Substance

Solving a Reinforced Concrete Beam Design Problem by Using the Method of This Blog

 Jsun Yui Wong

The computer program listed below seeks to solve the reinforced concrete beam design problem on pages 28-29 of Gandomi, Yang, and Alavi [5]--also see Gandomi, Yang, and Alavi [6].

 

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 IF RND < .091 THEN A(1) = 6.0 ELSE IF RND < .1 THEN A(1) = 6.16 ELSE IF RND < .111 THEN A(1) = 6.32 ELSE IF RND < .125 THEN A(1) = 6.6 ELSE IF RND < .143 THEN A(1) = 7 ELSE IF RND < .1666 THEN A(1) = 7.11 ELSE IF RND < .2 THEN A(1) = 7.2 ELSE IF RND < .25 THEN A(1) = 7.8 ELSE IF RND < .333 THEN A(1) = 7.9 ELSE IF RND < .5 THEN A(1) = 8 ELSE A(1) = 8.4

 

 

74 A(2) = 28 + FIX(RND * 13)

 

76 A(3) = 5 + RND * 5

 

128 FOR I = 1 TO 5000

 

129 FOR KKQQ = 1 TO 3

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

 

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

 

193 X(1) = A(1)

197 X(3) = (X(3))

 

198 X(2) = A(2)

 

200 X(4) = 4 - X(2) / X(3)

 

202 X(5) = X(1) * X(2) - 180 - 7.375 * (X(1) ^ 2 / X(3))

 

221 IF X(1) < 6 THEN 1670

222 IF X(1) > 8.4 THEN 1670
223 IF X(2) < 28 THEN 1670

224 IF X(2) > 40 THEN 1670
225 IF X(3) < 5 THEN 1670

226 IF X(3) > 10 THEN 1670

 

268 FOR J99 = 4 TO 5

 

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

270 NEXT J99

 

328 POBA = -29.4 * X(1) - .6 * X(2) * X(3) + 1000000 * X(4) + 1000000 * X(5)

 

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 5

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128

1670 NEXT I
1889 IF M < -359.2080 THEN 1999

 

1900 PRINT A(1), A(2), A(3), A(4), A(5)
1909 PRINT M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [19]. The complete output through JJJJ= -31995.81000000067 is shown below:

6.32       34        8.5       0       0
-359.208         -31999.98

6.32 34 8.5 0 0
-359.208 -31999.18000000013

6.32 34 8.5 0 0
-359.208 -31998.1300000003

6.32       34                               8.500000000000028          0
0
-359.2080000000005        -31995.81000000067

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [19], the wall-clock time for obtaining the output through
JJJJ= -31995.81000000067 was 30 seconds.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
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] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846.

[4] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.

[5] 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.

[6] 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.

[7] 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.

[8] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.

[9] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[10] 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/

[11] 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.

[12] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. www.springer.com/cda/content/document/cda.../

[13] 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.

[14] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).

[15] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.

[16] 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

[17] 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.

[18] 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.

[19] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[20] 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/

[21] 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.

 

Solving Thanedar and Vanderplaats' Stepped Cantilever Beam Problem Involving Discrete Non-Integer Variables, Integer Variables, and Continuous Variables Using the Method of This Blog, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve the following problem in Thanedar and Vanderplaats [12] and Mathworks [8]. Also see Nigdeli, Bekdas, and Yang [10, pp. 34-36].

 

Minimize (X(1) * X(6) * 100 + X(2) * X(7) * 100 + X(3) * X(8) * 100 + X(4) * X(9) * 100 + X(5) * X(10) * 100)

subject to

600 * 50000 / (X(5) * X(10) ^ 2)<=14000

1200 * 50000 / (X(4) * X(9) ^ 2)<=14000
1800 * 50000 / (X(3) * X(8) ^ 2)<=14000
2400 * 50000 / (X(2) * X(7) ^ 2)<=14000
3000 * 50000 / (X(1) * X(6) ^ 2)<=14000
(50000 * 1000000 / 6E+07) * (((12 * (1) / (X(5) * X(10) ^ 3)) + (12 * (7) / (X(4) * X(9) ^ 3)) + 12 * (19) / (X(3) * X(8) ^ 3)) + (12 * (37) / (X(2) * X(7) ^ 3)) + (12 * (61) / (X(1) * X(6) ^ 3)))<=2.7

X(10) / X(5)<=20

X(9) / X(4)<=20
X(8) / X(3)<=20
X(7) / X(2)<=20
X(6) / X(1)<=20

In this paper the bounds on the ten variables are from Mathworks [8, pp. 3-4] and are used in line 221 through line 240 of the following computer program. X(1) and X(6) are integers, Mathworks [8, p. 3]. Furthermore, there is the following restriction. Following Thanedar and Vanderplaats [12, Case 2, pp. 304-305] and Mathworks [8, p. 6], X(2) and X(3) choose from 2.4, 2.6, 2.8, and 3.1; X(7) and X(8) choose from 45, 50, 55, and 60. This restriction is handled by line 197 through line 200. The other variables are continuous.

The X(11) through X(21) below are slacks variables.

One notes line 269, which is 269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0.

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
22 REM FOR J44 = 1 TO 4

 

68 A(1) = 1 + RND * 4
69 REM A(2) = 2.4 + RND * .7
70 IF RND < .25 THEN A(2) = 2.4 ELSE IF RND < .333 THEN A(2) = 2.6 ELSE IF RND < .5 THEN A(2) = 2.8 ELSE A(2) = 3.1

72 REM A(3) = 2.4 + RND * .7
73 IF RND < .25 THEN A(3) = 2.4 ELSE IF RND < .333 THEN A(3) = 2.6 ELSE IF RND < .5 THEN A(3) = 2.8 ELSE A(3) = 3.1

 

74 A(4) = 1 + RND * 4

 

76 A(5) = 1 + RND * 4

78 A(6) = 30 + RND * 35

82 REM A(7) = 45 + RND * 15
83 IF RND < .25 THEN A(7) = 45 ELSE IF RND < .333 THEN A(7) = 50 ELSE IF RND < .5 THEN A(7) = 55 ELSE A(7) = 60

88 REM A(8) = 45 + RND * 15

89 IF RND < .25 THEN A(8) = 45 ELSE IF RND < .333 THEN A(8) = 50 ELSE IF RND < .5 THEN A(8) = 55 ELSE A(8) = 60

 

90 A(9) = 30 + RND * 35

92 A(10) = 30 + RND * 35

 

128 FOR I = 1 TO 10000

 

129 FOR KKQQ = 1 TO 10

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

 

181 J = 1 + FIX(RND * 10)

 

