Principles of Mathematics in Operations Research
258 Solutions Min (xia + x2a + x3a) H 1- {xip + x2p + x3p) s.t. (xla + xig) - (xlb + xlf) = 0\ (x2a + X2 9 ) - (x 2 b + x 2 f) = ...
Solutions 259 In our example instance, «i is node A and £3 is node /. If we enumerate paths (some of them is given in Figure S.1 ...
260 Solutions and the formulation will be Min 5/i + • • • + 10/ 20 s.t. A + • • • + /e = 10 h + • • • + hi = 10 /12 + • • • + /2 ...
Solutions 261 c=[557789477994666888 10 10|000000000000] 1 1 1 000 000 1 1 1 000 000 000 1 1 1 000 000 110 001 000 000 111 00000 ...
262 Solutions B, = 100 010 001 001 001 010 100 001 100 00 1 000 100 010 010 100 000 000 000 100 010 001 000 000 000 000 000 000 ...
Solutions 263 Bn 1 00 0 10 00 1 001 001 010 100 00 1 100 001 000 100 010 010 100 000 000 000 100 010 00 1 000 000 000 000 000 00 ...
264 Solutions Ba 100 0 10 00 1 101 101 010 000 001 000 001 000 100 010 010 000 000 000 000 100 0 10 001 000 000 000 000 000 000 ...
Solutions 265 Fig. S.16. The optimum solution for our multi-commodity flow instance In a sense, we partition the optimization pr ...
266 Solutions Problems of Chapter 9 9.1 Let a = inf A Then, Vx e A, a < x «=> -x < -a. Hence, (-A) in bounded above. Al ...
Solutions 267 b---b b---bb---b Case 1 : p > 0, q > 0, W+q = ^~" = ^-v-'^-' = ^69; P + <? p q Case 2 : p < 0, q > ...
268 Solutions Then, r + s G Q, r + s < x + y =$• br+s = bsbr G B(x + y) =^> brbs < bx+y. Keep s fixed. 6r < ^-, V6r ...
Solutions 269 Claim: bx = y. Proof: If not, bx < y or bx > y. If bx < y, by (d) Vn G N, bx+lln < y, x + 1/n G A. Con ...
270 Solutions Vx € R. Hence, ~ is not reflexive. b) The statement x ~ y =>• y ~ x, x ~ y and y ~ x => x ~ x; therefore, x ...
Solutions 271 Problems of Chapter 10 10.1 Fix x,y eM.k arbitrary. k k d 2 (x,y) = \^2{xi - Vi)^2 ]1/2, di(x,y) = ^ |a?i -j/j|, » ...
272 Solutions AUB = X =>AUB = X, AC\B = %=>B = (A)c A and B are separated and A\J B = X => A = Bc B is closed. i.e. B i ...
Solutions 273 Fig. S.17. Nearest neighbor (in h metric): initial solution -6- Delete (E, F) k, (L, A): gain=18-12 Improved Tour: ...
274 Solutions ^ i s-! -—§*•# , j»M 1 \ * •*- r te "j ^ L f M_. ' X Delete {I, J) & (G,L): gain=12~8 Improved Tour: Lengt ...
Solutions 275 Fig. S.20. Nearest neighbor (in £ 2 metric): initial solution The gain values are tabulated below. See Figure S.21 ...
276 Solutions c) Use loo norm: UABCDEFGHIJKL 0 3 1 3 0 2 1 2 0 5 2 4 6 3 5 3 3 2 5 5 4 5 5 4 6 6 5 7 7 6 8 8 7 8 8 7 5 2 4 0 1 2 ...
Solutions 277 The gain values are tabulated below. See Figure S.23 for the improvement. GAIN (AC) (C,B) (B,D) (D,E) (E,F) (F,H) ...
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