Jsun Yui Wong
The computer program listed below aims to find multiple solutions for the following nonlinear programming problem from Lin and Tsai [17, pp. 438-439]:
Minimize X(17) + X(18) + X(19) + X(20) + .1 * X(21) + .2 * X(22) + .3 * X(23) + .4 * X(24)
subject to
X(17) * X(1) + X(18) * X(2) + X(19) * X(3) + X(20) * X(4)>=9,
X(17) * X(5) + X(18) * X(6) + X(19) * X(7) + X(20) * X(8)>=7,
X(17) * X(9) + X(18) * X(10) + X(19) * X(11) + X(20) * X(12)>=12,
X(17) * X(13) + X(18) * X(14) + X(19) * X(15) + X(20) * X(16)>=11,
-( -1700 * X(21) + 330 * X(1) + 360 * X(5) + 385 * X(9) + 415 * X(13))<=0,
-( -1700 * X(22) + 330 * X(2) + 360 * X(6) + 385 * X(10) + 415 * X(14))<=0,
-( -1700 * X(23) + 330 * X(3) + 360 * X(7) + 385 * X(11) + 415 * X(15))<=0,
-(-1700 * X(24) + 330 * X(4) + 360 * X(8) + 385 * X(12) + 415 * X(16))<=0,
-(1900 * X(21) - 330 * X(1) - 360 * X(5) - 385 * X(9) - 415 * X(13))<=0,
-(1900 * X(22) - 330 * X(2) - 360 * X(6) - 385 * X(10) - 415 * X(14))<=0,
-(1900 * X(23) - 330 * X(3) - 360 * X(7) - 385 * X(11) - 415 * X(15))<=0,
-(1900 * X(24) - 330 * X(4) - 360 * X(8) - 385 * X(12) - 415 * X(16))<=0,
-( -X(21) + X(1) + X(5) + X(9) + X(13))<=0,
-( -X(22) + X(2) + X(6) + X(10) + X(14))<=0,
-( -X(23) + X(3) + X(7) + X(11) + X(15))<=0,
-( -X(24) + X(4) + X(8) + X(12) + X(16))<=0,
-( 5 * X(21) - X(1) - X(5) - X(9) - X(13))<=0,
-( 5 * X(22) - X(2) - X(6) - X(10) - X(14))<=0,
-( 5 * X(23) - X(3) - X(7) - X(11) - X(15))<=0,
-( 5 * X(24) - X(4) - X(8) - X(12) - X(16))<=0,
X(21) - X(17)<=0,
X(22) - X(18)<=0,
X(23) - X(19)<=0,
X(24) - X(20)<=0,
- 15 * X(21) + X(17)<=0,
- 12 * X(22) + X(18)<=0,
- 9 * X(23) + X(19)<=0,
-6 * X(24) + X(20)<=0,
X(17) + X(18) + X(19) + X(20)>=8,
X(21) - X(22)>=0,
X(22) - X(23)>=0,
X(23) - X(24)>=0,
X(17) - X(18)>=0
X(18) - X(19)>=0,
X(19) - X(20)>=0,
0<= X(i) <=5, i=1, 2, 3,..., 16,
0<= X(17) <= 15,
0<=X(18) <= 12,
0<= X(19) <= 9,
0<= X(20) <= 6,
X(21) through X(24) are 0-1 variables.
0<= X(i) <=5, i=1,2,3,..., 16,
0<= X(17) <= 15,
0<=X(18) <= 12,
0<= X(19) <= 9,
0<= X(20) <= 6,
X(21) through X(24) are 0-1 variables,
where X(1) through X(20) are integer variables and X(21) through X(24) are 0-1 variables.
X(25) through X(59) below are slack variables.
Whereas line 128 of the Third Edition is 128 FOR I = 1 TO 1000000, here line 128 is 128 FOR I = 1 TO 100000.
Then intent behind the placement of line 189 through line 194 is to induce domino effect.
