This paper applies the computer program in Wong [1] to a problem in the Kocis and Grossmann article "Global Optimization of Nonconvex Mixed-Integer Nonlinear Programming (MINLP) Problems in Process Synthesis" [2] in the journal Industrial and Engineering Chemistry Research. This problem is Example 9 of Salcedo [3], whose description of the problem is given on pages 268-269 of Salcedo [3, pp. 268-269]. His input data are given on pages 269 and 270 [3]; these data come from Kocis and Grossmann [2, p. 1417].
Benefitting from the computer program on pages 229-232 of Conley [4] and from the computer program in Wong [1], the following computer program follows Grossmann and Sargent's [5] and Kocis and Grossmann's [2] mathematical formulation.
2 DEFINT I,J,K
5 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111)
12 FOR JJJJ=-32000 TO -16000
14 RANDOMIZE JJJJ
16 M=-1D+17
25 B(1)=300:B(2)=300:B(3)=300:B(4)=300
26 B(5)=300:B(6)=300
27 B(7)=2.075:B(8)=1.7:B(9)=2.975:B(10)=.875:B(11)=1.05
28 B(12)=86.458333333333#:B(13)=42.5:B(14)=89.25
29 B(15)=23.333333333333#:B(16)=21
33 B(17)=1:B(18)=1:B(19)=1:B(20)=1
34 B(21)=1:B(22)=1
37 N(1)=3000:N(2)=3000:N(3)=3000:N(4)=3000
38 N(5)=3000:N(6)=3000
39 N(7)=8.3:N(8)=6.8:N(9)=11.9:N(10)=3.5:N(11)=4.2
41 N(12)=379.7468354#:N(13)=882.3529412#:N(14)=833.333333333#
42 N(15)=638.2978723#:N(16)=666.666666666#
43 N(17)=4:N(18)=4:N(19)=4:N(20)=4
44 N(21)=4:N(22)=4
61 FOR KLQ=1 TO 22
62 A(KLQ)=B(KLQ)+RND*(N(KLQ)-B(KLQ))
63 NEXT KLQ
71 FOR KLR=1 TO 22
72 H(KLR)=3
73 NEXT KLR
88 FOR JJJ=1 TO 1000 STEP 10
90 FOR INEW=1 TO 5
94 FOR J=INEW*2 TO 0 STEP -2
102 FOR JJ=0 TO 22
111 KKLL=FIX(1+22*RND)
128 FOR I=1 TO 15
129 FOR KKQQ=1 TO 22
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
132 FOR K=1 TO 22
133 IF RND<=.5 THEN 298 ELSE 230
230 IF A(K)-(N(K)-B(K))/H(K)^J
255 GOTO 265
260 L(K)=A(K)-(N(K)-B(K))/H(K)^J
265 IF A(K)+(N(K)-B(K))/H(K)^J>N(K) THEN 266 ELSE 268
266 U(K)=N(K)-L(K)
267 GOTO 272
268 U(K)=A(K)+(N(K)-B(K))/H(K)^J-L(K)
272 GOTO 300
298 X(K)=A(K)+2*RND*(2*RND-1)*(1/(1+RND*JJJ))*.05*A(K)
299 GOTO 302
300 X(K)=(L(K)+2*RND*RND*U(K))
302 NEXT K
304 X(JJ)=A(JJ)
305 KLPS=FIX(1+KKLL*RND)
307 FOR KLA1=1 TO KLPS
308 KLA2=FIX(1+22*RND)
309 X(KLA2)=A(KLA2)
310 NEXT KLA1
342 REM
343 IF RND>.1 GOTO 461
352 IF RND<=.9 THEN 353 ELSE 355
353 X(JJ)=B(JJ)
354 GOTO 461
355 X(JJ)=N(JJ)
461 IF X(17)>4 THEN 1670
462 IF X(17)<1 THEN 1670
463 IF X(18)>4 THEN 1670
464 IF X(18)<1 THEN 1670
465 IF X(19)>4 THEN 1670
466 IF X(19)<1 THEN 1670
467 IF X(20)>4 THEN 1670
468 IF X(20)<1 THEN 1670
469 IF X(21)>4 THEN 1670
470 IF X(21)<1 THEN 1670
471 IF X(22)>4 THEN 1670
472 IF X(22)<1 THEN 1670
483 IF X(7)>8.3 THEN 1670
484 IF X(7)<2.075# THEN 1670
485 IF X(8)>6.8 THEN 1670
486 IF X(8)<1.7# THEN 1670
487 IF X(9)>11.9 THEN 1670
488 IF X(9)<2.975# THEN 1670
