Substance

Searching for Multiple Solutions of a Signomial Integer/Discrete Programming Problem, Second Edition

Jsun Yui Wong

The computer program listed below aims to find multiple solutions for the following nonlinear programming problem from Lin and Tsai [17, pp. 438-439]:

Minimize X(17) + X(18) + X(19) + X(20) + .1 * X(21) + .2 * X(22) + .3 * X(23) + .4 * X(24)

subject to

X(17) * X(1) + X(18) * X(2) + X(19) * X(3) + X(20) * X(4)>=9,
X(17) * X(5) + X(18) * X(6) + X(19) * X(7) + X(20) * X(8)>=7,
X(17) * X(9) + X(18) * X(10) + X(19) * X(11) + X(20) * X(12)>=12,
X(17) * X(13) + X(18) * X(14) + X(19) * X(15) + X(20) * X(16)>=11,

 

-( -1700 * X(21) + 330 * X(1) + 360 * X(5) + 385 * X(9) + 415 * X(13))<=0,
-( -1700 * X(22) + 330 * X(2) + 360 * X(6) + 385 * X(10) + 415 * X(14))<=0,
-( -1700 * X(23) + 330 * X(3) + 360 * X(7) + 385 * X(11) + 415 * X(15))<=0,
-(-1700 * X(24) + 330 * X(4) + 360 * X(8) + 385 * X(12) + 415 * X(16))<=0,

-(1900 * X(21) - 330 * X(1) - 360 * X(5) - 385 * X(9) - 415 * X(13))<=0,
-(1900 * X(22) - 330 * X(2) - 360 * X(6) - 385 * X(10) - 415 * X(14))<=0,
-(1900 * X(23) - 330 * X(3) - 360 * X(7) - 385 * X(11) - 415 * X(15))<=0,
-(1900 * X(24) - 330 * X(4) - 360 * X(8) - 385 * X(12) - 415 * X(16))<=0,

 

-( -X(21) + X(1) + X(5) + X(9) + X(13))<=0,
-( -X(22) + X(2) + X(6) + X(10) + X(14))<=0,
-( -X(23) + X(3) + X(7) + X(11) + X(15))<=0,
-( -X(24) + X(4) + X(8) + X(12) + X(16))<=0,

-( 5 * X(21) - X(1) - X(5) - X(9) - X(13))<=0,
-( 5 * X(22) - X(2) - X(6) - X(10) - X(14))<=0,
-( 5 * X(23) - X(3) - X(7) - X(11) - X(15))<=0,
-( 5 * X(24) - X(4) - X(8) - X(12) - X(16))<=0,

 

X(21) - X(17)<=0,
X(22) - X(18)<=0,
X(23) - X(19)<=0,
X(24) - X(20)<=0,

 

- 15 * X(21) + X(17)<=0,
- 12 * X(22) + X(18)<=0,
- 9 * X(23) + X(19)<=0,

-6 * X(24) + X(20)<=0,

X(17) + X(18) + X(19) + X(20)>=8,

 

X(21) - X(22)>=0,
X(22) - X(23)>=0,
X(23) - X(24)>=0,

 

X(17) - X(18)>=0
X(18) - X(19)>=0,
X(19) - X(20)>=0,

 

0<= X(i) <=5, i=1, 2, 3,..., 16,

0<= X(17) <= 15,

0<=X(18) <= 12,

0<= X(19) <= 9,

0<= X(20) <= 6,

X(21) through X(24) are 0-1 variables.

0<= X(i) <=5, i=1,2,3,..., 16,

0<= X(17) <= 15,

0<=X(18) <= 12,

0<= X(19) <= 9,

0<= X(20) <= 6,

X(21) through X(24) are 0-1 variables,

where X(1) through X(20) are integer variables and X(21) through X(24) are 0-1 variables..

X(25) through X(59) below are slack variables.

One notes line 111and line 191, which are 111 IF RND < .5 THEN SOFMS = 8 ELSE SOFMS = 9 and 191 X(20) = SOFMS - X(17) - X(18) - X(19).

 

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(99)

 

12 FOR JJJJ = -32000 TO 31997.00 STEP .01

 

14 RANDOMIZE JJJJ
16 M = -1D+37

22 FOR J55 = 1 TO 16

37 A(J55) = INT(RND * 5)

 

38 NEXT J55
71 A(17) = 2 + INT(RND * 3)
72 A(18) = 2 + INT(RND * 3)

73 A(19) = 2 + INT(RND * 3)
74 A(20) = 2 + INT(RND * 3)
75 GOTO 91

 

81 A(21) = INT(RND * 1)
82 A(22) = INT(RND * 1)

 

83 A(23) = INT(RND * 1)
84 A(24) = INT(RND * 1)

 

91 IF RND < .5 THEN A(21) = 0 ELSE A(21) = 1

97 IF RND < .5 THEN A(22) = 0 ELSE A(22) = 1
98 IF RND < .5 THEN A(23) = 0 ELSE A(23) = 1
99 IF RND < .5 THEN A(24) = 0 ELSE A(24) = 1

111 IF RND < .5 THEN SOFMS = 8 ELSE SOFMS = 9

 

128 FOR I = 1 TO 8000000

 

129 FOR KKQQ = 1 TO 24

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 16))

151 J = 1 + FIX(RND * 16)

152 X(J) = INT(RND * 5)

 

153 NEXT IPP
155 IF RND < .25 THEN GOTO 160 ELSE IF RND < .333 THEN GOTO 164 ELSE IF RND < .5 THEN GOTO 168 ELSE GOTO 173

 

160 X(17) = INT(RND * 15)
163 IF RND < .5 THEN 164 ELSE GOTO 168

164 X(18) = INT(RND * 12)
167 IF RND < .5 THEN 168 ELSE GOTO 173

168 X(19) = INT(RND * 9)
169 IF RND < .5 THEN 173 ELSE GOTO 175

173 X(20) = INT(RND * 6)

175 FOR IPQ = 1 TO (1 + FIX(RND * 4))

177 J = 21 + FIX(RND * 4)

179 X(J) = INT(RND * 1)

 

181 NEXT IPQ

 