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
191 NEXT IPP

 

192 REM FOR J100 = 1 TO 10

 

193 X(1) = INT(X(1))

194 X(6) = INT(X(6))
195 REM NEXT J100

197 X(2) = A(2)

 

198 X(3) = A(3)

 

199 X(7) = A(7)

 

200 X(8) = A(8)

 

201 X(11) = 14000 - 600 * 50000 / (X(5) * X(10) ^ 2)

202 X(12) = 14000 - 1200 * 50000 / (X(4) * X(9) ^ 2)
203 X(13) = 14000 - 1800 * 50000 / (X(3) * X(8) ^ 2)
204 X(14) = 14000 - 2400 * 50000 / (X(2) * X(7) ^ 2)
205 X(15) = 14000 - 3000 * 50000 / (X(1) * X(6) ^ 2)
206 X(16) = 2.7 - (50000 * 1000000 / 6E+07) * (((12 * (1) / (X(5) * X(10) ^ 3)) + (12 * (7) / (X(4) * X(9) ^ 3)) + 12 * (19) / (X(3) * X(8) ^ 3)) + (12 * (37) / (X(2) * X(7) ^ 3)) + (12 * (61) / (X(1) * X(6) ^ 3)))

 

207 X(17) = 20 - X(10) / X(5)

 

208 X(18) = 20 - X(9) / X(4)
209 X(19) = 20 - X(8) / X(3)
210 X(20) = 20 - X(7) / X(2)
211 X(21) = 20 - X(6) / X(1)

220 REM FOR J44 = 1 TO 5
221 IF X(1) < 1 THEN 1670

222 IF X(1) > 5 THEN 1670
223 IF X(2) < 2.4 THEN 1670

224 IF X(2) > 3.1 THEN 1670
225 IF X(3) < 2.4 THEN 1670

226 IF X(3) > 3.1 THEN 1670

227 IF X(4) < 1 THEN 1670

228 IF X(4) > 5 THEN 1670
229 IF X(5) < 1 THEN 1670

230 IF X(5) > 5 THEN 1670
231 IF X(6) < 30 THEN 1670

232 IF X(6) > 65 THEN 1670

233 IF X(7) < 45 THEN 1670

234 IF X(7) > 60 THEN 1670
235 IF X(8) < 45 THEN 1670

236 IF X(8) > 60 THEN 1670
237 IF X(9) < 30 THEN 1670

238 IF X(9) > 65 THEN 1670
239 IF X(10) < 30 THEN 1670

240 IF X(10) > 65 THEN 1670

 

246 REM FOR J44 = 6 TO 10
247 REM IF X(J44) < 30 THEN 1670
248 REM IF X(J44) > 65 THEN 1670

249 REM NEXT J44

 

268 FOR J99 = 11 TO 21

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

270 NEXT J99
318 POBAprel1 = -(X(1) * X(6) * 100 + X(2) * X(7) * 100 + X(3) * X(8) * 100 + X(4) * X(9) * 100 + X(5) * X(10) * 100)

 

319 POBAprel2 = 1000000 * X(11) + 1000000 * X(12) + 1000000 * X(13) + 1000000 * X(14) + 1000000 * X(15) + 1000000 * X(16) + 1000000 * X(17) + 1000000 * X(18) + 1000000 * X(19) + 1000000 * X(20) + 1000000 * X(21)

 

328 POBA = POBAprel1 + POBAprel2

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 21

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128

1670 NEXT I
1889 IF M < -63893.6 THEN 1999

 

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

1901 PRINT A(6), A(7), A(8), A(9), A(10)

1902 PRINT A(11), A(12)

1903 PRINT A(13), A(14), A(15), A(16)

1904 PRINT A(17), A(18), A(19), A(20)

1909 PRINT A(21), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [17]. The complete output through JJJJ= -31989.86000000162 is shown below:

3       3.1       2.6       2.204591257008201
1.749804904496618
60       55       50       44.09075818047691
34.99466132300527
0 0
0 0 0 0
0 0 0 0
0       -63893.5930013735       -31996.06000000063

3       3.1       2.6       2.204598926059098
1.749771736757567
60       55       50       44.09068149202357
34.99499299149384
0 0
0 0 0 0
0 0 0 0
0       -63893.5518731074       -31989.86000000162

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 [17], the wall-clock time for obtaining the output through JJJJ= -31989.86000000162 was 7.5 minutes.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
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] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846.

[4] 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.

[5] 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.

[6] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.

[7] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[8] 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/

[9] 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.

[10] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. www.springer.com/cda/content/document/cda.../

[11] 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.

[12] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).

[13] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.

[14] 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

[15] 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.

[16] 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.

[17] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[18] 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/

[19] 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.

 

Applying the Method of This Blog To Solve Thanedar and Vanderplaats' Stepped Cantilever Beam Problem Where 2 of the 10 Design Variables Are Integers, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve the following problem in Mathworks [8]. Also see Thanedar and Vanderplaats [12] and Nigdeli, Bekdas, and Yang [10, pp. 38-40].

Minimize (X(1) * X(6) * 100 + X(2) * X(7) * 100 + X(3) * X(8) * 100 + X(4) * X(9) * 100 + X(5) * X(10) * 100)

subject to

600 * 50000 / (X(5) * X(10) ^ 2)<=14000

1200 * 50000 / (X(4) * X(9) ^ 2)<=14000
1800 * 50000 / (X(3) * X(8) ^ 2)<=14000
2400 * 50000 / (X(2) * X(7) ^ 2)<=14000
3000 * 50000 / (X(1) * X(6) ^ 2)<=14000
(50000 * 1000000 / 6E+07) * (((12 * (1) / (X(5) * X(10) ^ 3)) + (12 * (7) / (X(4) * X(9) ^ 3)) + 12 * (19) / (X(3) * X(8) ^ 3)) + (12 * (37) / (X(2) * X(7) ^ 3)) + (12 * (61) / (X(1) * X(6) ^ 3)))<=2.7

X(10) / X(5)<=20

X(9) / X(4)<=20
X(8) / X(3)<=20
X(7) / X(2)<=20
X(6) / X(1)<=20

In this paper the bounds on the ten variables are from Mathworks [8, pp. 3-4] and are used in line 221 through line 240 of the following computer program. X(1) and X(6) are integers, Mathworks [8, p. 3].

The X(11) through X(21) below are slack variables.

One notes line 269, which is 269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0.

