Jsun Yui Wong
1. First Stage
The computer program listed below seeks to solve the following nonlinear integer programming problem:
Minimize
443.09704013 / X(1)+ 270.715638 / X(2) + 204.9419482 / X(3) + 240.6056486 / X(4) + 10.7780449 / X(5) + 2.849638294 / X(6) + 3.415699137 / X(7) + 5.070828722 / X(8)
subject to
2.4 * X(1) + 3.4 * X(2) + 4 * X(3) + 4.6 * X(4) + 3 * X(5) + 4 * X(6) + 5 * X(7) + 6 * X(8) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4 + X(6) / 4 + X(7) / 4 + X(8) / 4) <= 5000
2<=X(1)<=1214
2<=X(2)<=822
2<=X(3)<=1028
2<=X(4)<=786
2<=X(5)<=40
2<=X(6)<=40
2<=X(7)<=40
2<=X(8)<=40
where X(1) through X(8) are integer variables.
The problem above is based on the formulation on p. 2459 in Varshney and Mradula [90, p. 2459, Example 1].
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
118 FOR J44 = 1 TO 8
119 A(J44) = 2 + (RND * 8)
120 NEXT J44
128 FOR I = 1 TO 30000
129 FOR KKQQ = 1 TO 8
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 5)
153 J = 1 + FIX(RND * 8)
155 GOTO 162
156 r = (1 - RND * 2) * A(J)
158 X(J) = A(J) + (RND ^ (RND * 15)) * r
161 GOTO 169
162 REM IF RND < .5 THEN X(J) = A(J) - INT(RND * 15) ELSE X(J) = A(J) + INT(RND * 15)
163 IF RND < .5 THEN X(J) = A(J) - INT(RND * 5) ELSE X(J) = A(J) + INT(RND * 5)
164 REM IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0
169 NEXT IPP
215 FOR J44 = 1 TO 8
216 X(J44) = INT(X(J44))
217 REM X(4) = INT(X(4))
218 IF X(J44) < 2 THEN 1670
219 NEXT J44
225 IF X(1) > 1214 THEN 1670
226 IF X(2) > 822 THEN 1670
227 IF X(3) > 1028 THEN 1670
228 IF X(4) > 786 THEN 1670
229 IF X(5) > 40 THEN 1670
230 IF X(6) > 40 THEN 1670
233 IF X(7) > 40 THEN 1670
234 IF X(8) > 40 THEN 1670
248 IF 2.4 * X(1) + 3.4 * X(2) + 4 * X(3) + 4.6 * X(4) + 3 * X(5) + 4 * X(6) + 5 * X(7) + 6 * X(8) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4 + X(6) / 4 + X(7) / 4 + X(8) / 4) > 5000 THEN 1670
326 FOR J44 = 1 TO 8
328 IF X(J44) < 2 THEN 1670
330 NEXT J44
478 P = -443.09704013 / X(1) - 270.715638 / X(2) - 204.9419482 / X(3) - 240.6056486 / X(4) - 10.7780449 / X(5) - 2.849638294 / X(6) - 3.415699137 / X(7) - 5.070828722 / X(8)
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 8
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -999999999 THEN 1999
1933 PRINT A(1), A(2), A(3), A(4), A(5)
1936 PRINT A(6), A(7), A(8), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [95]. The complete output of a single run through JJJJ= -31995 is shown below:
50 38 33 34 7
4 4 5 -33.3932561317882
-32000
50 37 32 35 8
4 4 5 -33.38521836622201
-31999
49 39 32 34 8
4 4 5 -33.39305167216097
-31998
50 37 32 35 8
4 4 5 -33.38521836622201
-31997
50 37 32 35 8
4 4 5 -33.38521836622201
-31996
50 37 32 35 8
4 4 5 -33.38521836622201
-31995
.
.
.
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 [95], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31995 was 4 seconds, not including the time for “Creating .EXE file.” One can compare the computational results above with those in Varshney and Mradula [90, p. 2459], where one can see the following numbers: 50, 37, 32, 35, 8, 4, 4, 5, 33.3852.
2. Second Stage
The computer program listed below seeks to solve the following nonlinear mixed-integer programming problem:
Minimize
746.921876 / X(1) + 329.7093584 / X(2) + 97.34568757 / X(3) + 270.4706345 / X(4) + 18.16836996 / X(5) + 3.470624826 / X(6) + 1.622428126 / X(7) + 5.700241329 / X(8)+ X(9)
subject to
443.09704013 / X(1) + 270.715638 / X(2) + 204.9419482 / X(3) + 240.6056486 / X(4) + 10.7780449 / X(5) + 2.849638294 / X(6) + 3.415699137 / X(7) + 5.070828722 / X(8) -X(9) <= 33.38521836622201, which comes from stage 1
2.4 * X(1) + 3.4 * X(2) + 4 * X(3) + 4.6 * X(4) + 3 * X(5) + 4 * X(6) + 5 * X(7) + 6 * X(8) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4 + X(6) / 4 + X(7) / 4 + X(8) / 4) <= 5000
2<=X(1)<=1214
2<=X(2)<=822
2<=X(3)<=1028
2<=X(4)<=786
2<=X(5)<=40
2<=X(6)<=40
2<=X(7)<=40
2<=X(8)<=40
X(9)>=0
where X(1) through X(8) are integer variables, and X(9) is a continuous variable.