183 REM r = (1 - RND * 2) * A(J)
187 REM X(J) = A(J) + (RND ^ (RND * 10)) * r

191 X(20) = SOFMS - X(17) - X(18) - X(19)
192 IF X(17) > 0 THEN X(21) = 1
193 IF X(18) > 0 THEN X(22) = 1
194 IF X(19) > 0 THEN X(23) = 1

195 IF X(20) > 0 THEN X(24) = 1

201 IF X(1) < 0 THEN 1670
202 IF X(1) > 5 THEN 1670
203 IF X(2) < 0 THEN 1670
204 IF X(2) > 5 THEN 1670

205 IF X(3) < 0 THEN 1670
206 IF X(3) > 5 THEN 1670

 

207 IF X(4) < 0 THEN 1670
208 IF X(4) > 5 THEN 1670
209 IF X(5) < 0 THEN 1670
210 IF X(5) > 5 THEN 1670

211 IF X(6) < 0 THEN 1670
212 IF X(6) > 5 THEN 1670

213 IF X(7) < 0 THEN 1670
214 IF X(7) > 5 THEN 1670
215 IF X(8) < 0 THEN 1670
216 IF X(8) > 5 THEN 1670

217 IF X(9) < 0 THEN 1670
218 IF X(9) > 5 THEN 1670

219 IF X(10) < 0 THEN 1670
220 IF X(10) > 5 THEN 1670
221 IF X(11) < 0 THEN 1670
222 IF X(11) > 5 THEN 1670

223 IF X(12) < 0 THEN 1670
224 IF X(12) > 5 THEN 1670

 

225 IF X(13) < 0 THEN 1670
226 IF X(13) > 5 THEN 1670

227 IF X(14) < 0 THEN 1670
228 IF X(14) > 5 THEN 1670
229 IF X(15) < 0 THEN 1670
230 IF X(15) > 5 THEN 1670

231 IF X(16) < 0 THEN 1670
232 IF X(16) > 5 THEN 1670

233 IF X(17) < 0 THEN 1670
234 IF X(17) > 15 THEN 1670
235 IF X(18) < 0 THEN 1670
236 IF X(18) > 12 THEN 1670

237 IF X(19) < 0 THEN 1670
238 IF X(19) > 9 THEN 1670

 

239 IF X(20) < 0 THEN 1670
240 IF X(20) > 6 THEN 1670
241 IF X(21) < 0 THEN 1670
242 IF X(21) > 1 THEN 1670

243 IF X(22) < 0 THEN 1670
244 IF X(22) > 1 THEN 1670

 

245 IF X(23) < 0 THEN 1670
246 IF X(23) > 1 THEN 1670

 

247 IF X(24) < 0 THEN 1670
248 IF X(24) > 1 THEN 1670

 

301 X(25) = -9 + X(17) * X(1) + X(18) * X(2) + X(19) * X(3) + X(20) * X(4)
302 X(26) = -7 + X(17) * X(5) + X(18) * X(6) + X(19) * X(7) + X(20) * X(8)
303 X(27) = -12 + X(17) * X(9) + X(18) * X(10) + X(19) * X(11) + X(20) * X(12)
304 X(28) = -11 + X(17) * X(13) + X(18) * X(14) + X(19) * X(15) + X(20) * X(16)

 

 

305 X(29) = -1700 * X(21) + 330 * X(1) + 360 * X(5) + 385 * X(9) + 415 * X(13)
306 X(30) = -1700 * X(22) + 330 * X(2) + 360 * X(6) + 385 * X(10) + 415 * X(14)
307 X(31) = -1700 * X(23) + 330 * X(3) + 360 * X(7) + 385 * X(11) + 415 * X(15)
308 X(32) = -1700 * X(24) + 330 * X(4) + 360 * X(8) + 385 * X(12) + 415 * X(16)

309 X(33) = 1900 * X(21) - 330 * X(1) - 360 * X(5) - 385 * X(9) - 415 * X(13)
310 X(34) = 1900 * X(22) - 330 * X(2) - 360 * X(6) - 385 * X(10) - 415 * X(14)
311 X(35) = 1900 * X(23) - 330 * X(3) - 360 * X(7) - 385 * X(11) - 415 * X(15)
312 X(36) = 1900 * X(24) - 330 * X(4) - 360 * X(8) - 385 * X(12) - 415 * X(16)

 

 

313 X(37) = -X(21) + X(1) + X(5) + X(9) + X(13)
314 X(38) = -X(22) + X(2) + X(6) + X(10) + X(14)
315 X(39) = -X(23) + X(3) + X(7) + X(11) + X(15)
316 X(40) = -X(24) + X(4) + X(8) + X(12) + X(16)

317 X(41) = 5 * X(21) - X(1) - X(5) - X(9) - X(13)
318 X(42) = 5 * X(22) - X(2) - X(6) - X(10) - X(14)
319 X(43) = 5 * X(23) - X(3) - X(7) - X(11) - X(15)
320 X(44) = 5 * X(24) - X(4) - X(8) - X(12) - X(16)

 

 

321 X(45) = -X(21) + X(17)
322 X(46) = -X(22) + X(18)
323 X(47) = -X(23) + X(19)
324 X(48) = -X(24) + X(20)

 

325 X(49) = 15 * X(21) - X(17)
326 X(50) = 12 * X(22) - X(18)
327 X(51) = 9 * X(23) - X(19)

328 X(52) = 6 * X(24) - X(20)

329 X(53) = -8 + X(17) + X(18) + X(19) + X(20)

 

330 X(54) = X(21) - X(22)
331 X(55) = X(22) - X(23)
332 X(56) = X(23) - X(24)

 

333 X(57) = X(17) - X(18)
334 X(58) = X(18) - X(19)
335 X(59) = X(19) - X(20)

 

425 FOR J99 = 25 TO 59

 

426 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0

427 NEXT J99

431 SUMP = 0
434 FOR J66 = 25 TO 59
437 SUMP = SUMP + X(J66)
440 NEXT J66

 

459 POBA = -X(17) - X(18) - X(19) - X(20) - .1 * X(21) - .2 * X(22) - .3 * X(23) - .4 * X(24) + 1000000 * SUMP

 

466 P = POBA

 

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR klx = 1 TO 59

 

