G5 min s+ 5 , 55xp + 3.5xf + 5.5xs − 0.067 5xg + s5 − s5 = 0. 3. x1 , x2 , x3 . Gi : min s.t. s− i , i = 1, 2, · · · , 5 + x1 + x2 + x3 + s− 1 − s1 = 1 200 + 2x1 + x2 − 2x3 + s− 2 − s2 = 0 + −0.1x1 − 0.1x2 + 0.9x3 + s− 3 − s3 = 0 + 0.125x1 − 0.05x2 − 0.556x3 + s− 4 − s4 = 0 + −0.2x1 + 0.8x2 − 0.2x3 + s− 5 − s5 = 0 5. xj j min s.t. , j = 1, 2, 3. + z = s− 1 + s1 + −100x1 + 40x2 − 80x3 + s− 1 − s1 = 0 4 x1 5, 10 x2 20, 3 x3 20 7.2A − − + + 1. min z = s− 1 + s2 + s3 + s4 + s5 xp = 0.020 1, xf = 0.045 7, xs = 0.058 2, xg = 2 , s+ 5 = 1.45 145 . 4. x1 , x2 , x3 . + − − + min z = s− 1 + s2 + s3 + s4 + s5 21 22 C() x1 = 166.08 , x2 = 2 778.56 , x3 = 3 055.36 , z = 0. . , 3 4 . 7. j xj , j = 1, 2. . − + + min z = 100s− 1 + 100s2 + s3 + s4 + x1 = 80, x2 = 60, s+ 3 = 100 , s4 = 120 . 1 100 , 2 120 , . 7.2B 2. G1 xp = 0.017 45, xf = 0.045 7, xs = 0.058 2, xg = 21.33, s+ 4 = 19.33, 0. G1 , G2 G3 , G4 . − − G4 G1 , s1 = 0, s2 = 0, s− 3 = 0. G4 xp = 0.020 1, xf = 0.045 7, xs = 0.058 2, xg = 2, s+ 5 = 1.45. 0. G5 . G5 G4 , s+ 4 = 0. (s+ G5 G4 , G5 5 = 1.45). 8 8.1A 3. xij i=2 j i , i = 1 , i = 3 . x11 + x12 + x13 = 7, x21 + x22 + x23 = 7, x31 + x32 + x33 = 7 x11 + 0.5x21 = 3.5, x12 + 0.5x22 = 3.5, x13 + 0.5x23 = 3.5 x11 + x21 + x31 = 7, x12 + x22 + x32 = 7, x13 + x23 + x33 = 7 . . 1 2 3 1 3 3 5 1 1 1 3 3 , C() . xj j . 6. y x4 23 , j = 1, 2, 3. min z = y s.t. 3x1 − y = 2, x1 + 3x2 − y = 2, x1 + x2 + 3x3 − y = 2 y − x1 − x2 − x3 − 3x4 = 1 y = 79 + 81n, n = 0, 1, 2, · · · 10. 5,6 8( 27 ). 1,2,3,4 7( 28 ). . 12. i j, xij = 1; 0. cij j . , Cj max z = s.t. 6 j=1 10 10 6 cij xij i=1 j=1 xij = 2, i = 1, 2, · · · , 10 xij Cj , j = 1, 2, · · · , 6 i=1 1, (2,4,9); 2,(2,8); 3,(5,6,7,9); 5,(1,3,8,10); 6,(1,3). 1 775. 4,(4,5,7,10); 8.1B 1. j, xj = 1; xj = 0. 10 + 32 + 4 + 15 + 9 = 80( ). (ABC,1,2,3,4,ABC) min z = 80x1 + 50x2 + 70x3 + 52x4 + 60x5 + 44x6 s.t. x1 + x3 + x5 + x6 1, x1 + x3 + x4 + x5 1, x1 + x2 + x4 + x6 1, x1 + x2 + x5 1, x2 + x3 + x4 + x6 1, j, xj = (0, 1). (1,4,2) (1,3,5), z = 104. 1 . 2. a, d, f . . t, xt = 1; xt = 0. c , xc = 1; 7. xc = 0. ct t . Sc c j . . Pj 24 C() max z = s.t. 15 Pc xc c=1 xt xc , c = 1, 2, · · · , 15, 7 ct xt 15 t=1 t∈Sc 2,4,5,6 7. 1 , . 8.1C j , j = 1, 2, 3. 2. xj j, yj = 1; yj = 0. min z = 2x1 + 10x2 + 5x3 + 300y1 + 100y2 + 200y3 s.t. x1 + x2 + x3 2 000, x1 − 600y1 0, x2 − 800y2 0, x3 − 1 200y3 0, x1 500, x2 500, x3 500 x1 , x2 , x3 , y1 , y2 , y3 = (0, 1). x1 = 600, x2 = 500, x3 = 900, z = 11 300(). 3. 1 1 2, 2 3 4. z = 18. 10. xe Eastern (), xu US Air Continental (). (), xc e1 e2 , u c . max z = 1 000(xe + 1.5xu + 1.8xc + 5e1 + 5e2 + 10u + 7c) s.t. e1 xe /2, e2 xe /6, u xu /6, c xc /5, xe + xu + xc = 12. 2 Eastern 39 000 . , 10 Continental . 8.1D 1. xij 0. (i, j) . , 3 3 xij = 15, i = 1, 2, 3, xij = 15, j = 1, 2, 3 j=1 i=1 x11 + x22 + x33 = 15, x31 + x22 + x13 = 15 x11 x12 + 1x11 x12 − 1 x11 x13 + 1x11 x13 − 1 x12 x13 + 1x12 x13 − 1 x11 x21 + 1x11 x21 − 1 C() 25 x11 x31 + 1x11 x31 − 1 x21 x31 + 1x21 x31 − 1 i j, xij = 1, 2, · · · , 9 2 9 4 7 5 3 6 1 8 3. xj j . . max z = 25x1 + 30x2 + 22x3 s.t. 3x1 + 4x2 + 5x3 100 4x1 + 3x2 + 6x3 100 x1 , x2 , x3 0 3x1 + 4x2 + 5x3 90 4x1 + 3x2 + 6x3 120 . 