 

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
22 REM FOR J44 = 1 TO 4

 

68 A(1) = 1 + RND * 4
69 A(2) = 2.4 + RND * .7
70 REM IF RND < .25 THEN A(2) = 2.4 ELSE IF RND < .333 THEN A(2) = 2.6 ELSE IF RND < .5 THEN A(2) = 2.8 ELSE A(2) = 3.1

72 A(3) = 2.4 + RND * .7
73 IF RND < .25 THEN A(3) = 2.4 ELSE IF RND < .333 THEN A(3) = 2.6 ELSE IF RND < .5 THEN A(3) = 2.8 ELSE A(3) = 3.1

 

74 A(4) = 1 + RND * 4

 

76 A(5) = 1 + RND * 4

78 A(6) = 30 + RND * 35

82 A(7) = 45 + RND * 15
83 IF RND < .25 THEN A(7) = 45 ELSE IF RND < .333 THEN A(7) = 50 ELSE IF RND < .5 THEN A(7) = 55 ELSE A(7) = 60

88 A(8) = 45 + RND * 15

89 REM IF RND < .25 THEN A(8) = 45 ELSE IF RND < .333 THEN A(8) = 50 ELSE IF RND < .5 THEN A(8) = 55 ELSE A(8) = 60

 

90 A(9) = 30 + RND * 35

92 A(10) = 30 + RND * 35

 

128 FOR I = 1 TO 10000

 

129 FOR KKQQ = 1 TO 10

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

 

181 J = 1 + FIX(RND * 10)

 

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
191 NEXT IPP

 

192 REM FOR J100 = 1 TO 10

 

193 X(1) = INT(X(1))

194 X(6) = INT(X(6))
195 REM NEXT J100

 

201 X(11) = 14000 - 600 * 50000 / (X(5) * X(10) ^ 2)

202 X(12) = 14000 - 1200 * 50000 / (X(4) * X(9) ^ 2)
203 X(13) = 14000 - 1800 * 50000 / (X(3) * X(8) ^ 2)
204 X(14) = 14000 - 2400 * 50000 / (X(2) * X(7) ^ 2)
205 X(15) = 14000 - 3000 * 50000 / (X(1) * X(6) ^ 2)
206 X(16) = 2.7 - (50000 * 1000000 / 6E+07) * (((12 * (1) / (X(5) * X(10) ^ 3)) + (12 * (7) / (X(4) * X(9) ^ 3)) + 12 * (19) / (X(3) * X(8) ^ 3)) + (12 * (37) / (X(2) * X(7) ^ 3)) + (12 * (61) / (X(1) * X(6) ^ 3)))

 

207 X(17) = 20 - X(10) / X(5)

 

208 X(18) = 20 - X(9) / X(4)
209 X(19) = 20 - X(8) / X(3)
210 X(20) = 20 - X(7) / X(2)
211 X(21) = 20 - X(6) / X(1)

220 REM FOR J44 = 1 TO 5
221 IF X(1) < 1 THEN 1670

222 IF X(1) > 5 THEN 1670
223 IF X(2) < 2.4 THEN 1670

224 IF X(2) > 3.1 THEN 1670
225 IF X(3) < 2.4 THEN 1670

226 IF X(3) > 3.1 THEN 1670

227 IF X(4) < 1 THEN 1670

228 IF X(4) > 5 THEN 1670
229 IF X(5) < 1 THEN 1670

230 IF X(5) > 5 THEN 1670
231 IF X(6) < 30 THEN 1670

232 IF X(6) > 65 THEN 1670

233 IF X(7) < 45 THEN 1670

234 IF X(7) > 60 THEN 1670
235 IF X(8) < 45 THEN 1670

236 IF X(8) > 60 THEN 1670
237 IF X(9) < 30 THEN 1670

238 IF X(9) > 65 THEN 1670
239 IF X(10) < 30 THEN 1670

240 IF X(10) > 65 THEN 1670

 

246 REM FOR J44 = 6 TO 10
247 REM IF X(J44) < 30 THEN 1670
248 REM IF X(J44) > 65 THEN 1670

249 REM NEXT J44

 

268 FOR J99 = 11 TO 21

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

270 NEXT J99
318 POBAprel1 = -(X(1) * X(6) * 100 + X(2) * X(7) * 100 + X(3) * X(8) * 100 + X(4) * X(9) * 100 + X(5) * X(10) * 100)

 

319 POBAprel2 = 1000000 * X(11) + 1000000 * X(12) + 1000000 * X(13) + 1000000 * X(14) + 1000000 * X(15) + 1000000 * X(16) + 1000000 * X(17) + 1000000 * X(18) + 1000000 * X(19) + 1000000 * X(20) + 1000000 * X(21)

 

328 POBA = POBAprel1 + POBAprel2

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 21

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128

1670 NEXT I
1889 IF M < -62010.4 THEN 1999

 

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

1901 PRINT A(6), A(7), A(8), A(9), A(10)

1902 PRINT A(11), A(12)

1903 PRINT A(13), A(14), A(15), A(16)

1904 PRINT A(17), A(18), A(19), A(20)

1909 PRINT A(21), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [17]. The complete output through JJJJ= -31999.63000000006 is shown below:

3          2.777569257024501           2.52359769495554
2.2045555339516123          1.74977312884507
60          55.55129107295518          50.47161200912944
44.09110735109591          34.99497907080316
0 0
0 0 0 0
0 0 0 0
0          -62010.22062489206          -31999.63000000006

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 [17], the wall-clock time for obtaining the output through JJJJ= -31999.63000000006 was 2 minutes.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
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] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846.

[4] 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.

[5] 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.

[6] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.

[7] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[8] 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/

[9] 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.

[10] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. www.springer.com/cda/content/document/cda.../

[11] 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.

[12] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).

[13] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.

[14] 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

[15] 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.

[16] 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.

[17] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[18] 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/

[19] 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.

Applying the Method of This Blog To Solve Thanedar and Vanderplaats' Stepped Cantilever Beam Problem Where All Design Variables Are Integers, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve the following problem in Thanedar and Vanderplaats [12], Mathworks [8]. Also see Nigdeli, Bekdas, and Yang [10, pp. 38-40].