1459 A(klx) = X(klx)
1460 NEXT klx
1557 REM GOTO 128

1670 NEXT I

1889 IF M < -8.9 THEN 1999

 

1900 PRINT A(1), A(2), A(3), A(4), A(5)
1950 PRINT A(6), A(7), A(8), A(9), A(10)
1953 PRINT A(11), A(12), A(13), A(14), A(15)
1954 PRINT A(16), A(17), A(18), A(19), A(20)

1955 PRINT A(21), A(22), A(23), A(24)

1957 PRINT M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [34]. The complete output through JJJJ = -31994.9800000008 is shown below:

1 1 2 0 0
2 1 0 3 0
1 0 1 2 1
0 4 3 1 0
1 1 1 0
-8.6          -31999.3900000001

1 1 2 0 0
2 1 0 3 0
1 0 1 2 1
0 4 3 1 0
1 1 1 0
-8.6          -31997.29000000043

1 1 2 0 0
2 2 0 3 0
0 0 1 2 1
0 4 3 1 0
1 1 1 0
-8.6          -31995.52000000072

1 1 2 0 0
2 1 0 3 0
0 0 1 2 2
0 4 3 1 0
1 1 1 0
-8.6          -31994.9800000008

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 [34], the wall-clock time for obtaining the output through JJJJ=
-31994.9800000008 was 7 hours. One can compare the computational results here with those in Table 4 of Lin and Tsai [17, p. 440].

Acknowledgment

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

References

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Solving a Compression Spring Design Problem in Discrete Variables

 

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear programming problem from Lin and Tsai [17, p. 437]:

Minimize    X(1) ^ 2 * X(2) * (X(3) + 2)

subject to

71785 * X(1) ^ 4 - X(2) ^ 3 * X(3)<=0,

20432 * X(1) ^ 2 * X(2) ^ 2 - X(1) * X(2) - 5108 * X(1) ^ 3 * X(2) + 4 * X(2) ^ 2 - 64187128 * (X(1) ^ 5 * X(2) - X(1) ^ 6)<=0,

X(2) ^ 2 * X(3) -140.45 * X(1)<=0,

X(1) + X(2)<=1.5,

.05 <= X(1) <= .15,

X(2) Epsilon { .263, .283, .307, .331, .363, .394 .4375, .5 },

2<= X(3) <= 15,

where X(1) is a discrete variable with discreteness of 0.001 and X(3) is an integer variable.

X(4) through X(7) below are slack variables.

 

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

37 A(1) = .05 + INT(RND * 105) * .001

72 IF RND < .125 THEN X(2) = .263 ELSE IF RND < .143 THEN X(2) = .283 ELSE IF RND < .167 THEN X(2) = .307 ELSE IF RND < .2 THEN X(2) = .331 ELSE IF RND < .25 THEN X(2) = .363 ELSE IF RND < .333 THEN X(2) = .394 ELSE IF RND < .5 THEN X(2) = .4375 ELSE X(2) = .5

77 A(3) = 2 + FIX(RND * 14)

128 FOR I = 1 TO 50000

 

129 FOR KKQQ = 1 TO 3

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

151 J = 1 + FIX(RND * 3)

 

155 IF J = 1 THEN GOTO 157 ELSE IF J = 2 THEN GOTO 170 ELSE GOTO 177
157 X(1) = .05 + INT(RND * 105) * .001

168 IF RND < .5 THEN GOTO 170 ELSE GOTO 177
170 IF RND < .125 THEN X(2) = .263 ELSE IF RND < .143 THEN X(2) = .283 ELSE IF RND < .167 THEN X(2) = .307 ELSE IF RND < .2 THEN X(2) = .331 ELSE IF RND < .25 THEN X(2) = .363 ELSE IF RND < .333 THEN X(2) = .394 ELSE IF RND < .5 THEN X(2) = .4375 ELSE X(2) = .5

 

171 IF RND < .5 THEN GOTO 177 ELSE GOTO 191

177 X(3) = 2 + FIX(RND * 14)

 

 

183 REM r = (1 - RND * 2) * A(J)
187 REM X(J) = A(J) + (RND ^ (RND * 10)) * r

 

191 NEXT IPP

201 IF X(1) < .05 THEN 1670
203 IF X(1) > .15 THEN 1670
211 IF X(2) < .263 THEN 1670
213 IF X(2) > .5 THEN 1670
231 IF X(3) < 2 THEN 1670
233 IF X(3) > 15 THEN 1670

 

305 X(4) = -71785 * X(1) ^ 4 + X(2) ^ 3 * X(3)

306 X(5) = -20432 * X(1) ^ 2 * X(2) ^ 2 + X(1) * X(2) + 5108 * X(1) ^ 3 * X(2) - 4 * X(2) ^ 2 + 64187128 * (X(1) ^ 5 * X(2) - X(1) ^ 6)

 

307 X(6) = -X(2) ^ 2 * X(3) + 140.45 * X(1)

317 X(7) = 1.5 - X(1) - X(2)

 

325 FOR J99 = 4 TO 7

 

330 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0

331 NEXT J99

 

359 POBA = -X(1) ^ 2 * X(2) * (X(3) + 2) + 1000000 * (X(4) + X(5) + X(6) + X(7))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 7

 

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 REM GOTO 128

1670 NEXT I

1889 REM IF M < -.016 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4)
1903 PRINT A(5), A(6), A(7), M, JJJJ
1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [34]. The complete output through JJJJ = -31999.95000000001 is shown below:

5.200000000000001D-02       .363       11       0
0       0       0       -.012760176       -32000

5.200000000000001D-02       .363       11       0
0       0       0       -.012760176       -31999.99

5.200000000000001D-02       .363       11       0
0       0       0       -.012760176        -31999.98

5.200000000000001D-02       .363        11       0
0       0       0       -.012760176       -31999.97000000001

5.200000000000001D-02      .363       11       0
0       0       0       -.012760176       -31999.96000000001

5.200000000000001D-02       .363       11       0
0       0       0       -.012760176       -31999.95000000001

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 [34], the wall-clock time for obtaining the output through
JJJJ= -31999.95000000001
was 10 seconds, total, including the time for creating the .EXE file. One can compare the computational results here with those in Table 3 of Lin and Tsai [17, p. 438].

