Jsun Yui Wong

Based on the computer program in [5], the computer program listed below seeks to solve the air transport model of Markowitz and Manne [1, pages 95-100].

In this paper line 12 is 12 FOR JJJJ = -32000 TO 32000 STEP .1. The intent is to increase the probability of reaching optimality with one computer run.

Whereas line 128, line 133, and line 461 of the older edition of 2017 April 6 are 128 FOR I = 1 TO 1000, 133 FOR IPP = 1 TO (1 + FIX(RND * 47)), and

461 POB3 = -999999999# * (1000 * PS(21) ^ 4 + 1000 * PS(22) ^ 4 + 1000 * PS(23) ^ 4 + 1000 * PS(24) ^ 4 + 1000 * PS(25) ^ 4 + 1000 * PS(26) ^ 4 + PS(27) ^ 4 + PS(28) ^ 4 + PS(29) ^ 4 + PS(30) ^ 4 + PS(31) ^ 4 + PS(32) ^ 4), respectively, here line 128, line 133, and line 461 are 128 FOR I = 1 TO 500, 133 FOR IPP = 1 TO (1 + FIX(RND * 3)), and 461 POB3 = -99999999999# * (PS(21) ^ 4 + PS(22) ^ 4 + PS(23) ^ 4 + PS(24) ^ 4 + PS(25) ^ 4 + PS(26) ^ 4 + PS(27) ^ 4 + PS(28) ^ 4 + PS(29) ^ 4 + PS(30) ^ 4 + PS(31) ^ 4 + PS(32) ^ 4), respectively.

0 REM DEFDBL A-Z

2 DEFINT I, 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 .1

15 RANDOMIZE JJJJ

16 M = -1D+37

41 FOR J44 = 1 TO 48

42 A(J44) = (RND * 5)

43 NEXT J44

51 FOR J44 = 37 TO 48

52 A(J44) = FIX(RND * 2)

53 NEXT J44

126 IMAR = 10 + FIX(RND * 5000)

128 FOR I = 1 TO 500

129 FOR KKQQ = 1 TO 48

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

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

181 J = 1 + FIX(RND * 48)

183 R = (1 - RND * 2) * A(J)

187 X(J) = A(J) + (RND ^ 3) * R

192 NEXT IPP

201 FOR J88 = 37 TO 48

202 IF X(J88) > 100 THEN X(J88) = A(J88)

203 X(J88) = CINT(X(J88))

204 NEXT J88

211 REM X(43) = 5 – X(44) – X(45)

212 X(48) = X(43) + X(44) + X(45) - X(38) - X(41)

266 X(46) = X(37) + X(38) + X(39) - X(40) - X(43)

267 X(47) = X(40) + X(41) + X(42) - X(37) - X(44)

301 X(1) = X(20) + X(29) - X(4) - X(7) + 6

302 X(5) = X(11) + X(30) - X(8) + 6 - X(2)

303 X(9) = X(12) + X(21) - X(6) + 6 - X(3)

304 X(10) = X(22) + X(31) - X(13) - X(16) + 5

305 X(14) = X(2) - X(11) - X(17) + 20 + X(33)

306 X(18) = X(24) + X(3) - X(12) - X(15) + 31

307 X(19) = X(34) + X(13) - X(22) - X(25) + 2

308 X(23) = X(4) + X(35) - X(20) - X(26) + 24

309 X(27) = X(6) + X(15) - X(21) - X(24) + 11

310 X(28) = X(16) + X(25) - X(31) - X(34) + 14

311 X(32) = X(7) + X(26) - X(29) - X(35) + 36

312 X(36) = X(8) + X(17) - X(30) - X(33) + 7

371 FOR J44 = 1 TO 48

372 IF X(J44) < 0 THEN X(J44) = A(J44)

373 NEXT J44

431 PS(21) = -7.5 * X(37) + X(1) + X(2) + X(3)

432 PS(22) = -7.2 * X(38) + X(4) + X(5) + X(6)

433 PS(23) = -7.5 * X(39) + X(7) + X(8) + X(9)

434 PS(24) = -7.5 * X(40) + X(10) + X(11) + X(12)

435 PS(25) = -7.5 * X(41) + X(13) + X(14) + X(15)

436 PS(26) = -7.5 * X(42) + X(16) + X(17) + X(18)

437 PS(27) = -7.2 * X(43) + X(19) + X(20) + X(21)

438 PS(28) = -7.5 * X(44) + X(22) + X(23) + X(24)

439 PS(29) = -5.6 * X(45) + X(25) + X(26) + X(27)

440 PS(30) = -7.5 * X(46) + X(28) + X(29) + X(30)

441 PS(31) = -7.5 * X(47) + X(31) + X(32) + X(33)

442 PS(32) = -5.6 * X(48) + X(34) + X(35) + X(36)

454 FOR J44 = 21 TO 32

455 IF (PS(J44)) > 0 THEN PS(J44) = PS(J44) ELSE PS(J44) = 0

456 NEXT J44

459 POB1 = -4.5 * X(37) - 8.3 * X(38) - 2.9 * X(39) - 4.5 * X(40) - 4.2 * X(41) - 6.9 * X(42) - 8.3 * X(43) - 4.2 * X(44) - 10.9 * X(45) - 2.9 * X(46) - 6.9 * X(47) - 10.9 * X(48)