The problem above is based on the formulation on p. 2460 in Varshney and Mradula [90, p. 2460, Example 1].
One notes line 238, which is 238 X(9) = -(-443.09704013 / X(1) - 270.715638 / X(2) - 204.9419482 / X(3) - 240.6056486 / X(4) - 10.7780449 / X(5) - 2.849638294 / X(6) - 3.415699137 / X(7) - 5.070828722 / X(8)) - 33.38521836622201 from the first long inequality constraint above. That is a search for an active constraint for the goal of optimization; see Bernau [11].
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
118 FOR J44 = 1 TO 8
119 A(J44) = 2 + (RND * 8)
120 NEXT J44
123 A(9) = RND
128 FOR I = 1 TO 30000
129 FOR KKQQ = 1 TO 9
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 5)
153 J = 1 + FIX(RND * 9)
154 IF J > 8.5 THEN GOTO 156 ELSE GOTO 163
155 REM GOTO 162
156 r = (1 - RND * 2) * A(J)
158 X(J) = A(J) + (RND ^ (RND * 15)) * r
161 GOTO 169
162 REM IF RND < .5 THEN X(J) = A(J) - INT(RND * 15) ELSE X(J) = A(J) + INT(RND * 15)
163 IF RND < .5 THEN X(J) = A(J) - INT(RND * 5) ELSE X(J) = A(J) + INT(RND * 5)
164 REM IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0
169 NEXT IPP
215 FOR J44 = 1 TO 8
216 X(J44) = INT(X(J44))
217 REM X(4) = INT(X(4))
218 IF X(J44) < 2 THEN 1670
219 NEXT J44
222 IF X(9) < 0## THEN 1670
225 IF X(1) > 1214 THEN 1670
226 IF X(2) > 822 THEN 1670
227 IF X(3) > 1028 THEN 1670
228 IF X(4) > 786 THEN 1670
229 IF X(5) > 40 THEN 1670
230 IF X(6) > 40 THEN 1670
233 IF X(7) > 40 THEN 1670
234 IF X(8) > 40 THEN 1670
238 X(9) = -(-443.09704013 / X(1) - 270.715638 / X(2) - 204.9419482 / X(3) - 240.6056486 / X(4) - 10.7780449 / X(5) - 2.849638294 / X(6) - 3.415699137 / X(7) - 5.070828722 / X(8)) - 33.38521836622201
248 IF 2.4 * X(1) + 3.4 * X(2) + 4 * X(3) + 4.6 * X(4) + 3 * X(5) + 4 * X(6) + 5 * X(7) + 6 * X(8) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4 + X(6) / 4 + X(7) / 4 + X(8) / 4) > 5000 THEN 1670
326 FOR J44 = 1 TO 8
328 IF X(J44) < 2 THEN 1670
330 NEXT J44
478 P = -746.921876 / X(1) - 329.7093584 / X(2) - 97.34568757 / X(3) - 270.4706345 / X(4) - 18.16836996 / X(5) - 3.470624826 / X(6) - 1.622428126 / X(7) - 5.700241329 / X(8) - X(9)
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 9
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1533 dol = 2.4 * X(1) + 3.4 * X(2) + 4 * X(3) + 4.6 * X(4) + 3 * X(5) + 4 * X(6) + 5 * X(7) + 6 * X(8) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4 + X(6) / 4 + X(7) / 4 + X(8) / 4)
1557 GOTO 128
1670 NEXT I
1889 IF M < -999999999 THEN 1999
1933 PRINT A(1), A(2), A(3), A(4)
1936 PRINT A(5), A(6), A(7), A(8), A(9), M, JJJJ, dol
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [95]. The complete output of a single run through JJJJ= -31995 is shown below:
54 38 27 35
8 4 4 5 .3370234572013086
-38.86295811196344 -32000 4992.8
54 38 27 35
8 4 4 5 .3370234572013086
-38.86295811196344 -31999 4992.8
54 38 27 35
8 4 4 5 .3370234572013086
-38.86295811196344 -31998 4992.8
54 38 27 35
8 4 4 5 .3370234572013086
-38.86295811196344 -31997 4992.8
54 38 27 35
8 4 4 5 .3370234572013086
-38.86295811196344 -31996 4992.8
54 38 27 35
8 4 4 5 .3370234572013086
-38.86295811196344 -31995 4992.8
.
.
.
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 [95], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31995 was 5 seconds, not including the time for “Creating .EXE file.” One can compare the computational results above with those in Varshney and Mradula [90, p. 2460; Table 4, p. 2465], where one can see the following numbers: 54, 38, 27, 35, 8, 4, 4, 5, 0.3516, 38.8776, 4992.8.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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