CORRECTIONS file page line replace by ---------------------------------------------------------------- dedication iv my our Contents v 10 52 54 vi 3 180 179 vi 7 210 211 vi 8 220 221 preface vii -1 Thus Thus, ch1 4 -15 5y = 7 5y = -7 ch1 5 14 ,, , ch1 6 -9 1.8 1.8. ch1 7 6 2 3 ch1 9 1 differents different ch1 9 -4 rediscoved rediscovered ch1 11 12 36-42 36-43 ch1 11 -10 57 58 ch1 11 -5 linear a linear form ch1 11 -6 58 59 ch1 11 -4 59 60 ch1 14 -1 simplex dual simplex ch1 14 -1 Chapter 4 Chapter 5 ch1 17 13 x_3 x_3 . ch1 17 (table) Aarea Area ch1 17 (table) .6 0.8 ch1 21 14 to worker for worker ch1 22 14 is not obvious it is not obvious ch1 23 2, 5 §1 of Chapter 2 §3 ch1 23 16 2.6 2.5 ch1 28 1 after Fig 3.7 §3 §12 ch1 30 -12 min max ch1 31 -1 §3 §12 ch1 33 -1 y + y y+ z ch1 38 18 that than ch2 39 14 1951) 1951): ch2 40 -8 30 31 ch2 43 -2 uses use ch2 46 -9 5.8.. 5.8. ch2 41 Ex.31 if and only if means that ch2 47 3 prededing preceding ch2 49 -2 multiple...by a number scalar multiple ch2 50 3 B A ch2 50 the last row of P 0 0 1 1 0 0 ch2 50 the last row of D 4 ch2 50 the last row of D^{-1} 8 2 ch2 51 matrices E and E^{-1} their transposes ch2 52 Ex.12-14 in the sense (see page 4) of Definition 1.3 ch2 53 Ex.44 !! ch2 54 5 system systems ch2 54 -17 correspond corresponds ch2 54 -10 set of linear linear ch2 56 19 cz -cz ch2 56 -4 a upper an upper ch2 56 -1 0, 0 ch2 57 -2 system systems ch2 58 4 system systems ch2 58 -14 time times ch2 58 -13 a an ch2 59 9 a an ch2 61 12 -3/2 1/2 (left of third matrix) ch2 61 -9 -71 7 ch2 61 -5 - 71 + 7 ch2 62 4,18 -71 7 ch2 62 7 cannot to cannot ch2 62 11 means exactly exactly means ch2 62 -14 - 7/8 ,,, + 7/8 ch2 64 -3 3 2 ch2 66 19 36-38 36-39 ch3 68 15 necessary necessarily ch3 70 -15 7,4 7.4 ch3 70 -13 equivalent the equivalent ch3 70 -5 (eliminate) (eliminates) ch3 71 3 Thick Trick ch3 71 16 constraint constraints ch3 71 18 adding by adding ch3 71 -5 in the is the ch3 72 1 rid off get rid of ch3 72 4 know any upper or lower bonds do not know any bounds ch3 72 -19 preeding preceeding ch3 73 18,20 >= 1 <=-1 ch3 75 10 method methods ch3 77 2 system systems ch3 78 4 one of one of the ch3 80 1,2,3 beta gamma ch3 85 -5 is are ch3 86 1 e.) e) ch3 90 18 ,z , z ch3 91 1st tableau =w =-w ch3 93 -13 this is this ch3 96 2,6,7,7 y u ch3 96 11 y u,v ch3 97 3 Insert period after the first tableau. ch3 97 9 Ax - u = b Ax - u = -b ch3 99 -4 $x_7$ $ = x_7$ ch4 105 4 isthe is the ch4 105 -15 -2and -2 and ch4 105 -8 2x_2 2x_3 ch4 105 -6 min= min = ch4 106 last row in 1 0 1 1 1 4 1 1 4th tableau ch4 107 11 x3 to - x3 to ch4 107 21 previosly previously ch4 111 -2 one two two ch4 112 last row in (11.2) tableaux tableau ch4 112 -9 column in row in ch4 112 -1 tableaux tableau ch4 115 18 necessary necessarily ch4 116 -15 in 10.4 in 10.10 ch4 118 above Remark x_4 = 1 x_1 = 1 ch4 116 -9 Phase 2 Phase 1 ch4 119 9 z be linear z by linear ch4 120 Ex.10 with that ch4 123 -13 As As an ch4 125 -12 t T ch4 125 -5 empty is empty ch4 127 11 of sign sign ch4 127 14 could be now now could be ch4 127 16 that is stated as ch4 128 -21 a adjacent an adjacent ch4 128 -15,-7 = u + b = u ch5 135 Definition associated to associated with ch5 135 -1 min max ch5 136 -9 previosly previously ch5 137 first paragraph Delete the second sentence. ch5 137 -9 (twice) superscript t superscript T ch5 138 14 dv d = v ch5 138 -6 equality an equality ch5 139 6 equality the equality ch5 140 -17 theorem in theorem on ch5 141 3 in 1st matrix -2 2 ch5 141 2 in 2nd matrix -2 2 ch5 143 last in (14.2) respectivel,y respectively, ch5 143 11 but , but ch5 143 last in (14.2) respectivel,y respectively, ch5 145 1st after (14.5) tableu tableau ch5 145 9th after (14.5) stay stays ch5 151 -12 row raw ch5 152 -9 and and, when parameters are in c (resp., in b), ch5 152 -9 minimun minimum (resp., maximum) ch5 152 -8 concave concave (resp., convex) ch5 153 -10 y >>= 0 y >= 0 ch5 156 -4 low lower ch5 157 5.5 allx all x ch5 159 6 Is Does ch5 159 7 follows follow ch5 159 15 15.2 15.2. ch5 160 -14 van you to can ch5 160 -10 answer to answer ch5 160 -5 2.1 2.2 ch5 163 tableaux 105 122 ch5 164 11 Remark. Remark ch5 165 3 +- - ch5 165 4 . . . ch5 165 (Reduce hight of brackets in Exercises 9 and 10) ch6 166 -6 transpcrtation transportation ch6 168 -3 2-by-3 table 2-by-2 table above the last table ch6 170 3 15.2 15.7 ch6 170 9 = >= ch6 170 10 = <= ch6 172 4 after row column 1st table ch6 172 9 first row first column ch 6 173 last 1 2 3 77 39 105 matrix 1 2 2 150 186 122 ch6 175 Table 16.6 15.2 15.7 ch6 176 1st line after Table 16.9 previosly previously ch6 180 4 30) = 50) = ch6 184 4 after 17.11 e epsilon ch6 189 top of 17.23 c = 25 c = 35 ch6 189 -1 ficticious fictitious ch6 192 7 then is then is ch6 193 8 .The . The ch6 193 9 problem we may problem may. ch6 193 3 in Solution ., the , the. ch6 194 8 * * * * * * * ch6 194 12 | * |* ch6 195 the third table * * * * last rows * * * * * * ch6 195 first line column row after third table four two ch6 195 -1 + 2 + 3 ch6 196 second row in first two tables (2) (3) (1) (1) (2) (0) ch6 196 second row in third table (3) (3) (2) (2) (2) (1) ch6 196 left margin third table 6 2 1 5 3 2 ch6 197 first two rows 4|1 (-1) 0-e 4|1-e (-1) 0 in first table 3|(0)(0) (-7) 2|(-1)e (-1)(-2) ch6 197 first line after first table (2, 3). Again +e=0. (2, 1), and e=1. ch6 197 second line after (1, 3) (no other choice this time). first table (2.4). ch6 197 first two rows 1 (0) (1) 0 0 (0) (1) 1 in second table (0) (1) 0 1 1 (1) 0 (1) ch6 197 first two rows * * in last table * * ch6 197 3 lines above The optimal value is Remark 18.8 min=10. ch6 197 -4 [ ( ch7 199 8 Webster's a Webster's ch7 200 13 hin him ch7 200 -7 "(paoff matrix).." "(payoff matrix)." . ch7 200 -2 as of as ch7 201 -8 an unilateral a unilateral ch7 202 -2 is is is ch7 204 under max min and min max switch p in P and q in Q ch7 205 matrices , row (2,3) 1 1 -1 1 0 0 (both matrices should be skew symmetric) ch7 208 3 previosly previously ch7 210 10 20 -6 20 0 -6 (The empty entry in the last line of Exercise 10 means 0) ch7 210 -10 (0. 3) (0, 3) ch7 211 20 ) ] ch7 211 -3 ] ]^T ch7 212 14 th the ch7 212 -11 -1/2by -1/2 by ch7 213 -5 nodes node. ch7 215 -2 .µ , µ ch7 216 4 in paragraph 3 problem solved problem is solved. ch7 216 4 in 3nd paragraph solved is solved ch7 217 2 after 1st matrix lows lows us. ch7 218 7 game the game ch7 218 -1 solutions strategies ch7 219 12 A M ch7 221 -12 game ? game? ch7 221 -6 loose lose ch7 223 2 from with r3 with c3 the last matrix ch7 226 17 c4,c5 c4, c5 ch7 226 -2 0.1 0, 1 ch7 227 2 [1,1,1] [1,2,1] ch7 227 2 [0 [1 ch7 227 5 0.1 0, 1 ch7 227 Ex. 5 games game ch8 229 6 we can can we ch8 229 8 it not it is not ch8 230 17 number numbers ch8 231 -4 tells that tells us that ch8 232 9 to with to do with ch8 232 13 suma sumo ch8 232 16 "Survival" TV show TV show "Survivor" ch8 232 -4 5h + 75 5h - 200 ch8 232 -4 6h + 76 6h - 254 ch8 232 -3 75 -200 ch8 232 -3 76 -254 ch8 233 -13 we can can we ch8 234 6 twice d dh ch8 234 7 function functions ch8 234 -6 NHI NIH ch8 236 10 l^1-approach the l^1-approach ch8 238 -10 solve system solve a system ch8 239 1 assumptions on assumptions of ch8 240 4, 10 literature the literature ch8 240 Ex. 9 p $p$ (italize p twice) ch8 240 -6 12 13 ch8 240 -4 13 14 ch8 241 18 Otherwise When the columns of A are linearly independent ch8 241 -4 A A^T ch8 241 -3 w w^T ch8 241 -1 Switch a and b ch8 242 5 A A^T ch8 242 8 b a ch8 242 8 a c ch8 242 -16, -12 a X ch8 243 -13, -8 a X ch8 243 15 consider considered ch8 243 -12 that e that |e ch8 247 11 etc , etc ch8 247 18 allowe allow ch8 247 -12,-2 ^t ^T ch8 248 4 ({{ {{ ch8 248 4 }) }} ch8 248 11 Maple : Maple: ch8 248 -9 of at ch8 249 6 $p=3;$ $p=2;$ ch8 249 14 in trash in the trash ch8 250 10 best$l best $l ch8 250 Remark $+1/\alpha $-(-1/\alpha) ch8 250 Ex. 14 . , (replace the first four periods by commas) ch8 251 4 problem , problem, ch8 251 16 questions question ch8 251 18 year the year ch8 251 22 those these ch8 252 -8 billions billion ch8 253 16 Example 24.3 Example 24.4 ch8 253 17 $55K of $55K ch8 253 -14 from from from ch8 253 -11 On the top On top ch8 253 -5 intitial initial ch8 253 -4 sufficiantly sufficiently ch8 255 6 liner linear ch8 255 11 date data ch8 256 Ex.3 24.3 24.4 ch8 256 -4 l_p l^p ap 258 -15 a more more ap 258 -14 then than ap 259 14 a a a ap 259 16 of ap 260 8 .) ). ap 260 10 .] ]. ap 262 -10 = 0.] = 0]. ap 262 -5 otherwize otherwise ap 263 -6, -3 Lipshitz Lipschitz ap 264 17 Lipshitz Lipschitz ap 264 17 or or is ap 264 -14 [V2] [V] ap 265 -7 Lipshitz Lipschitz ap 268 -8 )) ) ap 268 -5 1| 1)| ap 271 8 0 0 . ap 273 head A3. ... A4. ... ap 273 -11 to toward toward ap 276 -9 get better get a better ap 283 11 , The . The ap 285 head A6. ... A5. ... ap 285 8,9 )) ) ap 285 9 - ap 286 1, 2 Pertubation Perturbation ap 287 head A6. ... A7. Goal Programming ap 289 -5 , While . While ap 290 6 point ,and point, and ap 291 10 (ee (see ap 293 -5 by be ap 295 -15 sort s sort n ap 295 -14 n] = m[ i] = n[ ap 296 6 * and and ap 296 -18 previosly previously ap 296 -5 (11) (1) ap 297 head A11. ... A10. ... ap 297 3 tupple tuple ap 299 12 Transportation The transportation see the manual at http://www.math.psu.edu/vstein/LPbook/sol2.ps and http://www.math.psu.edu/vstein/LPbook/sol2.pdf for corrections to pages 305-317. bibliography, index: bai 301 [B1] New York , New York, bai 301 [C1] C. C., bai 301 [DL] .and . and bai 301 [DL] . Mathematics . Mathematics bai 302 [FMP] M.C. M. C. bai 302 [FSS] Forg Forgó bai 302 [FSS] Szp Szép bai 303 [K4] . Klower . Klower bai 303 [K5] . , ., bai 303 [L] D.G. D. G. bai 303 [NC] . Cambridge . Cambridge bai 304 [S3] S.M. S. M. bai 304 [VCS] V.I. V. I. bai 304 [V] V. N. bai 318 assignment problem,, assignment problem, bai 318 Dantzig , Dantzig, bai 319 diagonal diagonal matrices. 48 ,53 matrices, 48, 53 bai 319 elementary elementary matrices. 51 matrices, 51 bai 319 inconsistant inconsistent bai 319 Karush-Kuhn-Tucker, 280 Karush-Kuhn-Tucker, 260 bai 319 KKT conditions, 280 KKT conditions, 260 bai 319 klein Klein bai 319 Lagrange multiplies Lagrange multipliers bai 320 payoff matrix payoff matrix, bai 320 step. 79 step, 79 bai 321 points. 79 points, 79 bai 321 shadow prices, 1144, 146 shadow prices, 144, 146 bai 321 TCP TSP