Jsun Yui Wong
The following formulation is from Ignizio and Daniels [15, p. 265]:
Minimize z1' =2x1+x2+5x3,
minimize z2' =2x1+5x2+x3,
subject to
x1+x2+x3>=1,
x1, x2, and x3 are 0-1 integer variables.
In [15] Ignizio and Daniels show their procedure for transforming the problem above to the following problem [15, p. 267] in a format like the format in Ignizio [14, p. 486]:
Minimize lambda,
subject to
lambda>=(1/4)[-1-(-2x1-x2-5x3)],
lambda>=(1/4)[-1-(-2x1-5x2-x3)],
x1+x2+x3>=1,
x1, x2, and x3 are 0-1 integer variables.
Lambda is a dummy variable.
The following computer program uses the second formulation shown above.
0 DEFDBL A-Z
2 DEFINT K
3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32000 STEP .01
14 RANDOMIZE JJJJ
16 M = -1D+37
19 FOR J44 = 1 TO 4
22 A(J44) = RND
31 NEXT J44
128 FOR I = 1 TO 5000
129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))
181 j = 1 + FIX(RND * 4)
183 r = (1 - RND * 2) * A(j)
187 X(j) = A(j) + (RND ^ (RND * 10)) * r
222 NEXT IPP
223 FOR j41 = 1 TO 3
224 X(j41) = INT(X(j41))
225 IF X(j41) < 0 THEN 1670
227 IF X(j41) > 1 THEN 1670
235 NEXT j41
305 X(5) = -1 / 4 * (-1 - (-2 * X(1) - X(2) - 5 * X(3))) + X(4)
307 X(6) = -1 / 4 * (-1 - (-2 * X(1) - 5 * X(2) - X(3))) + X(4)
309 X(7) = -1 + X(2) + X(1) + X(3)
337 FOR J44 = 5 TO 7
338 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0
339 NEXT J44
445 POBA = -X(4) + 1000000 * (X(5) + X(6) + X(7))
466 P = POBA
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 7
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -111111 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4), A(5)
1901 PRINT A(6), A(7), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [45]. The complete output through JJJJ = -31999.15000000014 is shown below:
0 0 1 1.000000000000001
0
0 0 -1.000000000000001 -31999.53000000008
1 0 0 .2500000000000002
0
0 0 -.2500000000000002 -31999.31000000011
0 1 0 1.000000000000002
0
0 0 -1.000000000000002 -31999.24000000012
1 0 0 .250000000000002
0
0 0 -.250000000000002 -31999.16000000013
1 0 0 .25 0
0 0 -.25 -31999.15000000014
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and QB64v1000-win [45], the wall-clock time for obtaining the output through
JJJJ= -31999.15000000014 was 3 seconds, not including the time for “Creating .EXE file.” One can compare the computational results above to those on page 269 of Ignizio and Daniels [15].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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