Minimize (X(1) * X(6) * 100 + X(2) * X(7) * 100 + X(3) * X(8) * 100 + X(4) * X(9) * 100 + X(5) * X(10) * 100)

subject to

600 * 50000 / (X(5) * X(10) ^ 2)<=14000

1200 * 50000 / (X(4) * X(9) ^ 2)<=14000
1800 * 50000 / (X(3) * X(8) ^ 2)<=14000
2400 * 50000 / (X(2) * X(7) ^ 2)<=14000
3000 * 50000 / (X(1) * X(6) ^ 2)<=14000
(50000 * 1000000 / 6E+07) * (((12 * (1) / (X(5) * X(10) ^ 3)) + (12 * (7) / (X(4) * X(9) ^ 3)) + 12 * (19) / (X(3) * X(8) ^ 3)) + (12 * (37) / (X(2) * X(7) ^ 3)) + (12 * (61) / (X(1) * X(6) ^ 3)))<=2.7

X(10) / X(5)<=20

X(9) / X(4)<=20
X(8) / X(3)<=20
X(7) / X(2)<=20
X(6) / X(1)<=20

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

30<= X(j) <=65, j=6, 7, 8, 9, 10;

these ten design variables are integers,

and whereas the bounds on the variables in the earlier edition are from Nigdeli, Bekdas, and Yang [10, p. 39], here the bounds are from Mathworks [8, pp. 3-4] and are used in line 221 through line 240 of the following computer program.

The X(11) through X(21) below are slack variables.

It is important to note line 269, which is 269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0.

 

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
22 REM FOR J44 = 1 TO 4

 

68 A(1) = 1 + RND * 4
69 A(2) = 2.4 + RND * .7
70 REM IF RND < .25 THEN A(2) = 2.4 ELSE IF RND < .333 THEN A(2) = 2.6 ELSE IF RND < .5 THEN A(2) = 2.8 ELSE A(2) = 3.1

72 A(3) = 2.4 + RND * .7
73 IF RND < .25 THEN A(3) = 2.4 ELSE IF RND < .333 THEN A(3) = 2.6 ELSE IF RND < .5 THEN A(3) = 2.8 ELSE A(3) = 3.1

 

74 A(4) = 1 + RND * 4

 

76 A(5) = 1 + RND * 4

78 A(6) = 30 + RND * 35

82 A(7) = 45 + RND * 15
83 IF RND < .25 THEN A(7) = 45 ELSE IF RND < .333 THEN A(7) = 50 ELSE IF RND < .5 THEN A(7) = 55 ELSE A(7) = 60

88 A(8) = 45 + RND * 15

89 REM IF RND < .25 THEN A(8) = 45 ELSE IF RND < .333 THEN A(8) = 50 ELSE IF RND < .5 THEN A(8) = 55 ELSE A(8) = 60

 

90 A(9) = 30 + RND * 35

92 A(10) = 30 + RND * 35

 

128 FOR I = 1 TO 10000

 

129 FOR KKQQ = 1 TO 10

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

 

181 J = 1 + FIX(RND * 10)

 

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
191 NEXT IPP

 

192 FOR J100 = 1 TO 10

 

193 X(J100) = INT(X(J100))
195 NEXT J100

 

201 X(11) = 14000 - 600 * 50000 / (X(5) * X(10) ^ 2)

202 X(12) = 14000 - 1200 * 50000 / (X(4) * X(9) ^ 2)
203 X(13) = 14000 - 1800 * 50000 / (X(3) * X(8) ^ 2)
204 X(14) = 14000 - 2400 * 50000 / (X(2) * X(7) ^ 2)
205 X(15) = 14000 - 3000 * 50000 / (X(1) * X(6) ^ 2)
206 X(16) = 2.7 - (50000 * 1000000 / 6E+07) * (((12 * (1) / (X(5) * X(10) ^ 3)) + (12 * (7) / (X(4) * X(9) ^ 3)) + 12 * (19) / (X(3) * X(8) ^ 3)) + (12 * (37) / (X(2) * X(7) ^ 3)) + (12 * (61) / (X(1) * X(6) ^ 3)))

 

207 X(17) = 20 - X(10) / X(5)

 

208 X(18) = 20 - X(9) / X(4)
209 X(19) = 20 - X(8) / X(3)
210 X(20) = 20 - X(7) / X(2)
211 X(21) = 20 - X(6) / X(1)

220 REM FOR J44 = 1 TO 5
221 IF X(1) < 1 THEN 1670

222 IF X(1) > 5 THEN 1670
223 IF X(2) < 2.4 THEN 1670

224 IF X(2) > 3.1 THEN 1670
225 IF X(3) < 2.4 THEN 1670

226 IF X(3) > 3.1 THEN 1670

227 IF X(4) < 1 THEN 1670

228 IF X(4) > 5 THEN 1670
229 IF X(5) < 1 THEN 1670

230 IF X(5) > 5 THEN 1670
231 IF X(6) < 30 THEN 1670

232 IF X(6) > 65 THEN 1670

233 IF X(7) < 45 THEN 1670

234 IF X(7) > 60 THEN 1670
235 IF X(8) < 45 THEN 1670

236 IF X(8) > 60 THEN 1670
237 IF X(9) < 30 THEN 1670

238 IF X(9) > 65 THEN 1670
239 IF X(10) < 30 THEN 1670

240 IF X(10) > 65 THEN 1670

 

246 REM FOR J44 = 6 TO 10
247 REM IF X(J44) < 30 THEN 1670
248 REM IF X(J44) > 65 THEN 1670

249 REM NEXT J44

 

268 FOR J99 = 11 TO 21

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

270 NEXT J99
318 POBAprel1 = -(X(1) * X(6) * 100 + X(2) * X(7) * 100 + X(3) * X(8) * 100 + X(4) * X(9) * 100 + X(5) * X(10) * 100)

 

319 POBAprel2 = 1000000 * X(11) + 1000000 * X(12) + 1000000 * X(13) + 1000000 * X(14) + 1000000 * X(15) + 1000000 * X(16) + 1000000 * X(17) + 1000000 * X(18) + 1000000 * X(19) + 1000000 * X(20) + 1000000 * X(21)

 

328 POBA = POBAprel1 + POBAprel2

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 21

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128

1670 NEXT I
1889 IF M < -66301 THEN 1999

 

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

1901 PRINT A(6), A(7), A(8), A(9), A(10)

1902 PRINT A(11), A(12)

1903 PRINT A(13), A(14), A(15), A(16)

1904 PRINT A(17), A(18), A(19), A(20)

1909 PRINT A(21), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [17]. The complete output through JJJJ= -31999.97000000001 is shown below:

3       3        3       3       2
60       54       47       38       33
0       0
0       0       0       0
0       0       0       0
0       -66300       -32000

3       3       3       3       2
60       54       47       38       33
0       0
0       0       0       0
0       0       0       0
0       -66300       -31999.97000000001

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 [17], the wall-clock time for obtaining the output through JJJJ= -31999.97000000001 was eight seconds.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
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] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846.

[4] 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.

[5] 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.

[6] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.

[7] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[8] 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/

[9] 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.

[10] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. www.springer.com/cda/content/document/cda.../

[11] 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.

[12] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (3), 301-306 (1995).

[13] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.

[14] 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

[15] 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.

[16] 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.