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66. http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[4] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[5] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[6] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[7] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[8] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Sinulation 18 (2013) 89-98.
[9] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[11] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[12] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

[13] Ali Husseinzadeh Kashan (2015). An effective algorithm for constrained optimization based on optics inspired optimization (OIO). Computer-Aided Design 63 (2015) 52-71.

[14] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[15] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.
[16] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[17] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming programs with free variables, Optimization and Engineering (2011) 12:425-443.
[18] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[19] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[20] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[21] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.
[22] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[23] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[24] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[25] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[26] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[27] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[28] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[29] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[30] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[31] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[32] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.
[33] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[34] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[35] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/
[36] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[37] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms. https://arxiv.org/pdf/1403.7793.pdf.
[38] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[39] B. D. Youn, K. K. Choi (2004). A new responsesurface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

Solving Another Nonlinear Programming Problem with Continuous Signomial Terms

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear programming problem:

Minimize X(1) ^ 2 * X(2) * (X(3) + 2)

subject to

7178 * X(1) ^ 4 - X(2) ^ 3 * X(3)<=0,

4 * X(1) ^ 2 * X(2) ^ 2 + (1 / 5108) * (7458 * X(1) ^ 3 * X(2) - X(1) ^ 4) - 12566 * (X(1) ^ 5 * X(2)) + X(1) ^ 6<=0,

X(2) ^ 2 * X(3) - 140.45 * X(1)<=0,

-1.5 + X(1) + X(2)<=0,

.05 <= X(1) <= 2,

.25<= X(2) <= 1.3

2<= X(3) <= 15,

where all three variables are continuous.

The problem above is based on Li et al.'s compression spring design optimization problem, [16, p. 931], which is

minimize X(1) ^ 2 * X(2) * (X(3) + 2)

subject to

7178 * X(1) ^ 4 - X(2) ^ 3 * X(3)<=0,

4 * X(1) ^ 2 * X(2) ^ 2 + (1 / 5108) * (7458 * X(1) ^ 3 * X(2) - X(1) ^ 4) - 12566 * (X(1) ^ 5 * X(2)) + X(1) ^ 6<=0,

X(2) ^ 2 * X(3) - 140.45 * X(1)<=0,

-1.5 + X(1) + X(2)<=0,

.05 <= X(1) <= 2,

.25<= X(2) <= 1.3

2<= X(3) <= 15,

where all three variables are discrete.

X(4) through X(7) below are slack variables.

 

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

37 A(1) = .05 + RND * 1.95
39 A(2) = .25 + RND * 1.05
77 A(3) = 2 + RND * 13

128 FOR I = 1 TO 50000

 

129 FOR KKQQ = 1 TO 3

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

151 J = 1 + FIX(RND * 3)

 

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

 

191 NEXT IPP

201 IF X(1) < .05 THEN 1670
203 IF X(1) > 2 THEN 1670
211 IF X(2) < .25 THEN 1670
213 IF X(2) > 1.3 THEN 1670

231 IF X(3) < 2 THEN 1670
233 IF X(3) > 15 THEN 1670

 

305 X(4) = -7178 * X(1) ^ 4 + X(2) ^ 3 * X(3)

306 X(5) = -4 * X(1) ^ 2 * X(2) ^ 2 - (1 / 5108) * (7458 * X(1) ^ 3 * X(2) - X(1) ^ 4) + 12566 * (X(1) ^ 5 * X(2)) - X(1) ^ 6

 

307 X(6) = -X(2) ^ 2 * X(3) + 140.45 * X(1)

317 X(7) = 1.5 - X(1) - X(2)

 

325 FOR J99 = 4 TO 7

 

330 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0

331 NEXT J99

 

359 POBA = -X(1) ^ 2 * X(2) * (X(3) + 2) + 1000000 * (X(4) + X(5) + X(6) + X(7))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 7

 

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 REM GOTO 128

1670 NEXT I

1889 IF M < -.00283 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4), A(5)
1903 PRINT A(6), A(7), M, JJJJ
1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [33]. The complete output through JJJJ =-31999.98 is shown below:

5.000000077177234D-02          .2819257615046565          2.002070509583578
0          0
0          0          -2.820717027102251D-03          -32000

4.999999888755837D-02          .2818412042392773          2.00387199227901
0          0
0          0          -2.821140134275423D-03          -31999.99

.0499999988824131           .282010732030957          2.00026034351743
0          0
0          0          -2.82029074339759D-03          -31999.98

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= -31999.98 was 3 seconds, not including the time for creating the .EXE file. One can cautiously compare the computational results here with those in Table 4 of Li et al.[16, p. 931] because here the variables are continuous whereas there the variables are discrete.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66. http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[4] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[5] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[6] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[7] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[8] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Sinulation 18 (2013) 89-98.
[9] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[11] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[12] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

[13] Ali Husseinzadeh Kashan (2015). An effective algorithm for constrained optimization based on optics inspired optimization (OIO). Computer-Aided Design 63 (2015) 52-71.

[14] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[15] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.
[16] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[17] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[18] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[19] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[20] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.
[21] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[22] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[23] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[24] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[25] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[26] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[27] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[28] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[29] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[30] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[31] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.
[32] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[33] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[34] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/
[35] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[36] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms. https://arxiv.org/pdf/1403.7793.pdf.
[37] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[38] B. D. Youn, K. K. Choi (2004). A new responsesurface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

Solving a Nonlinear Programming Problem with Continuous Signomial Terms

 

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear programming problem:

Minimize X(1) ^ 3 * X(2) * X(3) ^ 3 * X(4) + (X(1) ^ 3 * X(2) * X(3) * X(4) ^ 2)

subject to

X(1) ^ 3 * X(2) * X(3) ^ 2+ X(3) * X(4)<= -500,

- X(1) ^ 3 * X(2) * X(3) + X(3) ^ 2 * X(4)<= 500,

-6 <= X(1) <= 6.75 ,

-6<= X(2) <= 6.75 ,

-1 <= X(3) <= 9.2 ,

-9 <= X(4) <= 6.3,

where all four variables are continuous.

