Jsun Yui Wong
I. Tsai's Example 3 [20, p. 408]
The computer program listed below seeks to solve the following problem from page 408 of Tsai [20].
Minimize (2 + X(5)) / (X(1) * X(2) * (2 * X(3) + X(4))) - X(5) ^ .5 * X(3) ^ 1.5 + 2 * X(2) + X(4)
subject to
8 / (X(1) * (X(2) + 3 * X(4)) ^ 2) + 1 / (X(5) ^ 3)<=2
- 2 * X(1) + X(3) - X(4)<=10
X(1) + X(3) + .5 * X(5)<=8
0.1<X(1), X(2), X(3), X(4), X(5)<=10.
"This program is a highly NFP program unsolvable by current fractional programming methods," Tsai [20, p. 408].
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 FOR J44 = 1 TO 5
72 A(J44) = .1 + RND * 9.9
73 NEXT J44
128 FOR I = 1 TO 1000
129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))
181 J = 1 + FIX(RND * 5)
183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r
191 NEXT IPP
196 FOR J99 = 1 TO 5
201 IF X(J99) < .1 THEN 1670
203 IF X(J99) > 10 THEN 1670
204 NEXT J99
305 X(6) = 2 - 8 / (X(1) * (X(2) + 3 * X(4)) ^ 2) - 1 / (X(5) ^ 3)
306 X(7) = 10 + 2 * X(1) - X(3) + X(4)
307 X(8) = 8 - X(1) - X(3) - .5 * X(5)
325 FOR J99 = 6 TO 8
327 XX(J99) = X(J99)
330 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0
331 NEXT J99
357 POBA = -(2 + X(5)) / (X(1) * X(2) * (2 * X(3) + X(4))) + X(5) ^ .5 * X(3) ^ 1.5 - 2 * X(2) - X(4) + 1000000 * (X(6) + X(7) + X(8))
466 P = POBA
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 8
1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < 22.82 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4), A(5)
1902 PRINT A(6), A(7), A(8)
1912 PRINT M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [24]. The complete output through JJJJ = -31999.52000000008 is shown below:
.294624862833304 .9357350062234917 5.932980600348823
.9232544671551575 3.544789073502854
0 0 0
22.84117291174397 -31999.55000000007
.295980801658799 .8495190820457937 5.999493655132955
.9500308216052291 3.409051086414363
0 0 0
22.8220604982404 -31999.52000000008
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 [24], the wall-clock time for obtaining the output through JJJJ= -31999.52000000008 was 15 seconds, including time for creating .EXE file.
II. A First Look at the Same Example 3 of Tsai [20, p. 408] Plus the Constraint That X(i)=1, 2, 3,..., 10 Where i=1, 2, 3, 4, 5
In the following computer program for this integer fractional nonlinear programming example, one notes line 199, line 327, and line 1456, which are 199 X(J99) = INT(X(J99)), 327 XX(J99) = X(J99), and 1456 XXX(KLX) = XX(KLX), 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
70 FOR J44 = 1 TO 5
72 A(J44) = .1 + RND * 9.9
73 NEXT J44
128 FOR I = 1 TO 1000
129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 4))
181 J = 1 + FIX(RND * 5)
183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r
191 NEXT IPP
196 FOR J99 = 1 TO 5
199 X(J99) = INT(X(J99))
201 IF X(J99) < .1 THEN 1670
203 IF X(J99) > 10 THEN 1670
204 NEXT J99
305 X(6) = 2 - 8 / (X(1) * (X(2) + 3 * X(4)) ^ 2) - 1 / (X(5) ^ 3)
306 X(7) = 10 + 2 * X(1) - X(3) + X(4)
307 X(8) = 8 - X(1) - X(3) - .5 * X(5)
325 FOR J99 = 6 TO 8
327 XX(J99) = X(J99)
330 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0
331 NEXT J99
357 POBA = -(2 + X(5)) / (X(1) * X(2) * (2 * X(3) + X(4))) + X(5) ^ .5 * X(3) ^ 1.5 - 2 * X(2) - X(4) + 1000000 * (X(6) + X(7) + X(8))
466 P = POBA
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 8
1456 XXX(KLX) = XX(KLX)
1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < 18 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4), A(5)
1902 PRINT A(6), A(7), A(8)
1903 PRINT XXX(6), XXX(7), XXX(8)
1912 PRINT M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [24]. The complete output through JJJJ = -31999.93000000001 is shown below:
1 1 5 1 4
0 0 0
1.484375 8 0
18.81522522954335 -31999.96000000001
1 1 5 1 4
0 0 0
1.484375 8 0
18.81522522954335 -31999.93000000001
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 [24], the wall-clock time for obtaining the output through JJJJ= -31999.93000000001 was 8 seconds--most of these seconds were for creating .EXE file.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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