[17] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[18] 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/

[19] 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.

 

Solving Thanedar and Vanderplaats' Stepped Cantilever Beam Problem Involving Integer Variables and Continuous Variables Using the Method of This Blog

 Jsun Yui Wong

The computer program listed below seeks to solve the following problem in Thanedar and Vanderplaats [12], and Mathorks [8]. Also see Nigdeli, Bekdas, and Yang [10, pp. 34-36].

Minimize (X(1) * X(6) * 100 + X(2) * X(7) * 100 + X(3) * X(8) * 100 + X(4) * X(9) * 100 + X(5) * X(10) * 100)

subject to

600 * 50000 / (X(5) * X(10) ^ 2)<=14000

1200 * 50000 / (X(4) * X(9) ^ 2)<=14000
1800 * 50000 / (X(3) * X(8) ^ 2)<=14000
2400 * 50000 / (X(2) * X(7) ^ 2)<=14000
3000 * 50000 / (X(1) * X(6) ^ 2)<=14000
(50000 * 1000000 / 6E+07) * (((12 * (1) / (X(5) * X(10) ^ 3)) + (12 * (7) / (X(4) * X(9) ^ 3)) + 12 * (19) / (X(3) * X(8) ^ 3)) + (12 * (37) / (X(2) * X(7) ^ 3)) + (12 * (61) / (X(1) * X(6) ^ 3)))<=2.7

X(10) / X(5)<=20

X(9) / X(4)<=20
X(8) / X(3)<=20
X(7) / X(2)<=20
X(6) / X(1)<=20

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

30<= X(j) <=65, j=6, 7, 8, 9, 10,

and there is the following restriction. Suppose "The first step of the beam can only be machined to the nearest centimeter," Mathworks [8, p. 3]. So X(1) and X(6) must be integers. This restriction is handled by line 193 and line 195, which are 193 X(1) = INT(X(1)) and 195 X(6) = INT(X(6)). The other variables are continuous.

The X(11) through X(21) below are slacks variables.

It is important to note line 269, which is 269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0.

 

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
22 REM FOR J44 = 1 TO 4

 

68 A(1) = 2 + RND * 2

70 A(2) = 2 + RND * 2

72 A(3) = 2 + RND * 2

 

74 A(4) = 2 + RND * 2

 

76 A(5) = 2 + RND * 2

78 A(6) = 40 + RND * 15

82 A(7) = 40 + RND * 15

88 A(8) = 40 + RND * 15

90 A(9) = 40 + RND * 15

92 A(10) = 40 + RND * 15

 

128 FOR I = 1 TO 50000

 

129 FOR KKQQ = 1 TO 10

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

 

 

181 J = 1 + FIX(RND * 10)

 

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
191 NEXT IPP

193 X(1) = INT(X(1))
195 X(6) = INT(X(6))

 

201 X(11) = 14000 - 600 * 50000 / (X(5) * X(10) ^ 2)

202 X(12) = 14000 - 1200 * 50000 / (X(4) * X(9) ^ 2)
203 X(13) = 14000 - 1800 * 50000 / (X(3) * X(8) ^ 2)
204 X(14) = 14000 - 2400 * 50000 / (X(2) * X(7) ^ 2)
205 X(15) = 14000 - 3000 * 50000 / (X(1) * X(6) ^ 2)
206 X(16) = 2.7 - (50000 * 1000000 / 6E+07) * (((12 * (1) / (X(5) * X(10) ^ 3)) + (12 * (7) / (X(4) * X(9) ^ 3)) + 12 * (19) / (X(3) * X(8) ^ 3)) + (12 * (37) / (X(2) * X(7) ^ 3)) + (12 * (61) / (X(1) * X(6) ^ 3)))

 

 

207 X(17) = 20 - X(10) / X(5)

 

208 X(18) = 20 - X(9) / X(4)
209 X(19) = 20 - X(8) / X(3)
210 X(20) = 20 - X(7) / X(2)
211 X(21) = 20 - X(6) / X(1)

221 FOR J44 = 1 TO 5
231 IF X(J44) < 1 THEN 1670
233 IF X(J44) > 5 THEN 1670

244 NEXT J44

 

246 FOR J44 = 6 TO 10
247 IF X(J44) < 30 THEN 1670
248 IF X(J44) > 65 THEN 1670

249 NEXT J44

 

268 FOR J99 = 11 TO 21

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

270 NEXT J99
318 POBAprel1 = -(X(1) * X(6) * 100 + X(2) * X(7) * 100 + X(3) * X(8) * 100 + X(4) * X(9) * 100 + X(5) * X(10) * 100)

 

319 POBAprel2 = 1000000 * X(11) + 1000000 * X(12) + 1000000 * X(13) + 1000000 * X(14) + 1000000 * X(15) + 1000000 * X(16) + 1000000 * X(17) + 1000000 * X(18) + 1000000 * X(19) + 1000000 * X(20) + 1000000 * X(21)

 

328 POBA = POBAprel1 + POBAprel2

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 21

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128

1670 NEXT I
1889 IF M < -62010.3 THEN 1999

 

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

1901 PRINT A(5), A(6), A(7), A(8)

1902 PRINT A(9), A(10), A(11), A(12)

1903 PRINT A(13), A(14), A(15), A(16)

1904 PRINT A(17), A(18), A(19), A(20)

1909 PRINT A(21), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [17]. The complete output through JJJJ= -31999.94000000001 is shown below:

3           2.777580128381755          2.52358705118924
2.204574291429354
1.749760944702395           60          55.55118235988586
50.47171844640813
44..09092783313881          34.99510091118248          0
0
0 0 0 0
0 0 0 0
0          -62010.24221700657          -31999.98

3          2.777578775147318          2.523600644664524
2.204555895592278
1.749794347128448          60          55.55119589213281
50.47158251226527
44.09111179033276          34.99476689283034          0
0
0 0 0 0
0 0 0 0
0          -62010.29065379788          -31999.94000000001

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 [17], the wall-clock time for obtaining the output through JJJJ= -31999.94000000001 was 70 seconds.

 

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
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] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846.

[4] 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.

[5] 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.

[6] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.

[7] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[8] 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/

[9] 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.

[10] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. www.springer.com/cda/content/document/cda.../

[11] 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.

[12] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (3), 301-306 (1995).

[13] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.

[14] 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

[15] 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.

[16] 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.

[17] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[18] 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/

[19] 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.

 

Vertical Deflection Minimization Problem of an I-Beam with the Method of This Blog, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve the following problem in Nigdeli, Bekdas, and Yang [9, pp. 33-34].

