Jsun Yui Wong
Using the product operator, Zimmermann has converted his Example 2.1 problem with two goals [49, p. 46] to the following problem, Zimmermann [49, p. 52]:
Maximize (1 / 238) * (-2 * X(1) ^ 2 + 3 * X(1) * X(2) + 13 * X(1) - 11 * X(2) + 2 * X(2) ^ 2 - 21)
subject to
- X(1) + 3 * X(2)<=21,
X(1) + 3 * X(2)<=27,
4 * X(1) + 3 * X(2)<=45,
3 * X(1) + X(2)<=30,
X(1), X(2)>=0.
The computer program listed below seeks to solve the model immediately above. X(3) through X(6) below are slack variables.
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
51 FOR J40 = 1 TO 2
55 A(J40) = RND * 2
56 NEXT J40
128 FOR I = 1 TO 1000
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))
181 j = 1 + FIX(RND * 2)
183 r = (1 - RND * 2) * A(j)
187 X(j) = A(j) + (RND ^ (RND * 10)) * r
222 NEXT IPP
246 FOR J47 = 1 TO 2
247 IF X(J47) < 0 THEN 1670
249 NEXT J47
318 X(3) = 21 + X(1) - 3 * X(2)
321 X(4) = 27 - X(1) - 3 * X(2)
322 X(5) = 45 - 4 * X(1) - 3 * X(2)
323 X(6) = 30 - 3 * X(1) - X(2)
337 FOR j44 = 3 TO 6
338 IF X(j44) < 0 THEN X(j44) = X(j44) ELSE X(j44) = 0
339 NEXT j44
433 POBA = (1 / 238) * (-2 * X(1) ^ 2 + 3 * X(1) * X(2) + 13 * X(1) - 11 * X(2) + 2 * X(2) ^ 2 - 21) + 1000000 * (X(3) + X(4) + X(5) + X(6))
466 P = POBA
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 6
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < .5556 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4), A(5)
1901 PRINT A(6), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [47]. The complete output through JJJJ = -31999.87000000002 is shown below:
5.630904744562785 7.123031751812368 0
0 0
0 .555616548152143 -31999.99
5.683612687394603 7.105462437530837 0
0 0
0 .5556691346397322 -31999.98
5.730704803906347 7.089765065363786 0
0 0
0 .555661265347894 -31999.87000000002
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 [47], the wall-clock time for obtaining the output through
JJJJ= -31999.87000000002 was 2 or 3 seconds, not including the time for “Creating .EXE file.” One can compare the computational results above to those on page 52 of Zimmermann [49].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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