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(99)
12 FOR JJJJ = -32000 TO 31997.00 STEP .01
14 RANDOMIZE JJJJ
16 M = -1D+37
22 FOR J55 = 1 TO 16
37 A(J55) = INT(RND * 5)
38 NEXT J55
71 A(17) = 2 + INT(RND * 3)
72 A(18) = 2 + INT(RND * 3)
73 A(19) = 2 + INT(RND * 3)
74 A(20) = 2 + INT(RND * 3)
75 GOTO 91
81 A(21) = INT(RND * 1)
82 A(22) = INT(RND * 1)
83 A(23) = INT(RND * 1)
84 A(24) = INT(RND * 1)
91 IF RND < .5 THEN A(21) = 0 ELSE A(21) = 1
97 IF RND < .5 THEN A(22) = 0 ELSE A(22) = 1
98 IF RND < .5 THEN A(23) = 0 ELSE A(23) = 1
99 IF RND < .5 THEN A(24) = 0 ELSE A(24) = 1
111 IF RND < .25 THEN SOFMS = 8 ELSE IF RND < .333 THEN SOFMS = 9 ELSE IF RND < .5 THEN SOFMS = 10 ELSE SOFMS = 11
128 FOR I = 1 TO 100000
129 FOR KKQQ = 1 TO 24
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 16))
151 J = 1 + FIX(RND * 16)
152 X(J) = INT(RND * 5)
153 NEXT IPP
155 IF RND < .25 THEN GOTO 160 ELSE IF RND < .333 THEN GOTO 164 ELSE IF RND < .5 THEN GOTO 168 ELSE GOTO 173
160 X(17) = INT(RND * 15)
163 IF RND < .5 THEN 164 ELSE GOTO 168
164 X(18) = INT(RND * 12)
167 IF RND < .5 THEN 168 ELSE GOTO 173
168 X(19) = INT(RND * 9)
169 IF RND < .5 THEN 173 ELSE GOTO 175
173 X(20) = INT(RND * 6)
175 FOR IPQ = 1 TO (1 + FIX(RND * 4))
177 J = 21 + FIX(RND * 4)
179 X(J) = INT(RND * 1)
181 NEXT IPQ
183 REM r = (1 - RND * 2) * A(J)
187 REM X(J) = A(J) + (RND ^ (RND * 10)) * r
188 X(20) = SOFMS - X(17) - X(18) - X(19)
189 X(54) = X(21) - X(22)
190 X(55) = X(22) - X(23)
191 X(56) = X(23) - X(24)
192 X(57) = X(17) - X(18)
193 X(58) = X(18) - X(19)
194 X(59) = X(19) - X(20)
195 IF X(17) > 0 THEN X(21) = 1
196 IF X(18) > 0 THEN X(22) = 1
197 IF X(19) > 0 THEN X(23) = 1
198 IF X(20) > 0 THEN X(24) = 1
201 IF X(1) < 0 THEN 1670
202 IF X(1) > 5 THEN 1670
203 IF X(2) < 0 THEN 1670
204 IF X(2) > 5 THEN 1670
205 IF X(3) < 0 THEN 1670
206 IF X(3) > 5 THEN 1670
207 IF X(4) < 0 THEN 1670
208 IF X(4) > 5 THEN 1670
209 IF X(5) < 0 THEN 1670
210 IF X(5) > 5 THEN 1670
211 IF X(6) < 0 THEN 1670
212 IF X(6) > 5 THEN 1670
213 IF X(7) < 0 THEN 1670
214 IF X(7) > 5 THEN 1670
215 IF X(8) < 0 THEN 1670
216 IF X(8) > 5 THEN 1670
217 IF X(9) < 0 THEN 1670
218 IF X(9) > 5 THEN 1670
219 IF X(10) < 0 THEN 1670
220 IF X(10) > 5 THEN 1670
221 IF X(11) < 0 THEN 1670
222 IF X(11) > 5 THEN 1670
223 IF X(12) < 0 THEN 1670
224 IF X(12) > 5 THEN 1670
225 IF X(13) < 0 THEN 1670
226 IF X(13) > 5 THEN 1670
227 IF X(14) < 0 THEN 1670
228 IF X(14) > 5 THEN 1670
229 IF X(15) < 0 THEN 1670
230 IF X(15) > 5 THEN 1670
231 IF X(16) < 0 THEN 1670
