Jsun Yui Wong
The computer program listed below seeks to solve the following 2-objective integer nonlinear programming problem from Sharma, Dahiya, and Verma [54, p. 1924, Example 1]:
Miniimize – (-2 * X(1) ^ 2 – 4 * X(1) * X(2) + X(1) – X(2)) / (2 * X(1) * X(2) + X(2) ^ 2 + X(1))
minimize (X(1) ^ 2 + X(2) ^ 2 + 2 * X(1) * X(2)) / (X(1) ^ 2 + X(1) * X(2) + X(2))
subject to
X(1) >= 0 and integer
X(2) >= 0 and integer
3 * X(1) + 2 * X(2) >= 6
4 * X(1) + 5 * X(2) <= 20.
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
88 epsi = RND * 10
92 A(1) = (RND * 14)
93 A(2) = (RND * 14)
128 FOR I = 1 TO 3000
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 2)
153 J = 1 + FIX(RND * 2)
154 REM GOTO 162
156 r = (1 – RND * 2) * A(J)
158 X(J) = A(J) + (RND ^ (RND * 15)) * r
161 REM GOTO 169
162 REM IF RND < .5 THEN X(J) = A(J) – 1 ELSE X(J) = A(J) + 1
169 NEXT IPP
172 X(1) = INT(X(1))
174 X(2) = INT(X(2))
188 IF X(1) < 0## THEN 1670
189 IF X(2) < 0## THEN 1670
226 IF 3 * X(1) + 2 * X(2) < 6 THEN 1670
227 IF 4 * X(1) + 5 * X(2) > 20 THEN 1670
228 IF X(1) ^ 2 + X(1) * X(2) + X(2) = 0## THEN 1670
229 IF (X(1) ^ 2 + X(2) ^ 2 + 2 * X(1) * X(2)) / (X(1) ^ 2 + X(1) * X(2) + X(2)) > epsi THEN 1670
431 PDU = (-2 * X(1) ^ 2 – 4 * X(1) * X(2) + X(1) – X(2)) / (2 * X(1) * X(2) + X(2) ^ 2 + X(1))
466 P = PDU
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 2
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1527 gg01star = (-2 * X(1) ^ 2 – 4 * X(1) * X(2) + X(1) – X(2)) / (2 * X(1) * X(2) + X(2) ^ 2 + X(1))
1529 gg02star = (X(1) ^ 2 + X(2) ^ 2 + 2 * X(1) * X(2)) / (X(1) ^ 2 + X(1) * X(2) + X(2))
1557 GOTO 128
1670 NEXT I
1889 IF M < -99999999999 THEN 1999
1924 PRINT -gg01star, gg02star, epsi
1956 PRINT A(1), A(2), -M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [60]. The output of a single run through JJJJ= -31980 is summarized below:
.25 4 9.554092884063721
0 4 .25 -31999
3 1 3.04885506629944
2 0 3 -31995
2.142857142857143 1.285714285714286 4.794933199882507
2 1 2.142857142857143 -31994
.3333333333333333 3 3.203887939453125
0 3 .3333333333333333 -31993
2.142857142857143 1.285714285714286 5.443581342697144
2 1 2.142857142857143 -31992
3 1 9.825533032417297
2 0 3 -31990
.25 4 8.194036483764648
0 4 .25 -31989
2.142857142857143 1.285714285714286 8.698806166648865
2 1 2.142857142857143 -31988
3 1 3.1596546649933
2 0 3 -31987
2.142857142857143 1.285714285714286 4.93649301528931
2 1 2.142857142857143 -31985
.25 4 8.634269833564758
0 4 .25 -31983
3 1 9.953094720840454
2 0 3 -31982
.25 4 4.089516997337341
0 4 .25 -31981
.3333333333333333 3 3.526153564453125
0 3 .3333333333333333 -31980
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 [60], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31980 was 1 second, not including the time for “Creating .EXE file” (8 seconds, total, including the time for “Creating .EXE file”). The (-1 3) shown above is a dominated point. One can compare the computational results above with those in Sharma, Dahiya, and Verma [54, pp. 1924-1926, Example 1].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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