Minimize (5000) / (((X(4) * (X(2) - 2 * X(3)) ^ 3 / 12) + (X(1) * X(3) ^ 3 / 6) + (2 * X(1) * X(3) * ((X(2) - X(3)) / 2) ^ 2)))

subject to

2 * X(1) * X(4) + X(4) * (X(2) - 2 * X(3))<=300

(180000 * X(2)) / (((X(4) * (X(2) - 2 * X(3)) ^ 3) + 2 * X(1) * X(4) * (4 * X(3) ^ 2 + 3 * X(2) * (X(2) - 2 * X(3)))) + (15000 * X(1)) / ((X(4) ^ 3 * (X(2) - 2 * X(3))) + (2 * X(4) * X(1) ^ 3)))<=6

10<=X(1) <= 50

10<=X(2) <= 80

.9<=X(3) <= 5

.9<=X(4) <= 5.

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

Whereas line 68 through line 74 of the earlier edition are

68 A(1) = 20 + RND * 20

70 A(2) = 30 + RND * 30

72 A(3) = 1.9 + RND * 2.1

74 A(4) = 1.9 + RND * 2.1,

here line 68 through line 74 are

68 A(1) = 25 + RND * 10

70 A(2) = 25 + RND * 40

72 A(3) = 2.4 + RND * 1.1

74 A(4) = 2.4 + RND * 1.1, respectively.

 

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
22 REM FOR J44 = 1 TO 4

 

68 A(1) = 25 + RND * 10

70 A(2) = 25 + RND * 40

72 A(3) = 2.4 + RND * 1.1

74 A(4) = 2.4 + RND * 1.1

71 REM NEXT J44

 

79 REM

 

128 FOR I = 1 TO 10000

 

129 FOR KKQQ = 1 TO 4

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

 

181 J = 1 + FIX(RND * 4)

 

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP

 

249 X(5) = 300 - 2 * X(1) * X(4) - X(4) * (X(2) - 2 * X(3))

 

251 X(6) = 6 - (180000 * X(2)) / (((X(4) * (X(2) - 2 * X(3)) ^ 3) + 2 * X(1) * X(4) * (4 * X(3) ^ 2 + 3 * X(2) * (X(2) - 2 * X(3)))) - (15000 * X(1)) / ((X(4) ^ 3 * (X(2) - 2 * X(3))) + (2 * X(4) * X(1) ^ 3)))

 

253 IF X(1) < 10 THEN GOTO 1670

255 IF X(2) < 10 THEN GOTO 1670
257 IF X(3) < .9 THEN GOTO 1670
259 IF X(4) < .9 THEN GOTO 1670

261 IF X(1) > 50 THEN GOTO 1670

 

263 IF X(2) > 80 THEN GOTO 1670

 

265 IF X(3) > 5 THEN GOTO 1670

 

267 IF X(4) > 5 THEN GOTO 1670

 

268 FOR J99 = 5 TO 6

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

270 NEXT J99
318 POBA = -(5000) / (((X(4) * (X(2) - 2 * X(3)) ^ 3 / 12) + (X(1) * X(3) ^ 3 / 6) + (2 * X(1) * X(3) * ((X(2) - X(3)) / 2) ^ 2))) + 1000000 * X(6) + 1000000 * X(5)

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 6

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128

1670 NEXT I
1889 REM IF M < -1.339964 THEN 1999

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

1901 PRINT A(5), A(6), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [15]. The complete output through JJJJ=-31999.98 is shown below:

49.99999995393713          79.99999999999871          5
1.764705883309277
0          0          -6.625958170975432D-03          -32000

49.9999999985057          79.99999999921377          5
1.764705882392126
0          0          -6.625958165838354D-03          -31999.99

49.99999998776885          80          5          1.764705882606874
0          0          -6.625958166967821D-03          -31999.98

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

The solution above at JJJJ=-31999.99 can be compared to the four solutions presented in Table 5 of Nigdeli, Bekdas, and Yang [9, p. 34]. .

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [15], the wall-clock time for obtaining the output through JJJJ= -31999.98 was 8 seconds.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
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] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846.

[4] 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.

[5] 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.

[6] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.

[7] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[8] 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.

[9] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algorithm in Structural Engineering. Springer International Publishing Switzerland 2016. www.springer.com/cda/content/document/cda.../

[10] 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.

[11] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.

[12] 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

[13] 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.

[14] 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.

[15] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[16] 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/

[17] 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.

 

Weight Optimization of a Cantilever Beam with the Method of This Blog

Jsun Yui Wong

The computer program listed below seeks to solve the following problem in Nigdeli, Bekdas, and Yang [9, pp. 34-36].

Minimize .0624 * (X(1) + X(2) + X(3) + X(4) + X(5))

subject to 61 / X(1) ^ 3 + 37 / X(2) ^ 3 + 19 / X(3) ^ 3 + 7 / X(4) ^ 3 + 1 / X(5) ^ 3 -1<=0

.1<= X(i) <=100, i=1, 2, 3, 4, 5.

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
22 FOR J44 = 1 TO 5

68 A(J44) = 1 + RND * 4

71 NEXT J44

 

79 REM

 

128 FOR I = 1 TO 10000

 

129 FOR KKQQ = 1 TO 5

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

 

181 J = 1 + FIX(RND * 5)

 

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP

 

249 X(6) = 1 - 61 / X(1) ^ 3 - 37 / X(2) ^ 3 - 19 / X(3) ^ 3 - 7 / X(4) ^ 3 - 1 / X(5) ^ 3
255 FOR J49 = 1 TO 5
257 IF X(J49) < .01 THEN GOTO 1670

 

260 IF X(J49) > 100 THEN GOTO 1670

 

261 NEXT J49

 

268 FOR J99 = 6 TO 6

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

270 NEXT J99
318 POBA = -.0624 * (X(1) + X(2) + X(3) + X(4) + X(5)) + 1000000 * X(6)

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 6

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128

1670 NEXT I
1889 IF M < -1.339964 THEN 1999

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

1901 PRINT A(5), A(6), M, JJJJ

 

1902 REM PRINT M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [15]. The complete output through JJJJ=-31997.95000000033 is shown below:

6.023896661955166          5.312695761698846          4.48158734864664
3.502135893370947
2.153442635081754          0       -1.33996251796701
-31999.31000000011

6.024555015295377          5.30267589953989          4.490121170715447
3.504891849893852
2.15147168792947          0          -1.33995985489854
-31997.95000000033

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

The solution above is comparable to the five solutions presented in Table 9 of Nigdeli, Bekdas, and Yang [9, p. 38]. .

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [15], the wall-clock time for obtaining the output through JJJJ= -31998.7500000002 was 50 seconds.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
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] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846.

[4] 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.