232 IF X(16) > 5 THEN 1670
233 IF X(17) < 0 THEN 1670
234 IF X(17) > 15 THEN 1670
235 IF X(18) < 0 THEN 1670
236 IF X(18) > 12 THEN 1670
237 IF X(19) < 0 THEN 1670
238 IF X(19) > 9 THEN 1670
239 IF X(20) < 0 THEN 1670
240 IF X(20) > 6 THEN 1670
241 IF X(21) < 0 THEN 1670
242 IF X(21) > 1 THEN 1670
243 IF X(22) < 0 THEN 1670
244 IF X(22) > 1 THEN 1670
245 IF X(23) < 0 THEN 1670
246 IF X(23) > 1 THEN 1670
247 IF X(24) < 0 THEN 1670
248 IF X(24) > 1 THEN 1670
301 X(25) = -9 + X(17) * X(1) + X(18) * X(2) + X(19) * X(3) + X(20) * X(4)
302 X(26) = -7 + X(17) * X(5) + X(18) * X(6) + X(19) * X(7) + X(20) * X(8)
303 X(27) = -12 + X(17) * X(9) + X(18) * X(10) + X(19) * X(11) + X(20) * X(12)
304 X(28) = -11 + X(17) * X(13) + X(18) * X(14) + X(19) * X(15) + X(20) * X(16)
305 X(29) = -1700 * X(21) + 330 * X(1) + 360 * X(5) + 385 * X(9) + 415 * X(13)
306 X(30) = -1700 * X(22) + 330 * X(2) + 360 * X(6) + 385 * X(10) + 415 * X(14)
307 X(31) = -1700 * X(23) + 330 * X(3) + 360 * X(7) + 385 * X(11) + 415 * X(15)
308 X(32) = -1700 * X(24) + 330 * X(4) + 360 * X(8) + 385 * X(12) + 415 * X(16)
309 X(33) = 1900 * X(21) - 330 * X(1) - 360 * X(5) - 385 * X(9) - 415 * X(13)
310 X(34) = 1900 * X(22) - 330 * X(2) - 360 * X(6) - 385 * X(10) - 415 * X(14)
311 X(35) = 1900 * X(23) - 330 * X(3) - 360 * X(7) - 385 * X(11) - 415 * X(15)
312 X(36) = 1900 * X(24) - 330 * X(4) - 360 * X(8) - 385 * X(12) - 415 * X(16)
313 X(37) = -X(21) + X(1) + X(5) + X(9) + X(13)
314 X(38) = -X(22) + X(2) + X(6) + X(10) + X(14)
315 X(39) = -X(23) + X(3) + X(7) + X(11) + X(15)
316 X(40) = -X(24) + X(4) + X(8) + X(12) + X(16)
317 X(41) = 5 * X(21) - X(1) - X(5) - X(9) - X(13)
318 X(42) = 5 * X(22) - X(2) - X(6) - X(10) - X(14)
319 X(43) = 5 * X(23) - X(3) - X(7) - X(11) - X(15)
320 X(44) = 5 * X(24) - X(4) - X(8) - X(12) - X(16)
321 X(45) = -X(21) + X(17)
322 X(46) = -X(22) + X(18)
323 X(47) = -X(23) + X(19)
324 X(48) = -X(24) + X(20)
325 X(49) = 15 * X(21) - X(17)
326 X(50) = 12 * X(22) - X(18)
327 X(51) = 9 * X(23) - X(19)
328 X(52) = 6 * X(24) - X(20)
329 X(53) = -8 + X(17) + X(18) + X(19) + X(20)
339 GOTO 425
340 X(54) = X(21) - X(22)
341 X(55) = X(22) - X(23)
342 X(56) = X(23) - X(24)
343 X(57) = X(17) - X(18)
344 X(58) = X(18) - X(19)
345 X(59) = X(19) - X(20)
425 FOR J99 = 25 TO 59
426 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0
427 NEXT J99
431 SUMP = 0
434 FOR J66 = 25 TO 59
437 SUMP = SUMP + X(J66)
440 NEXT J66
459 POBA = -X(17) - X(18) - X(19) - X(20) - .1 * X(21) - .2 * X(22) - .3 * X(23) - .4 * X(24) + 1000000 * SUMP
466 P = POBA
