A Computer Program for Optimizing a Widely-Known Design of a Speed Reducer

Jsun Yui Wong

The computer program listed below attempts to solve the following nonlinear programming formulation in Tsai, Lin, and Peng [, pp. 177-178]:

Minimize

-1 * ( -.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) - 7.477 * (X(6) ^ 3 + X(7) ^ 3) - .7854 * (X(4) * X(6) ^ 2 + X(5) * X(7) ^ 2) )

subject to

27 * X(1) ^ -1 * X(2) ^ -2 * X(3) ^ -1 <= 1

397.5 * X(1) ^ -1 * X(2) ^ -2 * X(3) ^ -2 <= 1

1.93 * X(2) ^ -1 * X(3) ^ -1 * X(4) ^ 3 * X(6) ^ -4 <= 1

1.93 * X(2) ^ -1 * X(3) ^ -1 * X(5) ^ 3 * X(7) ^ -4 <= 1

(((745 * X(4)) / (X(2) * X(3)) ^ 2 + 16.9 * 10 ^ 6)) ^ .5 / (.1 * X(6) ^ 3) <= 1100

(((745 * X(5)) / (X(2) * X(3)) ^ 2 + 157.5 * 10 ^ 6)) ^ .5 / (.1 * X(7) ^ 3) <= 850

X(2) * X(3) <= 40

X(1) / X(2) >= 5

X(1) / X(2) <= 12

(1.5 * X(6) + 1.9) * X(4) ^ -1 <= 1

(1.1 * X(7) + 1.9) * X(5) ^ -1 <= 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

5.0 <= X(7) <= 5.5.

It is expected that at least one of the eleven constraints shown above is active. Therefore

line 301 below tries X(1)=5*X(2), 301 X(1) = 5 * X(2) .

0 DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), G(128), J44(222), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), PN(22), NN(22)

85 FOR JJJJ = -32000 TO 32000

87 RANDOMIZE JJJJ

88 M = -3D+30

111 A(1) = 2.6 + RND * 1

112 A(2) = .7 + RND * .1

113 A(3) = 17 + RND * 11

114 A(4) = 7.3 + RND * 1

115 A(5) = 7.3 + RND * 1

116 A(6) = 2.9 + RND * 1

117 A(7) = 5 + RND * .5

128 FOR i = 1 TO 500000

129 FOR KKQQ = 1 TO 7

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

139 FOR IPP = 1 TO FIX(1 + RND * 7)

141 B = 1 + FIX(RND * 7)

160 r = (1 - RND * 2) * A(B)

164 X(B) = A(B) + (RND ^ (RND * 15)) * r

280 NEXT IPP

284 X(3) = INT(X(3))

288 X(1) = 2.6 + .1 * FIX(RND * 11)

291 X(2) = .7 + .1 * FIX(RND * 2)

292 X(4) = 7.3 + .1 * FIX(RND * 11)

294 X(5) = 7.3 + .1 * FIX(RND * 11)

296 X(6) = 2.9 + .01 * FIX(RND * 101)

298 X(7) = 5 + .01 * FIX(RND * 51)

301 X(1) = 5 * X(2)

302 IF 27 * X(1) ^ -1 * X(2) ^ -2 * X(3) ^ -1 > 1 THEN 1670

303 IF 397.5 * X(1) ^ -1 * X(2) ^ -2 * X(3) ^ -2 > 1 THEN 1670

304 IF 1.93 * X(2) ^ -1 * X(3) ^ -1 * X(4) ^ 3 * X(6) ^ -4 > 1 THEN 1670

305 IF 1.93 * X(2) ^ -1 * X(3) ^ -1 * X(5) ^ 3 * X(7) ^ -4 > 1 THEN 1670

306 IF (((745 * X(4)) / (X(2) * X(3)) ^ 2 + 16.9 * 10 ^ 6)) ^ .5 / (.1 * X(6) ^ 3) > 1100 THEN 1670

307 IF (((745 * X(5)) / (X(2) * X(3)) ^ 2 + 157.5 * 10 ^ 6)) ^ .5 / (.1 * X(7) ^ 3) > 850 THEN 1670

308 IF X(2) * X(3) > 40 THEN 1670

310 IF X(1) / X(2) > 12 THEN 1670

311 IF (1.5 * X(6) + 1.9) * X(4) ^ -1 > 1 THEN 1670

312 IF (1.1 * X(7) + 1.9) * X(5) ^ -1 > 1 THEN 1670

324 IF X(1) < 2.6 THEN 1670

325 IF X(1) > 3.6 THEN 1670

343 IF X(2) < .7 THEN 1670

345 IF X(2) > .8 THEN 1670

348 IF X(3) < 17 THEN 1670

349 IF X(3) > 28 THEN 1670

366 IF X(4) < 7.3 THEN 1670

367 IF X(4) > 8.3 THEN 1670

368 IF X(5) < 7.3 THEN 1670

369 IF X(5) > 8.3 THEN 1670

370 IF X(6) < 2.9 THEN 1670

371 IF X(6) > 3.9 THEN 1670

372 IF X(7) < 5 THEN 1670

373 IF X(7) > 5.5 THEN 1670

445 PD1 = -.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) - 7.477 * (X(6) ^ 3 + X(7) ^ 3) - .7854 * (X(4) * X(6) ^ 2 + X(5) * X(7) ^ 2)

469 P = PD1

1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 7

1455 A(KLX) = X(KLX)

1459 NEXT KLX

1670 NEXT i

1777 IF M < -3009 THEN 1999

1888 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102]. Its complete output of one run through JJJJ= -31997 is shown below:

**3.5 .7 17 7.4 7.8 ****3.35 5.29 -2999.1555527941 -32000**

3.5 .7 17 7.5 7.8

3.36 5.29 -3002.6029075311 -31999

**3.5 .7 17 7.4 7.8 ****3.35 5.29 -2999.1555527941 -31998**

**3.5 .7 17 7.4 7.8 ****3.35 5.29 -2999.1555527941 -31997**

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. By using the following computer system and QB64v1000-win [102], the wall-clock time **(not CPU time)** for obtaining the output through JJJJ = -31997 was 10 seconds, counting from "Starting program...". One can compare the computational results above with those in Tsai, Lin, and Peng [91, Table 6 on p. 179].

The computational results presented above were obtained from the following computer system:

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System.

**Acknowledgement**

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

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