489 IF X(10)>3.5 THEN 1670
490 IF X(10)<.875# THEN 1670
491 IF X(11)>4.2 THEN 1670
492 IF X(11)<1.05 THEN 1670
493 IF X(12)>379.7468354# THEN 1670
494 IF X(12)<86.4583333333# THEN 1670
495 IF X(13)>882.3529412# THEN 1670
496 IF X(13)<42.5 THEN 1670
497 IF X(14)>833.3333333333333# THEN 1670
498 IF X(14)<89.25 THEN 1670
499 IF X(15)>638.2978723# THEN 1670
500 IF X(15)<23.3333333333333# THEN 1670
501 IF X(16)>666.666666666666# THEN 1670
502 IF X(16)<21 THEN 1670
504 X(17)=FIX(X(17)):X(18)=FIX(X(18)):X(19)=FIX(X(19)):X(20)=FIX(X(20)):X(21)=FIX(X(21)):X(22)=FIX(X(22))
559 PE=(250000!*X(7)/X(12)+150000!*X(8)/X(13)+180000!*X(9)/X(14)+160000!*X(10)/X(15)+120000!*X(11)/X(16))-6000
560 IF PE>0 THEN PE=PE ELSE PE=0
561 P11=X(1)-7.9*X(12)
562 P12=X(2)-2!*X(12)
563 P13=X(3)-5.2*X(12)
564 P14=X(4)-4.9*X(12)
565 P15=X(5)-6.1*X(12)
566 P16=X(6)-4.2*X(12)
567 P17=X(1)-.7*X(13)
568 P18=X(2)-.8*X(13)
569 P19=X(3)-.9*X(13)
570 P20=X(4)-3.4*X(13)
571 P21=X(5)-2.1*X(13)
572 P22=X(6)-2.5*X(13)
573 P23=X(1)-.7*X(14)
574 P24=X(2)-2.6*X(14)
575 P25=X(3)-1.6*X(14)
576 P26=X(4)-3.6*X(14)
577 P27=X(5)-3.2*X(14)
578 P28=X(6)-2.9*X(14)
579 P29=X(1)-4.7*X(15)
580 P30=X(2)-2.3*X(15)
581 P31=X(3)-1.6*X(15)
582 P32=X(4)-2.7*X(15)
583 P33=X(5)-1.2*X(15)
584 P34=X(6)-2.5*X(15)
585 P35=X(1)-1.2*X(16)
586 P36=X(2)-3.6*X(16)
587 P37=X(3)-2.4*X(16)
588 P38=X(4)-4.5*X(16)
589 P39=X(5)-1.6*X(16)
590 P40=X(6)-2.1*X(16)
601 P51=X(17)*X(7)-6.4
602 P52=X(18)*X(7)-4.7
603 P53=X(19)*X(7)-8.3
604 P54=X(20)*X(7)-3.9
605 P55=X(21)*X(7)-2.1
606 P56=X(22)*X(7)-1.2
607 P57=X(17)*X(8)-6.8
608 P58=X(18)*X(8)-6.4
609 P59=X(19)*X(8)-6.5
610 P60=X(20)*X(8)-4.4
611 P61=X(21)*X(8)-2.3
612 P62=X(22)*X(8)-3.2
613 P63=X(17)*X(9)-1!
614 P64=X(18)*X(9)-6.3
615 P65=X(19)*X(9)-5.4
616 P66=X(20)*X(9)-11.9
617 P67=X(21)*X(9)-5.7
618 P68=X(22)*X(9)-6.2
619 P69=X(17)*X(10)-3.2
620 P70=X(18)*X(10)-3!
621 P71=X(19)*X(10)-3.5
622 P72=X(20)*X(10)-3.3
623 P73=X(21)*X(10)-2.8
624 P74=X(22)*X(10)-3.4
625 P75=X(17)*X(11)-2.1
626 P76=X(18)*X(11)-2.5
627 P77=X(19)*X(11)-4.2
628 P78=X(20)*X(11)-3.6
629 P79=X(21)*X(11)-3.7
630 P80=X(22)*X(11)-2.2
801 IF P11<0 THEN P11=P11 ELSE P11=0
802 IF P12<0 THEN P12=P12 ELSE P12=0
803 IF P13<0 THEN P13=P13 ELSE P13=0
804 IF P14<0 THEN P14=P14 ELSE P14=0
805 IF P15<0 THEN P15=P15 ELSE P15=0
806 IF P16<0 THEN P16=P16 ELSE P16=0
807 IF P17<0 THEN P17=P17 ELSE P17=0
808 IF P18<0 THEN P18=P18 ELSE P18=0
809 IF P19<0 THEN P19=P19 ELSE P19=0
810 IF P20<0 THEN P20=P20 ELSE P20=0
811 IF P21<0 THEN P21=P21 ELSE P21=0
812 IF P22<0 THEN P22=P22 ELSE P22=0
813 IF P23<0 THEN P23=P23 ELSE P23=0
814 IF P24<0 THEN P24=P24 ELSE P24=0
815 IF P25<0 THEN P25=P25 ELSE P25=0
816 IF P26<0 THEN P26=P26 ELSE P26=0
817 IF P27<0 THEN P27=P27 ELSE P27=0
818 IF P28<0 THEN P28=P28 ELSE P28=0