461 POB3 = -99999999999# * (PS(21) ^ 4 + PS(22) ^ 4 + PS(23) ^ 4 + PS(24) ^ 4 + PS(25) ^ 4 + PS(26) ^ 4 + PS(27) ^ 4 + PS(28) ^ 4 + PS(29) ^ 4 + PS(30) ^ 4 + PS(31) ^ 4 + PS(32) ^ 4)

463 P1NEWMAY = POB1 + POB3

466 P = P1NEWMAY

1111 IF P <= M THEN 1670

1452 M = P

1454 FOR KLX = 1 TO 48

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1511 PPOB1 = POB1

1522 PPOB3 = POB3

1557 GOTO 128

1670 NEXT I

1889 IF M < -153.4 THEN 1999

1900 REM GOTO 1945

1911 PRINT A(1), A(2), A(3), A(4), A(5)

1912 PRINT A(6), A(7), A(8), A(9), A(10)

1913 PRINT A(11), A(12), A(13), A(14), A(15)

1914 PRINT A(16), A(17), A(18), A(19), A(20)

1915 PRINT A(21), A(22), A(23), A(24), A(25)

1916 PRINT A(26), A(27), A(28), A(29), A(30)

1917 PRINT A(31), A(32), A(33), A(34), A(35)

1945 PRINT A(36), A(37), A(38), A(39), A(40)

1946 PRINT A(41), A(42), A(43), A(44), A(45)

1947 PRINT A(46), A(47), A(48), M, JJJJ

1977 PRINT PPOB1, PPOB3

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [3]. The complete output through JJJJ=26240.24 is shown below:

6.896548 1.240778E-04 9.09872E-05 9.5613385E-04

12.26378

8.615011E-11 .3208449 6.807605E-04

6.49625 7.001408

5.527465E-04 .4962863 1.087427E-07

22.13178 1.329673E-10

1.911307E-04 6.23411E-06 41.52423 2.026523E-06

1.199707E-10

5.447751E-05 1.999998 24.00096 11.02042

3.526359E-11

1.724698E-10 1.568836E-03 13.99859

1.218349 6.26403

1.600733E-03 35.1025 2.132214 3.967786E-08

1.992196E-07

1.626015E-04 1 2 1

1

4 6 0 5 0

3 5 0 -151.5058 -26771.3

-150.9 -.605845

5.556017 .1373026 .2760432 3.582755E-02

7.161482

5.110291E-10 .5017739 3.027503E-02

5.732894 7.453427

3.377694E-02 8.691414E-03 2.56073E-08

20.45518 3.837951E-10

3.688717E-02 .1422242 42.26685 1.34548E-05

7.081356E-10

2.46021E-04 1.999987 24.03588 10.9995 1.58506E-08

1.146586E-09 2.527516E-04 13.54656

9.361827E-02 1.295182

.4903278 36.4081 .4938829 2.321014E-09

5.42002E-05

5.383334 1 1 1 1

3 7 0 5 0

2 5 1 -153.3009 26240.24

-153.3 -8.615249E-04

The PPOB3=-.605845 at JJJJ=-26771.3 and the PPOB3=-8.615249E-04 at JJJJ=26240.24 show the latter violates the constraint/s less. Hence the second candidate solution shown above is a better solution.

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 [3], the wall-clock time for obtaining the output through JJJJ=26240.24 was 12 hours.

Acknowledgment

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

References

[1] Harry M. Markowitz and Alan S. Mann. On the Solution of Discrete Programming Problems. *Econometrica*, Vol. 25, No. 1 (Jan., 1957) pp. 84-110.

[2] 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.

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

[4] Jsun Yui Wong (2015, August 27). The Domino Method of General Integer Nonlinear Programming Applied to the Air Transport Model of Markowitz and Manne, Third Edition Extended. https://computerprogramsandresults.wordpress.com/2015/08/27/the-domino-method-of-general-integer-nonlinear-programming-applied-to-the-air-transport-model-of-markowitz-and-manne-third-edition-extended/

[5] Jsun Yui Wong (2016, July 6). A Computer Program for the Air Transport Model of Markowitz and Manne.

https://computerprogramsandresults.wordpress.com/2016/07/06/a-computer-program-for-the-air-transport-model-of-markowitz-and-manne/