The problem above is based on Lu's [20, p. 111] Example 2, which is

minimize X(1) ^ 3 * X(2) * X(3) ^ 3 * X(4) + (X(1) ^ 3 * X(2) * X(3) * X(4) ^ 2)

subject to

X(1) ^ 3 * X(2) * X(3) ^ 2+ X(3) * X(4)<= -500,

- X(1) ^ 3 * X(2) * X(3) + X(3) ^ 2 * X(4)<= 500,

-6 <= X(1) <= 6.75 ,

-6<= X(2) <= 6.75 ,

-1 <= X(3) <= 9.2 ,

-9 <= X(4) <= 6.3,

where all four variables are discrete.

X(5) and X(6) below are slack variables.

 

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) = -6 + RND * 12.75

89 A(2) = -6 + RND * 12.75

 

91 A(3) = -1 + RND * 10.2

94 A(4) = -9 + RND * 15.3

128 FOR I = 1 TO 200000

 

129 FOR KKQQ = 1 TO 4

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))

 

151 J = 1 + FIX(RND * 4)

 

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

 

191 NEXT IPP
201 IF X(1) < -6 THEN 1670
203 IF X(1) > 6.75 THEN 1670
211 IF X(2) < -6 THEN 1670
213 IF X(2) > 6.75 THEN 1670

 

231 IF X(3) < -1 THEN 1670
233 IF X(3) > 9.2 THEN 1670
235 IF X(4) < -9 THEN 1670
237 IF X(4) > 6.3 THEN 1670

 

305 X(5) = -500 - X(1) ^ 3 * X(2) * X(3) ^ 2 - X(3) * X(4)

306 X(6) = 500 + X(1) ^ 3 * X(2) * X(3) - X(3) ^ 2 * X(4)

335 FOR J99 = 5 TO 6

 

340 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0

341 NEXT J99

 

359 POBA = -X(1) ^ 3 * X(2) * X(3) ^ 3 * X(4) - (X(1) ^ 3 * X(2) * X(3) * X(4) ^ 2) + 1000000 * (X(5) + X(6))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 6

 

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 REM GOTO 128

1670 NEXT I

1889 IF M < 72816.2 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4)

1903 PRINT A(5), A(6), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [33]. The complete output through JJJJ =-31999.3900000001 is shown below:

4.378054821269147          -.531121206274107          6.04800556285877
6.300000224648621
0          0          72816.24141297565          -31999.94000000001

3.754343291412405          -.8441787863318144          6.042873913456674
6.299980226397954
0          0          72816.24618529718          -31999.3900000001

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= -31999.3900000001 was 40 seconds, total, including the time for creating the .EXE file. One can cautiously compare the computational results here with those in Table 5 of Lu [20, p. 114] because here the variables are continuous whereas there the variables are discrete.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66. http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[4] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[5] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[6] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[7] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[8] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[9] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[11] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[12] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

[13] Ali Husseinzadeh Kashan (2015). An effective algorithm for constrained optimization based on optics inspired optimization (OIO). Computer-Aided Design 63 (2015) 52-71.

[14] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[15] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.
[16] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[17] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[18] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[19] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[20] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.
[21] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[22] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[23] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[24] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[25] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[26] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[27] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[28] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[29] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[30] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[31] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.
[32] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[33] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[34] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/
[35] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[36] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms. https://arxiv.org/pdf/1403.7793.pdf.
[37] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[38] B. D. Youn, K. K. Choi (2004). A new responsesurface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

Solving a Speed Reducer Design Problem, Third Edition

Jsun Yui Wong

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].

One also notes line 87 through line 99--these lines are the changes from the Second Edition.

 

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 + RND * 1

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

 

91 A(3) = 17 + RND * 11

94 A(4) = 7.3 + RND * 1

96 A(5) = 7.3 + RND * 1

 

98 A(6) = 2.9 + RND * 1

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

 

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)

153 GOTO 183

 

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

 

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r
189 REM X(J) = .5 + FIX(RND * 505) * .01

 

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 = -31999.99 is shown below:

3.499999791502988       .69999995827728          17
7.29999995235192       7.715319923485348
3.350214667860891       5.28665446646302          0
0          0
0       0       0       0       0
0       0       0          -2994.470796966784
-32000

3.499999791907917       .699999958277962          17
7.29999995235496       7.715319912085655
3.35021466686005       5.286654465174181          0
0          0
0       0       0       0       0
0       0       0          -2994.47079580493
-31999.99

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=-31999.99 was 3 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. 

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66. http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[4] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[5] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[6] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[7] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[8] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[9] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[11] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[12] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

[13] Ali Husseinzadeh Kashan (2015). An effective algorithm for constrained optimization based on optics inspired optimization (OIO). Computer-Aided Design 63 (2015) 52-71.

[14] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[15] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.
[16] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[17] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[18] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[19] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[20] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.
[21] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[22] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[23] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[24] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[25] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[26] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[27] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[28] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[29] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[30] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[31] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.
[32] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[33] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[34] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/
[35] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[36] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms. https://arxiv.org/pdf/1403.7793.pdf.
[37] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[38] B. D. Youn, K. K. Choi (2004). A new response surface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

Solving a Speed Reducer Design Problem, Second Edition

Jsun Yui Wong

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].

One notes line 153, which is 153 GOTO 183 and is the change from the earlier edition.

 

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)

153 GOTO 183

 

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

 

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r
189 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 = -31999.99 is shown below:

3.49999979256057       .6999999584326087       17
7.299999980649554       7.7153199124325
3.350214667092591       5.286654465164057          0
0       0
0       0       0       0       0
0       0       0       -2994.470797069908
-32000

3.499999795036738       .699999958337472       17
7.299999952340041       7.71531991199833
3.350214666845967       5.286654465139653          0
0       0
0       0       0       0       0
0       0       0       -2994.470797275173
-31999.99

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=-31999.99 was 3 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.

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66. http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[4] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[5] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[6] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[7] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[8] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Sinulation 18 (2013) 89-98.
[9] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[11] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[12] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

[13] Ali Husseinzadeh Kashan (2015). An effective algorithm for constrained optimization based on optics inspired optimization (OIO). Computer-Aided Design 63 (2015) 52-71.