819 IF P29<0 THEN P29=P28 ELSE P29=0
820 IF P30<0 THEN P30=P30 ELSE P30=0
821 IF P31<0 THEN P31=P31 ELSE P31=0
822 IF P32<0 THEN P32=P32 ELSE P32=0
823 IF P33<0 THEN P33=P33 ELSE P33=0
824 IF P34<0 THEN P34=P34 ELSE P34=0
825 IF P35<0 THEN P35=P35 ELSE P35=0
826 IF P36<0 THEN P36=P36 ELSE P36=0
827 IF P37<0 THEN P37=P37 ELSE P37=0
828 IF P38<0 THEN P38=P38 ELSE P38=0
829 IF P39<0 THEN P39=P39 ELSE P39=0
830 IF P40<0 THEN P40=P40 ELSE P40=0
831 IF P51<0 THEN P51=P51 ELSE P51=0
832 IF P52<0 THEN P52=P52 ELSE P52=0
833 IF P53<0 THEN P53=P53 ELSE P53=0
834 IF P54<0 THEN P54=P53 ELSE P54=0
835 IF P55<0 THEN P55=P55 ELSE P55=0
836 IF P56<0 THEN P56=P56 ELSE P56=0
837 IF P57<0 THEN P57=P57 ELSE P57=0
838 IF P58<0 THEN P58=P58 ELSE P58=0
839 IF P59<0 THEN P59=P59 ELSE P59=0
840 IF P60<0 THEN P60=P60 ELSE P60=0
841 IF P61<0 THEN P61=P61 ELSE P61=0
842 IF P62<0 THEN P62=P62 ELSE P62=0
843 IF P63<0 THEN P63=P63 ELSE P63=0
844 IF P64<0 THEN P64=P64 ELSE P64=0
845 IF P65<0 THEN P65=P65 ELSE P65=0
846 IF P66<0 THEN P66=P66 ELSE P66=0
847 IF P67<0 THEN P67=P67 ELSE P67=0
848 IF P68<0 THEN P68=P68 ELSE P68=0
849 IF P69<0 THEN P69=P69 ELSE P69=0
850 IF P70<0 THEN P70=P70 ELSE P70=0
851 IF P71<0 THEN P71=P71 ELSE P71=0
852 IF P72<0 THEN P72=P72 ELSE P72=0
853 IF P73<0 THEN P73=P73 ELSE P73=0
854 IF P74<0 THEN P74=P74 ELSE P74=0
855 IF P75<0 THEN P75=P75 ELSE P75=0
856 IF P76<0 THEN P76=P76 ELSE P76=0
857 IF P77<0 THEN P77=P77 ELSE P77=0
858 IF P78<0 THEN P78=P78 ELSE P78=0
859 IF P79<0 THEN P79=P79 ELSE P79=0
860 IF P80<0 THEN P80=P80 ELSE P80=0
1201 PS1=555555!*P11+555555!*P12+555555!*P13+555555!*P14+555555!*P15+555555!*P16+555555!*P17+555555!*P18+555555!*P19+555555!*P20
1202 PS2=555555!*P21+555555!*P22+555555!*P23+555555!*P24+555555!*P25+555555!*P26+555555!*P27+555555!*P28+555555!*P29+555555!*P30
1203 PS3=555555!*P31+555555!*P32+555555!*P33+555555!*P34+555555!*P35+555555!*P36+555555!*P37+555555!*P38+555555!*P39+555555!*P40
1204 PS4=555555!*P51+555555!*P52+555555!*P53+555555!*P54+555555!*P55+555555!*P56+555555!*P57+555555!*P58+555555!*P59+555555!*P60
1205 PS5=555555!*P61+555555!*P62+555555!*P63+555555!*P64+555555!*P65+555555!*P66+555555!*P67+555555!*P68+555555!*P69+555555!*P70
1206 PS6=555555!*P71+555555!*P72+555555!*P73+555555!*P74+555555!*P75+555555!*P76+555555!*P77+555555!*P78+555555!*P79+555555!*P80
1447 P1=+250*X(17)*X(1)^.6+250*X(18)*X(2)^.6+250*X(19)*X(3)^.6+250*X(20)*X(4)^.6+250*X(21)*X(5)^.6+250*X(22)*X(6)^.6
1449 P=-P1-555555!*PE+PS1+PS2+PS3+PS4+PS5+PS6
1451 IF P<=M THEN 1670
1452 M=P
1453 PP1=P1
1454 FOR KLX=1 TO 22
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1457 GOTO 128
1670 NEXT I
1702 NEXT JJ
1706 NEXT J
1777 NEXT INEW
1888 NEXT JJJ
1890 IF M>-300033! THEN 1916 ELSE 1999
1916 PRINT JJJJ,A(1),A(2),A(3),A(4),A(5),A(6),A(7)
