Jsun Yui Wong
The computer program listed below aims to find all (potentially all) feasible solutions of the following multiple-objective integer nonlinear programming problem from Arora and Arora [9, p. 377]:
Maximize (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10)
maximize (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11)
maximize (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9)
subject to
3 * X(1) - 2 * X(2) + X(3) + X(4) <= 9
X(2) + 3 * X(3) + X(4) <= 7
X(1) + X(2) + X(3) + X(4) <= 12
2 * X(2) + X(3) <= 10
0 <=X(1) <=5
0<= X(2)<=3
0 <=X(3) <=3
0<= X(4)<=1
X(1) through X(4) are general integer variables.
The problem above has been considered also by Drici and Moulai [21, p. 14].
One notes line 128, which is 128 FOR I = 1 TO FIX(1 + RND * 10).
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
87 M = -3E+50
91 w1 = 10# + (RND * 201)
92 w2 = 10# + (RND * 201)
94 w3 = 10# + (RND * 201)
95 A(1) = FIX(RND * 6)
97 A(2) = FIX(RND * 4)
98 A(3) = FIX(RND * 4)
99 A(4) = FIX(RND * 2)
128 FOR I = 1 TO FIX(1 + RND * 10)
129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 4)
153 J = 1 + FIX(RND * 4)
154 GOTO 162
156 r = (1 - RND * 2) * A(J)
158 X(J) = A(J) + (RND ^ (RND * 15)) * r
161 REM GOTO 169
162 IF RND < .5 THEN X(J) = A(J) - 1 ELSE X(J) = A(J) + 1
169 NEXT IPP
171 X(1) = INT(X(1))
173 X(2) = INT(X(2))
174 X(3) = INT(X(3))
175 X(4) = INT(X(4))
177 IF X(1) < 0## THEN 1670
187 IF X(2) < 0## THEN 1670
188 IF X(3) < 0## THEN 1670
189 IF X(4) < 0## THEN 1670
192 IF X(1) > 5 THEN 1670
194 IF X(2) > 3 THEN 1670
196 IF X(3) > 3 THEN 1670
198 IF X(4) > 1 THEN 1670
226 IF 3 * X(1) - 2 * X(2) + X(3) + X(4) > 9 THEN 1670
228 IF X(2) + 3 * X(3) + X(4) > 7 THEN 1670
230 IF X(1) + X(2) + X(3) + X(4) > 12 THEN 1670
232 IF 2 * X(2) + X(3) > 10 THEN 1670
241 IF (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10) < w1 THEN 1670
244 IF (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11) < w2 THEN 1670
255 IF (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9) < w3 THEN 1670
417 PDU = (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10)
466 P = PDU
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 4
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1522 g01star = (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10)
1524 g02star = (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11)
1526 g03star = (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9)
1557 GOTO 128
1670 NEXT I
1889 IF M < -99999999999 THEN 1999
1922 PRINT g01star, g02star, g03star
1928 PRINT A(1), A(2), A(3), A(4), JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [73]. Including non-dominated solutions and dominated solutions, the complete output of a single run through JJJJ= -31800 is shown below:
180 165 108
3 0 0 0 -31993
171 160 99
3 1 0 0 -31981
165 169 110
2 0 1 0 -31966
171 187 90
4 2 1 0 -31961
165 196 105
3 1 2 0 -31957
165 169 110
2 0 1 0 -31955
150 156 78
1 1 1 0 -31946
156 180 117
2 0 2 0 -31945
153 165 60
2 2 1 0 -31926
180 165 108
3 0 0 0 -31914
160 143 90
1 0 0 0 -31913
153 165 105
3 2 0 1 -31907
130 132 91
0 1 0 1 -31876
143 168 75
1 1 2 0 -31875
156 180 117
2 0 2 0 -31828
162 150 88
2 1 0 0 -31821
130 170 150
2 0 2 1 -31815
160 158 78
3 2 0 0 -31808
170 154 99
2 0 0 0 -31800
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 [73], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31800 was 2 seconds, not including the time for “Creating .EXE file" (7 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Drici and Moulai [21, p. 14, Table 12] and with those in Arora and Arora [9, p. 380].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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