The computer program listed below seeks to solve the transform design model on page 258 of Papalambros and Wilde [17]. Unlike the preceding paper, here X(5) and X(6) are continuous; these represent flux density and current density, respectively; see Papalambros and Wilde [17, p. 258]. Line 235 through line 331 roughly describe the problem; for details, see Papalambros and Wilde [17].
(-2.07+.001*X(1)*X(2)*X(3)*X(4)*X(5)*X(6)) of line 235 triggers the domino process. Line 235 and line 236 show a column of two dominoes, X(7) and X(8), which are slack variables defined in line 235 and line 236, respectively.
0 REM DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),LHS(44),PLHS(44),LB(22),UB(22),PX(44),J44(44)
4 REM DIM PE(35,35)
5 REM DIM SD(35,35)
75 FOR J44=1 TO 6
76 LB(J44)=0
77 NEXT J44
78 FOR J44=1 TO 6
79 UB(J44)=20
80 NEXT J44
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3E+30
100 FOR J44=1 TO 6
101 A(J44)=FIX(RND*21)
102 NEXT J44
126 IMAR=10+FIX(RND*1000)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 6
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
159 FOR IPP=1 TO FIX(1+RND*3)
160 B=1+FIX(RND*6)
162 IF B>4 THEN 166 ELSE 163
163 IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*3) ELSE X(B)=A(B)+FIX(1+RND*3)
164 GOTO 235
166 R=(1-RND*2)*A(B)
167 IF RND<.25 THEN X(B)=CINT(A(B))-FIX(1+RND*3) ELSE IF RND<.33 THEN X(B)=CINT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
169 NEXT IPP
235 X(7)=-2.07+.001*X(1)*X(2)*X(3)*X(4)*X(5)*X(6)
236 X(8)=-.00062*X(1)*X(4)*X(5)^2 *( X(1)+X(2)+X(3) ) -.00058*X(2)*X(3) *X(6)^2 *(X(1)+1.57*X(2)+X(4) ) +1.2
281 FOR J78=1 TO 6
282 IF X(J78)<LB(J78) THEN 1670
283 NEXT J78
292 FOR J79=1 TO 6
293 IF X(J79)>UB(J79) THEN 1670
294 NEXT J79
303 FOR J44=7 TO 8
305 IF X(J44) <0 THEN PX(J44)=999999999#*(X(J44)) ELSE PX(J44)=0
307 NEXT J44
320 PXT=0
321 FOR J44=7 TO 8
322 PXT=PXT+PX(J44)
323 NEXT J44
331 PD1=-.0204*X(1)*X(4)*(X(1)+X(2)+X(3) ) -.0187*X(2)*X(3)*(X(1)+1.57*X(2)+X(4) ) -.0607*X(1)*X(4)*X(5)^2 *(X(1)+X(2)+X(3) ) -.0437*X(2)*X(3)*X(6)^2 *(X(1)+1.57*X(2)+X(4) ) +PXT
335 PDU=+PD1
466 P=PDU
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<-131.08 THEN 1999
1904 PRINT A(1),A(2),A(3)
1987 PRINT A(4),A(5),A(6),A(7)
1988 PRINT A(8),M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=-5914 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.
5 4 9
10 .9498243 1.210755 8.583069E-06
4.524076E-02 -131.0469 -18736
5 4 9
10 .9568129 1.201907 4.768372E-07
.0472908 -131.0602 -5914
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time from JJJJ=-32000 through JJJJ=-5914 was fifty minutes.
References
[1] William Conley, Computer Optimization Techniques. Petrocelli Books, Inc., 1980.
[2] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[3] G. Dantzig, R. Fulkerson, S. Johnson, Solution of a Large-Scale Traveling-Salesman Problem.
Operations Research, Vol. 2, No. 4 (Nov., 1954), pp. 393-410.
[4] M. A. Duran, I. E. Grossmann, An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs. Mathematical Programming, Vol. 36:307-339, 1986.
[5] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.
[6] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.
[7] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms. Springer-Verlag, 1990.
[8] Christodoulos A. Floudas, Nonlinear Mixed-Integer Optimization. Oxford University Press, 1995.
[9] C. A. Floudas, Deterministic Global Optimization. Kluwer Academic Publishers, 2000.
[10] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.
[11] G. R. Kocis, I. E. Grossmann, Relaxation Strategy for the Structural Optimization of Process Flow Sheets. I & EC Res., 26 (9):1869, 1987.
[12] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[13] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.
[14] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Springer, 2006.
[15] Harry M. Markowitz and Alan S. Mann. On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957) pp. 84-110.
[16] 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.
[17] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design. Cambridge University Press, 2000.
[18] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design. Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.
[19] Jsun Yui Wong (2009, July 18). An Integer Programming Computer Program Applied to One-Dimensional Space Allocation. Retrieved from http://wongsllllblog.blogspot.com/2009/07/
[20] Jsun Yui Wong (2009, December 18). A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations. Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/
[21] Jsun Yui Wong (2011, July 27). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/
[22] Jsun Yui Wong (2012, April 24). The Domino Method of General Integer Nonlinear Programming Applied to Problem 10 of Lawler and Bell. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/
[23] Lian-sheng Zhang, Feng Gao, Wen-xing Zhu, Nonlinear Integer Programming and Global Optimization. Journal of Computational Mathematics, Vol. 17. No. 2, 1999, pp. 179-190.