The computer program listed below seeks to solve the following speed-reducer weight minimization problem in Li et al. [16, p. 932], Kashan [13, p. 68], and Kashan [12, p. 1790].
Minimize
.7854 * X(1) * X(2) ^ 2 * (3.3333 * X(3) ^ 2 + 14.9334 * X(3) - 43.0934) - 1.508 * X(1) * (X(6) ^ 2 + X(7) ^ 2) + 7.4777 * (X(6) ^ 3 + X(7) ^ 3) + .7854 * (X(4) * X(6) ^ 2 + X(5) * X(7) ^ 2)
subject to
27 / (X(1) * X(2) ^ 2 * X(3)) <=1,
397.5 / (X(1) * X(2) ^ 2 * X(3) ^ 2) <=1,
1.93 * X(4) ^ 3 / (X(2) * X(3) * X(6) ^ 4) <=1,
1.93 * X(5) ^ 3 / (X(2) * X(3) * X(7) ^ 4) <=1,
(1 / (110 * X(6) ^ 3)) * (((((745 * X(4)) / (X(2) * X(3))) ^ 2 + 16900000))) ^ .5 <=1,
(1 / (85 * X(7) ^ 3)) * (((((745 * X(5)) / (X(2) * X(3))) ^ 2 + 157500000))) ^ .5 <=1,
X(2) * X(3) / 40 <=1,
5 * X(2) / X(1) <=1,
X(1) / (12 * X(2)) <=1,
(1.5 * X(6) + 1.9) / X(4) <=1,
(1.1 * X(7) + 1.9) / X(5) <=1,
2.6 <= X(1) <= 3.6,
.7 <= X(2) <= .8 ,
17 <= X(3) <= 28,
7.3 <= X(4) <= 8.3,
7.3 <= X(5) <= 8.3,
2.9 <= X(6) <= 3.9,
5<= X(7) <= 5.5,
where X(3), the number of teeth, is an integer, and the other six variables are continuous.
X(8) through X(18) below are slack variables.
One notes line 193, which is 193 X(3) = INT(X(3)), where X(3) is the number of teeth, Kashan [13, p. 68].
0 DEFDBL A-Z
2 DEFINT K
3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32000 STEP .01
14 RANDOMIZE JJJJ
16 M = -1D+37
87 A(1) = 2.6 + FIX(RND * 1000) * .001
89 A(2) = .7 + FIX(RND * 1000) * .0001
91 A(3) = 17 + FIX(RND * 1000) * .011
94 A(4) = 7.3 + FIX(RND * 1000) * .001
96 A(5) = 7.3 + FIX(RND * 1000) * .001
98 A(6) = 2.9 + FIX(RND * 1000) * .001
99 A(7) = 5 + FIX(RND * 1000) * .0005
128 FOR I = 1 TO 100000
129 FOR KKQQ = 1 TO 7
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 7))
151 J = 1 + FIX(RND * 7)
155 IF J = 1 GOTO 167 ELSE IF J = 2 THEN GOTO 169 ELSE IF J = 3 THEN GOTO 172 ELSE IF J = 4 THEN GOTO 174 ELSE IF J = 5 THEN GOTO 176 ELSE IF J = 6 THEN GOTO 178 ELSE IF J = 7 THEN GOTO 180
167 X(1) = 2.6 + FIX(RND * 1000) * .001
168 IF RND < .5 THEN 170
169 X(2) = .7 + FIX(RND * 1000) * .0001
170 IF RND < .5 THEN 173
172 X(3) = 17 + FIX(RND * 1000) * .011
173 IF RND < .5 THEN 175
174 X(4) = 7.3 + FIX(RND * 1000) * .001
175 IF RND < .5 THEN 177
176 X(5) = 7.3 + FIX(RND * 1000) * .001
177 IF RND < .5 THEN 179
178 X(6) = 2.9 + FIX(RND * 1000) * .001
179 IF RND < .5 THEN 181
180 X(7) = 5 + FIX(RND * 1000) * .0005
181 REM
191 NEXT IPP
193 X(3) = INT(X(3))
201 IF X(1) < 2.6 THEN 1670
203 IF X(1) > 3.6 THEN 1670
211 IF X(2) < .7 THEN 1670