26 1, 3 2, 3. 2 . 8.2A 2. (a) z = 6, x1 = 2, x2 = 0; (d) z = 12, x1 = 0, x2 = 3. 3. (a) z = 7.25, x1 = 1.75, x2 = 1; (b) z = 10.5, x1 = 0.5, x2 = 2. 9. 0 − 1 ILP max z = 18y11 + 36y12 + 14y21 + 28y22 + 8y31 + 16y32 + 32y33 s.t. 15y11 + 30y12 + 12y21 + 24y22 + 7y31 + 14y32 + 28y33 43 z = 50, y12 = 1, y21 = 1, 0. , x1 = 2, x2 = 1. 0 − 1 41 , 29 . 8.2B 1. (a) , , . LP . 6. (a) (x1 , x2 , x3 ) = (2, 1, 6), z = 26. (x1 , x2 , x3 ) = (3, 1, 6), . TORA B&B . 26 C() 8.3A 1. i j , . “”. 1 2 3 4 5 6 1 − 4 4 6 6 5 2 4 − 6 4 6 3 3 4 6 − 4 8 7 4 6 4 4 − 6 5 5 6 6 8 6 − 5 6 5 3 7 5 5 − 8.3C 2. C.11. C.11 9 9.1A 1. =21 . 1-3-5-7. C() 27 9.2A 3. =17 . 1-2-3-5-7. 9.3A 2. (a) =120. (m1 , m2 , m3 ) = (0, 0, 3), (0, 4, 1), (0, 2, 2) (0, 6, 0). 5. =250. I 2 , II 3 , , IV 1 . xj = 1; 0. 7. j , III 4 max z = 78x1 + 64x2 + 68x3 + 62x4 + 85x5 s.t. 7x1 + 4x2 + 6x3 + 5x4 + 8x5 23, xj = (0, 1), j = 1, 2, · · · , 5 . =279. 9.3B 1. (a) 1 6 , 2 1 , 3 2 , 4 3 , 5 2 . 3. 1 7 , 2 3 , 3 4 , 4 . 9.3C 2. 4 , , , . =$458. 9.3D 3. (a) xi , yi i , zi = xi + yi . fn (zn ) = max {pn yn } yn =zn fi (zi ) = max{pi yi + f i+1 yi zi (2zi − 2yi )}, i = 1, 2, · · · , n − 1 10 10.3A 2. (a) = $51.50; (b) = $50.20, y ∗ = 239.05 . 4. (a) 1, $2.17, (b) 10 , 2 $2.50. 100 . 28 C() 10.3B 2. 130 , 500 . = $258.50. . , 4. T CU1 (ym ) T CU2 (q), 0.934 4% . 10.3C 1. AMPL/Solver (y1 , y2 , y3 , y4 , y5 ) = (4.42, 6.78, 4.12, 7.2, 5.8), =$568.12. 4. 4 i=1 365Di yi 150. Solver/AMPL (y1 , y2 , y3 , y4 ) = (155.3, 118.82, 74.36, 90.09), =$54.71. 10.4A 1. (a) 1, 4, 7, 10 , 500 . 10.4B 3. 1 173 , , 2 180 , 3 240 , 4 110 5 203 . 10.4C 1. (a) , . (b) (i) 0 z1 5, 1 z2 5, 0 z3 4, x1 = 4, 1 x2 6, 0 x3 4. (ii) 5 z1 14, 0 z2 9, 0 z3 5, x1 = 0, 0 x2 9, 0 x3 5. 2. (a) z1 = 7, z2 = 0, z3 = 6, z4 = 0. =$33. 10.4D 1. 1 2 4 , , 4 0, 22, 90, 67. 2 112 , 4 67 . =$632. 10.4E 1. 1 210 , 4 255 , 7 210 , 10 165 . 11 11.1A 1. A, B, C =(0.442 14, 0.251 84, 0.306 02) C() 29 11.1B 2. A , CR > 0.1. (wS , wJ , wM ) = (0.331, 0.292, 0.377). Maisa. 4. . (wH , wP ) = (0.502, 0.489). H. 11.2A 2. (a) C.12; (b) EV ()=−$8 250, EV ()=$250. . 6. (a) C.13; C.12 (b) EV ()=−$0.025. C.13 . 30 C() 12. =8 . =$397.50. =49 /. 19. 99∼151 . 15. 11.2B 2. z 5 1 , P{A|z} = 0.607 9, P{B|z} = 0.390 3. 4. (a) (b) 7. (b) =$196 000. , =$163 000. , , . , B, A. , 11.2C =$5, . (b) 0 x < 10, U (x) = 0; x = 10, U (x) = 100. (c) . 1. (a) 11.3A 1. (a) ( (b) a1 ); a2 a3 . 11.4A (2, 3). 3. (a) 2 < v < 4. 2. (a) =4. 11.4B 1. 50:50 , 2. =0. 100%A 50%A:50%B A 100 50 100%B 0 B 0 30 100 50:50 100%A 100%B. Robin 50:50 A B. =$50(=Robin ). 11.4C 1. (a) 1 C() AB AC AD BC BD CD AB 1 0 0 0 0 −1 AC 0 1 0 0 −1 0 AD 0 0 1 −1 0 0 BC 0 0 −1 1 0 0 BD 0 −1 0 0 1 0 CD −1 0 0 0 0 1 31 50:50 AB CD. =0. 3. (a) (m, n) =( ! 1 , ! 2 ). , ! 1; , −1. Botto (1, 1) " (0, 3), ! 1 ! 2, 1 + (−1) = 0. Blotto 3,0 2,1 1,2 0,3 2,0 −1 −1 0 0 1,1 0 −1 −1 0 0,2 0 0 −1 −1 Blotto Blotto 50:50 (2, 0) (0, 2), (3, 0) (1, 2). " 50:50 =−0.5, Blotto . . 12 12.1A 1. (a) =71%; (b) . 12.2A 1. (a) # (b) (h) 12.3A 1. (b) (i) 6 , = 16 . /, =0.2 . 3. (a) f (t) = 20e−20t , t > 0; (b) P t > 15 60 = 0.006 74. 