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR klx = 1 TO 59
1459 A(klx) = X(klx)
1460 NEXT klx
1557 REM GOTO 128
1670 NEXT I
1889 IF M < -8.9 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4), A(5)
1950 PRINT A(6), A(7), A(8), A(9), A(10)
1953 PRINT A(11), A(12), A(13), A(14), A(15)
1954 PRINT A(16), A(17), A(18), A(19), A(20)
1955 PRINT A(21), A(22), A(23), A(24)
1957 PRINT M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [34]. The complete output through JJJJ =-30650.12000021608 is shown below:
1 1 2 0 0
2 2 0 3 0
0 0 1 2 1
0 4 3 1 0
1 1 1 0
-8.6 -31951.18000000782
1 1 2 0 0
2 1 0 3 0
1 0 1 2 1
0 4 3 1 0
1 1 1 0
-8.6 -31820.05000002881
2 1 0 0 0
0 4 0 1 3
0 0 2 1 1
0 3 3 2 0
1 1 1 0
-8.6 -31779.80000003525
1 1 2 0 0
2 1 0 3 0
0 0 1 2 2
0 4 3 1 0
1 1 1 0
-8.6 -31741.05000004145
1 1 2 0 0
2 1 0 3 0
1 0 1 2 1
0 4 3 1 0
1 1 1 0
-8.6 -31820.05000002881
1 2 0 0 1
0 3 0 2 1
1 0 1 2 1
0 4 3 1 0
1 1 1 0
-8.6 -31724.82000004405
1 2 0 0 1
0 3 0 2 1
1 0 1 2 1
0 4 3 1 0
1 1 1 0
-8.6 -31598.01000006453
1 1 2 0 0
2 1 0 3 0
0 0 1 2 2
0 4 3 1 0
1 1 1 0
-8.6 -31583.73000006663
1 2 0 0 1
0 3 0 2 1
1 0 1 2 1
0 4 3 1 0
1 1 1 0
-8.6 -31531.62000007497
1 2 1 0 1
0 2 0 2 1
0 0 1 2 2
0 5 2 1 0
1 1 1 0
-8.6 -31393.97000009701
1 1 2 0 0
2 1 0 3 0
0 0 1 2 2
0 4 3 1 0
1 1 1 0
-8.6 -31349.81000010408
1 1 2 0 0
2 2 0 3 0
0 0 1 2 1
0 4 3 1 0
1 1 1 0
-8.6 -31329.1000001074
1 1 3 0 0
2 1 0 3 0
0 0 1 2 1
0 4 3 1 0
1 1 1 0
-8.6 -31173.33000013233
1 2 0 0 1
0 3 0 2 1
1 0 1 2 1
0 4 3 1 0
1 1 1 0
-8.6 -31169.55000013293
1 2 0 0 0
0 4 0 3 1
0 0 1 2 1
0 3 3 2 0
1 1 1 0
-8.6 -31130.31000013921
1 2 0 0 0
0 4 0 3 1
0 0 1 2 1
0 3 3 2 0
1 1 1 0
-8.6 -31097.69000014443
1 1 2 0 0
2 2 0 3 0
0 0 1 2 1
0 4 3 1 0
1 1 1 0
-8.6 -31015.33000015762
1 1 3 0 0
2 1 0 3 0
0 0 1 2 1
0 4 3 1 0
1 1 1 0
-8.6 -30838.58000018951
1 1 2 0 2
0 0 0 0 3
3 0 2 1 0
0 4 3 1 0
1 1 1 0
-8.6 -30816.1800001895
1 1 2 0 0
2 2 0 3 0
0 0 1 2 1
0 4 3 1 0
1 1 1 0
-8.6 -30787.76000019405
1 2 1 0 1
0 2 0 2 1
0 0 1 2 2
0 5 2 1 0
1 1 1 0
-8.6 -30656.17000021511
1 2 0 0 1
0 3 0 2 1
1 0 1 2 1
0 4 3 1 0
1 1 1 0
-8.6 -30653.03000021561
1 2 0 0 0
0 4 0 3 1
0 0 1 2 1
0 3 3 2 0
1 1 1 0
-8.6 -30650.12000021608
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 [34], the wall-clock time for obtaining the output through JJJJ= -30650.12000021608
was 24 hours. One can compare the computational results here with those in Table 4 of Lin and Tsai [17, p. 440].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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