[5] 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.

[6] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.

[7] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[8] 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.

[9] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. www.springer.com/cda/content/document/cda.../

[10] 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.

[11] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.

[12] 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

[13] 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.

[14] 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.

[15] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[16] 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/

[17] 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.

 

Solving a Tubular Column Design Problem with the Method of This Blog

 Jsun Yui Wong

The computer program listed below seeks to solve the following problem in Nigdeli, Bekdas, and Yang [9, pp. 34-36].

Minimize 9.8 * X(1) * X(2) + 2 * X(1)

subject to

-1 + (2500) / (3.141592654 * X(1) * X(2) * 500)<=0

 

-1 + (8 * 2500 * 250 ^ 2) / (3.141592654 ^ 3 * .85D+06 * X(1) * X(2) * (X(1) ^ 2 + X(2) ^ 2))<=0

 

-1 + 2.0 / X(1)<=0

 

-1 + X(1) / 14<=0

-1 + .2 / X(2)<=0

-1 + X(2) / .8<=0

2<= X(1) <=14

.2<= X(2)<=.8.

One notes line 269, which is 269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0.

 

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

 

68 A(1) = 5 + RND * 9

 

69 A(2) = .3 + RND * .5

 

128 FOR I = 1 TO 10000

 

129 FOR KKQQ = 1 TO 2

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

 

181 J = 1 + FIX(RND * 2)

 

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP

239 X(3) = 1 - (2500) / (3.141592654 * X(1) * X(2) * 500)

 

243 X(4) = 1 - (8 * 2500 * 250 ^ 2) / (3.141592654 ^ 3 * .85D+06 * X(1) * X(2) * (X(1) ^ 2 + X(2) ^ 2))

 

247 X(5) =1 - 2.0 / X(1)

 

249 X(6) =1 - X(1) / 14

251 X(7) =1 - .2 / X(2)

253 X(8) =1 - X(2) / .8

 

260 IF X(1) < 2 THEN GOTO 1670

 

261 IF X(2) < .2 THEN GOTO 1670

 

264 IF X(1) > 14 THEN GOTO 1670

265 IF X(2) > .8 THEN GOTO 1670

 

268 FOR J99 = 3 TO 8

 

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

270 NEXT J99
318 POBA = -9.8 * X(1) * X(2) - 2 * X(1) + 1000000 * X(3) + 1000000 * X(4) + 1000000 * X(5) + 1000000 * X(6) + 1000000 * X(7) + 1000000 * X(8)

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 8

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128

1670 NEXT I
1889 IF M < -26.499497 THEN 1999

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

1901 PRINT A(5), A(6), A(7), A(8)

1902 PRINT M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [15]. The complete output through JJJJ= -31998.7500000002 is shown below:

5.451156238119991          .2919654768985374          0
0
0          0          0          0
-26.49949689720916          -31998.8000000002

5.451156240630642          .2919654767640663          0
0
0          0          0          0      
-26.49949690223046          -31998.7500000002

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

The solution above is comparable to the solutions presented in Table 11 of Gandomi, Yang, and Alavi [4, p. 25] and Table 7 and Table 8 of Nigdeli, Bekdas, and Yang[9, p. 36]. .

One notes that with the present algorithm none of the six constraints is violated because of line 269, which is 269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [15], the wall-clock time for obtaining the output through JJJJ= -31998.7500000002 was 30 seconds.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
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] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846.

[4] 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.

[5] 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.

[6] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.

[7] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[8] 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.

[9] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. www.springer.com/cda/content/document/cda.../

[10] 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.

[11] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.

[12] 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

[13] 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.

[14] 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.

[15] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[16] 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/

[17] 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.

Solving a Nonlinear Fractional Programming Problem with the Method of This Blog, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve the following problem in Tsai [11, pp. 406-408]; also see Nigdeli, Bekdas, and Yang [9].

Minimize 200 * 2 ^ .5 * X(1) + 100 * X(2)

subject to

-1 + (2 ^ .5 * X(1) + X(2)) / (2 ^ .5 * X(1) ^ 2 + 2 * X(1) * X(2))<=0

 

-1 + (1) / (2 ^ .5 * X(2) + X(1))<=0

-1 + (X(2)) / (2 ^ .5 * X(1) ^ 2 + 2 * X(1) * X(2))<=0

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

X(3) through X(5) below are slack variables.

One notes that line 269 is 269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0.

 

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

21 REM

67 REM

68 A(1) = .01 + RND

69 A(2) = .01 + RND

 

128 FOR I = 1 TO 10000

 

129 FOR KKQQ = 1 TO 2

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

 

181 J = 1 + FIX(RND * .2)

 

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP
225 REM

 

239 X(3) = 1 - (2 ^ .5 * X(1) + X(2)) / (2 ^ .5 * X(1) ^ 2 + 2 * X(1) * X(2))

 

241 REM

 

243 X(4) = 1 - (1) / (2 ^ .5 * X(2) + X(1))

 

245 REM

247 X(5) = 1 - (X(2)) / (2 ^ .5 * X(1) ^ 2 + 2 * X(1) * X(2))

 

249 REM

251 REM

253 REM

 

255 REM

 

260 IF X(1) < .01 THEN GOTO 1670

 

261 IF X(2) < .01 THEN GOTO 1670

 

262 REM

263 REM

 

264 IF X(1) > 1 THEN GOTO 1670

265 IF X(2) > 1 THEN GOTO 1670

266 REM

267 REM

 

268 FOR J99 = 3 TO 5

 

269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0
270 NEXT J99
318 POBA = -200 * 2 ^ .5 * X(1) - 100 * X(2) + 1000000 * X(4) + 1000000 * X(5) + 1000000 * X(3)

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 5

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128

1670 NEXT I
1889 IF M < -263.89585 THEN 1999

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

1902 PRINT A(5), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [15]. The complete output through JJJJ=-31952.68000000758 is shown below:

.7887039354961913          .408166835308075          0
0
0          -263.8958439859572          -31993.85000000098

.7886747466491203          .408249387741089          0
0
0          -263.895843376579          -31952.68000000758

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

The solution above is comparable to the six solutions presented in Table 4 of Nigdeli, Bekdas, and Yang [9, p. 32].

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [15], the wall-clock time for obtaining the output through JJJJ=-31952.68000000758 was three minutes and a half.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
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] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846.

[4] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (20130 29:17-35.

[5] 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.

[6] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.

[7] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[8] 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.

[9] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. www.springer.com/cda/content/document/cda.../

[10] 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.

[11] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.

[12] 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

[13] 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.

[14] 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.