213 IF X(2) > .8 THEN 1670
231 IF X(3) < 17 THEN 1670
233 IF X(3) > 28 THEN 1670
235 IF X(4) < 7.3 THEN 1670
237 IF X(4) > 8.3 THEN 1670
239 IF X(5) < 7.3 THEN 1670
241 IF X(5) > 8.3 THEN 1670
247 IF X(6) < 2.9 THEN 1670
249 IF X(6) > 3.9 THEN 1670
251 IF X(7) < 5 THEN 1670
253 IF X(7) > 5.5 THEN 1670
305 X(8) = 1 - 27 / (X(1) * X(2) ^ 2 * X(3))
306 X(9) = 1 - 397.5 / (X(1) * X(2) ^ 2 * X(3) ^ 2)
307 X(10) = 1 - 1.93 * X(4) ^ 3 / (X(2) * X(3) * X(6) ^ 4)
309 X(11) = 1 - 1.93 * X(5) ^ 3 / (X(2) * X(3) * X(7) ^ 4)
320 X(12) = 1 - (1 / (110 * X(6) ^ 3)) * (((((745 * X(4)) / (X(2) * X(3))) ^ 2 + 16900000))) ^ .5
322 X(13) = 1 - (1 / (85 * X(7) ^ 3)) * (((((745 * X(5)) / (X(2) * X(3))) ^ 2 + 157500000))) ^ .5
323 X(14) = 1 - X(2) * X(3) / 40
325 X(15) = 1 - 5 * X(2) / X(1)
327 X(16) = 1 - X(1) / (12 * X(2))
329 X(17) = 1 - (1.5 * X(6) + 1.9) / X(4)
330 X(18) = 1 - (1.1 * X(7) + 1.9) / X(5)
335 FOR J99 = 8 TO 18
340 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0
341 NEXT J99
359 POBA = -.7854 * X(1) * X(2) ^ 2 * (3.3333 * X(3) ^ 2 + 14.9334 * X(3) - 43.0934) + 1.508 * X(1) * (X(6) ^ 2 + X(7) ^ 2) - 7.4777 * (X(6) ^ 3 + X(7) ^ 3) - .7854 * (X(4) * X(6) ^ 2 + X(5) * X(7) ^ 2) + 1000000 * (X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15) + X(16) + X(17) + X(18))
466 P = POBA
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 18
1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 REM GOTO 128
1670 NEXT I
1889 IF M < -2996 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4), A(5)
1903 PRINT A(6), A(7), A(8), A(9), A(10)
1950 PRINT A(11), A(12), A(13), A(14), A(15), A(16), A(17), A(18), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [33]. The complete output through JJJJ = -31999.74000000004 is shown below:
3.5 .7 17 7.301 7.718
3.351 5.287 0 0 0
0 0 0 0 0
0 0 0 -2994.958416306808
-31999.99
3.5 .7 17 7.301 7.716
3.351 5.287 0 0 0
0 0 0 0 0
0 0 0 -2994.914508725582
-31999.98
3.5 .7 17 7.301 7.716
3.351 5.287 0 0 0
0 0 0 0 0
0 0 0 -2994.914508725582
-31999.78000000004
3.5 .7 17 7.301 7.718
3.352 5.287 0 0 0
0 0 0 0 0
0 0 0 -2995.213455221353
-31999.74000000004
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 [33], the wall-clock time for obtaining the output through
JJJJ=-31999.74000000004 was 15 seconds, not including the time for creating the .EXE file. One can compare the computational results here with those in Table 6 of Li et al. [16, p. 933].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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