7. Jim 2 P{t 1} = 0.486 6, 2 P{t 1} = 0.513 4. 8 , Jim Ann=17.15 . 10. (a) P{t 4} = 0.486 6; (b) =6.208. (c) µ = 5 32 C() 12.4A 1. pn5 (1) = 0.559 51. 4. (a)p2 (t = 7) = 0.241 67. 6. (a) λ = 1 10 + 17 , p2 (t = 5) = 0.219. 12.4B 2. (a) p0 (t = 3) = 0.005 32; (b) pn17 (t = 1) = 0.950 2. 5. p0 (4) = 0.371 16. 8. (a) =25 − 7.11 = 17.89 ; (b) p0 (t = 4) = 0.000 69. 12.5A 3. (a) pn3 = 0.444 5; (b) pn2 = 0.555 5. 6. (a) pj = 0.2, j = 0, 1, 2, 3, 4; (b) =2 ; (c) p4 = 0.2. 12.6A 1. (a) Lq = p6 + 2p7 + 3p8 = 0.191 7 . (b) λlost = 0.126 3 /, 8 (c) =c − (Ls − Lq ) = c − 8 n=0 npn + =1.01 . 8 (n − c)pn . n=c+1 12.6B 2. (a) p0 = 0.2; =$40 × Lq = $128. 5. (a) p0 = 0.4; (c) Wq = 2.25 ; 6. (d) 13 . (b) =$50 × µt = $375; (c) (b) Lq = 0.9 ; (d) pn11 = 0.003 6. 12.6C 1. P{τ > 1} = 0.659. 5. $37.95/. 12.6D 1. (a) p0 = 0.365 4; (b) Wq = 0.207 ; = 4 − Lq = 3.212; (d) p5 = 0.048 12. (e) Ws 40%, 9.6 (µ = 10 /). (c) C() 33 (b) Lq = 6.34 ; , 0.4. 60%. 7. (a) 1 − p5 = 0.962; (b) λlost = λp5 = 0.19 /. 4. (a) p8 = 0.6; (c) 12.6E 2. c = 2, Wq = 3.446 ; c = 4, Wq = 1.681 , 51%. 5. K $, TORA , p0 + p1 + · · · + pK+2 0.999, K 10. 7. (a) pn4 = 0.657 72; (b) =0.667 . 12.6F 2. (c) =81.8%; (d) p2 + p3 + p4 = 0.545. 4. (a) p40 = 0.000 14; (b) p30 + p31 + L + p39 = 0.024 53; # =Ls − Lq = 20.043 − 0.046 ≈ 20; (f) =1 − pn29 = 0.024 67. 8 4. (d) 12.6G 2. (a) 7 ; (b)pn8 = 0.291 1. 12.6H 1. (b) =2.01. } = p0 + p1 = 0.344 92. 4. (a) Ls = 1.25 ; (b) p0 = 0.333 42; (c) Ws = 0.25 . 6. λ = 2 /( · ), µ = 0.5 /, R = 5, K = 5. (a) =5 − Ls = 1 . (b) p5 = 0.327 68. (c) pn2 = 0.057 92. (d) P{23 12.7A 2. (a) E{t} = 14 , var{t}=12 2 . Ls = 7.867 2 . 4. λ = 0.062 5 /, E{t} = 15 , var{t} = 9.33 2 . (a) p0 = 0.062 5; (c) Ws = 132.17 . (b) Lq = 7.3 ; 34 C() 12.9A 2. (M/M/1) : (GD/10/10). $386.50. 4. (b) µ = λ + c2 λ c1 ; 1 $431.50, 2 (c) =2 725 /. 12.9B 2. (a) 2 $86.4, 3 $94.80. (b) 2 , =$30 × Ws = $121.11; 3 $94.62. 4. λ = 0.361 25/, µ = 10/. (M/M/1) : (GD/20/20) Ls = 0.705 29 . =$36.60, 3 =$60. 12.9C 1. (a) 5; (b) 4. C () 13 13.1A 2. (1, 0) (0, 2) Q , 0 < λ < 1, λ(1, 0) + (1 − λ)(0, 2) = (λ, 2 − 2λ) Q . 13.1B 2. (b) , x1 > 1, 0 < x2 < 1. C.14. (d) . (f ) . C.14 3. (a) , det(B) = −4. (d) , 3 . 13.1C 1. B −1 = 0.3 −0.2 0.1 0.1 x1 x2 x3 x4 z 1.5 −0.5 0 0 21.5 x3 x4 0 0.5 0.5 0 1 0 0 1 2 1.5 , . 4. z = 34. max z = 2x1 + 5x2 s.t. x1 4, x2 6, x1 + x2 8, x1 , x2 0. , * . 2 C() 13.2A 1. (a) P 1 . (b) B = (P 2 , P 4 ) . 2. X B , {zj − cj } = cB B −1 B − cB = cB I − cB = cB − cB = 0 7. , 10. , 11. (a) xj = 1 , α n − m. , xj . (b) xj = β , α . xj . 13.2B 2. (b) (x1 , x2 , x3 ) = (1.5, 2, 0), z = 5. 13.3A 2. (b) (x1 , x2 , x3 , x4 , x5 , x6 ) = (0, 1, 0.75, 1, 0, 1), z = 22. 13.4A 2. max w = Y b s.t. Y A c, Y 0. 13.4B 5. 1: (b1 , b2 , b3 ) = (4, 6, 8) ⇒ 2: (c1 , c2 ) = (2, 5) ⇒ = 34. = 34. 7. min w = Y b s.t. Y A = C, Y . 13.5A 1. − 27 t < 1. 2. (a) t 0t (x2 , x3 , x1 ) = ( 25 , 4 1 3 5 2 (x2 , x4 , x1 ) = 5. {zj − cj }j=1,4,5 = 4 − (x2 , x3 , x6 ) = (5, 30, 10) 3t 2 − 3t2 ,1 2 90 , 5) 4 5 ( 2 , 15, 20) − t2 , 2 − t 2 + t2 2 t 1 3 5 2 t < ∞ . 0 t 1, . 13.5B 1. (a) t1 = 10, B 1 = (P 2 , P 3 , P 4 ). 2. t = 0 , (x1 , x2 , x4 ) = (0.4, 1.8, 1). 