Jsun Yui Wong
The two computer programs listed below are about the following discrete variable nonlinear programming problem in Siwale [58, p. 10, Example 4]:
Minimize (.6224 * X(1) * X(3) * X(4) + 1.7781 * X(2) * X(3) ^ 2 + 3.1661 * X(4) * X(1) ^ 2 + 19.84 * X(3) * X(1) ^ 2)
subject to
X(4) – 240 <= 0
-X(1) + .0193 * X(3) <= 0
-X(2) + .00954 * X(3) <= 0
-3.141592654 * X(3) ^ 2 * X(4) – (4 / 3) * 3.141592654 * X(3) ^ 3 + 1296000 <=0
X(1) and X(2) are integer multiples of .0625
10<= X(i) <=200, i=3, 4.
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
89 FOR J44 = 1 TO 2
92 A(J44) = .0625 + .0625 * FIX(RND * 50)
93 NEXT J44
95 FOR J44 = 3 TO 4
96 A(J44) = 10 + (RND * 190)
97 NEXT J44
128 FOR I = 1 TO 50000
129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
141 FOR IPP = 1 TO FIX(1 + RND * 3)
143 J = 1 + FIX(RND * 4)
154 IF J > 2.5 THEN GOTO 156 ELSE GOTO 163
156 r = (1 – RND * 2) * A(J)
158 X(J) = A(J) + (RND ^ (RND * 15)) * r
161 GOTO 169
163 IF RND < .5 THEN X(J) = A(J) – FIX(1 + RND * 2) * .0625 ELSE X(J) = A(J) + FIX(RND * 2) * .0625
168 REM IF RND < .5 THEN X(J) = A(J) – 1 ELSE X(J) = A(J) + 1
169 NEXT IPP
171 GOTO 192
172 X(1) = INT(X(1))
174 X(2) = INT(X(2))
176 X(3) = INT(X(3))
177 X(4) = INT(X(4))
192 IF X(3) < 10## THEN 1670
193 IF X(4) < 10## THEN 1670
194 IF X(3) > 200## THEN 1670
195 IF X(4) > 200## THEN 1670
198 IF X(4) – 240 > 0## THEN 1670
199 IF -X(1) + .0193 * X(3) > 0## THEN 1670
222 REM
233 IF -X(2) + .00954 * X(3) > 0## THEN 1670
311 IF -3.141592654 * X(3) ^ 2 * X(4) – (4 / 3) * 3.141592654 * X(3) ^ 3 + 1296000 > 0## THEN 1670
447 PDU = -(.6224 * X(1) * X(3) * X(4) + 1.7781 * X(2) * X(3) ^ 2 + 3.1661 * X(4) * X(1) ^ 2 + 19.84 * X(3) * X(1) ^ 2)
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
1557 GOTO 128
1670 NEXT I
1889 IF M < -6059.72 THEN 1999
1924 PRINT A(1), A(2), A(3), A(4), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [64]. The complete output of a single run through JJJJ= -31703 is shown below:
.8125 .4375 42.09840370156255 176.6371149499273
-6059.719436480822 -31747
.8125 .4375 42.09844342169498 176.636622753362
-6059.714599120028 -31703
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 [64], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31703 was 100 seconds, total, including the time for “Creating .EXE file.” One can compare the computational results above with those in the last table of Siwale [58, p. 10, Example 4].
Part 2. What If a Given Inequality Constraint Is Changed to an Equality Constraint?
When a given problem has only several constraints, sometimes it is worthwhile to change an inequality constraint to an equality constraint as follows:
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
89 FOR J44 = 1 TO 2
92 A(J44) = .0625 + .0625 * FIX(RND * 50)
93 NEXT J44
95 FOR J44 = 3 TO 4
96 A(J44) = 10 + (RND * 190)
97 NEXT J44
128 FOR I = 1 TO 50000
129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
141 FOR IPP = 1 TO FIX(1 + RND * 3)
143 J = 1 + FIX(RND * 4)
154 IF J > 2.5 THEN GOTO 156 ELSE GOTO 163
156 r = (1 – RND * 2) * A(J)
158 X(J) = A(J) + (RND ^ (RND * 15)) * r
161 GOTO 169
163 IF RND < .5 THEN X(J) = A(J) – FIX(1 + RND * 2) * .0625 ELSE X(J) = A(J) + FIX(RND * 2) * .0625
168 REM IF RND < .5 THEN X(J) = A(J) – 1 ELSE X(J) = A(J) + 1
169 NEXT IPP
170 X(4) = ((4 / 3) * 3.141592654 * X(3) ^ 3 – 1296000) / (-3.141592654 * X(3) ^ 2)
171 GOTO 192
172 X(1) = INT(X(1))
174 X(2) = INT(X(2))
176 X(3) = INT(X(3))
177 X(4) = INT(X(4))
192 IF X(3) < 10## THEN 1670
193 IF X(4) < 10## THEN 1670
194 IF X(3) > 200## THEN 1670
195 IF X(4) > 200## THEN 1670
198 IF X(4) – 240 > 0## THEN 1670
199 IF -X(1) + .0193 * X(3) > 0## THEN 1670
222 REM
233 IF -X(2) + .00954 * X(3) > 0## THEN 1670
311 REM IF -3.141592654 * X(3) ^ 2 * X(4) – (4 / 3) * 3.141592654 * X(3) ^ 3 + 1296000 > 0## THEN 1670
447 PDU = -(.6224 * X(1) * X(3) * X(4) + 1.7781 * X(2) * X(3) ^ 2 + 3.1661 * X(4) * X(1) ^ 2 + 19.84 * X(3) * X(1) ^ 2)
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
1557 GOTO 128
1670 NEXT I
1889 IF M < -6059.72 THEN 1999
1924 PRINT A(1), A(2), A(3), A(4), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [64]. The complete output of a single run through JJJJ= -31997 is shown below:
.8125 .4375 42.09844559585492 176.6365958120462
-6059.714334337864 -32000
.8125 .4375 42.09844559585492 176.6365958120462
-6059.714334337864 -31997
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 [641], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31997 was 3 seconds, not including the time for “Creating .EXE file” (8 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those in the last table of Siwale [58, p. 10, Example 4].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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