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].
Whereas line 241 of the earlier edition is 241 IF (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10) < w1 THEN 1670, here line 241 is 241 REM IF (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10) < w1 THEN 1670.
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 REM 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= -31804 is shown below:
180 165 108
3 0 0 0 -31994
160 168 130
3 1 0 1 -31993.
140 169 85
2 2 1 1 -31992
130 170 150
2 0 2 1 -31982
150 156 117
2 1 0 1 -31981
150 182 102
3 2 1 1 -31968
153 192 76
4 3 1 1 -31966
144 165 104
1 0 2 0 -31950
154 156 99
1 0 1 0 -31943
162 176 75
3 2 1 0 -31942
154 182 135
3 1 1 1 -31941
153 165 105
3 2 0 1 -31938
170 180 104
3 1 1 0 -31933
162 176 120
4 2 0 1 -31924
144 165 104
1 0 2 0 -31918
104 136 120
0 0 2 1 -31917
117 153 135
1 0 2 1 -31914
160 143 90
1 0 0 0 -31908
165 196 105
3 1 2 0 -31907
168 162 91
4 2 0 0 -31897
153 165 60
2 2 1 0 -31886
153 192 76
4 3 1 1 -31882
150 156 117
2 1 0 1 -31877
153 165 105
3 2 0 1 -31875
165 196 105
3 1 2 0 -31872
156 180 117
2 0 2 0 -31851
154 156 99
1 0 1 0 -31848
153 165 105
3 2 0 1 -31838
153 165 105
3 2 0 1 -31837
170 154 99
2 0 0 0 -31830
171 187 90
4 2 1 0 -31829
150 182 102
3 2 1 1 -31817
170 154 99
2 0 0 0 -31815
143 143 121
1 0 0 1 -31814
168 162 91
4 2 0 0 -31811
144 180 57
3 3 1 1 -31810
156 180 117
2 0 2 0 -31807
180 165 108
3 0 0 0 -31804
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 = -31804 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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