0 t 1.5, . . t > 1.5, C() 3 14 14.1A 1. (a) 0.15 0.25. (b) 0.571. (c) 0.821. 2. n 23. 3. n > 253. 14.1B 3. 5 . 32 4. p =Liz , John 3p, 6p. 4 3 . 13 (a) (b) 7 . 13 (c) Jim . Ann , p + 3p + 3p + 3p + 6p = 1. 6 . 13 14.1C 3. (a) 0.375. (b) 0.6. 7. 0.954 5. 14.2A 2. (a) K = 20. 3. P{ 1 100} = 0.3. 14.3A 3. (a) P{50 (b) (c) 70} = 0.666 7. = 2.67. = $22.33. 14.3B 1. =3.667 =1.556. 14.3C 1. (a) P(x1 = 1) =P(x2 = 1) = 0.4, P(x1 = 2) = P(x2 = 2) = 0.2, P(x1 = 3) =P(x2 = 3) = 0.4. (b) 14.4A 1. 1 10 2 . 2. 0.054 7. P(x1 , x2 ) =P(x1 )P(x2 ). 4 C() 14.4B 1. 0.864 6. 3. (a) P{n = 0} = 0. (b) P{n 3}; 1. 14.4C 1. λ = 12 /. P{t 5} = 0.63. 14.4D 2. 0.001 435. 15 15.1A 1. (a) 537 , 1 000 . 15.1B 2. y ∗ = 317.82 , R∗ = 46.82 . 3. y ∗ = 316.85 , R∗ = 58.73 . 15.1-2 , y ∗ = 319.44 , R∗ = 93.61 . 15.1-2 , R∗ , . 15.2A 3. 0.43 p 0.82. 6. 32 . 15.2B 1. x < 4.53, 9 − x , . 15.3A 2. x < 4.61, 4.16 − x , . 16 16.1A 1. (a) P{H} =P{T } = 0.5. 0 R 0.5, Jim 10 0.5 R 1, Jan 10 . 7. 0 R 0.5, L = 1 0.5 R 1, L = 2 . 0 R 0.2, = 0 0.2 R 0.9, = 1 ; 0.9 R 1, = 2 . R L. L = 1 , C() R , L = 2, , 5 , . 16.2A 1. (a) . 16.3A 4. C.15. C.15 16.3B 1. t = − λ1 ln(1 − R), λ = 4 . R t( ) 1 2 3 4 – 0.058 9 0.673 3 0.479 9 – 0.015 176 0.279 678 0.163 434 0 0.015 176 0.294 855 0.458 288 2. t = a + (b − a)R. 4. (a) 0 R 0.2: d = 0; 0.2 R 0.5: d = 1; 0.5 R 0.9: d = 2; 0.9 R 1: d = 3; 9. 0 R p, x = 0; x =( ln(1−R) ln q ). 16.3C 1. y = − 15 ln(0.058 9 × 0.673 3 × 0.479 9 × 0.948 6) = 0.803 . 2. t = x1 + x2 + x3 + x4 , xi = 10 + 10Ri , i = 1, 2, 3, 4. 16.4A 1. 16.4-1 , 4. , 50 , 16.5A 2. (a) . 3. (a) 1.48 . (b) . (b) 7.4 . . C() 6 16.6A 2. 15.07 µ 23.27. 17 17.1A 2. S1: S2: S3: S4: S5: S1 S2 S1 0.4 0.1 S2 0.6 0.3 S3 0 0.6 S4 0 0 S5 0 0 S3 S4 S5 0.1 0.4 1 0 0 0 0.5 0 0 0.4 0 0 0 0.6 0 S1 S2 S3 S4 S5 0 0 1 0 0 17.2A 2. (2 2 S1 S2 S3 S4 S5 S1 S2 S3 0.4 0.1 0.1 0.6 0.3 0 0 0.6 0.5 0 0 0.4 0 0 0 S4 S5 0.4 1 0 0 0 0 0 0 0.6 0 ) (P 2 ) S1 S1 0.22 S2 0.42 S3 0.36 S4 0 S5 0 S2 S3 0.13 0.25 0.15 0.06 0.48 0.25 0.24 0.2 0 0.24 S4 S5 0.76 0.4 0.24 0.6 0 0 0 0 0 0 2 = (0 0 1 0 0)P 2 C() P{ , S4, 2 (2 ) S1 S2 S3 0.25 0.06 0.25 S4 S5 0.2 0.24 }=0.2. 17.3A excelMarkovChains.xls, 1. (a) (b) 1, 2, 3 , 4 , 3. . 17.4A 1. (a) S S 0.8 C 0.2 R 0 C R 0.3 0.1 0.5 0.1 0.2 0.8 (π1 , π2 , π3 ) = (π1 , π2 , π3 )P π1 + π2 + π3 = 1 S C R 2.0 4.0 4.0 =2 × 0.5 + 1.6 × 0.25 + 0.4 × 0.25 = $1 500 (b) µSS = 2 5. (a) 0.50 0.25 0.25 , . 0.95 0.06 0.04 0.9 0.01 0.04 0 0.1 0.9 (b) 44.12% 0.441 175 0.367 646 2.266 672 8 2.720 008 9 0.191 176 5.230 789 2 , 36.76% , 19.11% . 7 8 C() (c) =0.12($5 000 × 0.367 6 + 12 000 × 0.191 1) × 70 000 000 = $34 711 641 097.07 14. (a) =(i, j, k)=( −2 i, j, k = (0 , , −1 , 1) , (1-0-0) (0-0-1). 0-0-0 1-0-0 0-1-0 0-0-1 1-1-0 1-0-1 0-1-1 1-1-1 0-0-0 1-0-0 0-1-0 0.1 0.2 0 0 0 0.2 0 0 0 0.9 0.8 0 0 0 0 0 0 0.8 0 0 0 0 0 0 0-0-1 1-1-0 0 0 0 0.3 0.2 0 0 0 0 0 0 0.7 0.8 0 0 0 1-0-1 0-1-1 1-1-1 0 0 0 0 0 0 0.3 0 0 0 0 0 0 0.3 0.5 0 0 0 0.7 0 0 0 0.7 0.5 (b) 3 0-0-0 1-0-0 0.014 859 0.066 865 0-1-0 0-0-1 0.066 865 0.066 865 1-1-0 1-0-1 0-1-1 0.178 306 0.178 306 0.178 306 1-1-1 0.249 629 = 1(0.066 865 + 0.066 865 + 0.066 865) +2(0.178 306 + 0.178 306 + 0.178 306) +3(0.249 