Direct approach to solve signomial geometric programming problems: another illustration from the literature

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following heat exchange design problem in Lin and Tsai [52, p. 24, SGP 3]:

Minimize

X(1) + X(2) + X(3)

subject to

833.33252 * X(1) ^ -1 * X(4) * X(6) ^ -1 + 100 * X(6) ^ -1 - 83333.333 * X(1) ^ -1 * X(6) ^ -1 <= 1

1250 * X(2) ^ -1 * X(5) * X(7) ^ -1 + X(4) * X(7) ^ -1 - 1250 * X(2) ^ -1 * X(4) * X(7) ^ -1 <= 1

1250000 * X(3) ^ -1 * X(8) ^ -1 + X(5) * X(8) ^ -1 - 2500 * X(3) ^ -1 * X(5) * X(8) ^ -1 <= 1

.0025 * X(4) + .0025 * X(6) <= 1

-.0025 * X(4) + .0025 * X(5) + .0025 * X(7) <= 1

.01 * X(8) - .01 * X(5) <= 1

(100, 1000, 1000, 10, 10, 10, 10, 10)

<= ( X(1), X(2), X(3), X(4), X(5), X(6), X(7), X(8) ) <=

(10000, 10000, 10000, 1000, 1000, 1000, 1000, 1000).

0 DEFDBL A-Z

1 REM DEFINT I, J

2 DIM B(99), N(99), A(200), H(99), L(99), U(99), X(200), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

79 FOR JJJJ = -32000 TO 32000 STEP .01

89 RANDOMIZE JJJJ

90 M = -3D+30

91 A(1) = INT(100 * (100 + RND * 9900 + .000001)) / 100

92 A(2) = INT(100 * (1000 + RND * 9000 + .000001)) / 100

93 A(3) = INT(100 * (1000 + RND * 9000 + .000001)) / 100

97 FOR J44 = 4 TO 8

101 A(J44) = INT(100 * (10 + RND * 990 + .000001)) / 100

102 NEXT J44

124 FOR I = 1 TO 65000

129 FOR KKQQ = 1 TO 8

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

133 FOR IPP = 1 TO FIX(1 + RND * 5)

140 B = 1 + FIX(RND * 8)

144 IF RND < .5 THEN 160 ELSE GOTO 166

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

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

165 GOTO 168

166 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) * .01 ELSE X(B) = A(B) + FIX(RND * 2) * .01

168 NEXT IPP

170 IF X(1) < 100 THEN 1670

171 IF X(1) > 10000 THEN 1670

173 IF X(2) < 1000 THEN 1670

174 IF X(2) > 10000 THEN 1670

176 IF X(3) < 1000 THEN 1670

177 IF X(3) > 10000 THEN 1670

208 FOR J44 = 4 TO 8

210 IF X(J44) < 10 THEN 1670

211 IF X(J44) > 1000 THEN 1670

212 NEXT J44

346 X(4) = (1 - .0025 * X(6)) / (.0025)

349 X(8) = (1 + .01 * X(5)) / .01

373 IF -.0025 * X(4) + .0025 * X(5) + .0025 * X(7) > 1 THEN 1670

376 IF 833.33252 * X(1) ^ -1 * X(4) * X(6) ^ -1 + 100 * X(6) ^ -1 - 83333.333 * X(1) ^ -1 * X(6) ^ -1 > 1 THEN 1670

381 IF 1250000 * X(3) ^ -1 * X(8) ^ -1 + X(5) * X(8) ^ -1 - 2500 * X(3) ^ -1 * X(5) * X(8) ^ -1 > 1 THEN 1670

383 IF 1250 * X(2) ^ -1 * X(5) * X(7) ^ -1 + X(4) * X(7) ^ -1 - 1250 * X(2) ^ -1 * X(4) * X(7) ^ -1 > 1 THEN 1670

391 IF X(1) < 100 THEN 1670

392 IF X(1) > 10000 THEN 1670

393 IF X(2) < 1000 THEN 1670

394 IF X(2) > 10000 THEN 1670

395 IF X(3) < 1000 THEN 1670

396 IF X(3) > 10000 THEN 1670

397 FOR J44 = 4 TO 8

398 IF X(J44) < 10 THEN 1670

399 IF X(J44) > 1000 THEN 1670

400 NEXT J44

464 PD1 = -X(1) - X(2) - X(3)

466 P = PD1

1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 8

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1670 NEXT I

1886 IF M < -7200 THEN 1999

1899 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102]. Its complete output of one run through JJJJ= -31999.85000000002 is shown below:

**586.3200000000045 1331.710000000001 5131.509999999883****182.6000000000004 294.7399999999998 217.3999999999996 ****287.8599999999996 394.7399999999998 -7049.539999999888****-31999.99**

414.3600000000056 1000.000000000001 5747.429999999795

166.4200000000004 270.1099999999996 233.5799999999996

296.13 370.1099999999996 -7161.789999999801

-31999.85000000002

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with Processor Intel Pentium CPU G620@ 2.60GHz, 4.00 GB of RAM, 64-bit Operating System, and QB64v1000-win [102], the wall-clock time (**not CPU time**) for obtaining the output through JJJJ = -31999.85000000002 was 4 seconds, counting from "Starting program...". One can compare the computational results above with those in Lin and Tsai [52, p. 22, Table 3, SGP 3].

**Acknowledgement**

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

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