A Computer Program for Alkylation Process Optimization, Second Edition
Jsun Yui Wong
Similar to the computer program of the preceding paper, the computer program listed below aims to solve the nonlinear programming formulation on page 42 of Bracken and McCormick [15, p. 42], which is briefly summarized as follows:
Maximize
( .063 * X(4) * X(7) - 5.04 * X(1) - .035 * X(2) - 10 * X(3) - 3.36 * X(5) )
subject to
X(j) LB <= X(j) <= X(j) UB, j=1, ..., 10,
(X(1) * (1.12 + .13167 * X(8) - .00667 * X(8) ^ 2)) - (99 / 100) * X(4) >= 0
-(X(1) * (1.12 + .13167 * X(8) - .00667 * X(8) ^ 2)) + (100 / 99) * X(4) >= 0
(86.35 + 1.098 * X(8) - .038 * X(8) ^ 2 + .325 * (X(6) - 89)) - (99 / 100) * X(7) >= 0
-(86.35 + 1.098 * X(8) - .038 * X(8) ^ 2 + .325 * (X(6) - 89)) + (100 / 99) * X(7) >= 0
(35.82 - .222 * X(10)) - (9 / 10) * X(9) >= 0
-(35.82 - .222 * X(10)) + (10 / 9) * X(9) >= 0
(-133 + 3 * X(7)) - (99 / 100) * X(10) >= 0
-(-133 + 3 * X(7)) + (100 / 99) * X(10) >= 0
1.22 * X(4) -X(1) - X(5) = 0
(98000 * X(3)) / (X(4) * X(9) + 1000 * X(3)) - X(6) = 0 ; this is correct, according to p. 40 of Reference 15
( (X(2) + X(5)) / X(1) ) - X(8) = 0.
The "final element" on p. 42 of Bracken an d McCormick [15] and four of the starting values (1745, 12000, 3048, and 1974) of Bracken and McCormick [15, p. 43, Table 4.2] help reduce the starting ranges of line 91, line 92, line 94, and line 95 below, which involve the four longest ranges, respectively.
One notes line 2 below, which is 2 REM 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), PN(22), NN(22). One also notes that whereas line 128 of the earlier edition is 128 FOR i = 1 TO 100000, here line 128 is 128 FOR i = 1 TO 120000.
0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 REM 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), PN(22), NN(22)
85 FOR JJJJ = -32000 TO 31111
89 RANDOMIZE JJJJ
90 M = -3D+30
91 A(1) = 1700 + RND * 50
92 A(2) = 11900 + RND * 200
93 A(3) = 0 + RND * 120
94 A(4) = 3000 + RND * 50
95 A(5) = 1900 + RND * 50
96 A(6) = 85 + RND * 8
97 A(7) = 90 + RND * 5
98 A(8) = 3 + RND * 9
99 A(9) = 1.2 + RND * 2.8
100 A(10) = 145 + RND * 17
128 FOR i = 1 TO 120000
129 FOR KKQQ = 1 TO 10
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
139 FOR IPP = 1 TO FIX(1 + RND * 10)
141 B = 1 + FIX(RND * 10)
144 IF RND < .9999 THEN 160 ELSE GOTO 167
160 r = (1 - RND * 2) * A(B)
164 X(B) = A(B) + (RND ^ (RND * 15)) * r
165 GOTO 168
167 REM IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)
168 NEXT IPP
181 X(5) = -X(1) + 1.22 * X(4)
184 X(8) = (X(2) + X(5)) / X(1)
187 REM X(6) = (98000 * X(3)) / (X(3) * X(9) + 1000 * X(3))
197 X(6) = (98000 * X(3)) / (X(4) * X(9) + 1000 * X(3))
210 IF (-133 + 3 * X(7)) - (99 / 100) * X(10) < 0 THEN 1670
214 IF -(-133 + 3 * X(7)) + (100 / 99) * X(10) < 0 THEN 1670
220 IF (35.82 - .222 * X(10)) - (9 / 10) * X(9) < 0 THEN 1670
224 IF -(35.82 - .222 * X(10)) + (10 / 9) * X(9) < 0 THEN 1670
230 IF (X(1) * (1.12 + .13167 * X(8) - .00667 * X(8) ^ 2)) - (99 / 100) * X(4) < 0 THEN 1670
235 IF -(X(1) * (1.12 + .13167 * X(8) - .00667 * X(8) ^ 2)) + (100 / 99) * X(4) < 0 THEN 1670
240 IF (86.35 + 1.098 * X(8) - .038 * X(8) ^ 2 + .325 * (X(6) - 89)) - (99 / 100) * X(7) < 0 THEN 1670
245 IF -(86.35 + 1.098 * X(8) - .038 * X(8) ^ 2 + .325 * (X(6) - 89)) + (100 / 99) * X(7) < 0 THEN 1670
324 IF X(1) > 2000 THEN 1670
325 IF X(1) < 0 THEN 1670
342 IF X(2) > 16000 THEN 1670
345 IF X(2) < 0 THEN 1670
351 IF X(3) > 120 THEN 1670
352 IF X(3) < 0 THEN 1670
353 IF X(4) > 5000 THEN 1670
354 IF X(4) < 0 THEN 1670
355 IF X(5) > 2000 THEN 1670
356 IF X(5) < 0 THEN 1670
361 IF X(6) > 93 THEN 1670
362 IF X(6) < 85 THEN 1670
363 IF X(7) > 95 THEN 1670
364 IF X(7) < 90 THEN 1670
365 IF X(8) > 12 THEN 1670
366 IF X(8) < 3 THEN 1670
367 IF X(9) > 4 THEN 1670
368 IF X(9) < 1.2 THEN 1670
369 IF X(10) > 162 THEN 1670
370 IF X(10) < 145 THEN 1670
468 PD1 = .063 * X(4) * X(7) - 5.04 * X(1) - .035 * X(2) - 10 * X(3) - 3.36 * X(5)
469 P = PD1
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 10
1455 A(KLX) = X(KLX)
1459 NEXT KLX
1670 NEXT i
1777 IF M < 1765 THEN 1999
1888 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [102]. Its selected candidate solutions of one run through JJJJ= -28300 are shown below:
.
.
.
1697.856 15712.7 54.52645 3031.029 2000
90.17217 95 10.43239 1.561658 153.5353
1768.309 -30568
.
.
.
1699.033 156.9587 54.68921 3031.994 2000
90.1845 94.99876 10.41526 1.563142 153.5282
1766.874 -28507
.
.
.
1699.854 15872.39 54.01141 3032.667 2000
90.09627 95 10.51407 1.562378 153.5349
1767.601 -28300
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. By using the following computer system and QB64v1000-win [102], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -28300 was 18 minutes, counting from "Starting program...". One can compare the computational results above with those in Bracken and McCormick [15, p. 44, Table 4.5 ] and in Babu and Angira [10, p. 997, Table 6].
The computational results presented above were obtained from the following computer system:
Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz
Installed memory (RAM): 4.00GB (3.87 GB usable)
System type: 64-bit Operating System.
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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[104] Helen Wu, (2015), Geometric Programming
https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming
[105] G. Xu, (2014). Global optimization of signomial geometric programming problems, European J. of Operational Research 233 (2014) 500-510.
[106] James Yan. Signomial programs with equality constraints: numerical solution and applications. Ph. D. thesis. University of British Columbia, 1976.
One can easily view this dissertation on Google.
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