629) = 2.019 32 = 2.019 32/3 = 0.673 11 17.5A 1. (a) 1 2 3 4 5 1 0 0 0 0 : 1 0 2 0.333 3 3 0.333 3 4 0.333 3 0 5 0.333 3 0.333 3 0.5 0 0.333 3 0 0.333 3 0 0 0 0 0 0.333 3 0.333 3 0.5 0 0.333 3 0.333 3 0.333 3 0 ), C() (3 ) 1 2 3 0.074 07 0.296 3 0.296 3 0.214 286 0.214 286 0.214 286 4 5 0.259 26 0.074 07 0.142 857 0.214 286 (b) a5 = 0.074 07 (c) π5 = 0.214 286 (d) µ5 = 4.666 6 5. (a) (I − N)−1 Mu 1 2 1 2 1 2 1 1.625 3 1 0.875 5 0.666 7 0.333 3 4.666 6 3.833 3 3 4 1 1 0.875 0.5 1.625 0.5 0.333 3 1.333 3 3.833 3 3.333 3 A B A 0.75 0.2 B 0.1 0.75 C 0.15 0.05 C 0.125 0.125 0.75 (b) A 0.394 737 B C 0.307 018 0.298 246 A: 39.5%, B:30.7%, C:29.8% (c) (I − N)−1 A C C 3.428 57 5.714 29 A C B 9.142 86 8.571 43 A 1 5.882 35 2 2.352 94 A C 8.235 29 B 4.705 88 5.882 35 B 1.588 2 A→B: 9.14 A→C: 8.23 17.6A 2. (a) Mu A 5.714 29 2.857 14 1 , 2 , 3 , 9 10 C() P 1 2 1 0 0 2 0.3 0 3 0 0.1 0.7 0.9 3 0 0 0 0 0 0 1 1 (b) (I − N)−1 2 1 Mu 3 1 2 1 0 0.3 1 0.03 0.01 1 2 1.33 1.1 3 0 0 1 3 1 1.33 . 8. (a) 1 2 P 3 4 1 2 3 0.2 0 0 0.8 0.22 0 0 0.78 0.25 0 0 0.75 0 0 0 4 F 0 0 0 0 0 0 0.3 0 0.7 1 (b) (c) 1 1 1.25 (I − N)−1 2 3 1.282 1.333 4 1.429 1 Mu F 5.29 2 3 0 0 1.282 0 1.333 1.333 1.429 1.429 2 3 4.04 2.76 4 0 0 0 1.429 4 1.43 =5.29, (d) 10. (a) F , 16 (4 ) , (c). 0,1,2,3,D( ) . . P (b) 0 1 0 0.5 0.4 1 0.5 0 2 0 0.6 3 0 0 D 0 0 2 3 D 0.3 0.2 0 0 0 0 0 0 0 0.7 0 0 0 0.8 1 12 . C() (I − N)−1 11 Mu 0 1 0 5.952 3.952 1 2.976 2.976 2 1.786 1.786 3 1.25 1.25 0 1 D 12 9.96 2 3 2.619 1.19 1.31 0.595 1.786 0.357 1.25 1.25 2 3 6.96 3.39 (c) 6.96 . 18 18.1A 1. (a) . (b) x = 0 (e) x = 0 . , x = 0.63 4. (x1 , x2 ) = (−1, 1) , x = −0.63 . (2, 4). 18.2A 1. (b) (∂x1 , ∂x2 ) = (2.83, −2.5)∂x2 . 18.2B 3. 2(xi − √ n ∂f = 2δ C 2−n . x2 n ) xi = 0, i = 1, 2, · · · , n − 1. xi = 5 6. (b) (x1 , x2 , x3 , x4 ) = (− 74 , − 10 , 74 155 60 , ), 74 74 √ n C, i = 1, 2, · · · , n. . 18.2C 2. (x1 , x2 , x3 ) = (−14.4, 4.56, −1.44) (4.4, 0.44, 0.44). 19 19.1A 2. (c) x = 2.5, (e) x = 2, ∆=0.000 001. ∆=0.000 001. 19.1B 1. , ∇f (X ) = ∇f (X 0 ) + H (X − X 0 ). Hessie f (X ) . 0 , X = X 0 − H −1 ∇f (X 0 ). X , X . H X , , , . , ∇f (X ) = ∇f (X) = 0 , X 0 12 C() 19.2A 2. x1 = 0, x2 = 3, z = 17. 4. wj = xj + 1, j = 1, 2, 3,v1 = w1 w2 , v2 = w1 w3 , max z = v1 + v2 − 2w1 − w2 + 1 s.t. v1 + v2 − 2w1 − w2 9 ln v1 − ln w1 − ln w2 = 0 ln v2 − ln w1 − ln w3 = 0 19.2B 1. x1 = 1, x2 = 0, z = 4. 2. x1 = 0, x2 = 4, x3 = 0.7, z = −2.35. 19.2C 1. max z = x1 + 2x2 + 5x3 s.t. 2x1 + 3x2 + 5x3 + 1.28y 10 9x21 + 16x23 − y 2 = 0 7x1 + 5x2 + x3 12.4 x1 , x2 , x3 , y 0 20 20.1A 1. C.16. C.16 20.1B 1. 1: . C() x12 x13 x24 x32 x34 min z 1 5 3 4 6 1 1 1 2 3 4 −1 1 −1 1 −1 0 ∞ min z 1 2 3 4 1 −1 −1 2 = 50 30 40 10 ∞ 10 ∞ 0 ∞ x12 x13 x24 x32 x34 1 5 3 4 6 1 −1 1 1 −1 . −1 = 20 = −40 1 1 −1 ∞ ∞ −1 = −40 = 20 = −30 ∞ ∞ 10 = 40 = −20 20.1C 1. = 9 895 . 1 5. = 24 300 210 , 3 . 220 . . 1 2 1 0 500 2 3 450 0 0 300 1 000 0 0 1 000 1 2 20.2A 1. (c) x2 M . , (x1 , x2 ) = α1 (0, 0) + α2 (10, 0) + α3 (20, 10) + α4 (20, M ) + α5 (0, M ) α1 + α2 + α3 + α4 + α5 = 1, αj 0, j = 1, 2, · · · , 5 , 0) + α3 (0, 12) 2. 