[14] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[15] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.
[16] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[17] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[18] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[19] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[20] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.
[21] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[22] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[23] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[24] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[25] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[26] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[27] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[28] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[29] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[30] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[31] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.
[32] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[33] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[34] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/
[35] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[36] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms. https://arxiv.org/pdf/1403.7793.pdf.
[37] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[38] B. D. Youn, K. K. Choi (2004). A new responsesurface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

Solving an Integer Fractional Nonlinear Programming Problem, Third Edition

Jsun Yui Wong

The computer program listed below seeks to solve the following integer fractional nonlinear programming problem:

Minimize       ((2 + X(5)) / (X(1) * X(2) * (2 * X(3) + X(4)))) - X(5) ^ .5 * X(3) ^ 1.5 + 2 * X(2) + X(4)

subject to

8 / (X(1) * (X(2) + 3 * X(4)) ^ 2) + 1 / (X(5) ^ 3)<=2000,

-2 * X(1) + X(3) - X(4)<=10000,

X(1)+ X(3) + .5 * X(5)<=12000,

.1<= X(i) <= 10000, i=1, 2, 3, 4, 5,

X(1) through X(5) are integer variables.

This integer fractional nonlinear programming problem is based on Example 3 in Tsai [26], which is:

Minimize       ((2 + X(5)) / (X(1) * X(2) * (2 * X(3) + X(4)))) - X(5) ^ .5 * X(3) ^ 1.5 + 2 * X(2) + X(4)

subject to

8 / (X(1) * (X(2) + 3 * X(4)) ^ 2) + 1 / (X(5) ^ 3)<=2,

-2 * X(1) + X(3) - X(4)<=10,

X(1)+ X(3) + .5 * X(5)<=8,

.1<= X(i) <= 10,  i=1, 2, 3, 4, 5.

 

Referring to his Example 3 Tsai wrote that "This program is a highly NFP program unsolvable by current fractional programming methods," Tsai [26 (2005), p. 408].

Whereas line 307 of the preceding edition is 307 X(8) = 8000 - X(1) - X(3) - .5 * X(5), here line 307 is 307 X(8) = 12000 - X(1) - X(3) - .5 * X(5).

X(6) through X(8) below are slack variables.

 

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
70 FOR J44 = 1 TO 5

72 A(J44) = .1 + RND * 9.9

 

73 NEXT J44

 

128 FOR I = 1 TO 20000

 

129 FOR KKQQ = 1 TO 5

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))

 

181 J = 1 + FIX(RND * 5)

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

191 NEXT IPP

196 FOR J99 = 1 TO 5

199 X(J99) = INT(X(J99))

201 IF X(J99) < .1 THEN 1670
203 IF X(J99) > 10000 THEN 1670
204 NEXT J99

305 X(6) = 2000 - 8 / (X(1) * (X(2) + 3 * X(4)) ^ 2) - 1 / (X(5) ^ 3)

 

306 X(7) = 10000 + 2 * X(1) - X(3) + X(4)

 

307 X(8) = 12000 - X(1) - X(3) - .5 * X(5)

 

325 FOR J99 = 6 TO 8

327 XX(J99) = X(J99)

 

330 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0

331 NEXT J99

 

357 POBA = - ((2 + X(5)) / (X(1) * X(2) * (2 * X(3) + X(4)))) + X(5) ^ .5 * X(3) ^ 1.5 - 2 * X(2) - X(4) + 1000000 * (X(6) + X(7) + X(8))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 8

 

1456 XXX(KLX) = XX(KLX)

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128

1670 NEXT I

 

1889 IF M < 66125190 THEN 1999

 

 

1900 PRINT A(1), A(2), A(3), A(4), A(5)

1902 PRINT A(6), A(7), A(8)

 

1903 PRINT XXX(6), XXX(7), XXX(8)

1912 PRINT M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [30]. The complete output through JJJJ = -31998.70000000021 is shown below:

 1          1          8999          1          6000
0          0          0
1999.499999999996          1004          0
66125197.3240322          -31999.74000000004

1          1          8998          1          6002
0          0          0
1999.499999999996          1004          0
66125193.6503524          -31999.62000000006

1 1 8999 1 6000
0 0 0
1999.499999999996 1004 0
66125197.3240322 -31999.03000000016

 1 1 8999 1 6000
0 0 0
1999.499999999996 1004 0
66125197.3240322 -31998.70000000021

 Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. The computational results above can be cautiously compared with those on page 408 of Tsai [26]. The former is optimization in integers, and the latter is continuous optimization.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [30], the wall-clock time for obtaining the output through
JJJJ= -31998.70000000021 was 13 seconds, not including the time for creating the .EXE file.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66. http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[4] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[5] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[6] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[7] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[8] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Sinulation 18 (2013) 89-98.
[9] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[11] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[12] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[13] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.
[14] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[15] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[16] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[17] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.
[18] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[19] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[20] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[21] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[22] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[23] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[24] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[25] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[26] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[27] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[28] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.
[29] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[30] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[31] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/
[32] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[33] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms. https://arxiv.org/pdf/1403.7793.pdf.
[34] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[35] B. D. Youn, K. K. Choi (2004). A new responsesurface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

 

Solving an Integer Fractional Nonlinear Programming Problem, Second Edition

 Jsun Yui Wong

The computer program listed below seeks to solve the following integer fractional nonlinear programming problem:

Minimize    ((2 + X(5))   /   (X(1) * X(2) * (2 * X(3) + X(4)))) - X(5) ^ .5 * X(3) ^ 1.5 + 2 * X(2) + X(4)

subject to

8 / (X(1) * (X(2) + 3 * X(4)) ^ 2) + 1 / (X(5) ^ 3)<=2000,

-2 * X(1) + X(3) - X(4)<=10000,

X(1)+ X(3) + .5 * X(5)<=8000,

.1<= X(i) <= 10000, i=1, 2, 3, 4, 5,

X(1) through X(5) are integer variables.

This integer fractional nonlinear programming problem is based on Example 3 in Tsai [26]. About his Example 3 Tsai wrote that "This program is a highly NFP program unsolvable by current fractional programming methods," Tsai [26, p. 408].

X(6) through X(8) below are slack variables.

