Jsun Yui Wong
The computer program listed below seeks to solve the following nonlinear integer programming problem from Raghav, Ali, and Bari [65, pp. 32-33]:
Minimize
X(9)+X(10)
subject to
(.7893 + (X(5) - 1) * .06) / X(1) + (.4411 + (X(6) - 1) * .02) / X(2) + (.3576 + (X(7) - 1) * .02) / X(3) + (.3973 + (X(8) - 1) * .03) / X(4) - X(9) <= .00655
(.8316 + (X(5) - 1) * .06) / X(1) + (.3358 + (X(6) - 1) * .02) / X(2) + (.1062 + (X(7) - 1) * .01) / X(3) + (.2791 + (X(8) - 1) * .02) / X(4) - X(10) <= .00454
(2.4 +.9/X(5)* X(1) + (3.4 + .8 / X(6)) * X(2) + (4 + 1.25 / X(7)) * X(3) + (4.6 + 1.68 / X(8)) * X(4))) <=5000
2<=X(1)<=1214
2<=X(2)<=822
2<=X(3)<=1028
2<=X(4)<=786
where X(1) through X(4) are integer variables, X(4) through X(8) are continuous variables and >1, and X(9) and X(10) are >=0.
One notes line 234, line 239, and line 233, which are 234 X(9) = (.7893 + (X(5) - 1) * .06) / X(1) + (.4411 + (X(6) - 1) * .02) / X(2) + (.3576 + (X(7) - 1) * .02) / X(3) + (.3973 + (X(8) - 1) * .03) / X(4) - .00655,
239 X(10) = (.8316 + (X(5) - 1) * .06) / X(1) + (.3358 + (X(6) - 1) * .02) / X(2) + (.1062 + (X(7) - 1) * .01) / X(3) + (.2791 + (X(8) - 1) * .02) / X(4) - .00454, and
233 X(5) = .9 * X(1) * (1 / (5000 - 2.4 * X(1) - (3.4 + .8 / X(6)) * X(2) - (4 + 1.25 / X(7)) * X(3) - (4.6 + 1.68 / X(8)) * X(4))), respectively, from the long inequality constraints above, respectively. That is an attempt to find active constraints.
0 DEFDBL A-Z
1 DEFINT K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)
81 FOR JJJJ = -32000 TO 32000
85 RANDOMIZE JJJJ
86 M = -3E+50
111 REM FOR J44 = 1 TO 4
114 A(1) = 2 + FIX(RND * 1213)
115 A(2) = 2 + FIX(RND * 821)
116 A(3) = 2 + FIX(RND * 1027)
117 A(4) = 2 + FIX(RND * 785)
118 FOR J44 = 5 TO 10
119 A(J44) = 1 + (RND * 2)
120 NEXT J44
128 FOR I = 1 TO 50000
129 FOR KKQQ = 1 TO 10
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 5)
153 J = 1 + FIX(RND * 10)
154 IF J < 4.5 THEN GOTO 162 ELSE GOTO 156
155 REM GOTO 162
156 r = (1 - RND * 2) * A(J)
158 X(J) = A(J) + (RND ^ (RND * 15)) * r
161 GOTO 169
162 IF RND < .5 THEN X(J) = A(J) - INT(RND * 15) ELSE X(J) = A(J) + INT(RND * 15)
164 REM IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0
169 NEXT IPP
172 X(1) = INT(X(1))
174 X(2) = INT(X(2))
176 X(3) = INT(X(3))
178 X(4) = INT(X(4))
215 FOR J44 = 1 TO 10
216 IF X(J44) < 0 THEN 1670
219 NEXT J44
220 FOR J44 = 1 TO 4
221 IF X(J44) < 2 THEN 1670
223 NEXT J44
225 IF X(1) > 1214 THEN 1670
226 IF X(2) > 822 THEN 1670
227 IF X(3) > 1028 THEN 1670
229 IF X(4) > 786 THEN 1670
230 FOR J44 = 5 TO 8
231 IF X(J44) <= 1## THEN 1670
232 NEXT J44
233 X(5) = .9 * X(1) * (1 / (5000 - 2.4 * X(1) - (3.4 + .8 / X(6)) * X(2) - (4 + 1.25 / X(7)) * X(3) - (4.6 + 1.68 / X(8)) * X(4)))
234 X(9) = (.7893 + (X(5) - 1) * .06) / X(1) + (.4411 + (X(6) - 1) * .02) / X(2) + (.3576 + (X(7) - 1) * .02) / X(3) + (.3973 + (X(8) - 1) * .03) / X(4) - .00655
239 X(10) = (.8316 + (X(5) - 1) * .06) / X(1) + (.3358 + (X(6) - 1) * .02) / X(2) + (.1062 + (X(7) - 1) * .01) / X(3) + (.2791 + (X(8) - 1) * .02) / X(4) - .00454
326 FOR J44 = 1 TO 10
328 IF X(J44) < 0 THEN 1670
330 NEXT J44
333 FOR J44 = 1 TO 4
334 IF X(J44) < 2 THEN 1670
336 NEXT J44
471 P = -X(9) - X(10)
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 10
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -.00011997 THEN 1999
1933 PRINT A(1), A(2), A(3), A(4)
1936 PRINT A(5), A(6), A(7), A(8), A(9), A(10), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [78]. The complete output of a single run through JJJJ= -31965 is shown below:
527 311 220 247
2.161809919487455 2.08329930063022 2.127126519609546
2.144582945096825 4.343164689580447D-05 7.626989558445075D-05
-1.197015424802552D-04 -31978
529 311 220 246
2.170670355426797 2.08392912799284 2.127769788581231
2.136546310641485 4.449669608975741D-05 7.51952696422561D-05
-1.196919657320135D-04 -31971
529 311 220 246
2.170670355426797 2.08392912799284 2.12776979823337
2.136546310641485 4.449669657008732D-05 7.51952691619263D-05
-1.196919657320136D-04 -31965
.
.
.
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 [78], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31965 was 8 seconds, not including the time for “Creating .EXE file.” One can compare the computational results above with those in Raghav, Ali, and Bari [65, p. 33], where one can see the following numbers: 529, 311, 220, 246, 0.0001196, 0.000044, 0.000075.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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