1(x1 , x2 ) = α1 (0, 0) + α2 ( 12 5 2(x4 , x5 ) = β1 (5, 0) + β2 (50, 0) + β3 (0, 10) + β4 (0, 5) α1 = α2 = 0, α3 = 1 ⇒ x1 = 0, x2 = 12 β1 = 0.488 9, β2 = 0.511 1, β3 = β4 = 0 ⇒ x4 = 28, x5 = 0 6. , . (x1 , x2 , x3 , x4 ) = ( 53 , 15 , 0, 20), 3 z = 195. 13 14 C() 22 22.1A 2. 1 , . 2 , . 3 . 22.2A 1. 1 , $10 000. 2 , . 3 . 4 . $35 520 4. 2 1, 3 2, 3 3. 22.3A 3. 1 $1, 2 $1, 3 $1 . =0.109 375. 23 23.1A 2. , 1 , , 2 3 2 , , 3 , 1 2 . 23.2A 1. 1 2 3. , 2 23.3A 1. 1 . , . 3 . . , 1 3 D D.1 D.1.1 p1 , p2 , · · · , pn n , P , P = (p1 , p2 , · · · , pn ) P n ( ), P i pi . , P = (1, 2) . D.1.2 () n P = (p1 , p2 , · · · , pn ) Q = (q1 , q2 , · · · , qn ) R = (r1 , r2 , · · · , rn ) ri = pi ± qi . , R = P ± Q i D.1.3 P , Q, S, P +Q=Q+P () (P ± Q) ± S = P ± (Q ± S) ( P + (−P ) = 0 ( ) ) P ()θ, Q = θP = (θp1 , θp2 , · · · , θpn ) P D.1.4 θ . , P , S θ, γ, θ(P + S) = θP + θS () θ(γP ) = (θγ)P ( ) P 1 , P 2 , · · · , P n , n j=1 θj P j = 0 ⇒ θj = 0, j = 1, 2, · · · , n D 844 n θj P j = 0, θj = 0 j=1 . , P 1 = (1, 2), P 2 = (2, 4) , θ1 = 2 θ2 = −1, θ1 P 1 + θ2 P 2 = 0 D.2 D.2.1 m n A . m × n () ⎛ a11 ⎜ a21 ⎜ A=⎜ ⎝ a31 a41 D.2.2 a12 a22 a32 a42 (2) . , (4 × 3) ⎞ a13 a23 ⎟ ⎟ ⎟ = aij 4×3 a33 ⎠ a43 , m = n. , (3 × 3) ⎛ 1 ⎜ I3 = ⎝ 0 0 (3) 1 n . (4) m 1 . A T AT 1, 0 0 ⎛ 1 4 1 ⎟ 0⎠ 0 1 ⎞ ⎟ 5 ⎠ ⇒ AT = 6 A = aij , B = bij aij = bij . 0. ⎞ 1 2 3 4 5 6 B = 0 (zero matrix), B (7) A (transpose), i j, A aji . , ⎜ A = ⎝2 3 (6) i j . (1) (5) aij aij . , , i, j D.2 845 D.2.3 () . , ( D.2.6). () , A = aij B = bij , (m × n) , D = A + B . dij m×n = aij + bij m×n A, B, C , A+B =B+A () A ± (B ± C) = (A ± B) ± C ( T T (A ± B) = A ± B ) T A = aij B = bij , A D = AB . A (m × r) , m n , B (r × n) . , D dij = r B , , D (m × n) i j aik bkj , k=1 , A= D= = 7 9 2 4 6 8 23 31 9 0 1 34 3 46 5 1 3 2 4 = , B= 5 7 9 6 8 0 1×5+3×6 1×7+3×8 1×9+3×0 2×5+4×6 2×7+4×8 2×9+4×0 18 , AB = BA, BA . I m A = AI n = A, I m , I n (AB)C = A(BC) C(A ± B) = CA ± CB (A ± B)C = AC ± BC αAB = (αA)B = A(αB), α A (m × r) A= , B (r × n) A11 A12 A13 A21 A22 A23 ⎛ B 11 ⎜ , B = ⎝ B 21 B 31 B 12 , A B ⎞ ⎟ B 22 ⎠ B 32 D 846 , i, j, Aij B ij , A11 B 11 + A12 B 12 + A13 B 31 A11 B 12 + A12 B 22 + A13 B 32 A×B = A21 B 11 + A22 B 21 + A23 B 31 A21 B 12 + A22 B 22 + A23 B 32 , ⎛ 1 ⎜ ⎝1 2 D.2.4 2 0 5 ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ 1 4 30 ⎟ ⎜ ⎟ ⎜ (1)(4) + ( 2 3 ) 8 ⎟ ⎜ 4 + 2 + 24 ⎟ ⎜ ⎟ ⎟⎜ ⎟ ⎜ 5⎠⎝1⎠ = ⎜ ⎟ = ⎜ ⎟ = ⎝ 44 ⎠ 0 5 1 ⎠ ⎝ 4 40 ⎠ ⎝ 1 (4) + + 6 8 61 2 5 0 8 8 53 3 n ⎛ a11 ⎜ ⎜ a21 A=⎜ ⎜ .. ⎝ . an1 a12 ··· a22 .. . ··· an2 ··· a1n ⎞ ⎟ a2n ⎟ .. ⎟ ⎟ . ⎠ ann , Pj1 j2 ···jn = a1j1 a2j2 · · · anjn A ρ n! j1 , j2 , · · · , jn . 1, j1 j2 · · · jn ∈j1 j2 ···jn = 0, j1 j2 · · · jn , A ∈j1 j2 ···jn Pj1 j2 ···jn ρ detA |A|. ⎛ a11 ⎜ A = ⎝ a21 a31 a12 a13 ⎞ a22 ⎟ a23 ⎠ a32 a33 |A| = a11 (a22 a33 − a23 a32 ) − a12 (a21 a33 − a31 a23 ) + a13 (a21 a32 − a22 a31 ) (1) . , . T (2) |A| = |A |. (3) B A (4) A ( (5) ) α () , |B| = −|A|. , |A| = 0. , |A| . D.2 (6) α, 847 . (7) A B n , |AB| = |A||B| |A| aij Mij . , ⎛ a12 a11 ⎜ A = ⎝ a21 M11 a22 = a32 Aij = (−1) A a13 ⎟ a23 ⎠ a22 33 a13 ,··· a 31 33 Mij B ||Aij || , ⎛ A11 A21 ⎜ A A22 ⎜ 12 adj A = Aij T = ⎜ .. ⎜ .. . ⎝ . , ⎛ A1n A2n 1 3 ⎜ A = ⎝2 3 2 i j ⎞ a31 a32 a33 