 

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
70 FOR J44 = 1 TO 5

72 A(J44) = .1 + RND * 9.9

 

73 NEXT J44

 

128 FOR I = 1 TO 20000

 

 

129 FOR KKQQ = 1 TO 5

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))

 

181 J = 1 + FIX(RND * 5)

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

191 NEXT IPP

196 FOR J99 = 1 TO 5

199 X(J99) = INT(X(J99))

201 IF X(J99) < .1 THEN 1670
203 IF X(J99) > 10000 THEN 1670
204 NEXT J99

305 X(6) = 2000 - 8 / (X(1) * (X(2) + 3 * X(4)) ^ 2) - 1 / (X(5) ^ 3)

 

306 X(7) = 10000 + 2 * X(1) - X(3) + X(4)

 

307 X(8) = 8000 - X(1) - X(3) - .5 * X(5)

 

325 FOR J99 = 6 TO 8

327 XX(J99) = X(J99)

 

330 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0

331 NEXT J99

 

357 POBA = - ((2 + X(5)) / (X(1) * X(2) * (2 * X(3) + X(4)))) + X(5) ^ .5 * X(3) ^ 1.5 - 2 * X(2) - X(4) + 1000000 * (X(6) + X(7) + X(8))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 8

 

1456 XXX(KLX) = XX(KLX)

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128

1670 NEXT I

 

1889 IF M < 29386400 THEN 1999

 

 

1900 PRINT A(1), A(2), A(3), A(4), A(5)

1902 PRINT A(6), A(7), A(8)

 

1903 PRINT XXX(6), XXX(7), XXX(8)

1912 PRINT M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [30]. The complete output through JJJJ = -31999.10000000014 is shown below:

1       1       6005       1        3988
0       0       0
1999.499999999984       3998       0
29386444.48135456       -31999.57000000007

1       1       6001       1       3996
0       0       0
1999.499999999984       4002       0
29386518.06588446        -31999.4200000001

1       1       5999       1       4000
0       0       0
1999.499999999984        4004       0
29386525.41683672       -31999.4000000001

1       1       5999       1       4000
0       0       0
1999.499999999984       4004       0
29386525.41683672       -31999.33000000011

1       1       6000       1       3998
0       0       0
1999.499999999984       4003       0
29386524.19207587       -31999.10000000014

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. The computational results above can be cautiously compared with those on page 408 of Tsai [26]. The former is optimization in integers, and the latter is continuous optimization.
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [30], the wall-clock time for obtaining the output through JJJJ= -31999.10000000014 was 11 seconds, not including the time for creating the .EXE file.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66. http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[4] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[5] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[6] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[7] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[8] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Sinulation 18 (2013) 89-98.
[9] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[11] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[12] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[13] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.
[14] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[15] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[16] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[17] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.
[18] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[19] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[20] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[21] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[22] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[23] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[24] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[25] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[26] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[27] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[28] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.
[29] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[30] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[31] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/
[32] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[33] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms. https://arxiv.org/pdf/1403.7793.pdf.
[34] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[35] B. D. Youn, K. K. Choi (2004). A new responsesurface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

 

Enlarging and Using Integers in Chang's Mixed Integer Fractional Posynomial Programming Model To Make It More Challenging for Computers

 

Jsun Yui Wong

The computer program listed below seeks to solve the following integer fractional nonlinear programming problem:

Minimize       ups / dow

where ups = ABS(X(5) ^ 1.8) * X(1) * X(3) ^ .3 * X(4) ^ 1.5 + X(2) * X(3) ^ 1.8 * X(4) ^ 1.3 * X(5) + X(1) * X(2) * X(3) ^ 1.7

dow = X(1) * X(3) * X(4) + X(2) * X(3) ^ 2.3 * X(4) ^ 1.4

subject to

2 * X(1) + X(1) * X(3) ^ 1.6>=5,

X(1) + X(2)>=1,

X(2) + X(4)<=6,

X(1) + X(2) + X(5)>=3,

0<=X(1) <= 1,

0<=X(2) <=1,

1<=X(3) <= 32007,

1<=X(4) <= 32006,

1<= X(5) <= 32005,

where X(1) through X(5) are integer variables.

This integer fractional nonlinear programming problem is based on Example 4 in Chang [5, p. 383].

X(6) through X(9) below are slack variables.

 

0 DEFDBL A-Z

2 DEFINT K

3 DIM B(199), N(199), A(199), H(199), L(199), U(199), X(1111), D(111), P(111), PS(133)

12 FOR JJJJ = -32000 TO 32000 STEP .01

14 RANDOMIZE JJJJ
16 M = -1D+37
71 IF RND < .5 THEN A(1) = 0 ELSE A(1) = 1

73 IF RND < .5 THEN A(2) = 0 ELSE A(2) = 1

84 A(3) = 1 + INT(RND * 1117)
85 A(4) = 1 + INT(RND * 1116)
86 A(5) = 1 + INT(RND * 1115)

128 FOR I = 1 TO 5000

 

129 FOR KKQQ = 1 TO 5

 

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

132 REM GOTO 162

133 FOR IPP = 1 TO (1 + FIX(RND * 3))

 

151 J = 3 + FIX(RND * 3)

153 r = (1 - RND * 2) * A(J)
157 X(J) = A(J) + (RND ^ (RND * 10)) * r

159 REM IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0

161 NEXT IPP

 

166 FOR J44 = 3 TO 5

167 X(J44) = INT(X(J44))

169 NEXT J44

 

197 IF X(3) < 1 THEN 1670

199 IF X(3) > 32007 THEN 1670

201 IF X(4) < 1 THEN 1670

203 IF X(4) > 32006 THEN 1670

 

205 IF X(5) < 1 THEN 1670

207 IF X(5) > 32005 THEN 1670

261 X(6) = -5 + 2 * X(1) + X(1) * X(3) ^ 1.6

 

264 X(7) = -1 + X(1) + X(2)

267 X(8) = 6 - X(2) - X(4)

269 X(9) = -3 + X(1) + X(2) + X(5)

 

281 FOR J99 = 6 TO 9

283 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0

 