[15] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[16] 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

[17] 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/

 

Solving a Mixed-Discrete Structural Optimization Problem with the Method of This Blog, Third Edition

Jsun Yui Wong

The computer program listed below seeks to solve the following problem in Lin and Tsai [5, pp. 873-875]:

Minimize 0.25 * PIE ^ 2 * X(2) * X(1) ^ 2 * X(3) + .5 * PIE ^ 2 * X(2) * X(1) ^ 2

subject to

4 * X(2) ^ 2 + 1.46 * X(1) * X(2) - 2.46 * X(1) ^ 2 - .5 * ((PIE * S) / (FMAX)) * (X(1) ^ 3 * X(2)) + .5 * ((PIE * S) / (FMAX)) * X(1) ^ 4<=0

 

8 * ((FMAX / G)) * X(1) ^ (-4) * (X(2) ^ 3 * X(3)) + 1.05 * X(1) * X(3) + 2.1 * X(1) - LMAX<=0

 

DMIN - X(1)<=0

 

X(2) - DMAX<=0

 

3.0 - X(1) ^ (-1) * X(2)<=0

 

8 * ((FP / G)) * X(1) ^ (-4) * (X(2) ^ 3 * X(3)) - SIGMAPM<=0

 

SIGMAW - 8 * (((FMAX - FP) / G)) * X(1) ^ (-4) * (X(2) ^ 3 * X(3))<=0

 

where PIE = 3.141592654, FMAX = 1000, LMAX = 14.0, DMIN = .2, S = 189000.0, DMAX = 3.0, FP = 300.0, SIGMAPM = 6.0, SIGMAW = 1.25, and G = 11.5D+06.

 

X(1) = .207, .225, .244, .263, .283, .307, .331, .362, .394, .4375, or .500,

0.6<= X(2) <=3,

1<= X(3) <=70 and integer,

X(4) through X(10) below are slack variables.

One notes that here line 269 is 269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0; see Lin and Tsai [5, p. 875, Table 5].

Also, whereas line 0 and line 225 of the preceding paper are 0 REM DEFDBL A-Z and 225 X(3) = INT(A(3)), respectively, here line 0 and line 225 are 0 DEFDBL A-Z and 225 X(3) = INT(X(3)), respectively.

 

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

21 PIE = 3.141592654: FMAX = 1000: LMAX = 14.0: DMIN = .2: S = 189000.0: DMAX = 3.0: FP = 300.0: SIGMAPM = 6.0: SIGMAW = 1.25: G = 11.5D+06

67 IF RND < .091 THEN A(1) = .207 ELSE IF RND < .1 THEN A(1) = .225 ELSE IF RND < .111 THEN A(1) = .244 ELSE IF RND < .125 THEN A(1) = .263 ELSE IF RND < .143 THEN A(1) = .283 ELSE IF RND < .167 THEN A(1) = .307 ELSE IF RND < .2 THEN A(1) = .331 ELSE IF RND < .25 THEN A(1) = .362 ELSE IF RND < .333 THEN A(1) = .394 ELSE IF RND < .5 THEN A(1) = .4375 ELSE A(1) = .500

68 A(2) = .6 + RND * 2.4

69 A(3) = 1 + FIX(RND * 70)

 

128 FOR I = 1 TO 10000

 

129 FOR KKQQ = 1 TO 3

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

 

181 J = 2 + FIX(RND * .2)

 

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP
225 X(3) = INT(X(3))

 

239 X(4) = -4 * X(2) ^ 2 - 1.46 * X(1) * X(2) + 2.46 * X(1) ^ 2 + .5 * ((PIE * S) / (FMAX)) * (X(1) ^ 3 * X(2)) - .5 * ((PIE * S) / (FMAX)) * X(1) ^ 4

 

241 X(5) = -8 * ((FMAX / G)) * X(1) ^ (-4) * (X(2) ^ 3 * X(3)) - 1.05 * X(1) * X(3) - 2.1 * X(1) + LMAX

 

245 X(6) = -DMIN + X(1)

 

249 X(7) = -X(2) + DMAX

251 X(8) = -3.0 + X(1) ^ (-1) * X(2)

 

253 X(9) = -8 * ((FP / G)) * X(1) ^ (-4) * (X(2) ^ 3 * X(3)) + SIGMAPM

 

255 X(10) = -SIGMAW + 8 * (((FMAX - FP) / G)) * X(1) ^ (-4) * (X(2) ^ 3 * X(3))

 

260 IF X(1) < .2 THEN GOTO 1670

 

261 IF X(2) < .6 THEN GOTO 1670

 

262 IF X(3) < 1 THEN GOTO 1670

263 REM IF X(4) < 10 THEN GOTO 1670

 

264 IF X(1) > .5 THEN GOTO 1670

265 IF X(2) > 3 THEN GOTO 1670

266 IF X(3) > 70 THEN GOTO 1670

267 REM IF X(4) > 200 THEN GOTO 1670

 

268 FOR J99 = 4 TO 10

 

 

269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0
270 NEXT J99
318 POBA = -.25 * PIE ^ 2 * X(2) * X(1) ^ 2 * X(3) - .5 * PIE ^ 2 * X(2) * X(1) ^ 2 + 1000000 * X(4) + 1000000 * X(5) + 1000000 * X(6) + 1000000 * X(7) + 1000000 * X(8) + 1000000 * X(9) + 1000000 * X(10)

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 10

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128

1670 NEXT I
1889 IF M < -2.66 THEN 1999

 

1900 PRINT A(1), A(2), A(3), A(4)
1901 PRINT A(5), A(6), A(7), A(8)

1902 PRINT A(9), A(10), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [13]. The complete output through JJJJ=-31954.06000000735 is shown below:

.283       1.223041009963807       9       0
0       0       0       0
0       0       -2.658559166663872       -31988.65000000182

.283       1.223041009963807       9       0
0       0       0       0
0       0       -2.658559166663872       -31969.13000000494

.283       1.223041009963807       9       0
0       0       0       0
0       0       -2.658559166663872       -31954.06000000735

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

The (same) solution above is somewhat comparable to the 5 solutions presented in Table 5 of Lin and Tsai [5, p. 875], but the error in constraint is less here because of line 269, which is 269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [13], the wall-clock time for obtaining the output through JJJJ=-31954.06000000735 was four minutes and a half.

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66.
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] H. Chickermane, H. C. Gea (1996) Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39, pp. 829-846..

[4] 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.

[5] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.

[6] Harry Markowitz (1952). Portfolio Selection. The Journal of Finance 7 (2008) pp. 77-91.

[7] 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.

[8] 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.

[9] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.

[10] 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

[11] 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.

[12] 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.

[13] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[14] 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

[15] 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/