a a23 , M22 = 11 a a i+j A 2 ··· ··· ··· An1 aij (cofactor), ⎞ ⎟ An2 ⎟ .. ⎟ ⎟ . ⎠ Ann ⎞ 3 ⎟ 2⎠ 3 4 3 A11 = (−1) (3 × 4 − 2 × 3) = 6, A12 = (−1) (2 × 4 − 3 × 2) = −2, · · · , ⎛ ⎞ 6 1 −5 ⎜ ⎟ adj A = ⎝ −2 −5 4 ⎠ −3 D.2.5 3 −1 r, (full-rank) r. (nonsingular) ⎛ ⎞ 1 2 3 ⎜ ⎟ A = ⎝2 3 4⎠ 3 5 7 A (singular) . , |A| = 1 × (21 − 20) − 2 × (14 − 12) + 3 × (10 − 9) = 0 A r = 2, 1 2 2 = −1 = 0 3 D 848 D.2.6 B C n . , BC = CB = I, B C , C B B −1 C −1 . BC = I, B , C = B −1 , . BC = I B −1 BC = B −1 I IC = B −1 C = B −1 , (1) A B n (2) . , (AB)−1 = B −1 A−1 . A , AB = AC B = C. n . ⎞⎛ ⎞ ⎛ ⎞ ⎛ x1 b1 a11 a12 · · · a1n ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎜ a21 a22 · · · a2n ⎟ ⎜ x2 ⎟ ⎜ b2 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ . .. .. ⎟ ⎟⎜ . ⎟ = ⎜ . ⎟ ⎜ . . . ⎠ ⎝ .. ⎠ ⎝ .. ⎠ ⎝ . xn bn an1 an2 · · · ann xi , aij bi . n AX = b , A . A−1 AX = A−1 b X = A−1 b D.2.7 A n A−1 = , 1 1 adj A = |A| |A| ⎛ A11 ⎜ ⎜ A12 ⎜ . ⎜ . ⎝ . A1n TORA LU . Press A21 ··· A22 .. . ··· A2n ··· (1986). An1 ⎞ ⎟ An2 ⎟ .. ⎟ ⎟ . ⎠ Ann D.2 , ⎛ 1 2 ⎜ A = ⎝2 ⎛ 6 ⎜ adj A = ⎝ −2 −3 ⎛ A−1 = (–) 6 1 ⎜ ⎝ −2 −7 −3 3 1 −5 849 ⎞ 4 ⎞ ⎟ 4 ⎠ , |A| = −7 −1 ⎞ −5 ⎛ ⎟ ⎜ 4 ⎠=⎝ −1 3 3 3 −5 ⎟ 2⎠ 3 −5 1 3 − 76 − 71 2 7 3 7 5 7 − 73 (A|I), A . 5 7 − 47 1 7 ⎞ ⎟ ⎠ A−1 , (A−1 A|A−1 I) = (I|A−1 ) , , A I, I A−1 . ⎛ 1 2 ⎜ ⎝2 3 X 3 3 , ⎛ ⎞ 3 ⎟⎜ ⎟ ⎜ ⎟ 2 ⎠ ⎝ x2 ⎠ = ⎝ 4 ⎠ x3 5 4 3 ⎞⎛ x1 ⎞ A−1 (A|I|b) = (I|A−1 |A−1 b) 0 ⎛ 1 ⎜ A = ⎝2 3 1 ⎛ 1 ⎜ A=⎝ 0 0 2 ⎛ 1 ⎜ A=⎝ 0 3 ⎛ 2 −1 −3 0 1 0 0 1 0 ⎜ A=⎝ 0 0 1 0 2 3 3 3 1 2 0 4 0 1 0 3 1 −4 −2 −5 −3 −5 −3 4 2 7 3 0 − 67 2 0 7 3 1 7 ⎞ 0 3 ⎟ 0 4⎠ 1 5 0 0 1 0 2 −1 −3 − 17 5 7 − 37 ⎞ 3 0 ⎟ 0 −2 ⎠ 1 −4 ⎞ 0 −1 ⎟ 0 2 ⎠ 2 1 5 7 4 −7 1 7 3 7 6 7 2 7 ⎞ ⎟ ⎠ D 850 3 , 7 x1 = x2 = 67 , x3 = 2 . 7 A , . , B −1 B B next , B −1 next , . −1 B −1 next = EB E m B P j P r B next , E r ⎛ −(B −1 P ) ⎞ j 1 ⎜ −(B −1 P j )2 ⎜ ⎜ .. ⎜ . 1 ⎜ ξ= ⎜ (B −1 P j )r ⎜ +1 ⎜ .. ⎜ ⎝ . ⎟ ⎟ ⎟ ⎟ ⎟ , ⎟ ⎟ ← r ⎟ ⎟ ⎠ (B −1 P j )r = 0 −(B −1 P j )m (B −1 P j )r = 0, B −1 next . B −1 next . F m , r (B −1 P j ) , F = (e1 , · · · , er−1 , B −1 P j , er+1 , · · · , em ) B next B r B −1 P j , B next = BF , −1 = F −1 B −1 B −1 next = (BF ) E = F −1 , . B B 1 , B , . B 0 = I = B −1 0 , −1 B −1 1 = E 1B 0 = E 1I = E 1 , B i , i B i , −1 B −1 = E i B −1 i i−1 = E i E i−1 B i−2 = · · · = E i E i−1 · · · E 1 B, B −1 = E n E n−1 · · · E 1 . ⎛ ⎞ 2 1 0 ⎜ ⎟ B = ⎝0 2 0⎠ 4 0 1 D.2 0 ⎛ 1 0 ⎜ B 0 = B −1 0 = ⎝0 0 0 851 ⎞ 1 ⎟ 0⎠ 0 1 1 ⎛ 2 0 ⎜ B1 = ⎝ 0 ⎛ E1 = 2 ⎛ 2 ⎜ B2 = ⎝ 0 ⎛ 4 ⎜ B −1 1 P2 =⎝ ⎛ 1 0 1 0 0 1 0 0 1 2 −2 + 21 − −2 2 ⎟ 0⎠, 0 1 0 0 ⎛ ⎞←r=1 2 ⎜ ⎟ B −1 P = P = 0⎠ ⎝ 1 1 0 4 ⎛ 1 ⎞ 0 0 2 ⎜ ⎟ B −1 0 1 0 ⎠ 1 = ⎝ ⎞ 1 ⎟ 0⎠, 0 1 −2 ⎞⎛ ⎞ ⎛ 1 ⎞ 1 2 ⎟ ⎟⎜ ⎟ ⎜ 0 ⎠⎝2⎠ = ⎝ 2 ⎠ ← r = 2 0 −2 1 ⎞ ⎛ ⎞ 0 1 − 14 0 ⎟ ⎜ ⎟ 1 0⎠ = ⎝ 0 0 ⎠ 2 1 0 ⎛ 1 1 ⎜ −1 B −1 = B −1 2 = E 2B 1 = ⎝ 0 0 0 1 1 1 ⎞⎛ ⎞ (q × p) ⎛ 0 0 1 ⎟ ⎜ 0 ⎠=⎝ −2 0 1 ⎟⎜ 0 ⎠⎝ (q × q) B 11 ⎜ (p × p) ⎜ B=⎜ ⎝ B 21 (q × p) ⎛ 1 2 0 n A B, ⎛ ⎞ A12 A11 ⎜ (p × p) (p × q) ⎟ ⎜ ⎟ A=⎜ ⎟ , A11 ⎝ A21 A22 ⎠ B A 1 0 − 14 1 2 0 ⎞ ⎟ 0⎠ = B 1 2 ⎞ 1 + 21 ⎜ 0 ⎝ −2 − 24 2 −2 1 ⎜ E2 = ⎝ 0 0 4 0 B 12 ⎞ (p × q) ⎟ ⎟ ⎟, B 22 ⎠ (q × q) , AB = I n , A11 B 11 + A12 B 21 = I p A11 B 12 + B 12 A22 = 0 1 2 ⎞ − 14 0 0 1 2 ⎟ 0 ⎠ −2 1 1 D 