285 NEXT J99

311 ups = ABS(X(5) ^ 1.8) * X(1) * X(3) ^ .3 * X(4) ^ 1.5 + X(2) * X(3) ^ 1.8 * X(4) ^ 1.3 * X(5) + X(1) * X(2) * X(3) ^ 1.7

322 dow = X(1) * X(3) * X(4) + X(2) * X(3) ^ 2.3 * X(4) ^ 1.4

 

365 POBA = -ups / dow + 1000000 * (X(6) + X(7) + X(8) + X(9))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 9

 

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 REM GOTO 128

1670 NEXT I
1774 IF M < -.005 THEN 1999

 

1777 FOR J44 = 6 TO 9

1778 IF A(J44) < 0 THEN 1999
1779 NEXT J44

1889 REM IF M < -.0025 THEN 1999

1906 PRINT A(1), A(2), A(3), A(4), A(5)

1908 PRINT A(6), A(7), A(8), A(9), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [30]. The complete output through JJJJ = -31999.80000000003 is shown below:

1          1          32007          5          1
0          0          0          0         -4.96672290889686D-03
-31999.91000000001

1          0          32007          1          2
0          0          0          0          -2.444457626064133D-03
-31999.90000000002

1 0 32007 1 2
0 0 0 0 -2.444457626064133D-03
-31999.87000000002

1 1 32007 5 1
0 0 0 0 -4.96672290889686D-03
-31999.86000000002

1 1 32007 5 1
0 0 0 0 -4.96672290889686D-03
-31999.84000000003

1 1 32007 5 1
0 0 0 0 -4.96672290889686D-03
-31999.82000000003

1 0 32007 1 2
0 0 0 0 -2.444457626064133D-03
-31999.80000000003

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. The computational process here can be compared with that on pages 383-385 of Chang [5].

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [30], the wall-clock time for obtaining the output through
JJJJ=-31999.80000000003 was 2 seconds, not including the time for creating the .EXE file.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66. http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[4] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[5] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of  Operational Research 173 (2006) pp. 370-386. 

[6] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[7] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[8] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Sinulation 18 (2013) 89-98.
[9] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[11] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[12] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[13] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.
[14] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[15] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[16] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[17] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.
[18] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[19] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[20] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[21] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[22] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[23] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[24] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[25] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[26] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[27] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[28] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.
[29] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[30] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[31] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/
[32] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[33] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms. https://arxiv.org/pdf/1403.7793.pdf.
[34] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[35] B. D. Youn, K. K. Choi (2004). A new responsesurface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

Using Integers in and Expanding Chang's Posynomial Fractional Programming Model To Make It More Challenging for Computers

 Jsun Yui Wong

The computer program listed below seeks to solve the following integer fractional nonlinear programming problem:

Minimize (X(1) * X(2) ^ .5 * X(3) ^ 1.2) / (2 * X(2) * X(3) + X(2) ^ 2.1 * X(3) ^ 1.3)

subject to

2 * X(1) + 3 * X(2) + X(3) >= 8,

3 * X(1) + X(2) >= 5,

X(2) + 2 * X(3) >= 10,

2 * X(1) + X(3) <= 4,

0 <= X(1) <=30000,

1 <= X(2) <=30000,

1 <= X(3) <=30000,

X(1) through X(3) are integer variables.

This integer fractional nonlinear programming problem is based on the illustrative example in Chang [4, p. 49].

X(4) through X(7) below are slack variables.

One notes line 128, which is 128 FOR I = 1 TO 280.

 

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
70 FOR J44 = 2 TO 3

72 A(J44) = 1 + RND * 29999

73 NEXT J44

75 A(1) = RND * 30000

128 FOR I = 1 TO 280

 

129 FOR KKQQ = 1 TO 3

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))

181 J = 1 + FIX(RND * 3)

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

191 NEXT IPP

196 FOR J99 = 2 TO 3

199 X(J99) = INT(X(J99))

201 IF X(J99) < 1 THEN 1670
203 IF X(J99) > 30000 THEN 1670

204 NEXT J99
206 X(1) = INT(X(1))

 

209 IF X(1) < 0 THEN 1670

212 IF X(1) > 30000 THEN 1670

305 X(4) = -8 + 2 * X(1) + 3 * X(2) + X(3)

 

306 X(5) = -5 + 3 * X(1) + X(2)

 

307 X(6) = -10 + X(2) + 2 * X(3)

311 X(7) = 4 - 2 * X(1) - X(3)

325 FOR J99 = 4 TO 7

327 XX(J99) = X(J99)

 

330 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0

331 NEXT J99

 

357 POBA = -(X(1) * X(2) ^ .5 * X(3) ^ 1.2) / (2 * X(2) * X(3) + X(2) ^ 2.1 * X(3) ^ 1.3) + 1000000 * (X(6) + X(7) + X(4) + X(5))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 7

 

1456 XXX(KLX) = XX(KLX)

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 REM GOTO 128

1670 NEXT I

 

1889 REM IF M < -99 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), M, JJJJ

1903 REM PRINT XXX(4), XXX(5), XXX(6), XXX(7)

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [30]. The complete output through JJJJ = -31999.90000000002 is shown below:

5          25247          1          0          0
0          -7          -7000000.000000453          -32000

0          7207           1          0          0
0          0          0          -31999.99

0          13800          1          0          0
0          0          0          -31999.98

0 9872 1 0 0
0 0 0 -31999.97000000001

0 26158 4 0 0
0 0 0 -31999.96000000001

0 13889 1 0 0
0 0 0 -31999.95000000001

0 9367 4 0 0
0 0 0 -31999.94000000001

5 28475 12 0 0
0 -8 -8000000 -31999.93000000001

5 24939 23 0 0
0 -69 -69000000.00000168 -31999.92000000001

0 21739 1 0 0
0 0 0 -31999.91000000001

0 1815 1 0 0
0 0 0 -31999.90000000002

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. The computational results above can be cautiously compared with those on page 51 of Chang [4]; the former is optimization in integers and the latter is continuous optimization.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [30], the wall-clock time for obtaining the output through JJJJ= -31999.90000000002 was less than 2 seconds, not including the time for creating the .EXE file.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66. http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[4] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[5] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[6] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[7] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[8] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Sinulation 18 (2013) 89-98.
[9] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
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