1917 PRINT A(8),A(9),A(10),A(11),A(12),A(13),A(14),A(15),A(16)
1918 PRINT A(17),A(18),A(19),A(20),A(21),A(22),M,PP1
1999 NEXT JJJJ
This BASIC computer program was run with the IBM basica/D interpreter, and the best candidate solutions from JJJJ=-32000 through JJJJ=-31893 (in compressed form and to be interpreted in accordance with line 1916 through line 1918) are presented below.
-31935 2773.685 2003.521 1825.619 2787.92
2465.808 2234.618 3.200041
3.400418 6.200672 3.400841 3.700547 351.0785
819.9677 770.5147 638.2677 556.3699
2 2 3 2 1
1 -284984.5 284984.5
-31923 2875.431 2005.215 1892.686 2677.446
2377.128 2154.176 3.200021
3.400219 6.201046 3.40001 3.705313 363.9694
787.4606 742.7872 637.57 556.9416
2 2 3 2 1
1 -285222 285222
-31893 2669.909 2112.693 1757.276 2845.691
2529.677 2292.354 3.200002
3.403048 6.200127 3.400073 3.707986 337.9257
836.9243 790.4537 638.2486 586.6831
2 2 3 2 1
1 -285215.8 285215.8
These computational results were obtained in fourteen hours on a personal computer with an Intel 2.66 GHz. chip and the IBM interpreter. If one makes changes to the computer program, such as a change of changing line 128 above to, for example, 128 FOR I=1 TO 14, one may or may not obtain 284984.5, which may or may not be optimal. This point is relevant to the present problem AND beyond it. That leads to a general remedy that in general, in order to increase the chance of getting an optimal solution, one preferably should run several computers simultaneously, each with a different line 128. Running several computers simultaneously can mitigate the possible danger of missed optimality and can produce a usable solution faster than just running one computer.
Computational results from other methods are reported in Kocis and Grossmann [2, pp. 1417-1418], Floudas et al. [6, p. 1120], and Salcedo [3, pp. 270-272].
References
[1] J. Y. Wong, "A Computer Program for Mixed-Integer Nonlinear Programming Formulations,"
[2] G. R. Kocis and I. E. Grossmann, "Global optimization of nonconvex mixed-integer nonlinear programming (MINLP) problem in process synthesis," Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988, pp. 1407-1421.
[3] R. L. Salcedo, "Solving nonconvex nonlinear programming and mixed-integer nonlinear programming problems with adaptive random search," Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992, pp. 262-273.
[4] W. C. Conley, Optimization: A Simplified Approach. Petrocelli, Princeton, New Jersey, 1981, pp. 229-232.
[5] I. E. Grossmann and R. W. H. Sargent, "Optimal design of multipurpose chemical plants," Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979, pp. 343-348.
[6] C. A. Floudas, A. Aggarwal, and A. R. Ciric, "Global optimum search for nonconvex NLP and MINLP problems," Comput. Chem. Eng., Vol. 13, No. 10, 1989, pp. 1117-1132.