852 BA = I n , B 21 A11 + B 22 A21 = 0 B 21 A12 + B 22 A22 = I q A11 , A−1 11 . B 11 , B 12 , B 21 , B 22 , −1 −1 B 11 = A−1 (A21 A−1 11 + (A 11 A12 )D 11 ) −1 B 12 = −(A−1 11 A 12 )D B 21 = −D −1 (A21 A−1 11 ) B 22 = D −1 D = A22 − A21 (A−1 11 A 12 ) , ⎛ 1 ⎜ A = ⎝2 3 2 ⎟ 2⎠ 3 4 A11 = (1), A12 = (2, 3), A21 = , 2 3 , A22 = 3 2 3 4 6 B 11 = − 7 = 1, 3 2 2 −1 −4 D= − (1)(2, 3) = 3 −3 −5 3 4 5 −5 4 − 47 1 −1 7 D =− = 1 7 − 37 3 −1 7 , B 12 = 1 − 7 5 7 , B 21 = 2 7 3 7 , B 22 = 5 7 − 37 − 47 1 7 B = A−1 . D.2.8 A−1 11 ⎞ 3 3 Excel Excel (1) ; D.1 , E4:F6 , (2) ; (3) (2 × 3) ; . ( excelMatManip.xls). 1() , A (4) A4:C5. Transpose(A), AT , D.3 D.1 (1) E4 (2) () Excel 853 ( excel Mat Manip.xls) =TRANSPOSE(A4:C5); E4:F6; (3) F2 ; (4) CTRL+SHIFT+ENTER . 2 , A B A10:C13 A16:A18 , =MMULT(A10:C13, A16:A18), E10:E13. E10 1 2 4 ( E4:F6 E10:E13). A10:C13) . 3 , A22:C24 =MINVERSE(A22:C24) , 4 , E28 E22:G24 , MMULT(A16:A18, E22 1 2 =MDETERM(A28:C30), . D.3 X = (x1 , x2 , · · · , xn )T 4 . A28:C30 D 854 ⎛ a11 ⎜ ⎜ a21 A=⎜ ⎜ .. ⎝ . an1 a12 ··· a22 .. . ··· an2 ··· Q(X) = X T AX = a1n ⎞ ⎟ a2n ⎟ .. ⎟ ⎟ . ⎠ ann n n aij xi xj i=1 j=1 . aij +aji 2 A , Q(X) . . ⎛ 1 ⎛ ⎞⎛ x1 ⎞ 3 0 2 1 1 2 2 . 1 ⎟⎜ ⎟ 6 ⎠ ⎝ x2 ⎠ ⎜ Q(X) = (x1 , x2 , x3 ) ⎝ 1 , 2 , A 0 7 ⎜ Q(X) = (x1 , x2 , x3 ) ⎝ 2 aij aji (i = j) x3 ⎞⎛ x1 ⎞ 7 ⎟⎜ ⎟ 3 ⎠ ⎝ x2 ⎠ 3 2 x3 A . (1) , X = 0, Q(X) > 0. (2) , X, Q(X) 0, X = 0, Q(X) = 0. (3) , −Q(X) . (4) , −Q(X) . (5) , . , . (1) Q(X) (), A ( , (2) Q(X) ) . A (). , A k (−1)k , k = 1, 2, · · · , n. , (3) Q(X) A (). , A k 1, 2, · · · , n. An×n k a11 a12 · · · a21 a22 · · · . .. . . . a a ··· k1 k2 a1k a2k .. , k = 1, 2, · · · , n. . akk , (−1)k , k = D.4 D.4 855 f (X ) , X 1 X 2 , f (λX 1 + (1 − λ)X 2 ) < λf (X 1 ) + (1 − λ)f (X 2 ) 0 < λ < 1. , f (X ) , −f (X) . () ( D.3) f (X ) = CX + X T AX C , A , A . , A , f (X) , f (X) . 1. ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 −2 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (a) ⎝ −2 ⎠ ⎝ 4 ⎠ ⎝ −2 ⎠ −2 3 2. ⎛ 1 4 ⎜ A=⎝ 2 9 ⎛ −1 7 ⎟ ⎜ −8 ⎠ , B = ⎝ 9 5 3 ⎞ 7 ⎞ ⎛ 4 2 ⎟ 8 ⎠ 4 3 ⎞ 2 6 10 (a) A + 7B (c) (A + 7B)T (b) 2A − 3B 3. 2 , AB = BA. 4. ⎛ 1 ⎜ ⎜ 2 A=⎜ ⎜ 3 ⎝ 4 5 −6 7 9 ⎞ ⎛ 2 ⎟ ⎟ 9 ⎟ ⎜ , B = 1 ⎝ 2 ⎟ ⎠ 3 1 7 3 −4 2 6 1 0 5 ⎞ ⎟ 7 ⎠ 9 AB. 5. 2 , A−1 B −1 . (a) (c) 6. . . ⎛ 2 ⎜ B = ⎝0 4 1 2 ⎞ (b) . (d) . ⎛ 2 ⎜ ⎟ 1 ⎠ , B −1 = ⎝ 0 5 ⎞ ⎟ ⎟ ⎜ ⎜ ⎜ −3 ⎟ ⎜ −6 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ 4 ⎟ ⎜ 8 ⎟ ⎠ ⎠ ⎝ ⎝ 10 5 (b) −1 ⎛ 2 5 4 1 2 −1 − 85 1 4 1 2 − 83 ⎞ ⎟ − 41 ⎠ 1 2 856 D 3 P 3 V 3 = P 1 + 2P 2 7. (a) . . (b) x1 + 2x2 = 5 x1 + 2x2 = 3 −x1 − 2x2 = −5 x1 + 4x2 = 2 (c) , . x1 + 7x2 + x3 = 5 4x1 + x2 + 3x3 = 8 x1 + 3x2 − 2x3 = 3 8. B.2.7 9. A= 10. . 1 G H B , B Q(x1 , x2 ) = 6x1 + 3x2 − 4x1 x2 − 2x21 − 3x22 − 27 4 11. Q(x1 , x2 , x3 ) = 2x21 + 2x22 + 3x23 + 2x1 x2 + 2x2 x3 12. f (x) = ex . 13. f (x1 , x2 , x3 ) = 5x21 + 5x22 + 4x23 + 4x1 x2 + 2x2 x3 . 14. 13 , −f (x1 , x2 , x3 ) . Hadley, G., Matrix Algebra, Addison-Wesley, Reading, MA, 1961. Hohn, F., Elementary Matrix Algebra, 2nd ed., Macmillan, New York, 1964. Press, W., B. Flannery, B. Teukolsky, and W. Vetterling, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, England, 1986.